LMFDB - Embedded newform 7500.2.d.g.1249.11

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			1249.11
Character	$\chi$	$=$	7500.1249
Dual form			7500.2.d.g.1249.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +4.41540i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +4.41540i q^{7} -1.00000 q^{9} +4.45181 q^{11} +5.74778i q^{13} +6.57035i q^{17} +2.78315 q^{19} +4.41540 q^{21} -1.31495i q^{23} +1.00000i q^{27} +4.84521 q^{29} -0.197136 q^{31} -4.45181i q^{33} -8.20866i q^{37} +5.74778 q^{39} +7.84984 q^{41} +0.412792i q^{43} +7.79459i q^{47} -12.4958 q^{49} +6.57035 q^{51} +0.315455i q^{53} -2.78315i q^{57} -2.51902 q^{59} +9.33114 q^{61} -4.41540i q^{63} +11.9457i q^{67} -1.31495 q^{69} +0.509248 q^{71} -15.7708i q^{73} +19.6565i q^{77} +3.43153 q^{79} +1.00000 q^{81} +5.37155i q^{83} -4.84521i q^{87} -11.3452 q^{89} -25.3787 q^{91} +0.197136i q^{93} -5.07960i q^{97} -4.45181 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	4.41540i	1.66886i	0.551111	+	0.834432i	$0.314204\pi$
$7$	4.41540i	1.66886i	−0.551111	+	0.834432i	$0.685796\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.45181	1.34227	0.671135	−	0.741335i	$-0.265807\pi$
$11$	4.45181	1.34227	0.671135	+	0.741335i	$0.265807\pi$
$12$	0	0
$13$	5.74778i	1.59415i	0.603882	+	0.797074i	$0.293620\pi$
$13$	5.74778i	1.59415i	−0.603882	+	0.797074i	$0.706380\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.57035i	1.59355i	0.604279	+	0.796773i	$0.293461\pi$
$17$	6.57035i	1.59355i	−0.604279	+	0.796773i	$0.706539\pi$
$18$	0	0
$19$	2.78315	0.638499	0.319250	−	0.947671i	$-0.396569\pi$
$19$	2.78315	0.638499	0.319250	+	0.947671i	$0.396569\pi$
$20$	0	0
$21$	4.41540	0.963519
$22$	0	0
$23$	− 1.31495i	− 0.274186i	−0.990558	−	0.137093i	$-0.956224\pi$
$23$	− 1.31495i	− 0.274186i	0.990558	−	0.137093i	$-0.0437759\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	4.84521	0.899732	0.449866	−	0.893096i	$-0.351472\pi$
$29$	4.84521	0.899732	0.449866	+	0.893096i	$0.351472\pi$
$30$	0	0
$31$	−0.197136	−0.0354066	−0.0177033	−	0.999843i	$-0.505635\pi$
$31$	−0.197136	−0.0354066	−0.0177033	+	0.999843i	$0.505635\pi$
$32$	0	0
$33$	− 4.45181i	− 0.774960i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 8.20866i	− 1.34950i	−0.738048	−	0.674748i	$-0.764252\pi$
$37$	− 8.20866i	− 1.34950i	0.738048	−	0.674748i	$-0.235748\pi$
$38$	0	0
$39$	5.74778	0.920381
$40$	0	0
$41$	7.84984	1.22594	0.612970	−	0.790106i	$-0.289975\pi$
$41$	7.84984	1.22594	0.612970	+	0.790106i	$0.289975\pi$
$42$	0	0
$43$	0.412792i	0.0629502i	0.999505	+	0.0314751i	$0.0100205\pi$
$43$	0.412792i	0.0629502i	−0.999505	+	0.0314751i	$0.989980\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.79459i	1.13696i	0.822698	+	0.568479i	$0.192468\pi$
$47$	7.79459i	1.13696i	−0.822698	+	0.568479i	$0.807532\pi$
$48$	0	0
$49$	−12.4958	−1.78511
$50$	0	0
$51$	6.57035	0.920034
$52$	0	0
$53$	0.315455i	0.0433311i	0.999765	+	0.0216655i	$0.00689689\pi$
$53$	0.315455i	0.0433311i	−0.999765	+	0.0216655i	$0.993103\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.78315i	− 0.368638i
$58$	0	0
$59$	−2.51902	−0.327948	−0.163974	−	0.986465i	$-0.552431\pi$
$59$	−2.51902	−0.327948	−0.163974	+	0.986465i	$0.552431\pi$
$60$	0	0
$61$	9.33114	1.19473	0.597365	−	0.801970i	$-0.296214\pi$
$61$	9.33114	1.19473	0.597365	+	0.801970i	$0.296214\pi$
$62$	0	0
$63$	− 4.41540i	− 0.556288i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.9457i	1.45940i	0.683767	+	0.729700i	$0.260340\pi$
$67$	11.9457i	1.45940i	−0.683767	+	0.729700i	$0.739660\pi$
$68$	0	0
$69$	−1.31495	−0.158301
$70$	0	0
$71$	0.509248	0.0604366	0.0302183	−	0.999543i	$-0.490380\pi$
$71$	0.509248	0.0604366	0.0302183	+	0.999543i	$0.490380\pi$
$72$	0	0
$73$	− 15.7708i	− 1.84584i	−0.384996	−	0.922918i	$-0.625797\pi$
$73$	− 15.7708i	− 1.84584i	0.384996	−	0.922918i	$-0.374203\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	19.6565i	2.24007i
$78$	0	0
$79$	3.43153	0.386077	0.193039	−	0.981191i	$-0.438166\pi$
$79$	3.43153	0.386077	0.193039	+	0.981191i	$0.438166\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.37155i	0.589604i	0.955558	+	0.294802i	$0.0952538\pi$
$83$	5.37155i	0.589604i	−0.955558	+	0.294802i	$0.904746\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 4.84521i	− 0.519461i
$88$	0	0
$89$	−11.3452	−1.20258	−0.601292	−	0.799029i	$-0.705347\pi$
$89$	−11.3452	−1.20258	−0.601292	+	0.799029i	$0.705347\pi$
$90$	0	0
$91$	−25.3787	−2.66042
$92$	0	0
$93$	0.197136i	0.0204420i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 5.07960i	− 0.515755i	−0.966178	−	0.257878i	$-0.916977\pi$
$97$	− 5.07960i	− 0.515755i	0.966178	−	0.257878i	$-0.0830232\pi$
$98$	0	0
$99$	−4.45181	−0.447424
$100$	0	0
$101$	−11.1860	−1.11305	−0.556525	−	0.830831i	$-0.687866\pi$
$101$	−11.1860	−1.11305	−0.556525	+	0.830831i	$0.687866\pi$
$102$	0	0
$103$	3.14567i	0.309952i	0.987918	+	0.154976i	$0.0495301\pi$
$103$	3.14567i	0.309952i	−0.987918	+	0.154976i	$0.950470\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 7.74013i	− 0.748266i	−0.927375	−	0.374133i	$-0.877940\pi$
$107$	− 7.74013i	− 0.748266i	0.927375	−	0.374133i	$-0.122060\pi$
$108$	0	0
$109$	10.3708	0.993347	0.496673	−	0.867937i	$-0.334555\pi$
$109$	10.3708	0.993347	0.496673	+	0.867937i	$0.334555\pi$
$110$	0	0
$111$	−8.20866	−0.779132
$112$	0	0
$113$	10.2263i	0.962005i	0.876719	+	0.481002i	$0.159727\pi$
$113$	10.2263i	0.962005i	−0.876719	+	0.481002i	$0.840273\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 5.74778i	− 0.531382i
$118$	0	0
$119$	−29.0107	−2.65941
$120$	0	0
$121$	8.81860	0.801691
$122$	0	0
$123$	− 7.84984i	− 0.707797i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 11.3127i	− 1.00384i	−0.864914	−	0.501920i	$-0.832627\pi$
$127$	− 11.3127i	− 1.00384i	0.864914	−	0.501920i	$-0.167373\pi$
$128$	0	0
$129$	0.412792	0.0363443
$130$	0	0
$131$	−9.45365	−0.825969	−0.412985	−	0.910738i	$-0.635514\pi$
$131$	−9.45365	−0.825969	−0.412985	+	0.910738i	$0.635514\pi$
$132$	0	0
$133$	12.2887i	1.06557i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.41023i	0.718535i	0.933235	+	0.359267i	$0.116973\pi$
$137$	8.41023i	0.718535i	−0.933235	+	0.359267i	$0.883027\pi$
$138$	0	0
$139$	−11.8216	−1.00269	−0.501346	−	0.865247i	$-0.667162\pi$
$139$	−11.8216	−1.00269	−0.501346	+	0.865247i	$0.667162\pi$
$140$	0	0
$141$	7.79459	0.656422
$142$	0	0
$143$	25.5880i	2.13978i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	12.4958i	1.03063i
$148$	0	0
$149$	−10.6355	−0.871294	−0.435647	−	0.900118i	$-0.643480\pi$
$149$	−10.6355	−0.871294	−0.435647	+	0.900118i	$0.643480\pi$
$150$	0	0
$151$	4.41657	0.359415	0.179707	−	0.983720i	$-0.442485\pi$
$151$	4.41657	0.359415	0.179707	+	0.983720i	$0.442485\pi$
$152$	0	0
$153$	− 6.57035i	− 0.531182i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	13.5289i	1.07972i	0.841754	+	0.539861i	$0.181523\pi$
$157$	13.5289i	1.07972i	−0.841754	+	0.539861i	$0.818477\pi$
$158$	0	0
$159$	0.315455	0.0250172
$160$	0	0
$161$	5.80602	0.457579
$162$	0	0
$163$	− 14.5583i	− 1.14029i	−0.821543	−	0.570146i	$-0.806886\pi$
$163$	− 14.5583i	− 1.14029i	0.821543	−	0.570146i	$-0.193114\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 9.96824i	− 0.771366i	−0.922631	−	0.385683i	$-0.873966\pi$
$167$	− 9.96824i	− 0.771366i	0.922631	−	0.385683i	$-0.126034\pi$
$168$	0	0
$169$	−20.0370	−1.54131
$170$	0	0
$171$	−2.78315	−0.212833
$172$	0	0
$173$	10.8750i	0.826811i	0.910547	+	0.413406i	$0.135661\pi$
$173$	10.8750i	0.826811i	−0.910547	+	0.413406i	$0.864339\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.51902i	0.189341i
$178$	0	0
$179$	25.1740	1.88159	0.940795	−	0.338975i	$-0.110080\pi$
$179$	25.1740	1.88159	0.940795	+	0.338975i	$0.110080\pi$
$180$	0	0
$181$	−7.59173	−0.564289	−0.282145	−	0.959372i	$-0.591046\pi$
$181$	−7.59173	−0.564289	−0.282145	+	0.959372i	$0.591046\pi$
$182$	0	0
$183$	− 9.33114i	− 0.689777i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	29.2500i	2.13897i
$188$	0	0
$189$	−4.41540	−0.321173
$190$	0	0
$191$	−0.0120931	−0.000875025 0	−0.000437512	−	1.00000i	$-0.500139\pi$
$191$	−0.0120931	−0.000875025 0	−0.000437512		1.00000i	$0.500139\pi$
$192$	0	0
$193$	− 12.7841i	− 0.920219i	−0.887862	−	0.460110i	$-0.847810\pi$
$193$	− 12.7841i	− 0.920219i	0.887862	−	0.460110i	$-0.152190\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.86545i	0.702885i	0.936210	+	0.351442i	$0.114309\pi$
$197$	9.86545i	0.702885i	−0.936210	+	0.351442i	$0.885691\pi$
$198$	0	0
$199$	0.295640	0.0209573	0.0104787	−	0.999945i	$-0.496664\pi$
$199$	0.295640	0.0209573	0.0104787	+	0.999945i	$0.496664\pi$
$200$	0	0
$201$	11.9457	0.842585
$202$	0	0
$203$	21.3935i	1.50153i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.31495i	0.0913952i
$208$	0	0
$209$	12.3901	0.857039
$210$	0	0
$211$	−12.6477	−0.870705	−0.435352	−	0.900260i	$-0.643376\pi$
$211$	−12.6477	−0.870705	−0.435352	+	0.900260i	$0.643376\pi$
$212$	0	0
$213$	− 0.509248i	− 0.0348931i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 0.870433i	− 0.0590888i
$218$	0	0
$219$	−15.7708	−1.06569
$220$	0	0
$221$	−37.7650	−2.54035
$222$	0	0
$223$	− 5.23980i	− 0.350883i	−0.984490	−	0.175442i	$-0.943865\pi$
$223$	− 5.23980i	− 0.350883i	0.984490	−	0.175442i	$-0.0561353\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.76897i	0.515645i	0.966192	+	0.257822i	$0.0830049\pi$
$227$	7.76897i	0.515645i	−0.966192	+	0.257822i	$0.916995\pi$
$228$	0	0
$229$	5.99192	0.395958	0.197979	−	0.980206i	$-0.436562\pi$
$229$	5.99192	0.395958	0.197979	+	0.980206i	$0.436562\pi$
$230$	0	0
$231$	19.6565	1.29330
$232$	0	0
$233$	− 14.8631i	− 0.973714i	−0.873482	−	0.486857i	$-0.838143\pi$
$233$	− 14.8631i	− 0.973714i	0.873482	−	0.486857i	$-0.161857\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 3.43153i	− 0.222902i
$238$	0	0
$239$	21.2008	1.37136	0.685682	−	0.727902i	$-0.259504\pi$
$239$	21.2008	1.37136	0.685682	+	0.727902i	$0.259504\pi$
$240$	0	0
$241$	16.0562	1.03427	0.517134	−	0.855905i	$-0.326999\pi$
$241$	16.0562	1.03427	0.517134	+	0.855905i	$0.326999\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	15.9970i	1.01786i
$248$	0	0
$249$	5.37155	0.340408
$250$	0	0
$251$	−24.1371	−1.52352	−0.761761	−	0.647858i	$-0.775665\pi$
$251$	−24.1371	−1.52352	−0.761761	+	0.647858i	$0.775665\pi$
$252$	0	0
$253$	− 5.85390i	− 0.368031i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.22940i	0.263823i	0.991262	+	0.131911i	$0.0421114\pi$
$257$	4.22940i	0.263823i	−0.991262	+	0.131911i	$0.957889\pi$
$258$	0	0
$259$	36.2445	2.25213
$260$	0	0
$261$	−4.84521	−0.299911
$262$	0	0
$263$	− 8.38782i	− 0.517215i	−0.965982	−	0.258608i	$-0.916736\pi$
$263$	− 8.38782i	− 0.517215i	0.965982	−	0.258608i	$-0.0832636\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	11.3452i	0.694313i
$268$	0	0
$269$	5.53526	0.337490	0.168745	−	0.985660i	$-0.446028\pi$
$269$	5.53526	0.337490	0.168745	+	0.985660i	$0.446028\pi$
$270$	0	0
$271$	15.8497	0.962802	0.481401	−	0.876501i	$-0.340128\pi$
$271$	15.8497	0.962802	0.481401	+	0.876501i	$0.340128\pi$
$272$	0	0
$273$	25.3787i	1.53599i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 8.69243i	− 0.522277i	−0.965301	−	0.261139i	$-0.915902\pi$
$277$	− 8.69243i	− 0.522277i	0.965301	−	0.261139i	$-0.0840980\pi$
$278$	0	0
$279$	0.197136	0.0118022
$280$	0	0
$281$	10.7800	0.643081	0.321540	−	0.946896i	$-0.395799\pi$
$281$	10.7800	0.643081	0.321540	+	0.946896i	$0.395799\pi$
$282$	0	0
$283$	− 9.84520i	− 0.585236i	−0.956229	−	0.292618i	$-0.905473\pi$
$283$	− 9.84520i	− 0.585236i	0.956229	−	0.292618i	$-0.0945265\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	34.6602i	2.04593i
$288$	0	0
$289$	−26.1696	−1.53939
$290$	0	0
$291$	−5.07960	−0.297772
$292$	0	0
$293$	− 9.77733i	− 0.571198i	−0.958349	−	0.285599i	$-0.907808\pi$
$293$	− 9.77733i	− 0.571198i	0.958349	−	0.285599i	$-0.0921925\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.45181i	0.258320i
$298$	0	0
$299$	7.55803	0.437092
$300$	0	0
$301$	−1.82264	−0.105055
$302$	0	0
$303$	11.1860i	0.642620i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	32.7301i	1.86801i	0.357265	+	0.934003i	$0.383709\pi$
$307$	32.7301i	1.86801i	−0.357265	+	0.934003i	$0.616291\pi$
$308$	0	0
$309$	3.14567	0.178951
$310$	0	0
$311$	−24.6245	−1.39633	−0.698164	−	0.715938i	$-0.745999\pi$
$311$	−24.6245	−1.39633	−0.698164	+	0.715938i	$0.745999\pi$
$312$	0	0
$313$	22.5495i	1.27457i	0.770628	+	0.637285i	$0.219943\pi$
$313$	22.5495i	1.27457i	−0.770628	+	0.637285i	$0.780057\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.62435i	0.0912324i	0.998959	+	0.0456162i	$0.0145251\pi$
$317$	1.62435i	0.0912324i	−0.998959	+	0.0456162i	$0.985475\pi$
$318$	0	0
$319$	21.5699	1.20768
$320$	0	0
$321$	−7.74013	−0.432012
$322$	0	0
$323$	18.2863i	1.01748i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 10.3708i	− 0.573509i
$328$	0	0
$329$	−34.4162	−1.89743
$330$	0	0
$331$	−9.01314	−0.495407	−0.247703	−	0.968836i	$-0.579676\pi$
$331$	−9.01314	−0.495407	−0.247703	+	0.968836i	$0.579676\pi$
$332$	0	0
$333$	8.20866i	0.449832i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	26.9681i	1.46905i	0.678583	+	0.734524i	$0.262594\pi$
$337$	26.9681i	1.46905i	−0.678583	+	0.734524i	$0.737406\pi$
$338$	0	0
$339$	10.2263	0.555414
$340$	0	0
$341$	−0.877611	−0.0475253
$342$	0	0
$343$	− 24.2660i	− 1.31024i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 26.2448i	− 1.40889i	−0.709757	−	0.704447i	$-0.751195\pi$
$347$	− 26.2448i	− 1.40889i	0.709757	−	0.704447i	$-0.248805\pi$
$348$	0	0
$349$	−18.2310	−0.975885	−0.487943	−	0.872876i	$-0.662252\pi$
$349$	−18.2310	−0.975885	−0.487943	+	0.872876i	$0.662252\pi$
$350$	0	0
$351$	−5.74778	−0.306794
$352$	0	0
$353$	− 4.44962i	− 0.236829i	−0.992964	−	0.118415i	$-0.962219\pi$
$353$	− 4.44962i	− 0.236829i	0.992964	−	0.118415i	$-0.0377812\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	29.0107i	1.53541i
$358$	0	0
$359$	32.3271	1.70616	0.853081	−	0.521779i	$-0.174731\pi$
$359$	32.3271	1.70616	0.853081	+	0.521779i	$0.174731\pi$
$360$	0	0
$361$	−11.2540	−0.592318
$362$	0	0
$363$	− 8.81860i	− 0.462857i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 5.74002i	− 0.299627i	−0.988714	−	0.149813i	$-0.952133\pi$
$367$	− 5.74002i	− 0.299627i	0.988714	−	0.149813i	$-0.0478673\pi$
$368$	0	0
$369$	−7.84984	−0.408647
$370$	0	0
$371$	−1.39286	−0.0723137
$372$	0	0
$373$	− 1.04186i	− 0.0539455i	−0.999636	−	0.0269727i	$-0.991413\pi$
$373$	− 1.04186i	− 0.0539455i	0.999636	−	0.0269727i	$-0.00858673\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	27.8492i	1.43431i
$378$	0	0
$379$	4.74934	0.243957	0.121979	−	0.992533i	$-0.461076\pi$
$379$	4.74934	0.243957	0.121979	+	0.992533i	$0.461076\pi$
$380$	0	0
$381$	−11.3127	−0.579568
$382$	0	0
$383$	− 11.0200i	− 0.563095i	−0.959547	−	0.281548i	$-0.909152\pi$
$383$	− 11.0200i	− 0.563095i	0.959547	−	0.281548i	$-0.0908477\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 0.412792i	− 0.0209834i
$388$	0	0
$389$	33.2111	1.68387	0.841934	−	0.539581i	$-0.181417\pi$
$389$	33.2111	1.68387	0.841934	+	0.539581i	$0.181417\pi$
$390$	0	0
$391$	8.63968	0.436927
$392$	0	0
$393$	9.45365i	0.476873i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.74418i	0.137726i	0.997626	+	0.0688631i	$0.0219372\pi$
$397$	2.74418i	0.137726i	−0.997626	+	0.0688631i	$0.978063\pi$
$398$	0	0
$399$	12.2887	0.615207
$400$	0	0
$401$	−2.11503	−0.105619	−0.0528097	−	0.998605i	$-0.516818\pi$
$401$	−2.11503	−0.105619	−0.0528097	+	0.998605i	$0.516818\pi$
$402$	0	0
$403$	− 1.13309i	− 0.0564434i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 36.5434i	− 1.81139i
$408$	0	0
$409$	6.57251	0.324990	0.162495	−	0.986709i	$-0.448046\pi$
$409$	6.57251	0.324990	0.162495	+	0.986709i	$0.448046\pi$
$410$	0	0
$411$	8.41023	0.414846
$412$	0	0
$413$	− 11.1225i	− 0.547301i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	11.8216i	0.578905i
$418$	0	0
$419$	−35.2434	−1.72175	−0.860876	−	0.508815i	$-0.830084\pi$
$419$	−35.2434	−1.72175	−0.860876	+	0.508815i	$0.830084\pi$
$420$	0	0
$421$	6.27329	0.305742	0.152871	−	0.988246i	$-0.451148\pi$
$421$	6.27329	0.305742	0.152871	+	0.988246i	$0.451148\pi$
$422$	0	0
$423$	− 7.79459i	− 0.378986i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	41.2007i	1.99384i
$428$	0	0
$429$	25.5880	1.23540
$430$	0	0
$431$	−13.6721	−0.658561	−0.329280	−	0.944232i	$-0.606806\pi$
$431$	−13.6721	−0.658561	−0.329280	+	0.944232i	$0.606806\pi$
$432$	0	0
$433$	6.77819i	0.325739i	0.986648	+	0.162870i	$0.0520750\pi$
$433$	6.77819i	0.325739i	−0.986648	+	0.162870i	$0.947925\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3.65970i	− 0.175067i
$438$	0	0
$439$	16.9446	0.808723	0.404362	−	0.914599i	$-0.367494\pi$
$439$	16.9446	0.808723	0.404362	+	0.914599i	$0.367494\pi$
$440$	0	0
$441$	12.4958	0.595036
$442$	0	0
$443$	− 23.8927i	− 1.13517i	−0.823313	−	0.567587i	$-0.807877\pi$
$443$	− 23.8927i	− 1.13517i	0.823313	−	0.567587i	$-0.192123\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	10.6355i	0.503042i
$448$	0	0
$449$	−6.39281	−0.301695	−0.150848	−	0.988557i	$-0.548200\pi$
$449$	−6.39281	−0.301695	−0.150848	+	0.988557i	$0.548200\pi$
$450$	0	0
$451$	34.9460	1.64554
$452$	0	0
$453$	− 4.41657i	− 0.207508i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 5.89482i	− 0.275748i	−0.990450	−	0.137874i	$-0.955973\pi$
$457$	− 5.89482i	− 0.275748i	0.990450	−	0.137874i	$-0.0440269\pi$
$458$	0	0
$459$	−6.57035	−0.306678
$460$	0	0
$461$	−24.6200	−1.14667	−0.573335	−	0.819321i	$-0.694351\pi$
$461$	−24.6200	−1.14667	−0.573335	+	0.819321i	$0.694351\pi$
$462$	0	0
$463$	− 29.8611i	− 1.38776i	−0.720090	−	0.693880i	$-0.755900\pi$
$463$	− 29.8611i	− 1.38776i	0.720090	−	0.693880i	$-0.244100\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.1513i	1.21014i	0.796173	+	0.605069i	$0.206855\pi$
$467$	26.1513i	1.21014i	−0.796173	+	0.605069i	$0.793145\pi$
$468$	0	0
$469$	−52.7451	−2.43554
$470$	0	0
$471$	13.5289	0.623378
$472$	0	0
$473$	1.83767i	0.0844962i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 0.315455i	− 0.0144437i
$478$	0	0
$479$	−1.54690	−0.0706798	−0.0353399	−	0.999375i	$-0.511251\pi$
$479$	−1.54690	−0.0706798	−0.0353399	+	0.999375i	$0.511251\pi$
$480$	0	0
$481$	47.1816	2.15130
$482$	0	0
$483$	− 5.80602i	− 0.264183i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.4789i	0.565473i	0.959198	+	0.282736i	$0.0912422\pi$
$487$	12.4789i	0.565473i	−0.959198	+	0.282736i	$0.908758\pi$
$488$	0	0
$489$	−14.5583	−0.658348
$490$	0	0
$491$	−17.6969	−0.798649	−0.399325	−	0.916810i	$-0.630755\pi$
$491$	−17.6969	−0.798649	−0.399325	+	0.916810i	$0.630755\pi$
$492$	0	0
$493$	31.8347i	1.43376i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.24853i	0.100860i
$498$	0	0
$499$	30.9281	1.38453	0.692267	−	0.721642i	$-0.256612\pi$
$499$	30.9281	1.38453	0.692267	+	0.721642i	$0.256612\pi$
$500$	0	0
$501$	−9.96824	−0.445348
$502$	0	0
$503$	− 31.5535i	− 1.40690i	−0.710744	−	0.703450i	$-0.751642\pi$
$503$	− 31.5535i	− 1.40690i	0.710744	−	0.703450i	$-0.248358\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	20.0370i	0.889873i
$508$	0	0
$509$	7.22535	0.320258	0.160129	−	0.987096i	$-0.448809\pi$
$509$	7.22535	0.320258	0.160129	+	0.987096i	$0.448809\pi$
$510$	0	0
$511$	69.6345	3.08045
$512$	0	0
$513$	2.78315i	0.122879i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	34.7000i	1.52610i
$518$	0	0
$519$	10.8750	0.477360
$520$	0	0
$521$	−9.48574	−0.415578	−0.207789	−	0.978174i	$-0.566627\pi$
$521$	−9.48574	−0.415578	−0.207789	+	0.978174i	$0.566627\pi$
$522$	0	0
$523$	20.3551i	0.890067i	0.895514	+	0.445033i	$0.146808\pi$
$523$	20.3551i	0.890067i	−0.895514	+	0.445033i	$0.853192\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.29525i	− 0.0564220i
$528$	0	0
$529$	21.2709	0.924822
$530$	0	0
$531$	2.51902	0.109316
$532$	0	0
$533$	45.1192i	1.95433i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 25.1740i	− 1.08634i
$538$	0	0
$539$	−55.6287	−2.39610
$540$	0	0
$541$	30.8749	1.32741	0.663707	−	0.747992i	$-0.268982\pi$
$541$	30.8749	1.32741	0.663707	+	0.747992i	$0.268982\pi$
$542$	0	0
$543$	7.59173i	0.325792i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 15.3396i	− 0.655872i	−0.944700	−	0.327936i	$-0.893647\pi$
$547$	− 15.3396i	− 0.655872i	0.944700	−	0.327936i	$-0.106353\pi$
$548$	0	0
$549$	−9.33114	−0.398243
$550$	0	0
$551$	13.4850	0.574478
$552$	0	0
$553$	15.1516i	0.644310i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.6722i	1.21488i	0.794365	+	0.607441i	$0.207804\pi$
$557$	28.6722i	1.21488i	−0.794365	+	0.607441i	$0.792196\pi$
$558$	0	0
$559$	−2.37264	−0.100352
$560$	0	0
$561$	29.2500	1.23493
$562$	0	0
$563$	− 7.34999i	− 0.309765i	−0.987933	−	0.154883i	$-0.950500\pi$
$563$	− 7.34999i	− 0.309765i	0.987933	−	0.154883i	$-0.0494999\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.41540i	0.185429i
$568$	0	0
$569$	31.1279	1.30495	0.652474	−	0.757811i	$-0.273731\pi$
$569$	31.1279	1.30495	0.652474	+	0.757811i	$0.273731\pi$
$570$	0	0
$571$	12.9855	0.543426	0.271713	−	0.962378i	$-0.412410\pi$
$571$	12.9855	0.543426	0.271713	+	0.962378i	$0.412410\pi$
$572$	0	0
$573$	0.0120931i	0 0.000505196i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 21.7895i	− 0.907109i	−0.891229	−	0.453554i	$-0.850156\pi$
$577$	− 21.7895i	− 0.907109i	0.891229	−	0.453554i	$-0.149844\pi$
$578$	0	0
$579$	−12.7841	−0.531289
$580$	0	0
$581$	−23.7176	−0.983970
$582$	0	0
$583$	1.40434i	0.0581620i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 26.6239i	− 1.09889i	−0.835531	−	0.549443i	$-0.814840\pi$
$587$	− 26.6239i	− 1.09889i	0.835531	−	0.549443i	$-0.185160\pi$
$588$	0	0
$589$	−0.548659	−0.0226071
$590$	0	0
$591$	9.86545	0.405811
$592$	0	0
$593$	5.23169i	0.214840i	0.994214	+	0.107420i	$0.0342589\pi$
$593$	5.23169i	0.214840i	−0.994214	+	0.107420i	$0.965741\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 0.295640i	− 0.0120997i
$598$	0	0
$599$	−39.5405	−1.61558	−0.807790	−	0.589471i	$-0.799336\pi$
$599$	−39.5405	−1.61558	−0.807790	+	0.589471i	$0.799336\pi$
$600$	0	0
$601$	−45.5789	−1.85920	−0.929602	−	0.368565i	$-0.879849\pi$
$601$	−45.5789	−1.85920	−0.929602	+	0.368565i	$0.879849\pi$
$602$	0	0
$603$	− 11.9457i	− 0.486467i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	18.6524i	0.757078i	0.925585	+	0.378539i	$0.123573\pi$
$607$	18.6524i	0.757078i	−0.925585	+	0.378539i	$0.876427\pi$
$608$	0	0
$609$	21.3935	0.866909
$610$	0	0
$611$	−44.8016	−1.81248
$612$	0	0
$613$	10.7077i	0.432482i	0.976340	+	0.216241i	$0.0693796\pi$
$613$	10.7077i	0.432482i	−0.976340	+	0.216241i	$0.930620\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	16.7334i	0.673662i	0.941565	+	0.336831i	$0.109355\pi$
$617$	16.7334i	0.673662i	−0.941565	+	0.336831i	$0.890645\pi$
$618$	0	0
$619$	−22.9329	−0.921750	−0.460875	−	0.887465i	$-0.652464\pi$
$619$	−22.9329	−0.921750	−0.460875	+	0.887465i	$0.652464\pi$
$620$	0	0
$621$	1.31495	0.0527671
$622$	0	0
$623$	− 50.0934i	− 2.00695i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 12.3901i	− 0.494812i
$628$	0	0
$629$	53.9338	2.15048
$630$	0	0
$631$	14.5167	0.577903	0.288951	−	0.957344i	$-0.406693\pi$
$631$	14.5167	0.577903	0.288951	+	0.957344i	$0.406693\pi$
$632$	0	0
$633$	12.6477i	0.502701i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 71.8229i	− 2.84573i
$638$	0	0
$639$	−0.509248	−0.0201455
$640$	0	0
$641$	−31.7526	−1.25415	−0.627075	−	0.778959i	$-0.715748\pi$
$641$	−31.7526	−1.25415	−0.627075	+	0.778959i	$0.715748\pi$
$642$	0	0
$643$	− 3.63816i	− 0.143475i	−0.997424	−	0.0717376i	$-0.977146\pi$
$643$	− 3.63816i	− 0.143475i	0.997424	−	0.0717376i	$-0.0228544\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 11.2444i	− 0.442065i	−0.975266	−	0.221032i	$-0.929057\pi$
$647$	− 11.2444i	− 0.442065i	0.975266	−	0.221032i	$-0.0709426\pi$
$648$	0	0
$649$	−11.2142	−0.440195
$650$	0	0
$651$	−0.870433	−0.0341150
$652$	0	0
$653$	2.02226i	0.0791372i	0.999217	+	0.0395686i	$0.0125984\pi$
$653$	2.02226i	0.0791372i	−0.999217	+	0.0395686i	$0.987402\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	15.7708i	0.615279i
$658$	0	0
$659$	0.794810	0.0309614	0.0154807	−	0.999880i	$-0.495072\pi$
$659$	0.794810	0.0309614	0.0154807	+	0.999880i	$0.495072\pi$
$660$	0	0
$661$	40.9778	1.59385	0.796925	−	0.604078i	$-0.206459\pi$
$661$	40.9778	1.59385	0.796925	+	0.604078i	$0.206459\pi$
$662$	0	0
$663$	37.7650i	1.46667i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 6.37119i	− 0.246694i
$668$	0	0
$669$	−5.23980	−0.202583
$670$	0	0
$671$	41.5404	1.60365
$672$	0	0
$673$	35.1014i	1.35306i	0.736416	+	0.676529i	$0.236517\pi$
$673$	35.1014i	1.35306i	−0.736416	+	0.676529i	$0.763483\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.3867i	1.47532i	0.675173	+	0.737660i	$0.264069\pi$
$677$	38.3867i	1.47532i	−0.675173	+	0.737660i	$0.735931\pi$
$678$	0	0
$679$	22.4285	0.860726
$680$	0	0
$681$	7.76897	0.297708
$682$	0	0
$683$	− 8.36485i	− 0.320072i	−0.987111	−	0.160036i	$-0.948839\pi$
$683$	− 8.36485i	− 0.320072i	0.987111	−	0.160036i	$-0.0511610\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 5.99192i	− 0.228606i
$688$	0	0
$689$	−1.81317	−0.0690761
$690$	0	0
$691$	5.83085	0.221816	0.110908	−	0.993831i	$-0.464624\pi$
$691$	5.83085	0.221816	0.110908	+	0.993831i	$0.464624\pi$
$692$	0	0
$693$	− 19.6565i	− 0.746689i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	51.5763i	1.95359i
$698$	0	0
$699$	−14.8631	−0.562174
$700$	0	0
$701$	50.6649	1.91359	0.956793	−	0.290770i	$-0.0939114\pi$
$701$	50.6649	1.91359	0.956793	+	0.290770i	$0.0939114\pi$
$702$	0	0
$703$	− 22.8460i	− 0.861652i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 49.3907i	− 1.85753i
$708$	0	0
$709$	9.73196	0.365492	0.182746	−	0.983160i	$-0.441501\pi$
$709$	9.73196	0.365492	0.182746	+	0.983160i	$0.441501\pi$
$710$	0	0
$711$	−3.43153	−0.128692
$712$	0	0
$713$	0.259223i	0.00970799i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 21.2008i	− 0.791757i
$718$	0	0
$719$	−18.7362	−0.698742	−0.349371	−	0.936985i	$-0.613605\pi$
$719$	−18.7362	−0.698742	−0.349371	+	0.936985i	$0.613605\pi$
$720$	0	0
$721$	−13.8894	−0.517269
$722$	0	0
$723$	− 16.0562i	− 0.597135i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 37.9837i	− 1.40874i	−0.709834	−	0.704369i	$-0.751230\pi$
$727$	− 37.9837i	− 1.40874i	0.709834	−	0.704369i	$-0.248770\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−2.71219	−0.100314
$732$	0	0
$733$	26.3618i	0.973694i	0.873487	+	0.486847i	$0.161853\pi$
$733$	26.3618i	0.973694i	−0.873487	+	0.486847i	$0.838147\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	53.1800i	1.95891i
$738$	0	0
$739$	33.0342	1.21518	0.607592	−	0.794249i	$-0.292136\pi$
$739$	33.0342	1.21518	0.607592	+	0.794249i	$0.292136\pi$
$740$	0	0
$741$	15.9970	0.587663
$742$	0	0
$743$	− 40.2017i	− 1.47486i	−0.675425	−	0.737428i	$-0.736040\pi$
$743$	− 40.2017i	− 1.47486i	0.675425	−	0.737428i	$-0.263960\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 5.37155i	− 0.196535i
$748$	0	0
$749$	34.1758	1.24876
$750$	0	0
$751$	30.5937	1.11638	0.558190	−	0.829713i	$-0.311496\pi$
$751$	30.5937	1.11638	0.558190	+	0.829713i	$0.311496\pi$
$752$	0	0
$753$	24.1371i	0.879606i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 24.4003i	− 0.886845i	−0.896313	−	0.443422i	$-0.853764\pi$
$757$	− 24.4003i	− 0.886845i	0.896313	−	0.443422i	$-0.146236\pi$
$758$	0	0
$759$	−5.85390	−0.212483
$760$	0	0
$761$	28.3342	1.02711	0.513556	−	0.858056i	$-0.328328\pi$
$761$	28.3342	1.02711	0.513556	+	0.858056i	$0.328328\pi$
$762$	0	0
$763$	45.7914i	1.65776i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 14.4788i	− 0.522798i
$768$	0	0
$769$	41.1214	1.48288	0.741438	−	0.671021i	$-0.234144\pi$
$769$	41.1214	1.48288	0.741438	+	0.671021i	$0.234144\pi$
$770$	0	0
$771$	4.22940	0.152318
$772$	0	0
$773$	− 4.81707i	− 0.173258i	−0.996241	−	0.0866290i	$-0.972391\pi$
$773$	− 4.81707i	− 0.173258i	0.996241	−	0.0866290i	$-0.0276095\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 36.2445i	− 1.30027i
$778$	0	0
$779$	21.8473	0.782762
$780$	0	0
$781$	2.26707	0.0811223
$782$	0	0
$783$	4.84521i	0.173154i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 40.3180i	− 1.43718i	−0.695433	−	0.718591i	$-0.744787\pi$
$787$	− 40.3180i	− 1.43718i	0.695433	−	0.718591i	$-0.255213\pi$
$788$	0	0
$789$	−8.38782	−0.298614
$790$	0	0
$791$	−45.1530	−1.60546
$792$	0	0
$793$	53.6333i	1.90457i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 30.1958i	− 1.06959i	−0.844981	−	0.534796i	$-0.820389\pi$
$797$	− 30.1958i	− 1.06959i	0.844981	−	0.534796i	$-0.179611\pi$
$798$	0	0
$799$	−51.2132	−1.81179
$800$	0	0
$801$	11.3452	0.400862
$802$	0	0
$803$	− 70.2087i	− 2.47761i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 5.53526i	− 0.194850i
$808$	0	0
$809$	−10.8803	−0.382529	−0.191265	−	0.981539i	$-0.561259\pi$
$809$	−10.8803	−0.382529	−0.191265	+	0.981539i	$0.561259\pi$
$810$	0	0
$811$	−47.2580	−1.65945	−0.829727	−	0.558169i	$-0.811504\pi$
$811$	−47.2580	−1.65945	−0.829727	+	0.558169i	$0.811504\pi$
$812$	0	0
$813$	− 15.8497i	− 0.555874i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.14886i	0.0401936i
$818$	0	0
$819$	25.3787	0.886805
$820$	0	0
$821$	24.8696	0.867956	0.433978	−	0.900923i	$-0.357110\pi$
$821$	24.8696	0.867956	0.433978	+	0.900923i	$0.357110\pi$
$822$	0	0
$823$	22.7590i	0.793330i	0.917963	+	0.396665i	$0.129833\pi$
$823$	22.7590i	0.793330i	−0.917963	+	0.396665i	$0.870167\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	46.8458i	1.62899i	0.580172	+	0.814494i	$0.302985\pi$
$827$	46.8458i	1.62899i	−0.580172	+	0.814494i	$0.697015\pi$
$828$	0	0
$829$	13.8010	0.479328	0.239664	−	0.970856i	$-0.422963\pi$
$829$	13.8010	0.479328	0.239664	+	0.970856i	$0.422963\pi$
$830$	0	0
$831$	−8.69243	−0.301537
$832$	0	0
$833$	− 82.1015i	− 2.84465i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 0.197136i	− 0.00681401i
$838$	0	0
$839$	10.5887	0.365564	0.182782	−	0.983153i	$-0.441490\pi$
$839$	10.5887	0.365564	0.182782	+	0.983153i	$0.441490\pi$
$840$	0	0
$841$	−5.52399	−0.190482
$842$	0	0
$843$	− 10.7800i	− 0.371283i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	38.9377i	1.33791i
$848$	0	0
$849$	−9.84520	−0.337886
$850$	0	0
$851$	−10.7940	−0.370012
$852$	0	0
$853$	22.6497i	0.775510i	0.921762	+	0.387755i	$0.126749\pi$
$853$	22.6497i	0.775510i	−0.921762	+	0.387755i	$0.873251\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 39.4749i	− 1.34844i	−0.738533	−	0.674218i	$-0.764481\pi$
$857$	− 39.4749i	− 1.34844i	0.738533	−	0.674218i	$-0.235519\pi$
$858$	0	0
$859$	−44.3379	−1.51279	−0.756395	−	0.654115i	$-0.773041\pi$
$859$	−44.3379	−1.51279	−0.756395	+	0.654115i	$0.773041\pi$
$860$	0	0
$861$	34.6602	1.18122
$862$	0	0
$863$	5.03563i	0.171415i	0.996320	+	0.0857074i	$0.0273150\pi$
$863$	5.03563i	0.171415i	−0.996320	+	0.0857074i	$0.972685\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	26.1696i	0.888765i
$868$	0	0
$869$	15.2765	0.518220
$870$	0	0
$871$	−68.6613	−2.32650
$872$	0	0
$873$	5.07960i	0.171918i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	17.4943i	0.590739i	0.955383	+	0.295370i	$0.0954428\pi$
$877$	17.4943i	0.590739i	−0.955383	+	0.295370i	$0.904557\pi$
$878$	0	0
$879$	−9.77733	−0.329781
$880$	0	0
$881$	−54.2456	−1.82758	−0.913791	−	0.406185i	$-0.866859\pi$
$881$	−54.2456	−1.82758	−0.913791	+	0.406185i	$0.866859\pi$
$882$	0	0
$883$	− 30.3544i	− 1.02151i	−0.859727	−	0.510753i	$-0.829367\pi$
$883$	− 30.3544i	− 1.02151i	0.859727	−	0.510753i	$-0.170633\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	52.6472i	1.76772i	0.467751	+	0.883860i	$0.345065\pi$
$887$	52.6472i	1.76772i	−0.467751	+	0.883860i	$0.654935\pi$
$888$	0	0
$889$	49.9501	1.67527
$890$	0	0
$891$	4.45181	0.149141
$892$	0	0
$893$	21.6935i	0.725947i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 7.55803i	− 0.252355i
$898$	0	0
$899$	−0.955163	−0.0318565
$900$	0	0
$901$	−2.07265	−0.0690500
$902$	0	0
$903$	1.82264i	0.0606537i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	8.79799i	0.292133i	0.989275	+	0.146066i	$0.0466613\pi$
$907$	8.79799i	0.292133i	−0.989275	+	0.146066i	$0.953339\pi$
$908$	0	0
$909$	11.1860	0.371017
$910$	0	0
$911$	−31.5174	−1.04422	−0.522110	−	0.852878i	$-0.674855\pi$
$911$	−31.5174	−1.04422	−0.522110	+	0.852878i	$0.674855\pi$
$912$	0	0
$913$	23.9131i	0.791409i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 41.7416i	− 1.37843i
$918$	0	0
$919$	33.4799	1.10440	0.552200	−	0.833712i	$-0.313789\pi$
$919$	33.4799	1.10440	0.552200	+	0.833712i	$0.313789\pi$
$920$	0	0
$921$	32.7301	1.07849
$922$	0	0
$923$	2.92704i	0.0963448i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 3.14567i	− 0.103317i
$928$	0	0
$929$	1.87841	0.0616286	0.0308143	−	0.999525i	$-0.490190\pi$
$929$	1.87841	0.0616286	0.0308143	+	0.999525i	$0.490190\pi$
$930$	0	0
$931$	−34.7776	−1.13979
$932$	0	0
$933$	24.6245i	0.806170i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 2.28820i	− 0.0747523i	−0.999301	−	0.0373762i	$-0.988100\pi$
$937$	− 2.28820i	− 0.0747523i	0.999301	−	0.0373762i	$-0.0119000\pi$
$938$	0	0
$939$	22.5495	0.735874
$940$	0	0
$941$	−60.1381	−1.96044	−0.980222	−	0.197900i	$-0.936588\pi$
$941$	−60.1381	−1.96044	−0.980222	+	0.197900i	$0.936588\pi$
$942$	0	0
$943$	− 10.3221i	− 0.336135i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	43.2697i	1.40608i	0.711152	+	0.703039i	$0.248174\pi$
$947$	43.2697i	1.40608i	−0.711152	+	0.703039i	$0.751826\pi$
$948$	0	0
$949$	90.6473	2.94253
$950$	0	0
$951$	1.62435	0.0526731
$952$	0	0
$953$	34.8553i	1.12907i	0.825407	+	0.564537i	$0.190945\pi$
$953$	34.8553i	1.12907i	−0.825407	+	0.564537i	$0.809055\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 21.5699i	− 0.697257i
$958$	0	0
$959$	−37.1345	−1.19914
$960$	0	0
$961$	−30.9611	−0.998746
$962$	0	0
$963$	7.74013i	0.249422i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 34.9748i	− 1.12471i	−0.826895	−	0.562357i	$-0.809895\pi$
$967$	− 34.9748i	− 1.12471i	0.826895	−	0.562357i	$-0.190105\pi$
$968$	0	0
$969$	18.2863	0.587441
$970$	0	0
$971$	−59.4788	−1.90877	−0.954383	−	0.298586i	$-0.903485\pi$
$971$	−59.4788	−1.90877	−0.954383	+	0.298586i	$0.903485\pi$
$972$	0	0
$973$	− 52.1970i	− 1.67336i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	15.4162i	0.493207i	0.969117	+	0.246603i	$0.0793145\pi$
$977$	15.4162i	0.493207i	−0.969117	+	0.246603i	$0.920686\pi$
$978$	0	0
$979$	−50.5065	−1.61419
$980$	0	0
$981$	−10.3708	−0.331116
$982$	0	0
$983$	58.7500i	1.87383i	0.349552	+	0.936917i	$0.386334\pi$
$983$	58.7500i	1.87383i	−0.349552	+	0.936917i	$0.613666\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	34.4162i	1.09548i
$988$	0	0
$989$	0.542800	0.0172600
$990$	0	0
$991$	1.71246	0.0543982	0.0271991	−	0.999630i	$-0.491341\pi$
$991$	1.71246	0.0543982	0.0271991	+	0.999630i	$0.491341\pi$
$992$	0	0
$993$	9.01314i	0.286023i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	11.7966i	0.373603i	0.982398	+	0.186802i	$0.0598121\pi$
$997$	11.7966i	0.373603i	−0.982398	+	0.186802i	$0.940188\pi$
$998$	0	0
$999$	8.20866	0.259711

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.d.g.1249.11		24
5.2	odd	4		7500.2.a.m.1.2		12
5.3	odd	4		7500.2.a.n.1.11		12
5.4	even	2	inner	7500.2.d.g.1249.14		24
25.2	odd	20		1500.2.m.d.601.1		24
25.9	even	10		1500.2.o.c.349.3		24
25.11	even	5		1500.2.o.c.649.3		24
25.12	odd	20		1500.2.m.d.901.1		24
25.13	odd	20		1500.2.m.c.901.6		24
25.14	even	10		300.2.o.a.229.4	yes	24
25.16	even	5		300.2.o.a.169.4	✓	24
25.23	odd	20		1500.2.m.c.601.6		24
75.14	odd	10		900.2.w.c.829.5		24
75.41	odd	10		900.2.w.c.469.5		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.o.a.169.4	✓	24	25.16	even	5
300.2.o.a.229.4	yes	24	25.14	even	10
900.2.w.c.469.5		24	75.41	odd	10
900.2.w.c.829.5		24	75.14	odd	10
1500.2.m.c.601.6		24	25.23	odd	20
1500.2.m.c.901.6		24	25.13	odd	20
1500.2.m.d.601.1		24	25.2	odd	20
1500.2.m.d.901.1		24	25.12	odd	20
1500.2.o.c.349.3		24	25.9	even	10
1500.2.o.c.649.3		24	25.11	even	5
7500.2.a.m.1.2		12	5.2	odd	4
7500.2.a.n.1.11		12	5.3	odd	4
7500.2.d.g.1249.11		24	1.1	even	1	trivial
7500.2.d.g.1249.14		24	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +4.41540i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +4.41540i q^{7} -1.00000 q^{9} +4.45181 q^{11} +5.74778i q^{13} +6.57035i q^{17} +2.78315 q^{19} +4.41540 q^{21} -1.31495i q^{23} +1.00000i q^{27} +4.84521 q^{29} -0.197136 q^{31} -4.45181i q^{33} -8.20866i q^{37} +5.74778 q^{39} +7.84984 q^{41} +0.412792i q^{43} +7.79459i q^{47} -12.4958 q^{49} +6.57035 q^{51} +0.315455i q^{53} -2.78315i q^{57} -2.51902 q^{59} +9.33114 q^{61} -4.41540i q^{63} +11.9457i q^{67} -1.31495 q^{69} +0.509248 q^{71} -15.7708i q^{73} +19.6565i q^{77} +3.43153 q^{79} +1.00000 q^{81} +5.37155i q^{83} -4.84521i q^{87} -11.3452 q^{89} -25.3787 q^{91} +0.197136i q^{93} -5.07960i q^{97} -4.45181 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{9}+O(q^{10}) \) 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

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