LMFDB - Embedded newform 3528.2.k.b.881.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3528,2,Mod(881,3528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3528, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3528.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3528.k (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:	$28.1712218331$
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} + 10x^{14} + 61x^{12} + 266x^{10} + 852x^{8} + 1438x^{6} + 1933x^{4} + 3038x^{2} + 2401 $ x^16 + 10x^14 + 61x^12 + 266x^10 + 852x^8 + 1438x^6 + 1933x^4 + 3038*x^2 + 2401
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.4
Root			$-0.144868 - 1.25092i$ of defining polynomial
Character	$\chi$	$=$	3528.881
Dual form			3528.2.k.b.881.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.02179 q^{5} +O(q^{10})$	$q-2.02179 q^{5} +3.62576i q^{11} +4.74306i q^{13} +3.43871 q^{17} +5.01899i q^{19} -6.95235i q^{23} -0.912375 q^{25} -2.03630i q^{29} +0.307689i q^{31} -11.6968 q^{37} +8.77283 q^{41} +3.21155 q^{43} +0.384361 q^{47} -6.38680i q^{53} -7.33052i q^{55} -8.51485 q^{59} +1.22024i q^{61} -9.58946i q^{65} +0.0839475 q^{67} -8.41789i q^{71} +10.5228i q^{73} +1.64232 q^{79} -10.3595 q^{83} -6.95235 q^{85} -8.60311 q^{89} -10.1473i q^{95} -12.7477i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 24 q^{25} - 8 q^{37} + 8 q^{43} + 56 q^{67} + 64 q^{79} - 32 q^{85}+O(q^{100}) $ 16 * q - 24 * q^25 - 8 * q^37 + 8 * q^43 + 56 * q^67 + 64 * q^79 - 32 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$1765$	$2647$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$5$	−2.02179	−0.904171	−0.452085	−	0.891975i	$-0.649320\pi$
$5$	−2.02179	−0.904171	−0.452085	+	0.891975i	$0.649320\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.62576i	1.09321i	0.837391	+	0.546604i	$0.184080\pi$
$11$	3.62576i	1.09321i	−0.837391	+	0.546604i	$0.815920\pi$
$12$	0	0
$13$	4.74306i	1.31549i	0.753241	+	0.657744i	$0.228489\pi$
$13$	4.74306i	1.31549i	−0.753241	+	0.657744i	$0.771511\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.43871	0.834010	0.417005	−	0.908904i	$-0.363080\pi$
$17$	3.43871	0.834010	0.417005	+	0.908904i	$0.363080\pi$
$18$	0	0
$19$	5.01899i	1.15144i	0.817648	+	0.575718i	$0.195277\pi$
$19$	5.01899i	1.15144i	−0.817648	+	0.575718i	$0.804723\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 6.95235i	− 1.44966i	−0.688925	−	0.724832i	$-0.741917\pi$
$23$	− 6.95235i	− 1.44966i	0.688925	−	0.724832i	$-0.258083\pi$
$24$	0	0
$25$	−0.912375	−0.182475
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.03630i	− 0.378131i	−0.981965	−	0.189065i	$-0.939454\pi$
$29$	− 2.03630i	− 0.378131i	0.981965	−	0.189065i	$-0.0605458\pi$
$30$	0	0
$31$	0.307689i	0.0552626i	0.999618	+	0.0276313i	$0.00879644\pi$
$31$	0.307689i	0.0552626i	−0.999618	+	0.0276313i	$0.991204\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−11.6968	−1.92295	−0.961473	−	0.274898i	$-0.911356\pi$
$37$	−11.6968	−1.92295	−0.961473	+	0.274898i	$0.911356\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.77283	1.37009	0.685043	−	0.728503i	$-0.259783\pi$
$41$	8.77283	1.37009	0.685043	+	0.728503i	$0.259783\pi$
$42$	0	0
$43$	3.21155	0.489756	0.244878	−	0.969554i	$-0.421252\pi$
$43$	3.21155	0.489756	0.244878	+	0.969554i	$0.421252\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.384361	0.0560649	0.0280324	−	0.999607i	$-0.491076\pi$
$47$	0.384361	0.0560649	0.0280324	+	0.999607i	$0.491076\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 6.38680i	− 0.877294i	−0.898659	−	0.438647i	$-0.855458\pi$
$53$	− 6.38680i	− 0.877294i	0.898659	−	0.438647i	$-0.144542\pi$
$54$	0	0
$55$	− 7.33052i	− 0.988447i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.51485	−1.10854	−0.554270	−	0.832337i	$-0.687002\pi$
$59$	−8.51485	−1.10854	−0.554270	+	0.832337i	$0.687002\pi$
$60$	0	0
$61$	1.22024i	0.156236i	0.996944	+	0.0781179i	$0.0248911\pi$
$61$	1.22024i	0.156236i	−0.996944	+	0.0781179i	$0.975109\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 9.58946i	− 1.18943i
$66$	0	0
$67$	0.0839475	0.0102558	0.00512791	−	0.999987i	$-0.498368\pi$
$67$	0.0839475	0.0102558	0.00512791	+	0.999987i	$0.498368\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 8.41789i	− 0.999020i	−0.866308	−	0.499510i	$-0.833513\pi$
$71$	− 8.41789i	− 0.999020i	0.866308	−	0.499510i	$-0.166487\pi$
$72$	0	0
$73$	10.5228i	1.23160i	0.787901	+	0.615802i	$0.211168\pi$
$73$	10.5228i	1.23160i	−0.787901	+	0.615802i	$0.788832\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.64232	0.184775	0.0923876	−	0.995723i	$-0.470550\pi$
$79$	1.64232	0.184775	0.0923876	+	0.995723i	$0.470550\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.3595	−1.13710	−0.568550	−	0.822648i	$-0.692496\pi$
$83$	−10.3595	−1.13710	−0.568550	+	0.822648i	$0.692496\pi$
$84$	0	0
$85$	−6.95235	−0.754088
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−8.60311	−0.911928	−0.455964	−	0.889998i	$-0.650705\pi$
$89$	−8.60311	−0.911928	−0.455964	+	0.889998i	$0.650705\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 10.1473i	− 1.04109i
$96$	0	0
$97$	− 12.7477i	− 1.29433i	−0.762350	−	0.647165i	$-0.775954\pi$
$97$	− 12.7477i	− 1.29433i	0.762350	−	0.647165i	$-0.224046\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.6000	−1.35325	−0.676625	−	0.736327i	$-0.736558\pi$
$101$	−13.6000	−1.35325	−0.676625	+	0.736327i	$0.736558\pi$
$102$	0	0
$103$	17.9797i	1.77160i	0.464071	+	0.885798i	$0.346388\pi$
$103$	17.9797i	1.77160i	−0.464071	+	0.885798i	$0.653612\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.99600i	0.676329i	0.941087	+	0.338164i	$0.109806\pi$
$107$	6.99600i	0.676329i	−0.941087	+	0.338164i	$0.890194\pi$
$108$	0	0
$109$	−12.9197	−1.23749	−0.618743	−	0.785594i	$-0.712358\pi$
$109$	−12.9197	−1.23749	−0.618743	+	0.785594i	$0.712358\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.61440i	0.810375i	0.914234	+	0.405187i	$0.132794\pi$
$113$	8.61440i	0.810375i	−0.914234	+	0.405187i	$0.867206\pi$
$114$	0	0
$115$	14.0562i	1.31074i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2.14614	−0.195103
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.9536	1.06916
$126$	0	0
$127$	−18.8607	−1.67362	−0.836809	−	0.547495i	$-0.815582\pi$
$127$	−18.8607	−1.67362	−0.836809	+	0.547495i	$0.815582\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−13.1723	−1.15087	−0.575435	−	0.817848i	$-0.695167\pi$
$131$	−13.1723	−1.15087	−0.575435	+	0.817848i	$0.695167\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21.7442i	1.85773i	0.370420	+	0.928864i	$0.379214\pi$
$137$	21.7442i	1.85773i	−0.370420	+	0.928864i	$0.620786\pi$
$138$	0	0
$139$	− 6.59355i	− 0.559258i	−0.960108	−	0.279629i	$-0.909789\pi$
$139$	− 6.59355i	− 0.559258i	0.960108	−	0.279629i	$-0.0902114\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−17.1972	−1.43810
$144$	0	0
$145$	4.11696i	0.341895i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.9121i	0.975873i	0.872879	+	0.487937i	$0.162250\pi$
$149$	11.9121i	0.975873i	−0.872879	+	0.487937i	$0.837750\pi$
$150$	0	0
$151$	15.0400	1.22394	0.611968	−	0.790883i	$-0.290378\pi$
$151$	15.0400	1.22394	0.611968	+	0.790883i	$0.290378\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 0.622082i	− 0.0499668i
$156$	0	0
$157$	18.1251i	1.44654i	0.690564	+	0.723272i	$0.257363\pi$
$157$	18.1251i	1.44654i	−0.690564	+	0.723272i	$0.742637\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−20.3687	−1.59540	−0.797700	−	0.603054i	$-0.793950\pi$
$163$	−20.3687	−1.59540	−0.797700	+	0.603054i	$0.793950\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.1056	0.859377	0.429688	−	0.902977i	$-0.358623\pi$
$167$	11.1056	0.859377	0.429688	+	0.902977i	$0.358623\pi$
$168$	0	0
$169$	−9.49664	−0.730511
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−5.28920	−0.402131	−0.201065	−	0.979578i	$-0.564440\pi$
$173$	−5.28920	−0.402131	−0.201065	+	0.979578i	$0.564440\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 25.0271i	− 1.87061i	−0.353839	−	0.935306i	$-0.615124\pi$
$179$	− 25.0271i	− 1.87061i	0.353839	−	0.935306i	$-0.384876\pi$
$180$	0	0
$181$	− 1.10023i	− 0.0817793i	−0.999164	−	0.0408896i	$-0.986981\pi$
$181$	− 1.10023i	− 0.0817793i	0.999164	−	0.0408896i	$-0.0130192\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	23.6485	1.73867
$186$	0	0
$187$	12.4679i	0.911747i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 23.7829i	− 1.72087i	−0.509558	−	0.860436i	$-0.670191\pi$
$191$	− 23.7829i	− 1.72087i	0.509558	−	0.860436i	$-0.329809\pi$
$192$	0	0
$193$	−9.07259	−0.653059	−0.326530	−	0.945187i	$-0.605879\pi$
$193$	−9.07259	−0.653059	−0.326530	+	0.945187i	$0.605879\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5.34682i	− 0.380945i	−0.981692	−	0.190473i	$-0.938998\pi$
$197$	− 5.34682i	− 0.380945i	0.981692	−	0.190473i	$-0.0610021\pi$
$198$	0	0
$199$	− 13.0010i	− 0.921617i	−0.887500	−	0.460808i	$-0.847560\pi$
$199$	− 13.0010i	− 0.921617i	0.887500	−	0.460808i	$-0.152440\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−17.7368	−1.23879
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−18.1977	−1.25876
$210$	0	0
$211$	4.77710	0.328869	0.164434	−	0.986388i	$-0.447420\pi$
$211$	4.77710	0.328869	0.164434	+	0.986388i	$0.447420\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.49307	−0.442823
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	16.3100i	1.09713i
$222$	0	0
$223$	− 7.26700i	− 0.486634i	−0.969947	−	0.243317i	$-0.921764\pi$
$223$	− 7.26700i	− 0.486634i	0.969947	−	0.243317i	$-0.0782356\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.81751	−0.319749	−0.159875	−	0.987137i	$-0.551109\pi$
$227$	−4.81751	−0.319749	−0.159875	+	0.987137i	$0.551109\pi$
$228$	0	0
$229$	12.7772i	0.844342i	0.906516	+	0.422171i	$0.138732\pi$
$229$	12.7772i	0.844342i	−0.906516	+	0.422171i	$0.861268\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 18.0851i	− 1.18480i	−0.805645	−	0.592399i	$-0.798181\pi$
$233$	− 18.0851i	− 1.18480i	0.805645	−	0.592399i	$-0.201819\pi$
$234$	0	0
$235$	−0.777097	−0.0506922
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.01256i	0.388920i	0.980910	+	0.194460i	$0.0622954\pi$
$239$	6.01256i	0.388920i	−0.980910	+	0.194460i	$0.937705\pi$
$240$	0	0
$241$	9.76048i	0.628728i	0.949303	+	0.314364i	$0.101791\pi$
$241$	9.76048i	0.628728i	−0.949303	+	0.314364i	$0.898209\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−23.8054	−1.51470
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−14.2692	−0.900666	−0.450333	−	0.892861i	$-0.648695\pi$
$251$	−14.2692	−0.900666	−0.450333	+	0.892861i	$0.648695\pi$
$252$	0	0
$253$	25.2075	1.58478
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.5780	−0.909350	−0.454675	−	0.890657i	$-0.650245\pi$
$257$	−14.5780	−0.909350	−0.454675	+	0.890657i	$0.650245\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 15.1172i	− 0.932166i	−0.884741	−	0.466083i	$-0.845665\pi$
$263$	− 15.1172i	− 0.932166i	0.884741	−	0.466083i	$-0.154335\pi$
$264$	0	0
$265$	12.9127i	0.793224i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.70337	0.103856	0.0519282	−	0.998651i	$-0.483463\pi$
$269$	1.70337	0.103856	0.0519282	+	0.998651i	$0.483463\pi$
$270$	0	0
$271$	− 28.5571i	− 1.73472i	−0.497679	−	0.867361i	$-0.665814\pi$
$271$	− 28.5571i	− 1.73472i	0.497679	−	0.867361i	$-0.334186\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 3.30805i	− 0.199483i
$276$	0	0
$277$	−2.48928	−0.149566	−0.0747832	−	0.997200i	$-0.523826\pi$
$277$	−2.48928	−0.149566	−0.0747832	+	0.997200i	$0.523826\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 0.670935i	− 0.0400246i	−0.999800	−	0.0200123i	$-0.993629\pi$
$281$	− 0.670935i	− 0.0400246i	0.999800	−	0.0200123i	$-0.00637054\pi$
$282$	0	0
$283$	− 20.7672i	− 1.23448i	−0.786775	−	0.617240i	$-0.788251\pi$
$283$	− 20.7672i	− 1.23448i	0.786775	−	0.617240i	$-0.211749\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−5.17525	−0.304427
$290$	0	0
$291$	0	0
$292$	0	0
$293$	20.9828	1.22583	0.612914	−	0.790149i	$-0.289997\pi$
$293$	20.9828	1.22583	0.612914	+	0.790149i	$0.289997\pi$
$294$	0	0
$295$	17.2152	1.00231
$296$	0	0
$297$	0	0
$298$	0	0
$299$	32.9754	1.90702
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 2.46707i	− 0.141264i
$306$	0	0
$307$	4.77897i	0.272750i	0.990657	+	0.136375i	$0.0435452\pi$
$307$	4.77897i	0.272750i	−0.990657	+	0.136375i	$0.956455\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−25.3974	−1.44016	−0.720078	−	0.693893i	$-0.755894\pi$
$311$	−25.3974	−1.44016	−0.720078	+	0.693893i	$0.755894\pi$
$312$	0	0
$313$	28.3736i	1.60377i	0.597477	+	0.801886i	$0.296170\pi$
$313$	28.3736i	1.60377i	−0.597477	+	0.801886i	$0.703830\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 11.8684i	− 0.666596i	−0.942822	−	0.333298i	$-0.891839\pi$
$317$	− 11.8684i	− 0.666596i	0.942822	−	0.333298i	$-0.108161\pi$
$318$	0	0
$319$	7.38312	0.413375
$320$	0	0
$321$	0	0
$322$	0	0
$323$	17.2589i	0.960310i
$324$	0	0
$325$	− 4.32745i	− 0.240044i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	14.3465	0.788555	0.394278	−	0.918991i	$-0.370995\pi$
$331$	14.3465	0.788555	0.394278	+	0.918991i	$0.370995\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−0.169724	−0.00927301
$336$	0	0
$337$	−26.8167	−1.46080	−0.730401	−	0.683019i	$-0.760667\pi$
$337$	−26.8167	−1.46080	−0.730401	+	0.683019i	$0.760667\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.11561	−0.0604135
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.33947i	0.394003i	0.980403	+	0.197002i	$0.0631204\pi$
$347$	7.33947i	0.394003i	−0.980403	+	0.197002i	$0.936880\pi$
$348$	0	0
$349$	22.9882i	1.23053i	0.788320	+	0.615265i	$0.210951\pi$
$349$	22.9882i	1.23053i	−0.788320	+	0.615265i	$0.789049\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11.6426	0.619672	0.309836	−	0.950790i	$-0.399726\pi$
$353$	11.6426	0.619672	0.309836	+	0.950790i	$0.399726\pi$
$354$	0	0
$355$	17.0192i	0.903285i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 20.7953i	− 1.09753i	−0.835975	−	0.548767i	$-0.815097\pi$
$359$	− 20.7953i	− 1.09753i	0.835975	−	0.548767i	$-0.184903\pi$
$360$	0	0
$361$	−6.19028	−0.325804
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 21.2749i	− 1.11358i
$366$	0	0
$367$	− 4.67797i	− 0.244188i	−0.992519	−	0.122094i	$-0.961039\pi$
$367$	− 4.67797i	− 0.244188i	0.992519	−	0.122094i	$-0.0389610\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	18.8970	0.978451	0.489225	−	0.872157i	$-0.337280\pi$
$373$	18.8970	0.978451	0.489225	+	0.872157i	$0.337280\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.65828	0.497427
$378$	0	0
$379$	5.99600	0.307994	0.153997	−	0.988071i	$-0.450785\pi$
$379$	5.99600	0.307994	0.153997	+	0.988071i	$0.450785\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−26.0980	−1.33355	−0.666773	−	0.745261i	$-0.732325\pi$
$383$	−26.0980	−1.33355	−0.666773	+	0.745261i	$0.732325\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	18.6839i	0.947314i	0.880709	+	0.473657i	$0.157066\pi$
$389$	18.6839i	0.947314i	−0.880709	+	0.473657i	$0.842934\pi$
$390$	0	0
$391$	− 23.9071i	− 1.20904i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3.32042	−0.167068
$396$	0	0
$397$	− 9.71821i	− 0.487743i	−0.969808	−	0.243871i	$-0.921583\pi$
$397$	− 9.71821i	− 0.487743i	0.969808	−	0.243871i	$-0.0784175\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	29.0836i	1.45237i	0.687501	+	0.726183i	$0.258708\pi$
$401$	29.0836i	1.45237i	−0.687501	+	0.726183i	$0.741292\pi$
$402$	0	0
$403$	−1.45939	−0.0726973
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 42.4099i	− 2.10218i
$408$	0	0
$409$	25.6931i	1.27044i	0.772330	+	0.635221i	$0.219091\pi$
$409$	25.6931i	1.27044i	−0.772330	+	0.635221i	$0.780909\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	20.9447	1.02813
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6.33235	0.309356	0.154678	−	0.987965i	$-0.450566\pi$
$419$	6.33235	0.309356	0.154678	+	0.987965i	$0.450566\pi$
$420$	0	0
$421$	−21.0541	−1.02611	−0.513056	−	0.858355i	$-0.671487\pi$
$421$	−21.0541	−1.02611	−0.513056	+	0.858355i	$0.671487\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.13739	−0.152186
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.23914i	0.107855i	0.998545	+	0.0539277i	$0.0171741\pi$
$431$	2.23914i	0.107855i	−0.998545	+	0.0539277i	$0.982826\pi$
$432$	0	0
$433$	− 1.93460i	− 0.0929709i	−0.998919	−	0.0464855i	$-0.985198\pi$
$433$	− 1.93460i	− 0.0929709i	0.998919	−	0.0464855i	$-0.0148021\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	34.8938	1.66920
$438$	0	0
$439$	− 5.67878i	− 0.271033i	−0.990775	−	0.135517i	$-0.956731\pi$
$439$	− 5.67878i	− 0.271033i	0.990775	−	0.135517i	$-0.0432694\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	23.5839i	1.12051i	0.828321	+	0.560253i	$0.189296\pi$
$443$	23.5839i	1.12051i	−0.828321	+	0.560253i	$0.810704\pi$
$444$	0	0
$445$	17.3937	0.824538
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 0.991121i	− 0.0467739i	−0.999726	−	0.0233869i	$-0.992555\pi$
$449$	− 0.991121i	− 0.0467739i	0.999726	−	0.0233869i	$-0.00744497\pi$
$450$	0	0
$451$	31.8082i	1.49779i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16.6642	0.779519	0.389759	−	0.920917i	$-0.372558\pi$
$457$	16.6642	0.779519	0.389759	+	0.920917i	$0.372558\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	21.9318	1.02147	0.510734	−	0.859739i	$-0.329374\pi$
$461$	21.9318	1.02147	0.510734	+	0.859739i	$0.329374\pi$
$462$	0	0
$463$	14.5144	0.674541	0.337271	−	0.941408i	$-0.390496\pi$
$463$	14.5144	0.674541	0.337271	+	0.941408i	$0.390496\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	40.4995	1.87409	0.937046	−	0.349206i	$-0.113549\pi$
$467$	40.4995	1.87409	0.937046	+	0.349206i	$0.113549\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	11.6443i	0.535405i
$474$	0	0
$475$	− 4.57920i	− 0.210108i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.65610	0.0756690	0.0378345	−	0.999284i	$-0.487954\pi$
$479$	1.65610	0.0756690	0.0378345	+	0.999284i	$0.487954\pi$
$480$	0	0
$481$	− 55.4788i	− 2.52961i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	25.7731i	1.17030i
$486$	0	0
$487$	−7.48178	−0.339032	−0.169516	−	0.985527i	$-0.554220\pi$
$487$	−7.48178	−0.339032	−0.169516	+	0.985527i	$0.554220\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 0.518074i	− 0.0233804i	−0.999932	−	0.0116902i	$-0.996279\pi$
$491$	− 0.518074i	− 0.0233804i	0.999932	−	0.0116902i	$-0.00372119\pi$
$492$	0	0
$493$	− 7.00224i	− 0.315365i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	19.0619	0.853326	0.426663	−	0.904411i	$-0.359689\pi$
$499$	19.0619	0.853326	0.426663	+	0.904411i	$0.359689\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−16.7859	−0.748448	−0.374224	−	0.927338i	$-0.622091\pi$
$503$	−16.7859	−0.748448	−0.374224	+	0.927338i	$0.622091\pi$
$504$	0	0
$505$	27.4963	1.22357
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−13.4587	−0.596548	−0.298274	−	0.954480i	$-0.596411\pi$
$509$	−13.4587	−0.596548	−0.298274	+	0.954480i	$0.596411\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 36.3512i	− 1.60183i
$516$	0	0
$517$	1.39360i	0.0612906i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−42.1116	−1.84494	−0.922471	−	0.386067i	$-0.873833\pi$
$521$	−42.1116	−1.84494	−0.922471	+	0.386067i	$0.873833\pi$
$522$	0	0
$523$	18.8036i	0.822224i	0.911585	+	0.411112i	$0.134859\pi$
$523$	18.8036i	0.822224i	−0.911585	+	0.411112i	$0.865141\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.05805i	0.0460896i
$528$	0	0
$529$	−25.3351	−1.10153
$530$	0	0
$531$	0	0
$532$	0	0
$533$	41.6101i	1.80233i
$534$	0	0
$535$	− 14.1444i	− 0.611517i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	7.62424	0.327792	0.163896	−	0.986478i	$-0.447594\pi$
$541$	7.62424	0.327792	0.163896	+	0.986478i	$0.447594\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	26.1210	1.11890
$546$	0	0
$547$	−29.4627	−1.25974	−0.629868	−	0.776702i	$-0.716891\pi$
$547$	−29.4627	−1.25974	−0.629868	+	0.776702i	$0.716891\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	10.2202	0.435393
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.24296i	− 0.0950374i	−0.998870	−	0.0475187i	$-0.984869\pi$
$557$	− 2.24296i	− 0.0950374i	0.998870	−	0.0475187i	$-0.0151314\pi$
$558$	0	0
$559$	15.2326i	0.644269i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	45.4657	1.91615	0.958075	−	0.286518i	$-0.0924978\pi$
$563$	45.4657	1.91615	0.958075	+	0.286518i	$0.0924978\pi$
$564$	0	0
$565$	− 17.4165i	− 0.732717i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	19.1437i	0.802544i	0.915959	+	0.401272i	$0.131432\pi$
$569$	19.1437i	0.802544i	−0.915959	+	0.401272i	$0.868568\pi$
$570$	0	0
$571$	1.20419	0.0503938	0.0251969	−	0.999683i	$-0.491979\pi$
$571$	1.20419	0.0503938	0.0251969	+	0.999683i	$0.491979\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.34315i	0.264527i
$576$	0	0
$577$	− 20.1182i	− 0.837533i	−0.908094	−	0.418767i	$-0.862462\pi$
$577$	− 20.1182i	− 0.837533i	0.908094	−	0.418767i	$-0.137538\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	23.1570	0.959065
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−44.9416	−1.85494	−0.927470	−	0.373898i	$-0.878021\pi$
$587$	−44.9416	−1.85494	−0.927470	+	0.373898i	$0.878021\pi$
$588$	0	0
$589$	−1.54429	−0.0636313
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15.1110	0.620536	0.310268	−	0.950649i	$-0.399581\pi$
$593$	15.1110	0.620536	0.310268	+	0.950649i	$0.399581\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	36.4231i	1.48821i	0.668064	+	0.744104i	$0.267123\pi$
$599$	36.4231i	1.48821i	−0.668064	+	0.744104i	$0.732877\pi$
$600$	0	0
$601$	26.5993i	1.08501i	0.840053	+	0.542504i	$0.182524\pi$
$601$	26.5993i	1.08501i	−0.840053	+	0.542504i	$0.817476\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4.33903	0.176407
$606$	0	0
$607$	− 22.2512i	− 0.903150i	−0.892233	−	0.451575i	$-0.850862\pi$
$607$	− 22.2512i	− 0.903150i	0.892233	−	0.451575i	$-0.149138\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.82305i	0.0737527i
$612$	0	0
$613$	−16.5065	−0.666693	−0.333346	−	0.942804i	$-0.608178\pi$
$613$	−16.5065	−0.666693	−0.333346	+	0.942804i	$0.608178\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24.1547i	0.972431i	0.873839	+	0.486216i	$0.161623\pi$
$617$	24.1547i	0.972431i	−0.873839	+	0.486216i	$0.838377\pi$
$618$	0	0
$619$	− 27.9056i	− 1.12162i	−0.827944	−	0.560811i	$-0.810489\pi$
$619$	− 27.9056i	− 1.12162i	0.827944	−	0.560811i	$-0.189511\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19.6057	−0.784228
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−40.2220	−1.60376
$630$	0	0
$631$	−13.9089	−0.553703	−0.276852	−	0.960913i	$-0.589291\pi$
$631$	−13.9089	−0.553703	−0.276852	+	0.960913i	$0.589291\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	38.1324	1.51324
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 30.6417i	− 1.21027i	−0.796122	−	0.605137i	$-0.793118\pi$
$641$	− 30.6417i	− 1.21027i	0.796122	−	0.605137i	$-0.206882\pi$
$642$	0	0
$643$	− 15.4008i	− 0.607348i	−0.952776	−	0.303674i	$-0.901787\pi$
$643$	− 15.4008i	− 0.607348i	0.952776	−	0.303674i	$-0.0982133\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−41.8013	−1.64338	−0.821688	−	0.569937i	$-0.806968\pi$
$647$	−41.8013	−1.64338	−0.821688	+	0.569937i	$0.806968\pi$
$648$	0	0
$649$	− 30.8728i	− 1.21186i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 7.83698i	− 0.306685i	−0.988173	−	0.153342i	$-0.950996\pi$
$653$	− 7.83698i	− 0.306685i	0.988173	−	0.153342i	$-0.0490038\pi$
$654$	0	0
$655$	26.6316	1.04058
$656$	0	0
$657$	0	0
$658$	0	0
$659$	11.6341i	0.453202i	0.973988	+	0.226601i	$0.0727613\pi$
$659$	11.6341i	0.453202i	−0.973988	+	0.226601i	$0.927239\pi$
$660$	0	0
$661$	− 22.5647i	− 0.877663i	−0.898569	−	0.438832i	$-0.855392\pi$
$661$	− 22.5647i	− 0.877663i	0.898569	−	0.438832i	$-0.144608\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−14.1570	−0.548163
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.42430	−0.170798
$672$	0	0
$673$	24.6415	0.949860	0.474930	−	0.880024i	$-0.342473\pi$
$673$	24.6415	0.949860	0.474930	+	0.880024i	$0.342473\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.74835	0.297793	0.148897	−	0.988853i	$-0.452428\pi$
$677$	7.74835	0.297793	0.148897	+	0.988853i	$0.452428\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	11.0686i	0.423528i	0.977321	+	0.211764i	$0.0679208\pi$
$683$	11.0686i	0.423528i	−0.977321	+	0.211764i	$0.932079\pi$
$684$	0	0
$685$	− 43.9621i	− 1.67970i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	30.2930	1.15407
$690$	0	0
$691$	− 15.5571i	− 0.591821i	−0.955216	−	0.295911i	$-0.904377\pi$
$691$	− 15.5571i	− 0.591821i	0.955216	−	0.295911i	$-0.0956231\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.3308i	0.505665i
$696$	0	0
$697$	30.1673	1.14267
$698$	0	0
$699$	0	0
$700$	0	0
$701$	15.1353i	0.571652i	0.958282	+	0.285826i	$0.0922679\pi$
$701$	15.1353i	0.571652i	−0.958282	+	0.285826i	$0.907732\pi$
$702$	0	0
$703$	− 58.7063i	− 2.21415i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−0.861542	−0.0323559	−0.0161780	−	0.999869i	$-0.505150\pi$
$709$	−0.861542	−0.0323559	−0.0161780	+	0.999869i	$0.505150\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.13916	0.0801122
$714$	0	0
$715$	34.7691	1.30029
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−33.9107	−1.26466	−0.632328	−	0.774701i	$-0.717901\pi$
$719$	−33.9107	−1.26466	−0.632328	+	0.774701i	$0.717901\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.85786i	0.0689994i
$726$	0	0
$727$	26.5949i	0.986351i	0.869930	+	0.493176i	$0.164164\pi$
$727$	26.5949i	0.986351i	−0.869930	+	0.493176i	$0.835836\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	11.0436	0.408462
$732$	0	0
$733$	− 2.08044i	− 0.0768429i	−0.999262	−	0.0384214i	$-0.987767\pi$
$733$	− 2.08044i	− 0.0768429i	0.999262	−	0.0384214i	$-0.0122329\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.304373i	0.0112117i
$738$	0	0
$739$	39.8201	1.46481	0.732403	−	0.680871i	$-0.238399\pi$
$739$	39.8201	1.46481	0.732403	+	0.680871i	$0.238399\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12.2940i	0.451022i	0.974241	+	0.225511i	$0.0724052\pi$
$743$	12.2940i	0.451022i	−0.974241	+	0.225511i	$0.927595\pi$
$744$	0	0
$745$	− 24.0836i	− 0.882356i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	48.2544	1.76083	0.880414	−	0.474207i	$-0.157265\pi$
$751$	48.2544	1.76083	0.880414	+	0.474207i	$0.157265\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−30.4076	−1.10665
$756$	0	0
$757$	−9.77773	−0.355378	−0.177689	−	0.984087i	$-0.556862\pi$
$757$	−9.77773	−0.355378	−0.177689	+	0.984087i	$0.556862\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	51.6060	1.87072	0.935358	−	0.353703i	$-0.115078\pi$
$761$	51.6060	1.87072	0.935358	+	0.353703i	$0.115078\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 40.3865i	− 1.45827i
$768$	0	0
$769$	21.8208i	0.786877i	0.919351	+	0.393438i	$0.128715\pi$
$769$	21.8208i	0.786877i	−0.919351	+	0.393438i	$0.871285\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	37.9364	1.36448	0.682239	−	0.731129i	$-0.261006\pi$
$773$	37.9364	1.36448	0.682239	+	0.731129i	$0.261006\pi$
$774$	0	0
$775$	− 0.280728i	− 0.0100840i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	44.0308i	1.57757i
$780$	0	0
$781$	30.5213	1.09214
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 36.6452i	− 1.30792i
$786$	0	0
$787$	− 4.94540i	− 0.176284i	−0.996108	−	0.0881422i	$-0.971907\pi$
$787$	− 4.94540i	− 0.176284i	0.996108	−	0.0881422i	$-0.0280930\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−5.78768	−0.205526
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−47.6012	−1.68612	−0.843061	−	0.537819i	$-0.819248\pi$
$797$	−47.6012	−1.68612	−0.843061	+	0.537819i	$0.819248\pi$
$798$	0	0
$799$	1.32171	0.0467587
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−38.1533	−1.34640
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 4.55717i	− 0.160222i	−0.996786	−	0.0801108i	$-0.974473\pi$
$809$	− 4.55717i	− 0.160222i	0.996786	−	0.0801108i	$-0.0255274\pi$
$810$	0	0
$811$	15.6097i	0.548131i	0.961711	+	0.274065i	$0.0883685\pi$
$811$	15.6097i	0.548131i	−0.961711	+	0.274065i	$0.911632\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	41.1812	1.44252
$816$	0	0
$817$	16.1187i	0.563923i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 3.74968i	− 0.130865i	−0.997857	−	0.0654324i	$-0.979157\pi$
$821$	− 3.74968i	− 0.130865i	0.997857	−	0.0654324i	$-0.0208427\pi$
$822$	0	0
$823$	−23.8321	−0.830735	−0.415367	−	0.909654i	$-0.636347\pi$
$823$	−23.8321	−0.830735	−0.415367	+	0.909654i	$0.636347\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 6.30470i	− 0.219236i	−0.993974	−	0.109618i	$-0.965037\pi$
$827$	− 6.30470i	− 0.219236i	0.993974	−	0.109618i	$-0.0349627\pi$
$828$	0	0
$829$	15.4389i	0.536216i	0.963389	+	0.268108i	$0.0863983\pi$
$829$	15.4389i	0.536216i	−0.963389	+	0.268108i	$0.913602\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−22.4532	−0.777024
$836$	0	0
$837$	0	0
$838$	0	0
$839$	26.0853	0.900566	0.450283	−	0.892886i	$-0.351323\pi$
$839$	26.0853	0.900566	0.450283	+	0.892886i	$0.351323\pi$
$840$	0	0
$841$	24.8535	0.857017
$842$	0	0
$843$	0	0
$844$	0	0
$845$	19.2002	0.660506
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	81.3204i	2.78763i
$852$	0	0
$853$	24.7463i	0.847297i	0.905827	+	0.423649i	$0.139251\pi$
$853$	24.7463i	0.847297i	−0.905827	+	0.423649i	$0.860749\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−17.5806	−0.600543	−0.300272	−	0.953854i	$-0.597077\pi$
$857$	−17.5806	−0.600543	−0.300272	+	0.953854i	$0.597077\pi$
$858$	0	0
$859$	0.415613i	0.0141805i	0.999975	+	0.00709026i	$0.00225692\pi$
$859$	0.415613i	0.0141805i	−0.999975	+	0.00709026i	$0.997743\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 8.34080i	− 0.283924i	−0.989872	−	0.141962i	$-0.954659\pi$
$863$	− 8.34080i	− 0.283924i	0.989872	−	0.141962i	$-0.0453411\pi$
$864$	0	0
$865$	10.6936	0.363595
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.95465i	0.201998i
$870$	0	0
$871$	0.398168i	0.0134914i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46.8900	1.58336	0.791681	−	0.610935i	$-0.209206\pi$
$877$	46.8900	1.58336	0.791681	+	0.610935i	$0.209206\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−16.9745	−0.571885	−0.285942	−	0.958247i	$-0.592307\pi$
$881$	−16.9745	−0.571885	−0.285942	+	0.958247i	$0.592307\pi$
$882$	0	0
$883$	−25.2950	−0.851244	−0.425622	−	0.904901i	$-0.639945\pi$
$883$	−25.2950	−0.851244	−0.425622	+	0.904901i	$0.639945\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7.64887	−0.256824	−0.128412	−	0.991721i	$-0.540988\pi$
$887$	−7.64887	−0.256824	−0.128412	+	0.991721i	$0.540988\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.92911i	0.0645551i
$894$	0	0
$895$	50.5995i	1.69135i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.626546	0.0208965
$900$	0	0
$901$	− 21.9624i	− 0.731672i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.22443i	0.0739424i
$906$	0	0
$907$	−22.5553	−0.748937	−0.374469	−	0.927240i	$-0.622175\pi$
$907$	−22.5553	−0.748937	−0.374469	+	0.927240i	$0.622175\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	39.6484i	1.31361i	0.754060	+	0.656805i	$0.228093\pi$
$911$	39.6484i	1.31361i	−0.754060	+	0.656805i	$0.771907\pi$
$912$	0	0
$913$	− 37.5610i	− 1.24309i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	42.3465	1.39688	0.698441	−	0.715668i	$-0.253877\pi$
$919$	42.3465	1.39688	0.698441	+	0.715668i	$0.253877\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	39.9266	1.31420
$924$	0	0
$925$	10.6719	0.350890
$926$	0	0
$927$	0	0
$928$	0	0
$929$	2.07099	0.0679470	0.0339735	−	0.999423i	$-0.489184\pi$
$929$	2.07099	0.0679470	0.0339735	+	0.999423i	$0.489184\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 25.2075i	− 0.824375i
$936$	0	0
$937$	13.4301i	0.438741i	0.975642	+	0.219371i	$0.0704004\pi$
$937$	13.4301i	0.438741i	−0.975642	+	0.219371i	$0.929600\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	30.4203	0.991673	0.495836	−	0.868416i	$-0.334862\pi$
$941$	30.4203	0.991673	0.495836	+	0.868416i	$0.334862\pi$
$942$	0	0
$943$	− 60.9918i	− 1.98617i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	55.2161i	1.79428i	0.441745	+	0.897140i	$0.354359\pi$
$947$	55.2161i	1.79428i	−0.441745	+	0.897140i	$0.645641\pi$
$948$	0	0
$949$	−49.9105	−1.62016
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 23.7985i	− 0.770910i	−0.922727	−	0.385455i	$-0.874044\pi$
$953$	− 23.7985i	− 0.770910i	0.922727	−	0.385455i	$-0.125956\pi$
$954$	0	0
$955$	48.0840i	1.55596i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	30.9053	0.996946
$962$	0	0
$963$	0	0
$964$	0	0
$965$	18.3429	0.590477
$966$	0	0
$967$	35.2186	1.13255	0.566277	−	0.824215i	$-0.308383\pi$
$967$	35.2186	1.13255	0.566277	+	0.824215i	$0.308383\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	54.4522	1.74746	0.873728	−	0.486416i	$-0.161696\pi$
$971$	54.4522	1.74746	0.873728	+	0.486416i	$0.161696\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	39.6798i	1.26947i	0.772730	+	0.634735i	$0.218891\pi$
$977$	39.6798i	1.26947i	−0.772730	+	0.634735i	$0.781109\pi$
$978$	0	0
$979$	− 31.1928i	− 0.996926i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−30.9220	−0.986259	−0.493130	−	0.869956i	$-0.664147\pi$
$983$	−30.9220	−0.986259	−0.493130	+	0.869956i	$0.664147\pi$
$984$	0	0
$985$	10.8101i	0.344440i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 22.3278i	− 0.709982i
$990$	0	0
$991$	9.42355	0.299349	0.149674	−	0.988735i	$-0.452177\pi$
$991$	9.42355	0.299349	0.149674	+	0.988735i	$0.452177\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	26.2853i	0.833299i
$996$	0	0
$997$	− 13.4941i	− 0.427364i	−0.976903	−	0.213682i	$-0.931454\pi$
$997$	− 13.4941i	− 0.427364i	0.976903	−	0.213682i	$-0.0685456\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.k.b.881.4		16
3.2	odd	2	inner	3528.2.k.b.881.13		16
4.3	odd	2		7056.2.k.h.881.3		16
7.2	even	3		504.2.bl.a.17.7	yes	16
7.3	odd	6		504.2.bl.a.89.2	yes	16
7.4	even	3		3528.2.bl.a.1097.7		16
7.5	odd	6		3528.2.bl.a.521.2		16
7.6	odd	2	inner	3528.2.k.b.881.14		16
12.11	even	2		7056.2.k.h.881.14		16
21.2	odd	6		504.2.bl.a.17.2	✓	16
21.5	even	6		3528.2.bl.a.521.7		16
21.11	odd	6		3528.2.bl.a.1097.2		16
21.17	even	6		504.2.bl.a.89.7	yes	16
21.20	even	2	inner	3528.2.k.b.881.3		16
28.3	even	6		1008.2.bt.d.593.2		16
28.23	odd	6		1008.2.bt.d.17.7		16
28.27	even	2		7056.2.k.h.881.13		16
84.23	even	6		1008.2.bt.d.17.2		16
84.59	odd	6		1008.2.bt.d.593.7		16
84.83	odd	2		7056.2.k.h.881.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.bl.a.17.2	✓	16	21.2	odd	6
504.2.bl.a.17.7	yes	16	7.2	even	3
504.2.bl.a.89.2	yes	16	7.3	odd	6
504.2.bl.a.89.7	yes	16	21.17	even	6
1008.2.bt.d.17.2		16	84.23	even	6
1008.2.bt.d.17.7		16	28.23	odd	6
1008.2.bt.d.593.2		16	28.3	even	6
1008.2.bt.d.593.7		16	84.59	odd	6
3528.2.k.b.881.3		16	21.20	even	2	inner
3528.2.k.b.881.4		16	1.1	even	1	trivial
3528.2.k.b.881.13		16	3.2	odd	2	inner
3528.2.k.b.881.14		16	7.6	odd	2	inner
3528.2.bl.a.521.2		16	7.5	odd	6
3528.2.bl.a.521.7		16	21.5	even	6
3528.2.bl.a.1097.2		16	21.11	odd	6
3528.2.bl.a.1097.7		16	7.4	even	3
7056.2.k.h.881.3		16	4.3	odd	2
7056.2.k.h.881.4		16	84.83	odd	2
7056.2.k.h.881.13		16	28.27	even	2
7056.2.k.h.881.14		16	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3528.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.02179 q^{5} +O(q^{10})\)	\(q-2.02179 q^{5} +3.62576i q^{11} +4.74306i q^{13} +3.43871 q^{17} +5.01899i q^{19} -6.95235i q^{23} -0.912375 q^{25} -2.03630i q^{29} +0.307689i q^{31} -11.6968 q^{37} +8.77283 q^{41} +3.21155 q^{43} +0.384361 q^{47} -6.38680i q^{53} -7.33052i q^{55} -8.51485 q^{59} +1.22024i q^{61} -9.58946i q^{65} +0.0839475 q^{67} -8.41789i q^{71} +10.5228i q^{73} +1.64232 q^{79} -10.3595 q^{83} -6.95235 q^{85} -8.60311 q^{89} -10.1473i q^{95} -12.7477i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 24 q^{25} - 8 q^{37} + 8 q^{43} + 56 q^{67} + 64 q^{79} - 32 q^{85}+O(q^{100}) \) 16 * q - 24 * q^25 - 8 * q^37 + 8 * q^43 + 56 * q^67 + 64 * q^79 - 32 * q^85

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