LMFDB - Embedded newform 2400.3.j.d.799.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2400,3,Mod(799,2400)] 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("2400.799"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2400.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,-16,0,24,0,0,0,0,0,0,0,0,0,0,0,96] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	no
Analytic conductor:	$65.3952634465$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12960000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 $ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.6
Root			$-1.40126 + 0.809017i$ of defining polynomial
Character	$\chi$	$=$	2400.799
Dual form			2400.3.j.d.799.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} +0.456067 q^{7} +3.00000 q^{9} +12.6624i q^{11} +11.2261i q^{13} -33.6463i q^{17} +22.7219i q^{19} +0.789932 q^{21} +27.1581 q^{23} +5.19615 q^{27} -1.41641 q^{29} -39.9748i q^{31} +21.9319i q^{33} +45.4952i q^{37} +19.4442i q^{39} +48.6355 q^{41} -18.8961 q^{43} +4.48288 q^{47} -48.7920 q^{49} -58.2771i q^{51} +42.6769i q^{53} +39.3554i q^{57} +67.0825i q^{59} +11.2605 q^{61} +1.36820 q^{63} -23.6033 q^{67} +47.0392 q^{69} +98.9583i q^{71} +31.9784i q^{73} +5.77491i q^{77} +68.1003i q^{79} +9.00000 q^{81} -67.5883 q^{83} -2.45329 q^{87} -167.081 q^{89} +5.11987i q^{91} -69.2384i q^{93} -10.5744i q^{97} +37.9872i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{7} + 24 q^{9} + 96 q^{21} - 16 q^{23} + 96 q^{29} + 112 q^{41} - 128 q^{43} - 272 q^{47} + 184 q^{49} + 80 q^{61} - 48 q^{63} + 320 q^{67} + 144 q^{69} + 72 q^{81} - 320 q^{83} + 48 q^{89}+O(q^{100}) $ 8 * q - 16 * q^7 + 24 * q^9 + 96 * q^21 - 16 * q^23 + 96 * q^29 + 112 * q^41 - 128 * q^43 - 272 * q^47 + 184 * q^49 + 80 * q^61 - 48 * q^63 + 320 * q^67 + 144 * q^69 + 72 * q^81 - 320 * q^83 + 48 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$3$	1.73205	0.577350
$3$	1.73205	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.456067	0.0651525	0.0325762	−	0.999469i	$-0.489629\pi$
$7$	0.456067	0.0651525	0.0325762	+	0.999469i	$0.489629\pi$
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	12.6624i	1.15113i	0.817757	+	0.575564i	$0.195217\pi$
$11$	12.6624i	1.15113i	−0.817757	+	0.575564i	$0.804783\pi$
$12$	0	0
$13$	11.2261i	0.863549i	0.901982	+	0.431774i	$0.142112\pi$
$13$	11.2261i	0.863549i	−0.901982	+	0.431774i	$0.857888\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 33.6463i	− 1.97919i	−0.143872	−	0.989596i	$-0.545955\pi$
$17$	− 33.6463i	− 1.97919i	0.143872	−	0.989596i	$-0.454045\pi$
$18$	0	0
$19$	22.7219i	1.19589i	0.801538	+	0.597944i	$0.204015\pi$
$19$	22.7219i	1.19589i	−0.801538	+	0.597944i	$0.795985\pi$
$20$	0	0
$21$	0.789932	0.0376158
$22$	0	0
$23$	27.1581	1.18079	0.590393	−	0.807116i	$-0.298973\pi$
$23$	27.1581	1.18079	0.590393	+	0.807116i	$0.298973\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.19615	0.192450
$28$	0	0
$29$	−1.41641	−0.0488417	−0.0244208	−	0.999702i	$-0.507774\pi$
$29$	−1.41641	−0.0488417	−0.0244208	+	0.999702i	$0.507774\pi$
$30$	0	0
$31$	− 39.9748i	− 1.28951i	−0.764389	−	0.644755i	$-0.776959\pi$
$31$	− 39.9748i	− 1.28951i	0.764389	−	0.644755i	$-0.223041\pi$
$32$	0	0
$33$	21.9319i	0.664604i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	45.4952i	1.22960i	0.788683	+	0.614800i	$0.210763\pi$
$37$	45.4952i	1.22960i	−0.788683	+	0.614800i	$0.789237\pi$
$38$	0	0
$39$	19.4442i	0.498570i
$40$	0	0
$41$	48.6355	1.18623	0.593115	−	0.805117i	$-0.297898\pi$
$41$	48.6355	1.18623	0.593115	+	0.805117i	$0.297898\pi$
$42$	0	0
$43$	−18.8961	−0.439443	−0.219722	−	0.975563i	$-0.570515\pi$
$43$	−18.8961	−0.439443	−0.219722	+	0.975563i	$0.570515\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.48288	0.0953805	0.0476902	−	0.998862i	$-0.484814\pi$
$47$	4.48288	0.0953805	0.0476902	+	0.998862i	$0.484814\pi$
$48$	0	0
$49$	−48.7920	−0.995755
$50$	0	0
$51$	− 58.2771i	− 1.14269i
$52$	0	0
$53$	42.6769i	0.805225i	0.915370	+	0.402613i	$0.131898\pi$
$53$	42.6769i	0.805225i	−0.915370	+	0.402613i	$0.868102\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	39.3554i	0.690446i
$58$	0	0
$59$	67.0825i	1.13699i	0.822686	+	0.568496i	$0.192475\pi$
$59$	67.0825i	1.13699i	−0.822686	+	0.568496i	$0.807525\pi$
$60$	0	0
$61$	11.2605	0.184599	0.0922995	−	0.995731i	$-0.470578\pi$
$61$	11.2605	0.184599	0.0922995	+	0.995731i	$0.470578\pi$
$62$	0	0
$63$	1.36820	0.0217175
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−23.6033	−0.352287	−0.176144	−	0.984364i	$-0.556362\pi$
$67$	−23.6033	−0.352287	−0.176144	+	0.984364i	$0.556362\pi$
$68$	0	0
$69$	47.0392	0.681727
$70$	0	0
$71$	98.9583i	1.39378i	0.717179	+	0.696889i	$0.245433\pi$
$71$	98.9583i	1.39378i	−0.717179	+	0.696889i	$0.754567\pi$
$72$	0	0
$73$	31.9784i	0.438060i	0.975718	+	0.219030i	$0.0702893\pi$
$73$	31.9784i	0.438060i	−0.975718	+	0.219030i	$0.929711\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.77491i	0.0749988i
$78$	0	0
$79$	68.1003i	0.862029i	0.902345	+	0.431015i	$0.141844\pi$
$79$	68.1003i	0.862029i	−0.902345	+	0.431015i	$0.858156\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	−67.5883	−0.814317	−0.407159	−	0.913357i	$-0.633480\pi$
$83$	−67.5883	−0.814317	−0.407159	+	0.913357i	$0.633480\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.45329	−0.0281987
$88$	0	0
$89$	−167.081	−1.87731	−0.938655	−	0.344857i	$-0.887927\pi$
$89$	−167.081	−1.87731	−0.938655	+	0.344857i	$0.887927\pi$
$90$	0	0
$91$	5.11987i	0.0562623i
$92$	0	0
$93$	− 69.2384i	− 0.744499i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.5744i	− 0.109014i	−0.998513	−	0.0545071i	$-0.982641\pi$
$97$	− 10.5744i	− 0.109014i	0.998513	−	0.0545071i	$-0.0173587\pi$
$98$	0	0
$99$	37.9872i	0.383709i
$100$	0	0
$101$	−10.1097	−0.100096	−0.0500480	−	0.998747i	$-0.515937\pi$
$101$	−10.1097	−0.100096	−0.0500480	+	0.998747i	$0.515937\pi$
$102$	0	0
$103$	81.2417	0.788754	0.394377	−	0.918949i	$-0.370960\pi$
$103$	81.2417	0.788754	0.394377	+	0.918949i	$0.370960\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	154.027	1.43950	0.719752	−	0.694231i	$-0.244256\pi$
$107$	154.027	1.43950	0.719752	+	0.694231i	$0.244256\pi$
$108$	0	0
$109$	73.2784	0.672279	0.336139	−	0.941812i	$-0.390879\pi$
$109$	73.2784	0.672279	0.336139	+	0.941812i	$0.390879\pi$
$110$	0	0
$111$	78.8000i	0.709910i
$112$	0	0
$113$	138.985i	1.22996i	0.788544	+	0.614978i	$0.210835\pi$
$113$	138.985i	1.22996i	−0.788544	+	0.614978i	$0.789165\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	33.6784i	0.287850i
$118$	0	0
$119$	− 15.3450i	− 0.128949i
$120$	0	0
$121$	−39.3365	−0.325095
$122$	0	0
$123$	84.2391	0.684871
$124$	0	0
$125$	0	0
$126$	0	0
$127$	214.913	1.69223	0.846113	−	0.533004i	$-0.178937\pi$
$127$	214.913	1.69223	0.846113	+	0.533004i	$0.178937\pi$
$128$	0	0
$129$	−32.7289	−0.253713
$130$	0	0
$131$	− 64.2819i	− 0.490701i	−0.969434	−	0.245351i	$-0.921097\pi$
$131$	− 64.2819i	− 0.490701i	0.969434	−	0.245351i	$-0.0789031\pi$
$132$	0	0
$133$	10.3627i	0.0779150i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	180.562i	1.31797i	0.752156	+	0.658985i	$0.229014\pi$
$137$	180.562i	1.31797i	−0.752156	+	0.658985i	$0.770986\pi$
$138$	0	0
$139$	− 68.0034i	− 0.489233i	−0.969620	−	0.244617i	$-0.921338\pi$
$139$	− 68.0034i	− 0.489233i	0.969620	−	0.244617i	$-0.0786621\pi$
$140$	0	0
$141$	7.76458	0.0550679
$142$	0	0
$143$	−142.150	−0.994055
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−84.5102	−0.574900
$148$	0	0
$149$	85.1441	0.571437	0.285719	−	0.958314i	$-0.407768\pi$
$149$	85.1441	0.571437	0.285719	+	0.958314i	$0.407768\pi$
$150$	0	0
$151$	− 150.599i	− 0.997344i	−0.866791	−	0.498672i	$-0.833821\pi$
$151$	− 150.599i	− 0.997344i	0.866791	−	0.498672i	$-0.166179\pi$
$152$	0	0
$153$	− 100.939i	− 0.659731i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	205.592i	1.30950i	0.755845	+	0.654751i	$0.227227\pi$
$157$	205.592i	1.30950i	−0.755845	+	0.654751i	$0.772773\pi$
$158$	0	0
$159$	73.9186i	0.464897i
$160$	0	0
$161$	12.3859	0.0769311
$162$	0	0
$163$	22.0922	0.135535	0.0677675	−	0.997701i	$-0.478412\pi$
$163$	22.0922	0.135535	0.0677675	+	0.997701i	$0.478412\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	263.402	1.57726	0.788629	−	0.614870i	$-0.210792\pi$
$167$	263.402	1.57726	0.788629	+	0.614870i	$0.210792\pi$
$168$	0	0
$169$	42.9738	0.254283
$170$	0	0
$171$	68.1656i	0.398629i
$172$	0	0
$173$	136.479i	0.788895i	0.918918	+	0.394448i	$0.129064\pi$
$173$	136.479i	0.788895i	−0.918918	+	0.394448i	$0.870936\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	116.190i	0.656443i
$178$	0	0
$179$	64.7130i	0.361525i	0.983527	+	0.180763i	$0.0578565\pi$
$179$	64.7130i	0.361525i	−0.983527	+	0.180763i	$0.942143\pi$
$180$	0	0
$181$	−184.848	−1.02126	−0.510629	−	0.859801i	$-0.670587\pi$
$181$	−184.848	−1.02126	−0.510629	+	0.859801i	$0.670587\pi$
$182$	0	0
$183$	19.5038	0.106578
$184$	0	0
$185$	0	0
$186$	0	0
$187$	426.043	2.27830
$188$	0	0
$189$	2.36980	0.0125386
$190$	0	0
$191$	145.803i	0.763364i	0.924294	+	0.381682i	$0.124655\pi$
$191$	145.803i	0.763364i	−0.924294	+	0.381682i	$0.875345\pi$
$192$	0	0
$193$	− 282.448i	− 1.46346i	−0.681594	−	0.731731i	$-0.738713\pi$
$193$	− 282.448i	− 1.46346i	0.681594	−	0.731731i	$-0.261287\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	199.521i	1.01280i	0.862300	+	0.506398i	$0.169023\pi$
$197$	199.521i	1.01280i	−0.862300	+	0.506398i	$0.830977\pi$
$198$	0	0
$199$	327.579i	1.64612i	0.567952	+	0.823062i	$0.307736\pi$
$199$	327.579i	1.64612i	−0.567952	+	0.823062i	$0.692264\pi$
$200$	0	0
$201$	−40.8820	−0.203393
$202$	0	0
$203$	−0.645977	−0.00318215
$204$	0	0
$205$	0	0
$206$	0	0
$207$	81.4742	0.393595
$208$	0	0
$209$	−287.713	−1.37662
$210$	0	0
$211$	− 69.7171i	− 0.330413i	−0.986259	−	0.165206i	$-0.947171\pi$
$211$	− 69.7171i	− 0.330413i	0.986259	−	0.165206i	$-0.0528290\pi$
$212$	0	0
$213$	171.401i	0.804699i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 18.2312i	− 0.0840148i
$218$	0	0
$219$	55.3882i	0.252914i
$220$	0	0
$221$	377.718	1.70913
$222$	0	0
$223$	372.487	1.67034	0.835172	−	0.549989i	$-0.185368\pi$
$223$	372.487	1.67034	0.835172	+	0.549989i	$0.185368\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−85.6029	−0.377105	−0.188553	−	0.982063i	$-0.560380\pi$
$227$	−85.6029	−0.377105	−0.188553	+	0.982063i	$0.560380\pi$
$228$	0	0
$229$	−425.900	−1.85983	−0.929913	−	0.367780i	$-0.880118\pi$
$229$	−425.900	−1.85983	−0.929913	+	0.367780i	$0.880118\pi$
$230$	0	0
$231$	10.0024i	0.0433006i
$232$	0	0
$233$	− 303.947i	− 1.30450i	−0.758006	−	0.652248i	$-0.773826\pi$
$233$	− 303.947i	− 1.30450i	0.758006	−	0.652248i	$-0.226174\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	117.953i	0.497693i
$238$	0	0
$239$	− 277.656i	− 1.16174i	−0.813997	−	0.580870i	$-0.802713\pi$
$239$	− 277.656i	− 1.16174i	0.813997	−	0.580870i	$-0.197287\pi$
$240$	0	0
$241$	−342.452	−1.42096	−0.710482	−	0.703715i	$-0.751523\pi$
$241$	−342.452	−1.42096	−0.710482	+	0.703715i	$0.751523\pi$
$242$	0	0
$243$	15.5885	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−255.079	−1.03271
$248$	0	0
$249$	−117.066	−0.470146
$250$	0	0
$251$	60.0413i	0.239208i	0.992822	+	0.119604i	$0.0381625\pi$
$251$	60.0413i	0.239208i	−0.992822	+	0.119604i	$0.961837\pi$
$252$	0	0
$253$	343.887i	1.35924i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	309.309i	1.20354i	0.798671	+	0.601768i	$0.205537\pi$
$257$	309.309i	1.20354i	−0.798671	+	0.601768i	$0.794463\pi$
$258$	0	0
$259$	20.7489i	0.0801115i
$260$	0	0
$261$	−4.24922	−0.0162806
$262$	0	0
$263$	297.586	1.13150	0.565752	−	0.824575i	$-0.308586\pi$
$263$	297.586	1.13150	0.565752	+	0.824575i	$0.308586\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−289.392	−1.08387
$268$	0	0
$269$	−118.485	−0.440466	−0.220233	−	0.975447i	$-0.570682\pi$
$269$	−118.485	−0.440466	−0.220233	+	0.975447i	$0.570682\pi$
$270$	0	0
$271$	50.5786i	0.186637i	0.995636	+	0.0933185i	$0.0297475\pi$
$271$	50.5786i	0.186637i	−0.995636	+	0.0933185i	$0.970253\pi$
$272$	0	0
$273$	8.86788i	0.0324831i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 316.313i	− 1.14192i	−0.820977	−	0.570961i	$-0.806571\pi$
$277$	− 316.313i	− 1.14192i	0.820977	−	0.570961i	$-0.193429\pi$
$278$	0	0
$279$	− 119.924i	− 0.429837i
$280$	0	0
$281$	35.8840	0.127701	0.0638506	−	0.997959i	$-0.479662\pi$
$281$	35.8840	0.127701	0.0638506	+	0.997959i	$0.479662\pi$
$282$	0	0
$283$	−473.282	−1.67237	−0.836187	−	0.548445i	$-0.815220\pi$
$283$	−473.282	−1.67237	−0.836187	+	0.548445i	$0.815220\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	22.1810	0.0772859
$288$	0	0
$289$	−843.072	−2.91720
$290$	0	0
$291$	− 18.3154i	− 0.0629394i
$292$	0	0
$293$	215.927i	0.736951i	0.929637	+	0.368475i	$0.120120\pi$
$293$	215.927i	0.736951i	−0.929637	+	0.368475i	$0.879880\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	65.7958i	0.221535i
$298$	0	0
$299$	304.880i	1.01967i
$300$	0	0
$301$	−8.61788	−0.0286308
$302$	0	0
$303$	−17.5105	−0.0577905
$304$	0	0
$305$	0	0
$306$	0	0
$307$	332.823	1.08411	0.542057	−	0.840342i	$-0.317646\pi$
$307$	332.823	1.08411	0.542057	+	0.840342i	$0.317646\pi$
$308$	0	0
$309$	140.715	0.455388
$310$	0	0
$311$	− 308.920i	− 0.993313i	−0.867947	−	0.496657i	$-0.834561\pi$
$311$	− 308.920i	− 0.993313i	0.867947	−	0.496657i	$-0.165439\pi$
$312$	0	0
$313$	132.594i	0.423623i	0.977311	+	0.211811i	$0.0679363\pi$
$313$	132.594i	0.423623i	−0.977311	+	0.211811i	$0.932064\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 116.149i	− 0.366399i	−0.983076	−	0.183200i	$-0.941355\pi$
$317$	− 116.149i	− 0.366399i	0.983076	−	0.183200i	$-0.0586455\pi$
$318$	0	0
$319$	− 17.9351i	− 0.0562230i
$320$	0	0
$321$	266.782	0.831098
$322$	0	0
$323$	764.506	2.36689
$324$	0	0
$325$	0	0
$326$	0	0
$327$	126.922	0.388140
$328$	0	0
$329$	2.04450	0.00621427
$330$	0	0
$331$	− 548.146i	− 1.65603i	−0.560705	−	0.828016i	$-0.689470\pi$
$331$	− 548.146i	− 1.65603i	0.560705	−	0.828016i	$-0.310530\pi$
$332$	0	0
$333$	136.486i	0.409867i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	488.435i	1.44936i	0.689084	+	0.724682i	$0.258013\pi$
$337$	488.435i	1.44936i	−0.689084	+	0.724682i	$0.741987\pi$
$338$	0	0
$339$	240.729i	0.710115i
$340$	0	0
$341$	506.177	1.48439
$342$	0	0
$343$	−44.5997	−0.130028
$344$	0	0
$345$	0	0
$346$	0	0
$347$	254.751	0.734154	0.367077	−	0.930191i	$-0.380359\pi$
$347$	254.751	0.734154	0.367077	+	0.930191i	$0.380359\pi$
$348$	0	0
$349$	637.823	1.82757	0.913787	−	0.406194i	$-0.133144\pi$
$349$	637.823	1.82757	0.913787	+	0.406194i	$0.133144\pi$
$350$	0	0
$351$	58.3327i	0.166190i
$352$	0	0
$353$	− 322.849i	− 0.914585i	−0.889316	−	0.457293i	$-0.848819\pi$
$353$	− 322.849i	− 0.914585i	0.889316	−	0.457293i	$-0.151181\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 26.5783i	− 0.0744489i
$358$	0	0
$359$	504.941i	1.40652i	0.710933	+	0.703260i	$0.248273\pi$
$359$	504.941i	1.40652i	−0.710933	+	0.703260i	$0.751727\pi$
$360$	0	0
$361$	−155.283	−0.430147
$362$	0	0
$363$	−68.1329	−0.187694
$364$	0	0
$365$	0	0
$366$	0	0
$367$	346.691	0.944662	0.472331	−	0.881421i	$-0.343413\pi$
$367$	346.691	0.944662	0.472331	+	0.881421i	$0.343413\pi$
$368$	0	0
$369$	145.906	0.395410
$370$	0	0
$371$	19.4636i	0.0524624i
$372$	0	0
$373$	482.244i	1.29288i	0.762966	+	0.646439i	$0.223743\pi$
$373$	482.244i	1.29288i	−0.762966	+	0.646439i	$0.776257\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 15.9008i	− 0.0421772i
$378$	0	0
$379$	− 519.758i	− 1.37139i	−0.727888	−	0.685697i	$-0.759498\pi$
$379$	− 519.758i	− 1.37139i	0.727888	−	0.685697i	$-0.240502\pi$
$380$	0	0
$381$	372.240	0.977007
$382$	0	0
$383$	−244.680	−0.638851	−0.319425	−	0.947611i	$-0.603490\pi$
$383$	−244.680	−0.638851	−0.319425	+	0.947611i	$0.603490\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−56.6882	−0.146481
$388$	0	0
$389$	189.155	0.486260	0.243130	−	0.969994i	$-0.421826\pi$
$389$	189.155	0.486260	0.243130	+	0.969994i	$0.421826\pi$
$390$	0	0
$391$	− 913.768i	− 2.33700i
$392$	0	0
$393$	− 111.339i	− 0.283307i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 668.580i	− 1.68408i	−0.539415	−	0.842040i	$-0.681354\pi$
$397$	− 668.580i	− 1.68408i	0.539415	−	0.842040i	$-0.318646\pi$
$398$	0	0
$399$	17.9487i	0.0449843i
$400$	0	0
$401$	−440.649	−1.09887	−0.549437	−	0.835535i	$-0.685158\pi$
$401$	−440.649	−1.09887	−0.549437	+	0.835535i	$0.685158\pi$
$402$	0	0
$403$	448.763	1.11356
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−576.079	−1.41543
$408$	0	0
$409$	−479.570	−1.17254	−0.586271	−	0.810115i	$-0.699405\pi$
$409$	−479.570	−1.17254	−0.586271	+	0.810115i	$0.699405\pi$
$410$	0	0
$411$	312.742i	0.760930i
$412$	0	0
$413$	30.5942i	0.0740779i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 117.785i	− 0.282459i
$418$	0	0
$419$	751.523i	1.79361i	0.442424	+	0.896806i	$0.354118\pi$
$419$	751.523i	1.79361i	−0.442424	+	0.896806i	$0.645882\pi$
$420$	0	0
$421$	398.042	0.945469	0.472734	−	0.881205i	$-0.343267\pi$
$421$	398.042	0.945469	0.472734	+	0.881205i	$0.343267\pi$
$422$	0	0
$423$	13.4486	0.0317935
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.13556	0.0120271
$428$	0	0
$429$	−246.211	−0.573918
$430$	0	0
$431$	645.575i	1.49785i	0.662652	+	0.748927i	$0.269431\pi$
$431$	645.575i	1.49785i	−0.662652	+	0.748927i	$0.730569\pi$
$432$	0	0
$433$	− 315.231i	− 0.728016i	−0.931396	−	0.364008i	$-0.881408\pi$
$433$	− 315.231i	− 0.728016i	0.931396	−	0.364008i	$-0.118592\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	617.082i	1.41209i
$438$	0	0
$439$	− 213.497i	− 0.486326i	−0.969985	−	0.243163i	$-0.921815\pi$
$439$	− 213.497i	− 0.486326i	0.969985	−	0.243163i	$-0.0781849\pi$
$440$	0	0
$441$	−146.376	−0.331918
$442$	0	0
$443$	162.016	0.365725	0.182862	−	0.983139i	$-0.441464\pi$
$443$	162.016	0.365725	0.182862	+	0.983139i	$0.441464\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	147.474	0.329919
$448$	0	0
$449$	−39.4184	−0.0877916	−0.0438958	−	0.999036i	$-0.513977\pi$
$449$	−39.4184	−0.0877916	−0.0438958	+	0.999036i	$0.513977\pi$
$450$	0	0
$451$	615.842i	1.36550i
$452$	0	0
$453$	− 260.845i	− 0.575817i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 604.930i	− 1.32370i	−0.749637	−	0.661849i	$-0.769772\pi$
$457$	− 604.930i	− 1.32370i	0.749637	−	0.661849i	$-0.230228\pi$
$458$	0	0
$459$	− 174.831i	− 0.380896i
$460$	0	0
$461$	−832.683	−1.80625	−0.903127	−	0.429373i	$-0.858735\pi$
$461$	−832.683	−1.80625	−0.903127	+	0.429373i	$0.858735\pi$
$462$	0	0
$463$	−88.3867	−0.190900	−0.0954500	−	0.995434i	$-0.530429\pi$
$463$	−88.3867	−0.190900	−0.0954500	+	0.995434i	$0.530429\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−466.635	−0.999217	−0.499609	−	0.866251i	$-0.666523\pi$
$467$	−466.635	−0.999217	−0.499609	+	0.866251i	$0.666523\pi$
$468$	0	0
$469$	−10.7647	−0.0229524
$470$	0	0
$471$	356.096i	0.756041i
$472$	0	0
$473$	− 239.270i	− 0.505856i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	128.031i	0.268408i
$478$	0	0
$479$	165.351i	0.345200i	0.984992	+	0.172600i	$0.0552168\pi$
$479$	165.351i	0.345200i	−0.984992	+	0.172600i	$0.944783\pi$
$480$	0	0
$481$	−510.736	−1.06182
$482$	0	0
$483$	21.4530	0.0444162
$484$	0	0
$485$	0	0
$486$	0	0
$487$	923.214	1.89572	0.947858	−	0.318693i	$-0.103244\pi$
$487$	923.214	1.89572	0.947858	+	0.318693i	$0.103244\pi$
$488$	0	0
$489$	38.2648	0.0782512
$490$	0	0
$491$	− 9.31638i	− 0.0189743i	−0.999955	−	0.00948715i	$-0.996980\pi$
$491$	− 9.31638i	− 0.0189743i	0.999955	−	0.00948715i	$-0.00301990\pi$
$492$	0	0
$493$	47.6568i	0.0966670i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	45.1316i	0.0908081i
$498$	0	0
$499$	− 194.696i	− 0.390173i	−0.980786	−	0.195086i	$-0.937501\pi$
$499$	− 194.696i	− 0.390173i	0.980786	−	0.195086i	$-0.0624987\pi$
$500$	0	0
$501$	456.226	0.910630
$502$	0	0
$503$	−635.801	−1.26402	−0.632009	−	0.774961i	$-0.717769\pi$
$503$	−635.801	−1.26402	−0.632009	+	0.774961i	$0.717769\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	74.4329	0.146810
$508$	0	0
$509$	755.390	1.48407	0.742033	−	0.670363i	$-0.233862\pi$
$509$	755.390	1.48407	0.742033	+	0.670363i	$0.233862\pi$
$510$	0	0
$511$	14.5843i	0.0285407i
$512$	0	0
$513$	118.066i	0.230149i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	56.7641i	0.109795i
$518$	0	0
$519$	236.388i	0.455469i
$520$	0	0
$521$	−133.729	−0.256677	−0.128339	−	0.991730i	$-0.540964\pi$
$521$	−133.729	−0.256677	−0.128339	+	0.991730i	$0.540964\pi$
$522$	0	0
$523$	−600.534	−1.14825	−0.574124	−	0.818768i	$-0.694657\pi$
$523$	−600.534	−1.14825	−0.574124	+	0.818768i	$0.694657\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1345.00	−2.55219
$528$	0	0
$529$	208.561	0.394255
$530$	0	0
$531$	201.248i	0.378997i
$532$	0	0
$533$	545.988i	1.02437i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	112.086i	0.208727i
$538$	0	0
$539$	− 617.824i	− 1.14624i
$540$	0	0
$541$	−424.315	−0.784317	−0.392158	−	0.919898i	$-0.628271\pi$
$541$	−424.315	−0.784317	−0.392158	+	0.919898i	$0.628271\pi$
$542$	0	0
$543$	−320.166	−0.589624
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9.30220	0.0170059	0.00850293	−	0.999964i	$-0.497293\pi$
$547$	9.30220	0.0170059	0.00850293	+	0.999964i	$0.497293\pi$
$548$	0	0
$549$	33.7816	0.0615330
$550$	0	0
$551$	− 32.1834i	− 0.0584091i
$552$	0	0
$553$	31.0583i	0.0561633i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 147.599i	− 0.264989i	−0.991184	−	0.132495i	$-0.957701\pi$
$557$	− 147.599i	− 0.264989i	0.991184	−	0.132495i	$-0.0422987\pi$
$558$	0	0
$559$	− 212.130i	− 0.379481i
$560$	0	0
$561$	737.928	1.31538
$562$	0	0
$563$	−745.746	−1.32459	−0.662297	−	0.749242i	$-0.730418\pi$
$563$	−745.746	−1.32459	−0.662297	+	0.749242i	$0.730418\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.10461	0.00723916
$568$	0	0
$569$	371.156	0.652296	0.326148	−	0.945319i	$-0.394249\pi$
$569$	371.156	0.652296	0.326148	+	0.945319i	$0.394249\pi$
$570$	0	0
$571$	56.1196i	0.0982830i	0.998792	+	0.0491415i	$0.0156485\pi$
$571$	56.1196i	0.0982830i	−0.998792	+	0.0491415i	$0.984351\pi$
$572$	0	0
$573$	252.537i	0.440728i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 204.806i	− 0.354950i	−0.984125	−	0.177475i	$-0.943207\pi$
$577$	− 204.806i	− 0.354950i	0.984125	−	0.177475i	$-0.0567930\pi$
$578$	0	0
$579$	− 489.214i	− 0.844930i
$580$	0	0
$581$	−30.8248	−0.0530548
$582$	0	0
$583$	−540.393	−0.926917
$584$	0	0
$585$	0	0
$586$	0	0
$587$	564.490	0.961653	0.480826	−	0.876816i	$-0.340337\pi$
$587$	564.490	0.961653	0.480826	+	0.876816i	$0.340337\pi$
$588$	0	0
$589$	908.302	1.54211
$590$	0	0
$591$	345.580i	0.584738i
$592$	0	0
$593$	− 243.931i	− 0.411351i	−0.978620	−	0.205675i	$-0.934061\pi$
$593$	− 243.931i	− 0.411351i	0.978620	−	0.205675i	$-0.0659391\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	567.383i	0.950390i
$598$	0	0
$599$	− 757.177i	− 1.26407i	−0.774941	−	0.632034i	$-0.782220\pi$
$599$	− 757.177i	− 1.26407i	0.774941	−	0.632034i	$-0.217780\pi$
$600$	0	0
$601$	−291.181	−0.484495	−0.242247	−	0.970215i	$-0.577885\pi$
$601$	−291.181	−0.484495	−0.242247	+	0.970215i	$0.577885\pi$
$602$	0	0
$603$	−70.8098	−0.117429
$604$	0	0
$605$	0	0
$606$	0	0
$607$	121.521	0.200199	0.100099	−	0.994977i	$-0.468084\pi$
$607$	121.521	0.200199	0.100099	+	0.994977i	$0.468084\pi$
$608$	0	0
$609$	−1.11887	−0.00183722
$610$	0	0
$611$	50.3255i	0.0823657i
$612$	0	0
$613$	− 129.079i	− 0.210569i	−0.994442	−	0.105285i	$-0.966425\pi$
$613$	− 129.079i	− 0.210569i	0.994442	−	0.105285i	$-0.0335754\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	203.843i	0.330378i	0.986262	+	0.165189i	$0.0528234\pi$
$617$	203.843i	0.330378i	−0.986262	+	0.165189i	$0.947177\pi$
$618$	0	0
$619$	− 488.699i	− 0.789498i	−0.918789	−	0.394749i	$-0.870832\pi$
$619$	− 488.699i	− 0.789498i	0.918789	−	0.394749i	$-0.129168\pi$
$620$	0	0
$621$	141.117	0.227242
$622$	0	0
$623$	−76.2000	−0.122311
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−498.334	−0.794792
$628$	0	0
$629$	1530.74	2.43362
$630$	0	0
$631$	− 745.667i	− 1.18172i	−0.806773	−	0.590861i	$-0.798788\pi$
$631$	− 745.667i	− 1.18172i	0.806773	−	0.590861i	$-0.201212\pi$
$632$	0	0
$633$	− 120.754i	− 0.190764i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 547.746i	− 0.859883i
$638$	0	0
$639$	296.875i	0.464593i
$640$	0	0
$641$	1099.03	1.71456	0.857282	−	0.514848i	$-0.172152\pi$
$641$	1099.03	1.71456	0.857282	+	0.514848i	$0.172152\pi$
$642$	0	0
$643$	270.257	0.420307	0.210153	−	0.977668i	$-0.432604\pi$
$643$	270.257	0.420307	0.210153	+	0.977668i	$0.432604\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−433.409	−0.669875	−0.334937	−	0.942240i	$-0.608715\pi$
$647$	−433.409	−0.669875	−0.334937	+	0.942240i	$0.608715\pi$
$648$	0	0
$649$	−849.426	−1.30882
$650$	0	0
$651$	− 31.5774i	− 0.0485060i
$652$	0	0
$653$	− 373.983i	− 0.572715i	−0.958123	−	0.286357i	$-0.907556\pi$
$653$	− 373.983i	− 0.572715i	0.958123	−	0.286357i	$-0.0924444\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	95.9351i	0.146020i
$658$	0	0
$659$	307.739i	0.466978i	0.972359	+	0.233489i	$0.0750143\pi$
$659$	307.739i	0.466978i	−0.972359	+	0.233489i	$0.924986\pi$
$660$	0	0
$661$	−994.768	−1.50494	−0.752472	−	0.658624i	$-0.771139\pi$
$661$	−994.768	−1.50494	−0.752472	+	0.658624i	$0.771139\pi$
$662$	0	0
$663$	654.226	0.986767
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−38.4669	−0.0576715
$668$	0	0
$669$	645.166	0.964373
$670$	0	0
$671$	142.586i	0.212497i
$672$	0	0
$673$	− 2.40845i	− 0.00357867i	−0.999998	−	0.00178934i	$-0.999430\pi$
$673$	− 2.40845i	− 0.00357867i	0.999998	−	0.00178934i	$-0.000569564\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	257.587i	0.380483i	0.981737	+	0.190242i	$0.0609272\pi$
$677$	257.587i	0.380483i	−0.981737	+	0.190242i	$0.939073\pi$
$678$	0	0
$679$	− 4.82263i	− 0.00710254i
$680$	0	0
$681$	−148.269	−0.217722
$682$	0	0
$683$	1101.45	1.61267	0.806334	−	0.591461i	$-0.201448\pi$
$683$	1101.45	1.61267	0.806334	+	0.591461i	$0.201448\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−737.680	−1.07377
$688$	0	0
$689$	−479.097	−0.695352
$690$	0	0
$691$	− 569.738i	− 0.824512i	−0.911068	−	0.412256i	$-0.864741\pi$
$691$	− 569.738i	− 0.824512i	0.911068	−	0.412256i	$-0.135259\pi$
$692$	0	0
$693$	17.3247i	0.0249996i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1636.40i	− 2.34778i
$698$	0	0
$699$	− 526.452i	− 0.753151i
$700$	0	0
$701$	829.418	1.18319	0.591596	−	0.806234i	$-0.298498\pi$
$701$	829.418	1.18319	0.591596	+	0.806234i	$0.298498\pi$
$702$	0	0
$703$	−1033.74	−1.47046
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.61070	−0.00652150
$708$	0	0
$709$	201.355	0.283998	0.141999	−	0.989867i	$-0.454647\pi$
$709$	201.355	0.283998	0.141999	+	0.989867i	$0.454647\pi$
$710$	0	0
$711$	204.301i	0.287343i
$712$	0	0
$713$	− 1085.64i	− 1.52264i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 480.914i	− 0.670731i
$718$	0	0
$719$	− 703.851i	− 0.978930i	−0.872023	−	0.489465i	$-0.837192\pi$
$719$	− 703.851i	− 0.978930i	0.872023	−	0.489465i	$-0.162808\pi$
$720$	0	0
$721$	37.0517	0.0513893
$722$	0	0
$723$	−593.145	−0.820394
$724$	0	0
$725$	0	0
$726$	0	0
$727$	510.680	0.702448	0.351224	−	0.936291i	$-0.385766\pi$
$727$	510.680	0.702448	0.351224	+	0.936291i	$0.385766\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	635.782i	0.869743i
$732$	0	0
$733$	− 784.518i	− 1.07028i	−0.844762	−	0.535142i	$-0.820258\pi$
$733$	− 784.518i	− 1.07028i	0.844762	−	0.535142i	$-0.179742\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 298.874i	− 0.405528i
$738$	0	0
$739$	211.762i	0.286552i	0.989683	+	0.143276i	$0.0457637\pi$
$739$	211.762i	0.286552i	−0.989683	+	0.143276i	$0.954236\pi$
$740$	0	0
$741$	−441.809	−0.596234
$742$	0	0
$743$	538.981	0.725412	0.362706	−	0.931904i	$-0.381853\pi$
$743$	538.981	0.725412	0.362706	+	0.931904i	$0.381853\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−202.765	−0.271439
$748$	0	0
$749$	70.2466	0.0937872
$750$	0	0
$751$	157.900i	0.210253i	0.994459	+	0.105127i	$0.0335248\pi$
$751$	157.900i	0.210253i	−0.994459	+	0.105127i	$0.966475\pi$
$752$	0	0
$753$	103.995i	0.138107i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1508.92i	− 1.99328i	−0.0818854	−	0.996642i	$-0.526094\pi$
$757$	− 1508.92i	− 1.99328i	0.0818854	−	0.996642i	$-0.473906\pi$
$758$	0	0
$759$	595.629i	0.784755i
$760$	0	0
$761$	283.109	0.372022	0.186011	−	0.982548i	$-0.440444\pi$
$761$	283.109	0.372022	0.186011	+	0.982548i	$0.440444\pi$
$762$	0	0
$763$	33.4199	0.0438006
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−753.078	−0.981849
$768$	0	0
$769$	931.659	1.21152	0.605760	−	0.795647i	$-0.292869\pi$
$769$	931.659	1.21152	0.605760	+	0.795647i	$0.292869\pi$
$770$	0	0
$771$	535.739i	0.694862i
$772$	0	0
$773$	− 604.742i	− 0.782331i	−0.920320	−	0.391166i	$-0.872072\pi$
$773$	− 604.742i	− 0.782331i	0.920320	−	0.391166i	$-0.127928\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	35.9381i	0.0462524i
$778$	0	0
$779$	1105.09i	1.41860i
$780$	0	0
$781$	−1253.05	−1.60442
$782$	0	0
$783$	−7.35987	−0.00939958
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1357.83	−1.72533	−0.862663	−	0.505779i	$-0.831205\pi$
$787$	−1357.83	−1.72533	−0.862663	+	0.505779i	$0.831205\pi$
$788$	0	0
$789$	515.433	0.653274
$790$	0	0
$791$	63.3865i	0.0801346i
$792$	0	0
$793$	126.412i	0.159410i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 986.238i	− 1.23744i	−0.785613	−	0.618719i	$-0.787652\pi$
$797$	− 986.238i	− 1.23744i	0.785613	−	0.618719i	$-0.212348\pi$
$798$	0	0
$799$	− 150.832i	− 0.188776i
$800$	0	0
$801$	−501.242	−0.625770
$802$	0	0
$803$	−404.923	−0.504263
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−205.223	−0.254303
$808$	0	0
$809$	−36.2371	−0.0447925	−0.0223963	−	0.999749i	$-0.507130\pi$
$809$	−36.2371	−0.0447925	−0.0223963	+	0.999749i	$0.507130\pi$
$810$	0	0
$811$	− 1174.15i	− 1.44778i	−0.689915	−	0.723891i	$-0.742352\pi$
$811$	− 1174.15i	− 1.44778i	0.689915	−	0.723891i	$-0.257648\pi$
$812$	0	0
$813$	87.6047i	0.107755i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 429.354i	− 0.525525i
$818$	0	0
$819$	15.3596i	0.0187541i
$820$	0	0
$821$	429.051	0.522596	0.261298	−	0.965258i	$-0.415849\pi$
$821$	429.051	0.522596	0.261298	+	0.965258i	$0.415849\pi$
$822$	0	0
$823$	639.325	0.776822	0.388411	−	0.921486i	$-0.373024\pi$
$823$	639.325	0.776822	0.388411	+	0.921486i	$0.373024\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	295.749	0.357616	0.178808	−	0.983884i	$-0.442776\pi$
$827$	295.749	0.357616	0.178808	+	0.983884i	$0.442776\pi$
$828$	0	0
$829$	−858.061	−1.03505	−0.517527	−	0.855667i	$-0.673147\pi$
$829$	−858.061	−1.03505	−0.517527	+	0.855667i	$0.673147\pi$
$830$	0	0
$831$	− 547.870i	− 0.659289i
$832$	0	0
$833$	1641.67i	1.97079i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 207.715i	− 0.248166i
$838$	0	0
$839$	− 1638.40i	− 1.95280i	−0.215979	−	0.976398i	$-0.569294\pi$
$839$	− 1638.40i	− 1.95280i	0.215979	−	0.976398i	$-0.430706\pi$
$840$	0	0
$841$	−838.994	−0.997614
$842$	0	0
$843$	62.1529	0.0737283
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−17.9401	−0.0211808
$848$	0	0
$849$	−819.748	−0.965545
$850$	0	0
$851$	1235.56i	1.45189i
$852$	0	0
$853$	1109.39i	1.30058i	0.759687	+	0.650289i	$0.225352\pi$
$853$	1109.39i	1.30058i	−0.759687	+	0.650289i	$0.774648\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 482.816i	− 0.563380i	−0.959506	−	0.281690i	$-0.909105\pi$
$857$	− 482.816i	− 0.563380i	0.959506	−	0.281690i	$-0.0908949\pi$
$858$	0	0
$859$	− 26.3839i	− 0.0307147i	−0.999882	−	0.0153573i	$-0.995111\pi$
$859$	− 26.3839i	− 0.0307147i	0.999882	−	0.0153573i	$-0.00488859\pi$
$860$	0	0
$861$	38.4187	0.0446210
$862$	0	0
$863$	56.7068	0.0657089	0.0328545	−	0.999460i	$-0.489540\pi$
$863$	56.7068	0.0657089	0.0328545	+	0.999460i	$0.489540\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1460.24	−1.68425
$868$	0	0
$869$	−862.314	−0.992306
$870$	0	0
$871$	− 264.973i	− 0.304217i
$872$	0	0
$873$	− 31.7231i	− 0.0363381i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1000.90i	1.14127i	0.821203	+	0.570636i	$0.193303\pi$
$877$	1000.90i	1.14127i	−0.821203	+	0.570636i	$0.806697\pi$
$878$	0	0
$879$	373.996i	0.425479i
$880$	0	0
$881$	100.129	0.113654	0.0568269	−	0.998384i	$-0.481902\pi$
$881$	100.129	0.113654	0.0568269	+	0.998384i	$0.481902\pi$
$882$	0	0
$883$	−832.872	−0.943230	−0.471615	−	0.881804i	$-0.656329\pi$
$883$	−832.872	−0.943230	−0.471615	+	0.881804i	$0.656329\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−724.253	−0.816519	−0.408260	−	0.912866i	$-0.633864\pi$
$887$	−724.253	−0.816519	−0.408260	+	0.912866i	$0.633864\pi$
$888$	0	0
$889$	98.0146	0.110253
$890$	0	0
$891$	113.962i	0.127903i
$892$	0	0
$893$	101.859i	0.114064i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	528.068i	0.588705i
$898$	0	0
$899$	56.6206i	0.0629818i
$900$	0	0
$901$	1435.92	1.59370
$902$	0	0
$903$	−14.9266	−0.0165300
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1095.96	−1.20833	−0.604165	−	0.796859i	$-0.706493\pi$
$907$	−1095.96	−1.20833	−0.604165	+	0.796859i	$0.706493\pi$
$908$	0	0
$909$	−30.3291	−0.0333653
$910$	0	0
$911$	− 617.137i	− 0.677428i	−0.940889	−	0.338714i	$-0.890008\pi$
$911$	− 617.137i	− 0.677428i	0.940889	−	0.338714i	$-0.109992\pi$
$912$	0	0
$913$	− 855.831i	− 0.937383i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 29.3169i	− 0.0319704i
$918$	0	0
$919$	− 1740.12i	− 1.89349i	−0.321981	−	0.946746i	$-0.604349\pi$
$919$	− 1740.12i	− 1.89349i	0.321981	−	0.946746i	$-0.395651\pi$
$920$	0	0
$921$	576.466	0.625913
$922$	0	0
$923$	−1110.92	−1.20360
$924$	0	0
$925$	0	0
$926$	0	0
$927$	243.725	0.262918
$928$	0	0
$929$	−263.490	−0.283627	−0.141814	−	0.989893i	$-0.545293\pi$
$929$	−263.490	−0.283627	−0.141814	+	0.989893i	$0.545293\pi$
$930$	0	0
$931$	− 1108.65i	− 1.19081i
$932$	0	0
$933$	− 535.066i	− 0.573490i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 819.385i	− 0.874477i	−0.899346	−	0.437239i	$-0.855957\pi$
$937$	− 819.385i	− 0.874477i	0.899346	−	0.437239i	$-0.144043\pi$
$938$	0	0
$939$	229.660i	0.244579i
$940$	0	0
$941$	1644.48	1.74759	0.873794	−	0.486296i	$-0.161652\pi$
$941$	1644.48	1.74759	0.873794	+	0.486296i	$0.161652\pi$
$942$	0	0
$943$	1320.85	1.40068
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1010.06	1.06659	0.533296	−	0.845929i	$-0.320953\pi$
$947$	1010.06	1.06659	0.533296	+	0.845929i	$0.320953\pi$
$948$	0	0
$949$	−358.994	−0.378286
$950$	0	0
$951$	− 201.175i	− 0.211541i
$952$	0	0
$953$	241.305i	0.253205i	0.991954	+	0.126603i	$0.0404073\pi$
$953$	241.305i	0.253205i	−0.991954	+	0.126603i	$0.959593\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 31.0646i	− 0.0324604i
$958$	0	0
$959$	82.3483i	0.0858690i
$960$	0	0
$961$	−636.986	−0.662837
$962$	0	0
$963$	462.081	0.479835
$964$	0	0
$965$	0	0
$966$	0	0
$967$	754.537	0.780286	0.390143	−	0.920754i	$-0.372425\pi$
$967$	754.537	0.780286	0.390143	+	0.920754i	$0.372425\pi$
$968$	0	0
$969$	1324.16	1.36653
$970$	0	0
$971$	− 1800.78i	− 1.85456i	−0.374368	−	0.927280i	$-0.622140\pi$
$971$	− 1800.78i	− 1.85456i	0.374368	−	0.927280i	$-0.377860\pi$
$972$	0	0
$973$	− 31.0141i	− 0.0318748i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 435.605i	− 0.445860i	−0.974834	−	0.222930i	$-0.928438\pi$
$977$	− 435.605i	− 0.445860i	0.974834	−	0.222930i	$-0.0715621\pi$
$978$	0	0
$979$	− 2115.64i	− 2.16102i
$980$	0	0
$981$	219.835	0.224093
$982$	0	0
$983$	−376.746	−0.383262	−0.191631	−	0.981467i	$-0.561378\pi$
$983$	−376.746	−0.383262	−0.191631	+	0.981467i	$0.561378\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.54117	0.00358781
$988$	0	0
$989$	−513.181	−0.518888
$990$	0	0
$991$	1119.69i	1.12985i	0.825141	+	0.564927i	$0.191096\pi$
$991$	1119.69i	1.12985i	−0.825141	+	0.564927i	$0.808904\pi$
$992$	0	0
$993$	− 949.417i	− 0.956110i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	549.043i	0.550695i	0.961345	+	0.275347i	$0.0887929\pi$
$997$	549.043i	0.550695i	−0.961345	+	0.275347i	$0.911207\pi$
$998$	0	0
$999$	236.400i	0.236637i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.3.j.d.799.6		8
4.3	odd	2		2400.3.j.j.799.3		8
5.2	odd	4		480.3.e.a.31.1	✓	8
5.3	odd	4		2400.3.e.g.1951.8		8
5.4	even	2		2400.3.j.j.799.4		8
15.2	even	4		1440.3.e.e.991.7		8
20.3	even	4		2400.3.e.g.1951.1		8
20.7	even	4		480.3.e.a.31.6	yes	8
20.19	odd	2	inner	2400.3.j.d.799.5		8
40.27	even	4		960.3.e.d.511.4		8
40.37	odd	4		960.3.e.d.511.7		8
60.47	odd	4		1440.3.e.e.991.6		8
120.77	even	4		2880.3.e.n.2431.3		8
120.107	odd	4		2880.3.e.n.2431.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.3.e.a.31.1	✓	8	5.2	odd	4
480.3.e.a.31.6	yes	8	20.7	even	4
960.3.e.d.511.4		8	40.27	even	4
960.3.e.d.511.7		8	40.37	odd	4
1440.3.e.e.991.6		8	60.47	odd	4
1440.3.e.e.991.7		8	15.2	even	4
2400.3.e.g.1951.1		8	20.3	even	4
2400.3.e.g.1951.8		8	5.3	odd	4
2400.3.j.d.799.5		8	20.19	odd	2	inner
2400.3.j.d.799.6		8	1.1	even	1	trivial
2400.3.j.j.799.3		8	4.3	odd	2
2400.3.j.j.799.4		8	5.4	even	2
2880.3.e.n.2431.2		8	120.107	odd	4
2880.3.e.n.2431.3		8	120.77	even	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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2400.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} +0.456067 q^{7} +3.00000 q^{9} +12.6624i q^{11} +11.2261i q^{13} -33.6463i q^{17} +22.7219i q^{19} +0.789932 q^{21} +27.1581 q^{23} +5.19615 q^{27} -1.41641 q^{29} -39.9748i q^{31} +21.9319i q^{33} +45.4952i q^{37} +19.4442i q^{39} +48.6355 q^{41} -18.8961 q^{43} +4.48288 q^{47} -48.7920 q^{49} -58.2771i q^{51} +42.6769i q^{53} +39.3554i q^{57} +67.0825i q^{59} +11.2605 q^{61} +1.36820 q^{63} -23.6033 q^{67} +47.0392 q^{69} +98.9583i q^{71} +31.9784i q^{73} +5.77491i q^{77} +68.1003i q^{79} +9.00000 q^{81} -67.5883 q^{83} -2.45329 q^{87} -167.081 q^{89} +5.11987i q^{91} -69.2384i q^{93} -10.5744i q^{97} +37.9872i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{7} + 24 q^{9} + 96 q^{21} - 16 q^{23} + 96 q^{29} + 112 q^{41} - 128 q^{43} - 272 q^{47} + 184 q^{49} + 80 q^{61} - 48 q^{63} + 320 q^{67} + 144 q^{69} + 72 q^{81} - 320 q^{83} + 48 q^{89}+O(q^{100}) \) 8 * q - 16 * q^7 + 24 * q^9 + 96 * q^21 - 16 * q^23 + 96 * q^29 + 112 * q^41 - 128 * q^43 - 272 * q^47 + 184 * q^49 + 80 * q^61 - 48 * q^63 + 320 * q^67 + 144 * q^69 + 72 * q^81 - 320 * q^83 + 48 * q^89

Introduction

L-functions

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