LMFDB - Embedded newform 8820.2.a.bn.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8820,2,Mod(1,8820)] 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("8820.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8820, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8820.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,-3,0,0,0,0,0,-2,0,-5,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$70.4280545828$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 1260)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.86620$ of defining polynomial
Character	$\chi$	$=$	8820.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +2.38350 q^{11} +1.38350 q^{13} -0.831590 q^{17} -4.21509 q^{19} -5.59859 q^{23} +1.00000 q^{25} +10.0467 q^{29} -1.00000 q^{31} +4.59859 q^{37} +8.81369 q^{41} -1.83159 q^{43} -12.4302 q^{47} -1.66318 q^{53} -2.38350 q^{55} +8.38350 q^{59} +5.21509 q^{61} -1.38350 q^{65} +1.38350 q^{67} +3.61650 q^{71} -12.5986 q^{73} -11.8783 q^{79} -3.93541 q^{83} +0.831590 q^{85} -0.720322 q^{89} +4.21509 q^{95} -2.44809 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} - 2 q^{11} - 5 q^{13} + 2 q^{17} + q^{19} + 6 q^{23} + 3 q^{25} + 12 q^{29} - 3 q^{31} - 9 q^{37} - 10 q^{41} - q^{43} - 10 q^{47} + 4 q^{53} + 2 q^{55} + 16 q^{59} + 2 q^{61} + 5 q^{65}+ \cdots - 12 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 - 2 * q^11 - 5 * q^13 + 2 * q^17 + q^19 + 6 * q^23 + 3 * q^25 + 12 * q^29 - 3 * q^31 - 9 * q^37 - 10 * q^41 - q^43 - 10 * q^47 + 4 * q^53 + 2 * q^55 + 16 * q^59 + 2 * q^61 + 5 * q^65 - 5 * q^67 + 20 * q^71 - 15 * q^73 - 13 * q^79 + 2 * q^83 - 2 * q^85 - 2 * q^89 - q^95 - 12 * q^97

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.38350	0.718653	0.359326	−	0.933212i	$-0.383006\pi$
$11$	2.38350	0.718653	0.359326	+	0.933212i	$0.383006\pi$
$12$	0	0
$13$	1.38350	0.383715	0.191857	−	0.981423i	$-0.438549\pi$
$13$	1.38350	0.383715	0.191857	+	0.981423i	$0.438549\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.831590	−0.201690	−0.100845	−	0.994902i	$-0.532155\pi$
$17$	−0.831590	−0.201690	−0.100845	+	0.994902i	$0.532155\pi$
$18$	0	0
$19$	−4.21509	−0.967009	−0.483504	−	0.875342i	$-0.660636\pi$
$19$	−4.21509	−0.967009	−0.483504	+	0.875342i	$0.660636\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.59859	−1.16739	−0.583694	−	0.811974i	$-0.698393\pi$
$23$	−5.59859	−1.16739	−0.583694	+	0.811974i	$0.698393\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	10.0467	1.86562	0.932811	−	0.360366i	$-0.117348\pi$
$29$	10.0467	1.86562	0.932811	+	0.360366i	$0.117348\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.59859	0.756004	0.378002	−	0.925805i	$-0.376611\pi$
$37$	4.59859	0.756004	0.378002	+	0.925805i	$0.376611\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.81369	1.37647	0.688233	−	0.725489i	$-0.258387\pi$
$41$	8.81369	1.37647	0.688233	+	0.725489i	$0.258387\pi$
$42$	0	0
$43$	−1.83159	−0.279315	−0.139657	−	0.990200i	$-0.544600\pi$
$43$	−1.83159	−0.279315	−0.139657	+	0.990200i	$0.544600\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.4302	−1.81313	−0.906564	−	0.422067i	$-0.861305\pi$
$47$	−12.4302	−1.81313	−0.906564	+	0.422067i	$0.861305\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.66318	−0.228455	−0.114228	−	0.993455i	$-0.536439\pi$
$53$	−1.66318	−0.228455	−0.114228	+	0.993455i	$0.536439\pi$
$54$	0	0
$55$	−2.38350	−0.321391
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.38350	1.09144	0.545720	−	0.837968i	$-0.316256\pi$
$59$	8.38350	1.09144	0.545720	+	0.837968i	$0.316256\pi$
$60$	0	0
$61$	5.21509	0.667724	0.333862	−	0.942622i	$-0.391648\pi$
$61$	5.21509	0.667724	0.333862	+	0.942622i	$0.391648\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.38350	−0.171602
$66$	0	0
$67$	1.38350	0.169022	0.0845109	−	0.996423i	$-0.473067\pi$
$67$	1.38350	0.169022	0.0845109	+	0.996423i	$0.473067\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.61650	0.429199	0.214600	−	0.976702i	$-0.431155\pi$
$71$	3.61650	0.429199	0.214600	+	0.976702i	$0.431155\pi$
$72$	0	0
$73$	−12.5986	−1.47455	−0.737277	−	0.675591i	$-0.763889\pi$
$73$	−12.5986	−1.47455	−0.737277	+	0.675591i	$0.763889\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−11.8783	−1.33641	−0.668205	−	0.743977i	$-0.732937\pi$
$79$	−11.8783	−1.33641	−0.668205	+	0.743977i	$0.732937\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.93541	−0.431968	−0.215984	−	0.976397i	$-0.569296\pi$
$83$	−3.93541	−0.431968	−0.215984	+	0.976397i	$0.569296\pi$
$84$	0	0
$85$	0.831590	0.0901986
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.720322	−0.0763540	−0.0381770	−	0.999271i	$-0.512155\pi$
$89$	−0.720322	−0.0763540	−0.0381770	+	0.999271i	$0.512155\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.21509	0.432459
$96$	0	0
$97$	−2.44809	−0.248566	−0.124283	−	0.992247i	$-0.539663\pi$
$97$	−2.44809	−0.248566	−0.124283	+	0.992247i	$0.539663\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	13.1505	1.30852	0.654262	−	0.756268i	$-0.272979\pi$
$101$	13.1505	1.30852	0.654262	+	0.756268i	$0.272979\pi$
$102$	0	0
$103$	6.26178	0.616991	0.308496	−	0.951226i	$-0.400175\pi$
$103$	6.26178	0.616991	0.308496	+	0.951226i	$0.400175\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.5986	1.12128	0.560639	−	0.828060i	$-0.310555\pi$
$107$	11.5986	1.12128	0.560639	+	0.828060i	$0.310555\pi$
$108$	0	0
$109$	12.9821	1.24346	0.621730	−	0.783232i	$-0.286430\pi$
$109$	12.9821	1.24346	0.621730	+	0.783232i	$0.286430\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.7670	1.57731	0.788654	−	0.614838i	$-0.210779\pi$
$113$	16.7670	1.57731	0.788654	+	0.614838i	$0.210779\pi$
$114$	0	0
$115$	5.59859	0.522072
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.31892	−0.483538
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.8137	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$127$	13.8137	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.23300	−0.107727	−0.0538637	−	0.998548i	$-0.517154\pi$
$131$	−1.23300	−0.107727	−0.0538637	+	0.998548i	$0.517154\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.401405	0.0342944	0.0171472	−	0.999853i	$-0.494542\pi$
$137$	0.401405	0.0342944	0.0171472	+	0.999853i	$0.494542\pi$
$138$	0	0
$139$	−0.888732	−0.0753812	−0.0376906	−	0.999289i	$-0.512000\pi$
$139$	−0.888732	−0.0753812	−0.0376906	+	0.999289i	$0.512000\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.29758	0.275758
$144$	0	0
$145$	−10.0467	−0.834332
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−16.7491	−1.36302	−0.681511	−	0.731808i	$-0.738677\pi$
$151$	−16.7491	−1.36302	−0.681511	+	0.731808i	$0.738677\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	3.66318	0.292354	0.146177	−	0.989258i	$-0.453303\pi$
$157$	3.66318	0.292354	0.146177	+	0.989258i	$0.453303\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.0934	1.26053	0.630265	−	0.776380i	$-0.282946\pi$
$163$	16.0934	1.26053	0.630265	+	0.776380i	$0.282946\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−11.5986	−0.897526	−0.448763	−	0.893651i	$-0.648135\pi$
$167$	−11.5986	−0.897526	−0.448763	+	0.893651i	$0.648135\pi$
$168$	0	0
$169$	−11.0859	−0.852763
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−12.4302	−0.945049	−0.472525	−	0.881317i	$-0.656657\pi$
$173$	−12.4302	−0.945049	−0.472525	+	0.881317i	$0.656657\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3.10382	−0.231991	−0.115995	−	0.993250i	$-0.537006\pi$
$179$	−3.10382	−0.231991	−0.115995	+	0.993250i	$0.537006\pi$
$180$	0	0
$181$	−2.98210	−0.221658	−0.110829	−	0.993840i	$-0.535351\pi$
$181$	−2.98210	−0.221658	−0.110829	+	0.993840i	$0.535351\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.59859	−0.338095
$186$	0	0
$187$	−1.98210	−0.144945
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.1038	1.09287	0.546437	−	0.837500i	$-0.315984\pi$
$191$	15.1038	1.09287	0.546437	+	0.837500i	$0.315984\pi$
$192$	0	0
$193$	−17.7958	−1.28097	−0.640484	−	0.767971i	$-0.721266\pi$
$193$	−17.7958	−1.28097	−0.640484	+	0.767971i	$0.721266\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.1684	1.22320	0.611599	−	0.791168i	$-0.290526\pi$
$197$	17.1684	1.22320	0.611599	+	0.791168i	$0.290526\pi$
$198$	0	0
$199$	20.7491	1.47086	0.735432	−	0.677598i	$-0.236979\pi$
$199$	20.7491	1.47086	0.735432	+	0.677598i	$0.236979\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−8.81369	−0.615575
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−10.0467	−0.694944
$210$	0	0
$211$	3.12173	0.214909	0.107454	−	0.994210i	$-0.465730\pi$
$211$	3.12173	0.214909	0.107454	+	0.994210i	$0.465730\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.83159	0.124913
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.15051	−0.0773915
$222$	0	0
$223$	−7.32636	−0.490609	−0.245305	−	0.969446i	$-0.578888\pi$
$223$	−7.32636	−0.490609	−0.245305	+	0.969446i	$0.578888\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	25.2618	1.67668	0.838341	−	0.545145i	$-0.183526\pi$
$227$	25.2618	1.67668	0.838341	+	0.545145i	$0.183526\pi$
$228$	0	0
$229$	−10.3264	−0.682385	−0.341193	−	0.939993i	$-0.610831\pi$
$229$	−10.3264	−0.682385	−0.341193	+	0.939993i	$0.610831\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	26.0934	1.70943	0.854717	−	0.519095i	$-0.173731\pi$
$233$	26.0934	1.70943	0.854717	+	0.519095i	$0.173731\pi$
$234$	0	0
$235$	12.4302	0.810856
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.86037	−0.443760	−0.221880	−	0.975074i	$-0.571219\pi$
$239$	−6.86037	−0.443760	−0.221880	+	0.975074i	$0.571219\pi$
$240$	0	0
$241$	−2.87827	−0.185406	−0.0927029	−	0.995694i	$-0.529551\pi$
$241$	−2.87827	−0.185406	−0.0927029	+	0.995694i	$0.529551\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.83159	−0.371055
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.1400	−1.52371	−0.761853	−	0.647750i	$-0.775710\pi$
$251$	−24.1400	−1.52371	−0.761853	+	0.647750i	$0.775710\pi$
$252$	0	0
$253$	−13.3443	−0.838947
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.4590	1.52571	0.762854	−	0.646571i	$-0.223797\pi$
$257$	24.4590	1.52571	0.762854	+	0.646571i	$0.223797\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.8604	1.16298	0.581490	−	0.813553i	$-0.302470\pi$
$263$	18.8604	1.16298	0.581490	+	0.813553i	$0.302470\pi$
$264$	0	0
$265$	1.66318	0.102168
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5.19719	−0.316878	−0.158439	−	0.987369i	$-0.550646\pi$
$269$	−5.19719	−0.316878	−0.158439	+	0.987369i	$0.550646\pi$
$270$	0	0
$271$	27.9821	1.69979	0.849896	−	0.526951i	$-0.176665\pi$
$271$	27.9821	1.69979	0.849896	+	0.526951i	$0.176665\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.38350	0.143731
$276$	0	0
$277$	−6.16841	−0.370624	−0.185312	−	0.982680i	$-0.559330\pi$
$277$	−6.16841	−0.370624	−0.185312	+	0.982680i	$0.559330\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	11.5698	0.690197	0.345099	−	0.938566i	$-0.387845\pi$
$281$	11.5698	0.690197	0.345099	+	0.938566i	$0.387845\pi$
$282$	0	0
$283$	−29.7958	−1.77118	−0.885588	−	0.464472i	$-0.846244\pi$
$283$	−29.7958	−1.77118	−0.885588	+	0.464472i	$0.846244\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.3085	−0.959321
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−19.6632	−1.14874	−0.574368	−	0.818597i	$-0.694752\pi$
$293$	−19.6632	−1.14874	−0.574368	+	0.818597i	$0.694752\pi$
$294$	0	0
$295$	−8.38350	−0.488106
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7.74567	−0.447944
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−5.21509	−0.298615
$306$	0	0
$307$	19.8137	1.13083	0.565413	−	0.824808i	$-0.308717\pi$
$307$	19.8137	1.13083	0.565413	+	0.824808i	$0.308717\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	22.0467	1.25015	0.625076	−	0.780564i	$-0.285068\pi$
$311$	22.0467	1.25015	0.625076	+	0.780564i	$0.285068\pi$
$312$	0	0
$313$	−10.5052	−0.593791	−0.296895	−	0.954910i	$-0.595951\pi$
$313$	−10.5052	−0.593791	−0.296895	+	0.954910i	$0.595951\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.73822	0.266125	0.133063	−	0.991108i	$-0.457519\pi$
$317$	4.73822	0.266125	0.133063	+	0.991108i	$0.457519\pi$
$318$	0	0
$319$	23.9463	1.34073
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.50523	0.195036
$324$	0	0
$325$	1.38350	0.0767429
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	17.8604	0.981695	0.490847	−	0.871246i	$-0.336687\pi$
$331$	17.8604	0.981695	0.490847	+	0.871246i	$0.336687\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.38350	−0.0755888
$336$	0	0
$337$	−11.4769	−0.625185	−0.312592	−	0.949887i	$-0.601197\pi$
$337$	−11.4769	−0.625185	−0.312592	+	0.949887i	$0.601197\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.38350	−0.129074
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5.56982	−0.299003	−0.149502	−	0.988761i	$-0.547767\pi$
$347$	−5.56982	−0.299003	−0.149502	+	0.988761i	$0.547767\pi$
$348$	0	0
$349$	3.66318	0.196086	0.0980428	−	0.995182i	$-0.468742\pi$
$349$	3.66318	0.196086	0.0980428	+	0.995182i	$0.468742\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0.401405	0.0213646	0.0106823	−	0.999943i	$-0.496600\pi$
$353$	0.401405	0.0213646	0.0106823	+	0.999943i	$0.496600\pi$
$354$	0	0
$355$	−3.61650	−0.191944
$356$	0	0
$357$	0	0
$358$	0	0
$359$	26.8137	1.41517	0.707586	−	0.706627i	$-0.249784\pi$
$359$	26.8137	1.41517	0.707586	+	0.706627i	$0.249784\pi$
$360$	0	0
$361$	−1.23300	−0.0648945
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12.5986	0.659441
$366$	0	0
$367$	0.523132	0.0273073	0.0136536	−	0.999907i	$-0.495654\pi$
$367$	0.523132	0.0273073	0.0136536	+	0.999907i	$0.495654\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	32.2439	1.66952	0.834762	−	0.550611i	$-0.185605\pi$
$373$	32.2439	1.66952	0.834762	+	0.550611i	$0.185605\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	13.8996	0.715866
$378$	0	0
$379$	19.5236	1.00286	0.501429	−	0.865199i	$-0.332808\pi$
$379$	19.5236	1.00286	0.501429	+	0.865199i	$0.332808\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−0.802810	−0.0410217	−0.0205108	−	0.999790i	$-0.506529\pi$
$383$	−0.802810	−0.0410217	−0.0205108	+	0.999790i	$0.506529\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	29.9175	1.51688	0.758439	−	0.651744i	$-0.225962\pi$
$389$	29.9175	1.51688	0.758439	+	0.651744i	$0.225962\pi$
$390$	0	0
$391$	4.65574	0.235451
$392$	0	0
$393$	0	0
$394$	0	0
$395$	11.8783	0.597661
$396$	0	0
$397$	29.4590	1.47850	0.739252	−	0.673429i	$-0.235179\pi$
$397$	29.4590	1.47850	0.739252	+	0.673429i	$0.235179\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	26.5236	1.32452	0.662261	−	0.749273i	$-0.269597\pi$
$401$	26.5236	1.32452	0.662261	+	0.749273i	$0.269597\pi$
$402$	0	0
$403$	−1.38350	−0.0689172
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.9608	0.543305
$408$	0	0
$409$	31.5236	1.55874	0.779370	−	0.626564i	$-0.215539\pi$
$409$	31.5236	1.55874	0.779370	+	0.626564i	$0.215539\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	3.93541	0.193182
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−0.802810	−0.0392199	−0.0196099	−	0.999808i	$-0.506242\pi$
$419$	−0.802810	−0.0392199	−0.0196099	+	0.999808i	$0.506242\pi$
$420$	0	0
$421$	37.8425	1.84433	0.922164	−	0.386798i	$-0.126419\pi$
$421$	37.8425	1.84433	0.922164	+	0.386798i	$0.126419\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.831590	−0.0403380
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−17.1972	−0.828359	−0.414180	−	0.910195i	$-0.635932\pi$
$431$	−17.1972	−0.828359	−0.414180	+	0.910195i	$0.635932\pi$
$432$	0	0
$433$	12.1505	0.583916	0.291958	−	0.956431i	$-0.405693\pi$
$433$	12.1505	0.583916	0.291958	+	0.956431i	$0.405693\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	23.5986	1.12887
$438$	0	0
$439$	−31.2727	−1.49256	−0.746281	−	0.665631i	$-0.768163\pi$
$439$	−31.2727	−1.49256	−0.746281	+	0.665631i	$0.768163\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6.86037	−0.325946	−0.162973	−	0.986631i	$-0.552108\pi$
$443$	−6.86037	−0.325946	−0.162973	+	0.986631i	$0.552108\pi$
$444$	0	0
$445$	0.720322	0.0341465
$446$	0	0
$447$	0	0
$448$	0	0
$449$	27.1038	1.27911	0.639554	−	0.768746i	$-0.279119\pi$
$449$	27.1038	1.27911	0.639554	+	0.768746i	$0.279119\pi$
$450$	0	0
$451$	21.0074	0.989202
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.8137	0.926845	0.463423	−	0.886137i	$-0.346621\pi$
$457$	19.8137	0.926845	0.463423	+	0.886137i	$0.346621\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3.82415	−0.178108	−0.0890541	−	0.996027i	$-0.528384\pi$
$461$	−3.82415	−0.178108	−0.0890541	+	0.996027i	$0.528384\pi$
$462$	0	0
$463$	−8.95332	−0.416096	−0.208048	−	0.978119i	$-0.566711\pi$
$463$	−8.95332	−0.416096	−0.208048	+	0.978119i	$0.566711\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.3368	−0.478331	−0.239165	−	0.970979i	$-0.576874\pi$
$467$	−10.3368	−0.478331	−0.239165	+	0.970979i	$0.576874\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.36560	−0.200730
$474$	0	0
$475$	−4.21509	−0.193402
$476$	0	0
$477$	0	0
$478$	0	0
$479$	40.3944	1.84567	0.922833	−	0.385200i	$-0.125868\pi$
$479$	40.3944	1.84567	0.922833	+	0.385200i	$0.125868\pi$
$480$	0	0
$481$	6.36217	0.290090
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.44809	0.111162
$486$	0	0
$487$	−13.1400	−0.595432	−0.297716	−	0.954654i	$-0.596225\pi$
$487$	−13.1400	−0.595432	−0.297716	+	0.954654i	$0.596225\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17.1972	0.776098	0.388049	−	0.921639i	$-0.373149\pi$
$491$	17.1972	0.776098	0.388049	+	0.921639i	$0.373149\pi$
$492$	0	0
$493$	−8.35472	−0.376278
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−2.66318	−0.119220	−0.0596102	−	0.998222i	$-0.518986\pi$
$499$	−2.66318	−0.119220	−0.0596102	+	0.998222i	$0.518986\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	−13.1505	−0.585190
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−24.0576	−1.06633	−0.533166	−	0.846010i	$-0.678998\pi$
$509$	−24.0576	−1.06633	−0.533166	+	0.846010i	$0.678998\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6.26178	−0.275927
$516$	0	0
$517$	−29.6274	−1.30301
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−17.6274	−0.772269	−0.386135	−	0.922442i	$-0.626190\pi$
$521$	−17.6274	−0.772269	−0.386135	+	0.922442i	$0.626190\pi$
$522$	0	0
$523$	−16.1863	−0.707778	−0.353889	−	0.935287i	$-0.615141\pi$
$523$	−16.1863	−0.707778	−0.353889	+	0.935287i	$0.615141\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.831590	0.0362246
$528$	0	0
$529$	8.34426	0.362794
$530$	0	0
$531$	0	0
$532$	0	0
$533$	12.1938	0.528170
$534$	0	0
$535$	−11.5986	−0.501451
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−17.5594	−0.754936	−0.377468	−	0.926023i	$-0.623205\pi$
$541$	−17.5594	−0.754936	−0.377468	+	0.926023i	$0.623205\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−12.9821	−0.556092
$546$	0	0
$547$	−7.96419	−0.340524	−0.170262	−	0.985399i	$-0.554461\pi$
$547$	−7.96419	−0.340524	−0.170262	+	0.985399i	$0.554461\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−42.3477	−1.80407
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	27.5340	1.16665	0.583327	−	0.812238i	$-0.301751\pi$
$557$	27.5340	1.16665	0.583327	+	0.812238i	$0.301751\pi$
$558$	0	0
$559$	−2.53401	−0.107177
$560$	0	0
$561$	0	0
$562$	0	0
$563$	22.9747	0.968266	0.484133	−	0.874994i	$-0.339135\pi$
$563$	22.9747	0.968266	0.484133	+	0.874994i	$0.339135\pi$
$564$	0	0
$565$	−16.7670	−0.705393
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3.24387	−0.135990	−0.0679951	−	0.997686i	$-0.521660\pi$
$569$	−3.24387	−0.135990	−0.0679951	+	0.997686i	$0.521660\pi$
$570$	0	0
$571$	−28.2151	−1.18076	−0.590382	−	0.807124i	$-0.701023\pi$
$571$	−28.2151	−1.18076	−0.590382	+	0.807124i	$0.701023\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5.59859	−0.233478
$576$	0	0
$577$	−17.4769	−0.727572	−0.363786	−	0.931483i	$-0.618516\pi$
$577$	−17.4769	−0.727572	−0.363786	+	0.931483i	$0.618516\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−3.96419	−0.164180
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.1580	0.419264	0.209632	−	0.977780i	$-0.432773\pi$
$587$	10.1580	0.419264	0.209632	+	0.977780i	$0.432773\pi$
$588$	0	0
$589$	4.21509	0.173680
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5.39095	0.221380	0.110690	−	0.993855i	$-0.464694\pi$
$593$	5.39095	0.221380	0.110690	+	0.993855i	$0.464694\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−22.9895	−0.939327	−0.469664	−	0.882845i	$-0.655625\pi$
$599$	−22.9895	−0.939327	−0.469664	+	0.882845i	$0.655625\pi$
$600$	0	0
$601$	38.2727	1.56117	0.780587	−	0.625047i	$-0.214920\pi$
$601$	38.2727	1.56117	0.780587	+	0.625047i	$0.214920\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5.31892	0.216245
$606$	0	0
$607$	11.7203	0.475713	0.237857	−	0.971300i	$-0.423555\pi$
$607$	11.7203	0.475713	0.237857	+	0.971300i	$0.423555\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−17.1972	−0.695724
$612$	0	0
$613$	27.9821	1.13019	0.565093	−	0.825027i	$-0.308840\pi$
$613$	27.9821	1.13019	0.565093	+	0.825027i	$0.308840\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.0680	1.25075	0.625376	−	0.780324i	$-0.284946\pi$
$617$	31.0680	1.25075	0.625376	+	0.780324i	$0.284946\pi$
$618$	0	0
$619$	7.30101	0.293453	0.146726	−	0.989177i	$-0.453126\pi$
$619$	7.30101	0.293453	0.146726	+	0.989177i	$0.453126\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3.82415	−0.152479
$630$	0	0
$631$	−22.4877	−0.895223	−0.447611	−	0.894228i	$-0.647725\pi$
$631$	−22.4877	−0.895223	−0.447611	+	0.894228i	$0.647725\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−13.8137	−0.548179
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	44.0109	1.73833	0.869163	−	0.494526i	$-0.164659\pi$
$641$	44.0109	1.73833	0.869163	+	0.494526i	$0.164659\pi$
$642$	0	0
$643$	−10.1863	−0.401709	−0.200854	−	0.979621i	$-0.564372\pi$
$643$	−10.1863	−0.401709	−0.200854	+	0.979621i	$0.564372\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−32.5523	−1.27976	−0.639882	−	0.768473i	$-0.721017\pi$
$647$	−32.5523	−1.27976	−0.639882	+	0.768473i	$0.721017\pi$
$648$	0	0
$649$	19.9821	0.784366
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20.9250	−0.818857	−0.409428	−	0.912342i	$-0.634272\pi$
$653$	−20.9250	−0.818857	−0.409428	+	0.912342i	$0.634272\pi$
$654$	0	0
$655$	1.23300	0.0481771
$656$	0	0
$657$	0	0
$658$	0	0
$659$	37.7207	1.46939	0.734696	−	0.678397i	$-0.237325\pi$
$659$	37.7207	1.46939	0.734696	+	0.678397i	$0.237325\pi$
$660$	0	0
$661$	9.22555	0.358832	0.179416	−	0.983773i	$-0.442579\pi$
$661$	9.22555	0.358832	0.179416	+	0.983773i	$0.442579\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−56.2473	−2.17790
$668$	0	0
$669$	0	0
$670$	0	0
$671$	12.4302	0.479862
$672$	0	0
$673$	−30.2797	−1.16720	−0.583598	−	0.812043i	$-0.698356\pi$
$673$	−30.2797	−1.16720	−0.583598	+	0.812043i	$0.698356\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−33.7853	−1.29848	−0.649238	−	0.760586i	$-0.724912\pi$
$677$	−33.7853	−1.29848	−0.649238	+	0.760586i	$0.724912\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−33.1326	−1.26778	−0.633892	−	0.773422i	$-0.718544\pi$
$683$	−33.1326	−1.26778	−0.633892	+	0.773422i	$0.718544\pi$
$684$	0	0
$685$	−0.401405	−0.0153369
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.30101	−0.0876616
$690$	0	0
$691$	44.9642	1.71052	0.855259	−	0.518200i	$-0.173398\pi$
$691$	44.9642	1.71052	0.855259	+	0.518200i	$0.173398\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0.888732	0.0337115
$696$	0	0
$697$	−7.32938	−0.277620
$698$	0	0
$699$	0	0
$700$	0	0
$701$	42.5702	1.60786	0.803928	−	0.594727i	$-0.202740\pi$
$701$	42.5702	1.60786	0.803928	+	0.594727i	$0.202740\pi$
$702$	0	0
$703$	−19.3835	−0.731063
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	10.2046	0.383243	0.191622	−	0.981469i	$-0.438625\pi$
$709$	10.2046	0.383243	0.191622	+	0.981469i	$0.438625\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.59859	0.209669
$714$	0	0
$715$	−3.29758	−0.123323
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.09337	0.0780694	0.0390347	−	0.999238i	$-0.487572\pi$
$719$	2.09337	0.0780694	0.0390347	+	0.999238i	$0.487572\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	10.0467	0.373124
$726$	0	0
$727$	34.8211	1.29144	0.645722	−	0.763572i	$-0.276556\pi$
$727$	34.8211	1.29144	0.645722	+	0.763572i	$0.276556\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.52313	0.0563351
$732$	0	0
$733$	−47.4769	−1.75360	−0.876799	−	0.480857i	$-0.840326\pi$
$733$	−47.4769	−1.75360	−0.876799	+	0.480857i	$0.840326\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.29758	0.121468
$738$	0	0
$739$	−40.2727	−1.48145	−0.740727	−	0.671806i	$-0.765519\pi$
$739$	−40.2727	−1.48145	−0.740727	+	0.671806i	$0.765519\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	27.7853	1.01934	0.509672	−	0.860369i	$-0.329767\pi$
$743$	27.7853	1.01934	0.509672	+	0.860369i	$0.329767\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1.80281	−0.0657855	−0.0328927	−	0.999459i	$-0.510472\pi$
$751$	−1.80281	−0.0657855	−0.0328927	+	0.999459i	$0.510472\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	16.7491	0.609562
$756$	0	0
$757$	21.8708	0.794909	0.397454	−	0.917622i	$-0.369894\pi$
$757$	21.8708	0.794909	0.397454	+	0.917622i	$0.369894\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22.0467	0.799192	0.399596	−	0.916691i	$-0.369150\pi$
$761$	22.0467	0.799192	0.399596	+	0.916691i	$0.369150\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11.5986	0.418801
$768$	0	0
$769$	−28.2727	−1.01954	−0.509769	−	0.860311i	$-0.670269\pi$
$769$	−28.2727	−1.01954	−0.509769	+	0.860311i	$0.670269\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	39.9930	1.43845	0.719224	−	0.694779i	$-0.244498\pi$
$773$	39.9930	1.43845	0.719224	+	0.694779i	$0.244498\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−37.1505	−1.33106
$780$	0	0
$781$	8.61993	0.308445
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3.66318	−0.130745
$786$	0	0
$787$	35.1614	1.25337	0.626684	−	0.779273i	$-0.284412\pi$
$787$	35.1614	1.25337	0.626684	+	0.779273i	$0.284412\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	7.21509	0.256215
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−49.0968	−1.73910	−0.869549	−	0.493847i	$-0.835590\pi$
$797$	−49.0968	−1.73910	−0.869549	+	0.493847i	$0.835590\pi$
$798$	0	0
$799$	10.3368	0.365690
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−30.0288	−1.05969
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−13.2906	−0.467271	−0.233636	−	0.972324i	$-0.575062\pi$
$809$	−13.2906	−0.467271	−0.233636	+	0.972324i	$0.575062\pi$
$810$	0	0
$811$	−44.9358	−1.57791	−0.788955	−	0.614451i	$-0.789378\pi$
$811$	−44.9358	−1.57791	−0.788955	+	0.614451i	$0.789378\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−16.0934	−0.563726
$816$	0	0
$817$	7.72032	0.270100
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−14.4411	−0.503997	−0.251998	−	0.967728i	$-0.581088\pi$
$821$	−14.4411	−0.503997	−0.251998	+	0.967728i	$0.581088\pi$
$822$	0	0
$823$	26.1113	0.910182	0.455091	−	0.890445i	$-0.349607\pi$
$823$	26.1113	0.910182	0.455091	+	0.890445i	$0.349607\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32.5236	−1.13095	−0.565477	−	0.824764i	$-0.691308\pi$
$827$	−32.5236	−1.13095	−0.565477	+	0.824764i	$0.691308\pi$
$828$	0	0
$829$	−39.0397	−1.35590	−0.677952	−	0.735106i	$-0.737132\pi$
$829$	−39.0397	−1.35590	−0.677952	+	0.735106i	$0.737132\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	11.5986	0.401386
$836$	0	0
$837$	0	0
$838$	0	0
$839$	35.7099	1.23284	0.616421	−	0.787417i	$-0.288582\pi$
$839$	35.7099	1.23284	0.616421	+	0.787417i	$0.288582\pi$
$840$	0	0
$841$	71.9358	2.48055
$842$	0	0
$843$	0	0
$844$	0	0
$845$	11.0859	0.381367
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−25.7457	−0.882550
$852$	0	0
$853$	−44.0968	−1.50985	−0.754923	−	0.655814i	$-0.772326\pi$
$853$	−44.0968	−1.50985	−0.754923	+	0.655814i	$0.772326\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−4.30804	−0.147160	−0.0735799	−	0.997289i	$-0.523442\pi$
$857$	−4.30804	−0.147160	−0.0735799	+	0.997289i	$0.523442\pi$
$858$	0	0
$859$	23.2727	0.794053	0.397026	−	0.917807i	$-0.370042\pi$
$859$	23.2727	0.794053	0.397026	+	0.917807i	$0.370042\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	54.9179	1.86943	0.934714	−	0.355401i	$-0.115656\pi$
$863$	54.9179	1.86943	0.934714	+	0.355401i	$0.115656\pi$
$864$	0	0
$865$	12.4302	0.422639
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−28.3119	−0.960415
$870$	0	0
$871$	1.91408	0.0648561
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−6.09337	−0.205758	−0.102879	−	0.994694i	$-0.532805\pi$
$877$	−6.09337	−0.205758	−0.102879	+	0.994694i	$0.532805\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−21.3264	−0.718503	−0.359252	−	0.933241i	$-0.616968\pi$
$881$	−21.3264	−0.718503	−0.359252	+	0.933241i	$0.616968\pi$
$882$	0	0
$883$	−24.3761	−0.820320	−0.410160	−	0.912014i	$-0.634527\pi$
$883$	−24.3761	−0.820320	−0.410160	+	0.912014i	$0.634527\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−44.3298	−1.48845	−0.744224	−	0.667930i	$-0.767181\pi$
$887$	−44.3298	−1.48845	−0.744224	+	0.667930i	$0.767181\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	52.3944	1.75331
$894$	0	0
$895$	3.10382	0.103749
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−10.0467	−0.335076
$900$	0	0
$901$	1.38308	0.0460772
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.98210	0.0991283
$906$	0	0
$907$	10.5986	0.351921	0.175960	−	0.984397i	$-0.443697\pi$
$907$	10.5986	0.351921	0.175960	+	0.984397i	$0.443697\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−24.1400	−0.799795	−0.399898	−	0.916560i	$-0.630954\pi$
$911$	−24.1400	−0.799795	−0.399898	+	0.916560i	$0.630954\pi$
$912$	0	0
$913$	−9.38007	−0.310435
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	9.98954	0.329525	0.164762	−	0.986333i	$-0.447314\pi$
$919$	9.98954	0.329525	0.164762	+	0.986333i	$0.447314\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.00343	0.164690
$924$	0	0
$925$	4.59859	0.151201
$926$	0	0
$927$	0	0
$928$	0	0
$929$	45.6741	1.49852	0.749259	−	0.662278i	$-0.230410\pi$
$929$	45.6741	1.49852	0.749259	+	0.662278i	$0.230410\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.98210	0.0648215
$936$	0	0
$937$	−15.4948	−0.506192	−0.253096	−	0.967441i	$-0.581449\pi$
$937$	−15.4948	−0.506192	−0.253096	+	0.967441i	$0.581449\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	19.8032	0.645567	0.322783	−	0.946473i	$-0.395381\pi$
$941$	19.8032	0.645567	0.322783	+	0.946473i	$0.395381\pi$
$942$	0	0
$943$	−49.3443	−1.60687
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10.1580	0.330089	0.165045	−	0.986286i	$-0.447223\pi$
$947$	10.1580	0.330089	0.165045	+	0.986286i	$0.447223\pi$
$948$	0	0
$949$	−17.4302	−0.565808
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−8.09337	−0.262170	−0.131085	−	0.991371i	$-0.541846\pi$
$953$	−8.09337	−0.262170	−0.131085	+	0.991371i	$0.541846\pi$
$954$	0	0
$955$	−15.1038	−0.488748
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	0	0
$964$	0	0
$965$	17.7958	0.572867
$966$	0	0
$967$	29.2513	0.940659	0.470329	−	0.882491i	$-0.344135\pi$
$967$	29.2513	0.940659	0.470329	+	0.882491i	$0.344135\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	16.3944	0.526121	0.263060	−	0.964779i	$-0.415268\pi$
$971$	16.3944	0.526121	0.263060	+	0.964779i	$0.415268\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−31.0968	−0.994875	−0.497437	−	0.867500i	$-0.665726\pi$
$977$	−31.0968	−0.994875	−0.497437	+	0.867500i	$0.665726\pi$
$978$	0	0
$979$	−1.71689	−0.0548720
$980$	0	0
$981$	0	0
$982$	0	0
$983$	4.36560	0.139241	0.0696205	−	0.997574i	$-0.477821\pi$
$983$	4.36560	0.139241	0.0696205	+	0.997574i	$0.477821\pi$
$984$	0	0
$985$	−17.1684	−0.547031
$986$	0	0
$987$	0	0
$988$	0	0
$989$	10.2543	0.326069
$990$	0	0
$991$	27.2980	0.867150	0.433575	−	0.901118i	$-0.357252\pi$
$991$	27.2980	0.867150	0.433575	+	0.901118i	$0.357252\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−20.7491	−0.657791
$996$	0	0
$997$	30.4843	0.965448	0.482724	−	0.875773i	$-0.339647\pi$
$997$	30.4843	0.965448	0.482724	+	0.875773i	$0.339647\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8820.2.a.bn.1.3		3
3.2	odd	2		8820.2.a.bq.1.1		3
7.2	even	3		1260.2.s.h.361.1	yes	6
7.4	even	3		1260.2.s.h.541.1	yes	6
7.6	odd	2		8820.2.a.bp.1.3		3
21.2	odd	6		1260.2.s.g.361.1	✓	6
21.11	odd	6		1260.2.s.g.541.1	yes	6
21.20	even	2		8820.2.a.bo.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.s.g.361.1	✓	6	21.2	odd	6
1260.2.s.g.541.1	yes	6	21.11	odd	6
1260.2.s.h.361.1	yes	6	7.2	even	3
1260.2.s.h.541.1	yes	6	7.4	even	3
8820.2.a.bn.1.3		3	1.1	even	1	trivial
8820.2.a.bo.1.1		3	21.20	even	2
8820.2.a.bp.1.3		3	7.6	odd	2
8820.2.a.bq.1.1		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8820.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +2.38350 q^{11} +1.38350 q^{13} -0.831590 q^{17} -4.21509 q^{19} -5.59859 q^{23} +1.00000 q^{25} +10.0467 q^{29} -1.00000 q^{31} +4.59859 q^{37} +8.81369 q^{41} -1.83159 q^{43} -12.4302 q^{47} -1.66318 q^{53} -2.38350 q^{55} +8.38350 q^{59} +5.21509 q^{61} -1.38350 q^{65} +1.38350 q^{67} +3.61650 q^{71} -12.5986 q^{73} -11.8783 q^{79} -3.93541 q^{83} +0.831590 q^{85} -0.720322 q^{89} +4.21509 q^{95} -2.44809 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} - 2 q^{11} - 5 q^{13} + 2 q^{17} + q^{19} + 6 q^{23} + 3 q^{25} + 12 q^{29} - 3 q^{31} - 9 q^{37} - 10 q^{41} - q^{43} - 10 q^{47} + 4 q^{53} + 2 q^{55} + 16 q^{59} + 2 q^{61} + 5 q^{65}+ \cdots - 12 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 - 2 * q^11 - 5 * q^13 + 2 * q^17 + q^19 + 6 * q^23 + 3 * q^25 + 12 * q^29 - 3 * q^31 - 9 * q^37 - 10 * q^41 - q^43 - 10 * q^47 + 4 * q^53 + 2 * q^55 + 16 * q^59 + 2 * q^61 + 5 * q^65 - 5 * q^67 + 20 * q^71 - 15 * q^73 - 13 * q^79 + 2 * q^83 - 2 * q^85 - 2 * q^89 - q^95 - 12 * q^97

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