LMFDB - Embedded newform 252.8.k.c.37.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,8,Mod(37,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("252.37");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	252.k (of order $3$, degree $2$, 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:	$78.7210264220$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + 342 x^{8} + 2165 x^{7} + 113605 x^{6} + 319380 x^{5} + 1438128 x^{4} + 1705752 x^{3} + \cdots + 23619600 $ x^10 - x^9 + 342x^8 + 2165x^7 + 113605x^6 + 319380x^5 + 1438128x^4 + 1705752x^3 + 7544016x^2 + 8573040x + 23619600
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{8}\cdot 7^{5} $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.5
Root			$1.14872 - 1.98964i$ of defining polynomial
Character	$\chi$	$=$	252.37
Dual form			252.8.k.c.109.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(209.191 - 362.329i) q^{5} +(-505.918 - 753.386i) q^{7} +(128.291 + 222.207i) q^{11} -8020.94 q^{13} +(-12820.8 - 22206.2i) q^{17} +(-27563.8 + 47742.0i) q^{19} +(-18948.1 + 32819.1i) q^{23} +(-48459.1 - 83933.6i) q^{25} -125755. q^{29} +(80194.6 + 138901. i) q^{31} +(-378807. + 25707.3i) q^{35} +(-70502.5 + 122114. i) q^{37} +674106. q^{41} +755814. q^{43} +(102751. - 177969. i) q^{47} +(-311638. + 762302. i) q^{49} +(182888. + 316772. i) q^{53} +107349. q^{55} +(-962456. - 1.66702e6i) q^{59} +(1.35960e6 - 2.35489e6i) q^{61} +(-1.67791e6 + 2.90622e6i) q^{65} +(782715. + 1.35570e6i) q^{67} +2.29364e6 q^{71} +(586366. + 1.01562e6i) q^{73} +(102503. - 209071. i) q^{77} +(-360081. + 623678. i) q^{79} -2.39050e6 q^{83} -1.07279e7 q^{85} +(-6.20699e6 + 1.07508e7i) q^{89} +(4.05794e6 + 6.04287e6i) q^{91} +(1.15322e7 + 1.99744e7i) q^{95} -1.48005e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 249 q^{5} + 332 q^{7} - 6399 q^{11} - 26988 q^{13} - 3609 q^{17} - 12403 q^{19} + 13959 q^{23} - 162364 q^{25} - 26148 q^{29} - 20181 q^{31} - 791715 q^{35} - 54763 q^{37} + 1824468 q^{41} - 1938424 q^{43}+ \cdots + 285580 q^{97}+O(q^{100}) $ 10 * q - 249 * q^5 + 332 * q^7 - 6399 * q^11 - 26988 * q^13 - 3609 * q^17 - 12403 * q^19 + 13959 * q^23 - 162364 * q^25 - 26148 * q^29 - 20181 * q^31 - 791715 * q^35 - 54763 * q^37 + 1824468 * q^41 - 1938424 * q^43 - 1217253 * q^47 + 1722058 * q^49 + 2349159 * q^53 + 3123494 * q^55 - 2188611 * q^59 + 6107445 * q^61 + 3022014 * q^65 - 3171581 * q^67 + 18938976 * q^71 + 8034853 * q^73 - 24100191 * q^77 - 7589559 * q^79 + 13452408 * q^83 - 48717670 * q^85 - 29864757 * q^89 + 34267512 * q^91 + 25812009 * q^95 + 285580 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	209.191	−	362.329i	0.748424	−	1.29631i	−0.200154	−	0.979764i	$-0.564144\pi$
$5$	209.191	−	362.329i	0.748424	−	1.29631i	0.948578	−	0.316543i	$-0.102522\pi$
$6$		0			0
$7$	−505.918	−	753.386i	−0.557490	−	0.830184i
$7$	−505.918	−	753.386i	−0.557490	−	0.830184i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	128.291	+	222.207i	0.0290618	+	0.0503365i	0.880191	−	0.474621i	$-0.157415\pi$
$11$	128.291	+	222.207i	0.0290618	+	0.0503365i	−0.851129	+	0.524957i	$0.824081\pi$
$12$		0			0
$13$	−8020.94			−1.01257			−0.506283	−	0.862367i	$-0.668981\pi$
$13$	−8020.94			−1.01257			−0.506283	+	0.862367i	$0.668981\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−12820.8	−	22206.2i	−0.632911	−	1.09623i	−0.986954	−	0.161005i	$-0.948527\pi$
$17$	−12820.8	−	22206.2i	−0.632911	−	1.09623i	0.354043	−	0.935229i	$-0.384807\pi$
$18$		0			0
$19$	−27563.8	+	47742.0i	−0.921939	+	1.59684i	−0.125527	+	0.992090i	$0.540062\pi$
$19$	−27563.8	+	47742.0i	−0.921939	+	1.59684i	−0.796412	+	0.604755i	$0.793271\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−18948.1	+	32819.1i	−0.324727	+	0.562444i	−0.981457	−	0.191682i	$-0.938606\pi$
$23$	−18948.1	+	32819.1i	−0.324727	+	0.562444i	0.656730	+	0.754126i	$0.271939\pi$
$24$		0			0
$25$	−48459.1	−	83933.6i	−0.620276	−	1.07435i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−125755.			−0.957482			−0.478741	−	0.877956i	$-0.658907\pi$
$29$	−125755.			−0.957482			−0.478741	+	0.877956i	$0.658907\pi$
$30$		0			0
$31$	80194.6	+	138901.i	0.483481	+	0.837413i	0.999820	−	0.0189707i	$-0.00603893\pi$
$31$	80194.6	+	138901.i	0.483481	+	0.837413i	−0.516339	+	0.856384i	$0.672706\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−378807.	+	25707.3i	−1.49341	+	0.101349i
$36$		0			0
$37$	−70502.5	+	122114.i	−0.228822	+	0.396332i	−0.957459	−	0.288568i	$-0.906821\pi$
$37$	−70502.5	+	122114.i	−0.228822	+	0.396332i	0.728637	+	0.684900i	$0.240154\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	674106.			1.52751			0.763756	−	0.645505i	$-0.223353\pi$
$41$	674106.			1.52751			0.763756	+	0.645505i	$0.223353\pi$
$42$		0			0
$43$	755814.			1.44969			0.724845	−	0.688912i	$-0.241911\pi$
$43$	755814.			1.44969			0.724845	+	0.688912i	$0.241911\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	102751.	−	177969.i	0.144358	−	0.250036i	−0.784775	−	0.619781i	$-0.787222\pi$
$47$	102751.	−	177969.i	0.144358	−	0.250036i	0.929133	+	0.369745i	$0.120555\pi$
$48$		0			0
$49$	−311638.	+	762302.i	−0.378411	+	0.925638i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	182888.	+	316772.i	0.168741	+	0.292268i	0.937977	−	0.346696i	$-0.112697\pi$
$53$	182888.	+	316772.i	0.168741	+	0.292268i	−0.769237	+	0.638964i	$0.779363\pi$
$54$		0			0
$55$	107349.			0.0870022
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−962456.	−	1.66702e6i	−0.610097	−	1.05672i	−0.991224	−	0.132196i	$-0.957797\pi$
$59$	−962456.	−	1.66702e6i	−0.610097	−	1.05672i	0.381127	−	0.924523i	$-0.375536\pi$
$60$		0			0
$61$	1.35960e6	−	2.35489e6i	0.766931	−	1.32836i	−0.172288	−	0.985047i	$-0.555116\pi$
$61$	1.35960e6	−	2.35489e6i	0.766931	−	1.32836i	0.939220	−	0.343317i	$-0.111551\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.67791e6	+	2.90622e6i	−0.757829	+	1.31260i
$66$		0			0
$67$	782715.	+	1.35570e6i	0.317938	+	0.550684i	0.980058	−	0.198714i	$-0.0636764\pi$
$67$	782715.	+	1.35570e6i	0.317938	+	0.550684i	−0.662120	+	0.749398i	$0.730343\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.29364e6			0.760538			0.380269	−	0.924876i	$-0.375831\pi$
$71$	2.29364e6			0.760538			0.380269	+	0.924876i	$0.375831\pi$
$72$		0			0
$73$	586366.	+	1.01562e6i	0.176416	+	0.305562i	0.940650	−	0.339377i	$-0.110216\pi$
$73$	586366.	+	1.01562e6i	0.176416	+	0.305562i	−0.764234	+	0.644939i	$0.776883\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	102503.	−	209071.i	0.0255869	−	0.0521887i
$78$		0			0
$79$	−360081.	+	623678.i	−0.0821685	+	0.142320i	−0.904181	−	0.427149i	$-0.859518\pi$
$79$	−360081.	+	623678.i	−0.0821685	+	0.142320i	0.822013	+	0.569469i	$0.192851\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.39050e6			−0.458898			−0.229449	−	0.973321i	$-0.573692\pi$
$83$	−2.39050e6			−0.458898			−0.229449	+	0.973321i	$0.573692\pi$
$84$		0			0
$85$	−1.07279e7			−1.89474
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.20699e6	+	1.07508e7i	−0.933288	+	1.61650i	−0.155630	+	0.987815i	$0.549741\pi$
$89$	−6.20699e6	+	1.07508e7i	−0.933288	+	1.61650i	−0.777658	+	0.628688i	$0.783592\pi$
$90$		0			0
$91$	4.05794e6	+	6.04287e6i	0.564495	+	0.840617i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.15322e7	+	1.99744e7i	1.38000	+	2.39023i
$96$		0			0
$97$	−1.48005e7			−1.64655			−0.823276	−	0.567641i	$-0.807856\pi$
$97$	−1.48005e7			−1.64655			−0.823276	+	0.567641i	$0.807856\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.21061e6	+	5.56093e6i	0.310072	+	0.537060i	0.978378	−	0.206827i	$-0.0663136\pi$
$101$	3.21061e6	+	5.56093e6i	0.310072	+	0.537060i	−0.668306	+	0.743887i	$0.732980\pi$
$102$		0			0
$103$	−1.73226e6	+	3.00036e6i	−0.156201	+	0.270547i	−0.933496	−	0.358589i	$-0.883258\pi$
$103$	−1.73226e6	+	3.00036e6i	−0.156201	+	0.270547i	0.777295	+	0.629136i	$0.216591\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.94299e6	−	1.54897e7i	0.705732	−	1.22236i	−0.260695	−	0.965421i	$-0.583952\pi$
$107$	8.94299e6	−	1.54897e7i	0.705732	−	1.22236i	0.966427	−	0.256942i	$-0.0827149\pi$
$108$		0			0
$109$	2.28468e6	+	3.95719e6i	0.168979	+	0.292681i	0.938061	−	0.346469i	$-0.112620\pi$
$109$	2.28468e6	+	3.95719e6i	0.168979	+	0.292681i	−0.769082	+	0.639150i	$0.779286\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.46079e6			−0.356025			−0.178013	−	0.984028i	$-0.556967\pi$
$113$	−5.46079e6			−0.356025			−0.178013	+	0.984028i	$0.556967\pi$
$114$		0			0
$115$	7.92754e6	+	1.37309e7i	0.486067	+	0.841892i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.02436e7	+	2.08935e7i	−0.557235	+	1.13657i
$120$		0			0
$121$	9.71067e6	−	1.68194e7i	0.498311	−	0.863100i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.86270e6			−0.360070
$126$		0			0
$127$	−1.65990e7			−0.719066			−0.359533	−	0.933132i	$-0.617064\pi$
$127$	−1.65990e7			−0.719066			−0.359533	+	0.933132i	$0.617064\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.00485e7	+	3.47250e7i	−0.779169	+	1.34956i	0.153252	+	0.988187i	$0.451026\pi$
$131$	−2.00485e7	+	3.47250e7i	−0.779169	+	1.34956i	−0.932421	+	0.361374i	$0.882308\pi$
$132$		0			0
$133$	4.99132e7	−	3.38730e6i	1.83965	−	0.124845i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	445291.	+	771267.i	0.0147953	+	0.0256261i	0.873328	−	0.487132i	$-0.161957\pi$
$137$	445291.	+	771267.i	0.0147953	+	0.0256261i	−0.858533	+	0.512758i	$0.828624\pi$
$138$		0			0
$139$	1.42244e7			0.449243			0.224622	−	0.974446i	$-0.427885\pi$
$139$	1.42244e7			0.449243			0.224622	+	0.974446i	$0.427885\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.02902e6	−	1.78231e6i	−0.0294270	−	0.0509691i
$144$		0			0
$145$	−2.63067e7	+	4.55645e7i	−0.716602	+	1.24119i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2.49059e7	+	4.31383e7i	−0.616809	+	1.06834i	0.373256	+	0.927729i	$0.378241\pi$
$149$	−2.49059e7	+	4.31383e7i	−0.616809	+	1.06834i	−0.990064	+	0.140615i	$0.955092\pi$
$150$		0			0
$151$	2.60248e7	+	4.50763e7i	0.615132	+	1.06544i	0.990361	+	0.138508i	$0.0442308\pi$
$151$	2.60248e7	+	4.50763e7i	0.615132	+	1.06544i	−0.375229	+	0.926932i	$0.622436\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.71039e7			1.44739
$156$		0			0
$157$	−1.33569e7	−	2.31348e7i	−0.275459	−	0.477108i	0.694792	−	0.719211i	$-0.255496\pi$
$157$	−1.33569e7	−	2.31348e7i	−0.275459	−	0.477108i	−0.970251	+	0.242102i	$0.922163\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.43116e7	−	2.32852e6i	0.647964	−	0.0439733i
$162$		0			0
$163$	−8.57538e6	+	1.48530e7i	−0.155095	+	0.268632i	−0.933093	−	0.359634i	$-0.882902\pi$
$163$	−8.57538e6	+	1.48530e7i	−0.155095	+	0.268632i	0.777999	+	0.628266i	$0.216235\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.00218e7			−1.16339			−0.581695	−	0.813407i	$-0.697610\pi$
$167$	−7.00218e7			−1.16339			−0.581695	+	0.813407i	$0.697610\pi$
$168$		0			0
$169$	1.58701e6			0.0252916
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.58506e7	+	4.47745e7i	−0.379585	+	0.657460i	−0.991002	−	0.133848i	$-0.957267\pi$
$173$	−2.58506e7	+	4.47745e7i	−0.379585	+	0.657460i	0.611417	+	0.791309i	$0.290600\pi$
$174$		0			0
$175$	−3.87181e7	+	7.89718e7i	−0.546110	+	1.11388i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.66140e7	−	2.87764e7i	−0.216516	−	0.375017i	0.737225	−	0.675648i	$-0.236136\pi$
$179$	−1.66140e7	−	2.87764e7i	−0.216516	−	0.375017i	−0.953740	+	0.300631i	$0.902803\pi$
$180$		0			0
$181$	−1.23418e8			−1.54705			−0.773523	−	0.633769i	$-0.781507\pi$
$181$	−1.23418e8			−1.54705			−0.773523	+	0.633769i	$0.781507\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.94969e7	+	5.10902e7i	0.342512	+	0.593248i
$186$		0			0
$187$	3.28958e6	−	5.69773e6i	0.0367871	−	0.0637171i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.12287e7	+	7.14103e7i	−0.428137	+	0.741556i	−0.996708	−	0.0810789i	$-0.974163\pi$
$191$	−4.12287e7	+	7.14103e7i	−0.428137	+	0.741556i	0.568570	+	0.822635i	$0.307497\pi$
$192$		0			0
$193$	5.29252e7	+	9.16691e7i	0.529922	+	0.917851i	0.999391	+	0.0349023i	$0.0111120\pi$
$193$	5.29252e7	+	9.16691e7i	0.529922	+	0.917851i	−0.469469	+	0.882949i	$0.655555\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.23667e8			−1.15245			−0.576224	−	0.817292i	$-0.695475\pi$
$197$	−1.23667e8			−1.15245			−0.576224	+	0.817292i	$0.695475\pi$
$198$		0			0
$199$	−1.96291e7	−	3.39986e7i	−0.176569	−	0.305827i	0.764134	−	0.645058i	$-0.223167\pi$
$199$	−1.96291e7	−	3.39986e7i	−0.176569	−	0.305827i	−0.940703	+	0.339231i	$0.889833\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.36214e7	+	9.47417e7i	0.533786	+	0.794886i
$204$		0			0
$205$	1.41017e8	−	2.44248e8i	1.14323	−	1.98013i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.41448e7			−0.107173
$210$		0			0
$211$	−1.64534e7			−0.120578			−0.0602890	−	0.998181i	$-0.519202\pi$
$211$	−1.64534e7			−0.120578			−0.0602890	+	0.998181i	$0.519202\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.58109e8	−	2.73853e8i	1.08498	−	1.87925i
$216$		0			0
$217$	6.40743e7	−	1.30690e8i	0.425672	−	0.868227i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.02835e8	+	1.78115e8i	0.640865	+	1.11001i
$222$		0			0
$223$	−9.12765e7			−0.551178			−0.275589	−	0.961276i	$-0.588873\pi$
$223$	−9.12765e7			−0.551178			−0.275589	+	0.961276i	$0.588873\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.65348e8	−	2.86392e8i	−0.938230	−	1.62506i	−0.768771	−	0.639524i	$-0.779132\pi$
$227$	−1.65348e8	−	2.86392e8i	−0.938230	−	1.62506i	−0.169458	−	0.985537i	$-0.554202\pi$
$228$		0			0
$229$	−1.38078e8	+	2.39158e8i	−0.759802	+	1.31602i	0.183149	+	0.983085i	$0.441371\pi$
$229$	−1.38078e8	+	2.39158e8i	−0.759802	+	1.31602i	−0.942951	+	0.332931i	$0.891962\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.54182e8	−	2.67052e8i	0.798526	−	1.38309i	−0.122050	−	0.992524i	$-0.538947\pi$
$233$	1.54182e8	−	2.67052e8i	0.798526	−	1.38309i	0.920576	−	0.390563i	$-0.127720\pi$
$234$		0			0
$235$	−4.29889e7	−	7.44590e7i	−0.216082	−	0.374266i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	7.05871e7			0.334451			0.167225	−	0.985919i	$-0.446519\pi$
$239$	7.05871e7			0.334451			0.167225	+	0.985919i	$0.446519\pi$
$240$		0			0
$241$	−3.98671e6	−	6.90519e6i	−0.0183466	−	0.0317772i	0.856706	−	0.515804i	$-0.172507\pi$
$241$	−3.98671e6	−	6.90519e6i	−0.0183466	−	0.0317772i	−0.875053	+	0.484027i	$0.839174\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.11013e8	+	2.72382e8i	0.916700	+	1.18331i
$246$		0			0
$247$	2.21088e8	−	3.82936e8i	0.933525	−	1.61691i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.09068e8			0.435351			0.217675	−	0.976021i	$-0.430153\pi$
$251$	1.09068e8			0.435351			0.217675	+	0.976021i	$0.430153\pi$
$252$		0			0
$253$	−9.72351e6			−0.0377486
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.11507e7	−	5.39545e7i	0.114473	−	0.198272i	−0.803096	−	0.595849i	$-0.796816\pi$
$257$	3.11507e7	−	5.39545e7i	0.114473	−	0.198272i	0.917569	+	0.397577i	$0.130149\pi$
$258$		0			0
$259$	1.27667e8	−	8.66399e6i	0.456594	−	0.0309862i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4.47919e7	−	7.75818e7i	−0.151829	−	0.262975i	0.780071	−	0.625691i	$-0.215183\pi$
$263$	−4.47919e7	−	7.75818e7i	−0.151829	−	0.262975i	−0.931900	+	0.362716i	$0.881850\pi$
$264$		0			0
$265$	1.53034e8			0.505159
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.38780e8	−	2.40375e8i	−0.434706	−	0.752932i	0.562566	−	0.826752i	$-0.309814\pi$
$269$	−1.38780e8	−	2.40375e8i	−0.434706	−	0.752932i	−0.997272	+	0.0738203i	$0.976481\pi$
$270$		0			0
$271$	−953906.	+	1.65221e6i	−0.00291147	+	0.00504282i	−0.867477	−	0.497476i	$-0.834260\pi$
$271$	−953906.	+	1.65221e6i	−0.00291147	+	0.00504282i	0.864566	+	0.502519i	$0.167593\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.24337e7	−	2.15359e7i	0.0360527	−	0.0624451i
$276$		0			0
$277$	1.90770e8	+	3.30422e8i	0.539299	+	0.934094i	0.998942	+	0.0459897i	$0.0146441\pi$
$277$	1.90770e8	+	3.30422e8i	0.539299	+	0.934094i	−0.459643	+	0.888104i	$0.652023\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.58621e8			−0.426470			−0.213235	−	0.977001i	$-0.568400\pi$
$281$	−1.58621e8			−0.426470			−0.213235	+	0.977001i	$0.568400\pi$
$282$		0			0
$283$	−1.21585e8	−	2.10591e8i	−0.318879	−	0.552315i	0.661375	−	0.750055i	$-0.269973\pi$
$283$	−1.21585e8	−	2.10591e8i	−0.318879	−	0.552315i	−0.980255	+	0.197740i	$0.936640\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.41042e8	−	5.07862e8i	−0.851572	−	1.26812i
$288$		0			0
$289$	−1.23575e8	+	2.14037e8i	−0.301152	+	0.521611i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−5.34536e8			−1.24148			−0.620740	−	0.784016i	$-0.713168\pi$
$293$	−5.34536e8			−1.24148			−0.620740	+	0.784016i	$0.713168\pi$
$294$		0			0
$295$	−8.05348e8			−1.82644
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.51982e8	−	2.63240e8i	0.328808	−	0.569512i
$300$		0			0
$301$	−3.82380e8	−	5.69420e8i	−0.808187	−	1.20351i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5.68831e8	−	9.85244e8i	−1.14798	−	1.98836i
$306$		0			0
$307$	6.57215e8			1.29635			0.648176	−	0.761490i	$-0.275532\pi$
$307$	6.57215e8			1.29635			0.648176	+	0.761490i	$0.275532\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.77204e7	−	4.80132e7i	−0.0522563	−	0.0905105i	0.838714	−	0.544572i	$-0.183308\pi$
$311$	−2.77204e7	−	4.80132e7i	−0.0522563	−	0.0905105i	−0.890970	+	0.454062i	$0.849975\pi$
$312$		0			0
$313$	2.25674e8	−	3.90879e8i	0.415984	−	0.720506i	−0.579547	−	0.814939i	$-0.696770\pi$
$313$	2.25674e8	−	3.90879e8i	0.415984	−	0.720506i	0.995531	+	0.0944330i	$0.0301038\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.06940e7	+	5.31636e7i	−0.0541186	+	0.0937361i	−0.891816	−	0.452399i	$-0.850568\pi$
$317$	−3.06940e7	+	5.31636e7i	−0.0541186	+	0.0937361i	0.837697	+	0.546135i	$0.183902\pi$
$318$		0			0
$319$	−1.61332e7	−	2.79435e7i	−0.0278262	−	0.0481963i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.41356e9			2.33402
$324$		0			0
$325$	3.88687e8	+	6.73226e8i	0.628071	+	1.08785i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.86063e8	+	1.26269e7i	−0.288054	+	0.0195484i
$330$		0			0
$331$	−2.03880e8	+	3.53131e8i	−0.309013	+	0.535227i	−0.978147	−	0.207916i	$-0.933332\pi$
$331$	−2.03880e8	+	3.53131e8i	−0.309013	+	0.535227i	0.669134	+	0.743142i	$0.266665\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.54947e8			0.951808
$336$		0			0
$337$	−2.63519e8			−0.375066			−0.187533	−	0.982258i	$-0.560049\pi$
$337$	−2.63519e8			−0.375066			−0.187533	+	0.982258i	$0.560049\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.05765e7	+	3.56396e7i	−0.0281017	+	0.0486735i
$342$		0			0
$343$	7.31971e8	−	1.50879e8i	0.979410	−	0.201883i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.03799e8	+	3.52991e8i	0.261848	+	0.453534i	0.966733	−	0.255787i	$-0.0823347\pi$
$347$	2.03799e8	+	3.52991e8i	0.261848	+	0.453534i	−0.704885	+	0.709322i	$0.749001\pi$
$348$		0			0
$349$	−5.06614e7			−0.0637952			−0.0318976	−	0.999491i	$-0.510155\pi$
$349$	−5.06614e7			−0.0637952			−0.0318976	+	0.999491i	$0.510155\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.26375e8	−	7.38502e8i	−0.515917	−	0.893594i	−0.999829	−	0.0184778i	$-0.994118\pi$
$353$	−4.26375e8	−	7.38502e8i	−0.515917	−	0.893594i	0.483912	−	0.875116i	$-0.339215\pi$
$354$		0			0
$355$	4.79808e8	−	8.31052e8i	0.569205	−	0.985891i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7.51554e8	+	1.30173e9i	−0.857293	+	1.48488i	0.0172074	+	0.999852i	$0.494522\pi$
$359$	−7.51554e8	+	1.30173e9i	−0.857293	+	1.48488i	−0.874501	+	0.485024i	$0.838811\pi$
$360$		0			0
$361$	−1.07259e9	−	1.85779e9i	−1.19994	−	2.07836i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.90649e8			0.528136
$366$		0			0
$367$	5.49573e8	+	9.51888e8i	0.580355	+	1.00520i	0.995437	+	0.0954206i	$0.0304196\pi$
$367$	5.49573e8	+	9.51888e8i	0.580355	+	1.00520i	−0.415082	+	0.909784i	$0.636247\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.46125e8	−	2.98046e8i	0.148565	−	0.303022i
$372$		0			0
$373$	−5.80290e8	+	1.00509e9i	−0.578981	+	1.00282i	0.416616	+	0.909083i	$0.363216\pi$
$373$	−5.80290e8	+	1.00509e9i	−0.578981	+	1.00282i	−0.995597	+	0.0937415i	$0.970117\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.00867e9			0.969515
$378$		0			0
$379$	1.63326e9			1.54106			0.770528	−	0.637406i	$-0.219993\pi$
$379$	1.63326e9			1.54106			0.770528	+	0.637406i	$0.219993\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6.14600e7	+	1.06452e8i	−0.0558981	+	0.0968183i	−0.892620	−	0.450809i	$-0.851136\pi$
$383$	−6.14600e7	+	1.06452e8i	−0.0558981	+	0.0968183i	0.836722	+	0.547627i	$0.184469\pi$
$384$		0			0
$385$	−5.43100e7	−	8.08755e7i	−0.0485028	−	0.0722278i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.79054e8	+	4.83336e8i	0.240361	+	0.416318i	0.960817	−	0.277183i	$-0.0894008\pi$
$389$	2.79054e8	+	4.83336e8i	0.240361	+	0.416318i	−0.720456	+	0.693501i	$0.756067\pi$
$390$		0			0
$391$	9.71717e8			0.822093
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.50651e8	+	2.60936e8i	0.122994	+	0.213031i
$396$		0			0
$397$	−4.46085e8	+	7.72641e8i	−0.357808	+	0.619742i	−0.987594	−	0.157026i	$-0.949809\pi$
$397$	−4.46085e8	+	7.72641e8i	−0.357808	+	0.619742i	0.629786	+	0.776769i	$0.283143\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.35557e8	+	5.81202e8i	−0.259873	+	0.450114i	−0.966208	−	0.257765i	$-0.917014\pi$
$401$	−3.35557e8	+	5.81202e8i	−0.259873	+	0.450114i	0.706335	+	0.707878i	$0.250347\pi$
$402$		0			0
$403$	−6.43237e8	−	1.11412e9i	−0.489557	−	0.847937i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.61794e7			−0.0266000
$408$		0			0
$409$	−2.01665e8	−	3.49294e8i	−0.145747	−	0.252441i	0.783905	−	0.620881i	$-0.213225\pi$
$409$	−2.01665e8	−	3.49294e8i	−0.145747	−	0.252441i	−0.929651	+	0.368441i	$0.879892\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−7.68988e8	+	1.56848e9i	−0.537149	+	1.09560i
$414$		0			0
$415$	−5.00071e8	+	8.66149e8i	−0.343450	+	0.594873i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4.45684e8			−0.295990			−0.147995	−	0.988988i	$-0.547282\pi$
$419$	−4.45684e8			−0.295990			−0.147995	+	0.988988i	$0.547282\pi$
$420$		0			0
$421$	−2.97060e9			−1.94025			−0.970123	−	0.242614i	$-0.921995\pi$
$421$	−2.97060e9			−1.94025			−0.970123	+	0.242614i	$0.921995\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.24256e9	+	2.15218e9i	−0.785159	+	1.35994i
$426$		0			0
$427$	−2.46199e9	+	1.67080e8i	−1.53034	+	0.103855i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	5.68414e8	+	9.84522e8i	0.341975	+	0.592318i	0.984799	−	0.173695i	$-0.0555709\pi$
$431$	5.68414e8	+	9.84522e8i	0.341975	+	0.592318i	−0.642824	+	0.766014i	$0.722238\pi$
$432$		0			0
$433$	−1.17793e9			−0.697287			−0.348644	−	0.937255i	$-0.613358\pi$
$433$	−1.17793e9			−0.697287			−0.348644	+	0.937255i	$0.613358\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.04457e9	−	1.80924e9i	−0.598757	−	1.03708i
$438$		0			0
$439$	−5.73538e8	+	9.93398e8i	−0.323546	+	0.560399i	−0.981217	−	0.192907i	$-0.938209\pi$
$439$	−5.73538e8	+	9.93398e8i	−0.323546	+	0.560399i	0.657671	+	0.753306i	$0.271542\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3.69222e8	+	6.39512e8i	−0.201778	+	0.349490i	−0.949102	−	0.314970i	$-0.898005\pi$
$443$	−3.69222e8	+	6.39512e8i	−0.201778	+	0.349490i	0.747323	+	0.664461i	$0.231339\pi$
$444$		0			0
$445$	2.59689e9	+	4.49794e9i	1.39699	+	2.41966i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.95056e9			−1.01695			−0.508473	−	0.861078i	$-0.669790\pi$
$449$	−1.95056e9			−1.01695			−0.508473	+	0.861078i	$0.669790\pi$
$450$		0			0
$451$	8.64819e7	+	1.49791e8i	0.0443923	+	0.0768897i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.03839e9	−	2.06197e8i	1.51218	−	0.102622i
$456$		0			0
$457$	4.94545e6	−	8.56577e6i	0.00242381	−	0.00419817i	−0.864811	−	0.502098i	$-0.832562\pi$
$457$	4.94545e6	−	8.56577e6i	0.00242381	−	0.00419817i	0.867235	+	0.497899i	$0.165895\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	8.01965e8			0.381243			0.190622	−	0.981664i	$-0.438950\pi$
$461$	8.01965e8			0.381243			0.190622	+	0.981664i	$0.438950\pi$
$462$		0			0
$463$	−3.05301e8			−0.142953			−0.0714767	−	0.997442i	$-0.522771\pi$
$463$	−3.05301e8			−0.142953			−0.0714767	+	0.997442i	$0.522771\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2.68121e8	−	4.64399e8i	0.121821	−	0.211000i	−0.798665	−	0.601776i	$-0.794460\pi$
$467$	2.68121e8	−	4.64399e8i	0.121821	−	0.211000i	0.920486	+	0.390776i	$0.127793\pi$
$468$		0			0
$469$	6.25378e8	−	1.27556e9i	0.279922	−	0.570947i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	9.69643e7	+	1.67947e8i	0.0421306	+	0.0729724i
$474$		0			0
$475$	5.34287e9			2.28743
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9.93121e8	+	1.72014e9i	0.412884	+	0.715136i	0.995204	−	0.0978238i	$-0.0311882\pi$
$479$	9.93121e8	+	1.72014e9i	0.412884	+	0.715136i	−0.582320	+	0.812960i	$0.697855\pi$
$480$		0			0
$481$	5.65496e8	−	9.79468e8i	0.231698	−	0.401312i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.09613e9	+	5.36266e9i	−1.23232	+	2.13444i
$486$		0			0
$487$	−4.14128e8	−	7.17291e8i	−0.162474	−	0.281413i	0.773281	−	0.634063i	$-0.218614\pi$
$487$	−4.14128e8	−	7.17291e8i	−0.162474	−	0.281413i	−0.935755	+	0.352650i	$0.885281\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.29504e9			−0.493739			−0.246869	−	0.969049i	$-0.579402\pi$
$491$	−1.29504e9			−0.493739			−0.246869	+	0.969049i	$0.579402\pi$
$492$		0			0
$493$	1.61227e9	+	2.79253e9i	0.606001	+	1.04962i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.16039e9	−	1.72800e9i	−0.423992	−	0.631387i
$498$		0			0
$499$	3.73072e7	−	6.46180e7i	0.0134413	−	0.0232810i	−0.859227	−	0.511595i	$-0.829055\pi$
$499$	3.73072e7	−	6.46180e7i	0.0134413	−	0.0232810i	0.872668	+	0.488314i	$0.162388\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	2.88380e8			0.101036			0.0505181	−	0.998723i	$-0.483913\pi$
$503$	2.88380e8			0.101036			0.0505181	+	0.998723i	$0.483913\pi$
$504$		0			0
$505$	2.68652e9			0.928260
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4.82513e8	−	8.35738e8i	0.162180	−	0.280904i	−0.773470	−	0.633833i	$-0.781481\pi$
$509$	4.82513e8	−	8.35738e8i	0.162180	−	0.280904i	0.935650	+	0.352929i	$0.114814\pi$
$510$		0			0
$511$	4.68497e8	−	9.55577e8i	0.155322	−	0.316805i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.24745e8	+	1.25530e9i	0.233808	+	0.404968i
$516$		0			0
$517$	5.27280e7			0.0167813
$518$		0			0
$519$		0			0
$520$		0			0
$521$	2.55195e9	+	4.42011e9i	0.790570	+	1.36931i	0.925615	+	0.378468i	$0.123549\pi$
$521$	2.55195e9	+	4.42011e9i	0.790570	+	1.36931i	−0.135045	+	0.990840i	$0.543118\pi$
$522$		0			0
$523$	−1.68515e9	+	2.91877e9i	−0.515090	+	0.892162i	0.484757	+	0.874649i	$0.338908\pi$
$523$	−1.68515e9	+	2.91877e9i	−0.515090	+	0.892162i	−0.999847	+	0.0175130i	$0.994425\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.05631e9	−	3.56164e9i	0.612001	−	1.06002i
$528$		0			0
$529$	9.84351e8	+	1.70495e9i	0.289105	+	0.500744i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−5.40696e9			−1.54671
$534$		0			0
$535$	−3.74158e9	−	6.48061e9i	−1.05637	−	1.82969i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.09369e8	+	2.85487e7i	−0.0575907	+	0.00785282i
$540$		0			0
$541$	−1.67298e9	+	2.89769e9i	−0.454256	+	0.786794i	−0.998645	−	0.0520386i	$-0.983428\pi$
$541$	−1.67298e9	+	2.89769e9i	−0.454256	+	0.786794i	0.544389	+	0.838833i	$0.316761\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.91174e9			0.505872
$546$		0			0
$547$	4.98263e9			1.30168			0.650838	−	0.759216i	$-0.274418\pi$
$547$	4.98263e9			1.30168			0.650838	+	0.759216i	$0.274418\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3.46628e9	−	6.00377e9i	0.882740	−	1.52895i
$552$		0			0
$553$	6.52042e8	−	4.42500e7i	0.163960	−	0.0111269i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2.02726e8	+	3.51132e8i	0.0497069	+	0.0860948i	0.889808	−	0.456335i	$-0.150838\pi$
$557$	2.02726e8	+	3.51132e8i	0.0497069	+	0.0860948i	−0.840101	+	0.542429i	$0.817505\pi$
$558$		0			0
$559$	−6.06234e9			−1.46791
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.64670e9	−	4.58422e9i	−0.625065	−	1.08264i	−0.988528	−	0.151036i	$-0.951739\pi$
$563$	−2.64670e9	−	4.58422e9i	−0.625065	−	1.08264i	0.363463	−	0.931609i	$-0.381594\pi$
$564$		0			0
$565$	−1.14235e9	+	1.97860e9i	−0.266458	+	0.461518i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−8.50601e8	+	1.47328e9i	−0.193568	+	0.335269i	−0.946430	−	0.322909i	$-0.895339\pi$
$569$	−8.50601e8	+	1.47328e9i	−0.193568	+	0.335269i	0.752862	+	0.658178i	$0.228673\pi$
$570$		0			0
$571$	−1.74216e9	−	3.01751e9i	−0.391617	−	0.678301i	0.601046	−	0.799215i	$-0.294751\pi$
$571$	−1.74216e9	−	3.01751e9i	−0.391617	−	0.678301i	−0.992663	+	0.120914i	$0.961418\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.67283e9			0.805681
$576$		0			0
$577$	−3.11391e9	−	5.39345e9i	−0.674825	−	1.16883i	−0.976520	−	0.215426i	$-0.930886\pi$
$577$	−3.11391e9	−	5.39345e9i	−0.674825	−	1.16883i	0.301696	−	0.953404i	$-0.402447\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.20940e9	+	1.80097e9i	0.255831	+	0.380970i
$582$		0			0
$583$	−4.69259e7	+	8.12781e7i	−0.00980783	+	0.0169877i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.01119e9			0.206348			0.103174	−	0.994663i	$-0.467100\pi$
$587$	1.01119e9			0.206348			0.103174	+	0.994663i	$0.467100\pi$
$588$		0			0
$589$	−8.84189e9			−1.78296
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.95949e9	+	6.85803e9i	−0.779736	+	1.35054i	0.152358	+	0.988325i	$0.451313\pi$
$593$	−3.95949e9	+	6.85803e9i	−0.779736	+	1.35054i	−0.932094	+	0.362217i	$0.882020\pi$
$594$		0			0
$595$	5.42745e9	+	8.08228e9i	1.05630	+	1.57298i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.17062e9	−	2.02757e9i	−0.222546	−	0.385462i	0.733034	−	0.680192i	$-0.238104\pi$
$599$	−1.17062e9	−	2.02757e9i	−0.222546	−	0.385462i	−0.955581	+	0.294730i	$0.904770\pi$
$600$		0			0
$601$	1.13771e8			0.0213781			0.0106891	−	0.999943i	$-0.496598\pi$
$601$	1.13771e8			0.0213781			0.0106891	+	0.999943i	$0.496598\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.06276e9	−	7.03691e9i	−0.745895	−	1.29193i
$606$		0			0
$607$	2.01530e9	−	3.49060e9i	0.365746	−	0.633490i	−0.623150	−	0.782103i	$-0.714147\pi$
$607$	2.01530e9	−	3.49060e9i	0.365746	−	0.633490i	0.988896	+	0.148612i	$0.0474806\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.24156e8	+	1.42748e9i	−0.146172	+	0.253178i
$612$		0			0
$613$	−6.71782e8	−	1.16356e9i	−0.117792	−	0.204022i	0.801100	−	0.598530i	$-0.204248\pi$
$613$	−6.71782e8	−	1.16356e9i	−0.117792	−	0.204022i	−0.918892	+	0.394508i	$0.870915\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−6.24917e9			−1.07109			−0.535543	−	0.844508i	$-0.679893\pi$
$617$	−6.24917e9			−1.07109			−0.535543	+	0.844508i	$0.679893\pi$
$618$		0			0
$619$	−3.84513e9	−	6.65996e9i	−0.651619	−	1.12864i	−0.982730	−	0.185046i	$-0.940757\pi$
$619$	−3.84513e9	−	6.65996e9i	−0.651619	−	1.12864i	0.331111	−	0.943592i	$-0.392577\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.12397e10	−	7.62772e8i	1.86229	−	0.126382i
$624$		0			0
$625$	2.14106e9	−	3.70843e9i	0.350791	−	0.607588i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	3.61558e9			0.579296
$630$		0			0
$631$	6.83436e9			1.08292			0.541459	−	0.840727i	$-0.317872\pi$
$631$	6.83436e9			1.08292			0.541459	+	0.840727i	$0.317872\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.47236e9	+	6.01430e9i	−0.538166	+	0.932131i
$636$		0			0
$637$	2.49963e9	−	6.11438e9i	0.383166	−	0.937270i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.29088e9	+	3.96793e9i	0.343558	+	0.595060i	0.985091	−	0.172036i	$-0.0550345\pi$
$641$	2.29088e9	+	3.96793e9i	0.343558	+	0.595060i	−0.641533	+	0.767096i	$0.721701\pi$
$642$		0			0
$643$	−6.98580e9			−1.03628			−0.518140	−	0.855296i	$-0.673376\pi$
$643$	−6.98580e9			−1.03628			−0.518140	+	0.855296i	$0.673376\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−4.89690e9	−	8.48167e9i	−0.710814	−	1.23117i	−0.964552	−	0.263893i	$-0.914993\pi$
$647$	−4.89690e9	−	8.48167e9i	−0.710814	−	1.23117i	0.253738	−	0.967273i	$-0.418340\pi$
$648$		0			0
$649$	2.46949e8	−	4.27729e8i	0.0354611	−	0.0614203i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	6.10656e9	−	1.05769e10i	0.858223	−	1.48649i	−0.0153985	−	0.999881i	$-0.504902\pi$
$653$	6.10656e9	−	1.05769e10i	0.858223	−	1.48649i	0.873622	−	0.486605i	$-0.161765\pi$
$654$		0			0
$655$	8.38791e9	+	1.45283e10i	1.16630	+	2.02009i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6.39022e9			0.869795			0.434898	−	0.900480i	$-0.356785\pi$
$659$	6.39022e9			0.869795			0.434898	+	0.900480i	$0.356785\pi$
$660$		0			0
$661$	−9.69710e8	−	1.67959e9i	−0.130598	−	0.226203i	0.793309	−	0.608819i	$-0.208356\pi$
$661$	−9.69710e8	−	1.67959e9i	−0.130598	−	0.226203i	−0.923907	+	0.382616i	$0.875023\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	9.21406e9	−	1.87936e10i	1.21500	−	2.47819i
$666$		0			0
$667$	2.38281e9	−	4.12715e9i	0.310920	−	0.538530i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	6.97699e8			0.0891536
$672$		0			0
$673$	−9.25555e8			−0.117044			−0.0585221	−	0.998286i	$-0.518639\pi$
$673$	−9.25555e8			−0.117044			−0.0585221	+	0.998286i	$0.518639\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2.74684e9	+	4.75766e9i	−0.340230	+	0.589296i	−0.984475	−	0.175523i	$-0.943838\pi$
$677$	−2.74684e9	+	4.75766e9i	−0.340230	+	0.589296i	0.644245	+	0.764819i	$0.277172\pi$
$678$		0			0
$679$	7.48784e9	+	1.11505e10i	0.917936	+	1.36694i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2.07004e9	−	3.58541e9i	−0.248603	−	0.430592i	0.714536	−	0.699599i	$-0.246638\pi$
$683$	−2.07004e9	−	3.58541e9i	−0.248603	−	0.430592i	−0.963138	+	0.269007i	$0.913305\pi$
$684$		0			0
$685$	3.72603e8			0.0442925
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.46694e9	−	2.54081e9i	−0.170861	−	0.295941i
$690$		0			0
$691$	5.23468e9	−	9.06673e9i	0.603555	−	1.04539i	−0.388723	−	0.921355i	$-0.627084\pi$
$691$	5.23468e9	−	9.06673e9i	0.603555	−	1.04539i	0.992278	−	0.124034i	$-0.0395831\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.97561e9	−	5.15390e9i	0.336224	−	0.582357i
$696$		0			0
$697$	−8.64255e9	−	1.49693e10i	−0.966779	−	1.67451i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.45142e10			−1.59141			−0.795703	−	0.605686i	$-0.792899\pi$
$701$	−1.45142e10			−1.59141			−0.795703	+	0.605686i	$0.792899\pi$
$702$		0			0
$703$	−3.88664e9	−	6.73185e9i	−0.421920	−	0.730787i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.56523e9	−	5.23220e9i	0.272997	−	0.556822i
$708$		0			0
$709$	5.63771e9	−	9.76480e9i	0.594074	−	1.02897i	−0.399603	−	0.916688i	$-0.630852\pi$
$709$	5.63771e9	−	9.76480e9i	0.594074	−	1.02897i	0.993677	−	0.112278i	$-0.0358148\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−6.07814e9			−0.627997
$714$		0			0
$715$	−8.61043e8			−0.0880955
$716$		0			0
$717$		0			0
$718$		0			0
$719$	2.84071e9	−	4.92026e9i	0.285020	−	0.493670i	−0.687594	−	0.726096i	$-0.741333\pi$
$719$	2.84071e9	−	4.92026e9i	0.285020	−	0.493670i	0.972614	+	0.232426i	$0.0746663\pi$
$720$		0			0
$721$	3.13681e9	−	2.12876e8i	0.311684	−	0.0211521i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6.09395e9	+	1.05550e10i	0.593903	+	1.02867i
$726$		0			0
$727$	1.98501e9			0.191599			0.0957993	−	0.995401i	$-0.469459\pi$
$727$	1.98501e9			0.191599			0.0957993	+	0.995401i	$0.469459\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−9.69011e9	−	1.67838e10i	−0.917525	−	1.58920i
$732$		0			0
$733$	−2.51985e9	+	4.36450e9i	−0.236325	+	0.409327i	−0.959657	−	0.281173i	$-0.909276\pi$
$733$	−2.51985e9	+	4.36450e9i	−0.236325	+	0.409327i	0.723332	+	0.690501i	$0.242610\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.00831e8	+	3.47849e8i	−0.0184797	+	0.0320078i
$738$		0			0
$739$	5.50892e9	+	9.54173e9i	0.502124	+	0.869704i	0.999997	+	0.00245436i	$0.000781249\pi$
$739$	5.50892e9	+	9.54173e9i	0.502124	+	0.869704i	−0.497873	+	0.867250i	$0.665885\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−4.80549e9			−0.429810			−0.214905	−	0.976635i	$-0.568944\pi$
$743$	−4.80549e9			−0.429810			−0.214905	+	0.976635i	$0.568944\pi$
$744$		0			0
$745$	1.04202e10	+	1.80483e10i	0.923268	+	1.59915i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.61942e10	+	1.09900e9i	−1.40822	+	0.0955675i
$750$		0			0
$751$	−8.29288e9	+	1.43637e10i	−0.714440	+	1.23745i	0.248735	+	0.968572i	$0.419985\pi$
$751$	−8.29288e9	+	1.43637e10i	−0.714440	+	1.23745i	−0.963175	+	0.268875i	$0.913348\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.17766e10			1.84152
$756$		0			0
$757$	−1.01328e10			−0.848973			−0.424487	−	0.905434i	$-0.639545\pi$
$757$	−1.01328e10			−0.848973			−0.424487	+	0.905434i	$0.639545\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	8.15181e8	−	1.41194e9i	0.0670514	−	0.116136i	−0.830551	−	0.556943i	$-0.811974\pi$
$761$	8.15181e8	−	1.41194e9i	0.0670514	−	0.116136i	0.897602	+	0.440807i	$0.145308\pi$
$762$		0			0
$763$	1.82543e9	−	3.72326e9i	0.148775	−	0.303450i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	7.71981e9	+	1.33711e10i	0.617764	+	1.07000i
$768$		0			0
$769$	−3.58444e9			−0.284236			−0.142118	−	0.989850i	$-0.545391\pi$
$769$	−3.58444e9			−0.284236			−0.142118	+	0.989850i	$0.545391\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−5.03445e9	−	8.71993e9i	−0.392034	−	0.679023i	0.600683	−	0.799487i	$-0.294895\pi$
$773$	−5.03445e9	−	8.71993e9i	−0.392034	−	0.679023i	−0.992718	+	0.120464i	$0.961562\pi$
$774$		0			0
$775$	7.77231e9	−	1.34620e10i	0.599783	−	1.03885i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.85809e10	+	3.21831e10i	−1.40827	+	2.43920i
$780$		0			0
$781$	2.94254e8	+	5.09663e8i	0.0221026	+	0.0382829i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.11765e10			−0.824639
$786$		0			0
$787$	−8.69347e9	−	1.50575e10i	−0.635743	−	1.10114i	−0.986357	−	0.164619i	$-0.947361\pi$
$787$	−8.69347e9	−	1.50575e10i	−0.635743	−	1.10114i	0.350615	−	0.936520i	$-0.385973\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	2.76271e9	+	4.11408e9i	0.198480	+	0.295566i
$792$		0			0
$793$	−1.09053e10	+	1.88885e10i	−0.776569	+	1.34506i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	2.03291e10			1.42238			0.711189	−	0.703001i	$-0.248157\pi$
$797$	2.03291e10			1.42238			0.711189	+	0.703001i	$0.248157\pi$
$798$		0			0
$799$	−5.26936e9			−0.365464
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.50451e8	+	2.60589e8i	−0.0102539	+	0.0177604i
$804$		0			0
$805$	6.33398e9	−	1.29192e10i	0.427948	−	0.872871i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−2.78004e9	−	4.81518e9i	−0.184600	−	0.319737i	0.758842	−	0.651275i	$-0.225766\pi$
$809$	−2.78004e9	−	4.81518e9i	−0.184600	−	0.319737i	−0.943442	+	0.331538i	$0.892432\pi$
$810$		0			0
$811$	1.89473e10			1.24731			0.623655	−	0.781700i	$-0.285647\pi$
$811$	1.89473e10			1.24731			0.623655	+	0.781700i	$0.285647\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.58778e9	+	6.21422e9i	0.232153	+	0.402101i
$816$		0			0
$817$	−2.08331e10	+	3.60840e10i	−1.33653	+	2.31493i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−7.93335e9	+	1.37410e10i	−0.500329	+	0.866595i	0.499671	+	0.866215i	$0.333454\pi$
$821$	−7.93335e9	+	1.37410e10i	−0.500329	+	0.866595i	−1.00000		0.000379943i	$0.999879\pi$
$822$		0			0
$823$	−8.51170e9	−	1.47427e10i	−0.532251	−	0.921886i	−0.999291	−	0.0376500i	$-0.988013\pi$
$823$	−8.51170e9	−	1.47427e10i	−0.532251	−	0.921886i	0.467040	−	0.884236i	$-0.345321\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−3.25344e9			−0.200020			−0.100010	−	0.994986i	$-0.531888\pi$
$827$	−3.25344e9			−0.200020			−0.100010	+	0.994986i	$0.531888\pi$
$828$		0			0
$829$	−1.29422e10	−	2.24166e10i	−0.788985	−	1.36656i	−0.926589	−	0.376075i	$-0.877273\pi$
$829$	−1.29422e10	−	2.24166e10i	−0.788985	−	1.36656i	0.137604	−	0.990487i	$-0.456060\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.09233e10	−	2.85301e9i	1.25422	−	0.171019i
$834$		0			0
$835$	−1.46479e10	+	2.53709e10i	−0.870709	+	1.50811i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2.19214e10			1.28145			0.640724	−	0.767771i	$-0.278634\pi$
$839$	2.19214e10			1.28145			0.640724	+	0.767771i	$0.278634\pi$
$840$		0			0
$841$	−1.43567e9			−0.0832278
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.31988e8	−	5.75021e8i	0.0189289	−	0.0327857i
$846$		0			0
$847$	−1.75843e10	+	1.19334e9i	−0.994335	+	0.0674793i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.67178e9	−	4.62765e9i	−0.148610	−	0.257399i
$852$		0			0
$853$	3.22899e10			1.78133			0.890666	−	0.454657i	$-0.150238\pi$
$853$	3.22899e10			1.78133			0.890666	+	0.454657i	$0.150238\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.27172e10	+	2.20268e10i	0.690172	+	1.19541i	0.971781	+	0.235883i	$0.0757983\pi$
$857$	1.27172e10	+	2.20268e10i	0.690172	+	1.19541i	−0.281610	+	0.959529i	$0.590868\pi$
$858$		0			0
$859$	9.99623e9	−	1.73140e10i	0.538096	−	0.932010i	−0.460910	−	0.887447i	$-0.652477\pi$
$859$	9.99623e9	−	1.73140e10i	0.538096	−	0.932010i	0.999007	−	0.0445634i	$-0.0141897\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.19142e9	−	2.06361e9i	0.0630998	−	0.109292i	−0.832750	−	0.553650i	$-0.813235\pi$
$863$	1.19142e9	−	2.06361e9i	0.0630998	−	0.109292i	0.895850	+	0.444358i	$0.146568\pi$
$864$		0			0
$865$	1.08154e10	+	1.87328e10i	0.568181	+	0.984118i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.84781e8			−0.00955186
$870$		0			0
$871$	−6.27811e9	−	1.08740e10i	−0.321933	−	0.557604i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.97788e9	+	5.92364e9i	0.200735	+	0.298924i
$876$		0			0
$877$	1.01482e9	−	1.75773e9i	0.0508033	−	0.0879939i	−0.839505	−	0.543351i	$-0.817155\pi$
$877$	1.01482e9	−	1.75773e9i	0.0508033	−	0.0879939i	0.890309	+	0.455357i	$0.150489\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−2.02197e10			−0.996231			−0.498116	−	0.867111i	$-0.665974\pi$
$881$	−2.02197e10			−0.996231			−0.498116	+	0.867111i	$0.665974\pi$
$882$		0			0
$883$	2.38857e10			1.16755			0.583775	−	0.811916i	$-0.301575\pi$
$883$	2.38857e10			1.16755			0.583775	+	0.811916i	$0.301575\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.73532e10	+	3.00567e10i	−0.834926	+	1.44613i	0.0591654	+	0.998248i	$0.481156\pi$
$887$	−1.73532e10	+	3.00567e10i	−0.834926	+	1.44613i	−0.894091	+	0.447885i	$0.852177\pi$
$888$		0			0
$889$	8.39772e9	+	1.25054e10i	0.400872	+	0.596957i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	5.66440e9	+	9.81103e9i	0.266179	+	0.461036i
$894$		0			0
$895$	−1.39020e10			−0.648183
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.00848e10	−	1.74675e10i	−0.462924	−	0.801808i
$900$		0			0
$901$	4.68953e9	−	8.12251e9i	0.213596	−	0.369959i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.58179e10	+	4.47179e10i	−1.15785	+	2.00545i
$906$		0			0
$907$	6.76837e9	+	1.17232e10i	0.301203	+	0.521698i	0.976409	−	0.215931i	$-0.0692786\pi$
$907$	6.76837e9	+	1.17232e10i	0.301203	+	0.521698i	−0.675206	+	0.737629i	$0.735945\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.62909e10			−0.713889			−0.356945	−	0.934126i	$-0.616181\pi$
$911$	−1.62909e10			−0.713889			−0.356945	+	0.934126i	$0.616181\pi$
$912$		0			0
$913$	−3.06681e8	−	5.31187e8i	−0.0133364	−	0.0230993i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	3.63042e10	−	2.46374e9i	1.55476	−	0.105512i
$918$		0			0
$919$	−5.46627e9	+	9.46786e9i	−0.232320	+	0.402390i	−0.958491	−	0.285124i	$-0.907965\pi$
$919$	−5.46627e9	+	9.46786e9i	−0.232320	+	0.402390i	0.726170	+	0.687515i	$0.241298\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.83972e10			−0.770096
$924$		0			0
$925$	1.36659e10			0.567732
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.45284e10	−	2.51640e10i	0.594516	−	1.02973i	−0.399099	−	0.916908i	$-0.630677\pi$
$929$	1.45284e10	−	2.51640e10i	0.594516	−	1.02973i	0.993615	−	0.112824i	$-0.0359898\pi$
$930$		0			0
$931$	−2.78039e10	−	3.58902e10i	−1.12923	−	1.45764i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.37630e9	−	2.38382e9i	−0.0550646	−	0.0953748i
$936$		0			0
$937$	3.95737e10			1.57151			0.785757	−	0.618535i	$-0.212273\pi$
$937$	3.95737e10			1.57151			0.785757	+	0.618535i	$0.212273\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.38478e10	−	2.39851e10i	−0.541774	−	0.938380i	−0.998802	−	0.0489276i	$-0.984420\pi$
$941$	−1.38478e10	−	2.39851e10i	−0.541774	−	0.938380i	0.457029	−	0.889452i	$-0.348914\pi$
$942$		0			0
$943$	−1.27730e10	+	2.21235e10i	−0.496024	+	0.859139i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.77937e10	+	3.08196e10i	−0.680834	+	1.17924i	0.293893	+	0.955838i	$0.405049\pi$
$947$	−1.77937e10	+	3.08196e10i	−0.680834	+	1.17924i	−0.974727	+	0.223401i	$0.928284\pi$
$948$		0			0
$949$	−4.70321e9	−	8.14619e9i	−0.178633	−	0.309402i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	2.97100e10			1.11193			0.555964	−	0.831206i	$-0.312349\pi$
$953$	2.97100e10			1.11193			0.555964	+	0.831206i	$0.312349\pi$
$954$		0			0
$955$	1.72493e10	+	2.98767e10i	0.640856	+	1.11000i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.55781e8	−	7.25674e8i	0.0130262	−	0.0265691i
$960$		0			0
$961$	8.93952e8	−	1.54837e9i	0.0324924	−	0.0562786i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	4.42858e10			1.58642
$966$		0			0
$967$	−2.76167e10			−0.982152			−0.491076	−	0.871117i	$-0.663396\pi$
$967$	−2.76167e10			−0.982152			−0.491076	+	0.871117i	$0.663396\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.94914e10	+	3.37601e10i	−0.683245	+	1.18341i	0.290740	+	0.956802i	$0.406098\pi$
$971$	−1.94914e10	+	3.37601e10i	−0.683245	+	1.18341i	−0.973985	+	0.226613i	$0.927235\pi$
$972$		0			0
$973$	−7.19636e9	−	1.07164e10i	−0.250448	−	0.372954i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.68615e10	+	2.92050e10i	0.578450	+	1.00190i	0.995657	+	0.0930935i	$0.0296755\pi$
$977$	1.68615e10	+	2.92050e10i	0.578450	+	1.00190i	−0.417207	+	0.908811i	$0.636991\pi$
$978$		0			0
$979$	−3.18521e9			−0.108492
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.31204e10	−	2.27251e10i	−0.440563	−	0.763078i	0.557168	−	0.830400i	$-0.311888\pi$
$983$	−1.31204e10	−	2.27251e10i	−0.440563	−	0.763078i	−0.997731	+	0.0673220i	$0.978555\pi$
$984$		0			0
$985$	−2.58700e10	+	4.48081e10i	−0.862520	+	1.49393i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.43212e10	+	2.48051e10i	−0.470754	+	0.815369i
$990$		0			0
$991$	4.39651e9	+	7.61498e9i	0.143500	+	0.248548i	0.928812	−	0.370551i	$-0.120831\pi$
$991$	4.39651e9	+	7.61498e9i	0.143500	+	0.248548i	−0.785313	+	0.619099i	$0.787498\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.64249e10			−0.528595
$996$		0			0
$997$	1.97048e10	+	3.41297e10i	0.629707	+	1.09068i	0.987610	+	0.156926i	$0.0501585\pi$
$997$	1.97048e10	+	3.41297e10i	0.629707	+	1.09068i	−0.357903	+	0.933759i	$0.616508\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.8.k.c.37.5		10
3.2	odd	2		28.8.e.a.9.4	✓	10
7.4	even	3	inner	252.8.k.c.109.5		10
12.11	even	2		112.8.i.d.65.2		10
21.2	odd	6		196.8.a.d.1.2		5
21.5	even	6		196.8.a.e.1.4		5
21.11	odd	6		28.8.e.a.25.4	yes	10
21.17	even	6		196.8.e.f.165.2		10
21.20	even	2		196.8.e.f.177.2		10
84.11	even	6		112.8.i.d.81.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.8.e.a.9.4	✓	10	3.2	odd	2
28.8.e.a.25.4	yes	10	21.11	odd	6
112.8.i.d.65.2		10	12.11	even	2
112.8.i.d.81.2		10	84.11	even	6
196.8.a.d.1.2		5	21.2	odd	6
196.8.a.e.1.4		5	21.5	even	6
196.8.e.f.165.2		10	21.17	even	6
196.8.e.f.177.2		10	21.20	even	2
252.8.k.c.37.5		10	1.1	even	1	trivial
252.8.k.c.109.5		10	7.4	even	3	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.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(209.191 - 362.329i) q^{5} +(-505.918 - 753.386i) q^{7} +(128.291 + 222.207i) q^{11} -8020.94 q^{13} +(-12820.8 - 22206.2i) q^{17} +(-27563.8 + 47742.0i) q^{19} +(-18948.1 + 32819.1i) q^{23} +(-48459.1 - 83933.6i) q^{25} -125755. q^{29} +(80194.6 + 138901. i) q^{31} +(-378807. + 25707.3i) q^{35} +(-70502.5 + 122114. i) q^{37} +674106. q^{41} +755814. q^{43} +(102751. - 177969. i) q^{47} +(-311638. + 762302. i) q^{49} +(182888. + 316772. i) q^{53} +107349. q^{55} +(-962456. - 1.66702e6i) q^{59} +(1.35960e6 - 2.35489e6i) q^{61} +(-1.67791e6 + 2.90622e6i) q^{65} +(782715. + 1.35570e6i) q^{67} +2.29364e6 q^{71} +(586366. + 1.01562e6i) q^{73} +(102503. - 209071. i) q^{77} +(-360081. + 623678. i) q^{79} -2.39050e6 q^{83} -1.07279e7 q^{85} +(-6.20699e6 + 1.07508e7i) q^{89} +(4.05794e6 + 6.04287e6i) q^{91} +(1.15322e7 + 1.99744e7i) q^{95} -1.48005e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 249 q^{5} + 332 q^{7} - 6399 q^{11} - 26988 q^{13} - 3609 q^{17} - 12403 q^{19} + 13959 q^{23} - 162364 q^{25} - 26148 q^{29} - 20181 q^{31} - 791715 q^{35} - 54763 q^{37} + 1824468 q^{41} - 1938424 q^{43}+ \cdots + 285580 q^{97}+O(q^{100}) \) 10 * q - 249 * q^5 + 332 * q^7 - 6399 * q^11 - 26988 * q^13 - 3609 * q^17 - 12403 * q^19 + 13959 * q^23 - 162364 * q^25 - 26148 * q^29 - 20181 * q^31 - 791715 * q^35 - 54763 * q^37 + 1824468 * q^41 - 1938424 * q^43 - 1217253 * q^47 + 1722058 * q^49 + 2349159 * q^53 + 3123494 * q^55 - 2188611 * q^59 + 6107445 * q^61 + 3022014 * q^65 - 3171581 * q^67 + 18938976 * q^71 + 8034853 * q^73 - 24100191 * q^77 - 7589559 * q^79 + 13452408 * q^83 - 48717670 * q^85 - 29864757 * q^89 + 34267512 * q^91 + 25812009 * q^95 + 285580 * q^97

Introduction

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