LMFDB - Embedded newform 450.2.j.b.349.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,2,Mod(49,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("450.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	450.j (of order $6$, degree $2$, not 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:	$3.59326809096$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			349.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	450.349
Dual form			450.2.j.b.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 - 0.500000i) q^{2} +1.73205 q^{3} +(0.500000 + 0.866025i) q^{4} +(-1.50000 - 0.866025i) q^{6} +(-1.73205 - 1.00000i) q^{7} -1.00000i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+(-0.866025 - 0.500000i) q^{2} +1.73205 q^{3} +(0.500000 + 0.866025i) q^{4} +(-1.50000 - 0.866025i) q^{6} +(-1.73205 - 1.00000i) q^{7} -1.00000i q^{8} +3.00000 q^{9} +(0.866025 + 1.50000i) q^{12} +(3.46410 - 2.00000i) q^{13} +(1.00000 + 1.73205i) q^{14} +(-0.500000 + 0.866025i) q^{16} -6.00000i q^{17} +(-2.59808 - 1.50000i) q^{18} +7.00000 q^{19} +(-3.00000 - 1.73205i) q^{21} -1.73205i q^{24} -4.00000 q^{26} +5.19615 q^{27} -2.00000i q^{28} +(-3.00000 + 5.19615i) q^{29} +(5.00000 + 8.66025i) q^{31} +(0.866025 - 0.500000i) q^{32} +(-3.00000 + 5.19615i) q^{34} +(1.50000 + 2.59808i) q^{36} +2.00000i q^{37} +(-6.06218 - 3.50000i) q^{38} +(6.00000 - 3.46410i) q^{39} +(-4.50000 - 7.79423i) q^{41} +(1.73205 + 3.00000i) q^{42} +(-0.866025 - 0.500000i) q^{43} +(-5.19615 - 3.00000i) q^{47} +(-0.866025 + 1.50000i) q^{48} +(-1.50000 - 2.59808i) q^{49} -10.3923i q^{51} +(3.46410 + 2.00000i) q^{52} +12.0000i q^{53} +(-4.50000 - 2.59808i) q^{54} +(-1.00000 + 1.73205i) q^{56} +12.1244 q^{57} +(5.19615 - 3.00000i) q^{58} +(-4.50000 - 7.79423i) q^{59} +(2.00000 - 3.46410i) q^{61} -10.0000i q^{62} +(-5.19615 - 3.00000i) q^{63} -1.00000 q^{64} +(-11.2583 + 6.50000i) q^{67} +(5.19615 - 3.00000i) q^{68} +6.00000 q^{71} -3.00000i q^{72} +1.00000i q^{73} +(1.00000 - 1.73205i) q^{74} +(3.50000 + 6.06218i) q^{76} -6.92820 q^{78} +(1.00000 - 1.73205i) q^{79} +9.00000 q^{81} +9.00000i q^{82} +(7.79423 + 4.50000i) q^{83} -3.46410i q^{84} +(0.500000 + 0.866025i) q^{86} +(-5.19615 + 9.00000i) q^{87} -15.0000 q^{89} -8.00000 q^{91} +(8.66025 + 15.0000i) q^{93} +(3.00000 + 5.19615i) q^{94} +(1.50000 - 0.866025i) q^{96} +(-14.7224 - 8.50000i) q^{97} +3.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 6 q^{6} + 12 q^{9}+O(q^{10}) $ 4 * q + 2 * q^4 - 6 * q^6 + 12 * q^9	$ 4 q + 2 q^{4} - 6 q^{6} + 12 q^{9} + 4 q^{14} - 2 q^{16} + 28 q^{19} - 12 q^{21} - 16 q^{26} - 12 q^{29} + 20 q^{31} - 12 q^{34} + 6 q^{36} + 24 q^{39} - 18 q^{41} - 6 q^{49} - 18 q^{54} - 4 q^{56} - 18 q^{59} + 8 q^{61} - 4 q^{64} + 24 q^{71} + 4 q^{74} + 14 q^{76} + 4 q^{79} + 36 q^{81} + 2 q^{86} - 60 q^{89} - 32 q^{91} + 12 q^{94} + 6 q^{96}+O(q^{100}) $ 4 * q + 2 * q^4 - 6 * q^6 + 12 * q^9 + 4 * q^14 - 2 * q^16 + 28 * q^19 - 12 * q^21 - 16 * q^26 - 12 * q^29 + 20 * q^31 - 12 * q^34 + 6 * q^36 + 24 * q^39 - 18 * q^41 - 6 * q^49 - 18 * q^54 - 4 * q^56 - 18 * q^59 + 8 * q^61 - 4 * q^64 + 24 * q^71 + 4 * q^74 + 14 * q^76 + 4 * q^79 + 36 * q^81 + 2 * q^86 - 60 * q^89 - 32 * q^91 + 12 * q^94 + 6 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$-1$

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 $

$2$	−0.866025	−	0.500000i	−0.612372	−	0.353553i
$2$	−0.866025	−	0.500000i	−0.612372	−	0.353553i
$3$	1.73205			1.00000
$3$	1.73205			1.00000
$4$	0.500000	+	0.866025i	0.250000	+	0.433013i
$5$		0			0
$5$		0			0
$6$	−1.50000	−	0.866025i	−0.612372	−	0.353553i
$7$	−1.73205	−	1.00000i	−0.654654	−	0.377964i	0.135583	−	0.990766i	$-0.456709\pi$
$7$	−1.73205	−	1.00000i	−0.654654	−	0.377964i	−0.790237	+	0.612801i	$0.790043\pi$
$8$		−	1.00000i		−	0.353553i
$9$	3.00000			1.00000
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$	0.866025	+	1.50000i	0.250000	+	0.433013i
$13$	3.46410	−	2.00000i	0.960769	−	0.554700i	0.0643593	−	0.997927i	$-0.479500\pi$
$13$	3.46410	−	2.00000i	0.960769	−	0.554700i	0.896410	+	0.443227i	$0.146166\pi$
$14$	1.00000	+	1.73205i	0.267261	+	0.462910i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$		−	6.00000i		−	1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$		−	6.00000i		−	1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	−2.59808	−	1.50000i	−0.612372	−	0.353553i
$19$	7.00000			1.60591			0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000			1.60591			0.802955	+	0.596040i	$0.203260\pi$
$20$		0			0
$21$	−3.00000	−	1.73205i	−0.654654	−	0.377964i
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		−	1.73205i		−	0.353553i
$25$		0			0
$26$	−4.00000			−0.784465
$27$	5.19615			1.00000
$28$		−	2.00000i		−	0.377964i
$29$	−3.00000	+	5.19615i	−0.557086	+	0.964901i	0.440652	+	0.897678i	$0.354747\pi$
$29$	−3.00000	+	5.19615i	−0.557086	+	0.964901i	−0.997738	+	0.0672232i	$0.978586\pi$
$30$		0			0
$31$	5.00000	+	8.66025i	0.898027	+	1.55543i	0.830014	+	0.557743i	$0.188333\pi$
$31$	5.00000	+	8.66025i	0.898027	+	1.55543i	0.0680129	+	0.997684i	$0.478334\pi$
$32$	0.866025	−	0.500000i	0.153093	−	0.0883883i
$33$		0			0
$34$	−3.00000	+	5.19615i	−0.514496	+	0.891133i
$35$		0			0
$36$	1.50000	+	2.59808i	0.250000	+	0.433013i
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	−6.06218	−	3.50000i	−0.983415	−	0.567775i
$39$	6.00000	−	3.46410i	0.960769	−	0.554700i
$40$		0			0
$41$	−4.50000	−	7.79423i	−0.702782	−	1.21725i	−0.967486	−	0.252924i	$-0.918608\pi$
$41$	−4.50000	−	7.79423i	−0.702782	−	1.21725i	0.264704	−	0.964330i	$-0.414726\pi$
$42$	1.73205	+	3.00000i	0.267261	+	0.462910i
$43$	−0.866025	−	0.500000i	−0.132068	−	0.0762493i	0.432511	−	0.901629i	$-0.357628\pi$
$43$	−0.866025	−	0.500000i	−0.132068	−	0.0762493i	−0.564578	+	0.825380i	$0.690961\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.19615	−	3.00000i	−0.757937	−	0.437595i	0.0706177	−	0.997503i	$-0.477503\pi$
$47$	−5.19615	−	3.00000i	−0.757937	−	0.437595i	−0.828554	+	0.559908i	$0.810836\pi$
$48$	−0.866025	+	1.50000i	−0.125000	+	0.216506i
$49$	−1.50000	−	2.59808i	−0.214286	−	0.371154i
$50$		0			0
$51$		−	10.3923i		−	1.45521i
$52$	3.46410	+	2.00000i	0.480384	+	0.277350i
$53$			12.0000i			1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$			12.0000i			1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$	−4.50000	−	2.59808i	−0.612372	−	0.353553i
$55$		0			0
$56$	−1.00000	+	1.73205i	−0.133631	+	0.231455i
$57$	12.1244			1.60591
$58$	5.19615	−	3.00000i	0.682288	−	0.393919i
$59$	−4.50000	−	7.79423i	−0.585850	−	1.01472i	−0.994769	−	0.102151i	$-0.967427\pi$
$59$	−4.50000	−	7.79423i	−0.585850	−	1.01472i	0.408919	−	0.912571i	$-0.365906\pi$
$60$		0			0
$61$	2.00000	−	3.46410i	0.256074	−	0.443533i	−0.709113	−	0.705095i	$-0.750904\pi$
$61$	2.00000	−	3.46410i	0.256074	−	0.443533i	0.965187	+	0.261562i	$0.0842377\pi$
$62$		−	10.0000i		−	1.27000i
$63$	−5.19615	−	3.00000i	−0.654654	−	0.377964i
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$	−11.2583	+	6.50000i	−1.37542	+	0.794101i	−0.991605	−	0.129307i	$-0.958725\pi$
$67$	−11.2583	+	6.50000i	−1.37542	+	0.794101i	−0.383819	+	0.923408i	$0.625391\pi$
$68$	5.19615	−	3.00000i	0.630126	−	0.363803i
$69$		0			0
$70$		0			0
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		−	3.00000i		−	0.353553i
$73$			1.00000i			0.117041i	0.998286	+	0.0585206i	$0.0186383\pi$
$73$			1.00000i			0.117041i	−0.998286	+	0.0585206i	$0.981362\pi$
$74$	1.00000	−	1.73205i	0.116248	−	0.201347i
$75$		0			0
$76$	3.50000	+	6.06218i	0.401478	+	0.695379i
$77$		0			0
$78$	−6.92820			−0.784465
$79$	1.00000	−	1.73205i	0.112509	−	0.194871i	−0.804272	−	0.594261i	$-0.797445\pi$
$79$	1.00000	−	1.73205i	0.112509	−	0.194871i	0.916781	+	0.399390i	$0.130778\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$			9.00000i			0.993884i
$83$	7.79423	+	4.50000i	0.855528	+	0.493939i	0.862512	−	0.506036i	$-0.168890\pi$
$83$	7.79423	+	4.50000i	0.855528	+	0.493939i	−0.00698436	+	0.999976i	$0.502223\pi$
$84$		−	3.46410i		−	0.377964i
$85$		0			0
$86$	0.500000	+	0.866025i	0.0539164	+	0.0933859i
$87$	−5.19615	+	9.00000i	−0.557086	+	0.964901i
$88$		0			0
$89$	−15.0000			−1.59000			−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000			−1.59000			−0.794998	+	0.606612i	$0.792528\pi$
$90$		0			0
$91$	−8.00000			−0.838628
$92$		0			0
$93$	8.66025	+	15.0000i	0.898027	+	1.55543i
$94$	3.00000	+	5.19615i	0.309426	+	0.535942i
$95$		0			0
$96$	1.50000	−	0.866025i	0.153093	−	0.0883883i
$97$	−14.7224	−	8.50000i	−1.49484	−	0.863044i	−0.494854	−	0.868976i	$-0.664778\pi$
$97$	−14.7224	−	8.50000i	−1.49484	−	0.863044i	−0.999982	+	0.00593185i	$0.998112\pi$
$98$			3.00000i			0.303046i
$99$		0			0
$100$		0			0
$101$	−6.00000	+	10.3923i	−0.597022	+	1.03407i	0.396236	+	0.918149i	$0.370316\pi$
$101$	−6.00000	+	10.3923i	−0.597022	+	1.03407i	−0.993258	+	0.115924i	$0.963017\pi$
$102$	−5.19615	+	9.00000i	−0.514496	+	0.891133i
$103$	−1.73205	+	1.00000i	−0.170664	+	0.0985329i	−0.582899	−	0.812545i	$-0.698082\pi$
$103$	−1.73205	+	1.00000i	−0.170664	+	0.0985329i	0.412235	+	0.911078i	$0.364748\pi$
$104$	−2.00000	−	3.46410i	−0.196116	−	0.339683i
$105$		0			0
$106$	6.00000	−	10.3923i	0.582772	−	1.00939i
$107$			9.00000i			0.870063i	0.900415	+	0.435031i	$0.143263\pi$
$107$			9.00000i			0.870063i	−0.900415	+	0.435031i	$0.856737\pi$
$108$	2.59808	+	4.50000i	0.250000	+	0.433013i
$109$	−2.00000			−0.191565			−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000			−0.191565			−0.0957826	+	0.995402i	$0.530535\pi$
$110$		0			0
$111$			3.46410i			0.328798i
$112$	1.73205	−	1.00000i	0.163663	−	0.0944911i
$113$	−2.59808	+	1.50000i	−0.244406	+	0.141108i	−0.617200	−	0.786806i	$-0.711733\pi$
$113$	−2.59808	+	1.50000i	−0.244406	+	0.141108i	0.372794	+	0.927914i	$0.378400\pi$
$114$	−10.5000	−	6.06218i	−0.983415	−	0.567775i
$115$		0			0
$116$	−6.00000			−0.557086
$117$	10.3923	−	6.00000i	0.960769	−	0.554700i
$118$			9.00000i			0.828517i
$119$	−6.00000	+	10.3923i	−0.550019	+	0.952661i
$120$		0			0
$121$	5.50000	+	9.52628i	0.500000	+	0.866025i
$122$	−3.46410	+	2.00000i	−0.313625	+	0.181071i
$123$	−7.79423	−	13.5000i	−0.702782	−	1.21725i
$124$	−5.00000	+	8.66025i	−0.449013	+	0.777714i
$125$		0			0
$126$	3.00000	+	5.19615i	0.267261	+	0.462910i
$127$			20.0000i			1.77471i	0.461084	+	0.887357i	$0.347461\pi$
$127$			20.0000i			1.77471i	−0.461084	+	0.887357i	$0.652539\pi$
$128$	0.866025	+	0.500000i	0.0765466	+	0.0441942i
$129$	−1.50000	−	0.866025i	−0.132068	−	0.0762493i
$130$		0			0
$131$	6.00000	+	10.3923i	0.524222	+	0.907980i	0.999602	+	0.0281993i	$0.00897729\pi$
$131$	6.00000	+	10.3923i	0.524222	+	0.907980i	−0.475380	+	0.879781i	$0.657689\pi$
$132$		0			0
$133$	−12.1244	−	7.00000i	−1.05131	−	0.606977i
$134$	13.0000			1.12303
$135$		0			0
$136$	−6.00000			−0.514496
$137$	2.59808	+	1.50000i	0.221969	+	0.128154i	0.606861	−	0.794808i	$-0.292428\pi$
$137$	2.59808	+	1.50000i	0.221969	+	0.128154i	−0.384893	+	0.922961i	$0.625762\pi$
$138$		0			0
$139$	−2.00000	−	3.46410i	−0.169638	−	0.293821i	0.768655	−	0.639664i	$-0.220926\pi$
$139$	−2.00000	−	3.46410i	−0.169638	−	0.293821i	−0.938293	+	0.345843i	$0.887593\pi$
$140$		0			0
$141$	−9.00000	−	5.19615i	−0.757937	−	0.437595i
$142$	−5.19615	−	3.00000i	−0.436051	−	0.251754i
$143$		0			0
$144$	−1.50000	+	2.59808i	−0.125000	+	0.216506i
$145$		0			0
$146$	0.500000	−	0.866025i	0.0413803	−	0.0716728i
$147$	−2.59808	−	4.50000i	−0.214286	−	0.371154i
$148$	−1.73205	+	1.00000i	−0.142374	+	0.0821995i
$149$	−3.00000	−	5.19615i	−0.245770	−	0.425685i	0.716578	−	0.697507i	$-0.245707\pi$
$149$	−3.00000	−	5.19615i	−0.245770	−	0.425685i	−0.962348	+	0.271821i	$0.912374\pi$
$150$		0			0
$151$	2.00000	−	3.46410i	0.162758	−	0.281905i	−0.773099	−	0.634285i	$-0.781294\pi$
$151$	2.00000	−	3.46410i	0.162758	−	0.281905i	0.935857	+	0.352381i	$0.114628\pi$
$152$		−	7.00000i		−	0.567775i
$153$		−	18.0000i		−	1.45521i
$154$		0			0
$155$		0			0
$156$	6.00000	+	3.46410i	0.480384	+	0.277350i
$157$	12.1244	−	7.00000i	0.967629	−	0.558661i	0.0691164	−	0.997609i	$-0.477982\pi$
$157$	12.1244	−	7.00000i	0.967629	−	0.558661i	0.898513	+	0.438948i	$0.144649\pi$
$158$	−1.73205	+	1.00000i	−0.137795	+	0.0795557i
$159$			20.7846i			1.64833i
$160$		0			0
$161$		0			0
$162$	−7.79423	−	4.50000i	−0.612372	−	0.353553i
$163$			7.00000i			0.548282i	0.961689	+	0.274141i	$0.0883936\pi$
$163$			7.00000i			0.548282i	−0.961689	+	0.274141i	$0.911606\pi$
$164$	4.50000	−	7.79423i	0.351391	−	0.608627i
$165$		0			0
$166$	−4.50000	−	7.79423i	−0.349268	−	0.604949i
$167$	10.3923	−	6.00000i	0.804181	−	0.464294i	−0.0407502	−	0.999169i	$-0.512975\pi$
$167$	10.3923	−	6.00000i	0.804181	−	0.464294i	0.844931	+	0.534875i	$0.179641\pi$
$168$	−1.73205	+	3.00000i	−0.133631	+	0.231455i
$169$	1.50000	−	2.59808i	0.115385	−	0.199852i
$170$		0			0
$171$	21.0000			1.60591
$172$		−	1.00000i		−	0.0762493i
$173$	−10.3923	−	6.00000i	−0.790112	−	0.456172i	0.0498898	−	0.998755i	$-0.484113\pi$
$173$	−10.3923	−	6.00000i	−0.790112	−	0.456172i	−0.840002	+	0.542583i	$0.817446\pi$
$174$	9.00000	−	5.19615i	0.682288	−	0.393919i
$175$		0			0
$176$		0			0
$177$	−7.79423	−	13.5000i	−0.585850	−	1.01472i
$178$	12.9904	+	7.50000i	0.973670	+	0.562149i
$179$	9.00000			0.672692			0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000			0.672692			0.336346	+	0.941739i	$0.390809\pi$
$180$		0			0
$181$	−16.0000			−1.18927			−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000			−1.18927			−0.594635	+	0.803996i	$0.702704\pi$
$182$	6.92820	+	4.00000i	0.513553	+	0.296500i
$183$	3.46410	−	6.00000i	0.256074	−	0.443533i
$184$		0			0
$185$		0			0
$186$		−	17.3205i		−	1.27000i
$187$		0			0
$188$		−	6.00000i		−	0.437595i
$189$	−9.00000	−	5.19615i	−0.654654	−	0.377964i
$190$		0			0
$191$	−3.00000	+	5.19615i	−0.217072	+	0.375980i	−0.953912	−	0.300088i	$-0.902984\pi$
$191$	−3.00000	+	5.19615i	−0.217072	+	0.375980i	0.736839	+	0.676068i	$0.236317\pi$
$192$	−1.73205			−0.125000
$193$	−1.73205	+	1.00000i	−0.124676	+	0.0719816i	−0.561041	−	0.827788i	$-0.689599\pi$
$193$	−1.73205	+	1.00000i	−0.124676	+	0.0719816i	0.436365	+	0.899770i	$0.356266\pi$
$194$	8.50000	+	14.7224i	0.610264	+	1.05701i
$195$		0			0
$196$	1.50000	−	2.59808i	0.107143	−	0.185577i
$197$			18.0000i			1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$			18.0000i			1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$		0			0
$199$	16.0000			1.13421			0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000			1.13421			0.567105	+	0.823646i	$0.308063\pi$
$200$		0			0
$201$	−19.5000	+	11.2583i	−1.37542	+	0.794101i
$202$	10.3923	−	6.00000i	0.731200	−	0.422159i
$203$	10.3923	−	6.00000i	0.729397	−	0.421117i
$204$	9.00000	−	5.19615i	0.630126	−	0.363803i
$205$		0			0
$206$	2.00000			0.139347
$207$		0			0
$208$			4.00000i			0.277350i
$209$		0			0
$210$		0			0
$211$	−2.50000	−	4.33013i	−0.172107	−	0.298098i	0.767049	−	0.641588i	$-0.221724\pi$
$211$	−2.50000	−	4.33013i	−0.172107	−	0.298098i	−0.939156	+	0.343490i	$0.888391\pi$
$212$	−10.3923	+	6.00000i	−0.713746	+	0.412082i
$213$	10.3923			0.712069
$214$	4.50000	−	7.79423i	0.307614	−	0.532803i
$215$		0			0
$216$		−	5.19615i		−	0.353553i
$217$		−	20.0000i		−	1.35769i
$218$	1.73205	+	1.00000i	0.117309	+	0.0677285i
$219$			1.73205i			0.117041i
$220$		0			0
$221$	−12.0000	−	20.7846i	−0.807207	−	1.39812i
$222$	1.73205	−	3.00000i	0.116248	−	0.201347i
$223$	−13.8564	−	8.00000i	−0.927894	−	0.535720i	−0.0417488	−	0.999128i	$-0.513293\pi$
$223$	−13.8564	−	8.00000i	−0.927894	−	0.535720i	−0.886145	+	0.463409i	$0.846626\pi$
$224$	−2.00000			−0.133631
$225$		0			0
$226$	3.00000			0.199557
$227$	12.9904	+	7.50000i	0.862202	+	0.497792i	0.864749	−	0.502204i	$-0.167477\pi$
$227$	12.9904	+	7.50000i	0.862202	+	0.497792i	−0.00254715	+	0.999997i	$0.500811\pi$
$228$	6.06218	+	10.5000i	0.401478	+	0.695379i
$229$	−5.00000	−	8.66025i	−0.330409	−	0.572286i	0.652183	−	0.758062i	$-0.273853\pi$
$229$	−5.00000	−	8.66025i	−0.330409	−	0.572286i	−0.982592	+	0.185776i	$0.940520\pi$
$230$		0			0
$231$		0			0
$232$	5.19615	+	3.00000i	0.341144	+	0.196960i
$233$		−	3.00000i		−	0.196537i	−0.995160	−	0.0982683i	$-0.968670\pi$
$233$		−	3.00000i		−	0.196537i	0.995160	−	0.0982683i	$-0.0313303\pi$
$234$	−12.0000			−0.784465
$235$		0			0
$236$	4.50000	−	7.79423i	0.292925	−	0.507361i
$237$	1.73205	−	3.00000i	0.112509	−	0.194871i
$238$	10.3923	−	6.00000i	0.673633	−	0.388922i
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$		0			0
$241$	9.50000	−	16.4545i	0.611949	−	1.05993i	−0.378963	−	0.925412i	$-0.623719\pi$
$241$	9.50000	−	16.4545i	0.611949	−	1.05993i	0.990912	−	0.134515i	$-0.0429475\pi$
$242$		−	11.0000i		−	0.707107i
$243$	15.5885			1.00000
$244$	4.00000			0.256074
$245$		0			0
$246$			15.5885i			0.993884i
$247$	24.2487	−	14.0000i	1.54291	−	0.890799i
$248$	8.66025	−	5.00000i	0.549927	−	0.317500i
$249$	13.5000	+	7.79423i	0.855528	+	0.493939i
$250$		0			0
$251$	15.0000			0.946792			0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000			0.946792			0.473396	+	0.880850i	$0.343028\pi$
$252$		−	6.00000i		−	0.377964i
$253$		0			0
$254$	10.0000	−	17.3205i	0.627456	−	1.08679i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	2.59808	−	1.50000i	0.162064	−	0.0935674i	−0.416775	−	0.909010i	$-0.636840\pi$
$257$	2.59808	−	1.50000i	0.162064	−	0.0935674i	0.578838	+	0.815442i	$0.303506\pi$
$258$	0.866025	+	1.50000i	0.0539164	+	0.0933859i
$259$	2.00000	−	3.46410i	0.124274	−	0.215249i
$260$		0			0
$261$	−9.00000	+	15.5885i	−0.557086	+	0.964901i
$262$		−	12.0000i		−	0.741362i
$263$	−5.19615	−	3.00000i	−0.320408	−	0.184988i	0.331166	−	0.943572i	$-0.392558\pi$
$263$	−5.19615	−	3.00000i	−0.320408	−	0.184988i	−0.651575	+	0.758585i	$0.725891\pi$
$264$		0			0
$265$		0			0
$266$	7.00000	+	12.1244i	0.429198	+	0.743392i
$267$	−25.9808			−1.59000
$268$	−11.2583	−	6.50000i	−0.687712	−	0.397051i
$269$	30.0000			1.82913			0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000			1.82913			0.914566	+	0.404436i	$0.132532\pi$
$270$		0			0
$271$	−28.0000			−1.70088			−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000			−1.70088			−0.850439	+	0.526073i	$0.823664\pi$
$272$	5.19615	+	3.00000i	0.315063	+	0.181902i
$273$	−13.8564			−0.838628
$274$	−1.50000	−	2.59808i	−0.0906183	−	0.156956i
$275$		0			0
$276$		0			0
$277$	−1.73205	−	1.00000i	−0.104069	−	0.0600842i	0.447062	−	0.894503i	$-0.352470\pi$
$277$	−1.73205	−	1.00000i	−0.104069	−	0.0600842i	−0.551131	+	0.834419i	$0.685804\pi$
$278$			4.00000i			0.239904i
$279$	15.0000	+	25.9808i	0.898027	+	1.55543i
$280$		0			0
$281$	9.00000	−	15.5885i	0.536895	−	0.929929i	−0.462174	−	0.886789i	$-0.652930\pi$
$281$	9.00000	−	15.5885i	0.536895	−	0.929929i	0.999069	−	0.0431402i	$-0.0137362\pi$
$282$	5.19615	+	9.00000i	0.309426	+	0.535942i
$283$	16.4545	−	9.50000i	0.978117	−	0.564716i	0.0764162	−	0.997076i	$-0.475652\pi$
$283$	16.4545	−	9.50000i	0.978117	−	0.564716i	0.901701	+	0.432360i	$0.142319\pi$
$284$	3.00000	+	5.19615i	0.178017	+	0.308335i
$285$		0			0
$286$		0			0
$287$			18.0000i			1.06251i
$288$	2.59808	−	1.50000i	0.153093	−	0.0883883i
$289$	−19.0000			−1.11765
$290$		0			0
$291$	−25.5000	−	14.7224i	−1.49484	−	0.863044i
$292$	−0.866025	+	0.500000i	−0.0506803	+	0.0292603i
$293$	−10.3923	+	6.00000i	−0.607125	+	0.350524i	−0.771839	−	0.635818i	$-0.780663\pi$
$293$	−10.3923	+	6.00000i	−0.607125	+	0.350524i	0.164714	+	0.986341i	$0.447330\pi$
$294$			5.19615i			0.303046i
$295$		0			0
$296$	2.00000			0.116248
$297$		0			0
$298$			6.00000i			0.347571i
$299$		0			0
$300$		0			0
$301$	1.00000	+	1.73205i	0.0576390	+	0.0998337i
$302$	−3.46410	+	2.00000i	−0.199337	+	0.115087i
$303$	−10.3923	+	18.0000i	−0.597022	+	1.03407i
$304$	−3.50000	+	6.06218i	−0.200739	+	0.347690i
$305$		0			0
$306$	−9.00000	+	15.5885i	−0.514496	+	0.891133i
$307$			8.00000i			0.456584i	0.973593	+	0.228292i	$0.0733141\pi$
$307$			8.00000i			0.456584i	−0.973593	+	0.228292i	$0.926686\pi$
$308$		0			0
$309$	−3.00000	+	1.73205i	−0.170664	+	0.0985329i
$310$		0			0
$311$	15.0000	+	25.9808i	0.850572	+	1.47323i	0.880693	+	0.473688i	$0.157077\pi$
$311$	15.0000	+	25.9808i	0.850572	+	1.47323i	−0.0301210	+	0.999546i	$0.509589\pi$
$312$	−3.46410	−	6.00000i	−0.196116	−	0.339683i
$313$	−16.4545	−	9.50000i	−0.930062	−	0.536972i	−0.0432311	−	0.999065i	$-0.513765\pi$
$313$	−16.4545	−	9.50000i	−0.930062	−	0.536972i	−0.886831	+	0.462093i	$0.847098\pi$
$314$	−14.0000			−0.790066
$315$		0			0
$316$	2.00000			0.112509
$317$	−25.9808	−	15.0000i	−1.45922	−	0.842484i	−0.460252	−	0.887788i	$-0.652241\pi$
$317$	−25.9808	−	15.0000i	−1.45922	−	0.842484i	−0.998973	+	0.0453045i	$0.985574\pi$
$318$	10.3923	−	18.0000i	0.582772	−	1.00939i
$319$		0			0
$320$		0			0
$321$			15.5885i			0.870063i
$322$		0			0
$323$		−	42.0000i		−	2.33694i
$324$	4.50000	+	7.79423i	0.250000	+	0.433013i
$325$		0			0
$326$	3.50000	−	6.06218i	0.193847	−	0.335753i
$327$	−3.46410			−0.191565
$328$	−7.79423	+	4.50000i	−0.430364	+	0.248471i
$329$	6.00000	+	10.3923i	0.330791	+	0.572946i
$330$		0			0
$331$	−5.50000	+	9.52628i	−0.302307	+	0.523612i	−0.976658	−	0.214799i	$-0.931090\pi$
$331$	−5.50000	+	9.52628i	−0.302307	+	0.523612i	0.674351	+	0.738411i	$0.264424\pi$
$332$			9.00000i			0.493939i
$333$			6.00000i			0.328798i
$334$	−12.0000			−0.656611
$335$		0			0
$336$	3.00000	−	1.73205i	0.163663	−	0.0944911i
$337$	12.1244	−	7.00000i	0.660456	−	0.381314i	−0.131995	−	0.991250i	$-0.542138\pi$
$337$	12.1244	−	7.00000i	0.660456	−	0.381314i	0.792451	+	0.609936i	$0.208805\pi$
$338$	−2.59808	+	1.50000i	−0.141317	+	0.0815892i
$339$	−4.50000	+	2.59808i	−0.244406	+	0.141108i
$340$		0			0
$341$		0			0
$342$	−18.1865	−	10.5000i	−0.983415	−	0.567775i
$343$			20.0000i			1.07990i
$344$	−0.500000	+	0.866025i	−0.0269582	+	0.0466930i
$345$		0			0
$346$	6.00000	+	10.3923i	0.322562	+	0.558694i
$347$	−10.3923	+	6.00000i	−0.557888	+	0.322097i	−0.752297	−	0.658824i	$-0.771054\pi$
$347$	−10.3923	+	6.00000i	−0.557888	+	0.322097i	0.194409	+	0.980921i	$0.437721\pi$
$348$	−10.3923			−0.557086
$349$	−5.00000	+	8.66025i	−0.267644	+	0.463573i	−0.968253	−	0.249973i	$-0.919578\pi$
$349$	−5.00000	+	8.66025i	−0.267644	+	0.463573i	0.700609	+	0.713545i	$0.252912\pi$
$350$		0			0
$351$	18.0000	−	10.3923i	0.960769	−	0.554700i
$352$		0			0
$353$	12.9904	+	7.50000i	0.691408	+	0.399185i	0.804139	−	0.594441i	$-0.202627\pi$
$353$	12.9904	+	7.50000i	0.691408	+	0.399185i	−0.112731	+	0.993626i	$0.535960\pi$
$354$			15.5885i			0.828517i
$355$		0			0
$356$	−7.50000	−	12.9904i	−0.397499	−	0.688489i
$357$	−10.3923	+	18.0000i	−0.550019	+	0.952661i
$358$	−7.79423	−	4.50000i	−0.411938	−	0.237832i
$359$	−12.0000			−0.633336			−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000			−0.633336			−0.316668	+	0.948536i	$0.602564\pi$
$360$		0			0
$361$	30.0000			1.57895
$362$	13.8564	+	8.00000i	0.728277	+	0.420471i
$363$	9.52628	+	16.5000i	0.500000	+	0.866025i
$364$	−4.00000	−	6.92820i	−0.209657	−	0.363137i
$365$		0			0
$366$	−6.00000	+	3.46410i	−0.313625	+	0.181071i
$367$	8.66025	+	5.00000i	0.452062	+	0.260998i	0.708700	−	0.705509i	$-0.249282\pi$
$367$	8.66025	+	5.00000i	0.452062	+	0.260998i	−0.256639	+	0.966507i	$0.582615\pi$
$368$		0			0
$369$	−13.5000	−	23.3827i	−0.702782	−	1.21725i
$370$		0			0
$371$	12.0000	−	20.7846i	0.623009	−	1.07908i
$372$	−8.66025	+	15.0000i	−0.449013	+	0.777714i
$373$	3.46410	−	2.00000i	0.179364	−	0.103556i	−0.407630	−	0.913147i	$-0.633645\pi$
$373$	3.46410	−	2.00000i	0.179364	−	0.103556i	0.586994	+	0.809591i	$0.300311\pi$
$374$		0			0
$375$		0			0
$376$	−3.00000	+	5.19615i	−0.154713	+	0.267971i
$377$			24.0000i			1.23606i
$378$	5.19615	+	9.00000i	0.267261	+	0.462910i
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$		0			0
$381$			34.6410i			1.77471i
$382$	5.19615	−	3.00000i	0.265858	−	0.153493i
$383$	−20.7846	+	12.0000i	−1.06204	+	0.613171i	−0.925997	−	0.377531i	$-0.876773\pi$
$383$	−20.7846	+	12.0000i	−1.06204	+	0.613171i	−0.136047	+	0.990702i	$0.543440\pi$
$384$	1.50000	+	0.866025i	0.0765466	+	0.0441942i
$385$		0			0
$386$	2.00000			0.101797
$387$	−2.59808	−	1.50000i	−0.132068	−	0.0762493i
$388$		−	17.0000i		−	0.863044i
$389$	3.00000	−	5.19615i	0.152106	−	0.263455i	−0.779895	−	0.625910i	$-0.784728\pi$
$389$	3.00000	−	5.19615i	0.152106	−	0.263455i	0.932002	+	0.362454i	$0.118061\pi$
$390$		0			0
$391$		0			0
$392$	−2.59808	+	1.50000i	−0.131223	+	0.0757614i
$393$	10.3923	+	18.0000i	0.524222	+	0.907980i
$394$	9.00000	−	15.5885i	0.453413	−	0.785335i
$395$		0			0
$396$		0			0
$397$		−	22.0000i		−	1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$		−	22.0000i		−	1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$	−13.8564	−	8.00000i	−0.694559	−	0.401004i
$399$	−21.0000	−	12.1244i	−1.05131	−	0.606977i
$400$		0			0
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	0.548911	−	0.835881i	$-0.315043\pi$
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	−0.998350	+	0.0574304i	$0.981709\pi$
$402$	22.5167			1.12303
$403$	34.6410	+	20.0000i	1.72559	+	0.996271i
$404$	−12.0000			−0.597022
$405$		0			0
$406$	−12.0000			−0.595550
$407$		0			0
$408$	−10.3923			−0.514496
$409$	−15.5000	−	26.8468i	−0.766426	−	1.32749i	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−15.5000	−	26.8468i	−0.766426	−	1.32749i	0.173064	−	0.984911i	$-0.444633\pi$
$410$		0			0
$411$	4.50000	+	2.59808i	0.221969	+	0.128154i
$412$	−1.73205	−	1.00000i	−0.0853320	−	0.0492665i
$413$			18.0000i			0.885722i
$414$		0			0
$415$		0			0
$416$	2.00000	−	3.46410i	0.0980581	−	0.169842i
$417$	−3.46410	−	6.00000i	−0.169638	−	0.293821i
$418$		0			0
$419$	1.50000	+	2.59808i	0.0732798	+	0.126924i	0.900337	−	0.435194i	$-0.143320\pi$
$419$	1.50000	+	2.59808i	0.0732798	+	0.126924i	−0.827057	+	0.562118i	$0.809987\pi$
$420$		0			0
$421$	5.00000	−	8.66025i	0.243685	−	0.422075i	−0.718076	−	0.695965i	$-0.754977\pi$
$421$	5.00000	−	8.66025i	0.243685	−	0.422075i	0.961761	+	0.273890i	$0.0883103\pi$
$422$			5.00000i			0.243396i
$423$	−15.5885	−	9.00000i	−0.757937	−	0.437595i
$424$	12.0000			0.582772
$425$		0			0
$426$	−9.00000	−	5.19615i	−0.436051	−	0.251754i
$427$	−6.92820	+	4.00000i	−0.335279	+	0.193574i
$428$	−7.79423	+	4.50000i	−0.376748	+	0.217516i
$429$		0			0
$430$		0			0
$431$	6.00000			0.289010			0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000			0.289010			0.144505	+	0.989504i	$0.453841\pi$
$432$	−2.59808	+	4.50000i	−0.125000	+	0.216506i
$433$		−	2.00000i		−	0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$		−	2.00000i		−	0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	−10.0000	+	17.3205i	−0.480015	+	0.831411i
$435$		0			0
$436$	−1.00000	−	1.73205i	−0.0478913	−	0.0829502i
$437$		0			0
$438$	0.866025	−	1.50000i	0.0413803	−	0.0716728i
$439$	−2.00000	+	3.46410i	−0.0954548	+	0.165333i	−0.909798	−	0.415051i	$-0.863764\pi$
$439$	−2.00000	+	3.46410i	−0.0954548	+	0.165333i	0.814344	+	0.580383i	$0.197097\pi$
$440$		0			0
$441$	−4.50000	−	7.79423i	−0.214286	−	0.371154i
$442$			24.0000i			1.14156i
$443$	31.1769	+	18.0000i	1.48126	+	0.855206i	0.999774	−	0.0212481i	$-0.00676401\pi$
$443$	31.1769	+	18.0000i	1.48126	+	0.855206i	0.481486	+	0.876454i	$0.340097\pi$
$444$	−3.00000	+	1.73205i	−0.142374	+	0.0821995i
$445$		0			0
$446$	8.00000	+	13.8564i	0.378811	+	0.656120i
$447$	−5.19615	−	9.00000i	−0.245770	−	0.425685i
$448$	1.73205	+	1.00000i	0.0818317	+	0.0472456i
$449$	15.0000			0.707894			0.353947	−	0.935266i	$-0.384839\pi$
$449$	15.0000			0.707894			0.353947	+	0.935266i	$0.384839\pi$
$450$		0			0
$451$		0			0
$452$	−2.59808	−	1.50000i	−0.122203	−	0.0705541i
$453$	3.46410	−	6.00000i	0.162758	−	0.281905i
$454$	−7.50000	−	12.9904i	−0.351992	−	0.609669i
$455$		0			0
$456$		−	12.1244i		−	0.567775i
$457$	16.4545	+	9.50000i	0.769708	+	0.444391i	0.832771	−	0.553618i	$-0.186753\pi$
$457$	16.4545	+	9.50000i	0.769708	+	0.444391i	−0.0630623	+	0.998010i	$0.520087\pi$
$458$			10.0000i			0.467269i
$459$		−	31.1769i		−	1.45521i
$460$		0			0
$461$	−3.00000	+	5.19615i	−0.139724	+	0.242009i	−0.927392	−	0.374091i	$-0.877955\pi$
$461$	−3.00000	+	5.19615i	−0.139724	+	0.242009i	0.787668	+	0.616100i	$0.211288\pi$
$462$		0			0
$463$	−22.5167	+	13.0000i	−1.04644	+	0.604161i	−0.921650	−	0.388022i	$-0.873158\pi$
$463$	−22.5167	+	13.0000i	−1.04644	+	0.604161i	−0.124788	+	0.992183i	$0.539825\pi$
$464$	−3.00000	−	5.19615i	−0.139272	−	0.241225i
$465$		0			0
$466$	−1.50000	+	2.59808i	−0.0694862	+	0.120354i
$467$		−	15.0000i		−	0.694117i	−0.937843	−	0.347059i	$-0.887180\pi$
$467$		−	15.0000i		−	0.694117i	0.937843	−	0.347059i	$-0.112820\pi$
$468$	10.3923	+	6.00000i	0.480384	+	0.277350i
$469$	26.0000			1.20057
$470$		0			0
$471$	21.0000	−	12.1244i	0.967629	−	0.558661i
$472$	−7.79423	+	4.50000i	−0.358758	+	0.207129i
$473$		0			0
$474$	−3.00000	+	1.73205i	−0.137795	+	0.0795557i
$475$		0			0
$476$	−12.0000			−0.550019
$477$			36.0000i			1.64833i
$478$		0			0
$479$	18.0000	−	31.1769i	0.822441	−	1.42451i	−0.0814184	−	0.996680i	$-0.525945\pi$
$479$	18.0000	−	31.1769i	0.822441	−	1.42451i	0.903859	−	0.427830i	$-0.140722\pi$
$480$		0			0
$481$	4.00000	+	6.92820i	0.182384	+	0.315899i
$482$	−16.4545	+	9.50000i	−0.749481	+	0.432713i
$483$		0			0
$484$	−5.50000	+	9.52628i	−0.250000	+	0.433013i
$485$		0			0
$486$	−13.5000	−	7.79423i	−0.612372	−	0.353553i
$487$			2.00000i			0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$			2.00000i			0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$	−3.46410	−	2.00000i	−0.156813	−	0.0905357i
$489$			12.1244i			0.548282i
$490$		0			0
$491$	1.50000	+	2.59808i	0.0676941	+	0.117250i	0.897886	−	0.440228i	$-0.145102\pi$
$491$	1.50000	+	2.59808i	0.0676941	+	0.117250i	−0.830192	+	0.557478i	$0.811769\pi$
$492$	7.79423	−	13.5000i	0.351391	−	0.608627i
$493$	31.1769	+	18.0000i	1.40414	+	0.810679i
$494$	−28.0000			−1.25978
$495$		0			0
$496$	−10.0000			−0.449013
$497$	−10.3923	−	6.00000i	−0.466159	−	0.269137i
$498$	−7.79423	−	13.5000i	−0.349268	−	0.604949i
$499$	5.50000	+	9.52628i	0.246214	+	0.426455i	0.962472	−	0.271380i	$-0.0874801\pi$
$499$	5.50000	+	9.52628i	0.246214	+	0.426455i	−0.716258	+	0.697835i	$0.754147\pi$
$500$		0			0
$501$	18.0000	−	10.3923i	0.804181	−	0.464294i
$502$	−12.9904	−	7.50000i	−0.579789	−	0.334741i
$503$		−	24.0000i		−	1.07011i	−0.844818	−	0.535054i	$-0.820291\pi$
$503$		−	24.0000i		−	1.07011i	0.844818	−	0.535054i	$-0.179709\pi$
$504$	−3.00000	+	5.19615i	−0.133631	+	0.231455i
$505$		0			0
$506$		0			0
$507$	2.59808	−	4.50000i	0.115385	−	0.199852i
$508$	−17.3205	+	10.0000i	−0.768473	+	0.443678i
$509$	−18.0000	−	31.1769i	−0.797836	−	1.38189i	−0.921023	−	0.389509i	$-0.872645\pi$
$509$	−18.0000	−	31.1769i	−0.797836	−	1.38189i	0.123187	−	0.992384i	$-0.460689\pi$
$510$		0			0
$511$	1.00000	−	1.73205i	0.0442374	−	0.0766214i
$512$			1.00000i			0.0441942i
$513$	36.3731			1.60591
$514$	−3.00000			−0.132324
$515$		0			0
$516$		−	1.73205i		−	0.0762493i
$517$		0			0
$518$	−3.46410	+	2.00000i	−0.152204	+	0.0878750i
$519$	−18.0000	−	10.3923i	−0.790112	−	0.456172i
$520$		0			0
$521$	6.00000			0.262865			0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$	15.5885	−	9.00000i	0.682288	−	0.393919i
$523$			1.00000i			0.0437269i	0.999761	+	0.0218635i	$0.00695991\pi$
$523$			1.00000i			0.0437269i	−0.999761	+	0.0218635i	$0.993040\pi$
$524$	−6.00000	+	10.3923i	−0.262111	+	0.453990i
$525$		0			0
$526$	3.00000	+	5.19615i	0.130806	+	0.226563i
$527$	51.9615	−	30.0000i	2.26348	−	1.30682i
$528$		0			0
$529$	−11.5000	+	19.9186i	−0.500000	+	0.866025i
$530$		0			0
$531$	−13.5000	−	23.3827i	−0.585850	−	1.01472i
$532$		−	14.0000i		−	0.606977i
$533$	−31.1769	−	18.0000i	−1.35042	−	0.779667i
$534$	22.5000	+	12.9904i	0.973670	+	0.562149i
$535$		0			0
$536$	6.50000	+	11.2583i	0.280757	+	0.486286i
$537$	15.5885			0.672692
$538$	−25.9808	−	15.0000i	−1.12011	−	0.646696i
$539$		0			0
$540$		0			0
$541$	32.0000			1.37579			0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000			1.37579			0.687894	+	0.725811i	$0.258536\pi$
$542$	24.2487	+	14.0000i	1.04157	+	0.601351i
$543$	−27.7128			−1.18927
$544$	−3.00000	−	5.19615i	−0.128624	−	0.222783i
$545$		0			0
$546$	12.0000	+	6.92820i	0.513553	+	0.296500i
$547$	−4.33013	−	2.50000i	−0.185143	−	0.106892i	0.404564	−	0.914510i	$-0.367423\pi$
$547$	−4.33013	−	2.50000i	−0.185143	−	0.106892i	−0.589707	+	0.807617i	$0.700757\pi$
$548$			3.00000i			0.128154i
$549$	6.00000	−	10.3923i	0.256074	−	0.443533i
$550$		0			0
$551$	−21.0000	+	36.3731i	−0.894630	+	1.54954i
$552$		0			0
$553$	−3.46410	+	2.00000i	−0.147309	+	0.0850487i
$554$	1.00000	+	1.73205i	0.0424859	+	0.0735878i
$555$		0			0
$556$	2.00000	−	3.46410i	0.0848189	−	0.146911i
$557$		−	12.0000i		−	0.508456i	−0.967144	−	0.254228i	$-0.918179\pi$
$557$		−	12.0000i		−	0.508456i	0.967144	−	0.254228i	$-0.0818214\pi$
$558$		−	30.0000i		−	1.27000i
$559$	−4.00000			−0.169182
$560$		0			0
$561$		0			0
$562$	−15.5885	+	9.00000i	−0.657559	+	0.379642i
$563$	−7.79423	+	4.50000i	−0.328488	+	0.189652i	−0.655169	−	0.755482i	$-0.727403\pi$
$563$	−7.79423	+	4.50000i	−0.328488	+	0.189652i	0.326682	+	0.945134i	$0.394069\pi$
$564$		−	10.3923i		−	0.437595i
$565$		0			0
$566$	−19.0000			−0.798630
$567$	−15.5885	−	9.00000i	−0.654654	−	0.377964i
$568$		−	6.00000i		−	0.251754i
$569$	−21.0000	+	36.3731i	−0.880366	+	1.52484i	−0.0294311	+	0.999567i	$0.509370\pi$
$569$	−21.0000	+	36.3731i	−0.880366	+	1.52484i	−0.850935	+	0.525271i	$0.823964\pi$
$570$		0			0
$571$	−17.5000	−	30.3109i	−0.732352	−	1.26847i	−0.955875	−	0.293773i	$-0.905089\pi$
$571$	−17.5000	−	30.3109i	−0.732352	−	1.26847i	0.223523	−	0.974699i	$-0.428244\pi$
$572$		0			0
$573$	−5.19615	+	9.00000i	−0.217072	+	0.375980i
$574$	9.00000	−	15.5885i	0.375653	−	0.650650i
$575$		0			0
$576$	−3.00000			−0.125000
$577$			35.0000i			1.45707i	0.685009	+	0.728535i	$0.259798\pi$
$577$			35.0000i			1.45707i	−0.685009	+	0.728535i	$0.740202\pi$
$578$	16.4545	+	9.50000i	0.684416	+	0.395148i
$579$	−3.00000	+	1.73205i	−0.124676	+	0.0719816i
$580$		0			0
$581$	−9.00000	−	15.5885i	−0.373383	−	0.646718i
$582$	14.7224	+	25.5000i	0.610264	+	1.05701i
$583$		0			0
$584$	1.00000			0.0413803
$585$		0			0
$586$	12.0000			0.495715
$587$	−31.1769	−	18.0000i	−1.28681	−	0.742940i	−0.308725	−	0.951151i	$-0.599902\pi$
$587$	−31.1769	−	18.0000i	−1.28681	−	0.742940i	−0.978084	+	0.208212i	$0.933236\pi$
$588$	2.59808	−	4.50000i	0.107143	−	0.185577i
$589$	35.0000	+	60.6218i	1.44215	+	2.49788i
$590$		0			0
$591$			31.1769i			1.28245i
$592$	−1.73205	−	1.00000i	−0.0711868	−	0.0410997i
$593$			9.00000i			0.369586i	0.982777	+	0.184793i	$0.0591614\pi$
$593$			9.00000i			0.369586i	−0.982777	+	0.184793i	$0.940839\pi$
$594$		0			0
$595$		0			0
$596$	3.00000	−	5.19615i	0.122885	−	0.212843i
$597$	27.7128			1.13421
$598$		0			0
$599$	15.0000	+	25.9808i	0.612883	+	1.06155i	0.990752	+	0.135686i	$0.0433238\pi$
$599$	15.0000	+	25.9808i	0.612883	+	1.06155i	−0.377869	+	0.925859i	$0.623343\pi$
$600$		0			0
$601$	5.00000	−	8.66025i	0.203954	−	0.353259i	−0.745845	−	0.666120i	$-0.767954\pi$
$601$	5.00000	−	8.66025i	0.203954	−	0.353259i	0.949799	+	0.312861i	$0.101287\pi$
$602$		−	2.00000i		−	0.0815139i
$603$	−33.7750	+	19.5000i	−1.37542	+	0.794101i
$604$	4.00000			0.162758
$605$		0			0
$606$	18.0000	−	10.3923i	0.731200	−	0.422159i
$607$	−34.6410	+	20.0000i	−1.40604	+	0.811775i	−0.995003	−	0.0998457i	$-0.968165\pi$
$607$	−34.6410	+	20.0000i	−1.40604	+	0.811775i	−0.411033	+	0.911621i	$0.634832\pi$
$608$	6.06218	−	3.50000i	0.245854	−	0.141944i
$609$	18.0000	−	10.3923i	0.729397	−	0.421117i
$610$		0			0
$611$	−24.0000			−0.970936
$612$	15.5885	−	9.00000i	0.630126	−	0.363803i
$613$		−	44.0000i		−	1.77714i	−0.458738	−	0.888572i	$-0.651698\pi$
$613$		−	44.0000i		−	1.77714i	0.458738	−	0.888572i	$-0.348302\pi$
$614$	4.00000	−	6.92820i	0.161427	−	0.279600i
$615$		0			0
$616$		0			0
$617$	−7.79423	+	4.50000i	−0.313784	+	0.181163i	−0.648618	−	0.761114i	$-0.724653\pi$
$617$	−7.79423	+	4.50000i	−0.313784	+	0.181163i	0.334835	+	0.942277i	$0.391320\pi$
$618$	3.46410			0.139347
$619$	11.5000	−	19.9186i	0.462224	−	0.800595i	−0.536847	−	0.843679i	$-0.680385\pi$
$619$	11.5000	−	19.9186i	0.462224	−	0.800595i	0.999071	+	0.0430838i	$0.0137183\pi$
$620$		0			0
$621$		0			0
$622$		−	30.0000i		−	1.20289i
$623$	25.9808	+	15.0000i	1.04090	+	0.600962i
$624$			6.92820i			0.277350i
$625$		0			0
$626$	9.50000	+	16.4545i	0.379696	+	0.657653i
$627$		0			0
$628$	12.1244	+	7.00000i	0.483814	+	0.279330i
$629$	12.0000			0.478471
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$	−1.73205	−	1.00000i	−0.0688973	−	0.0397779i
$633$	−4.33013	−	7.50000i	−0.172107	−	0.298098i
$634$	15.0000	+	25.9808i	0.595726	+	1.03183i
$635$		0			0
$636$	−18.0000	+	10.3923i	−0.713746	+	0.412082i
$637$	−10.3923	−	6.00000i	−0.411758	−	0.237729i
$638$		0			0
$639$	18.0000			0.712069
$640$		0			0
$641$	−16.5000	+	28.5788i	−0.651711	+	1.12880i	0.330997	+	0.943632i	$0.392615\pi$
$641$	−16.5000	+	28.5788i	−0.651711	+	1.12880i	−0.982708	+	0.185164i	$0.940718\pi$
$642$	7.79423	−	13.5000i	0.307614	−	0.532803i
$643$	11.2583	−	6.50000i	0.443985	−	0.256335i	−0.261301	−	0.965257i	$-0.584152\pi$
$643$	11.2583	−	6.50000i	0.443985	−	0.256335i	0.705287	+	0.708922i	$0.250818\pi$
$644$		0			0
$645$		0			0
$646$	−21.0000	+	36.3731i	−0.826234	+	1.43108i
$647$		−	48.0000i		−	1.88707i	−0.331266	−	0.943537i	$-0.607476\pi$
$647$		−	48.0000i		−	1.88707i	0.331266	−	0.943537i	$-0.392524\pi$
$648$		−	9.00000i		−	0.353553i
$649$		0			0
$650$		0			0
$651$		−	34.6410i		−	1.35769i
$652$	−6.06218	+	3.50000i	−0.237413	+	0.137071i
$653$	31.1769	−	18.0000i	1.22005	−	0.704394i	0.255119	−	0.966910i	$-0.417885\pi$
$653$	31.1769	−	18.0000i	1.22005	−	0.704394i	0.964928	+	0.262515i	$0.0845520\pi$
$654$	3.00000	+	1.73205i	0.117309	+	0.0677285i
$655$		0			0
$656$	9.00000			0.351391
$657$			3.00000i			0.117041i
$658$		−	12.0000i		−	0.467809i
$659$	16.5000	−	28.5788i	0.642749	−	1.11327i	−0.342068	−	0.939675i	$-0.611127\pi$
$659$	16.5000	−	28.5788i	0.642749	−	1.11327i	0.984817	−	0.173598i	$-0.0555394\pi$
$660$		0			0
$661$	−1.00000	−	1.73205i	−0.0388955	−	0.0673690i	0.845922	−	0.533306i	$-0.179051\pi$
$661$	−1.00000	−	1.73205i	−0.0388955	−	0.0673690i	−0.884818	+	0.465937i	$0.845717\pi$
$662$	9.52628	−	5.50000i	0.370249	−	0.213764i
$663$	−20.7846	−	36.0000i	−0.807207	−	1.39812i
$664$	4.50000	−	7.79423i	0.174634	−	0.302475i
$665$		0			0
$666$	3.00000	−	5.19615i	0.116248	−	0.201347i
$667$		0			0
$668$	10.3923	+	6.00000i	0.402090	+	0.232147i
$669$	−24.0000	−	13.8564i	−0.927894	−	0.535720i
$670$		0			0
$671$		0			0
$672$	−3.46410			−0.133631
$673$	1.73205	+	1.00000i	0.0667657	+	0.0385472i	0.533011	−	0.846108i	$-0.321060\pi$
$673$	1.73205	+	1.00000i	0.0667657	+	0.0385472i	−0.466246	+	0.884655i	$0.654394\pi$
$674$	−14.0000			−0.539260
$675$		0			0
$676$	3.00000			0.115385
$677$	10.3923	+	6.00000i	0.399409	+	0.230599i	0.686229	−	0.727386i	$-0.259265\pi$
$677$	10.3923	+	6.00000i	0.399409	+	0.230599i	−0.286820	+	0.957984i	$0.592598\pi$
$678$	5.19615			0.199557
$679$	17.0000	+	29.4449i	0.652400	+	1.12999i
$680$		0			0
$681$	22.5000	+	12.9904i	0.862202	+	0.497792i
$682$		0			0
$683$			3.00000i			0.114792i	0.998351	+	0.0573959i	$0.0182797\pi$
$683$			3.00000i			0.114792i	−0.998351	+	0.0573959i	$0.981720\pi$
$684$	10.5000	+	18.1865i	0.401478	+	0.695379i
$685$		0			0
$686$	10.0000	−	17.3205i	0.381802	−	0.661300i
$687$	−8.66025	−	15.0000i	−0.330409	−	0.572286i
$688$	0.866025	−	0.500000i	0.0330169	−	0.0190623i
$689$	24.0000	+	41.5692i	0.914327	+	1.58366i
$690$		0			0
$691$	−17.5000	+	30.3109i	−0.665731	+	1.15308i	0.313355	+	0.949636i	$0.398547\pi$
$691$	−17.5000	+	30.3109i	−0.665731	+	1.15308i	−0.979086	+	0.203445i	$0.934786\pi$
$692$		−	12.0000i		−	0.456172i
$693$		0			0
$694$	12.0000			0.455514
$695$		0			0
$696$	9.00000	+	5.19615i	0.341144	+	0.196960i
$697$	−46.7654	+	27.0000i	−1.77136	+	1.02270i
$698$	8.66025	−	5.00000i	0.327795	−	0.189253i
$699$		−	5.19615i		−	0.196537i
$700$		0			0
$701$	−48.0000			−1.81293			−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000			−1.81293			−0.906467	+	0.422276i	$0.861231\pi$
$702$	−20.7846			−0.784465
$703$			14.0000i			0.528020i
$704$		0			0
$705$		0			0
$706$	−7.50000	−	12.9904i	−0.282266	−	0.488899i
$707$	20.7846	−	12.0000i	0.781686	−	0.451306i
$708$	7.79423	−	13.5000i	0.292925	−	0.507361i
$709$	−14.0000	+	24.2487i	−0.525781	+	0.910679i	0.473768	+	0.880650i	$0.342894\pi$
$709$	−14.0000	+	24.2487i	−0.525781	+	0.910679i	−0.999549	+	0.0300298i	$0.990440\pi$
$710$		0			0
$711$	3.00000	−	5.19615i	0.112509	−	0.194871i
$712$			15.0000i			0.562149i
$713$		0			0
$714$	18.0000	−	10.3923i	0.673633	−	0.388922i
$715$		0			0
$716$	4.50000	+	7.79423i	0.168173	+	0.291284i
$717$		0			0
$718$	10.3923	+	6.00000i	0.387837	+	0.223918i
$719$	−18.0000			−0.671287			−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000			−0.671287			−0.335643	+	0.941989i	$0.608954\pi$
$720$		0			0
$721$	4.00000			0.148968
$722$	−25.9808	−	15.0000i	−0.966904	−	0.558242i
$723$	16.4545	−	28.5000i	0.611949	−	1.05993i
$724$	−8.00000	−	13.8564i	−0.297318	−	0.514969i
$725$		0			0
$726$		−	19.0526i		−	0.707107i
$727$	34.6410	+	20.0000i	1.28476	+	0.741759i	0.977715	−	0.209935i	$-0.0673252\pi$
$727$	34.6410	+	20.0000i	1.28476	+	0.741759i	0.307049	+	0.951694i	$0.400659\pi$
$728$			8.00000i			0.296500i
$729$	27.0000			1.00000
$730$		0			0
$731$	−3.00000	+	5.19615i	−0.110959	+	0.192187i
$732$	6.92820			0.256074
$733$	19.0526	−	11.0000i	0.703722	−	0.406294i	−0.105010	−	0.994471i	$-0.533487\pi$
$733$	19.0526	−	11.0000i	0.703722	−	0.406294i	0.808732	+	0.588177i	$0.200154\pi$
$734$	−5.00000	−	8.66025i	−0.184553	−	0.319656i
$735$		0			0
$736$		0			0
$737$		0			0
$738$			27.0000i			0.993884i
$739$	25.0000			0.919640			0.459820	−	0.888012i	$-0.347914\pi$
$739$	25.0000			0.919640			0.459820	+	0.888012i	$0.347914\pi$
$740$		0			0
$741$	42.0000	−	24.2487i	1.54291	−	0.890799i
$742$	−20.7846	+	12.0000i	−0.763027	+	0.440534i
$743$	−5.19615	+	3.00000i	−0.190628	+	0.110059i	−0.592277	−	0.805735i	$-0.701771\pi$
$743$	−5.19615	+	3.00000i	−0.190628	+	0.110059i	0.401648	+	0.915794i	$0.368437\pi$
$744$	15.0000	−	8.66025i	0.549927	−	0.317500i
$745$		0			0
$746$	−4.00000			−0.146450
$747$	23.3827	+	13.5000i	0.855528	+	0.493939i
$748$		0			0
$749$	9.00000	−	15.5885i	0.328853	−	0.569590i
$750$		0			0
$751$	2.00000	+	3.46410i	0.0729810	+	0.126407i	0.900207	−	0.435463i	$-0.143415\pi$
$751$	2.00000	+	3.46410i	0.0729810	+	0.126407i	−0.827225	+	0.561870i	$0.810082\pi$
$752$	5.19615	−	3.00000i	0.189484	−	0.109399i
$753$	25.9808			0.946792
$754$	12.0000	−	20.7846i	0.437014	−	0.756931i
$755$		0			0
$756$		−	10.3923i		−	0.377964i
$757$		−	10.0000i		−	0.363456i	−0.983349	−	0.181728i	$-0.941831\pi$
$757$		−	10.0000i		−	0.363456i	0.983349	−	0.181728i	$-0.0581691\pi$
$758$	17.3205	+	10.0000i	0.629109	+	0.363216i
$759$		0			0
$760$		0			0
$761$	−10.5000	−	18.1865i	−0.380625	−	0.659261i	0.610527	−	0.791995i	$-0.290958\pi$
$761$	−10.5000	−	18.1865i	−0.380625	−	0.659261i	−0.991152	+	0.132734i	$0.957624\pi$
$762$	17.3205	−	30.0000i	0.627456	−	1.08679i
$763$	3.46410	+	2.00000i	0.125409	+	0.0724049i
$764$	−6.00000			−0.217072
$765$		0			0
$766$	24.0000			0.867155
$767$	−31.1769	−	18.0000i	−1.12573	−	0.649942i
$768$	−0.866025	−	1.50000i	−0.0312500	−	0.0541266i
$769$	2.50000	+	4.33013i	0.0901523	+	0.156148i	0.907575	−	0.419890i	$-0.137931\pi$
$769$	2.50000	+	4.33013i	0.0901523	+	0.156148i	−0.817423	+	0.576038i	$0.804598\pi$
$770$		0			0
$771$	4.50000	−	2.59808i	0.162064	−	0.0935674i
$772$	−1.73205	−	1.00000i	−0.0623379	−	0.0359908i
$773$		−	6.00000i		−	0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$		−	6.00000i		−	0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	1.50000	+	2.59808i	0.0539164	+	0.0933859i
$775$		0			0
$776$	−8.50000	+	14.7224i	−0.305132	+	0.528505i
$777$	3.46410	−	6.00000i	0.124274	−	0.215249i
$778$	−5.19615	+	3.00000i	−0.186291	+	0.107555i
$779$	−31.5000	−	54.5596i	−1.12860	−	1.95480i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−15.5885	+	27.0000i	−0.557086	+	0.964901i
$784$	3.00000			0.107143
$785$		0			0
$786$		−	20.7846i		−	0.741362i
$787$	17.3205	−	10.0000i	0.617409	−	0.356462i	−0.158450	−	0.987367i	$-0.550650\pi$
$787$	17.3205	−	10.0000i	0.617409	−	0.356462i	0.775860	+	0.630905i	$0.217316\pi$
$788$	−15.5885	+	9.00000i	−0.555316	+	0.320612i
$789$	−9.00000	−	5.19615i	−0.320408	−	0.184988i
$790$		0			0
$791$	6.00000			0.213335
$792$		0			0
$793$		−	16.0000i		−	0.568177i
$794$	−11.0000	+	19.0526i	−0.390375	+	0.676150i
$795$		0			0
$796$	8.00000	+	13.8564i	0.283552	+	0.491127i
$797$	−31.1769	+	18.0000i	−1.10434	+	0.637593i	−0.937358	−	0.348367i	$-0.886736\pi$
$797$	−31.1769	+	18.0000i	−1.10434	+	0.637593i	−0.166985	+	0.985959i	$0.553403\pi$
$798$	12.1244	+	21.0000i	0.429198	+	0.743392i
$799$	−18.0000	+	31.1769i	−0.636794	+	1.10296i
$800$		0			0
$801$	−45.0000			−1.59000
$802$			18.0000i			0.635602i
$803$		0			0
$804$	−19.5000	−	11.2583i	−0.687712	−	0.397051i
$805$		0			0
$806$	−20.0000	−	34.6410i	−0.704470	−	1.22018i
$807$	51.9615			1.82913
$808$	10.3923	+	6.00000i	0.365600	+	0.211079i
$809$	39.0000			1.37117			0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000			1.37117			0.685583	+	0.727994i	$0.259547\pi$
$810$		0			0
$811$	5.00000			0.175574			0.0877869	−	0.996139i	$-0.472021\pi$
$811$	5.00000			0.175574			0.0877869	+	0.996139i	$0.472021\pi$
$812$	10.3923	+	6.00000i	0.364698	+	0.210559i
$813$	−48.4974			−1.70088
$814$		0			0
$815$		0			0
$816$	9.00000	+	5.19615i	0.315063	+	0.181902i
$817$	−6.06218	−	3.50000i	−0.212089	−	0.122449i
$818$			31.0000i			1.08389i
$819$	−24.0000			−0.838628
$820$		0			0
$821$	12.0000	−	20.7846i	0.418803	−	0.725388i	−0.577016	−	0.816733i	$-0.695783\pi$
$821$	12.0000	−	20.7846i	0.418803	−	0.725388i	0.995819	+	0.0913446i	$0.0291165\pi$
$822$	−2.59808	−	4.50000i	−0.0906183	−	0.156956i
$823$	−12.1244	+	7.00000i	−0.422628	+	0.244005i	−0.696201	−	0.717847i	$-0.745128\pi$
$823$	−12.1244	+	7.00000i	−0.422628	+	0.244005i	0.273573	+	0.961851i	$0.411795\pi$
$824$	1.00000	+	1.73205i	0.0348367	+	0.0603388i
$825$		0			0
$826$	9.00000	−	15.5885i	0.313150	−	0.542392i
$827$		−	27.0000i		−	0.938882i	−0.882964	−	0.469441i	$-0.844455\pi$
$827$		−	27.0000i		−	0.938882i	0.882964	−	0.469441i	$-0.155545\pi$
$828$		0			0
$829$	−32.0000			−1.11141			−0.555703	−	0.831381i	$-0.687551\pi$
$829$	−32.0000			−1.11141			−0.555703	+	0.831381i	$0.687551\pi$
$830$		0			0
$831$	−3.00000	−	1.73205i	−0.104069	−	0.0600842i
$832$	−3.46410	+	2.00000i	−0.120096	+	0.0693375i
$833$	−15.5885	+	9.00000i	−0.540108	+	0.311832i
$834$			6.92820i			0.239904i
$835$		0			0
$836$		0			0
$837$	25.9808	+	45.0000i	0.898027	+	1.55543i
$838$		−	3.00000i		−	0.103633i
$839$	−21.0000	+	36.3731i	−0.725001	+	1.25574i	0.233973	+	0.972243i	$0.424827\pi$
$839$	−21.0000	+	36.3731i	−0.725001	+	1.25574i	−0.958974	+	0.283495i	$0.908506\pi$
$840$		0			0
$841$	−3.50000	−	6.06218i	−0.120690	−	0.209041i
$842$	−8.66025	+	5.00000i	−0.298452	+	0.172311i
$843$	15.5885	−	27.0000i	0.536895	−	0.929929i
$844$	2.50000	−	4.33013i	0.0860535	−	0.149049i
$845$		0			0
$846$	9.00000	+	15.5885i	0.309426	+	0.535942i
$847$		−	22.0000i		−	0.755929i
$848$	−10.3923	−	6.00000i	−0.356873	−	0.206041i
$849$	28.5000	−	16.4545i	0.978117	−	0.564716i
$850$		0			0
$851$		0			0
$852$	5.19615	+	9.00000i	0.178017	+	0.308335i
$853$	−8.66025	−	5.00000i	−0.296521	−	0.171197i	0.344358	−	0.938839i	$-0.388097\pi$
$853$	−8.66025	−	5.00000i	−0.296521	−	0.171197i	−0.640879	+	0.767642i	$0.721430\pi$
$854$	8.00000			0.273754
$855$		0			0
$856$	9.00000			0.307614
$857$	33.7750	+	19.5000i	1.15373	+	0.666107i	0.949794	−	0.312877i	$-0.101293\pi$
$857$	33.7750	+	19.5000i	1.15373	+	0.666107i	0.203938	+	0.978984i	$0.434626\pi$
$858$		0			0
$859$	−15.5000	−	26.8468i	−0.528853	−	0.916001i	−0.999434	−	0.0336436i	$-0.989289\pi$
$859$	−15.5000	−	26.8468i	−0.528853	−	0.916001i	0.470581	−	0.882357i	$-0.344044\pi$
$860$		0			0
$861$			31.1769i			1.06251i
$862$	−5.19615	−	3.00000i	−0.176982	−	0.102180i
$863$		−	12.0000i		−	0.408485i	−0.978920	−	0.204242i	$-0.934527\pi$
$863$		−	12.0000i		−	0.408485i	0.978920	−	0.204242i	$-0.0654731\pi$
$864$	4.50000	−	2.59808i	0.153093	−	0.0883883i
$865$		0			0
$866$	−1.00000	+	1.73205i	−0.0339814	+	0.0588575i
$867$	−32.9090			−1.11765
$868$	17.3205	−	10.0000i	0.587896	−	0.339422i
$869$		0			0
$870$		0			0
$871$	−26.0000	+	45.0333i	−0.880976	+	1.52590i
$872$			2.00000i			0.0677285i
$873$	−44.1673	−	25.5000i	−1.49484	−	0.863044i
$874$		0			0
$875$		0			0
$876$	−1.50000	+	0.866025i	−0.0506803	+	0.0292603i
$877$	−8.66025	+	5.00000i	−0.292436	+	0.168838i	−0.639040	−	0.769174i	$-0.720668\pi$
$877$	−8.66025	+	5.00000i	−0.292436	+	0.168838i	0.346604	+	0.938012i	$0.387335\pi$
$878$	3.46410	−	2.00000i	0.116908	−	0.0674967i
$879$	−18.0000	+	10.3923i	−0.607125	+	0.350524i
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$			9.00000i			0.303046i
$883$		−	20.0000i		−	0.673054i	−0.941674	−	0.336527i	$-0.890748\pi$
$883$		−	20.0000i		−	0.673054i	0.941674	−	0.336527i	$-0.109252\pi$
$884$	12.0000	−	20.7846i	0.403604	−	0.699062i
$885$		0			0
$886$	−18.0000	−	31.1769i	−0.604722	−	1.04741i
$887$	5.19615	−	3.00000i	0.174470	−	0.100730i	−0.410222	−	0.911986i	$-0.634549\pi$
$887$	5.19615	−	3.00000i	0.174470	−	0.100730i	0.584692	+	0.811256i	$0.301215\pi$
$888$	3.46410			0.116248
$889$	20.0000	−	34.6410i	0.670778	−	1.16182i
$890$		0			0
$891$		0			0
$892$		−	16.0000i		−	0.535720i
$893$	−36.3731	−	21.0000i	−1.21718	−	0.702738i
$894$			10.3923i			0.347571i
$895$		0			0
$896$	−1.00000	−	1.73205i	−0.0334077	−	0.0578638i
$897$		0			0
$898$	−12.9904	−	7.50000i	−0.433495	−	0.250278i
$899$	−60.0000			−2.00111
$900$		0			0
$901$	72.0000			2.39867
$902$		0			0
$903$	1.73205	+	3.00000i	0.0576390	+	0.0998337i
$904$	1.50000	+	2.59808i	0.0498893	+	0.0864107i
$905$		0			0
$906$	−6.00000	+	3.46410i	−0.199337	+	0.115087i
$907$	6.06218	+	3.50000i	0.201291	+	0.116216i	0.597258	−	0.802049i	$-0.296257\pi$
$907$	6.06218	+	3.50000i	0.201291	+	0.116216i	−0.395966	+	0.918265i	$0.629590\pi$
$908$			15.0000i			0.497792i
$909$	−18.0000	+	31.1769i	−0.597022	+	1.03407i
$910$		0			0
$911$	−6.00000	+	10.3923i	−0.198789	+	0.344312i	−0.948136	−	0.317865i	$-0.897034\pi$
$911$	−6.00000	+	10.3923i	−0.198789	+	0.344312i	0.749347	+	0.662177i	$0.230367\pi$
$912$	−6.06218	+	10.5000i	−0.200739	+	0.347690i
$913$		0			0
$914$	−9.50000	−	16.4545i	−0.314232	−	0.544266i
$915$		0			0
$916$	5.00000	−	8.66025i	0.165205	−	0.286143i
$917$		−	24.0000i		−	0.792550i
$918$	−15.5885	+	27.0000i	−0.514496	+	0.891133i
$919$	−8.00000			−0.263896			−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000			−0.263896			−0.131948	+	0.991257i	$0.542123\pi$
$920$		0			0
$921$			13.8564i			0.456584i
$922$	5.19615	−	3.00000i	0.171126	−	0.0987997i
$923$	20.7846	−	12.0000i	0.684134	−	0.394985i
$924$		0			0
$925$		0			0
$926$	26.0000			0.854413
$927$	−5.19615	+	3.00000i	−0.170664	+	0.0985329i
$928$			6.00000i			0.196960i
$929$	−27.0000	+	46.7654i	−0.885841	+	1.53432i	−0.0410949	+	0.999155i	$0.513085\pi$
$929$	−27.0000	+	46.7654i	−0.885841	+	1.53432i	−0.844746	+	0.535167i	$0.820249\pi$
$930$		0			0
$931$	−10.5000	−	18.1865i	−0.344124	−	0.596040i
$932$	2.59808	−	1.50000i	0.0851028	−	0.0491341i
$933$	25.9808	+	45.0000i	0.850572	+	1.47323i
$934$	−7.50000	+	12.9904i	−0.245407	+	0.425058i
$935$		0			0
$936$	−6.00000	−	10.3923i	−0.196116	−	0.339683i
$937$			23.0000i			0.751377i	0.926746	+	0.375689i	$0.122594\pi$
$937$			23.0000i			0.751377i	−0.926746	+	0.375689i	$0.877406\pi$
$938$	−22.5167	−	13.0000i	−0.735195	−	0.424465i
$939$	−28.5000	−	16.4545i	−0.930062	−	0.536972i
$940$		0			0
$941$	27.0000	+	46.7654i	0.880175	+	1.52451i	0.851146	+	0.524929i	$0.175908\pi$
$941$	27.0000	+	46.7654i	0.880175	+	1.52451i	0.0290288	+	0.999579i	$0.490759\pi$
$942$	−24.2487			−0.790066
$943$		0			0
$944$	9.00000			0.292925
$945$		0			0
$946$		0			0
$947$	18.1865	+	10.5000i	0.590983	+	0.341204i	0.765486	−	0.643452i	$-0.222499\pi$
$947$	18.1865	+	10.5000i	0.590983	+	0.341204i	−0.174503	+	0.984657i	$0.555832\pi$
$948$	3.46410			0.112509
$949$	2.00000	+	3.46410i	0.0649227	+	0.112449i
$950$		0			0
$951$	−45.0000	−	25.9808i	−1.45922	−	0.842484i
$952$	10.3923	+	6.00000i	0.336817	+	0.194461i
$953$			30.0000i			0.971795i	0.874016	+	0.485898i	$0.161507\pi$
$953$			30.0000i			0.971795i	−0.874016	+	0.485898i	$0.838493\pi$
$954$	18.0000	−	31.1769i	0.582772	−	1.00939i
$955$		0			0
$956$		0			0
$957$		0			0
$958$	−31.1769	+	18.0000i	−1.00728	+	0.581554i
$959$	−3.00000	−	5.19615i	−0.0968751	−	0.167793i
$960$		0			0
$961$	−34.5000	+	59.7558i	−1.11290	+	1.92760i
$962$		−	8.00000i		−	0.257930i
$963$			27.0000i			0.870063i
$964$	19.0000			0.611949
$965$		0			0
$966$		0			0
$967$	17.3205	−	10.0000i	0.556990	−	0.321578i	−0.194946	−	0.980814i	$-0.562453\pi$
$967$	17.3205	−	10.0000i	0.556990	−	0.321578i	0.751936	+	0.659236i	$0.229120\pi$
$968$	9.52628	−	5.50000i	0.306186	−	0.176777i
$969$		−	72.7461i		−	2.33694i
$970$		0			0
$971$	−21.0000			−0.673922			−0.336961	−	0.941519i	$-0.609399\pi$
$971$	−21.0000			−0.673922			−0.336961	+	0.941519i	$0.609399\pi$
$972$	7.79423	+	13.5000i	0.250000	+	0.433013i
$973$			8.00000i			0.256468i
$974$	1.00000	−	1.73205i	0.0320421	−	0.0554985i
$975$		0			0
$976$	2.00000	+	3.46410i	0.0640184	+	0.110883i
$977$	−12.9904	+	7.50000i	−0.415599	+	0.239946i	−0.693193	−	0.720752i	$-0.743796\pi$
$977$	−12.9904	+	7.50000i	−0.415599	+	0.239946i	0.277594	+	0.960699i	$0.410463\pi$
$978$	6.06218	−	10.5000i	0.193847	−	0.335753i
$979$		0			0
$980$		0			0
$981$	−6.00000			−0.191565
$982$		−	3.00000i		−	0.0957338i
$983$	−25.9808	−	15.0000i	−0.828658	−	0.478426i	0.0247352	−	0.999694i	$-0.492126\pi$
$983$	−25.9808	−	15.0000i	−0.828658	−	0.478426i	−0.853393	+	0.521268i	$0.825459\pi$
$984$	−13.5000	+	7.79423i	−0.430364	+	0.248471i
$985$		0			0
$986$	−18.0000	−	31.1769i	−0.573237	−	0.992875i
$987$	10.3923	+	18.0000i	0.330791	+	0.572946i
$988$	24.2487	+	14.0000i	0.771454	+	0.445399i
$989$		0			0
$990$		0			0
$991$	44.0000			1.39771			0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000			1.39771			0.698853	+	0.715265i	$0.253694\pi$
$992$	8.66025	+	5.00000i	0.274963	+	0.158750i
$993$	−9.52628	+	16.5000i	−0.302307	+	0.523612i
$994$	6.00000	+	10.3923i	0.190308	+	0.329624i
$995$		0			0
$996$			15.5885i			0.493939i
$997$	24.2487	+	14.0000i	0.767964	+	0.443384i	0.832148	−	0.554554i	$-0.187111\pi$
$997$	24.2487	+	14.0000i	0.767964	+	0.443384i	−0.0641836	+	0.997938i	$0.520444\pi$
$998$		−	11.0000i		−	0.348199i
$999$			10.3923i			0.328798i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.j.b.349.1		4
3.2	odd	2		1350.2.j.b.1099.2		4
5.2	odd	4		450.2.e.f.151.1	yes	2
5.3	odd	4		450.2.e.c.151.1	✓	2
5.4	even	2	inner	450.2.j.b.349.2		4
9.2	odd	6		4050.2.c.m.649.1		2
9.4	even	3	inner	450.2.j.b.49.2		4
9.5	odd	6		1350.2.j.b.199.1		4
9.7	even	3		4050.2.c.h.649.2		2
15.2	even	4		1350.2.e.d.451.1		2
15.8	even	4		1350.2.e.h.451.1		2
15.14	odd	2		1350.2.j.b.1099.1		4
45.2	even	12		4050.2.a.u.1.1		1
45.4	even	6	inner	450.2.j.b.49.1		4
45.7	odd	12		4050.2.a.d.1.1		1
45.13	odd	12		450.2.e.c.301.1	yes	2
45.14	odd	6		1350.2.j.b.199.2		4
45.22	odd	12		450.2.e.f.301.1	yes	2
45.23	even	12		1350.2.e.h.901.1		2
45.29	odd	6		4050.2.c.m.649.2		2
45.32	even	12		1350.2.e.d.901.1		2
45.34	even	6		4050.2.c.h.649.1		2
45.38	even	12		4050.2.a.o.1.1		1
45.43	odd	12		4050.2.a.bg.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.e.c.151.1	✓	2	5.3	odd	4
450.2.e.c.301.1	yes	2	45.13	odd	12
450.2.e.f.151.1	yes	2	5.2	odd	4
450.2.e.f.301.1	yes	2	45.22	odd	12
450.2.j.b.49.1		4	45.4	even	6	inner
450.2.j.b.49.2		4	9.4	even	3	inner
450.2.j.b.349.1		4	1.1	even	1	trivial
450.2.j.b.349.2		4	5.4	even	2	inner
1350.2.e.d.451.1		2	15.2	even	4
1350.2.e.d.901.1		2	45.32	even	12
1350.2.e.h.451.1		2	15.8	even	4
1350.2.e.h.901.1		2	45.23	even	12
1350.2.j.b.199.1		4	9.5	odd	6
1350.2.j.b.199.2		4	45.14	odd	6
1350.2.j.b.1099.1		4	15.14	odd	2
1350.2.j.b.1099.2		4	3.2	odd	2
4050.2.a.d.1.1		1	45.7	odd	12
4050.2.a.o.1.1		1	45.38	even	12
4050.2.a.u.1.1		1	45.2	even	12
4050.2.a.bg.1.1		1	45.43	odd	12
4050.2.c.h.649.1		2	45.34	even	6
4050.2.c.h.649.2		2	9.7	even	3
4050.2.c.m.649.1		2	9.2	odd	6
4050.2.c.m.649.2		2	45.29	odd	6

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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	450.j (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 - 0.500000i) q^{2} +1.73205 q^{3} +(0.500000 + 0.866025i) q^{4} +(-1.50000 - 0.866025i) q^{6} +(-1.73205 - 1.00000i) q^{7} -1.00000i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+(-0.866025 - 0.500000i) q^{2} +1.73205 q^{3} +(0.500000 + 0.866025i) q^{4} +(-1.50000 - 0.866025i) q^{6} +(-1.73205 - 1.00000i) q^{7} -1.00000i q^{8} +3.00000 q^{9} +(0.866025 + 1.50000i) q^{12} +(3.46410 - 2.00000i) q^{13} +(1.00000 + 1.73205i) q^{14} +(-0.500000 + 0.866025i) q^{16} -6.00000i q^{17} +(-2.59808 - 1.50000i) q^{18} +7.00000 q^{19} +(-3.00000 - 1.73205i) q^{21} -1.73205i q^{24} -4.00000 q^{26} +5.19615 q^{27} -2.00000i q^{28} +(-3.00000 + 5.19615i) q^{29} +(5.00000 + 8.66025i) q^{31} +(0.866025 - 0.500000i) q^{32} +(-3.00000 + 5.19615i) q^{34} +(1.50000 + 2.59808i) q^{36} +2.00000i q^{37} +(-6.06218 - 3.50000i) q^{38} +(6.00000 - 3.46410i) q^{39} +(-4.50000 - 7.79423i) q^{41} +(1.73205 + 3.00000i) q^{42} +(-0.866025 - 0.500000i) q^{43} +(-5.19615 - 3.00000i) q^{47} +(-0.866025 + 1.50000i) q^{48} +(-1.50000 - 2.59808i) q^{49} -10.3923i q^{51} +(3.46410 + 2.00000i) q^{52} +12.0000i q^{53} +(-4.50000 - 2.59808i) q^{54} +(-1.00000 + 1.73205i) q^{56} +12.1244 q^{57} +(5.19615 - 3.00000i) q^{58} +(-4.50000 - 7.79423i) q^{59} +(2.00000 - 3.46410i) q^{61} -10.0000i q^{62} +(-5.19615 - 3.00000i) q^{63} -1.00000 q^{64} +(-11.2583 + 6.50000i) q^{67} +(5.19615 - 3.00000i) q^{68} +6.00000 q^{71} -3.00000i q^{72} +1.00000i q^{73} +(1.00000 - 1.73205i) q^{74} +(3.50000 + 6.06218i) q^{76} -6.92820 q^{78} +(1.00000 - 1.73205i) q^{79} +9.00000 q^{81} +9.00000i q^{82} +(7.79423 + 4.50000i) q^{83} -3.46410i q^{84} +(0.500000 + 0.866025i) q^{86} +(-5.19615 + 9.00000i) q^{87} -15.0000 q^{89} -8.00000 q^{91} +(8.66025 + 15.0000i) q^{93} +(3.00000 + 5.19615i) q^{94} +(1.50000 - 0.866025i) q^{96} +(-14.7224 - 8.50000i) q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 6 q^{6} + 12 q^{9}+O(q^{10}) \) 4 * q + 2 * q^4 - 6 * q^6 + 12 * q^9	\( 4 q + 2 q^{4} - 6 q^{6} + 12 q^{9} + 4 q^{14} - 2 q^{16} + 28 q^{19} - 12 q^{21} - 16 q^{26} - 12 q^{29} + 20 q^{31} - 12 q^{34} + 6 q^{36} + 24 q^{39} - 18 q^{41} - 6 q^{49} - 18 q^{54} - 4 q^{56} - 18 q^{59} + 8 q^{61} - 4 q^{64} + 24 q^{71} + 4 q^{74} + 14 q^{76} + 4 q^{79} + 36 q^{81} + 2 q^{86} - 60 q^{89} - 32 q^{91} + 12 q^{94} + 6 q^{96}+O(q^{100}) \) 4 * q + 2 * q^4 - 6 * q^6 + 12 * q^9 + 4 * q^14 - 2 * q^16 + 28 * q^19 - 12 * q^21 - 16 * q^26 - 12 * q^29 + 20 * q^31 - 12 * q^34 + 6 * q^36 + 24 * q^39 - 18 * q^41 - 6 * q^49 - 18 * q^54 - 4 * q^56 - 18 * q^59 + 8 * q^61 - 4 * q^64 + 24 * q^71 + 4 * q^74 + 14 * q^76 + 4 * q^79 + 36 * q^81 + 2 * q^86 - 60 * q^89 - 32 * q^91 + 12 * q^94 + 6 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more