LMFDB - Embedded newform 84.12.f.b.41.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,12,Mod(41,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("84.41");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	84.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$64.5408271670$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			41.14
Character	$\chi$	$=$	84.41
Dual form			84.12.f.b.41.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-24.0407 + 420.201i) q^{3} +11925.0 q^{5} +(-37084.7 - 24536.8i) q^{7} +(-175991. - 20203.9i) q^{9} +O(q^{10})$	\(q+(-24.0407 + 420.201i) q^{3} +11925.0 q^{5} +(-37084.7 - 24536.8i) q^{7} +(-175991. - 20203.9i) q^{9} +258716. i q^{11} -1.86405e6i q^{13} +(-286686. + 5.01090e6i) q^{15} +8.63671e6 q^{17} +1.34269e7i q^{19} +(1.12019e7 - 1.49931e7i) q^{21} +3.97432e7i q^{23} +9.33778e7 q^{25} +(1.27207e7 - 7.34659e7i) q^{27} -1.49714e8i q^{29} -4.61001e7i q^{31} +(-1.08713e8 - 6.21972e6i) q^{33} +(-4.42235e8 - 2.92601e8i) q^{35} -9.06276e7 q^{37} +(7.83276e8 + 4.48131e7i) q^{39} +1.17043e9 q^{41} -3.46043e8 q^{43} +(-2.09870e9 - 2.40932e8i) q^{45} +2.49596e9 q^{47} +(7.73220e8 + 1.81988e9i) q^{49} +(-2.07633e8 + 3.62916e9i) q^{51} +1.36550e9i q^{53} +3.08519e9i q^{55} +(-5.64200e9 - 3.22792e8i) q^{57} +3.00475e9 q^{59} +6.74891e9i q^{61} +(6.03083e9 + 5.06751e9i) q^{63} -2.22288e10i q^{65} -2.90561e8 q^{67} +(-1.67001e10 - 9.55456e8i) q^{69} +1.33562e10i q^{71} -3.40236e9i q^{73} +(-2.24487e9 + 3.92375e10i) q^{75} +(6.34805e9 - 9.59439e9i) q^{77} +3.51198e10 q^{79} +(3.05647e10 + 7.11141e9i) q^{81} +5.87747e9 q^{83} +1.02993e11 q^{85} +(6.29101e10 + 3.59924e9i) q^{87} +9.42901e8 q^{89} +(-4.57378e10 + 6.91277e10i) q^{91} +(1.93713e10 + 1.10828e9i) q^{93} +1.60116e11i q^{95} +1.56389e11i q^{97} +(5.22707e9 - 4.55317e10i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 9632 q^{7} + 267660 q^{9}+O(q^{10}) $ 28 * q + 9632 * q^7 + 267660 * q^9	$ 28 q + 9632 q^{7} + 267660 q^{9} - 3434160 q^{15} - 18804156 q^{21} + 397876900 q^{25} - 2059460504 q^{37} + 2276313936 q^{39} + 607100560 q^{43} + 1145242588 q^{49} + 1424787216 q^{51} - 32512522344 q^{57} + 16390616256 q^{63} - 48876957136 q^{67} - 1293110368 q^{79} + 82706814108 q^{81} + 197440859760 q^{85} - 329206232880 q^{91} - 243855044280 q^{93} - 81383696064 q^{99}+O(q^{100}) $ 28 * q + 9632 * q^7 + 267660 * q^9 - 3434160 * q^15 - 18804156 * q^21 + 397876900 * q^25 - 2059460504 * q^37 + 2276313936 * q^39 + 607100560 * q^43 + 1145242588 * q^49 + 1424787216 * q^51 - 32512522344 * q^57 + 16390616256 * q^63 - 48876957136 * q^67 - 1293110368 * q^79 + 82706814108 * q^81 + 197440859760 * q^85 - 329206232880 * q^91 - 243855044280 * q^93 - 81383696064 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−24.0407	+	420.201i	−0.0571190	+	0.998367i
$3$	−24.0407	+	420.201i	−0.0571190	+	0.998367i
$4$		0			0
$5$	11925.0			1.70657			0.853284	−	0.521446i	$-0.174607\pi$
$5$	11925.0			1.70657			0.853284	+	0.521446i	$0.174607\pi$
$6$		0			0
$7$	−37084.7	−	24536.8i	−0.833979	−	0.551796i
$7$	−37084.7	−	24536.8i	−0.833979	−	0.551796i
$8$		0			0
$9$	−175991.	−	20203.9i	−0.993475	−	0.114052i
$10$		0			0
$11$			258716.i			0.484354i	0.970232	+	0.242177i	$0.0778615\pi$
$11$			258716.i			0.484354i	−0.970232	+	0.242177i	$0.922138\pi$
$12$		0			0
$13$		−	1.86405e6i		−	1.39242i	−0.717840	−	0.696209i	$-0.754869\pi$
$13$		−	1.86405e6i		−	1.39242i	0.717840	−	0.696209i	$-0.245131\pi$
$14$		0			0
$15$	−286686.	+	5.01090e6i	−0.0974775	+	1.70378i
$16$		0			0
$17$	8.63671e6			1.47530			0.737649	−	0.675185i	$-0.235936\pi$
$17$	8.63671e6			1.47530			0.737649	+	0.675185i	$0.235936\pi$
$18$		0			0
$19$			1.34269e7i			1.24403i	0.783005	+	0.622015i	$0.213686\pi$
$19$			1.34269e7i			1.24403i	−0.783005	+	0.622015i	$0.786314\pi$
$20$		0			0
$21$	1.12019e7	−	1.49931e7i	0.598531	−	0.801100i
$22$		0			0
$23$			3.97432e7i			1.28754i	0.765220	+	0.643769i	$0.222630\pi$
$23$			3.97432e7i			1.28754i	−0.765220	+	0.643769i	$0.777370\pi$
$24$		0			0
$25$	9.33778e7			1.91238
$26$		0			0
$27$	1.27207e7	−	7.34659e7i	0.170612	−	0.985338i
$28$		0			0
$29$		−	1.49714e8i		−	1.35542i	−0.735329	−	0.677710i	$-0.762972\pi$
$29$		−	1.49714e8i		−	1.35542i	0.735329	−	0.677710i	$-0.237028\pi$
$30$		0			0
$31$		−	4.61001e7i		−	0.289210i	−0.989489	−	0.144605i	$-0.953809\pi$
$31$		−	4.61001e7i		−	0.289210i	0.989489	−	0.144605i	$-0.0461911\pi$
$32$		0			0
$33$	−1.08713e8	−	6.21972e6i	−0.483564	−	0.0276658i
$34$		0			0
$35$	−4.42235e8	−	2.92601e8i	−1.42324	−	0.941677i
$36$		0			0
$37$	−9.06276e7			−0.214858			−0.107429	−	0.994213i	$-0.534262\pi$
$37$	−9.06276e7			−0.214858			−0.107429	+	0.994213i	$0.534262\pi$
$38$		0			0
$39$	7.83276e8	+	4.48131e7i	1.39014	+	0.0795335i
$40$		0			0
$41$	1.17043e9			1.57773			0.788867	−	0.614565i	$-0.210668\pi$
$41$	1.17043e9			1.57773			0.788867	+	0.614565i	$0.210668\pi$
$42$		0			0
$43$	−3.46043e8			−0.358966			−0.179483	−	0.983761i	$-0.557443\pi$
$43$	−3.46043e8			−0.358966			−0.179483	+	0.983761i	$0.557443\pi$
$44$		0			0
$45$	−2.09870e9	−	2.40932e8i	−1.69543	−	0.194637i
$46$		0			0
$47$	2.49596e9			1.58745			0.793723	−	0.608280i	$-0.208140\pi$
$47$	2.49596e9			1.58745			0.793723	+	0.608280i	$0.208140\pi$
$48$		0			0
$49$	7.73220e8	+	1.81988e9i	0.391043	+	0.920372i
$50$		0			0
$51$	−2.07633e8	+	3.62916e9i	−0.0842675	+	1.47289i
$52$		0			0
$53$			1.36550e9i			0.448513i	0.974530	+	0.224257i	$0.0719954\pi$
$53$			1.36550e9i			0.448513i	−0.974530	+	0.224257i	$0.928005\pi$
$54$		0			0
$55$			3.08519e9i			0.826584i
$56$		0			0
$57$	−5.64200e9	−	3.22792e8i	−1.24200	−	0.0710578i
$58$		0			0
$59$	3.00475e9			0.547170			0.273585	−	0.961848i	$-0.411791\pi$
$59$	3.00475e9			0.547170			0.273585	+	0.961848i	$0.411791\pi$
$60$		0			0
$61$			6.74891e9i			1.02310i	0.859252	+	0.511552i	$0.170929\pi$
$61$			6.74891e9i			1.02310i	−0.859252	+	0.511552i	$0.829071\pi$
$62$		0			0
$63$	6.03083e9	+	5.06751e9i	0.765604	+	0.643312i
$64$		0			0
$65$		−	2.22288e10i		−	2.37626i
$66$		0			0
$67$	−2.90561e8			−0.0262922			−0.0131461	−	0.999914i	$-0.504185\pi$
$67$	−2.90561e8			−0.0262922			−0.0131461	+	0.999914i	$0.504185\pi$
$68$		0			0
$69$	−1.67001e10	−	9.55456e8i	−1.28544	−	0.0735429i
$70$		0			0
$71$			1.33562e10i			0.878540i	0.898355	+	0.439270i	$0.144763\pi$
$71$			1.33562e10i			0.878540i	−0.898355	+	0.439270i	$0.855237\pi$
$72$		0			0
$73$		−	3.40236e9i		−	0.192090i	−0.995377	−	0.0960448i	$-0.969381\pi$
$73$		−	3.40236e9i		−	0.192090i	0.995377	−	0.0960448i	$-0.0306192\pi$
$74$		0			0
$75$	−2.24487e9	+	3.92375e10i	−0.109233	+	1.90925i
$76$		0			0
$77$	6.34805e9	−	9.59439e9i	0.267265	−	0.403942i
$78$		0			0
$79$	3.51198e10			1.28411			0.642056	−	0.766658i	$-0.278082\pi$
$79$	3.51198e10			1.28411			0.642056	+	0.766658i	$0.278082\pi$
$80$		0			0
$81$	3.05647e10	+	7.11141e9i	0.973984	+	0.226615i
$82$		0			0
$83$	5.87747e9			0.163780			0.0818900	−	0.996641i	$-0.473904\pi$
$83$	5.87747e9			0.163780			0.0818900	+	0.996641i	$0.473904\pi$
$84$		0			0
$85$	1.02993e11			2.51770
$86$		0			0
$87$	6.29101e10	+	3.59924e9i	1.35321	+	0.0774203i
$88$		0			0
$89$	9.42901e8			0.0178987			0.00894934	−	0.999960i	$-0.497151\pi$
$89$	9.42901e8			0.0178987			0.00894934	+	0.999960i	$0.497151\pi$
$90$		0			0
$91$	−4.57378e10	+	6.91277e10i	−0.768330	+	1.16125i
$92$		0			0
$93$	1.93713e10	+	1.10828e9i	0.288737	+	0.0165194i
$94$		0			0
$95$			1.60116e11i			2.12302i
$96$		0			0
$97$			1.56389e11i			1.84911i	0.381048	+	0.924555i	$0.375563\pi$
$97$			1.56389e11i			1.84911i	−0.381048	+	0.924555i	$0.624437\pi$
$98$		0			0
$99$	5.22707e9	−	4.55317e10i	0.0552414	−	0.481194i
$100$		0			0
$101$	−1.61093e11			−1.52514			−0.762568	−	0.646908i	$-0.776062\pi$
$101$	−1.61093e11			−1.52514			−0.762568	+	0.646908i	$0.776062\pi$
$102$		0			0
$103$		−	1.13586e10i		−	0.0965428i	−0.998834	−	0.0482714i	$-0.984629\pi$
$103$		−	1.13586e10i		−	0.0965428i	0.998834	−	0.0482714i	$-0.0153712\pi$
$104$		0			0
$105$	1.33583e11	−	1.78793e11i	1.02143	−	1.36713i
$106$		0			0
$107$		−	1.84938e11i		−	1.27472i	−0.770566	−	0.637361i	$-0.780026\pi$
$107$		−	1.84938e11i		−	1.27472i	0.770566	−	0.637361i	$-0.219974\pi$
$108$		0			0
$109$	8.41909e10			0.524106			0.262053	−	0.965053i	$-0.415600\pi$
$109$	8.41909e10			0.524106			0.262053	+	0.965053i	$0.415600\pi$
$110$		0			0
$111$	2.17875e9	−	3.80818e10i	0.0122725	−	0.214507i
$112$		0			0
$113$			1.14641e11i			0.585339i	0.956214	+	0.292669i	$0.0945436\pi$
$113$			1.14641e11i			0.585339i	−0.956214	+	0.292669i	$0.905456\pi$
$114$		0			0
$115$			4.73938e11i			2.19727i
$116$		0			0
$117$	−3.76611e10	+	3.28056e11i	−0.158807	+	1.38333i
$118$		0			0
$119$	−3.20290e11	−	2.11917e11i	−1.23037	−	0.814063i
$120$		0			0
$121$	2.18378e11			0.765401
$122$		0			0
$123$	−2.81379e10	+	4.91815e11i	−0.0901186	+	1.57516i
$124$		0			0
$125$	5.31255e11			1.55703
$126$		0			0
$127$	−5.67494e11			−1.52420			−0.762099	−	0.647461i	$-0.775831\pi$
$127$	−5.67494e11			−1.52420			−0.762099	+	0.647461i	$0.775831\pi$
$128$		0			0
$129$	8.31913e9	−	1.45408e11i	0.0205038	−	0.358380i
$130$		0			0
$131$	1.92379e10			0.0435679			0.0217839	−	0.999763i	$-0.493065\pi$
$131$	1.92379e10			0.0435679			0.0217839	+	0.999763i	$0.493065\pi$
$132$		0			0
$133$	3.29453e11	−	4.97932e11i	0.686450	−	1.03750i
$134$		0			0
$135$	1.51694e11	−	8.76082e11i	0.291161	−	1.68155i
$136$		0			0
$137$			1.02475e12i			1.81407i	0.421055	+	0.907035i	$0.361660\pi$
$137$			1.02475e12i			1.81407i	−0.421055	+	0.907035i	$0.638340\pi$
$138$		0			0
$139$		−	2.17838e10i		−	0.0356084i	−0.999841	−	0.0178042i	$-0.994332\pi$
$139$		−	2.17838e10i		−	0.0356084i	0.999841	−	0.0178042i	$-0.00566756\pi$
$140$		0			0
$141$	−6.00046e10	+	1.04880e12i	−0.0906733	+	1.58485i
$142$		0			0
$143$	4.82259e11			0.674423
$144$		0			0
$145$		−	1.78534e12i		−	2.31312i
$146$		0			0
$147$	−7.83303e11	+	2.81157e11i	−0.941206	+	0.337834i
$148$		0			0
$149$		−	1.25151e12i		−	1.39608i	−0.716061	−	0.698038i	$-0.754057\pi$
$149$		−	1.25151e12i		−	1.39608i	0.716061	−	0.698038i	$-0.245943\pi$
$150$		0			0
$151$	1.32168e12			1.37010			0.685049	−	0.728497i	$-0.259781\pi$
$151$	1.32168e12			1.37010			0.685049	+	0.728497i	$0.259781\pi$
$152$		0			0
$153$	−1.51998e12	−	1.74495e11i	−1.46567	−	0.168260i
$154$		0			0
$155$		−	5.49744e11i		−	0.493556i
$156$		0			0
$157$		−	2.88328e11i		−	0.241235i	−0.992699	−	0.120617i	$-0.961513\pi$
$157$		−	2.88328e11i		−	0.241235i	0.992699	−	0.120617i	$-0.0384874\pi$
$158$		0			0
$159$	−5.73786e11	−	3.28277e10i	−0.447781	−	0.0256186i
$160$		0			0
$161$	9.75170e11	−	1.47386e12i	0.710457	−	1.07378i
$162$		0			0
$163$	6.32411e11			0.430495			0.215247	−	0.976560i	$-0.430944\pi$
$163$	6.32411e11			0.430495			0.215247	+	0.976560i	$0.430944\pi$
$164$		0			0
$165$	−1.29640e12	−	7.41702e10i	−0.825235	−	0.0472137i
$166$		0			0
$167$	−1.71070e12			−1.01914			−0.509568	−	0.860430i	$-0.670195\pi$
$167$	−1.71070e12			−1.01914			−0.509568	+	0.860430i	$0.670195\pi$
$168$		0			0
$169$	−1.68253e12			−0.938825
$170$		0			0
$171$	2.71276e11	−	2.36301e12i	0.141884	−	1.23591i
$172$		0			0
$173$	1.62171e12			0.795648			0.397824	−	0.917462i	$-0.369766\pi$
$173$	1.62171e12			0.795648			0.397824	+	0.917462i	$0.369766\pi$
$174$		0			0
$175$	−3.46289e12	−	2.29119e12i	−1.59488	−	1.05524i
$176$		0			0
$177$	−7.22364e10	+	1.26260e12i	−0.0312538	+	0.546277i
$178$		0			0
$179$			3.08640e12i			1.25534i	0.778480	+	0.627670i	$0.215991\pi$
$179$			3.08640e12i			1.25534i	−0.778480	+	0.627670i	$0.784009\pi$
$180$		0			0
$181$		−	3.47401e12i		−	1.32923i	−0.747188	−	0.664613i	$-0.768597\pi$
$181$		−	3.47401e12i		−	1.32923i	0.747188	−	0.664613i	$-0.231403\pi$
$182$		0			0
$183$	−2.83590e12	−	1.62249e11i	−1.02143	−	0.0584387i
$184$		0			0
$185$	−1.08073e12			−0.366669
$186$		0			0
$187$			2.23445e12i			0.714567i
$188$		0			0
$189$	−2.27436e12	+	2.41234e12i	−0.685992	+	0.727609i
$190$		0			0
$191$			1.66588e12i			0.474199i	0.971485	+	0.237099i	$0.0761967\pi$
$191$			1.66588e12i			0.474199i	−0.971485	+	0.237099i	$0.923803\pi$
$192$		0			0
$193$	−4.52875e12			−1.21734			−0.608671	−	0.793423i	$-0.708297\pi$
$193$	−4.52875e12			−1.21734			−0.608671	+	0.793423i	$0.708297\pi$
$194$		0			0
$195$	9.34058e12	+	5.34397e11i	2.37238	+	0.135729i
$196$		0			0
$197$		−	2.62445e11i		−	0.0630194i	−0.999503	−	0.0315097i	$-0.989968\pi$
$197$		−	2.62445e11i		−	0.0630194i	0.999503	−	0.0315097i	$-0.0100315\pi$
$198$		0			0
$199$		−	2.96055e12i		−	0.672482i	−0.941776	−	0.336241i	$-0.890844\pi$
$199$		−	2.96055e12i		−	0.672482i	0.941776	−	0.336241i	$-0.109156\pi$
$200$		0			0
$201$	6.98531e9	−	1.22094e11i	0.00150178	−	0.0262492i
$202$		0			0
$203$	−3.67350e12	+	5.55210e12i	−0.747915	+	1.13039i
$204$		0			0
$205$	1.39574e13			2.69251
$206$		0			0
$207$	8.02967e11	−	6.99445e12i	0.146846	−	1.27914i
$208$		0			0
$209$	−3.47375e12			−0.602551
$210$		0			0
$211$	3.70312e12			0.609556			0.304778	−	0.952423i	$-0.401418\pi$
$211$	3.70312e12			0.609556			0.304778	+	0.952423i	$0.401418\pi$
$212$		0			0
$213$	−5.61228e12	−	3.21092e11i	−0.877105	−	0.0501813i
$214$		0			0
$215$	−4.12657e12			−0.612600
$216$		0			0
$217$	−1.13115e12	+	1.70961e12i	−0.159585	+	0.241195i
$218$		0			0
$219$	1.42967e12	+	8.17951e10i	0.191776	+	0.0109720i
$220$		0			0
$221$		−	1.60993e13i		−	2.05423i
$222$		0			0
$223$		−	4.88727e11i		−	0.0593457i	−0.999560	−	0.0296728i	$-0.990553\pi$
$223$		−	4.88727e11i		−	0.0593457i	0.999560	−	0.0296728i	$-0.00944655\pi$
$224$		0			0
$225$	−1.64337e13	−	1.88659e12i	−1.89990	−	0.218110i
$226$		0			0
$227$	5.54329e12			0.610416			0.305208	−	0.952286i	$-0.401274\pi$
$227$	5.54329e12			0.610416			0.305208	+	0.952286i	$0.401274\pi$
$228$		0			0
$229$		−	5.21655e12i		−	0.547379i	−0.961818	−	0.273689i	$-0.911756\pi$
$229$		−	5.21655e12i		−	0.547379i	0.961818	−	0.273689i	$-0.0882441\pi$
$230$		0			0
$231$	3.87896e12	+	2.89812e12i	0.388016	+	0.289901i
$232$		0			0
$233$			2.42945e12i			0.231766i	0.993263	+	0.115883i	$0.0369698\pi$
$233$			2.42945e12i			0.231766i	−0.993263	+	0.115883i	$0.963030\pi$
$234$		0			0
$235$	2.97643e13			2.70908
$236$		0			0
$237$	−8.44305e11	+	1.47574e13i	−0.0733472	+	1.28202i
$238$		0			0
$239$			2.74816e12i			0.227957i	0.993483	+	0.113979i	$0.0363595\pi$
$239$			2.74816e12i			0.227957i	−0.993483	+	0.113979i	$0.963640\pi$
$240$		0			0
$241$		−	2.78257e12i		−	0.220472i	−0.993905	−	0.110236i	$-0.964839\pi$
$241$		−	2.78257e12i		−	0.220472i	0.993905	−	0.110236i	$-0.0351606\pi$
$242$		0			0
$243$	−3.72302e12	+	1.26723e13i	−0.281878	+	0.959450i
$244$		0			0
$245$	9.22066e12	+	2.17021e13i	0.667342	+	1.57068i
$246$		0			0
$247$	2.50284e13			1.73221
$248$		0			0
$249$	−1.41299e11	+	2.46972e12i	−0.00935496	+	0.163513i
$250$		0			0
$251$	−1.03292e13			−0.654430			−0.327215	−	0.944950i	$-0.606110\pi$
$251$	−1.03292e13			−0.654430			−0.327215	+	0.944950i	$0.606110\pi$
$252$		0			0
$253$	−1.02822e13			−0.623624
$254$		0			0
$255$	−2.47603e12	+	4.32777e13i	−0.143808	+	2.51359i
$256$		0			0
$257$	2.25510e13			1.25468			0.627340	−	0.778745i	$-0.284143\pi$
$257$	2.25510e13			1.25468			0.627340	+	0.778745i	$0.284143\pi$
$258$		0			0
$259$	3.36089e12	+	2.22371e12i	0.179187	+	0.118558i
$260$		0			0
$261$	−3.02481e12	+	2.63484e13i	−0.154588	+	1.34658i
$262$		0			0
$263$		−	3.86991e13i		−	1.89646i	−0.317578	−	0.948232i	$-0.602870\pi$
$263$		−	3.86991e13i		−	1.89646i	0.317578	−	0.948232i	$-0.397130\pi$
$264$		0			0
$265$			1.62836e13i			0.765419i
$266$		0			0
$267$	−2.26680e10	+	3.96208e11i	−0.00102235	+	0.0178695i
$268$		0			0
$269$	2.04693e13			0.886064			0.443032	−	0.896506i	$-0.353903\pi$
$269$	2.04693e13			0.886064			0.443032	+	0.896506i	$0.353903\pi$
$270$		0			0
$271$			1.93773e13i			0.805309i	0.915352	+	0.402654i	$0.131912\pi$
$271$			1.93773e13i			0.805309i	−0.915352	+	0.402654i	$0.868088\pi$
$272$		0			0
$273$	−2.79480e13	−	2.08810e13i	−1.11546	−	0.833405i
$274$		0			0
$275$			2.41583e13i			0.926268i
$276$		0			0
$277$	4.98633e12			0.183714			0.0918570	−	0.995772i	$-0.470720\pi$
$277$	4.98633e12			0.183714			0.0918570	+	0.995772i	$0.470720\pi$
$278$		0			0
$279$	−9.31401e11	+	8.11321e12i	−0.0329848	+	0.287322i
$280$		0			0
$281$		−	3.55563e13i		−	1.21069i	−0.795964	−	0.605344i	$-0.793035\pi$
$281$		−	3.55563e13i		−	1.21069i	0.795964	−	0.605344i	$-0.206965\pi$
$282$		0			0
$283$		−	8.28192e12i		−	0.271210i	−0.990763	−	0.135605i	$-0.956702\pi$
$283$		−	8.28192e12i		−	0.271210i	0.990763	−	0.135605i	$-0.0432978\pi$
$284$		0			0
$285$	−6.72809e13	−	3.84930e12i	−2.11956	−	0.121265i
$286$		0			0
$287$	−4.34050e13	−	2.87185e13i	−1.31580	−	0.870586i
$288$		0			0
$289$	4.03209e13			1.17650
$290$		0			0
$291$	−6.57150e13	−	3.75972e12i	−1.84609	−	0.105619i
$292$		0			0
$293$	−5.32343e12			−0.144019			−0.0720095	−	0.997404i	$-0.522941\pi$
$293$	−5.32343e12			−0.144019			−0.0720095	+	0.997404i	$0.522941\pi$
$294$		0			0
$295$	3.58317e13			0.933783
$296$		0			0
$297$	1.90068e13	+	3.29103e12i	0.477253	+	0.0826365i
$298$		0			0
$299$	7.40834e13			1.79279
$300$		0			0
$301$	1.28329e13	+	8.49078e12i	0.299370	+	0.198076i
$302$		0			0
$303$	3.87279e12	−	6.76914e13i	0.0871143	−	1.52265i
$304$		0			0
$305$			8.04809e13i			1.74600i
$306$		0			0
$307$		−	6.94010e12i		−	0.145246i	−0.997359	−	0.0726230i	$-0.976863\pi$
$307$		−	6.94010e12i		−	0.145246i	0.997359	−	0.0726230i	$-0.0231370\pi$
$308$		0			0
$309$	4.77290e12	+	2.73069e11i	0.0963852	+	0.00551443i
$310$		0			0
$311$	−6.64344e13			−1.29482			−0.647412	−	0.762140i	$-0.724149\pi$
$311$	−6.64344e13			−1.29482			−0.647412	+	0.762140i	$0.724149\pi$
$312$		0			0
$313$		−	1.02277e14i		−	1.92436i	−0.272417	−	0.962179i	$-0.587823\pi$
$313$		−	1.02277e14i		−	1.92436i	0.272417	−	0.962179i	$-0.412177\pi$
$314$		0			0
$315$	7.19178e13	+	6.04301e13i	1.30656	+	1.09786i
$316$		0			0
$317$		−	2.67990e13i		−	0.470212i	−0.971970	−	0.235106i	$-0.924456\pi$
$317$		−	2.67990e13i		−	0.470212i	0.971970	−	0.235106i	$-0.0755437\pi$
$318$		0			0
$319$	3.87334e13			0.656504
$320$		0			0
$321$	7.77111e13	+	4.44604e12i	1.27264	+	0.0728108i
$322$		0			0
$323$			1.15964e14i			1.83531i
$324$		0			0
$325$		−	1.74061e14i		−	2.66283i
$326$		0			0
$327$	−2.02401e12	+	3.53771e13i	−0.0299364	+	0.523250i
$328$		0			0
$329$	−9.25617e13	−	6.12427e13i	−1.32390	−	0.875946i
$330$		0			0
$331$	−3.97539e13			−0.549953			−0.274977	−	0.961451i	$-0.588670\pi$
$331$	−3.97539e13			−0.549953			−0.274977	+	0.961451i	$0.588670\pi$
$332$		0			0
$333$	1.59496e13	+	1.83103e12i	0.213456	+	0.0245048i
$334$		0			0
$335$	−3.46495e12			−0.0448694
$336$		0			0
$337$	−2.62374e12			−0.0328818			−0.0164409	−	0.999865i	$-0.505234\pi$
$337$	−2.62374e12			−0.0328818			−0.0164409	+	0.999865i	$0.505234\pi$
$338$		0			0
$339$	−4.81722e13	−	2.75605e12i	−0.584383	−	0.0334340i
$340$		0			0
$341$	1.19268e13			0.140080
$342$		0			0
$343$	1.59793e13	−	8.64619e13i	0.181736	−	0.983347i
$344$		0			0
$345$	−1.99149e14	−	1.13938e13i	−2.19368	−	0.125506i
$346$		0			0
$347$			1.15638e14i			1.23393i	0.786992	+	0.616963i	$0.211637\pi$
$347$			1.15638e14i			1.23393i	−0.786992	+	0.616963i	$0.788363\pi$
$348$		0			0
$349$		−	1.53750e14i		−	1.58956i	−0.606900	−	0.794778i	$-0.707587\pi$
$349$		−	1.53750e14i		−	1.58956i	0.606900	−	0.794778i	$-0.292413\pi$
$350$		0			0
$351$	−1.36944e14	−	2.37119e13i	−1.37200	−	0.237563i
$352$		0			0
$353$	2.00075e13			0.194282			0.0971410	−	0.995271i	$-0.469030\pi$
$353$	2.00075e13			0.194282			0.0971410	+	0.995271i	$0.469030\pi$
$354$		0			0
$355$			1.59273e14i			1.49929i
$356$		0			0
$357$	9.67478e13	−	1.29492e14i	0.883011	−	1.18186i
$358$		0			0
$359$		−	1.79162e14i		−	1.58572i	−0.609404	−	0.792860i	$-0.708591\pi$
$359$		−	1.79162e14i		−	1.58572i	0.609404	−	0.792860i	$-0.291409\pi$
$360$		0			0
$361$	−6.37914e13			−0.547611
$362$		0			0
$363$	−5.24996e12	+	9.17626e13i	−0.0437189	+	0.764151i
$364$		0			0
$365$		−	4.05731e13i		−	0.327814i
$366$		0			0
$367$		−	5.90723e12i		−	0.0463149i	−0.999732	−	0.0231574i	$-0.992628\pi$
$367$		−	5.90723e12i		−	0.0463149i	0.999732	−	0.0231574i	$-0.00737190\pi$
$368$		0			0
$369$	−2.05985e14	−	2.36472e13i	−1.56744	−	0.179943i
$370$		0			0
$371$	3.35050e13	−	5.06392e13i	0.247488	−	0.374051i
$372$		0			0
$373$	−2.36923e14			−1.69906			−0.849529	−	0.527542i	$-0.823114\pi$
$373$	−2.36923e14			−1.69906			−0.849529	+	0.527542i	$0.823114\pi$
$374$		0			0
$375$	−1.27718e13	+	2.23234e14i	−0.0889363	+	1.55449i
$376$		0			0
$377$	−2.79075e14			−1.88731
$378$		0			0
$379$	1.99384e14			1.30971			0.654854	−	0.755755i	$-0.272730\pi$
$379$	1.99384e14			1.30971			0.654854	+	0.755755i	$0.272730\pi$
$380$		0			0
$381$	1.36430e13	−	2.38462e14i	0.0870606	−	1.52171i
$382$		0			0
$383$	1.75271e14			1.08672			0.543360	−	0.839500i	$-0.317152\pi$
$383$	1.75271e14			1.08672			0.543360	+	0.839500i	$0.317152\pi$
$384$		0			0
$385$	7.57006e13	−	1.14413e14i	0.456106	−	0.689354i
$386$		0			0
$387$	6.09005e13	+	6.99141e12i	0.356624	+	0.0409406i
$388$		0			0
$389$			4.71799e13i			0.268556i	0.990944	+	0.134278i	$0.0428715\pi$
$389$			4.71799e13i			0.268556i	−0.990944	+	0.134278i	$0.957129\pi$
$390$		0			0
$391$			3.43251e14i			1.89950i
$392$		0			0
$393$	−4.62494e11	+	8.08381e12i	−0.00248855	+	0.0434968i
$394$		0			0
$395$	4.18804e14			2.19143
$396$		0			0
$397$		−	2.22777e14i		−	1.13376i	−0.823799	−	0.566882i	$-0.808149\pi$
$397$		−	2.22777e14i		−	1.13376i	0.823799	−	0.566882i	$-0.191851\pi$
$398$		0			0
$399$	2.01311e14	+	1.50407e14i	0.996592	+	0.744591i
$400$		0			0
$401$			2.90661e14i			1.39989i	0.714198	+	0.699943i	$0.246791\pi$
$401$			2.90661e14i			1.39989i	−0.714198	+	0.699943i	$0.753209\pi$
$402$		0			0
$403$	−8.59330e13			−0.402700
$404$		0			0
$405$	3.64484e14	+	8.48036e13i	1.66217	+	0.386734i
$406$		0			0
$407$		−	2.34468e13i		−	0.104067i
$408$		0			0
$409$			2.81851e14i			1.21770i	0.793285	+	0.608851i	$0.208369\pi$
$409$			2.81851e14i			1.21770i	−0.793285	+	0.608851i	$0.791631\pi$
$410$		0			0
$411$	−4.30600e14	−	2.46357e13i	−1.81111	−	0.103618i
$412$		0			0
$413$	−1.11430e14	−	7.37269e13i	−0.456328	−	0.301926i
$414$		0			0
$415$	7.00889e13			0.279502
$416$		0			0
$417$	9.15359e12	+	5.23699e11i	0.0355503	+	0.00203392i
$418$		0			0
$419$	−1.71537e14			−0.648903			−0.324451	−	0.945902i	$-0.605180\pi$
$419$	−1.71537e14			−0.648903			−0.324451	+	0.945902i	$0.605180\pi$
$420$		0			0
$421$	2.24149e14			0.826010			0.413005	−	0.910729i	$-0.364479\pi$
$421$	2.24149e14			0.826010			0.413005	+	0.910729i	$0.364479\pi$
$422$		0			0
$423$	−4.39266e14	−	5.04280e13i	−1.57709	−	0.181051i
$424$		0			0
$425$	8.06477e14			2.82132
$426$		0			0
$427$	1.65597e14	−	2.50281e14i	0.564544	−	0.853248i
$428$		0			0
$429$	−1.15939e13	+	2.02646e14i	−0.0385224	+	0.673322i
$430$		0			0
$431$		−	4.54323e14i		−	1.47143i	−0.677290	−	0.735716i	$-0.736846\pi$
$431$		−	4.54323e14i		−	1.47143i	0.677290	−	0.735716i	$-0.263154\pi$
$432$		0			0
$433$			4.25544e14i			1.34357i	0.740745	+	0.671787i	$0.234473\pi$
$433$			4.25544e14i			1.34357i	−0.740745	+	0.671787i	$0.765527\pi$
$434$		0			0
$435$	7.50204e14	+	4.29210e13i	2.30934	+	0.132123i
$436$		0			0
$437$	−5.33628e14			−1.60174
$438$		0			0
$439$			3.72625e14i			1.09073i	0.838199	+	0.545365i	$0.183609\pi$
$439$			3.72625e14i			1.09073i	−0.838199	+	0.545365i	$0.816391\pi$
$440$		0			0
$441$	−9.93112e13	−	3.35904e14i	−0.283522	−	0.958966i
$442$		0			0
$443$		−	5.93845e14i		−	1.65368i	−0.562434	−	0.826842i	$-0.690135\pi$
$443$		−	5.93845e14i		−	1.65368i	0.562434	−	0.826842i	$-0.309865\pi$
$444$		0			0
$445$	1.12441e13			0.0305453
$446$		0			0
$447$	5.25885e14	+	3.00871e13i	1.39380	+	0.0797424i
$448$		0			0
$449$		−	1.58922e14i		−	0.410989i	−0.978658	−	0.205494i	$-0.934120\pi$
$449$		−	1.58922e14i		−	0.410989i	0.978658	−	0.205494i	$-0.0658802\pi$
$450$		0			0
$451$			3.02808e14i			0.764182i
$452$		0			0
$453$	−3.17741e13	+	5.55370e14i	−0.0782587	+	1.36786i
$454$		0			0
$455$	−5.45424e14	+	8.24349e14i	−1.31121	+	1.98175i
$456$		0			0
$457$	−4.39546e14			−1.03149			−0.515745	−	0.856742i	$-0.672485\pi$
$457$	−4.39546e14			−1.03149			−0.515745	+	0.856742i	$0.672485\pi$
$458$		0			0
$459$	1.09865e14	−	6.34504e14i	0.251703	−	1.45367i
$460$		0			0
$461$	−2.22140e14			−0.496903			−0.248451	−	0.968644i	$-0.579922\pi$
$461$	−2.22140e14			−0.496903			−0.248451	+	0.968644i	$0.579922\pi$
$462$		0			0
$463$	−9.76665e12			−0.0213329			−0.0106665	−	0.999943i	$-0.503395\pi$
$463$	−9.76665e12			−0.0213329			−0.0106665	+	0.999943i	$0.503395\pi$
$464$		0			0
$465$	2.31003e14	+	1.32163e13i	0.492750	+	0.0281914i
$466$		0			0
$467$	−4.16702e14			−0.868126			−0.434063	−	0.900882i	$-0.642921\pi$
$467$	−4.16702e14			−0.868126			−0.434063	+	0.900882i	$0.642921\pi$
$468$		0			0
$469$	1.07754e13	+	7.12944e12i	0.0219271	+	0.0145079i
$470$		0			0
$471$	1.21156e14	+	6.93163e12i	0.240841	+	0.0137791i
$472$		0			0
$473$		−	8.95268e13i		−	0.173867i
$474$		0			0
$475$			1.25377e15i			2.37905i
$476$		0			0
$477$	2.75885e13	−	2.40316e14i	0.0511536	−	0.445587i
$478$		0			0
$479$	9.38142e14			1.69990			0.849949	−	0.526864i	$-0.176632\pi$
$479$	9.38142e14			1.69990			0.849949	+	0.526864i	$0.176632\pi$
$480$		0			0
$481$			1.68934e14i			0.299171i
$482$		0			0
$483$	5.95876e14	+	4.45200e14i	1.03145	+	0.770631i
$484$		0			0
$485$			1.86495e15i			3.15563i
$486$		0			0
$487$	6.75315e13			0.111711			0.0558557	−	0.998439i	$-0.482211\pi$
$487$	6.75315e13			0.111711			0.0558557	+	0.998439i	$0.482211\pi$
$488$		0			0
$489$	−1.52036e13	+	2.65740e14i	−0.0245894	+	0.429792i
$490$		0			0
$491$			1.65170e14i			0.261206i	0.991435	+	0.130603i	$0.0416913\pi$
$491$			1.65170e14i			0.261206i	−0.991435	+	0.130603i	$0.958309\pi$
$492$		0			0
$493$		−	1.29304e15i		−	1.99965i
$494$		0			0
$495$	6.23328e13	−	5.42966e14i	0.0942732	−	0.821190i
$496$		0			0
$497$	3.27718e14	−	4.95310e14i	0.484774	−	0.732684i
$498$		0			0
$499$	−4.81581e14			−0.696813			−0.348406	−	0.937344i	$-0.613277\pi$
$499$	−4.81581e14			−0.696813			−0.348406	+	0.937344i	$0.613277\pi$
$500$		0			0
$501$	4.11264e13	−	7.18836e14i	0.0582120	−	1.01747i
$502$		0			0
$503$	−4.92615e14			−0.682156			−0.341078	−	0.940035i	$-0.610792\pi$
$503$	−4.92615e14			−0.682156			−0.341078	+	0.940035i	$0.610792\pi$
$504$		0			0
$505$	−1.92103e15			−2.60275
$506$		0			0
$507$	4.04491e13	−	7.06999e14i	0.0536248	−	0.937293i
$508$		0			0
$509$	−1.93559e14			−0.251111			−0.125555	−	0.992087i	$-0.540071\pi$
$509$	−1.93559e14			−0.251111			−0.125555	+	0.992087i	$0.540071\pi$
$510$		0			0
$511$	−8.34828e13	+	1.26175e14i	−0.105994	+	0.160199i
$512$		0			0
$513$	9.86420e14	+	1.70799e14i	1.22579	+	0.212246i
$514$		0			0
$515$		−	1.35451e14i		−	0.164757i
$516$		0			0
$517$			6.45743e14i			0.768886i
$518$		0			0
$519$	−3.89872e13	+	6.81446e14i	−0.0454466	+	0.794349i
$520$		0			0
$521$	−7.48384e14			−0.854116			−0.427058	−	0.904224i	$-0.640450\pi$
$521$	−7.48384e14			−0.854116			−0.427058	+	0.904224i	$0.640450\pi$
$522$		0			0
$523$			4.81775e14i			0.538375i	0.963088	+	0.269188i	$0.0867552\pi$
$523$			4.81775e14i			0.538375i	−0.963088	+	0.269188i	$0.913245\pi$
$524$		0			0
$525$	1.04601e15	−	1.40003e15i	1.14462	−	1.53200i
$526$		0			0
$527$		−	3.98153e14i		−	0.426670i
$528$		0			0
$529$	−6.26713e14			−0.657752
$530$		0			0
$531$	−5.28809e14	−	6.07076e13i	−0.543600	−	0.0624056i
$532$		0			0
$533$		−	2.18174e15i		−	2.19686i
$534$		0			0
$535$		−	2.20539e15i		−	2.17540i
$536$		0			0
$537$	−1.29691e15	−	7.41994e13i	−1.25329	−	0.0717037i
$538$		0			0
$539$	−4.70831e14	+	2.00044e14i	−0.445786	+	0.189403i
$540$		0			0
$541$	1.08948e15			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$541$	1.08948e15			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$542$		0			0
$543$	1.45978e15	+	8.35177e13i	1.32706	+	0.0759241i
$544$		0			0
$545$	1.00398e15			0.894423
$546$		0			0
$547$	4.44754e13			0.0388320			0.0194160	−	0.999811i	$-0.493819\pi$
$547$	4.44754e13			0.0388320			0.0194160	+	0.999811i	$0.493819\pi$
$548$		0			0
$549$	1.36354e14	−	1.18775e15i	0.116687	−	1.01643i
$550$		0			0
$551$	2.01020e15			1.68618
$552$		0			0
$553$	−1.30241e15	−	8.61726e14i	−1.07092	−	0.708567i
$554$		0			0
$555$	2.59817e13	−	4.54126e14i	0.0209438	−	0.366071i
$556$		0			0
$557$		−	2.37356e14i		−	0.187584i	−0.995592	−	0.0937922i	$-0.970101\pi$
$557$		−	2.37356e14i		−	0.187584i	0.995592	−	0.0937922i	$-0.0298989\pi$
$558$		0			0
$559$			6.45042e14i			0.499831i
$560$		0			0
$561$	−9.38920e14	−	5.37179e13i	−0.713400	−	0.0408153i
$562$		0			0
$563$	−1.71308e15			−1.27638			−0.638191	−	0.769878i	$-0.720317\pi$
$563$	−1.71308e15			−1.27638			−0.638191	+	0.769878i	$0.720317\pi$
$564$		0			0
$565$			1.36709e15i			0.998921i
$566$		0			0
$567$	−9.58990e14	−	1.01368e15i	−0.687238	−	0.726432i
$568$		0			0
$569$		−	1.15148e15i		−	0.809357i	−0.914459	−	0.404679i	$-0.867383\pi$
$569$		−	1.15148e15i		−	0.809357i	0.914459	−	0.404679i	$-0.132617\pi$
$570$		0			0
$571$	−2.47904e15			−1.70917			−0.854585	−	0.519312i	$-0.826188\pi$
$571$	−2.47904e15			−1.70917			−0.854585	+	0.519312i	$0.826188\pi$
$572$		0			0
$573$	−7.00005e14	−	4.00490e13i	−0.473425	−	0.0270858i
$574$		0			0
$575$			3.71113e15i			2.46226i
$576$		0			0
$577$		−	1.05131e15i		−	0.684330i	−0.939640	−	0.342165i	$-0.888840\pi$
$577$		−	1.05131e15i		−	0.684330i	0.939640	−	0.342165i	$-0.111160\pi$
$578$		0			0
$579$	1.08874e14	−	1.90298e15i	0.0695334	−	1.21535i
$580$		0			0
$581$	−2.17964e14	−	1.44214e14i	−0.136589	−	0.0903731i
$582$		0			0
$583$	−3.53277e14			−0.217239
$584$		0			0
$585$	−4.49109e14	+	3.91208e15i	−0.271016	+	2.36075i
$586$		0			0
$587$	2.39074e15			1.41587			0.707934	−	0.706278i	$-0.249627\pi$
$587$	2.39074e15			1.41587			0.707934	+	0.706278i	$0.249627\pi$
$588$		0			0
$589$	6.18982e14			0.359785
$590$		0			0
$591$	1.10280e14	+	6.30937e12i	0.0629165	+	0.00359960i
$592$		0			0
$593$	3.31592e13			0.0185696			0.00928481	−	0.999957i	$-0.497045\pi$
$593$	3.31592e13			0.0185696			0.00928481	+	0.999957i	$0.497045\pi$
$594$		0			0
$595$	−3.81946e15	−	2.52711e15i	−2.09971	−	1.38925i
$596$		0			0
$597$	1.24403e15	+	7.11739e13i	0.671384	+	0.0384115i
$598$		0			0
$599$			1.13726e15i			0.602576i	0.953533	+	0.301288i	$0.0974165\pi$
$599$			1.13726e15i			0.602576i	−0.953533	+	0.301288i	$0.902584\pi$
$600$		0			0
$601$			3.44545e15i			1.79241i	0.443644	+	0.896203i	$0.353685\pi$
$601$			3.44545e15i			1.79241i	−0.443644	+	0.896203i	$0.646315\pi$
$602$		0			0
$603$	5.11362e13	+	5.87047e12i	0.0261206	+	0.00299866i
$604$		0			0
$605$	2.60416e15			1.30621
$606$		0			0
$607$			7.03227e14i			0.346384i	0.984888	+	0.173192i	$0.0554081\pi$
$607$			7.03227e14i			0.346384i	−0.984888	+	0.173192i	$0.944592\pi$
$608$		0			0
$609$	−2.24469e15	−	1.67709e15i	−1.08583	−	0.811261i
$610$		0			0
$611$		−	4.65259e15i		−	2.21039i
$612$		0			0
$613$	3.41741e15			1.59465			0.797323	−	0.603553i	$-0.206249\pi$
$613$	3.41741e15			1.59465			0.797323	+	0.603553i	$0.206249\pi$
$614$		0			0
$615$	−3.35545e14	+	5.86490e15i	−0.153794	+	2.68811i
$616$		0			0
$617$			3.55550e14i			0.160078i	0.996792	+	0.0800391i	$0.0255045\pi$
$617$			3.55550e14i			0.160078i	−0.996792	+	0.0800391i	$0.974495\pi$
$618$		0			0
$619$		−	1.89530e15i		−	0.838261i	−0.907926	−	0.419130i	$-0.862335\pi$
$619$		−	1.89530e15i		−	0.838261i	0.907926	−	0.419130i	$-0.137665\pi$
$620$		0			0
$621$	2.91977e15	+	5.05559e14i	1.26866	+	0.219669i
$622$		0			0
$623$	−3.49672e13	−	2.31357e13i	−0.0149271	−	0.00987641i
$624$		0			0
$625$	1.77576e15			0.744809
$626$		0			0
$627$	8.35115e13	−	1.45967e15i	0.0344171	−	0.601568i
$628$		0			0
$629$	−7.82724e14			−0.316979
$630$		0			0
$631$	−3.04028e15			−1.20991			−0.604955	−	0.796260i	$-0.706809\pi$
$631$	−3.04028e15			−1.20991			−0.604955	+	0.796260i	$0.706809\pi$
$632$		0			0
$633$	−8.90256e13	+	1.55605e15i	−0.0348173	+	0.608561i
$634$		0			0
$635$	−6.76738e15			−2.60115
$636$		0			0
$637$	3.39234e15	−	1.44132e15i	1.28154	−	0.544495i
$638$		0			0
$639$	2.69847e14	−	2.35057e15i	0.100199	−	0.872807i
$640$		0			0
$641$			3.49938e15i			1.27724i	0.769522	+	0.638620i	$0.220494\pi$
$641$			3.49938e15i			1.27724i	−0.769522	+	0.638620i	$0.779506\pi$
$642$		0			0
$643$			3.13445e15i			1.12461i	0.826931	+	0.562304i	$0.190085\pi$
$643$			3.13445e15i			1.12461i	−0.826931	+	0.562304i	$0.809915\pi$
$644$		0			0
$645$	9.92057e13	−	1.73399e15i	0.0349911	−	0.611600i
$646$		0			0
$647$	−2.63020e15			−0.912041			−0.456021	−	0.889969i	$-0.650726\pi$
$647$	−2.63020e15			−0.912041			−0.456021	+	0.889969i	$0.650726\pi$
$648$		0			0
$649$			7.77376e14i			0.265024i
$650$		0			0
$651$	−6.91186e14	−	5.16410e14i	−0.231686	−	0.173101i
$652$		0			0
$653$			3.54296e15i			1.16773i	0.811850	+	0.583867i	$0.198461\pi$
$653$			3.54296e15i			1.16773i	−0.811850	+	0.583867i	$0.801539\pi$
$654$		0			0
$655$	2.29413e14			0.0743516
$656$		0			0
$657$	−6.87408e13	+	5.98784e14i	−0.0219081	+	0.190836i
$658$		0			0
$659$		−	1.29800e15i		−	0.406822i	−0.979093	−	0.203411i	$-0.934797\pi$
$659$		−	1.29800e15i		−	0.406822i	0.979093	−	0.203411i	$-0.0652027\pi$
$660$		0			0
$661$			8.05554e14i			0.248306i	0.992263	+	0.124153i	$0.0396213\pi$
$661$			8.05554e14i			0.248306i	−0.992263	+	0.124153i	$0.960379\pi$
$662$		0			0
$663$	6.76493e15	+	3.87038e14i	2.05088	+	0.117336i
$664$		0			0
$665$	3.92873e15	−	5.93785e15i	1.17147	−	1.77056i
$666$		0			0
$667$	5.95012e15			1.74515
$668$		0			0
$669$	2.05363e14	+	1.17493e13i	0.0592488	+	0.00338977i
$670$		0			0
$671$	−1.74605e15			−0.495545
$672$		0			0
$673$	−5.14455e14			−0.143636			−0.0718182	−	0.997418i	$-0.522880\pi$
$673$	−5.14455e14			−0.143636			−0.0718182	+	0.997418i	$0.522880\pi$
$674$		0			0
$675$	1.18783e15	−	6.86009e15i	0.326274	−	1.88434i
$676$		0			0
$677$	−3.19293e15			−0.862882			−0.431441	−	0.902141i	$-0.641995\pi$
$677$	−3.19293e15			−0.862882			−0.431441	+	0.902141i	$0.641995\pi$
$678$		0			0
$679$	3.83729e15	−	5.79965e15i	1.02033	−	1.54212i
$680$		0			0
$681$	−1.33265e14	+	2.32930e15i	−0.0348663	+	0.609419i
$682$		0			0
$683$			4.33828e15i			1.11687i	0.829547	+	0.558437i	$0.188599\pi$
$683$			4.33828e15i			1.11687i	−0.829547	+	0.558437i	$0.811401\pi$
$684$		0			0
$685$			1.22201e16i			3.09584i
$686$		0			0
$687$	2.19200e15	+	1.25410e14i	0.546485	+	0.0312658i
$688$		0			0
$689$	2.54537e15			0.624517
$690$		0			0
$691$			3.86535e15i			0.933381i	0.884421	+	0.466691i	$0.154554\pi$
$691$			3.86535e15i			0.933381i	−0.884421	+	0.466691i	$0.845446\pi$
$692$		0			0
$693$	−1.31104e15	+	1.56027e15i	−0.311591	+	0.370824i
$694$		0			0
$695$		−	2.59773e14i		−	0.0607683i
$696$		0			0
$697$	1.01087e16			2.32763
$698$		0			0
$699$	−1.02086e15	−	5.84057e13i	−0.231388	−	0.0132382i
$700$		0			0
$701$			5.62000e15i			1.25397i	0.779031	+	0.626985i	$0.215711\pi$
$701$			5.62000e15i			1.25397i	−0.779031	+	0.626985i	$0.784289\pi$
$702$		0			0
$703$		−	1.21685e15i		−	0.267289i
$704$		0			0
$705$	−7.15556e14	+	1.25070e16i	−0.154740	+	2.70466i
$706$		0			0
$707$	5.97408e15	+	3.95270e15i	1.27193	+	0.841564i
$708$		0			0
$709$	−5.72350e15			−1.19980			−0.599898	−	0.800077i	$-0.704792\pi$
$709$	−5.72350e15			−1.19980			−0.599898	+	0.800077i	$0.704792\pi$
$710$		0			0
$711$	−6.18077e15	−	7.09556e14i	−1.27573	−	0.146455i
$712$		0			0
$713$	1.83217e15			0.372368
$714$		0			0
$715$	5.75095e15			1.15095
$716$		0			0
$717$	−1.15478e15	−	6.60677e13i	−0.227585	−	0.0130207i
$718$		0			0
$719$	2.12288e14			0.0412018			0.0206009	−	0.999788i	$-0.493442\pi$
$719$	2.12288e14			0.0412018			0.0206009	+	0.999788i	$0.493442\pi$
$720$		0			0
$721$	−2.78703e14	+	4.21230e14i	−0.0532719	+	0.0805147i
$722$		0			0
$723$	1.16924e15	+	6.68951e13i	0.220112	+	0.0125931i
$724$		0			0
$725$		−	1.39800e16i		−	2.59208i
$726$		0			0
$727$		−	9.99322e14i		−	0.182501i	−0.995828	−	0.0912507i	$-0.970914\pi$
$727$		−	9.99322e14i		−	0.182501i	0.995828	−	0.0912507i	$-0.0290864\pi$
$728$		0			0
$729$	−5.23543e15	−	1.86907e15i	−0.941783	−	0.336220i
$730$		0			0
$731$	−2.98867e15			−0.529582
$732$		0			0
$733$		−	6.98249e15i		−	1.21882i	−0.792856	−	0.609408i	$-0.791407\pi$
$733$		−	6.98249e15i		−	1.21882i	0.792856	−	0.609408i	$-0.208593\pi$
$734$		0			0
$735$	−9.34090e15	+	3.35280e15i	−1.60623	+	0.576537i
$736$		0			0
$737$		−	7.51728e13i		−	0.0127347i
$738$		0			0
$739$	−7.17394e15			−1.19733			−0.598664	−	0.801000i	$-0.704302\pi$
$739$	−7.17394e15			−1.19733			−0.598664	+	0.801000i	$0.704302\pi$
$740$		0			0
$741$	−6.01702e14	+	1.05170e16i	−0.0989421	+	1.72938i
$742$		0			0
$743$		−	1.03756e16i		−	1.68102i	−0.541795	−	0.840511i	$-0.682255\pi$
$743$		−	1.03756e16i		−	1.68102i	0.541795	−	0.840511i	$-0.317745\pi$
$744$		0			0
$745$		−	1.49242e16i		−	2.38250i
$746$		0			0
$747$	−1.03438e15	−	1.18748e14i	−0.162711	−	0.0186794i
$748$		0			0
$749$	−4.53778e15	+	6.85836e15i	−0.703386	+	1.06309i
$750$		0			0
$751$	−7.41115e15			−1.13205			−0.566026	−	0.824388i	$-0.691520\pi$
$751$	−7.41115e15			−1.13205			−0.566026	+	0.824388i	$0.691520\pi$
$752$		0			0
$753$	2.48323e14	−	4.34036e15i	0.0373804	−	0.653361i
$754$		0			0
$755$	1.57610e16			2.33817
$756$		0			0
$757$	5.80914e15			0.849345			0.424673	−	0.905347i	$-0.360389\pi$
$757$	5.80914e15			0.849345			0.424673	+	0.905347i	$0.360389\pi$
$758$		0			0
$759$	2.47192e14	−	4.32059e15i	0.0356208	−	0.622606i
$760$		0			0
$761$	−4.59114e15			−0.652086			−0.326043	−	0.945355i	$-0.605715\pi$
$761$	−4.59114e15			−0.652086			−0.326043	+	0.945355i	$0.605715\pi$
$762$		0			0
$763$	−3.12219e15	−	2.06577e15i	−0.437094	−	0.289199i
$764$		0			0
$765$	−1.81258e16	−	2.08086e15i	−2.50127	−	0.287147i
$766$		0			0
$767$		−	5.60101e15i		−	0.761889i
$768$		0			0
$769$			8.85133e15i			1.18690i	0.804871	+	0.593449i	$0.202234\pi$
$769$			8.85133e15i			1.18690i	−0.804871	+	0.593449i	$0.797766\pi$
$770$		0			0
$771$	−5.42142e14	+	9.47595e15i	−0.0716661	+	1.25263i
$772$		0			0
$773$	−7.76807e15			−1.01234			−0.506169	−	0.862434i	$-0.668939\pi$
$773$	−7.76807e15			−1.01234			−0.506169	+	0.862434i	$0.668939\pi$
$774$		0			0
$775$		−	4.30473e15i		−	0.553078i
$776$		0			0
$777$	−1.01520e15	+	1.35879e15i	−0.128599	+	0.172122i
$778$		0			0
$779$			1.57152e16i			1.96275i
$780$		0			0
$781$	−3.45545e15			−0.425524
$782$		0			0
$783$	−1.09989e16	−	1.90446e15i	−1.33555	−	0.231251i
$784$		0			0
$785$		−	3.43832e15i		−	0.411683i
$786$		0			0
$787$		−	9.23523e15i		−	1.09040i	−0.838305	−	0.545201i	$-0.816453\pi$
$787$		−	9.23523e15i		−	1.09040i	0.838305	−	0.545201i	$-0.183547\pi$
$788$		0			0
$789$	1.62614e16	+	9.30355e14i	1.89337	+	0.108324i
$790$		0			0
$791$	2.81291e15	−	4.25141e15i	0.322987	−	0.488161i
$792$		0			0
$793$	1.25803e16			1.42459
$794$		0			0
$795$	−6.84240e15	−	3.91470e14i	−0.764169	−	0.0437200i
$796$		0			0
$797$	5.96486e15			0.657022			0.328511	−	0.944500i	$-0.393453\pi$
$797$	5.96486e15			0.657022			0.328511	+	0.944500i	$0.393453\pi$
$798$		0			0
$799$	2.15569e16			2.34195
$800$		0			0
$801$	−1.65942e14	−	1.90503e13i	−0.0177819	−	0.00204137i
$802$		0			0
$803$	8.80243e14			0.0930395
$804$		0			0
$805$	1.16289e16	−	1.75758e16i	1.21244	−	1.83248i
$806$		0			0
$807$	−4.92097e14	+	8.60122e15i	−0.0506111	+	0.884617i
$808$		0			0
$809$		−	1.87795e15i		−	0.190532i	−0.995452	−	0.0952659i	$-0.969630\pi$
$809$		−	1.87795e15i		−	0.190532i	0.995452	−	0.0952659i	$-0.0303701\pi$
$810$		0			0
$811$		−	7.58178e15i		−	0.758851i	−0.925222	−	0.379426i	$-0.876122\pi$
$811$		−	7.58178e15i		−	0.758851i	0.925222	−	0.379426i	$-0.123878\pi$
$812$		0			0
$813$	−8.14237e15	−	4.65845e14i	−0.803994	−	0.0459985i
$814$		0			0
$815$	7.54151e15			0.734669
$816$		0			0
$817$		−	4.64628e15i		−	0.446565i
$818$		0			0
$819$	9.44609e15	−	1.12418e16i	0.895758	−	1.06604i
$820$		0			0
$821$			4.86791e15i			0.455466i	0.973724	+	0.227733i	$0.0731313\pi$
$821$			4.86791e15i			0.455466i	−0.973724	+	0.227733i	$0.926869\pi$
$822$		0			0
$823$	−9.48326e15			−0.875505			−0.437753	−	0.899095i	$-0.644225\pi$
$823$	−9.48326e15			−0.875505			−0.437753	+	0.899095i	$0.644225\pi$
$824$		0			0
$825$	−1.01514e16	−	5.80783e14i	−0.924756	−	0.0529075i
$826$		0			0
$827$		−	3.49312e15i		−	0.314003i	−0.987598	−	0.157001i	$-0.949817\pi$
$827$		−	3.49312e15i		−	0.314003i	0.987598	−	0.157001i	$-0.0501827\pi$
$828$		0			0
$829$		−	5.99869e15i		−	0.532116i	−0.963957	−	0.266058i	$-0.914279\pi$
$829$		−	5.99869e15i		−	0.532116i	0.963957	−	0.266058i	$-0.0857213\pi$
$830$		0			0
$831$	−1.19875e14	+	2.09526e15i	−0.0104936	+	0.183414i
$832$		0			0
$833$	6.67808e15	+	1.57178e16i	0.576905	+	1.35782i
$834$		0			0
$835$	−2.04001e16			−1.73923
$836$		0			0
$837$	−3.38679e15	−	5.86423e14i	−0.284969	−	0.0493425i
$838$		0			0
$839$	−1.82844e16			−1.51841			−0.759207	−	0.650849i	$-0.774413\pi$
$839$	−1.82844e16			−1.51841			−0.759207	+	0.650849i	$0.774413\pi$
$840$		0			0
$841$	−1.02138e16			−0.837165
$842$		0			0
$843$	1.49408e16	+	8.54801e14i	1.20871	+	0.0691533i
$844$		0			0
$845$	−2.00641e16			−1.60217
$846$		0			0
$847$	−8.09847e15	−	5.35829e15i	−0.638328	−	0.422345i
$848$		0			0
$849$	3.48007e15	+	1.99103e14i	0.270767	+	0.0154912i
$850$		0			0
$851$		−	3.60183e15i		−	0.276637i
$852$		0			0
$853$			5.44096e14i			0.0412531i	0.999787	+	0.0206265i	$0.00656609\pi$
$853$			5.44096e14i			0.0412531i	−0.999787	+	0.0206265i	$0.993434\pi$
$854$		0			0
$855$	3.23496e15	−	2.81790e16i	0.242134	−	2.10917i
$856$		0			0
$857$	−6.83391e15			−0.504981			−0.252490	−	0.967599i	$-0.581250\pi$
$857$	−6.83391e15			−0.504981			−0.252490	+	0.967599i	$0.581250\pi$
$858$		0			0
$859$			9.85648e15i			0.719050i	0.933135	+	0.359525i	$0.117061\pi$
$859$			9.85648e15i			0.719050i	−0.933135	+	0.359525i	$0.882939\pi$
$860$		0			0
$861$	1.31110e16	−	1.75484e16i	0.944322	−	1.26392i
$862$		0			0
$863$		−	9.16466e15i		−	0.651714i	−0.945419	−	0.325857i	$-0.894347\pi$
$863$		−	9.16466e15i		−	0.651714i	0.945419	−	0.325857i	$-0.105653\pi$
$864$		0			0
$865$	1.93390e16			1.35783
$866$		0			0
$867$	−9.69345e14	+	1.69429e16i	−0.0672006	+	1.17458i
$868$		0			0
$869$			9.08605e15i			0.621965i
$870$		0			0
$871$			5.41621e14i			0.0366097i
$872$		0			0
$873$	3.15967e15	−	2.75231e16i	0.210894	−	1.83704i
$874$		0			0
$875$	−1.97014e16	−	1.30353e16i	−1.29853	−	0.859165i
$876$		0			0
$877$	−1.52346e16			−0.991593			−0.495796	−	0.868439i	$-0.665124\pi$
$877$	−1.52346e16			−0.991593			−0.495796	+	0.868439i	$0.665124\pi$
$878$		0			0
$879$	1.27979e14	−	2.23691e15i	0.00822623	−	0.143784i
$880$		0			0
$881$	8.72513e15			0.553866			0.276933	−	0.960889i	$-0.410682\pi$
$881$	8.72513e15			0.553866			0.276933	+	0.960889i	$0.410682\pi$
$882$		0			0
$883$	−1.38870e16			−0.870610			−0.435305	−	0.900283i	$-0.643360\pi$
$883$	−1.38870e16			−0.870610			−0.435305	+	0.900283i	$0.643360\pi$
$884$		0			0
$885$	−8.61420e14	+	1.50565e16i	−0.0533368	+	0.932259i
$886$		0			0
$887$	7.57707e15			0.463363			0.231682	−	0.972792i	$-0.425577\pi$
$887$	7.57707e15			0.463363			0.231682	+	0.972792i	$0.425577\pi$
$888$		0			0
$889$	2.10453e16	+	1.39245e16i	1.27115	+	0.841045i
$890$		0			0
$891$	−1.83983e15	+	7.90756e15i	−0.109762	+	0.471754i
$892$		0			0
$893$			3.35130e16i			1.97483i
$894$		0			0
$895$			3.68054e16i			2.14232i
$896$		0			0
$897$	−1.78102e15	+	3.11299e16i	−0.102402	+	1.78986i
$898$		0			0
$899$	−6.90184e15			−0.392001
$900$		0			0
$901$			1.17935e16i			0.661690i
$902$		0			0
$903$	−3.87635e15	+	5.18827e15i	−0.214852	+	0.287568i
$904$		0			0
$905$		−	4.14276e16i		−	2.26841i
$906$		0			0
$907$	2.02955e16			1.09789			0.548945	−	0.835858i	$-0.315030\pi$
$907$	2.02955e16			1.09789			0.548945	+	0.835858i	$0.315030\pi$
$908$		0			0
$909$	2.83509e16	+	3.25470e15i	1.51519	+	0.173944i
$910$		0			0
$911$			2.45742e16i			1.29756i	0.760975	+	0.648781i	$0.224721\pi$
$911$			2.45742e16i			1.29756i	−0.760975	+	0.648781i	$0.775279\pi$
$912$		0			0
$913$			1.52059e15i			0.0793276i
$914$		0			0
$915$	−3.38182e16	−	1.93482e15i	−1.74315	−	0.0997297i
$916$		0			0
$917$	−7.13433e14	−	4.72037e14i	−0.0363347	−	0.0240406i
$918$		0			0
$919$	8.80041e15			0.442861			0.221431	−	0.975176i	$-0.428927\pi$
$919$	8.80041e15			0.442861			0.221431	+	0.975176i	$0.428927\pi$
$920$		0			0
$921$	2.91624e15	+	1.66845e14i	0.145009	+	0.00829632i
$922$		0			0
$923$	2.48966e16			1.22329
$924$		0			0
$925$	−8.46260e15			−0.410889
$926$		0			0
$927$	−2.29488e14	+	1.99901e15i	−0.0110109	+	0.0959129i
$928$		0			0
$929$	3.11086e16			1.47501			0.737504	−	0.675343i	$-0.236004\pi$
$929$	3.11086e16			1.47501			0.737504	+	0.675343i	$0.236004\pi$
$930$		0			0
$931$	−2.44353e16	+	1.03819e16i	−1.14497	+	0.486469i
$932$		0			0
$933$	1.59713e15	−	2.79158e16i	0.0739591	−	1.29271i
$934$		0			0
$935$			2.66459e16i			1.21946i
$936$		0			0
$937$		−	4.16475e16i		−	1.88374i	−0.335978	−	0.941870i	$-0.609067\pi$
$937$		−	4.16475e16i		−	1.88374i	0.335978	−	0.941870i	$-0.390933\pi$
$938$		0			0
$939$	4.29771e16	+	2.45882e15i	1.92122	+	0.109917i
$940$		0			0
$941$	1.40239e16			0.619622			0.309811	−	0.950798i	$-0.399734\pi$
$941$	1.40239e16			0.619622			0.309811	+	0.950798i	$0.399734\pi$
$942$		0			0
$943$			4.65166e16i			2.03139i
$944$		0			0
$945$	−2.71218e16	+	2.87672e16i	−1.17069	+	1.24171i
$946$		0			0
$947$			3.51821e15i			0.150106i	0.997180	+	0.0750529i	$0.0239126\pi$
$947$			3.51821e15i			0.150106i	−0.997180	+	0.0750529i	$0.976087\pi$
$948$		0			0
$949$	−6.34216e15			−0.267469
$950$		0			0
$951$	1.12610e16	+	6.44269e14i	0.469444	+	0.0268580i
$952$		0			0
$953$		−	2.05278e16i		−	0.845925i	−0.906147	−	0.422962i	$-0.860990\pi$
$953$		−	2.05278e16i		−	0.845925i	0.906147	−	0.422962i	$-0.139010\pi$
$954$		0			0
$955$			1.98656e16i			0.809253i
$956$		0			0
$957$	−9.31180e14	+	1.62758e16i	−0.0374989	+	0.655432i
$958$		0			0
$959$	2.51440e16	−	3.80025e16i	1.00100	−	1.51290i
$960$		0			0
$961$	2.32833e16			0.916358
$962$		0			0
$963$	−3.73647e15	+	3.25474e16i	−0.145384	+	1.26640i
$964$		0			0
$965$	−5.40053e16			−2.07748
$966$		0			0
$967$	2.85142e16			1.08446			0.542232	−	0.840229i	$-0.317579\pi$
$967$	2.85142e16			1.08446			0.542232	+	0.840229i	$0.317579\pi$
$968$		0			0
$969$	−4.87283e16	−	2.78787e15i	−1.83232	−	0.104831i
$970$		0			0
$971$	2.19479e16			0.815994			0.407997	−	0.912983i	$-0.366227\pi$
$971$	2.19479e16			0.815994			0.407997	+	0.912983i	$0.366227\pi$
$972$		0			0
$973$	−5.34505e14	+	8.07847e14i	−0.0196486	+	0.0296967i
$974$		0			0
$975$	7.31406e16	+	4.18455e15i	2.65848	+	0.152098i
$976$		0			0
$977$		−	8.38554e15i		−	0.301378i	−0.988581	−	0.150689i	$-0.951851\pi$
$977$		−	8.38554e15i		−	0.301378i	0.988581	−	0.150689i	$-0.0481491\pi$
$978$		0			0
$979$			2.43943e14i			0.00866930i
$980$		0			0
$981$	−1.48168e16	−	1.70098e15i	−0.520686	−	0.0597751i
$982$		0			0
$983$	−1.80530e16			−0.627342			−0.313671	−	0.949532i	$-0.601559\pi$
$983$	−1.80530e16			−0.627342			−0.313671	+	0.949532i	$0.601559\pi$
$984$		0			0
$985$		−	3.12966e15i		−	0.107547i
$986$		0			0
$987$	2.79595e16	−	3.74222e16i	0.950135	−	1.27170i
$988$		0			0
$989$		−	1.37529e16i		−	0.462182i
$990$		0			0
$991$	−1.25401e16			−0.416768			−0.208384	−	0.978047i	$-0.566820\pi$
$991$	−1.25401e16			−0.416768			−0.208384	+	0.978047i	$0.566820\pi$
$992$		0			0
$993$	9.55713e14	−	1.67046e16i	0.0314128	−	0.549055i
$994$		0			0
$995$		−	3.53046e16i		−	1.14764i
$996$		0			0
$997$		−	1.97530e16i		−	0.635053i	−0.948249	−	0.317527i	$-0.897148\pi$
$997$		−	1.97530e16i		−	0.635053i	0.948249	−	0.317527i	$-0.102852\pi$
$998$		0			0
$999$	−1.15284e15	+	6.65804e15i	−0.0366572	+	0.211707i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.12.f.b.41.14	yes	28
3.2	odd	2	inner	84.12.f.b.41.16	yes	28
7.6	odd	2	inner	84.12.f.b.41.15	yes	28
21.20	even	2	inner	84.12.f.b.41.13	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.12.f.b.41.13	✓	28	21.20	even	2	inner
84.12.f.b.41.14	yes	28	1.1	even	1	trivial
84.12.f.b.41.15	yes	28	7.6	odd	2	inner
84.12.f.b.41.16	yes	28	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:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	84.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-24.0407 + 420.201i) q^{3} +11925.0 q^{5} +(-37084.7 - 24536.8i) q^{7} +(-175991. - 20203.9i) q^{9} +O(q^{10})\)	\(q+(-24.0407 + 420.201i) q^{3} +11925.0 q^{5} +(-37084.7 - 24536.8i) q^{7} +(-175991. - 20203.9i) q^{9} +258716. i q^{11} -1.86405e6i q^{13} +(-286686. + 5.01090e6i) q^{15} +8.63671e6 q^{17} +1.34269e7i q^{19} +(1.12019e7 - 1.49931e7i) q^{21} +3.97432e7i q^{23} +9.33778e7 q^{25} +(1.27207e7 - 7.34659e7i) q^{27} -1.49714e8i q^{29} -4.61001e7i q^{31} +(-1.08713e8 - 6.21972e6i) q^{33} +(-4.42235e8 - 2.92601e8i) q^{35} -9.06276e7 q^{37} +(7.83276e8 + 4.48131e7i) q^{39} +1.17043e9 q^{41} -3.46043e8 q^{43} +(-2.09870e9 - 2.40932e8i) q^{45} +2.49596e9 q^{47} +(7.73220e8 + 1.81988e9i) q^{49} +(-2.07633e8 + 3.62916e9i) q^{51} +1.36550e9i q^{53} +3.08519e9i q^{55} +(-5.64200e9 - 3.22792e8i) q^{57} +3.00475e9 q^{59} +6.74891e9i q^{61} +(6.03083e9 + 5.06751e9i) q^{63} -2.22288e10i q^{65} -2.90561e8 q^{67} +(-1.67001e10 - 9.55456e8i) q^{69} +1.33562e10i q^{71} -3.40236e9i q^{73} +(-2.24487e9 + 3.92375e10i) q^{75} +(6.34805e9 - 9.59439e9i) q^{77} +3.51198e10 q^{79} +(3.05647e10 + 7.11141e9i) q^{81} +5.87747e9 q^{83} +1.02993e11 q^{85} +(6.29101e10 + 3.59924e9i) q^{87} +9.42901e8 q^{89} +(-4.57378e10 + 6.91277e10i) q^{91} +(1.93713e10 + 1.10828e9i) q^{93} +1.60116e11i q^{95} +1.56389e11i q^{97} +(5.22707e9 - 4.55317e10i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 9632 q^{7} + 267660 q^{9}+O(q^{10}) \) 28 * q + 9632 * q^7 + 267660 * q^9	\( 28 q + 9632 q^{7} + 267660 q^{9} - 3434160 q^{15} - 18804156 q^{21} + 397876900 q^{25} - 2059460504 q^{37} + 2276313936 q^{39} + 607100560 q^{43} + 1145242588 q^{49} + 1424787216 q^{51} - 32512522344 q^{57} + 16390616256 q^{63} - 48876957136 q^{67} - 1293110368 q^{79} + 82706814108 q^{81} + 197440859760 q^{85} - 329206232880 q^{91} - 243855044280 q^{93} - 81383696064 q^{99}+O(q^{100}) \) 28 * q + 9632 * q^7 + 267660 * q^9 - 3434160 * q^15 - 18804156 * q^21 + 397876900 * q^25 - 2059460504 * q^37 + 2276313936 * q^39 + 607100560 * q^43 + 1145242588 * q^49 + 1424787216 * q^51 - 32512522344 * q^57 + 16390616256 * q^63 - 48876957136 * q^67 - 1293110368 * q^79 + 82706814108 * q^81 + 197440859760 * q^85 - 329206232880 * q^91 - 243855044280 * q^93 - 81383696064 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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