LMFDB - Embedded newform 7500.2.d.g.1249.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7500,2,Mod(1249,7500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7500, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("7500.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7500 = 2^{2} \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7500.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$59.8878015160$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 300)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.16
Character	$\chi$	$=$	7500.1249
Dual form			7500.2.d.g.1249.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -3.78808i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -3.78808i q^{7} -1.00000 q^{9} -0.807679 q^{11} +4.74950i q^{13} +1.14884i q^{17} -0.0150112 q^{19} +3.78808 q^{21} -6.26797i q^{23} -1.00000i q^{27} -3.70092 q^{29} -1.58209 q^{31} -0.807679i q^{33} +8.54179i q^{37} -4.74950 q^{39} +11.4989 q^{41} +10.2458i q^{43} +0.526282i q^{47} -7.34957 q^{49} -1.14884 q^{51} -2.94976i q^{53} -0.0150112i q^{57} -11.4436 q^{59} +3.14212 q^{61} +3.78808i q^{63} -13.2812i q^{67} +6.26797 q^{69} -4.91318 q^{71} -4.67860i q^{73} +3.05955i q^{77} -9.27155 q^{79} +1.00000 q^{81} +1.42370i q^{83} -3.70092i q^{87} +16.1067 q^{89} +17.9915 q^{91} -1.58209i q^{93} -8.06203i q^{97} +0.807679 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{9}+O(q^{10}) $ 24 * q - 24 * q^9	$ 24 q - 24 q^{9} + 4 q^{11} - 20 q^{19} + 16 q^{21} - 16 q^{29} - 4 q^{31} + 20 q^{41} - 56 q^{49} + 16 q^{51} + 4 q^{59} + 68 q^{61} - 36 q^{69} - 12 q^{79} + 24 q^{81} - 20 q^{89} + 40 q^{91} - 4 q^{99}+O(q^{100}) $ 24 * q - 24 * q^9 + 4 * q^11 - 20 * q^19 + 16 * q^21 - 16 * q^29 - 4 * q^31 + 20 * q^41 - 56 * q^49 + 16 * q^51 + 4 * q^59 + 68 * q^61 - 36 * q^69 - 12 * q^79 + 24 * q^81 - 20 * q^89 + 40 * q^91 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7500\mathbb{Z}\right)^\times$.

$n$	$2501$	$3751$	$6877$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.78808i	− 1.43176i	−0.698223	−	0.715880i	$-0.746026\pi$
$7$	− 3.78808i	− 1.43176i	0.698223	−	0.715880i	$-0.253974\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−0.807679	−0.243524	−0.121762	−	0.992559i	$-0.538855\pi$
$11$	−0.807679	−0.243524	−0.121762	+	0.992559i	$0.538855\pi$
$12$	0	0
$13$	4.74950i	1.31727i	0.752461	+	0.658637i	$0.228867\pi$
$13$	4.74950i	1.31727i	−0.752461	+	0.658637i	$0.771133\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.14884i	0.278636i	0.990248	+	0.139318i	$0.0444910\pi$
$17$	1.14884i	0.278636i	−0.990248	+	0.139318i	$0.955509\pi$
$18$	0	0
$19$	−0.0150112	−0.00344380	−0.00172190	−	0.999999i	$-0.500548\pi$
$19$	−0.0150112	−0.00344380	−0.00172190	+	0.999999i	$0.500548\pi$
$20$	0	0
$21$	3.78808	0.826627
$22$	0	0
$23$	− 6.26797i	− 1.30696i	−0.756943	−	0.653481i	$-0.773308\pi$
$23$	− 6.26797i	− 1.30696i	0.756943	−	0.653481i	$-0.226692\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−3.70092	−0.687243	−0.343621	−	0.939108i	$-0.611654\pi$
$29$	−3.70092	−0.687243	−0.343621	+	0.939108i	$0.611654\pi$
$30$	0	0
$31$	−1.58209	−0.284152	−0.142076	−	0.989856i	$-0.545378\pi$
$31$	−1.58209	−0.284152	−0.142076	+	0.989856i	$0.545378\pi$
$32$	0	0
$33$	− 0.807679i	− 0.140599i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.54179i	1.40426i	0.712048	+	0.702131i	$0.247768\pi$
$37$	8.54179i	1.40426i	−0.712048	+	0.702131i	$0.752232\pi$
$38$	0	0
$39$	−4.74950	−0.760528
$40$	0	0
$41$	11.4989	1.79582	0.897912	−	0.440176i	$-0.145084\pi$
$41$	11.4989	1.79582	0.897912	+	0.440176i	$0.145084\pi$
$42$	0	0
$43$	10.2458i	1.56247i	0.624238	+	0.781234i	$0.285409\pi$
$43$	10.2458i	1.56247i	−0.624238	+	0.781234i	$0.714591\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.526282i	0.0767662i	0.999263	+	0.0383831i	$0.0122207\pi$
$47$	0.526282i	0.0767662i	−0.999263	+	0.0383831i	$0.987779\pi$
$48$	0	0
$49$	−7.34957	−1.04994
$50$	0	0
$51$	−1.14884	−0.160870
$52$	0	0
$53$	− 2.94976i	− 0.405181i	−0.979264	−	0.202591i	$-0.935064\pi$
$53$	− 2.94976i	− 0.405181i	0.979264	−	0.202591i	$-0.0649361\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 0.0150112i	− 0.00198828i
$58$	0	0
$59$	−11.4436	−1.48982	−0.744912	−	0.667163i	$-0.767508\pi$
$59$	−11.4436	−1.48982	−0.744912	+	0.667163i	$0.767508\pi$
$60$	0	0
$61$	3.14212	0.402307	0.201154	−	0.979560i	$-0.435531\pi$
$61$	3.14212	0.402307	0.201154	+	0.979560i	$0.435531\pi$
$62$	0	0
$63$	3.78808i	0.477254i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 13.2812i	− 1.62256i	−0.584660	−	0.811278i	$-0.698772\pi$
$67$	− 13.2812i	− 1.62256i	0.584660	−	0.811278i	$-0.301228\pi$
$68$	0	0
$69$	6.26797	0.754574
$70$	0	0
$71$	−4.91318	−0.583088	−0.291544	−	0.956557i	$-0.594169\pi$
$71$	−4.91318	−0.583088	−0.291544	+	0.956557i	$0.594169\pi$
$72$	0	0
$73$	− 4.67860i	− 0.547589i	−0.961788	−	0.273795i	$-0.911721\pi$
$73$	− 4.67860i	− 0.547589i	0.961788	−	0.273795i	$-0.0882789\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.05955i	0.348668i
$78$	0	0
$79$	−9.27155	−1.04313	−0.521565	−	0.853211i	$-0.674652\pi$
$79$	−9.27155	−1.04313	−0.521565	+	0.853211i	$0.674652\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.42370i	0.156271i	0.996943	+	0.0781355i	$0.0248967\pi$
$83$	1.42370i	0.156271i	−0.996943	+	0.0781355i	$0.975103\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 3.70092i	− 0.396780i
$88$	0	0
$89$	16.1067	1.70731	0.853655	−	0.520839i	$-0.174381\pi$
$89$	16.1067	1.70731	0.853655	+	0.520839i	$0.174381\pi$
$90$	0	0
$91$	17.9915	1.88602
$92$	0	0
$93$	− 1.58209i	− 0.164055i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 8.06203i	− 0.818575i	−0.912406	−	0.409287i	$-0.865777\pi$
$97$	− 8.06203i	− 0.818575i	0.912406	−	0.409287i	$-0.134223\pi$
$98$	0	0
$99$	0.807679	0.0811748
$100$	0	0
$101$	11.6496	1.15918	0.579590	−	0.814908i	$-0.303213\pi$
$101$	11.6496	1.15918	0.579590	+	0.814908i	$0.303213\pi$
$102$	0	0
$103$	18.8439i	1.85674i	0.371657	+	0.928370i	$0.378790\pi$
$103$	18.8439i	1.85674i	−0.371657	+	0.928370i	$0.621210\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.3957i	1.39168i	0.718195	+	0.695842i	$0.244969\pi$
$107$	14.3957i	1.39168i	−0.718195	+	0.695842i	$0.755031\pi$
$108$	0	0
$109$	5.76186	0.551886	0.275943	−	0.961174i	$-0.411010\pi$
$109$	5.76186	0.551886	0.275943	+	0.961174i	$0.411010\pi$
$110$	0	0
$111$	−8.54179	−0.810751
$112$	0	0
$113$	17.9263i	1.68637i	0.537624	+	0.843184i	$0.319322\pi$
$113$	17.9263i	1.68637i	−0.537624	+	0.843184i	$0.680678\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.74950i	− 0.439091i
$118$	0	0
$119$	4.35192	0.398940
$120$	0	0
$121$	−10.3477	−0.940696
$122$	0	0
$123$	11.4989i	1.03682i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.28981i	0.203188i	0.994826	+	0.101594i	$0.0323943\pi$
$127$	2.28981i	0.203188i	−0.994826	+	0.101594i	$0.967606\pi$
$128$	0	0
$129$	−10.2458	−0.902091
$130$	0	0
$131$	−7.04946	−0.615914	−0.307957	−	0.951400i	$-0.599645\pi$
$131$	−7.04946	−0.615914	−0.307957	+	0.951400i	$0.599645\pi$
$132$	0	0
$133$	0.0568635i	0.00493069i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21.9329i	1.87385i	0.349525	+	0.936927i	$0.386343\pi$
$137$	21.9329i	1.87385i	−0.349525	+	0.936927i	$0.613657\pi$
$138$	0	0
$139$	−9.71761	−0.824237	−0.412118	−	0.911130i	$-0.635211\pi$
$139$	−9.71761	−0.824237	−0.412118	+	0.911130i	$0.635211\pi$
$140$	0	0
$141$	−0.526282	−0.0443210
$142$	0	0
$143$	− 3.83607i	− 0.320788i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 7.34957i	− 0.606182i
$148$	0	0
$149$	13.9712	1.14457	0.572284	−	0.820056i	$-0.306058\pi$
$149$	13.9712	1.14457	0.572284	+	0.820056i	$0.306058\pi$
$150$	0	0
$151$	−20.1871	−1.64280	−0.821400	−	0.570352i	$-0.806807\pi$
$151$	−20.1871	−1.64280	−0.821400	+	0.570352i	$0.806807\pi$
$152$	0	0
$153$	− 1.14884i	− 0.0928786i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 7.76546i	− 0.619751i	−0.950777	−	0.309876i	$-0.899713\pi$
$157$	− 7.76546i	− 0.619751i	0.950777	−	0.309876i	$-0.100287\pi$
$158$	0	0
$159$	2.94976	0.233932
$160$	0	0
$161$	−23.7436	−1.87126
$162$	0	0
$163$	− 13.8908i	− 1.08801i	−0.839082	−	0.544006i	$-0.816907\pi$
$163$	− 13.8908i	− 1.08801i	0.839082	−	0.544006i	$-0.183093\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.89787i	0.224244i	0.993694	+	0.112122i	$0.0357647\pi$
$167$	2.89787i	0.224244i	−0.993694	+	0.112122i	$0.964235\pi$
$168$	0	0
$169$	−9.55771	−0.735209
$170$	0	0
$171$	0.0150112	0.00114793
$172$	0	0
$173$	15.2338i	1.15821i	0.815254	+	0.579104i	$0.196597\pi$
$173$	15.2338i	1.15821i	−0.815254	+	0.579104i	$0.803403\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 11.4436i	− 0.860150i
$178$	0	0
$179$	23.3056	1.74194	0.870971	−	0.491335i	$-0.163491\pi$
$179$	23.3056	1.74194	0.870971	+	0.491335i	$0.163491\pi$
$180$	0	0
$181$	17.6985	1.31552	0.657758	−	0.753229i	$-0.271505\pi$
$181$	17.6985	1.31552	0.657758	+	0.753229i	$0.271505\pi$
$182$	0	0
$183$	3.14212i	0.232272i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 0.927897i	− 0.0678546i
$188$	0	0
$189$	−3.78808	−0.275542
$190$	0	0
$191$	9.36414	0.677566	0.338783	−	0.940865i	$-0.389985\pi$
$191$	9.36414	0.677566	0.338783	+	0.940865i	$0.389985\pi$
$192$	0	0
$193$	18.9309i	1.36268i	0.731969	+	0.681338i	$0.238602\pi$
$193$	18.9309i	1.36268i	−0.731969	+	0.681338i	$0.761398\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.54357i	0.537457i	0.963216	+	0.268729i	$0.0866035\pi$
$197$	7.54357i	0.537457i	−0.963216	+	0.268729i	$0.913397\pi$
$198$	0	0
$199$	−3.58560	−0.254176	−0.127088	−	0.991891i	$-0.540563\pi$
$199$	−3.58560	−0.254176	−0.127088	+	0.991891i	$0.540563\pi$
$200$	0	0
$201$	13.2812	0.936783
$202$	0	0
$203$	14.0194i	0.983967i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.26797i	0.435654i
$208$	0	0
$209$	0.0121242	0.000838648 0
$210$	0	0
$211$	2.08493	0.143532	0.0717662	−	0.997421i	$-0.477136\pi$
$211$	2.08493	0.143532	0.0717662	+	0.997421i	$0.477136\pi$
$212$	0	0
$213$	− 4.91318i	− 0.336646i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5.99309i	0.406838i
$218$	0	0
$219$	4.67860	0.316151
$220$	0	0
$221$	−5.45643	−0.367039
$222$	0	0
$223$	14.0373i	0.940007i	0.882665	+	0.470003i	$0.155747\pi$
$223$	14.0373i	0.940007i	−0.882665	+	0.470003i	$0.844253\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.53486i	0.168244i	0.996455	+	0.0841221i	$0.0268086\pi$
$227$	2.53486i	0.168244i	−0.996455	+	0.0841221i	$0.973191\pi$
$228$	0	0
$229$	16.5262	1.09208	0.546041	−	0.837759i	$-0.316134\pi$
$229$	16.5262	1.09208	0.546041	+	0.837759i	$0.316134\pi$
$230$	0	0
$231$	−3.05955	−0.201304
$232$	0	0
$233$	26.5208i	1.73744i	0.495305	+	0.868719i	$0.335056\pi$
$233$	26.5208i	1.73744i	−0.495305	+	0.868719i	$0.664944\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 9.27155i	− 0.602252i
$238$	0	0
$239$	7.46385	0.482797	0.241398	−	0.970426i	$-0.422394\pi$
$239$	7.46385	0.482797	0.241398	+	0.970426i	$0.422394\pi$
$240$	0	0
$241$	10.2978	0.663337	0.331669	−	0.943396i	$-0.392388\pi$
$241$	10.2978	0.663337	0.331669	+	0.943396i	$0.392388\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 0.0712954i	− 0.00453642i
$248$	0	0
$249$	−1.42370	−0.0902231
$250$	0	0
$251$	19.5809	1.23593	0.617967	−	0.786204i	$-0.287956\pi$
$251$	19.5809	1.23593	0.617967	+	0.786204i	$0.287956\pi$
$252$	0	0
$253$	5.06250i	0.318277i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.98030i	0.373041i	0.982451	+	0.186520i	$0.0597210\pi$
$257$	5.98030i	0.373041i	−0.982451	+	0.186520i	$0.940279\pi$
$258$	0	0
$259$	32.3570	2.01057
$260$	0	0
$261$	3.70092	0.229081
$262$	0	0
$263$	6.07038i	0.374315i	0.982330	+	0.187158i	$0.0599275\pi$
$263$	6.07038i	0.374315i	−0.982330	+	0.187158i	$0.940072\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	16.1067i	0.985716i
$268$	0	0
$269$	−9.61605	−0.586301	−0.293150	−	0.956066i	$-0.594704\pi$
$269$	−9.61605	−0.586301	−0.293150	+	0.956066i	$0.594704\pi$
$270$	0	0
$271$	−30.4276	−1.84834	−0.924171	−	0.381979i	$-0.875243\pi$
$271$	−30.4276	−1.84834	−0.924171	+	0.381979i	$0.875243\pi$
$272$	0	0
$273$	17.9915i	1.08889i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.20393i	0.432842i	0.976300	+	0.216421i	$0.0694384\pi$
$277$	7.20393i	0.432842i	−0.976300	+	0.216421i	$0.930562\pi$
$278$	0	0
$279$	1.58209	0.0947173
$280$	0	0
$281$	29.7964	1.77750	0.888751	−	0.458390i	$-0.151574\pi$
$281$	29.7964	1.77750	0.888751	+	0.458390i	$0.151574\pi$
$282$	0	0
$283$	8.82402i	0.524534i	0.964995	+	0.262267i	$0.0844701\pi$
$283$	8.82402i	0.524534i	−0.964995	+	0.262267i	$0.915530\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 43.5587i	− 2.57119i
$288$	0	0
$289$	15.6802	0.922362
$290$	0	0
$291$	8.06203	0.472604
$292$	0	0
$293$	− 4.20743i	− 0.245800i	−0.992419	−	0.122900i	$-0.960780\pi$
$293$	− 4.20743i	− 0.245800i	0.992419	−	0.122900i	$-0.0392195\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.807679i	0.0468663i
$298$	0	0
$299$	29.7697	1.72163
$300$	0	0
$301$	38.8119	2.23708
$302$	0	0
$303$	11.6496i	0.669253i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 4.41109i	− 0.251754i	−0.992046	−	0.125877i	$-0.959825\pi$
$307$	− 4.41109i	− 0.251754i	0.992046	−	0.125877i	$-0.0401745\pi$
$308$	0	0
$309$	−18.8439	−1.07199
$310$	0	0
$311$	−6.32058	−0.358407	−0.179204	−	0.983812i	$-0.557352\pi$
$311$	−6.32058	−0.358407	−0.179204	+	0.983812i	$0.557352\pi$
$312$	0	0
$313$	− 15.3245i	− 0.866193i	−0.901347	−	0.433097i	$-0.857421\pi$
$313$	− 15.3245i	− 0.866193i	0.901347	−	0.433097i	$-0.142579\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.3155i	1.19720i	0.801048	+	0.598600i	$0.204276\pi$
$317$	21.3155i	1.19720i	−0.801048	+	0.598600i	$0.795724\pi$
$318$	0	0
$319$	2.98915	0.167360
$320$	0	0
$321$	−14.3957	−0.803489
$322$	0	0
$323$	− 0.0172455i	0 0.000959564i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5.76186i	0.318631i
$328$	0	0
$329$	1.99360	0.109911
$330$	0	0
$331$	19.8124	1.08899	0.544494	−	0.838764i	$-0.316722\pi$
$331$	19.8124	1.08899	0.544494	+	0.838764i	$0.316722\pi$
$332$	0	0
$333$	− 8.54179i	− 0.468087i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	10.0339i	0.546583i	0.961931	+	0.273292i	$0.0881124\pi$
$337$	10.0339i	0.546583i	−0.961931	+	0.273292i	$0.911888\pi$
$338$	0	0
$339$	−17.9263	−0.973626
$340$	0	0
$341$	1.27782	0.0691979
$342$	0	0
$343$	1.32419i	0.0714997i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 13.8672i	− 0.744429i	−0.928147	−	0.372214i	$-0.878599\pi$
$347$	− 13.8672i	− 0.744429i	0.928147	−	0.372214i	$-0.121401\pi$
$348$	0	0
$349$	−27.2533	−1.45883	−0.729417	−	0.684069i	$-0.760209\pi$
$349$	−27.2533	−1.45883	−0.729417	+	0.684069i	$0.760209\pi$
$350$	0	0
$351$	4.74950	0.253509
$352$	0	0
$353$	− 12.4138i	− 0.660722i	−0.943855	−	0.330361i	$-0.892830\pi$
$353$	− 12.4138i	− 0.660722i	0.943855	−	0.330361i	$-0.107170\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.35192i	0.230328i
$358$	0	0
$359$	19.0244	1.00407	0.502034	−	0.864848i	$-0.332585\pi$
$359$	19.0244	1.00407	0.502034	+	0.864848i	$0.332585\pi$
$360$	0	0
$361$	−18.9998	−0.999988
$362$	0	0
$363$	− 10.3477i	− 0.543111i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 0.975194i	− 0.0509047i	−0.999676	−	0.0254524i	$-0.991897\pi$
$367$	− 0.975194i	− 0.0509047i	0.999676	−	0.0254524i	$-0.00810261\pi$
$368$	0	0
$369$	−11.4989	−0.598608
$370$	0	0
$371$	−11.1740	−0.580123
$372$	0	0
$373$	18.7723i	0.971994i	0.873960	+	0.485997i	$0.161543\pi$
$373$	18.7723i	0.971994i	−0.873960	+	0.485997i	$0.838457\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 17.5775i	− 0.905287i
$378$	0	0
$379$	3.44756	0.177089	0.0885447	−	0.996072i	$-0.471778\pi$
$379$	3.44756	0.177089	0.0885447	+	0.996072i	$0.471778\pi$
$380$	0	0
$381$	−2.28981	−0.117311
$382$	0	0
$383$	27.3045i	1.39519i	0.716490	+	0.697597i	$0.245747\pi$
$383$	27.3045i	1.39519i	−0.716490	+	0.697597i	$0.754253\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 10.2458i	− 0.520823i
$388$	0	0
$389$	−9.81725	−0.497754	−0.248877	−	0.968535i	$-0.580062\pi$
$389$	−9.81725	−0.497754	−0.248877	+	0.968535i	$0.580062\pi$
$390$	0	0
$391$	7.20092	0.364166
$392$	0	0
$393$	− 7.04946i	− 0.355598i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	11.6131i	0.582846i	0.956594	+	0.291423i	$0.0941287\pi$
$397$	11.6131i	0.582846i	−0.956594	+	0.291423i	$0.905871\pi$
$398$	0	0
$399$	−0.0568635	−0.00284674
$400$	0	0
$401$	−25.4145	−1.26914	−0.634570	−	0.772865i	$-0.718823\pi$
$401$	−25.4145	−1.26914	−0.634570	+	0.772865i	$0.718823\pi$
$402$	0	0
$403$	− 7.51414i	− 0.374306i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 6.89902i	− 0.341972i
$408$	0	0
$409$	−17.2515	−0.853032	−0.426516	−	0.904480i	$-0.640259\pi$
$409$	−17.2515	−0.853032	−0.426516	+	0.904480i	$0.640259\pi$
$410$	0	0
$411$	−21.9329	−1.08187
$412$	0	0
$413$	43.3491i	2.13307i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 9.71761i	− 0.475873i
$418$	0	0
$419$	20.4505	0.999074	0.499537	−	0.866293i	$-0.333503\pi$
$419$	20.4505	0.999074	0.499537	+	0.866293i	$0.333503\pi$
$420$	0	0
$421$	−20.9082	−1.01901	−0.509503	−	0.860469i	$-0.670171\pi$
$421$	−20.9082	−1.01901	−0.509503	+	0.860469i	$0.670171\pi$
$422$	0	0
$423$	− 0.526282i	− 0.0255887i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 11.9026i	− 0.576008i
$428$	0	0
$429$	3.83607	0.185207
$430$	0	0
$431$	−9.25191	−0.445649	−0.222824	−	0.974859i	$-0.571528\pi$
$431$	−9.25191	−0.445649	−0.222824	+	0.974859i	$0.571528\pi$
$432$	0	0
$433$	− 0.235240i	− 0.0113049i	−0.999984	−	0.00565246i	$-0.998201\pi$
$433$	− 0.235240i	− 0.0113049i	0.999984	−	0.00565246i	$-0.00179924\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.0940894i	0.00450091i
$438$	0	0
$439$	−31.2013	−1.48916	−0.744578	−	0.667536i	$-0.767349\pi$
$439$	−31.2013	−1.48916	−0.744578	+	0.667536i	$0.767349\pi$
$440$	0	0
$441$	7.34957	0.349979
$442$	0	0
$443$	24.3862i	1.15862i	0.815106	+	0.579311i	$0.196678\pi$
$443$	24.3862i	1.15862i	−0.815106	+	0.579311i	$0.803322\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	13.9712i	0.660816i
$448$	0	0
$449$	−23.9483	−1.13019	−0.565096	−	0.825025i	$-0.691161\pi$
$449$	−23.9483	−1.13019	−0.565096	+	0.825025i	$0.691161\pi$
$450$	0	0
$451$	−9.28740	−0.437327
$452$	0	0
$453$	− 20.1871i	− 0.948471i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 3.91244i	− 0.183016i	−0.995804	−	0.0915082i	$-0.970831\pi$
$457$	− 3.91244i	− 0.183016i	0.995804	−	0.0915082i	$-0.0291688\pi$
$458$	0	0
$459$	1.14884	0.0536235
$460$	0	0
$461$	−2.58548	−0.120418	−0.0602090	−	0.998186i	$-0.519177\pi$
$461$	−2.58548	−0.120418	−0.0602090	+	0.998186i	$0.519177\pi$
$462$	0	0
$463$	21.9089i	1.01819i	0.860710	+	0.509096i	$0.170020\pi$
$463$	21.9089i	1.01819i	−0.860710	+	0.509096i	$0.829980\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 8.19679i	− 0.379302i	−0.981852	−	0.189651i	$-0.939264\pi$
$467$	− 8.19679i	− 0.379302i	0.981852	−	0.189651i	$-0.0607357\pi$
$468$	0	0
$469$	−50.3103	−2.32311
$470$	0	0
$471$	7.76546	0.357814
$472$	0	0
$473$	− 8.27531i	− 0.380499i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.94976i	0.135060i
$478$	0	0
$479$	6.15176	0.281081	0.140540	−	0.990075i	$-0.455116\pi$
$479$	6.15176	0.281081	0.140540	+	0.990075i	$0.455116\pi$
$480$	0	0
$481$	−40.5692	−1.84980
$482$	0	0
$483$	− 23.7436i	− 1.08037i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 6.28171i	− 0.284651i	−0.989820	−	0.142326i	$-0.954542\pi$
$487$	− 6.28171i	− 0.284651i	0.989820	−	0.142326i	$-0.0454580\pi$
$488$	0	0
$489$	13.8908	0.628163
$490$	0	0
$491$	−2.75137	−0.124168	−0.0620839	−	0.998071i	$-0.519775\pi$
$491$	−2.75137	−0.124168	−0.0620839	+	0.998071i	$0.519775\pi$
$492$	0	0
$493$	− 4.25178i	− 0.191490i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	18.6115i	0.834842i
$498$	0	0
$499$	16.4263	0.735341	0.367670	−	0.929956i	$-0.380156\pi$
$499$	16.4263	0.735341	0.367670	+	0.929956i	$0.380156\pi$
$500$	0	0
$501$	−2.89787	−0.129467
$502$	0	0
$503$	− 12.2874i	− 0.547868i	−0.961749	−	0.273934i	$-0.911675\pi$
$503$	− 12.2874i	− 0.547868i	0.961749	−	0.273934i	$-0.0883249\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.55771i	− 0.424473i
$508$	0	0
$509$	10.5100	0.465848	0.232924	−	0.972495i	$-0.425171\pi$
$509$	10.5100	0.465848	0.232924	+	0.972495i	$0.425171\pi$
$510$	0	0
$511$	−17.7229	−0.784017
$512$	0	0
$513$	0.0150112i	0 0.000662759i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 0.425067i	− 0.0186944i
$518$	0	0
$519$	−15.2338	−0.668691
$520$	0	0
$521$	−7.82409	−0.342780	−0.171390	−	0.985203i	$-0.554826\pi$
$521$	−7.82409	−0.342780	−0.171390	+	0.985203i	$0.554826\pi$
$522$	0	0
$523$	22.0475i	0.964068i	0.876153	+	0.482034i	$0.160102\pi$
$523$	22.0475i	0.964068i	−0.876153	+	0.482034i	$0.839898\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.81758i	− 0.0791749i
$528$	0	0
$529$	−16.2874	−0.708148
$530$	0	0
$531$	11.4436	0.496608
$532$	0	0
$533$	54.6139i	2.36559i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	23.3056i	1.00571i
$538$	0	0
$539$	5.93609	0.255685
$540$	0	0
$541$	12.4670	0.535998	0.267999	−	0.963419i	$-0.413638\pi$
$541$	12.4670	0.535998	0.267999	+	0.963419i	$0.413638\pi$
$542$	0	0
$543$	17.6985i	0.759514i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 2.51606i	− 0.107579i	−0.998552	−	0.0537895i	$-0.982870\pi$
$547$	− 2.51606i	− 0.107579i	0.998552	−	0.0537895i	$-0.0171300\pi$
$548$	0	0
$549$	−3.14212	−0.134102
$550$	0	0
$551$	0.0555550	0.00236672
$552$	0	0
$553$	35.1214i	1.49351i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.262544i	0.0111244i	0.999985	+	0.00556218i	$0.00177051\pi$
$557$	0.262544i	0.0111244i	−0.999985	+	0.00556218i	$0.998229\pi$
$558$	0	0
$559$	−48.6623	−2.05820
$560$	0	0
$561$	0.927897	0.0391759
$562$	0	0
$563$	32.4826i	1.36898i	0.729023	+	0.684489i	$0.239975\pi$
$563$	32.4826i	1.36898i	−0.729023	+	0.684489i	$0.760025\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 3.78808i	− 0.159085i
$568$	0	0
$569$	38.7980	1.62650	0.813248	−	0.581917i	$-0.197697\pi$
$569$	38.7980	1.62650	0.813248	+	0.581917i	$0.197697\pi$
$570$	0	0
$571$	−35.6801	−1.49317	−0.746583	−	0.665292i	$-0.768307\pi$
$571$	−35.6801	−1.49317	−0.746583	+	0.665292i	$0.768307\pi$
$572$	0	0
$573$	9.36414i	0.391193i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	25.3430i	1.05504i	0.849542	+	0.527522i	$0.176879\pi$
$577$	25.3430i	1.05504i	−0.849542	+	0.527522i	$0.823121\pi$
$578$	0	0
$579$	−18.9309	−0.786741
$580$	0	0
$581$	5.39308	0.223743
$582$	0	0
$583$	2.38246i	0.0986715i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 0.515670i	− 0.0212840i	−0.999943	−	0.0106420i	$-0.996612\pi$
$587$	− 0.515670i	− 0.0212840i	0.999943	−	0.0106420i	$-0.00338751\pi$
$588$	0	0
$589$	0.0237490	0.000978561 0
$590$	0	0
$591$	−7.54357	−0.310301
$592$	0	0
$593$	7.14389i	0.293364i	0.989184	+	0.146682i	$0.0468595\pi$
$593$	7.14389i	0.293364i	−0.989184	+	0.146682i	$0.953141\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 3.58560i	− 0.146749i
$598$	0	0
$599$	−23.6627	−0.966833	−0.483417	−	0.875390i	$-0.660604\pi$
$599$	−23.6627	−0.966833	−0.483417	+	0.875390i	$0.660604\pi$
$600$	0	0
$601$	−7.98023	−0.325520	−0.162760	−	0.986666i	$-0.552040\pi$
$601$	−7.98023	−0.325520	−0.162760	+	0.986666i	$0.552040\pi$
$602$	0	0
$603$	13.2812i	0.540852i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17.2004i	0.698144i	0.937096	+	0.349072i	$0.113503\pi$
$607$	17.2004i	0.698144i	−0.937096	+	0.349072i	$0.886497\pi$
$608$	0	0
$609$	−14.0194	−0.568094
$610$	0	0
$611$	−2.49958	−0.101122
$612$	0	0
$613$	14.7641i	0.596315i	0.954517	+	0.298158i	$0.0963721\pi$
$613$	14.7641i	0.596315i	−0.954517	+	0.298158i	$0.903628\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 15.4714i	− 0.622854i	−0.950270	−	0.311427i	$-0.899193\pi$
$617$	− 15.4714i	− 0.622854i	0.950270	−	0.311427i	$-0.100807\pi$
$618$	0	0
$619$	37.4710	1.50608	0.753042	−	0.657972i	$-0.228586\pi$
$619$	37.4710	1.50608	0.753042	+	0.657972i	$0.228586\pi$
$620$	0	0
$621$	−6.26797	−0.251525
$622$	0	0
$623$	− 61.0136i	− 2.44446i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.0121242i	0 0.000484194i
$628$	0	0
$629$	−9.81319	−0.391277
$630$	0	0
$631$	−1.51672	−0.0603796	−0.0301898	−	0.999544i	$-0.509611\pi$
$631$	−1.51672	−0.0603796	−0.0301898	+	0.999544i	$0.509611\pi$
$632$	0	0
$633$	2.08493i	0.0828685i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 34.9067i	− 1.38306i
$638$	0	0
$639$	4.91318	0.194363
$640$	0	0
$641$	−3.86770	−0.152765	−0.0763825	−	0.997079i	$-0.524337\pi$
$641$	−3.86770	−0.152765	−0.0763825	+	0.997079i	$0.524337\pi$
$642$	0	0
$643$	23.2212i	0.915756i	0.889015	+	0.457878i	$0.151390\pi$
$643$	23.2212i	0.915756i	−0.889015	+	0.457878i	$0.848610\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 26.1598i	− 1.02845i	−0.857655	−	0.514225i	$-0.828080\pi$
$647$	− 26.1598i	− 1.02845i	0.857655	−	0.514225i	$-0.171920\pi$
$648$	0	0
$649$	9.24271	0.362808
$650$	0	0
$651$	−5.99309	−0.234888
$652$	0	0
$653$	− 8.55471i	− 0.334772i	−0.985891	−	0.167386i	$-0.946467\pi$
$653$	− 8.55471i	− 0.334772i	0.985891	−	0.167386i	$-0.0535326\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.67860i	0.182530i
$658$	0	0
$659$	41.9383	1.63368	0.816841	−	0.576862i	$-0.195723\pi$
$659$	41.9383	1.63368	0.816841	+	0.576862i	$0.195723\pi$
$660$	0	0
$661$	−17.5167	−0.681322	−0.340661	−	0.940186i	$-0.610651\pi$
$661$	−17.5167	−0.681322	−0.340661	+	0.940186i	$0.610651\pi$
$662$	0	0
$663$	− 5.45643i	− 0.211910i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	23.1972i	0.898200i
$668$	0	0
$669$	−14.0373	−0.542713
$670$	0	0
$671$	−2.53782	−0.0979716
$672$	0	0
$673$	− 29.0952i	− 1.12154i	−0.827972	−	0.560769i	$-0.810506\pi$
$673$	− 29.0952i	− 1.12154i	0.827972	−	0.560769i	$-0.189494\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29.8863i	1.14862i	0.818637	+	0.574311i	$0.194730\pi$
$677$	29.8863i	1.14862i	−0.818637	+	0.574311i	$0.805270\pi$
$678$	0	0
$679$	−30.5396	−1.17200
$680$	0	0
$681$	−2.53486	−0.0971359
$682$	0	0
$683$	− 37.2200i	− 1.42419i	−0.702086	−	0.712093i	$-0.747748\pi$
$683$	− 37.2200i	− 1.42419i	0.702086	−	0.712093i	$-0.252252\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	16.5262i	0.630514i
$688$	0	0
$689$	14.0099	0.533734
$690$	0	0
$691$	16.8596	0.641368	0.320684	−	0.947186i	$-0.396087\pi$
$691$	16.8596	0.641368	0.320684	+	0.947186i	$0.396087\pi$
$692$	0	0
$693$	− 3.05955i	− 0.116223i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	13.2104i	0.500380i
$698$	0	0
$699$	−26.5208	−1.00311
$700$	0	0
$701$	−23.3495	−0.881898	−0.440949	−	0.897532i	$-0.645358\pi$
$701$	−23.3495	−0.881898	−0.440949	+	0.897532i	$0.645358\pi$
$702$	0	0
$703$	− 0.128222i	− 0.00483599i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 44.1297i	− 1.65967i
$708$	0	0
$709$	−19.0378	−0.714978	−0.357489	−	0.933917i	$-0.616367\pi$
$709$	−19.0378	−0.714978	−0.357489	+	0.933917i	$0.616367\pi$
$710$	0	0
$711$	9.27155	0.347710
$712$	0	0
$713$	9.91650i	0.371376i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7.46385i	0.278743i
$718$	0	0
$719$	14.7399	0.549706	0.274853	−	0.961486i	$-0.411371\pi$
$719$	14.7399	0.549706	0.274853	+	0.961486i	$0.411371\pi$
$720$	0	0
$721$	71.3821	2.65841
$722$	0	0
$723$	10.2978i	0.382978i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 30.0617i	− 1.11492i	−0.830202	−	0.557462i	$-0.811775\pi$
$727$	− 30.0617i	− 1.11492i	0.830202	−	0.557462i	$-0.188225\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−11.7708	−0.435359
$732$	0	0
$733$	− 22.6438i	− 0.836368i	−0.908362	−	0.418184i	$-0.862667\pi$
$733$	− 22.6438i	− 0.836368i	0.908362	−	0.418184i	$-0.137333\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.7269i	0.395132i
$738$	0	0
$739$	25.6810	0.944692	0.472346	−	0.881413i	$-0.343407\pi$
$739$	25.6810	0.944692	0.472346	+	0.881413i	$0.343407\pi$
$740$	0	0
$741$	0.0712954	0.00261910
$742$	0	0
$743$	− 24.9796i	− 0.916411i	−0.888846	−	0.458205i	$-0.848492\pi$
$743$	− 24.9796i	− 0.916411i	0.888846	−	0.458205i	$-0.151508\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 1.42370i	− 0.0520904i
$748$	0	0
$749$	54.5321	1.99256
$750$	0	0
$751$	17.9383	0.654580	0.327290	−	0.944924i	$-0.393865\pi$
$751$	17.9383	0.654580	0.327290	+	0.944924i	$0.393865\pi$
$752$	0	0
$753$	19.5809i	0.713567i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	9.91474i	0.360357i	0.983634	+	0.180179i	$0.0576676\pi$
$757$	9.91474i	0.360357i	−0.983634	+	0.180179i	$0.942332\pi$
$758$	0	0
$759$	−5.06250	−0.183757
$760$	0	0
$761$	−31.6527	−1.14741	−0.573705	−	0.819062i	$-0.694494\pi$
$761$	−31.6527	−1.14741	−0.573705	+	0.819062i	$0.694494\pi$
$762$	0	0
$763$	− 21.8264i	− 0.790168i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 54.3511i	− 1.96250i
$768$	0	0
$769$	−22.3751	−0.806868	−0.403434	−	0.915009i	$-0.632184\pi$
$769$	−22.3751	−0.806868	−0.403434	+	0.915009i	$0.632184\pi$
$770$	0	0
$771$	−5.98030	−0.215375
$772$	0	0
$773$	− 47.8317i	− 1.72039i	−0.509968	−	0.860194i	$-0.670343\pi$
$773$	− 47.8317i	− 1.72039i	0.509968	−	0.860194i	$-0.329657\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	32.3570i	1.16080i
$778$	0	0
$779$	−0.172611	−0.00618445
$780$	0	0
$781$	3.96827	0.141996
$782$	0	0
$783$	3.70092i	0.132260i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	27.5675i	0.982677i	0.870969	+	0.491338i	$0.163492\pi$
$787$	27.5675i	0.982677i	−0.870969	+	0.491338i	$0.836508\pi$
$788$	0	0
$789$	−6.07038	−0.216111
$790$	0	0
$791$	67.9065	2.41448
$792$	0	0
$793$	14.9235i	0.529948i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23.1019i	0.818311i	0.912465	+	0.409155i	$0.134177\pi$
$797$	23.1019i	0.818311i	−0.912465	+	0.409155i	$0.865823\pi$
$798$	0	0
$799$	−0.604617	−0.0213898
$800$	0	0
$801$	−16.1067	−0.569103
$802$	0	0
$803$	3.77881i	0.133351i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 9.61605i	− 0.338501i
$808$	0	0
$809$	1.69292	0.0595198	0.0297599	−	0.999557i	$-0.490526\pi$
$809$	1.69292	0.0595198	0.0297599	+	0.999557i	$0.490526\pi$
$810$	0	0
$811$	20.5726	0.722400	0.361200	−	0.932488i	$-0.382367\pi$
$811$	20.5726	0.722400	0.361200	+	0.932488i	$0.382367\pi$
$812$	0	0
$813$	− 30.4276i	− 1.06714i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 0.153801i	− 0.00538082i
$818$	0	0
$819$	−17.9915	−0.628673
$820$	0	0
$821$	−46.0282	−1.60639	−0.803197	−	0.595713i	$-0.796869\pi$
$821$	−46.0282	−1.60639	−0.803197	+	0.595713i	$0.796869\pi$
$822$	0	0
$823$	55.9867i	1.95157i	0.218721	+	0.975787i	$0.429812\pi$
$823$	55.9867i	1.95157i	−0.218721	+	0.975787i	$0.570188\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 20.6399i	− 0.717721i	−0.933391	−	0.358860i	$-0.883165\pi$
$827$	− 20.6399i	− 0.717721i	0.933391	−	0.358860i	$-0.116835\pi$
$828$	0	0
$829$	1.93898	0.0673437	0.0336718	−	0.999433i	$-0.489280\pi$
$829$	1.93898	0.0673437	0.0336718	+	0.999433i	$0.489280\pi$
$830$	0	0
$831$	−7.20393	−0.249901
$832$	0	0
$833$	− 8.44351i	− 0.292550i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.58209i	0.0546851i
$838$	0	0
$839$	−52.2589	−1.80418	−0.902089	−	0.431551i	$-0.857966\pi$
$839$	−52.2589	−1.80418	−0.902089	+	0.431551i	$0.857966\pi$
$840$	0	0
$841$	−15.3032	−0.527697
$842$	0	0
$843$	29.7964i	1.02624i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	39.1978i	1.34685i
$848$	0	0
$849$	−8.82402	−0.302840
$850$	0	0
$851$	53.5397	1.83532
$852$	0	0
$853$	− 17.5605i	− 0.601260i	−0.953741	−	0.300630i	$-0.902803\pi$
$853$	− 17.5605i	− 0.601260i	0.953741	−	0.300630i	$-0.0971969\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 50.9890i	− 1.74175i	−0.491503	−	0.870876i	$-0.663552\pi$
$857$	− 50.9890i	− 1.74175i	0.491503	−	0.870876i	$-0.336448\pi$
$858$	0	0
$859$	−18.6336	−0.635770	−0.317885	−	0.948129i	$-0.602973\pi$
$859$	−18.6336	−0.635770	−0.317885	+	0.948129i	$0.602973\pi$
$860$	0	0
$861$	43.5587	1.48448
$862$	0	0
$863$	0.292141i	0.00994461i	0.999988	+	0.00497230i	$0.00158274\pi$
$863$	0.292141i	0.00994461i	−0.999988	+	0.00497230i	$0.998417\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	15.6802i	0.532526i
$868$	0	0
$869$	7.48843	0.254028
$870$	0	0
$871$	63.0790	2.13735
$872$	0	0
$873$	8.06203i	0.272858i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	37.9604i	1.28183i	0.767611	+	0.640916i	$0.221445\pi$
$877$	37.9604i	1.28183i	−0.767611	+	0.640916i	$0.778555\pi$
$878$	0	0
$879$	4.20743	0.141913
$880$	0	0
$881$	−36.6899	−1.23612	−0.618058	−	0.786133i	$-0.712080\pi$
$881$	−36.6899	−1.23612	−0.618058	+	0.786133i	$0.712080\pi$
$882$	0	0
$883$	− 22.6199i	− 0.761220i	−0.924736	−	0.380610i	$-0.875714\pi$
$883$	− 22.6199i	− 0.761220i	0.924736	−	0.380610i	$-0.124286\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.4380i	0.820548i	0.911962	+	0.410274i	$0.134567\pi$
$887$	24.4380i	0.820548i	−0.911962	+	0.410274i	$0.865433\pi$
$888$	0	0
$889$	8.67399	0.290916
$890$	0	0
$891$	−0.807679	−0.0270583
$892$	0	0
$893$	− 0.00790011i	0 0.000264367i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	29.7697i	0.993981i
$898$	0	0
$899$	5.85519	0.195281
$900$	0	0
$901$	3.38882	0.112898
$902$	0	0
$903$	38.8119i	1.29158i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 32.7792i	− 1.08842i	−0.838950	−	0.544208i	$-0.816830\pi$
$907$	− 32.7792i	− 1.08842i	0.838950	−	0.544208i	$-0.183170\pi$
$908$	0	0
$909$	−11.6496	−0.386393
$910$	0	0
$911$	27.2897	0.904146	0.452073	−	0.891981i	$-0.350685\pi$
$911$	27.2897	0.904146	0.452073	+	0.891981i	$0.350685\pi$
$912$	0	0
$913$	− 1.14989i	− 0.0380558i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	26.7039i	0.881842i
$918$	0	0
$919$	41.7202	1.37622	0.688112	−	0.725604i	$-0.258440\pi$
$919$	41.7202	1.37622	0.688112	+	0.725604i	$0.258440\pi$
$920$	0	0
$921$	4.41109	0.145350
$922$	0	0
$923$	− 23.3351i	− 0.768086i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 18.8439i	− 0.618913i
$928$	0	0
$929$	33.7038	1.10579	0.552893	−	0.833252i	$-0.313524\pi$
$929$	33.7038	1.10579	0.552893	+	0.833252i	$0.313524\pi$
$930$	0	0
$931$	0.110326	0.00361577
$932$	0	0
$933$	− 6.32058i	− 0.206927i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.04606i	0.0341732i	0.999854	+	0.0170866i	$0.00543910\pi$
$937$	1.04606i	0.0341732i	−0.999854	+	0.0170866i	$0.994561\pi$
$938$	0	0
$939$	15.3245	0.500097
$940$	0	0
$941$	5.72210	0.186535	0.0932676	−	0.995641i	$-0.470269\pi$
$941$	5.72210	0.186535	0.0932676	+	0.995641i	$0.470269\pi$
$942$	0	0
$943$	− 72.0746i	− 2.34707i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 12.5948i	− 0.409275i	−0.978838	−	0.204637i	$-0.934398\pi$
$947$	− 12.5948i	− 0.409275i	0.978838	−	0.204637i	$-0.0656016\pi$
$948$	0	0
$949$	22.2210	0.721325
$950$	0	0
$951$	−21.3155	−0.691204
$952$	0	0
$953$	57.3902i	1.85905i	0.368757	+	0.929526i	$0.379783\pi$
$953$	57.3902i	1.85905i	−0.368757	+	0.929526i	$0.620217\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.98915i	0.0966255i
$958$	0	0
$959$	83.0836	2.68291
$960$	0	0
$961$	−28.4970	−0.919258
$962$	0	0
$963$	− 14.3957i	− 0.463895i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 8.84212i	− 0.284343i	−0.989842	−	0.142172i	$-0.954591\pi$
$967$	− 8.84212i	− 0.284343i	0.989842	−	0.142172i	$-0.0454085\pi$
$968$	0	0
$969$	0.0172455	0.000554005 0
$970$	0	0
$971$	51.9365	1.66672	0.833361	−	0.552730i	$-0.186414\pi$
$971$	51.9365	1.66672	0.833361	+	0.552730i	$0.186414\pi$
$972$	0	0
$973$	36.8111i	1.18011i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 37.2345i	− 1.19124i	−0.803267	−	0.595619i	$-0.796907\pi$
$977$	− 37.2345i	− 1.19124i	0.803267	−	0.595619i	$-0.203093\pi$
$978$	0	0
$979$	−13.0091	−0.415772
$980$	0	0
$981$	−5.76186	−0.183962
$982$	0	0
$983$	35.7610i	1.14060i	0.821437	+	0.570299i	$0.193173\pi$
$983$	35.7610i	1.14060i	−0.821437	+	0.570299i	$0.806827\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.99360i	0.0634570i
$988$	0	0
$989$	64.2203	2.04209
$990$	0	0
$991$	−43.8192	−1.39196	−0.695981	−	0.718060i	$-0.745030\pi$
$991$	−43.8192	−1.39196	−0.695981	+	0.718060i	$0.745030\pi$
$992$	0	0
$993$	19.8124i	0.628728i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 39.5083i	− 1.25124i	−0.780128	−	0.625620i	$-0.784846\pi$
$997$	− 39.5083i	− 1.25124i	0.780128	−	0.625620i	$-0.215154\pi$
$998$	0	0
$999$	8.54179	0.270250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.d.g.1249.16		24
5.2	odd	4		7500.2.a.n.1.9		12
5.3	odd	4		7500.2.a.m.1.4		12
5.4	even	2	inner	7500.2.d.g.1249.9		24
25.3	odd	20		1500.2.m.d.1201.2		24
25.4	even	10		300.2.o.a.109.5	✓	24
25.6	even	5		300.2.o.a.289.5	yes	24
25.8	odd	20		1500.2.m.d.301.2		24
25.17	odd	20		1500.2.m.c.301.5		24
25.19	even	10		1500.2.o.c.949.1		24
25.21	even	5		1500.2.o.c.49.1		24
25.22	odd	20		1500.2.m.c.1201.5		24
75.29	odd	10		900.2.w.c.109.4		24
75.56	odd	10		900.2.w.c.289.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.o.a.109.5	✓	24	25.4	even	10
300.2.o.a.289.5	yes	24	25.6	even	5
900.2.w.c.109.4		24	75.29	odd	10
900.2.w.c.289.4		24	75.56	odd	10
1500.2.m.c.301.5		24	25.17	odd	20
1500.2.m.c.1201.5		24	25.22	odd	20
1500.2.m.d.301.2		24	25.8	odd	20
1500.2.m.d.1201.2		24	25.3	odd	20
1500.2.o.c.49.1		24	25.21	even	5
1500.2.o.c.949.1		24	25.19	even	10
7500.2.a.m.1.4		12	5.3	odd	4
7500.2.a.n.1.9		12	5.2	odd	4
7500.2.d.g.1249.9		24	5.4	even	2	inner
7500.2.d.g.1249.16		24	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -3.78808i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -3.78808i q^{7} -1.00000 q^{9} -0.807679 q^{11} +4.74950i q^{13} +1.14884i q^{17} -0.0150112 q^{19} +3.78808 q^{21} -6.26797i q^{23} -1.00000i q^{27} -3.70092 q^{29} -1.58209 q^{31} -0.807679i q^{33} +8.54179i q^{37} -4.74950 q^{39} +11.4989 q^{41} +10.2458i q^{43} +0.526282i q^{47} -7.34957 q^{49} -1.14884 q^{51} -2.94976i q^{53} -0.0150112i q^{57} -11.4436 q^{59} +3.14212 q^{61} +3.78808i q^{63} -13.2812i q^{67} +6.26797 q^{69} -4.91318 q^{71} -4.67860i q^{73} +3.05955i q^{77} -9.27155 q^{79} +1.00000 q^{81} +1.42370i q^{83} -3.70092i q^{87} +16.1067 q^{89} +17.9915 q^{91} -1.58209i q^{93} -8.06203i q^{97} +0.807679 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{9}+O(q^{10}) \) 24 * q - 24 * q^9	\( 24 q - 24 q^{9} + 4 q^{11} - 20 q^{19} + 16 q^{21} - 16 q^{29} - 4 q^{31} + 20 q^{41} - 56 q^{49} + 16 q^{51} + 4 q^{59} + 68 q^{61} - 36 q^{69} - 12 q^{79} + 24 q^{81} - 20 q^{89} + 40 q^{91} - 4 q^{99}+O(q^{100}) \) 24 * q - 24 * q^9 + 4 * q^11 - 20 * q^19 + 16 * q^21 - 16 * q^29 - 4 * q^31 + 20 * q^41 - 56 * q^49 + 16 * q^51 + 4 * q^59 + 68 * q^61 - 36 * q^69 - 12 * q^79 + 24 * q^81 - 20 * q^89 + 40 * q^91 - 4 * q^99

Introduction

L-functions

Modular forms

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