LMFDB - Embedded newform 7500.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7500,2,Mod(1,7500)]

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, 0]))

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

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

chi := DirichletCharacter("7500.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7500 = 2^{2} \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7500.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:	$59.8878015160$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} - 11 x^{10} + 94 x^{9} + 27 x^{8} - 460 x^{7} + 55 x^{6} + 812 x^{5} - 127 x^{4} + \cdots - 4 $ x^12 - 6x^11 - 11x^10 + 94x^9 + 27x^8 - 460x^7 + 55x^6 + 812x^5 - 127x^4 - 512x^3 + 40x^2 + 72*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 5^{3} $
Twist minimal:	no (minimal twist has level 300)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.0550688$ of defining polynomial
Character	$\chi$	$=$	7500.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -4.41540 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -4.41540 q^{7} +1.00000 q^{9} +4.45181 q^{11} +5.74778 q^{13} -6.57035 q^{17} -2.78315 q^{19} +4.41540 q^{21} -1.31495 q^{23} -1.00000 q^{27} -4.84521 q^{29} -0.197136 q^{31} -4.45181 q^{33} +8.20866 q^{37} -5.74778 q^{39} +7.84984 q^{41} +0.412792 q^{43} -7.79459 q^{47} +12.4958 q^{49} +6.57035 q^{51} +0.315455 q^{53} +2.78315 q^{57} +2.51902 q^{59} +9.33114 q^{61} -4.41540 q^{63} -11.9457 q^{67} +1.31495 q^{69} +0.509248 q^{71} -15.7708 q^{73} -19.6565 q^{77} -3.43153 q^{79} +1.00000 q^{81} +5.37155 q^{83} +4.84521 q^{87} +11.3452 q^{89} -25.3787 q^{91} +0.197136 q^{93} +5.07960 q^{97} +4.45181 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} - 8 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 - 8 * q^7 + 12 * q^9	$ 12 q - 12 q^{3} - 8 q^{7} + 12 q^{9} + 2 q^{11} - 8 q^{17} + 10 q^{19} + 8 q^{21} - 18 q^{23} - 12 q^{27} + 8 q^{29} - 2 q^{31} - 2 q^{33} - 4 q^{37} + 10 q^{41} - 28 q^{43} - 22 q^{47} + 28 q^{49} + 8 q^{51} - 16 q^{53} - 10 q^{57} - 2 q^{59} + 34 q^{61} - 8 q^{63} - 32 q^{67} + 18 q^{69} - 24 q^{73} - 18 q^{77} + 6 q^{79} + 12 q^{81} - 28 q^{83} - 8 q^{87} + 10 q^{89} + 20 q^{91} + 2 q^{93} - 16 q^{97} + 2 q^{99}+O(q^{100}) $ 12 * q - 12 * q^3 - 8 * q^7 + 12 * q^9 + 2 * q^11 - 8 * q^17 + 10 * q^19 + 8 * q^21 - 18 * q^23 - 12 * q^27 + 8 * q^29 - 2 * q^31 - 2 * q^33 - 4 * q^37 + 10 * q^41 - 28 * q^43 - 22 * q^47 + 28 * q^49 + 8 * q^51 - 16 * q^53 - 10 * q^57 - 2 * q^59 + 34 * q^61 - 8 * q^63 - 32 * q^67 + 18 * q^69 - 24 * q^73 - 18 * q^77 + 6 * q^79 + 12 * q^81 - 28 * q^83 - 8 * q^87 + 10 * q^89 + 20 * q^91 + 2 * q^93 - 16 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.41540	−1.66886	−0.834432	−	0.551111i	$-0.814204\pi$
$7$	−4.41540	−1.66886	−0.834432	+	0.551111i	$0.814204\pi$
$8$	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
$13$	5.74778	1.59415	0.797074	−	0.603882i	$-0.206380\pi$
$13$	5.74778	1.59415	0.797074	+	0.603882i	$0.206380\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.57035	−1.59355	−0.796773	−	0.604279i	$-0.793461\pi$
$17$	−6.57035	−1.59355	−0.796773	+	0.604279i	$0.793461\pi$
$18$	0	0
$19$	−2.78315	−0.638499	−0.319250	−	0.947671i	$-0.603431\pi$
$19$	−2.78315	−0.638499	−0.319250	+	0.947671i	$0.603431\pi$
$20$	0	0
$21$	4.41540	0.963519
$22$	0	0
$23$	−1.31495	−0.274186	−0.137093	−	0.990558i	$-0.543776\pi$
$23$	−1.31495	−0.274186	−0.137093	+	0.990558i	$0.543776\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.84521	−0.899732	−0.449866	−	0.893096i	$-0.648528\pi$
$29$	−4.84521	−0.899732	−0.449866	+	0.893096i	$0.648528\pi$
$30$	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
$33$	−4.45181	−0.774960
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.20866	1.34950	0.674748	−	0.738048i	$-0.264252\pi$
$37$	8.20866	1.34950	0.674748	+	0.738048i	$0.264252\pi$
$38$	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
$43$	0.412792	0.0629502	0.0314751	−	0.999505i	$-0.489980\pi$
$43$	0.412792	0.0629502	0.0314751	+	0.999505i	$0.489980\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.79459	−1.13696	−0.568479	−	0.822698i	$-0.692468\pi$
$47$	−7.79459	−1.13696	−0.568479	+	0.822698i	$0.692468\pi$
$48$	0	0
$49$	12.4958	1.78511
$50$	0	0
$51$	6.57035	0.920034
$52$	0	0
$53$	0.315455	0.0433311	0.0216655	−	0.999765i	$-0.493103\pi$
$53$	0.315455	0.0433311	0.0216655	+	0.999765i	$0.493103\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.78315	0.368638
$58$	0	0
$59$	2.51902	0.327948	0.163974	−	0.986465i	$-0.447569\pi$
$59$	2.51902	0.327948	0.163974	+	0.986465i	$0.447569\pi$
$60$	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
$63$	−4.41540	−0.556288
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11.9457	−1.45940	−0.729700	−	0.683767i	$-0.760340\pi$
$67$	−11.9457	−1.45940	−0.729700	+	0.683767i	$0.760340\pi$
$68$	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
$73$	−15.7708	−1.84584	−0.922918	−	0.384996i	$-0.874203\pi$
$73$	−15.7708	−1.84584	−0.922918	+	0.384996i	$0.874203\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−19.6565	−2.24007
$78$	0	0
$79$	−3.43153	−0.386077	−0.193039	−	0.981191i	$-0.561834\pi$
$79$	−3.43153	−0.386077	−0.193039	+	0.981191i	$0.561834\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.37155	0.589604	0.294802	−	0.955558i	$-0.404746\pi$
$83$	5.37155	0.589604	0.294802	+	0.955558i	$0.404746\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.84521	0.519461
$88$	0	0
$89$	11.3452	1.20258	0.601292	−	0.799029i	$-0.294653\pi$
$89$	11.3452	1.20258	0.601292	+	0.799029i	$0.294653\pi$
$90$	0	0
$91$	−25.3787	−2.66042
$92$	0	0
$93$	0.197136	0.0204420
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.07960	0.515755	0.257878	−	0.966178i	$-0.416977\pi$
$97$	5.07960	0.515755	0.257878	+	0.966178i	$0.416977\pi$
$98$	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
$103$	3.14567	0.309952	0.154976	−	0.987918i	$-0.450470\pi$
$103$	3.14567	0.309952	0.154976	+	0.987918i	$0.450470\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.74013	0.748266	0.374133	−	0.927375i	$-0.377940\pi$
$107$	7.74013	0.748266	0.374133	+	0.927375i	$0.377940\pi$
$108$	0	0
$109$	−10.3708	−0.993347	−0.496673	−	0.867937i	$-0.665445\pi$
$109$	−10.3708	−0.993347	−0.496673	+	0.867937i	$0.665445\pi$
$110$	0	0
$111$	−8.20866	−0.779132
$112$	0	0
$113$	10.2263	0.962005	0.481002	−	0.876719i	$-0.340273\pi$
$113$	10.2263	0.962005	0.481002	+	0.876719i	$0.340273\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.74778	0.531382
$118$	0	0
$119$	29.0107	2.65941
$120$	0	0
$121$	8.81860	0.801691
$122$	0	0
$123$	−7.84984	−0.707797
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.3127	1.00384	0.501920	−	0.864914i	$-0.332627\pi$
$127$	11.3127	1.00384	0.501920	+	0.864914i	$0.332627\pi$
$128$	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
$133$	12.2887	1.06557
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.41023	−0.718535	−0.359267	−	0.933235i	$-0.616973\pi$
$137$	−8.41023	−0.718535	−0.359267	+	0.933235i	$0.616973\pi$
$138$	0	0
$139$	11.8216	1.00269	0.501346	−	0.865247i	$-0.332838\pi$
$139$	11.8216	1.00269	0.501346	+	0.865247i	$0.332838\pi$
$140$	0	0
$141$	7.79459	0.656422
$142$	0	0
$143$	25.5880	2.13978
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−12.4958	−1.03063
$148$	0	0
$149$	10.6355	0.871294	0.435647	−	0.900118i	$-0.356520\pi$
$149$	10.6355	0.871294	0.435647	+	0.900118i	$0.356520\pi$
$150$	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
$153$	−6.57035	−0.531182
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−13.5289	−1.07972	−0.539861	−	0.841754i	$-0.681523\pi$
$157$	−13.5289	−1.07972	−0.539861	+	0.841754i	$0.681523\pi$
$158$	0	0
$159$	−0.315455	−0.0250172
$160$	0	0
$161$	5.80602	0.457579
$162$	0	0
$163$	−14.5583	−1.14029	−0.570146	−	0.821543i	$-0.693114\pi$
$163$	−14.5583	−1.14029	−0.570146	+	0.821543i	$0.693114\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.96824	0.771366	0.385683	−	0.922631i	$-0.373966\pi$
$167$	9.96824	0.771366	0.385683	+	0.922631i	$0.373966\pi$
$168$	0	0
$169$	20.0370	1.54131
$170$	0	0
$171$	−2.78315	−0.212833
$172$	0	0
$173$	10.8750	0.826811	0.413406	−	0.910547i	$-0.364339\pi$
$173$	10.8750	0.826811	0.413406	+	0.910547i	$0.364339\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.51902	−0.189341
$178$	0	0
$179$	−25.1740	−1.88159	−0.940795	−	0.338975i	$-0.889920\pi$
$179$	−25.1740	−1.88159	−0.940795	+	0.338975i	$0.889920\pi$
$180$	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
$183$	−9.33114	−0.689777
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−29.2500	−2.13897
$188$	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
$193$	−12.7841	−0.920219	−0.460110	−	0.887862i	$-0.652190\pi$
$193$	−12.7841	−0.920219	−0.460110	+	0.887862i	$0.652190\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.86545	−0.702885	−0.351442	−	0.936210i	$-0.614309\pi$
$197$	−9.86545	−0.702885	−0.351442	+	0.936210i	$0.614309\pi$
$198$	0	0
$199$	−0.295640	−0.0209573	−0.0104787	−	0.999945i	$-0.503336\pi$
$199$	−0.295640	−0.0209573	−0.0104787	+	0.999945i	$0.503336\pi$
$200$	0	0
$201$	11.9457	0.842585
$202$	0	0
$203$	21.3935	1.50153
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.31495	−0.0913952
$208$	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
$213$	−0.509248	−0.0348931
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.870433	0.0590888
$218$	0	0
$219$	15.7708	1.06569
$220$	0	0
$221$	−37.7650	−2.54035
$222$	0	0
$223$	−5.23980	−0.350883	−0.175442	−	0.984490i	$-0.556135\pi$
$223$	−5.23980	−0.350883	−0.175442	+	0.984490i	$0.556135\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.76897	−0.515645	−0.257822	−	0.966192i	$-0.583005\pi$
$227$	−7.76897	−0.515645	−0.257822	+	0.966192i	$0.583005\pi$
$228$	0	0
$229$	−5.99192	−0.395958	−0.197979	−	0.980206i	$-0.563438\pi$
$229$	−5.99192	−0.395958	−0.197979	+	0.980206i	$0.563438\pi$
$230$	0	0
$231$	19.6565	1.29330
$232$	0	0
$233$	−14.8631	−0.973714	−0.486857	−	0.873482i	$-0.661857\pi$
$233$	−14.8631	−0.973714	−0.486857	+	0.873482i	$0.661857\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3.43153	0.222902
$238$	0	0
$239$	−21.2008	−1.37136	−0.685682	−	0.727902i	$-0.740496\pi$
$239$	−21.2008	−1.37136	−0.685682	+	0.727902i	$0.740496\pi$
$240$	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
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−15.9970	−1.01786
$248$	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
$253$	−5.85390	−0.368031
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.22940	−0.263823	−0.131911	−	0.991262i	$-0.542111\pi$
$257$	−4.22940	−0.263823	−0.131911	+	0.991262i	$0.542111\pi$
$258$	0	0
$259$	−36.2445	−2.25213
$260$	0	0
$261$	−4.84521	−0.299911
$262$	0	0
$263$	−8.38782	−0.517215	−0.258608	−	0.965982i	$-0.583264\pi$
$263$	−8.38782	−0.517215	−0.258608	+	0.965982i	$0.583264\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−11.3452	−0.694313
$268$	0	0
$269$	−5.53526	−0.337490	−0.168745	−	0.985660i	$-0.553972\pi$
$269$	−5.53526	−0.337490	−0.168745	+	0.985660i	$0.553972\pi$
$270$	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
$273$	25.3787	1.53599
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8.69243	0.522277	0.261139	−	0.965301i	$-0.415902\pi$
$277$	8.69243	0.522277	0.261139	+	0.965301i	$0.415902\pi$
$278$	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
$283$	−9.84520	−0.585236	−0.292618	−	0.956229i	$-0.594527\pi$
$283$	−9.84520	−0.585236	−0.292618	+	0.956229i	$0.594527\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−34.6602	−2.04593
$288$	0	0
$289$	26.1696	1.53939
$290$	0	0
$291$	−5.07960	−0.297772
$292$	0	0
$293$	−9.77733	−0.571198	−0.285599	−	0.958349i	$-0.592192\pi$
$293$	−9.77733	−0.571198	−0.285599	+	0.958349i	$0.592192\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.45181	−0.258320
$298$	0	0
$299$	−7.55803	−0.437092
$300$	0	0
$301$	−1.82264	−0.105055
$302$	0	0
$303$	11.1860	0.642620
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−32.7301	−1.86801	−0.934003	−	0.357265i	$-0.883709\pi$
$307$	−32.7301	−1.86801	−0.934003	+	0.357265i	$0.883709\pi$
$308$	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
$313$	22.5495	1.27457	0.637285	−	0.770628i	$-0.280057\pi$
$313$	22.5495	1.27457	0.637285	+	0.770628i	$0.280057\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.62435	−0.0912324	−0.0456162	−	0.998959i	$-0.514525\pi$
$317$	−1.62435	−0.0912324	−0.0456162	+	0.998959i	$0.514525\pi$
$318$	0	0
$319$	−21.5699	−1.20768
$320$	0	0
$321$	−7.74013	−0.432012
$322$	0	0
$323$	18.2863	1.01748
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.3708	0.573509
$328$	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
$333$	8.20866	0.449832
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−26.9681	−1.46905	−0.734524	−	0.678583i	$-0.762594\pi$
$337$	−26.9681	−1.46905	−0.734524	+	0.678583i	$0.762594\pi$
$338$	0	0
$339$	−10.2263	−0.555414
$340$	0	0
$341$	−0.877611	−0.0475253
$342$	0	0
$343$	−24.2660	−1.31024
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.2448	1.40889	0.704447	−	0.709757i	$-0.251195\pi$
$347$	26.2448	1.40889	0.704447	+	0.709757i	$0.251195\pi$
$348$	0	0
$349$	18.2310	0.975885	0.487943	−	0.872876i	$-0.337748\pi$
$349$	18.2310	0.975885	0.487943	+	0.872876i	$0.337748\pi$
$350$	0	0
$351$	−5.74778	−0.306794
$352$	0	0
$353$	−4.44962	−0.236829	−0.118415	−	0.992964i	$-0.537781\pi$
$353$	−4.44962	−0.236829	−0.118415	+	0.992964i	$0.537781\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−29.0107	−1.53541
$358$	0	0
$359$	−32.3271	−1.70616	−0.853081	−	0.521779i	$-0.825269\pi$
$359$	−32.3271	−1.70616	−0.853081	+	0.521779i	$0.825269\pi$
$360$	0	0
$361$	−11.2540	−0.592318
$362$	0	0
$363$	−8.81860	−0.462857
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5.74002	0.299627	0.149813	−	0.988714i	$-0.452133\pi$
$367$	5.74002	0.299627	0.149813	+	0.988714i	$0.452133\pi$
$368$	0	0
$369$	7.84984	0.408647
$370$	0	0
$371$	−1.39286	−0.0723137
$372$	0	0
$373$	−1.04186	−0.0539455	−0.0269727	−	0.999636i	$-0.508587\pi$
$373$	−1.04186	−0.0539455	−0.0269727	+	0.999636i	$0.508587\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−27.8492	−1.43431
$378$	0	0
$379$	−4.74934	−0.243957	−0.121979	−	0.992533i	$-0.538924\pi$
$379$	−4.74934	−0.243957	−0.121979	+	0.992533i	$0.538924\pi$
$380$	0	0
$381$	−11.3127	−0.579568
$382$	0	0
$383$	−11.0200	−0.563095	−0.281548	−	0.959547i	$-0.590848\pi$
$383$	−11.0200	−0.563095	−0.281548	+	0.959547i	$0.590848\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.412792	0.0209834
$388$	0	0
$389$	−33.2111	−1.68387	−0.841934	−	0.539581i	$-0.818583\pi$
$389$	−33.2111	−1.68387	−0.841934	+	0.539581i	$0.818583\pi$
$390$	0	0
$391$	8.63968	0.436927
$392$	0	0
$393$	9.45365	0.476873
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.74418	−0.137726	−0.0688631	−	0.997626i	$-0.521937\pi$
$397$	−2.74418	−0.137726	−0.0688631	+	0.997626i	$0.521937\pi$
$398$	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
$403$	−1.13309	−0.0564434
$404$	0	0
$405$	0	0
$406$	0	0
$407$	36.5434	1.81139
$408$	0	0
$409$	−6.57251	−0.324990	−0.162495	−	0.986709i	$-0.551954\pi$
$409$	−6.57251	−0.324990	−0.162495	+	0.986709i	$0.551954\pi$
$410$	0	0
$411$	8.41023	0.414846
$412$	0	0
$413$	−11.1225	−0.547301
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−11.8216	−0.578905
$418$	0	0
$419$	35.2434	1.72175	0.860876	−	0.508815i	$-0.169916\pi$
$419$	35.2434	1.72175	0.860876	+	0.508815i	$0.169916\pi$
$420$	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
$423$	−7.79459	−0.378986
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−41.2007	−1.99384
$428$	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
$433$	6.77819	0.325739	0.162870	−	0.986648i	$-0.447925\pi$
$433$	6.77819	0.325739	0.162870	+	0.986648i	$0.447925\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.65970	0.175067
$438$	0	0
$439$	−16.9446	−0.808723	−0.404362	−	0.914599i	$-0.632506\pi$
$439$	−16.9446	−0.808723	−0.404362	+	0.914599i	$0.632506\pi$
$440$	0	0
$441$	12.4958	0.595036
$442$	0	0
$443$	−23.8927	−1.13517	−0.567587	−	0.823313i	$-0.692123\pi$
$443$	−23.8927	−1.13517	−0.567587	+	0.823313i	$0.692123\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−10.6355	−0.503042
$448$	0	0
$449$	6.39281	0.301695	0.150848	−	0.988557i	$-0.451800\pi$
$449$	6.39281	0.301695	0.150848	+	0.988557i	$0.451800\pi$
$450$	0	0
$451$	34.9460	1.64554
$452$	0	0
$453$	−4.41657	−0.207508
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.89482	0.275748	0.137874	−	0.990450i	$-0.455973\pi$
$457$	5.89482	0.275748	0.137874	+	0.990450i	$0.455973\pi$
$458$	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
$463$	−29.8611	−1.38776	−0.693880	−	0.720090i	$-0.744100\pi$
$463$	−29.8611	−1.38776	−0.693880	+	0.720090i	$0.744100\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−26.1513	−1.21014	−0.605069	−	0.796173i	$-0.706855\pi$
$467$	−26.1513	−1.21014	−0.605069	+	0.796173i	$0.706855\pi$
$468$	0	0
$469$	52.7451	2.43554
$470$	0	0
$471$	13.5289	0.623378
$472$	0	0
$473$	1.83767	0.0844962
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.315455	0.0144437
$478$	0	0
$479$	1.54690	0.0706798	0.0353399	−	0.999375i	$-0.488749\pi$
$479$	1.54690	0.0706798	0.0353399	+	0.999375i	$0.488749\pi$
$480$	0	0
$481$	47.1816	2.15130
$482$	0	0
$483$	−5.80602	−0.264183
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−12.4789	−0.565473	−0.282736	−	0.959198i	$-0.591242\pi$
$487$	−12.4789	−0.565473	−0.282736	+	0.959198i	$0.591242\pi$
$488$	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
$493$	31.8347	1.43376
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.24853	−0.100860
$498$	0	0
$499$	−30.9281	−1.38453	−0.692267	−	0.721642i	$-0.743388\pi$
$499$	−30.9281	−1.38453	−0.692267	+	0.721642i	$0.743388\pi$
$500$	0	0
$501$	−9.96824	−0.445348
$502$	0	0
$503$	−31.5535	−1.40690	−0.703450	−	0.710744i	$-0.748358\pi$
$503$	−31.5535	−1.40690	−0.703450	+	0.710744i	$0.748358\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−20.0370	−0.889873
$508$	0	0
$509$	−7.22535	−0.320258	−0.160129	−	0.987096i	$-0.551191\pi$
$509$	−7.22535	−0.320258	−0.160129	+	0.987096i	$0.551191\pi$
$510$	0	0
$511$	69.6345	3.08045
$512$	0	0
$513$	2.78315	0.122879
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−34.7000	−1.52610
$518$	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
$523$	20.3551	0.890067	0.445033	−	0.895514i	$-0.353192\pi$
$523$	20.3551	0.890067	0.445033	+	0.895514i	$0.353192\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.29525	0.0564220
$528$	0	0
$529$	−21.2709	−0.924822
$530$	0	0
$531$	2.51902	0.109316
$532$	0	0
$533$	45.1192	1.95433
$534$	0	0
$535$	0	0
$536$	0	0
$537$	25.1740	1.08634
$538$	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
$543$	7.59173	0.325792
$544$	0	0
$545$	0	0
$546$	0	0
$547$	15.3396	0.655872	0.327936	−	0.944700i	$-0.393647\pi$
$547$	15.3396	0.655872	0.327936	+	0.944700i	$0.393647\pi$
$548$	0	0
$549$	9.33114	0.398243
$550$	0	0
$551$	13.4850	0.574478
$552$	0	0
$553$	15.1516	0.644310
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−28.6722	−1.21488	−0.607441	−	0.794365i	$-0.707804\pi$
$557$	−28.6722	−1.21488	−0.607441	+	0.794365i	$0.707804\pi$
$558$	0	0
$559$	2.37264	0.100352
$560$	0	0
$561$	29.2500	1.23493
$562$	0	0
$563$	−7.34999	−0.309765	−0.154883	−	0.987933i	$-0.549500\pi$
$563$	−7.34999	−0.309765	−0.154883	+	0.987933i	$0.549500\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−4.41540	−0.185429
$568$	0	0
$569$	−31.1279	−1.30495	−0.652474	−	0.757811i	$-0.726269\pi$
$569$	−31.1279	−1.30495	−0.652474	+	0.757811i	$0.726269\pi$
$570$	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
$573$	0.0120931	0.000505196 0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	21.7895	0.907109	0.453554	−	0.891229i	$-0.350156\pi$
$577$	21.7895	0.907109	0.453554	+	0.891229i	$0.350156\pi$
$578$	0	0
$579$	12.7841	0.531289
$580$	0	0
$581$	−23.7176	−0.983970
$582$	0	0
$583$	1.40434	0.0581620
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26.6239	1.09889	0.549443	−	0.835531i	$-0.314840\pi$
$587$	26.6239	1.09889	0.549443	+	0.835531i	$0.314840\pi$
$588$	0	0
$589$	0.548659	0.0226071
$590$	0	0
$591$	9.86545	0.405811
$592$	0	0
$593$	5.23169	0.214840	0.107420	−	0.994214i	$-0.465741\pi$
$593$	5.23169	0.214840	0.107420	+	0.994214i	$0.465741\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.295640	0.0120997
$598$	0	0
$599$	39.5405	1.61558	0.807790	−	0.589471i	$-0.200664\pi$
$599$	39.5405	1.61558	0.807790	+	0.589471i	$0.200664\pi$
$600$	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
$603$	−11.9457	−0.486467
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−18.6524	−0.757078	−0.378539	−	0.925585i	$-0.623573\pi$
$607$	−18.6524	−0.757078	−0.378539	+	0.925585i	$0.623573\pi$
$608$	0	0
$609$	−21.3935	−0.866909
$610$	0	0
$611$	−44.8016	−1.81248
$612$	0	0
$613$	10.7077	0.432482	0.216241	−	0.976340i	$-0.430620\pi$
$613$	10.7077	0.432482	0.216241	+	0.976340i	$0.430620\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−16.7334	−0.673662	−0.336831	−	0.941565i	$-0.609355\pi$
$617$	−16.7334	−0.673662	−0.336831	+	0.941565i	$0.609355\pi$
$618$	0	0
$619$	22.9329	0.921750	0.460875	−	0.887465i	$-0.347536\pi$
$619$	22.9329	0.921750	0.460875	+	0.887465i	$0.347536\pi$
$620$	0	0
$621$	1.31495	0.0527671
$622$	0	0
$623$	−50.0934	−2.00695
$624$	0	0
$625$	0	0
$626$	0	0
$627$	12.3901	0.494812
$628$	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
$633$	12.6477	0.502701
$634$	0	0
$635$	0	0
$636$	0	0
$637$	71.8229	2.84573
$638$	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
$643$	−3.63816	−0.143475	−0.0717376	−	0.997424i	$-0.522854\pi$
$643$	−3.63816	−0.143475	−0.0717376	+	0.997424i	$0.522854\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11.2444	0.442065	0.221032	−	0.975266i	$-0.429057\pi$
$647$	11.2444	0.442065	0.221032	+	0.975266i	$0.429057\pi$
$648$	0	0
$649$	11.2142	0.440195
$650$	0	0
$651$	−0.870433	−0.0341150
$652$	0	0
$653$	2.02226	0.0791372	0.0395686	−	0.999217i	$-0.487402\pi$
$653$	2.02226	0.0791372	0.0395686	+	0.999217i	$0.487402\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−15.7708	−0.615279
$658$	0	0
$659$	−0.794810	−0.0309614	−0.0154807	−	0.999880i	$-0.504928\pi$
$659$	−0.794810	−0.0309614	−0.0154807	+	0.999880i	$0.504928\pi$
$660$	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
$663$	37.7650	1.46667
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.37119	0.246694
$668$	0	0
$669$	5.23980	0.202583
$670$	0	0
$671$	41.5404	1.60365
$672$	0	0
$673$	35.1014	1.35306	0.676529	−	0.736416i	$-0.263483\pi$
$673$	35.1014	1.35306	0.676529	+	0.736416i	$0.263483\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−38.3867	−1.47532	−0.737660	−	0.675173i	$-0.764069\pi$
$677$	−38.3867	−1.47532	−0.737660	+	0.675173i	$0.764069\pi$
$678$	0	0
$679$	−22.4285	−0.860726
$680$	0	0
$681$	7.76897	0.297708
$682$	0	0
$683$	−8.36485	−0.320072	−0.160036	−	0.987111i	$-0.551161\pi$
$683$	−8.36485	−0.320072	−0.160036	+	0.987111i	$0.551161\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.99192	0.228606
$688$	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
$693$	−19.6565	−0.746689
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−51.5763	−1.95359
$698$	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
$703$	−22.8460	−0.861652
$704$	0	0
$705$	0	0
$706$	0	0
$707$	49.3907	1.85753
$708$	0	0
$709$	−9.73196	−0.365492	−0.182746	−	0.983160i	$-0.558499\pi$
$709$	−9.73196	−0.365492	−0.182746	+	0.983160i	$0.558499\pi$
$710$	0	0
$711$	−3.43153	−0.128692
$712$	0	0
$713$	0.259223	0.00970799
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21.2008	0.791757
$718$	0	0
$719$	18.7362	0.698742	0.349371	−	0.936985i	$-0.386395\pi$
$719$	18.7362	0.698742	0.349371	+	0.936985i	$0.386395\pi$
$720$	0	0
$721$	−13.8894	−0.517269
$722$	0	0
$723$	−16.0562	−0.597135
$724$	0	0
$725$	0	0
$726$	0	0
$727$	37.9837	1.40874	0.704369	−	0.709834i	$-0.251230\pi$
$727$	37.9837	1.40874	0.704369	+	0.709834i	$0.251230\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.71219	−0.100314
$732$	0	0
$733$	26.3618	0.973694	0.486847	−	0.873487i	$-0.338147\pi$
$733$	26.3618	0.973694	0.486847	+	0.873487i	$0.338147\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−53.1800	−1.95891
$738$	0	0
$739$	−33.0342	−1.21518	−0.607592	−	0.794249i	$-0.707864\pi$
$739$	−33.0342	−1.21518	−0.607592	+	0.794249i	$0.707864\pi$
$740$	0	0
$741$	15.9970	0.587663
$742$	0	0
$743$	−40.2017	−1.47486	−0.737428	−	0.675425i	$-0.763960\pi$
$743$	−40.2017	−1.47486	−0.737428	+	0.675425i	$0.763960\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	5.37155	0.196535
$748$	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
$753$	24.1371	0.879606
$754$	0	0
$755$	0	0
$756$	0	0
$757$	24.4003	0.886845	0.443422	−	0.896313i	$-0.353764\pi$
$757$	24.4003	0.886845	0.443422	+	0.896313i	$0.353764\pi$
$758$	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
$763$	45.7914	1.65776
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14.4788	0.522798
$768$	0	0
$769$	−41.1214	−1.48288	−0.741438	−	0.671021i	$-0.765856\pi$
$769$	−41.1214	−1.48288	−0.741438	+	0.671021i	$0.765856\pi$
$770$	0	0
$771$	4.22940	0.152318
$772$	0	0
$773$	−4.81707	−0.173258	−0.0866290	−	0.996241i	$-0.527609\pi$
$773$	−4.81707	−0.173258	−0.0866290	+	0.996241i	$0.527609\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	36.2445	1.30027
$778$	0	0
$779$	−21.8473	−0.782762
$780$	0	0
$781$	2.26707	0.0811223
$782$	0	0
$783$	4.84521	0.173154
$784$	0	0
$785$	0	0
$786$	0	0
$787$	40.3180	1.43718	0.718591	−	0.695433i	$-0.244787\pi$
$787$	40.3180	1.43718	0.718591	+	0.695433i	$0.244787\pi$
$788$	0	0
$789$	8.38782	0.298614
$790$	0	0
$791$	−45.1530	−1.60546
$792$	0	0
$793$	53.6333	1.90457
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30.1958	1.06959	0.534796	−	0.844981i	$-0.320389\pi$
$797$	30.1958	1.06959	0.534796	+	0.844981i	$0.320389\pi$
$798$	0	0
$799$	51.2132	1.81179
$800$	0	0
$801$	11.3452	0.400862
$802$	0	0
$803$	−70.2087	−2.47761
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.53526	0.194850
$808$	0	0
$809$	10.8803	0.382529	0.191265	−	0.981539i	$-0.438741\pi$
$809$	10.8803	0.382529	0.191265	+	0.981539i	$0.438741\pi$
$810$	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
$813$	−15.8497	−0.555874
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−1.14886	−0.0401936
$818$	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
$823$	22.7590	0.793330	0.396665	−	0.917963i	$-0.370167\pi$
$823$	22.7590	0.793330	0.396665	+	0.917963i	$0.370167\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−46.8458	−1.62899	−0.814494	−	0.580172i	$-0.802985\pi$
$827$	−46.8458	−1.62899	−0.814494	+	0.580172i	$0.802985\pi$
$828$	0	0
$829$	−13.8010	−0.479328	−0.239664	−	0.970856i	$-0.577037\pi$
$829$	−13.8010	−0.479328	−0.239664	+	0.970856i	$0.577037\pi$
$830$	0	0
$831$	−8.69243	−0.301537
$832$	0	0
$833$	−82.1015	−2.84465
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0.197136	0.00681401
$838$	0	0
$839$	−10.5887	−0.365564	−0.182782	−	0.983153i	$-0.558510\pi$
$839$	−10.5887	−0.365564	−0.182782	+	0.983153i	$0.558510\pi$
$840$	0	0
$841$	−5.52399	−0.190482
$842$	0	0
$843$	−10.7800	−0.371283
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−38.9377	−1.33791
$848$	0	0
$849$	9.84520	0.337886
$850$	0	0
$851$	−10.7940	−0.370012
$852$	0	0
$853$	22.6497	0.775510	0.387755	−	0.921762i	$-0.373251\pi$
$853$	22.6497	0.775510	0.387755	+	0.921762i	$0.373251\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39.4749	1.34844	0.674218	−	0.738533i	$-0.264481\pi$
$857$	39.4749	1.34844	0.674218	+	0.738533i	$0.264481\pi$
$858$	0	0
$859$	44.3379	1.51279	0.756395	−	0.654115i	$-0.226959\pi$
$859$	44.3379	1.51279	0.756395	+	0.654115i	$0.226959\pi$
$860$	0	0
$861$	34.6602	1.18122
$862$	0	0
$863$	5.03563	0.171415	0.0857074	−	0.996320i	$-0.472685\pi$
$863$	5.03563	0.171415	0.0857074	+	0.996320i	$0.472685\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−26.1696	−0.888765
$868$	0	0
$869$	−15.2765	−0.518220
$870$	0	0
$871$	−68.6613	−2.32650
$872$	0	0
$873$	5.07960	0.171918
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−17.4943	−0.590739	−0.295370	−	0.955383i	$-0.595443\pi$
$877$	−17.4943	−0.590739	−0.295370	+	0.955383i	$0.595443\pi$
$878$	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
$883$	−30.3544	−1.02151	−0.510753	−	0.859727i	$-0.670633\pi$
$883$	−30.3544	−1.02151	−0.510753	+	0.859727i	$0.670633\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−52.6472	−1.76772	−0.883860	−	0.467751i	$-0.845065\pi$
$887$	−52.6472	−1.76772	−0.883860	+	0.467751i	$0.845065\pi$
$888$	0	0
$889$	−49.9501	−1.67527
$890$	0	0
$891$	4.45181	0.149141
$892$	0	0
$893$	21.6935	0.725947
$894$	0	0
$895$	0	0
$896$	0	0
$897$	7.55803	0.252355
$898$	0	0
$899$	0.955163	0.0318565
$900$	0	0
$901$	−2.07265	−0.0690500
$902$	0	0
$903$	1.82264	0.0606537
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−8.79799	−0.292133	−0.146066	−	0.989275i	$-0.546661\pi$
$907$	−8.79799	−0.292133	−0.146066	+	0.989275i	$0.546661\pi$
$908$	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
$913$	23.9131	0.791409
$914$	0	0
$915$	0	0
$916$	0	0
$917$	41.7416	1.37843
$918$	0	0
$919$	−33.4799	−1.10440	−0.552200	−	0.833712i	$-0.686211\pi$
$919$	−33.4799	−1.10440	−0.552200	+	0.833712i	$0.686211\pi$
$920$	0	0
$921$	32.7301	1.07849
$922$	0	0
$923$	2.92704	0.0963448
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.14567	0.103317
$928$	0	0
$929$	−1.87841	−0.0616286	−0.0308143	−	0.999525i	$-0.509810\pi$
$929$	−1.87841	−0.0616286	−0.0308143	+	0.999525i	$0.509810\pi$
$930$	0	0
$931$	−34.7776	−1.13979
$932$	0	0
$933$	24.6245	0.806170
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.28820	0.0747523	0.0373762	−	0.999301i	$-0.488100\pi$
$937$	2.28820	0.0747523	0.0373762	+	0.999301i	$0.488100\pi$
$938$	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
$943$	−10.3221	−0.336135
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−43.2697	−1.40608	−0.703039	−	0.711152i	$-0.748174\pi$
$947$	−43.2697	−1.40608	−0.703039	+	0.711152i	$0.748174\pi$
$948$	0	0
$949$	−90.6473	−2.94253
$950$	0	0
$951$	1.62435	0.0526731
$952$	0	0
$953$	34.8553	1.12907	0.564537	−	0.825407i	$-0.309055\pi$
$953$	34.8553	1.12907	0.564537	+	0.825407i	$0.309055\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	21.5699	0.697257
$958$	0	0
$959$	37.1345	1.19914
$960$	0	0
$961$	−30.9611	−0.998746
$962$	0	0
$963$	7.74013	0.249422
$964$	0	0
$965$	0	0
$966$	0	0
$967$	34.9748	1.12471	0.562357	−	0.826895i	$-0.309895\pi$
$967$	34.9748	1.12471	0.562357	+	0.826895i	$0.309895\pi$
$968$	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
$973$	−52.1970	−1.67336
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−15.4162	−0.493207	−0.246603	−	0.969117i	$-0.579314\pi$
$977$	−15.4162	−0.493207	−0.246603	+	0.969117i	$0.579314\pi$
$978$	0	0
$979$	50.5065	1.61419
$980$	0	0
$981$	−10.3708	−0.331116
$982$	0	0
$983$	58.7500	1.87383	0.936917	−	0.349552i	$-0.113666\pi$
$983$	58.7500	1.87383	0.936917	+	0.349552i	$0.113666\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−34.4162	−1.09548
$988$	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
$993$	9.01314	0.286023
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−11.7966	−0.373603	−0.186802	−	0.982398i	$-0.559812\pi$
$997$	−11.7966	−0.373603	−0.186802	+	0.982398i	$0.559812\pi$
$998$	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.a.m.1.2		12
5.2	odd	4		7500.2.d.g.1249.14		24
5.3	odd	4		7500.2.d.g.1249.11		24
5.4	even	2		7500.2.a.n.1.11		12
25.2	odd	20		300.2.o.a.229.4	yes	24
25.9	even	10		1500.2.m.c.901.6		24
25.11	even	5		1500.2.m.d.601.1		24
25.12	odd	20		1500.2.o.c.349.3		24
25.13	odd	20		300.2.o.a.169.4	✓	24
25.14	even	10		1500.2.m.c.601.6		24
25.16	even	5		1500.2.m.d.901.1		24
25.23	odd	20		1500.2.o.c.649.3		24
75.2	even	20		900.2.w.c.829.5		24
75.38	even	20		900.2.w.c.469.5		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.o.a.169.4	✓	24	25.13	odd	20
300.2.o.a.229.4	yes	24	25.2	odd	20
900.2.w.c.469.5		24	75.38	even	20
900.2.w.c.829.5		24	75.2	even	20
1500.2.m.c.601.6		24	25.14	even	10
1500.2.m.c.901.6		24	25.9	even	10
1500.2.m.d.601.1		24	25.11	even	5
1500.2.m.d.901.1		24	25.16	even	5
1500.2.o.c.349.3		24	25.12	odd	20
1500.2.o.c.649.3		24	25.23	odd	20
7500.2.a.m.1.2		12	1.1	even	1	trivial
7500.2.a.n.1.11		12	5.4	even	2
7500.2.d.g.1249.11		24	5.3	odd	4
7500.2.d.g.1249.14		24	5.2	odd	4

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.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -4.41540 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -4.41540 q^{7} +1.00000 q^{9} +4.45181 q^{11} +5.74778 q^{13} -6.57035 q^{17} -2.78315 q^{19} +4.41540 q^{21} -1.31495 q^{23} -1.00000 q^{27} -4.84521 q^{29} -0.197136 q^{31} -4.45181 q^{33} +8.20866 q^{37} -5.74778 q^{39} +7.84984 q^{41} +0.412792 q^{43} -7.79459 q^{47} +12.4958 q^{49} +6.57035 q^{51} +0.315455 q^{53} +2.78315 q^{57} +2.51902 q^{59} +9.33114 q^{61} -4.41540 q^{63} -11.9457 q^{67} +1.31495 q^{69} +0.509248 q^{71} -15.7708 q^{73} -19.6565 q^{77} -3.43153 q^{79} +1.00000 q^{81} +5.37155 q^{83} +4.84521 q^{87} +11.3452 q^{89} -25.3787 q^{91} +0.197136 q^{93} +5.07960 q^{97} +4.45181 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} - 8 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 - 8 * q^7 + 12 * q^9	\( 12 q - 12 q^{3} - 8 q^{7} + 12 q^{9} + 2 q^{11} - 8 q^{17} + 10 q^{19} + 8 q^{21} - 18 q^{23} - 12 q^{27} + 8 q^{29} - 2 q^{31} - 2 q^{33} - 4 q^{37} + 10 q^{41} - 28 q^{43} - 22 q^{47} + 28 q^{49} + 8 q^{51} - 16 q^{53} - 10 q^{57} - 2 q^{59} + 34 q^{61} - 8 q^{63} - 32 q^{67} + 18 q^{69} - 24 q^{73} - 18 q^{77} + 6 q^{79} + 12 q^{81} - 28 q^{83} - 8 q^{87} + 10 q^{89} + 20 q^{91} + 2 q^{93} - 16 q^{97} + 2 q^{99}+O(q^{100}) \) 12 * q - 12 * q^3 - 8 * q^7 + 12 * q^9 + 2 * q^11 - 8 * q^17 + 10 * q^19 + 8 * q^21 - 18 * q^23 - 12 * q^27 + 8 * q^29 - 2 * q^31 - 2 * q^33 - 4 * q^37 + 10 * q^41 - 28 * q^43 - 22 * q^47 + 28 * q^49 + 8 * q^51 - 16 * q^53 - 10 * q^57 - 2 * q^59 + 34 * q^61 - 8 * q^63 - 32 * q^67 + 18 * q^69 - 24 * q^73 - 18 * q^77 + 6 * q^79 + 12 * q^81 - 28 * q^83 - 8 * q^87 + 10 * q^89 + 20 * q^91 + 2 * q^93 - 16 * q^97 + 2 * q^99

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