LMFDB - Embedded newform 8820.2.d.c.881.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8820,2,Mod(881,8820)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8820, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8820.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8820.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$70.4280545828$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 60 x^{14} - 280 x^{13} + 1134 x^{12} - 3528 x^{11} + 9316 x^{10} - 19960 x^{9} + \cdots + 68 $ x^16 - 8x^15 + 60x^14 - 280x^13 + 1134x^12 - 3528x^11 + 9316x^10 - 19960x^9 + 35421x^8 - 51008x^7 + 58384x^6 - 51840x^5 + 34228x^4 - 15904x^3 + 4848x^2 - 864*x + 68
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.4
Root			$0.500000 + 2.03007i$ of defining polynomial
Character	$\chi$	$=$	8820.881
Dual form			8820.2.d.c.881.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +O(q^{10})$	$q-1.00000 q^{5} -3.51188i q^{11} -0.0229148i q^{13} +0.381766 q^{17} -3.64374i q^{19} +2.90791i q^{23} +1.00000 q^{25} +6.73275i q^{29} +6.02822i q^{31} -4.23615 q^{37} -3.37244 q^{41} -0.392652 q^{43} -4.14792 q^{47} +7.61259i q^{53} +3.51188i q^{55} -1.37092 q^{59} +1.03392i q^{61} +0.0229148i q^{65} +10.2908 q^{67} +6.95032i q^{71} -9.99591i q^{73} +2.20441 q^{79} -3.95259 q^{83} -0.381766 q^{85} +8.45072 q^{89} +3.64374i q^{95} +8.06301i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{5}+O(q^{10}) $ 16 * q - 16 * q^5	$ 16 q - 16 q^{5} + 16 q^{25} - 32 q^{41} - 32 q^{43} + 32 q^{47} + 32 q^{59} - 32 q^{67} + 64 q^{89}+O(q^{100}) $ 16 * q - 16 * q^5 + 16 * q^25 - 32 * q^41 - 32 * q^43 + 32 * q^47 + 32 * q^59 - 32 * q^67 + 64 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8820\mathbb{Z}\right)^\times$.

$n$	$1081$	$4411$	$7057$	$7841$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$5$	−1.00000	−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
$11$	− 3.51188i	− 1.05887i	−0.848350	−	0.529435i	$-0.822404\pi$
$11$	− 3.51188i	− 1.05887i	0.848350	−	0.529435i	$-0.177596\pi$
$12$	0	0
$13$	− 0.0229148i	− 0.00635541i	−0.999995	−	0.00317771i	$-0.998989\pi$
$13$	− 0.0229148i	− 0.00635541i	0.999995	−	0.00317771i	$-0.00101150\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.381766	0.0925918	0.0462959	−	0.998928i	$-0.485258\pi$
$17$	0.381766	0.0925918	0.0462959	+	0.998928i	$0.485258\pi$
$18$	0	0
$19$	− 3.64374i	− 0.835931i	−0.908463	−	0.417965i	$-0.862743\pi$
$19$	− 3.64374i	− 0.835931i	0.908463	−	0.417965i	$-0.137257\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.90791i	0.606340i	0.952936	+	0.303170i	$0.0980451\pi$
$23$	2.90791i	0.606340i	−0.952936	+	0.303170i	$0.901955\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.73275i	1.25024i	0.780528	+	0.625120i	$0.214950\pi$
$29$	6.73275i	1.25024i	−0.780528	+	0.625120i	$0.785050\pi$
$30$	0	0
$31$	6.02822i	1.08270i	0.840798	+	0.541350i	$0.182086\pi$
$31$	6.02822i	1.08270i	−0.840798	+	0.541350i	$0.817914\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.23615	−0.696418	−0.348209	−	0.937417i	$-0.613210\pi$
$37$	−4.23615	−0.696418	−0.348209	+	0.937417i	$0.613210\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.37244	−0.526687	−0.263343	−	0.964702i	$-0.584825\pi$
$41$	−3.37244	−0.526687	−0.263343	+	0.964702i	$0.584825\pi$
$42$	0	0
$43$	−0.392652	−0.0598789	−0.0299395	−	0.999552i	$-0.509531\pi$
$43$	−0.392652	−0.0598789	−0.0299395	+	0.999552i	$0.509531\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.14792	−0.605036	−0.302518	−	0.953144i	$-0.597827\pi$
$47$	−4.14792	−0.605036	−0.302518	+	0.953144i	$0.597827\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.61259i	1.04567i	0.852434	+	0.522835i	$0.175126\pi$
$53$	7.61259i	1.04567i	−0.852434	+	0.522835i	$0.824874\pi$
$54$	0	0
$55$	3.51188i	0.473541i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.37092	−0.178479	−0.0892393	−	0.996010i	$-0.528444\pi$
$59$	−1.37092	−0.178479	−0.0892393	+	0.996010i	$0.528444\pi$
$60$	0	0
$61$	1.03392i	0.132380i	0.997807	+	0.0661898i	$0.0210843\pi$
$61$	1.03392i	0.132380i	−0.997807	+	0.0661898i	$0.978916\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.0229148i	0.00284223i
$66$	0	0
$67$	10.2908	1.25722	0.628612	−	0.777719i	$-0.283623\pi$
$67$	10.2908	1.25722	0.628612	+	0.777719i	$0.283623\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.95032i	0.824852i	0.910991	+	0.412426i	$0.135318\pi$
$71$	6.95032i	0.824852i	−0.910991	+	0.412426i	$0.864682\pi$
$72$	0	0
$73$	− 9.99591i	− 1.16993i	−0.811058	−	0.584966i	$-0.801108\pi$
$73$	− 9.99591i	− 1.16993i	0.811058	−	0.584966i	$-0.198892\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.20441	0.248016	0.124008	−	0.992281i	$-0.460425\pi$
$79$	2.20441	0.248016	0.124008	+	0.992281i	$0.460425\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.95259	−0.433853	−0.216927	−	0.976188i	$-0.569603\pi$
$83$	−3.95259	−0.433853	−0.216927	+	0.976188i	$0.569603\pi$
$84$	0	0
$85$	−0.381766	−0.0414083
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.45072	0.895774	0.447887	−	0.894090i	$-0.352177\pi$
$89$	8.45072	0.895774	0.447887	+	0.894090i	$0.352177\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.64374i	0.373840i
$96$	0	0
$97$	8.06301i	0.818674i	0.912383	+	0.409337i	$0.134240\pi$
$97$	8.06301i	0.818674i	−0.912383	+	0.409337i	$0.865760\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.6591	−1.06062	−0.530310	−	0.847804i	$-0.677925\pi$
$101$	−10.6591	−1.06062	−0.530310	+	0.847804i	$0.677925\pi$
$102$	0	0
$103$	− 17.6334i	− 1.73747i	−0.495274	−	0.868737i	$-0.664932\pi$
$103$	− 17.6334i	− 1.73747i	0.495274	−	0.868737i	$-0.335068\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 5.99790i	− 0.579839i	−0.957051	−	0.289920i	$-0.906371\pi$
$107$	− 5.99790i	− 0.579839i	0.957051	−	0.289920i	$-0.0936286\pi$
$108$	0	0
$109$	−0.349355	−0.0334622	−0.0167311	−	0.999860i	$-0.505326\pi$
$109$	−0.349355	−0.0334622	−0.0167311	+	0.999860i	$0.505326\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.59458i	0.244078i	0.992525	+	0.122039i	$0.0389433\pi$
$113$	2.59458i	0.244078i	−0.992525	+	0.122039i	$0.961057\pi$
$114$	0	0
$115$	− 2.90791i	− 0.271164i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.33327	−0.121206
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	18.0908	1.60529	0.802647	−	0.596454i	$-0.203424\pi$
$127$	18.0908	1.60529	0.802647	+	0.596454i	$0.203424\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.81849	0.595734	0.297867	−	0.954607i	$-0.403725\pi$
$131$	6.81849	0.595734	0.297867	+	0.954607i	$0.403725\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.44111i	− 0.464866i	−0.972612	−	0.232433i	$-0.925331\pi$
$137$	− 5.44111i	− 0.464866i	0.972612	−	0.232433i	$-0.0746686\pi$
$138$	0	0
$139$	0.546760i	0.0463756i	0.999731	+	0.0231878i	$0.00738156\pi$
$139$	0.546760i	0.0463756i	−0.999731	+	0.0231878i	$0.992618\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.0804738	−0.00672956
$144$	0	0
$145$	− 6.73275i	− 0.559125i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 3.05347i	− 0.250150i	−0.992147	−	0.125075i	$-0.960083\pi$
$149$	− 3.05347i	− 0.250150i	0.992147	−	0.125075i	$-0.0399172\pi$
$150$	0	0
$151$	6.01454	0.489456	0.244728	−	0.969592i	$-0.421301\pi$
$151$	6.01454	0.489456	0.244728	+	0.969592i	$0.421301\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 6.02822i	− 0.484198i
$156$	0	0
$157$	7.75523i	0.618935i	0.950910	+	0.309467i	$0.100151\pi$
$157$	7.75523i	0.618935i	−0.950910	+	0.309467i	$0.899849\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	10.7151	0.839268	0.419634	−	0.907693i	$-0.362158\pi$
$163$	10.7151	0.839268	0.419634	+	0.907693i	$0.362158\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.826998	0.0639951	0.0319975	−	0.999488i	$-0.489813\pi$
$167$	0.826998	0.0639951	0.0319975	+	0.999488i	$0.489813\pi$
$168$	0	0
$169$	12.9995	0.999960
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.07183	0.613690	0.306845	−	0.951760i	$-0.400727\pi$
$173$	8.07183	0.613690	0.306845	+	0.951760i	$0.400727\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.47813i	0.484199i	0.970251	+	0.242099i	$0.0778360\pi$
$179$	6.47813i	0.484199i	−0.970251	+	0.242099i	$0.922164\pi$
$180$	0	0
$181$	19.5153i	1.45056i	0.688453	+	0.725281i	$0.258290\pi$
$181$	19.5153i	1.45056i	−0.688453	+	0.725281i	$0.741710\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.23615	0.311448
$186$	0	0
$187$	− 1.34071i	− 0.0980428i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.14754i	0.155390i	0.996977	+	0.0776951i	$0.0247561\pi$
$191$	2.14754i	0.155390i	−0.996977	+	0.0776951i	$0.975244\pi$
$192$	0	0
$193$	18.5889	1.33806	0.669029	−	0.743236i	$-0.266710\pi$
$193$	18.5889	1.33806	0.669029	+	0.743236i	$0.266710\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.7060i	0.905267i	0.891697	+	0.452634i	$0.149515\pi$
$197$	12.7060i	0.905267i	−0.891697	+	0.452634i	$0.850485\pi$
$198$	0	0
$199$	3.85503i	0.273276i	0.990621	+	0.136638i	$0.0436296\pi$
$199$	3.85503i	0.273276i	−0.990621	+	0.136638i	$0.956370\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3.37244	0.235541
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−12.7964	−0.885142
$210$	0	0
$211$	24.0629	1.65656	0.828279	−	0.560316i	$-0.189320\pi$
$211$	24.0629	1.65656	0.828279	+	0.560316i	$0.189320\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.392652	0.0267787
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 0.00874807i	0 0.000588459i
$222$	0	0
$223$	8.06200i	0.539871i	0.962878	+	0.269936i	$0.0870024\pi$
$223$	8.06200i	0.539871i	−0.962878	+	0.269936i	$0.912998\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.4651	−0.893708	−0.446854	−	0.894607i	$-0.647456\pi$
$227$	−13.4651	−0.893708	−0.446854	+	0.894607i	$0.647456\pi$
$228$	0	0
$229$	2.71140i	0.179174i	0.995979	+	0.0895872i	$0.0285548\pi$
$229$	2.71140i	0.179174i	−0.995979	+	0.0895872i	$0.971445\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 22.7239i	− 1.48870i	−0.667792	−	0.744348i	$-0.732761\pi$
$233$	− 22.7239i	− 1.48870i	0.667792	−	0.744348i	$-0.267239\pi$
$234$	0	0
$235$	4.14792	0.270580
$236$	0	0
$237$	0	0
$238$	0	0
$239$	9.61187i	0.621740i	0.950452	+	0.310870i	$0.100620\pi$
$239$	9.61187i	0.621740i	−0.950452	+	0.310870i	$0.899380\pi$
$240$	0	0
$241$	6.79247i	0.437541i	0.975776	+	0.218771i	$0.0702047\pi$
$241$	6.79247i	0.437541i	−0.975776	+	0.218771i	$0.929795\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.0834954	−0.00531268
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.48499	0.409329	0.204664	−	0.978832i	$-0.434390\pi$
$251$	6.48499	0.409329	0.204664	+	0.978832i	$0.434390\pi$
$252$	0	0
$253$	10.2122	0.642036
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.8006	1.29751	0.648753	−	0.760999i	$-0.275291\pi$
$257$	20.8006	1.29751	0.648753	+	0.760999i	$0.275291\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	27.5988i	1.70181i	0.525316	+	0.850907i	$0.323947\pi$
$263$	27.5988i	1.70181i	−0.525316	+	0.850907i	$0.676053\pi$
$264$	0	0
$265$	− 7.61259i	− 0.467638i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−11.1847	−0.681944	−0.340972	−	0.940073i	$-0.610756\pi$
$269$	−11.1847	−0.681944	−0.340972	+	0.940073i	$0.610756\pi$
$270$	0	0
$271$	4.68359i	0.284508i	0.989830	+	0.142254i	$0.0454350\pi$
$271$	4.68359i	0.284508i	−0.989830	+	0.142254i	$0.954565\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 3.51188i	− 0.211774i
$276$	0	0
$277$	−11.2581	−0.676434	−0.338217	−	0.941068i	$-0.609824\pi$
$277$	−11.2581	−0.676434	−0.338217	+	0.941068i	$0.609824\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.7580i	1.29797i	0.760800	+	0.648987i	$0.224807\pi$
$281$	21.7580i	1.29797i	−0.760800	+	0.648987i	$0.775193\pi$
$282$	0	0
$283$	12.1038i	0.719499i	0.933049	+	0.359750i	$0.117138\pi$
$283$	12.1038i	0.719499i	−0.933049	+	0.359750i	$0.882862\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.8543	−0.991427
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−29.9176	−1.74780	−0.873901	−	0.486103i	$-0.838418\pi$
$293$	−29.9176	−1.74780	−0.873901	+	0.486103i	$0.838418\pi$
$294$	0	0
$295$	1.37092	0.0798181
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.0666340	0.00385354
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 1.03392i	− 0.0592020i
$306$	0	0
$307$	9.46444i	0.540164i	0.962837	+	0.270082i	$0.0870509\pi$
$307$	9.46444i	0.540164i	−0.962837	+	0.270082i	$0.912949\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−0.439728	−0.0249347	−0.0124673	−	0.999922i	$-0.503969\pi$
$311$	−0.439728	−0.0249347	−0.0124673	+	0.999922i	$0.503969\pi$
$312$	0	0
$313$	− 22.0215i	− 1.24473i	−0.782727	−	0.622366i	$-0.786172\pi$
$313$	− 22.0215i	− 1.24473i	0.782727	−	0.622366i	$-0.213828\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.2850i	1.36398i	0.731361	+	0.681991i	$0.238885\pi$
$317$	24.2850i	1.36398i	−0.731361	+	0.681991i	$0.761115\pi$
$318$	0	0
$319$	23.6446	1.32384
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 1.39106i	− 0.0774004i
$324$	0	0
$325$	− 0.0229148i	− 0.00127108i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−16.5275	−0.908432	−0.454216	−	0.890891i	$-0.650081\pi$
$331$	−16.5275	−0.908432	−0.454216	+	0.890891i	$0.650081\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−10.2908	−0.562248
$336$	0	0
$337$	11.6073	0.632288	0.316144	−	0.948711i	$-0.397612\pi$
$337$	11.6073	0.632288	0.316144	+	0.948711i	$0.397612\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	21.1703	1.14644
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.9248i	0.801203i	0.916252	+	0.400602i	$0.131199\pi$
$347$	14.9248i	0.801203i	−0.916252	+	0.400602i	$0.868801\pi$
$348$	0	0
$349$	33.2809i	1.78148i	0.454510	+	0.890742i	$0.349814\pi$
$349$	33.2809i	1.78148i	−0.454510	+	0.890742i	$0.650186\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	31.3889	1.67066	0.835332	−	0.549746i	$-0.185275\pi$
$353$	31.3889	1.67066	0.835332	+	0.549746i	$0.185275\pi$
$354$	0	0
$355$	− 6.95032i	− 0.368885i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9.65280i	0.509455i	0.967013	+	0.254728i	$0.0819858\pi$
$359$	9.65280i	0.509455i	−0.967013	+	0.254728i	$0.918014\pi$
$360$	0	0
$361$	5.72317	0.301219
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.99591i	0.523210i
$366$	0	0
$367$	8.28594i	0.432522i	0.976336	+	0.216261i	$0.0693863\pi$
$367$	8.28594i	0.432522i	−0.976336	+	0.216261i	$0.930614\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−3.36379	−0.174171	−0.0870853	−	0.996201i	$-0.527755\pi$
$373$	−3.36379	−0.174171	−0.0870853	+	0.996201i	$0.527755\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.154279	0.00794579
$378$	0	0
$379$	13.9276	0.715414	0.357707	−	0.933834i	$-0.383559\pi$
$379$	13.9276	0.715414	0.357707	+	0.933834i	$0.383559\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−19.3064	−0.986512	−0.493256	−	0.869884i	$-0.664193\pi$
$383$	−19.3064	−0.986512	−0.493256	+	0.869884i	$0.664193\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.20364i	0.314537i	0.987556	+	0.157268i	$0.0502688\pi$
$389$	6.20364i	0.314537i	−0.987556	+	0.157268i	$0.949731\pi$
$390$	0	0
$391$	1.11014i	0.0561422i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.20441	−0.110916
$396$	0	0
$397$	− 31.4997i	− 1.58093i	−0.612510	−	0.790463i	$-0.709840\pi$
$397$	− 31.4997i	− 1.58093i	0.612510	−	0.790463i	$-0.290160\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	20.3235i	1.01491i	0.861679	+	0.507453i	$0.169413\pi$
$401$	20.3235i	1.01491i	−0.861679	+	0.507453i	$0.830587\pi$
$402$	0	0
$403$	0.138135	0.00688100
$404$	0	0
$405$	0	0
$406$	0	0
$407$	14.8768i	0.737416i
$408$	0	0
$409$	2.89277i	0.143038i	0.997439	+	0.0715190i	$0.0227847\pi$
$409$	2.89277i	0.143038i	−0.997439	+	0.0715190i	$0.977215\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	3.95259	0.194025
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−0.642616	−0.0313938	−0.0156969	−	0.999877i	$-0.504997\pi$
$419$	−0.642616	−0.0313938	−0.0156969	+	0.999877i	$0.504997\pi$
$420$	0	0
$421$	15.4567	0.753316	0.376658	−	0.926352i	$-0.377073\pi$
$421$	15.4567	0.753316	0.376658	+	0.926352i	$0.377073\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.381766	0.0185184
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.52200i	0.265986i	0.991117	+	0.132993i	$0.0424587\pi$
$431$	5.52200i	0.265986i	−0.991117	+	0.132993i	$0.957541\pi$
$432$	0	0
$433$	0.495856i	0.0238293i	0.999929	+	0.0119147i	$0.00379264\pi$
$433$	0.495856i	0.0238293i	−0.999929	+	0.0119147i	$0.996207\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.5957	0.506859
$438$	0	0
$439$	− 25.0809i	− 1.19704i	−0.801106	−	0.598522i	$-0.795755\pi$
$439$	− 25.0809i	− 1.19704i	0.801106	−	0.598522i	$-0.204245\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 2.22789i	− 0.105850i	−0.998598	−	0.0529250i	$-0.983146\pi$
$443$	− 2.22789i	− 0.105850i	0.998598	−	0.0529250i	$-0.0168544\pi$
$444$	0	0
$445$	−8.45072	−0.400602
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 8.70571i	− 0.410848i	−0.978673	−	0.205424i	$-0.934143\pi$
$449$	− 8.70571i	− 0.410848i	0.978673	−	0.205424i	$-0.0658573\pi$
$450$	0	0
$451$	11.8436i	0.557693i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.54476	0.212595	0.106297	−	0.994334i	$-0.466100\pi$
$457$	4.54476	0.212595	0.106297	+	0.994334i	$0.466100\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	11.5276	0.536896	0.268448	−	0.963294i	$-0.413489\pi$
$461$	11.5276	0.536896	0.268448	+	0.963294i	$0.413489\pi$
$462$	0	0
$463$	21.4845	0.998470	0.499235	−	0.866467i	$-0.333615\pi$
$463$	21.4845	0.998470	0.499235	+	0.866467i	$0.333615\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	29.9593	1.38635	0.693177	−	0.720768i	$-0.256211\pi$
$467$	29.9593	1.38635	0.693177	+	0.720768i	$0.256211\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.37895i	0.0634040i
$474$	0	0
$475$	− 3.64374i	− 0.167186i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	17.8294	0.814646	0.407323	−	0.913284i	$-0.366462\pi$
$479$	17.8294	0.814646	0.407323	+	0.913284i	$0.366462\pi$
$480$	0	0
$481$	0.0970702i	0.00442602i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 8.06301i	− 0.366122i
$486$	0	0
$487$	31.4662	1.42587	0.712934	−	0.701231i	$-0.247366\pi$
$487$	31.4662	1.42587	0.712934	+	0.701231i	$0.247366\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12.7933i	0.577352i	0.957427	+	0.288676i	$0.0932150\pi$
$491$	12.7933i	0.577352i	−0.957427	+	0.288676i	$0.906785\pi$
$492$	0	0
$493$	2.57033i	0.115762i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	12.5790	0.563112	0.281556	−	0.959545i	$-0.409149\pi$
$499$	12.5790	0.563112	0.281556	+	0.959545i	$0.409149\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−15.1999	−0.677732	−0.338866	−	0.940835i	$-0.610043\pi$
$503$	−15.1999	−0.677732	−0.338866	+	0.940835i	$0.610043\pi$
$504$	0	0
$505$	10.6591	0.474324
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.33524	0.0591833	0.0295916	−	0.999562i	$-0.490579\pi$
$509$	1.33524	0.0591833	0.0295916	+	0.999562i	$0.490579\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	17.6334i	0.777022i
$516$	0	0
$517$	14.5670i	0.640654i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0.377866	0.0165546	0.00827730	−	0.999966i	$-0.497365\pi$
$521$	0.377866	0.0165546	0.00827730	+	0.999966i	$0.497365\pi$
$522$	0	0
$523$	− 18.8082i	− 0.822425i	−0.911540	−	0.411212i	$-0.865105\pi$
$523$	− 18.8082i	− 0.822425i	0.911540	−	0.411212i	$-0.134895\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.30137i	0.100249i
$528$	0	0
$529$	14.5441	0.632351
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0.0772787i	0.00334731i
$534$	0	0
$535$	5.99790i	0.259312i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−34.1532	−1.46836	−0.734181	−	0.678954i	$-0.762434\pi$
$541$	−34.1532	−1.46836	−0.734181	+	0.678954i	$0.762434\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0.349355	0.0149647
$546$	0	0
$547$	−13.8192	−0.590865	−0.295433	−	0.955364i	$-0.595464\pi$
$547$	−13.8192	−0.590865	−0.295433	+	0.955364i	$0.595464\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	24.5324	1.04511
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.16193i	− 0.0916041i	−0.998951	−	0.0458021i	$-0.985416\pi$
$557$	− 2.16193i	− 0.0916041i	0.998951	−	0.0458021i	$-0.0145843\pi$
$558$	0	0
$559$	0.00899753i	0 0.000380555i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−34.8552	−1.46897	−0.734486	−	0.678624i	$-0.762577\pi$
$563$	−34.8552	−1.46897	−0.734486	+	0.678624i	$0.762577\pi$
$564$	0	0
$565$	− 2.59458i	− 0.109155i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 10.1515i	− 0.425572i	−0.977099	−	0.212786i	$-0.931746\pi$
$569$	− 10.1515i	− 0.425572i	0.977099	−	0.212786i	$-0.0682537\pi$
$570$	0	0
$571$	42.7761	1.79013	0.895063	−	0.445940i	$-0.147131\pi$
$571$	42.7761	1.79013	0.895063	+	0.445940i	$0.147131\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.90791i	0.121268i
$576$	0	0
$577$	− 4.94352i	− 0.205801i	−0.994692	−	0.102901i	$-0.967188\pi$
$577$	− 4.94352i	− 0.205801i	0.994692	−	0.102901i	$-0.0328124\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	26.7345	1.10723
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−37.7443	−1.55787	−0.778937	−	0.627102i	$-0.784241\pi$
$587$	−37.7443	−1.55787	−0.778937	+	0.627102i	$0.784241\pi$
$588$	0	0
$589$	21.9652	0.905062
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−34.4434	−1.41442	−0.707210	−	0.707003i	$-0.750047\pi$
$593$	−34.4434	−1.41442	−0.707210	+	0.707003i	$0.750047\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5.57402i	0.227748i	0.993495	+	0.113874i	$0.0363261\pi$
$599$	5.57402i	0.227748i	−0.993495	+	0.113874i	$0.963674\pi$
$600$	0	0
$601$	35.0801i	1.43095i	0.698640	+	0.715473i	$0.253789\pi$
$601$	35.0801i	1.43095i	−0.698640	+	0.715473i	$0.746211\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.33327	0.0542052
$606$	0	0
$607$	43.8200i	1.77860i	0.457327	+	0.889299i	$0.348807\pi$
$607$	43.8200i	1.77860i	−0.457327	+	0.889299i	$0.651193\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.0950485i	0.00384525i
$612$	0	0
$613$	−32.8674	−1.32750	−0.663752	−	0.747953i	$-0.731037\pi$
$613$	−32.8674	−1.32750	−0.663752	+	0.747953i	$0.731037\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24.7029i	0.994502i	0.867607	+	0.497251i	$0.165657\pi$
$617$	24.7029i	0.994502i	−0.867607	+	0.497251i	$0.834343\pi$
$618$	0	0
$619$	26.8563i	1.07944i	0.841843	+	0.539722i	$0.181471\pi$
$619$	26.8563i	1.07944i	−0.841843	+	0.539722i	$0.818529\pi$
$620$	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
$629$	−1.61722	−0.0644826
$630$	0	0
$631$	−12.6228	−0.502507	−0.251253	−	0.967921i	$-0.580843\pi$
$631$	−12.6228	−0.502507	−0.251253	+	0.967921i	$0.580843\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−18.0908	−0.717910
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 29.9627i	− 1.18345i	−0.806138	−	0.591727i	$-0.798446\pi$
$641$	− 29.9627i	− 1.18345i	0.806138	−	0.591727i	$-0.201554\pi$
$642$	0	0
$643$	− 18.0469i	− 0.711701i	−0.934543	−	0.355851i	$-0.884191\pi$
$643$	− 18.0469i	− 0.711701i	0.934543	−	0.355851i	$-0.115809\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.4542	−0.489625	−0.244812	−	0.969570i	$-0.578726\pi$
$647$	−12.4542	−0.489625	−0.244812	+	0.969570i	$0.578726\pi$
$648$	0	0
$649$	4.81450i	0.188986i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 17.2965i	− 0.676864i	−0.940991	−	0.338432i	$-0.890104\pi$
$653$	− 17.2965i	− 0.676864i	0.940991	−	0.338432i	$-0.109896\pi$
$654$	0	0
$655$	−6.81849	−0.266421
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24.0060i	0.935141i	0.883956	+	0.467570i	$0.154871\pi$
$659$	24.0060i	0.935141i	−0.883956	+	0.467570i	$0.845129\pi$
$660$	0	0
$661$	− 38.9737i	− 1.51590i	−0.652312	−	0.757951i	$-0.726201\pi$
$661$	− 38.9737i	− 1.51590i	0.652312	−	0.757951i	$-0.273799\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−19.5782	−0.758071
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3.63099	0.140173
$672$	0	0
$673$	43.6204	1.68144	0.840722	−	0.541467i	$-0.182131\pi$
$673$	43.6204	1.68144	0.840722	+	0.541467i	$0.182131\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17.3622	0.667285	0.333642	−	0.942700i	$-0.391722\pi$
$677$	17.3622	0.667285	0.333642	+	0.942700i	$0.391722\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	34.8123i	1.33206i	0.745926	+	0.666029i	$0.232007\pi$
$683$	34.8123i	1.33206i	−0.745926	+	0.666029i	$0.767993\pi$
$684$	0	0
$685$	5.44111i	0.207894i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0.174441	0.00664566
$690$	0	0
$691$	− 0.215376i	− 0.00819330i	−0.999992	−	0.00409665i	$-0.998696\pi$
$691$	− 0.215376i	− 0.00819330i	0.999992	−	0.00409665i	$-0.00130401\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 0.546760i	− 0.0207398i
$696$	0	0
$697$	−1.28748	−0.0487669
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 17.2532i	− 0.651643i	−0.945431	−	0.325822i	$-0.894359\pi$
$701$	− 17.2532i	− 0.651643i	0.945431	−	0.325822i	$-0.105641\pi$
$702$	0	0
$703$	15.4354i	0.582157i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−7.73349	−0.290437	−0.145219	−	0.989400i	$-0.546389\pi$
$709$	−7.73349	−0.290437	−0.145219	+	0.989400i	$0.546389\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−17.5295	−0.656484
$714$	0	0
$715$	0.0804738	0.00300955
$716$	0	0
$717$	0	0
$718$	0	0
$719$	40.1361	1.49683	0.748413	−	0.663233i	$-0.230816\pi$
$719$	40.1361	1.49683	0.748413	+	0.663233i	$0.230816\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.73275i	0.250048i
$726$	0	0
$727$	3.51975i	0.130540i	0.997868	+	0.0652702i	$0.0207909\pi$
$727$	3.51975i	0.130540i	−0.997868	+	0.0652702i	$0.979209\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.149901	−0.00554430
$732$	0	0
$733$	− 28.8462i	− 1.06546i	−0.846286	−	0.532729i	$-0.821167\pi$
$733$	− 28.8462i	− 1.06546i	0.846286	−	0.532729i	$-0.178833\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 36.1401i	− 1.33124i
$738$	0	0
$739$	−12.2017	−0.448846	−0.224423	−	0.974492i	$-0.572050\pi$
$739$	−12.2017	−0.448846	−0.224423	+	0.974492i	$0.572050\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	13.8883i	0.509511i	0.967005	+	0.254756i	$0.0819950\pi$
$743$	13.8883i	0.509511i	−0.967005	+	0.254756i	$0.918005\pi$
$744$	0	0
$745$	3.05347i	0.111870i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	38.6816	1.41151	0.705756	−	0.708455i	$-0.250608\pi$
$751$	38.6816	1.41151	0.705756	+	0.708455i	$0.250608\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−6.01454	−0.218891
$756$	0	0
$757$	3.79970	0.138102	0.0690512	−	0.997613i	$-0.478003\pi$
$757$	3.79970	0.138102	0.0690512	+	0.997613i	$0.478003\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	53.5492	1.94116	0.970578	−	0.240788i	$-0.0774058\pi$
$761$	53.5492	1.94116	0.970578	+	0.240788i	$0.0774058\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.0314143i	0.00113431i
$768$	0	0
$769$	6.57615i	0.237142i	0.992946	+	0.118571i	$0.0378313\pi$
$769$	6.57615i	0.237142i	−0.992946	+	0.118571i	$0.962169\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−42.1280	−1.51524	−0.757619	−	0.652698i	$-0.773637\pi$
$773$	−42.1280	−1.51524	−0.757619	+	0.652698i	$0.773637\pi$
$774$	0	0
$775$	6.02822i	0.216540i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12.2883i	0.440274i
$780$	0	0
$781$	24.4087	0.873411
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 7.75523i	− 0.276796i
$786$	0	0
$787$	− 8.18111i	− 0.291625i	−0.989312	−	0.145813i	$-0.953420\pi$
$787$	− 8.18111i	− 0.291625i	0.989312	−	0.145813i	$-0.0465796\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0.0236920	0.000841327 0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	35.1931	1.24661	0.623303	−	0.781981i	$-0.285790\pi$
$797$	35.1931	1.24661	0.623303	+	0.781981i	$0.285790\pi$
$798$	0	0
$799$	−1.58353	−0.0560214
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−35.1044	−1.23881
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 26.8683i	− 0.944639i	−0.881427	−	0.472320i	$-0.843417\pi$
$809$	− 26.8683i	− 0.944639i	0.881427	−	0.472320i	$-0.156583\pi$
$810$	0	0
$811$	− 3.85722i	− 0.135445i	−0.997704	−	0.0677227i	$-0.978427\pi$
$811$	− 3.85722i	− 0.135445i	0.997704	−	0.0677227i	$-0.0215733\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−10.7151	−0.375332
$816$	0	0
$817$	1.43072i	0.0500546i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.6682i	0.581723i	0.956765	+	0.290861i	$0.0939418\pi$
$821$	16.6682i	0.581723i	−0.956765	+	0.290861i	$0.906058\pi$
$822$	0	0
$823$	−31.5041	−1.09816	−0.549081	−	0.835769i	$-0.685022\pi$
$823$	−31.5041	−1.09816	−0.549081	+	0.835769i	$0.685022\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 23.9192i	− 0.831751i	−0.909421	−	0.415876i	$-0.863475\pi$
$827$	− 23.9192i	− 0.831751i	0.909421	−	0.415876i	$-0.136525\pi$
$828$	0	0
$829$	− 26.7079i	− 0.927603i	−0.885939	−	0.463802i	$-0.846485\pi$
$829$	− 26.7079i	− 0.927603i	0.885939	−	0.463802i	$-0.153515\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−0.826998	−0.0286195
$836$	0	0
$837$	0	0
$838$	0	0
$839$	49.3248	1.70288	0.851441	−	0.524450i	$-0.175729\pi$
$839$	49.3248	1.70288	0.851441	+	0.524450i	$0.175729\pi$
$840$	0	0
$841$	−16.3299	−0.563101
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12.9995	−0.447196
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 12.3183i	− 0.422266i
$852$	0	0
$853$	5.64424i	0.193255i	0.995321	+	0.0966276i	$0.0308056\pi$
$853$	5.64424i	0.193255i	−0.995321	+	0.0966276i	$0.969194\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−4.66747	−0.159438	−0.0797188	−	0.996817i	$-0.525402\pi$
$857$	−4.66747	−0.159438	−0.0797188	+	0.996817i	$0.525402\pi$
$858$	0	0
$859$	− 21.1567i	− 0.721856i	−0.932594	−	0.360928i	$-0.882460\pi$
$859$	− 21.1567i	− 0.721856i	0.932594	−	0.360928i	$-0.117540\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 40.8753i	− 1.39141i	−0.718327	−	0.695705i	$-0.755092\pi$
$863$	− 40.8753i	− 1.39141i	0.718327	−	0.695705i	$-0.244908\pi$
$864$	0	0
$865$	−8.07183	−0.274450
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 7.74163i	− 0.262617i
$870$	0	0
$871$	− 0.235812i	− 0.00799018i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−13.1266	−0.443253	−0.221627	−	0.975132i	$-0.571137\pi$
$877$	−13.1266	−0.443253	−0.221627	+	0.975132i	$0.571137\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.87545	−0.0631854	−0.0315927	−	0.999501i	$-0.510058\pi$
$881$	−1.87545	−0.0631854	−0.0315927	+	0.999501i	$0.510058\pi$
$882$	0	0
$883$	23.3683	0.786405	0.393203	−	0.919452i	$-0.371367\pi$
$883$	23.3683	0.786405	0.393203	+	0.919452i	$0.371367\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−25.5804	−0.858907	−0.429453	−	0.903089i	$-0.641294\pi$
$887$	−25.5804	−0.858907	−0.429453	+	0.903089i	$0.641294\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	15.1139i	0.505768i
$894$	0	0
$895$	− 6.47813i	− 0.216540i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−40.5865	−1.35363
$900$	0	0
$901$	2.90623i	0.0968205i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 19.5153i	− 0.648711i
$906$	0	0
$907$	51.1554	1.69859	0.849293	−	0.527921i	$-0.177028\pi$
$907$	51.1554	1.69859	0.849293	+	0.527921i	$0.177028\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	20.1810i	0.668627i	0.942462	+	0.334313i	$0.108504\pi$
$911$	20.1810i	0.668627i	−0.942462	+	0.334313i	$0.891496\pi$
$912$	0	0
$913$	13.8810i	0.459394i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	38.5942	1.27311	0.636553	−	0.771233i	$-0.280360\pi$
$919$	38.5942	1.27311	0.636553	+	0.771233i	$0.280360\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0.159265	0.00524227
$924$	0	0
$925$	−4.23615	−0.139284
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−27.5393	−0.903534	−0.451767	−	0.892136i	$-0.649206\pi$
$929$	−27.5393	−0.903534	−0.451767	+	0.892136i	$0.649206\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.34071i	0.0438461i
$936$	0	0
$937$	30.3595i	0.991803i	0.868379	+	0.495901i	$0.165162\pi$
$937$	30.3595i	0.991803i	−0.868379	+	0.495901i	$0.834838\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	38.7177	1.26216	0.631080	−	0.775717i	$-0.282612\pi$
$941$	38.7177	1.26216	0.631080	+	0.775717i	$0.282612\pi$
$942$	0	0
$943$	− 9.80674i	− 0.319351i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	51.7592i	1.68195i	0.541075	+	0.840974i	$0.318017\pi$
$947$	51.7592i	1.68195i	−0.541075	+	0.840974i	$0.681983\pi$
$948$	0	0
$949$	−0.229054	−0.00743540
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 14.2596i	− 0.461913i	−0.972964	−	0.230956i	$-0.925815\pi$
$953$	− 14.2596i	− 0.461913i	0.972964	−	0.230956i	$-0.0741855\pi$
$954$	0	0
$955$	− 2.14754i	− 0.0694926i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−5.33938	−0.172238
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−18.5889	−0.598398
$966$	0	0
$967$	9.46768	0.304460	0.152230	−	0.988345i	$-0.451355\pi$
$967$	9.46768	0.304460	0.152230	+	0.988345i	$0.451355\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	46.4153	1.48954	0.744769	−	0.667322i	$-0.232560\pi$
$971$	46.4153	1.48954	0.744769	+	0.667322i	$0.232560\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 24.1346i	− 0.772133i	−0.922471	−	0.386066i	$-0.873834\pi$
$977$	− 24.1346i	− 0.772133i	0.922471	−	0.386066i	$-0.126166\pi$
$978$	0	0
$979$	− 29.6779i	− 0.948509i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	26.7567	0.853406	0.426703	−	0.904392i	$-0.359675\pi$
$983$	26.7567	0.853406	0.426703	+	0.904392i	$0.359675\pi$
$984$	0	0
$985$	− 12.7060i	− 0.404848i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1.14180i	− 0.0363070i
$990$	0	0
$991$	44.1144	1.40134	0.700670	−	0.713486i	$-0.252885\pi$
$991$	44.1144	1.40134	0.700670	+	0.713486i	$0.252885\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 3.85503i	− 0.122213i
$996$	0	0
$997$	− 27.2079i	− 0.861681i	−0.902428	−	0.430841i	$-0.858217\pi$
$997$	− 27.2079i	− 0.861681i	0.902428	−	0.430841i	$-0.141783\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8820.2.d.c.881.4	✓	16
3.2	odd	2		8820.2.d.d.881.13	yes	16
7.6	odd	2		8820.2.d.d.881.4	yes	16
21.20	even	2	inner	8820.2.d.c.881.13	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8820.2.d.c.881.4	✓	16	1.1	even	1	trivial
8820.2.d.c.881.13	yes	16	21.20	even	2	inner
8820.2.d.d.881.4	yes	16	7.6	odd	2
8820.2.d.d.881.13	yes	16	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8820.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +O(q^{10})\)	\(q-1.00000 q^{5} -3.51188i q^{11} -0.0229148i q^{13} +0.381766 q^{17} -3.64374i q^{19} +2.90791i q^{23} +1.00000 q^{25} +6.73275i q^{29} +6.02822i q^{31} -4.23615 q^{37} -3.37244 q^{41} -0.392652 q^{43} -4.14792 q^{47} +7.61259i q^{53} +3.51188i q^{55} -1.37092 q^{59} +1.03392i q^{61} +0.0229148i q^{65} +10.2908 q^{67} +6.95032i q^{71} -9.99591i q^{73} +2.20441 q^{79} -3.95259 q^{83} -0.381766 q^{85} +8.45072 q^{89} +3.64374i q^{95} +8.06301i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{5}+O(q^{10}) \) 16 * q - 16 * q^5	\( 16 q - 16 q^{5} + 16 q^{25} - 32 q^{41} - 32 q^{43} + 32 q^{47} + 32 q^{59} - 32 q^{67} + 64 q^{89}+O(q^{100}) \) 16 * q - 16 * q^5 + 16 * q^25 - 32 * q^41 - 32 * q^43 + 32 * q^47 + 32 * q^59 - 32 * q^67 + 64 * q^89

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