LMFDB - Embedded newform 441.4.c.b.440.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,Mod(440,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("441.440");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	441.c (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:	$26.0198423125$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			440.3
Character	$\chi$	$=$	441.440
Dual form			441.4.c.b.440.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.65356i q^{2} -5.34851 q^{4} +17.3990 q^{5} -9.68739i q^{8} +O(q^{10})$	$q-3.65356i q^{2} -5.34851 q^{4} +17.3990 q^{5} -9.68739i q^{8} -63.5684i q^{10} -32.5650i q^{11} +42.4779i q^{13} -78.1815 q^{16} +85.6362 q^{17} -69.1820i q^{19} -93.0588 q^{20} -118.978 q^{22} +167.020i q^{23} +177.726 q^{25} +155.196 q^{26} -254.270i q^{29} -325.437i q^{31} +208.142i q^{32} -312.877i q^{34} +345.666 q^{37} -252.761 q^{38} -168.551i q^{40} -182.210 q^{41} +140.292 q^{43} +174.174i q^{44} +610.218 q^{46} -43.0015 q^{47} -649.332i q^{50} -227.193i q^{52} +193.045i q^{53} -566.600i q^{55} -928.992 q^{58} +111.274 q^{59} -90.5402i q^{61} -1189.00 q^{62} +135.007 q^{64} +739.074i q^{65} -1048.92 q^{67} -458.026 q^{68} +464.023i q^{71} -182.785i q^{73} -1262.91i q^{74} +370.020i q^{76} -238.947 q^{79} -1360.28 q^{80} +665.714i q^{82} -375.260 q^{83} +1489.99 q^{85} -512.565i q^{86} -315.470 q^{88} +1439.65 q^{89} -893.308i q^{92} +157.108i q^{94} -1203.70i q^{95} -638.698i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 96 q^{4}+O(q^{10}) $ 24 * q - 96 * q^4	$ 24 q - 96 q^{4} + 144 q^{16} + 624 q^{22} + 312 q^{25} - 864 q^{37} + 1248 q^{43} - 3888 q^{46} - 7440 q^{58} - 3360 q^{64} - 2688 q^{67} + 480 q^{79} + 13248 q^{85} - 7248 q^{88}+O(q^{100}) $ 24 * q - 96 * q^4 + 144 * q^16 + 624 * q^22 + 312 * q^25 - 864 * q^37 + 1248 * q^43 - 3888 * q^46 - 7440 * q^58 - 3360 * q^64 - 2688 * q^67 + 480 * q^79 + 13248 * q^85 - 7248 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

$n$	$199$	$344$
$\chi(n)$	$-1$	$-1$

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 $

$2$	− 3.65356i	− 1.29173i	−0.763452	−	0.645864i	$-0.776497\pi$
$2$	− 3.65356i	− 1.29173i	0.763452	−	0.645864i	$-0.223503\pi$
$3$	0	0
$3$	0	0
$4$	−5.34851	−0.668563
$5$	17.3990	1.55622	0.778108	−	0.628131i	$-0.216180\pi$
$5$	17.3990	1.55622	0.778108	+	0.628131i	$0.216180\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 9.68739i	− 0.428126i
$9$	0	0
$10$	− 63.5684i	− 2.01021i
$11$	− 32.5650i	− 0.892612i	−0.894881	−	0.446306i	$-0.852739\pi$
$11$	− 32.5650i	− 0.892612i	0.894881	−	0.446306i	$-0.147261\pi$
$12$	0	0
$13$	42.4779i	0.906250i	0.891447	+	0.453125i	$0.149691\pi$
$13$	42.4779i	0.906250i	−0.891447	+	0.453125i	$0.850309\pi$
$14$	0	0
$15$	0	0
$16$	−78.1815	−1.22159
$17$	85.6362	1.22176	0.610878	−	0.791725i	$-0.290817\pi$
$17$	85.6362	1.22176	0.610878	+	0.791725i	$0.290817\pi$
$18$	0	0
$19$	− 69.1820i	− 0.835338i	−0.908599	−	0.417669i	$-0.862847\pi$
$19$	− 69.1820i	− 0.835338i	0.908599	−	0.417669i	$-0.137153\pi$
$20$	−93.0588	−1.04043
$21$	0	0
$22$	−118.978	−1.15301
$23$	167.020i	1.51418i	0.653311	+	0.757089i	$0.273379\pi$
$23$	167.020i	1.51418i	−0.653311	+	0.757089i	$0.726621\pi$
$24$	0	0
$25$	177.726	1.42181
$26$	155.196	1.17063
$27$	0	0
$28$	0	0
$29$	− 254.270i	− 1.62817i	−0.580748	−	0.814083i	$-0.697240\pi$
$29$	− 254.270i	− 1.62817i	0.580748	−	0.814083i	$-0.302760\pi$
$30$	0	0
$31$	− 325.437i	− 1.88549i	−0.333513	−	0.942745i	$-0.608234\pi$
$31$	− 325.437i	− 1.88549i	0.333513	−	0.942745i	$-0.391766\pi$
$32$	208.142i	1.14983i
$33$	0	0
$34$	− 312.877i	− 1.57818i
$35$	0	0
$36$	0	0
$37$	345.666	1.53587	0.767934	−	0.640529i	$-0.221285\pi$
$37$	345.666	1.53587	0.767934	+	0.640529i	$0.221285\pi$
$38$	−252.761	−1.07903
$39$	0	0
$40$	− 168.551i	− 0.666257i
$41$	−182.210	−0.694058	−0.347029	−	0.937854i	$-0.612809\pi$
$41$	−182.210	−0.694058	−0.347029	+	0.937854i	$0.612809\pi$
$42$	0	0
$43$	140.292	0.497542	0.248771	−	0.968562i	$-0.419973\pi$
$43$	140.292	0.497542	0.248771	+	0.968562i	$0.419973\pi$
$44$	174.174i	0.596767i
$45$	0	0
$46$	610.218	1.95591
$47$	−43.0015	−0.133456	−0.0667278	−	0.997771i	$-0.521256\pi$
$47$	−43.0015	−0.133456	−0.0667278	+	0.997771i	$0.521256\pi$
$48$	0	0
$49$	0	0
$50$	− 649.332i	− 1.83659i
$51$	0	0
$52$	− 227.193i	− 0.605885i
$53$	193.045i	0.500315i	0.968205	+	0.250158i	$0.0804825\pi$
$53$	193.045i	0.500315i	−0.968205	+	0.250158i	$0.919518\pi$
$54$	0	0
$55$	− 566.600i	− 1.38910i
$56$	0	0
$57$	0	0
$58$	−928.992	−2.10315
$59$	111.274	0.245536	0.122768	−	0.992435i	$-0.460823\pi$
$59$	111.274	0.245536	0.122768	+	0.992435i	$0.460823\pi$
$60$	0	0
$61$	− 90.5402i	− 0.190041i	−0.995475	−	0.0950204i	$-0.969708\pi$
$61$	− 90.5402i	− 0.190041i	0.995475	−	0.0950204i	$-0.0302916\pi$
$62$	−1189.00	−2.43554
$63$	0	0
$64$	135.007	0.263685
$65$	739.074i	1.41032i
$66$	0	0
$67$	−1048.92	−1.91262	−0.956310	−	0.292355i	$-0.905561\pi$
$67$	−1048.92	−1.91262	−0.956310	+	0.292355i	$0.905561\pi$
$68$	−458.026	−0.816821
$69$	0	0
$70$	0	0
$71$	464.023i	0.775626i	0.921738	+	0.387813i	$0.126769\pi$
$71$	464.023i	0.775626i	−0.921738	+	0.387813i	$0.873231\pi$
$72$	0	0
$73$	− 182.785i	− 0.293060i	−0.989206	−	0.146530i	$-0.953190\pi$
$73$	− 182.785i	− 0.293060i	0.989206	−	0.146530i	$-0.0468105\pi$
$74$	− 1262.91i	− 1.98393i
$75$	0	0
$76$	370.020i	0.558477i
$77$	0	0
$78$	0	0
$79$	−238.947	−0.340299	−0.170149	−	0.985418i	$-0.554425\pi$
$79$	−238.947	−0.340299	−0.170149	+	0.985418i	$0.554425\pi$
$80$	−1360.28	−1.90105
$81$	0	0
$82$	665.714i	0.896535i
$83$	−375.260	−0.496267	−0.248134	−	0.968726i	$-0.579817\pi$
$83$	−375.260	−0.496267	−0.248134	+	0.968726i	$0.579817\pi$
$84$	0	0
$85$	1489.99	1.90131
$86$	− 512.565i	− 0.642689i
$87$	0	0
$88$	−315.470	−0.382151
$89$	1439.65	1.71463	0.857315	−	0.514792i	$-0.172131\pi$
$89$	1439.65	1.71463	0.857315	+	0.514792i	$0.172131\pi$
$90$	0	0
$91$	0	0
$92$	− 893.308i	− 1.01232i
$93$	0	0
$94$	157.108i	0.172388i
$95$	− 1203.70i	− 1.29997i
$96$	0	0
$97$	− 638.698i	− 0.668556i	−0.942474	−	0.334278i	$-0.891508\pi$
$97$	− 638.698i	− 0.668556i	0.942474	−	0.334278i	$-0.108492\pi$
$98$	0	0
$99$	0	0
$100$	−950.568	−0.950568
$101$	−1051.64	−1.03606	−0.518031	−	0.855362i	$-0.673335\pi$
$101$	−1051.64	−1.03606	−0.518031	+	0.855362i	$0.673335\pi$
$102$	0	0
$103$	1097.31i	1.04972i	0.851187	+	0.524862i	$0.175883\pi$
$103$	1097.31i	1.04972i	−0.851187	+	0.524862i	$0.824117\pi$
$104$	411.500	0.387989
$105$	0	0
$106$	705.300	0.646272
$107$	575.756i	0.520191i	0.965583	+	0.260096i	$0.0837541\pi$
$107$	575.756i	0.520191i	−0.965583	+	0.260096i	$0.916246\pi$
$108$	0	0
$109$	−430.582	−0.378369	−0.189185	−	0.981942i	$-0.560584\pi$
$109$	−430.582	−0.378369	−0.189185	+	0.981942i	$0.560584\pi$
$110$	−2070.11	−1.79434
$111$	0	0
$112$	0	0
$113$	583.877i	0.486075i	0.970017	+	0.243038i	$0.0781438\pi$
$113$	583.877i	0.486075i	−0.970017	+	0.243038i	$0.921856\pi$
$114$	0	0
$115$	2905.99i	2.35639i
$116$	1359.97i	1.08853i
$117$	0	0
$118$	− 406.546i	− 0.317166i
$119$	0	0
$120$	0	0
$121$	270.519	0.203245
$122$	−330.794	−0.245481
$123$	0	0
$124$	1740.60i	1.26057i
$125$	917.377	0.656422
$126$	0	0
$127$	−499.692	−0.349138	−0.174569	−	0.984645i	$-0.555853\pi$
$127$	−499.692	−0.349138	−0.174569	+	0.984645i	$0.555853\pi$
$128$	1171.88i	0.809223i
$129$	0	0
$130$	2700.25	1.82175
$131$	−196.720	−0.131203	−0.0656013	−	0.997846i	$-0.520897\pi$
$131$	−196.720	−0.131203	−0.0656013	+	0.997846i	$0.520897\pi$
$132$	0	0
$133$	0	0
$134$	3832.28i	2.47059i
$135$	0	0
$136$	− 829.592i	− 0.523066i
$137$	386.925i	0.241293i	0.992696	+	0.120647i	$0.0384968\pi$
$137$	386.925i	0.241293i	−0.992696	+	0.120647i	$0.961503\pi$
$138$	0	0
$139$	− 1507.09i	− 0.919636i	−0.888013	−	0.459818i	$-0.847915\pi$
$139$	− 1507.09i	− 0.919636i	0.888013	−	0.459818i	$-0.152085\pi$
$140$	0	0
$141$	0	0
$142$	1695.34	1.00190
$143$	1383.29	0.808929
$144$	0	0
$145$	− 4424.06i	− 2.53378i
$146$	−667.816	−0.378554
$147$	0	0
$148$	−1848.80	−1.02683
$149$	− 1480.80i	− 0.814176i	−0.913389	−	0.407088i	$-0.866544\pi$
$149$	− 1480.80i	− 0.814176i	0.913389	−	0.407088i	$-0.133456\pi$
$150$	0	0
$151$	−2677.34	−1.44291	−0.721453	−	0.692463i	$-0.756525\pi$
$151$	−2677.34	−1.44291	−0.721453	+	0.692463i	$0.756525\pi$
$152$	−670.193	−0.357630
$153$	0	0
$154$	0	0
$155$	− 5662.28i	− 2.93423i
$156$	0	0
$157$	1279.93i	0.650636i	0.945605	+	0.325318i	$0.105471\pi$
$157$	1279.93i	0.650636i	−0.945605	+	0.325318i	$0.894529\pi$
$158$	873.006i	0.439574i
$159$	0	0
$160$	3621.46i	1.78939i
$161$	0	0
$162$	0	0
$163$	1795.86	0.862961	0.431481	−	0.902122i	$-0.357991\pi$
$163$	1795.86	0.862961	0.431481	+	0.902122i	$0.357991\pi$
$164$	974.550	0.464022
$165$	0	0
$166$	1371.04i	0.641043i
$167$	3546.20	1.64319	0.821596	−	0.570070i	$-0.193084\pi$
$167$	3546.20	1.64319	0.821596	+	0.570070i	$0.193084\pi$
$168$	0	0
$169$	392.629	0.178711
$170$	− 5443.76i	− 2.45598i
$171$	0	0
$172$	−750.352	−0.332638
$173$	3856.69	1.69491	0.847453	−	0.530870i	$-0.178135\pi$
$173$	3856.69	1.69491	0.847453	+	0.530870i	$0.178135\pi$
$174$	0	0
$175$	0	0
$176$	2545.98i	1.09040i
$177$	0	0
$178$	− 5259.83i	− 2.21484i
$179$	2010.38i	0.839455i	0.907650	+	0.419728i	$0.137874\pi$
$179$	2010.38i	0.839455i	−0.907650	+	0.419728i	$0.862126\pi$
$180$	0	0
$181$	3081.71i	1.26554i	0.774341	+	0.632768i	$0.218081\pi$
$181$	3081.71i	1.26554i	−0.774341	+	0.632768i	$0.781919\pi$
$182$	0	0
$183$	0	0
$184$	1617.99	0.648260
$185$	6014.25	2.39014
$186$	0	0
$187$	− 2788.75i	− 1.09055i
$188$	229.994	0.0892234
$189$	0	0
$190$	−4397.78	−1.67920
$191$	1012.80i	0.383685i	0.981426	+	0.191842i	$0.0614463\pi$
$191$	1012.80i	0.383685i	−0.981426	+	0.191842i	$0.938554\pi$
$192$	0	0
$193$	1444.12	0.538603	0.269301	−	0.963056i	$-0.413207\pi$
$193$	1444.12	0.538603	0.269301	+	0.963056i	$0.413207\pi$
$194$	−2333.52	−0.863593
$195$	0	0
$196$	0	0
$197$	2503.72i	0.905494i	0.891639	+	0.452747i	$0.149556\pi$
$197$	2503.72i	0.905494i	−0.891639	+	0.452747i	$0.850444\pi$
$198$	0	0
$199$	− 2861.95i	− 1.01949i	−0.860326	−	0.509744i	$-0.829740\pi$
$199$	− 2861.95i	− 1.01949i	0.860326	−	0.509744i	$-0.170260\pi$
$200$	− 1721.70i	− 0.608713i
$201$	0	0
$202$	3842.23i	1.33831i
$203$	0	0
$204$	0	0
$205$	−3170.27	−1.08010
$206$	4009.11	1.35596
$207$	0	0
$208$	− 3320.99i	− 1.10706i
$209$	−2252.91	−0.745633
$210$	0	0
$211$	−4118.68	−1.34380	−0.671900	−	0.740642i	$-0.734521\pi$
$211$	−4118.68	−1.34380	−0.671900	+	0.740642i	$0.734521\pi$
$212$	− 1032.50i	− 0.334492i
$213$	0	0
$214$	2103.56	0.671946
$215$	2440.94	0.774282
$216$	0	0
$217$	0	0
$218$	1573.16i	0.488750i
$219$	0	0
$220$	3030.46i	0.928699i
$221$	3637.65i	1.10722i
$222$	0	0
$223$	− 3065.23i	− 0.920462i	−0.887799	−	0.460231i	$-0.847767\pi$
$223$	− 3065.23i	− 0.920462i	0.887799	−	0.460231i	$-0.152233\pi$
$224$	0	0
$225$	0	0
$226$	2133.23	0.627877
$227$	1331.63	0.389355	0.194678	−	0.980867i	$-0.437634\pi$
$227$	1331.63	0.389355	0.194678	+	0.980867i	$0.437634\pi$
$228$	0	0
$229$	4018.78i	1.15969i	0.814727	+	0.579844i	$0.196887\pi$
$229$	4018.78i	1.15969i	−0.814727	+	0.579844i	$0.803113\pi$
$230$	10617.2	3.04381
$231$	0	0
$232$	−2463.22	−0.697061
$233$	2275.44i	0.639783i	0.947454	+	0.319891i	$0.103646\pi$
$233$	2275.44i	0.639783i	−0.947454	+	0.319891i	$0.896354\pi$
$234$	0	0
$235$	−748.183	−0.207686
$236$	−595.150	−0.164157
$237$	0	0
$238$	0	0
$239$	1453.02i	0.393257i	0.980478	+	0.196628i	$0.0629992\pi$
$239$	1453.02i	0.393257i	−0.980478	+	0.196628i	$0.937001\pi$
$240$	0	0
$241$	4727.34i	1.26355i	0.775153	+	0.631774i	$0.217673\pi$
$241$	4727.34i	1.26355i	−0.775153	+	0.631774i	$0.782327\pi$
$242$	− 988.356i	− 0.262537i
$243$	0	0
$244$	484.255i	0.127054i
$245$	0	0
$246$	0	0
$247$	2938.70	0.757025
$248$	−3152.64	−0.807228
$249$	0	0
$250$	− 3351.69i	− 0.847919i
$251$	4674.94	1.17562	0.587808	−	0.809001i	$-0.299991\pi$
$251$	4674.94	1.17562	0.587808	+	0.809001i	$0.299991\pi$
$252$	0	0
$253$	5439.02	1.35157
$254$	1825.65i	0.450991i
$255$	0	0
$256$	5361.59	1.30898
$257$	4609.10	1.11871	0.559354	−	0.828929i	$-0.311049\pi$
$257$	4609.10	1.11871	0.559354	+	0.828929i	$0.311049\pi$
$258$	0	0
$259$	0	0
$260$	− 3952.94i	− 0.942888i
$261$	0	0
$262$	718.730i	0.169478i
$263$	2319.51i	0.543830i	0.962321	+	0.271915i	$0.0876570\pi$
$263$	2319.51i	0.543830i	−0.962321	+	0.271915i	$0.912343\pi$
$264$	0	0
$265$	3358.79i	0.778598i
$266$	0	0
$267$	0	0
$268$	5610.14	1.27871
$269$	−757.791	−0.171760	−0.0858798	−	0.996306i	$-0.527370\pi$
$269$	−757.791	−0.171760	−0.0858798	+	0.996306i	$0.527370\pi$
$270$	0	0
$271$	8619.60i	1.93212i	0.258326	+	0.966058i	$0.416829\pi$
$271$	8619.60i	1.93212i	−0.258326	+	0.966058i	$0.583171\pi$
$272$	−6695.17	−1.49248
$273$	0	0
$274$	1413.65	0.311686
$275$	− 5787.65i	− 1.26912i
$276$	0	0
$277$	−2055.64	−0.445889	−0.222945	−	0.974831i	$-0.571567\pi$
$277$	−2055.64	−0.445889	−0.222945	+	0.974831i	$0.571567\pi$
$278$	−5506.23	−1.18792
$279$	0	0
$280$	0	0
$281$	235.581i	0.0500128i	0.999687	+	0.0250064i	$0.00796062\pi$
$281$	235.581i	0.0500128i	−0.999687	+	0.0250064i	$0.992039\pi$
$282$	0	0
$283$	746.700i	0.156843i	0.996920	+	0.0784217i	$0.0249881\pi$
$283$	746.700i	0.156843i	−0.996920	+	0.0784217i	$0.975012\pi$
$284$	− 2481.83i	− 0.518555i
$285$	0	0
$286$	− 5053.95i	− 1.04492i
$287$	0	0
$288$	0	0
$289$	2420.56	0.492685
$290$	−16163.6	−3.27295
$291$	0	0
$292$	977.627i	0.195929i
$293$	−4087.13	−0.814923	−0.407461	−	0.913222i	$-0.633586\pi$
$293$	−4087.13	−0.814923	−0.407461	+	0.913222i	$0.633586\pi$
$294$	0	0
$295$	1936.06	0.382108
$296$	− 3348.60i	− 0.657546i
$297$	0	0
$298$	−5410.21	−1.05169
$299$	−7094.66	−1.37222
$300$	0	0
$301$	0	0
$302$	9781.82i	1.86384i
$303$	0	0
$304$	5408.75i	1.02044i
$305$	− 1575.31i	− 0.295744i
$306$	0	0
$307$	− 4572.60i	− 0.850073i	−0.905176	−	0.425036i	$-0.860261\pi$
$307$	− 4572.60i	− 0.850073i	0.905176	−	0.425036i	$-0.139739\pi$
$308$	0	0
$309$	0	0
$310$	−20687.5	−3.79023
$311$	9689.62	1.76671	0.883357	−	0.468700i	$-0.155278\pi$
$311$	9689.62	1.76671	0.883357	+	0.468700i	$0.155278\pi$
$312$	0	0
$313$	3918.42i	0.707610i	0.935319	+	0.353805i	$0.115112\pi$
$313$	3918.42i	0.707610i	−0.935319	+	0.353805i	$0.884888\pi$
$314$	4676.32	0.840446
$315$	0	0
$316$	1278.01	0.227511
$317$	9042.96i	1.60222i	0.598518	+	0.801109i	$0.295756\pi$
$317$	9042.96i	1.60222i	−0.598518	+	0.801109i	$0.704244\pi$
$318$	0	0
$319$	−8280.32	−1.45332
$320$	2348.98	0.410350
$321$	0	0
$322$	0	0
$323$	− 5924.48i	− 1.02058i
$324$	0	0
$325$	7549.42i	1.28851i
$326$	− 6561.29i	− 1.11471i
$327$	0	0
$328$	1765.14i	0.297145i
$329$	0	0
$330$	0	0
$331$	−3521.99	−0.584852	−0.292426	−	0.956288i	$-0.594463\pi$
$331$	−3521.99	−0.584852	−0.292426	+	0.956288i	$0.594463\pi$
$332$	2007.08	0.331786
$333$	0	0
$334$	− 12956.3i	− 2.12256i
$335$	−18250.1	−2.97645
$336$	0	0
$337$	10294.0	1.66395	0.831975	−	0.554813i	$-0.187210\pi$
$337$	10294.0	1.66395	0.831975	+	0.554813i	$0.187210\pi$
$338$	− 1434.49i	− 0.230846i
$339$	0	0
$340$	−7969.20	−1.27115
$341$	−10597.9	−1.68301
$342$	0	0
$343$	0	0
$344$	− 1359.06i	− 0.213011i
$345$	0	0
$346$	− 14090.7i	− 2.18936i
$347$	− 94.3815i	− 0.0146013i	−0.999973	−	0.00730067i	$-0.997676\pi$
$347$	− 94.3815i	− 0.0146013i	0.999973	−	0.00730067i	$-0.00232390\pi$
$348$	0	0
$349$	4242.54i	0.650711i	0.945592	+	0.325356i	$0.105484\pi$
$349$	4242.54i	0.650711i	−0.945592	+	0.325356i	$0.894516\pi$
$350$	0	0
$351$	0	0
$352$	6778.15	1.02635
$353$	−7378.16	−1.11246	−0.556232	−	0.831027i	$-0.687754\pi$
$353$	−7378.16	−1.11246	−0.556232	+	0.831027i	$0.687754\pi$
$354$	0	0
$355$	8073.55i	1.20704i
$356$	−7699.95	−1.14634
$357$	0	0
$358$	7345.03	1.08435
$359$	− 395.109i	− 0.0580866i	−0.999578	−	0.0290433i	$-0.990754\pi$
$359$	− 395.109i	− 0.0580866i	0.999578	−	0.0290433i	$-0.00924607\pi$
$360$	0	0
$361$	2072.86	0.302210
$362$	11259.2	1.63473
$363$	0	0
$364$	0	0
$365$	− 3180.28i	− 0.456064i
$366$	0	0
$367$	− 3599.49i	− 0.511967i	−0.966681	−	0.255984i	$-0.917601\pi$
$367$	− 3599.49i	− 0.511967i	0.966681	−	0.255984i	$-0.0823993\pi$
$368$	− 13057.9i	− 1.84970i
$369$	0	0
$370$	− 21973.4i	− 3.08742i
$371$	0	0
$372$	0	0
$373$	−9243.21	−1.28310	−0.641549	−	0.767082i	$-0.721708\pi$
$373$	−9243.21	−1.28310	−0.641549	+	0.767082i	$0.721708\pi$
$374$	−10188.9	−1.40870
$375$	0	0
$376$	416.572i	0.0571358i
$377$	10800.9	1.47553
$378$	0	0
$379$	5818.96	0.788654	0.394327	−	0.918970i	$-0.370978\pi$
$379$	5818.96	0.788654	0.394327	+	0.918970i	$0.370978\pi$
$380$	6437.99i	0.869110i
$381$	0	0
$382$	3700.34	0.495617
$383$	−922.647	−0.123094	−0.0615471	−	0.998104i	$-0.519603\pi$
$383$	−922.647	−0.123094	−0.0615471	+	0.998104i	$0.519603\pi$
$384$	0	0
$385$	0	0
$386$	− 5276.19i	− 0.695728i
$387$	0	0
$388$	3416.08i	0.446972i
$389$	2334.74i	0.304308i	0.988357	+	0.152154i	$0.0486210\pi$
$389$	2334.74i	0.304308i	−0.988357	+	0.152154i	$0.951379\pi$
$390$	0	0
$391$	14303.0i	1.84996i
$392$	0	0
$393$	0	0
$394$	9147.48	1.16965
$395$	−4157.44	−0.529578
$396$	0	0
$397$	5192.01i	0.656372i	0.944613	+	0.328186i	$0.106437\pi$
$397$	5192.01i	0.656372i	−0.944613	+	0.328186i	$0.893563\pi$
$398$	−10456.3	−1.31690
$399$	0	0
$400$	−13894.9	−1.73686
$401$	− 3181.40i	− 0.396189i	−0.980183	−	0.198094i	$-0.936525\pi$
$401$	− 3181.40i	− 0.396189i	0.980183	−	0.198094i	$-0.0634753\pi$
$402$	0	0
$403$	13823.9	1.70873
$404$	5624.71	0.692672
$405$	0	0
$406$	0	0
$407$	− 11256.6i	− 1.37093i
$408$	0	0
$409$	− 7651.38i	− 0.925028i	−0.886612	−	0.462514i	$-0.846948\pi$
$409$	− 7651.38i	− 0.925028i	0.886612	−	0.462514i	$-0.153052\pi$
$410$	11582.8i	1.39520i
$411$	0	0
$412$	− 5868.99i	− 0.701807i
$413$	0	0
$414$	0	0
$415$	−6529.16	−0.772299
$416$	−8841.43	−1.04204
$417$	0	0
$418$	8231.15i	0.963155i
$419$	−1033.91	−0.120548	−0.0602740	−	0.998182i	$-0.519197\pi$
$419$	−1033.91	−0.120548	−0.0602740	+	0.998182i	$0.519197\pi$
$420$	0	0
$421$	−10640.4	−1.23178	−0.615890	−	0.787832i	$-0.711204\pi$
$421$	−10640.4	−1.23178	−0.615890	+	0.787832i	$0.711204\pi$
$422$	15047.8i	1.73582i
$423$	0	0
$424$	1870.10	0.214198
$425$	15219.8	1.73710
$426$	0	0
$427$	0	0
$428$	− 3079.44i	− 0.347781i
$429$	0	0
$430$	− 8918.12i	− 1.00016i
$431$	11301.9i	1.26309i	0.775338	+	0.631547i	$0.217580\pi$
$431$	11301.9i	1.26309i	−0.775338	+	0.631547i	$0.782420\pi$
$432$	0	0
$433$	− 8291.73i	− 0.920266i	−0.887850	−	0.460133i	$-0.847802\pi$
$433$	− 8291.73i	− 0.920266i	0.887850	−	0.460133i	$-0.152198\pi$
$434$	0	0
$435$	0	0
$436$	2302.97	0.252964
$437$	11554.8	1.26485
$438$	0	0
$439$	− 14739.7i	− 1.60248i	−0.598342	−	0.801241i	$-0.704174\pi$
$439$	− 14739.7i	− 1.60248i	0.598342	−	0.801241i	$-0.295826\pi$
$440$	−5488.87	−0.594709
$441$	0	0
$442$	13290.4	1.43022
$443$	11997.0i	1.28667i	0.765585	+	0.643334i	$0.222449\pi$
$443$	11997.0i	1.28667i	−0.765585	+	0.643334i	$0.777551\pi$
$444$	0	0
$445$	25048.4	2.66833
$446$	−11199.0	−1.18899
$447$	0	0
$448$	0	0
$449$	− 3457.16i	− 0.363371i	−0.983357	−	0.181685i	$-0.941845\pi$
$449$	− 3457.16i	− 0.363371i	0.983357	−	0.181685i	$-0.0581552\pi$
$450$	0	0
$451$	5933.67i	0.619524i
$452$	− 3122.87i	− 0.324972i
$453$	0	0
$454$	− 4865.21i	− 0.502941i
$455$	0	0
$456$	0	0
$457$	−2161.73	−0.221272	−0.110636	−	0.993861i	$-0.535289\pi$
$457$	−2161.73	−0.221272	−0.110636	+	0.993861i	$0.535289\pi$
$458$	14682.9	1.49800
$459$	0	0
$460$	− 15542.7i	− 1.57539i
$461$	−1097.92	−0.110923	−0.0554613	−	0.998461i	$-0.517663\pi$
$461$	−1097.92	−0.110923	−0.0554613	+	0.998461i	$0.517663\pi$
$462$	0	0
$463$	−9514.27	−0.955001	−0.477501	−	0.878631i	$-0.658457\pi$
$463$	−9514.27	−0.955001	−0.477501	+	0.878631i	$0.658457\pi$
$464$	19879.2i	1.98895i
$465$	0	0
$466$	8313.48	0.826426
$467$	−1703.14	−0.168762	−0.0843810	−	0.996434i	$-0.526891\pi$
$467$	−1703.14	−0.168762	−0.0843810	+	0.996434i	$0.526891\pi$
$468$	0	0
$469$	0	0
$470$	2733.53i	0.268273i
$471$	0	0
$472$	− 1077.96i	− 0.105121i
$473$	− 4568.61i	− 0.444112i
$474$	0	0
$475$	− 12295.4i	− 1.18769i
$476$	0	0
$477$	0	0
$478$	5308.71	0.507981
$479$	4154.22	0.396266	0.198133	−	0.980175i	$-0.436512\pi$
$479$	4154.22	0.396266	0.198133	+	0.980175i	$0.436512\pi$
$480$	0	0
$481$	14683.2i	1.39188i
$482$	17271.6	1.63216
$483$	0	0
$484$	−1446.87	−0.135882
$485$	− 11112.7i	− 1.04042i
$486$	0	0
$487$	−19187.3	−1.78534	−0.892670	−	0.450712i	$-0.851170\pi$
$487$	−19187.3	−1.78534	−0.892670	+	0.450712i	$0.851170\pi$
$488$	−877.099	−0.0813615
$489$	0	0
$490$	0	0
$491$	4090.61i	0.375981i	0.982171	+	0.187990i	$0.0601973\pi$
$491$	4090.61i	0.375981i	−0.982171	+	0.187990i	$0.939803\pi$
$492$	0	0
$493$	− 21774.8i	− 1.98922i
$494$	− 10736.7i	− 0.977871i
$495$	0	0
$496$	25443.2i	2.30329i
$497$	0	0
$498$	0	0
$499$	−1169.92	−0.104955	−0.0524777	−	0.998622i	$-0.516712\pi$
$499$	−1169.92	−0.104955	−0.0524777	+	0.998622i	$0.516712\pi$
$500$	−4906.60	−0.438860
$501$	0	0
$502$	− 17080.2i	− 1.51858i
$503$	−12774.4	−1.13237	−0.566187	−	0.824277i	$-0.691582\pi$
$503$	−12774.4	−1.13237	−0.566187	+	0.824277i	$0.691582\pi$
$504$	0	0
$505$	−18297.5	−1.61233
$506$	− 19871.8i	− 1.74587i
$507$	0	0
$508$	2672.60	0.233421
$509$	12294.4	1.07061	0.535304	−	0.844660i	$-0.320197\pi$
$509$	12294.4	1.07061	0.535304	+	0.844660i	$0.320197\pi$
$510$	0	0
$511$	0	0
$512$	− 10213.8i	− 0.881626i
$513$	0	0
$514$	− 16839.6i	− 1.44507i
$515$	19092.2i	1.63360i
$516$	0	0
$517$	1400.34i	0.119124i
$518$	0	0
$519$	0	0
$520$	7159.70	0.603795
$521$	−6271.20	−0.527344	−0.263672	−	0.964612i	$-0.584934\pi$
$521$	−6271.20	−0.527344	−0.263672	+	0.964612i	$0.584934\pi$
$522$	0	0
$523$	− 17333.5i	− 1.44922i	−0.689158	−	0.724611i	$-0.742019\pi$
$523$	− 17333.5i	− 1.44922i	0.689158	−	0.724611i	$-0.257981\pi$
$524$	1052.16	0.0877173
$525$	0	0
$526$	8474.48	0.702481
$527$	− 27869.2i	− 2.30361i
$528$	0	0
$529$	−15728.7	−1.29274
$530$	12271.5	1.00574
$531$	0	0
$532$	0	0
$533$	− 7739.89i	− 0.628990i
$534$	0	0
$535$	10017.6i	0.809530i
$536$	10161.3i	0.818843i
$537$	0	0
$538$	2768.63i	0.221867i
$539$	0	0
$540$	0	0
$541$	−5637.41	−0.448005	−0.224003	−	0.974589i	$-0.571912\pi$
$541$	−5637.41	−0.448005	−0.224003	+	0.974589i	$0.571912\pi$
$542$	31492.2	2.49577
$543$	0	0
$544$	17824.5i	1.40481i
$545$	−7491.70	−0.588824
$546$	0	0
$547$	12354.9	0.965733	0.482867	−	0.875694i	$-0.339596\pi$
$547$	12354.9	0.965733	0.482867	+	0.875694i	$0.339596\pi$
$548$	− 2069.47i	− 0.161320i
$549$	0	0
$550$	−21145.5	−1.63936
$551$	−17590.9	−1.36007
$552$	0	0
$553$	0	0
$554$	7510.40i	0.575968i
$555$	0	0
$556$	8060.66i	0.614835i
$557$	− 7654.58i	− 0.582289i	−0.956679	−	0.291145i	$-0.905964\pi$
$557$	− 7654.58i	− 0.582289i	0.956679	−	0.291145i	$-0.0940361\pi$
$558$	0	0
$559$	5959.30i	0.450897i
$560$	0	0
$561$	0	0
$562$	860.711	0.0646030
$563$	−10673.4	−0.798987	−0.399493	−	0.916736i	$-0.630814\pi$
$563$	−10673.4	−0.798987	−0.399493	+	0.916736i	$0.630814\pi$
$564$	0	0
$565$	10158.9i	0.756438i
$566$	2728.11	0.202599
$567$	0	0
$568$	4495.18	0.332066
$569$	4382.00i	0.322852i	0.986885	+	0.161426i	$0.0516093\pi$
$569$	4382.00i	0.322852i	−0.986885	+	0.161426i	$0.948391\pi$
$570$	0	0
$571$	23865.0	1.74907	0.874536	−	0.484961i	$-0.161166\pi$
$571$	23865.0	1.74907	0.874536	+	0.484961i	$0.161166\pi$
$572$	−7398.56	−0.540820
$573$	0	0
$574$	0	0
$575$	29683.8i	2.15287i
$576$	0	0
$577$	12914.2i	0.931759i	0.884848	+	0.465879i	$0.154262\pi$
$577$	12914.2i	0.931759i	−0.884848	+	0.465879i	$0.845738\pi$
$578$	− 8843.67i	− 0.636416i
$579$	0	0
$580$	23662.1i	1.69399i
$581$	0	0
$582$	0	0
$583$	6286.50	0.446587
$584$	−1770.71	−0.125467
$585$	0	0
$586$	14932.6i	1.05266i
$587$	17218.9	1.21074	0.605368	−	0.795946i	$-0.293026\pi$
$587$	17218.9	1.21074	0.605368	+	0.795946i	$0.293026\pi$
$588$	0	0
$589$	−22514.4	−1.57502
$590$	− 7073.51i	− 0.493579i
$591$	0	0
$592$	−27024.7	−1.87620
$593$	−10658.5	−0.738096	−0.369048	−	0.929410i	$-0.620316\pi$
$593$	−10658.5	−0.738096	−0.369048	+	0.929410i	$0.620316\pi$
$594$	0	0
$595$	0	0
$596$	7920.09i	0.544328i
$597$	0	0
$598$	25920.8i	1.77254i
$599$	− 21125.2i	− 1.44099i	−0.693462	−	0.720493i	$-0.743915\pi$
$599$	− 21125.2i	− 1.44099i	0.693462	−	0.720493i	$-0.256085\pi$
$600$	0	0
$601$	10924.1i	0.741434i	0.928746	+	0.370717i	$0.120888\pi$
$601$	10924.1i	0.741434i	−0.928746	+	0.370717i	$0.879112\pi$
$602$	0	0
$603$	0	0
$604$	14319.8	0.964674
$605$	4706.76	0.316292
$606$	0	0
$607$	− 18925.7i	− 1.26552i	−0.774348	−	0.632760i	$-0.781922\pi$
$607$	− 18925.7i	− 1.26552i	0.774348	−	0.632760i	$-0.218078\pi$
$608$	14399.7	0.960499
$609$	0	0
$610$	−5755.49	−0.382022
$611$	− 1826.61i	− 0.120944i
$612$	0	0
$613$	−14531.3	−0.957444	−0.478722	−	0.877967i	$-0.658900\pi$
$613$	−14531.3	−0.957444	−0.478722	+	0.877967i	$0.658900\pi$
$614$	−16706.3	−1.09806
$615$	0	0
$616$	0	0
$617$	2146.97i	0.140087i	0.997544	+	0.0700435i	$0.0223138\pi$
$617$	2146.97i	0.140087i	−0.997544	+	0.0700435i	$0.977686\pi$
$618$	0	0
$619$	20075.0i	1.30353i	0.758422	+	0.651764i	$0.225971\pi$
$619$	20075.0i	1.30353i	−0.758422	+	0.651764i	$0.774029\pi$
$620$	30284.8i	1.96172i
$621$	0	0
$622$	− 35401.6i	− 2.28212i
$623$	0	0
$624$	0	0
$625$	−6254.26	−0.400273
$626$	14316.2	0.914041
$627$	0	0
$628$	− 6845.74i	− 0.434992i
$629$	29601.5	1.87645
$630$	0	0
$631$	22589.8	1.42518	0.712589	−	0.701582i	$-0.247523\pi$
$631$	22589.8	1.42518	0.712589	+	0.701582i	$0.247523\pi$
$632$	2314.77i	0.145691i
$633$	0	0
$634$	33039.0	2.06963
$635$	−8694.15	−0.543333
$636$	0	0
$637$	0	0
$638$	30252.7i	1.87730i
$639$	0	0
$640$	20389.6i	1.25933i
$641$	− 17576.9i	− 1.08307i	−0.840678	−	0.541535i	$-0.817843\pi$
$641$	− 17576.9i	− 1.08307i	0.840678	−	0.541535i	$-0.182157\pi$
$642$	0	0
$643$	2583.33i	0.158439i	0.996857	+	0.0792197i	$0.0252429\pi$
$643$	2583.33i	0.158439i	−0.996857	+	0.0792197i	$0.974757\pi$
$644$	0	0
$645$	0	0
$646$	−21645.5	−1.31831
$647$	−26853.5	−1.63172	−0.815859	−	0.578251i	$-0.803735\pi$
$647$	−26853.5	−1.63172	−0.815859	+	0.578251i	$0.803735\pi$
$648$	0	0
$649$	− 3623.64i	− 0.219169i
$650$	27582.3	1.66441
$651$	0	0
$652$	−9605.17	−0.576944
$653$	25348.2i	1.51907i	0.650468	+	0.759533i	$0.274573\pi$
$653$	25348.2i	1.51907i	−0.650468	+	0.759533i	$0.725427\pi$
$654$	0	0
$655$	−3422.74	−0.204180
$656$	14245.4	0.847852
$657$	0	0
$658$	0	0
$659$	16599.6i	0.981225i	0.871378	+	0.490612i	$0.163227\pi$
$659$	16599.6i	0.981225i	−0.871378	+	0.490612i	$0.836773\pi$
$660$	0	0
$661$	6695.25i	0.393972i	0.980406	+	0.196986i	$0.0631153\pi$
$661$	6695.25i	0.393972i	−0.980406	+	0.196986i	$0.936885\pi$
$662$	12867.8i	0.755470i
$663$	0	0
$664$	3635.29i	0.212465i
$665$	0	0
$666$	0	0
$667$	42468.3	2.46533
$668$	−18966.9	−1.09858
$669$	0	0
$670$	66677.9i	3.84476i
$671$	−2948.45	−0.169633
$672$	0	0
$673$	10886.2	0.623524	0.311762	−	0.950160i	$-0.399081\pi$
$673$	10886.2	0.623524	0.311762	+	0.950160i	$0.399081\pi$
$674$	− 37609.8i	− 2.14937i
$675$	0	0
$676$	−2099.98	−0.119480
$677$	19575.3	1.11129	0.555643	−	0.831421i	$-0.312472\pi$
$677$	19575.3	1.11129	0.555643	+	0.831421i	$0.312472\pi$
$678$	0	0
$679$	0	0
$680$	− 14434.1i	− 0.814003i
$681$	0	0
$682$	38720.0i	2.17399i
$683$	11899.0i	0.666622i	0.942817	+	0.333311i	$0.108166\pi$
$683$	11899.0i	0.666622i	−0.942817	+	0.333311i	$0.891834\pi$
$684$	0	0
$685$	6732.11i	0.375505i
$686$	0	0
$687$	0	0
$688$	−10968.2	−0.607790
$689$	−8200.13	−0.453411
$690$	0	0
$691$	18078.9i	0.995304i	0.867377	+	0.497652i	$0.165804\pi$
$691$	18078.9i	0.995304i	−0.867377	+	0.497652i	$0.834196\pi$
$692$	−20627.5	−1.13315
$693$	0	0
$694$	−344.829	−0.0188610
$695$	− 26221.8i	− 1.43115i
$696$	0	0
$697$	−15603.8	−0.847969
$698$	15500.4	0.840542
$699$	0	0
$700$	0	0
$701$	11922.5i	0.642378i	0.947015	+	0.321189i	$0.104082\pi$
$701$	11922.5i	0.642378i	−0.947015	+	0.321189i	$0.895918\pi$
$702$	0	0
$703$	− 23913.8i	− 1.28297i
$704$	− 4396.49i	− 0.235368i
$705$	0	0
$706$	26956.6i	1.43700i
$707$	0	0
$708$	0	0
$709$	25610.4	1.35659	0.678294	−	0.734791i	$-0.262720\pi$
$709$	25610.4	1.35659	0.678294	+	0.734791i	$0.262720\pi$
$710$	29497.2	1.55917
$711$	0	0
$712$	− 13946.4i	− 0.734078i
$713$	54354.5	2.85497
$714$	0	0
$715$	24068.0	1.25887
$716$	− 10752.5i	− 0.561229i
$717$	0	0
$718$	−1443.56	−0.0750321
$719$	−13553.9	−0.703025	−0.351513	−	0.936183i	$-0.614333\pi$
$719$	−13553.9	−0.703025	−0.351513	+	0.936183i	$0.614333\pi$
$720$	0	0
$721$	0	0
$722$	− 7573.30i	− 0.390373i
$723$	0	0
$724$	− 16482.6i	− 0.846091i
$725$	− 45190.4i	− 2.31494i
$726$	0	0
$727$	− 6238.21i	− 0.318243i	−0.987259	−	0.159121i	$-0.949134\pi$
$727$	− 6238.21i	− 0.318243i	0.987259	−	0.159121i	$-0.0508661\pi$
$728$	0	0
$729$	0	0
$730$	−11619.3	−0.589111
$731$	12014.1	0.607874
$732$	0	0
$733$	3179.35i	0.160208i	0.996787	+	0.0801038i	$0.0255252\pi$
$733$	3179.35i	0.160208i	−0.996787	+	0.0801038i	$0.974475\pi$
$734$	−13151.0	−0.661323
$735$	0	0
$736$	−34763.9	−1.74105
$737$	34158.0i	1.70723i
$738$	0	0
$739$	2794.15	0.139086	0.0695431	−	0.997579i	$-0.477846\pi$
$739$	2794.15	0.139086	0.0695431	+	0.997579i	$0.477846\pi$
$740$	−32167.2	−1.59796
$741$	0	0
$742$	0	0
$743$	− 8539.68i	− 0.421656i	−0.977523	−	0.210828i	$-0.932384\pi$
$743$	− 8539.68i	− 0.421656i	0.977523	−	0.210828i	$-0.0676160\pi$
$744$	0	0
$745$	− 25764.6i	− 1.26703i
$746$	33770.6i	1.65741i
$747$	0	0
$748$	14915.6i	0.729103i
$749$	0	0
$750$	0	0
$751$	−28818.3	−1.40026	−0.700130	−	0.714015i	$-0.746875\pi$
$751$	−28818.3	−1.40026	−0.700130	+	0.714015i	$0.746875\pi$
$752$	3361.92	0.163027
$753$	0	0
$754$	− 39461.6i	− 1.90598i
$755$	−46583.1	−2.24547
$756$	0	0
$757$	−20086.6	−0.964413	−0.482206	−	0.876058i	$-0.660164\pi$
$757$	−20086.6	−0.964413	−0.482206	+	0.876058i	$0.660164\pi$
$758$	− 21259.9i	− 1.01873i
$759$	0	0
$760$	−11660.7	−0.556550
$761$	−18832.4	−0.897076	−0.448538	−	0.893764i	$-0.648055\pi$
$761$	−18832.4	−0.897076	−0.448538	+	0.893764i	$0.648055\pi$
$762$	0	0
$763$	0	0
$764$	− 5416.98i	− 0.256518i
$765$	0	0
$766$	3370.95i	0.159004i
$767$	4726.69i	0.222517i
$768$	0	0
$769$	− 5882.35i	− 0.275842i	−0.990443	−	0.137921i	$-0.955958\pi$
$769$	− 5882.35i	− 0.275842i	0.990443	−	0.137921i	$-0.0440421\pi$
$770$	0	0
$771$	0	0
$772$	−7723.90	−0.360090
$773$	−12121.3	−0.564001	−0.282001	−	0.959414i	$-0.590998\pi$
$773$	−12121.3	−0.564001	−0.282001	+	0.959414i	$0.590998\pi$
$774$	0	0
$775$	− 57838.6i	− 2.68080i
$776$	−6187.32	−0.286227
$777$	0	0
$778$	8530.11	0.393084
$779$	12605.6i	0.579774i
$780$	0	0
$781$	15110.9	0.692333
$782$	52256.8	2.38964
$783$	0	0
$784$	0	0
$785$	22269.6i	1.01253i
$786$	0	0
$787$	− 15406.1i	− 0.697797i	−0.937160	−	0.348899i	$-0.886556\pi$
$787$	− 15406.1i	− 0.697797i	0.937160	−	0.348899i	$-0.113444\pi$
$788$	− 13391.1i	− 0.605380i
$789$	0	0
$790$	15189.4i	0.684071i
$791$	0	0
$792$	0	0
$793$	3845.96	0.172224
$794$	18969.3	0.847854
$795$	0	0
$796$	15307.1i	0.681592i
$797$	−12661.3	−0.562716	−0.281358	−	0.959603i	$-0.590785\pi$
$797$	−12661.3	−0.562716	−0.281358	+	0.959603i	$0.590785\pi$
$798$	0	0
$799$	−3682.48	−0.163050
$800$	36992.2i	1.63484i
$801$	0	0
$802$	−11623.5	−0.511769
$803$	−5952.40	−0.261589
$804$	0	0
$805$	0	0
$806$	− 50506.4i	− 2.20721i
$807$	0	0
$808$	10187.7i	0.443565i
$809$	− 29294.3i	− 1.27309i	−0.771239	−	0.636546i	$-0.780363\pi$
$809$	− 29294.3i	− 1.27309i	0.771239	−	0.636546i	$-0.219637\pi$
$810$	0	0
$811$	− 28420.8i	− 1.23057i	−0.788305	−	0.615284i	$-0.789041\pi$
$811$	− 28420.8i	− 1.23057i	0.788305	−	0.615284i	$-0.210959\pi$
$812$	0	0
$813$	0	0
$814$	−41126.8	−1.77087
$815$	31246.2	1.34295
$816$	0	0
$817$	− 9705.66i	− 0.415616i
$818$	−27954.8	−1.19488
$819$	0	0
$820$	16956.2	0.722118
$821$	− 25813.0i	− 1.09730i	−0.836053	−	0.548649i	$-0.815142\pi$
$821$	− 25813.0i	− 1.09730i	0.836053	−	0.548649i	$-0.184858\pi$
$822$	0	0
$823$	−32738.2	−1.38661	−0.693306	−	0.720643i	$-0.743847\pi$
$823$	−32738.2	−1.38661	−0.693306	+	0.720643i	$0.743847\pi$
$824$	10630.1	0.449415
$825$	0	0
$826$	0	0
$827$	652.139i	0.0274209i	0.999906	+	0.0137105i	$0.00436431\pi$
$827$	652.139i	0.0274209i	−0.999906	+	0.0137105i	$0.995636\pi$
$828$	0	0
$829$	− 30343.9i	− 1.27128i	−0.771987	−	0.635638i	$-0.780737\pi$
$829$	− 30343.9i	− 1.27128i	0.771987	−	0.635638i	$-0.219263\pi$
$830$	23854.7i	0.997600i
$831$	0	0
$832$	5734.79i	0.238964i
$833$	0	0
$834$	0	0
$835$	61700.4	2.55716
$836$	12049.7	0.498503
$837$	0	0
$838$	3777.44i	0.155715i
$839$	−2133.68	−0.0877982	−0.0438991	−	0.999036i	$-0.513978\pi$
$839$	−2133.68	−0.0877982	−0.0438991	+	0.999036i	$0.513978\pi$
$840$	0	0
$841$	−40264.4	−1.65093
$842$	38875.2i	1.59113i
$843$	0	0
$844$	22028.8	0.898415
$845$	6831.35	0.278113
$846$	0	0
$847$	0	0
$848$	− 15092.5i	− 0.611178i
$849$	0	0
$850$	− 55606.3i	− 2.24386i
$851$	57733.2i	2.32558i
$852$	0	0
$853$	− 42021.1i	− 1.68672i	−0.537346	−	0.843362i	$-0.680573\pi$
$853$	− 42021.1i	− 1.68672i	0.537346	−	0.843362i	$-0.319427\pi$
$854$	0	0
$855$	0	0
$856$	5577.58	0.222708
$857$	15672.7	0.624703	0.312351	−	0.949967i	$-0.398883\pi$
$857$	15672.7	0.624703	0.312351	+	0.949967i	$0.398883\pi$
$858$	0	0
$859$	− 32677.9i	− 1.29797i	−0.760802	−	0.648984i	$-0.775194\pi$
$859$	− 32677.9i	− 1.29797i	0.760802	−	0.648984i	$-0.224806\pi$
$860$	−13055.4	−0.517657
$861$	0	0
$862$	41292.2	1.63158
$863$	− 26356.3i	− 1.03960i	−0.854287	−	0.519802i	$-0.826006\pi$
$863$	− 26356.3i	− 1.03960i	0.854287	−	0.519802i	$-0.173994\pi$
$864$	0	0
$865$	67102.6	2.63764
$866$	−30294.3	−1.18873
$867$	0	0
$868$	0	0
$869$	7781.30i	0.303754i
$870$	0	0
$871$	− 44555.8i	− 1.73331i
$872$	4171.21i	0.161990i
$873$	0	0
$874$	− 42216.1i	− 1.63385i
$875$	0	0
$876$	0	0
$877$	3002.81	0.115619	0.0578095	−	0.998328i	$-0.481588\pi$
$877$	3002.81	0.115619	0.0578095	+	0.998328i	$0.481588\pi$
$878$	−53852.5	−2.06997
$879$	0	0
$880$	44297.6i	1.69690i
$881$	−12329.6	−0.471504	−0.235752	−	0.971813i	$-0.575755\pi$
$881$	−12329.6	−0.471504	−0.235752	+	0.971813i	$0.575755\pi$
$882$	0	0
$883$	−38324.4	−1.46061	−0.730305	−	0.683121i	$-0.760622\pi$
$883$	−38324.4	−1.46061	−0.730305	+	0.683121i	$0.760622\pi$
$884$	− 19456.0i	− 0.740243i
$885$	0	0
$886$	43831.7	1.66203
$887$	45531.5	1.72356	0.861780	−	0.507282i	$-0.169350\pi$
$887$	45531.5	1.72356	0.861780	+	0.507282i	$0.169350\pi$
$888$	0	0
$889$	0	0
$890$	− 91515.9i	− 3.44676i
$891$	0	0
$892$	16394.4i	0.615387i
$893$	2974.93i	0.111481i
$894$	0	0
$895$	34978.6i	1.30637i
$896$	0	0
$897$	0	0
$898$	−12630.9	−0.469376
$899$	−82749.0	−3.06989
$900$	0	0
$901$	16531.6i	0.611263i
$902$	21679.0	0.800257
$903$	0	0
$904$	5656.24	0.208102
$905$	53618.8i	1.96945i
$906$	0	0
$907$	−12642.5	−0.462829	−0.231414	−	0.972855i	$-0.574335\pi$
$907$	−12642.5	−0.462829	−0.231414	+	0.972855i	$0.574335\pi$
$908$	−7122.25	−0.260309
$909$	0	0
$910$	0	0
$911$	− 4016.70i	− 0.146080i	−0.997329	−	0.0730402i	$-0.976730\pi$
$911$	− 4016.70i	− 0.146080i	0.997329	−	0.0730402i	$-0.0232702\pi$
$912$	0	0
$913$	12220.4i	0.442974i
$914$	7898.01i	0.285824i
$915$	0	0
$916$	− 21494.5i	− 0.775325i
$917$	0	0
$918$	0	0
$919$	−688.102	−0.0246990	−0.0123495	−	0.999924i	$-0.503931\pi$
$919$	−688.102	−0.0246990	−0.0123495	+	0.999924i	$0.503931\pi$
$920$	28151.4	1.00883
$921$	0	0
$922$	4011.33i	0.143282i
$923$	−19710.7	−0.702911
$924$	0	0
$925$	61433.8	2.18371
$926$	34761.0i	1.23360i
$927$	0	0
$928$	52924.3	1.87212
$929$	48976.8	1.72968	0.864842	−	0.502045i	$-0.167419\pi$
$929$	48976.8	1.72968	0.864842	+	0.502045i	$0.167419\pi$
$930$	0	0
$931$	0	0
$932$	− 12170.2i	− 0.427735i
$933$	0	0
$934$	6222.52i	0.217995i
$935$	− 48521.5i	− 1.69713i
$936$	0	0
$937$	− 1602.26i	− 0.0558629i	−0.999610	−	0.0279315i	$-0.991108\pi$
$937$	− 1602.26i	− 0.0558629i	0.999610	−	0.0279315i	$-0.00889202\pi$
$938$	0	0
$939$	0	0
$940$	4001.66	0.138851
$941$	−56744.1	−1.96579	−0.982894	−	0.184174i	$-0.941039\pi$
$941$	−56744.1	−1.96579	−0.982894	+	0.184174i	$0.941039\pi$
$942$	0	0
$943$	− 30432.7i	− 1.05093i
$944$	−8699.58	−0.299944
$945$	0	0
$946$	−16691.7	−0.573672
$947$	29814.9i	1.02308i	0.859260	+	0.511538i	$0.170924\pi$
$947$	29814.9i	1.02308i	−0.859260	+	0.511538i	$0.829076\pi$
$948$	0	0
$949$	7764.32	0.265585
$950$	−44922.1	−1.53417
$951$	0	0
$952$	0	0
$953$	16073.0i	0.546335i	0.961966	+	0.273167i	$0.0880713\pi$
$953$	16073.0i	0.546335i	−0.961966	+	0.273167i	$0.911929\pi$
$954$	0	0
$955$	17621.8i	0.597096i
$956$	− 7771.51i	− 0.262917i
$957$	0	0
$958$	− 15177.7i	− 0.511868i
$959$	0	0
$960$	0	0
$961$	−76118.2	−2.55508
$962$	53645.8	1.79793
$963$	0	0
$964$	− 25284.2i	− 0.844762i
$965$	25126.3	0.838182
$966$	0	0
$967$	−4382.43	−0.145739	−0.0728694	−	0.997341i	$-0.523216\pi$
$967$	−4382.43	−0.145739	−0.0728694	+	0.997341i	$0.523216\pi$
$968$	− 2620.62i	− 0.0870144i
$969$	0	0
$970$	−40601.0	−1.34394
$971$	−58016.2	−1.91744	−0.958718	−	0.284360i	$-0.908219\pi$
$971$	−58016.2	−1.91744	−0.958718	+	0.284360i	$0.908219\pi$
$972$	0	0
$973$	0	0
$974$	70102.0i	2.30617i
$975$	0	0
$976$	7078.57i	0.232151i
$977$	− 6963.32i	− 0.228021i	−0.993480	−	0.114010i	$-0.963630\pi$
$977$	− 6963.32i	− 0.228021i	0.993480	−	0.114010i	$-0.0363697\pi$
$978$	0	0
$979$	− 46882.1i	− 1.53050i
$980$	0	0
$981$	0	0
$982$	14945.3	0.485665
$983$	42775.1	1.38791	0.693954	−	0.720019i	$-0.255867\pi$
$983$	42775.1	1.38791	0.693954	+	0.720019i	$0.255867\pi$
$984$	0	0
$985$	43562.2i	1.40914i
$986$	−79555.4	−2.56953
$987$	0	0
$988$	−15717.7	−0.506119
$989$	23431.6i	0.753367i
$990$	0	0
$991$	−19948.1	−0.639427	−0.319714	−	0.947514i	$-0.603587\pi$
$991$	−19948.1	−0.639427	−0.319714	+	0.947514i	$0.603587\pi$
$992$	67737.1	2.16800
$993$	0	0
$994$	0	0
$995$	− 49795.1i	− 1.58654i
$996$	0	0
$997$	41428.5i	1.31600i	0.753017	+	0.658001i	$0.228598\pi$
$997$	41428.5i	1.31600i	−0.753017	+	0.658001i	$0.771402\pi$
$998$	4274.37i	0.135574i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.c.b.440.3	✓	24
3.2	odd	2	inner	441.4.c.b.440.22	yes	24
7.2	even	3		441.4.p.d.80.5		48
7.3	odd	6		441.4.p.d.215.20		48
7.4	even	3		441.4.p.d.215.19		48
7.5	odd	6		441.4.p.d.80.6		48
7.6	odd	2	inner	441.4.c.b.440.21	yes	24
21.2	odd	6		441.4.p.d.80.20		48
21.5	even	6		441.4.p.d.80.19		48
21.11	odd	6		441.4.p.d.215.6		48
21.17	even	6		441.4.p.d.215.5		48
21.20	even	2	inner	441.4.c.b.440.4	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.4.c.b.440.3	✓	24	1.1	even	1	trivial
441.4.c.b.440.4	yes	24	21.20	even	2	inner
441.4.c.b.440.21	yes	24	7.6	odd	2	inner
441.4.c.b.440.22	yes	24	3.2	odd	2	inner
441.4.p.d.80.5		48	7.2	even	3
441.4.p.d.80.6		48	7.5	odd	6
441.4.p.d.80.19		48	21.5	even	6
441.4.p.d.80.20		48	21.2	odd	6
441.4.p.d.215.5		48	21.17	even	6
441.4.p.d.215.6		48	21.11	odd	6
441.4.p.d.215.19		48	7.4	even	3
441.4.p.d.215.20		48	7.3	odd	6

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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.65356i q^{2} -5.34851 q^{4} +17.3990 q^{5} -9.68739i q^{8} +O(q^{10})\)	\(q-3.65356i q^{2} -5.34851 q^{4} +17.3990 q^{5} -9.68739i q^{8} -63.5684i q^{10} -32.5650i q^{11} +42.4779i q^{13} -78.1815 q^{16} +85.6362 q^{17} -69.1820i q^{19} -93.0588 q^{20} -118.978 q^{22} +167.020i q^{23} +177.726 q^{25} +155.196 q^{26} -254.270i q^{29} -325.437i q^{31} +208.142i q^{32} -312.877i q^{34} +345.666 q^{37} -252.761 q^{38} -168.551i q^{40} -182.210 q^{41} +140.292 q^{43} +174.174i q^{44} +610.218 q^{46} -43.0015 q^{47} -649.332i q^{50} -227.193i q^{52} +193.045i q^{53} -566.600i q^{55} -928.992 q^{58} +111.274 q^{59} -90.5402i q^{61} -1189.00 q^{62} +135.007 q^{64} +739.074i q^{65} -1048.92 q^{67} -458.026 q^{68} +464.023i q^{71} -182.785i q^{73} -1262.91i q^{74} +370.020i q^{76} -238.947 q^{79} -1360.28 q^{80} +665.714i q^{82} -375.260 q^{83} +1489.99 q^{85} -512.565i q^{86} -315.470 q^{88} +1439.65 q^{89} -893.308i q^{92} +157.108i q^{94} -1203.70i q^{95} -638.698i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 96 q^{4}+O(q^{10}) \) 24 * q - 96 * q^4	\( 24 q - 96 q^{4} + 144 q^{16} + 624 q^{22} + 312 q^{25} - 864 q^{37} + 1248 q^{43} - 3888 q^{46} - 7440 q^{58} - 3360 q^{64} - 2688 q^{67} + 480 q^{79} + 13248 q^{85} - 7248 q^{88}+O(q^{100}) \) 24 * q - 96 * q^4 + 144 * q^16 + 624 * q^22 + 312 * q^25 - 864 * q^37 + 1248 * q^43 - 3888 * q^46 - 7440 * q^58 - 3360 * q^64 - 2688 * q^67 + 480 * q^79 + 13248 * q^85 - 7248 * q^88

Introduction

L-functions

Modular forms

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