LMFDB - Embedded newform 7225.2.a.by.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, 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("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.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:	$57.6919154604$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	7225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.248918 q^{2} -1.64368 q^{3} -1.93804 q^{4} +0.409143 q^{6} -1.43109 q^{7} +0.980251 q^{8} -0.298307 q^{9} +2.69361 q^{11} +3.18552 q^{12} +0.174664 q^{13} +0.356224 q^{14} +3.63208 q^{16} +0.0742541 q^{18} -3.80994 q^{19} +2.35225 q^{21} -0.670490 q^{22} -2.75801 q^{23} -1.61122 q^{24} -0.0434772 q^{26} +5.42137 q^{27} +2.77350 q^{28} -6.43383 q^{29} -0.492358 q^{31} -2.86459 q^{32} -4.42745 q^{33} +0.578130 q^{36} +5.74291 q^{37} +0.948365 q^{38} -0.287093 q^{39} -5.99645 q^{41} -0.585519 q^{42} +11.3138 q^{43} -5.22033 q^{44} +0.686519 q^{46} -2.65291 q^{47} -5.96998 q^{48} -4.95199 q^{49} -0.338506 q^{52} -12.3532 q^{53} -1.34948 q^{54} -1.40282 q^{56} +6.26234 q^{57} +1.60150 q^{58} +8.09258 q^{59} -6.94789 q^{61} +0.122557 q^{62} +0.426903 q^{63} -6.55110 q^{64} +1.10207 q^{66} +12.3901 q^{67} +4.53329 q^{69} +12.2263 q^{71} -0.292415 q^{72} +15.6637 q^{73} -1.42952 q^{74} +7.38382 q^{76} -3.85480 q^{77} +0.0714627 q^{78} +12.9096 q^{79} -8.01609 q^{81} +1.49263 q^{82} -8.57565 q^{83} -4.55876 q^{84} -2.81621 q^{86} +10.5752 q^{87} +2.64042 q^{88} +11.4907 q^{89} -0.249960 q^{91} +5.34513 q^{92} +0.809280 q^{93} +0.660358 q^{94} +4.70848 q^{96} +2.29892 q^{97} +1.23264 q^{98} -0.803523 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 8 q^{4} - 8 q^{9} - 8 q^{16} - 16 q^{19} - 32 q^{21} - 48 q^{26} + 8 q^{36} - 56 q^{49} - 64 q^{59} - 104 q^{64} - 16 q^{66} - 32 q^{69} - 48 q^{76} - 88 q^{81} - 96 q^{84} - 112 q^{89} - 128 q^{94}+O(q^{100}) $ 24 * q + 8 * q^4 - 8 * q^9 - 8 * q^16 - 16 * q^19 - 32 * q^21 - 48 * q^26 + 8 * q^36 - 56 * q^49 - 64 * q^59 - 104 * q^64 - 16 * q^66 - 32 * q^69 - 48 * q^76 - 88 * q^81 - 96 * q^84 - 112 * q^89 - 128 * q^94

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 $

$2$	−0.248918	−0.176012	−0.0880060	−	0.996120i	$-0.528049\pi$
$2$	−0.248918	−0.176012	−0.0880060	+	0.996120i	$0.528049\pi$
$3$	−1.64368	−0.948981	−0.474490	−	0.880261i	$-0.657368\pi$
$3$	−1.64368	−0.948981	−0.474490	+	0.880261i	$0.657368\pi$
$4$	−1.93804	−0.969020
$5$	0	0
$5$	0	0
$6$	0.409143	0.167032
$7$	−1.43109	−0.540900	−0.270450	−	0.962734i	$-0.587172\pi$
$7$	−1.43109	−0.540900	−0.270450	+	0.962734i	$0.587172\pi$
$8$	0.980251	0.346571
$9$	−0.298307	−0.0994356
$10$	0	0
$11$	2.69361	0.812155	0.406077	−	0.913839i	$-0.366896\pi$
$11$	2.69361	0.812155	0.406077	+	0.913839i	$0.366896\pi$
$12$	3.18552	0.919581
$13$	0.174664	0.0484432	0.0242216	−	0.999707i	$-0.492289\pi$
$13$	0.174664	0.0484432	0.0242216	+	0.999707i	$0.492289\pi$
$14$	0.356224	0.0952049
$15$	0	0
$16$	3.63208	0.908019
$17$	0	0
$17$	0	0
$18$	0.0742541	0.0175019
$19$	−3.80994	−0.874061	−0.437031	−	0.899447i	$-0.643970\pi$
$19$	−3.80994	−0.874061	−0.437031	+	0.899447i	$0.643970\pi$
$20$	0	0
$21$	2.35225	0.513304
$22$	−0.670490	−0.142949
$23$	−2.75801	−0.575085	−0.287542	−	0.957768i	$-0.592838\pi$
$23$	−2.75801	−0.575085	−0.287542	+	0.957768i	$0.592838\pi$
$24$	−1.61122	−0.328889
$25$	0	0
$26$	−0.0434772	−0.00852658
$27$	5.42137	1.04334
$28$	2.77350	0.524143
$29$	−6.43383	−1.19473	−0.597366	−	0.801969i	$-0.703786\pi$
$29$	−6.43383	−1.19473	−0.597366	+	0.801969i	$0.703786\pi$
$30$	0	0
$31$	−0.492358	−0.0884300	−0.0442150	−	0.999022i	$-0.514079\pi$
$31$	−0.492358	−0.0884300	−0.0442150	+	0.999022i	$0.514079\pi$
$32$	−2.86459	−0.506393
$33$	−4.42745	−0.770719
$34$	0	0
$35$	0	0
$36$	0.578130	0.0963551
$37$	5.74291	0.944128	0.472064	−	0.881564i	$-0.343509\pi$
$37$	5.74291	0.944128	0.472064	+	0.881564i	$0.343509\pi$
$38$	0.948365	0.153845
$39$	−0.287093	−0.0459716
$40$	0	0
$41$	−5.99645	−0.936489	−0.468244	−	0.883599i	$-0.655113\pi$
$41$	−5.99645	−0.936489	−0.468244	+	0.883599i	$0.655113\pi$
$42$	−0.585519	−0.0903476
$43$	11.3138	1.72533	0.862667	−	0.505772i	$-0.168792\pi$
$43$	11.3138	1.72533	0.862667	+	0.505772i	$0.168792\pi$
$44$	−5.22033	−0.786994
$45$	0	0
$46$	0.686519	0.101222
$47$	−2.65291	−0.386967	−0.193483	−	0.981104i	$-0.561979\pi$
$47$	−2.65291	−0.386967	−0.193483	+	0.981104i	$0.561979\pi$
$48$	−5.96998	−0.861693
$49$	−4.95199	−0.707427
$50$	0	0
$51$	0	0
$52$	−0.338506	−0.0469424
$53$	−12.3532	−1.69684	−0.848419	−	0.529326i	$-0.822445\pi$
$53$	−12.3532	−1.69684	−0.848419	+	0.529326i	$0.822445\pi$
$54$	−1.34948	−0.183641
$55$	0	0
$56$	−1.40282	−0.187460
$57$	6.26234	0.829467
$58$	1.60150	0.210287
$59$	8.09258	1.05356	0.526782	−	0.850000i	$-0.323398\pi$
$59$	8.09258	1.05356	0.526782	+	0.850000i	$0.323398\pi$
$60$	0	0
$61$	−6.94789	−0.889587	−0.444793	−	0.895633i	$-0.646723\pi$
$61$	−6.94789	−0.889587	−0.444793	+	0.895633i	$0.646723\pi$
$62$	0.122557	0.0155647
$63$	0.426903	0.0537847
$64$	−6.55110	−0.818888
$65$	0	0
$66$	1.10207	0.135656
$67$	12.3901	1.51369	0.756845	−	0.653594i	$-0.226740\pi$
$67$	12.3901	1.51369	0.756845	+	0.653594i	$0.226740\pi$
$68$	0	0
$69$	4.53329	0.545744
$70$	0	0
$71$	12.2263	1.45099	0.725494	−	0.688228i	$-0.241611\pi$
$71$	12.2263	1.45099	0.725494	+	0.688228i	$0.241611\pi$
$72$	−0.292415	−0.0344615
$73$	15.6637	1.83330	0.916651	−	0.399690i	$-0.130882\pi$
$73$	15.6637	1.83330	0.916651	+	0.399690i	$0.130882\pi$
$74$	−1.42952	−0.166178
$75$	0	0
$76$	7.38382	0.846983
$77$	−3.85480	−0.439295
$78$	0.0714627	0.00809156
$79$	12.9096	1.45245	0.726224	−	0.687458i	$-0.241273\pi$
$79$	12.9096	1.45245	0.726224	+	0.687458i	$0.241273\pi$
$80$	0	0
$81$	−8.01609	−0.890677
$82$	1.49263	0.164833
$83$	−8.57565	−0.941300	−0.470650	−	0.882320i	$-0.655981\pi$
$83$	−8.57565	−0.941300	−0.470650	+	0.882320i	$0.655981\pi$
$84$	−4.55876	−0.497401
$85$	0	0
$86$	−2.81621	−0.303679
$87$	10.5752	1.13378
$88$	2.64042	0.281469
$89$	11.4907	1.21801	0.609006	−	0.793165i	$-0.291568\pi$
$89$	11.4907	1.21801	0.609006	+	0.793165i	$0.291568\pi$
$90$	0	0
$91$	−0.249960	−0.0262029
$92$	5.34513	0.557268
$93$	0.809280	0.0839184
$94$	0.660358	0.0681108
$95$	0	0
$96$	4.70848	0.480557
$97$	2.29892	0.233419	0.116710	−	0.993166i	$-0.462765\pi$
$97$	2.29892	0.233419	0.116710	+	0.993166i	$0.462765\pi$
$98$	1.23264	0.124516
$99$	−0.803523	−0.0807571
$100$	0	0
$101$	0.890917	0.0886495	0.0443248	−	0.999017i	$-0.485886\pi$
$101$	0.890917	0.0886495	0.0443248	+	0.999017i	$0.485886\pi$
$102$	0	0
$103$	−0.367137	−0.0361751	−0.0180875	−	0.999836i	$-0.505758\pi$
$103$	−0.367137	−0.0361751	−0.0180875	+	0.999836i	$0.505758\pi$
$104$	0.171215	0.0167890
$105$	0	0
$106$	3.07493	0.298664
$107$	6.64750	0.642638	0.321319	−	0.946971i	$-0.395874\pi$
$107$	6.64750	0.642638	0.321319	+	0.946971i	$0.395874\pi$
$108$	−10.5068	−1.01102
$109$	8.83952	0.846672	0.423336	−	0.905973i	$-0.360859\pi$
$109$	8.83952	0.846672	0.423336	+	0.905973i	$0.360859\pi$
$110$	0	0
$111$	−9.43952	−0.895960
$112$	−5.19782	−0.491148
$113$	5.71314	0.537447	0.268723	−	0.963217i	$-0.413398\pi$
$113$	5.71314	0.537447	0.268723	+	0.963217i	$0.413398\pi$
$114$	−1.55881	−0.145996
$115$	0	0
$116$	12.4690	1.15772
$117$	−0.0521036	−0.00481698
$118$	−2.01439	−0.185440
$119$	0	0
$120$	0	0
$121$	−3.74445	−0.340404
$122$	1.72946	0.156578
$123$	9.85627	0.888710
$124$	0.954208	0.0856904
$125$	0	0
$126$	−0.106264	−0.00946675
$127$	−16.3468	−1.45055	−0.725273	−	0.688461i	$-0.758287\pi$
$127$	−16.3468	−1.45055	−0.725273	+	0.688461i	$0.758287\pi$
$128$	7.35987	0.650527
$129$	−18.5963	−1.63731
$130$	0	0
$131$	−9.63890	−0.842155	−0.421077	−	0.907025i	$-0.638348\pi$
$131$	−9.63890	−0.842155	−0.421077	+	0.907025i	$0.638348\pi$
$132$	8.58057	0.746842
$133$	5.45236	0.472780
$134$	−3.08412	−0.266427
$135$	0	0
$136$	0	0
$137$	6.60524	0.564324	0.282162	−	0.959367i	$-0.408948\pi$
$137$	6.60524	0.564324	0.282162	+	0.959367i	$0.408948\pi$
$138$	−1.12842	−0.0960575
$139$	−4.59708	−0.389919	−0.194960	−	0.980811i	$-0.562458\pi$
$139$	−4.59708	−0.389919	−0.194960	+	0.980811i	$0.562458\pi$
$140$	0	0
$141$	4.36054	0.367224
$142$	−3.04334	−0.255391
$143$	0.470478	0.0393434
$144$	−1.08347	−0.0902894
$145$	0	0
$146$	−3.89899	−0.322683
$147$	8.13950	0.671335
$148$	−11.1300	−0.914879
$149$	−11.6404	−0.953619	−0.476810	−	0.879007i	$-0.658207\pi$
$149$	−11.6404	−0.953619	−0.476810	+	0.879007i	$0.658207\pi$
$150$	0	0
$151$	0.137778	0.0112122	0.00560609	−	0.999984i	$-0.498216\pi$
$151$	0.137778	0.0112122	0.00560609	+	0.999984i	$0.498216\pi$
$152$	−3.73470	−0.302924
$153$	0	0
$154$	0.959530	0.0773211
$155$	0	0
$156$	0.556397	0.0445474
$157$	9.66489	0.771343	0.385671	−	0.922636i	$-0.373970\pi$
$157$	9.66489	0.771343	0.385671	+	0.922636i	$0.373970\pi$
$158$	−3.21345	−0.255648
$159$	20.3047	1.61027
$160$	0	0
$161$	3.94695	0.311063
$162$	1.99535	0.156770
$163$	12.4150	0.972415	0.486207	−	0.873843i	$-0.338380\pi$
$163$	12.4150	0.972415	0.486207	+	0.873843i	$0.338380\pi$
$164$	11.6214	0.907476
$165$	0	0
$166$	2.13464	0.165680
$167$	−4.18747	−0.324036	−0.162018	−	0.986788i	$-0.551800\pi$
$167$	−4.18747	−0.324036	−0.162018	+	0.986788i	$0.551800\pi$
$168$	2.30580	0.177896
$169$	−12.9695	−0.997653
$170$	0	0
$171$	1.13653	0.0869128
$172$	−21.9265	−1.67188
$173$	2.95493	0.224659	0.112329	−	0.993671i	$-0.464169\pi$
$173$	2.95493	0.224659	0.112329	+	0.993671i	$0.464169\pi$
$174$	−2.63235	−0.199558
$175$	0	0
$176$	9.78341	0.737452
$177$	−13.3016	−0.999812
$178$	−2.86025	−0.214385
$179$	−23.2764	−1.73976	−0.869878	−	0.493267i	$-0.835803\pi$
$179$	−23.2764	−1.73976	−0.869878	+	0.493267i	$0.835803\pi$
$180$	0	0
$181$	11.4203	0.848866	0.424433	−	0.905459i	$-0.360474\pi$
$181$	11.4203	0.848866	0.424433	+	0.905459i	$0.360474\pi$
$182$	0.0622196	0.00461203
$183$	11.4201	0.844200
$184$	−2.70354	−0.199308
$185$	0	0
$186$	−0.201445	−0.0147706
$187$	0	0
$188$	5.14145	0.374978
$189$	−7.75845	−0.564344
$190$	0	0
$191$	3.08201	0.223007	0.111503	−	0.993764i	$-0.464433\pi$
$191$	3.08201	0.223007	0.111503	+	0.993764i	$0.464433\pi$
$192$	10.7679	0.777109
$193$	15.9856	1.15067	0.575335	−	0.817918i	$-0.304872\pi$
$193$	15.9856	1.15067	0.575335	+	0.817918i	$0.304872\pi$
$194$	−0.572242	−0.0410846
$195$	0	0
$196$	9.59715	0.685511
$197$	10.6736	0.760460	0.380230	−	0.924892i	$-0.375845\pi$
$197$	10.6736	0.760460	0.380230	+	0.924892i	$0.375845\pi$
$198$	0.200012	0.0142142
$199$	−15.8394	−1.12282	−0.561412	−	0.827536i	$-0.689742\pi$
$199$	−15.8394	−1.12282	−0.561412	+	0.827536i	$0.689742\pi$
$200$	0	0
$201$	−20.3654	−1.43646
$202$	−0.221766	−0.0156034
$203$	9.20737	0.646230
$204$	0	0
$205$	0	0
$206$	0.0913871	0.00636724
$207$	0.822733	0.0571839
$208$	0.634394	0.0439873
$209$	−10.2625	−0.709873
$210$	0	0
$211$	4.91960	0.338679	0.169340	−	0.985558i	$-0.445836\pi$
$211$	4.91960	0.338679	0.169340	+	0.985558i	$0.445836\pi$
$212$	23.9409	1.64427
$213$	−20.0961	−1.37696
$214$	−1.65468	−0.113112
$215$	0	0
$216$	5.31430	0.361592
$217$	0.704606	0.0478318
$218$	−2.20032	−0.149024
$219$	−25.7462	−1.73977
$220$	0	0
$221$	0	0
$222$	2.34967	0.157700
$223$	10.0439	0.672589	0.336294	−	0.941757i	$-0.390826\pi$
$223$	10.0439	0.672589	0.336294	+	0.941757i	$0.390826\pi$
$224$	4.09948	0.273908
$225$	0	0
$226$	−1.42211	−0.0945971
$227$	−4.47736	−0.297173	−0.148586	−	0.988899i	$-0.547472\pi$
$227$	−4.47736	−0.297173	−0.148586	+	0.988899i	$0.547472\pi$
$228$	−12.1367	−0.803770
$229$	1.19796	0.0791636	0.0395818	−	0.999216i	$-0.487397\pi$
$229$	1.19796	0.0791636	0.0395818	+	0.999216i	$0.487397\pi$
$230$	0	0
$231$	6.33606	0.416882
$232$	−6.30676	−0.414059
$233$	25.4544	1.66757	0.833785	−	0.552089i	$-0.186169\pi$
$233$	25.4544	1.66757	0.833785	+	0.552089i	$0.186169\pi$
$234$	0.0129695	0.000847845 0
$235$	0	0
$236$	−15.6837	−1.02092
$237$	−21.2194	−1.37835
$238$	0	0
$239$	−6.21407	−0.401955	−0.200977	−	0.979596i	$-0.564412\pi$
$239$	−6.21407	−0.401955	−0.200977	+	0.979596i	$0.564412\pi$
$240$	0	0
$241$	−26.8044	−1.72662	−0.863312	−	0.504671i	$-0.831614\pi$
$241$	−26.8044	−1.72662	−0.863312	+	0.504671i	$0.831614\pi$
$242$	0.932062	0.0599152
$243$	−3.08820	−0.198108
$244$	13.4653	0.862027
$245$	0	0
$246$	−2.45341	−0.156423
$247$	−0.665461	−0.0423423
$248$	−0.482634	−0.0306473
$249$	14.0957	0.893276
$250$	0	0
$251$	−10.6052	−0.669394	−0.334697	−	0.942326i	$-0.608634\pi$
$251$	−10.6052	−0.669394	−0.334697	+	0.942326i	$0.608634\pi$
$252$	−0.827355	−0.0521185
$253$	−7.42901	−0.467058
$254$	4.06903	0.255313
$255$	0	0
$256$	11.2702	0.704387
$257$	1.54024	0.0960773	0.0480386	−	0.998845i	$-0.484703\pi$
$257$	1.54024	0.0960773	0.0480386	+	0.998845i	$0.484703\pi$
$258$	4.62895	0.288186
$259$	−8.21860	−0.510679
$260$	0	0
$261$	1.91925	0.118799
$262$	2.39930	0.148229
$263$	0.426435	0.0262951	0.0131476	−	0.999914i	$-0.495815\pi$
$263$	0.426435	0.0262951	0.0131476	+	0.999914i	$0.495815\pi$
$264$	−4.34001	−0.267109
$265$	0	0
$266$	−1.35719	−0.0832149
$267$	−18.8871	−1.15587
$268$	−24.0125	−1.46680
$269$	17.1047	1.04289	0.521447	−	0.853284i	$-0.325392\pi$
$269$	17.1047	1.04289	0.521447	+	0.853284i	$0.325392\pi$
$270$	0	0
$271$	−16.0704	−0.976208	−0.488104	−	0.872785i	$-0.662311\pi$
$271$	−16.0704	−0.976208	−0.488104	+	0.872785i	$0.662311\pi$
$272$	0	0
$273$	0.410855	0.0248661
$274$	−1.64417	−0.0993277
$275$	0	0
$276$	−8.78570	−0.528837
$277$	−16.7628	−1.00718	−0.503588	−	0.863944i	$-0.667987\pi$
$277$	−16.7628	−1.00718	−0.503588	+	0.863944i	$0.667987\pi$
$278$	1.14430	0.0686305
$279$	0.146874	0.00879309
$280$	0	0
$281$	−7.83833	−0.467596	−0.233798	−	0.972285i	$-0.575115\pi$
$281$	−7.83833	−0.467596	−0.233798	+	0.972285i	$0.575115\pi$
$282$	−1.08542	−0.0646358
$283$	−6.58054	−0.391173	−0.195586	−	0.980686i	$-0.562661\pi$
$283$	−6.58054	−0.391173	−0.195586	+	0.980686i	$0.562661\pi$
$284$	−23.6950	−1.40604
$285$	0	0
$286$	−0.117111	−0.00692490
$287$	8.58145	0.506547
$288$	0.854527	0.0503535
$289$	0	0
$290$	0	0
$291$	−3.77869	−0.221511
$292$	−30.3569	−1.77651
$293$	3.15115	0.184092	0.0920461	−	0.995755i	$-0.470659\pi$
$293$	3.15115	0.184092	0.0920461	+	0.995755i	$0.470659\pi$
$294$	−2.02607	−0.118163
$295$	0	0
$296$	5.62949	0.327207
$297$	14.6031	0.847356
$298$	2.89751	0.167848
$299$	−0.481726	−0.0278589
$300$	0	0
$301$	−16.1910	−0.933233
$302$	−0.0342954	−0.00197348
$303$	−1.46438	−0.0841267
$304$	−13.8380	−0.793664
$305$	0	0
$306$	0	0
$307$	−14.9905	−0.855551	−0.427775	−	0.903885i	$-0.640703\pi$
$307$	−14.9905	−0.855551	−0.427775	+	0.903885i	$0.640703\pi$
$308$	7.47075	0.425685
$309$	0.603457	0.0343294
$310$	0	0
$311$	−10.0288	−0.568678	−0.284339	−	0.958724i	$-0.591774\pi$
$311$	−10.0288	−0.568678	−0.284339	+	0.958724i	$0.591774\pi$
$312$	−0.281423	−0.0159324
$313$	3.54087	0.200142	0.100071	−	0.994980i	$-0.468093\pi$
$313$	3.54087	0.200142	0.100071	+	0.994980i	$0.468093\pi$
$314$	−2.40577	−0.135765
$315$	0	0
$316$	−25.0194	−1.40745
$317$	−12.2108	−0.685829	−0.342914	−	0.939367i	$-0.611414\pi$
$317$	−12.2108	−0.685829	−0.342914	+	0.939367i	$0.611414\pi$
$318$	−5.05421	−0.283426
$319$	−17.3302	−0.970307
$320$	0	0
$321$	−10.9264	−0.609851
$322$	−0.982469	−0.0547508
$323$	0	0
$324$	15.5355	0.863084
$325$	0	0
$326$	−3.09031	−0.171157
$327$	−14.5294	−0.803475
$328$	−5.87803	−0.324560
$329$	3.79655	0.209310
$330$	0	0
$331$	−18.4933	−1.01649	−0.508243	−	0.861214i	$-0.669705\pi$
$331$	−18.4933	−1.01649	−0.508243	+	0.861214i	$0.669705\pi$
$332$	16.6200	0.912139
$333$	−1.71315	−0.0938800
$334$	1.04234	0.0570343
$335$	0	0
$336$	8.54356	0.466090
$337$	−18.4415	−1.00457	−0.502287	−	0.864701i	$-0.667508\pi$
$337$	−18.4415	−1.00457	−0.502287	+	0.864701i	$0.667508\pi$
$338$	3.22835	0.175599
$339$	−9.39059	−0.510027
$340$	0	0
$341$	−1.32622	−0.0718189
$342$	−0.282904	−0.0152977
$343$	17.1043	0.923547
$344$	11.0903	0.597951
$345$	0	0
$346$	−0.735536	−0.0395427
$347$	5.07535	0.272459	0.136229	−	0.990677i	$-0.456502\pi$
$347$	5.07535	0.272459	0.136229	+	0.990677i	$0.456502\pi$
$348$	−20.4951	−1.09865
$349$	−24.4627	−1.30946	−0.654730	−	0.755863i	$-0.727218\pi$
$349$	−24.4627	−1.30946	−0.654730	+	0.755863i	$0.727218\pi$
$350$	0	0
$351$	0.946920	0.0505429
$352$	−7.71610	−0.411270
$353$	17.5325	0.933161	0.466580	−	0.884479i	$-0.345486\pi$
$353$	17.5325	0.933161	0.466580	+	0.884479i	$0.345486\pi$
$354$	3.31102	0.175979
$355$	0	0
$356$	−22.2695	−1.18028
$357$	0	0
$358$	5.79391	0.306218
$359$	−2.01791	−0.106501	−0.0532505	−	0.998581i	$-0.516958\pi$
$359$	−2.01791	−0.106501	−0.0532505	+	0.998581i	$0.516958\pi$
$360$	0	0
$361$	−4.48433	−0.236017
$362$	−2.84273	−0.149410
$363$	6.15468	0.323037
$364$	0.484432	0.0253911
$365$	0	0
$366$	−2.84268	−0.148589
$367$	−32.4048	−1.69152	−0.845758	−	0.533567i	$-0.820851\pi$
$367$	−32.4048	−1.69152	−0.845758	+	0.533567i	$0.820851\pi$
$368$	−10.0173	−0.522188
$369$	1.78878	0.0931203
$370$	0	0
$371$	17.6785	0.917819
$372$	−1.56842	−0.0813186
$373$	20.6108	1.06719	0.533593	−	0.845741i	$-0.320841\pi$
$373$	20.6108	1.06719	0.533593	+	0.845741i	$0.320841\pi$
$374$	0	0
$375$	0	0
$376$	−2.60052	−0.134111
$377$	−1.12376	−0.0578766
$378$	1.93122	0.0993313
$379$	16.6023	0.852803	0.426402	−	0.904534i	$-0.359781\pi$
$379$	16.6023	0.852803	0.426402	+	0.904534i	$0.359781\pi$
$380$	0	0
$381$	26.8690	1.37654
$382$	−0.767169	−0.0392518
$383$	−23.2319	−1.18709	−0.593546	−	0.804800i	$-0.702273\pi$
$383$	−23.2319	−1.18709	−0.593546	+	0.804800i	$0.702273\pi$
$384$	−12.0973	−0.617338
$385$	0	0
$386$	−3.97912	−0.202532
$387$	−3.37498	−0.171560
$388$	−4.45539	−0.226188
$389$	−36.6648	−1.85898	−0.929490	−	0.368849i	$-0.879752\pi$
$389$	−36.6648	−1.85898	−0.929490	+	0.368849i	$0.879752\pi$
$390$	0	0
$391$	0	0
$392$	−4.85419	−0.245174
$393$	15.8433	0.799189
$394$	−2.65685	−0.133850
$395$	0	0
$396$	1.55726	0.0782553
$397$	−9.81212	−0.492456	−0.246228	−	0.969212i	$-0.579191\pi$
$397$	−9.81212	−0.492456	−0.246228	+	0.969212i	$0.579191\pi$
$398$	3.94272	0.197630
$399$	−8.96195	−0.448659
$400$	0	0
$401$	21.4897	1.07314	0.536572	−	0.843854i	$-0.319719\pi$
$401$	21.4897	1.07314	0.536572	+	0.843854i	$0.319719\pi$
$402$	5.06932	0.252835
$403$	−0.0859973	−0.00428383
$404$	−1.72663	−0.0859032
$405$	0	0
$406$	−2.29188	−0.113744
$407$	15.4692	0.766779
$408$	0	0
$409$	31.2735	1.54638	0.773189	−	0.634176i	$-0.218661\pi$
$409$	31.2735	1.54638	0.773189	+	0.634176i	$0.218661\pi$
$410$	0	0
$411$	−10.8569	−0.535532
$412$	0.711526	0.0350544
$413$	−11.5812	−0.569873
$414$	−0.204793	−0.0100650
$415$	0	0
$416$	−0.500342	−0.0245313
$417$	7.55614	0.370026
$418$	2.55453	0.124946
$419$	20.3462	0.993976	0.496988	−	0.867757i	$-0.334439\pi$
$419$	20.3462	0.993976	0.496988	+	0.867757i	$0.334439\pi$
$420$	0	0
$421$	1.80862	0.0881467	0.0440733	−	0.999028i	$-0.485966\pi$
$421$	1.80862	0.0881467	0.0440733	+	0.999028i	$0.485966\pi$
$422$	−1.22458	−0.0596116
$423$	0.791381	0.0384783
$424$	−12.1092	−0.588075
$425$	0	0
$426$	5.00228	0.242361
$427$	9.94304	0.481177
$428$	−12.8831	−0.622729
$429$	−0.773317	−0.0373361
$430$	0	0
$431$	−16.6638	−0.802665	−0.401333	−	0.915932i	$-0.631453\pi$
$431$	−16.6638	−0.802665	−0.401333	+	0.915932i	$0.631453\pi$
$432$	19.6908	0.947376
$433$	16.3183	0.784209	0.392104	−	0.919921i	$-0.371747\pi$
$433$	16.3183	0.784209	0.392104	+	0.919921i	$0.371747\pi$
$434$	−0.175390	−0.00841897
$435$	0	0
$436$	−17.1313	−0.820442
$437$	10.5079	0.502659
$438$	6.40870	0.306220
$439$	−24.0274	−1.14677	−0.573383	−	0.819288i	$-0.694369\pi$
$439$	−24.0274	−1.14677	−0.573383	+	0.819288i	$0.694369\pi$
$440$	0	0
$441$	1.47721	0.0703434
$442$	0	0
$443$	−21.2757	−1.01084	−0.505419	−	0.862874i	$-0.668662\pi$
$443$	−21.2757	−1.01084	−0.505419	+	0.862874i	$0.668662\pi$
$444$	18.2942	0.868203
$445$	0	0
$446$	−2.50011	−0.118384
$447$	19.1331	0.904966
$448$	9.37520	0.442937
$449$	29.2011	1.37808	0.689042	−	0.724721i	$-0.258031\pi$
$449$	29.2011	1.37808	0.689042	+	0.724721i	$0.258031\pi$
$450$	0	0
$451$	−16.1521	−0.760574
$452$	−11.0723	−0.520797
$453$	−0.226463	−0.0106402
$454$	1.11450	0.0523059
$455$	0	0
$456$	6.13866	0.287469
$457$	1.03811	0.0485608	0.0242804	−	0.999705i	$-0.492271\pi$
$457$	1.03811	0.0485608	0.0242804	+	0.999705i	$0.492271\pi$
$458$	−0.298195	−0.0139337
$459$	0	0
$460$	0	0
$461$	−0.0701975	−0.00326943	−0.00163471	−	0.999999i	$-0.500520\pi$
$461$	−0.0701975	−0.00326943	−0.00163471	+	0.999999i	$0.500520\pi$
$462$	−1.57716	−0.0733762
$463$	−40.0649	−1.86197	−0.930987	−	0.365051i	$-0.881051\pi$
$463$	−40.0649	−1.86197	−0.930987	+	0.365051i	$0.881051\pi$
$464$	−23.3682	−1.08484
$465$	0	0
$466$	−6.33606	−0.293512
$467$	−22.7070	−1.05076	−0.525378	−	0.850869i	$-0.676076\pi$
$467$	−22.7070	−1.05076	−0.525378	+	0.850869i	$0.676076\pi$
$468$	0.100979	0.00466775
$469$	−17.7313	−0.818755
$470$	0	0
$471$	−15.8860	−0.731989
$472$	7.93276	0.365135
$473$	30.4749	1.40124
$474$	5.28189	0.242605
$475$	0	0
$476$	0	0
$477$	3.68503	0.168726
$478$	1.54680	0.0707488
$479$	−23.2606	−1.06280	−0.531402	−	0.847120i	$-0.678335\pi$
$479$	−23.2606	−1.06280	−0.531402	+	0.847120i	$0.678335\pi$
$480$	0	0
$481$	1.00308	0.0457366
$482$	6.67211	0.303906
$483$	−6.48753	−0.295193
$484$	7.25689	0.329859
$485$	0	0
$486$	0.768709	0.0348694
$487$	16.9121	0.766362	0.383181	−	0.923673i	$-0.374829\pi$
$487$	16.9121	0.766362	0.383181	+	0.923673i	$0.374829\pi$
$488$	−6.81068	−0.308305
$489$	−20.4063	−0.922803
$490$	0	0
$491$	−21.0598	−0.950416	−0.475208	−	0.879874i	$-0.657627\pi$
$491$	−21.0598	−0.950416	−0.475208	+	0.879874i	$0.657627\pi$
$492$	−19.1018	−0.861177
$493$	0	0
$494$	0.165646	0.00745275
$495$	0	0
$496$	−1.78828	−0.0802962
$497$	−17.4968	−0.784840
$498$	−3.50867	−0.157227
$499$	22.5259	1.00840	0.504198	−	0.863588i	$-0.331788\pi$
$499$	22.5259	1.00840	0.504198	+	0.863588i	$0.331788\pi$
$500$	0	0
$501$	6.88288	0.307504
$502$	2.63983	0.117821
$503$	−29.0221	−1.29403	−0.647015	−	0.762477i	$-0.723983\pi$
$503$	−29.0221	−1.29403	−0.647015	+	0.762477i	$0.723983\pi$
$504$	0.418472	0.0186402
$505$	0	0
$506$	1.84922	0.0822077
$507$	21.3177	0.946754
$508$	31.6808	1.40561
$509$	−4.38401	−0.194318	−0.0971589	−	0.995269i	$-0.530976\pi$
$509$	−4.38401	−0.194318	−0.0971589	+	0.995269i	$0.530976\pi$
$510$	0	0
$511$	−22.4162	−0.991633
$512$	−17.5251	−0.774508
$513$	−20.6551	−0.911946
$514$	−0.383393	−0.0169107
$515$	0	0
$516$	36.0403	1.58658
$517$	−7.14592	−0.314277
$518$	2.04576	0.0898856
$519$	−4.85696	−0.213197
$520$	0	0
$521$	−3.19433	−0.139946	−0.0699732	−	0.997549i	$-0.522291\pi$
$521$	−3.19433	−0.139946	−0.0699732	+	0.997549i	$0.522291\pi$
$522$	−0.477738	−0.0209100
$523$	−34.0520	−1.48899	−0.744494	−	0.667629i	$-0.767309\pi$
$523$	−34.0520	−1.48899	−0.744494	+	0.667629i	$0.767309\pi$
$524$	18.6806	0.816065
$525$	0	0
$526$	−0.106148	−0.00462825
$527$	0	0
$528$	−16.0808	−0.699828
$529$	−15.3934	−0.669278
$530$	0	0
$531$	−2.41407	−0.104762
$532$	−10.5669	−0.458133
$533$	−1.04737	−0.0453665
$534$	4.70134	0.203447
$535$	0	0
$536$	12.1454	0.524601
$537$	38.2589	1.65100
$538$	−4.25768	−0.183562
$539$	−13.3387	−0.574540
$540$	0	0
$541$	−27.0686	−1.16377	−0.581885	−	0.813271i	$-0.697685\pi$
$541$	−27.0686	−1.16377	−0.581885	+	0.813271i	$0.697685\pi$
$542$	4.00022	0.171824
$543$	−18.7714	−0.805557
$544$	0	0
$545$	0	0
$546$	−0.102269	−0.00437672
$547$	13.5320	0.578588	0.289294	−	0.957240i	$-0.406579\pi$
$547$	13.5320	0.578588	0.289294	+	0.957240i	$0.406579\pi$
$548$	−12.8012	−0.546841
$549$	2.07260	0.0884566
$550$	0	0
$551$	24.5125	1.04427
$552$	4.44376	0.189139
$553$	−18.4748	−0.785629
$554$	4.17256	0.177275
$555$	0	0
$556$	8.90933	0.377840
$557$	−30.5182	−1.29310	−0.646548	−	0.762873i	$-0.723788\pi$
$557$	−30.5182	−1.29310	−0.646548	+	0.762873i	$0.723788\pi$
$558$	−0.0365595	−0.00154769
$559$	1.97611	0.0835807
$560$	0	0
$561$	0	0
$562$	1.95111	0.0823024
$563$	8.10847	0.341731	0.170866	−	0.985294i	$-0.445344\pi$
$563$	8.10847	0.341731	0.170866	+	0.985294i	$0.445344\pi$
$564$	−8.45091	−0.355847
$565$	0	0
$566$	1.63802	0.0688510
$567$	11.4717	0.481767
$568$	11.9848	0.502871
$569$	3.83428	0.160741	0.0803706	−	0.996765i	$-0.474390\pi$
$569$	3.83428	0.160741	0.0803706	+	0.996765i	$0.474390\pi$
$570$	0	0
$571$	6.56720	0.274829	0.137414	−	0.990514i	$-0.456121\pi$
$571$	6.56720	0.274829	0.137414	+	0.990514i	$0.456121\pi$
$572$	−0.911805	−0.0381245
$573$	−5.06585	−0.211629
$574$	−2.13608	−0.0891583
$575$	0	0
$576$	1.95424	0.0814266
$577$	23.2880	0.969493	0.484746	−	0.874655i	$-0.338912\pi$
$577$	23.2880	0.969493	0.484746	+	0.874655i	$0.338912\pi$
$578$	0	0
$579$	−26.2753	−1.09196
$580$	0	0
$581$	12.2725	0.509149
$582$	0.940585	0.0389885
$583$	−33.2746	−1.37809
$584$	15.3544	0.635369
$585$	0	0
$586$	−0.784379	−0.0324024
$587$	−36.7815	−1.51814	−0.759068	−	0.651011i	$-0.774345\pi$
$587$	−36.7815	−1.51814	−0.759068	+	0.651011i	$0.774345\pi$
$588$	−15.7747	−0.650537
$589$	1.87585	0.0772932
$590$	0	0
$591$	−17.5440	−0.721662
$592$	20.8587	0.857287
$593$	27.0504	1.11083	0.555414	−	0.831574i	$-0.312560\pi$
$593$	27.0504	1.11083	0.555414	+	0.831574i	$0.312560\pi$
$594$	−3.63497	−0.149145
$595$	0	0
$596$	22.5596	0.924076
$597$	26.0349	1.06554
$598$	0.119910	0.00490350
$599$	12.2973	0.502456	0.251228	−	0.967928i	$-0.419166\pi$
$599$	12.2973	0.502456	0.251228	+	0.967928i	$0.419166\pi$
$600$	0	0
$601$	8.07150	0.329243	0.164622	−	0.986357i	$-0.447360\pi$
$601$	8.07150	0.329243	0.164622	+	0.986357i	$0.447360\pi$
$602$	4.03024	0.164260
$603$	−3.69605	−0.150515
$604$	−0.267019	−0.0108648
$605$	0	0
$606$	0.364512	0.0148073
$607$	24.1032	0.978317	0.489158	−	0.872195i	$-0.337304\pi$
$607$	24.1032	0.978317	0.489158	+	0.872195i	$0.337304\pi$
$608$	10.9139	0.442619
$609$	−15.1340	−0.613260
$610$	0	0
$611$	−0.463369	−0.0187459
$612$	0	0
$613$	39.8359	1.60896	0.804478	−	0.593983i	$-0.202445\pi$
$613$	39.8359	1.60896	0.804478	+	0.593983i	$0.202445\pi$
$614$	3.73140	0.150587
$615$	0	0
$616$	−3.77867	−0.152247
$617$	−25.7956	−1.03849	−0.519246	−	0.854625i	$-0.673787\pi$
$617$	−25.7956	−1.03849	−0.519246	+	0.854625i	$0.673787\pi$
$618$	−0.150211	−0.00604239
$619$	−3.76840	−0.151465	−0.0757323	−	0.997128i	$-0.524129\pi$
$619$	−3.76840	−0.151465	−0.0757323	+	0.997128i	$0.524129\pi$
$620$	0	0
$621$	−14.9522	−0.600011
$622$	2.49634	0.100094
$623$	−16.4442	−0.658823
$624$	−1.04274	−0.0417431
$625$	0	0
$626$	−0.881389	−0.0352274
$627$	16.8683	0.673656
$628$	−18.7309	−0.747446
$629$	0	0
$630$	0	0
$631$	−19.3142	−0.768887	−0.384444	−	0.923148i	$-0.625607\pi$
$631$	−19.3142	−0.768887	−0.384444	+	0.923148i	$0.625607\pi$
$632$	12.6547	0.503376
$633$	−8.08627	−0.321400
$634$	3.03950	0.120714
$635$	0	0
$636$	−39.3513	−1.56038
$637$	−0.864936	−0.0342700
$638$	4.31382	0.170786
$639$	−3.64717	−0.144280
$640$	0	0
$641$	20.6563	0.815875	0.407938	−	0.913010i	$-0.366248\pi$
$641$	20.6563	0.815875	0.407938	+	0.913010i	$0.366248\pi$
$642$	2.71978	0.107341
$643$	−29.0196	−1.14442	−0.572211	−	0.820107i	$-0.693914\pi$
$643$	−29.0196	−1.14442	−0.572211	+	0.820107i	$0.693914\pi$
$644$	−7.64935	−0.301426
$645$	0	0
$646$	0	0
$647$	41.2848	1.62307	0.811536	−	0.584302i	$-0.198632\pi$
$647$	41.2848	1.62307	0.811536	+	0.584302i	$0.198632\pi$
$648$	−7.85778	−0.308683
$649$	21.7983	0.855657
$650$	0	0
$651$	−1.15815	−0.0453915
$652$	−24.0607	−0.942289
$653$	24.5583	0.961039	0.480519	−	0.876984i	$-0.340448\pi$
$653$	24.5583	0.961039	0.480519	+	0.876984i	$0.340448\pi$
$654$	3.61663	0.141421
$655$	0	0
$656$	−21.7796	−0.850350
$657$	−4.67260	−0.182295
$658$	−0.945030	−0.0368411
$659$	17.7024	0.689588	0.344794	−	0.938678i	$-0.387949\pi$
$659$	17.7024	0.689588	0.344794	+	0.938678i	$0.387949\pi$
$660$	0	0
$661$	14.1853	0.551743	0.275871	−	0.961195i	$-0.411034\pi$
$661$	14.1853	0.551743	0.275871	+	0.961195i	$0.411034\pi$
$662$	4.60333	0.178914
$663$	0	0
$664$	−8.40629	−0.326227
$665$	0	0
$666$	0.426434	0.0165240
$667$	17.7445	0.687072
$668$	8.11549	0.313998
$669$	−16.5090	−0.638274
$670$	0	0
$671$	−18.7149	−0.722482
$672$	−6.73825	−0.259933
$673$	−0.325055	−0.0125300	−0.00626498	−	0.999980i	$-0.501994\pi$
$673$	−0.325055	−0.0125300	−0.00626498	+	0.999980i	$0.501994\pi$
$674$	4.59044	0.176817
$675$	0	0
$676$	25.1354	0.966746
$677$	17.2971	0.664783	0.332392	−	0.943141i	$-0.392144\pi$
$677$	17.2971	0.664783	0.332392	+	0.943141i	$0.392144\pi$
$678$	2.33749	0.0897708
$679$	−3.28995	−0.126257
$680$	0	0
$681$	7.35936	0.282011
$682$	0.330121	0.0126410
$683$	−34.4818	−1.31941	−0.659705	−	0.751525i	$-0.729319\pi$
$683$	−34.4818	−1.31941	−0.659705	+	0.751525i	$0.729319\pi$
$684$	−2.20264	−0.0842202
$685$	0	0
$686$	−4.25758	−0.162555
$687$	−1.96907	−0.0751248
$688$	41.0925	1.56664
$689$	−2.15766	−0.0822002
$690$	0	0
$691$	42.1292	1.60267	0.801335	−	0.598216i	$-0.204124\pi$
$691$	42.1292	1.60267	0.801335	+	0.598216i	$0.204124\pi$
$692$	−5.72677	−0.217699
$693$	1.14991	0.0436815
$694$	−1.26335	−0.0479560
$695$	0	0
$696$	10.3663	0.392934
$697$	0	0
$698$	6.08922	0.230481
$699$	−41.8389	−1.58249
$700$	0	0
$701$	29.6266	1.11898	0.559491	−	0.828836i	$-0.310997\pi$
$701$	29.6266	1.11898	0.559491	+	0.828836i	$0.310997\pi$
$702$	−0.235706	−0.00889614
$703$	−21.8802	−0.825226
$704$	−17.6461	−0.665064
$705$	0	0
$706$	−4.36416	−0.164247
$707$	−1.27498	−0.0479505
$708$	25.7791	0.968838
$709$	−16.2739	−0.611181	−0.305590	−	0.952163i	$-0.598854\pi$
$709$	−16.2739	−0.611181	−0.305590	+	0.952163i	$0.598854\pi$
$710$	0	0
$711$	−3.85103	−0.144425
$712$	11.2638	0.422128
$713$	1.35793	0.0508547
$714$	0	0
$715$	0	0
$716$	45.1105	1.68586
$717$	10.2140	0.381447
$718$	0.502294	0.0187454
$719$	−31.4210	−1.17181	−0.585903	−	0.810381i	$-0.699260\pi$
$719$	−31.4210	−1.17181	−0.585903	+	0.810381i	$0.699260\pi$
$720$	0	0
$721$	0.525405	0.0195671
$722$	1.11623	0.0415418
$723$	44.0579	1.63853
$724$	−22.1330	−0.822568
$725$	0	0
$726$	−1.53201	−0.0568584
$727$	21.6544	0.803118	0.401559	−	0.915833i	$-0.368468\pi$
$727$	21.6544	0.803118	0.401559	+	0.915833i	$0.368468\pi$
$728$	−0.245023	−0.00908117
$729$	29.1243	1.07868
$730$	0	0
$731$	0	0
$732$	−22.1327	−0.818047
$733$	−31.4552	−1.16182	−0.580912	−	0.813966i	$-0.697304\pi$
$733$	−31.4552	−1.16182	−0.580912	+	0.813966i	$0.697304\pi$
$734$	8.06615	0.297727
$735$	0	0
$736$	7.90057	0.291219
$737$	33.3741	1.22935
$738$	−0.445261	−0.0163903
$739$	13.7032	0.504079	0.252040	−	0.967717i	$-0.418899\pi$
$739$	13.7032	0.504079	0.252040	+	0.967717i	$0.418899\pi$
$740$	0	0
$741$	1.09381	0.0401820
$742$	−4.40049	−0.161547
$743$	−16.2215	−0.595109	−0.297554	−	0.954705i	$-0.596171\pi$
$743$	−16.2215	−0.595109	−0.297554	+	0.954705i	$0.596171\pi$
$744$	0.793297	0.0290837
$745$	0	0
$746$	−5.13041	−0.187838
$747$	2.55818	0.0935988
$748$	0	0
$749$	−9.51315	−0.347603
$750$	0	0
$751$	11.1467	0.406751	0.203375	−	0.979101i	$-0.434809\pi$
$751$	11.1467	0.406751	0.203375	+	0.979101i	$0.434809\pi$
$752$	−9.63558	−0.351373
$753$	17.4316	0.635242
$754$	0.279725	0.0101870
$755$	0	0
$756$	15.0362	0.546861
$757$	−19.8933	−0.723034	−0.361517	−	0.932365i	$-0.617741\pi$
$757$	−19.8933	−0.723034	−0.361517	+	0.932365i	$0.617741\pi$
$758$	−4.13262	−0.150104
$759$	12.2109	0.443229
$760$	0	0
$761$	−45.4255	−1.64667	−0.823337	−	0.567553i	$-0.807890\pi$
$761$	−45.4255	−1.64667	−0.823337	+	0.567553i	$0.807890\pi$
$762$	−6.68819	−0.242288
$763$	−12.6501	−0.457965
$764$	−5.97306	−0.216098
$765$	0	0
$766$	5.78284	0.208942
$767$	1.41349	0.0510380
$768$	−18.5246	−0.668450
$769$	−5.98158	−0.215701	−0.107851	−	0.994167i	$-0.534397\pi$
$769$	−5.98158	−0.215701	−0.107851	+	0.994167i	$0.534397\pi$
$770$	0	0
$771$	−2.53166	−0.0911755
$772$	−30.9808	−1.11502
$773$	−13.1506	−0.472995	−0.236497	−	0.971632i	$-0.575999\pi$
$773$	−13.1506	−0.472995	−0.236497	+	0.971632i	$0.575999\pi$
$774$	0.840094	0.0301965
$775$	0	0
$776$	2.25351	0.0808964
$777$	13.5088	0.484625
$778$	9.12654	0.327202
$779$	22.8462	0.818548
$780$	0	0
$781$	32.9328	1.17843
$782$	0	0
$783$	−34.8802	−1.24652
$784$	−17.9860	−0.642357
$785$	0	0
$786$	−3.94369	−0.140667
$787$	−25.6968	−0.915993	−0.457997	−	0.888954i	$-0.651433\pi$
$787$	−25.6968	−0.915993	−0.457997	+	0.888954i	$0.651433\pi$
$788$	−20.6858	−0.736901
$789$	−0.700924	−0.0249536
$790$	0	0
$791$	−8.17600	−0.290705
$792$	−0.787654	−0.0279881
$793$	−1.21355	−0.0430944
$794$	2.44242	0.0866781
$795$	0	0
$796$	30.6974	1.08804
$797$	−50.4438	−1.78681	−0.893405	−	0.449251i	$-0.851691\pi$
$797$	−50.4438	−1.78681	−0.893405	+	0.449251i	$0.851691\pi$
$798$	2.23080	0.0789693
$799$	0	0
$800$	0	0
$801$	−3.42776	−0.121114
$802$	−5.34918	−0.188886
$803$	42.1920	1.48892
$804$	39.4689	1.39196
$805$	0	0
$806$	0.0214063	0.000754005 0
$807$	−28.1148	−0.989686
$808$	0.873322	0.0307234
$809$	9.20169	0.323514	0.161757	−	0.986831i	$-0.448284\pi$
$809$	9.20169	0.323514	0.161757	+	0.986831i	$0.448284\pi$
$810$	0	0
$811$	10.7139	0.376214	0.188107	−	0.982148i	$-0.439765\pi$
$811$	10.7139	0.376214	0.188107	+	0.982148i	$0.439765\pi$
$812$	−17.8442	−0.626210
$813$	26.4147	0.926403
$814$	−3.85056	−0.134962
$815$	0	0
$816$	0	0
$817$	−43.1048	−1.50805
$818$	−7.78456	−0.272181
$819$	0.0745647	0.00260550
$820$	0	0
$821$	−28.1783	−0.983431	−0.491715	−	0.870756i	$-0.663630\pi$
$821$	−28.1783	−0.983431	−0.491715	+	0.870756i	$0.663630\pi$
$822$	2.70249	0.0942600
$823$	−35.3425	−1.23196	−0.615981	−	0.787761i	$-0.711240\pi$
$823$	−35.3425	−1.23196	−0.615981	+	0.787761i	$0.711240\pi$
$824$	−0.359886	−0.0125372
$825$	0	0
$826$	2.88277	0.100304
$827$	−39.2434	−1.36463	−0.682314	−	0.731059i	$-0.739026\pi$
$827$	−39.2434	−1.36463	−0.682314	+	0.731059i	$0.739026\pi$
$828$	−1.59449	−0.0554123
$829$	−17.4130	−0.604779	−0.302389	−	0.953184i	$-0.597784\pi$
$829$	−17.4130	−0.604779	−0.302389	+	0.953184i	$0.597784\pi$
$830$	0	0
$831$	27.5526	0.955791
$832$	−1.14424	−0.0396695
$833$	0	0
$834$	−1.88086	−0.0651290
$835$	0	0
$836$	19.8892	0.687881
$837$	−2.66925	−0.0922629
$838$	−5.06454	−0.174952
$839$	−26.9046	−0.928851	−0.464426	−	0.885612i	$-0.653739\pi$
$839$	−26.9046	−0.928851	−0.464426	+	0.885612i	$0.653739\pi$
$840$	0	0
$841$	12.3941	0.427384
$842$	−0.450198	−0.0155149
$843$	12.8837	0.443739
$844$	−9.53439	−0.328187
$845$	0	0
$846$	−0.196989	−0.00677263
$847$	5.35863	0.184125
$848$	−44.8676	−1.54076
$849$	10.8163	0.371215
$850$	0	0
$851$	−15.8390	−0.542954
$852$	38.9470	1.33430
$853$	16.3461	0.559681	0.279841	−	0.960046i	$-0.409718\pi$
$853$	16.3461	0.559681	0.279841	+	0.960046i	$0.409718\pi$
$854$	−2.47501	−0.0846930
$855$	0	0
$856$	6.51622	0.222720
$857$	3.86719	0.132101	0.0660503	−	0.997816i	$-0.478960\pi$
$857$	3.86719	0.132101	0.0660503	+	0.997816i	$0.478960\pi$
$858$	0.192493	0.00657160
$859$	39.0324	1.33177	0.665883	−	0.746056i	$-0.268055\pi$
$859$	39.0324	1.33177	0.665883	+	0.746056i	$0.268055\pi$
$860$	0	0
$861$	−14.1052	−0.480703
$862$	4.14792	0.141279
$863$	−22.1279	−0.753243	−0.376621	−	0.926367i	$-0.622914\pi$
$863$	−22.1279	−0.753243	−0.376621	+	0.926367i	$0.622914\pi$
$864$	−15.5300	−0.528342
$865$	0	0
$866$	−4.06193	−0.138030
$867$	0	0
$868$	−1.36556	−0.0463500
$869$	34.7736	1.17961
$870$	0	0
$871$	2.16411	0.0733280
$872$	8.66494	0.293432
$873$	−0.685782	−0.0232102
$874$	−2.61560	−0.0884740
$875$	0	0
$876$	49.8972	1.68587
$877$	−19.7374	−0.666484	−0.333242	−	0.942841i	$-0.608143\pi$
$877$	−19.7374	−0.666484	−0.333242	+	0.942841i	$0.608143\pi$
$878$	5.98086	0.201844
$879$	−5.17949	−0.174700
$880$	0	0
$881$	11.4321	0.385159	0.192579	−	0.981281i	$-0.438315\pi$
$881$	11.4321	0.385159	0.192579	+	0.981281i	$0.438315\pi$
$882$	−0.367705	−0.0123813
$883$	−13.2658	−0.446431	−0.223216	−	0.974769i	$-0.571655\pi$
$883$	−13.2658	−0.446431	−0.223216	+	0.974769i	$0.571655\pi$
$884$	0	0
$885$	0	0
$886$	5.29591	0.177919
$887$	30.6070	1.02768	0.513842	−	0.857885i	$-0.328222\pi$
$887$	30.6070	1.02768	0.513842	+	0.857885i	$0.328222\pi$
$888$	−9.25310	−0.310514
$889$	23.3937	0.784601
$890$	0	0
$891$	−21.5923	−0.723368
$892$	−19.4655	−0.651752
$893$	10.1074	0.338233
$894$	−4.76259	−0.159285
$895$	0	0
$896$	−10.5326	−0.351870
$897$	0.791804	0.0264376
$898$	−7.26869	−0.242559
$899$	3.16774	0.105650
$900$	0	0
$901$	0	0
$902$	4.02056	0.133870
$903$	26.6129	0.885620
$904$	5.60031	0.186263
$905$	0	0
$906$	0.0563708	0.00187279
$907$	−0.766017	−0.0254352	−0.0127176	−	0.999919i	$-0.504048\pi$
$907$	−0.766017	−0.0254352	−0.0127176	+	0.999919i	$0.504048\pi$
$908$	8.67730	0.287966
$909$	−0.265767	−0.00881492
$910$	0	0
$911$	8.43534	0.279475	0.139738	−	0.990189i	$-0.455374\pi$
$911$	8.43534	0.279475	0.139738	+	0.990189i	$0.455374\pi$
$912$	22.7453	0.753172
$913$	−23.0995	−0.764482
$914$	−0.258405	−0.00854727
$915$	0	0
$916$	−2.32170	−0.0767111
$917$	13.7941	0.455522
$918$	0	0
$919$	13.0143	0.429302	0.214651	−	0.976691i	$-0.431139\pi$
$919$	13.0143	0.429302	0.214651	+	0.976691i	$0.431139\pi$
$920$	0	0
$921$	24.6396	0.811901
$922$	0.0174735	0.000575458 0
$923$	2.13549	0.0702905
$924$	−12.2795	−0.403967
$925$	0	0
$926$	9.97290	0.327730
$927$	0.109519	0.00359709
$928$	18.4303	0.605004
$929$	−25.8490	−0.848078	−0.424039	−	0.905644i	$-0.639388\pi$
$929$	−25.8490	−0.848078	−0.424039	+	0.905644i	$0.639388\pi$
$930$	0	0
$931$	18.8668	0.618335
$932$	−49.3316	−1.61591
$933$	16.4841	0.539665
$934$	5.65220	0.184946
$935$	0	0
$936$	−0.0510746	−0.00166942
$937$	−0.870730	−0.0284455	−0.0142228	−	0.999899i	$-0.504527\pi$
$937$	−0.870730	−0.0284455	−0.0142228	+	0.999899i	$0.504527\pi$
$938$	4.41365	0.144111
$939$	−5.82007	−0.189931
$940$	0	0
$941$	−53.3440	−1.73896	−0.869482	−	0.493965i	$-0.835547\pi$
$941$	−53.3440	−1.73896	−0.869482	+	0.493965i	$0.835547\pi$
$942$	3.95432	0.128839
$943$	16.5383	0.538560
$944$	29.3929	0.956657
$945$	0	0
$946$	−7.58577	−0.246635
$947$	−13.6405	−0.443255	−0.221628	−	0.975131i	$-0.571137\pi$
$947$	−13.6405	−0.443255	−0.221628	+	0.975131i	$0.571137\pi$
$948$	41.1240	1.33564
$949$	2.73590	0.0888109
$950$	0	0
$951$	20.0707	0.650838
$952$	0	0
$953$	−10.3660	−0.335788	−0.167894	−	0.985805i	$-0.553697\pi$
$953$	−10.3660	−0.335788	−0.167894	+	0.985805i	$0.553697\pi$
$954$	−0.917273	−0.0296978
$955$	0	0
$956$	12.0431	0.389502
$957$	28.4854	0.920803
$958$	5.78999	0.187066
$959$	−9.45267	−0.305243
$960$	0	0
$961$	−30.7576	−0.992180
$962$	−0.249685	−0.00805018
$963$	−1.98299	−0.0639011
$964$	51.9480	1.67313
$965$	0	0
$966$	1.61487	0.0519575
$967$	−14.3299	−0.460817	−0.230408	−	0.973094i	$-0.574006\pi$
$967$	−14.3299	−0.460817	−0.230408	+	0.973094i	$0.574006\pi$
$968$	−3.67050	−0.117974
$969$	0	0
$970$	0	0
$971$	−29.7422	−0.954471	−0.477236	−	0.878775i	$-0.658361\pi$
$971$	−29.7422	−0.954471	−0.477236	+	0.878775i	$0.658361\pi$
$972$	5.98505	0.191971
$973$	6.57882	0.210907
$974$	−4.20974	−0.134889
$975$	0	0
$976$	−25.2353	−0.807762
$977$	37.1142	1.18739	0.593695	−	0.804690i	$-0.297669\pi$
$977$	37.1142	1.18739	0.593695	+	0.804690i	$0.297669\pi$
$978$	5.07949	0.162424
$979$	30.9515	0.989215
$980$	0	0
$981$	−2.63689	−0.0841894
$982$	5.24217	0.167284
$983$	−46.4624	−1.48192	−0.740961	−	0.671549i	$-0.765629\pi$
$983$	−46.4624	−1.48192	−0.740961	+	0.671549i	$0.765629\pi$
$984$	9.66161	0.308001
$985$	0	0
$986$	0	0
$987$	−6.24032	−0.198631
$988$	1.28969	0.0410305
$989$	−31.2035	−0.992213
$990$	0	0
$991$	44.3013	1.40728	0.703638	−	0.710559i	$-0.251558\pi$
$991$	44.3013	1.40728	0.703638	+	0.710559i	$0.251558\pi$
$992$	1.41040	0.0447804
$993$	30.3972	0.964625
$994$	4.35528	0.138141
$995$	0	0
$996$	−27.3179	−0.865602
$997$	−24.6310	−0.780072	−0.390036	−	0.920800i	$-0.627538\pi$
$997$	−24.6310	−0.780072	−0.390036	+	0.920800i	$0.627538\pi$
$998$	−5.60711	−0.177490
$999$	31.1344	0.985050

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.by.1.11		24
5.2	odd	4		1445.2.b.i.579.12		24
5.3	odd	4		1445.2.b.i.579.13		24
5.4	even	2	inner	7225.2.a.by.1.14		24
17.5	odd	16		425.2.m.e.76.4		24
17.7	odd	16		425.2.m.e.151.4		24
17.16	even	2	inner	7225.2.a.by.1.12		24
85.7	even	16		85.2.m.a.49.3	✓	24
85.22	even	16		85.2.m.a.59.4	yes	24
85.24	odd	16		425.2.m.e.151.3		24
85.33	odd	4		1445.2.b.i.579.14		24
85.39	odd	16		425.2.m.e.76.3		24
85.58	even	16		85.2.m.a.49.4	yes	24
85.67	odd	4		1445.2.b.i.579.11		24
85.73	even	16		85.2.m.a.59.3	yes	24
85.84	even	2	inner	7225.2.a.by.1.13		24
255.92	odd	16		765.2.bh.b.559.4		24
255.107	odd	16		765.2.bh.b.739.3		24
255.143	odd	16		765.2.bh.b.559.3		24
255.158	odd	16		765.2.bh.b.739.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.m.a.49.3	✓	24	85.7	even	16
85.2.m.a.49.4	yes	24	85.58	even	16
85.2.m.a.59.3	yes	24	85.73	even	16
85.2.m.a.59.4	yes	24	85.22	even	16
425.2.m.e.76.3		24	85.39	odd	16
425.2.m.e.76.4		24	17.5	odd	16
425.2.m.e.151.3		24	85.24	odd	16
425.2.m.e.151.4		24	17.7	odd	16
765.2.bh.b.559.3		24	255.143	odd	16
765.2.bh.b.559.4		24	255.92	odd	16
765.2.bh.b.739.3		24	255.107	odd	16
765.2.bh.b.739.4		24	255.158	odd	16
1445.2.b.i.579.11		24	85.67	odd	4
1445.2.b.i.579.12		24	5.2	odd	4
1445.2.b.i.579.13		24	5.3	odd	4
1445.2.b.i.579.14		24	85.33	odd	4
7225.2.a.by.1.11		24	1.1	even	1	trivial
7225.2.a.by.1.12		24	17.16	even	2	inner
7225.2.a.by.1.13		24	85.84	even	2	inner
7225.2.a.by.1.14		24	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7225.a (trivial)

\(f(q)\)	\(=\)	\(q-0.248918 q^{2} -1.64368 q^{3} -1.93804 q^{4} +0.409143 q^{6} -1.43109 q^{7} +0.980251 q^{8} -0.298307 q^{9} +2.69361 q^{11} +3.18552 q^{12} +0.174664 q^{13} +0.356224 q^{14} +3.63208 q^{16} +0.0742541 q^{18} -3.80994 q^{19} +2.35225 q^{21} -0.670490 q^{22} -2.75801 q^{23} -1.61122 q^{24} -0.0434772 q^{26} +5.42137 q^{27} +2.77350 q^{28} -6.43383 q^{29} -0.492358 q^{31} -2.86459 q^{32} -4.42745 q^{33} +0.578130 q^{36} +5.74291 q^{37} +0.948365 q^{38} -0.287093 q^{39} -5.99645 q^{41} -0.585519 q^{42} +11.3138 q^{43} -5.22033 q^{44} +0.686519 q^{46} -2.65291 q^{47} -5.96998 q^{48} -4.95199 q^{49} -0.338506 q^{52} -12.3532 q^{53} -1.34948 q^{54} -1.40282 q^{56} +6.26234 q^{57} +1.60150 q^{58} +8.09258 q^{59} -6.94789 q^{61} +0.122557 q^{62} +0.426903 q^{63} -6.55110 q^{64} +1.10207 q^{66} +12.3901 q^{67} +4.53329 q^{69} +12.2263 q^{71} -0.292415 q^{72} +15.6637 q^{73} -1.42952 q^{74} +7.38382 q^{76} -3.85480 q^{77} +0.0714627 q^{78} +12.9096 q^{79} -8.01609 q^{81} +1.49263 q^{82} -8.57565 q^{83} -4.55876 q^{84} -2.81621 q^{86} +10.5752 q^{87} +2.64042 q^{88} +11.4907 q^{89} -0.249960 q^{91} +5.34513 q^{92} +0.809280 q^{93} +0.660358 q^{94} +4.70848 q^{96} +2.29892 q^{97} +1.23264 q^{98} -0.803523 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 8 q^{4} - 8 q^{9} - 8 q^{16} - 16 q^{19} - 32 q^{21} - 48 q^{26} + 8 q^{36} - 56 q^{49} - 64 q^{59} - 104 q^{64} - 16 q^{66} - 32 q^{69} - 48 q^{76} - 88 q^{81} - 96 q^{84} - 112 q^{89} - 128 q^{94}+O(q^{100}) \) 24 * q + 8 * q^4 - 8 * q^9 - 8 * q^16 - 16 * q^19 - 32 * q^21 - 48 * q^26 + 8 * q^36 - 56 * q^49 - 64 * q^59 - 104 * q^64 - 16 * q^66 - 32 * q^69 - 48 * q^76 - 88 * q^81 - 96 * q^84 - 112 * q^89 - 128 * q^94

Introduction

L-functions

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Database

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