LMFDB - Embedded newform 450.3.g.d.307.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [450,3,Mod(307,450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("450.307"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	450.g (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,2,0,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$12.2616118962$
Analytic rank:	$0$
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:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			307.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	450.307
Dual form			450.3.g.d.343.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +(-8.00000 - 8.00000i) q^{7} +(-2.00000 + 2.00000i) q^{8} +4.00000 q^{11} +(3.00000 - 3.00000i) q^{13} -16.0000i q^{14} -4.00000 q^{16} +(-19.0000 - 19.0000i) q^{17} -8.00000i q^{19} +(4.00000 + 4.00000i) q^{22} +(20.0000 - 20.0000i) q^{23} +6.00000 q^{26} +(16.0000 - 16.0000i) q^{28} -38.0000i q^{29} -44.0000 q^{31} +(-4.00000 - 4.00000i) q^{32} -38.0000i q^{34} +(3.00000 + 3.00000i) q^{37} +(8.00000 - 8.00000i) q^{38} +70.0000 q^{41} +(-36.0000 + 36.0000i) q^{43} +8.00000i q^{44} +40.0000 q^{46} +79.0000i q^{49} +(6.00000 + 6.00000i) q^{52} +(-17.0000 + 17.0000i) q^{53} +32.0000 q^{56} +(38.0000 - 38.0000i) q^{58} -92.0000i q^{59} +72.0000 q^{61} +(-44.0000 - 44.0000i) q^{62} -8.00000i q^{64} +(-44.0000 - 44.0000i) q^{67} +(38.0000 - 38.0000i) q^{68} -88.0000 q^{71} +(-55.0000 + 55.0000i) q^{73} +6.00000i q^{74} +16.0000 q^{76} +(-32.0000 - 32.0000i) q^{77} +12.0000i q^{79} +(70.0000 + 70.0000i) q^{82} +(24.0000 - 24.0000i) q^{83} -72.0000 q^{86} +(-8.00000 + 8.00000i) q^{88} +26.0000i q^{89} -48.0000 q^{91} +(40.0000 + 40.0000i) q^{92} +(57.0000 + 57.0000i) q^{97} +(-79.0000 + 79.0000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 16 q^{7} - 4 q^{8} + 8 q^{11} + 6 q^{13} - 8 q^{16} - 38 q^{17} + 8 q^{22} + 40 q^{23} + 12 q^{26} + 32 q^{28} - 88 q^{31} - 8 q^{32} + 6 q^{37} + 16 q^{38} + 140 q^{41} - 72 q^{43} + 80 q^{46}+ \cdots - 158 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 16 * q^7 - 4 * q^8 + 8 * q^11 + 6 * q^13 - 8 * q^16 - 38 * q^17 + 8 * q^22 + 40 * q^23 + 12 * q^26 + 32 * q^28 - 88 * q^31 - 8 * q^32 + 6 * q^37 + 16 * q^38 + 140 * q^41 - 72 * q^43 + 80 * q^46 + 12 * q^52 - 34 * q^53 + 64 * q^56 + 76 * q^58 + 144 * q^61 - 88 * q^62 - 88 * q^67 + 76 * q^68 - 176 * q^71 - 110 * q^73 + 32 * q^76 - 64 * q^77 + 140 * q^82 + 48 * q^83 - 144 * q^86 - 16 * q^88 - 96 * q^91 + 80 * q^92 + 114 * q^97 - 158 * q^98

Character values

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

$n$	$101$	$127$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

Coefficient data

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

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

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

$2$	1.00000	+	1.00000i	0.500000	+	0.500000i
$2$	1.00000	+	1.00000i	0.500000	+	0.500000i
$3$		0			0
$3$		0			0
$4$			2.00000i			0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−8.00000	−	8.00000i	−1.14286	−	1.14286i	−0.987925	−	0.154932i	$-0.950484\pi$
$7$	−8.00000	−	8.00000i	−1.14286	−	1.14286i	−0.154932	−	0.987925i	$-0.549516\pi$
$8$	−2.00000	+	2.00000i	−0.250000	+	0.250000i
$9$		0			0
$10$		0			0
$11$	4.00000			0.363636			0.181818	−	0.983332i	$-0.441802\pi$
$11$	4.00000			0.363636			0.181818	+	0.983332i	$0.441802\pi$
$12$		0			0
$13$	3.00000	−	3.00000i	0.230769	−	0.230769i	−0.582245	−	0.813014i	$-0.697825\pi$
$13$	3.00000	−	3.00000i	0.230769	−	0.230769i	0.813014	+	0.582245i	$0.197825\pi$
$14$		−	16.0000i		−	1.14286i
$15$		0			0
$16$	−4.00000			−0.250000
$17$	−19.0000	−	19.0000i	−1.11765	−	1.11765i	−0.992086	−	0.125561i	$-0.959927\pi$
$17$	−19.0000	−	19.0000i	−1.11765	−	1.11765i	−0.125561	−	0.992086i	$-0.540073\pi$
$18$		0			0
$19$		−	8.00000i		−	0.421053i	−0.977588	−	0.210526i	$-0.932482\pi$
$19$		−	8.00000i		−	0.421053i	0.977588	−	0.210526i	$-0.0675178\pi$
$20$		0			0
$21$		0			0
$22$	4.00000	+	4.00000i	0.181818	+	0.181818i
$23$	20.0000	−	20.0000i	0.869565	−	0.869565i	−0.122859	−	0.992424i	$-0.539206\pi$
$23$	20.0000	−	20.0000i	0.869565	−	0.869565i	0.992424	+	0.122859i	$0.0392063\pi$
$24$		0			0
$25$		0			0
$26$	6.00000			0.230769
$27$		0			0
$28$	16.0000	−	16.0000i	0.571429	−	0.571429i
$29$		−	38.0000i		−	1.31034i	−0.755479	−	0.655172i	$-0.772596\pi$
$29$		−	38.0000i		−	1.31034i	0.755479	−	0.655172i	$-0.227404\pi$
$30$		0			0
$31$	−44.0000			−1.41935			−0.709677	−	0.704527i	$-0.751159\pi$
$31$	−44.0000			−1.41935			−0.709677	+	0.704527i	$0.751159\pi$
$32$	−4.00000	−	4.00000i	−0.125000	−	0.125000i
$33$		0			0
$34$		−	38.0000i		−	1.11765i
$35$		0			0
$36$		0			0
$37$	3.00000	+	3.00000i	0.0810811	+	0.0810811i	0.746484	−	0.665403i	$-0.231740\pi$
$37$	3.00000	+	3.00000i	0.0810811	+	0.0810811i	−0.665403	+	0.746484i	$0.731740\pi$
$38$	8.00000	−	8.00000i	0.210526	−	0.210526i
$39$		0			0
$40$		0			0
$41$	70.0000			1.70732			0.853659	−	0.520833i	$-0.174379\pi$
$41$	70.0000			1.70732			0.853659	+	0.520833i	$0.174379\pi$
$42$		0			0
$43$	−36.0000	+	36.0000i	−0.837209	+	0.837209i	−0.988491	−	0.151281i	$-0.951660\pi$
$43$	−36.0000	+	36.0000i	−0.837209	+	0.837209i	0.151281	+	0.988491i	$0.451660\pi$
$44$			8.00000i			0.181818i
$45$		0			0
$46$	40.0000			0.869565
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$			79.0000i			1.61224i
$50$		0			0
$51$		0			0
$52$	6.00000	+	6.00000i	0.115385	+	0.115385i
$53$	−17.0000	+	17.0000i	−0.320755	+	0.320755i	−0.849057	−	0.528302i	$-0.822829\pi$
$53$	−17.0000	+	17.0000i	−0.320755	+	0.320755i	0.528302	+	0.849057i	$0.322829\pi$
$54$		0			0
$55$		0			0
$56$	32.0000			0.571429
$57$		0			0
$58$	38.0000	−	38.0000i	0.655172	−	0.655172i
$59$		−	92.0000i		−	1.55932i	−0.626202	−	0.779661i	$-0.715391\pi$
$59$		−	92.0000i		−	1.55932i	0.626202	−	0.779661i	$-0.284609\pi$
$60$		0			0
$61$	72.0000			1.18033			0.590164	−	0.807283i	$-0.299063\pi$
$61$	72.0000			1.18033			0.590164	+	0.807283i	$0.299063\pi$
$62$	−44.0000	−	44.0000i	−0.709677	−	0.709677i
$63$		0			0
$64$		−	8.00000i		−	0.125000i
$65$		0			0
$66$		0			0
$67$	−44.0000	−	44.0000i	−0.656716	−	0.656716i	0.297885	−	0.954602i	$-0.403719\pi$
$67$	−44.0000	−	44.0000i	−0.656716	−	0.656716i	−0.954602	+	0.297885i	$0.903719\pi$
$68$	38.0000	−	38.0000i	0.558824	−	0.558824i
$69$		0			0
$70$		0			0
$71$	−88.0000			−1.23944			−0.619718	−	0.784824i	$-0.712753\pi$
$71$	−88.0000			−1.23944			−0.619718	+	0.784824i	$0.712753\pi$
$72$		0			0
$73$	−55.0000	+	55.0000i	−0.753425	+	0.753425i	−0.975117	−	0.221692i	$-0.928842\pi$
$73$	−55.0000	+	55.0000i	−0.753425	+	0.753425i	0.221692	+	0.975117i	$0.428842\pi$
$74$			6.00000i			0.0810811i
$75$		0			0
$76$	16.0000			0.210526
$77$	−32.0000	−	32.0000i	−0.415584	−	0.415584i
$78$		0			0
$79$			12.0000i			0.151899i	0.997112	+	0.0759494i	$0.0241987\pi$
$79$			12.0000i			0.151899i	−0.997112	+	0.0759494i	$0.975801\pi$
$80$		0			0
$81$		0			0
$82$	70.0000	+	70.0000i	0.853659	+	0.853659i
$83$	24.0000	−	24.0000i	0.289157	−	0.289157i	−0.547590	−	0.836747i	$-0.684455\pi$
$83$	24.0000	−	24.0000i	0.289157	−	0.289157i	0.836747	+	0.547590i	$0.184455\pi$
$84$		0			0
$85$		0			0
$86$	−72.0000			−0.837209
$87$		0			0
$88$	−8.00000	+	8.00000i	−0.0909091	+	0.0909091i
$89$			26.0000i			0.292135i	0.989275	+	0.146067i	$0.0466616\pi$
$89$			26.0000i			0.292135i	−0.989275	+	0.146067i	$0.953338\pi$
$90$		0			0
$91$	−48.0000			−0.527473
$92$	40.0000	+	40.0000i	0.434783	+	0.434783i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	57.0000	+	57.0000i	0.587629	+	0.587629i	0.936989	−	0.349360i	$-0.113601\pi$
$97$	57.0000	+	57.0000i	0.587629	+	0.587629i	−0.349360	+	0.936989i	$0.613601\pi$
$98$	−79.0000	+	79.0000i	−0.806122	+	0.806122i
$99$		0			0
$100$		0			0
$101$	56.0000			0.554455			0.277228	−	0.960804i	$-0.410584\pi$
$101$	56.0000			0.554455			0.277228	+	0.960804i	$0.410584\pi$
$102$		0			0
$103$	−4.00000	+	4.00000i	−0.0388350	+	0.0388350i	−0.726258	−	0.687423i	$-0.758742\pi$
$103$	−4.00000	+	4.00000i	−0.0388350	+	0.0388350i	0.687423	+	0.726258i	$0.258742\pi$
$104$			12.0000i			0.115385i
$105$		0			0
$106$	−34.0000			−0.320755
$107$	68.0000	+	68.0000i	0.635514	+	0.635514i	0.949446	−	0.313932i	$-0.101646\pi$
$107$	68.0000	+	68.0000i	0.635514	+	0.635514i	−0.313932	+	0.949446i	$0.601646\pi$
$108$		0			0
$109$		−	46.0000i		−	0.422018i	−0.977484	−	0.211009i	$-0.932325\pi$
$109$		−	46.0000i		−	0.422018i	0.977484	−	0.211009i	$-0.0676750\pi$
$110$		0			0
$111$		0			0
$112$	32.0000	+	32.0000i	0.285714	+	0.285714i
$113$	−53.0000	+	53.0000i	−0.469027	+	0.469027i	−0.901599	−	0.432573i	$-0.857606\pi$
$113$	−53.0000	+	53.0000i	−0.469027	+	0.469027i	0.432573	+	0.901599i	$0.357606\pi$
$114$		0			0
$115$		0			0
$116$	76.0000			0.655172
$117$		0			0
$118$	92.0000	−	92.0000i	0.779661	−	0.779661i
$119$			304.000i			2.55462i
$120$		0			0
$121$	−105.000			−0.867769
$122$	72.0000	+	72.0000i	0.590164	+	0.590164i
$123$		0			0
$124$		−	88.0000i		−	0.709677i
$125$		0			0
$126$		0			0
$127$	−68.0000	−	68.0000i	−0.535433	−	0.535433i	0.386751	−	0.922184i	$-0.373597\pi$
$127$	−68.0000	−	68.0000i	−0.535433	−	0.535433i	−0.922184	+	0.386751i	$0.873597\pi$
$128$	8.00000	−	8.00000i	0.0625000	−	0.0625000i
$129$		0			0
$130$		0			0
$131$	−44.0000			−0.335878			−0.167939	−	0.985797i	$-0.553711\pi$
$131$	−44.0000			−0.335878			−0.167939	+	0.985797i	$0.553711\pi$
$132$		0			0
$133$	−64.0000	+	64.0000i	−0.481203	+	0.481203i
$134$		−	88.0000i		−	0.656716i
$135$		0			0
$136$	76.0000			0.558824
$137$	69.0000	+	69.0000i	0.503650	+	0.503650i	0.912570	−	0.408920i	$-0.134095\pi$
$137$	69.0000	+	69.0000i	0.503650	+	0.503650i	−0.408920	+	0.912570i	$0.634095\pi$
$138$		0			0
$139$			80.0000i			0.575540i	0.957700	+	0.287770i	$0.0929138\pi$
$139$			80.0000i			0.575540i	−0.957700	+	0.287770i	$0.907086\pi$
$140$		0			0
$141$		0			0
$142$	−88.0000	−	88.0000i	−0.619718	−	0.619718i
$143$	12.0000	−	12.0000i	0.0839161	−	0.0839161i
$144$		0			0
$145$		0			0
$146$	−110.000			−0.753425
$147$		0			0
$148$	−6.00000	+	6.00000i	−0.0405405	+	0.0405405i
$149$			168.000i			1.12752i	0.825940	+	0.563758i	$0.190645\pi$
$149$			168.000i			1.12752i	−0.825940	+	0.563758i	$0.809355\pi$
$150$		0			0
$151$	4.00000			0.0264901			0.0132450	−	0.999912i	$-0.495784\pi$
$151$	4.00000			0.0264901			0.0132450	+	0.999912i	$0.495784\pi$
$152$	16.0000	+	16.0000i	0.105263	+	0.105263i
$153$		0			0
$154$		−	64.0000i		−	0.415584i
$155$		0			0
$156$		0			0
$157$	−99.0000	−	99.0000i	−0.630573	−	0.630573i	0.317639	−	0.948212i	$-0.397110\pi$
$157$	−99.0000	−	99.0000i	−0.630573	−	0.630573i	−0.948212	+	0.317639i	$0.897110\pi$
$158$	−12.0000	+	12.0000i	−0.0759494	+	0.0759494i
$159$		0			0
$160$		0			0
$161$	−320.000			−1.98758
$162$		0			0
$163$	160.000	−	160.000i	0.981595	−	0.981595i	−0.0182386	−	0.999834i	$-0.505806\pi$
$163$	160.000	−	160.000i	0.981595	−	0.981595i	0.999834	+	0.0182386i	$0.00580584\pi$
$164$			140.000i			0.853659i
$165$		0			0
$166$	48.0000			0.289157
$167$	−56.0000	−	56.0000i	−0.335329	−	0.335329i	0.519277	−	0.854606i	$-0.326201\pi$
$167$	−56.0000	−	56.0000i	−0.335329	−	0.335329i	−0.854606	+	0.519277i	$0.826201\pi$
$168$		0			0
$169$			151.000i			0.893491i
$170$		0			0
$171$		0			0
$172$	−72.0000	−	72.0000i	−0.418605	−	0.418605i
$173$	41.0000	−	41.0000i	0.236994	−	0.236994i	−0.578610	−	0.815604i	$-0.696405\pi$
$173$	41.0000	−	41.0000i	0.236994	−	0.236994i	0.815604	+	0.578610i	$0.196405\pi$
$174$		0			0
$175$		0			0
$176$	−16.0000			−0.0909091
$177$		0			0
$178$	−26.0000	+	26.0000i	−0.146067	+	0.146067i
$179$		−	172.000i		−	0.960894i	−0.877024	−	0.480447i	$-0.840474\pi$
$179$		−	172.000i		−	0.960894i	0.877024	−	0.480447i	$-0.159526\pi$
$180$		0			0
$181$	62.0000			0.342541			0.171271	−	0.985224i	$-0.445213\pi$
$181$	62.0000			0.342541			0.171271	+	0.985224i	$0.445213\pi$
$182$	−48.0000	−	48.0000i	−0.263736	−	0.263736i
$183$		0			0
$184$			80.0000i			0.434783i
$185$		0			0
$186$		0			0
$187$	−76.0000	−	76.0000i	−0.406417	−	0.406417i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	248.000			1.29843			0.649215	−	0.760605i	$-0.275098\pi$
$191$	248.000			1.29843			0.649215	+	0.760605i	$0.275098\pi$
$192$		0			0
$193$	−135.000	+	135.000i	−0.699482	+	0.699482i	−0.964299	−	0.264817i	$-0.914688\pi$
$193$	−135.000	+	135.000i	−0.699482	+	0.699482i	0.264817	+	0.964299i	$0.414688\pi$
$194$			114.000i			0.587629i
$195$		0			0
$196$	−158.000			−0.806122
$197$	153.000	+	153.000i	0.776650	+	0.776650i	0.979260	−	0.202610i	$-0.0649423\pi$
$197$	153.000	+	153.000i	0.776650	+	0.776650i	−0.202610	+	0.979260i	$0.564942\pi$
$198$		0			0
$199$		−	252.000i		−	1.26633i	−0.774016	−	0.633166i	$-0.781755\pi$
$199$		−	252.000i		−	1.26633i	0.774016	−	0.633166i	$-0.218245\pi$
$200$		0			0
$201$		0			0
$202$	56.0000	+	56.0000i	0.277228	+	0.277228i
$203$	−304.000	+	304.000i	−1.49754	+	1.49754i
$204$		0			0
$205$		0			0
$206$	−8.00000			−0.0388350
$207$		0			0
$208$	−12.0000	+	12.0000i	−0.0576923	+	0.0576923i
$209$		−	32.0000i		−	0.153110i
$210$		0			0
$211$	−64.0000			−0.303318			−0.151659	−	0.988433i	$-0.548461\pi$
$211$	−64.0000			−0.303318			−0.151659	+	0.988433i	$0.548461\pi$
$212$	−34.0000	−	34.0000i	−0.160377	−	0.160377i
$213$		0			0
$214$			136.000i			0.635514i
$215$		0			0
$216$		0			0
$217$	352.000	+	352.000i	1.62212	+	1.62212i
$218$	46.0000	−	46.0000i	0.211009	−	0.211009i
$219$		0			0
$220$		0			0
$221$	−114.000			−0.515837
$222$		0			0
$223$	228.000	−	228.000i	1.02242	−	1.02242i	0.0226787	−	0.999743i	$-0.492781\pi$
$223$	228.000	−	228.000i	1.02242	−	1.02242i	0.999743	−	0.0226787i	$-0.00721948\pi$
$224$			64.0000i			0.285714i
$225$		0			0
$226$	−106.000			−0.469027
$227$	100.000	+	100.000i	0.440529	+	0.440529i	0.892190	−	0.451661i	$-0.149168\pi$
$227$	100.000	+	100.000i	0.440529	+	0.440529i	−0.451661	+	0.892190i	$0.649168\pi$
$228$		0			0
$229$		−	312.000i		−	1.36245i	−0.732076	−	0.681223i	$-0.761449\pi$
$229$		−	312.000i		−	1.36245i	0.732076	−	0.681223i	$-0.238551\pi$
$230$		0			0
$231$		0			0
$232$	76.0000	+	76.0000i	0.327586	+	0.327586i
$233$	93.0000	−	93.0000i	0.399142	−	0.399142i	−0.478789	−	0.877930i	$-0.658924\pi$
$233$	93.0000	−	93.0000i	0.399142	−	0.399142i	0.877930	+	0.478789i	$0.158924\pi$
$234$		0			0
$235$		0			0
$236$	184.000			0.779661
$237$		0			0
$238$	−304.000	+	304.000i	−1.27731	+	1.27731i
$239$		−	96.0000i		−	0.401674i	−0.979625	−	0.200837i	$-0.935634\pi$
$239$		−	96.0000i		−	0.401674i	0.979625	−	0.200837i	$-0.0643661\pi$
$240$		0			0
$241$	160.000			0.663900			0.331950	−	0.943297i	$-0.392293\pi$
$241$	160.000			0.663900			0.331950	+	0.943297i	$0.392293\pi$
$242$	−105.000	−	105.000i	−0.433884	−	0.433884i
$243$		0			0
$244$			144.000i			0.590164i
$245$		0			0
$246$		0			0
$247$	−24.0000	−	24.0000i	−0.0971660	−	0.0971660i
$248$	88.0000	−	88.0000i	0.354839	−	0.354839i
$249$		0			0
$250$		0			0
$251$	12.0000			0.0478088			0.0239044	−	0.999714i	$-0.492390\pi$
$251$	12.0000			0.0478088			0.0239044	+	0.999714i	$0.492390\pi$
$252$		0			0
$253$	80.0000	−	80.0000i	0.316206	−	0.316206i
$254$		−	136.000i		−	0.535433i
$255$		0			0
$256$	16.0000			0.0625000
$257$	−101.000	−	101.000i	−0.392996	−	0.392996i	0.482758	−	0.875754i	$-0.339635\pi$
$257$	−101.000	−	101.000i	−0.392996	−	0.392996i	−0.875754	+	0.482758i	$0.839635\pi$
$258$		0			0
$259$		−	48.0000i		−	0.185328i
$260$		0			0
$261$		0			0
$262$	−44.0000	−	44.0000i	−0.167939	−	0.167939i
$263$	208.000	−	208.000i	0.790875	−	0.790875i	−0.190762	−	0.981636i	$-0.561096\pi$
$263$	208.000	−	208.000i	0.790875	−	0.790875i	0.981636	+	0.190762i	$0.0610958\pi$
$264$		0			0
$265$		0			0
$266$	−128.000			−0.481203
$267$		0			0
$268$	88.0000	−	88.0000i	0.328358	−	0.328358i
$269$		−	296.000i		−	1.10037i	−0.835042	−	0.550186i	$-0.814557\pi$
$269$		−	296.000i		−	1.10037i	0.835042	−	0.550186i	$-0.185443\pi$
$270$		0			0
$271$	108.000			0.398524			0.199262	−	0.979946i	$-0.436146\pi$
$271$	108.000			0.398524			0.199262	+	0.979946i	$0.436146\pi$
$272$	76.0000	+	76.0000i	0.279412	+	0.279412i
$273$		0			0
$274$			138.000i			0.503650i
$275$		0			0
$276$		0			0
$277$	243.000	+	243.000i	0.877256	+	0.877256i	0.993250	−	0.115994i	$-0.0370052\pi$
$277$	243.000	+	243.000i	0.877256	+	0.877256i	−0.115994	+	0.993250i	$0.537005\pi$
$278$	−80.0000	+	80.0000i	−0.287770	+	0.287770i
$279$		0			0
$280$		0			0
$281$	378.000			1.34520			0.672598	−	0.740008i	$-0.265178\pi$
$281$	378.000			1.34520			0.672598	+	0.740008i	$0.265178\pi$
$282$		0			0
$283$	−92.0000	+	92.0000i	−0.325088	+	0.325088i	−0.850715	−	0.525627i	$-0.823831\pi$
$283$	−92.0000	+	92.0000i	−0.325088	+	0.325088i	0.525627	+	0.850715i	$0.323831\pi$
$284$		−	176.000i		−	0.619718i
$285$		0			0
$286$	24.0000			0.0839161
$287$	−560.000	−	560.000i	−1.95122	−	1.95122i
$288$		0			0
$289$			433.000i			1.49827i
$290$		0			0
$291$		0			0
$292$	−110.000	−	110.000i	−0.376712	−	0.376712i
$293$	279.000	−	279.000i	0.952218	−	0.952218i	−0.0466910	−	0.998909i	$-0.514868\pi$
$293$	279.000	−	279.000i	0.952218	−	0.952218i	0.998909	+	0.0466910i	$0.0148676\pi$
$294$		0			0
$295$		0			0
$296$	−12.0000			−0.0405405
$297$		0			0
$298$	−168.000	+	168.000i	−0.563758	+	0.563758i
$299$		−	120.000i		−	0.401338i
$300$		0			0
$301$	576.000			1.91362
$302$	4.00000	+	4.00000i	0.0132450	+	0.0132450i
$303$		0			0
$304$			32.0000i			0.105263i
$305$		0			0
$306$		0			0
$307$	−216.000	−	216.000i	−0.703583	−	0.703583i	0.261595	−	0.965178i	$-0.415752\pi$
$307$	−216.000	−	216.000i	−0.703583	−	0.703583i	−0.965178	+	0.261595i	$0.915752\pi$
$308$	64.0000	−	64.0000i	0.207792	−	0.207792i
$309$		0			0
$310$		0			0
$311$	272.000			0.874598			0.437299	−	0.899316i	$-0.355935\pi$
$311$	272.000			0.874598			0.437299	+	0.899316i	$0.355935\pi$
$312$		0			0
$313$	−15.0000	+	15.0000i	−0.0479233	+	0.0479233i	−0.730662	−	0.682739i	$-0.760789\pi$
$313$	−15.0000	+	15.0000i	−0.0479233	+	0.0479233i	0.682739	+	0.730662i	$0.260789\pi$
$314$		−	198.000i		−	0.630573i
$315$		0			0
$316$	−24.0000			−0.0759494
$317$	−87.0000	−	87.0000i	−0.274448	−	0.274448i	0.556440	−	0.830888i	$-0.312167\pi$
$317$	−87.0000	−	87.0000i	−0.274448	−	0.274448i	−0.830888	+	0.556440i	$0.812167\pi$
$318$		0			0
$319$		−	152.000i		−	0.476489i
$320$		0			0
$321$		0			0
$322$	−320.000	−	320.000i	−0.993789	−	0.993789i
$323$	−152.000	+	152.000i	−0.470588	+	0.470588i
$324$		0			0
$325$		0			0
$326$	320.000			0.981595
$327$		0			0
$328$	−140.000	+	140.000i	−0.426829	+	0.426829i
$329$		0			0
$330$		0			0
$331$	−584.000			−1.76435			−0.882175	−	0.470921i	$-0.843922\pi$
$331$	−584.000			−1.76435			−0.882175	+	0.470921i	$0.843922\pi$
$332$	48.0000	+	48.0000i	0.144578	+	0.144578i
$333$		0			0
$334$		−	112.000i		−	0.335329i
$335$		0			0
$336$		0			0
$337$	129.000	+	129.000i	0.382789	+	0.382789i	0.872106	−	0.489317i	$-0.162754\pi$
$337$	129.000	+	129.000i	0.382789	+	0.382789i	−0.489317	+	0.872106i	$0.662754\pi$
$338$	−151.000	+	151.000i	−0.446746	+	0.446746i
$339$		0			0
$340$		0			0
$341$	−176.000			−0.516129
$342$		0			0
$343$	240.000	−	240.000i	0.699708	−	0.699708i
$344$		−	144.000i		−	0.418605i
$345$		0			0
$346$	82.0000			0.236994
$347$	260.000	+	260.000i	0.749280	+	0.749280i	0.974344	−	0.225064i	$-0.0722592\pi$
$347$	260.000	+	260.000i	0.749280	+	0.749280i	−0.225064	+	0.974344i	$0.572259\pi$
$348$		0			0
$349$			136.000i			0.389685i	0.980835	+	0.194842i	$0.0624195\pi$
$349$			136.000i			0.389685i	−0.980835	+	0.194842i	$0.937580\pi$
$350$		0			0
$351$		0			0
$352$	−16.0000	−	16.0000i	−0.0454545	−	0.0454545i
$353$	75.0000	−	75.0000i	0.212465	−	0.212465i	−0.592849	−	0.805314i	$-0.701997\pi$
$353$	75.0000	−	75.0000i	0.212465	−	0.212465i	0.805314	+	0.592849i	$0.201997\pi$
$354$		0			0
$355$		0			0
$356$	−52.0000			−0.146067
$357$		0			0
$358$	172.000	−	172.000i	0.480447	−	0.480447i
$359$			32.0000i			0.0891365i	0.999006	+	0.0445682i	$0.0141912\pi$
$359$			32.0000i			0.0891365i	−0.999006	+	0.0445682i	$0.985809\pi$
$360$		0			0
$361$	297.000			0.822715
$362$	62.0000	+	62.0000i	0.171271	+	0.171271i
$363$		0			0
$364$		−	96.0000i		−	0.263736i
$365$		0			0
$366$		0			0
$367$	16.0000	+	16.0000i	0.0435967	+	0.0435967i	0.728569	−	0.684972i	$-0.240186\pi$
$367$	16.0000	+	16.0000i	0.0435967	+	0.0435967i	−0.684972	+	0.728569i	$0.740186\pi$
$368$	−80.0000	+	80.0000i	−0.217391	+	0.217391i
$369$		0			0
$370$		0			0
$371$	272.000			0.733154
$372$		0			0
$373$	251.000	−	251.000i	0.672922	−	0.672922i	−0.285466	−	0.958389i	$-0.592149\pi$
$373$	251.000	−	251.000i	0.672922	−	0.672922i	0.958389	+	0.285466i	$0.0921485\pi$
$374$		−	152.000i		−	0.406417i
$375$		0			0
$376$		0			0
$377$	−114.000	−	114.000i	−0.302387	−	0.302387i
$378$		0			0
$379$		−	560.000i		−	1.47757i	−0.673940	−	0.738786i	$-0.735399\pi$
$379$		−	560.000i		−	1.47757i	0.673940	−	0.738786i	$-0.264601\pi$
$380$		0			0
$381$		0			0
$382$	248.000	+	248.000i	0.649215	+	0.649215i
$383$	300.000	−	300.000i	0.783290	−	0.783290i	−0.197095	−	0.980384i	$-0.563151\pi$
$383$	300.000	−	300.000i	0.783290	−	0.783290i	0.980384	+	0.197095i	$0.0631506\pi$
$384$		0			0
$385$		0			0
$386$	−270.000			−0.699482
$387$		0			0
$388$	−114.000	+	114.000i	−0.293814	+	0.293814i
$389$		−	24.0000i		−	0.0616967i	−0.999524	−	0.0308483i	$-0.990179\pi$
$389$		−	24.0000i		−	0.0616967i	0.999524	−	0.0308483i	$-0.00982089\pi$
$390$		0			0
$391$	−760.000			−1.94373
$392$	−158.000	−	158.000i	−0.403061	−	0.403061i
$393$		0			0
$394$			306.000i			0.776650i
$395$		0			0
$396$		0			0
$397$	299.000	+	299.000i	0.753149	+	0.753149i	0.975066	−	0.221917i	$-0.0712314\pi$
$397$	299.000	+	299.000i	0.753149	+	0.753149i	−0.221917	+	0.975066i	$0.571231\pi$
$398$	252.000	−	252.000i	0.633166	−	0.633166i
$399$		0			0
$400$		0			0
$401$	−144.000			−0.359102			−0.179551	−	0.983749i	$-0.557465\pi$
$401$	−144.000			−0.359102			−0.179551	+	0.983749i	$0.557465\pi$
$402$		0			0
$403$	−132.000	+	132.000i	−0.327543	+	0.327543i
$404$			112.000i			0.277228i
$405$		0			0
$406$	−608.000			−1.49754
$407$	12.0000	+	12.0000i	0.0294840	+	0.0294840i
$408$		0			0
$409$		−	354.000i		−	0.865526i	−0.901508	−	0.432763i	$-0.857539\pi$
$409$		−	354.000i		−	0.865526i	0.901508	−	0.432763i	$-0.142461\pi$
$410$		0			0
$411$		0			0
$412$	−8.00000	−	8.00000i	−0.0194175	−	0.0194175i
$413$	−736.000	+	736.000i	−1.78208	+	1.78208i
$414$		0			0
$415$		0			0
$416$	−24.0000			−0.0576923
$417$		0			0
$418$	32.0000	−	32.0000i	0.0765550	−	0.0765550i
$419$		−	468.000i		−	1.11695i	−0.829523	−	0.558473i	$-0.811388\pi$
$419$		−	468.000i		−	1.11695i	0.829523	−	0.558473i	$-0.188612\pi$
$420$		0			0
$421$	104.000			0.247031			0.123515	−	0.992343i	$-0.460583\pi$
$421$	104.000			0.247031			0.123515	+	0.992343i	$0.460583\pi$
$422$	−64.0000	−	64.0000i	−0.151659	−	0.151659i
$423$		0			0
$424$		−	68.0000i		−	0.160377i
$425$		0			0
$426$		0			0
$427$	−576.000	−	576.000i	−1.34895	−	1.34895i
$428$	−136.000	+	136.000i	−0.317757	+	0.317757i
$429$		0			0
$430$		0			0
$431$	−680.000			−1.57773			−0.788863	−	0.614569i	$-0.789330\pi$
$431$	−680.000			−1.57773			−0.788863	+	0.614569i	$0.789330\pi$
$432$		0			0
$433$	−41.0000	+	41.0000i	−0.0946882	+	0.0946882i	−0.752864	−	0.658176i	$-0.771328\pi$
$433$	−41.0000	+	41.0000i	−0.0946882	+	0.0946882i	0.658176	+	0.752864i	$0.271328\pi$
$434$			704.000i			1.62212i
$435$		0			0
$436$	92.0000			0.211009
$437$	−160.000	−	160.000i	−0.366133	−	0.366133i
$438$		0			0
$439$		−	364.000i		−	0.829157i	−0.910014	−	0.414579i	$-0.863929\pi$
$439$		−	364.000i		−	0.829157i	0.910014	−	0.414579i	$-0.136071\pi$
$440$		0			0
$441$		0			0
$442$	−114.000	−	114.000i	−0.257919	−	0.257919i
$443$	372.000	−	372.000i	0.839729	−	0.839729i	−0.149094	−	0.988823i	$-0.547636\pi$
$443$	372.000	−	372.000i	0.839729	−	0.839729i	0.988823	+	0.149094i	$0.0476357\pi$
$444$		0			0
$445$		0			0
$446$	456.000			1.02242
$447$		0			0
$448$	−64.0000	+	64.0000i	−0.142857	+	0.142857i
$449$		−	176.000i		−	0.391982i	−0.980606	−	0.195991i	$-0.937208\pi$
$449$		−	176.000i		−	0.391982i	0.980606	−	0.195991i	$-0.0627924\pi$
$450$		0			0
$451$	280.000			0.620843
$452$	−106.000	−	106.000i	−0.234513	−	0.234513i
$453$		0			0
$454$			200.000i			0.440529i
$455$		0			0
$456$		0			0
$457$	−129.000	−	129.000i	−0.282276	−	0.282276i	0.551740	−	0.834016i	$-0.313964\pi$
$457$	−129.000	−	129.000i	−0.282276	−	0.282276i	−0.834016	+	0.551740i	$0.813964\pi$
$458$	312.000	−	312.000i	0.681223	−	0.681223i
$459$		0			0
$460$		0			0
$461$	568.000			1.23210			0.616052	−	0.787705i	$-0.288731\pi$
$461$	568.000			1.23210			0.616052	+	0.787705i	$0.288731\pi$
$462$		0			0
$463$	568.000	−	568.000i	1.22678	−	1.22678i	0.261608	−	0.965174i	$-0.415747\pi$
$463$	568.000	−	568.000i	1.22678	−	1.22678i	0.965174	−	0.261608i	$-0.0842526\pi$
$464$			152.000i			0.327586i
$465$		0			0
$466$	186.000			0.399142
$467$	272.000	+	272.000i	0.582441	+	0.582441i	0.935573	−	0.353132i	$-0.114883\pi$
$467$	272.000	+	272.000i	0.582441	+	0.582441i	−0.353132	+	0.935573i	$0.614883\pi$
$468$		0			0
$469$			704.000i			1.50107i
$470$		0			0
$471$		0			0
$472$	184.000	+	184.000i	0.389831	+	0.389831i
$473$	−144.000	+	144.000i	−0.304440	+	0.304440i
$474$		0			0
$475$		0			0
$476$	−608.000			−1.27731
$477$		0			0
$478$	96.0000	−	96.0000i	0.200837	−	0.200837i
$479$			928.000i			1.93737i	0.248294	+	0.968685i	$0.420130\pi$
$479$			928.000i			1.93737i	−0.248294	+	0.968685i	$0.579870\pi$
$480$		0			0
$481$	18.0000			0.0374220
$482$	160.000	+	160.000i	0.331950	+	0.331950i
$483$		0			0
$484$		−	210.000i		−	0.433884i
$485$		0			0
$486$		0			0
$487$	−252.000	−	252.000i	−0.517454	−	0.517454i	0.399346	−	0.916800i	$-0.369237\pi$
$487$	−252.000	−	252.000i	−0.517454	−	0.517454i	−0.916800	+	0.399346i	$0.869237\pi$
$488$	−144.000	+	144.000i	−0.295082	+	0.295082i
$489$		0			0
$490$		0			0
$491$	−844.000			−1.71894			−0.859470	−	0.511185i	$-0.829207\pi$
$491$	−844.000			−1.71894			−0.859470	+	0.511185i	$0.829207\pi$
$492$		0			0
$493$	−722.000	+	722.000i	−1.46450	+	1.46450i
$494$		−	48.0000i		−	0.0971660i
$495$		0			0
$496$	176.000			0.354839
$497$	704.000	+	704.000i	1.41650	+	1.41650i
$498$		0			0
$499$			872.000i			1.74749i	0.486380	+	0.873747i	$0.338317\pi$
$499$			872.000i			1.74749i	−0.486380	+	0.873747i	$0.661683\pi$
$500$		0			0
$501$		0			0
$502$	12.0000	+	12.0000i	0.0239044	+	0.0239044i
$503$	−480.000	+	480.000i	−0.954274	+	0.954274i	−0.998999	−	0.0447250i	$-0.985759\pi$
$503$	−480.000	+	480.000i	−0.954274	+	0.954274i	0.0447250	+	0.998999i	$0.485759\pi$
$504$		0			0
$505$		0			0
$506$	160.000			0.316206
$507$		0			0
$508$	136.000	−	136.000i	0.267717	−	0.267717i
$509$			694.000i			1.36346i	0.731605	+	0.681729i	$0.238772\pi$
$509$			694.000i			1.36346i	−0.731605	+	0.681729i	$0.761228\pi$
$510$		0			0
$511$	880.000			1.72211
$512$	16.0000	+	16.0000i	0.0312500	+	0.0312500i
$513$		0			0
$514$		−	202.000i		−	0.392996i
$515$		0			0
$516$		0			0
$517$		0			0
$518$	48.0000	−	48.0000i	0.0926641	−	0.0926641i
$519$		0			0
$520$		0			0
$521$	528.000			1.01344			0.506718	−	0.862112i	$-0.330859\pi$
$521$	528.000			1.01344			0.506718	+	0.862112i	$0.330859\pi$
$522$		0			0
$523$	−552.000	+	552.000i	−1.05545	+	1.05545i	−0.0570797	+	0.998370i	$0.518179\pi$
$523$	−552.000	+	552.000i	−1.05545	+	1.05545i	−0.998370	+	0.0570797i	$0.981821\pi$
$524$		−	88.0000i		−	0.167939i
$525$		0			0
$526$	416.000			0.790875
$527$	836.000	+	836.000i	1.58634	+	1.58634i
$528$		0			0
$529$		−	271.000i		−	0.512287i
$530$		0			0
$531$		0			0
$532$	−128.000	−	128.000i	−0.240602	−	0.240602i
$533$	210.000	−	210.000i	0.393996	−	0.393996i
$534$		0			0
$535$		0			0
$536$	176.000			0.328358
$537$		0			0
$538$	296.000	−	296.000i	0.550186	−	0.550186i
$539$			316.000i			0.586271i
$540$		0			0
$541$	−782.000			−1.44547			−0.722736	−	0.691125i	$-0.757116\pi$
$541$	−782.000			−1.44547			−0.722736	+	0.691125i	$0.757116\pi$
$542$	108.000	+	108.000i	0.199262	+	0.199262i
$543$		0			0
$544$			152.000i			0.279412i
$545$		0			0
$546$		0			0
$547$	420.000	+	420.000i	0.767824	+	0.767824i	0.977723	−	0.209899i	$-0.0673134\pi$
$547$	420.000	+	420.000i	0.767824	+	0.767824i	−0.209899	+	0.977723i	$0.567313\pi$
$548$	−138.000	+	138.000i	−0.251825	+	0.251825i
$549$		0			0
$550$		0			0
$551$	−304.000			−0.551724
$552$		0			0
$553$	96.0000	−	96.0000i	0.173599	−	0.173599i
$554$			486.000i			0.877256i
$555$		0			0
$556$	−160.000			−0.287770
$557$	−417.000	−	417.000i	−0.748654	−	0.748654i	0.225573	−	0.974226i	$-0.427575\pi$
$557$	−417.000	−	417.000i	−0.748654	−	0.748654i	−0.974226	+	0.225573i	$0.927575\pi$
$558$		0			0
$559$			216.000i			0.386404i
$560$		0			0
$561$		0			0
$562$	378.000	+	378.000i	0.672598	+	0.672598i
$563$	−228.000	+	228.000i	−0.404973	+	0.404973i	−0.879981	−	0.475008i	$-0.842445\pi$
$563$	−228.000	+	228.000i	−0.404973	+	0.404973i	0.475008	+	0.879981i	$0.342445\pi$
$564$		0			0
$565$		0			0
$566$	−184.000			−0.325088
$567$		0			0
$568$	176.000	−	176.000i	0.309859	−	0.309859i
$569$			368.000i			0.646749i	0.946271	+	0.323374i	$0.104817\pi$
$569$			368.000i			0.646749i	−0.946271	+	0.323374i	$0.895183\pi$
$570$		0			0
$571$	−736.000			−1.28897			−0.644483	−	0.764618i	$-0.722928\pi$
$571$	−736.000			−1.28897			−0.644483	+	0.764618i	$0.722928\pi$
$572$	24.0000	+	24.0000i	0.0419580	+	0.0419580i
$573$		0			0
$574$		−	1120.00i		−	1.95122i
$575$		0			0
$576$		0			0
$577$	113.000	+	113.000i	0.195841	+	0.195841i	0.798214	−	0.602374i	$-0.205778\pi$
$577$	113.000	+	113.000i	0.195841	+	0.195841i	−0.602374	+	0.798214i	$0.705778\pi$
$578$	−433.000	+	433.000i	−0.749135	+	0.749135i
$579$		0			0
$580$		0			0
$581$	−384.000			−0.660929
$582$		0			0
$583$	−68.0000	+	68.0000i	−0.116638	+	0.116638i
$584$		−	220.000i		−	0.376712i
$585$		0			0
$586$	558.000			0.952218
$587$	−684.000	−	684.000i	−1.16525	−	1.16525i	−0.983310	−	0.181937i	$-0.941763\pi$
$587$	−684.000	−	684.000i	−1.16525	−	1.16525i	−0.181937	−	0.983310i	$-0.558237\pi$
$588$		0			0
$589$			352.000i			0.597623i
$590$		0			0
$591$		0			0
$592$	−12.0000	−	12.0000i	−0.0202703	−	0.0202703i
$593$	149.000	−	149.000i	0.251265	−	0.251265i	−0.570224	−	0.821489i	$-0.693144\pi$
$593$	149.000	−	149.000i	0.251265	−	0.251265i	0.821489	+	0.570224i	$0.193144\pi$
$594$		0			0
$595$		0			0
$596$	−336.000			−0.563758
$597$		0			0
$598$	120.000	−	120.000i	0.200669	−	0.200669i
$599$		−	152.000i		−	0.253756i	−0.991918	−	0.126878i	$-0.959504\pi$
$599$		−	152.000i		−	0.253756i	0.991918	−	0.126878i	$-0.0404957\pi$
$600$		0			0
$601$	320.000			0.532446			0.266223	−	0.963911i	$-0.414224\pi$
$601$	320.000			0.532446			0.266223	+	0.963911i	$0.414224\pi$
$602$	576.000	+	576.000i	0.956811	+	0.956811i
$603$		0			0
$604$			8.00000i			0.0132450i
$605$		0			0
$606$		0			0
$607$	528.000	+	528.000i	0.869852	+	0.869852i	0.992456	−	0.122604i	$-0.0391245\pi$
$607$	528.000	+	528.000i	0.869852	+	0.869852i	−0.122604	+	0.992456i	$0.539124\pi$
$608$	−32.0000	+	32.0000i	−0.0526316	+	0.0526316i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−771.000	+	771.000i	−1.25775	+	1.25775i	−0.305583	+	0.952165i	$0.598852\pi$
$613$	−771.000	+	771.000i	−1.25775	+	1.25775i	−0.952165	+	0.305583i	$0.901148\pi$
$614$		−	432.000i		−	0.703583i
$615$		0			0
$616$	128.000			0.207792
$617$	−675.000	−	675.000i	−1.09400	−	1.09400i	−0.995097	−	0.0989065i	$-0.968466\pi$
$617$	−675.000	−	675.000i	−1.09400	−	1.09400i	−0.0989065	−	0.995097i	$-0.531534\pi$
$618$		0			0
$619$			600.000i			0.969305i	0.874707	+	0.484653i	$0.161054\pi$
$619$			600.000i			0.969305i	−0.874707	+	0.484653i	$0.838946\pi$
$620$		0			0
$621$		0			0
$622$	272.000	+	272.000i	0.437299	+	0.437299i
$623$	208.000	−	208.000i	0.333868	−	0.333868i
$624$		0			0
$625$		0			0
$626$	−30.0000			−0.0479233
$627$		0			0
$628$	198.000	−	198.000i	0.315287	−	0.315287i
$629$		−	114.000i		−	0.181240i
$630$		0			0
$631$	−20.0000			−0.0316957			−0.0158479	−	0.999874i	$-0.505045\pi$
$631$	−20.0000			−0.0316957			−0.0158479	+	0.999874i	$0.505045\pi$
$632$	−24.0000	−	24.0000i	−0.0379747	−	0.0379747i
$633$		0			0
$634$		−	174.000i		−	0.274448i
$635$		0			0
$636$		0			0
$637$	237.000	+	237.000i	0.372057	+	0.372057i
$638$	152.000	−	152.000i	0.238245	−	0.238245i
$639$		0			0
$640$		0			0
$641$	−694.000			−1.08268			−0.541342	−	0.840803i	$-0.682083\pi$
$641$	−694.000			−1.08268			−0.541342	+	0.840803i	$0.682083\pi$
$642$		0			0
$643$	168.000	−	168.000i	0.261275	−	0.261275i	−0.564297	−	0.825572i	$-0.690853\pi$
$643$	168.000	−	168.000i	0.261275	−	0.261275i	0.825572	+	0.564297i	$0.190853\pi$
$644$		−	640.000i		−	0.993789i
$645$		0			0
$646$	−304.000			−0.470588
$647$	328.000	+	328.000i	0.506955	+	0.506955i	0.913591	−	0.406635i	$-0.133298\pi$
$647$	328.000	+	328.000i	0.506955	+	0.506955i	−0.406635	+	0.913591i	$0.633298\pi$
$648$		0			0
$649$		−	368.000i		−	0.567026i
$650$		0			0
$651$		0			0
$652$	320.000	+	320.000i	0.490798	+	0.490798i
$653$	−81.0000	+	81.0000i	−0.124043	+	0.124043i	−0.766403	−	0.642360i	$-0.777955\pi$
$653$	−81.0000	+	81.0000i	−0.124043	+	0.124043i	0.642360	+	0.766403i	$0.277955\pi$
$654$		0			0
$655$		0			0
$656$	−280.000			−0.426829
$657$		0			0
$658$		0			0
$659$			500.000i			0.758725i	0.925248	+	0.379363i	$0.123857\pi$
$659$			500.000i			0.758725i	−0.925248	+	0.379363i	$0.876143\pi$
$660$		0			0
$661$	−568.000			−0.859304			−0.429652	−	0.902995i	$-0.641364\pi$
$661$	−568.000			−0.859304			−0.429652	+	0.902995i	$0.641364\pi$
$662$	−584.000	−	584.000i	−0.882175	−	0.882175i
$663$		0			0
$664$			96.0000i			0.144578i
$665$		0			0
$666$		0			0
$667$	−760.000	−	760.000i	−1.13943	−	1.13943i
$668$	112.000	−	112.000i	0.167665	−	0.167665i
$669$		0			0
$670$		0			0
$671$	288.000			0.429210
$672$		0			0
$673$	−73.0000	+	73.0000i	−0.108470	+	0.108470i	−0.759259	−	0.650789i	$-0.774438\pi$
$673$	−73.0000	+	73.0000i	−0.108470	+	0.108470i	0.650789	+	0.759259i	$0.274438\pi$
$674$			258.000i			0.382789i
$675$		0			0
$676$	−302.000			−0.446746
$677$	839.000	+	839.000i	1.23929	+	1.23929i	0.960291	+	0.279000i	$0.0900029\pi$
$677$	839.000	+	839.000i	1.23929	+	1.23929i	0.279000	+	0.960291i	$0.409997\pi$
$678$		0			0
$679$		−	912.000i		−	1.34315i
$680$		0			0
$681$		0			0
$682$	−176.000	−	176.000i	−0.258065	−	0.258065i
$683$	744.000	−	744.000i	1.08931	−	1.08931i	0.0937126	−	0.995599i	$-0.470127\pi$
$683$	744.000	−	744.000i	1.08931	−	1.08931i	0.995599	−	0.0937126i	$-0.0298735\pi$
$684$		0			0
$685$		0			0
$686$	480.000			0.699708
$687$		0			0
$688$	144.000	−	144.000i	0.209302	−	0.209302i
$689$			102.000i			0.148041i
$690$		0			0
$691$	504.000			0.729378			0.364689	−	0.931129i	$-0.381175\pi$
$691$	504.000			0.729378			0.364689	+	0.931129i	$0.381175\pi$
$692$	82.0000	+	82.0000i	0.118497	+	0.118497i
$693$		0			0
$694$			520.000i			0.749280i
$695$		0			0
$696$		0			0
$697$	−1330.00	−	1330.00i	−1.90818	−	1.90818i
$698$	−136.000	+	136.000i	−0.194842	+	0.194842i
$699$		0			0
$700$		0			0
$701$	298.000			0.425107			0.212553	−	0.977149i	$-0.431822\pi$
$701$	298.000			0.425107			0.212553	+	0.977149i	$0.431822\pi$
$702$		0			0
$703$	24.0000	−	24.0000i	0.0341394	−	0.0341394i
$704$		−	32.0000i		−	0.0454545i
$705$		0			0
$706$	150.000			0.212465
$707$	−448.000	−	448.000i	−0.633663	−	0.633663i
$708$		0			0
$709$			472.000i			0.665726i	0.942975	+	0.332863i	$0.108015\pi$
$709$			472.000i			0.665726i	−0.942975	+	0.332863i	$0.891985\pi$
$710$		0			0
$711$		0			0
$712$	−52.0000	−	52.0000i	−0.0730337	−	0.0730337i
$713$	−880.000	+	880.000i	−1.23422	+	1.23422i
$714$		0			0
$715$		0			0
$716$	344.000			0.480447
$717$		0			0
$718$	−32.0000	+	32.0000i	−0.0445682	+	0.0445682i
$719$		−	872.000i		−	1.21280i	−0.795161	−	0.606398i	$-0.792614\pi$
$719$		−	872.000i		−	1.21280i	0.795161	−	0.606398i	$-0.207386\pi$
$720$		0			0
$721$	64.0000			0.0887656
$722$	297.000	+	297.000i	0.411357	+	0.411357i
$723$		0			0
$724$			124.000i			0.171271i
$725$		0			0
$726$		0			0
$727$	60.0000	+	60.0000i	0.0825309	+	0.0825309i	0.747167	−	0.664636i	$-0.231413\pi$
$727$	60.0000	+	60.0000i	0.0825309	+	0.0825309i	−0.664636	+	0.747167i	$0.731413\pi$
$728$	96.0000	−	96.0000i	0.131868	−	0.131868i
$729$		0			0
$730$		0			0
$731$	1368.00			1.87141
$732$		0			0
$733$	581.000	−	581.000i	0.792633	−	0.792633i	−0.189289	−	0.981922i	$-0.560618\pi$
$733$	581.000	−	581.000i	0.792633	−	0.792633i	0.981922	+	0.189289i	$0.0606181\pi$
$734$			32.0000i			0.0435967i
$735$		0			0
$736$	−160.000			−0.217391
$737$	−176.000	−	176.000i	−0.238806	−	0.238806i
$738$		0			0
$739$			1160.00i			1.56969i	0.619693	+	0.784844i	$0.287257\pi$
$739$			1160.00i			1.56969i	−0.619693	+	0.784844i	$0.712743\pi$
$740$		0			0
$741$		0			0
$742$	272.000	+	272.000i	0.366577	+	0.366577i
$743$	−380.000	+	380.000i	−0.511440	+	0.511440i	−0.914968	−	0.403527i	$-0.867784\pi$
$743$	−380.000	+	380.000i	−0.511440	+	0.511440i	0.403527	+	0.914968i	$0.367784\pi$
$744$		0			0
$745$		0			0
$746$	502.000			0.672922
$747$		0			0
$748$	152.000	−	152.000i	0.203209	−	0.203209i
$749$		−	1088.00i		−	1.45260i
$750$		0			0
$751$	780.000			1.03862			0.519308	−	0.854587i	$-0.326190\pi$
$751$	780.000			1.03862			0.519308	+	0.854587i	$0.326190\pi$
$752$		0			0
$753$		0			0
$754$		−	228.000i		−	0.302387i
$755$		0			0
$756$		0			0
$757$	−285.000	−	285.000i	−0.376486	−	0.376486i	0.493347	−	0.869833i	$-0.335773\pi$
$757$	−285.000	−	285.000i	−0.376486	−	0.376486i	−0.869833	+	0.493347i	$0.835773\pi$
$758$	560.000	−	560.000i	0.738786	−	0.738786i
$759$		0			0
$760$		0			0
$761$	304.000			0.399474			0.199737	−	0.979850i	$-0.435991\pi$
$761$	304.000			0.399474			0.199737	+	0.979850i	$0.435991\pi$
$762$		0			0
$763$	−368.000	+	368.000i	−0.482307	+	0.482307i
$764$			496.000i			0.649215i
$765$		0			0
$766$	600.000			0.783290
$767$	−276.000	−	276.000i	−0.359844	−	0.359844i
$768$		0			0
$769$		−	1072.00i		−	1.39402i	−0.717062	−	0.697009i	$-0.754514\pi$
$769$		−	1072.00i		−	1.39402i	0.717062	−	0.697009i	$-0.245486\pi$
$770$		0			0
$771$		0			0
$772$	−270.000	−	270.000i	−0.349741	−	0.349741i
$773$	897.000	−	897.000i	1.16041	−	1.16041i	0.176029	−	0.984385i	$-0.443675\pi$
$773$	897.000	−	897.000i	1.16041	−	1.16041i	0.984385	−	0.176029i	$-0.0563252\pi$
$774$		0			0
$775$		0			0
$776$	−228.000			−0.293814
$777$		0			0
$778$	24.0000	−	24.0000i	0.0308483	−	0.0308483i
$779$		−	560.000i		−	0.718870i
$780$		0			0
$781$	−352.000			−0.450704
$782$	−760.000	−	760.000i	−0.971867	−	0.971867i
$783$		0			0
$784$		−	316.000i		−	0.403061i
$785$		0			0
$786$		0			0
$787$	−328.000	−	328.000i	−0.416773	−	0.416773i	0.467317	−	0.884090i	$-0.345221\pi$
$787$	−328.000	−	328.000i	−0.416773	−	0.416773i	−0.884090	+	0.467317i	$0.845221\pi$
$788$	−306.000	+	306.000i	−0.388325	+	0.388325i
$789$		0			0
$790$		0			0
$791$	848.000			1.07206
$792$		0			0
$793$	216.000	−	216.000i	0.272383	−	0.272383i
$794$			598.000i			0.753149i
$795$		0			0
$796$	504.000			0.633166
$797$	87.0000	+	87.0000i	0.109159	+	0.109159i	0.759577	−	0.650418i	$-0.225406\pi$
$797$	87.0000	+	87.0000i	0.109159	+	0.109159i	−0.650418	+	0.759577i	$0.725406\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$	−144.000	−	144.000i	−0.179551	−	0.179551i
$803$	−220.000	+	220.000i	−0.273973	+	0.273973i
$804$		0			0
$805$		0			0
$806$	−264.000			−0.327543
$807$		0			0
$808$	−112.000	+	112.000i	−0.138614	+	0.138614i
$809$		−	922.000i		−	1.13968i	−0.821756	−	0.569839i	$-0.807005\pi$
$809$		−	922.000i		−	1.13968i	0.821756	−	0.569839i	$-0.192995\pi$
$810$		0			0
$811$	−1312.00			−1.61776			−0.808878	−	0.587977i	$-0.799925\pi$
$811$	−1312.00			−1.61776			−0.808878	+	0.587977i	$0.799925\pi$
$812$	−608.000	−	608.000i	−0.748768	−	0.748768i
$813$		0			0
$814$			24.0000i			0.0294840i
$815$		0			0
$816$		0			0
$817$	288.000	+	288.000i	0.352509	+	0.352509i
$818$	354.000	−	354.000i	0.432763	−	0.432763i
$819$		0			0
$820$		0			0
$821$	−950.000			−1.15713			−0.578563	−	0.815638i	$-0.696386\pi$
$821$	−950.000			−1.15713			−0.578563	+	0.815638i	$0.696386\pi$
$822$		0			0
$823$	−68.0000	+	68.0000i	−0.0826245	+	0.0826245i	−0.747211	−	0.664587i	$-0.768608\pi$
$823$	−68.0000	+	68.0000i	−0.0826245	+	0.0826245i	0.664587	+	0.747211i	$0.268608\pi$
$824$		−	16.0000i		−	0.0194175i
$825$		0			0
$826$	−1472.00			−1.78208
$827$	−264.000	−	264.000i	−0.319226	−	0.319226i	0.529244	−	0.848470i	$-0.322476\pi$
$827$	−264.000	−	264.000i	−0.319226	−	0.319226i	−0.848470	+	0.529244i	$0.822476\pi$
$828$		0			0
$829$		−	178.000i		−	0.214717i	−0.994220	−	0.107358i	$-0.965761\pi$
$829$		−	178.000i		−	0.214717i	0.994220	−	0.107358i	$-0.0342392\pi$
$830$		0			0
$831$		0			0
$832$	−24.0000	−	24.0000i	−0.0288462	−	0.0288462i
$833$	1501.00	−	1501.00i	1.80192	−	1.80192i
$834$		0			0
$835$		0			0
$836$	64.0000			0.0765550
$837$		0			0
$838$	468.000	−	468.000i	0.558473	−	0.558473i
$839$		−	1576.00i		−	1.87843i	−0.343334	−	0.939213i	$-0.611556\pi$
$839$		−	1576.00i		−	1.87843i	0.343334	−	0.939213i	$-0.388444\pi$
$840$		0			0
$841$	−603.000			−0.717004
$842$	104.000	+	104.000i	0.123515	+	0.123515i
$843$		0			0
$844$		−	128.000i		−	0.151659i
$845$		0			0
$846$		0			0
$847$	840.000	+	840.000i	0.991736	+	0.991736i
$848$	68.0000	−	68.0000i	0.0801887	−	0.0801887i
$849$		0			0
$850$		0			0
$851$	120.000			0.141011
$852$		0			0
$853$	509.000	−	509.000i	0.596717	−	0.596717i	−0.342720	−	0.939438i	$-0.611348\pi$
$853$	509.000	−	509.000i	0.596717	−	0.596717i	0.939438	+	0.342720i	$0.111348\pi$
$854$		−	1152.00i		−	1.34895i
$855$		0			0
$856$	−272.000			−0.317757
$857$	−357.000	−	357.000i	−0.416569	−	0.416569i	0.467450	−	0.884019i	$-0.345173\pi$
$857$	−357.000	−	357.000i	−0.416569	−	0.416569i	−0.884019	+	0.467450i	$0.845173\pi$
$858$		0			0
$859$		−	520.000i		−	0.605355i	−0.953093	−	0.302678i	$-0.902119\pi$
$859$		−	520.000i		−	0.605355i	0.953093	−	0.302678i	$-0.0978805\pi$
$860$		0			0
$861$		0			0
$862$	−680.000	−	680.000i	−0.788863	−	0.788863i
$863$	−952.000	+	952.000i	−1.10313	+	1.10313i	−0.109098	+	0.994031i	$0.534796\pi$
$863$	−952.000	+	952.000i	−1.10313	+	1.10313i	−0.994031	+	0.109098i	$0.965204\pi$
$864$		0			0
$865$		0			0
$866$	−82.0000			−0.0946882
$867$		0			0
$868$	−704.000	+	704.000i	−0.811060	+	0.811060i
$869$			48.0000i			0.0552359i
$870$		0			0
$871$	−264.000			−0.303100
$872$	92.0000	+	92.0000i	0.105505	+	0.105505i
$873$		0			0
$874$		−	320.000i		−	0.366133i
$875$		0			0
$876$		0			0
$877$	717.000	+	717.000i	0.817560	+	0.817560i	0.985754	−	0.168194i	$-0.0537935\pi$
$877$	717.000	+	717.000i	0.817560	+	0.817560i	−0.168194	+	0.985754i	$0.553794\pi$
$878$	364.000	−	364.000i	0.414579	−	0.414579i
$879$		0			0
$880$		0			0
$881$	−554.000			−0.628831			−0.314415	−	0.949285i	$-0.601808\pi$
$881$	−554.000			−0.628831			−0.314415	+	0.949285i	$0.601808\pi$
$882$		0			0
$883$	−952.000	+	952.000i	−1.07814	+	1.07814i	−0.0814666	+	0.996676i	$0.525960\pi$
$883$	−952.000	+	952.000i	−1.07814	+	1.07814i	−0.996676	+	0.0814666i	$0.974040\pi$
$884$		−	228.000i		−	0.257919i
$885$		0			0
$886$	744.000			0.839729
$887$	−648.000	−	648.000i	−0.730552	−	0.730552i	0.240177	−	0.970729i	$-0.422795\pi$
$887$	−648.000	−	648.000i	−0.730552	−	0.730552i	−0.970729	+	0.240177i	$0.922795\pi$
$888$		0			0
$889$			1088.00i			1.22385i
$890$		0			0
$891$		0			0
$892$	456.000	+	456.000i	0.511211	+	0.511211i
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−128.000			−0.142857
$897$		0			0
$898$	176.000	−	176.000i	0.195991	−	0.195991i
$899$			1672.00i			1.85984i
$900$		0			0
$901$	646.000			0.716981
$902$	280.000	+	280.000i	0.310421	+	0.310421i
$903$		0			0
$904$		−	212.000i		−	0.234513i
$905$		0			0
$906$		0			0
$907$	740.000	+	740.000i	0.815877	+	0.815877i	0.985508	−	0.169631i	$-0.0542576\pi$
$907$	740.000	+	740.000i	0.815877	+	0.815877i	−0.169631	+	0.985508i	$0.554258\pi$
$908$	−200.000	+	200.000i	−0.220264	+	0.220264i
$909$		0			0
$910$		0			0
$911$	−520.000			−0.570801			−0.285401	−	0.958408i	$-0.592127\pi$
$911$	−520.000			−0.570801			−0.285401	+	0.958408i	$0.592127\pi$
$912$		0			0
$913$	96.0000	−	96.0000i	0.105148	−	0.105148i
$914$		−	258.000i		−	0.282276i
$915$		0			0
$916$	624.000			0.681223
$917$	352.000	+	352.000i	0.383860	+	0.383860i
$918$		0			0
$919$		−	844.000i		−	0.918390i	−0.888336	−	0.459195i	$-0.848138\pi$
$919$		−	844.000i		−	0.918390i	0.888336	−	0.459195i	$-0.151862\pi$
$920$		0			0
$921$		0			0
$922$	568.000	+	568.000i	0.616052	+	0.616052i
$923$	−264.000	+	264.000i	−0.286024	+	0.286024i
$924$		0			0
$925$		0			0
$926$	1136.00			1.22678
$927$		0			0
$928$	−152.000	+	152.000i	−0.163793	+	0.163793i
$929$			208.000i			0.223897i	0.993714	+	0.111948i	$0.0357091\pi$
$929$			208.000i			0.223897i	−0.993714	+	0.111948i	$0.964291\pi$
$930$		0			0
$931$	632.000			0.678840
$932$	186.000	+	186.000i	0.199571	+	0.199571i
$933$		0			0
$934$			544.000i			0.582441i
$935$		0			0
$936$		0			0
$937$	−7.00000	−	7.00000i	−0.00747065	−	0.00747065i	0.703362	−	0.710832i	$-0.251681\pi$
$937$	−7.00000	−	7.00000i	−0.00747065	−	0.00747065i	−0.710832	+	0.703362i	$0.751681\pi$
$938$	−704.000	+	704.000i	−0.750533	+	0.750533i
$939$		0			0
$940$		0			0
$941$	1752.00			1.86185			0.930925	−	0.365212i	$-0.119003\pi$
$941$	1752.00			1.86185			0.930925	+	0.365212i	$0.119003\pi$
$942$		0			0
$943$	1400.00	−	1400.00i	1.48462	−	1.48462i
$944$			368.000i			0.389831i
$945$		0			0
$946$	−288.000			−0.304440
$947$	−224.000	−	224.000i	−0.236536	−	0.236536i	0.578878	−	0.815414i	$-0.303491\pi$
$947$	−224.000	−	224.000i	−0.236536	−	0.236536i	−0.815414	+	0.578878i	$0.803491\pi$
$948$		0			0
$949$			330.000i			0.347734i
$950$		0			0
$951$		0			0
$952$	−608.000	−	608.000i	−0.638655	−	0.638655i
$953$	−645.000	+	645.000i	−0.676810	+	0.676810i	−0.959277	−	0.282467i	$-0.908847\pi$
$953$	−645.000	+	645.000i	−0.676810	+	0.676810i	0.282467	+	0.959277i	$0.408847\pi$
$954$		0			0
$955$		0			0
$956$	192.000			0.200837
$957$		0			0
$958$	−928.000	+	928.000i	−0.968685	+	0.968685i
$959$		−	1104.00i		−	1.15120i
$960$		0			0
$961$	975.000			1.01457
$962$	18.0000	+	18.0000i	0.0187110	+	0.0187110i
$963$		0			0
$964$			320.000i			0.331950i
$965$		0			0
$966$		0			0
$967$	92.0000	+	92.0000i	0.0951396	+	0.0951396i	0.753075	−	0.657935i	$-0.228570\pi$
$967$	92.0000	+	92.0000i	0.0951396	+	0.0951396i	−0.657935	+	0.753075i	$0.728570\pi$
$968$	210.000	−	210.000i	0.216942	−	0.216942i
$969$		0			0
$970$		0			0
$971$	−4.00000			−0.00411946			−0.00205973	−	0.999998i	$-0.500656\pi$
$971$	−4.00000			−0.00411946			−0.00205973	+	0.999998i	$0.500656\pi$
$972$		0			0
$973$	640.000	−	640.000i	0.657760	−	0.657760i
$974$		−	504.000i		−	0.517454i
$975$		0			0
$976$	−288.000			−0.295082
$977$	1011.00	+	1011.00i	1.03480	+	1.03480i	0.999372	+	0.0354282i	$0.0112795\pi$
$977$	1011.00	+	1011.00i	1.03480	+	1.03480i	0.0354282	+	0.999372i	$0.488720\pi$
$978$		0			0
$979$			104.000i			0.106231i
$980$		0			0
$981$		0			0
$982$	−844.000	−	844.000i	−0.859470	−	0.859470i
$983$	604.000	−	604.000i	0.614446	−	0.614446i	−0.329656	−	0.944101i	$-0.606933\pi$
$983$	604.000	−	604.000i	0.614446	−	0.614446i	0.944101	+	0.329656i	$0.106933\pi$
$984$		0			0
$985$		0			0
$986$	−1444.00			−1.46450
$987$		0			0
$988$	48.0000	−	48.0000i	0.0485830	−	0.0485830i
$989$			1440.00i			1.45602i
$990$		0			0
$991$	652.000			0.657921			0.328961	−	0.944344i	$-0.393302\pi$
$991$	652.000			0.657921			0.328961	+	0.944344i	$0.393302\pi$
$992$	176.000	+	176.000i	0.177419	+	0.177419i
$993$		0			0
$994$			1408.00i			1.41650i
$995$		0			0
$996$		0			0
$997$	−1051.00	−	1051.00i	−1.05416	−	1.05416i	−0.998447	−	0.0557158i	$-0.982256\pi$
$997$	−1051.00	−	1051.00i	−1.05416	−	1.05416i	−0.0557158	−	0.998447i	$-0.517744\pi$
$998$	−872.000	+	872.000i	−0.873747	+	0.873747i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.3.g.d.307.1		2
3.2	odd	2		450.3.g.a.307.1		2
5.2	odd	4		90.3.g.a.73.1	yes	2
5.3	odd	4	inner	450.3.g.d.343.1		2
5.4	even	2		90.3.g.a.37.1	✓	2
15.2	even	4		90.3.g.c.73.1	yes	2
15.8	even	4		450.3.g.a.343.1		2
15.14	odd	2		90.3.g.c.37.1	yes	2
20.7	even	4		720.3.bh.b.433.1		2
20.19	odd	2		720.3.bh.b.577.1		2
60.47	odd	4		720.3.bh.d.433.1		2
60.59	even	2		720.3.bh.d.577.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.3.g.a.37.1	✓	2	5.4	even	2
90.3.g.a.73.1	yes	2	5.2	odd	4
90.3.g.c.37.1	yes	2	15.14	odd	2
90.3.g.c.73.1	yes	2	15.2	even	4
450.3.g.a.307.1		2	3.2	odd	2
450.3.g.a.343.1		2	15.8	even	4
450.3.g.d.307.1		2	1.1	even	1	trivial
450.3.g.d.343.1		2	5.3	odd	4	inner
720.3.bh.b.433.1		2	20.7	even	4
720.3.bh.b.577.1		2	20.19	odd	2
720.3.bh.d.433.1		2	60.47	odd	4
720.3.bh.d.577.1		2	60.59	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	450.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +(-8.00000 - 8.00000i) q^{7} +(-2.00000 + 2.00000i) q^{8} +4.00000 q^{11} +(3.00000 - 3.00000i) q^{13} -16.0000i q^{14} -4.00000 q^{16} +(-19.0000 - 19.0000i) q^{17} -8.00000i q^{19} +(4.00000 + 4.00000i) q^{22} +(20.0000 - 20.0000i) q^{23} +6.00000 q^{26} +(16.0000 - 16.0000i) q^{28} -38.0000i q^{29} -44.0000 q^{31} +(-4.00000 - 4.00000i) q^{32} -38.0000i q^{34} +(3.00000 + 3.00000i) q^{37} +(8.00000 - 8.00000i) q^{38} +70.0000 q^{41} +(-36.0000 + 36.0000i) q^{43} +8.00000i q^{44} +40.0000 q^{46} +79.0000i q^{49} +(6.00000 + 6.00000i) q^{52} +(-17.0000 + 17.0000i) q^{53} +32.0000 q^{56} +(38.0000 - 38.0000i) q^{58} -92.0000i q^{59} +72.0000 q^{61} +(-44.0000 - 44.0000i) q^{62} -8.00000i q^{64} +(-44.0000 - 44.0000i) q^{67} +(38.0000 - 38.0000i) q^{68} -88.0000 q^{71} +(-55.0000 + 55.0000i) q^{73} +6.00000i q^{74} +16.0000 q^{76} +(-32.0000 - 32.0000i) q^{77} +12.0000i q^{79} +(70.0000 + 70.0000i) q^{82} +(24.0000 - 24.0000i) q^{83} -72.0000 q^{86} +(-8.00000 + 8.00000i) q^{88} +26.0000i q^{89} -48.0000 q^{91} +(40.0000 + 40.0000i) q^{92} +(57.0000 + 57.0000i) q^{97} +(-79.0000 + 79.0000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 16 q^{7} - 4 q^{8} + 8 q^{11} + 6 q^{13} - 8 q^{16} - 38 q^{17} + 8 q^{22} + 40 q^{23} + 12 q^{26} + 32 q^{28} - 88 q^{31} - 8 q^{32} + 6 q^{37} + 16 q^{38} + 140 q^{41} - 72 q^{43} + 80 q^{46}+ \cdots - 158 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 16 * q^7 - 4 * q^8 + 8 * q^11 + 6 * q^13 - 8 * q^16 - 38 * q^17 + 8 * q^22 + 40 * q^23 + 12 * q^26 + 32 * q^28 - 88 * q^31 - 8 * q^32 + 6 * q^37 + 16 * q^38 + 140 * q^41 - 72 * q^43 + 80 * q^46 + 12 * q^52 - 34 * q^53 + 64 * q^56 + 76 * q^58 + 144 * q^61 - 88 * q^62 - 88 * q^67 + 76 * q^68 - 176 * q^71 - 110 * q^73 + 32 * q^76 - 64 * q^77 + 140 * q^82 + 48 * q^83 - 144 * q^86 - 16 * q^88 - 96 * q^91 + 80 * q^92 + 114 * q^97 - 158 * q^98

Introduction

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