LMFDB - Embedded newform 3721.2.a.k.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3721,2,Mod(1,3721)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3721, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3721.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3721 = 61^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3721.a (trivial)

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:	yes
Analytic conductor:	$29.7123345921$
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} - 17x^{14} + 111x^{12} - 361x^{10} + 624x^{8} - 558x^{6} + 229x^{4} - 34x^{2} + 1 $ x^16 - 17x^14 + 111x^12 - 361x^10 + 624x^8 - 558x^6 + 229x^4 - 34*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 61)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.196205$ of defining polynomial
Character	$\chi$	$=$	3721.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.196205 q^{2} +2.79938 q^{3} -1.96150 q^{4} +3.62428 q^{5} +0.549253 q^{6} -0.234105 q^{7} -0.777267 q^{8} +4.83656 q^{9} +O(q^{10})$	\(q+0.196205 q^{2} +2.79938 q^{3} -1.96150 q^{4} +3.62428 q^{5} +0.549253 q^{6} -0.234105 q^{7} -0.777267 q^{8} +4.83656 q^{9} +0.711102 q^{10} -5.67170 q^{11} -5.49100 q^{12} +2.04020 q^{13} -0.0459325 q^{14} +10.1458 q^{15} +3.77050 q^{16} +2.98626 q^{17} +0.948956 q^{18} +0.842783 q^{19} -7.10904 q^{20} -0.655349 q^{21} -1.11282 q^{22} +3.51206 q^{23} -2.17587 q^{24} +8.13540 q^{25} +0.400298 q^{26} +5.14123 q^{27} +0.459197 q^{28} -3.91037 q^{29} +1.99065 q^{30} +3.59760 q^{31} +2.29433 q^{32} -15.8773 q^{33} +0.585918 q^{34} -0.848461 q^{35} -9.48692 q^{36} +1.19305 q^{37} +0.165358 q^{38} +5.71131 q^{39} -2.81703 q^{40} +3.84070 q^{41} -0.128583 q^{42} +9.57128 q^{43} +11.1251 q^{44} +17.5290 q^{45} +0.689084 q^{46} +6.21907 q^{47} +10.5551 q^{48} -6.94520 q^{49} +1.59621 q^{50} +8.35968 q^{51} -4.00186 q^{52} -2.31070 q^{53} +1.00873 q^{54} -20.5558 q^{55} +0.181962 q^{56} +2.35927 q^{57} -0.767234 q^{58} +5.00296 q^{59} -19.9009 q^{60} +0.705868 q^{62} -1.13226 q^{63} -7.09085 q^{64} +7.39426 q^{65} -3.11520 q^{66} -3.39714 q^{67} -5.85755 q^{68} +9.83162 q^{69} -0.166472 q^{70} +1.92117 q^{71} -3.75929 q^{72} +5.12218 q^{73} +0.234082 q^{74} +22.7741 q^{75} -1.65312 q^{76} +1.32777 q^{77} +1.12059 q^{78} +10.3008 q^{79} +13.6654 q^{80} -0.117398 q^{81} +0.753565 q^{82} -9.80217 q^{83} +1.28547 q^{84} +10.8230 q^{85} +1.87793 q^{86} -10.9466 q^{87} +4.40843 q^{88} +7.01825 q^{89} +3.43928 q^{90} -0.477620 q^{91} -6.88892 q^{92} +10.0711 q^{93} +1.22021 q^{94} +3.05448 q^{95} +6.42270 q^{96} -11.6170 q^{97} -1.36268 q^{98} -27.4315 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 6 q^{3} + 2 q^{4} + 10 q^{5} + 6 q^{9}+O(q^{10}) $ 16 * q + 6 * q^3 + 2 * q^4 + 10 * q^5 + 6 * q^9	$ 16 q + 6 q^{3} + 2 q^{4} + 10 q^{5} + 6 q^{9} + 10 q^{12} - 12 q^{13} + 18 q^{14} + 32 q^{15} - 6 q^{16} + 32 q^{19} + 2 q^{20} - 6 q^{22} - 2 q^{25} + 24 q^{27} + 16 q^{34} - 12 q^{36} - 12 q^{39} + 38 q^{41} + 40 q^{42} + 104 q^{45} + 28 q^{46} + 6 q^{47} + 20 q^{48} + 20 q^{49} - 54 q^{52} + 32 q^{56} + 64 q^{57} - 26 q^{58} - 14 q^{60} + 16 q^{62} + 18 q^{64} - 60 q^{65} - 18 q^{66} - 52 q^{70} + 54 q^{73} + 24 q^{74} + 88 q^{75} + 66 q^{76} - 42 q^{77} + 14 q^{80} + 16 q^{81} - 64 q^{83} - 20 q^{86} + 88 q^{88} + 68 q^{95} - 30 q^{97}+O(q^{100}) $ 16 * q + 6 * q^3 + 2 * q^4 + 10 * q^5 + 6 * q^9 + 10 * q^12 - 12 * q^13 + 18 * q^14 + 32 * q^15 - 6 * q^16 + 32 * q^19 + 2 * q^20 - 6 * q^22 - 2 * q^25 + 24 * q^27 + 16 * q^34 - 12 * q^36 - 12 * q^39 + 38 * q^41 + 40 * q^42 + 104 * q^45 + 28 * q^46 + 6 * q^47 + 20 * q^48 + 20 * q^49 - 54 * q^52 + 32 * q^56 + 64 * q^57 - 26 * q^58 - 14 * q^60 + 16 * q^62 + 18 * q^64 - 60 * q^65 - 18 * q^66 - 52 * q^70 + 54 * q^73 + 24 * q^74 + 88 * q^75 + 66 * q^76 - 42 * q^77 + 14 * q^80 + 16 * q^81 - 64 * q^83 - 20 * q^86 + 88 * q^88 + 68 * q^95 - 30 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	0.196205	0.138738	0.0693690	−	0.997591i	$-0.477901\pi$
$2$	0.196205	0.138738	0.0693690	+	0.997591i	$0.477901\pi$
$3$	2.79938	1.61623	0.808113	−	0.589028i	$-0.200489\pi$
$3$	2.79938	1.61623	0.808113	+	0.589028i	$0.200489\pi$
$4$	−1.96150	−0.980752
$5$	3.62428	1.62083	0.810414	−	0.585858i	$-0.199242\pi$
$5$	3.62428	1.62083	0.810414	+	0.585858i	$0.199242\pi$
$6$	0.549253	0.224232
$7$	−0.234105	−0.0884832	−0.0442416	−	0.999021i	$-0.514087\pi$
$7$	−0.234105	−0.0884832	−0.0442416	+	0.999021i	$0.514087\pi$
$8$	−0.777267	−0.274805
$9$	4.83656	1.61219
$10$	0.711102	0.224870
$11$	−5.67170	−1.71008	−0.855041	−	0.518560i	$-0.826468\pi$
$11$	−5.67170	−1.71008	−0.855041	+	0.518560i	$0.826468\pi$
$12$	−5.49100	−1.58512
$13$	2.04020	0.565850	0.282925	−	0.959142i	$-0.408695\pi$
$13$	2.04020	0.565850	0.282925	+	0.959142i	$0.408695\pi$
$14$	−0.0459325	−0.0122760
$15$	10.1458	2.61962
$16$	3.77050	0.942626
$17$	2.98626	0.724273	0.362137	−	0.932125i	$-0.382047\pi$
$17$	2.98626	0.724273	0.362137	+	0.932125i	$0.382047\pi$
$18$	0.948956	0.223671
$19$	0.842783	0.193348	0.0966739	−	0.995316i	$-0.469180\pi$
$19$	0.842783	0.193348	0.0966739	+	0.995316i	$0.469180\pi$
$20$	−7.10904	−1.58963
$21$	−0.655349	−0.143009
$22$	−1.11282	−0.237253
$23$	3.51206	0.732316	0.366158	−	0.930553i	$-0.380673\pi$
$23$	3.51206	0.732316	0.366158	+	0.930553i	$0.380673\pi$
$24$	−2.17587	−0.444147
$25$	8.13540	1.62708
$26$	0.400298	0.0785048
$27$	5.14123	0.989429
$28$	0.459197	0.0867801
$29$	−3.91037	−0.726137	−0.363068	−	0.931762i	$-0.618271\pi$
$29$	−3.91037	−0.726137	−0.363068	+	0.931762i	$0.618271\pi$
$30$	1.99065	0.363441
$31$	3.59760	0.646149	0.323074	−	0.946374i	$-0.395284\pi$
$31$	3.59760	0.646149	0.323074	+	0.946374i	$0.395284\pi$
$32$	2.29433	0.405583
$33$	−15.8773	−2.76388
$34$	0.585918	0.100484
$35$	−0.848461	−0.143416
$36$	−9.48692	−1.58115
$37$	1.19305	0.196136	0.0980680	−	0.995180i	$-0.468734\pi$
$37$	1.19305	0.196136	0.0980680	+	0.995180i	$0.468734\pi$
$38$	0.165358	0.0268247
$39$	5.71131	0.914541
$40$	−2.81703	−0.445412
$41$	3.84070	0.599817	0.299908	−	0.953968i	$-0.403044\pi$
$41$	3.84070	0.599817	0.299908	+	0.953968i	$0.403044\pi$
$42$	−0.128583	−0.0198407
$43$	9.57128	1.45961	0.729804	−	0.683657i	$-0.239612\pi$
$43$	9.57128	1.45961	0.729804	+	0.683657i	$0.239612\pi$
$44$	11.1251	1.67717
$45$	17.5290	2.61307
$46$	0.689084	0.101600
$47$	6.21907	0.907145	0.453573	−	0.891219i	$-0.350149\pi$
$47$	6.21907	0.907145	0.453573	+	0.891219i	$0.350149\pi$
$48$	10.5551	1.52350
$49$	−6.94520	−0.992171
$50$	1.59621	0.225738
$51$	8.35968	1.17059
$52$	−4.00186	−0.554958
$53$	−2.31070	−0.317399	−0.158700	−	0.987327i	$-0.550730\pi$
$53$	−2.31070	−0.317399	−0.158700	+	0.987327i	$0.550730\pi$
$54$	1.00873	0.137271
$55$	−20.5558	−2.77175
$56$	0.181962	0.0243157
$57$	2.35927	0.312494
$58$	−0.767234	−0.100743
$59$	5.00296	0.651329	0.325665	−	0.945485i	$-0.394412\pi$
$59$	5.00296	0.651329	0.325665	+	0.945485i	$0.394412\pi$
$60$	−19.9009	−2.56920
$61$	0	0
$61$	0	0
$62$	0.705868	0.0896453
$63$	−1.13226	−0.142651
$64$	−7.09085	−0.886356
$65$	7.39426	0.917145
$66$	−3.11520	−0.383455
$67$	−3.39714	−0.415027	−0.207513	−	0.978232i	$-0.566537\pi$
$67$	−3.39714	−0.415027	−0.207513	+	0.978232i	$0.566537\pi$
$68$	−5.85755	−0.710333
$69$	9.83162	1.18359
$70$	−0.166472	−0.0198972
$71$	1.92117	0.228001	0.114000	−	0.993481i	$-0.463633\pi$
$71$	1.92117	0.228001	0.114000	+	0.993481i	$0.463633\pi$
$72$	−3.75929	−0.443037
$73$	5.12218	0.599506	0.299753	−	0.954017i	$-0.403096\pi$
$73$	5.12218	0.599506	0.299753	+	0.954017i	$0.403096\pi$
$74$	0.234082	0.0272115
$75$	22.7741	2.62973
$76$	−1.65312	−0.189626
$77$	1.32777	0.151314
$78$	1.12059	0.126881
$79$	10.3008	1.15893	0.579463	−	0.814999i	$-0.303262\pi$
$79$	10.3008	1.15893	0.579463	+	0.814999i	$0.303262\pi$
$80$	13.6654	1.52783
$81$	−0.117398	−0.0130442
$82$	0.753565	0.0832173
$83$	−9.80217	−1.07593	−0.537964	−	0.842968i	$-0.680806\pi$
$83$	−9.80217	−1.07593	−0.537964	+	0.842968i	$0.680806\pi$
$84$	1.28547	0.140256
$85$	10.8230	1.17392
$86$	1.87793	0.202503
$87$	−10.9466	−1.17360
$88$	4.40843	0.469940
$89$	7.01825	0.743933	0.371967	−	0.928246i	$-0.378684\pi$
$89$	7.01825	0.743933	0.371967	+	0.928246i	$0.378684\pi$
$90$	3.43928	0.362532
$91$	−0.477620	−0.0500682
$92$	−6.88892	−0.718220
$93$	10.0711	1.04432
$94$	1.22021	0.125855
$95$	3.05448	0.313383
$96$	6.42270	0.655514
$97$	−11.6170	−1.17953	−0.589766	−	0.807575i	$-0.700780\pi$
$97$	−11.6170	−1.17953	−0.589766	+	0.807575i	$0.700780\pi$
$98$	−1.36268	−0.137652
$99$	−27.4315	−2.75697
$100$	−15.9576	−1.59576
$101$	9.61575	0.956803	0.478402	−	0.878141i	$-0.341216\pi$
$101$	9.61575	0.956803	0.478402	+	0.878141i	$0.341216\pi$
$102$	1.64021	0.162405
$103$	1.34519	0.132545	0.0662727	−	0.997802i	$-0.478889\pi$
$103$	1.34519	0.132545	0.0662727	+	0.997802i	$0.478889\pi$
$104$	−1.58578	−0.155499
$105$	−2.37517	−0.231793
$106$	−0.453371	−0.0440353
$107$	18.0408	1.74407	0.872037	−	0.489439i	$-0.162799\pi$
$107$	18.0408	1.74407	0.872037	+	0.489439i	$0.162799\pi$
$108$	−10.0845	−0.970385
$109$	−13.3997	−1.28346	−0.641731	−	0.766930i	$-0.721783\pi$
$109$	−13.3997	−1.28346	−0.641731	+	0.766930i	$0.721783\pi$
$110$	−4.03316	−0.384547
$111$	3.33980	0.317000
$112$	−0.882692	−0.0834066
$113$	−11.6734	−1.09814	−0.549068	−	0.835777i	$-0.685017\pi$
$113$	−11.6734	−1.09814	−0.549068	+	0.835777i	$0.685017\pi$
$114$	0.462901	0.0433547
$115$	12.7287	1.18696
$116$	7.67020	0.712160
$117$	9.86754	0.912255
$118$	0.981605	0.0903641
$119$	−0.699096	−0.0640861
$120$	−7.88596	−0.719886
$121$	21.1682	1.92438
$122$	0	0
$123$	10.7516	0.969439
$124$	−7.05671	−0.633712
$125$	11.3636	1.01639
$126$	−0.222155	−0.0197911
$127$	10.3694	0.920133	0.460066	−	0.887885i	$-0.347826\pi$
$127$	10.3694	0.920133	0.460066	+	0.887885i	$0.347826\pi$
$128$	−5.97991	−0.528554
$129$	26.7937	2.35905
$130$	1.45079	0.127243
$131$	1.30267	0.113815	0.0569075	−	0.998379i	$-0.481876\pi$
$131$	1.30267	0.113815	0.0569075	+	0.998379i	$0.481876\pi$
$132$	31.1433	2.71068
$133$	−0.197299	−0.0171080
$134$	−0.666536	−0.0575799
$135$	18.6332	1.60369
$136$	−2.32112	−0.199034
$137$	7.96556	0.680544	0.340272	−	0.940327i	$-0.389481\pi$
$137$	7.96556	0.680544	0.340272	+	0.940327i	$0.389481\pi$
$138$	1.92901	0.164208
$139$	−14.8041	−1.25567	−0.627834	−	0.778348i	$-0.716058\pi$
$139$	−14.8041	−1.25567	−0.627834	+	0.778348i	$0.716058\pi$
$140$	1.66426	0.140656
$141$	17.4096	1.46615
$142$	0.376943	0.0316323
$143$	−11.5714	−0.967650
$144$	18.2362	1.51969
$145$	−14.1723	−1.17694
$146$	1.00500	0.0831742
$147$	−19.4423	−1.60357
$148$	−2.34017	−0.192361
$149$	−15.9223	−1.30441	−0.652203	−	0.758044i	$-0.726155\pi$
$149$	−15.9223	−1.30441	−0.652203	+	0.758044i	$0.726155\pi$
$150$	4.46840	0.364843
$151$	−14.2978	−1.16354	−0.581769	−	0.813354i	$-0.697639\pi$
$151$	−14.2978	−1.16354	−0.581769	+	0.813354i	$0.697639\pi$
$152$	−0.655067	−0.0531330
$153$	14.4432	1.16766
$154$	0.260516	0.0209929
$155$	13.0387	1.04730
$156$	−11.2027	−0.896938
$157$	−1.53473	−0.122485	−0.0612424	−	0.998123i	$-0.519506\pi$
$157$	−1.53473	−0.122485	−0.0612424	+	0.998123i	$0.519506\pi$
$158$	2.02106	0.160787
$159$	−6.46855	−0.512989
$160$	8.31528	0.657380
$161$	−0.822190	−0.0647977
$162$	−0.0230340	−0.00180972
$163$	12.5848	0.985715	0.492857	−	0.870110i	$-0.335952\pi$
$163$	12.5848	0.985715	0.492857	+	0.870110i	$0.335952\pi$
$164$	−7.53355	−0.588271
$165$	−57.5437	−4.47977
$166$	−1.92323	−0.149272
$167$	−10.6120	−0.821179	−0.410590	−	0.911820i	$-0.634677\pi$
$167$	−10.6120	−0.821179	−0.410590	+	0.911820i	$0.634677\pi$
$168$	0.509381	0.0392996
$169$	−8.83758	−0.679814
$170$	2.12353	0.162867
$171$	4.07617	0.311712
$172$	−18.7741	−1.43151
$173$	19.5586	1.48701	0.743507	−	0.668728i	$-0.233161\pi$
$173$	19.5586	1.48701	0.743507	+	0.668728i	$0.233161\pi$
$174$	−2.14778	−0.162823
$175$	−1.90454	−0.143969
$176$	−21.3852	−1.61197
$177$	14.0052	1.05270
$178$	1.37702	0.103212
$179$	0.480787	0.0359357	0.0179678	−	0.999839i	$-0.494280\pi$
$179$	0.480787	0.0359357	0.0179678	+	0.999839i	$0.494280\pi$
$180$	−34.3833	−2.56278
$181$	−20.0414	−1.48967	−0.744833	−	0.667251i	$-0.767471\pi$
$181$	−20.0414	−1.48967	−0.744833	+	0.667251i	$0.767471\pi$
$182$	−0.0937115	−0.00694636
$183$	0	0
$184$	−2.72981	−0.201244
$185$	4.32394	0.317903
$186$	1.97600	0.144887
$187$	−16.9372	−1.23857
$188$	−12.1987	−0.889684
$189$	−1.20358	−0.0875479
$190$	0.599305	0.0434781
$191$	−12.8968	−0.933178	−0.466589	−	0.884474i	$-0.654517\pi$
$191$	−12.8968	−0.933178	−0.466589	+	0.884474i	$0.654517\pi$
$192$	−19.8500	−1.43255
$193$	−20.9850	−1.51053	−0.755267	−	0.655418i	$-0.772493\pi$
$193$	−20.9850	−1.51053	−0.755267	+	0.655418i	$0.772493\pi$
$194$	−2.27932	−0.163646
$195$	20.6994	1.48231
$196$	13.6230	0.973073
$197$	−6.13245	−0.436919	−0.218459	−	0.975846i	$-0.570103\pi$
$197$	−6.13245	−0.436919	−0.218459	+	0.975846i	$0.570103\pi$
$198$	−5.38220	−0.382496
$199$	−17.2126	−1.22017	−0.610083	−	0.792338i	$-0.708864\pi$
$199$	−17.2126	−1.22017	−0.610083	+	0.792338i	$0.708864\pi$
$200$	−6.32338	−0.447130
$201$	−9.50990	−0.670777
$202$	1.88666	0.132745
$203$	0.915435	0.0642509
$204$	−16.3975	−1.14806
$205$	13.9198	0.972199
$206$	0.263933	0.0183891
$207$	16.9863	1.18063
$208$	7.69258	0.533385
$209$	−4.78002	−0.330641
$210$	−0.466020	−0.0321584
$211$	5.90651	0.406621	0.203310	−	0.979114i	$-0.434830\pi$
$211$	5.90651	0.406621	0.203310	+	0.979114i	$0.434830\pi$
$212$	4.53245	0.311290
$213$	5.37809	0.368501
$214$	3.53971	0.241969
$215$	34.6890	2.36577
$216$	−3.99610	−0.271900
$217$	−0.842216	−0.0571733
$218$	−2.62909	−0.178065
$219$	14.3390	0.968937
$220$	40.3204	2.71840
$221$	6.09256	0.409830
$222$	0.655286	0.0439799
$223$	−27.4756	−1.83990	−0.919951	−	0.392034i	$-0.871772\pi$
$223$	−27.4756	−1.83990	−0.919951	+	0.392034i	$0.871772\pi$
$224$	−0.537112	−0.0358873
$225$	39.3473	2.62316
$226$	−2.29037	−0.152353
$227$	2.00687	0.133201	0.0666003	−	0.997780i	$-0.478785\pi$
$227$	2.00687	0.133201	0.0666003	+	0.997780i	$0.478785\pi$
$228$	−4.62773	−0.306479
$229$	8.35343	0.552010	0.276005	−	0.961156i	$-0.410989\pi$
$229$	8.35343	0.552010	0.276005	+	0.961156i	$0.410989\pi$
$230$	2.49743	0.164676
$231$	3.71694	0.244557
$232$	3.03940	0.199546
$233$	−18.5006	−1.21201	−0.606006	−	0.795460i	$-0.707229\pi$
$233$	−18.5006	−1.21201	−0.606006	+	0.795460i	$0.707229\pi$
$234$	1.93606	0.126564
$235$	22.5397	1.47033
$236$	−9.81332	−0.638792
$237$	28.8358	1.87308
$238$	−0.137166	−0.00889116
$239$	10.5010	0.679255	0.339628	−	0.940560i	$-0.389699\pi$
$239$	10.5010	0.679255	0.339628	+	0.940560i	$0.389699\pi$
$240$	38.2546	2.46932
$241$	14.1358	0.910567	0.455283	−	0.890347i	$-0.349538\pi$
$241$	14.1358	0.910567	0.455283	+	0.890347i	$0.349538\pi$
$242$	4.15331	0.266985
$243$	−15.7523	−1.01051
$244$	0	0
$245$	−25.1713	−1.60814
$246$	2.10952	0.134498
$247$	1.71945	0.109406
$248$	−2.79630	−0.177565
$249$	−27.4400	−1.73894
$250$	2.22959	0.141012
$251$	22.7852	1.43819	0.719093	−	0.694914i	$-0.244557\pi$
$251$	22.7852	1.43819	0.719093	+	0.694914i	$0.244557\pi$
$252$	2.22093	0.139906
$253$	−19.9194	−1.25232
$254$	2.03452	0.127657
$255$	30.2978	1.89732
$256$	13.0084	0.813026
$257$	−7.98748	−0.498245	−0.249123	−	0.968472i	$-0.580142\pi$
$257$	−7.98748	−0.498245	−0.249123	+	0.968472i	$0.580142\pi$
$258$	5.25706	0.327290
$259$	−0.279298	−0.0173547
$260$	−14.5039	−0.899491
$261$	−18.9127	−1.17067
$262$	0.255591	0.0157904
$263$	−18.2648	−1.12626	−0.563129	−	0.826369i	$-0.690403\pi$
$263$	−18.2648	−1.12626	−0.563129	+	0.826369i	$0.690403\pi$
$264$	12.3409	0.759529
$265$	−8.37463	−0.514450
$266$	−0.0387111	−0.00237353
$267$	19.6468	1.20236
$268$	6.66350	0.407038
$269$	−19.1636	−1.16842	−0.584212	−	0.811601i	$-0.698597\pi$
$269$	−19.1636	−1.16842	−0.584212	+	0.811601i	$0.698597\pi$
$270$	3.65594	0.222493
$271$	17.0415	1.03520	0.517599	−	0.855624i	$-0.326826\pi$
$271$	17.0415	1.03520	0.517599	+	0.855624i	$0.326826\pi$
$272$	11.2597	0.682719
$273$	−1.33704	−0.0809215
$274$	1.56288	0.0944172
$275$	−46.1416	−2.78244
$276$	−19.2847	−1.16081
$277$	8.07286	0.485051	0.242526	−	0.970145i	$-0.422024\pi$
$277$	8.07286	0.485051	0.242526	+	0.970145i	$0.422024\pi$
$278$	−2.90464	−0.174209
$279$	17.4000	1.04171
$280$	0.659480	0.0394115
$281$	−18.0705	−1.07799	−0.538997	−	0.842308i	$-0.681197\pi$
$281$	−18.0705	−1.07799	−0.538997	+	0.842308i	$0.681197\pi$
$282$	3.41585	0.203411
$283$	13.6413	0.810888	0.405444	−	0.914120i	$-0.367117\pi$
$283$	13.6413	0.810888	0.405444	+	0.914120i	$0.367117\pi$
$284$	−3.76838	−0.223612
$285$	8.55067	0.506498
$286$	−2.27037	−0.134250
$287$	−0.899126	−0.0530737
$288$	11.0966	0.653875
$289$	−8.08227	−0.475428
$290$	−2.78067	−0.163286
$291$	−32.5205	−1.90639
$292$	−10.0472	−0.587967
$293$	4.43755	0.259245	0.129622	−	0.991563i	$-0.458624\pi$
$293$	4.43755	0.259245	0.129622	+	0.991563i	$0.458624\pi$
$294$	−3.81467	−0.222476
$295$	18.1321	1.05569
$296$	−0.927317	−0.0538992
$297$	−29.1595	−1.69201
$298$	−3.12404	−0.180971
$299$	7.16531	0.414381
$300$	−44.6715	−2.57911
$301$	−2.24068	−0.129151
$302$	−2.80530	−0.161427
$303$	26.9182	1.54641
$304$	3.17772	0.182255
$305$	0	0
$306$	2.83383	0.161999
$307$	32.1885	1.83709	0.918546	−	0.395314i	$-0.129364\pi$
$307$	32.1885	1.83709	0.918546	+	0.395314i	$0.129364\pi$
$308$	−2.60443	−0.148401
$309$	3.76570	0.214223
$310$	2.55826	0.145300
$311$	−18.7388	−1.06258	−0.531291	−	0.847189i	$-0.678293\pi$
$311$	−18.7388	−1.06258	−0.531291	+	0.847189i	$0.678293\pi$
$312$	−4.43921	−0.251321
$313$	1.94064	0.109691	0.0548457	−	0.998495i	$-0.482533\pi$
$313$	1.94064	0.109691	0.0548457	+	0.998495i	$0.482533\pi$
$314$	−0.301122	−0.0169933
$315$	−4.10363	−0.231213
$316$	−20.2050	−1.13662
$317$	−3.03414	−0.170414	−0.0852072	−	0.996363i	$-0.527155\pi$
$317$	−3.03414	−0.170414	−0.0852072	+	0.996363i	$0.527155\pi$
$318$	−1.26916	−0.0711710
$319$	22.1784	1.24175
$320$	−25.6992	−1.43663
$321$	50.5033	2.81882
$322$	−0.161318	−0.00898989
$323$	2.51677	0.140037
$324$	0.230276	0.0127931
$325$	16.5978	0.920683
$326$	2.46919	0.136756
$327$	−37.5110	−2.07436
$328$	−2.98525	−0.164833
$329$	−1.45591	−0.0802671
$330$	−11.2904	−0.621514
$331$	28.2308	1.55170	0.775852	−	0.630915i	$-0.217320\pi$
$331$	28.2308	1.55170	0.775852	+	0.630915i	$0.217320\pi$
$332$	19.2270	1.05522
$333$	5.77025	0.316208
$334$	−2.08212	−0.113929
$335$	−12.3122	−0.672686
$336$	−2.47100	−0.134804
$337$	−28.0471	−1.52782	−0.763911	−	0.645322i	$-0.776723\pi$
$337$	−28.0471	−1.52782	−0.763911	+	0.645322i	$0.776723\pi$
$338$	−1.73398	−0.0943160
$339$	−32.6782	−1.77484
$340$	−21.2294	−1.15133
$341$	−20.4045	−1.10497
$342$	0.799765	0.0432463
$343$	3.26463	0.176274
$344$	−7.43944	−0.401108
$345$	35.6325	1.91839
$346$	3.83750	0.206305
$347$	−16.9032	−0.907412	−0.453706	−	0.891152i	$-0.649898\pi$
$347$	−16.9032	−0.907412	−0.453706	+	0.891152i	$0.649898\pi$
$348$	21.4718	1.15101
$349$	−28.7109	−1.53686	−0.768431	−	0.639933i	$-0.778962\pi$
$349$	−28.7109	−1.53686	−0.768431	+	0.639933i	$0.778962\pi$
$350$	−0.373679	−0.0199740
$351$	10.4891	0.559868
$352$	−13.0127	−0.693581
$353$	25.0218	1.33178	0.665888	−	0.746052i	$-0.268053\pi$
$353$	25.0218	1.33178	0.665888	+	0.746052i	$0.268053\pi$
$354$	2.74789	0.146049
$355$	6.96285	0.369550
$356$	−13.7663	−0.729614
$357$	−1.95704	−0.103578
$358$	0.0943327	0.00498564
$359$	−10.9081	−0.575709	−0.287854	−	0.957674i	$-0.592942\pi$
$359$	−10.9081	−0.575709	−0.287854	+	0.957674i	$0.592942\pi$
$360$	−13.6247	−0.718087
$361$	−18.2897	−0.962617
$362$	−3.93222	−0.206673
$363$	59.2580	3.11024
$364$	0.936854	0.0491045
$365$	18.5642	0.971696
$366$	0	0
$367$	22.4321	1.17094	0.585472	−	0.810693i	$-0.300909\pi$
$367$	22.4321	1.17094	0.585472	+	0.810693i	$0.300909\pi$
$368$	13.2422	0.690300
$369$	18.5758	0.967016
$370$	0.848379	0.0441051
$371$	0.540946	0.0280845
$372$	−19.7545	−1.02422
$373$	19.7242	1.02128	0.510640	−	0.859795i	$-0.329409\pi$
$373$	19.7242	1.02128	0.510640	+	0.859795i	$0.329409\pi$
$374$	−3.32316	−0.171836
$375$	31.8110	1.64271
$376$	−4.83388	−0.249288
$377$	−7.97793	−0.410884
$378$	−0.236149	−0.0121462
$379$	1.63979	0.0842305	0.0421152	−	0.999113i	$-0.486590\pi$
$379$	1.63979	0.0842305	0.0421152	+	0.999113i	$0.486590\pi$
$380$	−5.99138	−0.307351
$381$	29.0279	1.48714
$382$	−2.53041	−0.129467
$383$	14.8655	0.759592	0.379796	−	0.925070i	$-0.375994\pi$
$383$	14.8655	0.759592	0.379796	+	0.925070i	$0.375994\pi$
$384$	−16.7401	−0.854263
$385$	4.81222	0.245253
$386$	−4.11736	−0.209568
$387$	46.2920	2.35316
$388$	22.7869	1.15683
$389$	−24.7974	−1.25728	−0.628638	−	0.777698i	$-0.716387\pi$
$389$	−24.7974	−1.25728	−0.628638	+	0.777698i	$0.716387\pi$
$390$	4.06132	0.205653
$391$	10.4879	0.530397
$392$	5.39827	0.272654
$393$	3.64668	0.183951
$394$	−1.20322	−0.0606172
$395$	37.3328	1.87842
$396$	53.8070	2.70390
$397$	−10.8604	−0.545067	−0.272534	−	0.962146i	$-0.587862\pi$
$397$	−10.8604	−0.545067	−0.272534	+	0.962146i	$0.587862\pi$
$398$	−3.37719	−0.169283
$399$	−0.552317	−0.0276504
$400$	30.6746	1.53373
$401$	−20.3982	−1.01864	−0.509318	−	0.860579i	$-0.670102\pi$
$401$	−20.3982	−1.01864	−0.509318	+	0.860579i	$0.670102\pi$
$402$	−1.86589	−0.0930622
$403$	7.33983	0.365623
$404$	−18.8613	−0.938386
$405$	−0.425482	−0.0211424
$406$	0.179613	0.00891404
$407$	−6.76662	−0.335409
$408$	−6.49770	−0.321684
$409$	12.0509	0.595877	0.297938	−	0.954585i	$-0.403701\pi$
$409$	12.0509	0.595877	0.297938	+	0.954585i	$0.403701\pi$
$410$	2.73113	0.134881
$411$	22.2987	1.09991
$412$	−2.63859	−0.129994
$413$	−1.17122	−0.0576317
$414$	3.33279	0.163798
$415$	−35.5258	−1.74389
$416$	4.68088	0.229499
$417$	−41.4424	−2.02944
$418$	−0.937863	−0.0458724
$419$	20.2457	0.989069	0.494535	−	0.869158i	$-0.335338\pi$
$419$	20.2457	0.989069	0.494535	+	0.869158i	$0.335338\pi$
$420$	4.65890	0.227331
$421$	16.5944	0.808762	0.404381	−	0.914591i	$-0.367487\pi$
$421$	16.5944	0.808762	0.404381	+	0.914591i	$0.367487\pi$
$422$	1.15889	0.0564137
$423$	30.0789	1.46249
$424$	1.79603	0.0872230
$425$	24.2944	1.17845
$426$	1.05521	0.0511250
$427$	0	0
$428$	−35.3872	−1.71050
$429$	−32.3928	−1.56394
$430$	6.80616	0.328222
$431$	12.4852	0.601391	0.300695	−	0.953720i	$-0.402781\pi$
$431$	12.4852	0.601391	0.300695	+	0.953720i	$0.402781\pi$
$432$	19.3850	0.932662
$433$	30.5118	1.46631	0.733153	−	0.680064i	$-0.238048\pi$
$433$	30.5118	1.46631	0.733153	+	0.680064i	$0.238048\pi$
$434$	−0.165247	−0.00793211
$435$	−39.6736	−1.90220
$436$	26.2836	1.25876
$437$	2.95991	0.141592
$438$	2.81338	0.134428
$439$	−1.94801	−0.0929736	−0.0464868	−	0.998919i	$-0.514803\pi$
$439$	−1.94801	−0.0929736	−0.0464868	+	0.998919i	$0.514803\pi$
$440$	15.9774	0.761691
$441$	−33.5908	−1.59956
$442$	1.19539	0.0568589
$443$	−11.8482	−0.562927	−0.281463	−	0.959572i	$-0.590820\pi$
$443$	−11.8482	−0.562927	−0.281463	+	0.959572i	$0.590820\pi$
$444$	−6.55104	−0.310898
$445$	25.4361	1.20579
$446$	−5.39085	−0.255264
$447$	−44.5726	−2.10821
$448$	1.66000	0.0784276
$449$	−22.8075	−1.07635	−0.538177	−	0.842832i	$-0.680887\pi$
$449$	−22.8075	−1.07635	−0.538177	+	0.842832i	$0.680887\pi$
$450$	7.72014	0.363931
$451$	−21.7833	−1.02574
$452$	22.8973	1.07700
$453$	−40.0251	−1.88054
$454$	0.393758	0.0184800
$455$	−1.73103	−0.0811519
$456$	−1.83379	−0.0858749
$457$	−2.98428	−0.139599	−0.0697994	−	0.997561i	$-0.522236\pi$
$457$	−2.98428	−0.139599	−0.0697994	+	0.997561i	$0.522236\pi$
$458$	1.63898	0.0765847
$459$	15.3530	0.716617
$460$	−24.9674	−1.16411
$461$	−10.3667	−0.482827	−0.241413	−	0.970422i	$-0.577611\pi$
$461$	−10.3667	−0.482827	−0.241413	+	0.970422i	$0.577611\pi$
$462$	0.729283	0.0339293
$463$	1.81767	0.0844741	0.0422371	−	0.999108i	$-0.486552\pi$
$463$	1.81767	0.0844741	0.0422371	+	0.999108i	$0.486552\pi$
$464$	−14.7441	−0.684475
$465$	36.5004	1.69267
$466$	−3.62990	−0.168152
$467$	13.3142	0.616110	0.308055	−	0.951369i	$-0.400322\pi$
$467$	13.3142	0.616110	0.308055	+	0.951369i	$0.400322\pi$
$468$	−19.3552	−0.894695
$469$	0.795286	0.0367229
$470$	4.42240	0.203990
$471$	−4.29630	−0.197963
$472$	−3.88863	−0.178989
$473$	−54.2855	−2.49605
$474$	5.65772	0.259868
$475$	6.85638	0.314592
$476$	1.37128	0.0628525
$477$	−11.1758	−0.511707
$478$	2.06036	0.0942385
$479$	1.16825	0.0533787	0.0266894	−	0.999644i	$-0.491504\pi$
$479$	1.16825	0.0533787	0.0266894	+	0.999644i	$0.491504\pi$
$480$	23.2777	1.06248
$481$	2.43406	0.110984
$482$	2.77352	0.126330
$483$	−2.30163	−0.104728
$484$	−41.5215	−1.88734
$485$	−42.1034	−1.91182
$486$	−3.09068	−0.140196
$487$	−19.7482	−0.894875	−0.447438	−	0.894315i	$-0.647663\pi$
$487$	−19.7482	−0.894875	−0.447438	+	0.894315i	$0.647663\pi$
$488$	0	0
$489$	35.2296	1.59314
$490$	−4.93874	−0.223110
$491$	−13.2462	−0.597793	−0.298897	−	0.954286i	$-0.596619\pi$
$491$	−13.2462	−0.597793	−0.298897	+	0.954286i	$0.596619\pi$
$492$	−21.0893	−0.950779
$493$	−11.6774	−0.525922
$494$	0.337364	0.0151787
$495$	−99.4195	−4.46857
$496$	13.5648	0.609077
$497$	−0.449754	−0.0201742
$498$	−5.38387	−0.241257
$499$	−8.33441	−0.373100	−0.186550	−	0.982446i	$-0.559731\pi$
$499$	−8.33441	−0.373100	−0.186550	+	0.982446i	$0.559731\pi$
$500$	−22.2897	−0.996825
$501$	−29.7070	−1.32721
$502$	4.47056	0.199531
$503$	39.2623	1.75062	0.875310	−	0.483562i	$-0.160657\pi$
$503$	39.2623	1.75062	0.875310	+	0.483562i	$0.160657\pi$
$504$	0.880068	0.0392014
$505$	34.8502	1.55081
$506$	−3.90828	−0.173744
$507$	−24.7398	−1.09873
$508$	−20.3396	−0.902422
$509$	−40.3821	−1.78990	−0.894952	−	0.446163i	$-0.852790\pi$
$509$	−40.3821	−1.78990	−0.894952	+	0.446163i	$0.852790\pi$
$510$	5.94458	0.263231
$511$	−1.19913	−0.0530462
$512$	14.5121	0.641352
$513$	4.33294	0.191304
$514$	−1.56718	−0.0691255
$515$	4.87534	0.214833
$516$	−52.5560	−2.31365
$517$	−35.2727	−1.55129
$518$	−0.0547997	−0.00240776
$519$	54.7521	2.40335
$520$	−5.74731	−0.252036
$521$	−6.22502	−0.272723	−0.136361	−	0.990659i	$-0.543541\pi$
$521$	−6.22502	−0.272723	−0.136361	+	0.990659i	$0.543541\pi$
$522$	−3.71077	−0.162416
$523$	−19.7661	−0.864311	−0.432155	−	0.901799i	$-0.642247\pi$
$523$	−19.7661	−0.864311	−0.432155	+	0.901799i	$0.642247\pi$
$524$	−2.55519	−0.111624
$525$	−5.33153	−0.232687
$526$	−3.58365	−0.156255
$527$	10.7434	0.467989
$528$	−59.8653	−2.60530
$529$	−10.6654	−0.463714
$530$	−1.64314	−0.0713736
$531$	24.1971	1.05006
$532$	0.387004	0.0167787
$533$	7.83580	0.339406
$534$	3.85480	0.166813
$535$	65.3851	2.82684
$536$	2.64048	0.114052
$537$	1.34591	0.0580802
$538$	−3.75999	−0.162105
$539$	39.3911	1.69669
$540$	−36.5492	−1.57283
$541$	−9.07124	−0.390003	−0.195002	−	0.980803i	$-0.562471\pi$
$541$	−9.07124	−0.390003	−0.195002	+	0.980803i	$0.562471\pi$
$542$	3.34363	0.143621
$543$	−56.1036	−2.40764
$544$	6.85144	0.293753
$545$	−48.5644	−2.08027
$546$	−0.262335	−0.0112269
$547$	−12.5053	−0.534690	−0.267345	−	0.963601i	$-0.586146\pi$
$547$	−12.5053	−0.534690	−0.267345	+	0.963601i	$0.586146\pi$
$548$	−15.6245	−0.667445
$549$	0	0
$550$	−9.05321	−0.386030
$551$	−3.29559	−0.140397
$552$	−7.64179	−0.325256
$553$	−2.41145	−0.102545
$554$	1.58394	0.0672950
$555$	12.1044	0.513802
$556$	29.0383	1.23150
$557$	37.2325	1.57759	0.788795	−	0.614657i	$-0.210705\pi$
$557$	37.2325	1.57759	0.788795	+	0.614657i	$0.210705\pi$
$558$	3.41397	0.144525
$559$	19.5273	0.825918
$560$	−3.19912	−0.135188
$561$	−47.4136	−2.00180
$562$	−3.54552	−0.149559
$563$	−36.6641	−1.54521	−0.772603	−	0.634890i	$-0.781046\pi$
$563$	−36.6641	−1.54521	−0.772603	+	0.634890i	$0.781046\pi$
$564$	−34.1490	−1.43793
$565$	−42.3075	−1.77989
$566$	2.67648	0.112501
$567$	0.0274833	0.00115419
$568$	−1.49326	−0.0626558
$569$	26.1128	1.09470	0.547352	−	0.836902i	$-0.315636\pi$
$569$	26.1128	1.09470	0.547352	+	0.836902i	$0.315636\pi$
$570$	1.67768	0.0702705
$571$	5.74100	0.240253	0.120127	−	0.992759i	$-0.461670\pi$
$571$	5.74100	0.240253	0.120127	+	0.992759i	$0.461670\pi$
$572$	22.6974	0.949024
$573$	−36.1030	−1.50823
$574$	−0.176413	−0.00736334
$575$	28.5720	1.19154
$576$	−34.2953	−1.42897
$577$	8.96761	0.373326	0.186663	−	0.982424i	$-0.440233\pi$
$577$	8.96761	0.373326	0.186663	+	0.982424i	$0.440233\pi$
$578$	−1.58578	−0.0659599
$579$	−58.7451	−2.44136
$580$	27.7989	1.15429
$581$	2.29473	0.0952016
$582$	−6.38070	−0.264488
$583$	13.1056	0.542779
$584$	−3.98130	−0.164747
$585$	35.7627	1.47861
$586$	0.870670	0.0359670
$587$	−7.84121	−0.323641	−0.161821	−	0.986820i	$-0.551737\pi$
$587$	−7.84121	−0.323641	−0.161821	+	0.986820i	$0.551737\pi$
$588$	38.1361	1.57271
$589$	3.03200	0.124931
$590$	3.55761	0.146465
$591$	−17.1671	−0.706159
$592$	4.49840	0.184883
$593$	43.3235	1.77908	0.889541	−	0.456855i	$-0.151024\pi$
$593$	43.3235	1.77908	0.889541	+	0.456855i	$0.151024\pi$
$594$	−5.72124	−0.234745
$595$	−2.53372	−0.103872
$596$	31.2317	1.27930
$597$	−48.1846	−1.97206
$598$	1.40587	0.0574903
$599$	25.6723	1.04894	0.524472	−	0.851428i	$-0.324263\pi$
$599$	25.6723	1.04894	0.524472	+	0.851428i	$0.324263\pi$
$600$	−17.7016	−0.722664
$601$	33.8710	1.38163	0.690814	−	0.723033i	$-0.257253\pi$
$601$	33.8710	1.38163	0.690814	+	0.723033i	$0.257253\pi$
$602$	−0.439633	−0.0179181
$603$	−16.4305	−0.669100
$604$	28.0452	1.14114
$605$	76.7196	3.11909
$606$	5.28148	0.214546
$607$	2.79148	0.113303	0.0566514	−	0.998394i	$-0.481958\pi$
$607$	2.79148	0.113303	0.0566514	+	0.998394i	$0.481958\pi$
$608$	1.93362	0.0784186
$609$	2.56265	0.103844
$610$	0	0
$611$	12.6882	0.513308
$612$	−28.3304	−1.14519
$613$	−17.2830	−0.698052	−0.349026	−	0.937113i	$-0.613488\pi$
$613$	−17.2830	−0.698052	−0.349026	+	0.937113i	$0.613488\pi$
$614$	6.31554	0.254874
$615$	38.9668	1.57129
$616$	−1.03203	−0.0415818
$617$	−26.2673	−1.05748	−0.528740	−	0.848784i	$-0.677335\pi$
$617$	−26.2673	−1.05748	−0.528740	+	0.848784i	$0.677335\pi$
$618$	0.738850	0.0297209
$619$	−22.6165	−0.909032	−0.454516	−	0.890739i	$-0.650188\pi$
$619$	−22.6165	−0.909032	−0.454516	+	0.890739i	$0.650188\pi$
$620$	−25.5755	−1.02714
$621$	18.0563	0.724575
$622$	−3.67666	−0.147420
$623$	−1.64301	−0.0658256
$624$	21.5345	0.862070
$625$	0.507761	0.0203104
$626$	0.380763	0.0152183
$627$	−13.3811	−0.534390
$628$	3.01038	0.120127
$629$	3.56275	0.142056
$630$	−0.805152	−0.0320780
$631$	42.1014	1.67603	0.838015	−	0.545648i	$-0.183716\pi$
$631$	42.1014	1.67603	0.838015	+	0.545648i	$0.183716\pi$
$632$	−8.00644	−0.318479
$633$	16.5346	0.657191
$634$	−0.595314	−0.0236429
$635$	37.5815	1.49138
$636$	12.6881	0.503115
$637$	−14.1696	−0.561420
$638$	4.35152	0.172278
$639$	9.29184	0.367579
$640$	−21.6729	−0.856695
$641$	−17.0235	−0.672389	−0.336195	−	0.941793i	$-0.609140\pi$
$641$	−17.0235	−0.672389	−0.336195	+	0.941793i	$0.609140\pi$
$642$	9.90900	0.391077
$643$	−10.8824	−0.429159	−0.214579	−	0.976707i	$-0.568838\pi$
$643$	−10.8824	−0.429159	−0.214579	+	0.976707i	$0.568838\pi$
$644$	1.61273	0.0635504
$645$	97.1079	3.82362
$646$	0.493802	0.0194284
$647$	−7.47074	−0.293705	−0.146853	−	0.989158i	$-0.546914\pi$
$647$	−7.47074	−0.293705	−0.146853	+	0.989158i	$0.546914\pi$
$648$	0.0912493	0.00358461
$649$	−28.3753	−1.11383
$650$	3.25658	0.127734
$651$	−2.35769	−0.0924050
$652$	−24.6851	−0.966742
$653$	15.2254	0.595814	0.297907	−	0.954595i	$-0.403711\pi$
$653$	15.2254	0.595814	0.297907	+	0.954595i	$0.403711\pi$
$654$	−7.35985	−0.287793
$655$	4.72124	0.184474
$656$	14.4814	0.565403
$657$	24.7737	0.966515
$658$	−0.285658	−0.0111361
$659$	14.5975	0.568639	0.284320	−	0.958730i	$-0.408232\pi$
$659$	14.5975	0.568639	0.284320	+	0.958730i	$0.408232\pi$
$660$	112.872	4.39354
$661$	−19.4472	−0.756409	−0.378204	−	0.925722i	$-0.623458\pi$
$661$	−19.4472	−0.756409	−0.378204	+	0.925722i	$0.623458\pi$
$662$	5.53902	0.215280
$663$	17.0554	0.662378
$664$	7.61890	0.295671
$665$	−0.715068	−0.0277292
$666$	1.13215	0.0438700
$667$	−13.7335	−0.531761
$668$	20.8154	0.805373
$669$	−76.9147	−2.97370
$670$	−2.41571	−0.0933271
$671$	0	0
$672$	−1.50358	−0.0580020
$673$	6.62577	0.255405	0.127702	−	0.991813i	$-0.459240\pi$
$673$	6.62577	0.255405	0.127702	+	0.991813i	$0.459240\pi$
$674$	−5.50298	−0.211967
$675$	41.8259	1.60988
$676$	17.3350	0.666729
$677$	31.2484	1.20097	0.600487	−	0.799635i	$-0.294973\pi$
$677$	31.2484	1.20097	0.600487	+	0.799635i	$0.294973\pi$
$678$	−6.41163	−0.246237
$679$	2.71960	0.104369
$680$	−8.41238	−0.322600
$681$	5.61800	0.215282
$682$	−4.00347	−0.153301
$683$	19.5956	0.749805	0.374903	−	0.927064i	$-0.377676\pi$
$683$	19.5956	0.749805	0.374903	+	0.927064i	$0.377676\pi$
$684$	−7.99542	−0.305712
$685$	28.8694	1.10304
$686$	0.640538	0.0244558
$687$	23.3845	0.892173
$688$	36.0886	1.37586
$689$	−4.71430	−0.179600
$690$	6.99128	0.266153
$691$	−4.33324	−0.164844	−0.0824222	−	0.996598i	$-0.526266\pi$
$691$	−4.33324	−0.164844	−0.0824222	+	0.996598i	$0.526266\pi$
$692$	−38.3643	−1.45839
$693$	6.42184	0.243946
$694$	−3.31649	−0.125892
$695$	−53.6542	−2.03522
$696$	8.50845	0.322512
$697$	11.4693	0.434431
$698$	−5.63323	−0.213221
$699$	−51.7902	−1.95889
$700$	3.73575	0.141198
$701$	−20.5019	−0.774346	−0.387173	−	0.922007i	$-0.626548\pi$
$701$	−20.5019	−0.774346	−0.387173	+	0.922007i	$0.626548\pi$
$702$	2.05802	0.0776750
$703$	1.00548	0.0379225
$704$	40.2172	1.51574
$705$	63.0972	2.37638
$706$	4.90940	0.184768
$707$	−2.25109	−0.0846610
$708$	−27.4712	−1.03243
$709$	−49.6785	−1.86572	−0.932858	−	0.360244i	$-0.882693\pi$
$709$	−49.6785	−1.86572	−0.932858	+	0.360244i	$0.882693\pi$
$710$	1.36615	0.0512706
$711$	49.8202	1.86840
$712$	−5.45506	−0.204437
$713$	12.6350	0.473185
$714$	−0.383981	−0.0143701
$715$	−41.9380	−1.56839
$716$	−0.943065	−0.0352440
$717$	29.3964	1.09783
$718$	−2.14023	−0.0798726
$719$	15.3966	0.574196	0.287098	−	0.957901i	$-0.407309\pi$
$719$	15.3966	0.574196	0.287098	+	0.957901i	$0.407309\pi$
$720$	66.0933	2.46315
$721$	−0.314915	−0.0117280
$722$	−3.58853	−0.133551
$723$	39.5715	1.47168
$724$	39.3113	1.46099
$725$	−31.8124	−1.18148
$726$	11.6267	0.431508
$727$	−17.9932	−0.667331	−0.333665	−	0.942692i	$-0.608286\pi$
$727$	−17.9932	−0.667331	−0.333665	+	0.942692i	$0.608286\pi$
$728$	0.371238	0.0137590
$729$	−43.7446	−1.62017
$730$	3.64239	0.134811
$731$	28.5823	1.05715
$732$	0	0
$733$	6.80549	0.251366	0.125683	−	0.992070i	$-0.459888\pi$
$733$	6.80549	0.251366	0.125683	+	0.992070i	$0.459888\pi$
$734$	4.40128	0.162454
$735$	−70.4642	−2.59911
$736$	8.05782	0.297015
$737$	19.2676	0.709730
$738$	3.64466	0.134162
$739$	−21.6213	−0.795354	−0.397677	−	0.917525i	$-0.630184\pi$
$739$	−21.6213	−0.795354	−0.397677	+	0.917525i	$0.630184\pi$
$740$	−8.48143	−0.311784
$741$	4.81339	0.176824
$742$	0.106136	0.00389639
$743$	−40.0247	−1.46836	−0.734182	−	0.678952i	$-0.762434\pi$
$743$	−40.0247	−1.46836	−0.734182	+	0.678952i	$0.762434\pi$
$744$	−7.82792	−0.286985
$745$	−57.7069	−2.11422
$746$	3.86998	0.141690
$747$	−47.4087	−1.73459
$748$	33.2223	1.21473
$749$	−4.22345	−0.154321
$750$	6.24148	0.227907
$751$	33.6534	1.22803	0.614015	−	0.789294i	$-0.289553\pi$
$751$	33.6534	1.22803	0.614015	+	0.789294i	$0.289553\pi$
$752$	23.4490	0.855099
$753$	63.7844	2.32443
$754$	−1.56531	−0.0570052
$755$	−51.8192	−1.88590
$756$	2.36084	0.0858628
$757$	−28.9549	−1.05238	−0.526192	−	0.850366i	$-0.676381\pi$
$757$	−28.9549	−1.05238	−0.526192	+	0.850366i	$0.676381\pi$
$758$	0.321735	0.0116860
$759$	−55.7620	−2.02403
$760$	−2.37415	−0.0861194
$761$	−2.00614	−0.0727226	−0.0363613	−	0.999339i	$-0.511577\pi$
$761$	−2.00614	−0.0727226	−0.0363613	+	0.999339i	$0.511577\pi$
$762$	5.69541	0.206323
$763$	3.13694	0.113565
$764$	25.2971	0.915216
$765$	52.3462	1.89258
$766$	2.91669	0.105384
$767$	10.2070	0.368555
$768$	36.4155	1.31403
$769$	−17.0304	−0.614131	−0.307065	−	0.951688i	$-0.599347\pi$
$769$	−17.0304	−0.614131	−0.307065	+	0.951688i	$0.599347\pi$
$770$	0.944181	0.0340259
$771$	−22.3600	−0.805276
$772$	41.1622	1.48146
$773$	0.686305	0.0246847	0.0123423	−	0.999924i	$-0.496071\pi$
$773$	0.686305	0.0246847	0.0123423	+	0.999924i	$0.496071\pi$
$774$	9.08273	0.326472
$775$	29.2680	1.05134
$776$	9.02954	0.324141
$777$	−0.781863	−0.0280492
$778$	−4.86537	−0.174432
$779$	3.23688	0.115973
$780$	−40.6019	−1.45378
$781$	−10.8963	−0.389900
$782$	2.05778	0.0735861
$783$	−20.1041	−0.718461
$784$	−26.1869	−0.935246
$785$	−5.56229	−0.198527
$786$	0.715497	0.0255209
$787$	−13.2002	−0.470537	−0.235268	−	0.971930i	$-0.575597\pi$
$787$	−13.2002	−0.470537	−0.235268	+	0.971930i	$0.575597\pi$
$788$	12.0288	0.428509
$789$	−51.1303	−1.82029
$790$	7.32489	0.260608
$791$	2.73279	0.0971667
$792$	21.3216	0.757630
$793$	0	0
$794$	−2.13086	−0.0756215
$795$	−23.4438	−0.831466
$796$	33.7625	1.19668
$797$	−4.46285	−0.158082	−0.0790411	−	0.996871i	$-0.525186\pi$
$797$	−4.46285	−0.158082	−0.0790411	+	0.996871i	$0.525186\pi$
$798$	−0.108367	−0.00383616
$799$	18.5717	0.657021
$800$	18.6653	0.659917
$801$	33.9442	1.19936
$802$	−4.00222	−0.141323
$803$	−29.0515	−1.02521
$804$	18.6537	0.657865
$805$	−2.97985	−0.105026
$806$	1.44011	0.0507258
$807$	−53.6463	−1.88844
$808$	−7.47401	−0.262935
$809$	−37.0125	−1.30129	−0.650645	−	0.759382i	$-0.725501\pi$
$809$	−37.0125	−1.30129	−0.650645	+	0.759382i	$0.725501\pi$
$810$	−0.0834817	−0.00293325
$811$	−11.0980	−0.389703	−0.194852	−	0.980833i	$-0.562423\pi$
$811$	−11.0980	−0.389703	−0.194852	+	0.980833i	$0.562423\pi$
$812$	−1.79563	−0.0630142
$813$	47.7057	1.67311
$814$	−1.32764	−0.0465339
$815$	45.6107	1.59767
$816$	31.5202	1.10343
$817$	8.06652	0.282212
$818$	2.36444	0.0826707
$819$	−2.31004	−0.0807192
$820$	−27.3037	−0.953486
$821$	46.2494	1.61411	0.807057	−	0.590473i	$-0.201059\pi$
$821$	46.2494	1.61411	0.807057	+	0.590473i	$0.201059\pi$
$822$	4.37511	0.152600
$823$	15.0936	0.526131	0.263066	−	0.964778i	$-0.415266\pi$
$823$	15.0936	0.526131	0.263066	+	0.964778i	$0.415266\pi$
$824$	−1.04557	−0.0364242
$825$	−129.168	−4.49706
$826$	−0.229798	−0.00799570
$827$	−3.23824	−0.112605	−0.0563023	−	0.998414i	$-0.517931\pi$
$827$	−3.23824	−0.112605	−0.0563023	+	0.998414i	$0.517931\pi$
$828$	−33.3187	−1.15790
$829$	−19.7957	−0.687533	−0.343767	−	0.939055i	$-0.611703\pi$
$829$	−19.7957	−0.687533	−0.343767	+	0.939055i	$0.611703\pi$
$830$	−6.97034	−0.241944
$831$	22.5990	0.783952
$832$	−14.4668	−0.501544
$833$	−20.7401	−0.718603
$834$	−8.13120	−0.281560
$835$	−38.4608	−1.33099
$836$	9.37602	0.324276
$837$	18.4961	0.639319
$838$	3.97232	0.137221
$839$	0.0338001	0.00116691	0.000583455	−	1.00000i	$-0.499814\pi$
$839$	0.0338001	0.00116691	0.000583455		1.00000i	$0.499814\pi$
$840$	1.84614	0.0636979
$841$	−13.7090	−0.472725
$842$	3.25591	0.112206
$843$	−50.5862	−1.74228
$844$	−11.5856	−0.398794
$845$	−32.0299	−1.10186
$846$	5.90163	0.202902
$847$	−4.95558	−0.170276
$848$	−8.71251	−0.299189
$849$	38.1871	1.31058
$850$	4.76668	0.163496
$851$	4.19006	0.143634
$852$	−10.5491	−0.361408
$853$	−11.0513	−0.378390	−0.189195	−	0.981940i	$-0.560588\pi$
$853$	−11.0513	−0.378390	−0.189195	+	0.981940i	$0.560588\pi$
$854$	0	0
$855$	14.7732	0.505232
$856$	−14.0226	−0.479281
$857$	17.9801	0.614190	0.307095	−	0.951679i	$-0.400643\pi$
$857$	17.9801	0.614190	0.307095	+	0.951679i	$0.400643\pi$
$858$	−6.35564	−0.216978
$859$	−22.9580	−0.783316	−0.391658	−	0.920111i	$-0.628098\pi$
$859$	−22.9580	−0.783316	−0.391658	+	0.920111i	$0.628098\pi$
$860$	−68.0426	−2.32023
$861$	−2.51700	−0.0857791
$862$	2.44966	0.0834357
$863$	21.4206	0.729165	0.364582	−	0.931171i	$-0.381212\pi$
$863$	21.4206	0.729165	0.364582	+	0.931171i	$0.381212\pi$
$864$	11.7956	0.401296
$865$	70.8859	2.41019
$866$	5.98658	0.203432
$867$	−22.6254	−0.768399
$868$	1.65201	0.0560729
$869$	−58.4228	−1.98186
$870$	−7.78416	−0.263908
$871$	−6.93084	−0.234843
$872$	10.4152	0.352702
$873$	−56.1864	−1.90162
$874$	0.580749	0.0196441
$875$	−2.66027	−0.0899334
$876$	−28.1259	−0.950287
$877$	−28.5292	−0.963363	−0.481681	−	0.876346i	$-0.659974\pi$
$877$	−28.5292	−0.963363	−0.481681	+	0.876346i	$0.659974\pi$
$878$	−0.382210	−0.0128990
$879$	12.4224	0.418998
$880$	−77.5059	−2.61272
$881$	45.5418	1.53434	0.767172	−	0.641442i	$-0.221663\pi$
$881$	45.5418	1.53434	0.767172	+	0.641442i	$0.221663\pi$
$882$	−6.59069	−0.221920
$883$	17.7299	0.596658	0.298329	−	0.954463i	$-0.403571\pi$
$883$	17.7299	0.596658	0.298329	+	0.954463i	$0.403571\pi$
$884$	−11.9506	−0.401941
$885$	50.7588	1.70624
$886$	−2.32468	−0.0780993
$887$	23.6539	0.794221	0.397111	−	0.917771i	$-0.370013\pi$
$887$	23.6539	0.794221	0.397111	+	0.917771i	$0.370013\pi$
$888$	−2.59592	−0.0871133
$889$	−2.42752	−0.0814163
$890$	4.99069	0.167288
$891$	0.665845	0.0223066
$892$	53.8935	1.80449
$893$	5.24133	0.175394
$894$	−8.74538	−0.292489
$895$	1.74250	0.0582455
$896$	1.39992	0.0467682
$897$	20.0585	0.669733
$898$	−4.47495	−0.149331
$899$	−14.0680	−0.469193
$900$	−77.1799	−2.57266
$901$	−6.90035	−0.229884
$902$	−4.27400	−0.142309
$903$	−6.27253	−0.208737
$904$	9.07331	0.301774
$905$	−72.6356	−2.41449
$906$	−7.85312	−0.260902
$907$	31.8713	1.05827	0.529135	−	0.848538i	$-0.322517\pi$
$907$	31.8713	1.05827	0.529135	+	0.848538i	$0.322517\pi$
$908$	−3.93648	−0.130637
$909$	46.5071	1.54254
$910$	−0.339637	−0.0112588
$911$	−21.8627	−0.724344	−0.362172	−	0.932111i	$-0.617965\pi$
$911$	−21.8627	−0.724344	−0.362172	+	0.932111i	$0.617965\pi$
$912$	8.89565	0.294564
$913$	55.5950	1.83993
$914$	−0.585531	−0.0193676
$915$	0	0
$916$	−16.3853	−0.541385
$917$	−0.304961	−0.0100707
$918$	3.01234	0.0994220
$919$	−3.69859	−0.122005	−0.0610026	−	0.998138i	$-0.519430\pi$
$919$	−3.69859	−0.122005	−0.0610026	+	0.998138i	$0.519430\pi$
$920$	−9.89360	−0.326182
$921$	90.1079	2.96916
$922$	−2.03400	−0.0669864
$923$	3.91957	0.129014
$924$	−7.29080	−0.239850
$925$	9.70593	0.319129
$926$	0.356635	0.0117198
$927$	6.50608	0.213688
$928$	−8.97165	−0.294509
$929$	−39.8361	−1.30698	−0.653491	−	0.756934i	$-0.726696\pi$
$929$	−39.8361	−1.30698	−0.653491	+	0.756934i	$0.726696\pi$
$930$	7.16156	0.234837
$931$	−5.85329	−0.191834
$932$	36.2889	1.18868
$933$	−52.4572	−1.71737
$934$	2.61232	0.0854778
$935$	−61.3850	−2.00750
$936$	−7.66971	−0.250692
$937$	−44.5580	−1.45565	−0.727823	−	0.685766i	$-0.759468\pi$
$937$	−44.5580	−1.45565	−0.727823	+	0.685766i	$0.759468\pi$
$938$	0.156039	0.00509486
$939$	5.43259	0.177286
$940$	−44.2116	−1.44202
$941$	−19.7035	−0.642317	−0.321158	−	0.947026i	$-0.604072\pi$
$941$	−19.7035	−0.642317	−0.321158	+	0.947026i	$0.604072\pi$
$942$	−0.842955	−0.0274650
$943$	13.4888	0.439255
$944$	18.8637	0.613960
$945$	−4.36213	−0.141900
$946$	−10.6511	−0.346297
$947$	13.1491	0.427288	0.213644	−	0.976912i	$-0.431467\pi$
$947$	13.1491	0.427288	0.213644	+	0.976912i	$0.431467\pi$
$948$	−56.5615	−1.83703
$949$	10.4503	0.339230
$950$	1.34526	0.0436459
$951$	−8.49373	−0.275428
$952$	0.543384	0.0176112
$953$	27.4098	0.887891	0.443946	−	0.896054i	$-0.353578\pi$
$953$	27.4098	0.887891	0.443946	+	0.896054i	$0.353578\pi$
$954$	−2.19276	−0.0709931
$955$	−46.7415	−1.51252
$956$	−20.5978	−0.666181
$957$	62.0860	2.00695
$958$	0.229217	0.00740565
$959$	−1.86477	−0.0602167
$960$	−71.9420	−2.32192
$961$	−18.0572	−0.582492
$962$	0.477575	0.0153976
$963$	87.2556	2.81177
$964$	−27.7274	−0.893040
$965$	−76.0555	−2.44831
$966$	−0.451591	−0.0145297
$967$	−31.2006	−1.00334	−0.501671	−	0.865058i	$-0.667281\pi$
$967$	−31.2006	−1.00334	−0.501671	+	0.865058i	$0.667281\pi$
$968$	−16.4534	−0.528831
$969$	7.04540	0.226331
$970$	−8.26089	−0.265241
$971$	−21.5538	−0.691695	−0.345847	−	0.938291i	$-0.612408\pi$
$971$	−21.5538	−0.691695	−0.345847	+	0.938291i	$0.612408\pi$
$972$	30.8982	0.991061
$973$	3.46571	0.111105
$974$	−3.87469	−0.124153
$975$	46.4638	1.48803
$976$	0	0
$977$	−55.0844	−1.76231	−0.881153	−	0.472831i	$-0.843232\pi$
$977$	−55.0844	−1.76231	−0.881153	+	0.472831i	$0.843232\pi$
$978$	6.91222	0.221029
$979$	−39.8055	−1.27219
$980$	49.3737	1.57718
$981$	−64.8085	−2.06918
$982$	−2.59897	−0.0829366
$983$	−41.9508	−1.33802	−0.669011	−	0.743253i	$-0.733282\pi$
$983$	−41.9508	−1.33802	−0.669011	+	0.743253i	$0.733282\pi$
$984$	−8.35686	−0.266407
$985$	−22.2257	−0.708170
$986$	−2.29116	−0.0729653
$987$	−4.07566	−0.129730
$988$	−3.37270	−0.107300
$989$	33.6150	1.06889
$990$	−19.5066	−0.619960
$991$	8.61164	0.273558	0.136779	−	0.990602i	$-0.456325\pi$
$991$	8.61164	0.273558	0.136779	+	0.990602i	$0.456325\pi$
$992$	8.25408	0.262067
$993$	79.0288	2.50790
$994$	−0.0882441	−0.00279893
$995$	−62.3831	−1.97768
$996$	53.8237	1.70547
$997$	7.81495	0.247502	0.123751	−	0.992313i	$-0.460508\pi$
$997$	7.81495	0.247502	0.123751	+	0.992313i	$0.460508\pi$
$998$	−1.63525	−0.0517630
$999$	6.13373	0.194063

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3721.2.a.k.1.9		16
61.38	odd	20		61.2.g.a.41.3	yes	16
61.53	odd	20		61.2.g.a.3.3	✓	16
61.60	even	2	inner	3721.2.a.k.1.8		16
183.38	even	20		549.2.y.b.163.2		16
183.53	even	20		549.2.y.b.64.2		16
244.99	even	20		976.2.bd.b.529.4		16
244.175	even	20		976.2.bd.b.369.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.2.g.a.3.3	✓	16	61.53	odd	20
61.2.g.a.41.3	yes	16	61.38	odd	20
549.2.y.b.64.2		16	183.53	even	20
549.2.y.b.163.2		16	183.38	even	20
976.2.bd.b.369.4		16	244.175	even	20
976.2.bd.b.529.4		16	244.99	even	20
3721.2.a.k.1.8		16	61.60	even	2	inner
3721.2.a.k.1.9		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3721 = 61^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3721.a (trivial)

\(f(q)\)	\(=\)	\(q+0.196205 q^{2} +2.79938 q^{3} -1.96150 q^{4} +3.62428 q^{5} +0.549253 q^{6} -0.234105 q^{7} -0.777267 q^{8} +4.83656 q^{9} +O(q^{10})\)	\(q+0.196205 q^{2} +2.79938 q^{3} -1.96150 q^{4} +3.62428 q^{5} +0.549253 q^{6} -0.234105 q^{7} -0.777267 q^{8} +4.83656 q^{9} +0.711102 q^{10} -5.67170 q^{11} -5.49100 q^{12} +2.04020 q^{13} -0.0459325 q^{14} +10.1458 q^{15} +3.77050 q^{16} +2.98626 q^{17} +0.948956 q^{18} +0.842783 q^{19} -7.10904 q^{20} -0.655349 q^{21} -1.11282 q^{22} +3.51206 q^{23} -2.17587 q^{24} +8.13540 q^{25} +0.400298 q^{26} +5.14123 q^{27} +0.459197 q^{28} -3.91037 q^{29} +1.99065 q^{30} +3.59760 q^{31} +2.29433 q^{32} -15.8773 q^{33} +0.585918 q^{34} -0.848461 q^{35} -9.48692 q^{36} +1.19305 q^{37} +0.165358 q^{38} +5.71131 q^{39} -2.81703 q^{40} +3.84070 q^{41} -0.128583 q^{42} +9.57128 q^{43} +11.1251 q^{44} +17.5290 q^{45} +0.689084 q^{46} +6.21907 q^{47} +10.5551 q^{48} -6.94520 q^{49} +1.59621 q^{50} +8.35968 q^{51} -4.00186 q^{52} -2.31070 q^{53} +1.00873 q^{54} -20.5558 q^{55} +0.181962 q^{56} +2.35927 q^{57} -0.767234 q^{58} +5.00296 q^{59} -19.9009 q^{60} +0.705868 q^{62} -1.13226 q^{63} -7.09085 q^{64} +7.39426 q^{65} -3.11520 q^{66} -3.39714 q^{67} -5.85755 q^{68} +9.83162 q^{69} -0.166472 q^{70} +1.92117 q^{71} -3.75929 q^{72} +5.12218 q^{73} +0.234082 q^{74} +22.7741 q^{75} -1.65312 q^{76} +1.32777 q^{77} +1.12059 q^{78} +10.3008 q^{79} +13.6654 q^{80} -0.117398 q^{81} +0.753565 q^{82} -9.80217 q^{83} +1.28547 q^{84} +10.8230 q^{85} +1.87793 q^{86} -10.9466 q^{87} +4.40843 q^{88} +7.01825 q^{89} +3.43928 q^{90} -0.477620 q^{91} -6.88892 q^{92} +10.0711 q^{93} +1.22021 q^{94} +3.05448 q^{95} +6.42270 q^{96} -11.6170 q^{97} -1.36268 q^{98} -27.4315 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 6 q^{3} + 2 q^{4} + 10 q^{5} + 6 q^{9}+O(q^{10}) \) 16 * q + 6 * q^3 + 2 * q^4 + 10 * q^5 + 6 * q^9	\( 16 q + 6 q^{3} + 2 q^{4} + 10 q^{5} + 6 q^{9} + 10 q^{12} - 12 q^{13} + 18 q^{14} + 32 q^{15} - 6 q^{16} + 32 q^{19} + 2 q^{20} - 6 q^{22} - 2 q^{25} + 24 q^{27} + 16 q^{34} - 12 q^{36} - 12 q^{39} + 38 q^{41} + 40 q^{42} + 104 q^{45} + 28 q^{46} + 6 q^{47} + 20 q^{48} + 20 q^{49} - 54 q^{52} + 32 q^{56} + 64 q^{57} - 26 q^{58} - 14 q^{60} + 16 q^{62} + 18 q^{64} - 60 q^{65} - 18 q^{66} - 52 q^{70} + 54 q^{73} + 24 q^{74} + 88 q^{75} + 66 q^{76} - 42 q^{77} + 14 q^{80} + 16 q^{81} - 64 q^{83} - 20 q^{86} + 88 q^{88} + 68 q^{95} - 30 q^{97}+O(q^{100}) \) 16 * q + 6 * q^3 + 2 * q^4 + 10 * q^5 + 6 * q^9 + 10 * q^12 - 12 * q^13 + 18 * q^14 + 32 * q^15 - 6 * q^16 + 32 * q^19 + 2 * q^20 - 6 * q^22 - 2 * q^25 + 24 * q^27 + 16 * q^34 - 12 * q^36 - 12 * q^39 + 38 * q^41 + 40 * q^42 + 104 * q^45 + 28 * q^46 + 6 * q^47 + 20 * q^48 + 20 * q^49 - 54 * q^52 + 32 * q^56 + 64 * q^57 - 26 * q^58 - 14 * q^60 + 16 * q^62 + 18 * q^64 - 60 * q^65 - 18 * q^66 - 52 * q^70 + 54 * q^73 + 24 * q^74 + 88 * q^75 + 66 * q^76 - 42 * q^77 + 14 * q^80 + 16 * q^81 - 64 * q^83 - 20 * q^86 + 88 * q^88 + 68 * q^95 - 30 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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