LMFDB - Embedded newform 840.2.j.b.589.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [840,2,Mod(589,840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 1, 0]))

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

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

chi := DirichletCharacter("840.589");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	840.j (of order $2$, degree $1$, 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:	$6.70743376979$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			589.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	840.589
Dual form			840.2.j.b.589.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.00000i) q^{2} +1.00000 q^{3} -2.00000i q^{4} +(-2.00000 - 1.00000i) q^{5} +(-1.00000 + 1.00000i) q^{6} -1.00000i q^{7} +(2.00000 + 2.00000i) q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+(-1.00000 + 1.00000i) q^{2} +1.00000 q^{3} -2.00000i q^{4} +(-2.00000 - 1.00000i) q^{5} +(-1.00000 + 1.00000i) q^{6} -1.00000i q^{7} +(2.00000 + 2.00000i) q^{8} +1.00000 q^{9} +(3.00000 - 1.00000i) q^{10} -2.00000i q^{12} -6.00000 q^{13} +(1.00000 + 1.00000i) q^{14} +(-2.00000 - 1.00000i) q^{15} -4.00000 q^{16} +2.00000i q^{17} +(-1.00000 + 1.00000i) q^{18} -4.00000i q^{19} +(-2.00000 + 4.00000i) q^{20} -1.00000i q^{21} +4.00000i q^{23} +(2.00000 + 2.00000i) q^{24} +(3.00000 + 4.00000i) q^{25} +(6.00000 - 6.00000i) q^{26} +1.00000 q^{27} -2.00000 q^{28} +6.00000i q^{29} +(3.00000 - 1.00000i) q^{30} -8.00000 q^{31} +(4.00000 - 4.00000i) q^{32} +(-2.00000 - 2.00000i) q^{34} +(-1.00000 + 2.00000i) q^{35} -2.00000i q^{36} -2.00000 q^{37} +(4.00000 + 4.00000i) q^{38} -6.00000 q^{39} +(-2.00000 - 6.00000i) q^{40} -8.00000 q^{41} +(1.00000 + 1.00000i) q^{42} -6.00000 q^{43} +(-2.00000 - 1.00000i) q^{45} +(-4.00000 - 4.00000i) q^{46} +2.00000i q^{47} -4.00000 q^{48} -1.00000 q^{49} +(-7.00000 - 1.00000i) q^{50} +2.00000i q^{51} +12.0000i q^{52} -6.00000 q^{53} +(-1.00000 + 1.00000i) q^{54} +(2.00000 - 2.00000i) q^{56} -4.00000i q^{57} +(-6.00000 - 6.00000i) q^{58} +6.00000i q^{59} +(-2.00000 + 4.00000i) q^{60} -10.0000i q^{61} +(8.00000 - 8.00000i) q^{62} -1.00000i q^{63} +8.00000i q^{64} +(12.0000 + 6.00000i) q^{65} -2.00000 q^{67} +4.00000 q^{68} +4.00000i q^{69} +(-1.00000 - 3.00000i) q^{70} -8.00000 q^{71} +(2.00000 + 2.00000i) q^{72} -6.00000i q^{73} +(2.00000 - 2.00000i) q^{74} +(3.00000 + 4.00000i) q^{75} -8.00000 q^{76} +(6.00000 - 6.00000i) q^{78} +10.0000 q^{79} +(8.00000 + 4.00000i) q^{80} +1.00000 q^{81} +(8.00000 - 8.00000i) q^{82} +4.00000 q^{83} -2.00000 q^{84} +(2.00000 - 4.00000i) q^{85} +(6.00000 - 6.00000i) q^{86} +6.00000i q^{87} +(3.00000 - 1.00000i) q^{90} +6.00000i q^{91} +8.00000 q^{92} -8.00000 q^{93} +(-2.00000 - 2.00000i) q^{94} +(-4.00000 + 8.00000i) q^{95} +(4.00000 - 4.00000i) q^{96} +2.00000i q^{97} +(1.00000 - 1.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} - 4 q^{5} - 2 q^{6} + 4 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 - 4 * q^5 - 2 * q^6 + 4 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} - 4 q^{5} - 2 q^{6} + 4 q^{8} + 2 q^{9} + 6 q^{10} - 12 q^{13} + 2 q^{14} - 4 q^{15} - 8 q^{16} - 2 q^{18} - 4 q^{20} + 4 q^{24} + 6 q^{25} + 12 q^{26} + 2 q^{27} - 4 q^{28} + 6 q^{30} - 16 q^{31} + 8 q^{32} - 4 q^{34} - 2 q^{35} - 4 q^{37} + 8 q^{38} - 12 q^{39} - 4 q^{40} - 16 q^{41} + 2 q^{42} - 12 q^{43} - 4 q^{45} - 8 q^{46} - 8 q^{48} - 2 q^{49} - 14 q^{50} - 12 q^{53} - 2 q^{54} + 4 q^{56} - 12 q^{58} - 4 q^{60} + 16 q^{62} + 24 q^{65} - 4 q^{67} + 8 q^{68} - 2 q^{70} - 16 q^{71} + 4 q^{72} + 4 q^{74} + 6 q^{75} - 16 q^{76} + 12 q^{78} + 20 q^{79} + 16 q^{80} + 2 q^{81} + 16 q^{82} + 8 q^{83} - 4 q^{84} + 4 q^{85} + 12 q^{86} + 6 q^{90} + 16 q^{92} - 16 q^{93} - 4 q^{94} - 8 q^{95} + 8 q^{96} + 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 - 4 * q^5 - 2 * q^6 + 4 * q^8 + 2 * q^9 + 6 * q^10 - 12 * q^13 + 2 * q^14 - 4 * q^15 - 8 * q^16 - 2 * q^18 - 4 * q^20 + 4 * q^24 + 6 * q^25 + 12 * q^26 + 2 * q^27 - 4 * q^28 + 6 * q^30 - 16 * q^31 + 8 * q^32 - 4 * q^34 - 2 * q^35 - 4 * q^37 + 8 * q^38 - 12 * q^39 - 4 * q^40 - 16 * q^41 + 2 * q^42 - 12 * q^43 - 4 * q^45 - 8 * q^46 - 8 * q^48 - 2 * q^49 - 14 * q^50 - 12 * q^53 - 2 * q^54 + 4 * q^56 - 12 * q^58 - 4 * q^60 + 16 * q^62 + 24 * q^65 - 4 * q^67 + 8 * q^68 - 2 * q^70 - 16 * q^71 + 4 * q^72 + 4 * q^74 + 6 * q^75 - 16 * q^76 + 12 * q^78 + 20 * q^79 + 16 * q^80 + 2 * q^81 + 16 * q^82 + 8 * q^83 - 4 * q^84 + 4 * q^85 + 12 * q^86 + 6 * q^90 + 16 * q^92 - 16 * q^93 - 4 * q^94 - 8 * q^95 + 8 * q^96 + 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/840\mathbb{Z}\right)^\times$.

$n$	$241$	$281$	$337$	$421$	$631$
$\chi(n)$	$1$	$1$	$-1$	$-1$	$1$

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 $

$2$	−1.00000	+	1.00000i	−0.707107	+	0.707107i
$2$	−1.00000	+	1.00000i	−0.707107	+	0.707107i
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		−	2.00000i		−	1.00000i
$5$	−2.00000	−	1.00000i	−0.894427	−	0.447214i
$5$	−2.00000	−	1.00000i	−0.894427	−	0.447214i
$6$	−1.00000	+	1.00000i	−0.408248	+	0.408248i
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$	2.00000	+	2.00000i	0.707107	+	0.707107i
$9$	1.00000			0.333333
$10$	3.00000	−	1.00000i	0.948683	−	0.316228i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		−	2.00000i		−	0.577350i
$13$	−6.00000			−1.66410			−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000			−1.66410			−0.832050	+	0.554700i	$0.812833\pi$
$14$	1.00000	+	1.00000i	0.267261	+	0.267261i
$15$	−2.00000	−	1.00000i	−0.516398	−	0.258199i
$16$	−4.00000			−1.00000
$17$			2.00000i			0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$			2.00000i			0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	−1.00000	+	1.00000i	−0.235702	+	0.235702i
$19$		−	4.00000i		−	0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$		−	4.00000i		−	0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	−2.00000	+	4.00000i	−0.447214	+	0.894427i
$21$		−	1.00000i		−	0.218218i
$22$		0			0
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	2.00000	+	2.00000i	0.408248	+	0.408248i
$25$	3.00000	+	4.00000i	0.600000	+	0.800000i
$26$	6.00000	−	6.00000i	1.17670	−	1.17670i
$27$	1.00000			0.192450
$28$	−2.00000			−0.377964
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	3.00000	−	1.00000i	0.547723	−	0.182574i
$31$	−8.00000			−1.43684			−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000			−1.43684			−0.718421	+	0.695608i	$0.755135\pi$
$32$	4.00000	−	4.00000i	0.707107	−	0.707107i
$33$		0			0
$34$	−2.00000	−	2.00000i	−0.342997	−	0.342997i
$35$	−1.00000	+	2.00000i	−0.169031	+	0.338062i
$36$		−	2.00000i		−	0.333333i
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$	4.00000	+	4.00000i	0.648886	+	0.648886i
$39$	−6.00000			−0.960769
$40$	−2.00000	−	6.00000i	−0.316228	−	0.948683i
$41$	−8.00000			−1.24939			−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000			−1.24939			−0.624695	+	0.780869i	$0.714777\pi$
$42$	1.00000	+	1.00000i	0.154303	+	0.154303i
$43$	−6.00000			−0.914991			−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000			−0.914991			−0.457496	+	0.889212i	$0.651253\pi$
$44$		0			0
$45$	−2.00000	−	1.00000i	−0.298142	−	0.149071i
$46$	−4.00000	−	4.00000i	−0.589768	−	0.589768i
$47$			2.00000i			0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$			2.00000i			0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$	−4.00000			−0.577350
$49$	−1.00000			−0.142857
$50$	−7.00000	−	1.00000i	−0.989949	−	0.141421i
$51$			2.00000i			0.280056i
$52$			12.0000i			1.66410i
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$	−1.00000	+	1.00000i	−0.136083	+	0.136083i
$55$		0			0
$56$	2.00000	−	2.00000i	0.267261	−	0.267261i
$57$		−	4.00000i		−	0.529813i
$58$	−6.00000	−	6.00000i	−0.787839	−	0.787839i
$59$			6.00000i			0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$			6.00000i			0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$	−2.00000	+	4.00000i	−0.258199	+	0.516398i
$61$		−	10.0000i		−	1.28037i	−0.768221	−	0.640184i	$-0.778858\pi$
$61$		−	10.0000i		−	1.28037i	0.768221	−	0.640184i	$-0.221142\pi$
$62$	8.00000	−	8.00000i	1.01600	−	1.01600i
$63$		−	1.00000i		−	0.125988i
$64$			8.00000i			1.00000i
$65$	12.0000	+	6.00000i	1.48842	+	0.744208i
$66$		0			0
$67$	−2.00000			−0.244339			−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000			−0.244339			−0.122169	+	0.992509i	$0.538985\pi$
$68$	4.00000			0.485071
$69$			4.00000i			0.481543i
$70$	−1.00000	−	3.00000i	−0.119523	−	0.358569i
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$	2.00000	+	2.00000i	0.235702	+	0.235702i
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	2.00000	−	2.00000i	0.232495	−	0.232495i
$75$	3.00000	+	4.00000i	0.346410	+	0.461880i
$76$	−8.00000			−0.917663
$77$		0			0
$78$	6.00000	−	6.00000i	0.679366	−	0.679366i
$79$	10.0000			1.12509			0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000			1.12509			0.562544	+	0.826767i	$0.309823\pi$
$80$	8.00000	+	4.00000i	0.894427	+	0.447214i
$81$	1.00000			0.111111
$82$	8.00000	−	8.00000i	0.883452	−	0.883452i
$83$	4.00000			0.439057			0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000			0.439057			0.219529	+	0.975606i	$0.429548\pi$
$84$	−2.00000			−0.218218
$85$	2.00000	−	4.00000i	0.216930	−	0.433861i
$86$	6.00000	−	6.00000i	0.646997	−	0.646997i
$87$			6.00000i			0.643268i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	3.00000	−	1.00000i	0.316228	−	0.105409i
$91$			6.00000i			0.628971i
$92$	8.00000			0.834058
$93$	−8.00000			−0.829561
$94$	−2.00000	−	2.00000i	−0.206284	−	0.206284i
$95$	−4.00000	+	8.00000i	−0.410391	+	0.820783i
$96$	4.00000	−	4.00000i	0.408248	−	0.408248i
$97$			2.00000i			0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$			2.00000i			0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	1.00000	−	1.00000i	0.101015	−	0.101015i
$99$		0			0
$100$	8.00000	−	6.00000i	0.800000	−	0.600000i
$101$		−	10.0000i		−	0.995037i	−0.867453	−	0.497519i	$-0.834245\pi$
$101$		−	10.0000i		−	0.995037i	0.867453	−	0.497519i	$-0.165755\pi$
$102$	−2.00000	−	2.00000i	−0.198030	−	0.198030i
$103$		−	16.0000i		−	1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$		−	16.0000i		−	1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	−12.0000	−	12.0000i	−1.17670	−	1.17670i
$105$	−1.00000	+	2.00000i	−0.0975900	+	0.195180i
$106$	6.00000	−	6.00000i	0.582772	−	0.582772i
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		−	2.00000i		−	0.192450i
$109$			16.0000i			1.53252i	0.642529	+	0.766261i	$0.277885\pi$
$109$			16.0000i			1.53252i	−0.642529	+	0.766261i	$0.722115\pi$
$110$		0			0
$111$	−2.00000			−0.189832
$112$			4.00000i			0.377964i
$113$		−	6.00000i		−	0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$		−	6.00000i		−	0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	4.00000	+	4.00000i	0.374634	+	0.374634i
$115$	4.00000	−	8.00000i	0.373002	−	0.746004i
$116$	12.0000			1.11417
$117$	−6.00000			−0.554700
$118$	−6.00000	−	6.00000i	−0.552345	−	0.552345i
$119$	2.00000			0.183340
$120$	−2.00000	−	6.00000i	−0.182574	−	0.547723i
$121$	11.0000			1.00000
$122$	10.0000	+	10.0000i	0.905357	+	0.905357i
$123$	−8.00000			−0.721336
$124$			16.0000i			1.43684i
$125$	−2.00000	−	11.0000i	−0.178885	−	0.983870i
$126$	1.00000	+	1.00000i	0.0890871	+	0.0890871i
$127$		−	8.00000i		−	0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$		−	8.00000i		−	0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	−8.00000	−	8.00000i	−0.707107	−	0.707107i
$129$	−6.00000			−0.528271
$130$	−18.0000	+	6.00000i	−1.57870	+	0.526235i
$131$			10.0000i			0.873704i	0.899533	+	0.436852i	$0.143907\pi$
$131$			10.0000i			0.873704i	−0.899533	+	0.436852i	$0.856093\pi$
$132$		0			0
$133$	−4.00000			−0.346844
$134$	2.00000	−	2.00000i	0.172774	−	0.172774i
$135$	−2.00000	−	1.00000i	−0.172133	−	0.0860663i
$136$	−4.00000	+	4.00000i	−0.342997	+	0.342997i
$137$		−	18.0000i		−	1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$		−	18.0000i		−	1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	−4.00000	−	4.00000i	−0.340503	−	0.340503i
$139$			16.0000i			1.35710i	0.734553	+	0.678551i	$0.237392\pi$
$139$			16.0000i			1.35710i	−0.734553	+	0.678551i	$0.762608\pi$
$140$	4.00000	+	2.00000i	0.338062	+	0.169031i
$141$			2.00000i			0.168430i
$142$	8.00000	−	8.00000i	0.671345	−	0.671345i
$143$		0			0
$144$	−4.00000			−0.333333
$145$	6.00000	−	12.0000i	0.498273	−	0.996546i
$146$	6.00000	+	6.00000i	0.496564	+	0.496564i
$147$	−1.00000			−0.0824786
$148$			4.00000i			0.328798i
$149$		−	14.0000i		−	1.14692i	−0.819232	−	0.573462i	$-0.805600\pi$
$149$		−	14.0000i		−	1.14692i	0.819232	−	0.573462i	$-0.194400\pi$
$150$	−7.00000	−	1.00000i	−0.571548	−	0.0816497i
$151$	2.00000			0.162758			0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000			0.162758			0.0813788	+	0.996683i	$0.474068\pi$
$152$	8.00000	−	8.00000i	0.648886	−	0.648886i
$153$			2.00000i			0.161690i
$154$		0			0
$155$	16.0000	+	8.00000i	1.28515	+	0.642575i
$156$			12.0000i			0.960769i
$157$	−22.0000			−1.75579			−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000			−1.75579			−0.877896	+	0.478852i	$0.841053\pi$
$158$	−10.0000	+	10.0000i	−0.795557	+	0.795557i
$159$	−6.00000			−0.475831
$160$	−12.0000	+	4.00000i	−0.948683	+	0.316228i
$161$	4.00000			0.315244
$162$	−1.00000	+	1.00000i	−0.0785674	+	0.0785674i
$163$	−6.00000			−0.469956			−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000			−0.469956			−0.234978	+	0.972001i	$0.575502\pi$
$164$			16.0000i			1.24939i
$165$		0			0
$166$	−4.00000	+	4.00000i	−0.310460	+	0.310460i
$167$			22.0000i			1.70241i	0.524832	+	0.851206i	$0.324128\pi$
$167$			22.0000i			1.70241i	−0.524832	+	0.851206i	$0.675872\pi$
$168$	2.00000	−	2.00000i	0.154303	−	0.154303i
$169$	23.0000			1.76923
$170$	2.00000	+	6.00000i	0.153393	+	0.460179i
$171$		−	4.00000i		−	0.305888i
$172$			12.0000i			0.914991i
$173$	24.0000			1.82469			0.912343	−	0.409426i	$-0.134271\pi$
$173$	24.0000			1.82469			0.912343	+	0.409426i	$0.134271\pi$
$174$	−6.00000	−	6.00000i	−0.454859	−	0.454859i
$175$	4.00000	−	3.00000i	0.302372	−	0.226779i
$176$		0			0
$177$			6.00000i			0.450988i
$178$		0			0
$179$			16.0000i			1.19590i	0.801535	+	0.597948i	$0.204017\pi$
$179$			16.0000i			1.19590i	−0.801535	+	0.597948i	$0.795983\pi$
$180$	−2.00000	+	4.00000i	−0.149071	+	0.298142i
$181$			10.0000i			0.743294i	0.928374	+	0.371647i	$0.121207\pi$
$181$			10.0000i			0.743294i	−0.928374	+	0.371647i	$0.878793\pi$
$182$	−6.00000	−	6.00000i	−0.444750	−	0.444750i
$183$		−	10.0000i		−	0.739221i
$184$	−8.00000	+	8.00000i	−0.589768	+	0.589768i
$185$	4.00000	+	2.00000i	0.294086	+	0.147043i
$186$	8.00000	−	8.00000i	0.586588	−	0.586588i
$187$		0			0
$188$	4.00000			0.291730
$189$		−	1.00000i		−	0.0727393i
$190$	−4.00000	−	12.0000i	−0.290191	−	0.870572i
$191$	−8.00000			−0.578860			−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000			−0.578860			−0.289430	+	0.957199i	$0.593466\pi$
$192$			8.00000i			0.577350i
$193$		−	16.0000i		−	1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$		−	16.0000i		−	1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$	−2.00000	−	2.00000i	−0.143592	−	0.143592i
$195$	12.0000	+	6.00000i	0.859338	+	0.429669i
$196$			2.00000i			0.142857i
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	20.0000			1.41776			0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000			1.41776			0.708881	+	0.705328i	$0.249200\pi$
$200$	−2.00000	+	14.0000i	−0.141421	+	0.989949i
$201$	−2.00000			−0.141069
$202$	10.0000	+	10.0000i	0.703598	+	0.703598i
$203$	6.00000			0.421117
$204$	4.00000			0.280056
$205$	16.0000	+	8.00000i	1.11749	+	0.558744i
$206$	16.0000	+	16.0000i	1.11477	+	1.11477i
$207$			4.00000i			0.278019i
$208$	24.0000			1.66410
$209$		0			0
$210$	−1.00000	−	3.00000i	−0.0690066	−	0.207020i
$211$			20.0000i			1.37686i	0.725304	+	0.688428i	$0.241699\pi$
$211$			20.0000i			1.37686i	−0.725304	+	0.688428i	$0.758301\pi$
$212$			12.0000i			0.824163i
$213$	−8.00000			−0.548151
$214$	12.0000	−	12.0000i	0.820303	−	0.820303i
$215$	12.0000	+	6.00000i	0.818393	+	0.409197i
$216$	2.00000	+	2.00000i	0.136083	+	0.136083i
$217$			8.00000i			0.543075i
$218$	−16.0000	−	16.0000i	−1.08366	−	1.08366i
$219$		−	6.00000i		−	0.405442i
$220$		0			0
$221$		−	12.0000i		−	0.807207i
$222$	2.00000	−	2.00000i	0.134231	−	0.134231i
$223$			4.00000i			0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$			4.00000i			0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$	−4.00000	−	4.00000i	−0.267261	−	0.267261i
$225$	3.00000	+	4.00000i	0.200000	+	0.266667i
$226$	6.00000	+	6.00000i	0.399114	+	0.399114i
$227$	−12.0000			−0.796468			−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000			−0.796468			−0.398234	+	0.917284i	$0.630377\pi$
$228$	−8.00000			−0.529813
$229$		−	14.0000i		−	0.925146i	−0.886581	−	0.462573i	$-0.846926\pi$
$229$		−	14.0000i		−	0.925146i	0.886581	−	0.462573i	$-0.153074\pi$
$230$	4.00000	+	12.0000i	0.263752	+	0.791257i
$231$		0			0
$232$	−12.0000	+	12.0000i	−0.787839	+	0.787839i
$233$		−	6.00000i		−	0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$		−	6.00000i		−	0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	6.00000	−	6.00000i	0.392232	−	0.392232i
$235$	2.00000	−	4.00000i	0.130466	−	0.260931i
$236$	12.0000			0.781133
$237$	10.0000			0.649570
$238$	−2.00000	+	2.00000i	−0.129641	+	0.129641i
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	8.00000	+	4.00000i	0.516398	+	0.258199i
$241$	−18.0000			−1.15948			−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000			−1.15948			−0.579741	+	0.814801i	$0.696846\pi$
$242$	−11.0000	+	11.0000i	−0.707107	+	0.707107i
$243$	1.00000			0.0641500
$244$	−20.0000			−1.28037
$245$	2.00000	+	1.00000i	0.127775	+	0.0638877i
$246$	8.00000	−	8.00000i	0.510061	−	0.510061i
$247$			24.0000i			1.52708i
$248$	−16.0000	−	16.0000i	−1.01600	−	1.01600i
$249$	4.00000			0.253490
$250$	13.0000	+	9.00000i	0.822192	+	0.569210i
$251$		−	10.0000i		−	0.631194i	−0.948893	−	0.315597i	$-0.897795\pi$
$251$		−	10.0000i		−	0.631194i	0.948893	−	0.315597i	$-0.102205\pi$
$252$	−2.00000			−0.125988
$253$		0			0
$254$	8.00000	+	8.00000i	0.501965	+	0.501965i
$255$	2.00000	−	4.00000i	0.125245	−	0.250490i
$256$	16.0000			1.00000
$257$		−	18.0000i		−	1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$		−	18.0000i		−	1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	6.00000	−	6.00000i	0.373544	−	0.373544i
$259$			2.00000i			0.124274i
$260$	12.0000	−	24.0000i	0.744208	−	1.48842i
$261$			6.00000i			0.371391i
$262$	−10.0000	−	10.0000i	−0.617802	−	0.617802i
$263$			24.0000i			1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$			24.0000i			1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$		0			0
$265$	12.0000	+	6.00000i	0.737154	+	0.368577i
$266$	4.00000	−	4.00000i	0.245256	−	0.245256i
$267$		0			0
$268$			4.00000i			0.244339i
$269$		−	14.0000i		−	0.853595i	−0.904347	−	0.426798i	$-0.859642\pi$
$269$		−	14.0000i		−	0.853595i	0.904347	−	0.426798i	$-0.140358\pi$
$270$	3.00000	−	1.00000i	0.182574	−	0.0608581i
$271$	−28.0000			−1.70088			−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000			−1.70088			−0.850439	+	0.526073i	$0.823664\pi$
$272$		−	8.00000i		−	0.485071i
$273$			6.00000i			0.363137i
$274$	18.0000	+	18.0000i	1.08742	+	1.08742i
$275$		0			0
$276$	8.00000			0.481543
$277$	18.0000			1.08152			0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000			1.08152			0.540758	+	0.841178i	$0.318138\pi$
$278$	−16.0000	−	16.0000i	−0.959616	−	0.959616i
$279$	−8.00000			−0.478947
$280$	−6.00000	+	2.00000i	−0.358569	+	0.119523i
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$	−2.00000	−	2.00000i	−0.119098	−	0.119098i
$283$	4.00000			0.237775			0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000			0.237775			0.118888	+	0.992908i	$0.462067\pi$
$284$			16.0000i			0.949425i
$285$	−4.00000	+	8.00000i	−0.236940	+	0.473879i
$286$		0			0
$287$			8.00000i			0.472225i
$288$	4.00000	−	4.00000i	0.235702	−	0.235702i
$289$	13.0000			0.764706
$290$	6.00000	+	18.0000i	0.352332	+	1.05700i
$291$			2.00000i			0.117242i
$292$	−12.0000			−0.702247
$293$	−16.0000			−0.934730			−0.467365	−	0.884064i	$-0.654797\pi$
$293$	−16.0000			−0.934730			−0.467365	+	0.884064i	$0.654797\pi$
$294$	1.00000	−	1.00000i	0.0583212	−	0.0583212i
$295$	6.00000	−	12.0000i	0.349334	−	0.698667i
$296$	−4.00000	−	4.00000i	−0.232495	−	0.232495i
$297$		0			0
$298$	14.0000	+	14.0000i	0.810998	+	0.810998i
$299$		−	24.0000i		−	1.38796i
$300$	8.00000	−	6.00000i	0.461880	−	0.346410i
$301$			6.00000i			0.345834i
$302$	−2.00000	+	2.00000i	−0.115087	+	0.115087i
$303$		−	10.0000i		−	0.574485i
$304$			16.0000i			0.917663i
$305$	−10.0000	+	20.0000i	−0.572598	+	1.14520i
$306$	−2.00000	−	2.00000i	−0.114332	−	0.114332i
$307$	28.0000			1.59804			0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000			1.59804			0.799022	+	0.601302i	$0.205351\pi$
$308$		0			0
$309$		−	16.0000i		−	0.910208i
$310$	−24.0000	+	8.00000i	−1.36311	+	0.454369i
$311$	−8.00000			−0.453638			−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000			−0.453638			−0.226819	+	0.973937i	$0.572833\pi$
$312$	−12.0000	−	12.0000i	−0.679366	−	0.679366i
$313$			14.0000i			0.791327i	0.918396	+	0.395663i	$0.129485\pi$
$313$			14.0000i			0.791327i	−0.918396	+	0.395663i	$0.870515\pi$
$314$	22.0000	−	22.0000i	1.24153	−	1.24153i
$315$	−1.00000	+	2.00000i	−0.0563436	+	0.112687i
$316$		−	20.0000i		−	1.12509i
$317$	18.0000			1.01098			0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000			1.01098			0.505490	+	0.862832i	$0.331312\pi$
$318$	6.00000	−	6.00000i	0.336463	−	0.336463i
$319$		0			0
$320$	8.00000	−	16.0000i	0.447214	−	0.894427i
$321$	−12.0000			−0.669775
$322$	−4.00000	+	4.00000i	−0.222911	+	0.222911i
$323$	8.00000			0.445132
$324$		−	2.00000i		−	0.111111i
$325$	−18.0000	−	24.0000i	−0.998460	−	1.33128i
$326$	6.00000	−	6.00000i	0.332309	−	0.332309i
$327$			16.0000i			0.884802i
$328$	−16.0000	−	16.0000i	−0.883452	−	0.883452i
$329$	2.00000			0.110264
$330$		0			0
$331$		−	20.0000i		−	1.09930i	−0.835395	−	0.549650i	$-0.814761\pi$
$331$		−	20.0000i		−	1.09930i	0.835395	−	0.549650i	$-0.185239\pi$
$332$		−	8.00000i		−	0.439057i
$333$	−2.00000			−0.109599
$334$	−22.0000	−	22.0000i	−1.20379	−	1.20379i
$335$	4.00000	+	2.00000i	0.218543	+	0.109272i
$336$			4.00000i			0.218218i
$337$		−	8.00000i		−	0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$337$		−	8.00000i		−	0.435788i	0.975972	−	0.217894i	$-0.0699187\pi$
$338$	−23.0000	+	23.0000i	−1.25104	+	1.25104i
$339$		−	6.00000i		−	0.325875i
$340$	−8.00000	−	4.00000i	−0.433861	−	0.216930i
$341$		0			0
$342$	4.00000	+	4.00000i	0.216295	+	0.216295i
$343$			1.00000i			0.0539949i
$344$	−12.0000	−	12.0000i	−0.646997	−	0.646997i
$345$	4.00000	−	8.00000i	0.215353	−	0.430706i
$346$	−24.0000	+	24.0000i	−1.29025	+	1.29025i
$347$	−12.0000			−0.644194			−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000			−0.644194			−0.322097	+	0.946707i	$0.604388\pi$
$348$	12.0000			0.643268
$349$			6.00000i			0.321173i	0.987022	+	0.160586i	$0.0513385\pi$
$349$			6.00000i			0.321173i	−0.987022	+	0.160586i	$0.948662\pi$
$350$	−1.00000	+	7.00000i	−0.0534522	+	0.374166i
$351$	−6.00000			−0.320256
$352$		0			0
$353$			14.0000i			0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$			14.0000i			0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$	−6.00000	−	6.00000i	−0.318896	−	0.318896i
$355$	16.0000	+	8.00000i	0.849192	+	0.424596i
$356$		0			0
$357$	2.00000			0.105851
$358$	−16.0000	−	16.0000i	−0.845626	−	0.845626i
$359$	−20.0000			−1.05556			−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000			−1.05556			−0.527780	+	0.849381i	$0.676975\pi$
$360$	−2.00000	−	6.00000i	−0.105409	−	0.316228i
$361$	3.00000			0.157895
$362$	−10.0000	−	10.0000i	−0.525588	−	0.525588i
$363$	11.0000			0.577350
$364$	12.0000			0.628971
$365$	−6.00000	+	12.0000i	−0.314054	+	0.628109i
$366$	10.0000	+	10.0000i	0.522708	+	0.522708i
$367$		−	28.0000i		−	1.46159i	−0.682598	−	0.730794i	$-0.739150\pi$
$367$		−	28.0000i		−	1.46159i	0.682598	−	0.730794i	$-0.260850\pi$
$368$		−	16.0000i		−	0.834058i
$369$	−8.00000			−0.416463
$370$	−6.00000	+	2.00000i	−0.311925	+	0.103975i
$371$			6.00000i			0.311504i
$372$			16.0000i			0.829561i
$373$	−26.0000			−1.34623			−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000			−1.34623			−0.673114	+	0.739538i	$0.735044\pi$
$374$		0			0
$375$	−2.00000	−	11.0000i	−0.103280	−	0.568038i
$376$	−4.00000	+	4.00000i	−0.206284	+	0.206284i
$377$		−	36.0000i		−	1.85409i
$378$	1.00000	+	1.00000i	0.0514344	+	0.0514344i
$379$		−	4.00000i		−	0.205466i	−0.994709	−	0.102733i	$-0.967241\pi$
$379$		−	4.00000i		−	0.205466i	0.994709	−	0.102733i	$-0.0327588\pi$
$380$	16.0000	+	8.00000i	0.820783	+	0.410391i
$381$		−	8.00000i		−	0.409852i
$382$	8.00000	−	8.00000i	0.409316	−	0.409316i
$383$		−	26.0000i		−	1.32854i	−0.747494	−	0.664269i	$-0.768743\pi$
$383$		−	26.0000i		−	1.32854i	0.747494	−	0.664269i	$-0.231257\pi$
$384$	−8.00000	−	8.00000i	−0.408248	−	0.408248i
$385$		0			0
$386$	16.0000	+	16.0000i	0.814379	+	0.814379i
$387$	−6.00000			−0.304997
$388$	4.00000			0.203069
$389$			26.0000i			1.31825i	0.752032	+	0.659126i	$0.229074\pi$
$389$			26.0000i			1.31825i	−0.752032	+	0.659126i	$0.770926\pi$
$390$	−18.0000	+	6.00000i	−0.911465	+	0.303822i
$391$	−8.00000			−0.404577
$392$	−2.00000	−	2.00000i	−0.101015	−	0.101015i
$393$			10.0000i			0.504433i
$394$	−18.0000	+	18.0000i	−0.906827	+	0.906827i
$395$	−20.0000	−	10.0000i	−1.00631	−	0.503155i
$396$		0			0
$397$	38.0000			1.90717			0.953583	−	0.301131i	$-0.0973643\pi$
$397$	38.0000			1.90717			0.953583	+	0.301131i	$0.0973643\pi$
$398$	−20.0000	+	20.0000i	−1.00251	+	1.00251i
$399$	−4.00000			−0.200250
$400$	−12.0000	−	16.0000i	−0.600000	−	0.800000i
$401$	2.00000			0.0998752			0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000			0.0998752			0.0499376	+	0.998752i	$0.484098\pi$
$402$	2.00000	−	2.00000i	0.0997509	−	0.0997509i
$403$	48.0000			2.39105
$404$	−20.0000			−0.995037
$405$	−2.00000	−	1.00000i	−0.0993808	−	0.0496904i
$406$	−6.00000	+	6.00000i	−0.297775	+	0.297775i
$407$		0			0
$408$	−4.00000	+	4.00000i	−0.198030	+	0.198030i
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$	−24.0000	+	8.00000i	−1.18528	+	0.395092i
$411$		−	18.0000i		−	0.887875i
$412$	−32.0000			−1.57653
$413$	6.00000			0.295241
$414$	−4.00000	−	4.00000i	−0.196589	−	0.196589i
$415$	−8.00000	−	4.00000i	−0.392705	−	0.196352i
$416$	−24.0000	+	24.0000i	−1.17670	+	1.17670i
$417$			16.0000i			0.783523i
$418$		0			0
$419$		−	34.0000i		−	1.66101i	−0.557012	−	0.830504i	$-0.688052\pi$
$419$		−	34.0000i		−	1.66101i	0.557012	−	0.830504i	$-0.311948\pi$
$420$	4.00000	+	2.00000i	0.195180	+	0.0975900i
$421$		−	20.0000i		−	0.974740i	−0.873195	−	0.487370i	$-0.837956\pi$
$421$		−	20.0000i		−	0.974740i	0.873195	−	0.487370i	$-0.162044\pi$
$422$	−20.0000	−	20.0000i	−0.973585	−	0.973585i
$423$			2.00000i			0.0972433i
$424$	−12.0000	−	12.0000i	−0.582772	−	0.582772i
$425$	−8.00000	+	6.00000i	−0.388057	+	0.291043i
$426$	8.00000	−	8.00000i	0.387601	−	0.387601i
$427$	−10.0000			−0.483934
$428$			24.0000i			1.16008i
$429$		0			0
$430$	−18.0000	+	6.00000i	−0.868037	+	0.289346i
$431$	12.0000			0.578020			0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000			0.578020			0.289010	+	0.957326i	$0.406674\pi$
$432$	−4.00000			−0.192450
$433$		−	6.00000i		−	0.288342i	−0.989553	−	0.144171i	$-0.953949\pi$
$433$		−	6.00000i		−	0.288342i	0.989553	−	0.144171i	$-0.0460515\pi$
$434$	−8.00000	−	8.00000i	−0.384012	−	0.384012i
$435$	6.00000	−	12.0000i	0.287678	−	0.575356i
$436$	32.0000			1.53252
$437$	16.0000			0.765384
$438$	6.00000	+	6.00000i	0.286691	+	0.286691i
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$	12.0000	+	12.0000i	0.570782	+	0.570782i
$443$	−36.0000			−1.71041			−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000			−1.71041			−0.855206	+	0.518289i	$0.826569\pi$
$444$			4.00000i			0.189832i
$445$		0			0
$446$	−4.00000	−	4.00000i	−0.189405	−	0.189405i
$447$		−	14.0000i		−	0.662177i
$448$	8.00000			0.377964
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$	−7.00000	−	1.00000i	−0.329983	−	0.0471405i
$451$		0			0
$452$	−12.0000			−0.564433
$453$	2.00000			0.0939682
$454$	12.0000	−	12.0000i	0.563188	−	0.563188i
$455$	6.00000	−	12.0000i	0.281284	−	0.562569i
$456$	8.00000	−	8.00000i	0.374634	−	0.374634i
$457$			12.0000i			0.561336i	0.959805	+	0.280668i	$0.0905560\pi$
$457$			12.0000i			0.561336i	−0.959805	+	0.280668i	$0.909444\pi$
$458$	14.0000	+	14.0000i	0.654177	+	0.654177i
$459$			2.00000i			0.0933520i
$460$	−16.0000	−	8.00000i	−0.746004	−	0.373002i
$461$			30.0000i			1.39724i	0.715493	+	0.698620i	$0.246202\pi$
$461$			30.0000i			1.39724i	−0.715493	+	0.698620i	$0.753798\pi$
$462$		0			0
$463$		−	16.0000i		−	0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$		−	16.0000i		−	0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$		−	24.0000i		−	1.11417i
$465$	16.0000	+	8.00000i	0.741982	+	0.370991i
$466$	6.00000	+	6.00000i	0.277945	+	0.277945i
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$			12.0000i			0.554700i
$469$			2.00000i			0.0923514i
$470$	2.00000	+	6.00000i	0.0922531	+	0.276759i
$471$	−22.0000			−1.01371
$472$	−12.0000	+	12.0000i	−0.552345	+	0.552345i
$473$		0			0
$474$	−10.0000	+	10.0000i	−0.459315	+	0.459315i
$475$	16.0000	−	12.0000i	0.734130	−	0.550598i
$476$		−	4.00000i		−	0.183340i
$477$	−6.00000			−0.274721
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$	−12.0000	+	4.00000i	−0.547723	+	0.182574i
$481$	12.0000			0.547153
$482$	18.0000	−	18.0000i	0.819878	−	0.819878i
$483$	4.00000			0.182006
$484$		−	22.0000i		−	1.00000i
$485$	2.00000	−	4.00000i	0.0908153	−	0.181631i
$486$	−1.00000	+	1.00000i	−0.0453609	+	0.0453609i
$487$			32.0000i			1.45006i	0.688718	+	0.725029i	$0.258174\pi$
$487$			32.0000i			1.45006i	−0.688718	+	0.725029i	$0.741826\pi$
$488$	20.0000	−	20.0000i	0.905357	−	0.905357i
$489$	−6.00000			−0.271329
$490$	−3.00000	+	1.00000i	−0.135526	+	0.0451754i
$491$			20.0000i			0.902587i	0.892375	+	0.451294i	$0.149037\pi$
$491$			20.0000i			0.902587i	−0.892375	+	0.451294i	$0.850963\pi$
$492$			16.0000i			0.721336i
$493$	−12.0000			−0.540453
$494$	−24.0000	−	24.0000i	−1.07981	−	1.07981i
$495$		0			0
$496$	32.0000			1.43684
$497$			8.00000i			0.358849i
$498$	−4.00000	+	4.00000i	−0.179244	+	0.179244i
$499$		−	4.00000i		−	0.179065i	−0.995984	−	0.0895323i	$-0.971463\pi$
$499$		−	4.00000i		−	0.179065i	0.995984	−	0.0895323i	$-0.0285372\pi$
$500$	−22.0000	+	4.00000i	−0.983870	+	0.178885i
$501$			22.0000i			0.982888i
$502$	10.0000	+	10.0000i	0.446322	+	0.446322i
$503$		−	6.00000i		−	0.267527i	−0.991013	−	0.133763i	$-0.957294\pi$
$503$		−	6.00000i		−	0.267527i	0.991013	−	0.133763i	$-0.0427062\pi$
$504$	2.00000	−	2.00000i	0.0890871	−	0.0890871i
$505$	−10.0000	+	20.0000i	−0.444994	+	0.889988i
$506$		0			0
$507$	23.0000			1.02147
$508$	−16.0000			−0.709885
$509$		−	34.0000i		−	1.50702i	−0.657434	−	0.753512i	$-0.728358\pi$
$509$		−	34.0000i		−	1.50702i	0.657434	−	0.753512i	$-0.271642\pi$
$510$	2.00000	+	6.00000i	0.0885615	+	0.265684i
$511$	−6.00000			−0.265424
$512$	−16.0000	+	16.0000i	−0.707107	+	0.707107i
$513$		−	4.00000i		−	0.176604i
$514$	18.0000	+	18.0000i	0.793946	+	0.793946i
$515$	−16.0000	+	32.0000i	−0.705044	+	1.41009i
$516$			12.0000i			0.528271i
$517$		0			0
$518$	−2.00000	−	2.00000i	−0.0878750	−	0.0878750i
$519$	24.0000			1.05348
$520$	12.0000	+	36.0000i	0.526235	+	1.57870i
$521$	−28.0000			−1.22670			−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000			−1.22670			−0.613351	+	0.789810i	$0.710179\pi$
$522$	−6.00000	−	6.00000i	−0.262613	−	0.262613i
$523$	−16.0000			−0.699631			−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000			−0.699631			−0.349816	+	0.936819i	$0.613756\pi$
$524$	20.0000			0.873704
$525$	4.00000	−	3.00000i	0.174574	−	0.130931i
$526$	−24.0000	−	24.0000i	−1.04645	−	1.04645i
$527$		−	16.0000i		−	0.696971i
$528$		0			0
$529$	7.00000			0.304348
$530$	−18.0000	+	6.00000i	−0.781870	+	0.260623i
$531$			6.00000i			0.260378i
$532$			8.00000i			0.346844i
$533$	48.0000			2.07911
$534$		0			0
$535$	24.0000	+	12.0000i	1.03761	+	0.518805i
$536$	−4.00000	−	4.00000i	−0.172774	−	0.172774i
$537$			16.0000i			0.690451i
$538$	14.0000	+	14.0000i	0.603583	+	0.603583i
$539$		0			0
$540$	−2.00000	+	4.00000i	−0.0860663	+	0.172133i
$541$		−	20.0000i		−	0.859867i	−0.902861	−	0.429934i	$-0.858537\pi$
$541$		−	20.0000i		−	0.859867i	0.902861	−	0.429934i	$-0.141463\pi$
$542$	28.0000	−	28.0000i	1.20270	−	1.20270i
$543$			10.0000i			0.429141i
$544$	8.00000	+	8.00000i	0.342997	+	0.342997i
$545$	16.0000	−	32.0000i	0.685365	−	1.37073i
$546$	−6.00000	−	6.00000i	−0.256776	−	0.256776i
$547$	−2.00000			−0.0855138			−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000			−0.0855138			−0.0427569	+	0.999086i	$0.513614\pi$
$548$	−36.0000			−1.53784
$549$		−	10.0000i		−	0.426790i
$550$		0			0
$551$	24.0000			1.02243
$552$	−8.00000	+	8.00000i	−0.340503	+	0.340503i
$553$		−	10.0000i		−	0.425243i
$554$	−18.0000	+	18.0000i	−0.764747	+	0.764747i
$555$	4.00000	+	2.00000i	0.169791	+	0.0848953i
$556$	32.0000			1.35710
$557$	−2.00000			−0.0847427			−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000			−0.0847427			−0.0423714	+	0.999102i	$0.513491\pi$
$558$	8.00000	−	8.00000i	0.338667	−	0.338667i
$559$	36.0000			1.52264
$560$	4.00000	−	8.00000i	0.169031	−	0.338062i
$561$		0			0
$562$	18.0000	−	18.0000i	0.759284	−	0.759284i
$563$	4.00000			0.168580			0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000			0.168580			0.0842900	+	0.996441i	$0.473138\pi$
$564$	4.00000			0.168430
$565$	−6.00000	+	12.0000i	−0.252422	+	0.504844i
$566$	−4.00000	+	4.00000i	−0.168133	+	0.168133i
$567$		−	1.00000i		−	0.0419961i
$568$	−16.0000	−	16.0000i	−0.671345	−	0.671345i
$569$	−30.0000			−1.25767			−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000			−1.25767			−0.628833	+	0.777541i	$0.716467\pi$
$570$	−4.00000	−	12.0000i	−0.167542	−	0.502625i
$571$		−	20.0000i		−	0.836974i	−0.908223	−	0.418487i	$-0.862561\pi$
$571$		−	20.0000i		−	0.836974i	0.908223	−	0.418487i	$-0.137439\pi$
$572$		0			0
$573$	−8.00000			−0.334205
$574$	−8.00000	−	8.00000i	−0.333914	−	0.333914i
$575$	−16.0000	+	12.0000i	−0.667246	+	0.500435i
$576$			8.00000i			0.333333i
$577$		−	38.0000i		−	1.58196i	−0.611842	−	0.790980i	$-0.709571\pi$
$577$		−	38.0000i		−	1.58196i	0.611842	−	0.790980i	$-0.290429\pi$
$578$	−13.0000	+	13.0000i	−0.540729	+	0.540729i
$579$		−	16.0000i		−	0.664937i
$580$	−24.0000	−	12.0000i	−0.996546	−	0.498273i
$581$		−	4.00000i		−	0.165948i
$582$	−2.00000	−	2.00000i	−0.0829027	−	0.0829027i
$583$		0			0
$584$	12.0000	−	12.0000i	0.496564	−	0.496564i
$585$	12.0000	+	6.00000i	0.496139	+	0.248069i
$586$	16.0000	−	16.0000i	0.660954	−	0.660954i
$587$	−12.0000			−0.495293			−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000			−0.495293			−0.247647	+	0.968850i	$0.579657\pi$
$588$			2.00000i			0.0824786i
$589$			32.0000i			1.31854i
$590$	6.00000	+	18.0000i	0.247016	+	0.741048i
$591$	18.0000			0.740421
$592$	8.00000			0.328798
$593$			14.0000i			0.574911i	0.957794	+	0.287456i	$0.0928094\pi$
$593$			14.0000i			0.574911i	−0.957794	+	0.287456i	$0.907191\pi$
$594$		0			0
$595$	−4.00000	−	2.00000i	−0.163984	−	0.0819920i
$596$	−28.0000			−1.14692
$597$	20.0000			0.818546
$598$	24.0000	+	24.0000i	0.981433	+	0.981433i
$599$	40.0000			1.63436			0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000			1.63436			0.817178	+	0.576386i	$0.195537\pi$
$600$	−2.00000	+	14.0000i	−0.0816497	+	0.571548i
$601$	2.00000			0.0815817			0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000			0.0815817			0.0407909	+	0.999168i	$0.487012\pi$
$602$	−6.00000	−	6.00000i	−0.244542	−	0.244542i
$603$	−2.00000			−0.0814463
$604$		−	4.00000i		−	0.162758i
$605$	−22.0000	−	11.0000i	−0.894427	−	0.447214i
$606$	10.0000	+	10.0000i	0.406222	+	0.406222i
$607$		−	8.00000i		−	0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$		−	8.00000i		−	0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$	−16.0000	−	16.0000i	−0.648886	−	0.648886i
$609$	6.00000			0.243132
$610$	−10.0000	−	30.0000i	−0.404888	−	1.21466i
$611$		−	12.0000i		−	0.485468i
$612$	4.00000			0.161690
$613$	−26.0000			−1.05013			−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000			−1.05013			−0.525065	+	0.851062i	$0.675959\pi$
$614$	−28.0000	+	28.0000i	−1.12999	+	1.12999i
$615$	16.0000	+	8.00000i	0.645182	+	0.322591i
$616$		0			0
$617$		−	18.0000i		−	0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$		−	18.0000i		−	0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	16.0000	+	16.0000i	0.643614	+	0.643614i
$619$		−	44.0000i		−	1.76851i	−0.467005	−	0.884255i	$-0.654667\pi$
$619$		−	44.0000i		−	1.76851i	0.467005	−	0.884255i	$-0.345333\pi$
$620$	16.0000	−	32.0000i	0.642575	−	1.28515i
$621$			4.00000i			0.160514i
$622$	8.00000	−	8.00000i	0.320771	−	0.320771i
$623$		0			0
$624$	24.0000			0.960769
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$	−14.0000	−	14.0000i	−0.559553	−	0.559553i
$627$		0			0
$628$			44.0000i			1.75579i
$629$		−	4.00000i		−	0.159490i
$630$	−1.00000	−	3.00000i	−0.0398410	−	0.119523i
$631$	−18.0000			−0.716569			−0.358284	−	0.933613i	$-0.616638\pi$
$631$	−18.0000			−0.716569			−0.358284	+	0.933613i	$0.616638\pi$
$632$	20.0000	+	20.0000i	0.795557	+	0.795557i
$633$			20.0000i			0.794929i
$634$	−18.0000	+	18.0000i	−0.714871	+	0.714871i
$635$	−8.00000	+	16.0000i	−0.317470	+	0.634941i
$636$			12.0000i			0.475831i
$637$	6.00000			0.237729
$638$		0			0
$639$	−8.00000			−0.316475
$640$	8.00000	+	24.0000i	0.316228	+	0.948683i
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$	12.0000	−	12.0000i	0.473602	−	0.473602i
$643$	−16.0000			−0.630978			−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000			−0.630978			−0.315489	+	0.948929i	$0.602169\pi$
$644$		−	8.00000i		−	0.315244i
$645$	12.0000	+	6.00000i	0.472500	+	0.236250i
$646$	−8.00000	+	8.00000i	−0.314756	+	0.314756i
$647$			42.0000i			1.65119i	0.564263	+	0.825595i	$0.309160\pi$
$647$			42.0000i			1.65119i	−0.564263	+	0.825595i	$0.690840\pi$
$648$	2.00000	+	2.00000i	0.0785674	+	0.0785674i
$649$		0			0
$650$	42.0000	+	6.00000i	1.64738	+	0.235339i
$651$			8.00000i			0.313545i
$652$			12.0000i			0.469956i
$653$	14.0000			0.547862			0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000			0.547862			0.273931	+	0.961749i	$0.411676\pi$
$654$	−16.0000	−	16.0000i	−0.625650	−	0.625650i
$655$	10.0000	−	20.0000i	0.390732	−	0.781465i
$656$	32.0000			1.24939
$657$		−	6.00000i		−	0.234082i
$658$	−2.00000	+	2.00000i	−0.0779681	+	0.0779681i
$659$			36.0000i			1.40236i	0.712984	+	0.701180i	$0.247343\pi$
$659$			36.0000i			1.40236i	−0.712984	+	0.701180i	$0.752657\pi$
$660$		0			0
$661$		−	30.0000i		−	1.16686i	−0.812162	−	0.583432i	$-0.801709\pi$
$661$		−	30.0000i		−	1.16686i	0.812162	−	0.583432i	$-0.198291\pi$
$662$	20.0000	+	20.0000i	0.777322	+	0.777322i
$663$		−	12.0000i		−	0.466041i
$664$	8.00000	+	8.00000i	0.310460	+	0.310460i
$665$	8.00000	+	4.00000i	0.310227	+	0.155113i
$666$	2.00000	−	2.00000i	0.0774984	−	0.0774984i
$667$	−24.0000			−0.929284
$668$	44.0000			1.70241
$669$			4.00000i			0.154649i
$670$	−6.00000	+	2.00000i	−0.231800	+	0.0772667i
$671$		0			0
$672$	−4.00000	−	4.00000i	−0.154303	−	0.154303i
$673$			44.0000i			1.69608i	0.529936	+	0.848038i	$0.322216\pi$
$673$			44.0000i			1.69608i	−0.529936	+	0.848038i	$0.677784\pi$
$674$	8.00000	+	8.00000i	0.308148	+	0.308148i
$675$	3.00000	+	4.00000i	0.115470	+	0.153960i
$676$		−	46.0000i		−	1.76923i
$677$	−32.0000			−1.22986			−0.614930	−	0.788582i	$-0.710816\pi$
$677$	−32.0000			−1.22986			−0.614930	+	0.788582i	$0.710816\pi$
$678$	6.00000	+	6.00000i	0.230429	+	0.230429i
$679$	2.00000			0.0767530
$680$	12.0000	−	4.00000i	0.460179	−	0.153393i
$681$	−12.0000			−0.459841
$682$		0			0
$683$	44.0000			1.68361			0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000			1.68361			0.841807	+	0.539779i	$0.181492\pi$
$684$	−8.00000			−0.305888
$685$	−18.0000	+	36.0000i	−0.687745	+	1.37549i
$686$	−1.00000	−	1.00000i	−0.0381802	−	0.0381802i
$687$		−	14.0000i		−	0.534133i
$688$	24.0000			0.914991
$689$	36.0000			1.37149
$690$	4.00000	+	12.0000i	0.152277	+	0.456832i
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		−	48.0000i		−	1.82469i
$693$		0			0
$694$	12.0000	−	12.0000i	0.455514	−	0.455514i
$695$	16.0000	−	32.0000i	0.606915	−	1.21383i
$696$	−12.0000	+	12.0000i	−0.454859	+	0.454859i
$697$		−	16.0000i		−	0.606043i
$698$	−6.00000	−	6.00000i	−0.227103	−	0.227103i
$699$		−	6.00000i		−	0.226941i
$700$	−6.00000	−	8.00000i	−0.226779	−	0.302372i
$701$			50.0000i			1.88847i	0.329267	+	0.944237i	$0.393198\pi$
$701$			50.0000i			1.88847i	−0.329267	+	0.944237i	$0.606802\pi$
$702$	6.00000	−	6.00000i	0.226455	−	0.226455i
$703$			8.00000i			0.301726i
$704$		0			0
$705$	2.00000	−	4.00000i	0.0753244	−	0.150649i
$706$	−14.0000	−	14.0000i	−0.526897	−	0.526897i
$707$	−10.0000			−0.376089
$708$	12.0000			0.450988
$709$			16.0000i			0.600893i	0.953799	+	0.300446i	$0.0971356\pi$
$709$			16.0000i			0.600893i	−0.953799	+	0.300446i	$0.902864\pi$
$710$	−24.0000	+	8.00000i	−0.900704	+	0.300235i
$711$	10.0000			0.375029
$712$		0			0
$713$		−	32.0000i		−	1.19841i
$714$	−2.00000	+	2.00000i	−0.0748481	+	0.0748481i
$715$		0			0
$716$	32.0000			1.19590
$717$		0			0
$718$	20.0000	−	20.0000i	0.746393	−	0.746393i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	8.00000	+	4.00000i	0.298142	+	0.149071i
$721$	−16.0000			−0.595871
$722$	−3.00000	+	3.00000i	−0.111648	+	0.111648i
$723$	−18.0000			−0.669427
$724$	20.0000			0.743294
$725$	−24.0000	+	18.0000i	−0.891338	+	0.668503i
$726$	−11.0000	+	11.0000i	−0.408248	+	0.408248i
$727$			52.0000i			1.92857i	0.264861	+	0.964287i	$0.414674\pi$
$727$			52.0000i			1.92857i	−0.264861	+	0.964287i	$0.585326\pi$
$728$	−12.0000	+	12.0000i	−0.444750	+	0.444750i
$729$	1.00000			0.0370370
$730$	−6.00000	−	18.0000i	−0.222070	−	0.666210i
$731$		−	12.0000i		−	0.443836i
$732$	−20.0000			−0.739221
$733$	14.0000			0.517102			0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000			0.517102			0.258551	+	0.965998i	$0.416755\pi$
$734$	28.0000	+	28.0000i	1.03350	+	1.03350i
$735$	2.00000	+	1.00000i	0.0737711	+	0.0368856i
$736$	16.0000	+	16.0000i	0.589768	+	0.589768i
$737$		0			0
$738$	8.00000	−	8.00000i	0.294484	−	0.294484i
$739$		−	4.00000i		−	0.147142i	−0.997290	−	0.0735712i	$-0.976560\pi$
$739$		−	4.00000i		−	0.147142i	0.997290	−	0.0735712i	$-0.0234396\pi$
$740$	4.00000	−	8.00000i	0.147043	−	0.294086i
$741$			24.0000i			0.881662i
$742$	−6.00000	−	6.00000i	−0.220267	−	0.220267i
$743$			24.0000i			0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$			24.0000i			0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	−16.0000	−	16.0000i	−0.586588	−	0.586588i
$745$	−14.0000	+	28.0000i	−0.512920	+	1.02584i
$746$	26.0000	−	26.0000i	0.951928	−	0.951928i
$747$	4.00000			0.146352
$748$		0			0
$749$			12.0000i			0.438470i
$750$	13.0000	+	9.00000i	0.474693	+	0.328634i
$751$	−38.0000			−1.38664			−0.693320	−	0.720630i	$-0.743853\pi$
$751$	−38.0000			−1.38664			−0.693320	+	0.720630i	$0.743853\pi$
$752$		−	8.00000i		−	0.291730i
$753$		−	10.0000i		−	0.364420i
$754$	36.0000	+	36.0000i	1.31104	+	1.31104i
$755$	−4.00000	−	2.00000i	−0.145575	−	0.0727875i
$756$	−2.00000			−0.0727393
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$	4.00000	+	4.00000i	0.145287	+	0.145287i
$759$		0			0
$760$	−24.0000	+	8.00000i	−0.870572	+	0.290191i
$761$	−8.00000			−0.290000			−0.145000	−	0.989432i	$-0.546318\pi$
$761$	−8.00000			−0.290000			−0.145000	+	0.989432i	$0.546318\pi$
$762$	8.00000	+	8.00000i	0.289809	+	0.289809i
$763$	16.0000			0.579239
$764$			16.0000i			0.578860i
$765$	2.00000	−	4.00000i	0.0723102	−	0.144620i
$766$	26.0000	+	26.0000i	0.939418	+	0.939418i
$767$		−	36.0000i		−	1.29988i
$768$	16.0000			0.577350
$769$	−50.0000			−1.80305			−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000			−1.80305			−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$		−	18.0000i		−	0.648254i
$772$	−32.0000			−1.15171
$773$	24.0000			0.863220			0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000			0.863220			0.431610	+	0.902060i	$0.357946\pi$
$774$	6.00000	−	6.00000i	0.215666	−	0.215666i
$775$	−24.0000	−	32.0000i	−0.862105	−	1.14947i
$776$	−4.00000	+	4.00000i	−0.143592	+	0.143592i
$777$			2.00000i			0.0717496i
$778$	−26.0000	−	26.0000i	−0.932145	−	0.932145i
$779$			32.0000i			1.14652i
$780$	12.0000	−	24.0000i	0.429669	−	0.859338i
$781$		0			0
$782$	8.00000	−	8.00000i	0.286079	−	0.286079i
$783$			6.00000i			0.214423i
$784$	4.00000			0.142857
$785$	44.0000	+	22.0000i	1.57043	+	0.785214i
$786$	−10.0000	−	10.0000i	−0.356688	−	0.356688i
$787$	−52.0000			−1.85360			−0.926800	−	0.375555i	$-0.877452\pi$
$787$	−52.0000			−1.85360			−0.926800	+	0.375555i	$0.877452\pi$
$788$		−	36.0000i		−	1.28245i
$789$			24.0000i			0.854423i
$790$	30.0000	−	10.0000i	1.06735	−	0.355784i
$791$	−6.00000			−0.213335
$792$		0			0
$793$			60.0000i			2.13066i
$794$	−38.0000	+	38.0000i	−1.34857	+	1.34857i
$795$	12.0000	+	6.00000i	0.425596	+	0.212798i
$796$		−	40.0000i		−	1.41776i
$797$	8.00000			0.283375			0.141687	−	0.989911i	$-0.454747\pi$
$797$	8.00000			0.283375			0.141687	+	0.989911i	$0.454747\pi$
$798$	4.00000	−	4.00000i	0.141598	−	0.141598i
$799$	−4.00000			−0.141510
$800$	28.0000	+	4.00000i	0.989949	+	0.141421i
$801$		0			0
$802$	−2.00000	+	2.00000i	−0.0706225	+	0.0706225i
$803$		0			0
$804$			4.00000i			0.141069i
$805$	−8.00000	−	4.00000i	−0.281963	−	0.140981i
$806$	−48.0000	+	48.0000i	−1.69073	+	1.69073i
$807$		−	14.0000i		−	0.492823i
$808$	20.0000	−	20.0000i	0.703598	−	0.703598i
$809$	−30.0000			−1.05474			−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000			−1.05474			−0.527372	+	0.849635i	$0.676823\pi$
$810$	3.00000	−	1.00000i	0.105409	−	0.0351364i
$811$			20.0000i			0.702295i	0.936320	+	0.351147i	$0.114208\pi$
$811$			20.0000i			0.702295i	−0.936320	+	0.351147i	$0.885792\pi$
$812$		−	12.0000i		−	0.421117i
$813$	−28.0000			−0.982003
$814$		0			0
$815$	12.0000	+	6.00000i	0.420342	+	0.210171i
$816$		−	8.00000i		−	0.280056i
$817$			24.0000i			0.839654i
$818$	−10.0000	+	10.0000i	−0.349642	+	0.349642i
$819$			6.00000i			0.209657i
$820$	16.0000	−	32.0000i	0.558744	−	1.11749i
$821$		−	10.0000i		−	0.349002i	−0.984657	−	0.174501i	$-0.944169\pi$
$821$		−	10.0000i		−	0.349002i	0.984657	−	0.174501i	$-0.0558313\pi$
$822$	18.0000	+	18.0000i	0.627822	+	0.627822i
$823$		−	16.0000i		−	0.557725i	−0.960331	−	0.278862i	$-0.910043\pi$
$823$		−	16.0000i		−	0.557725i	0.960331	−	0.278862i	$-0.0899574\pi$
$824$	32.0000	−	32.0000i	1.11477	−	1.11477i
$825$		0			0
$826$	−6.00000	+	6.00000i	−0.208767	+	0.208767i
$827$	−52.0000			−1.80822			−0.904109	−	0.427303i	$-0.859464\pi$
$827$	−52.0000			−1.80822			−0.904109	+	0.427303i	$0.859464\pi$
$828$	8.00000			0.278019
$829$			46.0000i			1.59765i	0.601566	+	0.798823i	$0.294544\pi$
$829$			46.0000i			1.59765i	−0.601566	+	0.798823i	$0.705456\pi$
$830$	12.0000	−	4.00000i	0.416526	−	0.138842i
$831$	18.0000			0.624413
$832$		−	48.0000i		−	1.66410i
$833$		−	2.00000i		−	0.0692959i
$834$	−16.0000	−	16.0000i	−0.554035	−	0.554035i
$835$	22.0000	−	44.0000i	0.761341	−	1.52268i
$836$		0			0
$837$	−8.00000			−0.276520
$838$	34.0000	+	34.0000i	1.17451	+	1.17451i
$839$	40.0000			1.38095			0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000			1.38095			0.690477	+	0.723355i	$0.257401\pi$
$840$	−6.00000	+	2.00000i	−0.207020	+	0.0690066i
$841$	−7.00000			−0.241379
$842$	20.0000	+	20.0000i	0.689246	+	0.689246i
$843$	−18.0000			−0.619953
$844$	40.0000			1.37686
$845$	−46.0000	−	23.0000i	−1.58245	−	0.791224i
$846$	−2.00000	−	2.00000i	−0.0687614	−	0.0687614i
$847$		−	11.0000i		−	0.377964i
$848$	24.0000			0.824163
$849$	4.00000			0.137280
$850$	2.00000	−	14.0000i	0.0685994	−	0.480196i
$851$		−	8.00000i		−	0.274236i
$852$			16.0000i			0.548151i
$853$	−26.0000			−0.890223			−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000			−0.890223			−0.445112	+	0.895475i	$0.646836\pi$
$854$	10.0000	−	10.0000i	0.342193	−	0.342193i
$855$	−4.00000	+	8.00000i	−0.136797	+	0.273594i
$856$	−24.0000	−	24.0000i	−0.820303	−	0.820303i
$857$		−	18.0000i		−	0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$		−	18.0000i		−	0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$		0			0
$859$			36.0000i			1.22830i	0.789188	+	0.614152i	$0.210502\pi$
$859$			36.0000i			1.22830i	−0.789188	+	0.614152i	$0.789498\pi$
$860$	12.0000	−	24.0000i	0.409197	−	0.818393i
$861$			8.00000i			0.272639i
$862$	−12.0000	+	12.0000i	−0.408722	+	0.408722i
$863$		−	36.0000i		−	1.22545i	−0.790295	−	0.612727i	$-0.790072\pi$
$863$		−	36.0000i		−	1.22545i	0.790295	−	0.612727i	$-0.209928\pi$
$864$	4.00000	−	4.00000i	0.136083	−	0.136083i
$865$	−48.0000	−	24.0000i	−1.63205	−	0.816024i
$866$	6.00000	+	6.00000i	0.203888	+	0.203888i
$867$	13.0000			0.441503
$868$	16.0000			0.543075
$869$		0			0
$870$	6.00000	+	18.0000i	0.203419	+	0.610257i
$871$	12.0000			0.406604
$872$	−32.0000	+	32.0000i	−1.08366	+	1.08366i
$873$			2.00000i			0.0676897i
$874$	−16.0000	+	16.0000i	−0.541208	+	0.541208i
$875$	−11.0000	+	2.00000i	−0.371868	+	0.0676123i
$876$	−12.0000			−0.405442
$877$	18.0000			0.607817			0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000			0.607817			0.303908	+	0.952701i	$0.401708\pi$
$878$		0			0
$879$	−16.0000			−0.539667
$880$		0			0
$881$	−28.0000			−0.943344			−0.471672	−	0.881774i	$-0.656349\pi$
$881$	−28.0000			−0.943344			−0.471672	+	0.881774i	$0.656349\pi$
$882$	1.00000	−	1.00000i	0.0336718	−	0.0336718i
$883$	−26.0000			−0.874970			−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000			−0.874970			−0.437485	+	0.899226i	$0.644131\pi$
$884$	−24.0000			−0.807207
$885$	6.00000	−	12.0000i	0.201688	−	0.403376i
$886$	36.0000	−	36.0000i	1.20944	−	1.20944i
$887$			42.0000i			1.41022i	0.709097	+	0.705111i	$0.249103\pi$
$887$			42.0000i			1.41022i	−0.709097	+	0.705111i	$0.750897\pi$
$888$	−4.00000	−	4.00000i	−0.134231	−	0.134231i
$889$	−8.00000			−0.268311
$890$		0			0
$891$		0			0
$892$	8.00000			0.267860
$893$	8.00000			0.267710
$894$	14.0000	+	14.0000i	0.468230	+	0.468230i
$895$	16.0000	−	32.0000i	0.534821	−	1.06964i
$896$	−8.00000	+	8.00000i	−0.267261	+	0.267261i
$897$		−	24.0000i		−	0.801337i
$898$	30.0000	−	30.0000i	1.00111	−	1.00111i
$899$		−	48.0000i		−	1.60089i
$900$	8.00000	−	6.00000i	0.266667	−	0.200000i
$901$		−	12.0000i		−	0.399778i
$902$		0			0
$903$			6.00000i			0.199667i
$904$	12.0000	−	12.0000i	0.399114	−	0.399114i
$905$	10.0000	−	20.0000i	0.332411	−	0.664822i
$906$	−2.00000	+	2.00000i	−0.0664455	+	0.0664455i
$907$	18.0000			0.597680			0.298840	−	0.954303i	$-0.403400\pi$
$907$	18.0000			0.597680			0.298840	+	0.954303i	$0.403400\pi$
$908$			24.0000i			0.796468i
$909$		−	10.0000i		−	0.331679i
$910$	6.00000	+	18.0000i	0.198898	+	0.596694i
$911$	−28.0000			−0.927681			−0.463841	−	0.885919i	$-0.653529\pi$
$911$	−28.0000			−0.927681			−0.463841	+	0.885919i	$0.653529\pi$
$912$			16.0000i			0.529813i
$913$		0			0
$914$	−12.0000	−	12.0000i	−0.396925	−	0.396925i
$915$	−10.0000	+	20.0000i	−0.330590	+	0.661180i
$916$	−28.0000			−0.925146
$917$	10.0000			0.330229
$918$	−2.00000	−	2.00000i	−0.0660098	−	0.0660098i
$919$	50.0000			1.64935			0.824674	−	0.565608i	$-0.191359\pi$
$919$	50.0000			1.64935			0.824674	+	0.565608i	$0.191359\pi$
$920$	24.0000	−	8.00000i	0.791257	−	0.263752i
$921$	28.0000			0.922631
$922$	−30.0000	−	30.0000i	−0.987997	−	0.987997i
$923$	48.0000			1.57994
$924$		0			0
$925$	−6.00000	−	8.00000i	−0.197279	−	0.263038i
$926$	16.0000	+	16.0000i	0.525793	+	0.525793i
$927$		−	16.0000i		−	0.525509i
$928$	24.0000	+	24.0000i	0.787839	+	0.787839i
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$	−24.0000	+	8.00000i	−0.786991	+	0.262330i
$931$			4.00000i			0.131095i
$932$	−12.0000			−0.393073
$933$	−8.00000			−0.261908
$934$	12.0000	−	12.0000i	0.392652	−	0.392652i
$935$		0			0
$936$	−12.0000	−	12.0000i	−0.392232	−	0.392232i
$937$		−	38.0000i		−	1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$		−	38.0000i		−	1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$	−2.00000	−	2.00000i	−0.0653023	−	0.0653023i
$939$			14.0000i			0.456873i
$940$	−8.00000	−	4.00000i	−0.260931	−	0.130466i
$941$			50.0000i			1.62995i	0.579494	+	0.814977i	$0.303250\pi$
$941$			50.0000i			1.62995i	−0.579494	+	0.814977i	$0.696750\pi$
$942$	22.0000	−	22.0000i	0.716799	−	0.716799i
$943$		−	32.0000i		−	1.04206i
$944$		−	24.0000i		−	0.781133i
$945$	−1.00000	+	2.00000i	−0.0325300	+	0.0650600i
$946$		0			0
$947$	−12.0000			−0.389948			−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000			−0.389948			−0.194974	+	0.980808i	$0.562462\pi$
$948$		−	20.0000i		−	0.649570i
$949$			36.0000i			1.16861i
$950$	−4.00000	+	28.0000i	−0.129777	+	0.908440i
$951$	18.0000			0.583690
$952$	4.00000	+	4.00000i	0.129641	+	0.129641i
$953$			34.0000i			1.10137i	0.834714	+	0.550684i	$0.185633\pi$
$953$			34.0000i			1.10137i	−0.834714	+	0.550684i	$0.814367\pi$
$954$	6.00000	−	6.00000i	0.194257	−	0.194257i
$955$	16.0000	+	8.00000i	0.517748	+	0.258874i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−18.0000			−0.581250
$960$	8.00000	−	16.0000i	0.258199	−	0.516398i
$961$	33.0000			1.06452
$962$	−12.0000	+	12.0000i	−0.386896	+	0.386896i
$963$	−12.0000			−0.386695
$964$			36.0000i			1.15948i
$965$	−16.0000	+	32.0000i	−0.515058	+	1.03012i
$966$	−4.00000	+	4.00000i	−0.128698	+	0.128698i
$967$		−	8.00000i		−	0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$		−	8.00000i		−	0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$	22.0000	+	22.0000i	0.707107	+	0.707107i
$969$	8.00000			0.256997
$970$	2.00000	+	6.00000i	0.0642161	+	0.192648i
$971$		−	10.0000i		−	0.320915i	−0.987043	−	0.160458i	$-0.948703\pi$
$971$		−	10.0000i		−	0.320915i	0.987043	−	0.160458i	$-0.0512970\pi$
$972$		−	2.00000i		−	0.0641500i
$973$	16.0000			0.512936
$974$	−32.0000	−	32.0000i	−1.02535	−	1.02535i
$975$	−18.0000	−	24.0000i	−0.576461	−	0.768615i
$976$			40.0000i			1.28037i
$977$		−	18.0000i		−	0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$		−	18.0000i		−	0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	6.00000	−	6.00000i	0.191859	−	0.191859i
$979$		0			0
$980$	2.00000	−	4.00000i	0.0638877	−	0.127775i
$981$			16.0000i			0.510841i
$982$	−20.0000	−	20.0000i	−0.638226	−	0.638226i
$983$			14.0000i			0.446531i	0.974758	+	0.223265i	$0.0716716\pi$
$983$			14.0000i			0.446531i	−0.974758	+	0.223265i	$0.928328\pi$
$984$	−16.0000	−	16.0000i	−0.510061	−	0.510061i
$985$	−36.0000	−	18.0000i	−1.14706	−	0.573528i
$986$	12.0000	−	12.0000i	0.382158	−	0.382158i
$987$	2.00000			0.0636607
$988$	48.0000			1.52708
$989$		−	24.0000i		−	0.763156i
$990$		0			0
$991$	22.0000			0.698853			0.349427	−	0.936964i	$-0.386376\pi$
$991$	22.0000			0.698853			0.349427	+	0.936964i	$0.386376\pi$
$992$	−32.0000	+	32.0000i	−1.01600	+	1.01600i
$993$		−	20.0000i		−	0.634681i
$994$	−8.00000	−	8.00000i	−0.253745	−	0.253745i
$995$	−40.0000	−	20.0000i	−1.26809	−	0.634043i
$996$		−	8.00000i		−	0.253490i
$997$	−2.00000			−0.0633406			−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000			−0.0633406			−0.0316703	+	0.999498i	$0.510083\pi$
$998$	4.00000	+	4.00000i	0.126618	+	0.126618i
$999$	−2.00000			−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.2.j.b.589.2	yes	2
4.3	odd	2		3360.2.j.a.1009.1		2
5.4	even	2		840.2.j.c.589.1	yes	2
8.3	odd	2		3360.2.j.d.1009.2		2
8.5	even	2		840.2.j.c.589.2	yes	2
20.19	odd	2		3360.2.j.d.1009.1		2
40.19	odd	2		3360.2.j.a.1009.2		2
40.29	even	2	inner	840.2.j.b.589.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.j.b.589.1	✓	2	40.29	even	2	inner
840.2.j.b.589.2	yes	2	1.1	even	1	trivial
840.2.j.c.589.1	yes	2	5.4	even	2
840.2.j.c.589.2	yes	2	8.5	even	2
3360.2.j.a.1009.1		2	4.3	odd	2
3360.2.j.a.1009.2		2	40.19	odd	2
3360.2.j.d.1009.1		2	20.19	odd	2
3360.2.j.d.1009.2		2	8.3	odd	2

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:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	840.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{2} +1.00000 q^{3} -2.00000i q^{4} +(-2.00000 - 1.00000i) q^{5} +(-1.00000 + 1.00000i) q^{6} -1.00000i q^{7} +(2.00000 + 2.00000i) q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+(-1.00000 + 1.00000i) q^{2} +1.00000 q^{3} -2.00000i q^{4} +(-2.00000 - 1.00000i) q^{5} +(-1.00000 + 1.00000i) q^{6} -1.00000i q^{7} +(2.00000 + 2.00000i) q^{8} +1.00000 q^{9} +(3.00000 - 1.00000i) q^{10} -2.00000i q^{12} -6.00000 q^{13} +(1.00000 + 1.00000i) q^{14} +(-2.00000 - 1.00000i) q^{15} -4.00000 q^{16} +2.00000i q^{17} +(-1.00000 + 1.00000i) q^{18} -4.00000i q^{19} +(-2.00000 + 4.00000i) q^{20} -1.00000i q^{21} +4.00000i q^{23} +(2.00000 + 2.00000i) q^{24} +(3.00000 + 4.00000i) q^{25} +(6.00000 - 6.00000i) q^{26} +1.00000 q^{27} -2.00000 q^{28} +6.00000i q^{29} +(3.00000 - 1.00000i) q^{30} -8.00000 q^{31} +(4.00000 - 4.00000i) q^{32} +(-2.00000 - 2.00000i) q^{34} +(-1.00000 + 2.00000i) q^{35} -2.00000i q^{36} -2.00000 q^{37} +(4.00000 + 4.00000i) q^{38} -6.00000 q^{39} +(-2.00000 - 6.00000i) q^{40} -8.00000 q^{41} +(1.00000 + 1.00000i) q^{42} -6.00000 q^{43} +(-2.00000 - 1.00000i) q^{45} +(-4.00000 - 4.00000i) q^{46} +2.00000i q^{47} -4.00000 q^{48} -1.00000 q^{49} +(-7.00000 - 1.00000i) q^{50} +2.00000i q^{51} +12.0000i q^{52} -6.00000 q^{53} +(-1.00000 + 1.00000i) q^{54} +(2.00000 - 2.00000i) q^{56} -4.00000i q^{57} +(-6.00000 - 6.00000i) q^{58} +6.00000i q^{59} +(-2.00000 + 4.00000i) q^{60} -10.0000i q^{61} +(8.00000 - 8.00000i) q^{62} -1.00000i q^{63} +8.00000i q^{64} +(12.0000 + 6.00000i) q^{65} -2.00000 q^{67} +4.00000 q^{68} +4.00000i q^{69} +(-1.00000 - 3.00000i) q^{70} -8.00000 q^{71} +(2.00000 + 2.00000i) q^{72} -6.00000i q^{73} +(2.00000 - 2.00000i) q^{74} +(3.00000 + 4.00000i) q^{75} -8.00000 q^{76} +(6.00000 - 6.00000i) q^{78} +10.0000 q^{79} +(8.00000 + 4.00000i) q^{80} +1.00000 q^{81} +(8.00000 - 8.00000i) q^{82} +4.00000 q^{83} -2.00000 q^{84} +(2.00000 - 4.00000i) q^{85} +(6.00000 - 6.00000i) q^{86} +6.00000i q^{87} +(3.00000 - 1.00000i) q^{90} +6.00000i q^{91} +8.00000 q^{92} -8.00000 q^{93} +(-2.00000 - 2.00000i) q^{94} +(-4.00000 + 8.00000i) q^{95} +(4.00000 - 4.00000i) q^{96} +2.00000i q^{97} +(1.00000 - 1.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} - 4 q^{5} - 2 q^{6} + 4 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 - 4 * q^5 - 2 * q^6 + 4 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} - 4 q^{5} - 2 q^{6} + 4 q^{8} + 2 q^{9} + 6 q^{10} - 12 q^{13} + 2 q^{14} - 4 q^{15} - 8 q^{16} - 2 q^{18} - 4 q^{20} + 4 q^{24} + 6 q^{25} + 12 q^{26} + 2 q^{27} - 4 q^{28} + 6 q^{30} - 16 q^{31} + 8 q^{32} - 4 q^{34} - 2 q^{35} - 4 q^{37} + 8 q^{38} - 12 q^{39} - 4 q^{40} - 16 q^{41} + 2 q^{42} - 12 q^{43} - 4 q^{45} - 8 q^{46} - 8 q^{48} - 2 q^{49} - 14 q^{50} - 12 q^{53} - 2 q^{54} + 4 q^{56} - 12 q^{58} - 4 q^{60} + 16 q^{62} + 24 q^{65} - 4 q^{67} + 8 q^{68} - 2 q^{70} - 16 q^{71} + 4 q^{72} + 4 q^{74} + 6 q^{75} - 16 q^{76} + 12 q^{78} + 20 q^{79} + 16 q^{80} + 2 q^{81} + 16 q^{82} + 8 q^{83} - 4 q^{84} + 4 q^{85} + 12 q^{86} + 6 q^{90} + 16 q^{92} - 16 q^{93} - 4 q^{94} - 8 q^{95} + 8 q^{96} + 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 - 4 * q^5 - 2 * q^6 + 4 * q^8 + 2 * q^9 + 6 * q^10 - 12 * q^13 + 2 * q^14 - 4 * q^15 - 8 * q^16 - 2 * q^18 - 4 * q^20 + 4 * q^24 + 6 * q^25 + 12 * q^26 + 2 * q^27 - 4 * q^28 + 6 * q^30 - 16 * q^31 + 8 * q^32 - 4 * q^34 - 2 * q^35 - 4 * q^37 + 8 * q^38 - 12 * q^39 - 4 * q^40 - 16 * q^41 + 2 * q^42 - 12 * q^43 - 4 * q^45 - 8 * q^46 - 8 * q^48 - 2 * q^49 - 14 * q^50 - 12 * q^53 - 2 * q^54 + 4 * q^56 - 12 * q^58 - 4 * q^60 + 16 * q^62 + 24 * q^65 - 4 * q^67 + 8 * q^68 - 2 * q^70 - 16 * q^71 + 4 * q^72 + 4 * q^74 + 6 * q^75 - 16 * q^76 + 12 * q^78 + 20 * q^79 + 16 * q^80 + 2 * q^81 + 16 * q^82 + 8 * q^83 - 4 * q^84 + 4 * q^85 + 12 * q^86 + 6 * q^90 + 16 * q^92 - 16 * q^93 - 4 * q^94 - 8 * q^95 + 8 * q^96 + 2 * q^98

Introduction

L-functions

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Database

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