LMFDB - Embedded newform 225.8.a.w.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [225,8,Mod(1,225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("225.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,20,0,96,0,0,100] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$70.2866307339$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{19}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 19 $ x^2 - 19
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.35890$ of defining polynomial
Character	$\chi$	$=$	225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+18.7178 q^{2} +222.356 q^{4} -438.197 q^{7} +1766.14 q^{8} -5759.12 q^{11} +3530.42 q^{13} -8202.08 q^{14} +4596.61 q^{16} -23991.9 q^{17} +16590.3 q^{19} -107798. q^{22} -65626.9 q^{23} +66081.7 q^{26} -97435.6 q^{28} -134041. q^{29} +129002. q^{31} -140027. q^{32} -449075. q^{34} -161108. q^{37} +310534. q^{38} +362989. q^{41} -588189. q^{43} -1.28057e6 q^{44} -1.22839e6 q^{46} +343895. q^{47} -631527. q^{49} +785010. q^{52} -1.66139e6 q^{53} -773915. q^{56} -2.50896e6 q^{58} +2.54214e6 q^{59} +2.52337e6 q^{61} +2.41464e6 q^{62} -3.20936e6 q^{64} -1.56618e6 q^{67} -5.33474e6 q^{68} +299354. q^{71} -312494. q^{73} -3.01558e6 q^{74} +3.68895e6 q^{76} +2.52363e6 q^{77} -1.95247e6 q^{79} +6.79435e6 q^{82} -621372. q^{83} -1.10096e7 q^{86} -1.01714e7 q^{88} -5.78298e6 q^{89} -1.54702e6 q^{91} -1.45925e7 q^{92} +6.43696e6 q^{94} -7.20152e6 q^{97} -1.18208e7 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 20 q^{2} + 96 q^{4} + 100 q^{7} + 1440 q^{8} - 4544 q^{11} - 3540 q^{13} - 7512 q^{14} + 20352 q^{16} - 27340 q^{17} + 38760 q^{19} - 106240 q^{22} - 124140 q^{23} + 57016 q^{26} - 165440 q^{28} + 72260 q^{29}+ \cdots - 12505340 q^{98}+O(q^{100}) $ 2 * q + 20 * q^2 + 96 * q^4 + 100 * q^7 + 1440 * q^8 - 4544 * q^11 - 3540 * q^13 - 7512 * q^14 + 20352 * q^16 - 27340 * q^17 + 38760 * q^19 - 106240 * q^22 - 124140 * q^23 + 57016 * q^26 - 165440 * q^28 + 72260 * q^29 + 306824 * q^31 - 78080 * q^32 - 453368 * q^34 + 123020 * q^37 + 338960 * q^38 - 264364 * q^41 - 423300 * q^43 - 1434112 * q^44 - 1303416 * q^46 - 105460 * q^47 - 1165414 * q^49 + 1678400 * q^52 - 2391580 * q^53 - 949440 * q^56 - 2244440 * q^58 + 1120120 * q^59 + 2257044 * q^61 + 2642640 * q^62 - 5146624 * q^64 - 4516460 * q^67 - 4911680 * q^68 - 621784 * q^71 - 4569060 * q^73 - 2651272 * q^74 + 887680 * q^76 + 3177600 * q^77 + 4333040 * q^79 + 5989960 * q^82 - 9793020 * q^83 - 10798184 * q^86 - 10567680 * q^88 - 6025620 * q^89 - 5352296 * q^91 - 7199040 * q^92 + 5860792 * q^94 - 4609540 * q^97 - 12505340 * 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 $

$2$	18.7178	1.65444	0.827218	−	0.561882i	$-0.189922\pi$
$2$	18.7178	1.65444	0.827218	+	0.561882i	$0.189922\pi$
$3$	0	0
$3$	0	0
$4$	222.356	1.73716
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−438.197	−0.482865	−0.241433	−	0.970418i	$-0.577617\pi$
$7$	−438.197	−0.482865	−0.241433	+	0.970418i	$0.577617\pi$
$8$	1766.14	1.21958
$9$	0	0
$10$	0	0
$11$	−5759.12	−1.30461	−0.652306	−	0.757955i	$-0.726198\pi$
$11$	−5759.12	−1.30461	−0.652306	+	0.757955i	$0.726198\pi$
$12$	0	0
$13$	3530.42	0.445682	0.222841	−	0.974855i	$-0.428467\pi$
$13$	3530.42	0.445682	0.222841	+	0.974855i	$0.428467\pi$
$14$	−8202.08	−0.798869
$15$	0	0
$16$	4596.61	0.280555
$17$	−23991.9	−1.18439	−0.592193	−	0.805797i	$-0.701738\pi$
$17$	−23991.9	−1.18439	−0.592193	+	0.805797i	$0.701738\pi$
$18$	0	0
$19$	16590.3	0.554903	0.277451	−	0.960740i	$-0.410510\pi$
$19$	16590.3	0.554903	0.277451	+	0.960740i	$0.410510\pi$
$20$	0	0
$21$	0	0
$22$	−107798.	−2.15840
$23$	−65626.9	−1.12469	−0.562347	−	0.826902i	$-0.690101\pi$
$23$	−65626.9	−1.12469	−0.562347	+	0.826902i	$0.690101\pi$
$24$	0	0
$25$	0	0
$26$	66081.7	0.737351
$27$	0	0
$28$	−97435.6	−0.838812
$29$	−134041.	−1.02058	−0.510289	−	0.860003i	$-0.670461\pi$
$29$	−134041.	−1.02058	−0.510289	+	0.860003i	$0.670461\pi$
$30$	0	0
$31$	129002.	0.777734	0.388867	−	0.921294i	$-0.372867\pi$
$31$	129002.	0.777734	0.388867	+	0.921294i	$0.372867\pi$
$32$	−140027.	−0.755417
$33$	0	0
$34$	−449075.	−1.95949
$35$	0	0
$36$	0	0
$37$	−161108.	−0.522890	−0.261445	−	0.965218i	$-0.584199\pi$
$37$	−161108.	−0.522890	−0.261445	+	0.965218i	$0.584199\pi$
$38$	310534.	0.918050
$39$	0	0
$40$	0	0
$41$	362989.	0.822526	0.411263	−	0.911517i	$-0.365088\pi$
$41$	362989.	0.822526	0.411263	+	0.911517i	$0.365088\pi$
$42$	0	0
$43$	−588189.	−1.12818	−0.564089	−	0.825714i	$-0.690772\pi$
$43$	−588189.	−1.12818	−0.564089	+	0.825714i	$0.690772\pi$
$44$	−1.28057e6	−2.26632
$45$	0	0
$46$	−1.22839e6	−1.86073
$47$	343895.	0.483151	0.241576	−	0.970382i	$-0.422336\pi$
$47$	343895.	0.483151	0.241576	+	0.970382i	$0.422336\pi$
$48$	0	0
$49$	−631527.	−0.766841
$50$	0	0
$51$	0	0
$52$	785010.	0.774219
$53$	−1.66139e6	−1.53287	−0.766436	−	0.642320i	$-0.777972\pi$
$53$	−1.66139e6	−1.53287	−0.766436	+	0.642320i	$0.777972\pi$
$54$	0	0
$55$	0	0
$56$	−773915.	−0.588891
$57$	0	0
$58$	−2.50896e6	−1.68848
$59$	2.54214e6	1.61145	0.805726	−	0.592289i	$-0.201776\pi$
$59$	2.54214e6	1.61145	0.805726	+	0.592289i	$0.201776\pi$
$60$	0	0
$61$	2.52337e6	1.42340	0.711699	−	0.702484i	$-0.247926\pi$
$61$	2.52337e6	1.42340	0.711699	+	0.702484i	$0.247926\pi$
$62$	2.41464e6	1.28671
$63$	0	0
$64$	−3.20936e6	−1.53034
$65$	0	0
$66$	0	0
$67$	−1.56618e6	−0.636178	−0.318089	−	0.948061i	$-0.603041\pi$
$67$	−1.56618e6	−0.636178	−0.318089	+	0.948061i	$0.603041\pi$
$68$	−5.33474e6	−2.05746
$69$	0	0
$70$	0	0
$71$	299354.	0.0992615	0.0496307	−	0.998768i	$-0.484196\pi$
$71$	299354.	0.0992615	0.0496307	+	0.998768i	$0.484196\pi$
$72$	0	0
$73$	−312494.	−0.0940183	−0.0470091	−	0.998894i	$-0.514969\pi$
$73$	−312494.	−0.0940183	−0.0470091	+	0.998894i	$0.514969\pi$
$74$	−3.01558e6	−0.865087
$75$	0	0
$76$	3.68895e6	0.963952
$77$	2.52363e6	0.629952
$78$	0	0
$79$	−1.95247e6	−0.445542	−0.222771	−	0.974871i	$-0.571510\pi$
$79$	−1.95247e6	−0.445542	−0.222771	+	0.974871i	$0.571510\pi$
$80$	0	0
$81$	0	0
$82$	6.79435e6	1.36082
$83$	−621372.	−0.119283	−0.0596414	−	0.998220i	$-0.518996\pi$
$83$	−621372.	−0.119283	−0.0596414	+	0.998220i	$0.518996\pi$
$84$	0	0
$85$	0	0
$86$	−1.10096e7	−1.86650
$87$	0	0
$88$	−1.01714e7	−1.59108
$89$	−5.78298e6	−0.869534	−0.434767	−	0.900543i	$-0.643169\pi$
$89$	−5.78298e6	−0.869534	−0.434767	+	0.900543i	$0.643169\pi$
$90$	0	0
$91$	−1.54702e6	−0.215204
$92$	−1.45925e7	−1.95377
$93$	0	0
$94$	6.43696e6	0.799343
$95$	0	0
$96$	0	0
$97$	−7.20152e6	−0.801167	−0.400584	−	0.916260i	$-0.631193\pi$
$97$	−7.20152e6	−0.801167	−0.400584	+	0.916260i	$0.631193\pi$
$98$	−1.18208e7	−1.26869
$99$	0	0
$100$	0	0
$101$	2.91989e6	0.281995	0.140997	−	0.990010i	$-0.454969\pi$
$101$	2.91989e6	0.281995	0.140997	+	0.990010i	$0.454969\pi$
$102$	0	0
$103$	−3.94639e6	−0.355852	−0.177926	−	0.984044i	$-0.556939\pi$
$103$	−3.94639e6	−0.355852	−0.177926	+	0.984044i	$0.556939\pi$
$104$	6.23520e6	0.543543
$105$	0	0
$106$	−3.10976e7	−2.53604
$107$	−3.81991e6	−0.301446	−0.150723	−	0.988576i	$-0.548160\pi$
$107$	−3.81991e6	−0.301446	−0.150723	+	0.988576i	$0.548160\pi$
$108$	0	0
$109$	−8.82259e6	−0.652534	−0.326267	−	0.945278i	$-0.605791\pi$
$109$	−8.82259e6	−0.652534	−0.326267	+	0.945278i	$0.605791\pi$
$110$	0	0
$111$	0	0
$112$	−2.01422e6	−0.135470
$113$	2.12074e7	1.38265	0.691324	−	0.722545i	$-0.257028\pi$
$113$	2.12074e7	1.38265	0.691324	+	0.722545i	$0.257028\pi$
$114$	0	0
$115$	0	0
$116$	−2.98049e7	−1.77290
$117$	0	0
$118$	4.75832e7	2.66604
$119$	1.05132e7	0.571898
$120$	0	0
$121$	1.36803e7	0.702015
$122$	4.72319e7	2.35492
$123$	0	0
$124$	2.86844e7	1.35105
$125$	0	0
$126$	0	0
$127$	2.55822e7	1.10822	0.554108	−	0.832445i	$-0.313059\pi$
$127$	2.55822e7	1.10822	0.554108	+	0.832445i	$0.313059\pi$
$128$	−4.21487e7	−1.77644
$129$	0	0
$130$	0	0
$131$	−1.30640e7	−0.507722	−0.253861	−	0.967241i	$-0.581701\pi$
$131$	−1.30640e7	−0.507722	−0.253861	+	0.967241i	$0.581701\pi$
$132$	0	0
$133$	−7.26982e6	−0.267943
$134$	−2.93154e7	−1.05252
$135$	0	0
$136$	−4.23729e7	−1.44445
$137$	2.14021e7	0.711106	0.355553	−	0.934656i	$-0.384293\pi$
$137$	2.14021e7	0.711106	0.355553	+	0.934656i	$0.384293\pi$
$138$	0	0
$139$	4.00656e7	1.26538	0.632688	−	0.774406i	$-0.281951\pi$
$139$	4.00656e7	1.26538	0.632688	+	0.774406i	$0.281951\pi$
$140$	0	0
$141$	0	0
$142$	5.60324e6	0.164222
$143$	−2.03321e7	−0.581442
$144$	0	0
$145$	0	0
$146$	−5.84921e6	−0.155547
$147$	0	0
$148$	−3.58233e7	−0.908341
$149$	5.96142e7	1.47638	0.738190	−	0.674593i	$-0.235681\pi$
$149$	5.96142e7	1.47638	0.738190	+	0.674593i	$0.235681\pi$
$150$	0	0
$151$	−5.21166e6	−0.123185	−0.0615924	−	0.998101i	$-0.519618\pi$
$151$	−5.21166e6	−0.123185	−0.0615924	+	0.998101i	$0.519618\pi$
$152$	2.93007e7	0.676746
$153$	0	0
$154$	4.72367e7	1.04222
$155$	0	0
$156$	0	0
$157$	1.10197e7	0.227259	0.113630	−	0.993523i	$-0.463752\pi$
$157$	1.10197e7	0.227259	0.113630	+	0.993523i	$0.463752\pi$
$158$	−3.65458e7	−0.737120
$159$	0	0
$160$	0	0
$161$	2.87575e7	0.543075
$162$	0	0
$163$	−2.32415e7	−0.420346	−0.210173	−	0.977664i	$-0.567403\pi$
$163$	−2.32415e7	−0.420346	−0.210173	+	0.977664i	$0.567403\pi$
$164$	8.07127e7	1.42886
$165$	0	0
$166$	−1.16307e7	−0.197346
$167$	−5.84152e7	−0.970550	−0.485275	−	0.874361i	$-0.661281\pi$
$167$	−5.84152e7	−0.970550	−0.485275	+	0.874361i	$0.661281\pi$
$168$	0	0
$169$	−5.02846e7	−0.801368
$170$	0	0
$171$	0	0
$172$	−1.30787e8	−1.95982
$173$	1.18828e6	0.0174485	0.00872427	−	0.999962i	$-0.497223\pi$
$173$	1.18828e6	0.0174485	0.00872427	+	0.999962i	$0.497223\pi$
$174$	0	0
$175$	0	0
$176$	−2.64724e7	−0.366015
$177$	0	0
$178$	−1.08245e8	−1.43859
$179$	−1.28635e8	−1.67638	−0.838191	−	0.545377i	$-0.816387\pi$
$179$	−1.28635e8	−1.67638	−0.838191	+	0.545377i	$0.816387\pi$
$180$	0	0
$181$	1.40320e8	1.75892	0.879458	−	0.475976i	$-0.157905\pi$
$181$	1.40320e8	1.75892	0.879458	+	0.475976i	$0.157905\pi$
$182$	−2.89568e7	−0.356041
$183$	0	0
$184$	−1.15906e8	−1.37165
$185$	0	0
$186$	0	0
$187$	1.38172e8	1.54516
$188$	7.64671e7	0.839309
$189$	0	0
$190$	0	0
$191$	3.81784e7	0.396461	0.198231	−	0.980155i	$-0.436481\pi$
$191$	3.81784e7	0.396461	0.198231	+	0.980155i	$0.436481\pi$
$192$	0	0
$193$	1.35915e8	1.36087	0.680436	−	0.732807i	$-0.261790\pi$
$193$	1.35915e8	1.36087	0.680436	+	0.732807i	$0.261790\pi$
$194$	−1.34797e8	−1.32548
$195$	0	0
$196$	−1.40424e8	−1.33212
$197$	6.16154e7	0.574193	0.287096	−	0.957902i	$-0.407310\pi$
$197$	6.16154e7	0.574193	0.287096	+	0.957902i	$0.407310\pi$
$198$	0	0
$199$	−1.84377e8	−1.65852	−0.829261	−	0.558862i	$-0.811238\pi$
$199$	−1.84377e8	−1.65852	−0.829261	+	0.558862i	$0.811238\pi$
$200$	0	0
$201$	0	0
$202$	5.46539e7	0.466542
$203$	5.87365e7	0.492801
$204$	0	0
$205$	0	0
$206$	−7.38677e7	−0.588734
$207$	0	0
$208$	1.62280e7	0.125038
$209$	−9.55455e7	−0.723933
$210$	0	0
$211$	1.71174e8	1.25444	0.627218	−	0.778844i	$-0.284194\pi$
$211$	1.71174e8	1.25444	0.627218	+	0.778844i	$0.284194\pi$
$212$	−3.69420e8	−2.66284
$213$	0	0
$214$	−7.15003e7	−0.498723
$215$	0	0
$216$	0	0
$217$	−5.65283e7	−0.375541
$218$	−1.65139e8	−1.07958
$219$	0	0
$220$	0	0
$221$	−8.47014e7	−0.527859
$222$	0	0
$223$	−2.67014e8	−1.61238	−0.806190	−	0.591657i	$-0.798474\pi$
$223$	−2.67014e8	−1.61238	−0.806190	+	0.591657i	$0.798474\pi$
$224$	6.13594e7	0.364765
$225$	0	0
$226$	3.96955e8	2.28750
$227$	378641.	0.00214851	0.00107425	−	0.999999i	$-0.499658\pi$
$227$	378641.	0.00214851	0.00107425	+	0.999999i	$0.499658\pi$
$228$	0	0
$229$	−2.24429e8	−1.23497	−0.617483	−	0.786584i	$-0.711848\pi$
$229$	−2.24429e8	−1.23497	−0.617483	+	0.786584i	$0.711848\pi$
$230$	0	0
$231$	0	0
$232$	−2.36735e8	−1.24467
$233$	1.55173e8	0.803656	0.401828	−	0.915715i	$-0.368375\pi$
$233$	1.55173e8	0.803656	0.401828	+	0.915715i	$0.368375\pi$
$234$	0	0
$235$	0	0
$236$	5.65260e8	2.79934
$237$	0	0
$238$	1.96783e8	0.946169
$239$	−4.07160e7	−0.192918	−0.0964588	−	0.995337i	$-0.530752\pi$
$239$	−4.07160e7	−0.192918	−0.0964588	+	0.995337i	$0.530752\pi$
$240$	0	0
$241$	−3.06501e8	−1.41050	−0.705249	−	0.708960i	$-0.749165\pi$
$241$	−3.06501e8	−1.41050	−0.705249	+	0.708960i	$0.749165\pi$
$242$	2.56065e8	1.16144
$243$	0	0
$244$	5.61086e8	2.47266
$245$	0	0
$246$	0	0
$247$	5.85708e7	0.247310
$248$	2.27835e8	0.948506
$249$	0	0
$250$	0	0
$251$	1.30381e8	0.520421	0.260211	−	0.965552i	$-0.416208\pi$
$251$	1.30381e8	0.520421	0.260211	+	0.965552i	$0.416208\pi$
$252$	0	0
$253$	3.77953e8	1.46729
$254$	4.78842e8	1.83347
$255$	0	0
$256$	−3.78133e8	−1.40866
$257$	3.23514e8	1.18885	0.594426	−	0.804151i	$-0.297380\pi$
$257$	3.23514e8	1.18885	0.594426	+	0.804151i	$0.297380\pi$
$258$	0	0
$259$	7.05969e7	0.252485
$260$	0	0
$261$	0	0
$262$	−2.44529e8	−0.839994
$263$	2.39895e7	0.0813159	0.0406579	−	0.999173i	$-0.487055\pi$
$263$	2.39895e7	0.0813159	0.0406579	+	0.999173i	$0.487055\pi$
$264$	0	0
$265$	0	0
$266$	−1.36075e8	−0.443295
$267$	0	0
$268$	−3.48249e8	−1.10514
$269$	1.73612e8	0.543809	0.271905	−	0.962324i	$-0.412346\pi$
$269$	1.73612e8	0.543809	0.271905	+	0.962324i	$0.412346\pi$
$270$	0	0
$271$	5.08478e8	1.55196	0.775978	−	0.630760i	$-0.217257\pi$
$271$	5.08478e8	1.55196	0.775978	+	0.630760i	$0.217257\pi$
$272$	−1.10281e8	−0.332285
$273$	0	0
$274$	4.00600e8	1.17648
$275$	0	0
$276$	0	0
$277$	6.01050e8	1.69915	0.849575	−	0.527468i	$-0.176859\pi$
$277$	6.01050e8	1.69915	0.849575	+	0.527468i	$0.176859\pi$
$278$	7.49940e8	2.09348
$279$	0	0
$280$	0	0
$281$	6.36212e8	1.71053	0.855264	−	0.518193i	$-0.173395\pi$
$281$	6.36212e8	1.71053	0.855264	+	0.518193i	$0.173395\pi$
$282$	0	0
$283$	−5.46181e7	−0.143247	−0.0716233	−	0.997432i	$-0.522818\pi$
$283$	−5.46181e7	−0.143247	−0.0716233	+	0.997432i	$0.522818\pi$
$284$	6.65631e7	0.172433
$285$	0	0
$286$	−3.80572e8	−0.961958
$287$	−1.59061e8	−0.397169
$288$	0	0
$289$	1.65271e8	0.402768
$290$	0	0
$291$	0	0
$292$	−6.94850e7	−0.163324
$293$	−1.22481e8	−0.284467	−0.142234	−	0.989833i	$-0.545428\pi$
$293$	−1.22481e8	−0.284467	−0.142234	+	0.989833i	$0.545428\pi$
$294$	0	0
$295$	0	0
$296$	−2.84538e8	−0.637704
$297$	0	0
$298$	1.11585e9	2.44257
$299$	−2.31690e8	−0.501255
$300$	0	0
$301$	2.57743e8	0.544758
$302$	−9.75509e7	−0.203801
$303$	0	0
$304$	7.62591e7	0.155681
$305$	0	0
$306$	0	0
$307$	−5.58187e8	−1.10102	−0.550510	−	0.834829i	$-0.685567\pi$
$307$	−5.58187e8	−1.10102	−0.550510	+	0.834829i	$0.685567\pi$
$308$	5.61143e8	1.09433
$309$	0	0
$310$	0	0
$311$	−8.94564e7	−0.168636	−0.0843180	−	0.996439i	$-0.526871\pi$
$311$	−8.94564e7	−0.168636	−0.0843180	+	0.996439i	$0.526871\pi$
$312$	0	0
$313$	2.75895e8	0.508556	0.254278	−	0.967131i	$-0.418162\pi$
$313$	2.75895e8	0.508556	0.254278	+	0.967131i	$0.418162\pi$
$314$	2.06265e8	0.375986
$315$	0	0
$316$	−4.34142e8	−0.773976
$317$	−4.40449e8	−0.776584	−0.388292	−	0.921536i	$-0.626935\pi$
$317$	−4.40449e8	−0.776584	−0.388292	+	0.921536i	$0.626935\pi$
$318$	0	0
$319$	7.71960e8	1.33146
$320$	0	0
$321$	0	0
$322$	5.38277e8	0.898483
$323$	−3.98032e8	−0.657218
$324$	0	0
$325$	0	0
$326$	−4.35030e8	−0.695436
$327$	0	0
$328$	6.41088e8	1.00313
$329$	−1.50694e8	−0.233297
$330$	0	0
$331$	1.68079e8	0.254751	0.127376	−	0.991855i	$-0.459345\pi$
$331$	1.68079e8	0.254751	0.127376	+	0.991855i	$0.459345\pi$
$332$	−1.38166e8	−0.207213
$333$	0	0
$334$	−1.09340e9	−1.60571
$335$	0	0
$336$	0	0
$337$	−8.38651e8	−1.19365	−0.596824	−	0.802372i	$-0.703571\pi$
$337$	−8.38651e8	−1.19365	−0.596824	+	0.802372i	$0.703571\pi$
$338$	−9.41218e8	−1.32581
$339$	0	0
$340$	0	0
$341$	−7.42939e8	−1.01464
$342$	0	0
$343$	6.37607e8	0.853146
$344$	−1.03882e9	−1.37590
$345$	0	0
$346$	2.22421e7	0.0288675
$347$	−1.16128e9	−1.49205	−0.746023	−	0.665921i	$-0.768039\pi$
$347$	−1.16128e9	−1.49205	−0.746023	+	0.665921i	$0.768039\pi$
$348$	0	0
$349$	−8.37482e8	−1.05460	−0.527298	−	0.849680i	$-0.676795\pi$
$349$	−8.37482e8	−1.05460	−0.527298	+	0.849680i	$0.676795\pi$
$350$	0	0
$351$	0	0
$352$	8.06432e8	0.985527
$353$	7.61561e8	0.921496	0.460748	−	0.887531i	$-0.347581\pi$
$353$	7.61561e8	0.921496	0.460748	+	0.887531i	$0.347581\pi$
$354$	0	0
$355$	0	0
$356$	−1.28588e9	−1.51052
$357$	0	0
$358$	−2.40776e9	−2.77347
$359$	−3.19728e8	−0.364712	−0.182356	−	0.983233i	$-0.558372\pi$
$359$	−3.19728e8	−0.364712	−0.182356	+	0.983233i	$0.558372\pi$
$360$	0	0
$361$	−6.18634e8	−0.692083
$362$	2.62649e9	2.91001
$363$	0	0
$364$	−3.43989e8	−0.373843
$365$	0	0
$366$	0	0
$367$	−1.25415e8	−0.132440	−0.0662199	−	0.997805i	$-0.521094\pi$
$367$	−1.25415e8	−0.132440	−0.0662199	+	0.997805i	$0.521094\pi$
$368$	−3.01661e8	−0.315538
$369$	0	0
$370$	0	0
$371$	7.28016e8	0.740171
$372$	0	0
$373$	−1.71505e9	−1.71118	−0.855588	−	0.517657i	$-0.826804\pi$
$373$	−1.71505e9	−1.71118	−0.855588	+	0.517657i	$0.826804\pi$
$374$	2.58628e9	2.55637
$375$	0	0
$376$	6.07365e8	0.589240
$377$	−4.73223e8	−0.454853
$378$	0	0
$379$	−1.07297e8	−0.101239	−0.0506196	−	0.998718i	$-0.516120\pi$
$379$	−1.07297e8	−0.101239	−0.0506196	+	0.998718i	$0.516120\pi$
$380$	0	0
$381$	0	0
$382$	7.14615e8	0.655919
$383$	−6.77468e8	−0.616160	−0.308080	−	0.951361i	$-0.599686\pi$
$383$	−6.77468e8	−0.616160	−0.308080	+	0.951361i	$0.599686\pi$
$384$	0	0
$385$	0	0
$386$	2.54403e9	2.25148
$387$	0	0
$388$	−1.60130e9	−1.39175
$389$	−1.94836e9	−1.67821	−0.839104	−	0.543971i	$-0.816920\pi$
$389$	−1.94836e9	−1.67821	−0.839104	+	0.543971i	$0.816920\pi$
$390$	0	0
$391$	1.57451e9	1.33207
$392$	−1.11536e9	−0.935222
$393$	0	0
$394$	1.15331e9	0.949965
$395$	0	0
$396$	0	0
$397$	1.11752e9	0.896369	0.448185	−	0.893941i	$-0.352071\pi$
$397$	1.11752e9	0.896369	0.448185	+	0.893941i	$0.352071\pi$
$398$	−3.45113e9	−2.74392
$399$	0	0
$400$	0	0
$401$	−1.61315e9	−1.24931	−0.624655	−	0.780901i	$-0.714760\pi$
$401$	−1.61315e9	−1.24931	−0.624655	+	0.780901i	$0.714760\pi$
$402$	0	0
$403$	4.55432e8	0.346622
$404$	6.49254e8	0.489869
$405$	0	0
$406$	1.09942e9	0.815308
$407$	9.27838e8	0.682169
$408$	0	0
$409$	1.32866e7	0.00960248	0.00480124	−	0.999988i	$-0.498472\pi$
$409$	1.32866e7	0.00960248	0.00480124	+	0.999988i	$0.498472\pi$
$410$	0	0
$411$	0	0
$412$	−8.77503e8	−0.618170
$413$	−1.11396e9	−0.778114
$414$	0	0
$415$	0	0
$416$	−4.94354e8	−0.336676
$417$	0	0
$418$	−1.78840e9	−1.19770
$419$	5.93249e7	0.0393992	0.0196996	−	0.999806i	$-0.493729\pi$
$419$	5.93249e7	0.0393992	0.0196996	+	0.999806i	$0.493729\pi$
$420$	0	0
$421$	−2.93348e9	−1.91600	−0.958000	−	0.286769i	$-0.907419\pi$
$421$	−2.93348e9	−1.91600	−0.958000	+	0.286769i	$0.907419\pi$
$422$	3.20399e9	2.07538
$423$	0	0
$424$	−2.93424e9	−1.86946
$425$	0	0
$426$	0	0
$427$	−1.10573e9	−0.687310
$428$	−8.49380e8	−0.523659
$429$	0	0
$430$	0	0
$431$	1.96307e8	0.118104	0.0590520	−	0.998255i	$-0.481192\pi$
$431$	1.96307e8	0.118104	0.0590520	+	0.998255i	$0.481192\pi$
$432$	0	0
$433$	7.68256e8	0.454777	0.227389	−	0.973804i	$-0.426981\pi$
$433$	7.68256e8	0.454777	0.227389	+	0.973804i	$0.426981\pi$
$434$	−1.05809e9	−0.621308
$435$	0	0
$436$	−1.96175e9	−1.13355
$437$	−1.08877e9	−0.624095
$438$	0	0
$439$	−1.35112e9	−0.762197	−0.381099	−	0.924534i	$-0.624454\pi$
$439$	−1.35112e9	−0.762197	−0.381099	+	0.924534i	$0.624454\pi$
$440$	0	0
$441$	0	0
$442$	−1.58542e9	−0.873308
$443$	2.20956e9	1.20752	0.603759	−	0.797167i	$-0.293669\pi$
$443$	2.20956e9	1.20752	0.603759	+	0.797167i	$0.293669\pi$
$444$	0	0
$445$	0	0
$446$	−4.99792e9	−2.66758
$447$	0	0
$448$	1.40633e9	0.738950
$449$	−3.16264e9	−1.64887	−0.824436	−	0.565955i	$-0.808507\pi$
$449$	−3.16264e9	−1.64887	−0.824436	+	0.565955i	$0.808507\pi$
$450$	0	0
$451$	−2.09050e9	−1.07308
$452$	4.71558e9	2.40188
$453$	0	0
$454$	7.08732e6	0.00355457
$455$	0	0
$456$	0	0
$457$	−5.97325e8	−0.292755	−0.146377	−	0.989229i	$-0.546761\pi$
$457$	−5.97325e8	−0.292755	−0.146377	+	0.989229i	$0.546761\pi$
$458$	−4.20082e9	−2.04317
$459$	0	0
$460$	0	0
$461$	2.01803e9	0.959346	0.479673	−	0.877447i	$-0.340755\pi$
$461$	2.01803e9	0.959346	0.479673	+	0.877447i	$0.340755\pi$
$462$	0	0
$463$	−1.32264e8	−0.0619310	−0.0309655	−	0.999520i	$-0.509858\pi$
$463$	−1.32264e8	−0.0619310	−0.0309655	+	0.999520i	$0.509858\pi$
$464$	−6.16136e8	−0.286328
$465$	0	0
$466$	2.90449e9	1.32960
$467$	−3.62018e9	−1.64483	−0.822416	−	0.568887i	$-0.807374\pi$
$467$	−3.62018e9	−1.64483	−0.822416	+	0.568887i	$0.807374\pi$
$468$	0	0
$469$	6.86293e8	0.307188
$470$	0	0
$471$	0	0
$472$	4.48976e9	1.96529
$473$	3.38745e9	1.47183
$474$	0	0
$475$	0	0
$476$	2.33766e9	0.993477
$477$	0	0
$478$	−7.62113e8	−0.319170
$479$	2.77900e9	1.15535	0.577675	−	0.816267i	$-0.303960\pi$
$479$	2.77900e9	1.15535	0.577675	+	0.816267i	$0.303960\pi$
$480$	0	0
$481$	−5.68778e8	−0.233042
$482$	−5.73703e9	−2.33358
$483$	0	0
$484$	3.04189e9	1.21951
$485$	0	0
$486$	0	0
$487$	−1.86684e9	−0.732414	−0.366207	−	0.930533i	$-0.619344\pi$
$487$	−1.86684e9	−0.732414	−0.366207	+	0.930533i	$0.619344\pi$
$488$	4.45661e9	1.73594
$489$	0	0
$490$	0	0
$491$	−5.06515e9	−1.93111	−0.965555	−	0.260198i	$-0.916212\pi$
$491$	−5.06515e9	−1.93111	−0.965555	+	0.260198i	$0.916212\pi$
$492$	0	0
$493$	3.21590e9	1.20876
$494$	1.09632e9	0.409158
$495$	0	0
$496$	5.92973e8	0.218197
$497$	−1.31176e8	−0.0479299
$498$	0	0
$499$	−5.56606e8	−0.200538	−0.100269	−	0.994960i	$-0.531970\pi$
$499$	−5.56606e8	−0.200538	−0.100269	+	0.994960i	$0.531970\pi$
$500$	0	0
$501$	0	0
$502$	2.44044e9	0.861004
$503$	−7.79533e8	−0.273116	−0.136558	−	0.990632i	$-0.543604\pi$
$503$	−7.79533e8	−0.273116	−0.136558	+	0.990632i	$0.543604\pi$
$504$	0	0
$505$	0	0
$506$	7.07445e9	2.42754
$507$	0	0
$508$	5.68835e9	1.92515
$509$	1.12351e9	0.377629	0.188814	−	0.982013i	$-0.439536\pi$
$509$	1.12351e9	0.377629	0.188814	+	0.982013i	$0.439536\pi$
$510$	0	0
$511$	1.36934e8	0.0453981
$512$	−1.68278e9	−0.554094
$513$	0	0
$514$	6.05547e9	1.96688
$515$	0	0
$516$	0	0
$517$	−1.98053e9	−0.630326
$518$	1.32142e9	0.417721
$519$	0	0
$520$	0	0
$521$	6.43550e8	0.199366	0.0996828	−	0.995019i	$-0.468217\pi$
$521$	6.43550e8	0.199366	0.0996828	+	0.995019i	$0.468217\pi$
$522$	0	0
$523$	−6.30863e9	−1.92832	−0.964159	−	0.265324i	$-0.914521\pi$
$523$	−6.30863e9	−1.92832	−0.964159	+	0.265324i	$0.914521\pi$
$524$	−2.90486e9	−0.881993
$525$	0	0
$526$	4.49030e8	0.134532
$527$	−3.09500e9	−0.921137
$528$	0	0
$529$	9.02060e8	0.264936
$530$	0	0
$531$	0	0
$532$	−1.61649e9	−0.465459
$533$	1.28150e9	0.366585
$534$	0	0
$535$	0	0
$536$	−2.76608e9	−0.775868
$537$	0	0
$538$	3.24963e9	0.899697
$539$	3.63704e9	1.00043
$540$	0	0
$541$	−3.71277e9	−1.00811	−0.504055	−	0.863672i	$-0.668159\pi$
$541$	−3.71277e9	−1.00811	−0.504055	+	0.863672i	$0.668159\pi$
$542$	9.51758e9	2.56761
$543$	0	0
$544$	3.35951e9	0.894705
$545$	0	0
$546$	0	0
$547$	−4.19241e9	−1.09524	−0.547619	−	0.836728i	$-0.684466\pi$
$547$	−4.19241e9	−1.09524	−0.547619	+	0.836728i	$0.684466\pi$
$548$	4.75888e9	1.23530
$549$	0	0
$550$	0	0
$551$	−2.22379e9	−0.566321
$552$	0	0
$553$	8.55564e8	0.215137
$554$	1.12503e10	2.81113
$555$	0	0
$556$	8.90883e9	2.19816
$557$	2.52750e9	0.619723	0.309861	−	0.950782i	$-0.399717\pi$
$557$	2.52750e9	0.619723	0.309861	+	0.950782i	$0.399717\pi$
$558$	0	0
$559$	−2.07656e9	−0.502808
$560$	0	0
$561$	0	0
$562$	1.19085e10	2.82996
$563$	−4.12649e9	−0.974543	−0.487272	−	0.873250i	$-0.662008\pi$
$563$	−4.12649e9	−0.974543	−0.487272	+	0.873250i	$0.662008\pi$
$564$	0	0
$565$	0	0
$566$	−1.02233e9	−0.236992
$567$	0	0
$568$	5.28700e8	0.121057
$569$	−5.27044e8	−0.119937	−0.0599686	−	0.998200i	$-0.519100\pi$
$569$	−5.27044e8	−0.119937	−0.0599686	+	0.998200i	$0.519100\pi$
$570$	0	0
$571$	−3.18083e9	−0.715014	−0.357507	−	0.933911i	$-0.616373\pi$
$571$	−3.18083e9	−0.715014	−0.357507	+	0.933911i	$0.616373\pi$
$572$	−4.52097e9	−1.01006
$573$	0	0
$574$	−2.97726e9	−0.657091
$575$	0	0
$576$	0	0
$577$	−4.88332e9	−1.05828	−0.529139	−	0.848535i	$-0.677485\pi$
$577$	−4.88332e9	−1.05828	−0.529139	+	0.848535i	$0.677485\pi$
$578$	3.09351e9	0.666354
$579$	0	0
$580$	0	0
$581$	2.72283e8	0.0575976
$582$	0	0
$583$	9.56814e9	1.99981
$584$	−5.51907e8	−0.114662
$585$	0	0
$586$	−2.29257e9	−0.470632
$587$	8.66750e8	0.176873	0.0884363	−	0.996082i	$-0.471813\pi$
$587$	8.66750e8	0.176873	0.0884363	+	0.996082i	$0.471813\pi$
$588$	0	0
$589$	2.14019e9	0.431567
$590$	0	0
$591$	0	0
$592$	−7.40549e8	−0.146699
$593$	−1.43710e9	−0.283007	−0.141503	−	0.989938i	$-0.545194\pi$
$593$	−1.43710e9	−0.283007	−0.141503	+	0.989938i	$0.545194\pi$
$594$	0	0
$595$	0	0
$596$	1.32556e10	2.56470
$597$	0	0
$598$	−4.33674e9	−0.829294
$599$	−7.50069e9	−1.42596	−0.712980	−	0.701185i	$-0.752655\pi$
$599$	−7.50069e9	−1.42596	−0.712980	+	0.701185i	$0.752655\pi$
$600$	0	0
$601$	5.05436e9	0.949741	0.474870	−	0.880056i	$-0.342495\pi$
$601$	5.05436e9	0.949741	0.474870	+	0.880056i	$0.342495\pi$
$602$	4.82437e9	0.901266
$603$	0	0
$604$	−1.15884e9	−0.213991
$605$	0	0
$606$	0	0
$607$	9.94598e8	0.180504	0.0902521	−	0.995919i	$-0.471233\pi$
$607$	9.94598e8	0.180504	0.0902521	+	0.995919i	$0.471233\pi$
$608$	−2.32309e9	−0.419183
$609$	0	0
$610$	0	0
$611$	1.21409e9	0.215332
$612$	0	0
$613$	−1.58283e9	−0.277539	−0.138769	−	0.990325i	$-0.544315\pi$
$613$	−1.58283e9	−0.277539	−0.138769	+	0.990325i	$0.544315\pi$
$614$	−1.04480e10	−1.82157
$615$	0	0
$616$	4.45707e9	0.768275
$617$	−8.35079e9	−1.43130	−0.715648	−	0.698461i	$-0.753868\pi$
$617$	−8.35079e9	−1.43130	−0.715648	+	0.698461i	$0.753868\pi$
$618$	0	0
$619$	1.12049e9	0.189886	0.0949428	−	0.995483i	$-0.469733\pi$
$619$	1.12049e9	0.189886	0.0949428	+	0.995483i	$0.469733\pi$
$620$	0	0
$621$	0	0
$622$	−1.67443e9	−0.278997
$623$	2.53408e9	0.419868
$624$	0	0
$625$	0	0
$626$	5.16415e9	0.841372
$627$	0	0
$628$	2.45030e9	0.394785
$629$	3.86528e9	0.619303
$630$	0	0
$631$	3.72303e9	0.589921	0.294961	−	0.955509i	$-0.404693\pi$
$631$	3.72303e9	0.589921	0.294961	+	0.955509i	$0.404693\pi$
$632$	−3.44832e9	−0.543372
$633$	0	0
$634$	−8.24424e9	−1.28481
$635$	0	0
$636$	0	0
$637$	−2.22956e9	−0.341767
$638$	1.44494e10	2.20281
$639$	0	0
$640$	0	0
$641$	1.08733e9	0.163065	0.0815323	−	0.996671i	$-0.474019\pi$
$641$	1.08733e9	0.163065	0.0815323	+	0.996671i	$0.474019\pi$
$642$	0	0
$643$	−9.30287e9	−1.38000	−0.689999	−	0.723810i	$-0.742389\pi$
$643$	−9.30287e9	−1.38000	−0.689999	+	0.723810i	$0.742389\pi$
$644$	6.39440e9	0.943407
$645$	0	0
$646$	−7.45029e9	−1.08733
$647$	−3.75633e9	−0.545254	−0.272627	−	0.962120i	$-0.587892\pi$
$647$	−3.75633e9	−0.545254	−0.272627	+	0.962120i	$0.587892\pi$
$648$	0	0
$649$	−1.46405e10	−2.10232
$650$	0	0
$651$	0	0
$652$	−5.16788e9	−0.730207
$653$	6.47262e9	0.909671	0.454835	−	0.890576i	$-0.349698\pi$
$653$	6.47262e9	0.909671	0.454835	+	0.890576i	$0.349698\pi$
$654$	0	0
$655$	0	0
$656$	1.66852e9	0.230764
$657$	0	0
$658$	−2.82065e9	−0.385975
$659$	−8.31257e9	−1.13145	−0.565726	−	0.824593i	$-0.691404\pi$
$659$	−8.31257e9	−1.13145	−0.565726	+	0.824593i	$0.691404\pi$
$660$	0	0
$661$	−1.42966e9	−0.192543	−0.0962714	−	0.995355i	$-0.530692\pi$
$661$	−1.42966e9	−0.192543	−0.0962714	+	0.995355i	$0.530692\pi$
$662$	3.14608e9	0.421470
$663$	0	0
$664$	−1.09743e9	−0.145475
$665$	0	0
$666$	0	0
$667$	8.79672e9	1.14784
$668$	−1.29890e10	−1.68600
$669$	0	0
$670$	0	0
$671$	−1.45324e10	−1.85698
$672$	0	0
$673$	−7.03224e9	−0.889285	−0.444642	−	0.895708i	$-0.646669\pi$
$673$	−7.03224e9	−0.889285	−0.444642	+	0.895708i	$0.646669\pi$
$674$	−1.56977e10	−1.97481
$675$	0	0
$676$	−1.11811e10	−1.39210
$677$	−5.75322e8	−0.0712608	−0.0356304	−	0.999365i	$-0.511344\pi$
$677$	−5.75322e8	−0.0712608	−0.0356304	+	0.999365i	$0.511344\pi$
$678$	0	0
$679$	3.15568e9	0.386856
$680$	0	0
$681$	0	0
$682$	−1.39062e10	−1.67866
$683$	7.67069e9	0.921217	0.460609	−	0.887603i	$-0.347631\pi$
$683$	7.67069e9	0.921217	0.460609	+	0.887603i	$0.347631\pi$
$684$	0	0
$685$	0	0
$686$	1.19346e10	1.41148
$687$	0	0
$688$	−2.70368e9	−0.316516
$689$	−5.86541e9	−0.683173
$690$	0	0
$691$	3.62855e9	0.418369	0.209185	−	0.977876i	$-0.432919\pi$
$691$	3.62855e9	0.418369	0.209185	+	0.977876i	$0.432919\pi$
$692$	2.64222e8	0.0303108
$693$	0	0
$694$	−2.17365e10	−2.46849
$695$	0	0
$696$	0	0
$697$	−8.70878e9	−0.974188
$698$	−1.56758e10	−1.74476
$699$	0	0
$700$	0	0
$701$	1.71375e10	1.87904	0.939518	−	0.342499i	$-0.111273\pi$
$701$	1.71375e10	1.87904	0.939518	+	0.342499i	$0.111273\pi$
$702$	0	0
$703$	−2.67283e9	−0.290153
$704$	1.84831e10	1.99651
$705$	0	0
$706$	1.42547e10	1.52456
$707$	−1.27948e9	−0.136166
$708$	0	0
$709$	1.34055e10	1.41260	0.706302	−	0.707911i	$-0.250362\pi$
$709$	1.34055e10	1.41260	0.706302	+	0.707911i	$0.250362\pi$
$710$	0	0
$711$	0	0
$712$	−1.02135e10	−1.06046
$713$	−8.46601e9	−0.874712
$714$	0	0
$715$	0	0
$716$	−2.86027e10	−2.91214
$717$	0	0
$718$	−5.98460e9	−0.603392
$719$	5.99982e9	0.601987	0.300994	−	0.953626i	$-0.402682\pi$
$719$	5.99982e9	0.601987	0.300994	+	0.953626i	$0.402682\pi$
$720$	0	0
$721$	1.72929e9	0.171829
$722$	−1.15795e10	−1.14501
$723$	0	0
$724$	3.12010e10	3.05551
$725$	0	0
$726$	0	0
$727$	1.84312e10	1.77903	0.889513	−	0.456909i	$-0.151044\pi$
$727$	1.84312e10	1.77903	0.889513	+	0.456909i	$0.151044\pi$
$728$	−2.73225e9	−0.262458
$729$	0	0
$730$	0	0
$731$	1.41118e10	1.33620
$732$	0	0
$733$	7.01069e9	0.657502	0.328751	−	0.944417i	$-0.393372\pi$
$733$	7.01069e9	0.657502	0.328751	+	0.944417i	$0.393372\pi$
$734$	−2.34749e9	−0.219113
$735$	0	0
$736$	9.18953e9	0.849613
$737$	9.01980e9	0.829966
$738$	0	0
$739$	3.58081e9	0.326382	0.163191	−	0.986595i	$-0.447821\pi$
$739$	3.58081e9	0.326382	0.163191	+	0.986595i	$0.447821\pi$
$740$	0	0
$741$	0	0
$742$	1.36268e10	1.22456
$743$	−4.23347e9	−0.378648	−0.189324	−	0.981915i	$-0.560630\pi$
$743$	−4.23347e9	−0.378648	−0.189324	+	0.981915i	$0.560630\pi$
$744$	0	0
$745$	0	0
$746$	−3.21019e10	−2.83103
$747$	0	0
$748$	3.07234e10	2.68419
$749$	1.67387e9	0.145558
$750$	0	0
$751$	3.94072e9	0.339497	0.169749	−	0.985487i	$-0.445704\pi$
$751$	3.94072e9	0.339497	0.169749	+	0.985487i	$0.445704\pi$
$752$	1.58075e9	0.135550
$753$	0	0
$754$	−8.85769e9	−0.752524
$755$	0	0
$756$	0	0
$757$	8.01225e8	0.0671303	0.0335652	−	0.999437i	$-0.489314\pi$
$757$	8.01225e8	0.0671303	0.0335652	+	0.999437i	$0.489314\pi$
$758$	−2.00836e9	−0.167494
$759$	0	0
$760$	0	0
$761$	3.84688e9	0.316419	0.158209	−	0.987406i	$-0.449428\pi$
$761$	3.84688e9	0.316419	0.158209	+	0.987406i	$0.449428\pi$
$762$	0	0
$763$	3.86603e9	0.315086
$764$	8.48919e9	0.688715
$765$	0	0
$766$	−1.26807e10	−1.01940
$767$	8.97482e9	0.718194
$768$	0	0
$769$	4.40578e9	0.349366	0.174683	−	0.984625i	$-0.444110\pi$
$769$	4.40578e9	0.349366	0.174683	+	0.984625i	$0.444110\pi$
$770$	0	0
$771$	0	0
$772$	3.02216e10	2.36405
$773$	−1.54451e10	−1.20272	−0.601359	−	0.798979i	$-0.705374\pi$
$773$	−1.54451e10	−1.20272	−0.601359	+	0.798979i	$0.705374\pi$
$774$	0	0
$775$	0	0
$776$	−1.27189e10	−0.977085
$777$	0	0
$778$	−3.64690e10	−2.77649
$779$	6.02210e9	0.456422
$780$	0	0
$781$	−1.72401e9	−0.129498
$782$	2.94714e10	2.20382
$783$	0	0
$784$	−2.90288e9	−0.215141
$785$	0	0
$786$	0	0
$787$	2.40331e10	1.75751	0.878755	−	0.477274i	$-0.158375\pi$
$787$	2.40331e10	1.75751	0.878755	+	0.477274i	$0.158375\pi$
$788$	1.37006e10	0.997463
$789$	0	0
$790$	0	0
$791$	−9.29299e9	−0.667633
$792$	0	0
$793$	8.90856e9	0.634383
$794$	2.09174e10	1.48299
$795$	0	0
$796$	−4.09973e10	−2.88111
$797$	1.37624e10	0.962918	0.481459	−	0.876469i	$-0.340107\pi$
$797$	1.37624e10	0.962918	0.481459	+	0.876469i	$0.340107\pi$
$798$	0	0
$799$	−8.25068e9	−0.572237
$800$	0	0
$801$	0	0
$802$	−3.01947e10	−2.06690
$803$	1.79969e9	0.122657
$804$	0	0
$805$	0	0
$806$	8.52468e9	0.573463
$807$	0	0
$808$	5.15692e9	0.343914
$809$	−5.00163e9	−0.332118	−0.166059	−	0.986116i	$-0.553104\pi$
$809$	−5.00163e9	−0.332118	−0.166059	+	0.986116i	$0.553104\pi$
$810$	0	0
$811$	8.14966e9	0.536496	0.268248	−	0.963350i	$-0.413555\pi$
$811$	8.14966e9	0.536496	0.268248	+	0.963350i	$0.413555\pi$
$812$	1.30604e10	0.856073
$813$	0	0
$814$	1.73671e10	1.12860
$815$	0	0
$816$	0	0
$817$	−9.75824e9	−0.626029
$818$	2.48697e8	0.0158867
$819$	0	0
$820$	0	0
$821$	−1.30820e10	−0.825035	−0.412518	−	0.910950i	$-0.635350\pi$
$821$	−1.30820e10	−0.825035	−0.412518	+	0.910950i	$0.635350\pi$
$822$	0	0
$823$	3.12969e10	1.95705	0.978524	−	0.206133i	$-0.0660881\pi$
$823$	3.12969e10	1.95705	0.978524	+	0.206133i	$0.0660881\pi$
$824$	−6.96985e9	−0.433989
$825$	0	0
$826$	−2.08508e10	−1.28734
$827$	−1.98645e10	−1.22126	−0.610630	−	0.791916i	$-0.709084\pi$
$827$	−1.98645e10	−1.22126	−0.610630	+	0.791916i	$0.709084\pi$
$828$	0	0
$829$	1.06183e10	0.647310	0.323655	−	0.946175i	$-0.395088\pi$
$829$	1.06183e10	0.647310	0.323655	+	0.946175i	$0.395088\pi$
$830$	0	0
$831$	0	0
$832$	−1.13304e10	−0.682046
$833$	1.51515e10	0.908235
$834$	0	0
$835$	0	0
$836$	−2.12451e10	−1.25758
$837$	0	0
$838$	1.11043e9	0.0651834
$839$	−4.19640e9	−0.245307	−0.122654	−	0.992450i	$-0.539140\pi$
$839$	−4.19640e9	−0.245307	−0.122654	+	0.992450i	$0.539140\pi$
$840$	0	0
$841$	7.17225e8	0.0415785
$842$	−5.49082e10	−3.16990
$843$	0	0
$844$	3.80615e10	2.17915
$845$	0	0
$846$	0	0
$847$	−5.99465e9	−0.338979
$848$	−7.63676e9	−0.430055
$849$	0	0
$850$	0	0
$851$	1.05730e10	0.588091
$852$	0	0
$853$	−2.52606e10	−1.39355	−0.696775	−	0.717289i	$-0.745383\pi$
$853$	−2.52606e10	−1.39355	−0.696775	+	0.717289i	$0.745383\pi$
$854$	−2.06969e10	−1.13711
$855$	0	0
$856$	−6.74648e9	−0.367637
$857$	1.07248e10	0.582046	0.291023	−	0.956716i	$-0.406004\pi$
$857$	1.07248e10	0.582046	0.291023	+	0.956716i	$0.406004\pi$
$858$	0	0
$859$	−3.02643e10	−1.62913	−0.814563	−	0.580075i	$-0.803023\pi$
$859$	−3.02643e10	−1.62913	−0.814563	+	0.580075i	$0.803023\pi$
$860$	0	0
$861$	0	0
$862$	3.67443e9	0.195396
$863$	−1.39177e10	−0.737106	−0.368553	−	0.929607i	$-0.620147\pi$
$863$	−1.39177e10	−0.737106	−0.368553	+	0.929607i	$0.620147\pi$
$864$	0	0
$865$	0	0
$866$	1.43801e10	0.752399
$867$	0	0
$868$	−1.25694e10	−0.652373
$869$	1.12445e10	0.581260
$870$	0	0
$871$	−5.52926e9	−0.283533
$872$	−1.55819e10	−0.795815
$873$	0	0
$874$	−2.03794e10	−1.03253
$875$	0	0
$876$	0	0
$877$	1.81702e10	0.909622	0.454811	−	0.890588i	$-0.349707\pi$
$877$	1.81702e10	0.909622	0.454811	+	0.890588i	$0.349707\pi$
$878$	−2.52900e10	−1.26101
$879$	0	0
$880$	0	0
$881$	3.14352e10	1.54882	0.774409	−	0.632685i	$-0.218047\pi$
$881$	3.14352e10	1.54882	0.774409	+	0.632685i	$0.218047\pi$
$882$	0	0
$883$	2.11722e10	1.03491	0.517457	−	0.855709i	$-0.326879\pi$
$883$	2.11722e10	1.03491	0.517457	+	0.855709i	$0.326879\pi$
$884$	−1.88339e10	−0.916973
$885$	0	0
$886$	4.13582e10	1.99776
$887$	3.72771e10	1.79353	0.896766	−	0.442505i	$-0.145910\pi$
$887$	3.72771e10	1.79353	0.896766	+	0.442505i	$0.145910\pi$
$888$	0	0
$889$	−1.12100e10	−0.535119
$890$	0	0
$891$	0	0
$892$	−5.93722e10	−2.80095
$893$	5.70532e9	0.268102
$894$	0	0
$895$	0	0
$896$	1.84694e10	0.857780
$897$	0	0
$898$	−5.91976e10	−2.72795
$899$	−1.72916e10	−0.793738
$900$	0	0
$901$	3.98599e10	1.81551
$902$	−3.91295e10	−1.77534
$903$	0	0
$904$	3.74551e10	1.68625
$905$	0	0
$906$	0	0
$907$	−1.00189e10	−0.445854	−0.222927	−	0.974835i	$-0.571561\pi$
$907$	−1.00189e10	−0.445854	−0.222927	+	0.974835i	$0.571561\pi$
$908$	8.41930e7	0.00373229
$909$	0	0
$910$	0	0
$911$	−3.16088e10	−1.38514	−0.692571	−	0.721350i	$-0.743522\pi$
$911$	−3.16088e10	−1.38514	−0.692571	+	0.721350i	$0.743522\pi$
$912$	0	0
$913$	3.57855e9	0.155618
$914$	−1.11806e10	−0.484344
$915$	0	0
$916$	−4.99032e10	−2.14533
$917$	5.72460e9	0.245161
$918$	0	0
$919$	7.18397e8	0.0305324	0.0152662	−	0.999883i	$-0.495140\pi$
$919$	7.18397e8	0.0305324	0.0152662	+	0.999883i	$0.495140\pi$
$920$	0	0
$921$	0	0
$922$	3.77732e10	1.58718
$923$	1.05685e9	0.0442390
$924$	0	0
$925$	0	0
$926$	−2.47569e9	−0.102461
$927$	0	0
$928$	1.87694e10	0.770962
$929$	1.63954e10	0.670915	0.335458	−	0.942055i	$-0.391109\pi$
$929$	1.63954e10	0.670915	0.335458	+	0.942055i	$0.391109\pi$
$930$	0	0
$931$	−1.04772e10	−0.425522
$932$	3.45036e10	1.39608
$933$	0	0
$934$	−6.77618e10	−2.72127
$935$	0	0
$936$	0	0
$937$	9.79890e9	0.389125	0.194562	−	0.980890i	$-0.437671\pi$
$937$	9.79890e9	0.389125	0.194562	+	0.980890i	$0.437671\pi$
$938$	1.28459e10	0.508223
$939$	0	0
$940$	0	0
$941$	−1.68542e10	−0.659394	−0.329697	−	0.944087i	$-0.606947\pi$
$941$	−1.68542e10	−0.659394	−0.329697	+	0.944087i	$0.606947\pi$
$942$	0	0
$943$	−2.38218e10	−0.925090
$944$	1.16852e10	0.452100
$945$	0	0
$946$	6.34056e10	2.43506
$947$	2.47225e10	0.945948	0.472974	−	0.881076i	$-0.343180\pi$
$947$	2.47225e10	0.945948	0.472974	+	0.881076i	$0.343180\pi$
$948$	0	0
$949$	−1.10324e9	−0.0419022
$950$	0	0
$951$	0	0
$952$	1.85677e10	0.697474
$953$	−4.36738e9	−0.163454	−0.0817271	−	0.996655i	$-0.526044\pi$
$953$	−4.36738e9	−0.163454	−0.0817271	+	0.996655i	$0.526044\pi$
$954$	0	0
$955$	0	0
$956$	−9.05343e9	−0.335128
$957$	0	0
$958$	5.20167e10	1.91145
$959$	−9.37832e9	−0.343368
$960$	0	0
$961$	−1.08711e10	−0.395130
$962$	−1.06463e10	−0.385554
$963$	0	0
$964$	−6.81523e10	−2.45025
$965$	0	0
$966$	0	0
$967$	1.00818e10	0.358547	0.179274	−	0.983799i	$-0.442625\pi$
$967$	1.00818e10	0.358547	0.179274	+	0.983799i	$0.442625\pi$
$968$	2.41612e10	0.856161
$969$	0	0
$970$	0	0
$971$	1.19066e10	0.417369	0.208685	−	0.977983i	$-0.433082\pi$
$971$	1.19066e10	0.417369	0.208685	+	0.977983i	$0.433082\pi$
$972$	0	0
$973$	−1.75566e10	−0.611006
$974$	−3.49432e10	−1.21173
$975$	0	0
$976$	1.15989e10	0.399341
$977$	1.99598e10	0.684741	0.342370	−	0.939565i	$-0.388770\pi$
$977$	1.99598e10	0.684741	0.342370	+	0.939565i	$0.388770\pi$
$978$	0	0
$979$	3.33049e10	1.13441
$980$	0	0
$981$	0	0
$982$	−9.48085e10	−3.19490
$983$	−3.22193e10	−1.08188	−0.540940	−	0.841061i	$-0.681931\pi$
$983$	−3.22193e10	−1.08188	−0.540940	+	0.841061i	$0.681931\pi$
$984$	0	0
$985$	0	0
$986$	6.01947e10	1.99981
$987$	0	0
$988$	1.30236e10	0.429616
$989$	3.86010e10	1.26885
$990$	0	0
$991$	−1.42552e10	−0.465281	−0.232640	−	0.972563i	$-0.574737\pi$
$991$	−1.42552e10	−0.465281	−0.232640	+	0.972563i	$0.574737\pi$
$992$	−1.80638e10	−0.587513
$993$	0	0
$994$	−2.45532e9	−0.0792969
$995$	0	0
$996$	0	0
$997$	2.73327e10	0.873475	0.436737	−	0.899589i	$-0.356134\pi$
$997$	2.73327e10	0.873475	0.436737	+	0.899589i	$0.356134\pi$
$998$	−1.04184e10	−0.331776
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.8.a.w.1.2		2
3.2	odd	2		25.8.a.b.1.1		2
5.2	odd	4		225.8.b.m.199.4		4
5.3	odd	4		225.8.b.m.199.1		4
5.4	even	2		45.8.a.h.1.1		2
12.11	even	2		400.8.a.bb.1.1		2
15.2	even	4		25.8.b.c.24.1		4
15.8	even	4		25.8.b.c.24.4		4
15.14	odd	2		5.8.a.b.1.2	✓	2
60.23	odd	4		400.8.c.m.49.2		4
60.47	odd	4		400.8.c.m.49.3		4
60.59	even	2		80.8.a.g.1.2		2
105.104	even	2		245.8.a.c.1.2		2
120.29	odd	2		320.8.a.l.1.2		2
120.59	even	2		320.8.a.u.1.1		2
165.164	even	2		605.8.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.8.a.b.1.2	✓	2	15.14	odd	2
25.8.a.b.1.1		2	3.2	odd	2
25.8.b.c.24.1		4	15.2	even	4
25.8.b.c.24.4		4	15.8	even	4
45.8.a.h.1.1		2	5.4	even	2
80.8.a.g.1.2		2	60.59	even	2
225.8.a.w.1.2		2	1.1	even	1	trivial
225.8.b.m.199.1		4	5.3	odd	4
225.8.b.m.199.4		4	5.2	odd	4
245.8.a.c.1.2		2	105.104	even	2
320.8.a.l.1.2		2	120.29	odd	2
320.8.a.u.1.1		2	120.59	even	2
400.8.a.bb.1.1		2	12.11	even	2
400.8.c.m.49.2		4	60.23	odd	4
400.8.c.m.49.3		4	60.47	odd	4
605.8.a.d.1.1		2	165.164	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q+18.7178 q^{2} +222.356 q^{4} -438.197 q^{7} +1766.14 q^{8} -5759.12 q^{11} +3530.42 q^{13} -8202.08 q^{14} +4596.61 q^{16} -23991.9 q^{17} +16590.3 q^{19} -107798. q^{22} -65626.9 q^{23} +66081.7 q^{26} -97435.6 q^{28} -134041. q^{29} +129002. q^{31} -140027. q^{32} -449075. q^{34} -161108. q^{37} +310534. q^{38} +362989. q^{41} -588189. q^{43} -1.28057e6 q^{44} -1.22839e6 q^{46} +343895. q^{47} -631527. q^{49} +785010. q^{52} -1.66139e6 q^{53} -773915. q^{56} -2.50896e6 q^{58} +2.54214e6 q^{59} +2.52337e6 q^{61} +2.41464e6 q^{62} -3.20936e6 q^{64} -1.56618e6 q^{67} -5.33474e6 q^{68} +299354. q^{71} -312494. q^{73} -3.01558e6 q^{74} +3.68895e6 q^{76} +2.52363e6 q^{77} -1.95247e6 q^{79} +6.79435e6 q^{82} -621372. q^{83} -1.10096e7 q^{86} -1.01714e7 q^{88} -5.78298e6 q^{89} -1.54702e6 q^{91} -1.45925e7 q^{92} +6.43696e6 q^{94} -7.20152e6 q^{97} -1.18208e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 20 q^{2} + 96 q^{4} + 100 q^{7} + 1440 q^{8} - 4544 q^{11} - 3540 q^{13} - 7512 q^{14} + 20352 q^{16} - 27340 q^{17} + 38760 q^{19} - 106240 q^{22} - 124140 q^{23} + 57016 q^{26} - 165440 q^{28} + 72260 q^{29}+ \cdots - 12505340 q^{98}+O(q^{100}) \) 2 * q + 20 * q^2 + 96 * q^4 + 100 * q^7 + 1440 * q^8 - 4544 * q^11 - 3540 * q^13 - 7512 * q^14 + 20352 * q^16 - 27340 * q^17 + 38760 * q^19 - 106240 * q^22 - 124140 * q^23 + 57016 * q^26 - 165440 * q^28 + 72260 * q^29 + 306824 * q^31 - 78080 * q^32 - 453368 * q^34 + 123020 * q^37 + 338960 * q^38 - 264364 * q^41 - 423300 * q^43 - 1434112 * q^44 - 1303416 * q^46 - 105460 * q^47 - 1165414 * q^49 + 1678400 * q^52 - 2391580 * q^53 - 949440 * q^56 - 2244440 * q^58 + 1120120 * q^59 + 2257044 * q^61 + 2642640 * q^62 - 5146624 * q^64 - 4516460 * q^67 - 4911680 * q^68 - 621784 * q^71 - 4569060 * q^73 - 2651272 * q^74 + 887680 * q^76 + 3177600 * q^77 + 4333040 * q^79 + 5989960 * q^82 - 9793020 * q^83 - 10798184 * q^86 - 10567680 * q^88 - 6025620 * q^89 - 5352296 * q^91 - 7199040 * q^92 + 5860792 * q^94 - 4609540 * q^97 - 12505340 * q^98

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