LMFDB - Embedded newform 5625.2.a.bb.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, 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("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 15x^{6} + 70x^{4} - 105x^{2} + 45 $ x^8 - 15x^6 + 70x^4 - 105*x^2 + 45
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.23762$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.23762 q^{2} -0.468306 q^{4} -2.08634 q^{7} +3.05482 q^{8} +O(q^{10})$	$q-1.23762 q^{2} -0.468306 q^{4} -2.08634 q^{7} +3.05482 q^{8} +2.65287 q^{11} +0.149728 q^{13} +2.58209 q^{14} -2.84408 q^{16} +5.12810 q^{17} -3.08634 q^{19} -3.28323 q^{22} -2.76265 q^{23} -0.185305 q^{26} +0.977047 q^{28} -6.94530 q^{29} +1.52550 q^{31} -2.58976 q^{32} -6.34662 q^{34} +8.52550 q^{37} +3.81970 q^{38} +9.82722 q^{41} +3.43296 q^{43} -1.24236 q^{44} +3.41911 q^{46} +4.00501 q^{47} -2.64718 q^{49} -0.0701184 q^{52} +0.945455 q^{53} -6.37339 q^{56} +8.59562 q^{58} -8.18766 q^{59} -4.47450 q^{61} -1.88798 q^{62} +8.89328 q^{64} -3.34042 q^{67} -2.40152 q^{68} +12.4801 q^{71} +11.6018 q^{73} -10.5513 q^{74} +1.44535 q^{76} -5.53479 q^{77} +11.3696 q^{79} -12.1623 q^{82} -3.45676 q^{83} -4.24869 q^{86} +8.10403 q^{88} -14.8875 q^{89} -0.312383 q^{91} +1.29377 q^{92} -4.95666 q^{94} -0.0571908 q^{97} +3.27620 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 14 q^{4} + 10 q^{7}+O(q^{10}) $ 8 * q + 14 * q^4 + 10 * q^7	$ 8 q + 14 q^{4} + 10 q^{7} + 10 q^{13} + 22 q^{16} + 2 q^{19} + 10 q^{22} + 70 q^{28} - 6 q^{31} - 50 q^{34} + 50 q^{37} + 14 q^{49} + 80 q^{52} - 30 q^{58} - 54 q^{61} + 36 q^{64} + 10 q^{67} + 30 q^{73} + 56 q^{76} + 28 q^{79} + 20 q^{88} + 60 q^{91} - 40 q^{94}+O(q^{100}) $ 8 * q + 14 * q^4 + 10 * q^7 + 10 * q^13 + 22 * q^16 + 2 * q^19 + 10 * q^22 + 70 * q^28 - 6 * q^31 - 50 * q^34 + 50 * q^37 + 14 * q^49 + 80 * q^52 - 30 * q^58 - 54 * q^61 + 36 * q^64 + 10 * q^67 + 30 * q^73 + 56 * q^76 + 28 * q^79 + 20 * q^88 + 60 * q^91 - 40 * q^94

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 $

$2$	−1.23762	−0.875127	−0.437563	−	0.899188i	$-0.644158\pi$
$2$	−1.23762	−0.875127	−0.437563	+	0.899188i	$0.644158\pi$
$3$	0	0
$3$	0	0
$4$	−0.468306	−0.234153
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.08634	−0.788563	−0.394281	−	0.918990i	$-0.629006\pi$
$7$	−2.08634	−0.788563	−0.394281	+	0.918990i	$0.629006\pi$
$8$	3.05482	1.08004
$9$	0	0
$10$	0	0
$11$	2.65287	0.799870	0.399935	−	0.916543i	$-0.369033\pi$
$11$	2.65287	0.799870	0.399935	+	0.916543i	$0.369033\pi$
$12$	0	0
$13$	0.149728	0.0415269	0.0207635	−	0.999784i	$-0.493390\pi$
$13$	0.149728	0.0415269	0.0207635	+	0.999784i	$0.493390\pi$
$14$	2.58209	0.690092
$15$	0	0
$16$	−2.84408	−0.711019
$17$	5.12810	1.24375	0.621874	−	0.783118i	$-0.286372\pi$
$17$	5.12810	1.24375	0.621874	+	0.783118i	$0.286372\pi$
$18$	0	0
$19$	−3.08634	−0.708055	−0.354028	−	0.935235i	$-0.615188\pi$
$19$	−3.08634	−0.708055	−0.354028	+	0.935235i	$0.615188\pi$
$20$	0	0
$21$	0	0
$22$	−3.28323	−0.699988
$23$	−2.76265	−0.576053	−0.288027	−	0.957622i	$-0.592999\pi$
$23$	−2.76265	−0.576053	−0.288027	+	0.957622i	$0.592999\pi$
$24$	0	0
$25$	0	0
$26$	−0.185305	−0.0363413
$27$	0	0
$28$	0.977047	0.184644
$29$	−6.94530	−1.28971	−0.644855	−	0.764305i	$-0.723082\pi$
$29$	−6.94530	−1.28971	−0.644855	+	0.764305i	$0.723082\pi$
$30$	0	0
$31$	1.52550	0.273987	0.136994	−	0.990572i	$-0.456256\pi$
$31$	1.52550	0.273987	0.136994	+	0.990572i	$0.456256\pi$
$32$	−2.58976	−0.457809
$33$	0	0
$34$	−6.34662	−1.08844
$35$	0	0
$36$	0	0
$37$	8.52550	1.40158	0.700792	−	0.713366i	$-0.252830\pi$
$37$	8.52550	1.40158	0.700792	+	0.713366i	$0.252830\pi$
$38$	3.81970	0.619638
$39$	0	0
$40$	0	0
$41$	9.82722	1.53475	0.767377	−	0.641196i	$-0.221562\pi$
$41$	9.82722	1.53475	0.767377	+	0.641196i	$0.221562\pi$
$42$	0	0
$43$	3.43296	0.523522	0.261761	−	0.965133i	$-0.415697\pi$
$43$	3.43296	0.523522	0.261761	+	0.965133i	$0.415697\pi$
$44$	−1.24236	−0.187292
$45$	0	0
$46$	3.41911	0.504120
$47$	4.00501	0.584191	0.292095	−	0.956389i	$-0.405648\pi$
$47$	4.00501	0.584191	0.292095	+	0.956389i	$0.405648\pi$
$48$	0	0
$49$	−2.64718	−0.378169
$50$	0	0
$51$	0	0
$52$	−0.0701184	−0.00972367
$53$	0.945455	0.129868	0.0649341	−	0.997890i	$-0.479316\pi$
$53$	0.945455	0.129868	0.0649341	+	0.997890i	$0.479316\pi$
$54$	0	0
$55$	0	0
$56$	−6.37339	−0.851679
$57$	0	0
$58$	8.59562	1.12866
$59$	−8.18766	−1.06594	−0.532971	−	0.846133i	$-0.678925\pi$
$59$	−8.18766	−1.06594	−0.532971	+	0.846133i	$0.678925\pi$
$60$	0	0
$61$	−4.47450	−0.572901	−0.286451	−	0.958095i	$-0.592475\pi$
$61$	−4.47450	−0.572901	−0.286451	+	0.958095i	$0.592475\pi$
$62$	−1.88798	−0.239774
$63$	0	0
$64$	8.89328	1.11166
$65$	0	0
$66$	0	0
$67$	−3.34042	−0.408098	−0.204049	−	0.978961i	$-0.565410\pi$
$67$	−3.34042	−0.408098	−0.204049	+	0.978961i	$0.565410\pi$
$68$	−2.40152	−0.291227
$69$	0	0
$70$	0	0
$71$	12.4801	1.48111	0.740557	−	0.671994i	$-0.234562\pi$
$71$	12.4801	1.48111	0.740557	+	0.671994i	$0.234562\pi$
$72$	0	0
$73$	11.6018	1.35789	0.678945	−	0.734189i	$-0.262438\pi$
$73$	11.6018	1.35789	0.678945	+	0.734189i	$0.262438\pi$
$74$	−10.5513	−1.22656
$75$	0	0
$76$	1.44535	0.165793
$77$	−5.53479	−0.630748
$78$	0	0
$79$	11.3696	1.27918	0.639588	−	0.768717i	$-0.279105\pi$
$79$	11.3696	1.27918	0.639588	+	0.768717i	$0.279105\pi$
$80$	0	0
$81$	0	0
$82$	−12.1623	−1.34310
$83$	−3.45676	−0.379429	−0.189715	−	0.981839i	$-0.560756\pi$
$83$	−3.45676	−0.379429	−0.189715	+	0.981839i	$0.560756\pi$
$84$	0	0
$85$	0	0
$86$	−4.24869	−0.458148
$87$	0	0
$88$	8.10403	0.863892
$89$	−14.8875	−1.57807	−0.789034	−	0.614349i	$-0.789419\pi$
$89$	−14.8875	−1.57807	−0.789034	+	0.614349i	$0.789419\pi$
$90$	0	0
$91$	−0.312383	−0.0327466
$92$	1.29377	0.134885
$93$	0	0
$94$	−4.95666	−0.511241
$95$	0	0
$96$	0	0
$97$	−0.0571908	−0.00580685	−0.00290342	−	0.999996i	$-0.500924\pi$
$97$	−0.0571908	−0.00580685	−0.00290342	+	0.999996i	$0.500924\pi$
$98$	3.27620	0.330946
$99$	0	0
$100$	0	0
$101$	−16.2980	−1.62171	−0.810855	−	0.585247i	$-0.800997\pi$
$101$	−16.2980	−1.62171	−0.810855	+	0.585247i	$0.800997\pi$
$102$	0	0
$103$	5.37814	0.529924	0.264962	−	0.964259i	$-0.414641\pi$
$103$	5.37814	0.529924	0.264962	+	0.964259i	$0.414641\pi$
$104$	0.457390	0.0448508
$105$	0	0
$106$	−1.17011	−0.113651
$107$	−20.2611	−1.95871	−0.979355	−	0.202147i	$-0.935208\pi$
$107$	−20.2611	−1.95871	−0.979355	+	0.202147i	$0.935208\pi$
$108$	0	0
$109$	−16.0778	−1.53997	−0.769986	−	0.638061i	$-0.779737\pi$
$109$	−16.0778	−1.53997	−0.769986	+	0.638061i	$0.779737\pi$
$110$	0	0
$111$	0	0
$112$	5.93371	0.560683
$113$	−8.40723	−0.790885	−0.395443	−	0.918491i	$-0.629409\pi$
$113$	−8.40723	−0.790885	−0.395443	+	0.918491i	$0.629409\pi$
$114$	0	0
$115$	0	0
$116$	3.25253	0.301990
$117$	0	0
$118$	10.1332	0.932834
$119$	−10.6990	−0.980772
$120$	0	0
$121$	−3.96229	−0.360208
$122$	5.53772	0.501361
$123$	0	0
$124$	−0.714400	−0.0641550
$125$	0	0
$126$	0	0
$127$	21.8709	1.94072	0.970362	−	0.241654i	$-0.0776899\pi$
$127$	21.8709	1.94072	0.970362	+	0.241654i	$0.0776899\pi$
$128$	−5.82695	−0.515034
$129$	0	0
$130$	0	0
$131$	−14.1197	−1.23364	−0.616820	−	0.787104i	$-0.711579\pi$
$131$	−14.1197	−1.23364	−0.616820	+	0.787104i	$0.711579\pi$
$132$	0	0
$133$	6.43916	0.558346
$134$	4.13416	0.357137
$135$	0	0
$136$	15.6654	1.34330
$137$	9.59817	0.820027	0.410013	−	0.912079i	$-0.365524\pi$
$137$	9.59817	0.820027	0.410013	+	0.912079i	$0.365524\pi$
$138$	0	0
$139$	17.7945	1.50931	0.754657	−	0.656120i	$-0.227803\pi$
$139$	17.7945	1.50931	0.754657	+	0.656120i	$0.227803\pi$
$140$	0	0
$141$	0	0
$142$	−15.4456	−1.29616
$143$	0.397208	0.0332162
$144$	0	0
$145$	0	0
$146$	−14.3586	−1.18833
$147$	0	0
$148$	−3.99255	−0.328185
$149$	1.01331	0.0830132	0.0415066	−	0.999138i	$-0.486784\pi$
$149$	1.01331	0.0830132	0.0415066	+	0.999138i	$0.486784\pi$
$150$	0	0
$151$	−13.2809	−1.08078	−0.540391	−	0.841414i	$-0.681724\pi$
$151$	−13.2809	−1.08078	−0.540391	+	0.841414i	$0.681724\pi$
$152$	−9.42820	−0.764728
$153$	0	0
$154$	6.84994	0.551984
$155$	0	0
$156$	0	0
$157$	−21.5579	−1.72051	−0.860254	−	0.509866i	$-0.829695\pi$
$157$	−21.5579	−1.72051	−0.860254	+	0.509866i	$0.829695\pi$
$158$	−14.0712	−1.11944
$159$	0	0
$160$	0	0
$161$	5.76384	0.454254
$162$	0	0
$163$	4.99144	0.390960	0.195480	−	0.980708i	$-0.437374\pi$
$163$	4.99144	0.390960	0.195480	+	0.980708i	$0.437374\pi$
$164$	−4.60215	−0.359368
$165$	0	0
$166$	4.27815	0.332048
$167$	22.9559	1.77638	0.888189	−	0.459478i	$-0.151964\pi$
$167$	22.9559	1.77638	0.888189	+	0.459478i	$0.151964\pi$
$168$	0	0
$169$	−12.9776	−0.998276
$170$	0	0
$171$	0	0
$172$	−1.60768	−0.122584
$173$	6.17748	0.469665	0.234833	−	0.972036i	$-0.424546\pi$
$173$	6.17748	0.469665	0.234833	+	0.972036i	$0.424546\pi$
$174$	0	0
$175$	0	0
$176$	−7.54496	−0.568723
$177$	0	0
$178$	18.4250	1.38101
$179$	14.6584	1.09562	0.547811	−	0.836602i	$-0.315461\pi$
$179$	14.6584	1.09562	0.547811	+	0.836602i	$0.315461\pi$
$180$	0	0
$181$	−2.87559	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$181$	−2.87559	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$182$	0.386610	0.0286574
$183$	0	0
$184$	−8.43940	−0.622161
$185$	0	0
$186$	0	0
$187$	13.6042	0.994836
$188$	−1.87557	−0.136790
$189$	0	0
$190$	0	0
$191$	12.6216	0.913270	0.456635	−	0.889654i	$-0.349054\pi$
$191$	12.6216	0.913270	0.456635	+	0.889654i	$0.349054\pi$
$192$	0	0
$193$	7.02059	0.505353	0.252676	−	0.967551i	$-0.418689\pi$
$193$	7.02059	0.505353	0.252676	+	0.967551i	$0.418689\pi$
$194$	0.0707803	0.00508173
$195$	0	0
$196$	1.23969	0.0885495
$197$	11.3056	0.805489	0.402745	−	0.915312i	$-0.368056\pi$
$197$	11.3056	0.805489	0.402745	+	0.915312i	$0.368056\pi$
$198$	0	0
$199$	13.4827	0.955763	0.477882	−	0.878424i	$-0.341405\pi$
$199$	13.4827	0.955763	0.477882	+	0.878424i	$0.341405\pi$
$200$	0	0
$201$	0	0
$202$	20.1706	1.41920
$203$	14.4903	1.01702
$204$	0	0
$205$	0	0
$206$	−6.65607	−0.463750
$207$	0	0
$208$	−0.425837	−0.0295265
$209$	−8.18766	−0.566352
$210$	0	0
$211$	−10.1297	−0.697356	−0.348678	−	0.937243i	$-0.613369\pi$
$211$	−10.1297	−0.697356	−0.348678	+	0.937243i	$0.613369\pi$
$212$	−0.442762	−0.0304091
$213$	0	0
$214$	25.0754	1.71412
$215$	0	0
$216$	0	0
$217$	−3.18271	−0.216056
$218$	19.8981	1.34767
$219$	0	0
$220$	0	0
$221$	0.767818	0.0516490
$222$	0	0
$223$	22.6330	1.51562	0.757809	−	0.652477i	$-0.226270\pi$
$223$	22.6330	1.51562	0.757809	+	0.652477i	$0.226270\pi$
$224$	5.40311	0.361011
$225$	0	0
$226$	10.4049	0.692125
$227$	−5.12810	−0.340364	−0.170182	−	0.985413i	$-0.554436\pi$
$227$	−5.12810	−0.340364	−0.170182	+	0.985413i	$0.554436\pi$
$228$	0	0
$229$	20.8709	1.37919	0.689593	−	0.724198i	$-0.257790\pi$
$229$	20.8709	1.37919	0.689593	+	0.724198i	$0.257790\pi$
$230$	0	0
$231$	0	0
$232$	−21.2166	−1.39294
$233$	17.3988	1.13983	0.569916	−	0.821703i	$-0.306976\pi$
$233$	17.3988	1.13983	0.569916	+	0.821703i	$0.306976\pi$
$234$	0	0
$235$	0	0
$236$	3.83433	0.249594
$237$	0	0
$238$	13.2412	0.858300
$239$	−4.66304	−0.301627	−0.150814	−	0.988562i	$-0.548189\pi$
$239$	−4.66304	−0.301627	−0.150814	+	0.988562i	$0.548189\pi$
$240$	0	0
$241$	13.3242	0.858287	0.429144	−	0.903236i	$-0.358815\pi$
$241$	13.3242	0.858287	0.429144	+	0.903236i	$0.358815\pi$
$242$	4.90379	0.315228
$243$	0	0
$244$	2.09544	0.134147
$245$	0	0
$246$	0	0
$247$	−0.462110	−0.0294034
$248$	4.66011	0.295917
$249$	0	0
$250$	0	0
$251$	13.1063	0.827265	0.413633	−	0.910444i	$-0.364260\pi$
$251$	13.1063	0.827265	0.413633	+	0.910444i	$0.364260\pi$
$252$	0	0
$253$	−7.32896	−0.460768
$254$	−27.0677	−1.69838
$255$	0	0
$256$	−10.5750	−0.660939
$257$	5.19595	0.324115	0.162057	−	0.986781i	$-0.448187\pi$
$257$	5.19595	0.324115	0.162057	+	0.986781i	$0.448187\pi$
$258$	0	0
$259$	−17.7871	−1.10524
$260$	0	0
$261$	0	0
$262$	17.4747	1.07959
$263$	0.145861	0.00899420	0.00449710	−	0.999990i	$-0.498569\pi$
$263$	0.145861	0.00899420	0.00449710	+	0.999990i	$0.498569\pi$
$264$	0	0
$265$	0	0
$266$	−7.96920	−0.488623
$267$	0	0
$268$	1.56434	0.0955574
$269$	−13.8640	−0.845303	−0.422652	−	0.906292i	$-0.638901\pi$
$269$	−13.8640	−0.845303	−0.422652	+	0.906292i	$0.638901\pi$
$270$	0	0
$271$	−3.09196	−0.187823	−0.0939117	−	0.995581i	$-0.529937\pi$
$271$	−3.09196	−0.187823	−0.0939117	+	0.995581i	$0.529937\pi$
$272$	−14.5847	−0.884328
$273$	0	0
$274$	−11.8788	−0.717627
$275$	0	0
$276$	0	0
$277$	11.3301	0.680758	0.340379	−	0.940288i	$-0.389445\pi$
$277$	11.3301	0.680758	0.340379	+	0.940288i	$0.389445\pi$
$278$	−22.0228	−1.32084
$279$	0	0
$280$	0	0
$281$	3.42069	0.204061	0.102031	−	0.994781i	$-0.467466\pi$
$281$	3.42069	0.204061	0.102031	+	0.994781i	$0.467466\pi$
$282$	0	0
$283$	27.4747	1.63320	0.816601	−	0.577203i	$-0.195856\pi$
$283$	27.4747	1.63320	0.816601	+	0.577203i	$0.195856\pi$
$284$	−5.84451	−0.346808
$285$	0	0
$286$	−0.491590	−0.0290684
$287$	−20.5029	−1.21025
$288$	0	0
$289$	9.29742	0.546907
$290$	0	0
$291$	0	0
$292$	−5.43320	−0.317954
$293$	−23.5916	−1.37824	−0.689118	−	0.724649i	$-0.742002\pi$
$293$	−23.5916	−1.37824	−0.689118	+	0.724649i	$0.742002\pi$
$294$	0	0
$295$	0	0
$296$	26.0438	1.51377
$297$	0	0
$298$	−1.25408	−0.0726471
$299$	−0.413645	−0.0239217
$300$	0	0
$301$	−7.16232	−0.412830
$302$	16.4366	0.945821
$303$	0	0
$304$	8.77779	0.503441
$305$	0	0
$306$	0	0
$307$	4.34390	0.247919	0.123960	−	0.992287i	$-0.460441\pi$
$307$	4.34390	0.247919	0.123960	+	0.992287i	$0.460441\pi$
$308$	2.59198	0.147692
$309$	0	0
$310$	0	0
$311$	−6.69981	−0.379912	−0.189956	−	0.981793i	$-0.560834\pi$
$311$	−6.69981	−0.379912	−0.189956	+	0.981793i	$0.560834\pi$
$312$	0	0
$313$	16.0033	0.904558	0.452279	−	0.891877i	$-0.350611\pi$
$313$	16.0033	0.904558	0.452279	+	0.891877i	$0.350611\pi$
$314$	26.6804	1.50566
$315$	0	0
$316$	−5.32444	−0.299523
$317$	−21.0650	−1.18313	−0.591563	−	0.806259i	$-0.701489\pi$
$317$	−21.0650	−1.18313	−0.591563	+	0.806259i	$0.701489\pi$
$318$	0	0
$319$	−18.4250	−1.03160
$320$	0	0
$321$	0	0
$322$	−7.13342	−0.397530
$323$	−15.8271	−0.880641
$324$	0	0
$325$	0	0
$326$	−6.17748	−0.342139
$327$	0	0
$328$	30.0203	1.65760
$329$	−8.35581	−0.460671
$330$	0	0
$331$	30.4432	1.67331	0.836655	−	0.547731i	$-0.184508\pi$
$331$	30.4432	1.67331	0.836655	+	0.547731i	$0.184508\pi$
$332$	1.61882	0.0888445
$333$	0	0
$334$	−28.4105	−1.55456
$335$	0	0
$336$	0	0
$337$	28.9925	1.57932	0.789662	−	0.613542i	$-0.210256\pi$
$337$	28.9925	1.57932	0.789662	+	0.613542i	$0.210256\pi$
$338$	16.0613	0.873618
$339$	0	0
$340$	0	0
$341$	4.04694	0.219154
$342$	0	0
$343$	20.1273	1.08677
$344$	10.4871	0.565424
$345$	0	0
$346$	−7.64535	−0.411017
$347$	−5.55708	−0.298320	−0.149160	−	0.988813i	$-0.547657\pi$
$347$	−5.55708	−0.298320	−0.149160	+	0.988813i	$0.547657\pi$
$348$	0	0
$349$	−13.9703	−0.747812	−0.373906	−	0.927467i	$-0.621982\pi$
$349$	−13.9703	−0.747812	−0.373906	+	0.927467i	$0.621982\pi$
$350$	0	0
$351$	0	0
$352$	−6.87029	−0.366187
$353$	19.7642	1.05194	0.525972	−	0.850502i	$-0.323702\pi$
$353$	19.7642	1.05194	0.525972	+	0.850502i	$0.323702\pi$
$354$	0	0
$355$	0	0
$356$	6.97190	0.369510
$357$	0	0
$358$	−18.1415	−0.958808
$359$	11.6083	0.612665	0.306332	−	0.951925i	$-0.400898\pi$
$359$	11.6083	0.612665	0.306332	+	0.951925i	$0.400898\pi$
$360$	0	0
$361$	−9.47450	−0.498658
$362$	3.55888	0.187051
$363$	0	0
$364$	0.146291	0.00766772
$365$	0	0
$366$	0	0
$367$	−0.0182198	−0.000951065 0	−0.000475533	−	1.00000i	$-0.500151\pi$
$367$	−0.0182198	−0.000951065 0	−0.000475533		1.00000i	$0.500151\pi$
$368$	7.85720	0.409585
$369$	0	0
$370$	0	0
$371$	−1.97254	−0.102409
$372$	0	0
$373$	16.5665	0.857779	0.428890	−	0.903357i	$-0.358905\pi$
$373$	16.5665	0.857779	0.428890	+	0.903357i	$0.358905\pi$
$374$	−16.8368	−0.870608
$375$	0	0
$376$	12.2346	0.630950
$377$	−1.03990	−0.0535577
$378$	0	0
$379$	−2.52370	−0.129634	−0.0648170	−	0.997897i	$-0.520646\pi$
$379$	−2.52370	−0.129634	−0.0648170	+	0.997897i	$0.520646\pi$
$380$	0	0
$381$	0	0
$382$	−15.6208	−0.799227
$383$	15.7910	0.806882	0.403441	−	0.915006i	$-0.367814\pi$
$383$	15.7910	0.806882	0.403441	+	0.915006i	$0.367814\pi$
$384$	0	0
$385$	0	0
$386$	−8.68879	−0.442248
$387$	0	0
$388$	0.0267828	0.00135969
$389$	−4.36976	−0.221556	−0.110778	−	0.993845i	$-0.535334\pi$
$389$	−4.36976	−0.221556	−0.110778	+	0.993845i	$0.535334\pi$
$390$	0	0
$391$	−14.1672	−0.716465
$392$	−8.08666	−0.408438
$393$	0	0
$394$	−13.9920	−0.704905
$395$	0	0
$396$	0	0
$397$	11.6729	0.585844	0.292922	−	0.956136i	$-0.405372\pi$
$397$	11.6729	0.585844	0.292922	+	0.956136i	$0.405372\pi$
$398$	−16.6864	−0.836414
$399$	0	0
$400$	0	0
$401$	19.8999	0.993755	0.496877	−	0.867821i	$-0.334480\pi$
$401$	19.8999	0.993755	0.496877	+	0.867821i	$0.334480\pi$
$402$	0	0
$403$	0.228409	0.0113779
$404$	7.63245	0.379729
$405$	0	0
$406$	−17.9334	−0.890019
$407$	22.6170	1.12108
$408$	0	0
$409$	−10.2084	−0.504772	−0.252386	−	0.967627i	$-0.581215\pi$
$409$	−10.2084	−0.504772	−0.252386	+	0.967627i	$0.581215\pi$
$410$	0	0
$411$	0	0
$412$	−2.51862	−0.124083
$413$	17.0822	0.840562
$414$	0	0
$415$	0	0
$416$	−0.387758	−0.0190114
$417$	0	0
$418$	10.1332	0.495630
$419$	37.8375	1.84848	0.924241	−	0.381810i	$-0.124699\pi$
$419$	37.8375	1.84848	0.924241	+	0.381810i	$0.124699\pi$
$420$	0	0
$421$	10.1730	0.495802	0.247901	−	0.968785i	$-0.420259\pi$
$421$	10.1730	0.495802	0.247901	+	0.968785i	$0.420259\pi$
$422$	12.5366	0.610275
$423$	0	0
$424$	2.88819	0.140263
$425$	0	0
$426$	0	0
$427$	9.33534	0.451769
$428$	9.48838	0.458638
$429$	0	0
$430$	0	0
$431$	37.9626	1.82859	0.914297	−	0.405045i	$-0.132744\pi$
$431$	37.9626	1.82859	0.914297	+	0.405045i	$0.132744\pi$
$432$	0	0
$433$	16.4745	0.791714	0.395857	−	0.918312i	$-0.370448\pi$
$433$	16.4745	0.791714	0.395857	+	0.918312i	$0.370448\pi$
$434$	3.93897	0.189077
$435$	0	0
$436$	7.52933	0.360589
$437$	8.52649	0.407877
$438$	0	0
$439$	16.4231	0.783834	0.391917	−	0.920001i	$-0.371812\pi$
$439$	16.4231	0.783834	0.391917	+	0.920001i	$0.371812\pi$
$440$	0	0
$441$	0	0
$442$	−0.950264	−0.0451994
$443$	−34.6321	−1.64542	−0.822709	−	0.568462i	$-0.807539\pi$
$443$	−34.6321	−1.64542	−0.822709	+	0.568462i	$0.807539\pi$
$444$	0	0
$445$	0	0
$446$	−28.0110	−1.32636
$447$	0	0
$448$	−18.5544	−0.876613
$449$	14.7835	0.697678	0.348839	−	0.937183i	$-0.386576\pi$
$449$	14.7835	0.697678	0.348839	+	0.937183i	$0.386576\pi$
$450$	0	0
$451$	26.0703	1.22760
$452$	3.93716	0.185188
$453$	0	0
$454$	6.34662	0.297862
$455$	0	0
$456$	0	0
$457$	−23.8075	−1.11367	−0.556833	−	0.830624i	$-0.687984\pi$
$457$	−23.8075	−1.11367	−0.556833	+	0.830624i	$0.687984\pi$
$458$	−25.8301	−1.20696
$459$	0	0
$460$	0	0
$461$	23.1017	1.07595	0.537977	−	0.842959i	$-0.319189\pi$
$461$	23.1017	1.07595	0.537977	+	0.842959i	$0.319189\pi$
$462$	0	0
$463$	−0.884510	−0.0411067	−0.0205533	−	0.999789i	$-0.506543\pi$
$463$	−0.884510	−0.0411067	−0.0205533	+	0.999789i	$0.506543\pi$
$464$	19.7530	0.917008
$465$	0	0
$466$	−21.5330	−0.997497
$467$	−19.3835	−0.896959	−0.448480	−	0.893793i	$-0.648034\pi$
$467$	−19.3835	−0.896959	−0.448480	+	0.893793i	$0.648034\pi$
$468$	0	0
$469$	6.96926	0.321811
$470$	0	0
$471$	0	0
$472$	−25.0118	−1.15126
$473$	9.10719	0.418749
$474$	0	0
$475$	0	0
$476$	5.01039	0.229651
$477$	0	0
$478$	5.77106	0.263962
$479$	32.8149	1.49935	0.749674	−	0.661807i	$-0.230210\pi$
$479$	32.8149	1.49935	0.749674	+	0.661807i	$0.230210\pi$
$480$	0	0
$481$	1.27650	0.0582035
$482$	−16.4902	−0.751110
$483$	0	0
$484$	1.85556	0.0843439
$485$	0	0
$486$	0	0
$487$	18.5956	0.842648	0.421324	−	0.906910i	$-0.361566\pi$
$487$	18.5956	0.842648	0.421324	+	0.906910i	$0.361566\pi$
$488$	−13.6688	−0.618757
$489$	0	0
$490$	0	0
$491$	−27.3676	−1.23508	−0.617540	−	0.786539i	$-0.711871\pi$
$491$	−27.3676	−1.23508	−0.617540	+	0.786539i	$0.711871\pi$
$492$	0	0
$493$	−35.6162	−1.60407
$494$	0.571915	0.0257317
$495$	0	0
$496$	−4.33863	−0.194810
$497$	−26.0377	−1.16795
$498$	0	0
$499$	−0.852639	−0.0381694	−0.0190847	−	0.999818i	$-0.506075\pi$
$499$	−0.852639	−0.0381694	−0.0190847	+	0.999818i	$0.506075\pi$
$500$	0	0
$501$	0	0
$502$	−16.2206	−0.723962
$503$	−30.4176	−1.35626	−0.678128	−	0.734944i	$-0.737208\pi$
$503$	−30.4176	−1.35626	−0.678128	+	0.734944i	$0.737208\pi$
$504$	0	0
$505$	0	0
$506$	9.07044	0.403230
$507$	0	0
$508$	−10.2423	−0.454427
$509$	4.30887	0.190987	0.0954936	−	0.995430i	$-0.469557\pi$
$509$	4.30887	0.190987	0.0954936	+	0.995430i	$0.469557\pi$
$510$	0	0
$511$	−24.2053	−1.07078
$512$	24.7417	1.09344
$513$	0	0
$514$	−6.43059	−0.283641
$515$	0	0
$516$	0	0
$517$	10.6248	0.467277
$518$	22.0136	0.967222
$519$	0	0
$520$	0	0
$521$	−9.99538	−0.437905	−0.218953	−	0.975735i	$-0.570264\pi$
$521$	−9.99538	−0.437905	−0.218953	+	0.975735i	$0.570264\pi$
$522$	0	0
$523$	13.1913	0.576814	0.288407	−	0.957508i	$-0.406874\pi$
$523$	13.1913	0.576814	0.288407	+	0.957508i	$0.406874\pi$
$524$	6.61232	0.288861
$525$	0	0
$526$	−0.180521	−0.00787107
$527$	7.82290	0.340771
$528$	0	0
$529$	−15.3677	−0.668163
$530$	0	0
$531$	0	0
$532$	−3.01550	−0.130738
$533$	1.47141	0.0637336
$534$	0	0
$535$	0	0
$536$	−10.2044	−0.440762
$537$	0	0
$538$	17.1583	0.739747
$539$	−7.02263	−0.302486
$540$	0	0
$541$	35.6710	1.53362	0.766809	−	0.641876i	$-0.221844\pi$
$541$	35.6710	1.53362	0.766809	+	0.641876i	$0.221844\pi$
$542$	3.82666	0.164369
$543$	0	0
$544$	−13.2805	−0.569398
$545$	0	0
$546$	0	0
$547$	−16.6018	−0.709842	−0.354921	−	0.934896i	$-0.615492\pi$
$547$	−16.6018	−0.709842	−0.354921	+	0.934896i	$0.615492\pi$
$548$	−4.49488	−0.192012
$549$	0	0
$550$	0	0
$551$	21.4356	0.913186
$552$	0	0
$553$	−23.7208	−1.00871
$554$	−14.0223	−0.595749
$555$	0	0
$556$	−8.33330	−0.353411
$557$	14.8875	0.630802	0.315401	−	0.948958i	$-0.397861\pi$
$557$	14.8875	0.630802	0.315401	+	0.948958i	$0.397861\pi$
$558$	0	0
$559$	0.514009	0.0217403
$560$	0	0
$561$	0	0
$562$	−4.23350	−0.178579
$563$	−46.4541	−1.95781	−0.978904	−	0.204322i	$-0.934501\pi$
$563$	−46.4541	−1.95781	−0.978904	+	0.204322i	$0.934501\pi$
$564$	0	0
$565$	0	0
$566$	−34.0031	−1.42926
$567$	0	0
$568$	38.1244	1.59966
$569$	9.57157	0.401261	0.200631	−	0.979667i	$-0.435701\pi$
$569$	9.57157	0.401261	0.200631	+	0.979667i	$0.435701\pi$
$570$	0	0
$571$	0.921854	0.0385784	0.0192892	−	0.999814i	$-0.493860\pi$
$571$	0.921854	0.0385784	0.0192892	+	0.999814i	$0.493860\pi$
$572$	−0.186015	−0.00777767
$573$	0	0
$574$	25.3747	1.05912
$575$	0	0
$576$	0	0
$577$	20.5519	0.855589	0.427794	−	0.903876i	$-0.359291\pi$
$577$	20.5519	0.855589	0.427794	+	0.903876i	$0.359291\pi$
$578$	−11.5066	−0.478613
$579$	0	0
$580$	0	0
$581$	7.21198	0.299204
$582$	0	0
$583$	2.50817	0.103878
$584$	35.4414	1.46658
$585$	0	0
$586$	29.1973	1.20613
$587$	0.487355	0.0201153	0.0100576	−	0.999949i	$-0.496798\pi$
$587$	0.487355	0.0201153	0.0100576	+	0.999949i	$0.496798\pi$
$588$	0	0
$589$	−4.70820	−0.193998
$590$	0	0
$591$	0	0
$592$	−24.2472	−0.996552
$593$	34.3654	1.41122	0.705608	−	0.708602i	$-0.250674\pi$
$593$	34.3654	1.41122	0.705608	+	0.708602i	$0.250674\pi$
$594$	0	0
$595$	0	0
$596$	−0.474538	−0.0194378
$597$	0	0
$598$	0.511934	0.0209345
$599$	39.2856	1.60517	0.802583	−	0.596540i	$-0.203458\pi$
$599$	39.2856	1.60517	0.802583	+	0.596540i	$0.203458\pi$
$600$	0	0
$601$	−48.6333	−1.98379	−0.991897	−	0.127044i	$-0.959451\pi$
$601$	−48.6333	−1.98379	−0.991897	+	0.127044i	$0.959451\pi$
$602$	8.86421	0.361278
$603$	0	0
$604$	6.21952	0.253068
$605$	0	0
$606$	0	0
$607$	37.4916	1.52174	0.760869	−	0.648906i	$-0.224773\pi$
$607$	37.4916	1.52174	0.760869	+	0.648906i	$0.224773\pi$
$608$	7.99287	0.324154
$609$	0	0
$610$	0	0
$611$	0.599660	0.0242597
$612$	0	0
$613$	−37.7346	−1.52409	−0.762043	−	0.647526i	$-0.775804\pi$
$613$	−37.7346	−1.52409	−0.762043	+	0.647526i	$0.775804\pi$
$614$	−5.37608	−0.216961
$615$	0	0
$616$	−16.9078	−0.681233
$617$	−41.3260	−1.66372	−0.831861	−	0.554984i	$-0.812724\pi$
$617$	−41.3260	−1.66372	−0.831861	+	0.554984i	$0.812724\pi$
$618$	0	0
$619$	11.7256	0.471294	0.235647	−	0.971839i	$-0.424279\pi$
$619$	11.7256	0.471294	0.235647	+	0.971839i	$0.424279\pi$
$620$	0	0
$621$	0	0
$622$	8.29180	0.332471
$623$	31.0603	1.24441
$624$	0	0
$625$	0	0
$626$	−19.8059	−0.791602
$627$	0	0
$628$	10.0957	0.402862
$629$	43.7196	1.74322
$630$	0	0
$631$	24.4523	0.973430	0.486715	−	0.873561i	$-0.338195\pi$
$631$	24.4523	0.973430	0.486715	+	0.873561i	$0.338195\pi$
$632$	34.7320	1.38156
$633$	0	0
$634$	26.0703	1.03538
$635$	0	0
$636$	0	0
$637$	−0.396356	−0.0157042
$638$	22.8030	0.902781
$639$	0	0
$640$	0	0
$641$	−20.8359	−0.822969	−0.411484	−	0.911417i	$-0.634990\pi$
$641$	−20.8359	−0.822969	−0.411484	+	0.911417i	$0.634990\pi$
$642$	0	0
$643$	−20.6793	−0.815510	−0.407755	−	0.913091i	$-0.633688\pi$
$643$	−20.6793	−0.815510	−0.407755	+	0.913091i	$0.633688\pi$
$644$	−2.69924	−0.106365
$645$	0	0
$646$	19.5878	0.770673
$647$	−15.5302	−0.610554	−0.305277	−	0.952264i	$-0.598749\pi$
$647$	−15.5302	−0.610554	−0.305277	+	0.952264i	$0.598749\pi$
$648$	0	0
$649$	−21.7208	−0.852615
$650$	0	0
$651$	0	0
$652$	−2.33752	−0.0915444
$653$	28.4323	1.11264	0.556320	−	0.830968i	$-0.312213\pi$
$653$	28.4323	1.11264	0.556320	+	0.830968i	$0.312213\pi$
$654$	0	0
$655$	0	0
$656$	−27.9494	−1.09124
$657$	0	0
$658$	10.3413	0.403145
$659$	−17.9375	−0.698748	−0.349374	−	0.936983i	$-0.613606\pi$
$659$	−17.9375	−0.698748	−0.349374	+	0.936983i	$0.613606\pi$
$660$	0	0
$661$	−5.32444	−0.207097	−0.103548	−	0.994624i	$-0.533020\pi$
$661$	−5.32444	−0.207097	−0.103548	+	0.994624i	$0.533020\pi$
$662$	−37.6770	−1.46436
$663$	0	0
$664$	−10.5598	−0.409799
$665$	0	0
$666$	0	0
$667$	19.1875	0.742942
$668$	−10.7504	−0.415945
$669$	0	0
$670$	0	0
$671$	−11.8703	−0.458247
$672$	0	0
$673$	−7.66248	−0.295367	−0.147683	−	0.989035i	$-0.547182\pi$
$673$	−7.66248	−0.295367	−0.147683	+	0.989035i	$0.547182\pi$
$674$	−35.8816	−1.38211
$675$	0	0
$676$	6.07749	0.233749
$677$	37.4111	1.43783	0.718914	−	0.695099i	$-0.244640\pi$
$677$	37.4111	1.43783	0.718914	+	0.695099i	$0.244640\pi$
$678$	0	0
$679$	0.119320	0.00457906
$680$	0	0
$681$	0	0
$682$	−5.00856	−0.191788
$683$	−18.3023	−0.700318	−0.350159	−	0.936690i	$-0.613872\pi$
$683$	−18.3023	−0.700318	−0.350159	+	0.936690i	$0.613872\pi$
$684$	0	0
$685$	0	0
$686$	−24.9099	−0.951064
$687$	0	0
$688$	−9.76360	−0.372234
$689$	0.141561	0.00539303
$690$	0	0
$691$	19.9505	0.758952	0.379476	−	0.925202i	$-0.376104\pi$
$691$	19.9505	0.758952	0.379476	+	0.925202i	$0.376104\pi$
$692$	−2.89295	−0.109974
$693$	0	0
$694$	6.87754	0.261068
$695$	0	0
$696$	0	0
$697$	50.3950	1.90885
$698$	17.2898	0.654430
$699$	0	0
$700$	0	0
$701$	41.9220	1.58337	0.791687	−	0.610927i	$-0.209203\pi$
$701$	41.9220	1.58337	0.791687	+	0.610927i	$0.209203\pi$
$702$	0	0
$703$	−26.3126	−0.992398
$704$	23.5927	0.889183
$705$	0	0
$706$	−24.4605	−0.920584
$707$	34.0031	1.27882
$708$	0	0
$709$	6.12755	0.230125	0.115063	−	0.993358i	$-0.463293\pi$
$709$	6.12755	0.230125	0.115063	+	0.993358i	$0.463293\pi$
$710$	0	0
$711$	0	0
$712$	−45.4785	−1.70438
$713$	−4.21442	−0.157831
$714$	0	0
$715$	0	0
$716$	−6.86463	−0.256543
$717$	0	0
$718$	−14.3667	−0.536159
$719$	−40.2449	−1.50088	−0.750440	−	0.660939i	$-0.770158\pi$
$719$	−40.2449	−1.50088	−0.750440	+	0.660939i	$0.770158\pi$
$720$	0	0
$721$	−11.2206	−0.417878
$722$	11.7258	0.436389
$723$	0	0
$724$	1.34666	0.0500482
$725$	0	0
$726$	0	0
$727$	−9.61564	−0.356624	−0.178312	−	0.983974i	$-0.557064\pi$
$727$	−9.61564	−0.356624	−0.178312	+	0.983974i	$0.557064\pi$
$728$	−0.954271	−0.0353676
$729$	0	0
$730$	0	0
$731$	17.6046	0.651128
$732$	0	0
$733$	9.23751	0.341195	0.170598	−	0.985341i	$-0.445430\pi$
$733$	9.23751	0.341195	0.170598	+	0.985341i	$0.445430\pi$
$734$	0.0225491	0.000832303 0
$735$	0	0
$736$	7.15460	0.263722
$737$	−8.86171	−0.326425
$738$	0	0
$739$	11.7962	0.433931	0.216966	−	0.976179i	$-0.430384\pi$
$739$	11.7962	0.433931	0.216966	+	0.976179i	$0.430384\pi$
$740$	0	0
$741$	0	0
$742$	2.44125	0.0896210
$743$	−7.23858	−0.265558	−0.132779	−	0.991146i	$-0.542390\pi$
$743$	−7.23858	−0.265558	−0.132779	+	0.991146i	$0.542390\pi$
$744$	0	0
$745$	0	0
$746$	−20.5029	−0.750665
$747$	0	0
$748$	−6.37092	−0.232944
$749$	42.2715	1.54457
$750$	0	0
$751$	4.76976	0.174051	0.0870255	−	0.996206i	$-0.472264\pi$
$751$	4.76976	0.174051	0.0870255	+	0.996206i	$0.472264\pi$
$752$	−11.3906	−0.415371
$753$	0	0
$754$	1.28700	0.0468698
$755$	0	0
$756$	0	0
$757$	15.7834	0.573658	0.286829	−	0.957982i	$-0.407399\pi$
$757$	15.7834	0.573658	0.286829	+	0.957982i	$0.407399\pi$
$758$	3.12338	0.113446
$759$	0	0
$760$	0	0
$761$	−28.6365	−1.03807	−0.519036	−	0.854752i	$-0.673709\pi$
$761$	−28.6365	−1.03807	−0.519036	+	0.854752i	$0.673709\pi$
$762$	0	0
$763$	33.5437	1.21436
$764$	−5.91080	−0.213845
$765$	0	0
$766$	−19.5432	−0.706124
$767$	−1.22592	−0.0442653
$768$	0	0
$769$	44.2898	1.59713	0.798566	−	0.601907i	$-0.205592\pi$
$769$	44.2898	1.59713	0.798566	+	0.601907i	$0.205592\pi$
$770$	0	0
$771$	0	0
$772$	−3.28779	−0.118330
$773$	−14.3392	−0.515746	−0.257873	−	0.966179i	$-0.583022\pi$
$773$	−14.3392	−0.515746	−0.257873	+	0.966179i	$0.583022\pi$
$774$	0	0
$775$	0	0
$776$	−0.174707	−0.00627163
$777$	0	0
$778$	5.40809	0.193889
$779$	−30.3301	−1.08669
$780$	0	0
$781$	33.1080	1.18470
$782$	17.5335	0.626997
$783$	0	0
$784$	7.52879	0.268885
$785$	0	0
$786$	0	0
$787$	−49.6574	−1.77009	−0.885047	−	0.465501i	$-0.845874\pi$
$787$	−49.6574	−1.77009	−0.885047	+	0.465501i	$0.845874\pi$
$788$	−5.29448	−0.188608
$789$	0	0
$790$	0	0
$791$	17.5403	0.623663
$792$	0	0
$793$	−0.669956	−0.0237908
$794$	−14.4465	−0.512688
$795$	0	0
$796$	−6.31403	−0.223795
$797$	−3.33054	−0.117974	−0.0589869	−	0.998259i	$-0.518787\pi$
$797$	−3.33054	−0.117974	−0.0589869	+	0.998259i	$0.518787\pi$
$798$	0	0
$799$	20.5381	0.726586
$800$	0	0
$801$	0	0
$802$	−24.6285	−0.869661
$803$	30.7781	1.08614
$804$	0	0
$805$	0	0
$806$	−0.282683	−0.00995707
$807$	0	0
$808$	−49.7873	−1.75151
$809$	22.9978	0.808559	0.404280	−	0.914635i	$-0.367522\pi$
$809$	22.9978	0.808559	0.404280	+	0.914635i	$0.367522\pi$
$810$	0	0
$811$	1.70528	0.0598804	0.0299402	−	0.999552i	$-0.490468\pi$
$811$	1.70528	0.0598804	0.0299402	+	0.999552i	$0.490468\pi$
$812$	−6.78588	−0.238138
$813$	0	0
$814$	−27.9912	−0.981091
$815$	0	0
$816$	0	0
$817$	−10.5953	−0.370682
$818$	12.6341	0.441739
$819$	0	0
$820$	0	0
$821$	19.5935	0.683819	0.341910	−	0.939733i	$-0.388926\pi$
$821$	19.5935	0.683819	0.341910	+	0.939733i	$0.388926\pi$
$822$	0	0
$823$	6.41237	0.223521	0.111761	−	0.993735i	$-0.464351\pi$
$823$	6.41237	0.223521	0.111761	+	0.993735i	$0.464351\pi$
$824$	16.4292	0.572339
$825$	0	0
$826$	−21.1413	−0.735598
$827$	31.6309	1.09991	0.549957	−	0.835193i	$-0.314644\pi$
$827$	31.6309	1.09991	0.549957	+	0.835193i	$0.314644\pi$
$828$	0	0
$829$	−3.08634	−0.107193	−0.0535965	−	0.998563i	$-0.517068\pi$
$829$	−3.08634	−0.107193	−0.0535965	+	0.998563i	$0.517068\pi$
$830$	0	0
$831$	0	0
$832$	1.33157	0.0461638
$833$	−13.5750	−0.470347
$834$	0	0
$835$	0	0
$836$	3.83433	0.132613
$837$	0	0
$838$	−46.8283	−1.61766
$839$	15.2136	0.525233	0.262616	−	0.964900i	$-0.415415\pi$
$839$	15.2136	0.525233	0.262616	+	0.964900i	$0.415415\pi$
$840$	0	0
$841$	19.2372	0.663352
$842$	−12.5903	−0.433890
$843$	0	0
$844$	4.74379	0.163288
$845$	0	0
$846$	0	0
$847$	8.26668	0.284046
$848$	−2.68894	−0.0923387
$849$	0	0
$850$	0	0
$851$	−23.5530	−0.807386
$852$	0	0
$853$	−25.1005	−0.859426	−0.429713	−	0.902966i	$-0.641385\pi$
$853$	−25.1005	−0.859426	−0.429713	+	0.902966i	$0.641385\pi$
$854$	−11.5536	−0.395355
$855$	0	0
$856$	−61.8938	−2.11549
$857$	−34.7042	−1.18547	−0.592737	−	0.805396i	$-0.701952\pi$
$857$	−34.7042	−1.18547	−0.592737	+	0.805396i	$0.701952\pi$
$858$	0	0
$859$	38.5260	1.31449	0.657246	−	0.753676i	$-0.271721\pi$
$859$	38.5260	1.31449	0.657246	+	0.753676i	$0.271721\pi$
$860$	0	0
$861$	0	0
$862$	−46.9831	−1.60025
$863$	45.6656	1.55447	0.777237	−	0.629208i	$-0.216621\pi$
$863$	45.6656	1.55447	0.777237	+	0.629208i	$0.216621\pi$
$864$	0	0
$865$	0	0
$866$	−20.3891	−0.692850
$867$	0	0
$868$	1.49048	0.0505903
$869$	30.1620	1.02318
$870$	0	0
$871$	−0.500153	−0.0169471
$872$	−49.1146	−1.66323
$873$	0	0
$874$	−10.5525	−0.356944
$875$	0	0
$876$	0	0
$877$	−6.58359	−0.222312	−0.111156	−	0.993803i	$-0.535455\pi$
$877$	−6.58359	−0.222312	−0.111156	+	0.993803i	$0.535455\pi$
$878$	−20.3255	−0.685954
$879$	0	0
$880$	0	0
$881$	42.9142	1.44581	0.722907	−	0.690945i	$-0.242805\pi$
$881$	42.9142	1.44581	0.722907	+	0.690945i	$0.242805\pi$
$882$	0	0
$883$	35.4942	1.19447	0.597237	−	0.802065i	$-0.296265\pi$
$883$	35.4942	1.19447	0.597237	+	0.802065i	$0.296265\pi$
$884$	−0.359574	−0.0120938
$885$	0	0
$886$	42.8612	1.43995
$887$	24.5790	0.825282	0.412641	−	0.910894i	$-0.364606\pi$
$887$	24.5790	0.825282	0.412641	+	0.910894i	$0.364606\pi$
$888$	0	0
$889$	−45.6301	−1.53038
$890$	0	0
$891$	0	0
$892$	−10.5992	−0.354887
$893$	−12.3608	−0.413639
$894$	0	0
$895$	0	0
$896$	12.1570	0.406137
$897$	0	0
$898$	−18.2963	−0.610557
$899$	−10.5950	−0.353364
$900$	0	0
$901$	4.84839	0.161523
$902$	−32.2651	−1.07431
$903$	0	0
$904$	−25.6825	−0.854188
$905$	0	0
$906$	0	0
$907$	46.5140	1.54447	0.772236	−	0.635336i	$-0.219138\pi$
$907$	46.5140	1.54447	0.772236	+	0.635336i	$0.219138\pi$
$908$	2.40152	0.0796973
$909$	0	0
$910$	0	0
$911$	15.2581	0.505523	0.252761	−	0.967529i	$-0.418661\pi$
$911$	15.2581	0.505523	0.252761	+	0.967529i	$0.418661\pi$
$912$	0	0
$913$	−9.17034	−0.303494
$914$	29.4645	0.974599
$915$	0	0
$916$	−9.77396	−0.322941
$917$	29.4584	0.972802
$918$	0	0
$919$	−12.0177	−0.396427	−0.198213	−	0.980159i	$-0.563514\pi$
$919$	−12.0177	−0.396427	−0.198213	+	0.980159i	$0.563514\pi$
$920$	0	0
$921$	0	0
$922$	−28.5911	−0.941597
$923$	1.86861	0.0615061
$924$	0	0
$925$	0	0
$926$	1.09468	0.0359735
$927$	0	0
$928$	17.9866	0.590440
$929$	1.48156	0.0486086	0.0243043	−	0.999705i	$-0.492263\pi$
$929$	1.48156	0.0486086	0.0243043	+	0.999705i	$0.492263\pi$
$930$	0	0
$931$	8.17011	0.267765
$932$	−8.14796	−0.266895
$933$	0	0
$934$	23.9893	0.784953
$935$	0	0
$936$	0	0
$937$	38.5057	1.25793	0.628963	−	0.777435i	$-0.283480\pi$
$937$	38.5057	1.25793	0.628963	+	0.777435i	$0.283480\pi$
$938$	−8.62527	−0.281625
$939$	0	0
$940$	0	0
$941$	−24.8562	−0.810291	−0.405145	−	0.914252i	$-0.632779\pi$
$941$	−24.8562	−0.810291	−0.405145	+	0.914252i	$0.632779\pi$
$942$	0	0
$943$	−27.1492	−0.884100
$944$	23.2863	0.757905
$945$	0	0
$946$	−11.2712	−0.366459
$947$	10.6476	0.345999	0.172999	−	0.984922i	$-0.444654\pi$
$947$	10.6476	0.345999	0.172999	+	0.984922i	$0.444654\pi$
$948$	0	0
$949$	1.73711	0.0563890
$950$	0	0
$951$	0	0
$952$	−32.6834	−1.05927
$953$	−27.2483	−0.882659	−0.441329	−	0.897345i	$-0.645493\pi$
$953$	−27.2483	−0.882659	−0.441329	+	0.897345i	$0.645493\pi$
$954$	0	0
$955$	0	0
$956$	2.18373	0.0706270
$957$	0	0
$958$	−40.6122	−1.31212
$959$	−20.0250	−0.646643
$960$	0	0
$961$	−28.6729	−0.924931
$962$	−1.57982	−0.0509354
$963$	0	0
$964$	−6.23981	−0.200971
$965$	0	0
$966$	0	0
$967$	37.9832	1.22146	0.610729	−	0.791840i	$-0.290877\pi$
$967$	37.9832	1.22146	0.610729	+	0.791840i	$0.290877\pi$
$968$	−12.1041	−0.389039
$969$	0	0
$970$	0	0
$971$	4.14071	0.132882	0.0664409	−	0.997790i	$-0.478836\pi$
$971$	4.14071	0.132882	0.0664409	+	0.997790i	$0.478836\pi$
$972$	0	0
$973$	−37.1255	−1.19019
$974$	−23.0142	−0.737424
$975$	0	0
$976$	12.7258	0.407344
$977$	−50.3811	−1.61183	−0.805917	−	0.592028i	$-0.798327\pi$
$977$	−50.3811	−1.61183	−0.805917	+	0.592028i	$0.798327\pi$
$978$	0	0
$979$	−39.4945	−1.26225
$980$	0	0
$981$	0	0
$982$	33.8705	1.08085
$983$	30.0565	0.958654	0.479327	−	0.877637i	$-0.340881\pi$
$983$	30.0565	0.958654	0.479327	+	0.877637i	$0.340881\pi$
$984$	0	0
$985$	0	0
$986$	44.0792	1.40377
$987$	0	0
$988$	0.216409	0.00688489
$989$	−9.48408	−0.301576
$990$	0	0
$991$	4.37287	0.138909	0.0694544	−	0.997585i	$-0.477874\pi$
$991$	4.37287	0.138909	0.0694544	+	0.997585i	$0.477874\pi$
$992$	−3.95067	−0.125434
$993$	0	0
$994$	32.2247	1.02211
$995$	0	0
$996$	0	0
$997$	−2.57051	−0.0814090	−0.0407045	−	0.999171i	$-0.512960\pi$
$997$	−2.57051	−0.0814090	−0.0407045	+	0.999171i	$0.512960\pi$
$998$	1.05524	0.0334030
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.bb.1.3	yes	8
3.2	odd	2	inner	5625.2.a.bb.1.6	yes	8
5.4	even	2		5625.2.a.z.1.6	yes	8
15.14	odd	2		5625.2.a.z.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5625.2.a.z.1.3	✓	8	15.14	odd	2
5625.2.a.z.1.6	yes	8	5.4	even	2
5625.2.a.bb.1.3	yes	8	1.1	even	1	trivial
5625.2.a.bb.1.6	yes	8	3.2	odd	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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q-1.23762 q^{2} -0.468306 q^{4} -2.08634 q^{7} +3.05482 q^{8} +O(q^{10})\)	\(q-1.23762 q^{2} -0.468306 q^{4} -2.08634 q^{7} +3.05482 q^{8} +2.65287 q^{11} +0.149728 q^{13} +2.58209 q^{14} -2.84408 q^{16} +5.12810 q^{17} -3.08634 q^{19} -3.28323 q^{22} -2.76265 q^{23} -0.185305 q^{26} +0.977047 q^{28} -6.94530 q^{29} +1.52550 q^{31} -2.58976 q^{32} -6.34662 q^{34} +8.52550 q^{37} +3.81970 q^{38} +9.82722 q^{41} +3.43296 q^{43} -1.24236 q^{44} +3.41911 q^{46} +4.00501 q^{47} -2.64718 q^{49} -0.0701184 q^{52} +0.945455 q^{53} -6.37339 q^{56} +8.59562 q^{58} -8.18766 q^{59} -4.47450 q^{61} -1.88798 q^{62} +8.89328 q^{64} -3.34042 q^{67} -2.40152 q^{68} +12.4801 q^{71} +11.6018 q^{73} -10.5513 q^{74} +1.44535 q^{76} -5.53479 q^{77} +11.3696 q^{79} -12.1623 q^{82} -3.45676 q^{83} -4.24869 q^{86} +8.10403 q^{88} -14.8875 q^{89} -0.312383 q^{91} +1.29377 q^{92} -4.95666 q^{94} -0.0571908 q^{97} +3.27620 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 14 q^{4} + 10 q^{7}+O(q^{10}) \) 8 * q + 14 * q^4 + 10 * q^7	\( 8 q + 14 q^{4} + 10 q^{7} + 10 q^{13} + 22 q^{16} + 2 q^{19} + 10 q^{22} + 70 q^{28} - 6 q^{31} - 50 q^{34} + 50 q^{37} + 14 q^{49} + 80 q^{52} - 30 q^{58} - 54 q^{61} + 36 q^{64} + 10 q^{67} + 30 q^{73} + 56 q^{76} + 28 q^{79} + 20 q^{88} + 60 q^{91} - 40 q^{94}+O(q^{100}) \) 8 * q + 14 * q^4 + 10 * q^7 + 10 * q^13 + 22 * q^16 + 2 * q^19 + 10 * q^22 + 70 * q^28 - 6 * q^31 - 50 * q^34 + 50 * q^37 + 14 * q^49 + 80 * q^52 - 30 * q^58 - 54 * q^61 + 36 * q^64 + 10 * q^67 + 30 * q^73 + 56 * q^76 + 28 * q^79 + 20 * q^88 + 60 * q^91 - 40 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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