LMFDB - Embedded newform 5220.2.b.f.289.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5220,2,Mod(289,5220)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5220, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5220.289"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5220.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,48,0,0,0,0, 0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)

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

Self dual:	no
Analytic conductor:	$41.6819098551$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.13
Character	$\chi$	$=$	5220.289
Dual form			5220.2.b.f.289.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.56286 - 1.59920i) q^{5} +4.48349i q^{7} +2.24386i q^{11} -1.60476i q^{13} -5.17062 q^{17} +4.76203i q^{19} -0.697164i q^{23} +(-0.114908 - 4.99868i) q^{25} +(-1.00347 - 5.29084i) q^{29} +10.0713i q^{31} +(7.17002 + 7.00709i) q^{35} -0.559917 q^{37} +1.09060i q^{41} +8.56135 q^{43} -8.49054 q^{47} -13.1017 q^{49} -5.61954i q^{53} +(3.58838 + 3.50684i) q^{55} -4.60368 q^{59} +1.70446i q^{61} +(-2.56634 - 2.50802i) q^{65} +2.30435i q^{67} -9.73636 q^{71} -11.3099 q^{73} -10.0603 q^{77} +4.76203i q^{79} +9.92487i q^{83} +(-8.08098 + 8.26887i) q^{85} +12.6007i q^{89} +7.19493 q^{91} +(7.61546 + 7.44241i) q^{95} -13.6139 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 48 q^{25} - 104 q^{49} - 80 q^{91}+O(q^{100}) $ 24 * q + 48 * q^25 - 104 * q^49 - 80 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5220\mathbb{Z}\right)^\times$.

$n$	$901$	$2611$	$4061$	$4177$
$\chi(n)$	$-1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.56286	−	1.59920i	0.698934	−	0.715186i
$5$	1.56286	−	1.59920i	0.698934	−	0.715186i
$6$		0			0
$7$			4.48349i			1.69460i	0.531114	+	0.847300i	$0.321774\pi$
$7$			4.48349i			1.69460i	−0.531114	+	0.847300i	$0.678226\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			2.24386i			0.676548i	0.941048	+	0.338274i	$0.109843\pi$
$11$			2.24386i			0.676548i	−0.941048	+	0.338274i	$0.890157\pi$
$12$		0			0
$13$		−	1.60476i		−	0.445080i	−0.974924	−	0.222540i	$-0.928565\pi$
$13$		−	1.60476i		−	0.445080i	0.974924	−	0.222540i	$-0.0714349\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.17062			−1.25406			−0.627030	−	0.778995i	$-0.715730\pi$
$17$	−5.17062			−1.25406			−0.627030	+	0.778995i	$0.715730\pi$
$18$		0			0
$19$			4.76203i			1.09249i	0.837627	+	0.546243i	$0.183942\pi$
$19$			4.76203i			1.09249i	−0.837627	+	0.546243i	$0.816058\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	0.697164i		−	0.145369i	−0.997355	−	0.0726844i	$-0.976843\pi$
$23$		−	0.697164i		−	0.145369i	0.997355	−	0.0726844i	$-0.0231566\pi$
$24$		0			0
$25$	−0.114908	−	4.99868i	−0.0229815	−	0.999736i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.00347	−	5.29084i	−0.186340	−	0.982485i
$29$	−1.00347	−	5.29084i	−0.186340	−	0.982485i
$30$		0			0
$31$			10.0713i			1.80885i	0.426631	+	0.904426i	$0.359700\pi$
$31$			10.0713i			1.80885i	−0.426631	+	0.904426i	$0.640300\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	7.17002	+	7.00709i	1.21195	+	1.18441i
$36$		0			0
$37$	−0.559917			−0.0920497			−0.0460249	−	0.998940i	$-0.514655\pi$
$37$	−0.559917			−0.0920497			−0.0460249	+	0.998940i	$0.514655\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			1.09060i			0.170323i	0.996367	+	0.0851614i	$0.0271406\pi$
$41$			1.09060i			0.170323i	−0.996367	+	0.0851614i	$0.972859\pi$
$42$		0			0
$43$	8.56135			1.30559			0.652796	−	0.757533i	$-0.273596\pi$
$43$	8.56135			1.30559			0.652796	+	0.757533i	$0.273596\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8.49054			−1.23847			−0.619236	−	0.785205i	$-0.712558\pi$
$47$	−8.49054			−1.23847			−0.619236	+	0.785205i	$0.712558\pi$
$48$		0			0
$49$	−13.1017			−1.87167
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	5.61954i		−	0.771904i	−0.922519	−	0.385952i	$-0.873873\pi$
$53$		−	5.61954i		−	0.771904i	0.922519	−	0.385952i	$-0.126127\pi$
$54$		0			0
$55$	3.58838	+	3.50684i	0.483857	+	0.472862i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.60368			−0.599348			−0.299674	−	0.954042i	$-0.596878\pi$
$59$	−4.60368			−0.599348			−0.299674	+	0.954042i	$0.596878\pi$
$60$		0			0
$61$			1.70446i			0.218233i	0.994029	+	0.109117i	$0.0348022\pi$
$61$			1.70446i			0.218233i	−0.994029	+	0.109117i	$0.965198\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.56634	−	2.50802i	−0.318315	−	0.311082i
$66$		0			0
$67$			2.30435i			0.281521i	0.990044	+	0.140760i	$0.0449547\pi$
$67$			2.30435i			0.281521i	−0.990044	+	0.140760i	$0.955045\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−9.73636			−1.15549			−0.577747	−	0.816216i	$-0.696068\pi$
$71$	−9.73636			−1.15549			−0.577747	+	0.816216i	$0.696068\pi$
$72$		0			0
$73$	−11.3099			−1.32372			−0.661859	−	0.749628i	$-0.730232\pi$
$73$	−11.3099			−1.32372			−0.661859	+	0.749628i	$0.730232\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.0603			−1.14648
$78$		0			0
$79$			4.76203i			0.535771i	0.963451	+	0.267885i	$0.0863248\pi$
$79$			4.76203i			0.535771i	−0.963451	+	0.267885i	$0.913675\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			9.92487i			1.08940i	0.838632	+	0.544698i	$0.183356\pi$
$83$			9.92487i			1.08940i	−0.838632	+	0.544698i	$0.816644\pi$
$84$		0			0
$85$	−8.08098	+	8.26887i	−0.876505	+	0.896885i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			12.6007i			1.33568i	0.744307	+	0.667838i	$0.232780\pi$
$89$			12.6007i			1.33568i	−0.744307	+	0.667838i	$0.767220\pi$
$90$		0			0
$91$	7.19493			0.754234
$92$		0			0
$93$		0			0
$94$		0			0
$95$	7.61546	+	7.44241i	0.781330	+	0.763576i
$96$		0			0
$97$	−13.6139			−1.38228			−0.691139	−	0.722722i	$-0.742891\pi$
$97$	−13.6139			−1.38228			−0.691139	+	0.722722i	$0.742891\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	4.90049i		−	0.487617i	−0.969823	−	0.243809i	$-0.921603\pi$
$101$		−	4.90049i		−	0.487617i	0.969823	−	0.243809i	$-0.0783969\pi$
$102$		0			0
$103$		−	6.08825i		−	0.599893i	−0.953956	−	0.299947i	$-0.903031\pi$
$103$		−	6.08825i		−	0.599893i	0.953956	−	0.299947i	$-0.0969689\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	13.7268i		−	1.32702i	−0.748169	−	0.663509i	$-0.769067\pi$
$107$		−	13.7268i		−	1.32702i	0.748169	−	0.663509i	$-0.230933\pi$
$108$		0			0
$109$	11.1949			1.07228			0.536140	−	0.844129i	$-0.319882\pi$
$109$	11.1949			1.07228			0.536140	+	0.844129i	$0.319882\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	13.3802			1.25871			0.629353	−	0.777120i	$-0.283320\pi$
$113$	13.3802			1.25871			0.629353	+	0.777120i	$0.283320\pi$
$114$		0			0
$115$	−1.11491	−	1.08957i	−0.103966	−	0.101603i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	23.1824i		−	2.12513i
$120$		0			0
$121$	5.96511			0.542283
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.17349	−	7.62850i	−0.731059	−	0.682314i
$126$		0			0
$127$	−7.37717			−0.654618			−0.327309	−	0.944917i	$-0.606142\pi$
$127$	−7.37717			−0.654618			−0.327309	+	0.944917i	$0.606142\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	12.1253i		−	1.05939i	−0.848187	−	0.529696i	$-0.822306\pi$
$131$		−	12.1253i		−	1.05939i	0.848187	−	0.529696i	$-0.177694\pi$
$132$		0			0
$133$	−21.3505			−1.85133
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−18.5508			−1.58491			−0.792453	−	0.609933i	$-0.791196\pi$
$137$	−18.5508			−1.58491			−0.792453	+	0.609933i	$0.791196\pi$
$138$		0			0
$139$	−12.3315			−1.04595			−0.522973	−	0.852349i	$-0.675177\pi$
$139$	−12.3315			−1.04595			−0.522973	+	0.852349i	$0.675177\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.60085			0.301118
$144$		0			0
$145$	−10.0294	−	6.66411i	−0.832899	−	0.553425i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	17.8249			1.46028			0.730138	−	0.683300i	$-0.239456\pi$
$149$	17.8249			1.46028			0.730138	+	0.683300i	$0.239456\pi$
$150$		0			0
$151$	−17.2966			−1.40758			−0.703790	−	0.710408i	$-0.748510\pi$
$151$	−17.2966			−1.40758			−0.703790	+	0.710408i	$0.748510\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	16.1060	+	15.7400i	1.29367	+	1.26427i
$156$		0			0
$157$	4.42826			0.353413			0.176707	−	0.984264i	$-0.443456\pi$
$157$	4.42826			0.353413			0.176707	+	0.984264i	$0.443456\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.12573			0.246342
$162$		0			0
$163$	−17.5465			−1.37435			−0.687175	−	0.726492i	$-0.741150\pi$
$163$	−17.5465			−1.37435			−0.687175	+	0.726492i	$0.741150\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			12.3460i			0.955362i	0.878533	+	0.477681i	$0.158523\pi$
$167$			12.3460i			0.955362i	−0.878533	+	0.477681i	$0.841477\pi$
$168$		0			0
$169$	10.4247			0.801903
$170$		0			0
$171$		0			0
$172$		0			0
$173$			2.78866i			0.212018i	0.994365	+	0.106009i	$0.0338072\pi$
$173$			2.78866i			0.212018i	−0.994365	+	0.106009i	$0.966193\pi$
$174$		0			0
$175$	22.4115	−	0.515187i	1.69415	−	0.0389445i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	22.9576			1.71593			0.857966	−	0.513706i	$-0.171728\pi$
$179$	22.9576			1.71593			0.857966	+	0.513706i	$0.171728\pi$
$180$		0			0
$181$	−12.5613			−0.933677			−0.466838	−	0.884343i	$-0.654607\pi$
$181$	−12.5613			−0.933677			−0.466838	+	0.884343i	$0.654607\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.875074	+	0.895421i	−0.0643367	+	0.0658327i
$186$		0			0
$187$		−	11.6021i		−	0.848431i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			23.1198i			1.67289i	0.548052	+	0.836444i	$0.315369\pi$
$191$			23.1198i			1.67289i	−0.548052	+	0.836444i	$0.684631\pi$
$192$		0			0
$193$	−0.760326			−0.0547294			−0.0273647	−	0.999626i	$-0.508712\pi$
$193$	−0.760326			−0.0547294			−0.0273647	+	0.999626i	$0.508712\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	9.01170i		−	0.642057i	−0.947070	−	0.321028i	$-0.895972\pi$
$197$		−	9.01170i		−	0.642057i	0.947070	−	0.321028i	$-0.104028\pi$
$198$		0			0
$199$	−14.1017			−0.999643			−0.499822	−	0.866128i	$-0.666601\pi$
$199$	−14.1017			−0.999643			−0.499822	+	0.866128i	$0.666601\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	23.7215	−	4.49907i	1.66492	−	0.315773i
$204$		0			0
$205$	1.74409	+	1.70446i	0.121812	+	0.119044i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−10.6853			−0.739118
$210$		0			0
$211$			6.46649i			0.445172i	0.974913	+	0.222586i	$0.0714497\pi$
$211$			6.46649i			0.445172i	−0.974913	+	0.222586i	$0.928550\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	13.3802	−	13.6913i	0.912524	−	0.933742i
$216$		0			0
$217$	−45.1544			−3.06528
$218$		0			0
$219$		0			0
$220$		0			0
$221$			8.29760i			0.558157i
$222$		0			0
$223$		−	3.33472i		−	0.223309i	−0.993747	−	0.111655i	$-0.964385\pi$
$223$		−	3.33472i		−	0.223309i	0.993747	−	0.111655i	$-0.0356151\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	11.3192i		−	0.751281i	−0.926765	−	0.375641i	$-0.877423\pi$
$227$		−	11.3192i		−	0.751281i	0.926765	−	0.375641i	$-0.122577\pi$
$228$		0			0
$229$			26.0618i			1.72221i	0.508424	+	0.861107i	$0.330228\pi$
$229$			26.0618i			1.72221i	−0.508424	+	0.861107i	$0.669772\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			23.8454i			1.56216i	0.624429	+	0.781082i	$0.285332\pi$
$233$			23.8454i			1.56216i	−0.624429	+	0.781082i	$0.714668\pi$
$234$		0			0
$235$	−13.2696	+	13.5781i	−0.865611	+	0.885738i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−10.3261			−0.667943			−0.333971	−	0.942583i	$-0.608389\pi$
$239$	−10.3261			−0.667943			−0.333971	+	0.942583i	$0.608389\pi$
$240$		0			0
$241$	8.96511			0.577494			0.288747	−	0.957405i	$-0.406761\pi$
$241$	8.96511			0.577494			0.288747	+	0.957405i	$0.406761\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−20.4762	+	20.9523i	−1.30818	+	1.33859i
$246$		0			0
$247$	7.64192			0.486244
$248$		0			0
$249$		0			0
$250$		0			0
$251$			2.75953i			0.174180i	0.996200	+	0.0870899i	$0.0277567\pi$
$251$			2.75953i			0.174180i	−0.996200	+	0.0870899i	$0.972243\pi$
$252$		0			0
$253$	1.56433			0.0983489
$254$		0			0
$255$		0			0
$256$		0			0
$257$			9.24126i			0.576454i	0.957562	+	0.288227i	$0.0930658\pi$
$257$			9.24126i			0.576454i	−0.957562	+	0.288227i	$0.906934\pi$
$258$		0			0
$259$		−	2.51038i		−	0.155988i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−16.9811			−1.04710			−0.523549	−	0.851996i	$-0.675392\pi$
$263$	−16.9811			−1.04710			−0.523549	+	0.851996i	$0.675392\pi$
$264$		0			0
$265$	−8.98680	−	8.78259i	−0.552055	−	0.539510i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	18.8793i		−	1.15109i	−0.817770	−	0.575545i	$-0.804790\pi$
$269$		−	18.8793i		−	1.15109i	0.817770	−	0.575545i	$-0.195210\pi$
$270$		0			0
$271$		−	13.6760i		−	0.830760i	−0.909648	−	0.415380i	$-0.863649\pi$
$271$		−	13.6760i		−	0.830760i	0.909648	−	0.415380i	$-0.136351\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	11.2163	−	0.257836i	0.676369	−	0.0155481i
$276$		0			0
$277$		−	11.6021i		−	0.697104i	−0.937289	−	0.348552i	$-0.886674\pi$
$277$		−	11.6021i		−	0.697104i	0.937289	−	0.348552i	$-0.113326\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	15.8180			0.943622			0.471811	−	0.881700i	$-0.343600\pi$
$281$	15.8180			0.943622			0.471811	+	0.881700i	$0.343600\pi$
$282$		0			0
$283$			10.6969i			0.635867i	0.948113	+	0.317934i	$0.102989\pi$
$283$			10.6969i			0.635867i	−0.948113	+	0.317934i	$0.897011\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4.88969			−0.288629
$288$		0			0
$289$	9.73530			0.572665
$290$		0			0
$291$		0			0
$292$		0			0
$293$	17.0816			0.997919			0.498960	−	0.866625i	$-0.333716\pi$
$293$	17.0816			0.997919			0.498960	+	0.866625i	$0.333716\pi$
$294$		0			0
$295$	−7.19493	+	7.36222i	−0.418905	+	0.428645i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.11878			−0.0647008
$300$		0			0
$301$			38.3847i			2.21246i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.72577	+	2.66384i	0.156077	+	0.152531i
$306$		0			0
$307$	−5.29661			−0.302293			−0.151147	−	0.988511i	$-0.548297\pi$
$307$	−5.29661			−0.302293			−0.151147	+	0.988511i	$0.548297\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			19.4352i			1.10207i	0.834483	+	0.551034i	$0.185767\pi$
$311$			19.4352i			1.10207i	−0.834483	+	0.551034i	$0.814233\pi$
$312$		0			0
$313$			6.21345i			0.351205i	0.984461	+	0.175603i	$0.0561874\pi$
$313$			6.21345i			0.351205i	−0.984461	+	0.175603i	$0.943813\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.280931			−0.0157787			−0.00788933	−	0.999969i	$-0.502511\pi$
$317$	−0.280931			−0.0157787			−0.00788933	+	0.999969i	$0.502511\pi$
$318$		0			0
$319$	11.8719	−	2.25165i	0.664698	−	0.126068i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	24.6227i		−	1.37004i
$324$		0			0
$325$	−8.02168	+	0.184399i	−0.444963	+	0.0102286i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	38.0673i		−	2.09872i
$330$		0			0
$331$			16.1864i			0.889686i	0.895609	+	0.444843i	$0.146740\pi$
$331$			16.1864i			0.889686i	−0.895609	+	0.444843i	$0.853260\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3.68512	+	3.60138i	0.201340	+	0.196764i
$336$		0			0
$337$	12.0702			0.657504			0.328752	−	0.944416i	$-0.393372\pi$
$337$	12.0702			0.657504			0.328752	+	0.944416i	$0.393372\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−22.5984			−1.22377
$342$		0			0
$343$		−	27.3569i		−	1.47714i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			27.5185i			1.47727i	0.674104	+	0.738636i	$0.264530\pi$
$347$			27.5185i			1.47727i	−0.674104	+	0.738636i	$0.735470\pi$
$348$		0			0
$349$	−28.2034			−1.50969			−0.754847	−	0.655901i	$-0.772289\pi$
$349$	−28.2034			−1.50969			−0.754847	+	0.655901i	$0.772289\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	22.2286i		−	1.18311i	−0.806264	−	0.591556i	$-0.798514\pi$
$353$		−	22.2286i		−	1.18311i	0.806264	−	0.591556i	$-0.201486\pi$
$354$		0			0
$355$	−15.2166	+	15.5704i	−0.807614	+	0.826392i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	15.1096i		−	0.797456i	−0.917069	−	0.398728i	$-0.869452\pi$
$359$		−	15.1096i		−	0.797456i	0.917069	−	0.398728i	$-0.130548\pi$
$360$		0			0
$361$	−3.67696			−0.193524
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−17.6758	+	18.0868i	−0.925192	+	0.946704i
$366$		0			0
$367$	30.7159			1.60336			0.801678	−	0.597756i	$-0.203941\pi$
$367$	30.7159			1.60336			0.801678	+	0.597756i	$0.203941\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	25.1952			1.30807
$372$		0			0
$373$			12.5453i			0.649571i	0.945788	+	0.324786i	$0.105292\pi$
$373$			12.5453i			0.649571i	−0.945788	+	0.324786i	$0.894708\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−8.49054	+	1.61034i	−0.437285	+	0.0829365i
$378$		0			0
$379$			13.4802i			0.692430i	0.938155	+	0.346215i	$0.112533\pi$
$379$			13.4802i			0.692430i	−0.938155	+	0.346215i	$0.887467\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			11.8805i			0.607063i	0.952821	+	0.303532i	$0.0981658\pi$
$383$			11.8805i			0.607063i	−0.952821	+	0.303532i	$0.901834\pi$
$384$		0			0
$385$	−15.7229	+	16.0885i	−0.801313	+	0.819945i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			0.928453i			0.0470744i	0.999723	+	0.0235372i	$0.00749282\pi$
$389$			0.928453i			0.0470744i	−0.999723	+	0.0235372i	$0.992507\pi$
$390$		0			0
$391$			3.60477i			0.182301i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	7.61546	+	7.44241i	0.383176	+	0.374468i
$396$		0			0
$397$		−	25.3834i		−	1.27396i	−0.770882	−	0.636978i	$-0.780184\pi$
$397$		−	25.3834i		−	1.27396i	0.770882	−	0.636978i	$-0.219816\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	35.4605			1.77081			0.885407	−	0.464816i	$-0.153880\pi$
$401$	35.4605			1.77081			0.885407	+	0.464816i	$0.153880\pi$
$402$		0			0
$403$	16.1620			0.805084
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	1.25637i		−	0.0622760i
$408$		0			0
$409$			38.3847i			1.89800i	0.315272	+	0.949001i	$0.397904\pi$
$409$			38.3847i			1.89800i	−0.315272	+	0.949001i	$0.602096\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	20.6406i		−	1.01566i
$414$		0			0
$415$	15.8719	+	15.5112i	0.779120	+	0.761416i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−36.0503			−1.76117			−0.880587	−	0.473885i	$-0.842851\pi$
$419$	−36.0503			−1.76117			−0.880587	+	0.473885i	$0.842851\pi$
$420$		0			0
$421$			25.4517i			1.24044i	0.784427	+	0.620221i	$0.212957\pi$
$421$			25.4517i			1.24044i	−0.784427	+	0.620221i	$0.787043\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.594143	+	25.8463i	0.0288202	+	1.25373i
$426$		0			0
$427$	−7.64192			−0.369818
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−23.4866			−1.13131			−0.565655	−	0.824642i	$-0.691377\pi$
$431$	−23.4866			−1.13131			−0.565655	+	0.824642i	$0.691377\pi$
$432$		0			0
$433$	−22.7145			−1.09159			−0.545794	−	0.837920i	$-0.683772\pi$
$433$	−22.7145			−1.09159			−0.545794	+	0.837920i	$0.683772\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.31992			0.158813
$438$		0			0
$439$	19.2383			0.918194			0.459097	−	0.888386i	$-0.348173\pi$
$439$	19.2383			0.918194			0.459097	+	0.888386i	$0.348173\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−32.3126			−1.53522			−0.767608	−	0.640920i	$-0.778553\pi$
$443$	−32.3126			−1.53522			−0.767608	+	0.640920i	$0.778553\pi$
$444$		0			0
$445$	20.1512	+	19.6933i	0.955256	+	0.933550i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	1.37806i		−	0.0650347i	−0.999471	−	0.0325174i	$-0.989648\pi$
$449$		−	1.37806i		−	0.0650347i	0.999471	−	0.0325174i	$-0.0103524\pi$
$450$		0			0
$451$	−2.44714			−0.115232
$452$		0			0
$453$		0			0
$454$		0			0
$455$	11.2447	−	11.5062i	0.527160	−	0.539417i
$456$		0			0
$457$			2.63513i			0.123266i	0.998099	+	0.0616332i	$0.0196309\pi$
$457$			2.63513i			0.123266i	−0.998099	+	0.0616332i	$0.980369\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	1.21592i		−	0.0566309i	−0.999599	−	0.0283154i	$-0.990986\pi$
$461$		−	1.21592i		−	0.0566309i	0.999599	−	0.0283154i	$-0.00901428\pi$
$462$		0			0
$463$			27.8061i			1.29226i	0.763227	+	0.646130i	$0.223614\pi$
$463$			27.8061i			1.29226i	−0.763227	+	0.646130i	$0.776386\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	21.8708			1.01206			0.506029	−	0.862516i	$-0.331113\pi$
$467$	21.8708			1.01206			0.506029	+	0.862516i	$0.331113\pi$
$468$		0			0
$469$	−10.3315			−0.477065
$470$		0			0
$471$		0			0
$472$		0			0
$473$			19.2104i			0.883296i
$474$		0			0
$475$	23.8039	−	0.547194i	1.09220	−	0.0251070i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	0.740476i		−	0.0338332i	−0.999857	−	0.0169166i	$-0.994615\pi$
$479$		−	0.740476i		−	0.0338332i	0.999857	−	0.0169166i	$-0.00538498\pi$
$480$		0			0
$481$			0.898532i			0.0409695i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−21.2766	+	21.7713i	−0.966121	+	0.988586i
$486$		0			0
$487$			22.8735i			1.03649i	0.855231	+	0.518247i	$0.173415\pi$
$487$			22.8735i			1.03649i	−0.855231	+	0.518247i	$0.826585\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	5.53808i		−	0.249930i	−0.992161	−	0.124965i	$-0.960118\pi$
$491$		−	5.53808i		−	0.249930i	0.992161	−	0.124965i	$-0.0398818\pi$
$492$		0			0
$493$	5.18858	+	27.3569i	0.233682	+	1.23209i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	43.6529i		−	1.95810i
$498$		0			0
$499$	31.3983			1.40558			0.702791	−	0.711396i	$-0.251937\pi$
$499$	31.3983			1.40558			0.702791	+	0.711396i	$0.251937\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−35.8128			−1.59682			−0.798408	−	0.602117i	$-0.794324\pi$
$503$	−35.8128			−1.59682			−0.798408	+	0.602117i	$0.794324\pi$
$504$		0			0
$505$	−7.83689	−	7.65880i	−0.348737	−	0.340812i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	23.1274			1.02511			0.512553	−	0.858656i	$-0.328700\pi$
$509$	23.1274			1.02511			0.512553	+	0.858656i	$0.328700\pi$
$510$		0			0
$511$		−	50.7076i		−	2.24317i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−9.73636	−	9.51512i	−0.429035	−	0.419286i
$516$		0			0
$517$		−	19.0515i		−	0.837885i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−19.8319			−0.868851			−0.434425	−	0.900708i	$-0.643049\pi$
$521$	−19.8319			−0.868851			−0.434425	+	0.900708i	$0.643049\pi$
$522$		0			0
$523$		−	35.6243i		−	1.55774i	−0.627183	−	0.778872i	$-0.715792\pi$
$523$		−	35.6243i		−	1.55774i	0.627183	−	0.778872i	$-0.284208\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	52.0746i		−	2.26841i
$528$		0			0
$529$	22.5140			0.978868
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.75015			0.0758073
$534$		0			0
$535$	−21.9519	−	21.4531i	−0.949064	−	0.927498i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	29.3983i		−	1.26628i
$540$		0			0
$541$			26.0618i			1.12048i	0.828329	+	0.560242i	$0.189292\pi$
$541$			26.0618i			1.12048i	−0.828329	+	0.560242i	$0.810708\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	17.4962	−	17.9030i	0.749453	−	0.766879i
$546$		0			0
$547$		−	2.67314i		−	0.114295i	−0.998366	−	0.0571477i	$-0.981799\pi$
$547$		−	2.67314i		−	0.114295i	0.998366	−	0.0571477i	$-0.0182006\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	25.1952	−	4.77858i	1.07335	−	0.203574i
$552$		0			0
$553$	−21.3505			−0.907917
$554$		0			0
$555$		0			0
$556$		0			0
$557$			27.3978i			1.16088i	0.814303	+	0.580440i	$0.197119\pi$
$557$			27.3978i			1.16088i	−0.814303	+	0.580440i	$0.802881\pi$
$558$		0			0
$559$		−	13.7389i		−	0.581094i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7.30225			0.307753			0.153877	−	0.988090i	$-0.450824\pi$
$563$	7.30225			0.307753			0.153877	+	0.988090i	$0.450824\pi$
$564$		0			0
$565$	20.9115	−	21.3977i	0.879753	−	0.900208i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	33.7641i		−	1.41547i	−0.706480	−	0.707733i	$-0.749718\pi$
$569$		−	33.7641i		−	1.41547i	0.706480	−	0.707733i	$-0.250282\pi$
$570$		0			0
$571$	11.3664			0.475669			0.237835	−	0.971306i	$-0.423562\pi$
$571$	11.3664			0.475669			0.237835	+	0.971306i	$0.423562\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.48490	+	0.0801094i	−0.145330	+	0.00334079i
$576$		0			0
$577$	−0.760326			−0.0316528			−0.0158264	−	0.999875i	$-0.505038\pi$
$577$	−0.760326			−0.0316528			−0.0158264	+	0.999875i	$0.505038\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−44.4981			−1.84609
$582$		0			0
$583$	12.6094			0.522230
$584$		0			0
$585$		0			0
$586$		0			0
$587$			31.7924i			1.31221i	0.754668	+	0.656106i	$0.227798\pi$
$587$			31.7924i			1.31221i	−0.754668	+	0.656106i	$0.772202\pi$
$588$		0			0
$589$	−47.9597			−1.97614
$590$		0			0
$591$		0			0
$592$		0			0
$593$			45.3103i			1.86067i	0.366709	+	0.930336i	$0.380484\pi$
$593$			45.3103i			1.86067i	−0.366709	+	0.930336i	$0.619516\pi$
$594$		0			0
$595$	−37.0734	−	36.2310i	−1.51986	−	1.48533i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	9.03808i		−	0.369286i	−0.982806	−	0.184643i	$-0.940887\pi$
$599$		−	9.03808i		−	0.369286i	0.982806	−	0.184643i	$-0.0591129\pi$
$600$		0			0
$601$		−	27.1562i		−	1.10772i	−0.832608	−	0.553862i	$-0.813153\pi$
$601$		−	27.1562i		−	1.10772i	0.832608	−	0.553862i	$-0.186847\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	9.32267	−	9.53944i	0.379020	−	0.387833i
$606$		0			0
$607$	−32.6604			−1.32564			−0.662822	−	0.748777i	$-0.730641\pi$
$607$	−32.6604			−1.32564			−0.662822	+	0.748777i	$0.730641\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			13.6253i			0.551220i
$612$		0			0
$613$		−	10.4533i		−	0.422207i	−0.977464	−	0.211103i	$-0.932294\pi$
$613$		−	10.4533i		−	0.422207i	0.977464	−	0.211103i	$-0.0677057\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	10.3412			0.416323			0.208161	−	0.978095i	$-0.433252\pi$
$617$	10.3412			0.416323			0.208161	+	0.978095i	$0.433252\pi$
$618$		0			0
$619$		−	2.86172i		−	0.115022i	−0.998345	−	0.0575111i	$-0.981684\pi$
$619$		−	2.86172i		−	0.115022i	0.998345	−	0.0575111i	$-0.0183165\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−56.4953			−2.26344
$624$		0			0
$625$	−24.9736	+	1.14877i	−0.998944	+	0.0459509i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.89512			0.115436
$630$		0			0
$631$	−15.6546			−0.623198			−0.311599	−	0.950214i	$-0.600865\pi$
$631$	−15.6546			−0.623198			−0.311599	+	0.950214i	$0.600865\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−11.5295	+	11.7976i	−0.457535	+	0.468174i
$636$		0			0
$637$			21.0251i			0.833044i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			14.0012i			0.553015i	0.961012	+	0.276508i	$0.0891771\pi$
$641$			14.0012i			0.553015i	−0.961012	+	0.276508i	$0.910823\pi$
$642$		0			0
$643$		−	22.4175i		−	0.884058i	−0.897001	−	0.442029i	$-0.854259\pi$
$643$		−	22.4175i		−	0.884058i	0.897001	−	0.442029i	$-0.145741\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			33.7838i			1.32818i	0.747653	+	0.664089i	$0.231181\pi$
$647$			33.7838i			1.32818i	−0.747653	+	0.664089i	$0.768819\pi$
$648$		0			0
$649$		−	10.3300i		−	0.405488i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	7.92867			0.310273			0.155137	−	0.987893i	$-0.450418\pi$
$653$	7.92867			0.310273			0.155137	+	0.987893i	$0.450418\pi$
$654$		0			0
$655$	−19.3908	−	18.9502i	−0.757662	−	0.740446i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	25.7137i		−	1.00167i	−0.865544	−	0.500833i	$-0.833027\pi$
$659$		−	25.7137i		−	1.00167i	0.865544	−	0.500833i	$-0.166973\pi$
$660$		0			0
$661$	19.6421			0.763988			0.381994	−	0.924165i	$-0.375237\pi$
$661$	19.6421			0.763988			0.381994	+	0.924165i	$0.375237\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−33.3680	+	34.1439i	−1.29396	+	1.32404i
$666$		0			0
$667$	−3.68859	+	0.699586i	−0.142823	+	0.0270881i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.82455			−0.147645
$672$		0			0
$673$		−	20.5691i		−	0.792881i	−0.918060	−	0.396440i	$-0.870245\pi$
$673$		−	20.5691i		−	0.792881i	0.918060	−	0.396440i	$-0.129755\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	34.0627			1.30914			0.654568	−	0.756003i	$-0.272851\pi$
$677$	34.0627			1.30914			0.654568	+	0.756003i	$0.272851\pi$
$678$		0			0
$679$		−	61.0376i		−	2.34241i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	14.5511i		−	0.556783i	−0.960468	−	0.278392i	$-0.910199\pi$
$683$		−	14.5511i		−	0.556783i	0.960468	−	0.278392i	$-0.0898013\pi$
$684$		0			0
$685$	−28.9925	+	29.6666i	−1.10774	+	1.13350i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−9.01802			−0.343559
$690$		0			0
$691$	−14.2757			−0.543072			−0.271536	−	0.962428i	$-0.587532\pi$
$691$	−14.2757			−0.543072			−0.271536	+	0.962428i	$0.587532\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−19.2725	+	19.7206i	−0.731047	+	0.748046i
$696$		0			0
$697$		−	5.63907i		−	0.213595i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	12.6923			0.479380			0.239690	−	0.970849i	$-0.422954\pi$
$701$	12.6923			0.479380			0.239690	+	0.970849i	$0.422954\pi$
$702$		0			0
$703$		−	2.66634i		−	0.100563i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	21.9713			0.826316
$708$		0			0
$709$	29.1227			1.09372			0.546862	−	0.837223i	$-0.315822\pi$
$709$	29.1227			1.09372			0.546862	+	0.837223i	$0.315822\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	7.02132			0.262950
$714$		0			0
$715$	5.62764	−	5.75849i	0.210462	−	0.215355i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	44.6071			1.66356			0.831782	−	0.555102i	$-0.187321\pi$
$719$	44.6071			1.66356			0.831782	+	0.555102i	$0.187321\pi$
$720$		0			0
$721$	27.2966			1.01658
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−26.3319	+	5.62400i	−0.977943	+	0.208870i
$726$		0			0
$727$	7.30544			0.270944			0.135472	−	0.990781i	$-0.456745\pi$
$727$	7.30544			0.270944			0.135472	+	0.990781i	$0.456745\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−44.2675			−1.63729
$732$		0			0
$733$	23.0436			0.851133			0.425566	−	0.904927i	$-0.360075\pi$
$733$	23.0436			0.851133			0.425566	+	0.904927i	$0.360075\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.17062			−0.190462
$738$		0			0
$739$		−	15.3805i		−	0.565780i	−0.959152	−	0.282890i	$-0.908707\pi$
$739$		−	15.3805i		−	0.565780i	0.959152	−	0.282890i	$-0.0912932\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−6.74039			−0.247281			−0.123640	−	0.992327i	$-0.539457\pi$
$743$	−6.74039			−0.247281			−0.123640	+	0.992327i	$0.539457\pi$
$744$		0			0
$745$	27.8580	−	28.5057i	1.02064	−	1.04437i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	61.5439			2.24876
$750$		0			0
$751$		−	50.1604i		−	1.83038i	−0.403022	−	0.915190i	$-0.632040\pi$
$751$		−	50.1604i		−	1.83038i	0.403022	−	0.915190i	$-0.367960\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−27.0323	+	27.6608i	−0.983806	+	1.00668i
$756$		0			0
$757$	6.50143			0.236299			0.118149	−	0.992996i	$-0.462304\pi$
$757$	6.50143			0.236299			0.118149	+	0.992996i	$0.462304\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.759611			−0.0275359			−0.0137679	−	0.999905i	$-0.504383\pi$
$761$	−0.759611			−0.0275359			−0.0137679	+	0.999905i	$0.504383\pi$
$762$		0			0
$763$			50.1924i			1.81709i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			7.38780i			0.266758i
$768$		0			0
$769$			0.610070i			0.0219997i	0.999939	+	0.0109998i	$0.00350143\pi$
$769$			0.610070i			0.0219997i	−0.999939	+	0.0109998i	$0.996499\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−14.8494			−0.534097			−0.267049	−	0.963683i	$-0.586048\pi$
$773$	−14.8494			−0.534097			−0.267049	+	0.963683i	$0.586048\pi$
$774$		0			0
$775$	50.3430	−	1.15726i	1.80837	−	0.0415701i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−5.19346			−0.186075
$780$		0			0
$781$		−	21.8470i		−	0.781746i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	6.92077	−	7.08169i	0.247013	−	0.252756i
$786$		0			0
$787$		−	5.42668i		−	0.193440i	−0.995312	−	0.0967201i	$-0.969165\pi$
$787$		−	5.42668i		−	0.193440i	0.995312	−	0.0967201i	$-0.0308352\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			59.9901i			2.13300i
$792$		0			0
$793$	2.73524			0.0971314
$794$		0			0
$795$		0			0
$796$		0			0
$797$	27.3223			0.967806			0.483903	−	0.875122i	$-0.339219\pi$
$797$	27.3223			0.967806			0.483903	+	0.875122i	$0.339219\pi$
$798$		0			0
$799$	43.9013			1.55312
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	25.3777i		−	0.895558i
$804$		0			0
$805$	4.88509	−	4.99868i	0.172177	−	0.176180i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	43.1299i		−	1.51637i	−0.652042	−	0.758183i	$-0.726087\pi$
$809$		−	43.1299i		−	1.51637i	0.652042	−	0.758183i	$-0.273913\pi$
$810$		0			0
$811$	−20.7911			−0.730076			−0.365038	−	0.930993i	$-0.618944\pi$
$811$	−20.7911			−0.730076			−0.365038	+	0.930993i	$0.618944\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−27.4229	+	28.0605i	−0.960581	+	0.982916i
$816$		0			0
$817$			40.7694i			1.42634i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−9.20736			−0.321339			−0.160670	−	0.987008i	$-0.551365\pi$
$821$	−9.20736			−0.321339			−0.160670	+	0.987008i	$0.551365\pi$
$822$		0			0
$823$	10.0407			0.349996			0.174998	−	0.984569i	$-0.444008\pi$
$823$	10.0407			0.349996			0.174998	+	0.984569i	$0.444008\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	25.5722			0.889231			0.444616	−	0.895721i	$-0.353340\pi$
$827$	25.5722			0.889231			0.444616	+	0.895721i	$0.353340\pi$
$828$		0			0
$829$			39.1907i			1.36115i	0.732680	+	0.680574i	$0.238269\pi$
$829$			39.1907i			1.36115i	−0.732680	+	0.680574i	$0.761731\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	67.7439			2.34719
$834$		0			0
$835$	19.7438	+	19.2951i	0.683262	+	0.667736i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	23.9229i		−	0.825910i	−0.910751	−	0.412955i	$-0.864497\pi$
$839$		−	23.9229i		−	0.825910i	0.910751	−	0.412955i	$-0.135503\pi$
$840$		0			0
$841$	−26.9861	+	10.6185i	−0.930554	+	0.366154i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	16.2925	−	16.6713i	0.560478	−	0.573510i
$846$		0			0
$847$			26.7445i			0.918953i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			0.390354i			0.0133812i
$852$		0			0
$853$	29.8912			1.02346			0.511728	−	0.859147i	$-0.329006\pi$
$853$	29.8912			1.02346			0.511728	+	0.859147i	$0.329006\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			25.9883i			0.887744i	0.896090	+	0.443872i	$0.146395\pi$
$857$			25.9883i			0.887744i	−0.896090	+	0.443872i	$0.853605\pi$
$858$		0			0
$859$		−	34.2328i		−	1.16801i	−0.811751	−	0.584003i	$-0.801485\pi$
$859$		−	34.2328i		−	1.16801i	0.811751	−	0.584003i	$-0.198515\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	22.3087i		−	0.759399i	−0.925110	−	0.379699i	$-0.876027\pi$
$863$		−	22.3087i		−	0.759399i	0.925110	−	0.379699i	$-0.123973\pi$
$864$		0			0
$865$	4.45963	+	4.35829i	0.151632	+	0.148186i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−10.6853			−0.362474
$870$		0			0
$871$	3.69792			0.125299
$872$		0			0
$873$		0			0
$874$		0			0
$875$	34.2023	−	36.6458i	1.15625	−	1.23885i
$876$		0			0
$877$			49.3252i			1.66559i	0.553578	+	0.832797i	$0.313262\pi$
$877$			49.3252i			1.66559i	−0.553578	+	0.832797i	$0.686738\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	7.59736i		−	0.255962i	−0.991777	−	0.127981i	$-0.959150\pi$
$881$		−	7.59736i		−	0.255962i	0.991777	−	0.127981i	$-0.0408496\pi$
$882$		0			0
$883$			4.02750i			0.135536i	0.997701	+	0.0677682i	$0.0215878\pi$
$883$			4.02750i			0.135536i	−0.997701	+	0.0677682i	$0.978412\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	3.60085			0.120905			0.0604523	−	0.998171i	$-0.480746\pi$
$887$	3.60085			0.120905			0.0604523	+	0.998171i	$0.480746\pi$
$888$		0			0
$889$		−	33.0755i		−	1.10932i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	40.4322i		−	1.35301i
$894$		0			0
$895$	35.8796	−	36.7139i	1.19932	−	1.22721i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	53.2855	−	10.1062i	1.77717	−	0.337062i
$900$		0			0
$901$			29.0565i			0.968013i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−19.6317	+	20.0881i	−0.652579	+	0.667752i
$906$		0			0
$907$	51.6632			1.71545			0.857725	−	0.514109i	$-0.171877\pi$
$907$	51.6632			1.71545			0.857725	+	0.514109i	$0.171877\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	0.453013i		−	0.0150090i	−0.999972	−	0.00750449i	$-0.997611\pi$
$911$		−	0.453013i		−	0.0150090i	0.999972	−	0.00750449i	$-0.00238878\pi$
$912$		0			0
$913$	−22.2700			−0.737028
$914$		0			0
$915$		0			0
$916$		0			0
$917$	54.3637			1.79525
$918$		0			0
$919$	2.34401			0.0773216			0.0386608	−	0.999252i	$-0.487691\pi$
$919$	2.34401			0.0773216			0.0386608	+	0.999252i	$0.487691\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			15.6245i			0.514287i
$924$		0			0
$925$	0.0643387	+	2.79884i	0.00211544	+	0.0920254i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−2.23756			−0.0734120			−0.0367060	−	0.999326i	$-0.511687\pi$
$929$	−2.23756			−0.0734120			−0.0367060	+	0.999326i	$0.511687\pi$
$930$		0			0
$931$		−	62.3907i		−	2.04477i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−18.5542	−	18.1325i	−0.606786	−	0.592998i
$936$		0			0
$937$			27.7257i			0.905760i	0.891571	+	0.452880i	$0.149603\pi$
$937$			27.7257i			0.905760i	−0.891571	+	0.452880i	$0.850397\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−42.8308			−1.39624			−0.698122	−	0.715979i	$-0.745981\pi$
$941$	−42.8308			−1.39624			−0.698122	+	0.715979i	$0.745981\pi$
$942$		0			0
$943$	0.760326			0.0247596
$944$		0			0
$945$		0			0
$946$		0			0
$947$	23.1596			0.752586			0.376293	−	0.926501i	$-0.377199\pi$
$947$	23.1596			0.752586			0.376293	+	0.926501i	$0.377199\pi$
$948$		0			0
$949$			18.1496i			0.589161i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	20.6892i		−	0.670189i	−0.942184	−	0.335095i	$-0.891232\pi$
$953$		−	20.6892i		−	0.670189i	0.942184	−	0.335095i	$-0.108768\pi$
$954$		0			0
$955$	36.9732	+	36.1331i	1.19643	+	1.16924i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	83.1726i		−	2.68578i
$960$		0			0
$961$	−70.4303			−2.27194
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.18829	+	1.21592i	−0.0382523	+	0.0391417i
$966$		0			0
$967$	17.7056			0.569375			0.284687	−	0.958620i	$-0.408110\pi$
$967$	17.7056			0.569375			0.284687	+	0.958620i	$0.408110\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			56.5160i			1.81368i	0.421471	+	0.906842i	$0.361514\pi$
$971$			56.5160i			1.81368i	−0.421471	+	0.906842i	$0.638486\pi$
$972$		0			0
$973$		−	55.2883i		−	1.77246i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	9.17191i		−	0.293436i	−0.989178	−	0.146718i	$-0.953129\pi$
$977$		−	9.17191i		−	0.293436i	0.989178	−	0.146718i	$-0.0468709\pi$
$978$		0			0
$979$	−28.2742			−0.903648
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−46.2546			−1.47529			−0.737647	−	0.675187i	$-0.764063\pi$
$983$	−46.2546			−1.47529			−0.737647	+	0.675187i	$0.764063\pi$
$984$		0			0
$985$	−14.4115	−	14.0841i	−0.459190	−	0.448756i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	5.96866i		−	0.189792i
$990$		0			0
$991$	0.171474			0.00544706			0.00272353	−	0.999996i	$-0.499133\pi$
$991$	0.171474			0.00544706			0.00272353	+	0.999996i	$0.499133\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−22.0391	+	22.5515i	−0.698685	+	0.714931i
$996$		0			0
$997$	−20.8550			−0.660483			−0.330242	−	0.943896i	$-0.607130\pi$
$997$	−20.8550			−0.660483			−0.330242	+	0.943896i	$0.607130\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5220.2.b.f.289.13	yes	24
3.2	odd	2	inner	5220.2.b.f.289.12	yes	24
5.4	even	2	inner	5220.2.b.f.289.16	yes	24
15.14	odd	2	inner	5220.2.b.f.289.9	✓	24
29.28	even	2	inner	5220.2.b.f.289.14	yes	24
87.86	odd	2	inner	5220.2.b.f.289.11	yes	24
145.144	even	2	inner	5220.2.b.f.289.15	yes	24
435.434	odd	2	inner	5220.2.b.f.289.10	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5220.2.b.f.289.9	✓	24	15.14	odd	2	inner
5220.2.b.f.289.10	yes	24	435.434	odd	2	inner
5220.2.b.f.289.11	yes	24	87.86	odd	2	inner
5220.2.b.f.289.12	yes	24	3.2	odd	2	inner
5220.2.b.f.289.13	yes	24	1.1	even	1	trivial
5220.2.b.f.289.14	yes	24	29.28	even	2	inner
5220.2.b.f.289.15	yes	24	145.144	even	2	inner
5220.2.b.f.289.16	yes	24	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5220.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.56286 - 1.59920i) q^{5} +4.48349i q^{7} +2.24386i q^{11} -1.60476i q^{13} -5.17062 q^{17} +4.76203i q^{19} -0.697164i q^{23} +(-0.114908 - 4.99868i) q^{25} +(-1.00347 - 5.29084i) q^{29} +10.0713i q^{31} +(7.17002 + 7.00709i) q^{35} -0.559917 q^{37} +1.09060i q^{41} +8.56135 q^{43} -8.49054 q^{47} -13.1017 q^{49} -5.61954i q^{53} +(3.58838 + 3.50684i) q^{55} -4.60368 q^{59} +1.70446i q^{61} +(-2.56634 - 2.50802i) q^{65} +2.30435i q^{67} -9.73636 q^{71} -11.3099 q^{73} -10.0603 q^{77} +4.76203i q^{79} +9.92487i q^{83} +(-8.08098 + 8.26887i) q^{85} +12.6007i q^{89} +7.19493 q^{91} +(7.61546 + 7.44241i) q^{95} -13.6139 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 48 q^{25} - 104 q^{49} - 80 q^{91}+O(q^{100}) \) 24 * q + 48 * q^25 - 104 * q^49 - 80 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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