LMFDB - Embedded newform 9025.2.a.ct.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, 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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.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:	$72.0649878242$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	9025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.78468 q^{2} +2.38377 q^{3} +1.18508 q^{4} +4.25426 q^{6} -4.23911 q^{7} -1.45438 q^{8} +2.68235 q^{9} -0.490889 q^{11} +2.82495 q^{12} +4.16199 q^{13} -7.56544 q^{14} -4.96575 q^{16} +2.03619 q^{17} +4.78713 q^{18} -10.1050 q^{21} -0.876080 q^{22} -4.39525 q^{23} -3.46689 q^{24} +7.42782 q^{26} -0.757211 q^{27} -5.02367 q^{28} +3.26270 q^{29} -4.08833 q^{31} -5.95351 q^{32} -1.17017 q^{33} +3.63395 q^{34} +3.17879 q^{36} -2.14440 q^{37} +9.92123 q^{39} -4.36602 q^{41} -18.0343 q^{42} -10.6241 q^{43} -0.581742 q^{44} -7.84410 q^{46} -2.62142 q^{47} -11.8372 q^{48} +10.9700 q^{49} +4.85381 q^{51} +4.93228 q^{52} +11.4341 q^{53} -1.35138 q^{54} +6.16526 q^{56} +5.82287 q^{58} +0.542306 q^{59} -13.6423 q^{61} -7.29635 q^{62} -11.3708 q^{63} -0.693606 q^{64} -2.08837 q^{66} -7.15699 q^{67} +2.41305 q^{68} -10.4772 q^{69} -6.03858 q^{71} -3.90114 q^{72} -2.05419 q^{73} -3.82707 q^{74} +2.08093 q^{77} +17.7062 q^{78} -5.34029 q^{79} -9.85206 q^{81} -7.79193 q^{82} -8.11578 q^{83} -11.9753 q^{84} -18.9605 q^{86} +7.77752 q^{87} +0.713937 q^{88} +4.34099 q^{89} -17.6431 q^{91} -5.20871 q^{92} -9.74562 q^{93} -4.67839 q^{94} -14.1918 q^{96} +5.64669 q^{97} +19.5780 q^{98} -1.31674 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 18 q^{4} - 12 q^{6} + 12 q^{9} + 12 q^{11} - 24 q^{14} + 6 q^{16} - 6 q^{21} - 42 q^{24} - 12 q^{26} - 36 q^{29} - 42 q^{31} - 6 q^{34} - 6 q^{36} + 24 q^{39} - 60 q^{41} - 30 q^{44} - 6 q^{46} + 12 q^{49}+ \cdots - 120 q^{96}+O(q^{100}) $ 24 * q + 18 * q^4 - 12 * q^6 + 12 * q^9 + 12 * q^11 - 24 * q^14 + 6 * q^16 - 6 * q^21 - 42 * q^24 - 12 * q^26 - 36 * q^29 - 42 * q^31 - 6 * q^34 - 6 * q^36 + 24 * q^39 - 60 * q^41 - 30 * q^44 - 6 * q^46 + 12 * q^49 - 30 * q^51 - 24 * q^54 - 18 * q^56 - 60 * q^59 + 30 * q^61 + 36 * q^66 - 66 * q^69 - 96 * q^71 + 24 * q^74 - 72 * q^79 - 96 * q^81 + 54 * q^84 - 108 * q^86 - 84 * q^89 - 96 * q^91 - 36 * q^94 - 120 * q^96

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 $

$2$	1.78468	1.26196	0.630979	−	0.775800i	$-0.282653\pi$
$2$	1.78468	1.26196	0.630979	+	0.775800i	$0.282653\pi$
$3$	2.38377	1.37627	0.688134	−	0.725583i	$-0.258430\pi$
$3$	2.38377	1.37627	0.688134	+	0.725583i	$0.258430\pi$
$4$	1.18508	0.592538
$5$	0	0
$5$	0	0
$6$	4.25426	1.73679
$7$	−4.23911	−1.60223	−0.801116	−	0.598509i	$-0.795760\pi$
$7$	−4.23911	−1.60223	−0.801116	+	0.598509i	$0.795760\pi$
$8$	−1.45438	−0.514199
$9$	2.68235	0.894116
$10$	0	0
$11$	−0.490889	−0.148009	−0.0740043	−	0.997258i	$-0.523578\pi$
$11$	−0.490889	−0.148009	−0.0740043	+	0.997258i	$0.523578\pi$
$12$	2.82495	0.815492
$13$	4.16199	1.15433	0.577165	−	0.816628i	$-0.304159\pi$
$13$	4.16199	1.15433	0.577165	+	0.816628i	$0.304159\pi$
$14$	−7.56544	−2.02195
$15$	0	0
$16$	−4.96575	−1.24144
$17$	2.03619	0.493850	0.246925	−	0.969035i	$-0.420580\pi$
$17$	2.03619	0.493850	0.246925	+	0.969035i	$0.420580\pi$
$18$	4.78713	1.12834
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−10.1050	−2.20510
$22$	−0.876080	−0.186781
$23$	−4.39525	−0.916472	−0.458236	−	0.888830i	$-0.651519\pi$
$23$	−4.39525	−0.916472	−0.458236	+	0.888830i	$0.651519\pi$
$24$	−3.46689	−0.707677
$25$	0	0
$26$	7.42782	1.45672
$27$	−0.757211	−0.145725
$28$	−5.02367	−0.949384
$29$	3.26270	0.605869	0.302934	−	0.953011i	$-0.402034\pi$
$29$	3.26270	0.605869	0.302934	+	0.953011i	$0.402034\pi$
$30$	0	0
$31$	−4.08833	−0.734285	−0.367143	−	0.930165i	$-0.619664\pi$
$31$	−4.08833	−0.734285	−0.367143	+	0.930165i	$0.619664\pi$
$32$	−5.95351	−1.05244
$33$	−1.17017	−0.203700
$34$	3.63395	0.623218
$35$	0	0
$36$	3.17879	0.529798
$37$	−2.14440	−0.352538	−0.176269	−	0.984342i	$-0.556403\pi$
$37$	−2.14440	−0.352538	−0.176269	+	0.984342i	$0.556403\pi$
$38$	0	0
$39$	9.92123	1.58867
$40$	0	0
$41$	−4.36602	−0.681857	−0.340929	−	0.940089i	$-0.610741\pi$
$41$	−4.36602	−0.681857	−0.340929	+	0.940089i	$0.610741\pi$
$42$	−18.0343	−2.78275
$43$	−10.6241	−1.62015	−0.810077	−	0.586324i	$-0.800575\pi$
$43$	−10.6241	−1.62015	−0.810077	+	0.586324i	$0.800575\pi$
$44$	−0.581742	−0.0877008
$45$	0	0
$46$	−7.84410	−1.15655
$47$	−2.62142	−0.382373	−0.191187	−	0.981554i	$-0.561234\pi$
$47$	−2.62142	−0.382373	−0.191187	+	0.981554i	$0.561234\pi$
$48$	−11.8372	−1.70855
$49$	10.9700	1.56715
$50$	0	0
$51$	4.85381	0.679670
$52$	4.93228	0.683985
$53$	11.4341	1.57059	0.785296	−	0.619121i	$-0.212511\pi$
$53$	11.4341	1.57059	0.785296	+	0.619121i	$0.212511\pi$
$54$	−1.35138	−0.183899
$55$	0	0
$56$	6.16526	0.823867
$57$	0	0
$58$	5.82287	0.764581
$59$	0.542306	0.0706022	0.0353011	−	0.999377i	$-0.488761\pi$
$59$	0.542306	0.0706022	0.0353011	+	0.999377i	$0.488761\pi$
$60$	0	0
$61$	−13.6423	−1.74671	−0.873356	−	0.487083i	$-0.838061\pi$
$61$	−13.6423	−1.74671	−0.873356	+	0.487083i	$0.838061\pi$
$62$	−7.29635	−0.926637
$63$	−11.3708	−1.43258
$64$	−0.693606	−0.0867007
$65$	0	0
$66$	−2.08837	−0.257061
$67$	−7.15699	−0.874365	−0.437182	−	0.899373i	$-0.644024\pi$
$67$	−7.15699	−0.874365	−0.437182	+	0.899373i	$0.644024\pi$
$68$	2.41305	0.292625
$69$	−10.4772	−1.26131
$70$	0	0
$71$	−6.03858	−0.716647	−0.358324	−	0.933597i	$-0.616652\pi$
$71$	−6.03858	−0.716647	−0.358324	+	0.933597i	$0.616652\pi$
$72$	−3.90114	−0.459754
$73$	−2.05419	−0.240425	−0.120212	−	0.992748i	$-0.538358\pi$
$73$	−2.05419	−0.240425	−0.120212	+	0.992748i	$0.538358\pi$
$74$	−3.82707	−0.444888
$75$	0	0
$76$	0	0
$77$	2.08093	0.237144
$78$	17.7062	2.00483
$79$	−5.34029	−0.600830	−0.300415	−	0.953809i	$-0.597125\pi$
$79$	−5.34029	−0.600830	−0.300415	+	0.953809i	$0.597125\pi$
$80$	0	0
$81$	−9.85206	−1.09467
$82$	−7.79193	−0.860475
$83$	−8.11578	−0.890823	−0.445411	−	0.895326i	$-0.646943\pi$
$83$	−8.11578	−0.890823	−0.445411	+	0.895326i	$0.646943\pi$
$84$	−11.9753	−1.30661
$85$	0	0
$86$	−18.9605	−2.04457
$87$	7.77752	0.833838
$88$	0.713937	0.0761060
$89$	4.34099	0.460144	0.230072	−	0.973174i	$-0.426104\pi$
$89$	4.34099	0.460144	0.230072	+	0.973174i	$0.426104\pi$
$90$	0	0
$91$	−17.6431	−1.84950
$92$	−5.20871	−0.543045
$93$	−9.74562	−1.01057
$94$	−4.67839	−0.482539
$95$	0	0
$96$	−14.1918	−1.44844
$97$	5.64669	0.573335	0.286667	−	0.958030i	$-0.407453\pi$
$97$	5.64669	0.573335	0.286667	+	0.958030i	$0.407453\pi$
$98$	19.5780	1.97768
$99$	−1.31674	−0.132337
$100$	0	0
$101$	−3.30082	−0.328444	−0.164222	−	0.986423i	$-0.552511\pi$
$101$	−3.30082	−0.328444	−0.164222	+	0.986423i	$0.552511\pi$
$102$	8.66250	0.857715
$103$	−3.41567	−0.336556	−0.168278	−	0.985740i	$-0.553821\pi$
$103$	−3.41567	−0.336556	−0.168278	+	0.985740i	$0.553821\pi$
$104$	−6.05310	−0.593556
$105$	0	0
$106$	20.4062	1.98202
$107$	−1.75252	−0.169422	−0.0847110	−	0.996406i	$-0.526997\pi$
$107$	−1.75252	−0.169422	−0.0847110	+	0.996406i	$0.526997\pi$
$108$	−0.897354	−0.0863479
$109$	3.51923	0.337081	0.168541	−	0.985695i	$-0.446095\pi$
$109$	3.51923	0.337081	0.168541	+	0.985695i	$0.446095\pi$
$110$	0	0
$111$	−5.11176	−0.485186
$112$	21.0503	1.98907
$113$	13.2583	1.24723	0.623616	−	0.781731i	$-0.285663\pi$
$113$	13.2583	1.24723	0.623616	+	0.781731i	$0.285663\pi$
$114$	0	0
$115$	0	0
$116$	3.86655	0.359000
$117$	11.1639	1.03210
$118$	0.967842	0.0890971
$119$	−8.63165	−0.791262
$120$	0	0
$121$	−10.7590	−0.978093
$122$	−24.3470	−2.20428
$123$	−10.4076	−0.938419
$124$	−4.84498	−0.435092
$125$	0	0
$126$	−20.2931	−1.80786
$127$	−19.8082	−1.75769	−0.878846	−	0.477105i	$-0.841686\pi$
$127$	−19.8082	−1.75769	−0.878846	+	0.477105i	$0.841686\pi$
$128$	10.6692	0.943029
$129$	−25.3253	−2.22977
$130$	0	0
$131$	−3.43004	−0.299684	−0.149842	−	0.988710i	$-0.547876\pi$
$131$	−3.43004	−0.299684	−0.149842	+	0.988710i	$0.547876\pi$
$132$	−1.38674	−0.120700
$133$	0	0
$134$	−12.7729	−1.10341
$135$	0	0
$136$	−2.96139	−0.253937
$137$	−0.608296	−0.0519702	−0.0259851	−	0.999662i	$-0.508272\pi$
$137$	−0.608296	−0.0519702	−0.0259851	+	0.999662i	$0.508272\pi$
$138$	−18.6985	−1.59172
$139$	−8.45799	−0.717397	−0.358699	−	0.933453i	$-0.616779\pi$
$139$	−8.45799	−0.717397	−0.358699	+	0.933453i	$0.616779\pi$
$140$	0	0
$141$	−6.24885	−0.526248
$142$	−10.7769	−0.904379
$143$	−2.04308	−0.170851
$144$	−13.3199	−1.10999
$145$	0	0
$146$	−3.66607	−0.303406
$147$	26.1500	2.15682
$148$	−2.54128	−0.208892
$149$	−4.63735	−0.379907	−0.189953	−	0.981793i	$-0.560834\pi$
$149$	−4.63735	−0.379907	−0.189953	+	0.981793i	$0.560834\pi$
$150$	0	0
$151$	−14.2109	−1.15646	−0.578232	−	0.815872i	$-0.696257\pi$
$151$	−14.2109	−1.15646	−0.578232	+	0.815872i	$0.696257\pi$
$152$	0	0
$153$	5.46178	0.441559
$154$	3.71380	0.299266
$155$	0	0
$156$	11.7574	0.941347
$157$	−10.7281	−0.856194	−0.428097	−	0.903733i	$-0.640816\pi$
$157$	−10.7281	−0.856194	−0.428097	+	0.903733i	$0.640816\pi$
$158$	−9.53071	−0.758222
$159$	27.2562	2.16156
$160$	0	0
$161$	18.6319	1.46840
$162$	−17.5828	−1.38143
$163$	16.9366	1.32658	0.663289	−	0.748363i	$-0.269160\pi$
$163$	16.9366	1.32658	0.663289	+	0.748363i	$0.269160\pi$
$164$	−5.17406	−0.404027
$165$	0	0
$166$	−14.4841	−1.12418
$167$	16.2567	1.25798	0.628991	−	0.777412i	$-0.283468\pi$
$167$	16.2567	1.25798	0.628991	+	0.777412i	$0.283468\pi$
$168$	14.6965	1.13386
$169$	4.32219	0.332476
$170$	0	0
$171$	0	0
$172$	−12.5903	−0.960003
$173$	8.59112	0.653171	0.326585	−	0.945168i	$-0.394102\pi$
$173$	8.59112	0.653171	0.326585	+	0.945168i	$0.394102\pi$
$174$	13.8804	1.05227
$175$	0	0
$176$	2.43763	0.183743
$177$	1.29273	0.0971677
$178$	7.74728	0.580683
$179$	23.5276	1.75853	0.879267	−	0.476330i	$-0.158033\pi$
$179$	23.5276	1.75853	0.879267	+	0.476330i	$0.158033\pi$
$180$	0	0
$181$	16.6135	1.23487	0.617434	−	0.786623i	$-0.288172\pi$
$181$	16.6135	1.23487	0.617434	+	0.786623i	$0.288172\pi$
$182$	−31.4873	−2.33400
$183$	−32.5200	−2.40394
$184$	6.39234	0.471250
$185$	0	0
$186$	−17.3928	−1.27530
$187$	−0.999546	−0.0730941
$188$	−3.10658	−0.226571
$189$	3.20990	0.233486
$190$	0	0
$191$	−12.8109	−0.926965	−0.463482	−	0.886106i	$-0.653400\pi$
$191$	−12.8109	−0.926965	−0.463482	+	0.886106i	$0.653400\pi$
$192$	−1.65339	−0.119323
$193$	9.63983	0.693890	0.346945	−	0.937885i	$-0.387219\pi$
$193$	9.63983	0.693890	0.346945	+	0.937885i	$0.387219\pi$
$194$	10.0775	0.723525
$195$	0	0
$196$	13.0003	0.928596
$197$	−3.10241	−0.221038	−0.110519	−	0.993874i	$-0.535251\pi$
$197$	−3.10241	−0.221038	−0.110519	+	0.993874i	$0.535251\pi$
$198$	−2.34995	−0.167004
$199$	0.524290	0.0371659	0.0185830	−	0.999827i	$-0.494085\pi$
$199$	0.524290	0.0371659	0.0185830	+	0.999827i	$0.494085\pi$
$200$	0	0
$201$	−17.0606	−1.20336
$202$	−5.89091	−0.414483
$203$	−13.8309	−0.970742
$204$	5.75214	0.402731
$205$	0	0
$206$	−6.09586	−0.424719
$207$	−11.7896	−0.819432
$208$	−20.6674	−1.43303
$209$	0	0
$210$	0	0
$211$	19.0637	1.31240	0.656201	−	0.754586i	$-0.272162\pi$
$211$	19.0637	1.31240	0.656201	+	0.754586i	$0.272162\pi$
$212$	13.5503	0.930636
$213$	−14.3946	−0.986299
$214$	−3.12768	−0.213804
$215$	0	0
$216$	1.10127	0.0749319
$217$	17.3309	1.17650
$218$	6.28069	0.425382
$219$	−4.89671	−0.330889
$220$	0	0
$221$	8.47463	0.570065
$222$	−9.12284	−0.612285
$223$	−3.16283	−0.211799	−0.105899	−	0.994377i	$-0.533772\pi$
$223$	−3.16283	−0.211799	−0.105899	+	0.994377i	$0.533772\pi$
$224$	25.2376	1.68626
$225$	0	0
$226$	23.6617	1.57395
$227$	−9.04654	−0.600440	−0.300220	−	0.953870i	$-0.597060\pi$
$227$	−9.04654	−0.600440	−0.300220	+	0.953870i	$0.597060\pi$
$228$	0	0
$229$	−17.8920	−1.18234	−0.591168	−	0.806549i	$-0.701333\pi$
$229$	−17.8920	−1.18234	−0.591168	+	0.806549i	$0.701333\pi$
$230$	0	0
$231$	4.96046	0.326374
$232$	−4.74519	−0.311537
$233$	−6.44330	−0.422114	−0.211057	−	0.977474i	$-0.567691\pi$
$233$	−6.44330	−0.422114	−0.211057	+	0.977474i	$0.567691\pi$
$234$	19.9240	1.30247
$235$	0	0
$236$	0.642674	0.0418345
$237$	−12.7300	−0.826903
$238$	−15.4047	−0.998539
$239$	−13.4742	−0.871571	−0.435785	−	0.900051i	$-0.643529\pi$
$239$	−13.4742	−0.871571	−0.435785	+	0.900051i	$0.643529\pi$
$240$	0	0
$241$	17.6351	1.13598	0.567990	−	0.823036i	$-0.307721\pi$
$241$	17.6351	1.13598	0.567990	+	0.823036i	$0.307721\pi$
$242$	−19.2014	−1.23431
$243$	−21.2134	−1.36084
$244$	−16.1671	−1.03499
$245$	0	0
$246$	−18.5742	−1.18424
$247$	0	0
$248$	5.94596	0.377569
$249$	−19.3461	−1.22601
$250$	0	0
$251$	17.7326	1.11927	0.559636	−	0.828738i	$-0.310941\pi$
$251$	17.7326	1.11927	0.559636	+	0.828738i	$0.310941\pi$
$252$	−13.4752	−0.848859
$253$	2.15758	0.135646
$254$	−35.3513	−2.21813
$255$	0	0
$256$	20.4282	1.27676
$257$	27.7475	1.73084	0.865421	−	0.501046i	$-0.167051\pi$
$257$	27.7475	1.73084	0.865421	+	0.501046i	$0.167051\pi$
$258$	−45.1975	−2.81387
$259$	9.09035	0.564847
$260$	0	0
$261$	8.75170	0.541716
$262$	−6.12152	−0.378189
$263$	5.56032	0.342864	0.171432	−	0.985196i	$-0.445161\pi$
$263$	5.56032	0.342864	0.171432	+	0.985196i	$0.445161\pi$
$264$	1.70186	0.104742
$265$	0	0
$266$	0	0
$267$	10.3479	0.633282
$268$	−8.48158	−0.518095
$269$	4.62765	0.282153	0.141076	−	0.989999i	$-0.454944\pi$
$269$	4.62765	0.282153	0.141076	+	0.989999i	$0.454944\pi$
$270$	0	0
$271$	−7.60838	−0.462176	−0.231088	−	0.972933i	$-0.574229\pi$
$271$	−7.60838	−0.462176	−0.231088	+	0.972933i	$0.574229\pi$
$272$	−10.1112	−0.613083
$273$	−42.0571	−2.54541
$274$	−1.08561	−0.0655843
$275$	0	0
$276$	−12.4163	−0.747376
$277$	7.53544	0.452761	0.226380	−	0.974039i	$-0.427311\pi$
$277$	7.53544	0.452761	0.226380	+	0.974039i	$0.427311\pi$
$278$	−15.0948	−0.905325
$279$	−10.9663	−0.656536
$280$	0	0
$281$	18.2549	1.08900	0.544499	−	0.838762i	$-0.316720\pi$
$281$	18.2549	1.08900	0.544499	+	0.838762i	$0.316720\pi$
$282$	−11.1522	−0.664103
$283$	−30.9333	−1.83880	−0.919398	−	0.393328i	$-0.871324\pi$
$283$	−30.9333	−1.83880	−0.919398	+	0.393328i	$0.871324\pi$
$284$	−7.15618	−0.424641
$285$	0	0
$286$	−3.64624	−0.215607
$287$	18.5080	1.09249
$288$	−15.9694	−0.941004
$289$	−12.8539	−0.756112
$290$	0	0
$291$	13.4604	0.789063
$292$	−2.43437	−0.142461
$293$	−21.8992	−1.27937	−0.639683	−	0.768639i	$-0.720934\pi$
$293$	−21.8992	−1.27937	−0.639683	+	0.768639i	$0.720934\pi$
$294$	46.6694	2.72181
$295$	0	0
$296$	3.11877	0.181275
$297$	0.371707	0.0215686
$298$	−8.27618	−0.479426
$299$	−18.2930	−1.05791
$300$	0	0
$301$	45.0365	2.59586
$302$	−25.3618	−1.45941
$303$	−7.86840	−0.452028
$304$	0	0
$305$	0	0
$306$	9.74752	0.557229
$307$	5.45543	0.311358	0.155679	−	0.987808i	$-0.450243\pi$
$307$	5.45543	0.311358	0.155679	+	0.987808i	$0.450243\pi$
$308$	2.46607	0.140517
$309$	−8.14215	−0.463191
$310$	0	0
$311$	6.69721	0.379764	0.189882	−	0.981807i	$-0.439189\pi$
$311$	6.69721	0.379764	0.189882	+	0.981807i	$0.439189\pi$
$312$	−14.4292	−0.816892
$313$	4.58279	0.259035	0.129517	−	0.991577i	$-0.458657\pi$
$313$	4.58279	0.259035	0.129517	+	0.991577i	$0.458657\pi$
$314$	−19.1462	−1.08048
$315$	0	0
$316$	−6.32866	−0.356015
$317$	27.6794	1.55463	0.777314	−	0.629113i	$-0.216582\pi$
$317$	27.6794	1.55463	0.777314	+	0.629113i	$0.216582\pi$
$318$	48.6435	2.72779
$319$	−1.60163	−0.0896738
$320$	0	0
$321$	−4.17759	−0.233170
$322$	33.2520	1.85306
$323$	0	0
$324$	−11.6754	−0.648636
$325$	0	0
$326$	30.2264	1.67409
$327$	8.38903	0.463914
$328$	6.34983	0.350611
$329$	11.1125	0.612651
$330$	0	0
$331$	14.1302	0.776665	0.388332	−	0.921519i	$-0.373051\pi$
$331$	14.1302	0.776665	0.388332	+	0.921519i	$0.373051\pi$
$332$	−9.61782	−0.527847
$333$	−5.75203	−0.315209
$334$	29.0130	1.58752
$335$	0	0
$336$	50.1791	2.73749
$337$	−14.9974	−0.816962	−0.408481	−	0.912767i	$-0.633941\pi$
$337$	−14.9974	−0.816962	−0.408481	+	0.912767i	$0.633941\pi$
$338$	7.71372	0.419571
$339$	31.6046	1.71653
$340$	0	0
$341$	2.00692	0.108681
$342$	0	0
$343$	−16.8294	−0.908703
$344$	15.4514	0.833082
$345$	0	0
$346$	15.3324	0.824274
$347$	−20.9133	−1.12268	−0.561342	−	0.827584i	$-0.689715\pi$
$347$	−20.9133	−1.12268	−0.561342	+	0.827584i	$0.689715\pi$
$348$	9.21696	0.494081
$349$	−3.13055	−0.167574	−0.0837872	−	0.996484i	$-0.526702\pi$
$349$	−3.13055	−0.167574	−0.0837872	+	0.996484i	$0.526702\pi$
$350$	0	0
$351$	−3.15151	−0.168215
$352$	2.92251	0.155771
$353$	8.04538	0.428212	0.214106	−	0.976810i	$-0.431316\pi$
$353$	8.04538	0.428212	0.214106	+	0.976810i	$0.431316\pi$
$354$	2.30711	0.122622
$355$	0	0
$356$	5.14441	0.272653
$357$	−20.5758	−1.08899
$358$	41.9891	2.21920
$359$	30.5842	1.61417	0.807087	−	0.590432i	$-0.201043\pi$
$359$	30.5842	1.61417	0.807087	+	0.590432i	$0.201043\pi$
$360$	0	0
$361$	0	0
$362$	29.6497	1.55835
$363$	−25.6470	−1.34612
$364$	−20.9085	−1.09590
$365$	0	0
$366$	−58.0377	−3.03368
$367$	−9.95175	−0.519477	−0.259739	−	0.965679i	$-0.583636\pi$
$367$	−9.95175	−0.519477	−0.259739	+	0.965679i	$0.583636\pi$
$368$	21.8257	1.13774
$369$	−11.7112	−0.609659
$370$	0	0
$371$	−48.4703	−2.51645
$372$	−11.5493	−0.598804
$373$	9.27611	0.480299	0.240149	−	0.970736i	$-0.422804\pi$
$373$	9.27611	0.480299	0.240149	+	0.970736i	$0.422804\pi$
$374$	−1.78387	−0.0922416
$375$	0	0
$376$	3.81253	0.196616
$377$	13.5793	0.699372
$378$	5.72864	0.294649
$379$	21.1472	1.08626	0.543129	−	0.839649i	$-0.317239\pi$
$379$	21.1472	1.08626	0.543129	+	0.839649i	$0.317239\pi$
$380$	0	0
$381$	−47.2181	−2.41906
$382$	−22.8634	−1.16979
$383$	15.1501	0.774136	0.387068	−	0.922051i	$-0.373488\pi$
$383$	15.1501	0.774136	0.387068	+	0.922051i	$0.373488\pi$
$384$	25.4328	1.29786
$385$	0	0
$386$	17.2040	0.875660
$387$	−28.4974	−1.44860
$388$	6.69177	0.339723
$389$	−31.4483	−1.59449	−0.797247	−	0.603653i	$-0.793711\pi$
$389$	−31.4483	−1.59449	−0.797247	+	0.603653i	$0.793711\pi$
$390$	0	0
$391$	−8.94958	−0.452600
$392$	−15.9546	−0.805827
$393$	−8.17642	−0.412446
$394$	−5.53681	−0.278940
$395$	0	0
$396$	−1.56043	−0.0784147
$397$	−9.38559	−0.471049	−0.235525	−	0.971868i	$-0.575681\pi$
$397$	−9.38559	−0.471049	−0.235525	+	0.971868i	$0.575681\pi$
$398$	0.935689	0.0469019
$399$	0	0
$400$	0	0
$401$	−6.16049	−0.307640	−0.153820	−	0.988099i	$-0.549158\pi$
$401$	−6.16049	−0.307640	−0.153820	+	0.988099i	$0.549158\pi$
$402$	−30.4477	−1.51859
$403$	−17.0156	−0.847607
$404$	−3.91173	−0.194616
$405$	0	0
$406$	−24.6838	−1.22504
$407$	1.05266	0.0521786
$408$	−7.05927	−0.349486
$409$	−2.28482	−0.112977	−0.0564885	−	0.998403i	$-0.517990\pi$
$409$	−2.28482	−0.112977	−0.0564885	+	0.998403i	$0.517990\pi$
$410$	0	0
$411$	−1.45004	−0.0715250
$412$	−4.04783	−0.199422
$413$	−2.29889	−0.113121
$414$	−21.0406	−1.03409
$415$	0	0
$416$	−24.7785	−1.21486
$417$	−20.1619	−0.987332
$418$	0	0
$419$	−22.6494	−1.10649	−0.553247	−	0.833017i	$-0.686611\pi$
$419$	−22.6494	−1.10649	−0.553247	+	0.833017i	$0.686611\pi$
$420$	0	0
$421$	22.9969	1.12080	0.560400	−	0.828222i	$-0.310647\pi$
$421$	22.9969	1.12080	0.560400	+	0.828222i	$0.310647\pi$
$422$	34.0226	1.65620
$423$	−7.03155	−0.341886
$424$	−16.6294	−0.807597
$425$	0	0
$426$	−25.6897	−1.24467
$427$	57.8310	2.79864
$428$	−2.07687	−0.100389
$429$	−4.87022	−0.235137
$430$	0	0
$431$	−25.6319	−1.23465	−0.617323	−	0.786710i	$-0.711783\pi$
$431$	−25.6319	−1.23465	−0.617323	+	0.786710i	$0.711783\pi$
$432$	3.76012	0.180909
$433$	2.36330	0.113573	0.0567864	−	0.998386i	$-0.481915\pi$
$433$	2.36330	0.113573	0.0567864	+	0.998386i	$0.481915\pi$
$434$	30.9300	1.48469
$435$	0	0
$436$	4.17056	0.199734
$437$	0	0
$438$	−8.73906	−0.417568
$439$	6.31659	0.301474	0.150737	−	0.988574i	$-0.451835\pi$
$439$	6.31659	0.301474	0.150737	+	0.988574i	$0.451835\pi$
$440$	0	0
$441$	29.4254	1.40121
$442$	15.1245	0.719399
$443$	−31.5246	−1.49778	−0.748890	−	0.662694i	$-0.769413\pi$
$443$	−31.5246	−1.49778	−0.748890	+	0.662694i	$0.769413\pi$
$444$	−6.05782	−0.287492
$445$	0	0
$446$	−5.64463	−0.267281
$447$	−11.0544	−0.522854
$448$	2.94027	0.138915
$449$	15.0828	0.711803	0.355902	−	0.934523i	$-0.384174\pi$
$449$	15.0828	0.711803	0.355902	+	0.934523i	$0.384174\pi$
$450$	0	0
$451$	2.14323	0.100921
$452$	15.7121	0.739033
$453$	−33.8754	−1.59161
$454$	−16.1452	−0.757730
$455$	0	0
$456$	0	0
$457$	−20.1347	−0.941861	−0.470930	−	0.882170i	$-0.656082\pi$
$457$	−20.1347	−0.941861	−0.470930	+	0.882170i	$0.656082\pi$
$458$	−31.9314	−1.49206
$459$	−1.54183	−0.0719664
$460$	0	0
$461$	−7.48489	−0.348606	−0.174303	−	0.984692i	$-0.555767\pi$
$461$	−7.48489	−0.348606	−0.174303	+	0.984692i	$0.555767\pi$
$462$	8.85283	0.411871
$463$	21.5973	1.00371	0.501855	−	0.864952i	$-0.332651\pi$
$463$	21.5973	1.00371	0.501855	+	0.864952i	$0.332651\pi$
$464$	−16.2017	−0.752147
$465$	0	0
$466$	−11.4992	−0.532691
$467$	27.0503	1.25174	0.625870	−	0.779927i	$-0.284744\pi$
$467$	27.0503	1.25174	0.625870	+	0.779927i	$0.284744\pi$
$468$	13.2301	0.611561
$469$	30.3392	1.40094
$470$	0	0
$471$	−25.5732	−1.17835
$472$	−0.788717	−0.0363036
$473$	5.21524	0.239797
$474$	−22.7190	−1.04352
$475$	0	0
$476$	−10.2292	−0.468853
$477$	30.6702	1.40429
$478$	−24.0470	−1.09989
$479$	−38.0534	−1.73870	−0.869352	−	0.494193i	$-0.835464\pi$
$479$	−38.0534	−1.73870	−0.869352	+	0.494193i	$0.835464\pi$
$480$	0	0
$481$	−8.92499	−0.406944
$482$	31.4731	1.43356
$483$	44.4142	2.02092
$484$	−12.7503	−0.579558
$485$	0	0
$486$	−37.8591	−1.71732
$487$	−18.3353	−0.830851	−0.415426	−	0.909627i	$-0.636367\pi$
$487$	−18.3353	−0.830851	−0.415426	+	0.909627i	$0.636367\pi$
$488$	19.8410	0.898158
$489$	40.3730	1.82573
$490$	0	0
$491$	−15.0675	−0.679986	−0.339993	−	0.940428i	$-0.610425\pi$
$491$	−15.0675	−0.679986	−0.339993	+	0.940428i	$0.610425\pi$
$492$	−12.3338	−0.556049
$493$	6.64350	0.299208
$494$	0	0
$495$	0	0
$496$	20.3016	0.911568
$497$	25.5982	1.14824
$498$	−34.5266	−1.54717
$499$	34.5299	1.54577	0.772886	−	0.634545i	$-0.218812\pi$
$499$	34.5299	1.54577	0.772886	+	0.634545i	$0.218812\pi$
$500$	0	0
$501$	38.7522	1.73132
$502$	31.6470	1.41247
$503$	30.8704	1.37644	0.688221	−	0.725501i	$-0.258392\pi$
$503$	30.8704	1.37644	0.688221	+	0.725501i	$0.258392\pi$
$504$	16.5374	0.736632
$505$	0	0
$506$	3.85059	0.171179
$507$	10.3031	0.457577
$508$	−23.4742	−1.04150
$509$	−10.1777	−0.451118	−0.225559	−	0.974230i	$-0.572421\pi$
$509$	−10.1777	−0.451118	−0.225559	+	0.974230i	$0.572421\pi$
$510$	0	0
$511$	8.70794	0.385217
$512$	15.1195	0.668194
$513$	0	0
$514$	49.5204	2.18425
$515$	0	0
$516$	−30.0124	−1.32122
$517$	1.28683	0.0565946
$518$	16.2234	0.712813
$519$	20.4792	0.898938
$520$	0	0
$521$	−16.5423	−0.724732	−0.362366	−	0.932036i	$-0.618031\pi$
$521$	−16.5423	−0.724732	−0.362366	+	0.932036i	$0.618031\pi$
$522$	15.6190	0.683624
$523$	−7.99801	−0.349729	−0.174864	−	0.984593i	$-0.555949\pi$
$523$	−7.99801	−0.349729	−0.174864	+	0.984593i	$0.555949\pi$
$524$	−4.06486	−0.177574
$525$	0	0
$526$	9.92338	0.432680
$527$	−8.32463	−0.362626
$528$	5.81075	0.252880
$529$	−3.68180	−0.160078
$530$	0	0
$531$	1.45465	0.0631266
$532$	0	0
$533$	−18.1713	−0.787088
$534$	18.4677	0.799176
$535$	0	0
$536$	10.4089	0.449598
$537$	56.0843	2.42021
$538$	8.25886	0.356065
$539$	−5.38507	−0.231952
$540$	0	0
$541$	5.20048	0.223586	0.111793	−	0.993732i	$-0.464341\pi$
$541$	5.20048	0.223586	0.111793	+	0.993732i	$0.464341\pi$
$542$	−13.5785	−0.583247
$543$	39.6026	1.69951
$544$	−12.1225	−0.519748
$545$	0	0
$546$	−75.0585	−3.21221
$547$	11.0339	0.471777	0.235889	−	0.971780i	$-0.424200\pi$
$547$	11.0339	0.471777	0.235889	+	0.971780i	$0.424200\pi$
$548$	−0.720878	−0.0307944
$549$	−36.5933	−1.56176
$550$	0	0
$551$	0	0
$552$	15.2379	0.648566
$553$	22.6381	0.962669
$554$	13.4483	0.571365
$555$	0	0
$556$	−10.0234	−0.425086
$557$	21.2817	0.901735	0.450867	−	0.892591i	$-0.351115\pi$
$557$	21.2817	0.901735	0.450867	+	0.892591i	$0.351115\pi$
$558$	−19.5713	−0.828521
$559$	−44.2173	−1.87019
$560$	0	0
$561$	−2.38269	−0.100597
$562$	32.5792	1.37427
$563$	14.7072	0.619833	0.309917	−	0.950764i	$-0.399699\pi$
$563$	14.7072	0.619833	0.309917	+	0.950764i	$0.399699\pi$
$564$	−7.40537	−0.311822
$565$	0	0
$566$	−55.2061	−2.32048
$567$	41.7639	1.75392
$568$	8.78236	0.368500
$569$	−40.6551	−1.70435	−0.852174	−	0.523258i	$-0.824716\pi$
$569$	−40.6551	−1.70435	−0.852174	+	0.523258i	$0.824716\pi$
$570$	0	0
$571$	−24.3240	−1.01793	−0.508964	−	0.860788i	$-0.669972\pi$
$571$	−24.3240	−1.01793	−0.508964	+	0.860788i	$0.669972\pi$
$572$	−2.42120	−0.101236
$573$	−30.5382	−1.27575
$574$	33.0308	1.37868
$575$	0	0
$576$	−1.86049	−0.0775204
$577$	−33.6876	−1.40243	−0.701216	−	0.712949i	$-0.747359\pi$
$577$	−33.6876	−1.40243	−0.701216	+	0.712949i	$0.747359\pi$
$578$	−22.9401	−0.954182
$579$	22.9791	0.954979
$580$	0	0
$581$	34.4037	1.42730
$582$	24.0225	0.995764
$583$	−5.61287	−0.232461
$584$	2.98757	0.123626
$585$	0	0
$586$	−39.0830	−1.61451
$587$	2.24977	0.0928580	0.0464290	−	0.998922i	$-0.485216\pi$
$587$	2.24977	0.0928580	0.0464290	+	0.998922i	$0.485216\pi$
$588$	30.9898	1.27800
$589$	0	0
$590$	0	0
$591$	−7.39543	−0.304207
$592$	10.6486	0.437653
$593$	31.3871	1.28891	0.644457	−	0.764640i	$-0.277083\pi$
$593$	31.3871	1.28891	0.644457	+	0.764640i	$0.277083\pi$
$594$	0.663377	0.0272187
$595$	0	0
$596$	−5.49562	−0.225109
$597$	1.24979	0.0511503
$598$	−32.6471	−1.33504
$599$	−10.4068	−0.425212	−0.212606	−	0.977138i	$-0.568195\pi$
$599$	−10.4068	−0.425212	−0.212606	+	0.977138i	$0.568195\pi$
$600$	0	0
$601$	1.29592	0.0528616	0.0264308	−	0.999651i	$-0.491586\pi$
$601$	1.29592	0.0528616	0.0264308	+	0.999651i	$0.491586\pi$
$602$	80.3757	3.27587
$603$	−19.1975	−0.781783
$604$	−16.8410	−0.685249
$605$	0	0
$606$	−14.0426	−0.570440
$607$	5.50902	0.223604	0.111802	−	0.993730i	$-0.464338\pi$
$607$	5.50902	0.223604	0.111802	+	0.993730i	$0.464338\pi$
$608$	0	0
$609$	−32.9698	−1.33600
$610$	0	0
$611$	−10.9103	−0.441385
$612$	6.47263	0.261641
$613$	40.6727	1.64275	0.821376	−	0.570386i	$-0.193207\pi$
$613$	40.6727	1.64275	0.821376	+	0.570386i	$0.193207\pi$
$614$	9.73619	0.392921
$615$	0	0
$616$	−3.02646	−0.121939
$617$	−30.5053	−1.22810	−0.614049	−	0.789268i	$-0.710460\pi$
$617$	−30.5053	−1.22810	−0.614049	+	0.789268i	$0.710460\pi$
$618$	−14.5311	−0.584527
$619$	5.80411	0.233287	0.116643	−	0.993174i	$-0.462787\pi$
$619$	5.80411	0.233287	0.116643	+	0.993174i	$0.462787\pi$
$620$	0	0
$621$	3.32813	0.133553
$622$	11.9524	0.479246
$623$	−18.4019	−0.737258
$624$	−49.2663	−1.97223
$625$	0	0
$626$	8.17881	0.326891
$627$	0	0
$628$	−12.7136	−0.507328
$629$	−4.36642	−0.174101
$630$	0	0
$631$	−23.3642	−0.930116	−0.465058	−	0.885280i	$-0.653967\pi$
$631$	−23.3642	−0.930116	−0.465058	+	0.885280i	$0.653967\pi$
$632$	7.76679	0.308946
$633$	45.4435	1.80622
$634$	49.3987	1.96187
$635$	0	0
$636$	32.3007	1.28080
$637$	45.6572	1.80901
$638$	−2.85839	−0.113165
$639$	−16.1976	−0.640766
$640$	0	0
$641$	−18.3459	−0.724621	−0.362311	−	0.932057i	$-0.618012\pi$
$641$	−18.3459	−0.724621	−0.362311	+	0.932057i	$0.618012\pi$
$642$	−7.45565	−0.294251
$643$	−15.7630	−0.621633	−0.310817	−	0.950470i	$-0.600603\pi$
$643$	−15.7630	−0.621633	−0.310817	+	0.950470i	$0.600603\pi$
$644$	22.0803	0.870084
$645$	0	0
$646$	0	0
$647$	25.0443	0.984592	0.492296	−	0.870428i	$-0.336158\pi$
$647$	25.0443	0.984592	0.492296	+	0.870428i	$0.336158\pi$
$648$	14.3286	0.562880
$649$	−0.266212	−0.0104497
$650$	0	0
$651$	41.3127	1.61917
$652$	20.0712	0.786049
$653$	−2.98852	−0.116950	−0.0584749	−	0.998289i	$-0.518624\pi$
$653$	−2.98852	−0.116950	−0.0584749	+	0.998289i	$0.518624\pi$
$654$	14.9717	0.585440
$655$	0	0
$656$	21.6805	0.846482
$657$	−5.51005	−0.214968
$658$	19.8322	0.773140
$659$	35.7695	1.39338	0.696691	−	0.717372i	$-0.254655\pi$
$659$	35.7695	1.39338	0.696691	+	0.717372i	$0.254655\pi$
$660$	0	0
$661$	−3.22147	−0.125301	−0.0626503	−	0.998036i	$-0.519955\pi$
$661$	−3.22147	−0.125301	−0.0626503	+	0.998036i	$0.519955\pi$
$662$	25.2178	0.980118
$663$	20.2015	0.784563
$664$	11.8034	0.458061
$665$	0	0
$666$	−10.2655	−0.397781
$667$	−14.3404	−0.555262
$668$	19.2655	0.745403
$669$	−7.53945	−0.291492
$670$	0	0
$671$	6.69684	0.258529
$672$	60.1605	2.32074
$673$	−0.639706	−0.0246588	−0.0123294	−	0.999924i	$-0.503925\pi$
$673$	−0.639706	−0.0246588	−0.0123294	+	0.999924i	$0.503925\pi$
$674$	−26.7656	−1.03097
$675$	0	0
$676$	5.12213	0.197005
$677$	25.8005	0.991592	0.495796	−	0.868439i	$-0.334876\pi$
$677$	25.8005	0.991592	0.495796	+	0.868439i	$0.334876\pi$
$678$	56.4041	2.16618
$679$	−23.9369	−0.918616
$680$	0	0
$681$	−21.5648	−0.826366
$682$	3.58170	0.137150
$683$	5.33696	0.204213	0.102107	−	0.994773i	$-0.467442\pi$
$683$	5.33696	0.204213	0.102107	+	0.994773i	$0.467442\pi$
$684$	0	0
$685$	0	0
$686$	−30.0351	−1.14675
$687$	−42.6503	−1.62721
$688$	52.7564	2.01132
$689$	47.5886	1.81298
$690$	0	0
$691$	13.2507	0.504079	0.252040	−	0.967717i	$-0.418899\pi$
$691$	13.2507	0.504079	0.252040	+	0.967717i	$0.418899\pi$
$692$	10.1811	0.387029
$693$	5.58178	0.212034
$694$	−37.3235	−1.41678
$695$	0	0
$696$	−11.3114	−0.428759
$697$	−8.89006	−0.336735
$698$	−5.58702	−0.211472
$699$	−15.3593	−0.580943
$700$	0	0
$701$	40.8699	1.54363	0.771817	−	0.635845i	$-0.219348\pi$
$701$	40.8699	1.54363	0.771817	+	0.635845i	$0.219348\pi$
$702$	−5.62443	−0.212280
$703$	0	0
$704$	0.340484	0.0128325
$705$	0	0
$706$	14.3584	0.540386
$707$	13.9925	0.526244
$708$	1.53199	0.0575756
$709$	7.35745	0.276315	0.138157	−	0.990410i	$-0.455882\pi$
$709$	7.35745	0.276315	0.138157	+	0.990410i	$0.455882\pi$
$710$	0	0
$711$	−14.3245	−0.537211
$712$	−6.31344	−0.236606
$713$	17.9692	0.672952
$714$	−36.7213	−1.37426
$715$	0	0
$716$	27.8820	1.04200
$717$	−32.1193	−1.19952
$718$	54.5830	2.03702
$719$	−31.9597	−1.19190	−0.595948	−	0.803023i	$-0.703224\pi$
$719$	−31.9597	−1.19190	−0.595948	+	0.803023i	$0.703224\pi$
$720$	0	0
$721$	14.4794	0.539240
$722$	0	0
$723$	42.0381	1.56341
$724$	19.6882	0.731707
$725$	0	0
$726$	−45.7717	−1.69875
$727$	29.3883	1.08995	0.544976	−	0.838452i	$-0.316539\pi$
$727$	29.3883	1.08995	0.544976	+	0.838452i	$0.316539\pi$
$728$	25.6598	0.951014
$729$	−21.0116	−0.778207
$730$	0	0
$731$	−21.6326	−0.800112
$732$	−38.5387	−1.42443
$733$	50.8285	1.87739	0.938696	−	0.344745i	$-0.112035\pi$
$733$	50.8285	1.87739	0.938696	+	0.344745i	$0.112035\pi$
$734$	−17.7607	−0.655559
$735$	0	0
$736$	26.1671	0.964534
$737$	3.51329	0.129414
$738$	−20.9007	−0.769364
$739$	14.7245	0.541650	0.270825	−	0.962629i	$-0.412704\pi$
$739$	14.7245	0.541650	0.270825	+	0.962629i	$0.412704\pi$
$740$	0	0
$741$	0	0
$742$	−86.5039	−3.17566
$743$	34.6568	1.27144	0.635718	−	0.771922i	$-0.280704\pi$
$743$	34.6568	1.27144	0.635718	+	0.771922i	$0.280704\pi$
$744$	14.1738	0.519636
$745$	0	0
$746$	16.5549	0.606117
$747$	−21.7693	−0.796498
$748$	−1.18454	−0.0433110
$749$	7.42910	0.271453
$750$	0	0
$751$	8.35539	0.304893	0.152446	−	0.988312i	$-0.451285\pi$
$751$	8.35539	0.304893	0.152446	+	0.988312i	$0.451285\pi$
$752$	13.0173	0.474692
$753$	42.2704	1.54042
$754$	24.2348	0.882578
$755$	0	0
$756$	3.80398	0.138349
$757$	−13.8709	−0.504146	−0.252073	−	0.967708i	$-0.581112\pi$
$757$	−13.8709	−0.504146	−0.252073	+	0.967708i	$0.581112\pi$
$758$	37.7410	1.37081
$759$	5.14317	0.186685
$760$	0	0
$761$	44.3970	1.60939	0.804696	−	0.593687i	$-0.202328\pi$
$761$	44.3970	1.60939	0.804696	+	0.593687i	$0.202328\pi$
$762$	−84.2692	−3.05275
$763$	−14.9184	−0.540082
$764$	−15.1819	−0.549262
$765$	0	0
$766$	27.0381	0.976927
$767$	2.25707	0.0814983
$768$	48.6961	1.75717
$769$	19.0001	0.685160	0.342580	−	0.939489i	$-0.388699\pi$
$769$	19.0001	0.685160	0.342580	+	0.939489i	$0.388699\pi$
$770$	0	0
$771$	66.1436	2.38210
$772$	11.4239	0.411157
$773$	−1.86738	−0.0671650	−0.0335825	−	0.999436i	$-0.510692\pi$
$773$	−1.86738	−0.0671650	−0.0335825	+	0.999436i	$0.510692\pi$
$774$	−50.8587	−1.82808
$775$	0	0
$776$	−8.21241	−0.294808
$777$	21.6693	0.777381
$778$	−56.1252	−2.01218
$779$	0	0
$780$	0	0
$781$	2.96427	0.106070
$782$	−15.9721	−0.571162
$783$	−2.47055	−0.0882904
$784$	−54.4744	−1.94552
$785$	0	0
$786$	−14.5923	−0.520489
$787$	12.9689	0.462291	0.231146	−	0.972919i	$-0.425753\pi$
$787$	12.9689	0.462291	0.231146	+	0.972919i	$0.425753\pi$
$788$	−3.67660	−0.130973
$789$	13.2545	0.471873
$790$	0	0
$791$	−56.2032	−1.99836
$792$	1.91503	0.0680476
$793$	−56.7790	−2.01628
$794$	−16.7503	−0.594444
$795$	0	0
$796$	0.621324	0.0220222
$797$	−29.9226	−1.05991	−0.529956	−	0.848025i	$-0.677791\pi$
$797$	−29.9226	−1.05991	−0.529956	+	0.848025i	$0.677791\pi$
$798$	0	0
$799$	−5.33772	−0.188835
$800$	0	0
$801$	11.6441	0.411422
$802$	−10.9945	−0.388229
$803$	1.00838	0.0355850
$804$	−20.2181	−0.713038
$805$	0	0
$806$	−30.3674	−1.06964
$807$	11.0312	0.388318
$808$	4.80064	0.168886
$809$	42.4578	1.49274	0.746369	−	0.665532i	$-0.231795\pi$
$809$	42.4578	1.49274	0.746369	+	0.665532i	$0.231795\pi$
$810$	0	0
$811$	31.1514	1.09387	0.546936	−	0.837174i	$-0.315794\pi$
$811$	31.1514	1.09387	0.546936	+	0.837174i	$0.315794\pi$
$812$	−16.3907	−0.575202
$813$	−18.1366	−0.636079
$814$	1.87867	0.0658472
$815$	0	0
$816$	−24.1028	−0.843767
$817$	0	0
$818$	−4.07767	−0.142572
$819$	−47.3250	−1.65367
$820$	0	0
$821$	−6.20740	−0.216640	−0.108320	−	0.994116i	$-0.534547\pi$
$821$	−6.20740	−0.216640	−0.108320	+	0.994116i	$0.534547\pi$
$822$	−2.58785	−0.0902616
$823$	−35.9702	−1.25384	−0.626921	−	0.779083i	$-0.715685\pi$
$823$	−35.9702	−1.25384	−0.626921	+	0.779083i	$0.715685\pi$
$824$	4.96766	0.173057
$825$	0	0
$826$	−4.10279	−0.142754
$827$	−15.3837	−0.534945	−0.267473	−	0.963565i	$-0.586189\pi$
$827$	−15.3837	−0.534945	−0.267473	+	0.963565i	$0.586189\pi$
$828$	−13.9716	−0.485545
$829$	24.1941	0.840295	0.420148	−	0.907456i	$-0.361978\pi$
$829$	24.1941	0.840295	0.420148	+	0.907456i	$0.361978\pi$
$830$	0	0
$831$	17.9627	0.623120
$832$	−2.88678	−0.100081
$833$	22.3371	0.773936
$834$	−35.9825	−1.24597
$835$	0	0
$836$	0	0
$837$	3.09573	0.107004
$838$	−40.4218	−1.39635
$839$	−12.4265	−0.429012	−0.214506	−	0.976723i	$-0.568814\pi$
$839$	−12.4265	−0.429012	−0.214506	+	0.976723i	$0.568814\pi$
$840$	0	0
$841$	−18.3548	−0.632923
$842$	41.0420	1.41440
$843$	43.5155	1.49875
$844$	22.5920	0.777649
$845$	0	0
$846$	−12.5491	−0.431446
$847$	45.6087	1.56713
$848$	−56.7787	−1.94979
$849$	−73.7379	−2.53068
$850$	0	0
$851$	9.42518	0.323091
$852$	−17.0587	−0.584420
$853$	23.1527	0.792734	0.396367	−	0.918092i	$-0.370271\pi$
$853$	23.1527	0.792734	0.396367	+	0.918092i	$0.370271\pi$
$854$	103.210	3.53176
$855$	0	0
$856$	2.54882	0.0871167
$857$	2.84272	0.0971054	0.0485527	−	0.998821i	$-0.484539\pi$
$857$	2.84272	0.0971054	0.0485527	+	0.998821i	$0.484539\pi$
$858$	−8.69178	−0.296733
$859$	−25.8715	−0.882725	−0.441363	−	0.897329i	$-0.645505\pi$
$859$	−25.8715	−0.882725	−0.441363	+	0.897329i	$0.645505\pi$
$860$	0	0
$861$	44.1188	1.50356
$862$	−45.7447	−1.55807
$863$	38.0032	1.29365	0.646823	−	0.762641i	$-0.276097\pi$
$863$	38.0032	1.29365	0.646823	+	0.762641i	$0.276097\pi$
$864$	4.50806	0.153367
$865$	0	0
$866$	4.21773	0.143324
$867$	−30.6407	−1.04061
$868$	20.5384	0.697119
$869$	2.62149	0.0889281
$870$	0	0
$871$	−29.7873	−1.00931
$872$	−5.11828	−0.173327
$873$	15.1464	0.512628
$874$	0	0
$875$	0	0
$876$	−5.80298	−0.196065
$877$	−49.0943	−1.65780	−0.828898	−	0.559400i	$-0.811032\pi$
$877$	−49.0943	−1.65780	−0.828898	+	0.559400i	$0.811032\pi$
$878$	11.2731	0.380448
$879$	−52.2026	−1.76075
$880$	0	0
$881$	−55.7494	−1.87825	−0.939123	−	0.343581i	$-0.888360\pi$
$881$	−55.7494	−1.87825	−0.939123	+	0.343581i	$0.888360\pi$
$882$	52.5150	1.76827
$883$	−16.6355	−0.559828	−0.279914	−	0.960025i	$-0.590306\pi$
$883$	−16.6355	−0.559828	−0.279914	+	0.960025i	$0.590306\pi$
$884$	10.0431	0.337786
$885$	0	0
$886$	−56.2613	−1.89014
$887$	−43.3883	−1.45684	−0.728418	−	0.685133i	$-0.759744\pi$
$887$	−43.3883	−1.45684	−0.728418	+	0.685133i	$0.759744\pi$
$888$	7.43441	0.249483
$889$	83.9691	2.81623
$890$	0	0
$891$	4.83627	0.162021
$892$	−3.74820	−0.125499
$893$	0	0
$894$	−19.7285	−0.659819
$895$	0	0
$896$	−45.2277	−1.51095
$897$	−43.6062	−1.45597
$898$	26.9180	0.898266
$899$	−13.3390	−0.444880
$900$	0	0
$901$	23.2820	0.775636
$902$	3.82498	0.127358
$903$	107.357	3.57260
$904$	−19.2825	−0.641326
$905$	0	0
$906$	−60.4567	−2.00854
$907$	−19.8364	−0.658656	−0.329328	−	0.944216i	$-0.606822\pi$
$907$	−19.8364	−0.658656	−0.329328	+	0.944216i	$0.606822\pi$
$908$	−10.7208	−0.355784
$909$	−8.85395	−0.293667
$910$	0	0
$911$	−15.4076	−0.510478	−0.255239	−	0.966878i	$-0.582154\pi$
$911$	−15.4076	−0.510478	−0.255239	+	0.966878i	$0.582154\pi$
$912$	0	0
$913$	3.98395	0.131849
$914$	−35.9339	−1.18859
$915$	0	0
$916$	−21.2034	−0.700579
$917$	14.5403	0.480163
$918$	−2.75167	−0.0908186
$919$	−13.2561	−0.437278	−0.218639	−	0.975806i	$-0.570162\pi$
$919$	−13.2561	−0.437278	−0.218639	+	0.975806i	$0.570162\pi$
$920$	0	0
$921$	13.0045	0.428512
$922$	−13.3581	−0.439926
$923$	−25.1325	−0.827247
$924$	5.87853	0.193389
$925$	0	0
$926$	38.5442	1.26664
$927$	−9.16200	−0.300920
$928$	−19.4245	−0.637641
$929$	31.2016	1.02369	0.511846	−	0.859077i	$-0.328962\pi$
$929$	31.2016	1.02369	0.511846	+	0.859077i	$0.328962\pi$
$930$	0	0
$931$	0	0
$932$	−7.63580	−0.250119
$933$	15.9646	0.522658
$934$	48.2762	1.57964
$935$	0	0
$936$	−16.2365	−0.530707
$937$	−16.2187	−0.529840	−0.264920	−	0.964270i	$-0.585346\pi$
$937$	−16.2187	−0.529840	−0.264920	+	0.964270i	$0.585346\pi$
$938$	54.1458	1.76792
$939$	10.9243	0.356502
$940$	0	0
$941$	−35.1748	−1.14666	−0.573332	−	0.819323i	$-0.694350\pi$
$941$	−35.1748	−1.14666	−0.573332	+	0.819323i	$0.694350\pi$
$942$	−45.6400	−1.48703
$943$	19.1897	0.624903
$944$	−2.69295	−0.0876482
$945$	0	0
$946$	9.30752	0.302614
$947$	−22.2784	−0.723950	−0.361975	−	0.932188i	$-0.617897\pi$
$947$	−22.2784	−0.723950	−0.361975	+	0.932188i	$0.617897\pi$
$948$	−15.0860	−0.489972
$949$	−8.54953	−0.277530
$950$	0	0
$951$	65.9811	2.13958
$952$	12.5537	0.406866
$953$	4.25996	0.137994	0.0689968	−	0.997617i	$-0.478020\pi$
$953$	4.25996	0.137994	0.0689968	+	0.997617i	$0.478020\pi$
$954$	54.7364	1.77216
$955$	0	0
$956$	−15.9679	−0.516439
$957$	−3.81790	−0.123415
$958$	−67.9131	−2.19417
$959$	2.57863	0.0832684
$960$	0	0
$961$	−14.2856	−0.460825
$962$	−15.9282	−0.513547
$963$	−4.70085	−0.151483
$964$	20.8990	0.673111
$965$	0	0
$966$	79.2650	2.55031
$967$	−1.21906	−0.0392023	−0.0196012	−	0.999808i	$-0.506240\pi$
$967$	−1.21906	−0.0392023	−0.0196012	+	0.999808i	$0.506240\pi$
$968$	15.6477	0.502935
$969$	0	0
$970$	0	0
$971$	4.34003	0.139278	0.0696391	−	0.997572i	$-0.477815\pi$
$971$	4.34003	0.139278	0.0696391	+	0.997572i	$0.477815\pi$
$972$	−25.1395	−0.806349
$973$	35.8543	1.14944
$974$	−32.7226	−1.04850
$975$	0	0
$976$	67.7440	2.16843
$977$	−21.5270	−0.688710	−0.344355	−	0.938840i	$-0.611902\pi$
$977$	−21.5270	−0.688710	−0.344355	+	0.938840i	$0.611902\pi$
$978$	72.0527	2.30399
$979$	−2.13095	−0.0681054
$980$	0	0
$981$	9.43980	0.301389
$982$	−26.8906	−0.858114
$983$	−14.5998	−0.465661	−0.232830	−	0.972517i	$-0.574799\pi$
$983$	−14.5998	−0.465661	−0.232830	+	0.972517i	$0.574799\pi$
$984$	15.1365	0.482534
$985$	0	0
$986$	11.8565	0.377588
$987$	26.4896	0.843172
$988$	0	0
$989$	46.6954	1.48483
$990$	0	0
$991$	−31.9286	−1.01425	−0.507123	−	0.861874i	$-0.669291\pi$
$991$	−31.9286	−1.01425	−0.507123	+	0.861874i	$0.669291\pi$
$992$	24.3399	0.772792
$993$	33.6831	1.06890
$994$	45.6845	1.44903
$995$	0	0
$996$	−22.9267	−0.726459
$997$	55.1812	1.74761	0.873803	−	0.486280i	$-0.161647\pi$
$997$	55.1812	1.74761	0.873803	+	0.486280i	$0.161647\pi$
$998$	61.6248	1.95070
$999$	1.62377	0.0513737

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.ct.1.20		24
5.2	odd	4		1805.2.b.l.1084.20		24
5.3	odd	4		1805.2.b.l.1084.5		24
5.4	even	2	inner	9025.2.a.ct.1.5		24
19.14	odd	18		475.2.l.f.101.2		48
19.15	odd	18		475.2.l.f.301.2		48
19.18	odd	2		9025.2.a.cu.1.5		24
95.14	odd	18		475.2.l.f.101.7		48
95.18	even	4		1805.2.b.k.1084.20		24
95.33	even	36		95.2.p.a.44.7	yes	48
95.34	odd	18		475.2.l.f.301.7		48
95.37	even	4		1805.2.b.k.1084.5		24
95.52	even	36		95.2.p.a.44.2	✓	48
95.53	even	36		95.2.p.a.54.2	yes	48
95.72	even	36		95.2.p.a.54.7	yes	48
95.94	odd	2		9025.2.a.cu.1.20		24
285.53	odd	36		855.2.da.b.244.7		48
285.128	odd	36		855.2.da.b.424.2		48
285.167	odd	36		855.2.da.b.244.2		48
285.242	odd	36		855.2.da.b.424.7		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.p.a.44.2	✓	48	95.52	even	36
95.2.p.a.44.7	yes	48	95.33	even	36
95.2.p.a.54.2	yes	48	95.53	even	36
95.2.p.a.54.7	yes	48	95.72	even	36
475.2.l.f.101.2		48	19.14	odd	18
475.2.l.f.101.7		48	95.14	odd	18
475.2.l.f.301.2		48	19.15	odd	18
475.2.l.f.301.7		48	95.34	odd	18
855.2.da.b.244.2		48	285.167	odd	36
855.2.da.b.244.7		48	285.53	odd	36
855.2.da.b.424.2		48	285.128	odd	36
855.2.da.b.424.7		48	285.242	odd	36
1805.2.b.k.1084.5		24	95.37	even	4
1805.2.b.k.1084.20		24	95.18	even	4
1805.2.b.l.1084.5		24	5.3	odd	4
1805.2.b.l.1084.20		24	5.2	odd	4
9025.2.a.ct.1.5		24	5.4	even	2	inner
9025.2.a.ct.1.20		24	1.1	even	1	trivial
9025.2.a.cu.1.5		24	19.18	odd	2
9025.2.a.cu.1.20		24	95.94	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q+1.78468 q^{2} +2.38377 q^{3} +1.18508 q^{4} +4.25426 q^{6} -4.23911 q^{7} -1.45438 q^{8} +2.68235 q^{9} -0.490889 q^{11} +2.82495 q^{12} +4.16199 q^{13} -7.56544 q^{14} -4.96575 q^{16} +2.03619 q^{17} +4.78713 q^{18} -10.1050 q^{21} -0.876080 q^{22} -4.39525 q^{23} -3.46689 q^{24} +7.42782 q^{26} -0.757211 q^{27} -5.02367 q^{28} +3.26270 q^{29} -4.08833 q^{31} -5.95351 q^{32} -1.17017 q^{33} +3.63395 q^{34} +3.17879 q^{36} -2.14440 q^{37} +9.92123 q^{39} -4.36602 q^{41} -18.0343 q^{42} -10.6241 q^{43} -0.581742 q^{44} -7.84410 q^{46} -2.62142 q^{47} -11.8372 q^{48} +10.9700 q^{49} +4.85381 q^{51} +4.93228 q^{52} +11.4341 q^{53} -1.35138 q^{54} +6.16526 q^{56} +5.82287 q^{58} +0.542306 q^{59} -13.6423 q^{61} -7.29635 q^{62} -11.3708 q^{63} -0.693606 q^{64} -2.08837 q^{66} -7.15699 q^{67} +2.41305 q^{68} -10.4772 q^{69} -6.03858 q^{71} -3.90114 q^{72} -2.05419 q^{73} -3.82707 q^{74} +2.08093 q^{77} +17.7062 q^{78} -5.34029 q^{79} -9.85206 q^{81} -7.79193 q^{82} -8.11578 q^{83} -11.9753 q^{84} -18.9605 q^{86} +7.77752 q^{87} +0.713937 q^{88} +4.34099 q^{89} -17.6431 q^{91} -5.20871 q^{92} -9.74562 q^{93} -4.67839 q^{94} -14.1918 q^{96} +5.64669 q^{97} +19.5780 q^{98} -1.31674 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 18 q^{4} - 12 q^{6} + 12 q^{9} + 12 q^{11} - 24 q^{14} + 6 q^{16} - 6 q^{21} - 42 q^{24} - 12 q^{26} - 36 q^{29} - 42 q^{31} - 6 q^{34} - 6 q^{36} + 24 q^{39} - 60 q^{41} - 30 q^{44} - 6 q^{46} + 12 q^{49}+ \cdots - 120 q^{96}+O(q^{100}) \) 24 * q + 18 * q^4 - 12 * q^6 + 12 * q^9 + 12 * q^11 - 24 * q^14 + 6 * q^16 - 6 * q^21 - 42 * q^24 - 12 * q^26 - 36 * q^29 - 42 * q^31 - 6 * q^34 - 6 * q^36 + 24 * q^39 - 60 * q^41 - 30 * q^44 - 6 * q^46 + 12 * q^49 - 30 * q^51 - 24 * q^54 - 18 * q^56 - 60 * q^59 + 30 * q^61 + 36 * q^66 - 66 * q^69 - 96 * q^71 + 24 * q^74 - 72 * q^79 - 96 * q^81 + 54 * q^84 - 108 * q^86 - 84 * q^89 - 96 * q^91 - 36 * q^94 - 120 * q^96

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