LMFDB - Embedded newform 252.8.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,8,Mod(1,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("252.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	252.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:	$78.7210264220$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 14292x^{2} + 540043x + 5027477 $ x^4 - x^3 - 14292x^2 + 540043x + 5027477
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-133.838$ of defining polynomial
Character	$\chi$	$=$	252.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-537.098 q^{5} -343.000 q^{7} +2419.02 q^{11} +5190.96 q^{13} +25751.6 q^{17} -34136.3 q^{19} +93525.7 q^{23} +210349. q^{25} +170556. q^{29} -285569. q^{31} +184225. q^{35} -540970. q^{37} -219356. q^{41} +876643. q^{43} -532161. q^{47} +117649. q^{49} -1.70818e6 q^{53} -1.29925e6 q^{55} -200256. q^{59} -941914. q^{61} -2.78805e6 q^{65} -2.43568e6 q^{67} +4.15568e6 q^{71} +5.31689e6 q^{73} -829724. q^{77} -3.41181e6 q^{79} +5.76711e6 q^{83} -1.38311e7 q^{85} -5.37254e6 q^{89} -1.78050e6 q^{91} +1.83346e7 q^{95} -7.26128e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 1372 q^{7} + 13272 q^{13} - 9184 q^{19} + 264524 q^{25} - 445536 q^{31} - 852808 q^{37} + 150224 q^{43} + 470596 q^{49} - 2627296 q^{55} - 1302840 q^{61} - 6071696 q^{67} + 3631768 q^{73} - 16898688 q^{79}+ \cdots - 36492008 q^{97}+O(q^{100}) $ 4 * q - 1372 * q^7 + 13272 * q^13 - 9184 * q^19 + 264524 * q^25 - 445536 * q^31 - 852808 * q^37 + 150224 * q^43 + 470596 * q^49 - 2627296 * q^55 - 1302840 * q^61 - 6071696 * q^67 + 3631768 * q^73 - 16898688 * q^79 - 27267520 * q^85 - 4552296 * q^91 - 36492008 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−537.098	−1.92158	−0.960790	−	0.277278i	$-0.910568\pi$
$5$	−537.098	−1.92158	−0.960790	+	0.277278i	$0.910568\pi$
$6$	0	0
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2419.02	0.547980	0.273990	−	0.961733i	$-0.411656\pi$
$11$	2419.02	0.547980	0.273990	+	0.961733i	$0.411656\pi$
$12$	0	0
$13$	5190.96	0.655309	0.327654	−	0.944798i	$-0.393742\pi$
$13$	5190.96	0.655309	0.327654	+	0.944798i	$0.393742\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	25751.6	1.27126	0.635628	−	0.771996i	$-0.280741\pi$
$17$	25751.6	1.27126	0.635628	+	0.771996i	$0.280741\pi$
$18$	0	0
$19$	−34136.3	−1.14177	−0.570886	−	0.821029i	$-0.693400\pi$
$19$	−34136.3	−1.14177	−0.570886	+	0.821029i	$0.693400\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	93525.7	1.60282	0.801408	−	0.598118i	$-0.204085\pi$
$23$	93525.7	1.60282	0.801408	+	0.598118i	$0.204085\pi$
$24$	0	0
$25$	210349.	2.69247
$26$	0	0
$27$	0	0
$28$	0	0
$29$	170556.	1.29859	0.649296	−	0.760536i	$-0.275064\pi$
$29$	170556.	1.29859	0.649296	+	0.760536i	$0.275064\pi$
$30$	0	0
$31$	−285569.	−1.72165	−0.860827	−	0.508898i	$-0.830053\pi$
$31$	−285569.	−1.72165	−0.860827	+	0.508898i	$0.830053\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	184225.	0.726289
$36$	0	0
$37$	−540970.	−1.75577	−0.877884	−	0.478873i	$-0.841046\pi$
$37$	−540970.	−1.75577	−0.877884	+	0.478873i	$0.841046\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−219356.	−0.497056	−0.248528	−	0.968625i	$-0.579947\pi$
$41$	−219356.	−0.497056	−0.248528	+	0.968625i	$0.579947\pi$
$42$	0	0
$43$	876643.	1.68145	0.840723	−	0.541465i	$-0.182130\pi$
$43$	876643.	1.68145	0.840723	+	0.541465i	$0.182130\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−532161.	−0.747653	−0.373827	−	0.927499i	$-0.621955\pi$
$47$	−532161.	−0.747653	−0.373827	+	0.927499i	$0.621955\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.70818e6	−1.57604	−0.788022	−	0.615647i	$-0.788895\pi$
$53$	−1.70818e6	−1.57604	−0.788022	+	0.615647i	$0.788895\pi$
$54$	0	0
$55$	−1.29925e6	−1.05299
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−200256.	−0.126941	−0.0634706	−	0.997984i	$-0.520217\pi$
$59$	−200256.	−0.126941	−0.0634706	+	0.997984i	$0.520217\pi$
$60$	0	0
$61$	−941914.	−0.531321	−0.265660	−	0.964067i	$-0.585590\pi$
$61$	−941914.	−0.531321	−0.265660	+	0.964067i	$0.585590\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.78805e6	−1.25923
$66$	0	0
$67$	−2.43568e6	−0.989367	−0.494684	−	0.869073i	$-0.664716\pi$
$67$	−2.43568e6	−0.989367	−0.494684	+	0.869073i	$0.664716\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.15568e6	1.37796	0.688981	−	0.724779i	$-0.258058\pi$
$71$	4.15568e6	1.37796	0.688981	+	0.724779i	$0.258058\pi$
$72$	0	0
$73$	5.31689e6	1.59966	0.799830	−	0.600226i	$-0.204923\pi$
$73$	5.31689e6	1.59966	0.799830	+	0.600226i	$0.204923\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−829724.	−0.207117
$78$	0	0
$79$	−3.41181e6	−0.778556	−0.389278	−	0.921120i	$-0.627275\pi$
$79$	−3.41181e6	−0.778556	−0.389278	+	0.921120i	$0.627275\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.76711e6	1.10709	0.553547	−	0.832818i	$-0.313274\pi$
$83$	5.76711e6	1.10709	0.553547	+	0.832818i	$0.313274\pi$
$84$	0	0
$85$	−1.38311e7	−2.44282
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.37254e6	−0.807820	−0.403910	−	0.914799i	$-0.632349\pi$
$89$	−5.37254e6	−0.807820	−0.403910	+	0.914799i	$0.632349\pi$
$90$	0	0
$91$	−1.78050e6	−0.247683
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.83346e7	2.19401
$96$	0	0
$97$	−7.26128e6	−0.807815	−0.403908	−	0.914800i	$-0.632348\pi$
$97$	−7.26128e6	−0.807815	−0.403908	+	0.914800i	$0.632348\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.59557e6	0.347251	0.173625	−	0.984812i	$-0.444452\pi$
$101$	3.59557e6	0.347251	0.173625	+	0.984812i	$0.444452\pi$
$102$	0	0
$103$	3.05688e6	0.275644	0.137822	−	0.990457i	$-0.455990\pi$
$103$	3.05688e6	0.275644	0.137822	+	0.990457i	$0.455990\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.93900e6	−0.310844	−0.155422	−	0.987848i	$-0.549674\pi$
$107$	−3.93900e6	−0.310844	−0.155422	+	0.987848i	$0.549674\pi$
$108$	0	0
$109$	−5.07994e6	−0.375722	−0.187861	−	0.982196i	$-0.560155\pi$
$109$	−5.07994e6	−0.375722	−0.187861	+	0.982196i	$0.560155\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.96196e7	1.27913	0.639566	−	0.768736i	$-0.279114\pi$
$113$	1.96196e7	1.27913	0.639566	+	0.768736i	$0.279114\pi$
$114$	0	0
$115$	−5.02324e7	−3.07994
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−8.83279e6	−0.480489
$120$	0	0
$121$	−1.36355e7	−0.699718
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−7.10172e7	−3.25221
$126$	0	0
$127$	6.82658e6	0.295726	0.147863	−	0.989008i	$-0.452760\pi$
$127$	6.82658e6	0.295726	0.147863	+	0.989008i	$0.452760\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.45906e7	−1.34434	−0.672169	−	0.740397i	$-0.734637\pi$
$131$	−3.45906e7	−1.34434	−0.672169	+	0.740397i	$0.734637\pi$
$132$	0	0
$133$	1.17088e7	0.431549
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.64855e6	−0.0547748	−0.0273874	−	0.999625i	$-0.508719\pi$
$137$	−1.64855e6	−0.0547748	−0.0273874	+	0.999625i	$0.508719\pi$
$138$	0	0
$139$	−7.49730e6	−0.236784	−0.118392	−	0.992967i	$-0.537774\pi$
$139$	−7.49730e6	−0.236784	−0.118392	+	0.992967i	$0.537774\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.25570e7	0.359096
$144$	0	0
$145$	−9.16050e7	−2.49535
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.88012e7	−0.713277	−0.356639	−	0.934242i	$-0.616077\pi$
$149$	−2.88012e7	−0.713277	−0.356639	+	0.934242i	$0.616077\pi$
$150$	0	0
$151$	−4.72224e7	−1.11617	−0.558083	−	0.829785i	$-0.688463\pi$
$151$	−4.72224e7	−1.11617	−0.558083	+	0.829785i	$0.688463\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.53379e8	3.30829
$156$	0	0
$157$	2.49497e7	0.514538	0.257269	−	0.966340i	$-0.417177\pi$
$157$	2.49497e7	0.514538	0.257269	+	0.966340i	$0.417177\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.20793e7	−0.605807
$162$	0	0
$163$	−2.79194e7	−0.504950	−0.252475	−	0.967603i	$-0.581245\pi$
$163$	−2.79194e7	−0.504950	−0.252475	+	0.967603i	$0.581245\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.44866e6	−0.107143	−0.0535713	−	0.998564i	$-0.517060\pi$
$167$	−6.44866e6	−0.107143	−0.0535713	+	0.998564i	$0.517060\pi$
$168$	0	0
$169$	−3.58024e7	−0.570570
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.48961e7	−0.659246	−0.329623	−	0.944113i	$-0.606922\pi$
$173$	−4.48961e7	−0.659246	−0.329623	+	0.944113i	$0.606922\pi$
$174$	0	0
$175$	−7.21497e7	−1.01766
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−6.52042e6	−0.0849748	−0.0424874	−	0.999097i	$-0.513528\pi$
$179$	−6.52042e6	−0.0849748	−0.0424874	+	0.999097i	$0.513528\pi$
$180$	0	0
$181$	−5.31522e7	−0.666264	−0.333132	−	0.942880i	$-0.608105\pi$
$181$	−5.31522e7	−0.666264	−0.333132	+	0.942880i	$0.608105\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.90554e8	3.37385
$186$	0	0
$187$	6.22936e7	0.696623
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.52197e7	−0.884959	−0.442479	−	0.896779i	$-0.645901\pi$
$191$	−8.52197e7	−0.884959	−0.442479	+	0.896779i	$0.645901\pi$
$192$	0	0
$193$	−2.18951e7	−0.219228	−0.109614	−	0.993974i	$-0.534961\pi$
$193$	−2.18951e7	−0.219228	−0.109614	+	0.993974i	$0.534961\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.67401e8	1.56000	0.780001	−	0.625778i	$-0.215218\pi$
$197$	1.67401e8	1.56000	0.780001	+	0.625778i	$0.215218\pi$
$198$	0	0
$199$	6.16647e7	0.554691	0.277346	−	0.960770i	$-0.410545\pi$
$199$	6.16647e7	0.554691	0.277346	+	0.960770i	$0.410545\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.85006e7	−0.490822
$204$	0	0
$205$	1.17815e8	0.955132
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−8.25765e7	−0.625668
$210$	0	0
$211$	−4.83314e7	−0.354193	−0.177097	−	0.984193i	$-0.556671\pi$
$211$	−4.83314e7	−0.354193	−0.177097	+	0.984193i	$0.556671\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4.70843e8	−3.23103
$216$	0	0
$217$	9.79503e7	0.650724
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.33675e8	0.833065
$222$	0	0
$223$	−9.42998e7	−0.569435	−0.284717	−	0.958612i	$-0.591900\pi$
$223$	−9.42998e7	−0.569435	−0.284717	+	0.958612i	$0.591900\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.83633e7	−0.274426	−0.137213	−	0.990542i	$-0.543815\pi$
$227$	−4.83633e7	−0.274426	−0.137213	+	0.990542i	$0.543815\pi$
$228$	0	0
$229$	1.12295e8	0.617925	0.308962	−	0.951074i	$-0.400018\pi$
$229$	1.12295e8	0.617925	0.308962	+	0.951074i	$0.400018\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.27460e8	−1.17804	−0.589020	−	0.808118i	$-0.700486\pi$
$233$	−2.27460e8	−1.17804	−0.589020	+	0.808118i	$0.700486\pi$
$234$	0	0
$235$	2.85822e8	1.43668
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.09223e8	0.991328	0.495664	−	0.868514i	$-0.334925\pi$
$239$	2.09223e8	0.991328	0.495664	+	0.868514i	$0.334925\pi$
$240$	0	0
$241$	−1.65415e8	−0.761227	−0.380614	−	0.924734i	$-0.624287\pi$
$241$	−1.65415e8	−0.761227	−0.380614	+	0.924734i	$0.624287\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.31890e7	−0.274511
$246$	0	0
$247$	−1.77200e8	−0.748213
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4.17188e8	−1.66523	−0.832614	−	0.553854i	$-0.813156\pi$
$251$	−4.17188e8	−1.66523	−0.832614	+	0.553854i	$0.813156\pi$
$252$	0	0
$253$	2.26240e8	0.878311
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.89850e8	1.43262	0.716311	−	0.697781i	$-0.245829\pi$
$257$	3.89850e8	1.43262	0.716311	+	0.697781i	$0.245829\pi$
$258$	0	0
$259$	1.85553e8	0.663618
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.22631e7	0.0415675	0.0207838	−	0.999784i	$-0.493384\pi$
$263$	1.22631e7	0.0415675	0.0207838	+	0.999784i	$0.493384\pi$
$264$	0	0
$265$	9.17460e8	3.02849
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3.04005e8	−0.952243	−0.476121	−	0.879380i	$-0.657958\pi$
$269$	−3.04005e8	−0.952243	−0.476121	+	0.879380i	$0.657958\pi$
$270$	0	0
$271$	−1.30857e8	−0.399397	−0.199699	−	0.979857i	$-0.563996\pi$
$271$	−1.30857e8	−0.399397	−0.199699	+	0.979857i	$0.563996\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.08838e8	1.47542
$276$	0	0
$277$	3.68396e7	0.104144	0.0520722	−	0.998643i	$-0.483417\pi$
$277$	3.68396e7	0.104144	0.0520722	+	0.998643i	$0.483417\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.26456e8	0.877714	0.438857	−	0.898557i	$-0.355383\pi$
$281$	3.26456e8	0.877714	0.438857	+	0.898557i	$0.355383\pi$
$282$	0	0
$283$	4.48742e8	1.17691	0.588456	−	0.808529i	$-0.299736\pi$
$283$	4.48742e8	1.17691	0.588456	+	0.808529i	$0.299736\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.52390e7	0.187869
$288$	0	0
$289$	2.52806e8	0.616090
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.24224e8	0.288515	0.144257	−	0.989540i	$-0.453921\pi$
$293$	1.24224e8	0.288515	0.144257	+	0.989540i	$0.453921\pi$
$294$	0	0
$295$	1.07557e8	0.243928
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.85488e8	1.05034
$300$	0	0
$301$	−3.00688e8	−0.635527
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.05900e8	1.02098
$306$	0	0
$307$	−8.34331e8	−1.64571	−0.822856	−	0.568249i	$-0.807621\pi$
$307$	−8.34331e8	−1.64571	−0.822856	+	0.568249i	$0.807621\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.30334e8	0.245695	0.122848	−	0.992426i	$-0.460797\pi$
$311$	1.30334e8	0.245695	0.122848	+	0.992426i	$0.460797\pi$
$312$	0	0
$313$	9.25098e8	1.70523	0.852615	−	0.522540i	$-0.175016\pi$
$313$	9.25098e8	1.70523	0.852615	+	0.522540i	$0.175016\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.54884e8	−0.802036	−0.401018	−	0.916070i	$-0.631343\pi$
$317$	−4.54884e8	−0.802036	−0.401018	+	0.916070i	$0.631343\pi$
$318$	0	0
$319$	4.12577e8	0.711603
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−8.79065e8	−1.45148
$324$	0	0
$325$	1.09191e9	1.76440
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.82531e8	0.282586
$330$	0	0
$331$	1.24612e9	1.88870	0.944349	−	0.328945i	$-0.106693\pi$
$331$	1.24612e9	1.88870	0.944349	+	0.328945i	$0.106693\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.30820e9	1.90115
$336$	0	0
$337$	3.89702e7	0.0554661	0.0277331	−	0.999615i	$-0.491171\pi$
$337$	3.89702e7	0.0554661	0.0277331	+	0.999615i	$0.491171\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−6.90798e8	−0.943432
$342$	0	0
$343$	−4.03536e7	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.16172e9	−1.49262	−0.746309	−	0.665600i	$-0.768176\pi$
$347$	−1.16172e9	−1.49262	−0.746309	+	0.665600i	$0.768176\pi$
$348$	0	0
$349$	−7.69763e8	−0.969323	−0.484661	−	0.874702i	$-0.661057\pi$
$349$	−7.69763e8	−0.969323	−0.484661	+	0.874702i	$0.661057\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.43421e9	−1.73541	−0.867705	−	0.497080i	$-0.834406\pi$
$353$	−1.43421e9	−1.73541	−0.867705	+	0.497080i	$0.834406\pi$
$354$	0	0
$355$	−2.23200e9	−2.64786
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.77386e8	0.316413	0.158206	−	0.987406i	$-0.449429\pi$
$359$	2.77386e8	0.316413	0.158206	+	0.987406i	$0.449429\pi$
$360$	0	0
$361$	2.71418e8	0.303643
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.85569e9	−3.07388
$366$	0	0
$367$	−1.04550e9	−1.10406	−0.552030	−	0.833824i	$-0.686147\pi$
$367$	−1.04550e9	−1.10406	−0.552030	+	0.833824i	$0.686147\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5.85906e8	0.595689
$372$	0	0
$373$	1.00446e8	0.100219	0.0501097	−	0.998744i	$-0.484043\pi$
$373$	1.00446e8	0.100219	0.0501097	+	0.998744i	$0.484043\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.85347e8	0.850979
$378$	0	0
$379$	−4.16517e7	−0.0393002	−0.0196501	−	0.999807i	$-0.506255\pi$
$379$	−4.16517e7	−0.0393002	−0.0196501	+	0.999807i	$0.506255\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6.91160e8	0.628612	0.314306	−	0.949322i	$-0.398228\pi$
$383$	6.91160e8	0.628612	0.314306	+	0.949322i	$0.398228\pi$
$384$	0	0
$385$	4.45643e8	0.397992
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.19534e9	−1.02960	−0.514799	−	0.857311i	$-0.672133\pi$
$389$	−1.19534e9	−1.02960	−0.514799	+	0.857311i	$0.672133\pi$
$390$	0	0
$391$	2.40844e9	2.03759
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.83247e9	1.49606
$396$	0	0
$397$	1.05066e8	0.0842745	0.0421373	−	0.999112i	$-0.486583\pi$
$397$	1.05066e8	0.0842745	0.0421373	+	0.999112i	$0.486583\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.95595e9	1.51479	0.757395	−	0.652957i	$-0.226472\pi$
$401$	1.95595e9	1.51479	0.757395	+	0.652957i	$0.226472\pi$
$402$	0	0
$403$	−1.48238e9	−1.12821
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.30862e9	−0.962127
$408$	0	0
$409$	−1.70570e9	−1.23274	−0.616370	−	0.787457i	$-0.711397\pi$
$409$	−1.70570e9	−1.23274	−0.616370	+	0.787457i	$0.711397\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.86877e7	0.0479793
$414$	0	0
$415$	−3.09750e9	−2.12737
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.64569e9	1.09294	0.546472	−	0.837477i	$-0.315970\pi$
$419$	1.64569e9	1.09294	0.546472	+	0.837477i	$0.315970\pi$
$420$	0	0
$421$	−1.01405e9	−0.662327	−0.331164	−	0.943573i	$-0.607441\pi$
$421$	−1.01405e9	−0.662327	−0.331164	+	0.943573i	$0.607441\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.41682e9	3.42281
$426$	0	0
$427$	3.23077e8	0.200820
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.44415e8	−0.327537	−0.163768	−	0.986499i	$-0.552365\pi$
$431$	−5.44415e8	−0.327537	−0.163768	+	0.986499i	$0.552365\pi$
$432$	0	0
$433$	−7.16016e8	−0.423853	−0.211927	−	0.977286i	$-0.567974\pi$
$433$	−7.16016e8	−0.423853	−0.211927	+	0.977286i	$0.567974\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.19263e9	−1.83005
$438$	0	0
$439$	−1.77093e9	−0.999025	−0.499513	−	0.866307i	$-0.666488\pi$
$439$	−1.77093e9	−0.999025	−0.499513	+	0.866307i	$0.666488\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5.61387e7	−0.0306796	−0.0153398	−	0.999882i	$-0.504883\pi$
$443$	−5.61387e7	−0.0306796	−0.0153398	+	0.999882i	$0.504883\pi$
$444$	0	0
$445$	2.88558e9	1.55229
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.41414e8	0.230136	0.115068	−	0.993358i	$-0.463291\pi$
$449$	4.41414e8	0.230136	0.115068	+	0.993358i	$0.463291\pi$
$450$	0	0
$451$	−5.30625e8	−0.272377
$452$	0	0
$453$	0	0
$454$	0	0
$455$	9.56302e8	0.475943
$456$	0	0
$457$	2.27735e9	1.11615	0.558075	−	0.829790i	$-0.311540\pi$
$457$	2.27735e9	1.11615	0.558075	+	0.829790i	$0.311540\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3.02054e9	−1.43593	−0.717963	−	0.696081i	$-0.754925\pi$
$461$	−3.02054e9	−1.43593	−0.717963	+	0.696081i	$0.754925\pi$
$462$	0	0
$463$	−4.14855e9	−1.94251	−0.971253	−	0.238050i	$-0.923492\pi$
$463$	−4.14855e9	−1.94251	−0.971253	+	0.238050i	$0.923492\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8.64631e8	−0.392845	−0.196423	−	0.980519i	$-0.562932\pi$
$467$	−8.64631e8	−0.392845	−0.196423	+	0.980519i	$0.562932\pi$
$468$	0	0
$469$	8.35437e8	0.373946
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.12062e9	0.921399
$474$	0	0
$475$	−7.18055e9	−3.07418
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−8.88155e8	−0.369245	−0.184622	−	0.982810i	$-0.559106\pi$
$479$	−8.88155e8	−0.369245	−0.184622	+	0.982810i	$0.559106\pi$
$480$	0	0
$481$	−2.80816e9	−1.15057
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.90002e9	1.55228
$486$	0	0
$487$	−3.71672e9	−1.45817	−0.729086	−	0.684422i	$-0.760055\pi$
$487$	−3.71672e9	−1.45817	−0.729086	+	0.684422i	$0.760055\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.38376e9	−1.67133	−0.835664	−	0.549241i	$-0.814917\pi$
$491$	−4.38376e9	−1.67133	−0.835664	+	0.549241i	$0.814917\pi$
$492$	0	0
$493$	4.39208e9	1.65084
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.42540e9	−0.520821
$498$	0	0
$499$	1.44123e9	0.519256	0.259628	−	0.965709i	$-0.416400\pi$
$499$	1.44123e9	0.519256	0.259628	+	0.965709i	$0.416400\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.09180e8	−0.0382521	−0.0191260	−	0.999817i	$-0.506088\pi$
$503$	−1.09180e8	−0.0382521	−0.0191260	+	0.999817i	$0.506088\pi$
$504$	0	0
$505$	−1.93117e9	−0.667270
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2.59191e9	−0.871179	−0.435590	−	0.900145i	$-0.643460\pi$
$509$	−2.59191e9	−0.871179	−0.435590	+	0.900145i	$0.643460\pi$
$510$	0	0
$511$	−1.82369e9	−0.604615
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.64184e9	−0.529671
$516$	0	0
$517$	−1.28731e9	−0.409699
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.15106e9	0.356589	0.178294	−	0.983977i	$-0.442942\pi$
$521$	1.15106e9	0.356589	0.178294	+	0.983977i	$0.442942\pi$
$522$	0	0
$523$	−1.69209e8	−0.0517212	−0.0258606	−	0.999666i	$-0.508233\pi$
$523$	−1.69209e8	−0.0517212	−0.0258606	+	0.999666i	$0.508233\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.35387e9	−2.18866
$528$	0	0
$529$	5.34223e9	1.56902
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.13867e9	−0.325725
$534$	0	0
$535$	2.11563e9	0.597311
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.84595e8	0.0782829
$540$	0	0
$541$	−5.89209e8	−0.159985	−0.0799925	−	0.996795i	$-0.525490\pi$
$541$	−5.89209e8	−0.159985	−0.0799925	+	0.996795i	$0.525490\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.72843e9	0.721979
$546$	0	0
$547$	3.40768e7	0.00890231	0.00445116	−	0.999990i	$-0.498583\pi$
$547$	3.40768e7	0.00890231	0.00445116	+	0.999990i	$0.498583\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.82214e9	−1.48270
$552$	0	0
$553$	1.17025e9	0.294266
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.03397e9	1.96987	0.984933	−	0.172934i	$-0.0553246\pi$
$557$	8.03397e9	1.96987	0.984933	+	0.172934i	$0.0553246\pi$
$558$	0	0
$559$	4.55062e9	1.10187
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.65930e9	0.391874	0.195937	−	0.980616i	$-0.437225\pi$
$563$	1.65930e9	0.391874	0.195937	+	0.980616i	$0.437225\pi$
$564$	0	0
$565$	−1.05376e10	−2.45795
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.28966e9	−0.293482	−0.146741	−	0.989175i	$-0.546878\pi$
$569$	−1.28966e9	−0.293482	−0.146741	+	0.989175i	$0.546878\pi$
$570$	0	0
$571$	1.02422e9	0.230233	0.115116	−	0.993352i	$-0.463276\pi$
$571$	1.02422e9	0.230233	0.115116	+	0.993352i	$0.463276\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.96730e10	4.31553
$576$	0	0
$577$	1.92556e9	0.417294	0.208647	−	0.977991i	$-0.433094\pi$
$577$	1.92556e9	0.417294	0.208647	+	0.977991i	$0.433094\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.97812e9	−0.418442
$582$	0	0
$583$	−4.13212e9	−0.863641
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.95122e9	0.398174	0.199087	−	0.979982i	$-0.436202\pi$
$587$	1.95122e9	0.398174	0.199087	+	0.979982i	$0.436202\pi$
$588$	0	0
$589$	9.74829e9	1.96574
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7.05091e7	0.0138853	0.00694263	−	0.999976i	$-0.497790\pi$
$593$	7.05091e7	0.0138853	0.00694263	+	0.999976i	$0.497790\pi$
$594$	0	0
$595$	4.74407e9	0.923298
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−7.94791e9	−1.51098	−0.755491	−	0.655159i	$-0.772601\pi$
$599$	−7.94791e9	−1.51098	−0.755491	+	0.655159i	$0.772601\pi$
$600$	0	0
$601$	7.79115e9	1.46400	0.731999	−	0.681305i	$-0.238587\pi$
$601$	7.79115e9	1.46400	0.731999	+	0.681305i	$0.238587\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.32361e9	1.34456
$606$	0	0
$607$	−3.61289e9	−0.655683	−0.327841	−	0.944733i	$-0.606321\pi$
$607$	−3.61289e9	−0.655683	−0.327841	+	0.944733i	$0.606321\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.76243e9	−0.489944
$612$	0	0
$613$	5.86399e9	1.02821	0.514104	−	0.857728i	$-0.328124\pi$
$613$	5.86399e9	1.02821	0.514104	+	0.857728i	$0.328124\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.16695e9	1.22839	0.614195	−	0.789154i	$-0.289481\pi$
$617$	7.16695e9	1.22839	0.614195	+	0.789154i	$0.289481\pi$
$618$	0	0
$619$	−1.02581e10	−1.73839	−0.869196	−	0.494468i	$-0.835363\pi$
$619$	−1.02581e10	−1.73839	−0.869196	+	0.494468i	$0.835363\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.84278e9	0.305327
$624$	0	0
$625$	2.17097e10	3.55691
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.39308e10	−2.23203
$630$	0	0
$631$	−9.65432e9	−1.52974	−0.764872	−	0.644182i	$-0.777198\pi$
$631$	−9.65432e9	−1.52974	−0.764872	+	0.644182i	$0.777198\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.66654e9	−0.568262
$636$	0	0
$637$	6.10711e8	0.0936156
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5.02362e9	0.753380	0.376690	−	0.926339i	$-0.377062\pi$
$641$	5.02362e9	0.753380	0.376690	+	0.926339i	$0.377062\pi$
$642$	0	0
$643$	−9.98500e9	−1.48119	−0.740593	−	0.671953i	$-0.765455\pi$
$643$	−9.98500e9	−1.48119	−0.740593	+	0.671953i	$0.765455\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.71598e9	0.539397	0.269698	−	0.962945i	$-0.413076\pi$
$647$	3.71598e9	0.539397	0.269698	+	0.962945i	$0.413076\pi$
$648$	0	0
$649$	−4.84422e8	−0.0695613
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3.16574e9	−0.444917	−0.222459	−	0.974942i	$-0.571408\pi$
$653$	−3.16574e9	−0.444917	−0.222459	+	0.974942i	$0.571408\pi$
$654$	0	0
$655$	1.85785e10	2.58325
$656$	0	0
$657$	0	0
$658$	0	0
$659$	9.98585e9	1.35921	0.679604	−	0.733579i	$-0.262151\pi$
$659$	9.98585e9	1.35921	0.679604	+	0.733579i	$0.262151\pi$
$660$	0	0
$661$	1.21710e10	1.63916	0.819578	−	0.572968i	$-0.194208\pi$
$661$	1.21710e10	1.63916	0.819578	+	0.572968i	$0.194208\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−6.28875e9	−0.829256
$666$	0	0
$667$	1.59513e10	2.08140
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2.27851e9	−0.291153
$672$	0	0
$673$	−8.03702e9	−1.01635	−0.508174	−	0.861254i	$-0.669679\pi$
$673$	−8.03702e9	−1.01635	−0.508174	+	0.861254i	$0.669679\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.71271e9	−1.07918	−0.539589	−	0.841929i	$-0.681420\pi$
$677$	−8.71271e9	−1.07918	−0.539589	+	0.841929i	$0.681420\pi$
$678$	0	0
$679$	2.49062e9	0.305325
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.51030e10	−1.81380	−0.906900	−	0.421345i	$-0.861558\pi$
$683$	−1.51030e10	−1.81380	−0.906900	+	0.421345i	$0.861558\pi$
$684$	0	0
$685$	8.85434e8	0.105254
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−8.86710e9	−1.03280
$690$	0	0
$691$	−8.19095e9	−0.944411	−0.472206	−	0.881488i	$-0.656542\pi$
$691$	−8.19095e9	−0.944411	−0.472206	+	0.881488i	$0.656542\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.02678e9	0.455000
$696$	0	0
$697$	−5.64875e9	−0.631885
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6.88854e9	−0.755290	−0.377645	−	0.925950i	$-0.623266\pi$
$701$	−6.88854e9	−0.755290	−0.377645	+	0.925950i	$0.623266\pi$
$702$	0	0
$703$	1.84667e10	2.00469
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.23328e9	−0.131249
$708$	0	0
$709$	−6.06034e9	−0.638609	−0.319304	−	0.947652i	$-0.603449\pi$
$709$	−6.06034e9	−0.638609	−0.319304	+	0.947652i	$0.603449\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.67081e10	−2.75949
$714$	0	0
$715$	−6.74436e9	−0.690032
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.21081e10	−1.21485	−0.607427	−	0.794376i	$-0.707798\pi$
$719$	−1.21081e10	−1.21485	−0.607427	+	0.794376i	$0.707798\pi$
$720$	0	0
$721$	−1.04851e9	−0.104183
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.58762e10	3.49642
$726$	0	0
$727$	5.51884e9	0.532694	0.266347	−	0.963877i	$-0.414183\pi$
$727$	5.51884e9	0.532694	0.266347	+	0.963877i	$0.414183\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.25749e10	2.13755
$732$	0	0
$733$	9.47965e9	0.889054	0.444527	−	0.895765i	$-0.353372\pi$
$733$	9.47965e9	0.889054	0.444527	+	0.895765i	$0.353372\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.89194e9	−0.542154
$738$	0	0
$739$	9.44295e9	0.860701	0.430351	−	0.902662i	$-0.358390\pi$
$739$	9.44295e9	0.860701	0.430351	+	0.902662i	$0.358390\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7.43094e9	0.664635	0.332317	−	0.943168i	$-0.392169\pi$
$743$	7.43094e9	0.664635	0.332317	+	0.943168i	$0.392169\pi$
$744$	0	0
$745$	1.54690e10	1.37062
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.35108e9	0.117488
$750$	0	0
$751$	9.42171e9	0.811689	0.405845	−	0.913942i	$-0.366977\pi$
$751$	9.42171e9	0.811689	0.405845	+	0.913942i	$0.366977\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.53631e10	2.14480
$756$	0	0
$757$	1.93994e9	0.162537	0.0812684	−	0.996692i	$-0.474103\pi$
$757$	1.93994e9	0.162537	0.0812684	+	0.996692i	$0.474103\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.66028e10	1.36563	0.682817	−	0.730589i	$-0.260754\pi$
$761$	1.66028e10	1.36563	0.682817	+	0.730589i	$0.260754\pi$
$762$	0	0
$763$	1.74242e9	0.142009
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.03952e9	−0.0831857
$768$	0	0
$769$	−7.78402e9	−0.617251	−0.308625	−	0.951184i	$-0.599869\pi$
$769$	−7.78402e9	−0.617251	−0.308625	+	0.951184i	$0.599869\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.37574e10	−1.07130	−0.535648	−	0.844441i	$-0.679933\pi$
$773$	−1.37574e10	−1.07130	−0.535648	+	0.844441i	$0.679933\pi$
$774$	0	0
$775$	−6.00692e10	−4.63550
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.48800e9	0.567524
$780$	0	0
$781$	1.00527e10	0.755096
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.34004e10	−0.988725
$786$	0	0
$787$	4.66786e9	0.341355	0.170678	−	0.985327i	$-0.445404\pi$
$787$	4.66786e9	0.341355	0.170678	+	0.985327i	$0.445404\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.72952e9	−0.483467
$792$	0	0
$793$	−4.88944e9	−0.348179
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.78171e10	1.24662	0.623308	−	0.781976i	$-0.285788\pi$
$797$	1.78171e10	1.24662	0.623308	+	0.781976i	$0.285788\pi$
$798$	0	0
$799$	−1.37040e10	−0.950458
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.28617e10	0.876582
$804$	0	0
$805$	1.72297e10	1.16411
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−7.65416e9	−0.508250	−0.254125	−	0.967171i	$-0.581788\pi$
$809$	−7.65416e9	−0.508250	−0.254125	+	0.967171i	$0.581788\pi$
$810$	0	0
$811$	−1.69961e10	−1.11886	−0.559431	−	0.828877i	$-0.688980\pi$
$811$	−1.69961e10	−1.11886	−0.559431	+	0.828877i	$0.688980\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.49954e10	0.970302
$816$	0	0
$817$	−2.99254e10	−1.91983
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−7.21555e8	−0.0455060	−0.0227530	−	0.999741i	$-0.507243\pi$
$821$	−7.21555e8	−0.0455060	−0.0227530	+	0.999741i	$0.507243\pi$
$822$	0	0
$823$	2.18708e10	1.36762	0.683808	−	0.729662i	$-0.260322\pi$
$823$	2.18708e10	1.36762	0.683808	+	0.729662i	$0.260322\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.17686e10	−1.33832	−0.669161	−	0.743117i	$-0.733346\pi$
$827$	−2.17686e10	−1.33832	−0.669161	+	0.743117i	$0.733346\pi$
$828$	0	0
$829$	−2.57497e10	−1.56975	−0.784875	−	0.619654i	$-0.787273\pi$
$829$	−2.57497e10	−1.56975	−0.784875	+	0.619654i	$0.787273\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.02965e9	0.181608
$834$	0	0
$835$	3.46356e9	0.205883
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.99796e9	0.409077	0.204538	−	0.978859i	$-0.434431\pi$
$839$	6.99796e9	0.409077	0.204538	+	0.978859i	$0.434431\pi$
$840$	0	0
$841$	1.18393e10	0.686343
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.92294e10	1.09640
$846$	0	0
$847$	4.67698e9	0.264468
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−5.05946e10	−2.81417
$852$	0	0
$853$	−3.00829e10	−1.65958	−0.829790	−	0.558076i	$-0.811540\pi$
$853$	−3.00829e10	−1.65958	−0.829790	+	0.558076i	$0.811540\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.98054e10	−1.61757	−0.808783	−	0.588107i	$-0.799873\pi$
$857$	−2.98054e10	−1.61757	−0.808783	+	0.588107i	$0.799873\pi$
$858$	0	0
$859$	1.44457e9	0.0777612	0.0388806	−	0.999244i	$-0.487621\pi$
$859$	1.44457e9	0.0777612	0.0388806	+	0.999244i	$0.487621\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.26360e9	0.119884	0.0599421	−	0.998202i	$-0.480908\pi$
$863$	2.26360e9	0.119884	0.0599421	+	0.998202i	$0.480908\pi$
$864$	0	0
$865$	2.41136e10	1.26679
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8.25323e9	−0.426633
$870$	0	0
$871$	−1.26435e10	−0.648341
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.43589e10	1.22922
$876$	0	0
$877$	−1.22601e9	−0.0613754	−0.0306877	−	0.999529i	$-0.509770\pi$
$877$	−1.22601e9	−0.0613754	−0.0306877	+	0.999529i	$0.509770\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−9.59512e9	−0.472753	−0.236377	−	0.971661i	$-0.575960\pi$
$881$	−9.59512e9	−0.472753	−0.236377	+	0.971661i	$0.575960\pi$
$882$	0	0
$883$	−1.76041e10	−0.860500	−0.430250	−	0.902710i	$-0.641575\pi$
$883$	−1.76041e10	−0.860500	−0.430250	+	0.902710i	$0.641575\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.33540e10	−1.12364	−0.561821	−	0.827259i	$-0.689899\pi$
$887$	−2.33540e10	−1.12364	−0.561821	+	0.827259i	$0.689899\pi$
$888$	0	0
$889$	−2.34152e9	−0.111774
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.81660e10	0.853650
$894$	0	0
$895$	3.50210e9	0.163286
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4.87055e10	−2.23573
$900$	0	0
$901$	−4.39884e10	−2.00355
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.85479e10	1.28028
$906$	0	0
$907$	−2.41519e10	−1.07480	−0.537398	−	0.843329i	$-0.680593\pi$
$907$	−2.41519e10	−1.07480	−0.537398	+	0.843329i	$0.680593\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.95549e10	−0.856923	−0.428462	−	0.903560i	$-0.640944\pi$
$911$	−1.95549e10	−0.856923	−0.428462	+	0.903560i	$0.640944\pi$
$912$	0	0
$913$	1.39507e10	0.606666
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.18646e10	0.508112
$918$	0	0
$919$	3.05841e10	1.29984	0.649921	−	0.760002i	$-0.274802\pi$
$919$	3.05841e10	1.29984	0.649921	+	0.760002i	$0.274802\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.15720e10	0.902991
$924$	0	0
$925$	−1.13793e11	−4.72735
$926$	0	0
$927$	0	0
$928$	0	0
$929$	2.53417e10	1.03700	0.518502	−	0.855076i	$-0.326490\pi$
$929$	2.53417e10	1.03700	0.518502	+	0.855076i	$0.326490\pi$
$930$	0	0
$931$	−4.01611e9	−0.163110
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3.34577e10	−1.33862
$936$	0	0
$937$	3.21831e9	0.127802	0.0639012	−	0.997956i	$-0.479646\pi$
$937$	3.21831e9	0.127802	0.0639012	+	0.997956i	$0.479646\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.54966e10	0.606278	0.303139	−	0.952946i	$-0.401965\pi$
$941$	1.54966e10	0.606278	0.303139	+	0.952946i	$0.401965\pi$
$942$	0	0
$943$	−2.05154e10	−0.796689
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.26112e10	−1.24779	−0.623896	−	0.781508i	$-0.714451\pi$
$947$	−3.26112e10	−1.24779	−0.623896	+	0.781508i	$0.714451\pi$
$948$	0	0
$949$	2.75998e10	1.04827
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.30210e10	−1.23585	−0.617925	−	0.786237i	$-0.712026\pi$
$953$	−3.30210e10	−1.23585	−0.617925	+	0.786237i	$0.712026\pi$
$954$	0	0
$955$	4.57713e10	1.70052
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.65454e8	0.0207029
$960$	0	0
$961$	5.40373e10	1.96409
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.17598e10	0.421264
$966$	0	0
$967$	1.70652e9	0.0606904	0.0303452	−	0.999539i	$-0.490339\pi$
$967$	1.70652e9	0.0606904	0.0303452	+	0.999539i	$0.490339\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.51748e10	0.882469	0.441234	−	0.897392i	$-0.354541\pi$
$971$	2.51748e10	0.882469	0.441234	+	0.897392i	$0.354541\pi$
$972$	0	0
$973$	2.57157e9	0.0894961
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.44905e10	1.18323	0.591615	−	0.806221i	$-0.298491\pi$
$977$	3.44905e10	1.18323	0.591615	+	0.806221i	$0.298491\pi$
$978$	0	0
$979$	−1.29963e10	−0.442670
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.14368e10	0.719819	0.359909	−	0.932987i	$-0.382808\pi$
$983$	2.14368e10	0.719819	0.359909	+	0.932987i	$0.382808\pi$
$984$	0	0
$985$	−8.99105e10	−2.99767
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.19886e10	2.69505
$990$	0	0
$991$	−3.90017e10	−1.27299	−0.636497	−	0.771280i	$-0.719617\pi$
$991$	−3.90017e10	−1.27299	−0.636497	+	0.771280i	$0.719617\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3.31200e10	−1.06588
$996$	0	0
$997$	−2.29029e8	−0.00731911	−0.00365955	−	0.999993i	$-0.501165\pi$
$997$	−2.29029e8	−0.00731911	−0.00365955	+	0.999993i	$0.501165\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.8.a.g.1.1	✓	4
3.2	odd	2	inner	252.8.a.g.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.8.a.g.1.1	✓	4	1.1	even	1	trivial
252.8.a.g.1.4	yes	4	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	252.a (trivial)

\(f(q)\)	\(=\)	\(q-537.098 q^{5} -343.000 q^{7} +2419.02 q^{11} +5190.96 q^{13} +25751.6 q^{17} -34136.3 q^{19} +93525.7 q^{23} +210349. q^{25} +170556. q^{29} -285569. q^{31} +184225. q^{35} -540970. q^{37} -219356. q^{41} +876643. q^{43} -532161. q^{47} +117649. q^{49} -1.70818e6 q^{53} -1.29925e6 q^{55} -200256. q^{59} -941914. q^{61} -2.78805e6 q^{65} -2.43568e6 q^{67} +4.15568e6 q^{71} +5.31689e6 q^{73} -829724. q^{77} -3.41181e6 q^{79} +5.76711e6 q^{83} -1.38311e7 q^{85} -5.37254e6 q^{89} -1.78050e6 q^{91} +1.83346e7 q^{95} -7.26128e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 1372 q^{7} + 13272 q^{13} - 9184 q^{19} + 264524 q^{25} - 445536 q^{31} - 852808 q^{37} + 150224 q^{43} + 470596 q^{49} - 2627296 q^{55} - 1302840 q^{61} - 6071696 q^{67} + 3631768 q^{73} - 16898688 q^{79}+ \cdots - 36492008 q^{97}+O(q^{100}) \) 4 * q - 1372 * q^7 + 13272 * q^13 - 9184 * q^19 + 264524 * q^25 - 445536 * q^31 - 852808 * q^37 + 150224 * q^43 + 470596 * q^49 - 2627296 * q^55 - 1302840 * q^61 - 6071696 * q^67 + 3631768 * q^73 - 16898688 * q^79 - 27267520 * q^85 - 4552296 * q^91 - 36492008 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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