LMFDB - Embedded newform 825.2.k.f.518.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(518,825)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("825.518");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	825.k (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.58765816676$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			518.1
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	825.518
Dual form			825.2.k.f.782.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.292893 + 0.292893i) q^{2} +(1.00000 - 1.41421i) q^{3} -1.82843i q^{4} +(0.707107 - 0.121320i) q^{6} +(3.41421 - 3.41421i) q^{7} +(1.12132 - 1.12132i) q^{8} +(-1.00000 - 2.82843i) q^{9} +O(q^{10})$	\(q+(0.292893 + 0.292893i) q^{2} +(1.00000 - 1.41421i) q^{3} -1.82843i q^{4} +(0.707107 - 0.121320i) q^{6} +(3.41421 - 3.41421i) q^{7} +(1.12132 - 1.12132i) q^{8} +(-1.00000 - 2.82843i) q^{9} +1.00000i q^{11} +(-2.58579 - 1.82843i) q^{12} +(2.00000 + 2.00000i) q^{13} +2.00000 q^{14} -3.00000 q^{16} +(2.82843 + 2.82843i) q^{17} +(0.535534 - 1.12132i) q^{18} +2.82843i q^{19} +(-1.41421 - 8.24264i) q^{21} +(-0.292893 + 0.292893i) q^{22} +(-3.24264 + 3.24264i) q^{23} +(-0.464466 - 2.70711i) q^{24} +1.17157i q^{26} +(-5.00000 - 1.41421i) q^{27} +(-6.24264 - 6.24264i) q^{28} +0.828427 q^{29} -3.17157 q^{31} +(-3.12132 - 3.12132i) q^{32} +(1.41421 + 1.00000i) q^{33} +1.65685i q^{34} +(-5.17157 + 1.82843i) q^{36} +(0.171573 - 0.171573i) q^{37} +(-0.828427 + 0.828427i) q^{38} +(4.82843 - 0.828427i) q^{39} +7.65685i q^{41} +(2.00000 - 2.82843i) q^{42} +(-0.242641 - 0.242641i) q^{43} +1.82843 q^{44} -1.89949 q^{46} +(7.24264 + 7.24264i) q^{47} +(-3.00000 + 4.24264i) q^{48} -16.3137i q^{49} +(6.82843 - 1.17157i) q^{51} +(3.65685 - 3.65685i) q^{52} +(-1.00000 + 1.00000i) q^{53} +(-1.05025 - 1.87868i) q^{54} -7.65685i q^{56} +(4.00000 + 2.82843i) q^{57} +(0.242641 + 0.242641i) q^{58} -4.00000 q^{59} -6.48528 q^{61} +(-0.928932 - 0.928932i) q^{62} +(-13.0711 - 6.24264i) q^{63} +4.17157i q^{64} +(0.121320 + 0.707107i) q^{66} +(6.41421 - 6.41421i) q^{67} +(5.17157 - 5.17157i) q^{68} +(1.34315 + 7.82843i) q^{69} +2.48528i q^{71} +(-4.29289 - 2.05025i) q^{72} +0.100505 q^{74} +5.17157 q^{76} +(3.41421 + 3.41421i) q^{77} +(1.65685 + 1.17157i) q^{78} -10.8284i q^{79} +(-7.00000 + 5.65685i) q^{81} +(-2.24264 + 2.24264i) q^{82} +(9.07107 - 9.07107i) q^{83} +(-15.0711 + 2.58579i) q^{84} -0.142136i q^{86} +(0.828427 - 1.17157i) q^{87} +(1.12132 + 1.12132i) q^{88} -9.65685 q^{89} +13.6569 q^{91} +(5.92893 + 5.92893i) q^{92} +(-3.17157 + 4.48528i) q^{93} +4.24264i q^{94} +(-7.53553 + 1.29289i) q^{96} +(-10.6569 + 10.6569i) q^{97} +(4.77817 - 4.77817i) q^{98} +(2.82843 - 1.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{3} + 8 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^3 + 8 * q^7 - 4 * q^8 - 4 * q^9	$ 4 q + 4 q^{2} + 4 q^{3} + 8 q^{7} - 4 q^{8} - 4 q^{9} - 16 q^{12} + 8 q^{13} + 8 q^{14} - 12 q^{16} - 12 q^{18} - 4 q^{22} + 4 q^{23} - 16 q^{24} - 20 q^{27} - 8 q^{28} - 8 q^{29} - 24 q^{31} - 4 q^{32} - 32 q^{36} + 12 q^{37} + 8 q^{38} + 8 q^{39} + 8 q^{42} + 16 q^{43} - 4 q^{44} + 32 q^{46} + 12 q^{47} - 12 q^{48} + 16 q^{51} - 8 q^{52} - 4 q^{53} - 24 q^{54} + 16 q^{57} - 16 q^{58} - 16 q^{59} + 8 q^{61} - 32 q^{62} - 24 q^{63} - 8 q^{66} + 20 q^{67} + 32 q^{68} + 28 q^{69} - 20 q^{72} + 40 q^{74} + 32 q^{76} + 8 q^{77} - 16 q^{78} - 28 q^{81} + 8 q^{82} + 8 q^{83} - 32 q^{84} - 8 q^{87} - 4 q^{88} - 16 q^{89} + 32 q^{91} + 52 q^{92} - 24 q^{93} - 16 q^{96} - 20 q^{97} - 12 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^3 + 8 * q^7 - 4 * q^8 - 4 * q^9 - 16 * q^12 + 8 * q^13 + 8 * q^14 - 12 * q^16 - 12 * q^18 - 4 * q^22 + 4 * q^23 - 16 * q^24 - 20 * q^27 - 8 * q^28 - 8 * q^29 - 24 * q^31 - 4 * q^32 - 32 * q^36 + 12 * q^37 + 8 * q^38 + 8 * q^39 + 8 * q^42 + 16 * q^43 - 4 * q^44 + 32 * q^46 + 12 * q^47 - 12 * q^48 + 16 * q^51 - 8 * q^52 - 4 * q^53 - 24 * q^54 + 16 * q^57 - 16 * q^58 - 16 * q^59 + 8 * q^61 - 32 * q^62 - 24 * q^63 - 8 * q^66 + 20 * q^67 + 32 * q^68 + 28 * q^69 - 20 * q^72 + 40 * q^74 + 32 * q^76 + 8 * q^77 - 16 * q^78 - 28 * q^81 + 8 * q^82 + 8 * q^83 - 32 * q^84 - 8 * q^87 - 4 * q^88 - 16 * q^89 + 32 * q^91 + 52 * q^92 - 24 * q^93 - 16 * q^96 - 20 * q^97 - 12 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.292893	+	0.292893i	0.207107	+	0.207107i	0.803037	−	0.595930i	$-0.203216\pi$
$2$	0.292893	+	0.292893i	0.207107	+	0.207107i	−0.595930	+	0.803037i	$0.703216\pi$
$3$	1.00000	−	1.41421i	0.577350	−	0.816497i
$3$	1.00000	−	1.41421i	0.577350	−	0.816497i
$4$		−	1.82843i		−	0.914214i
$5$		0			0
$5$		0			0
$6$	0.707107	−	0.121320i	0.288675	−	0.0495288i
$7$	3.41421	−	3.41421i	1.29045	−	1.29045i	0.355944	−	0.934507i	$-0.384159\pi$
$7$	3.41421	−	3.41421i	1.29045	−	1.29045i	0.934507	−	0.355944i	$-0.115841\pi$
$8$	1.12132	−	1.12132i	0.396447	−	0.396447i
$9$	−1.00000	−	2.82843i	−0.333333	−	0.942809i
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$	−2.58579	−	1.82843i	−0.746452	−	0.527821i
$13$	2.00000	+	2.00000i	0.554700	+	0.554700i	0.927794	−	0.373094i	$-0.121703\pi$
$13$	2.00000	+	2.00000i	0.554700	+	0.554700i	−0.373094	+	0.927794i	$0.621703\pi$
$14$	2.00000			0.534522
$15$		0			0
$16$	−3.00000			−0.750000
$17$	2.82843	+	2.82843i	0.685994	+	0.685994i	0.961344	−	0.275350i	$-0.0887937\pi$
$17$	2.82843	+	2.82843i	0.685994	+	0.685994i	−0.275350	+	0.961344i	$0.588794\pi$
$18$	0.535534	−	1.12132i	0.126227	−	0.264298i
$19$			2.82843i			0.648886i	0.945905	+	0.324443i	$0.105177\pi$
$19$			2.82843i			0.648886i	−0.945905	+	0.324443i	$0.894823\pi$
$20$		0			0
$21$	−1.41421	−	8.24264i	−0.308607	−	1.79869i
$22$	−0.292893	+	0.292893i	−0.0624450	+	0.0624450i
$23$	−3.24264	+	3.24264i	−0.676137	+	0.676137i	−0.959124	−	0.282987i	$-0.908675\pi$
$23$	−3.24264	+	3.24264i	−0.676137	+	0.676137i	0.282987	+	0.959124i	$0.408675\pi$
$24$	−0.464466	−	2.70711i	−0.0948087	−	0.552586i
$25$		0			0
$26$			1.17157i			0.229764i
$27$	−5.00000	−	1.41421i	−0.962250	−	0.272166i
$28$	−6.24264	−	6.24264i	−1.17975	−	1.17975i
$29$	0.828427			0.153835			0.0769175	−	0.997037i	$-0.475492\pi$
$29$	0.828427			0.153835			0.0769175	+	0.997037i	$0.475492\pi$
$30$		0			0
$31$	−3.17157			−0.569631			−0.284816	−	0.958582i	$-0.591932\pi$
$31$	−3.17157			−0.569631			−0.284816	+	0.958582i	$0.591932\pi$
$32$	−3.12132	−	3.12132i	−0.551777	−	0.551777i
$33$	1.41421	+	1.00000i	0.246183	+	0.174078i
$34$			1.65685i			0.284148i
$35$		0			0
$36$	−5.17157	+	1.82843i	−0.861929	+	0.304738i
$37$	0.171573	−	0.171573i	0.0282064	−	0.0282064i	−0.692863	−	0.721069i	$-0.743651\pi$
$37$	0.171573	−	0.171573i	0.0282064	−	0.0282064i	0.721069	+	0.692863i	$0.243651\pi$
$38$	−0.828427	+	0.828427i	−0.134389	+	0.134389i
$39$	4.82843	−	0.828427i	0.773167	−	0.132655i
$40$		0			0
$41$			7.65685i			1.19580i	0.801571	+	0.597900i	$0.203998\pi$
$41$			7.65685i			1.19580i	−0.801571	+	0.597900i	$0.796002\pi$
$42$	2.00000	−	2.82843i	0.308607	−	0.436436i
$43$	−0.242641	−	0.242641i	−0.0370024	−	0.0370024i	0.688364	−	0.725366i	$-0.258329\pi$
$43$	−0.242641	−	0.242641i	−0.0370024	−	0.0370024i	−0.725366	+	0.688364i	$0.758329\pi$
$44$	1.82843			0.275646
$45$		0			0
$46$	−1.89949			−0.280065
$47$	7.24264	+	7.24264i	1.05645	+	1.05645i	0.998308	+	0.0581392i	$0.0185167\pi$
$47$	7.24264	+	7.24264i	1.05645	+	1.05645i	0.0581392	+	0.998308i	$0.481483\pi$
$48$	−3.00000	+	4.24264i	−0.433013	+	0.612372i
$49$		−	16.3137i		−	2.33053i
$50$		0			0
$51$	6.82843	−	1.17157i	0.956171	−	0.164053i
$52$	3.65685	−	3.65685i	0.507114	−	0.507114i
$53$	−1.00000	+	1.00000i	−0.137361	+	0.137361i	−0.772444	−	0.635083i	$-0.780966\pi$
$53$	−1.00000	+	1.00000i	−0.137361	+	0.137361i	0.635083	+	0.772444i	$0.280966\pi$
$54$	−1.05025	−	1.87868i	−0.142921	−	0.255656i
$55$		0			0
$56$		−	7.65685i		−	1.02319i
$57$	4.00000	+	2.82843i	0.529813	+	0.374634i
$58$	0.242641	+	0.242641i	0.0318603	+	0.0318603i
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$	−6.48528			−0.830355			−0.415178	−	0.909740i	$-0.636281\pi$
$61$	−6.48528			−0.830355			−0.415178	+	0.909740i	$0.636281\pi$
$62$	−0.928932	−	0.928932i	−0.117975	−	0.117975i
$63$	−13.0711	−	6.24264i	−1.64680	−	0.786499i
$64$			4.17157i			0.521447i
$65$		0			0
$66$	0.121320	+	0.707107i	0.0149335	+	0.0870388i
$67$	6.41421	−	6.41421i	0.783621	−	0.783621i	−0.196819	−	0.980440i	$-0.563061\pi$
$67$	6.41421	−	6.41421i	0.783621	−	0.783621i	0.980440	+	0.196819i	$0.0630611\pi$
$68$	5.17157	−	5.17157i	0.627145	−	0.627145i
$69$	1.34315	+	7.82843i	0.161696	+	0.942432i
$70$		0			0
$71$			2.48528i			0.294949i	0.989066	+	0.147474i	$0.0471144\pi$
$71$			2.48528i			0.294949i	−0.989066	+	0.147474i	$0.952886\pi$
$72$	−4.29289	−	2.05025i	−0.505922	−	0.241625i
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$	0.100505			0.0116835
$75$		0			0
$76$	5.17157			0.593220
$77$	3.41421	+	3.41421i	0.389086	+	0.389086i
$78$	1.65685	+	1.17157i	0.187602	+	0.132655i
$79$		−	10.8284i		−	1.21829i	−0.793058	−	0.609147i	$-0.791512\pi$
$79$		−	10.8284i		−	1.21829i	0.793058	−	0.609147i	$-0.208488\pi$
$80$		0			0
$81$	−7.00000	+	5.65685i	−0.777778	+	0.628539i
$82$	−2.24264	+	2.24264i	−0.247658	+	0.247658i
$83$	9.07107	−	9.07107i	0.995679	−	0.995679i	−0.00431166	−	0.999991i	$-0.501372\pi$
$83$	9.07107	−	9.07107i	0.995679	−	0.995679i	0.999991	+	0.00431166i	$0.00137245\pi$
$84$	−15.0711	+	2.58579i	−1.64439	+	0.282132i
$85$		0			0
$86$		−	0.142136i		−	0.0153269i
$87$	0.828427	−	1.17157i	0.0888167	−	0.125606i
$88$	1.12132	+	1.12132i	0.119533	+	0.119533i
$89$	−9.65685			−1.02362			−0.511812	−	0.859097i	$-0.671026\pi$
$89$	−9.65685			−1.02362			−0.511812	+	0.859097i	$0.671026\pi$
$90$		0			0
$91$	13.6569			1.43163
$92$	5.92893	+	5.92893i	0.618134	+	0.618134i
$93$	−3.17157	+	4.48528i	−0.328877	+	0.465102i
$94$			4.24264i			0.437595i
$95$		0			0
$96$	−7.53553	+	1.29289i	−0.769092	+	0.131955i
$97$	−10.6569	+	10.6569i	−1.08204	+	1.08204i	−0.0857204	+	0.996319i	$0.527319\pi$
$97$	−10.6569	+	10.6569i	−1.08204	+	1.08204i	−0.996319	+	0.0857204i	$0.972681\pi$
$98$	4.77817	−	4.77817i	0.482669	−	0.482669i
$99$	2.82843	−	1.00000i	0.284268	−	0.100504i
$100$		0			0
$101$			7.17157i			0.713598i	0.934181	+	0.356799i	$0.116132\pi$
$101$			7.17157i			0.713598i	−0.934181	+	0.356799i	$0.883868\pi$
$102$	2.34315	+	1.65685i	0.232006	+	0.164053i
$103$	6.41421	+	6.41421i	0.632011	+	0.632011i	0.948572	−	0.316561i	$-0.102528\pi$
$103$	6.41421	+	6.41421i	0.632011	+	0.632011i	−0.316561	+	0.948572i	$0.602528\pi$
$104$	4.48528			0.439818
$105$		0			0
$106$	−0.585786			−0.0568966
$107$	−0.242641	−	0.242641i	−0.0234570	−	0.0234570i	0.695281	−	0.718738i	$-0.255280\pi$
$107$	−0.242641	−	0.242641i	−0.0234570	−	0.0234570i	−0.718738	+	0.695281i	$0.755280\pi$
$108$	−2.58579	+	9.14214i	−0.248817	+	0.879702i
$109$		−	3.17157i		−	0.303782i	−0.988397	−	0.151891i	$-0.951464\pi$
$109$		−	3.17157i		−	0.303782i	0.988397	−	0.151891i	$-0.0485362\pi$
$110$		0			0
$111$	−0.0710678	−	0.414214i	−0.00674546	−	0.0393154i
$112$	−10.2426	+	10.2426i	−0.967839	+	0.967839i
$113$	5.82843	−	5.82843i	0.548292	−	0.548292i	−0.377654	−	0.925947i	$-0.623269\pi$
$113$	5.82843	−	5.82843i	0.548292	−	0.548292i	0.925947	+	0.377654i	$0.123269\pi$
$114$	0.343146	+	2.00000i	0.0321385	+	0.187317i
$115$		0			0
$116$		−	1.51472i		−	0.140638i
$117$	3.65685	−	7.65685i	0.338076	−	0.707876i
$118$	−1.17157	−	1.17157i	−0.107852	−	0.107852i
$119$	19.3137			1.77048
$120$		0			0
$121$	−1.00000			−0.0909091
$122$	−1.89949	−	1.89949i	−0.171972	−	0.171972i
$123$	10.8284	+	7.65685i	0.976366	+	0.690395i
$124$			5.79899i			0.520765i
$125$		0			0
$126$	−2.00000	−	5.65685i	−0.178174	−	0.503953i
$127$	−1.41421	+	1.41421i	−0.125491	+	0.125491i	−0.767063	−	0.641572i	$-0.778283\pi$
$127$	−1.41421	+	1.41421i	−0.125491	+	0.125491i	0.641572	+	0.767063i	$0.278283\pi$
$128$	−7.46447	+	7.46447i	−0.659772	+	0.659772i
$129$	−0.585786	+	0.100505i	−0.0515756	+	0.00884898i
$130$		0			0
$131$		−	6.82843i		−	0.596602i	−0.954472	−	0.298301i	$-0.903580\pi$
$131$		−	6.82843i		−	0.596602i	0.954472	−	0.298301i	$-0.0964200\pi$
$132$	1.82843	−	2.58579i	0.159144	−	0.225064i
$133$	9.65685	+	9.65685i	0.837355	+	0.837355i
$134$	3.75736			0.324586
$135$		0			0
$136$	6.34315			0.543920
$137$	−0.171573	−	0.171573i	−0.0146585	−	0.0146585i	0.699740	−	0.714398i	$-0.253299\pi$
$137$	−0.171573	−	0.171573i	−0.0146585	−	0.0146585i	−0.714398	+	0.699740i	$0.753299\pi$
$138$	−1.89949	+	2.68629i	−0.161696	+	0.228672i
$139$			17.6569i			1.49763i	0.662776	+	0.748817i	$0.269378\pi$
$139$			17.6569i			1.49763i	−0.662776	+	0.748817i	$0.730622\pi$
$140$		0			0
$141$	17.4853	−	3.00000i	1.47253	−	0.252646i
$142$	−0.727922	+	0.727922i	−0.0610859	+	0.0610859i
$143$	−2.00000	+	2.00000i	−0.167248	+	0.167248i
$144$	3.00000	+	8.48528i	0.250000	+	0.707107i
$145$		0			0
$146$		0			0
$147$	−23.0711	−	16.3137i	−1.90287	−	1.34553i
$148$	−0.313708	−	0.313708i	−0.0257867	−	0.0257867i
$149$	−10.0000			−0.819232			−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000			−0.819232			−0.409616	+	0.912258i	$0.634337\pi$
$150$		0			0
$151$	10.1421			0.825355			0.412678	−	0.910877i	$-0.364594\pi$
$151$	10.1421			0.825355			0.412678	+	0.910877i	$0.364594\pi$
$152$	3.17157	+	3.17157i	0.257249	+	0.257249i
$153$	5.17157	−	10.8284i	0.418097	−	0.875426i
$154$			2.00000i			0.161165i
$155$		0			0
$156$	−1.51472	−	8.82843i	−0.121275	−	0.706840i
$157$	9.48528	−	9.48528i	0.757008	−	0.757008i	−0.218769	−	0.975777i	$-0.570204\pi$
$157$	9.48528	−	9.48528i	0.757008	−	0.757008i	0.975777	+	0.218769i	$0.0702041\pi$
$158$	3.17157	−	3.17157i	0.252317	−	0.252317i
$159$	0.414214	+	2.41421i	0.0328493	+	0.191460i
$160$		0			0
$161$			22.1421i			1.74504i
$162$	−3.70711	−	0.393398i	−0.291258	−	0.0309083i
$163$	−3.58579	−	3.58579i	−0.280860	−	0.280860i	0.552592	−	0.833452i	$-0.313639\pi$
$163$	−3.58579	−	3.58579i	−0.280860	−	0.280860i	−0.833452	+	0.552592i	$0.813639\pi$
$164$	14.0000			1.09322
$165$		0			0
$166$	5.31371			0.412424
$167$	−14.7279	−	14.7279i	−1.13968	−	1.13968i	−0.988507	−	0.151174i	$-0.951695\pi$
$167$	−14.7279	−	14.7279i	−1.13968	−	1.13968i	−0.151174	−	0.988507i	$-0.548305\pi$
$168$	−10.8284	−	7.65685i	−0.835431	−	0.590739i
$169$		−	5.00000i		−	0.384615i
$170$		0			0
$171$	8.00000	−	2.82843i	0.611775	−	0.216295i
$172$	−0.443651	+	0.443651i	−0.0338281	+	0.0338281i
$173$	−8.48528	+	8.48528i	−0.645124	+	0.645124i	−0.951811	−	0.306687i	$-0.900780\pi$
$173$	−8.48528	+	8.48528i	−0.645124	+	0.645124i	0.306687	+	0.951811i	$0.400780\pi$
$174$	0.585786	−	0.100505i	0.0444084	−	0.00761927i
$175$		0			0
$176$		−	3.00000i		−	0.226134i
$177$	−4.00000	+	5.65685i	−0.300658	+	0.425195i
$178$	−2.82843	−	2.82843i	−0.212000	−	0.212000i
$179$	24.1421			1.80447			0.902234	−	0.431247i	$-0.141926\pi$
$179$	24.1421			1.80447			0.902234	+	0.431247i	$0.141926\pi$
$180$		0			0
$181$	5.65685			0.420471			0.210235	−	0.977651i	$-0.432577\pi$
$181$	5.65685			0.420471			0.210235	+	0.977651i	$0.432577\pi$
$182$	4.00000	+	4.00000i	0.296500	+	0.296500i
$183$	−6.48528	+	9.17157i	−0.479406	+	0.677982i
$184$			7.27208i			0.536105i
$185$		0			0
$186$	−2.24264	+	0.384776i	−0.164438	+	0.0282132i
$187$	−2.82843	+	2.82843i	−0.206835	+	0.206835i
$188$	13.2426	−	13.2426i	0.965819	−	0.965819i
$189$	−21.8995	+	12.2426i	−1.59295	+	0.890521i
$190$		0			0
$191$		−	11.3137i		−	0.818631i	−0.912393	−	0.409316i	$-0.865768\pi$
$191$		−	11.3137i		−	0.818631i	0.912393	−	0.409316i	$-0.134232\pi$
$192$	5.89949	+	4.17157i	0.425759	+	0.301057i
$193$	−8.00000	−	8.00000i	−0.575853	−	0.575853i	0.357905	−	0.933758i	$-0.383491\pi$
$193$	−8.00000	−	8.00000i	−0.575853	−	0.575853i	−0.933758	+	0.357905i	$0.883491\pi$
$194$	−6.24264			−0.448195
$195$		0			0
$196$	−29.8284			−2.13060
$197$	14.1421	+	14.1421i	1.00759	+	1.00759i	0.999971	+	0.00761443i	$0.00242377\pi$
$197$	14.1421	+	14.1421i	1.00759	+	1.00759i	0.00761443	+	0.999971i	$0.497576\pi$
$198$	1.12132	+	0.535534i	0.0796888	+	0.0380587i
$199$			14.4853i			1.02683i	0.858139	+	0.513417i	$0.171621\pi$
$199$			14.4853i			1.02683i	−0.858139	+	0.513417i	$0.828379\pi$
$200$		0			0
$201$	−2.65685	−	15.4853i	−0.187400	−	1.09225i
$202$	−2.10051	+	2.10051i	−0.147791	+	0.147791i
$203$	2.82843	−	2.82843i	0.198517	−	0.198517i
$204$	−2.14214	−	12.4853i	−0.149979	−	0.874145i
$205$		0			0
$206$			3.75736i			0.261788i
$207$	12.4142	+	5.92893i	0.862847	+	0.412089i
$208$	−6.00000	−	6.00000i	−0.416025	−	0.416025i
$209$	−2.82843			−0.195646
$210$		0			0
$211$	−4.48528			−0.308780			−0.154390	−	0.988010i	$-0.549341\pi$
$211$	−4.48528			−0.308780			−0.154390	+	0.988010i	$0.549341\pi$
$212$	1.82843	+	1.82843i	0.125577	+	0.125577i
$213$	3.51472	+	2.48528i	0.240825	+	0.170289i
$214$		−	0.142136i		−	0.00971619i
$215$		0			0
$216$	−7.19239	+	4.02082i	−0.489380	+	0.273582i
$217$	−10.8284	+	10.8284i	−0.735082	+	0.735082i
$218$	0.928932	−	0.928932i	0.0629152	−	0.0629152i
$219$		0			0
$220$		0			0
$221$			11.3137i			0.761042i
$222$	0.100505	−	0.142136i	0.00674546	−	0.00953952i
$223$	16.5563	+	16.5563i	1.10870	+	1.10870i	0.993322	+	0.115373i	$0.0368063\pi$
$223$	16.5563	+	16.5563i	1.10870	+	1.10870i	0.115373	+	0.993322i	$0.463194\pi$
$224$	−21.3137			−1.42408
$225$		0			0
$226$	3.41421			0.227110
$227$	−10.2426	−	10.2426i	−0.679828	−	0.679828i	0.280133	−	0.959961i	$-0.409621\pi$
$227$	−10.2426	−	10.2426i	−0.679828	−	0.679828i	−0.959961	+	0.280133i	$0.909621\pi$
$228$	5.17157	−	7.31371i	0.342496	−	0.484362i
$229$			9.65685i			0.638143i	0.947731	+	0.319071i	$0.103371\pi$
$229$			9.65685i			0.638143i	−0.947731	+	0.319071i	$0.896629\pi$
$230$		0			0
$231$	8.24264	−	1.41421i	0.542326	−	0.0930484i
$232$	0.928932	−	0.928932i	0.0609874	−	0.0609874i
$233$	−19.6569	+	19.6569i	−1.28776	+	1.28776i	−0.351621	+	0.936143i	$0.614369\pi$
$233$	−19.6569	+	19.6569i	−1.28776	+	1.28776i	−0.936143	+	0.351621i	$0.885631\pi$
$234$	3.31371	−	1.17157i	0.216624	−	0.0765881i
$235$		0			0
$236$			7.31371i			0.476082i
$237$	−15.3137	−	10.8284i	−0.994732	−	0.703382i
$238$	5.65685	+	5.65685i	0.366679	+	0.366679i
$239$	−23.7990			−1.53943			−0.769714	−	0.638388i	$-0.779601\pi$
$239$	−23.7990			−1.53943			−0.769714	+	0.638388i	$0.779601\pi$
$240$		0			0
$241$	0.142136			0.00915576			0.00457788	−	0.999990i	$-0.498543\pi$
$241$	0.142136			0.00915576			0.00457788	+	0.999990i	$0.498543\pi$
$242$	−0.292893	−	0.292893i	−0.0188279	−	0.0188279i
$243$	1.00000	+	15.5563i	0.0641500	+	0.997940i
$244$			11.8579i			0.759122i
$245$		0			0
$246$	0.928932	+	5.41421i	0.0592266	+	0.345198i
$247$	−5.65685	+	5.65685i	−0.359937	+	0.359937i
$248$	−3.55635	+	3.55635i	−0.225828	+	0.225828i
$249$	−3.75736	−	21.8995i	−0.238113	−	1.38782i
$250$		0			0
$251$			16.1421i			1.01888i	0.860505	+	0.509441i	$0.170148\pi$
$251$			16.1421i			1.01888i	−0.860505	+	0.509441i	$0.829852\pi$
$252$	−11.4142	+	23.8995i	−0.719028	+	1.50553i
$253$	−3.24264	−	3.24264i	−0.203863	−	0.203863i
$254$	−0.828427			−0.0519801
$255$		0			0
$256$	3.97056			0.248160
$257$	1.82843	+	1.82843i	0.114054	+	0.114054i	0.761831	−	0.647776i	$-0.224301\pi$
$257$	1.82843	+	1.82843i	0.114054	+	0.114054i	−0.647776	+	0.761831i	$0.724301\pi$
$258$	−0.201010	−	0.142136i	−0.0125143	−	0.00884898i
$259$		−	1.17157i		−	0.0727980i
$260$		0			0
$261$	−0.828427	−	2.34315i	−0.0512784	−	0.145037i
$262$	2.00000	−	2.00000i	0.123560	−	0.123560i
$263$	1.75736	−	1.75736i	0.108363	−	0.108363i	−0.650846	−	0.759210i	$-0.725586\pi$
$263$	1.75736	−	1.75736i	0.108363	−	0.108363i	0.759210	+	0.650846i	$0.225586\pi$
$264$	2.70711	−	0.464466i	0.166611	−	0.0285859i
$265$		0			0
$266$			5.65685i			0.346844i
$267$	−9.65685	+	13.6569i	−0.590990	+	0.835786i
$268$	−11.7279	−	11.7279i	−0.716397	−	0.716397i
$269$	−14.0000			−0.853595			−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000			−0.853595			−0.426798	+	0.904347i	$0.640358\pi$
$270$		0			0
$271$	8.48528			0.515444			0.257722	−	0.966219i	$-0.417028\pi$
$271$	8.48528			0.515444			0.257722	+	0.966219i	$0.417028\pi$
$272$	−8.48528	−	8.48528i	−0.514496	−	0.514496i
$273$	13.6569	−	19.3137i	0.826550	−	1.16892i
$274$		−	0.100505i		−	0.00607173i
$275$		0			0
$276$	14.3137	−	2.45584i	0.861584	−	0.147824i
$277$	18.4853	−	18.4853i	1.11067	−	1.11067i	0.117613	−	0.993059i	$-0.462476\pi$
$277$	18.4853	−	18.4853i	1.11067	−	1.11067i	0.993059	−	0.117613i	$-0.0375244\pi$
$278$	−5.17157	+	5.17157i	−0.310170	+	0.310170i
$279$	3.17157	+	8.97056i	0.189877	+	0.537054i
$280$		0			0
$281$		−	28.6274i		−	1.70777i	−0.520463	−	0.853884i	$-0.674241\pi$
$281$		−	28.6274i		−	1.70777i	0.520463	−	0.853884i	$-0.325759\pi$
$282$	6.00000	+	4.24264i	0.357295	+	0.252646i
$283$	15.0711	+	15.0711i	0.895882	+	0.895882i	0.995069	−	0.0991868i	$-0.0316241\pi$
$283$	15.0711	+	15.0711i	0.895882	+	0.895882i	−0.0991868	+	0.995069i	$0.531624\pi$
$284$	4.54416			0.269646
$285$		0			0
$286$	−1.17157			−0.0692766
$287$	26.1421	+	26.1421i	1.54312	+	1.54312i
$288$	−5.70711	+	11.9497i	−0.336294	+	0.704146i
$289$		−	1.00000i		−	0.0588235i
$290$		0			0
$291$	4.41421	+	25.7279i	0.258766	+	1.50820i
$292$		0			0
$293$	3.65685	−	3.65685i	0.213636	−	0.213636i	−0.592174	−	0.805810i	$-0.701730\pi$
$293$	3.65685	−	3.65685i	0.213636	−	0.213636i	0.805810	+	0.592174i	$0.201730\pi$
$294$	−1.97918	−	11.5355i	−0.115428	−	0.672766i
$295$		0			0
$296$		−	0.384776i		−	0.0223647i
$297$	1.41421	−	5.00000i	0.0820610	−	0.290129i
$298$	−2.92893	−	2.92893i	−0.169668	−	0.169668i
$299$	−12.9706			−0.750107
$300$		0			0
$301$	−1.65685			−0.0954995
$302$	2.97056	+	2.97056i	0.170937	+	0.170937i
$303$	10.1421	+	7.17157i	0.582650	+	0.411996i
$304$		−	8.48528i		−	0.486664i
$305$		0			0
$306$	4.68629	−	1.65685i	0.267897	−	0.0947161i
$307$	11.8995	−	11.8995i	0.679140	−	0.679140i	−0.280666	−	0.959806i	$-0.590555\pi$
$307$	11.8995	−	11.8995i	0.679140	−	0.679140i	0.959806	+	0.280666i	$0.0905552\pi$
$308$	6.24264	−	6.24264i	0.355707	−	0.355707i
$309$	15.4853	−	2.65685i	0.880927	−	0.151143i
$310$		0			0
$311$		−	28.1421i		−	1.59579i	−0.602794	−	0.797897i	$-0.705946\pi$
$311$		−	28.1421i		−	1.59579i	0.602794	−	0.797897i	$-0.294054\pi$
$312$	4.48528	−	6.34315i	0.253929	−	0.359110i
$313$	−12.6569	−	12.6569i	−0.715408	−	0.715408i	0.252253	−	0.967661i	$-0.418828\pi$
$313$	−12.6569	−	12.6569i	−0.715408	−	0.715408i	−0.967661	+	0.252253i	$0.918828\pi$
$314$	5.55635			0.313563
$315$		0			0
$316$	−19.7990			−1.11378
$317$	−2.31371	−	2.31371i	−0.129951	−	0.129951i	0.639140	−	0.769091i	$-0.279291\pi$
$317$	−2.31371	−	2.31371i	−0.129951	−	0.129951i	−0.769091	+	0.639140i	$0.779291\pi$
$318$	−0.585786	+	0.828427i	−0.0328493	+	0.0464559i
$319$			0.828427i			0.0463830i
$320$		0			0
$321$	−0.585786	+	0.100505i	−0.0326954	+	0.00560965i
$322$	−6.48528	+	6.48528i	−0.361411	+	0.361411i
$323$	−8.00000	+	8.00000i	−0.445132	+	0.445132i
$324$	10.3431	+	12.7990i	0.574619	+	0.711055i
$325$		0			0
$326$		−	2.10051i		−	0.116336i
$327$	−4.48528	−	3.17157i	−0.248037	−	0.175388i
$328$	8.58579	+	8.58579i	0.474071	+	0.474071i
$329$	49.4558			2.72659
$330$		0			0
$331$	−6.48528			−0.356463			−0.178232	−	0.983989i	$-0.557038\pi$
$331$	−6.48528			−0.356463			−0.178232	+	0.983989i	$0.557038\pi$
$332$	−16.5858	−	16.5858i	−0.910263	−	0.910263i
$333$	−0.656854	−	0.313708i	−0.0359954	−	0.0171911i
$334$		−	8.62742i		−	0.472071i
$335$		0			0
$336$	4.24264	+	24.7279i	0.231455	+	1.34902i
$337$	10.8284	−	10.8284i	0.589862	−	0.589862i	−0.347732	−	0.937594i	$-0.613048\pi$
$337$	10.8284	−	10.8284i	0.589862	−	0.589862i	0.937594	+	0.347732i	$0.113048\pi$
$338$	1.46447	−	1.46447i	0.0796565	−	0.0796565i
$339$	−2.41421	−	14.0711i	−0.131122	−	0.764235i
$340$		0			0
$341$		−	3.17157i		−	0.171750i
$342$	3.17157	+	1.51472i	0.171499	+	0.0819066i
$343$	−31.7990	−	31.7990i	−1.71698	−	1.71698i
$344$	−0.544156			−0.0293389
$345$		0			0
$346$	−4.97056			−0.267219
$347$	−1.89949	−	1.89949i	−0.101970	−	0.101970i	0.654281	−	0.756251i	$-0.272971\pi$
$347$	−1.89949	−	1.89949i	−0.101970	−	0.101970i	−0.756251	+	0.654281i	$0.772971\pi$
$348$	−2.14214	−	1.51472i	−0.114831	−	0.0811974i
$349$		−	30.0000i		−	1.60586i	−0.596071	−	0.802932i	$-0.703272\pi$
$349$		−	30.0000i		−	1.60586i	0.596071	−	0.802932i	$-0.296728\pi$
$350$		0			0
$351$	−7.17157	−	12.8284i	−0.382790	−	0.684731i
$352$	3.12132	−	3.12132i	0.166367	−	0.166367i
$353$	−7.82843	+	7.82843i	−0.416665	+	0.416665i	−0.884053	−	0.467387i	$-0.845195\pi$
$353$	−7.82843	+	7.82843i	−0.416665	+	0.416665i	0.467387	+	0.884053i	$0.345195\pi$
$354$	−2.82843	+	0.485281i	−0.150329	+	0.0257924i
$355$		0			0
$356$			17.6569i			0.935811i
$357$	19.3137	−	27.3137i	1.02219	−	1.44559i
$358$	7.07107	+	7.07107i	0.373718	+	0.373718i
$359$	−22.1421			−1.16862			−0.584309	−	0.811532i	$-0.698634\pi$
$359$	−22.1421			−1.16862			−0.584309	+	0.811532i	$0.698634\pi$
$360$		0			0
$361$	11.0000			0.578947
$362$	1.65685	+	1.65685i	0.0870823	+	0.0870823i
$363$	−1.00000	+	1.41421i	−0.0524864	+	0.0742270i
$364$		−	24.9706i		−	1.30881i
$365$		0			0
$366$	−4.58579	+	0.786797i	−0.239703	+	0.0411265i
$367$	1.10051	−	1.10051i	0.0574459	−	0.0574459i	−0.677800	−	0.735246i	$-0.737067\pi$
$367$	1.10051	−	1.10051i	0.0574459	−	0.0574459i	0.735246	+	0.677800i	$0.237067\pi$
$368$	9.72792	−	9.72792i	0.507103	−	0.507103i
$369$	21.6569	−	7.65685i	1.12741	−	0.398600i
$370$		0			0
$371$			6.82843i			0.354514i
$372$	8.20101	+	5.79899i	0.425203	+	0.300664i
$373$	−3.51472	−	3.51472i	−0.181985	−	0.181985i	0.610235	−	0.792220i	$-0.291075\pi$
$373$	−3.51472	−	3.51472i	−0.181985	−	0.181985i	−0.792220	+	0.610235i	$0.791075\pi$
$374$	−1.65685			−0.0856739
$375$		0			0
$376$	16.2426			0.837650
$377$	1.65685	+	1.65685i	0.0853323	+	0.0853323i
$378$	−10.0000	−	2.82843i	−0.514344	−	0.145479i
$379$			28.1421i			1.44556i	0.691076	+	0.722782i	$0.257137\pi$
$379$			28.1421i			1.44556i	−0.691076	+	0.722782i	$0.742863\pi$
$380$		0			0
$381$	0.585786	+	3.41421i	0.0300107	+	0.174915i
$382$	3.31371	−	3.31371i	0.169544	−	0.169544i
$383$	6.07107	−	6.07107i	0.310217	−	0.310217i	−0.534776	−	0.844994i	$-0.679604\pi$
$383$	6.07107	−	6.07107i	0.310217	−	0.310217i	0.844994	+	0.534776i	$0.179604\pi$
$384$	3.09188	+	18.0208i	0.157782	+	0.919621i
$385$		0			0
$386$		−	4.68629i		−	0.238526i
$387$	−0.443651	+	0.928932i	−0.0225520	+	0.0472203i
$388$	19.4853	+	19.4853i	0.989215	+	0.989215i
$389$	−17.3137			−0.877840			−0.438920	−	0.898526i	$-0.644639\pi$
$389$	−17.3137			−0.877840			−0.438920	+	0.898526i	$0.644639\pi$
$390$		0			0
$391$	−18.3431			−0.927653
$392$	−18.2929	−	18.2929i	−0.923931	−	0.923931i
$393$	−9.65685	−	6.82843i	−0.487124	−	0.344449i
$394$			8.28427i			0.417356i
$395$		0			0
$396$	−1.82843	−	5.17157i	−0.0918819	−	0.259881i
$397$	−6.17157	+	6.17157i	−0.309742	+	0.309742i	−0.844810	−	0.535067i	$-0.820286\pi$
$397$	−6.17157	+	6.17157i	−0.309742	+	0.309742i	0.535067	+	0.844810i	$0.320286\pi$
$398$	−4.24264	+	4.24264i	−0.212664	+	0.212664i
$399$	23.3137	−	4.00000i	1.16715	−	0.200250i
$400$		0			0
$401$		−	35.6569i		−	1.78062i	−0.455357	−	0.890309i	$-0.650488\pi$
$401$		−	35.6569i		−	1.78062i	0.455357	−	0.890309i	$-0.349512\pi$
$402$	3.75736	−	5.31371i	0.187400	−	0.265024i
$403$	−6.34315	−	6.34315i	−0.315975	−	0.315975i
$404$	13.1127			0.652381
$405$		0			0
$406$	1.65685			0.0822283
$407$	0.171573	+	0.171573i	0.00850455	+	0.00850455i
$408$	6.34315	−	8.97056i	0.314033	−	0.444109i
$409$		−	18.4853i		−	0.914038i	−0.889457	−	0.457019i	$-0.848917\pi$
$409$		−	18.4853i		−	0.914038i	0.889457	−	0.457019i	$-0.151083\pi$
$410$		0			0
$411$	−0.414214	+	0.0710678i	−0.0204316	+	0.00350552i
$412$	11.7279	−	11.7279i	0.577793	−	0.577793i
$413$	−13.6569	+	13.6569i	−0.672010	+	0.672010i
$414$	1.89949	+	5.37258i	0.0933551	+	0.264048i
$415$		0			0
$416$		−	12.4853i		−	0.612141i
$417$	24.9706	+	17.6569i	1.22281	+	0.864660i
$418$	−0.828427	−	0.828427i	−0.0405197	−	0.0405197i
$419$	15.4558			0.755067			0.377534	−	0.925996i	$-0.376772\pi$
$419$	15.4558			0.755067			0.377534	+	0.925996i	$0.376772\pi$
$420$		0			0
$421$	−4.00000			−0.194948			−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000			−0.194948			−0.0974740	+	0.995238i	$0.531076\pi$
$422$	−1.31371	−	1.31371i	−0.0639503	−	0.0639503i
$423$	13.2426	−	27.7279i	0.643879	−	1.34818i
$424$			2.24264i			0.108912i
$425$		0			0
$426$	0.301515	+	1.75736i	0.0146085	+	0.0851443i
$427$	−22.1421	+	22.1421i	−1.07153	+	1.07153i
$428$	−0.443651	+	0.443651i	−0.0214447	+	0.0214447i
$429$	0.828427	+	4.82843i	0.0399968	+	0.233119i
$430$		0			0
$431$			6.34315i			0.305539i	0.988262	+	0.152769i	$0.0488191\pi$
$431$			6.34315i			0.305539i	−0.988262	+	0.152769i	$0.951181\pi$
$432$	15.0000	+	4.24264i	0.721688	+	0.204124i
$433$	0.313708	+	0.313708i	0.0150759	+	0.0150759i	0.714605	−	0.699529i	$-0.246607\pi$
$433$	0.313708	+	0.313708i	0.0150759	+	0.0150759i	−0.699529	+	0.714605i	$0.746607\pi$
$434$	−6.34315			−0.304481
$435$		0			0
$436$	−5.79899			−0.277721
$437$	−9.17157	−	9.17157i	−0.438736	−	0.438736i
$438$		0			0
$439$		−	7.31371i		−	0.349064i	−0.984651	−	0.174532i	$-0.944159\pi$
$439$		−	7.31371i		−	0.349064i	0.984651	−	0.174532i	$-0.0558413\pi$
$440$		0			0
$441$	−46.1421	+	16.3137i	−2.19724	+	0.776843i
$442$	−3.31371	+	3.31371i	−0.157617	+	0.157617i
$443$	20.7574	−	20.7574i	0.986212	−	0.986212i	−0.0136943	−	0.999906i	$-0.504359\pi$
$443$	20.7574	−	20.7574i	0.986212	−	0.986212i	0.999906	+	0.0136943i	$0.00435918\pi$
$444$	−0.757359	+	0.129942i	−0.0359427	+	0.00616679i
$445$		0			0
$446$			9.69848i			0.459237i
$447$	−10.0000	+	14.1421i	−0.472984	+	0.668900i
$448$	14.2426	+	14.2426i	0.672902	+	0.672902i
$449$	−3.02944			−0.142968			−0.0714840	−	0.997442i	$-0.522773\pi$
$449$	−3.02944			−0.142968			−0.0714840	+	0.997442i	$0.522773\pi$
$450$		0			0
$451$	−7.65685			−0.360547
$452$	−10.6569	−	10.6569i	−0.501256	−	0.501256i
$453$	10.1421	−	14.3431i	0.476519	−	0.673900i
$454$		−	6.00000i		−	0.281594i
$455$		0			0
$456$	7.65685	−	1.31371i	0.358565	−	0.0615200i
$457$	6.48528	−	6.48528i	0.303369	−	0.303369i	−0.538962	−	0.842330i	$-0.681183\pi$
$457$	6.48528	−	6.48528i	0.303369	−	0.303369i	0.842330	+	0.538962i	$0.181183\pi$
$458$	−2.82843	+	2.82843i	−0.132164	+	0.132164i
$459$	−10.1421	−	18.1421i	−0.473394	−	0.846802i
$460$		0			0
$461$			13.5147i			0.629443i	0.949184	+	0.314722i	$0.101911\pi$
$461$			13.5147i			0.629443i	−0.949184	+	0.314722i	$0.898089\pi$
$462$	2.82843	+	2.00000i	0.131590	+	0.0930484i
$463$	−11.7279	−	11.7279i	−0.545043	−	0.545043i	0.379960	−	0.925003i	$-0.375938\pi$
$463$	−11.7279	−	11.7279i	−0.545043	−	0.545043i	−0.925003	+	0.379960i	$0.875938\pi$
$464$	−2.48528			−0.115376
$465$		0			0
$466$	−11.5147			−0.533409
$467$	0.414214	+	0.414214i	0.0191675	+	0.0191675i	0.716626	−	0.697458i	$-0.245686\pi$
$467$	0.414214	+	0.414214i	0.0191675	+	0.0191675i	−0.697458	+	0.716626i	$0.745686\pi$
$468$	−14.0000	−	6.68629i	−0.647150	−	0.309074i
$469$		−	43.7990i		−	2.02245i
$470$		0			0
$471$	−3.92893	−	22.8995i	−0.181036	−	1.05515i
$472$	−4.48528	+	4.48528i	−0.206452	+	0.206452i
$473$	0.242641	−	0.242641i	0.0111566	−	0.0111566i
$474$	−1.31371	−	7.65685i	−0.0603406	−	0.351691i
$475$		0			0
$476$		−	35.3137i		−	1.61860i
$477$	3.82843	+	1.82843i	0.175292	+	0.0837179i
$478$	−6.97056	−	6.97056i	−0.318826	−	0.318826i
$479$	−30.1421			−1.37723			−0.688615	−	0.725127i	$-0.741781\pi$
$479$	−30.1421			−1.37723			−0.688615	+	0.725127i	$0.741781\pi$
$480$		0			0
$481$	0.686292			0.0312922
$482$	0.0416306	+	0.0416306i	0.00189622	+	0.00189622i
$483$	31.3137	+	22.1421i	1.42482	+	1.00750i
$484$			1.82843i			0.0831103i
$485$		0			0
$486$	−4.26346	+	4.84924i	−0.193394	+	0.219966i
$487$	−11.7279	+	11.7279i	−0.531443	+	0.531443i	−0.921002	−	0.389559i	$-0.872628\pi$
$487$	−11.7279	+	11.7279i	−0.531443	+	0.531443i	0.389559	+	0.921002i	$0.372628\pi$
$488$	−7.27208	+	7.27208i	−0.329192	+	0.329192i
$489$	−8.65685	+	1.48528i	−0.391476	+	0.0671667i
$490$		0			0
$491$			20.0000i			0.902587i	0.892375	+	0.451294i	$0.149037\pi$
$491$			20.0000i			0.902587i	−0.892375	+	0.451294i	$0.850963\pi$
$492$	14.0000	−	19.7990i	0.631169	−	0.892607i
$493$	2.34315	+	2.34315i	0.105530	+	0.105530i
$494$	−3.31371			−0.149091
$495$		0			0
$496$	9.51472			0.427223
$497$	8.48528	+	8.48528i	0.380617	+	0.380617i
$498$	5.31371	−	7.51472i	0.238113	−	0.336743i
$499$			24.8284i			1.11147i	0.831358	+	0.555737i	$0.187564\pi$
$499$			24.8284i			1.11147i	−0.831358	+	0.555737i	$0.812436\pi$
$500$		0			0
$501$	−35.5563	+	6.10051i	−1.58854	+	0.272550i
$502$	−4.72792	+	4.72792i	−0.211017	+	0.211017i
$503$	−0.928932	+	0.928932i	−0.0414190	+	0.0414190i	−0.727513	−	0.686094i	$-0.759324\pi$
$503$	−0.928932	+	0.928932i	−0.0414190	+	0.0414190i	0.686094	+	0.727513i	$0.259324\pi$
$504$	−21.6569	+	7.65685i	−0.964673	+	0.341063i
$505$		0			0
$506$		−	1.89949i		−	0.0844428i
$507$	−7.07107	−	5.00000i	−0.314037	−	0.222058i
$508$	2.58579	+	2.58579i	0.114726	+	0.114726i
$509$	35.6569			1.58046			0.790231	−	0.612809i	$-0.209960\pi$
$509$	35.6569			1.58046			0.790231	+	0.612809i	$0.209960\pi$
$510$		0			0
$511$		0			0
$512$	16.0919	+	16.0919i	0.711167	+	0.711167i
$513$	4.00000	−	14.1421i	0.176604	−	0.624391i
$514$			1.07107i			0.0472428i
$515$		0			0
$516$	0.183766	+	1.07107i	0.00808986	+	0.0471511i
$517$	−7.24264	+	7.24264i	−0.318531	+	0.318531i
$518$	0.343146	−	0.343146i	0.0150770	−	0.0150770i
$519$	3.51472	+	20.4853i	0.154279	+	0.899204i
$520$		0			0
$521$			27.6569i			1.21167i	0.795591	+	0.605834i	$0.207161\pi$
$521$			27.6569i			1.21167i	−0.795591	+	0.605834i	$0.792839\pi$
$522$	0.443651	−	0.928932i	0.0194181	−	0.0406583i
$523$	27.2132	+	27.2132i	1.18995	+	1.18995i	0.977081	+	0.212870i	$0.0682810\pi$
$523$	27.2132	+	27.2132i	1.18995	+	1.18995i	0.212870	+	0.977081i	$0.431719\pi$
$524$	−12.4853			−0.545422
$525$		0			0
$526$	1.02944			0.0448856
$527$	−8.97056	−	8.97056i	−0.390764	−	0.390764i
$528$	−4.24264	−	3.00000i	−0.184637	−	0.130558i
$529$			1.97056i			0.0856766i
$530$		0			0
$531$	4.00000	+	11.3137i	0.173585	+	0.490973i
$532$	17.6569	−	17.6569i	0.765522	−	0.765522i
$533$	−15.3137	+	15.3137i	−0.663310	+	0.663310i
$534$	−6.82843	+	1.17157i	−0.295495	+	0.0506989i
$535$		0			0
$536$		−	14.3848i		−	0.621328i
$537$	24.1421	−	34.1421i	1.04181	−	1.47334i
$538$	−4.10051	−	4.10051i	−0.176785	−	0.176785i
$539$	16.3137			0.702681
$540$		0			0
$541$	−10.6863			−0.459440			−0.229720	−	0.973257i	$-0.573781\pi$
$541$	−10.6863			−0.459440			−0.229720	+	0.973257i	$0.573781\pi$
$542$	2.48528	+	2.48528i	0.106752	+	0.106752i
$543$	5.65685	−	8.00000i	0.242759	−	0.343313i
$544$		−	17.6569i		−	0.757031i
$545$		0			0
$546$	9.65685	−	1.65685i	0.413275	−	0.0709068i
$547$	−2.10051	+	2.10051i	−0.0898111	+	0.0898111i	−0.750585	−	0.660774i	$-0.770228\pi$
$547$	−2.10051	+	2.10051i	−0.0898111	+	0.0898111i	0.660774	+	0.750585i	$0.270228\pi$
$548$	−0.313708	+	0.313708i	−0.0134010	+	0.0134010i
$549$	6.48528	+	18.3431i	0.276785	+	0.782866i
$550$		0			0
$551$			2.34315i			0.0998214i
$552$	10.2843	+	7.27208i	0.437728	+	0.309520i
$553$	−36.9706	−	36.9706i	−1.57215	−	1.57215i
$554$	10.8284			0.460056
$555$		0			0
$556$	32.2843			1.36916
$557$	−8.97056	−	8.97056i	−0.380095	−	0.380095i	0.491041	−	0.871136i	$-0.336616\pi$
$557$	−8.97056	−	8.97056i	−0.380095	−	0.380095i	−0.871136	+	0.491041i	$0.836616\pi$
$558$	−1.69848	+	3.55635i	−0.0719026	+	0.150552i
$559$		−	0.970563i		−	0.0410504i
$560$		0			0
$561$	1.17157	+	6.82843i	0.0494638	+	0.288296i
$562$	8.38478	−	8.38478i	0.353690	−	0.353690i
$563$	9.89949	−	9.89949i	0.417214	−	0.417214i	−0.467028	−	0.884242i	$-0.654675\pi$
$563$	9.89949	−	9.89949i	0.417214	−	0.417214i	0.884242	+	0.467028i	$0.154675\pi$
$564$	−5.48528	−	31.9706i	−0.230972	−	1.34620i
$565$		0			0
$566$			8.82843i			0.371086i
$567$	−4.58579	+	43.2132i	−0.192585	+	1.81478i
$568$	2.78680	+	2.78680i	0.116931	+	0.116931i
$569$	−28.3431			−1.18821			−0.594103	−	0.804389i	$-0.702493\pi$
$569$	−28.3431			−1.18821			−0.594103	+	0.804389i	$0.702493\pi$
$570$		0			0
$571$	29.6569			1.24110			0.620550	−	0.784167i	$-0.286909\pi$
$571$	29.6569			1.24110			0.620550	+	0.784167i	$0.286909\pi$
$572$	3.65685	+	3.65685i	0.152901	+	0.152901i
$573$	−16.0000	−	11.3137i	−0.668410	−	0.472637i
$574$			15.3137i			0.639182i
$575$		0			0
$576$	11.7990	−	4.17157i	0.491625	−	0.173816i
$577$	−17.0000	+	17.0000i	−0.707719	+	0.707719i	−0.966055	−	0.258336i	$-0.916826\pi$
$577$	−17.0000	+	17.0000i	−0.707719	+	0.707719i	0.258336	+	0.966055i	$0.416826\pi$
$578$	0.292893	−	0.292893i	0.0121828	−	0.0121828i
$579$	−19.3137	+	3.31371i	−0.802650	+	0.137713i
$580$		0			0
$581$		−	61.9411i		−	2.56975i
$582$	−6.24264	+	8.82843i	−0.258766	+	0.365950i
$583$	−1.00000	−	1.00000i	−0.0414158	−	0.0414158i
$584$		0			0
$585$		0			0
$586$	2.14214			0.0884908
$587$	0.414214	+	0.414214i	0.0170964	+	0.0170964i	0.715603	−	0.698507i	$-0.246152\pi$
$587$	0.414214	+	0.414214i	0.0170964	+	0.0170964i	−0.698507	+	0.715603i	$0.746152\pi$
$588$	−29.8284	+	42.1838i	−1.23010	+	1.73963i
$589$		−	8.97056i		−	0.369626i
$590$		0			0
$591$	34.1421	−	5.85786i	1.40442	−	0.240960i
$592$	−0.514719	+	0.514719i	−0.0211548	+	0.0211548i
$593$	−5.85786	+	5.85786i	−0.240554	+	0.240554i	−0.817079	−	0.576525i	$-0.804408\pi$
$593$	−5.85786	+	5.85786i	−0.240554	+	0.240554i	0.576525	+	0.817079i	$0.304408\pi$
$594$	1.87868	−	1.05025i	0.0770832	−	0.0430924i
$595$		0			0
$596$			18.2843i			0.748953i
$597$	20.4853	+	14.4853i	0.838407	+	0.592843i
$598$	−3.79899	−	3.79899i	−0.155352	−	0.155352i
$599$	−3.45584			−0.141202			−0.0706010	−	0.997505i	$-0.522492\pi$
$599$	−3.45584			−0.141202			−0.0706010	+	0.997505i	$0.522492\pi$
$600$		0			0
$601$	−27.4558			−1.11995			−0.559974	−	0.828510i	$-0.689189\pi$
$601$	−27.4558			−1.11995			−0.559974	+	0.828510i	$0.689189\pi$
$602$	−0.485281	−	0.485281i	−0.0197786	−	0.0197786i
$603$	−24.5563	−	11.7279i	−1.00001	−	0.477598i
$604$		−	18.5442i		−	0.754551i
$605$		0			0
$606$	0.870058	+	5.07107i	0.0353437	+	0.205998i
$607$	21.8995	−	21.8995i	0.888873	−	0.888873i	−0.105542	−	0.994415i	$-0.533658\pi$
$607$	21.8995	−	21.8995i	0.888873	−	0.888873i	0.994415	+	0.105542i	$0.0336577\pi$
$608$	8.82843	−	8.82843i	0.358040	−	0.358040i
$609$	−1.17157	−	6.82843i	−0.0474745	−	0.276702i
$610$		0			0
$611$			28.9706i			1.17202i
$612$	−19.7990	−	9.45584i	−0.800327	−	0.382230i
$613$	16.0000	+	16.0000i	0.646234	+	0.646234i	0.952081	−	0.305847i	$-0.0989395\pi$
$613$	16.0000	+	16.0000i	0.646234	+	0.646234i	−0.305847	+	0.952081i	$0.598940\pi$
$614$	6.97056			0.281309
$615$		0			0
$616$	7.65685			0.308503
$617$	−29.8284	−	29.8284i	−1.20085	−	1.20085i	−0.973909	−	0.226938i	$-0.927129\pi$
$617$	−29.8284	−	29.8284i	−1.20085	−	1.20085i	−0.226938	−	0.973909i	$-0.572871\pi$
$618$	5.31371	+	3.75736i	0.213749	+	0.151143i
$619$		−	0.686292i		−	0.0275844i	−0.999905	−	0.0137922i	$-0.995610\pi$
$619$		−	0.686292i		−	0.0275844i	0.999905	−	0.0137922i	$-0.00439033\pi$
$620$		0			0
$621$	20.7990	−	11.6274i	0.834635	−	0.466592i
$622$	8.24264	−	8.24264i	0.330500	−	0.330500i
$623$	−32.9706	+	32.9706i	−1.32094	+	1.32094i
$624$	−14.4853	+	2.48528i	−0.579875	+	0.0994909i
$625$		0			0
$626$		−	7.41421i		−	0.296332i
$627$	−2.82843	+	4.00000i	−0.112956	+	0.159745i
$628$	−17.3431	−	17.3431i	−0.692067	−	0.692067i
$629$	0.970563			0.0386989
$630$		0			0
$631$	−16.9706			−0.675587			−0.337794	−	0.941220i	$-0.609681\pi$
$631$	−16.9706			−0.675587			−0.337794	+	0.941220i	$0.609681\pi$
$632$	−12.1421	−	12.1421i	−0.482988	−	0.482988i
$633$	−4.48528	+	6.34315i	−0.178274	+	0.252117i
$634$		−	1.35534i		−	0.0538274i
$635$		0			0
$636$	4.41421	−	0.757359i	0.175035	−	0.0300313i
$637$	32.6274	−	32.6274i	1.29275	−	1.29275i
$638$	−0.242641	+	0.242641i	−0.00960624	+	0.00960624i
$639$	7.02944	−	2.48528i	0.278080	−	0.0983162i
$640$		0			0
$641$			32.6274i			1.28871i	0.764728	+	0.644353i	$0.222873\pi$
$641$			32.6274i			1.28871i	−0.764728	+	0.644353i	$0.777127\pi$
$642$	−0.201010	−	0.142136i	−0.00793324	−	0.00560965i
$643$	9.24264	+	9.24264i	0.364494	+	0.364494i	0.865464	−	0.500970i	$-0.167023\pi$
$643$	9.24264	+	9.24264i	0.364494	+	0.364494i	−0.500970	+	0.865464i	$0.667023\pi$
$644$	40.4853			1.59534
$645$		0			0
$646$	−4.68629			−0.184380
$647$	−11.7279	−	11.7279i	−0.461072	−	0.461072i	0.437935	−	0.899007i	$-0.355710\pi$
$647$	−11.7279	−	11.7279i	−0.461072	−	0.461072i	−0.899007	+	0.437935i	$0.855710\pi$
$648$	−1.50610	+	14.1924i	−0.0591651	+	0.557530i
$649$		−	4.00000i		−	0.157014i
$650$		0			0
$651$	4.48528	+	26.1421i	0.175792	+	1.02459i
$652$	−6.55635	+	6.55635i	−0.256766	+	0.256766i
$653$	−8.65685	+	8.65685i	−0.338769	+	0.338769i	−0.855904	−	0.517135i	$-0.826999\pi$
$653$	−8.65685	+	8.65685i	−0.338769	+	0.338769i	0.517135	+	0.855904i	$0.326999\pi$
$654$	−0.384776	−	2.24264i	−0.0150459	−	0.0876942i
$655$		0			0
$656$		−	22.9706i		−	0.896850i
$657$		0			0
$658$	14.4853	+	14.4853i	0.564695	+	0.564695i
$659$	−24.9706			−0.972715			−0.486358	−	0.873760i	$-0.661675\pi$
$659$	−24.9706			−0.972715			−0.486358	+	0.873760i	$0.661675\pi$
$660$		0			0
$661$	14.0000			0.544537			0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000			0.544537			0.272268	+	0.962221i	$0.412226\pi$
$662$	−1.89949	−	1.89949i	−0.0738260	−	0.0738260i
$663$	16.0000	+	11.3137i	0.621389	+	0.439388i
$664$		−	20.3431i		−	0.789467i
$665$		0			0
$666$	−0.100505	−	0.284271i	−0.00389449	−	0.0110153i
$667$	−2.68629	+	2.68629i	−0.104014	+	0.104014i
$668$	−26.9289	+	26.9289i	−1.04191	+	1.04191i
$669$	39.9706	−	6.85786i	1.54535	−	0.265140i
$670$		0			0
$671$		−	6.48528i		−	0.250362i
$672$	−21.3137	+	30.1421i	−0.822194	+	1.16276i
$673$	22.3431	+	22.3431i	0.861265	+	0.861265i	0.991485	−	0.130220i	$-0.0415684\pi$
$673$	22.3431	+	22.3431i	0.861265	+	0.861265i	−0.130220	+	0.991485i	$0.541568\pi$
$674$	6.34315			0.244329
$675$		0			0
$676$	−9.14214			−0.351621
$677$	−19.6569	−	19.6569i	−0.755474	−	0.755474i	0.220021	−	0.975495i	$-0.429387\pi$
$677$	−19.6569	−	19.6569i	−0.755474	−	0.755474i	−0.975495	+	0.220021i	$0.929387\pi$
$678$	3.41421	−	4.82843i	0.131122	−	0.185435i
$679$			72.7696i			2.79264i
$680$		0			0
$681$	−24.7279	+	4.24264i	−0.947576	+	0.162578i
$682$	0.928932	−	0.928932i	0.0355707	−	0.0355707i
$683$	−7.72792	+	7.72792i	−0.295701	+	0.295701i	−0.839327	−	0.543627i	$-0.817051\pi$
$683$	−7.72792	+	7.72792i	−0.295701	+	0.295701i	0.543627	+	0.839327i	$0.317051\pi$
$684$	−5.17157	−	14.6274i	−0.197740	−	0.559293i
$685$		0			0
$686$		−	18.6274i		−	0.711198i
$687$	13.6569	+	9.65685i	0.521041	+	0.368432i
$688$	0.727922	+	0.727922i	0.0277518	+	0.0277518i
$689$	−4.00000			−0.152388
$690$		0			0
$691$	20.0000			0.760836			0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000			0.760836			0.380418	+	0.924815i	$0.375780\pi$
$692$	15.5147	+	15.5147i	0.589781	+	0.589781i
$693$	6.24264	−	13.0711i	0.237138	−	0.496529i
$694$		−	1.11270i		−	0.0422375i
$695$		0			0
$696$	−0.384776	−	2.24264i	−0.0145849	−	0.0850071i
$697$	−21.6569	+	21.6569i	−0.820312	+	0.820312i
$698$	8.78680	−	8.78680i	0.332585	−	0.332585i
$699$	8.14214	+	47.4558i	0.307964	+	1.79494i
$700$		0			0
$701$		−	1.31371i		−	0.0496181i	−0.999692	−	0.0248090i	$-0.992102\pi$
$701$		−	1.31371i		−	0.0496181i	0.999692	−	0.0248090i	$-0.00789777\pi$
$702$	1.65685	−	5.85786i	0.0625339	−	0.221091i
$703$	0.485281	+	0.485281i	0.0183027	+	0.0183027i
$704$	−4.17157			−0.157222
$705$		0			0
$706$	−4.58579			−0.172588
$707$	24.4853	+	24.4853i	0.920864	+	0.920864i
$708$	10.3431	+	7.31371i	0.388719	+	0.274866i
$709$			29.3137i			1.10090i	0.834868	+	0.550450i	$0.185544\pi$
$709$			29.3137i			1.10090i	−0.834868	+	0.550450i	$0.814456\pi$
$710$		0			0
$711$	−30.6274	+	10.8284i	−1.14862	+	0.406098i
$712$	−10.8284	+	10.8284i	−0.405812	+	0.405812i
$713$	10.2843	−	10.2843i	0.385149	−	0.385149i
$714$	13.6569	−	2.34315i	0.511095	−	0.0876900i
$715$		0			0
$716$		−	44.1421i		−	1.64967i
$717$	−23.7990	+	33.6569i	−0.888790	+	1.25694i
$718$	−6.48528	−	6.48528i	−0.242029	−	0.242029i
$719$	−22.7696			−0.849161			−0.424581	−	0.905390i	$-0.639578\pi$
$719$	−22.7696			−0.849161			−0.424581	+	0.905390i	$0.639578\pi$
$720$		0			0
$721$	43.7990			1.63116
$722$	3.22183	+	3.22183i	0.119904	+	0.119904i
$723$	0.142136	−	0.201010i	0.00528608	−	0.00747565i
$724$		−	10.3431i		−	0.384400i
$725$		0			0
$726$	−0.707107	+	0.121320i	−0.0262432	+	0.00450262i
$727$	−20.2132	+	20.2132i	−0.749666	+	0.749666i	−0.974416	−	0.224750i	$-0.927843\pi$
$727$	−20.2132	+	20.2132i	−0.749666	+	0.749666i	0.224750	+	0.974416i	$0.427843\pi$
$728$	15.3137	−	15.3137i	0.567564	−	0.567564i
$729$	23.0000	+	14.1421i	0.851852	+	0.523783i
$730$		0			0
$731$		−	1.37258i		−	0.0507668i
$732$	16.7696	+	11.8579i	0.619821	+	0.438279i
$733$	−3.79899	−	3.79899i	−0.140319	−	0.140319i	0.633458	−	0.773777i	$-0.281635\pi$
$733$	−3.79899	−	3.79899i	−0.140319	−	0.140319i	−0.773777	+	0.633458i	$0.781635\pi$
$734$	0.644661			0.0237949
$735$		0			0
$736$	20.2426			0.746154
$737$	6.41421	+	6.41421i	0.236271	+	0.236271i
$738$	8.58579	+	4.10051i	0.316047	+	0.150942i
$739$		−	14.6274i		−	0.538078i	−0.963129	−	0.269039i	$-0.913294\pi$
$739$		−	14.6274i		−	0.538078i	0.963129	−	0.269039i	$-0.0867061\pi$
$740$		0			0
$741$	2.34315	+	13.6569i	0.0860776	+	0.501697i
$742$	−2.00000	+	2.00000i	−0.0734223	+	0.0734223i
$743$	26.0416	−	26.0416i	0.955375	−	0.955375i	−0.0436712	−	0.999046i	$-0.513905\pi$
$743$	26.0416	−	26.0416i	0.955375	−	0.955375i	0.999046	+	0.0436712i	$0.0139054\pi$
$744$	1.47309	+	8.58579i	0.0540060	+	0.314770i
$745$		0			0
$746$		−	2.05887i		−	0.0753808i
$747$	−34.7279	−	16.5858i	−1.27063	−	0.606842i
$748$	5.17157	+	5.17157i	0.189091	+	0.189091i
$749$	−1.65685			−0.0605401
$750$		0			0
$751$	−46.6274			−1.70146			−0.850729	−	0.525604i	$-0.823839\pi$
$751$	−46.6274			−1.70146			−0.850729	+	0.525604i	$0.823839\pi$
$752$	−21.7279	−	21.7279i	−0.792336	−	0.792336i
$753$	22.8284	+	16.1421i	0.831914	+	0.588252i
$754$			0.970563i			0.0353458i
$755$		0			0
$756$	22.3848	+	40.0416i	0.814126	+	1.45630i
$757$	−0.171573	+	0.171573i	−0.00623592	+	0.00623592i	−0.710218	−	0.703982i	$-0.751404\pi$
$757$	−0.171573	+	0.171573i	−0.00623592	+	0.00623592i	0.703982	+	0.710218i	$0.251404\pi$
$758$	−8.24264	+	8.24264i	−0.299386	+	0.299386i
$759$	−7.82843	+	1.34315i	−0.284154	+	0.0487531i
$760$		0			0
$761$		−	1.51472i		−	0.0549085i	−0.999623	−	0.0274543i	$-0.991260\pi$
$761$		−	1.51472i		−	0.0549085i	0.999623	−	0.0274543i	$-0.00874006\pi$
$762$	−0.828427	+	1.17157i	−0.0300107	+	0.0424416i
$763$	−10.8284	−	10.8284i	−0.392015	−	0.392015i
$764$	−20.6863			−0.748404
$765$		0			0
$766$	3.55635			0.128496
$767$	−8.00000	−	8.00000i	−0.288863	−	0.288863i
$768$	3.97056	−	5.61522i	0.143275	−	0.202622i
$769$			52.6274i			1.89779i	0.315587	+	0.948897i	$0.397799\pi$
$769$			52.6274i			1.89779i	−0.315587	+	0.948897i	$0.602201\pi$
$770$		0			0
$771$	4.41421	−	0.757359i	0.158974	−	0.0272756i
$772$	−14.6274	+	14.6274i	−0.526452	+	0.526452i
$773$	−10.6569	+	10.6569i	−0.383300	+	0.383300i	−0.872290	−	0.488989i	$-0.837366\pi$
$773$	−10.6569	+	10.6569i	−0.383300	+	0.383300i	0.488989	+	0.872290i	$0.337366\pi$
$774$	−0.402020	+	0.142136i	−0.0144503	+	0.00510896i
$775$		0			0
$776$			23.8995i			0.857942i
$777$	−1.65685	−	1.17157i	−0.0594393	−	0.0420299i
$778$	−5.07107	−	5.07107i	−0.181807	−	0.181807i
$779$	−21.6569			−0.775937
$780$		0			0
$781$	−2.48528			−0.0889304
$782$	−5.37258	−	5.37258i	−0.192123	−	0.192123i
$783$	−4.14214	−	1.17157i	−0.148028	−	0.0418686i
$784$			48.9411i			1.74790i
$785$		0			0
$786$	−0.828427	−	4.82843i	−0.0295490	−	0.172224i
$787$	28.5858	−	28.5858i	1.01897	−	1.01897i	0.0191567	−	0.999816i	$-0.493902\pi$
$787$	28.5858	−	28.5858i	1.01897	−	1.01897i	0.999816	−	0.0191567i	$-0.00609815\pi$
$788$	25.8579	−	25.8579i	0.921148	−	0.921148i
$789$	−0.727922	−	4.24264i	−0.0259147	−	0.151042i
$790$		0			0
$791$		−	39.7990i		−	1.41509i
$792$	2.05025	−	4.29289i	0.0728526	−	0.152541i
$793$	−12.9706	−	12.9706i	−0.460598	−	0.460598i
$794$	−3.61522			−0.128299
$795$		0			0
$796$	26.4853			0.938746
$797$	−25.9706	−	25.9706i	−0.919925	−	0.919925i	0.0770989	−	0.997023i	$-0.475434\pi$
$797$	−25.9706	−	25.9706i	−0.919925	−	0.919925i	−0.997023	+	0.0770989i	$0.975434\pi$
$798$	8.00000	+	5.65685i	0.283197	+	0.200250i
$799$			40.9706i			1.44943i
$800$		0			0
$801$	9.65685	+	27.3137i	0.341208	+	0.965082i
$802$	10.4437	−	10.4437i	0.368778	−	0.368778i
$803$		0			0
$804$	−28.3137	+	4.85786i	−0.998548	+	0.171324i
$805$		0			0
$806$		−	3.71573i		−	0.130881i
$807$	−14.0000	+	19.7990i	−0.492823	+	0.696957i
$808$	8.04163	+	8.04163i	0.282904	+	0.282904i
$809$	−15.8579			−0.557533			−0.278766	−	0.960359i	$-0.589926\pi$
$809$	−15.8579			−0.557533			−0.278766	+	0.960359i	$0.589926\pi$
$810$		0			0
$811$	−41.6569			−1.46277			−0.731385	−	0.681965i	$-0.761126\pi$
$811$	−41.6569			−1.46277			−0.731385	+	0.681965i	$0.761126\pi$
$812$	−5.17157	−	5.17157i	−0.181487	−	0.181487i
$813$	8.48528	−	12.0000i	0.297592	−	0.420858i
$814$			0.100505i			0.00352270i
$815$		0			0
$816$	−20.4853	+	3.51472i	−0.717128	+	0.123040i
$817$	0.686292	−	0.686292i	0.0240103	−	0.0240103i
$818$	5.41421	−	5.41421i	0.189304	−	0.189304i
$819$	−13.6569	−	38.6274i	−0.477209	−	1.34975i
$820$		0			0
$821$			0.343146i			0.0119759i	0.999982	+	0.00598793i	$0.00190603\pi$
$821$			0.343146i			0.0119759i	−0.999982	+	0.00598793i	$0.998094\pi$
$822$	−0.142136	−	0.100505i	−0.00495755	−	0.00350552i
$823$	18.4142	+	18.4142i	0.641879	+	0.641879i	0.951017	−	0.309138i	$-0.100041\pi$
$823$	18.4142	+	18.4142i	0.641879	+	0.641879i	−0.309138	+	0.951017i	$0.600041\pi$
$824$	14.3848			0.501117
$825$		0			0
$826$	−8.00000			−0.278356
$827$	28.0416	+	28.0416i	0.975103	+	0.975103i	0.999698	−	0.0245945i	$-0.00782945\pi$
$827$	28.0416	+	28.0416i	0.975103	+	0.975103i	−0.0245945	+	0.999698i	$0.507829\pi$
$828$	10.8406	−	22.6985i	0.376738	−	0.788827i
$829$		−	10.9706i		−	0.381023i	−0.981685	−	0.190512i	$-0.938985\pi$
$829$		−	10.9706i		−	0.381023i	0.981685	−	0.190512i	$-0.0610147\pi$
$830$		0			0
$831$	−7.65685	−	44.6274i	−0.265613	−	1.54811i
$832$	−8.34315	+	8.34315i	−0.289247	+	0.289247i
$833$	46.1421	−	46.1421i	1.59873	−	1.59873i
$834$	2.14214	+	12.4853i	0.0741761	+	0.432330i
$835$		0			0
$836$			5.17157i			0.178863i
$837$	15.8579	+	4.48528i	0.548128	+	0.155034i
$838$	4.52691	+	4.52691i	0.156380	+	0.156380i
$839$	−21.6569			−0.747678			−0.373839	−	0.927494i	$-0.621959\pi$
$839$	−21.6569			−0.747678			−0.373839	+	0.927494i	$0.621959\pi$
$840$		0			0
$841$	−28.3137			−0.976335
$842$	−1.17157	−	1.17157i	−0.0403751	−	0.0403751i
$843$	−40.4853	−	28.6274i	−1.39439	−	0.985981i
$844$			8.20101i			0.282290i
$845$		0			0
$846$	12.0000	−	4.24264i	0.412568	−	0.145865i
$847$	−3.41421	+	3.41421i	−0.117314	+	0.117314i
$848$	3.00000	−	3.00000i	0.103020	−	0.103020i
$849$	36.3848	−	6.24264i	1.24872	−	0.214247i
$850$		0			0
$851$			1.11270i			0.0381428i
$852$	4.54416	−	6.42641i	0.155680	−	0.220165i
$853$	−18.1421	−	18.1421i	−0.621175	−	0.621175i	0.324657	−	0.945832i	$-0.394751\pi$
$853$	−18.1421	−	18.1421i	−0.621175	−	0.621175i	−0.945832	+	0.324657i	$0.894751\pi$
$854$	−12.9706			−0.443844
$855$		0			0
$856$	−0.544156			−0.0185989
$857$	16.4853	+	16.4853i	0.563126	+	0.563126i	0.930194	−	0.367068i	$-0.119638\pi$
$857$	16.4853	+	16.4853i	0.563126	+	0.563126i	−0.367068	+	0.930194i	$0.619638\pi$
$858$	−1.17157	+	1.65685i	−0.0399968	+	0.0565641i
$859$		−	36.0000i		−	1.22830i	−0.789188	−	0.614152i	$-0.789498\pi$
$859$		−	36.0000i		−	1.22830i	0.789188	−	0.614152i	$-0.210502\pi$
$860$		0			0
$861$	63.1127	−	10.8284i	2.15088	−	0.369032i
$862$	−1.85786	+	1.85786i	−0.0632791	+	0.0632791i
$863$	−12.5563	+	12.5563i	−0.427423	+	0.427423i	−0.887750	−	0.460327i	$-0.847732\pi$
$863$	−12.5563	+	12.5563i	−0.427423	+	0.427423i	0.460327	+	0.887750i	$0.347732\pi$
$864$	11.1924	+	20.0208i	0.380773	+	0.681122i
$865$		0			0
$866$			0.183766i			0.00624463i
$867$	−1.41421	−	1.00000i	−0.0480292	−	0.0339618i
$868$	19.7990	+	19.7990i	0.672022	+	0.672022i
$869$	10.8284			0.367329
$870$		0			0
$871$	25.6569			0.869349
$872$	−3.55635	−	3.55635i	−0.120433	−	0.120433i
$873$	40.7990	+	19.4853i	1.38084	+	0.659477i
$874$		−	5.37258i		−	0.181730i
$875$		0			0
$876$		0			0
$877$	−6.62742	+	6.62742i	−0.223792	+	0.223792i	−0.810093	−	0.586301i	$-0.800584\pi$
$877$	−6.62742	+	6.62742i	−0.223792	+	0.223792i	0.586301	+	0.810093i	$0.300584\pi$
$878$	2.14214	−	2.14214i	0.0722936	−	0.0722936i
$879$	−1.51472	−	8.82843i	−0.0510902	−	0.297775i
$880$		0			0
$881$		−	12.9706i		−	0.436989i	−0.975838	−	0.218495i	$-0.929885\pi$
$881$		−	12.9706i		−	0.436989i	0.975838	−	0.218495i	$-0.0701146\pi$
$882$	−18.2929	−	8.73654i	−0.615954	−	0.294175i
$883$	3.58579	+	3.58579i	0.120671	+	0.120671i	0.764864	−	0.644192i	$-0.222806\pi$
$883$	3.58579	+	3.58579i	0.120671	+	0.120671i	−0.644192	+	0.764864i	$0.722806\pi$
$884$	20.6863			0.695755
$885$		0			0
$886$	12.1594			0.408502
$887$	9.75736	+	9.75736i	0.327620	+	0.327620i	0.851681	−	0.524061i	$-0.175584\pi$
$887$	9.75736	+	9.75736i	0.327620	+	0.327620i	−0.524061	+	0.851681i	$0.675584\pi$
$888$	−0.544156	−	0.384776i	−0.0182607	−	0.0129122i
$889$			9.65685i			0.323880i
$890$		0			0
$891$	−5.65685	−	7.00000i	−0.189512	−	0.234509i
$892$	30.2721	−	30.2721i	1.01358	−	1.01358i
$893$	−20.4853	+	20.4853i	−0.685514	+	0.685514i
$894$	−7.07107	+	1.21320i	−0.236492	+	0.0405756i
$895$		0			0
$896$			50.9706i			1.70281i
$897$	−12.9706	+	18.3431i	−0.433074	+	0.612460i
$898$	−0.887302	−	0.887302i	−0.0296096	−	0.0296096i
$899$	−2.62742			−0.0876293
$900$		0			0
$901$	−5.65685			−0.188457
$902$	−2.24264	−	2.24264i	−0.0746718	−	0.0746718i
$903$	−1.65685	+	2.34315i	−0.0551367	+	0.0779750i
$904$		−	13.0711i		−	0.434737i
$905$		0			0
$906$	7.17157	−	1.23045i	0.238260	−	0.0408789i
$907$	−29.0416	+	29.0416i	−0.964312	+	0.964312i	−0.999385	−	0.0350732i	$-0.988834\pi$
$907$	−29.0416	+	29.0416i	−0.964312	+	0.964312i	0.0350732	+	0.999385i	$0.488834\pi$
$908$	−18.7279	+	18.7279i	−0.621508	+	0.621508i
$909$	20.2843	−	7.17157i	0.672787	−	0.237866i
$910$		0			0
$911$			56.0000i			1.85536i	0.373373	+	0.927681i	$0.378201\pi$
$911$			56.0000i			1.85536i	−0.373373	+	0.927681i	$0.621799\pi$
$912$	−12.0000	−	8.48528i	−0.397360	−	0.280976i
$913$	9.07107	+	9.07107i	0.300209	+	0.300209i
$914$	3.79899			0.125659
$915$		0			0
$916$	17.6569			0.583399
$917$	−23.3137	−	23.3137i	−0.769886	−	0.769886i
$918$	2.34315	−	8.28427i	0.0773353	−	0.273422i
$919$			14.6274i			0.482514i	0.970461	+	0.241257i	$0.0775597\pi$
$919$			14.6274i			0.482514i	−0.970461	+	0.241257i	$0.922440\pi$
$920$		0			0
$921$	−4.92893	−	28.7279i	−0.162414	−	0.946617i
$922$	−3.95837	+	3.95837i	−0.130362	+	0.130362i
$923$	−4.97056	+	4.97056i	−0.163608	+	0.163608i
$924$	−2.58579	−	15.0711i	−0.0850661	−	0.495802i
$925$		0			0
$926$		−	6.87006i		−	0.225764i
$927$	11.7279	−	24.5563i	0.385195	−	0.806536i
$928$	−2.58579	−	2.58579i	−0.0848826	−	0.0848826i
$929$	18.0000			0.590561			0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000			0.590561			0.295280	+	0.955411i	$0.404587\pi$
$930$		0			0
$931$	46.1421			1.51225
$932$	35.9411	+	35.9411i	1.17729	+	1.17729i
$933$	−39.7990	−	28.1421i	−1.30296	−	0.921332i
$934$			0.242641i			0.00793945i
$935$		0			0
$936$	−4.48528	−	12.6863i	−0.146606	−	0.414664i
$937$	10.9706	−	10.9706i	0.358393	−	0.358393i	−0.504828	−	0.863220i	$-0.668444\pi$
$937$	10.9706	−	10.9706i	0.358393	−	0.358393i	0.863220	+	0.504828i	$0.168444\pi$
$938$	12.8284	−	12.8284i	0.418863	−	0.418863i
$939$	−30.5563	+	5.24264i	−0.997169	+	0.171087i
$940$		0			0
$941$		−	37.1127i		−	1.20984i	−0.796286	−	0.604920i	$-0.793205\pi$
$941$		−	37.1127i		−	1.20984i	0.796286	−	0.604920i	$-0.206795\pi$
$942$	5.55635	−	7.85786i	0.181036	−	0.256023i
$943$	−24.8284	−	24.8284i	−0.808525	−	0.808525i
$944$	12.0000			0.390567
$945$		0			0
$946$	0.142136			0.00462123
$947$	17.2426	+	17.2426i	0.560311	+	0.560311i	0.929396	−	0.369085i	$-0.120329\pi$
$947$	17.2426	+	17.2426i	0.560311	+	0.560311i	−0.369085	+	0.929396i	$0.620329\pi$
$948$	−19.7990	+	28.0000i	−0.643041	+	0.909398i
$949$		0			0
$950$		0			0
$951$	−5.58579	+	0.958369i	−0.181132	+	0.0310773i
$952$	21.6569	−	21.6569i	0.701903	−	0.701903i
$953$	−5.79899	+	5.79899i	−0.187848	+	0.187848i	−0.794765	−	0.606917i	$-0.792406\pi$
$953$	−5.79899	+	5.79899i	−0.187848	+	0.187848i	0.606917	+	0.794765i	$0.292406\pi$
$954$	0.585786	+	1.65685i	0.0189655	+	0.0536426i
$955$		0			0
$956$			43.5147i			1.40737i
$957$	1.17157	+	0.828427i	0.0378716	+	0.0267792i
$958$	−8.82843	−	8.82843i	−0.285234	−	0.285234i
$959$	−1.17157			−0.0378321
$960$		0			0
$961$	−20.9411			−0.675520
$962$	0.201010	+	0.201010i	0.00648083	+	0.00648083i
$963$	−0.443651	+	0.928932i	−0.0142964	+	0.0299344i
$964$		−	0.259885i		−	0.00837032i
$965$		0			0
$966$	2.68629	+	15.6569i	0.0864300	+	0.503751i
$967$	−38.3848	+	38.3848i	−1.23437	+	1.23437i	−0.272103	+	0.962268i	$0.587719\pi$
$967$	−38.3848	+	38.3848i	−1.23437	+	1.23437i	−0.962268	+	0.272103i	$0.912281\pi$
$968$	−1.12132	+	1.12132i	−0.0360406	+	0.0360406i
$969$	3.31371	+	19.3137i	0.106452	+	0.620446i
$970$		0			0
$971$			20.0000i			0.641831i	0.947108	+	0.320915i	$0.103990\pi$
$971$			20.0000i			0.641831i	−0.947108	+	0.320915i	$0.896010\pi$
$972$	28.4437	−	1.82843i	0.912331	−	0.0586468i
$973$	60.2843	+	60.2843i	1.93263	+	1.93263i
$974$	−6.87006			−0.220131
$975$		0			0
$976$	19.4558			0.622766
$977$	38.1127	+	38.1127i	1.21933	+	1.21933i	0.967866	+	0.251468i	$0.0809132\pi$
$977$	38.1127	+	38.1127i	1.21933	+	1.21933i	0.251468	+	0.967866i	$0.419087\pi$
$978$	−2.97056	−	2.10051i	−0.0949881	−	0.0671667i
$979$		−	9.65685i		−	0.308634i
$980$		0			0
$981$	−8.97056	+	3.17157i	−0.286408	+	0.101261i
$982$	−5.85786	+	5.85786i	−0.186932	+	0.186932i
$983$	−23.7279	+	23.7279i	−0.756803	+	0.756803i	−0.975739	−	0.218936i	$-0.929741\pi$
$983$	−23.7279	+	23.7279i	−0.756803	+	0.756803i	0.218936	+	0.975739i	$0.429741\pi$
$984$	20.7279	−	3.55635i	0.660782	−	0.113372i
$985$		0			0
$986$			1.37258i			0.0437119i
$987$	49.4558	−	69.9411i	1.57420	−	2.22625i
$988$	10.3431	+	10.3431i	0.329059	+	0.329059i
$989$	1.57359			0.0500374
$990$		0			0
$991$	24.8284			0.788701			0.394350	−	0.918960i	$-0.370970\pi$
$991$	24.8284			0.788701			0.394350	+	0.918960i	$0.370970\pi$
$992$	9.89949	+	9.89949i	0.314309	+	0.314309i
$993$	−6.48528	+	9.17157i	−0.205804	+	0.291051i
$994$			4.97056i			0.157657i
$995$		0			0
$996$	−40.0416	+	6.87006i	−1.26877	+	0.217686i
$997$	−36.7696	+	36.7696i	−1.16450	+	1.16450i	−0.181025	+	0.983479i	$0.557941\pi$
$997$	−36.7696	+	36.7696i	−1.16450	+	1.16450i	−0.983479	+	0.181025i	$0.942059\pi$
$998$	−7.27208	+	7.27208i	−0.230194	+	0.230194i
$999$	−1.10051	+	0.615224i	−0.0348184	+	0.0194648i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.2.k.f.518.1		4
3.2	odd	2		825.2.k.c.518.2		4
5.2	odd	4		825.2.k.c.782.2		4
5.3	odd	4		165.2.k.b.122.1	yes	4
5.4	even	2		165.2.k.a.23.2	✓	4
15.2	even	4	inner	825.2.k.f.782.1		4
15.8	even	4		165.2.k.a.122.2	yes	4
15.14	odd	2		165.2.k.b.23.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.k.a.23.2	✓	4	5.4	even	2
165.2.k.a.122.2	yes	4	15.8	even	4
165.2.k.b.23.1	yes	4	15.14	odd	2
165.2.k.b.122.1	yes	4	5.3	odd	4
825.2.k.c.518.2		4	3.2	odd	2
825.2.k.c.782.2		4	5.2	odd	4
825.2.k.f.518.1		4	1.1	even	1	trivial
825.2.k.f.782.1		4	15.2	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.292893 + 0.292893i) q^{2} +(1.00000 - 1.41421i) q^{3} -1.82843i q^{4} +(0.707107 - 0.121320i) q^{6} +(3.41421 - 3.41421i) q^{7} +(1.12132 - 1.12132i) q^{8} +(-1.00000 - 2.82843i) q^{9} +O(q^{10})\)	\(q+(0.292893 + 0.292893i) q^{2} +(1.00000 - 1.41421i) q^{3} -1.82843i q^{4} +(0.707107 - 0.121320i) q^{6} +(3.41421 - 3.41421i) q^{7} +(1.12132 - 1.12132i) q^{8} +(-1.00000 - 2.82843i) q^{9} +1.00000i q^{11} +(-2.58579 - 1.82843i) q^{12} +(2.00000 + 2.00000i) q^{13} +2.00000 q^{14} -3.00000 q^{16} +(2.82843 + 2.82843i) q^{17} +(0.535534 - 1.12132i) q^{18} +2.82843i q^{19} +(-1.41421 - 8.24264i) q^{21} +(-0.292893 + 0.292893i) q^{22} +(-3.24264 + 3.24264i) q^{23} +(-0.464466 - 2.70711i) q^{24} +1.17157i q^{26} +(-5.00000 - 1.41421i) q^{27} +(-6.24264 - 6.24264i) q^{28} +0.828427 q^{29} -3.17157 q^{31} +(-3.12132 - 3.12132i) q^{32} +(1.41421 + 1.00000i) q^{33} +1.65685i q^{34} +(-5.17157 + 1.82843i) q^{36} +(0.171573 - 0.171573i) q^{37} +(-0.828427 + 0.828427i) q^{38} +(4.82843 - 0.828427i) q^{39} +7.65685i q^{41} +(2.00000 - 2.82843i) q^{42} +(-0.242641 - 0.242641i) q^{43} +1.82843 q^{44} -1.89949 q^{46} +(7.24264 + 7.24264i) q^{47} +(-3.00000 + 4.24264i) q^{48} -16.3137i q^{49} +(6.82843 - 1.17157i) q^{51} +(3.65685 - 3.65685i) q^{52} +(-1.00000 + 1.00000i) q^{53} +(-1.05025 - 1.87868i) q^{54} -7.65685i q^{56} +(4.00000 + 2.82843i) q^{57} +(0.242641 + 0.242641i) q^{58} -4.00000 q^{59} -6.48528 q^{61} +(-0.928932 - 0.928932i) q^{62} +(-13.0711 - 6.24264i) q^{63} +4.17157i q^{64} +(0.121320 + 0.707107i) q^{66} +(6.41421 - 6.41421i) q^{67} +(5.17157 - 5.17157i) q^{68} +(1.34315 + 7.82843i) q^{69} +2.48528i q^{71} +(-4.29289 - 2.05025i) q^{72} +0.100505 q^{74} +5.17157 q^{76} +(3.41421 + 3.41421i) q^{77} +(1.65685 + 1.17157i) q^{78} -10.8284i q^{79} +(-7.00000 + 5.65685i) q^{81} +(-2.24264 + 2.24264i) q^{82} +(9.07107 - 9.07107i) q^{83} +(-15.0711 + 2.58579i) q^{84} -0.142136i q^{86} +(0.828427 - 1.17157i) q^{87} +(1.12132 + 1.12132i) q^{88} -9.65685 q^{89} +13.6569 q^{91} +(5.92893 + 5.92893i) q^{92} +(-3.17157 + 4.48528i) q^{93} +4.24264i q^{94} +(-7.53553 + 1.29289i) q^{96} +(-10.6569 + 10.6569i) q^{97} +(4.77817 - 4.77817i) q^{98} +(2.82843 - 1.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{3} + 8 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^3 + 8 * q^7 - 4 * q^8 - 4 * q^9	\( 4 q + 4 q^{2} + 4 q^{3} + 8 q^{7} - 4 q^{8} - 4 q^{9} - 16 q^{12} + 8 q^{13} + 8 q^{14} - 12 q^{16} - 12 q^{18} - 4 q^{22} + 4 q^{23} - 16 q^{24} - 20 q^{27} - 8 q^{28} - 8 q^{29} - 24 q^{31} - 4 q^{32} - 32 q^{36} + 12 q^{37} + 8 q^{38} + 8 q^{39} + 8 q^{42} + 16 q^{43} - 4 q^{44} + 32 q^{46} + 12 q^{47} - 12 q^{48} + 16 q^{51} - 8 q^{52} - 4 q^{53} - 24 q^{54} + 16 q^{57} - 16 q^{58} - 16 q^{59} + 8 q^{61} - 32 q^{62} - 24 q^{63} - 8 q^{66} + 20 q^{67} + 32 q^{68} + 28 q^{69} - 20 q^{72} + 40 q^{74} + 32 q^{76} + 8 q^{77} - 16 q^{78} - 28 q^{81} + 8 q^{82} + 8 q^{83} - 32 q^{84} - 8 q^{87} - 4 q^{88} - 16 q^{89} + 32 q^{91} + 52 q^{92} - 24 q^{93} - 16 q^{96} - 20 q^{97} - 12 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^3 + 8 * q^7 - 4 * q^8 - 4 * q^9 - 16 * q^12 + 8 * q^13 + 8 * q^14 - 12 * q^16 - 12 * q^18 - 4 * q^22 + 4 * q^23 - 16 * q^24 - 20 * q^27 - 8 * q^28 - 8 * q^29 - 24 * q^31 - 4 * q^32 - 32 * q^36 + 12 * q^37 + 8 * q^38 + 8 * q^39 + 8 * q^42 + 16 * q^43 - 4 * q^44 + 32 * q^46 + 12 * q^47 - 12 * q^48 + 16 * q^51 - 8 * q^52 - 4 * q^53 - 24 * q^54 + 16 * q^57 - 16 * q^58 - 16 * q^59 + 8 * q^61 - 32 * q^62 - 24 * q^63 - 8 * q^66 + 20 * q^67 + 32 * q^68 + 28 * q^69 - 20 * q^72 + 40 * q^74 + 32 * q^76 + 8 * q^77 - 16 * q^78 - 28 * q^81 + 8 * q^82 + 8 * q^83 - 32 * q^84 - 8 * q^87 - 4 * q^88 - 16 * q^89 + 32 * q^91 + 52 * q^92 - 24 * q^93 - 16 * q^96 - 20 * q^97 - 12 * q^98

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