LMFDB - Embedded newform 7500.2.a.n.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7500,2,Mod(1,7500)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7500.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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:traces = [12,0,12,0,0,0,8,0,12,0,2,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$59.8878015160$
Analytic rank:	$0$
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			1.9
Root			$3.89742$ 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)

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

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
$7$	3.78808	1.43176	0.715880	−	0.698223i	$-0.246026\pi$
$7$	3.78808	1.43176	0.715880	+	0.698223i	$0.246026\pi$
$8$	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
$13$	4.74950	1.31727	0.658637	−	0.752461i	$-0.271133\pi$
$13$	4.74950	1.31727	0.658637	+	0.752461i	$0.271133\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.14884	−0.278636	−0.139318	−	0.990248i	$-0.544491\pi$
$17$	−1.14884	−0.278636	−0.139318	+	0.990248i	$0.544491\pi$
$18$	0	0
$19$	0.0150112	0.00344380	0.00172190	−	0.999999i	$-0.499452\pi$
$19$	0.0150112	0.00344380	0.00172190	+	0.999999i	$0.499452\pi$
$20$	0	0
$21$	3.78808	0.826627
$22$	0	0
$23$	−6.26797	−1.30696	−0.653481	−	0.756943i	$-0.726692\pi$
$23$	−6.26797	−1.30696	−0.653481	+	0.756943i	$0.726692\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.70092	0.687243	0.343621	−	0.939108i	$-0.388346\pi$
$29$	3.70092	0.687243	0.343621	+	0.939108i	$0.388346\pi$
$30$	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
$33$	−0.807679	−0.140599
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.54179	−1.40426	−0.702131	−	0.712048i	$-0.747768\pi$
$37$	−8.54179	−1.40426	−0.702131	+	0.712048i	$0.747768\pi$
$38$	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
$43$	10.2458	1.56247	0.781234	−	0.624238i	$-0.214591\pi$
$43$	10.2458	1.56247	0.781234	+	0.624238i	$0.214591\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.526282	−0.0767662	−0.0383831	−	0.999263i	$-0.512221\pi$
$47$	−0.526282	−0.0767662	−0.0383831	+	0.999263i	$0.512221\pi$
$48$	0	0
$49$	7.34957	1.04994
$50$	0	0
$51$	−1.14884	−0.160870
$52$	0	0
$53$	−2.94976	−0.405181	−0.202591	−	0.979264i	$-0.564936\pi$
$53$	−2.94976	−0.405181	−0.202591	+	0.979264i	$0.564936\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.0150112	0.00198828
$58$	0	0
$59$	11.4436	1.48982	0.744912	−	0.667163i	$-0.232492\pi$
$59$	11.4436	1.48982	0.744912	+	0.667163i	$0.232492\pi$
$60$	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
$63$	3.78808	0.477254
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.2812	1.62256	0.811278	−	0.584660i	$-0.198772\pi$
$67$	13.2812	1.62256	0.811278	+	0.584660i	$0.198772\pi$
$68$	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
$73$	−4.67860	−0.547589	−0.273795	−	0.961788i	$-0.588279\pi$
$73$	−4.67860	−0.547589	−0.273795	+	0.961788i	$0.588279\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.05955	−0.348668
$78$	0	0
$79$	9.27155	1.04313	0.521565	−	0.853211i	$-0.325348\pi$
$79$	9.27155	1.04313	0.521565	+	0.853211i	$0.325348\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.42370	0.156271	0.0781355	−	0.996943i	$-0.475103\pi$
$83$	1.42370	0.156271	0.0781355	+	0.996943i	$0.475103\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.70092	0.396780
$88$	0	0
$89$	−16.1067	−1.70731	−0.853655	−	0.520839i	$-0.825619\pi$
$89$	−16.1067	−1.70731	−0.853655	+	0.520839i	$0.825619\pi$
$90$	0	0
$91$	17.9915	1.88602
$92$	0	0
$93$	−1.58209	−0.164055
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.06203	0.818575	0.409287	−	0.912406i	$-0.365777\pi$
$97$	8.06203	0.818575	0.409287	+	0.912406i	$0.365777\pi$
$98$	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
$103$	18.8439	1.85674	0.928370	−	0.371657i	$-0.121210\pi$
$103$	18.8439	1.85674	0.928370	+	0.371657i	$0.121210\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.3957	−1.39168	−0.695842	−	0.718195i	$-0.744969\pi$
$107$	−14.3957	−1.39168	−0.695842	+	0.718195i	$0.744969\pi$
$108$	0	0
$109$	−5.76186	−0.551886	−0.275943	−	0.961174i	$-0.588990\pi$
$109$	−5.76186	−0.551886	−0.275943	+	0.961174i	$0.588990\pi$
$110$	0	0
$111$	−8.54179	−0.810751
$112$	0	0
$113$	17.9263	1.68637	0.843184	−	0.537624i	$-0.180678\pi$
$113$	17.9263	1.68637	0.843184	+	0.537624i	$0.180678\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.74950	0.439091
$118$	0	0
$119$	−4.35192	−0.398940
$120$	0	0
$121$	−10.3477	−0.940696
$122$	0	0
$123$	11.4989	1.03682
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2.28981	−0.203188	−0.101594	−	0.994826i	$-0.532394\pi$
$127$	−2.28981	−0.203188	−0.101594	+	0.994826i	$0.532394\pi$
$128$	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
$133$	0.0568635	0.00493069
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−21.9329	−1.87385	−0.936927	−	0.349525i	$-0.886343\pi$
$137$	−21.9329	−1.87385	−0.936927	+	0.349525i	$0.886343\pi$
$138$	0	0
$139$	9.71761	0.824237	0.412118	−	0.911130i	$-0.364789\pi$
$139$	9.71761	0.824237	0.412118	+	0.911130i	$0.364789\pi$
$140$	0	0
$141$	−0.526282	−0.0443210
$142$	0	0
$143$	−3.83607	−0.320788
$144$	0	0
$145$	0	0
$146$	0	0
$147$	7.34957	0.606182
$148$	0	0
$149$	−13.9712	−1.14457	−0.572284	−	0.820056i	$-0.693942\pi$
$149$	−13.9712	−1.14457	−0.572284	+	0.820056i	$0.693942\pi$
$150$	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
$153$	−1.14884	−0.0928786
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.76546	0.619751	0.309876	−	0.950777i	$-0.399713\pi$
$157$	7.76546	0.619751	0.309876	+	0.950777i	$0.399713\pi$
$158$	0	0
$159$	−2.94976	−0.233932
$160$	0	0
$161$	−23.7436	−1.87126
$162$	0	0
$163$	−13.8908	−1.08801	−0.544006	−	0.839082i	$-0.683093\pi$
$163$	−13.8908	−1.08801	−0.544006	+	0.839082i	$0.683093\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.89787	−0.224244	−0.112122	−	0.993694i	$-0.535765\pi$
$167$	−2.89787	−0.224244	−0.112122	+	0.993694i	$0.535765\pi$
$168$	0	0
$169$	9.55771	0.735209
$170$	0	0
$171$	0.0150112	0.00114793
$172$	0	0
$173$	15.2338	1.15821	0.579104	−	0.815254i	$-0.303403\pi$
$173$	15.2338	1.15821	0.579104	+	0.815254i	$0.303403\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	11.4436	0.860150
$178$	0	0
$179$	−23.3056	−1.74194	−0.870971	−	0.491335i	$-0.836509\pi$
$179$	−23.3056	−1.74194	−0.870971	+	0.491335i	$0.836509\pi$
$180$	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
$183$	3.14212	0.232272
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.927897	0.0678546
$188$	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
$193$	18.9309	1.36268	0.681338	−	0.731969i	$-0.261398\pi$
$193$	18.9309	1.36268	0.681338	+	0.731969i	$0.261398\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−7.54357	−0.537457	−0.268729	−	0.963216i	$-0.586603\pi$
$197$	−7.54357	−0.537457	−0.268729	+	0.963216i	$0.586603\pi$
$198$	0	0
$199$	3.58560	0.254176	0.127088	−	0.991891i	$-0.459437\pi$
$199$	3.58560	0.254176	0.127088	+	0.991891i	$0.459437\pi$
$200$	0	0
$201$	13.2812	0.936783
$202$	0	0
$203$	14.0194	0.983967
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−6.26797	−0.435654
$208$	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
$213$	−4.91318	−0.336646
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.99309	−0.406838
$218$	0	0
$219$	−4.67860	−0.316151
$220$	0	0
$221$	−5.45643	−0.367039
$222$	0	0
$223$	14.0373	0.940007	0.470003	−	0.882665i	$-0.344253\pi$
$223$	14.0373	0.940007	0.470003	+	0.882665i	$0.344253\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.53486	−0.168244	−0.0841221	−	0.996455i	$-0.526809\pi$
$227$	−2.53486	−0.168244	−0.0841221	+	0.996455i	$0.526809\pi$
$228$	0	0
$229$	−16.5262	−1.09208	−0.546041	−	0.837759i	$-0.683866\pi$
$229$	−16.5262	−1.09208	−0.546041	+	0.837759i	$0.683866\pi$
$230$	0	0
$231$	−3.05955	−0.201304
$232$	0	0
$233$	26.5208	1.73744	0.868719	−	0.495305i	$-0.164944\pi$
$233$	26.5208	1.73744	0.868719	+	0.495305i	$0.164944\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	9.27155	0.602252
$238$	0	0
$239$	−7.46385	−0.482797	−0.241398	−	0.970426i	$-0.577606\pi$
$239$	−7.46385	−0.482797	−0.241398	+	0.970426i	$0.577606\pi$
$240$	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
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.0712954	0.00453642
$248$	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
$253$	5.06250	0.318277
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.98030	−0.373041	−0.186520	−	0.982451i	$-0.559721\pi$
$257$	−5.98030	−0.373041	−0.186520	+	0.982451i	$0.559721\pi$
$258$	0	0
$259$	−32.3570	−2.01057
$260$	0	0
$261$	3.70092	0.229081
$262$	0	0
$263$	6.07038	0.374315	0.187158	−	0.982330i	$-0.440072\pi$
$263$	6.07038	0.374315	0.187158	+	0.982330i	$0.440072\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−16.1067	−0.985716
$268$	0	0
$269$	9.61605	0.586301	0.293150	−	0.956066i	$-0.405296\pi$
$269$	9.61605	0.586301	0.293150	+	0.956066i	$0.405296\pi$
$270$	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
$273$	17.9915	1.08889
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7.20393	−0.432842	−0.216421	−	0.976300i	$-0.569438\pi$
$277$	−7.20393	−0.432842	−0.216421	+	0.976300i	$0.569438\pi$
$278$	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
$283$	8.82402	0.524534	0.262267	−	0.964995i	$-0.415530\pi$
$283$	8.82402	0.524534	0.262267	+	0.964995i	$0.415530\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	43.5587	2.57119
$288$	0	0
$289$	−15.6802	−0.922362
$290$	0	0
$291$	8.06203	0.472604
$292$	0	0
$293$	−4.20743	−0.245800	−0.122900	−	0.992419i	$-0.539220\pi$
$293$	−4.20743	−0.245800	−0.122900	+	0.992419i	$0.539220\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−0.807679	−0.0468663
$298$	0	0
$299$	−29.7697	−1.72163
$300$	0	0
$301$	38.8119	2.23708
$302$	0	0
$303$	11.6496	0.669253
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.41109	0.251754	0.125877	−	0.992046i	$-0.459825\pi$
$307$	4.41109	0.251754	0.125877	+	0.992046i	$0.459825\pi$
$308$	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
$313$	−15.3245	−0.866193	−0.433097	−	0.901347i	$-0.642579\pi$
$313$	−15.3245	−0.866193	−0.433097	+	0.901347i	$0.642579\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.3155	−1.19720	−0.598600	−	0.801048i	$-0.704276\pi$
$317$	−21.3155	−1.19720	−0.598600	+	0.801048i	$0.704276\pi$
$318$	0	0
$319$	−2.98915	−0.167360
$320$	0	0
$321$	−14.3957	−0.803489
$322$	0	0
$323$	−0.0172455	−0.000959564 0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5.76186	−0.318631
$328$	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
$333$	−8.54179	−0.468087
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−10.0339	−0.546583	−0.273292	−	0.961931i	$-0.588112\pi$
$337$	−10.0339	−0.546583	−0.273292	+	0.961931i	$0.588112\pi$
$338$	0	0
$339$	17.9263	0.973626
$340$	0	0
$341$	1.27782	0.0691979
$342$	0	0
$343$	1.32419	0.0714997
$344$	0	0
$345$	0	0
$346$	0	0
$347$	13.8672	0.744429	0.372214	−	0.928147i	$-0.378599\pi$
$347$	13.8672	0.744429	0.372214	+	0.928147i	$0.378599\pi$
$348$	0	0
$349$	27.2533	1.45883	0.729417	−	0.684069i	$-0.239791\pi$
$349$	27.2533	1.45883	0.729417	+	0.684069i	$0.239791\pi$
$350$	0	0
$351$	4.74950	0.253509
$352$	0	0
$353$	−12.4138	−0.660722	−0.330361	−	0.943855i	$-0.607170\pi$
$353$	−12.4138	−0.660722	−0.330361	+	0.943855i	$0.607170\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.35192	−0.230328
$358$	0	0
$359$	−19.0244	−1.00407	−0.502034	−	0.864848i	$-0.667415\pi$
$359$	−19.0244	−1.00407	−0.502034	+	0.864848i	$0.667415\pi$
$360$	0	0
$361$	−18.9998	−0.999988
$362$	0	0
$363$	−10.3477	−0.543111
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0.975194	0.0509047	0.0254524	−	0.999676i	$-0.491897\pi$
$367$	0.975194	0.0509047	0.0254524	+	0.999676i	$0.491897\pi$
$368$	0	0
$369$	11.4989	0.598608
$370$	0	0
$371$	−11.1740	−0.580123
$372$	0	0
$373$	18.7723	0.971994	0.485997	−	0.873960i	$-0.338457\pi$
$373$	18.7723	0.971994	0.485997	+	0.873960i	$0.338457\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	17.5775	0.905287
$378$	0	0
$379$	−3.44756	−0.177089	−0.0885447	−	0.996072i	$-0.528222\pi$
$379$	−3.44756	−0.177089	−0.0885447	+	0.996072i	$0.528222\pi$
$380$	0	0
$381$	−2.28981	−0.117311
$382$	0	0
$383$	27.3045	1.39519	0.697597	−	0.716490i	$-0.254253\pi$
$383$	27.3045	1.39519	0.697597	+	0.716490i	$0.254253\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.2458	0.520823
$388$	0	0
$389$	9.81725	0.497754	0.248877	−	0.968535i	$-0.419938\pi$
$389$	9.81725	0.497754	0.248877	+	0.968535i	$0.419938\pi$
$390$	0	0
$391$	7.20092	0.364166
$392$	0	0
$393$	−7.04946	−0.355598
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−11.6131	−0.582846	−0.291423	−	0.956594i	$-0.594129\pi$
$397$	−11.6131	−0.582846	−0.291423	+	0.956594i	$0.594129\pi$
$398$	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
$403$	−7.51414	−0.374306
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.89902	0.341972
$408$	0	0
$409$	17.2515	0.853032	0.426516	−	0.904480i	$-0.359741\pi$
$409$	17.2515	0.853032	0.426516	+	0.904480i	$0.359741\pi$
$410$	0	0
$411$	−21.9329	−1.08187
$412$	0	0
$413$	43.3491	2.13307
$414$	0	0
$415$	0	0
$416$	0	0
$417$	9.71761	0.475873
$418$	0	0
$419$	−20.4505	−0.999074	−0.499537	−	0.866293i	$-0.666497\pi$
$419$	−20.4505	−0.999074	−0.499537	+	0.866293i	$0.666497\pi$
$420$	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
$423$	−0.526282	−0.0255887
$424$	0	0
$425$	0	0
$426$	0	0
$427$	11.9026	0.576008
$428$	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
$433$	−0.235240	−0.0113049	−0.00565246	−	0.999984i	$-0.501799\pi$
$433$	−0.235240	−0.0113049	−0.00565246	+	0.999984i	$0.501799\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.0940894	−0.00450091
$438$	0	0
$439$	31.2013	1.48916	0.744578	−	0.667536i	$-0.232651\pi$
$439$	31.2013	1.48916	0.744578	+	0.667536i	$0.232651\pi$
$440$	0	0
$441$	7.34957	0.349979
$442$	0	0
$443$	24.3862	1.15862	0.579311	−	0.815106i	$-0.303322\pi$
$443$	24.3862	1.15862	0.579311	+	0.815106i	$0.303322\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−13.9712	−0.660816
$448$	0	0
$449$	23.9483	1.13019	0.565096	−	0.825025i	$-0.308839\pi$
$449$	23.9483	1.13019	0.565096	+	0.825025i	$0.308839\pi$
$450$	0	0
$451$	−9.28740	−0.437327
$452$	0	0
$453$	−20.1871	−0.948471
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.91244	0.183016	0.0915082	−	0.995804i	$-0.470831\pi$
$457$	3.91244	0.183016	0.0915082	+	0.995804i	$0.470831\pi$
$458$	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
$463$	21.9089	1.01819	0.509096	−	0.860710i	$-0.329980\pi$
$463$	21.9089	1.01819	0.509096	+	0.860710i	$0.329980\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.19679	0.379302	0.189651	−	0.981852i	$-0.439264\pi$
$467$	8.19679	0.379302	0.189651	+	0.981852i	$0.439264\pi$
$468$	0	0
$469$	50.3103	2.32311
$470$	0	0
$471$	7.76546	0.357814
$472$	0	0
$473$	−8.27531	−0.380499
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.94976	−0.135060
$478$	0	0
$479$	−6.15176	−0.281081	−0.140540	−	0.990075i	$-0.544884\pi$
$479$	−6.15176	−0.281081	−0.140540	+	0.990075i	$0.544884\pi$
$480$	0	0
$481$	−40.5692	−1.84980
$482$	0	0
$483$	−23.7436	−1.08037
$484$	0	0
$485$	0	0
$486$	0	0
$487$	6.28171	0.284651	0.142326	−	0.989820i	$-0.454542\pi$
$487$	6.28171	0.284651	0.142326	+	0.989820i	$0.454542\pi$
$488$	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
$493$	−4.25178	−0.191490
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−18.6115	−0.834842
$498$	0	0
$499$	−16.4263	−0.735341	−0.367670	−	0.929956i	$-0.619844\pi$
$499$	−16.4263	−0.735341	−0.367670	+	0.929956i	$0.619844\pi$
$500$	0	0
$501$	−2.89787	−0.129467
$502$	0	0
$503$	−12.2874	−0.547868	−0.273934	−	0.961749i	$-0.588325\pi$
$503$	−12.2874	−0.547868	−0.273934	+	0.961749i	$0.588325\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9.55771	0.424473
$508$	0	0
$509$	−10.5100	−0.465848	−0.232924	−	0.972495i	$-0.574829\pi$
$509$	−10.5100	−0.465848	−0.232924	+	0.972495i	$0.574829\pi$
$510$	0	0
$511$	−17.7229	−0.784017
$512$	0	0
$513$	0.0150112	0.000662759 0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0.425067	0.0186944
$518$	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
$523$	22.0475	0.964068	0.482034	−	0.876153i	$-0.339898\pi$
$523$	22.0475	0.964068	0.482034	+	0.876153i	$0.339898\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.81758	0.0791749
$528$	0	0
$529$	16.2874	0.708148
$530$	0	0
$531$	11.4436	0.496608
$532$	0	0
$533$	54.6139	2.36559
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−23.3056	−1.00571
$538$	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
$543$	17.6985	0.759514
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.51606	0.107579	0.0537895	−	0.998552i	$-0.482870\pi$
$547$	2.51606	0.107579	0.0537895	+	0.998552i	$0.482870\pi$
$548$	0	0
$549$	3.14212	0.134102
$550$	0	0
$551$	0.0555550	0.00236672
$552$	0	0
$553$	35.1214	1.49351
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−0.262544	−0.0111244	−0.00556218	−	0.999985i	$-0.501771\pi$
$557$	−0.262544	−0.0111244	−0.00556218	+	0.999985i	$0.501771\pi$
$558$	0	0
$559$	48.6623	2.05820
$560$	0	0
$561$	0.927897	0.0391759
$562$	0	0
$563$	32.4826	1.36898	0.684489	−	0.729023i	$-0.260025\pi$
$563$	32.4826	1.36898	0.684489	+	0.729023i	$0.260025\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.78808	0.159085
$568$	0	0
$569$	−38.7980	−1.62650	−0.813248	−	0.581917i	$-0.802303\pi$
$569$	−38.7980	−1.62650	−0.813248	+	0.581917i	$0.802303\pi$
$570$	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
$573$	9.36414	0.391193
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−25.3430	−1.05504	−0.527522	−	0.849542i	$-0.676879\pi$
$577$	−25.3430	−1.05504	−0.527522	+	0.849542i	$0.676879\pi$
$578$	0	0
$579$	18.9309	0.786741
$580$	0	0
$581$	5.39308	0.223743
$582$	0	0
$583$	2.38246	0.0986715
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0.515670	0.0212840	0.0106420	−	0.999943i	$-0.496612\pi$
$587$	0.515670	0.0212840	0.0106420	+	0.999943i	$0.496612\pi$
$588$	0	0
$589$	−0.0237490	−0.000978561 0
$590$	0	0
$591$	−7.54357	−0.310301
$592$	0	0
$593$	7.14389	0.293364	0.146682	−	0.989184i	$-0.453141\pi$
$593$	7.14389	0.293364	0.146682	+	0.989184i	$0.453141\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.58560	0.146749
$598$	0	0
$599$	23.6627	0.966833	0.483417	−	0.875390i	$-0.339396\pi$
$599$	23.6627	0.966833	0.483417	+	0.875390i	$0.339396\pi$
$600$	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
$603$	13.2812	0.540852
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−17.2004	−0.698144	−0.349072	−	0.937096i	$-0.613503\pi$
$607$	−17.2004	−0.698144	−0.349072	+	0.937096i	$0.613503\pi$
$608$	0	0
$609$	14.0194	0.568094
$610$	0	0
$611$	−2.49958	−0.101122
$612$	0	0
$613$	14.7641	0.596315	0.298158	−	0.954517i	$-0.403628\pi$
$613$	14.7641	0.596315	0.298158	+	0.954517i	$0.403628\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15.4714	0.622854	0.311427	−	0.950270i	$-0.399193\pi$
$617$	15.4714	0.622854	0.311427	+	0.950270i	$0.399193\pi$
$618$	0	0
$619$	−37.4710	−1.50608	−0.753042	−	0.657972i	$-0.771414\pi$
$619$	−37.4710	−1.50608	−0.753042	+	0.657972i	$0.771414\pi$
$620$	0	0
$621$	−6.26797	−0.251525
$622$	0	0
$623$	−61.0136	−2.44446
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−0.0121242	−0.000484194 0
$628$	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
$633$	2.08493	0.0828685
$634$	0	0
$635$	0	0
$636$	0	0
$637$	34.9067	1.38306
$638$	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
$643$	23.2212	0.915756	0.457878	−	0.889015i	$-0.348610\pi$
$643$	23.2212	0.915756	0.457878	+	0.889015i	$0.348610\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26.1598	1.02845	0.514225	−	0.857655i	$-0.328080\pi$
$647$	26.1598	1.02845	0.514225	+	0.857655i	$0.328080\pi$
$648$	0	0
$649$	−9.24271	−0.362808
$650$	0	0
$651$	−5.99309	−0.234888
$652$	0	0
$653$	−8.55471	−0.334772	−0.167386	−	0.985891i	$-0.553533\pi$
$653$	−8.55471	−0.334772	−0.167386	+	0.985891i	$0.553533\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−4.67860	−0.182530
$658$	0	0
$659$	−41.9383	−1.63368	−0.816841	−	0.576862i	$-0.804277\pi$
$659$	−41.9383	−1.63368	−0.816841	+	0.576862i	$0.804277\pi$
$660$	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
$663$	−5.45643	−0.211910
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−23.1972	−0.898200
$668$	0	0
$669$	14.0373	0.542713
$670$	0	0
$671$	−2.53782	−0.0979716
$672$	0	0
$673$	−29.0952	−1.12154	−0.560769	−	0.827972i	$-0.689494\pi$
$673$	−29.0952	−1.12154	−0.560769	+	0.827972i	$0.689494\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−29.8863	−1.14862	−0.574311	−	0.818637i	$-0.694730\pi$
$677$	−29.8863	−1.14862	−0.574311	+	0.818637i	$0.694730\pi$
$678$	0	0
$679$	30.5396	1.17200
$680$	0	0
$681$	−2.53486	−0.0971359
$682$	0	0
$683$	−37.2200	−1.42419	−0.712093	−	0.702086i	$-0.752252\pi$
$683$	−37.2200	−1.42419	−0.712093	+	0.702086i	$0.752252\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−16.5262	−0.630514
$688$	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
$693$	−3.05955	−0.116223
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−13.2104	−0.500380
$698$	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
$703$	−0.128222	−0.00483599
$704$	0	0
$705$	0	0
$706$	0	0
$707$	44.1297	1.65967
$708$	0	0
$709$	19.0378	0.714978	0.357489	−	0.933917i	$-0.383633\pi$
$709$	19.0378	0.714978	0.357489	+	0.933917i	$0.383633\pi$
$710$	0	0
$711$	9.27155	0.347710
$712$	0	0
$713$	9.91650	0.371376
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−7.46385	−0.278743
$718$	0	0
$719$	−14.7399	−0.549706	−0.274853	−	0.961486i	$-0.588629\pi$
$719$	−14.7399	−0.549706	−0.274853	+	0.961486i	$0.588629\pi$
$720$	0	0
$721$	71.3821	2.65841
$722$	0	0
$723$	10.2978	0.382978
$724$	0	0
$725$	0	0
$726$	0	0
$727$	30.0617	1.11492	0.557462	−	0.830202i	$-0.311775\pi$
$727$	30.0617	1.11492	0.557462	+	0.830202i	$0.311775\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−11.7708	−0.435359
$732$	0	0
$733$	−22.6438	−0.836368	−0.418184	−	0.908362i	$-0.637333\pi$
$733$	−22.6438	−0.836368	−0.418184	+	0.908362i	$0.637333\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.7269	−0.395132
$738$	0	0
$739$	−25.6810	−0.944692	−0.472346	−	0.881413i	$-0.656593\pi$
$739$	−25.6810	−0.944692	−0.472346	+	0.881413i	$0.656593\pi$
$740$	0	0
$741$	0.0712954	0.00261910
$742$	0	0
$743$	−24.9796	−0.916411	−0.458205	−	0.888846i	$-0.651508\pi$
$743$	−24.9796	−0.916411	−0.458205	+	0.888846i	$0.651508\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.42370	0.0520904
$748$	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
$753$	19.5809	0.713567
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−9.91474	−0.360357	−0.180179	−	0.983634i	$-0.557668\pi$
$757$	−9.91474	−0.360357	−0.180179	+	0.983634i	$0.557668\pi$
$758$	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
$763$	−21.8264	−0.790168
$764$	0	0
$765$	0	0
$766$	0	0
$767$	54.3511	1.96250
$768$	0	0
$769$	22.3751	0.806868	0.403434	−	0.915009i	$-0.367816\pi$
$769$	22.3751	0.806868	0.403434	+	0.915009i	$0.367816\pi$
$770$	0	0
$771$	−5.98030	−0.215375
$772$	0	0
$773$	−47.8317	−1.72039	−0.860194	−	0.509968i	$-0.829657\pi$
$773$	−47.8317	−1.72039	−0.860194	+	0.509968i	$0.829657\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−32.3570	−1.16080
$778$	0	0
$779$	0.172611	0.00618445
$780$	0	0
$781$	3.96827	0.141996
$782$	0	0
$783$	3.70092	0.132260
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−27.5675	−0.982677	−0.491338	−	0.870969i	$-0.663492\pi$
$787$	−27.5675	−0.982677	−0.491338	+	0.870969i	$0.663492\pi$
$788$	0	0
$789$	6.07038	0.216111
$790$	0	0
$791$	67.9065	2.41448
$792$	0	0
$793$	14.9235	0.529948
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−23.1019	−0.818311	−0.409155	−	0.912465i	$-0.634177\pi$
$797$	−23.1019	−0.818311	−0.409155	+	0.912465i	$0.634177\pi$
$798$	0	0
$799$	0.604617	0.0213898
$800$	0	0
$801$	−16.1067	−0.569103
$802$	0	0
$803$	3.77881	0.133351
$804$	0	0
$805$	0	0
$806$	0	0
$807$	9.61605	0.338501
$808$	0	0
$809$	−1.69292	−0.0595198	−0.0297599	−	0.999557i	$-0.509474\pi$
$809$	−1.69292	−0.0595198	−0.0297599	+	0.999557i	$0.509474\pi$
$810$	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
$813$	−30.4276	−1.06714
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0.153801	0.00538082
$818$	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
$823$	55.9867	1.95157	0.975787	−	0.218721i	$-0.0701884\pi$
$823$	55.9867	1.95157	0.975787	+	0.218721i	$0.0701884\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.6399	0.717721	0.358860	−	0.933391i	$-0.383165\pi$
$827$	20.6399	0.717721	0.358860	+	0.933391i	$0.383165\pi$
$828$	0	0
$829$	−1.93898	−0.0673437	−0.0336718	−	0.999433i	$-0.510720\pi$
$829$	−1.93898	−0.0673437	−0.0336718	+	0.999433i	$0.510720\pi$
$830$	0	0
$831$	−7.20393	−0.249901
$832$	0	0
$833$	−8.44351	−0.292550
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.58209	−0.0546851
$838$	0	0
$839$	52.2589	1.80418	0.902089	−	0.431551i	$-0.142034\pi$
$839$	52.2589	1.80418	0.902089	+	0.431551i	$0.142034\pi$
$840$	0	0
$841$	−15.3032	−0.527697
$842$	0	0
$843$	29.7964	1.02624
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−39.1978	−1.34685
$848$	0	0
$849$	8.82402	0.302840
$850$	0	0
$851$	53.5397	1.83532
$852$	0	0
$853$	−17.5605	−0.601260	−0.300630	−	0.953741i	$-0.597197\pi$
$853$	−17.5605	−0.601260	−0.300630	+	0.953741i	$0.597197\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	50.9890	1.74175	0.870876	−	0.491503i	$-0.163552\pi$
$857$	50.9890	1.74175	0.870876	+	0.491503i	$0.163552\pi$
$858$	0	0
$859$	18.6336	0.635770	0.317885	−	0.948129i	$-0.397027\pi$
$859$	18.6336	0.635770	0.317885	+	0.948129i	$0.397027\pi$
$860$	0	0
$861$	43.5587	1.48448
$862$	0	0
$863$	0.292141	0.00994461	0.00497230	−	0.999988i	$-0.498417\pi$
$863$	0.292141	0.00994461	0.00497230	+	0.999988i	$0.498417\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−15.6802	−0.532526
$868$	0	0
$869$	−7.48843	−0.254028
$870$	0	0
$871$	63.0790	2.13735
$872$	0	0
$873$	8.06203	0.272858
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−37.9604	−1.28183	−0.640916	−	0.767611i	$-0.721445\pi$
$877$	−37.9604	−1.28183	−0.640916	+	0.767611i	$0.721445\pi$
$878$	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
$883$	−22.6199	−0.761220	−0.380610	−	0.924736i	$-0.624286\pi$
$883$	−22.6199	−0.761220	−0.380610	+	0.924736i	$0.624286\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24.4380	−0.820548	−0.410274	−	0.911962i	$-0.634567\pi$
$887$	−24.4380	−0.820548	−0.410274	+	0.911962i	$0.634567\pi$
$888$	0	0
$889$	−8.67399	−0.290916
$890$	0	0
$891$	−0.807679	−0.0270583
$892$	0	0
$893$	−0.00790011	−0.000264367 0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−29.7697	−0.993981
$898$	0	0
$899$	−5.85519	−0.195281
$900$	0	0
$901$	3.38882	0.112898
$902$	0	0
$903$	38.8119	1.29158
$904$	0	0
$905$	0	0
$906$	0	0
$907$	32.7792	1.08842	0.544208	−	0.838950i	$-0.316830\pi$
$907$	32.7792	1.08842	0.544208	+	0.838950i	$0.316830\pi$
$908$	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
$913$	−1.14989	−0.0380558
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−26.7039	−0.881842
$918$	0	0
$919$	−41.7202	−1.37622	−0.688112	−	0.725604i	$-0.741560\pi$
$919$	−41.7202	−1.37622	−0.688112	+	0.725604i	$0.741560\pi$
$920$	0	0
$921$	4.41109	0.145350
$922$	0	0
$923$	−23.3351	−0.768086
$924$	0	0
$925$	0	0
$926$	0	0
$927$	18.8439	0.618913
$928$	0	0
$929$	−33.7038	−1.10579	−0.552893	−	0.833252i	$-0.686476\pi$
$929$	−33.7038	−1.10579	−0.552893	+	0.833252i	$0.686476\pi$
$930$	0	0
$931$	0.110326	0.00361577
$932$	0	0
$933$	−6.32058	−0.206927
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−1.04606	−0.0341732	−0.0170866	−	0.999854i	$-0.505439\pi$
$937$	−1.04606	−0.0341732	−0.0170866	+	0.999854i	$0.505439\pi$
$938$	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
$943$	−72.0746	−2.34707
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12.5948	0.409275	0.204637	−	0.978838i	$-0.434398\pi$
$947$	12.5948	0.409275	0.204637	+	0.978838i	$0.434398\pi$
$948$	0	0
$949$	−22.2210	−0.721325
$950$	0	0
$951$	−21.3155	−0.691204
$952$	0	0
$953$	57.3902	1.85905	0.929526	−	0.368757i	$-0.120217\pi$
$953$	57.3902	1.85905	0.929526	+	0.368757i	$0.120217\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.98915	−0.0966255
$958$	0	0
$959$	−83.0836	−2.68291
$960$	0	0
$961$	−28.4970	−0.919258
$962$	0	0
$963$	−14.3957	−0.463895
$964$	0	0
$965$	0	0
$966$	0	0
$967$	8.84212	0.284343	0.142172	−	0.989842i	$-0.454591\pi$
$967$	8.84212	0.284343	0.142172	+	0.989842i	$0.454591\pi$
$968$	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
$973$	36.8111	1.18011
$974$	0	0
$975$	0	0
$976$	0	0
$977$	37.2345	1.19124	0.595619	−	0.803267i	$-0.296907\pi$
$977$	37.2345	1.19124	0.595619	+	0.803267i	$0.296907\pi$
$978$	0	0
$979$	13.0091	0.415772
$980$	0	0
$981$	−5.76186	−0.183962
$982$	0	0
$983$	35.7610	1.14060	0.570299	−	0.821437i	$-0.306827\pi$
$983$	35.7610	1.14060	0.570299	+	0.821437i	$0.306827\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.99360	−0.0634570
$988$	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
$993$	19.8124	0.628728
$994$	0	0
$995$	0	0
$996$	0	0
$997$	39.5083	1.25124	0.625620	−	0.780128i	$-0.284846\pi$
$997$	39.5083	1.25124	0.625620	+	0.780128i	$0.284846\pi$
$998$	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.a.n.1.9		12
5.2	odd	4		7500.2.d.g.1249.9		24
5.3	odd	4		7500.2.d.g.1249.16		24
5.4	even	2		7500.2.a.m.1.4		12
25.3	odd	20		1500.2.o.c.49.1		24
25.4	even	10		1500.2.m.d.1201.2		24
25.6	even	5		1500.2.m.c.301.5		24
25.8	odd	20		300.2.o.a.289.5	yes	24
25.17	odd	20		1500.2.o.c.949.1		24
25.19	even	10		1500.2.m.d.301.2		24
25.21	even	5		1500.2.m.c.1201.5		24
25.22	odd	20		300.2.o.a.109.5	✓	24
75.8	even	20		900.2.w.c.289.4		24
75.47	even	20		900.2.w.c.109.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.o.a.109.5	✓	24	25.22	odd	20
300.2.o.a.289.5	yes	24	25.8	odd	20
900.2.w.c.109.4		24	75.47	even	20
900.2.w.c.289.4		24	75.8	even	20
1500.2.m.c.301.5		24	25.6	even	5
1500.2.m.c.1201.5		24	25.21	even	5
1500.2.m.d.301.2		24	25.19	even	10
1500.2.m.d.1201.2		24	25.4	even	10
1500.2.o.c.49.1		24	25.3	odd	20
1500.2.o.c.949.1		24	25.17	odd	20
7500.2.a.m.1.4		12	5.4	even	2
7500.2.a.n.1.9		12	1.1	even	1	trivial
7500.2.d.g.1249.9		24	5.2	odd	4
7500.2.d.g.1249.16		24	5.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.78808 q^{7} +1.00000 q^{9} -0.807679 q^{11} +4.74950 q^{13} -1.14884 q^{17} +0.0150112 q^{19} +3.78808 q^{21} -6.26797 q^{23} +1.00000 q^{27} +3.70092 q^{29} -1.58209 q^{31} -0.807679 q^{33} -8.54179 q^{37} +4.74950 q^{39} +11.4989 q^{41} +10.2458 q^{43} -0.526282 q^{47} +7.34957 q^{49} -1.14884 q^{51} -2.94976 q^{53} +0.0150112 q^{57} +11.4436 q^{59} +3.14212 q^{61} +3.78808 q^{63} +13.2812 q^{67} -6.26797 q^{69} -4.91318 q^{71} -4.67860 q^{73} -3.05955 q^{77} +9.27155 q^{79} +1.00000 q^{81} +1.42370 q^{83} +3.70092 q^{87} -16.1067 q^{89} +17.9915 q^{91} -1.58209 q^{93} +8.06203 q^{97} -0.807679 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 8 q^{7} + 12 q^{9} + 2 q^{11} + 8 q^{17} + 10 q^{19} + 8 q^{21} + 18 q^{23} + 12 q^{27} + 8 q^{29} - 2 q^{31} + 2 q^{33} + 4 q^{37} + 10 q^{41} + 28 q^{43} + 22 q^{47} + 28 q^{49} + 8 q^{51}+ \cdots + 2 q^{99}+O(q^{100}) \) 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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