LMFDB - Embedded newform 8649.2.a.bu.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8649,2,Mod(1,8649)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8649, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8649.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8649 = 3^{2} \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8649.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:	$69.0626127082$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 2883)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.33041 q^{2} +3.43080 q^{4} +4.15417 q^{5} +1.65165 q^{7} +3.33434 q^{8} +9.68090 q^{10} -3.57216 q^{11} +5.72597 q^{13} +3.84901 q^{14} +0.908780 q^{16} +0.565884 q^{17} +4.78254 q^{19} +14.2521 q^{20} -8.32459 q^{22} +1.56241 q^{23} +12.2571 q^{25} +13.3438 q^{26} +5.66647 q^{28} -1.53197 q^{29} -4.55086 q^{32} +1.31874 q^{34} +6.86122 q^{35} -2.46370 q^{37} +11.1453 q^{38} +13.8514 q^{40} -5.37388 q^{41} +2.70459 q^{43} -12.2554 q^{44} +3.64106 q^{46} +0.768928 q^{47} -4.27206 q^{49} +28.5640 q^{50} +19.6446 q^{52} -9.50473 q^{53} -14.8394 q^{55} +5.50716 q^{56} -3.57010 q^{58} +5.75881 q^{59} -7.16953 q^{61} -12.4229 q^{64} +23.7866 q^{65} -1.90301 q^{67} +1.94144 q^{68} +15.9894 q^{70} -10.0456 q^{71} -6.61798 q^{73} -5.74143 q^{74} +16.4079 q^{76} -5.89995 q^{77} -15.3513 q^{79} +3.77522 q^{80} -12.5233 q^{82} +12.7125 q^{83} +2.35078 q^{85} +6.30281 q^{86} -11.9108 q^{88} -0.417559 q^{89} +9.45728 q^{91} +5.36032 q^{92} +1.79192 q^{94} +19.8675 q^{95} +2.34660 q^{97} -9.95564 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8} - 8 q^{10} - 16 q^{11} + 32 q^{13} + 24 q^{14} + 48 q^{16} - 32 q^{17} + 32 q^{19} + 24 q^{20} + 32 q^{22} - 32 q^{23} + 40 q^{25} - 16 q^{26} + 8 q^{28} - 48 q^{29}+ \cdots + 24 q^{98}+O(q^{100}) $ 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8 - 8 * q^10 - 16 * q^11 + 32 * q^13 + 24 * q^14 + 48 * q^16 - 32 * q^17 + 32 * q^19 + 24 * q^20 + 32 * q^22 - 32 * q^23 + 40 * q^25 - 16 * q^26 + 8 * q^28 - 48 * q^29 + 48 * q^32 + 48 * q^35 + 64 * q^37 + 24 * q^38 + 32 * q^43 - 48 * q^44 + 32 * q^46 + 48 * q^47 + 56 * q^49 + 24 * q^50 + 64 * q^52 - 80 * q^53 + 48 * q^56 + 32 * q^58 + 32 * q^61 + 56 * q^64 - 16 * q^65 - 16 * q^67 - 80 * q^68 + 8 * q^70 + 32 * q^73 + 56 * q^76 - 96 * q^77 + 32 * q^79 + 72 * q^80 + 8 * q^82 - 48 * q^83 + 96 * q^85 + 32 * q^86 + 96 * q^88 + 16 * q^89 - 32 * q^92 + 48 * q^94 + 48 * q^95 + 16 * q^97 + 24 * q^98

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 $

$2$	2.33041	1.64785	0.823923	−	0.566701i	$-0.191781\pi$
$2$	2.33041	1.64785	0.823923	+	0.566701i	$0.191781\pi$
$3$	0	0
$3$	0	0
$4$	3.43080	1.71540
$5$	4.15417	1.85780	0.928900	−	0.370331i	$-0.120756\pi$
$5$	4.15417	1.85780	0.928900	+	0.370331i	$0.120756\pi$
$6$	0	0
$7$	1.65165	0.624264	0.312132	−	0.950039i	$-0.398957\pi$
$7$	1.65165	0.624264	0.312132	+	0.950039i	$0.398957\pi$
$8$	3.33434	1.17887
$9$	0	0
$10$	9.68090	3.06137
$11$	−3.57216	−1.07705	−0.538524	−	0.842610i	$-0.681018\pi$
$11$	−3.57216	−1.07705	−0.538524	+	0.842610i	$0.681018\pi$
$12$	0	0
$13$	5.72597	1.58810	0.794049	−	0.607854i	$-0.207970\pi$
$13$	5.72597	1.58810	0.794049	+	0.607854i	$0.207970\pi$
$14$	3.84901	1.02869
$15$	0	0
$16$	0.908780	0.227195
$17$	0.565884	0.137247	0.0686236	−	0.997643i	$-0.478139\pi$
$17$	0.565884	0.137247	0.0686236	+	0.997643i	$0.478139\pi$
$18$	0	0
$19$	4.78254	1.09719	0.548595	−	0.836088i	$-0.315163\pi$
$19$	4.78254	1.09719	0.548595	+	0.836088i	$0.315163\pi$
$20$	14.2521	3.18687
$21$	0	0
$22$	−8.32459	−1.77481
$23$	1.56241	0.325785	0.162893	−	0.986644i	$-0.447918\pi$
$23$	1.56241	0.325785	0.162893	+	0.986644i	$0.447918\pi$
$24$	0	0
$25$	12.2571	2.45142
$26$	13.3438	2.61694
$27$	0	0
$28$	5.66647	1.07086
$29$	−1.53197	−0.284479	−0.142239	−	0.989832i	$-0.545430\pi$
$29$	−1.53197	−0.284479	−0.142239	+	0.989832i	$0.545430\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	−4.55086	−0.804486
$33$	0	0
$34$	1.31874	0.226162
$35$	6.86122	1.15976
$36$	0	0
$37$	−2.46370	−0.405030	−0.202515	−	0.979279i	$-0.564912\pi$
$37$	−2.46370	−0.405030	−0.202515	+	0.979279i	$0.564912\pi$
$38$	11.1453	1.80800
$39$	0	0
$40$	13.8514	2.19010
$41$	−5.37388	−0.839260	−0.419630	−	0.907695i	$-0.637840\pi$
$41$	−5.37388	−0.839260	−0.419630	+	0.907695i	$0.637840\pi$
$42$	0	0
$43$	2.70459	0.412447	0.206223	−	0.978505i	$-0.433883\pi$
$43$	2.70459	0.412447	0.206223	+	0.978505i	$0.433883\pi$
$44$	−12.2554	−1.84757
$45$	0	0
$46$	3.64106	0.536844
$47$	0.768928	0.112160	0.0560799	−	0.998426i	$-0.482140\pi$
$47$	0.768928	0.112160	0.0560799	+	0.998426i	$0.482140\pi$
$48$	0	0
$49$	−4.27206	−0.610294
$50$	28.5640	4.03956
$51$	0	0
$52$	19.6446	2.72422
$53$	−9.50473	−1.30557	−0.652787	−	0.757541i	$-0.726401\pi$
$53$	−9.50473	−1.30557	−0.652787	+	0.757541i	$0.726401\pi$
$54$	0	0
$55$	−14.8394	−2.00094
$56$	5.50716	0.735925
$57$	0	0
$58$	−3.57010	−0.468777
$59$	5.75881	0.749734	0.374867	−	0.927079i	$-0.377688\pi$
$59$	5.75881	0.749734	0.374867	+	0.927079i	$0.377688\pi$
$60$	0	0
$61$	−7.16953	−0.917964	−0.458982	−	0.888446i	$-0.651786\pi$
$61$	−7.16953	−0.917964	−0.458982	+	0.888446i	$0.651786\pi$
$62$	0	0
$63$	0	0
$64$	−12.4229	−1.55286
$65$	23.7866	2.95037
$66$	0	0
$67$	−1.90301	−0.232489	−0.116245	−	0.993221i	$-0.537086\pi$
$67$	−1.90301	−0.232489	−0.116245	+	0.993221i	$0.537086\pi$
$68$	1.94144	0.235434
$69$	0	0
$70$	15.9894	1.91110
$71$	−10.0456	−1.19220	−0.596098	−	0.802912i	$-0.703283\pi$
$71$	−10.0456	−1.19220	−0.596098	+	0.802912i	$0.703283\pi$
$72$	0	0
$73$	−6.61798	−0.774576	−0.387288	−	0.921959i	$-0.626588\pi$
$73$	−6.61798	−0.774576	−0.387288	+	0.921959i	$0.626588\pi$
$74$	−5.74143	−0.667428
$75$	0	0
$76$	16.4079	1.88212
$77$	−5.89995	−0.672362
$78$	0	0
$79$	−15.3513	−1.72715	−0.863576	−	0.504220i	$-0.831780\pi$
$79$	−15.3513	−1.72715	−0.863576	+	0.504220i	$0.831780\pi$
$80$	3.77522	0.422083
$81$	0	0
$82$	−12.5233	−1.38297
$83$	12.7125	1.39538	0.697691	−	0.716399i	$-0.254211\pi$
$83$	12.7125	1.39538	0.697691	+	0.716399i	$0.254211\pi$
$84$	0	0
$85$	2.35078	0.254978
$86$	6.30281	0.679649
$87$	0	0
$88$	−11.9108	−1.26970
$89$	−0.417559	−0.0442612	−0.0221306	−	0.999755i	$-0.507045\pi$
$89$	−0.417559	−0.0442612	−0.0221306	+	0.999755i	$0.507045\pi$
$90$	0	0
$91$	9.45728	0.991392
$92$	5.36032	0.558852
$93$	0	0
$94$	1.79192	0.184822
$95$	19.8675	2.03836
$96$	0	0
$97$	2.34660	0.238261	0.119130	−	0.992879i	$-0.461989\pi$
$97$	2.34660	0.238261	0.119130	+	0.992879i	$0.461989\pi$
$98$	−9.95564	−1.00567
$99$	0	0
$100$	42.0516	4.20516
$101$	−17.3169	−1.72309	−0.861547	−	0.507678i	$-0.830504\pi$
$101$	−17.3169	−1.72309	−0.861547	+	0.507678i	$0.830504\pi$
$102$	0	0
$103$	5.92115	0.583428	0.291714	−	0.956506i	$-0.405774\pi$
$103$	5.92115	0.583428	0.291714	+	0.956506i	$0.405774\pi$
$104$	19.0923	1.87216
$105$	0	0
$106$	−22.1499	−2.15139
$107$	4.72697	0.456973	0.228487	−	0.973547i	$-0.426622\pi$
$107$	4.72697	0.456973	0.228487	+	0.973547i	$0.426622\pi$
$108$	0	0
$109$	13.9699	1.33807	0.669035	−	0.743231i	$-0.266708\pi$
$109$	13.9699	1.33807	0.669035	+	0.743231i	$0.266708\pi$
$110$	−34.5817	−3.29724
$111$	0	0
$112$	1.50098	0.141830
$113$	−8.81880	−0.829603	−0.414801	−	0.909912i	$-0.636149\pi$
$113$	−8.81880	−0.829603	−0.414801	+	0.909912i	$0.636149\pi$
$114$	0	0
$115$	6.49052	0.605244
$116$	−5.25586	−0.487995
$117$	0	0
$118$	13.4204	1.23545
$119$	0.934642	0.0856785
$120$	0	0
$121$	1.76034	0.160031
$122$	−16.7079	−1.51266
$123$	0	0
$124$	0	0
$125$	30.1472	2.69645
$126$	0	0
$127$	20.1800	1.79069	0.895344	−	0.445376i	$-0.146930\pi$
$127$	20.1800	1.79069	0.895344	+	0.445376i	$0.146930\pi$
$128$	−19.8487	−1.75440
$129$	0	0
$130$	55.4325	4.86175
$131$	8.02193	0.700879	0.350439	−	0.936585i	$-0.386032\pi$
$131$	8.02193	0.700879	0.350439	+	0.936585i	$0.386032\pi$
$132$	0	0
$133$	7.89908	0.684937
$134$	−4.43478	−0.383107
$135$	0	0
$136$	1.88685	0.161796
$137$	−0.581602	−0.0496896	−0.0248448	−	0.999691i	$-0.507909\pi$
$137$	−0.581602	−0.0496896	−0.0248448	+	0.999691i	$0.507909\pi$
$138$	0	0
$139$	−4.35014	−0.368974	−0.184487	−	0.982835i	$-0.559062\pi$
$139$	−4.35014	−0.368974	−0.184487	+	0.982835i	$0.559062\pi$
$140$	23.5395	1.98945
$141$	0	0
$142$	−23.4104	−1.96456
$143$	−20.4541	−1.71046
$144$	0	0
$145$	−6.36404	−0.528505
$146$	−15.4226	−1.27638
$147$	0	0
$148$	−8.45247	−0.694789
$149$	21.3209	1.74668	0.873340	−	0.487112i	$-0.161950\pi$
$149$	21.3209	1.74668	0.873340	+	0.487112i	$0.161950\pi$
$150$	0	0
$151$	10.7008	0.870815	0.435407	−	0.900234i	$-0.356604\pi$
$151$	10.7008	0.870815	0.435407	+	0.900234i	$0.356604\pi$
$152$	15.9466	1.29344
$153$	0	0
$154$	−13.7493	−1.10795
$155$	0	0
$156$	0	0
$157$	5.19490	0.414598	0.207299	−	0.978278i	$-0.433533\pi$
$157$	5.19490	0.414598	0.207299	+	0.978278i	$0.433533\pi$
$158$	−35.7747	−2.84608
$159$	0	0
$160$	−18.9050	−1.49457
$161$	2.58055	0.203376
$162$	0	0
$163$	−15.3104	−1.19920	−0.599602	−	0.800298i	$-0.704675\pi$
$163$	−15.3104	−1.19920	−0.599602	+	0.800298i	$0.704675\pi$
$164$	−18.4367	−1.43967
$165$	0	0
$166$	29.6254	2.29937
$167$	−7.70280	−0.596061	−0.298030	−	0.954556i	$-0.596330\pi$
$167$	−7.70280	−0.596061	−0.298030	+	0.954556i	$0.596330\pi$
$168$	0	0
$169$	19.7867	1.52205
$170$	5.47827	0.420164
$171$	0	0
$172$	9.27892	0.707511
$173$	−17.3804	−1.32140	−0.660702	−	0.750648i	$-0.729741\pi$
$173$	−17.3804	−1.32140	−0.660702	+	0.750648i	$0.729741\pi$
$174$	0	0
$175$	20.2444	1.53033
$176$	−3.24631	−0.244700
$177$	0	0
$178$	−0.973083	−0.0729357
$179$	13.7099	1.02473	0.512365	−	0.858768i	$-0.328770\pi$
$179$	13.7099	1.02473	0.512365	+	0.858768i	$0.328770\pi$
$180$	0	0
$181$	7.42967	0.552243	0.276122	−	0.961123i	$-0.410951\pi$
$181$	7.42967	0.552243	0.276122	+	0.961123i	$0.410951\pi$
$182$	22.0393	1.63366
$183$	0	0
$184$	5.20962	0.384058
$185$	−10.2346	−0.752465
$186$	0	0
$187$	−2.02143	−0.147822
$188$	2.63804	0.192399
$189$	0	0
$190$	46.2993	3.35891
$191$	−15.0365	−1.08800	−0.544002	−	0.839084i	$-0.683091\pi$
$191$	−15.0365	−1.08800	−0.544002	+	0.839084i	$0.683091\pi$
$192$	0	0
$193$	−11.2247	−0.807968	−0.403984	−	0.914766i	$-0.632375\pi$
$193$	−11.2247	−0.807968	−0.403984	+	0.914766i	$0.632375\pi$
$194$	5.46853	0.392617
$195$	0	0
$196$	−14.6566	−1.04690
$197$	5.78297	0.412020	0.206010	−	0.978550i	$-0.433952\pi$
$197$	5.78297	0.412020	0.206010	+	0.978550i	$0.433952\pi$
$198$	0	0
$199$	19.3866	1.37428	0.687141	−	0.726524i	$-0.258865\pi$
$199$	19.3866	1.37428	0.687141	+	0.726524i	$0.258865\pi$
$200$	40.8694	2.88990
$201$	0	0
$202$	−40.3554	−2.83940
$203$	−2.53027	−0.177590
$204$	0	0
$205$	−22.3240	−1.55918
$206$	13.7987	0.961400
$207$	0	0
$208$	5.20364	0.360808
$209$	−17.0840	−1.18173
$210$	0	0
$211$	−9.12288	−0.628045	−0.314022	−	0.949416i	$-0.601677\pi$
$211$	−9.12288	−0.628045	−0.314022	+	0.949416i	$0.601677\pi$
$212$	−32.6088	−2.23958
$213$	0	0
$214$	11.0158	0.753022
$215$	11.2353	0.766244
$216$	0	0
$217$	0	0
$218$	32.5554	2.20493
$219$	0	0
$220$	−50.9108	−3.43241
$221$	3.24023	0.217962
$222$	0	0
$223$	19.0384	1.27491	0.637453	−	0.770490i	$-0.279988\pi$
$223$	19.0384	1.27491	0.637453	+	0.770490i	$0.279988\pi$
$224$	−7.51642	−0.502212
$225$	0	0
$226$	−20.5514	−1.36706
$227$	8.74272	0.580275	0.290137	−	0.956985i	$-0.406299\pi$
$227$	8.74272	0.580275	0.290137	+	0.956985i	$0.406299\pi$
$228$	0	0
$229$	−2.37374	−0.156861	−0.0784306	−	0.996920i	$-0.524991\pi$
$229$	−2.37374	−0.156861	−0.0784306	+	0.996920i	$0.524991\pi$
$230$	15.1256	0.997349
$231$	0	0
$232$	−5.10810	−0.335363
$233$	−4.15967	−0.272509	−0.136254	−	0.990674i	$-0.543506\pi$
$233$	−4.15967	−0.272509	−0.136254	+	0.990674i	$0.543506\pi$
$234$	0	0
$235$	3.19426	0.208370
$236$	19.7573	1.28609
$237$	0	0
$238$	2.17810	0.141185
$239$	3.94657	0.255282	0.127641	−	0.991820i	$-0.459259\pi$
$239$	3.94657	0.255282	0.127641	+	0.991820i	$0.459259\pi$
$240$	0	0
$241$	−9.20017	−0.592635	−0.296317	−	0.955090i	$-0.595759\pi$
$241$	−9.20017	−0.592635	−0.296317	+	0.955090i	$0.595759\pi$
$242$	4.10230	0.263706
$243$	0	0
$244$	−24.5972	−1.57467
$245$	−17.7468	−1.13380
$246$	0	0
$247$	27.3847	1.74245
$248$	0	0
$249$	0	0
$250$	70.2552	4.44333
$251$	−4.51770	−0.285155	−0.142577	−	0.989784i	$-0.545539\pi$
$251$	−4.51770	−0.285155	−0.142577	+	0.989784i	$0.545539\pi$
$252$	0	0
$253$	−5.58119	−0.350886
$254$	47.0277	2.95078
$255$	0	0
$256$	−21.4098	−1.33811
$257$	16.6467	1.03839	0.519197	−	0.854655i	$-0.326231\pi$
$257$	16.6467	1.03839	0.519197	+	0.854655i	$0.326231\pi$
$258$	0	0
$259$	−4.06917	−0.252846
$260$	81.6071	5.06106
$261$	0	0
$262$	18.6944	1.15494
$263$	−8.39304	−0.517537	−0.258769	−	0.965939i	$-0.583317\pi$
$263$	−8.39304	−0.517537	−0.258769	+	0.965939i	$0.583317\pi$
$264$	0	0
$265$	−39.4842	−2.42550
$266$	18.4081	1.12867
$267$	0	0
$268$	−6.52884	−0.398812
$269$	−27.0941	−1.65196	−0.825980	−	0.563700i	$-0.809377\pi$
$269$	−27.0941	−1.65196	−0.825980	+	0.563700i	$0.809377\pi$
$270$	0	0
$271$	−7.57443	−0.460114	−0.230057	−	0.973177i	$-0.573891\pi$
$271$	−7.57443	−0.460114	−0.230057	+	0.973177i	$0.573891\pi$
$272$	0.514264	0.0311818
$273$	0	0
$274$	−1.35537	−0.0818809
$275$	−43.7843	−2.64029
$276$	0	0
$277$	24.3277	1.46171	0.730856	−	0.682532i	$-0.239121\pi$
$277$	24.3277	1.46171	0.730856	+	0.682532i	$0.239121\pi$
$278$	−10.1376	−0.608013
$279$	0	0
$280$	22.8777	1.36720
$281$	−21.2225	−1.26603	−0.633013	−	0.774141i	$-0.718182\pi$
$281$	−21.2225	−1.26603	−0.633013	+	0.774141i	$0.718182\pi$
$282$	0	0
$283$	−15.6565	−0.930680	−0.465340	−	0.885132i	$-0.654068\pi$
$283$	−15.6565	−0.930680	−0.465340	+	0.885132i	$0.654068\pi$
$284$	−34.4645	−2.04509
$285$	0	0
$286$	−47.6663	−2.81857
$287$	−8.87576	−0.523920
$288$	0	0
$289$	−16.6798	−0.981163
$290$	−14.8308	−0.870895
$291$	0	0
$292$	−22.7049	−1.32871
$293$	−10.6217	−0.620524	−0.310262	−	0.950651i	$-0.600417\pi$
$293$	−10.6217	−0.620524	−0.310262	+	0.950651i	$0.600417\pi$
$294$	0	0
$295$	23.9231	1.39285
$296$	−8.21483	−0.477477
$297$	0	0
$298$	49.6865	2.87826
$299$	8.94632	0.517379
$300$	0	0
$301$	4.46704	0.257476
$302$	24.9371	1.43497
$303$	0	0
$304$	4.34628	0.249276
$305$	−29.7834	−1.70539
$306$	0	0
$307$	26.5520	1.51540	0.757700	−	0.652603i	$-0.226323\pi$
$307$	26.5520	1.51540	0.757700	+	0.652603i	$0.226323\pi$
$308$	−20.2415	−1.15337
$309$	0	0
$310$	0	0
$311$	−12.7370	−0.722250	−0.361125	−	0.932517i	$-0.617607\pi$
$311$	−12.7370	−0.722250	−0.361125	+	0.932517i	$0.617607\pi$
$312$	0	0
$313$	13.4833	0.762122	0.381061	−	0.924550i	$-0.375559\pi$
$313$	13.4833	0.762122	0.381061	+	0.924550i	$0.375559\pi$
$314$	12.1062	0.683194
$315$	0	0
$316$	−52.6671	−2.96275
$317$	19.0901	1.07221	0.536104	−	0.844152i	$-0.319896\pi$
$317$	19.0901	1.07221	0.536104	+	0.844152i	$0.319896\pi$
$318$	0	0
$319$	5.47243	0.306397
$320$	−51.6068	−2.88491
$321$	0	0
$322$	6.01374	0.335133
$323$	2.70637	0.150586
$324$	0	0
$325$	70.1837	3.89309
$326$	−35.6795	−1.97611
$327$	0	0
$328$	−17.9184	−0.989376
$329$	1.27000	0.0700173
$330$	0	0
$331$	22.5320	1.23847	0.619236	−	0.785205i	$-0.287442\pi$
$331$	22.5320	1.23847	0.619236	+	0.785205i	$0.287442\pi$
$332$	43.6141	2.39364
$333$	0	0
$334$	−17.9507	−0.982217
$335$	−7.90541	−0.431919
$336$	0	0
$337$	−1.41197	−0.0769147	−0.0384573	−	0.999260i	$-0.512244\pi$
$337$	−1.41197	−0.0769147	−0.0384573	+	0.999260i	$0.512244\pi$
$338$	46.1110	2.50811
$339$	0	0
$340$	8.06504	0.437388
$341$	0	0
$342$	0	0
$343$	−18.6175	−1.00525
$344$	9.01804	0.486220
$345$	0	0
$346$	−40.5033	−2.17747
$347$	−22.5146	−1.20865	−0.604323	−	0.796739i	$-0.706556\pi$
$347$	−22.5146	−1.20865	−0.604323	+	0.796739i	$0.706556\pi$
$348$	0	0
$349$	−28.2486	−1.51211	−0.756057	−	0.654506i	$-0.772877\pi$
$349$	−28.2486	−1.51211	−0.756057	+	0.654506i	$0.772877\pi$
$350$	47.1777	2.52176
$351$	0	0
$352$	16.2564	0.866469
$353$	15.5403	0.827126	0.413563	−	0.910475i	$-0.364284\pi$
$353$	15.5403	0.827126	0.413563	+	0.910475i	$0.364284\pi$
$354$	0	0
$355$	−41.7312	−2.21486
$356$	−1.43256	−0.0759256
$357$	0	0
$358$	31.9498	1.68860
$359$	−0.177558	−0.00937116	−0.00468558	−	0.999989i	$-0.501491\pi$
$359$	−0.177558	−0.00937116	−0.00468558	+	0.999989i	$0.501491\pi$
$360$	0	0
$361$	3.87273	0.203828
$362$	17.3142	0.910012
$363$	0	0
$364$	32.4460	1.70063
$365$	−27.4922	−1.43901
$366$	0	0
$367$	15.1788	0.792328	0.396164	−	0.918180i	$-0.370341\pi$
$367$	15.1788	0.792328	0.396164	+	0.918180i	$0.370341\pi$
$368$	1.41989	0.0740168
$369$	0	0
$370$	−23.8509	−1.23995
$371$	−15.6985	−0.815024
$372$	0	0
$373$	3.26402	0.169005	0.0845024	−	0.996423i	$-0.473070\pi$
$373$	3.26402	0.169005	0.0845024	+	0.996423i	$0.473070\pi$
$374$	−4.71076	−0.243587
$375$	0	0
$376$	2.56387	0.132221
$377$	−8.77198	−0.451780
$378$	0	0
$379$	37.5117	1.92685	0.963423	−	0.267984i	$-0.0863576\pi$
$379$	37.5117	1.92685	0.963423	+	0.267984i	$0.0863576\pi$
$380$	68.1613	3.49660
$381$	0	0
$382$	−35.0412	−1.79286
$383$	−38.2598	−1.95499	−0.977493	−	0.210970i	$-0.932338\pi$
$383$	−38.2598	−1.95499	−0.977493	+	0.210970i	$0.932338\pi$
$384$	0	0
$385$	−24.5094	−1.24911
$386$	−26.1580	−1.33141
$387$	0	0
$388$	8.05070	0.408712
$389$	17.1797	0.871046	0.435523	−	0.900178i	$-0.356563\pi$
$389$	17.1797	0.871046	0.435523	+	0.900178i	$0.356563\pi$
$390$	0	0
$391$	0.884144	0.0447131
$392$	−14.2445	−0.719456
$393$	0	0
$394$	13.4767	0.678946
$395$	−63.7717	−3.20870
$396$	0	0
$397$	−10.1959	−0.511718	−0.255859	−	0.966714i	$-0.582358\pi$
$397$	−10.1959	−0.511718	−0.255859	+	0.966714i	$0.582358\pi$
$398$	45.1788	2.26461
$399$	0	0
$400$	11.1390	0.556950
$401$	−24.3178	−1.21437	−0.607186	−	0.794560i	$-0.707702\pi$
$401$	−24.3178	−1.21437	−0.607186	+	0.794560i	$0.707702\pi$
$402$	0	0
$403$	0	0
$404$	−59.4107	−2.95579
$405$	0	0
$406$	−5.89655	−0.292641
$407$	8.80075	0.436237
$408$	0	0
$409$	−36.5945	−1.80948	−0.904740	−	0.425965i	$-0.859935\pi$
$409$	−36.5945	−1.80948	−0.904740	+	0.425965i	$0.859935\pi$
$410$	−52.0240	−2.56928
$411$	0	0
$412$	20.3143	1.00081
$413$	9.51153	0.468032
$414$	0	0
$415$	52.8100	2.59234
$416$	−26.0581	−1.27760
$417$	0	0
$418$	−39.8127	−1.94730
$419$	−22.2202	−1.08553	−0.542763	−	0.839886i	$-0.682622\pi$
$419$	−22.2202	−1.08553	−0.542763	+	0.839886i	$0.682622\pi$
$420$	0	0
$421$	9.82044	0.478619	0.239309	−	0.970943i	$-0.423079\pi$
$421$	9.82044	0.478619	0.239309	+	0.970943i	$0.423079\pi$
$422$	−21.2600	−1.03492
$423$	0	0
$424$	−31.6920	−1.53910
$425$	6.93610	0.336450
$426$	0	0
$427$	−11.8415	−0.573052
$428$	16.2173	0.783892
$429$	0	0
$430$	26.1829	1.26265
$431$	2.56171	0.123393	0.0616966	−	0.998095i	$-0.480349\pi$
$431$	2.56171	0.123393	0.0616966	+	0.998095i	$0.480349\pi$
$432$	0	0
$433$	−7.35747	−0.353577	−0.176789	−	0.984249i	$-0.556571\pi$
$433$	−7.35747	−0.353577	−0.176789	+	0.984249i	$0.556571\pi$
$434$	0	0
$435$	0	0
$436$	47.9277	2.29532
$437$	7.47230	0.357449
$438$	0	0
$439$	−28.5778	−1.36395	−0.681973	−	0.731377i	$-0.738878\pi$
$439$	−28.5778	−1.36395	−0.681973	+	0.731377i	$0.738878\pi$
$440$	−49.4795	−2.35884
$441$	0	0
$442$	7.55107	0.359168
$443$	2.83340	0.134619	0.0673094	−	0.997732i	$-0.478559\pi$
$443$	2.83340	0.134619	0.0673094	+	0.997732i	$0.478559\pi$
$444$	0	0
$445$	−1.73461	−0.0822284
$446$	44.3672	2.10085
$447$	0	0
$448$	−20.5183	−0.969398
$449$	24.3383	1.14859	0.574297	−	0.818647i	$-0.305276\pi$
$449$	24.3383	1.14859	0.574297	+	0.818647i	$0.305276\pi$
$450$	0	0
$451$	19.1964	0.903922
$452$	−30.2555	−1.42310
$453$	0	0
$454$	20.3741	0.956204
$455$	39.2871	1.84181
$456$	0	0
$457$	21.8877	1.02387	0.511933	−	0.859026i	$-0.328930\pi$
$457$	21.8877	1.02387	0.511933	+	0.859026i	$0.328930\pi$
$458$	−5.53178	−0.258483
$459$	0	0
$460$	22.2677	1.03823
$461$	24.8682	1.15823	0.579113	−	0.815248i	$-0.303399\pi$
$461$	24.8682	1.15823	0.579113	+	0.815248i	$0.303399\pi$
$462$	0	0
$463$	2.97304	0.138169	0.0690845	−	0.997611i	$-0.477992\pi$
$463$	2.97304	0.138169	0.0690845	+	0.997611i	$0.477992\pi$
$464$	−1.39222	−0.0646321
$465$	0	0
$466$	−9.69372	−0.449053
$467$	10.6715	0.493816	0.246908	−	0.969039i	$-0.420585\pi$
$467$	10.6715	0.493816	0.246908	+	0.969039i	$0.420585\pi$
$468$	0	0
$469$	−3.14310	−0.145135
$470$	7.44392	0.343362
$471$	0	0
$472$	19.2019	0.883837
$473$	−9.66125	−0.444225
$474$	0	0
$475$	58.6201	2.68968
$476$	3.20657	0.146973
$477$	0	0
$478$	9.19711	0.420666
$479$	17.3984	0.794952	0.397476	−	0.917613i	$-0.369886\pi$
$479$	17.3984	0.794952	0.397476	+	0.917613i	$0.369886\pi$
$480$	0	0
$481$	−14.1071	−0.643227
$482$	−21.4401	−0.976571
$483$	0	0
$484$	6.03936	0.274516
$485$	9.74815	0.442641
$486$	0	0
$487$	29.1998	1.32317	0.661585	−	0.749870i	$-0.269884\pi$
$487$	29.1998	1.32317	0.661585	+	0.749870i	$0.269884\pi$
$488$	−23.9057	−1.08216
$489$	0	0
$490$	−41.3574	−1.86834
$491$	6.45569	0.291341	0.145671	−	0.989333i	$-0.453466\pi$
$491$	6.45569	0.291341	0.145671	+	0.989333i	$0.453466\pi$
$492$	0	0
$493$	−0.866915	−0.0390439
$494$	63.8175	2.87128
$495$	0	0
$496$	0	0
$497$	−16.5918	−0.744246
$498$	0	0
$499$	−3.30497	−0.147951	−0.0739753	−	0.997260i	$-0.523569\pi$
$499$	−3.30497	−0.147951	−0.0739753	+	0.997260i	$0.523569\pi$
$500$	103.429	4.62548
$501$	0	0
$502$	−10.5281	−0.469891
$503$	4.41674	0.196933	0.0984664	−	0.995140i	$-0.468606\pi$
$503$	4.41674	0.196933	0.0984664	+	0.995140i	$0.468606\pi$
$504$	0	0
$505$	−71.9372	−3.20116
$506$	−13.0064	−0.578207
$507$	0	0
$508$	69.2336	3.07174
$509$	−2.09660	−0.0929303	−0.0464652	−	0.998920i	$-0.514796\pi$
$509$	−2.09660	−0.0929303	−0.0464652	+	0.998920i	$0.514796\pi$
$510$	0	0
$511$	−10.9306	−0.483540
$512$	−10.1961	−0.450608
$513$	0	0
$514$	38.7936	1.71111
$515$	24.5974	1.08389
$516$	0	0
$517$	−2.74674	−0.120801
$518$	−9.48283	−0.416651
$519$	0	0
$520$	79.3127	3.47809
$521$	−36.1620	−1.58429	−0.792144	−	0.610335i	$-0.791035\pi$
$521$	−36.1620	−1.58429	−0.792144	+	0.610335i	$0.791035\pi$
$522$	0	0
$523$	−0.412008	−0.0180158	−0.00900792	−	0.999959i	$-0.502867\pi$
$523$	−0.412008	−0.0180158	−0.00900792	+	0.999959i	$0.502867\pi$
$524$	27.5216	1.20229
$525$	0	0
$526$	−19.5592	−0.852822
$527$	0	0
$528$	0	0
$529$	−20.5589	−0.893864
$530$	−92.0143	−3.99685
$531$	0	0
$532$	27.1002	1.17494
$533$	−30.7707	−1.33283
$534$	0	0
$535$	19.6366	0.848965
$536$	−6.34528	−0.274074
$537$	0	0
$538$	−63.1404	−2.72218
$539$	15.2605	0.657316
$540$	0	0
$541$	17.4868	0.751815	0.375908	−	0.926657i	$-0.377331\pi$
$541$	17.4868	0.751815	0.375908	+	0.926657i	$0.377331\pi$
$542$	−17.6515	−0.758198
$543$	0	0
$544$	−2.57526	−0.110413
$545$	58.0331	2.48586
$546$	0	0
$547$	35.9041	1.53515	0.767575	−	0.640959i	$-0.221463\pi$
$547$	35.9041	1.53515	0.767575	+	0.640959i	$0.221463\pi$
$548$	−1.99536	−0.0852375
$549$	0	0
$550$	−102.035	−4.35080
$551$	−7.32669	−0.312128
$552$	0	0
$553$	−25.3549	−1.07820
$554$	56.6935	2.40868
$555$	0	0
$556$	−14.9245	−0.632938
$557$	−14.2983	−0.605838	−0.302919	−	0.953016i	$-0.597961\pi$
$557$	−14.2983	−0.605838	−0.302919	+	0.953016i	$0.597961\pi$
$558$	0	0
$559$	15.4864	0.655006
$560$	6.23534	0.263491
$561$	0	0
$562$	−49.4570	−2.08622
$563$	18.5366	0.781226	0.390613	−	0.920555i	$-0.372263\pi$
$563$	18.5366	0.781226	0.390613	+	0.920555i	$0.372263\pi$
$564$	0	0
$565$	−36.6348	−1.54124
$566$	−36.4859	−1.53362
$567$	0	0
$568$	−33.4955	−1.40544
$569$	6.63855	0.278302	0.139151	−	0.990271i	$-0.455563\pi$
$569$	6.63855	0.278302	0.139151	+	0.990271i	$0.455563\pi$
$570$	0	0
$571$	−6.93232	−0.290109	−0.145054	−	0.989424i	$-0.546336\pi$
$571$	−6.93232	−0.290109	−0.145054	+	0.989424i	$0.546336\pi$
$572$	−70.1738	−2.93411
$573$	0	0
$574$	−20.6841	−0.863340
$575$	19.1506	0.798637
$576$	0	0
$577$	−20.1163	−0.837451	−0.418726	−	0.908113i	$-0.637523\pi$
$577$	−20.1163	−0.837451	−0.418726	+	0.908113i	$0.637523\pi$
$578$	−38.8707	−1.61681
$579$	0	0
$580$	−21.8337	−0.906596
$581$	20.9966	0.871087
$582$	0	0
$583$	33.9524	1.40617
$584$	−22.0666	−0.913122
$585$	0	0
$586$	−24.7528	−1.02253
$587$	−16.1167	−0.665208	−0.332604	−	0.943067i	$-0.607927\pi$
$587$	−16.1167	−0.665208	−0.332604	+	0.943067i	$0.607927\pi$
$588$	0	0
$589$	0	0
$590$	55.7505	2.29521
$591$	0	0
$592$	−2.23896	−0.0920208
$593$	23.7429	0.975005	0.487502	−	0.873122i	$-0.337908\pi$
$593$	23.7429	0.975005	0.487502	+	0.873122i	$0.337908\pi$
$594$	0	0
$595$	3.88266	0.159173
$596$	73.1478	2.99625
$597$	0	0
$598$	20.8486	0.852561
$599$	45.7722	1.87020	0.935101	−	0.354381i	$-0.115308\pi$
$599$	45.7722	1.87020	0.935101	+	0.354381i	$0.115308\pi$
$600$	0	0
$601$	−36.9040	−1.50534	−0.752672	−	0.658395i	$-0.771236\pi$
$601$	−36.9040	−1.50534	−0.752672	+	0.658395i	$0.771236\pi$
$602$	10.4100	0.424281
$603$	0	0
$604$	36.7121	1.49379
$605$	7.31273	0.297305
$606$	0	0
$607$	22.1292	0.898197	0.449099	−	0.893482i	$-0.351745\pi$
$607$	22.1292	0.898197	0.449099	+	0.893482i	$0.351745\pi$
$608$	−21.7647	−0.882674
$609$	0	0
$610$	−69.4075	−2.81023
$611$	4.40286	0.178121
$612$	0	0
$613$	1.00830	0.0407250	0.0203625	−	0.999793i	$-0.493518\pi$
$613$	1.00830	0.0407250	0.0203625	+	0.999793i	$0.493518\pi$
$614$	61.8769	2.49715
$615$	0	0
$616$	−19.6725	−0.792626
$617$	−5.94272	−0.239245	−0.119623	−	0.992819i	$-0.538168\pi$
$617$	−5.94272	−0.239245	−0.119623	+	0.992819i	$0.538168\pi$
$618$	0	0
$619$	−34.6733	−1.39364	−0.696819	−	0.717247i	$-0.745402\pi$
$619$	−34.6733	−1.39364	−0.696819	+	0.717247i	$0.745402\pi$
$620$	0	0
$621$	0	0
$622$	−29.6824	−1.19016
$623$	−0.689661	−0.0276307
$624$	0	0
$625$	63.9510	2.55804
$626$	31.4216	1.25586
$627$	0	0
$628$	17.8227	0.711201
$629$	−1.39417	−0.0555892
$630$	0	0
$631$	−40.6967	−1.62011	−0.810054	−	0.586355i	$-0.800562\pi$
$631$	−40.6967	−1.62011	−0.810054	+	0.586355i	$0.800562\pi$
$632$	−51.1863	−2.03608
$633$	0	0
$634$	44.4877	1.76683
$635$	83.8312	3.32674
$636$	0	0
$637$	−24.4617	−0.969206
$638$	12.7530	0.504895
$639$	0	0
$640$	−82.4549	−3.25932
$641$	−43.6515	−1.72413	−0.862066	−	0.506797i	$-0.830830\pi$
$641$	−43.6515	−1.72413	−0.862066	+	0.506797i	$0.830830\pi$
$642$	0	0
$643$	−31.4817	−1.24152	−0.620759	−	0.784002i	$-0.713175\pi$
$643$	−31.4817	−1.24152	−0.620759	+	0.784002i	$0.713175\pi$
$644$	8.85336	0.348871
$645$	0	0
$646$	6.30694	0.248143
$647$	49.2323	1.93552	0.967760	−	0.251873i	$-0.0810466\pi$
$647$	49.2323	1.93552	0.967760	+	0.251873i	$0.0810466\pi$
$648$	0	0
$649$	−20.5714	−0.807499
$650$	163.557	6.41522
$651$	0	0
$652$	−52.5270	−2.05711
$653$	−8.05179	−0.315091	−0.157545	−	0.987512i	$-0.550358\pi$
$653$	−8.05179	−0.315091	−0.157545	+	0.987512i	$0.550358\pi$
$654$	0	0
$655$	33.3244	1.30209
$656$	−4.88368	−0.190675
$657$	0	0
$658$	2.95961	0.115378
$659$	−26.3006	−1.02452	−0.512262	−	0.858829i	$-0.671192\pi$
$659$	−26.3006	−1.02452	−0.512262	+	0.858829i	$0.671192\pi$
$660$	0	0
$661$	−12.5448	−0.487938	−0.243969	−	0.969783i	$-0.578449\pi$
$661$	−12.5448	−0.487938	−0.243969	+	0.969783i	$0.578449\pi$
$662$	52.5088	2.04081
$663$	0	0
$664$	42.3879	1.64497
$665$	32.8141	1.27248
$666$	0	0
$667$	−2.39356	−0.0926790
$668$	−26.4268	−1.02248
$669$	0	0
$670$	−18.4228	−0.711736
$671$	25.6107	0.988690
$672$	0	0
$673$	26.2972	1.01368	0.506840	−	0.862040i	$-0.330813\pi$
$673$	26.2972	1.01368	0.506840	+	0.862040i	$0.330813\pi$
$674$	−3.29045	−0.126744
$675$	0	0
$676$	67.8841	2.61093
$677$	−44.5434	−1.71194	−0.855970	−	0.517025i	$-0.827039\pi$
$677$	−44.5434	−1.71194	−0.855970	+	0.517025i	$0.827039\pi$
$678$	0	0
$679$	3.87575	0.148738
$680$	7.83830	0.300585
$681$	0	0
$682$	0	0
$683$	−0.986898	−0.0377626	−0.0188813	−	0.999822i	$-0.506010\pi$
$683$	−0.986898	−0.0377626	−0.0188813	+	0.999822i	$0.506010\pi$
$684$	0	0
$685$	−2.41607	−0.0923133
$686$	−43.3863	−1.65650
$687$	0	0
$688$	2.45788	0.0937058
$689$	−54.4237	−2.07338
$690$	0	0
$691$	28.6534	1.09003	0.545013	−	0.838428i	$-0.316525\pi$
$691$	28.6534	1.09003	0.545013	+	0.838428i	$0.316525\pi$
$692$	−59.6285	−2.26674
$693$	0	0
$694$	−52.4682	−1.99166
$695$	−18.0712	−0.685480
$696$	0	0
$697$	−3.04100	−0.115186
$698$	−65.8308	−2.49173
$699$	0	0
$700$	69.4545	2.62513
$701$	27.1957	1.02717	0.513584	−	0.858040i	$-0.328318\pi$
$701$	27.1957	1.02717	0.513584	+	0.858040i	$0.328318\pi$
$702$	0	0
$703$	−11.7828	−0.444396
$704$	44.3766	1.67251
$705$	0	0
$706$	36.2152	1.36298
$707$	−28.6014	−1.07567
$708$	0	0
$709$	28.1085	1.05564	0.527819	−	0.849357i	$-0.323010\pi$
$709$	28.1085	1.05564	0.527819	+	0.849357i	$0.323010\pi$
$710$	−97.2507	−3.64975
$711$	0	0
$712$	−1.39229	−0.0521781
$713$	0	0
$714$	0	0
$715$	−84.9696	−3.17768
$716$	47.0361	1.75782
$717$	0	0
$718$	−0.413783	−0.0154422
$719$	−20.8740	−0.778468	−0.389234	−	0.921139i	$-0.627260\pi$
$719$	−20.8740	−0.778468	−0.389234	+	0.921139i	$0.627260\pi$
$720$	0	0
$721$	9.77965	0.364213
$722$	9.02504	0.335877
$723$	0	0
$724$	25.4897	0.947317
$725$	−18.7774	−0.697377
$726$	0	0
$727$	−5.96155	−0.221102	−0.110551	−	0.993870i	$-0.535261\pi$
$727$	−5.96155	−0.221102	−0.110551	+	0.993870i	$0.535261\pi$
$728$	31.5338	1.16872
$729$	0	0
$730$	−64.0680	−2.37126
$731$	1.53049	0.0566071
$732$	0	0
$733$	23.4155	0.864873	0.432436	−	0.901664i	$-0.357654\pi$
$733$	23.4155	0.864873	0.432436	+	0.901664i	$0.357654\pi$
$734$	35.3728	1.30563
$735$	0	0
$736$	−7.11031	−0.262090
$737$	6.79785	0.250402
$738$	0	0
$739$	−2.25362	−0.0829008	−0.0414504	−	0.999141i	$-0.513198\pi$
$739$	−2.25362	−0.0829008	−0.0414504	+	0.999141i	$0.513198\pi$
$740$	−35.1130	−1.29078
$741$	0	0
$742$	−36.5838	−1.34303
$743$	−45.7086	−1.67689	−0.838443	−	0.544990i	$-0.816534\pi$
$743$	−45.7086	−1.67689	−0.838443	+	0.544990i	$0.816534\pi$
$744$	0	0
$745$	88.5707	3.24498
$746$	7.60650	0.278494
$747$	0	0
$748$	−6.93512	−0.253573
$749$	7.80729	0.285272
$750$	0	0
$751$	−35.6244	−1.29995	−0.649977	−	0.759954i	$-0.725221\pi$
$751$	−35.6244	−1.29995	−0.649977	+	0.759954i	$0.725221\pi$
$752$	0.698786	0.0254821
$753$	0	0
$754$	−20.4423	−0.744464
$755$	44.4527	1.61780
$756$	0	0
$757$	13.6771	0.497102	0.248551	−	0.968619i	$-0.420046\pi$
$757$	13.6771	0.497102	0.248551	+	0.968619i	$0.420046\pi$
$758$	87.4175	3.17515
$759$	0	0
$760$	66.2450	2.40296
$761$	29.8950	1.08369	0.541847	−	0.840477i	$-0.317725\pi$
$761$	29.8950	1.08369	0.541847	+	0.840477i	$0.317725\pi$
$762$	0	0
$763$	23.0733	0.835309
$764$	−51.5872	−1.86636
$765$	0	0
$766$	−89.1609	−3.22152
$767$	32.9748	1.19065
$768$	0	0
$769$	−28.4216	−1.02491	−0.512455	−	0.858714i	$-0.671264\pi$
$769$	−28.4216	−1.02491	−0.512455	+	0.858714i	$0.671264\pi$
$770$	−57.1169	−2.05835
$771$	0	0
$772$	−38.5095	−1.38599
$773$	−9.66694	−0.347696	−0.173848	−	0.984773i	$-0.555620\pi$
$773$	−9.66694	−0.347696	−0.173848	+	0.984773i	$0.555620\pi$
$774$	0	0
$775$	0	0
$776$	7.82436	0.280878
$777$	0	0
$778$	40.0357	1.43535
$779$	−25.7008	−0.920828
$780$	0	0
$781$	35.8846	1.28405
$782$	2.06042	0.0736803
$783$	0	0
$784$	−3.88236	−0.138656
$785$	21.5805	0.770240
$786$	0	0
$787$	12.6164	0.449725	0.224862	−	0.974391i	$-0.427807\pi$
$787$	12.6164	0.449725	0.224862	+	0.974391i	$0.427807\pi$
$788$	19.8402	0.706778
$789$	0	0
$790$	−148.614	−5.28745
$791$	−14.5656	−0.517892
$792$	0	0
$793$	−41.0525	−1.45782
$794$	−23.7606	−0.843233
$795$	0	0
$796$	66.5116	2.35744
$797$	0.567057	0.0200862	0.0100431	−	0.999950i	$-0.496803\pi$
$797$	0.567057	0.0200862	0.0100431	+	0.999950i	$0.496803\pi$
$798$	0	0
$799$	0.435124	0.0153936
$800$	−55.7803	−1.97213
$801$	0	0
$802$	−56.6703	−2.00110
$803$	23.6405	0.834254
$804$	0	0
$805$	10.7201	0.377832
$806$	0	0
$807$	0	0
$808$	−57.7404	−2.03130
$809$	−2.36052	−0.0829913	−0.0414956	−	0.999139i	$-0.513212\pi$
$809$	−2.36052	−0.0829913	−0.0414956	+	0.999139i	$0.513212\pi$
$810$	0	0
$811$	40.8306	1.43376	0.716878	−	0.697199i	$-0.245571\pi$
$811$	40.8306	1.43376	0.716878	+	0.697199i	$0.245571\pi$
$812$	−8.68084	−0.304638
$813$	0	0
$814$	20.5093	0.718851
$815$	−63.6020	−2.22788
$816$	0	0
$817$	12.9348	0.452533
$818$	−85.2800	−2.98174
$819$	0	0
$820$	−76.5892	−2.67461
$821$	−34.7565	−1.21301	−0.606505	−	0.795080i	$-0.707429\pi$
$821$	−34.7565	−1.21301	−0.606505	+	0.795080i	$0.707429\pi$
$822$	0	0
$823$	38.8988	1.35593	0.677964	−	0.735095i	$-0.262862\pi$
$823$	38.8988	1.35593	0.677964	+	0.735095i	$0.262862\pi$
$824$	19.7431	0.687785
$825$	0	0
$826$	22.1657	0.771245
$827$	29.9815	1.04256	0.521279	−	0.853386i	$-0.325455\pi$
$827$	29.9815	1.04256	0.521279	+	0.853386i	$0.325455\pi$
$828$	0	0
$829$	−9.02926	−0.313599	−0.156800	−	0.987630i	$-0.550118\pi$
$829$	−9.02926	−0.313599	−0.156800	+	0.987630i	$0.550118\pi$
$830$	123.069	4.27178
$831$	0	0
$832$	−71.1332	−2.46610
$833$	−2.41749	−0.0837611
$834$	0	0
$835$	−31.9987	−1.10736
$836$	−58.6118	−2.02713
$837$	0	0
$838$	−51.7820	−1.78878
$839$	17.6163	0.608184	0.304092	−	0.952643i	$-0.401647\pi$
$839$	17.6163	0.608184	0.304092	+	0.952643i	$0.401647\pi$
$840$	0	0
$841$	−26.6531	−0.919072
$842$	22.8856	0.788690
$843$	0	0
$844$	−31.2988	−1.07735
$845$	82.1972	2.82767
$846$	0	0
$847$	2.90746	0.0999014
$848$	−8.63770	−0.296620
$849$	0	0
$850$	16.1639	0.554419
$851$	−3.84932	−0.131953
$852$	0	0
$853$	34.3306	1.17546	0.587728	−	0.809058i	$-0.300022\pi$
$853$	34.3306	1.17546	0.587728	+	0.809058i	$0.300022\pi$
$854$	−27.5956	−0.944302
$855$	0	0
$856$	15.7613	0.538711
$857$	−37.3423	−1.27559	−0.637795	−	0.770206i	$-0.720153\pi$
$857$	−37.3423	−1.27559	−0.637795	+	0.770206i	$0.720153\pi$
$858$	0	0
$859$	−15.3156	−0.522563	−0.261282	−	0.965263i	$-0.584145\pi$
$859$	−15.3156	−0.522563	−0.261282	+	0.965263i	$0.584145\pi$
$860$	38.5462	1.31441
$861$	0	0
$862$	5.96982	0.203333
$863$	56.7996	1.93348	0.966740	−	0.255761i	$-0.0823260\pi$
$863$	56.7996	1.93348	0.966740	+	0.255761i	$0.0823260\pi$
$864$	0	0
$865$	−72.2009	−2.45491
$866$	−17.1459	−0.582641
$867$	0	0
$868$	0	0
$869$	54.8372	1.86022
$870$	0	0
$871$	−10.8966	−0.369216
$872$	46.5803	1.57741
$873$	0	0
$874$	17.4135	0.589021
$875$	49.7925	1.68330
$876$	0	0
$877$	−36.7407	−1.24065	−0.620323	−	0.784347i	$-0.712998\pi$
$877$	−36.7407	−1.24065	−0.620323	+	0.784347i	$0.712998\pi$
$878$	−66.5980	−2.24757
$879$	0	0
$880$	−13.4857	−0.454603
$881$	−7.35814	−0.247902	−0.123951	−	0.992288i	$-0.539557\pi$
$881$	−7.35814	−0.247902	−0.123951	+	0.992288i	$0.539557\pi$
$882$	0	0
$883$	31.1461	1.04815	0.524075	−	0.851672i	$-0.324411\pi$
$883$	31.1461	1.04815	0.524075	+	0.851672i	$0.324411\pi$
$884$	11.1166	0.373891
$885$	0	0
$886$	6.60297	0.221831
$887$	−15.1283	−0.507957	−0.253979	−	0.967210i	$-0.581739\pi$
$887$	−15.1283	−0.507957	−0.253979	+	0.967210i	$0.581739\pi$
$888$	0	0
$889$	33.3303	1.11786
$890$	−4.04235	−0.135500
$891$	0	0
$892$	65.3169	2.18697
$893$	3.67743	0.123061
$894$	0	0
$895$	56.9534	1.90374
$896$	−32.7831	−1.09521
$897$	0	0
$898$	56.7180	1.89271
$899$	0	0
$900$	0	0
$901$	−5.37858	−0.179186
$902$	44.7354	1.48953
$903$	0	0
$904$	−29.4049	−0.977992
$905$	30.8641	1.02596
$906$	0	0
$907$	4.72657	0.156943	0.0784716	−	0.996916i	$-0.474996\pi$
$907$	4.72657	0.156943	0.0784716	+	0.996916i	$0.474996\pi$
$908$	29.9945	0.995403
$909$	0	0
$910$	91.5550	3.03502
$911$	−39.1885	−1.29837	−0.649187	−	0.760629i	$-0.724891\pi$
$911$	−39.1885	−1.29837	−0.649187	+	0.760629i	$0.724891\pi$
$912$	0	0
$913$	−45.4112	−1.50289
$914$	51.0073	1.68717
$915$	0	0
$916$	−8.14382	−0.269079
$917$	13.2494	0.437534
$918$	0	0
$919$	−20.2830	−0.669075	−0.334538	−	0.942382i	$-0.608580\pi$
$919$	−20.2830	−0.669075	−0.334538	+	0.942382i	$0.608580\pi$
$920$	21.6416	0.713503
$921$	0	0
$922$	57.9529	1.90858
$923$	−57.5209	−1.89332
$924$	0	0
$925$	−30.1979	−0.992899
$926$	6.92840	0.227681
$927$	0	0
$928$	6.97176	0.228859
$929$	12.6462	0.414909	0.207455	−	0.978245i	$-0.433482\pi$
$929$	12.6462	0.414909	0.207455	+	0.978245i	$0.433482\pi$
$930$	0	0
$931$	−20.4313	−0.669609
$932$	−14.2710	−0.467462
$933$	0	0
$934$	24.8688	0.813733
$935$	−8.39736	−0.274623
$936$	0	0
$937$	−0.926058	−0.0302530	−0.0151265	−	0.999886i	$-0.504815\pi$
$937$	−0.926058	−0.0302530	−0.0151265	+	0.999886i	$0.504815\pi$
$938$	−7.32470	−0.239160
$939$	0	0
$940$	10.9588	0.357438
$941$	23.7883	0.775475	0.387738	−	0.921770i	$-0.373257\pi$
$941$	23.7883	0.775475	0.387738	+	0.921770i	$0.373257\pi$
$942$	0	0
$943$	−8.39622	−0.273419
$944$	5.23349	0.170336
$945$	0	0
$946$	−22.5146	−0.732014
$947$	−20.1610	−0.655144	−0.327572	−	0.944826i	$-0.606230\pi$
$947$	−20.1610	−0.655144	−0.327572	+	0.944826i	$0.606230\pi$
$948$	0	0
$949$	−37.8943	−1.23010
$950$	136.609	4.43217
$951$	0	0
$952$	3.11642	0.101004
$953$	4.41316	0.142956	0.0714781	−	0.997442i	$-0.477228\pi$
$953$	4.41316	0.142956	0.0714781	+	0.997442i	$0.477228\pi$
$954$	0	0
$955$	−62.4641	−2.02129
$956$	13.5399	0.437911
$957$	0	0
$958$	40.5453	1.30996
$959$	−0.960602	−0.0310194
$960$	0	0
$961$	0	0
$962$	−32.8752	−1.05994
$963$	0	0
$964$	−31.5639	−1.01661
$965$	−46.6291	−1.50104
$966$	0	0
$967$	−52.5227	−1.68902	−0.844509	−	0.535542i	$-0.820107\pi$
$967$	−52.5227	−1.68902	−0.844509	+	0.535542i	$0.820107\pi$
$968$	5.86957	0.188655
$969$	0	0
$970$	22.7172	0.729404
$971$	−15.2898	−0.490674	−0.245337	−	0.969438i	$-0.578899\pi$
$971$	−15.2898	−0.490674	−0.245337	+	0.969438i	$0.578899\pi$
$972$	0	0
$973$	−7.18490	−0.230337
$974$	68.0475	2.18038
$975$	0	0
$976$	−6.51552	−0.208557
$977$	45.2365	1.44725	0.723623	−	0.690196i	$-0.242476\pi$
$977$	45.2365	1.44725	0.723623	+	0.690196i	$0.242476\pi$
$978$	0	0
$979$	1.49159	0.0476714
$980$	−60.8858	−1.94493
$981$	0	0
$982$	15.0444	0.480086
$983$	34.9761	1.11556	0.557782	−	0.829987i	$-0.311652\pi$
$983$	34.9761	1.11556	0.557782	+	0.829987i	$0.311652\pi$
$984$	0	0
$985$	24.0234	0.765450
$986$	−2.02027	−0.0643384
$987$	0	0
$988$	93.9513	2.98899
$989$	4.22569	0.134369
$990$	0	0
$991$	−3.85931	−0.122595	−0.0612974	−	0.998120i	$-0.519524\pi$
$991$	−3.85931	−0.122595	−0.0612974	+	0.998120i	$0.519524\pi$
$992$	0	0
$993$	0	0
$994$	−38.6657	−1.22640
$995$	80.5353	2.55314
$996$	0	0
$997$	−39.2889	−1.24429	−0.622146	−	0.782901i	$-0.713739\pi$
$997$	−39.2889	−1.24429	−0.622146	+	0.782901i	$0.713739\pi$
$998$	−7.70192	−0.243800
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.bu.1.20		24
3.2	odd	2		2883.2.a.v.1.5	yes	24
31.30	odd	2		8649.2.a.bv.1.20		24
93.92	even	2		2883.2.a.u.1.5	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2883.2.a.u.1.5	✓	24	93.92	even	2
2883.2.a.v.1.5	yes	24	3.2	odd	2
8649.2.a.bu.1.20		24	1.1	even	1	trivial
8649.2.a.bv.1.20		24	31.30	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q+2.33041 q^{2} +3.43080 q^{4} +4.15417 q^{5} +1.65165 q^{7} +3.33434 q^{8} +9.68090 q^{10} -3.57216 q^{11} +5.72597 q^{13} +3.84901 q^{14} +0.908780 q^{16} +0.565884 q^{17} +4.78254 q^{19} +14.2521 q^{20} -8.32459 q^{22} +1.56241 q^{23} +12.2571 q^{25} +13.3438 q^{26} +5.66647 q^{28} -1.53197 q^{29} -4.55086 q^{32} +1.31874 q^{34} +6.86122 q^{35} -2.46370 q^{37} +11.1453 q^{38} +13.8514 q^{40} -5.37388 q^{41} +2.70459 q^{43} -12.2554 q^{44} +3.64106 q^{46} +0.768928 q^{47} -4.27206 q^{49} +28.5640 q^{50} +19.6446 q^{52} -9.50473 q^{53} -14.8394 q^{55} +5.50716 q^{56} -3.57010 q^{58} +5.75881 q^{59} -7.16953 q^{61} -12.4229 q^{64} +23.7866 q^{65} -1.90301 q^{67} +1.94144 q^{68} +15.9894 q^{70} -10.0456 q^{71} -6.61798 q^{73} -5.74143 q^{74} +16.4079 q^{76} -5.89995 q^{77} -15.3513 q^{79} +3.77522 q^{80} -12.5233 q^{82} +12.7125 q^{83} +2.35078 q^{85} +6.30281 q^{86} -11.9108 q^{88} -0.417559 q^{89} +9.45728 q^{91} +5.36032 q^{92} +1.79192 q^{94} +19.8675 q^{95} +2.34660 q^{97} -9.95564 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8} - 8 q^{10} - 16 q^{11} + 32 q^{13} + 24 q^{14} + 48 q^{16} - 32 q^{17} + 32 q^{19} + 24 q^{20} + 32 q^{22} - 32 q^{23} + 40 q^{25} - 16 q^{26} + 8 q^{28} - 48 q^{29}+ \cdots + 24 q^{98}+O(q^{100}) \) 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8 - 8 * q^10 - 16 * q^11 + 32 * q^13 + 24 * q^14 + 48 * q^16 - 32 * q^17 + 32 * q^19 + 24 * q^20 + 32 * q^22 - 32 * q^23 + 40 * q^25 - 16 * q^26 + 8 * q^28 - 48 * q^29 + 48 * q^32 + 48 * q^35 + 64 * q^37 + 24 * q^38 + 32 * q^43 - 48 * q^44 + 32 * q^46 + 48 * q^47 + 56 * q^49 + 24 * q^50 + 64 * q^52 - 80 * q^53 + 48 * q^56 + 32 * q^58 + 32 * q^61 + 56 * q^64 - 16 * q^65 - 16 * q^67 - 80 * q^68 + 8 * q^70 + 32 * q^73 + 56 * q^76 - 96 * q^77 + 32 * q^79 + 72 * q^80 + 8 * q^82 - 48 * q^83 + 96 * q^85 + 32 * q^86 + 96 * q^88 + 16 * q^89 - 32 * q^92 + 48 * q^94 + 48 * q^95 + 16 * q^97 + 24 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more