LMFDB - Embedded newform 256.2.e.b.193.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,2,Mod(65,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("256.65");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	256.e (of order $4$, 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:	$2.04417029174$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			193.1
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	256.193
Dual form			256.2.e.b.65.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.93185 + 1.93185i) q^{3} +(-1.73205 - 1.73205i) q^{5} -1.03528i q^{7} -4.46410i q^{9} +O(q^{10})$	\(q+(-1.93185 + 1.93185i) q^{3} +(-1.73205 - 1.73205i) q^{5} -1.03528i q^{7} -4.46410i q^{9} +(-0.896575 - 0.896575i) q^{11} +(3.73205 - 3.73205i) q^{13} +6.69213 q^{15} -3.46410 q^{17} +(0.896575 - 0.896575i) q^{19} +(2.00000 + 2.00000i) q^{21} -6.69213i q^{23} +1.00000i q^{25} +(2.82843 + 2.82843i) q^{27} +(-1.73205 + 1.73205i) q^{29} -5.65685 q^{31} +3.46410 q^{33} +(-1.79315 + 1.79315i) q^{35} +(-0.267949 - 0.267949i) q^{37} +14.4195i q^{39} +6.92820i q^{41} +(-5.79555 - 5.79555i) q^{43} +(-7.73205 + 7.73205i) q^{45} -9.79796 q^{47} +5.92820 q^{49} +(6.69213 - 6.69213i) q^{51} +(-4.26795 - 4.26795i) q^{53} +3.10583i q^{55} +3.46410i q^{57} +(7.58871 + 7.58871i) q^{59} +(-0.267949 + 0.267949i) q^{61} -4.62158 q^{63} -12.9282 q^{65} +(2.96713 - 2.96713i) q^{67} +(12.9282 + 12.9282i) q^{69} -6.69213i q^{71} +9.46410i q^{73} +(-1.93185 - 1.93185i) q^{75} +(-0.928203 + 0.928203i) q^{77} +15.4548 q^{79} +2.46410 q^{81} +(5.79555 - 5.79555i) q^{83} +(6.00000 + 6.00000i) q^{85} -6.69213i q^{87} -9.46410i q^{89} +(-3.86370 - 3.86370i) q^{91} +(10.9282 - 10.9282i) q^{93} -3.10583 q^{95} -3.46410 q^{97} +(-4.00240 + 4.00240i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 16 q^{13} + 16 q^{21} - 16 q^{37} - 48 q^{45} - 8 q^{49} - 48 q^{53} - 16 q^{61} - 48 q^{65} + 48 q^{69} + 48 q^{77} - 8 q^{81} + 48 q^{85} + 32 q^{93}+O(q^{100}) $ 8 * q + 16 * q^13 + 16 * q^21 - 16 * q^37 - 48 * q^45 - 8 * q^49 - 48 * q^53 - 16 * q^61 - 48 * q^65 + 48 * q^69 + 48 * q^77 - 8 * q^81 + 48 * q^85 + 32 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$

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
$3$	−1.93185	+	1.93185i	−1.11536	+	1.11536i	−0.122941	+	0.992414i	$0.539233\pi$
$3$	−1.93185	+	1.93185i	−1.11536	+	1.11536i	−0.992414	+	0.122941i	$0.960767\pi$
$4$		0			0
$5$	−1.73205	−	1.73205i	−0.774597	−	0.774597i	0.204310	−	0.978906i	$-0.434505\pi$
$5$	−1.73205	−	1.73205i	−0.774597	−	0.774597i	−0.978906	+	0.204310i	$0.934505\pi$
$6$		0			0
$7$		−	1.03528i		−	0.391298i	−0.980674	−	0.195649i	$-0.937319\pi$
$7$		−	1.03528i		−	0.391298i	0.980674	−	0.195649i	$-0.0626813\pi$
$8$		0			0
$9$		−	4.46410i		−	1.48803i
$10$		0			0
$11$	−0.896575	−	0.896575i	−0.270328	−	0.270328i	0.558904	−	0.829232i	$-0.311222\pi$
$11$	−0.896575	−	0.896575i	−0.270328	−	0.270328i	−0.829232	+	0.558904i	$0.811222\pi$
$12$		0			0
$13$	3.73205	−	3.73205i	1.03508	−	1.03508i	0.0357229	−	0.999362i	$-0.488627\pi$
$13$	3.73205	−	3.73205i	1.03508	−	1.03508i	0.999362	−	0.0357229i	$-0.0113734\pi$
$14$		0			0
$15$	6.69213			1.72790
$16$		0			0
$17$	−3.46410			−0.840168			−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410			−0.840168			−0.420084	+	0.907485i	$0.637999\pi$
$18$		0			0
$19$	0.896575	−	0.896575i	0.205689	−	0.205689i	−0.596744	−	0.802432i	$-0.703539\pi$
$19$	0.896575	−	0.896575i	0.205689	−	0.205689i	0.802432	+	0.596744i	$0.203539\pi$
$20$		0			0
$21$	2.00000	+	2.00000i	0.436436	+	0.436436i
$22$		0			0
$23$		−	6.69213i		−	1.39541i	−0.716387	−	0.697703i	$-0.754206\pi$
$23$		−	6.69213i		−	1.39541i	0.716387	−	0.697703i	$-0.245794\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$	2.82843	+	2.82843i	0.544331	+	0.544331i
$28$		0			0
$29$	−1.73205	+	1.73205i	−0.321634	+	0.321634i	−0.849394	−	0.527760i	$-0.823032\pi$
$29$	−1.73205	+	1.73205i	−0.321634	+	0.321634i	0.527760	+	0.849394i	$0.323032\pi$
$30$		0			0
$31$	−5.65685			−1.01600			−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685			−1.01600			−0.508001	+	0.861357i	$0.669615\pi$
$32$		0			0
$33$	3.46410			0.603023
$34$		0			0
$35$	−1.79315	+	1.79315i	−0.303098	+	0.303098i
$36$		0			0
$37$	−0.267949	−	0.267949i	−0.0440506	−	0.0440506i	0.684738	−	0.728789i	$-0.259916\pi$
$37$	−0.267949	−	0.267949i	−0.0440506	−	0.0440506i	−0.728789	+	0.684738i	$0.759916\pi$
$38$		0			0
$39$			14.4195i			2.30897i
$40$		0			0
$41$			6.92820i			1.08200i	0.841021	+	0.541002i	$0.181955\pi$
$41$			6.92820i			1.08200i	−0.841021	+	0.541002i	$0.818045\pi$
$42$		0			0
$43$	−5.79555	−	5.79555i	−0.883814	−	0.883814i	0.110106	−	0.993920i	$-0.464881\pi$
$43$	−5.79555	−	5.79555i	−0.883814	−	0.883814i	−0.993920	+	0.110106i	$0.964881\pi$
$44$		0			0
$45$	−7.73205	+	7.73205i	−1.15263	+	1.15263i
$46$		0			0
$47$	−9.79796			−1.42918			−0.714590	−	0.699544i	$-0.753387\pi$
$47$	−9.79796			−1.42918			−0.714590	+	0.699544i	$0.753387\pi$
$48$		0			0
$49$	5.92820			0.846886
$50$		0			0
$51$	6.69213	−	6.69213i	0.937086	−	0.937086i
$52$		0			0
$53$	−4.26795	−	4.26795i	−0.586248	−	0.586248i	0.350365	−	0.936613i	$-0.386057\pi$
$53$	−4.26795	−	4.26795i	−0.586248	−	0.586248i	−0.936613	+	0.350365i	$0.886057\pi$
$54$		0			0
$55$			3.10583i			0.418790i
$56$		0			0
$57$			3.46410i			0.458831i
$58$		0			0
$59$	7.58871	+	7.58871i	0.987965	+	0.987965i	0.999928	−	0.0119631i	$-0.00380806\pi$
$59$	7.58871	+	7.58871i	0.987965	+	0.987965i	−0.0119631	+	0.999928i	$0.503808\pi$
$60$		0			0
$61$	−0.267949	+	0.267949i	−0.0343074	+	0.0343074i	−0.724052	−	0.689745i	$-0.757723\pi$
$61$	−0.267949	+	0.267949i	−0.0343074	+	0.0343074i	0.689745	+	0.724052i	$0.257723\pi$
$62$		0			0
$63$	−4.62158			−0.582264
$64$		0			0
$65$	−12.9282			−1.60355
$66$		0			0
$67$	2.96713	−	2.96713i	0.362492	−	0.362492i	−0.502237	−	0.864730i	$-0.667490\pi$
$67$	2.96713	−	2.96713i	0.362492	−	0.362492i	0.864730	+	0.502237i	$0.167490\pi$
$68$		0			0
$69$	12.9282	+	12.9282i	1.55637	+	1.55637i
$70$		0			0
$71$		−	6.69213i		−	0.794210i	−0.917773	−	0.397105i	$-0.870015\pi$
$71$		−	6.69213i		−	0.794210i	0.917773	−	0.397105i	$-0.129985\pi$
$72$		0			0
$73$			9.46410i			1.10769i	0.832620	+	0.553845i	$0.186840\pi$
$73$			9.46410i			1.10769i	−0.832620	+	0.553845i	$0.813160\pi$
$74$		0			0
$75$	−1.93185	−	1.93185i	−0.223071	−	0.223071i
$76$		0			0
$77$	−0.928203	+	0.928203i	−0.105779	+	0.105779i
$78$		0			0
$79$	15.4548			1.73880			0.869401	−	0.494107i	$-0.164505\pi$
$79$	15.4548			1.73880			0.869401	+	0.494107i	$0.164505\pi$
$80$		0			0
$81$	2.46410			0.273789
$82$		0			0
$83$	5.79555	−	5.79555i	0.636145	−	0.636145i	−0.313457	−	0.949602i	$-0.601487\pi$
$83$	5.79555	−	5.79555i	0.636145	−	0.636145i	0.949602	+	0.313457i	$0.101487\pi$
$84$		0			0
$85$	6.00000	+	6.00000i	0.650791	+	0.650791i
$86$		0			0
$87$		−	6.69213i		−	0.717472i
$88$		0			0
$89$		−	9.46410i		−	1.00319i	−0.865102	−	0.501596i	$-0.832746\pi$
$89$		−	9.46410i		−	1.00319i	0.865102	−	0.501596i	$-0.167254\pi$
$90$		0			0
$91$	−3.86370	−	3.86370i	−0.405026	−	0.405026i
$92$		0			0
$93$	10.9282	−	10.9282i	1.13320	−	1.13320i
$94$		0			0
$95$	−3.10583			−0.318651
$96$		0			0
$97$	−3.46410			−0.351726			−0.175863	−	0.984415i	$-0.556272\pi$
$97$	−3.46410			−0.351726			−0.175863	+	0.984415i	$0.556272\pi$
$98$		0			0
$99$	−4.00240	+	4.00240i	−0.402257	+	0.402257i
$100$		0			0
$101$	−11.1962	−	11.1962i	−1.11406	−	1.11406i	−0.992596	−	0.121463i	$-0.961241\pi$
$101$	−11.1962	−	11.1962i	−1.11406	−	1.11406i	−0.121463	−	0.992596i	$-0.538759\pi$
$102$		0			0
$103$			4.62158i			0.455378i	0.973734	+	0.227689i	$0.0731169\pi$
$103$			4.62158i			0.455378i	−0.973734	+	0.227689i	$0.926883\pi$
$104$		0			0
$105$		−	6.92820i		−	0.676123i
$106$		0			0
$107$	8.90138	+	8.90138i	0.860529	+	0.860529i	0.991400	−	0.130870i	$-0.0417771\pi$
$107$	8.90138	+	8.90138i	0.860529	+	0.860529i	−0.130870	+	0.991400i	$0.541777\pi$
$108$		0			0
$109$	−2.80385	+	2.80385i	−0.268560	+	0.268560i	−0.828520	−	0.559960i	$-0.810817\pi$
$109$	−2.80385	+	2.80385i	−0.268560	+	0.268560i	0.559960	+	0.828520i	$0.310817\pi$
$110$		0			0
$111$	1.03528			0.0982641
$112$		0			0
$113$	−0.928203			−0.0873180			−0.0436590	−	0.999046i	$-0.513902\pi$
$113$	−0.928203			−0.0873180			−0.0436590	+	0.999046i	$0.513902\pi$
$114$		0			0
$115$	−11.5911	+	11.5911i	−1.08088	+	1.08088i
$116$		0			0
$117$	−16.6603	−	16.6603i	−1.54024	−	1.54024i
$118$		0			0
$119$			3.58630i			0.328756i
$120$		0			0
$121$		−	9.39230i		−	0.853846i
$122$		0			0
$123$	−13.3843	−	13.3843i	−1.20682	−	1.20682i
$124$		0			0
$125$	−6.92820	+	6.92820i	−0.619677	+	0.619677i
$126$		0			0
$127$	5.65685			0.501965			0.250982	−	0.967992i	$-0.419246\pi$
$127$	5.65685			0.501965			0.250982	+	0.967992i	$0.419246\pi$
$128$		0			0
$129$	22.3923			1.97153
$130$		0			0
$131$	−12.4877	+	12.4877i	−1.09105	+	1.09105i	−0.0956380	+	0.995416i	$0.530489\pi$
$131$	−12.4877	+	12.4877i	−1.09105	+	1.09105i	−0.995416	+	0.0956380i	$0.969511\pi$
$132$		0			0
$133$	−0.928203	−	0.928203i	−0.0804854	−	0.0804854i
$134$		0			0
$135$		−	9.79796i		−	0.843274i
$136$		0			0
$137$		−	12.0000i		−	1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$		−	12.0000i		−	1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$		0			0
$139$	−0.138701	−	0.138701i	−0.0117644	−	0.0117644i	0.701200	−	0.712965i	$-0.252648\pi$
$139$	−0.138701	−	0.138701i	−0.0117644	−	0.0117644i	−0.712965	+	0.701200i	$0.752648\pi$
$140$		0			0
$141$	18.9282	−	18.9282i	1.59404	−	1.59404i
$142$		0			0
$143$	−6.69213			−0.559624
$144$		0			0
$145$	6.00000			0.498273
$146$		0			0
$147$	−11.4524	+	11.4524i	−0.944579	+	0.944579i
$148$		0			0
$149$	10.2679	+	10.2679i	0.841183	+	0.841183i	0.989013	−	0.147830i	$-0.0472287\pi$
$149$	10.2679	+	10.2679i	0.841183	+	0.841183i	−0.147830	+	0.989013i	$0.547229\pi$
$150$		0			0
$151$			4.62158i			0.376099i	0.982160	+	0.188049i	$0.0602165\pi$
$151$			4.62158i			0.376099i	−0.982160	+	0.188049i	$0.939784\pi$
$152$		0			0
$153$			15.4641i			1.25020i
$154$		0			0
$155$	9.79796	+	9.79796i	0.786991	+	0.786991i
$156$		0			0
$157$	−3.19615	+	3.19615i	−0.255081	+	0.255081i	−0.823050	−	0.567969i	$-0.807729\pi$
$157$	−3.19615	+	3.19615i	−0.255081	+	0.255081i	0.567969	+	0.823050i	$0.307729\pi$
$158$		0			0
$159$	16.4901			1.30775
$160$		0			0
$161$	−6.92820			−0.546019
$162$		0			0
$163$	9.38186	−	9.38186i	0.734844	−	0.734844i	−0.236731	−	0.971575i	$-0.576076\pi$
$163$	9.38186	−	9.38186i	0.734844	−	0.734844i	0.971575	+	0.236731i	$0.0760761\pi$
$164$		0			0
$165$	−6.00000	−	6.00000i	−0.467099	−	0.467099i
$166$		0			0
$167$		−	23.6627i		−	1.83107i	−0.402234	−	0.915537i	$-0.631766\pi$
$167$		−	23.6627i		−	1.83107i	0.402234	−	0.915537i	$-0.368234\pi$
$168$		0			0
$169$		−	14.8564i		−	1.14280i
$170$		0			0
$171$	−4.00240	−	4.00240i	−0.306071	−	0.306071i
$172$		0			0
$173$	−4.26795	+	4.26795i	−0.324486	+	0.324486i	−0.850485	−	0.525999i	$-0.823692\pi$
$173$	−4.26795	+	4.26795i	−0.324486	+	0.324486i	0.525999	+	0.850485i	$0.323692\pi$
$174$		0			0
$175$	1.03528			0.0782595
$176$		0			0
$177$	−29.3205			−2.20386
$178$		0			0
$179$	5.79555	−	5.79555i	0.433180	−	0.433180i	−0.456529	−	0.889709i	$-0.650907\pi$
$179$	5.79555	−	5.79555i	0.433180	−	0.433180i	0.889709	+	0.456529i	$0.150907\pi$
$180$		0			0
$181$	8.12436	+	8.12436i	0.603879	+	0.603879i	0.941340	−	0.337461i	$-0.109568\pi$
$181$	8.12436	+	8.12436i	0.603879	+	0.603879i	−0.337461	+	0.941340i	$0.609568\pi$
$182$		0			0
$183$		−	1.03528i		−	0.0765298i
$184$		0			0
$185$			0.928203i			0.0682429i
$186$		0			0
$187$	3.10583	+	3.10583i	0.227121	+	0.227121i
$188$		0			0
$189$	2.92820	−	2.92820i	0.212995	−	0.212995i
$190$		0			0
$191$	16.9706			1.22795			0.613973	−	0.789327i	$-0.289570\pi$
$191$	16.9706			1.22795			0.613973	+	0.789327i	$0.289570\pi$
$192$		0			0
$193$	2.39230			0.172202			0.0861009	−	0.996286i	$-0.472559\pi$
$193$	2.39230			0.172202			0.0861009	+	0.996286i	$0.472559\pi$
$194$		0			0
$195$	24.9754	−	24.9754i	1.78852	−	1.78852i
$196$		0			0
$197$	−1.73205	−	1.73205i	−0.123404	−	0.123404i	0.642708	−	0.766111i	$-0.277811\pi$
$197$	−1.73205	−	1.73205i	−0.123404	−	0.123404i	−0.766111	+	0.642708i	$0.777811\pi$
$198$		0			0
$199$			18.5606i			1.31573i	0.753136	+	0.657865i	$0.228540\pi$
$199$			18.5606i			1.31573i	−0.753136	+	0.657865i	$0.771460\pi$
$200$		0			0
$201$			11.4641i			0.808615i
$202$		0			0
$203$	1.79315	+	1.79315i	0.125855	+	0.125855i
$204$		0			0
$205$	12.0000	−	12.0000i	0.838116	−	0.838116i
$206$		0			0
$207$	−29.8744			−2.07641
$208$		0			0
$209$	−1.60770			−0.111207
$210$		0			0
$211$	0.138701	−	0.138701i	0.00954855	−	0.00954855i	−0.702316	−	0.711865i	$-0.747851\pi$
$211$	0.138701	−	0.138701i	0.00954855	−	0.00954855i	0.711865	+	0.702316i	$0.247851\pi$
$212$		0			0
$213$	12.9282	+	12.9282i	0.885826	+	0.885826i
$214$		0			0
$215$			20.0764i			1.36920i
$216$		0			0
$217$			5.85641i			0.397559i
$218$		0			0
$219$	−18.2832	−	18.2832i	−1.23547	−	1.23547i
$220$		0			0
$221$	−12.9282	+	12.9282i	−0.869645	+	0.869645i
$222$		0			0
$223$	−5.65685			−0.378811			−0.189405	−	0.981899i	$-0.560656\pi$
$223$	−5.65685			−0.378811			−0.189405	+	0.981899i	$0.560656\pi$
$224$		0			0
$225$	4.46410			0.297607
$226$		0			0
$227$	2.20925	−	2.20925i	0.146633	−	0.146633i	−0.629979	−	0.776612i	$-0.716936\pi$
$227$	2.20925	−	2.20925i	0.146633	−	0.146633i	0.776612	+	0.629979i	$0.216936\pi$
$228$		0			0
$229$	1.19615	+	1.19615i	0.0790440	+	0.0790440i	0.745523	−	0.666479i	$-0.232200\pi$
$229$	1.19615	+	1.19615i	0.0790440	+	0.0790440i	−0.666479	+	0.745523i	$0.732200\pi$
$230$		0			0
$231$		−	3.58630i		−	0.235961i
$232$		0			0
$233$			9.46410i			0.620014i	0.950734	+	0.310007i	$0.100331\pi$
$233$			9.46410i			0.620014i	−0.950734	+	0.310007i	$0.899669\pi$
$234$		0			0
$235$	16.9706	+	16.9706i	1.10704	+	1.10704i
$236$		0			0
$237$	−29.8564	+	29.8564i	−1.93938	+	1.93938i
$238$		0			0
$239$	−7.17260			−0.463957			−0.231979	−	0.972721i	$-0.574520\pi$
$239$	−7.17260			−0.463957			−0.231979	+	0.972721i	$0.574520\pi$
$240$		0			0
$241$	18.3923			1.18475			0.592376	−	0.805661i	$-0.298190\pi$
$241$	18.3923			1.18475			0.592376	+	0.805661i	$0.298190\pi$
$242$		0			0
$243$	−13.2456	+	13.2456i	−0.849703	+	0.849703i
$244$		0			0
$245$	−10.2679	−	10.2679i	−0.655995	−	0.655995i
$246$		0			0
$247$		−	6.69213i		−	0.425810i
$248$		0			0
$249$			22.3923i			1.41905i
$250$		0			0
$251$	−2.20925	−	2.20925i	−0.139447	−	0.139447i	0.633937	−	0.773384i	$-0.281438\pi$
$251$	−2.20925	−	2.20925i	−0.139447	−	0.139447i	−0.773384	+	0.633937i	$0.781438\pi$
$252$		0			0
$253$	−6.00000	+	6.00000i	−0.377217	+	0.377217i
$254$		0			0
$255$	−23.1822			−1.45173
$256$		0			0
$257$	−0.928203			−0.0578997			−0.0289499	−	0.999581i	$-0.509216\pi$
$257$	−0.928203			−0.0578997			−0.0289499	+	0.999581i	$0.509216\pi$
$258$		0			0
$259$	−0.277401	+	0.277401i	−0.0172369	+	0.0172369i
$260$		0			0
$261$	7.73205	+	7.73205i	0.478602	+	0.478602i
$262$		0			0
$263$			12.9038i			0.795682i	0.917454	+	0.397841i	$0.130240\pi$
$263$			12.9038i			0.795682i	−0.917454	+	0.397841i	$0.869760\pi$
$264$		0			0
$265$			14.7846i			0.908211i
$266$		0			0
$267$	18.2832	+	18.2832i	1.11892	+	1.11892i
$268$		0			0
$269$	5.19615	−	5.19615i	0.316815	−	0.316815i	−0.530728	−	0.847543i	$-0.678081\pi$
$269$	5.19615	−	5.19615i	0.316815	−	0.316815i	0.847543	+	0.530728i	$0.178081\pi$
$270$		0			0
$271$	−21.1117			−1.28244			−0.641221	−	0.767356i	$-0.721572\pi$
$271$	−21.1117			−1.28244			−0.641221	+	0.767356i	$0.721572\pi$
$272$		0			0
$273$	14.9282			0.903496
$274$		0			0
$275$	0.896575	−	0.896575i	0.0540655	−	0.0540655i
$276$		0			0
$277$	−7.19615	−	7.19615i	−0.432375	−	0.432375i	0.457061	−	0.889436i	$-0.348902\pi$
$277$	−7.19615	−	7.19615i	−0.432375	−	0.432375i	−0.889436	+	0.457061i	$0.848902\pi$
$278$		0			0
$279$			25.2528i			1.51184i
$280$		0			0
$281$		−	4.39230i		−	0.262023i	−0.991381	−	0.131011i	$-0.958178\pi$
$281$		−	4.39230i		−	0.262023i	0.991381	−	0.131011i	$-0.0418225\pi$
$282$		0			0
$283$	−8.62398	−	8.62398i	−0.512643	−	0.512643i	0.402693	−	0.915335i	$-0.368074\pi$
$283$	−8.62398	−	8.62398i	−0.512643	−	0.512643i	−0.915335	+	0.402693i	$0.868074\pi$
$284$		0			0
$285$	6.00000	−	6.00000i	0.355409	−	0.355409i
$286$		0			0
$287$	7.17260			0.423385
$288$		0			0
$289$	−5.00000			−0.294118
$290$		0			0
$291$	6.69213	−	6.69213i	0.392300	−	0.392300i
$292$		0			0
$293$	12.1244	+	12.1244i	0.708312	+	0.708312i	0.966180	−	0.257868i	$-0.0830199\pi$
$293$	12.1244	+	12.1244i	0.708312	+	0.708312i	−0.257868	+	0.966180i	$0.583020\pi$
$294$		0			0
$295$		−	26.2880i		−	1.53055i
$296$		0			0
$297$		−	5.07180i		−	0.294295i
$298$		0			0
$299$	−24.9754	−	24.9754i	−1.44436	−	1.44436i
$300$		0			0
$301$	−6.00000	+	6.00000i	−0.345834	+	0.345834i
$302$		0			0
$303$	43.2586			2.48514
$304$		0			0
$305$	0.928203			0.0531488
$306$		0			0
$307$	14.8356	−	14.8356i	0.846715	−	0.846715i	−0.143007	−	0.989722i	$-0.545677\pi$
$307$	14.8356	−	14.8356i	0.846715	−	0.846715i	0.989722	+	0.143007i	$0.0456772\pi$
$308$		0			0
$309$	−8.92820	−	8.92820i	−0.507908	−	0.507908i
$310$		0			0
$311$			10.2784i			0.582836i	0.956596	+	0.291418i	$0.0941271\pi$
$311$			10.2784i			0.582836i	−0.956596	+	0.291418i	$0.905873\pi$
$312$		0			0
$313$		−	28.7846i		−	1.62700i	−0.581563	−	0.813501i	$-0.697559\pi$
$313$		−	28.7846i		−	1.62700i	0.581563	−	0.813501i	$-0.302441\pi$
$314$		0			0
$315$	8.00481	+	8.00481i	0.451020	+	0.451020i
$316$		0			0
$317$	17.1962	−	17.1962i	0.965832	−	0.965832i	−0.0336031	−	0.999435i	$-0.510698\pi$
$317$	17.1962	−	17.1962i	0.965832	−	0.965832i	0.999435	+	0.0336031i	$0.0106982\pi$
$318$		0			0
$319$	3.10583			0.173893
$320$		0			0
$321$	−34.3923			−1.91959
$322$		0			0
$323$	−3.10583	+	3.10583i	−0.172813	+	0.172813i
$324$		0			0
$325$	3.73205	+	3.73205i	0.207017	+	0.207017i
$326$		0			0
$327$		−	10.8332i		−	0.599079i
$328$		0			0
$329$			10.1436i			0.559234i
$330$		0			0
$331$	3.24453	+	3.24453i	0.178335	+	0.178335i	0.790630	−	0.612294i	$-0.209753\pi$
$331$	3.24453	+	3.24453i	0.178335	+	0.178335i	−0.612294	+	0.790630i	$0.709753\pi$
$332$		0			0
$333$	−1.19615	+	1.19615i	−0.0655487	+	0.0655487i
$334$		0			0
$335$	−10.2784			−0.561571
$336$		0			0
$337$	−33.7128			−1.83645			−0.918227	−	0.396055i	$-0.870379\pi$
$337$	−33.7128			−1.83645			−0.918227	+	0.396055i	$0.870379\pi$
$338$		0			0
$339$	1.79315	−	1.79315i	0.0973906	−	0.0973906i
$340$		0			0
$341$	5.07180	+	5.07180i	0.274653	+	0.274653i
$342$		0			0
$343$		−	13.3843i		−	0.722682i
$344$		0			0
$345$		−	44.7846i		−	2.41112i
$346$		0			0
$347$	7.58871	+	7.58871i	0.407383	+	0.407383i	0.880825	−	0.473442i	$-0.156989\pi$
$347$	7.58871	+	7.58871i	0.407383	+	0.407383i	−0.473442	+	0.880825i	$0.656989\pi$
$348$		0			0
$349$	20.1244	−	20.1244i	1.07723	−	1.07723i	0.0804755	−	0.996757i	$-0.474356\pi$
$349$	20.1244	−	20.1244i	1.07723	−	1.07723i	0.996757	−	0.0804755i	$-0.0256439\pi$
$350$		0			0
$351$	21.1117			1.12686
$352$		0			0
$353$	12.9282			0.688099			0.344049	−	0.938952i	$-0.388201\pi$
$353$	12.9282			0.688099			0.344049	+	0.938952i	$0.388201\pi$
$354$		0			0
$355$	−11.5911	+	11.5911i	−0.615192	+	0.615192i
$356$		0			0
$357$	−6.92820	−	6.92820i	−0.366679	−	0.366679i
$358$		0			0
$359$		−	26.2880i		−	1.38743i	−0.720250	−	0.693715i	$-0.755973\pi$
$359$		−	26.2880i		−	1.38743i	0.720250	−	0.693715i	$-0.244027\pi$
$360$		0			0
$361$			17.3923i			0.915384i
$362$		0			0
$363$	18.1445	+	18.1445i	0.952341	+	0.952341i
$364$		0			0
$365$	16.3923	−	16.3923i	0.858012	−	0.858012i
$366$		0			0
$367$	12.8295			0.669692			0.334846	−	0.942273i	$-0.391316\pi$
$367$	12.8295			0.669692			0.334846	+	0.942273i	$0.391316\pi$
$368$		0			0
$369$	30.9282			1.61006
$370$		0			0
$371$	−4.41851	+	4.41851i	−0.229397	+	0.229397i
$372$		0			0
$373$	−10.1244	−	10.1244i	−0.524219	−	0.524219i	0.394624	−	0.918843i	$-0.370875\pi$
$373$	−10.1244	−	10.1244i	−0.524219	−	0.524219i	−0.918843	+	0.394624i	$0.870875\pi$
$374$		0			0
$375$		−	26.7685i		−	1.38232i
$376$		0			0
$377$			12.9282i			0.665836i
$378$		0			0
$379$	−19.1798	−	19.1798i	−0.985201	−	0.985201i	0.0146911	−	0.999892i	$-0.495324\pi$
$379$	−19.1798	−	19.1798i	−0.985201	−	0.985201i	−0.999892	+	0.0146911i	$0.995324\pi$
$380$		0			0
$381$	−10.9282	+	10.9282i	−0.559869	+	0.559869i
$382$		0			0
$383$	19.5959			1.00130			0.500652	−	0.865648i	$-0.333094\pi$
$383$	19.5959			1.00130			0.500652	+	0.865648i	$0.333094\pi$
$384$		0			0
$385$	3.21539			0.163871
$386$		0			0
$387$	−25.8719	+	25.8719i	−1.31514	+	1.31514i
$388$		0			0
$389$	7.73205	+	7.73205i	0.392031	+	0.392031i	0.875411	−	0.483380i	$-0.160591\pi$
$389$	7.73205	+	7.73205i	0.392031	+	0.392031i	−0.483380	+	0.875411i	$0.660591\pi$
$390$		0			0
$391$			23.1822i			1.17238i
$392$		0			0
$393$		−	48.2487i		−	2.43383i
$394$		0			0
$395$	−26.7685	−	26.7685i	−1.34687	−	1.34687i
$396$		0			0
$397$	−11.5885	+	11.5885i	−0.581608	+	0.581608i	−0.935345	−	0.353737i	$-0.884911\pi$
$397$	−11.5885	+	11.5885i	−0.581608	+	0.581608i	0.353737	+	0.935345i	$0.384911\pi$
$398$		0			0
$399$	3.58630			0.179540
$400$		0			0
$401$	34.3923			1.71747			0.858735	−	0.512420i	$-0.171251\pi$
$401$	34.3923			1.71747			0.858735	+	0.512420i	$0.171251\pi$
$402$		0			0
$403$	−21.1117	+	21.1117i	−1.05165	+	1.05165i
$404$		0			0
$405$	−4.26795	−	4.26795i	−0.212076	−	0.212076i
$406$		0			0
$407$			0.480473i			0.0238162i
$408$		0			0
$409$			30.9282i			1.52930i	0.644445	+	0.764651i	$0.277088\pi$
$409$			30.9282i			1.52930i	−0.644445	+	0.764651i	$0.722912\pi$
$410$		0			0
$411$	23.1822	+	23.1822i	1.14349	+	1.14349i
$412$		0			0
$413$	7.85641	−	7.85641i	0.386588	−	0.386588i
$414$		0			0
$415$	−20.0764			−0.985511
$416$		0			0
$417$	0.535898			0.0262431
$418$		0			0
$419$	−2.68973	+	2.68973i	−0.131402	+	0.131402i	−0.769749	−	0.638347i	$-0.779619\pi$
$419$	−2.68973	+	2.68973i	−0.131402	+	0.131402i	0.638347	+	0.769749i	$0.279619\pi$
$420$		0			0
$421$	−26.5167	−	26.5167i	−1.29234	−	1.29234i	−0.933334	−	0.359009i	$-0.883115\pi$
$421$	−26.5167	−	26.5167i	−1.29234	−	1.29234i	−0.359009	−	0.933334i	$-0.616885\pi$
$422$		0			0
$423$			43.7391i			2.12667i
$424$		0			0
$425$		−	3.46410i		−	0.168034i
$426$		0			0
$427$	0.277401	+	0.277401i	0.0134244	+	0.0134244i
$428$		0			0
$429$	12.9282	−	12.9282i	0.624180	−	0.624180i
$430$		0			0
$431$	−7.17260			−0.345492			−0.172746	−	0.984966i	$-0.555264\pi$
$431$	−7.17260			−0.345492			−0.172746	+	0.984966i	$0.555264\pi$
$432$		0			0
$433$	−13.6077			−0.653944			−0.326972	−	0.945034i	$-0.606028\pi$
$433$	−13.6077			−0.653944			−0.326972	+	0.945034i	$0.606028\pi$
$434$		0			0
$435$	−11.5911	+	11.5911i	−0.555751	+	0.555751i
$436$		0			0
$437$	−6.00000	−	6.00000i	−0.287019	−	0.287019i
$438$		0			0
$439$		−	31.9449i		−	1.52465i	−0.647196	−	0.762324i	$-0.724059\pi$
$439$		−	31.9449i		−	1.52465i	0.647196	−	0.762324i	$-0.275941\pi$
$440$		0			0
$441$		−	26.4641i		−	1.26020i
$442$		0			0
$443$	2.68973	+	2.68973i	0.127793	+	0.127793i	0.768110	−	0.640318i	$-0.221197\pi$
$443$	2.68973	+	2.68973i	0.127793	+	0.127793i	−0.640318	+	0.768110i	$0.721197\pi$
$444$		0			0
$445$	−16.3923	+	16.3923i	−0.777070	+	0.777070i
$446$		0			0
$447$	−39.6723			−1.87644
$448$		0			0
$449$	24.2487			1.14437			0.572184	−	0.820125i	$-0.306096\pi$
$449$	24.2487			1.14437			0.572184	+	0.820125i	$0.306096\pi$
$450$		0			0
$451$	6.21166	−	6.21166i	0.292496	−	0.292496i
$452$		0			0
$453$	−8.92820	−	8.92820i	−0.419484	−	0.419484i
$454$		0			0
$455$			13.3843i			0.627464i
$456$		0			0
$457$		−	20.7846i		−	0.972263i	−0.873886	−	0.486132i	$-0.838408\pi$
$457$		−	20.7846i		−	0.972263i	0.873886	−	0.486132i	$-0.161592\pi$
$458$		0			0
$459$	−9.79796	−	9.79796i	−0.457330	−	0.457330i
$460$		0			0
$461$	0.124356	−	0.124356i	0.00579182	−	0.00579182i	−0.704205	−	0.709997i	$-0.748696\pi$
$461$	0.124356	−	0.124356i	0.00579182	−	0.00579182i	0.709997	+	0.704205i	$0.248696\pi$
$462$		0			0
$463$	18.4863			0.859132			0.429566	−	0.903036i	$-0.358667\pi$
$463$	18.4863			0.859132			0.429566	+	0.903036i	$0.358667\pi$
$464$		0			0
$465$	−37.8564			−1.75555
$466$		0			0
$467$	17.8671	−	17.8671i	0.826793	−	0.826793i	−0.160279	−	0.987072i	$-0.551239\pi$
$467$	17.8671	−	17.8671i	0.826793	−	0.826793i	0.987072	+	0.160279i	$0.0512394\pi$
$468$		0			0
$469$	−3.07180	−	3.07180i	−0.141842	−	0.141842i
$470$		0			0
$471$		−	12.3490i		−	0.569011i
$472$		0			0
$473$			10.3923i			0.477839i
$474$		0			0
$475$	0.896575	+	0.896575i	0.0411377	+	0.0411377i
$476$		0			0
$477$	−19.0526	+	19.0526i	−0.872357	+	0.872357i
$478$		0			0
$479$	2.62536			0.119956			0.0599778	−	0.998200i	$-0.480897\pi$
$479$	2.62536			0.119956			0.0599778	+	0.998200i	$0.480897\pi$
$480$		0			0
$481$	−2.00000			−0.0911922
$482$		0			0
$483$	13.3843	−	13.3843i	0.609005	−	0.609005i
$484$		0			0
$485$	6.00000	+	6.00000i	0.272446	+	0.272446i
$486$		0			0
$487$			7.24693i			0.328390i	0.986428	+	0.164195i	$0.0525026\pi$
$487$			7.24693i			0.328390i	−0.986428	+	0.164195i	$0.947497\pi$
$488$		0			0
$489$			36.2487i			1.63922i
$490$		0			0
$491$	−22.7661	−	22.7661i	−1.02742	−	1.02742i	−0.999613	−	0.0278072i	$-0.991148\pi$
$491$	−22.7661	−	22.7661i	−1.02742	−	1.02742i	−0.0278072	−	0.999613i	$-0.508852\pi$
$492$		0			0
$493$	6.00000	−	6.00000i	0.270226	−	0.270226i
$494$		0			0
$495$	13.8647			0.623173
$496$		0			0
$497$	−6.92820			−0.310772
$498$		0			0
$499$	11.4524	−	11.4524i	0.512680	−	0.512680i	−0.402667	−	0.915347i	$-0.631917\pi$
$499$	11.4524	−	11.4524i	0.512680	−	0.512680i	0.915347	+	0.402667i	$0.131917\pi$
$500$		0			0
$501$	45.7128	+	45.7128i	2.04230	+	2.04230i
$502$		0			0
$503$			32.4997i			1.44909i	0.689227	+	0.724545i	$0.257950\pi$
$503$			32.4997i			1.44909i	−0.689227	+	0.724545i	$0.742050\pi$
$504$		0			0
$505$			38.7846i			1.72589i
$506$		0			0
$507$	28.7004	+	28.7004i	1.27463	+	1.27463i
$508$		0			0
$509$	21.5885	−	21.5885i	0.956892	−	0.956892i	−0.0422169	−	0.999108i	$-0.513442\pi$
$509$	21.5885	−	21.5885i	0.956892	−	0.956892i	0.999108	+	0.0422169i	$0.0134421\pi$
$510$		0			0
$511$	9.79796			0.433436
$512$		0			0
$513$	5.07180			0.223925
$514$		0			0
$515$	8.00481	−	8.00481i	0.352734	−	0.352734i
$516$		0			0
$517$	8.78461	+	8.78461i	0.386347	+	0.386347i
$518$		0			0
$519$		−	16.4901i		−	0.723835i
$520$		0			0
$521$		−	12.0000i		−	0.525730i	−0.964833	−	0.262865i	$-0.915333\pi$
$521$		−	12.0000i		−	0.525730i	0.964833	−	0.262865i	$-0.0846673\pi$
$522$		0			0
$523$	16.0740	+	16.0740i	0.702866	+	0.702866i	0.965025	−	0.262158i	$-0.0844343\pi$
$523$	16.0740	+	16.0740i	0.702866	+	0.702866i	−0.262158	+	0.965025i	$0.584434\pi$
$524$		0			0
$525$	−2.00000	+	2.00000i	−0.0872872	+	0.0872872i
$526$		0			0
$527$	19.5959			0.853612
$528$		0			0
$529$	−21.7846			−0.947157
$530$		0			0
$531$	33.8768	−	33.8768i	1.47013	−	1.47013i
$532$		0			0
$533$	25.8564	+	25.8564i	1.11997	+	1.11997i
$534$		0			0
$535$		−	30.8353i		−	1.33313i
$536$		0			0
$537$			22.3923i			0.966299i
$538$		0			0
$539$	−5.31508	−	5.31508i	−0.228937	−	0.228937i
$540$		0			0
$541$	−14.1244	+	14.1244i	−0.607253	+	0.607253i	−0.942227	−	0.334974i	$-0.891272\pi$
$541$	−14.1244	+	14.1244i	−0.607253	+	0.607253i	0.334974	+	0.942227i	$0.391272\pi$
$542$		0			0
$543$	−31.3901			−1.34708
$544$		0			0
$545$	9.71281			0.416051
$546$		0			0
$547$	−12.4877	+	12.4877i	−0.533935	+	0.533935i	−0.921741	−	0.387806i	$-0.873233\pi$
$547$	−12.4877	+	12.4877i	−0.533935	+	0.533935i	0.387806	+	0.921741i	$0.373233\pi$
$548$		0			0
$549$	1.19615	+	1.19615i	0.0510505	+	0.0510505i
$550$		0			0
$551$			3.10583i			0.132313i
$552$		0			0
$553$		−	16.0000i		−	0.680389i
$554$		0			0
$555$	−1.79315	−	1.79315i	−0.0761150	−	0.0761150i
$556$		0			0
$557$	0.803848	−	0.803848i	0.0340601	−	0.0340601i	−0.689872	−	0.723932i	$-0.742333\pi$
$557$	0.803848	−	0.803848i	0.0340601	−	0.0340601i	0.723932	+	0.689872i	$0.242333\pi$
$558$		0			0
$559$	−43.2586			−1.82964
$560$		0			0
$561$	−12.0000			−0.506640
$562$		0			0
$563$	27.6651	−	27.6651i	1.16594	−	1.16594i	0.182794	−	0.983151i	$-0.441486\pi$
$563$	27.6651	−	27.6651i	1.16594	−	1.16594i	0.983151	−	0.182794i	$-0.0585140\pi$
$564$		0			0
$565$	1.60770	+	1.60770i	0.0676362	+	0.0676362i
$566$		0			0
$567$		−	2.55103i		−	0.107133i
$568$		0			0
$569$			12.0000i			0.503066i	0.967849	+	0.251533i	$0.0809347\pi$
$569$			12.0000i			0.503066i	−0.967849	+	0.251533i	$0.919065\pi$
$570$		0			0
$571$	14.0034	+	14.0034i	0.586026	+	0.586026i	0.936553	−	0.350527i	$-0.113998\pi$
$571$	14.0034	+	14.0034i	0.586026	+	0.586026i	−0.350527	+	0.936553i	$0.613998\pi$
$572$		0			0
$573$	−32.7846	+	32.7846i	−1.36960	+	1.36960i
$574$		0			0
$575$	6.69213			0.279081
$576$		0			0
$577$	−16.1436			−0.672067			−0.336033	−	0.941850i	$-0.609085\pi$
$577$	−16.1436			−0.672067			−0.336033	+	0.941850i	$0.609085\pi$
$578$		0			0
$579$	−4.62158	+	4.62158i	−0.192066	+	0.192066i
$580$		0			0
$581$	−6.00000	−	6.00000i	−0.248922	−	0.248922i
$582$		0			0
$583$			7.65308i			0.316958i
$584$		0			0
$585$			57.7128i			2.38613i
$586$		0			0
$587$	−10.6945	−	10.6945i	−0.441411	−	0.441411i	0.451075	−	0.892486i	$-0.351041\pi$
$587$	−10.6945	−	10.6945i	−0.441411	−	0.441411i	−0.892486	+	0.451075i	$0.851041\pi$
$588$		0			0
$589$	−5.07180	+	5.07180i	−0.208980	+	0.208980i
$590$		0			0
$591$	6.69213			0.275277
$592$		0			0
$593$	7.85641			0.322624			0.161312	−	0.986903i	$-0.448427\pi$
$593$	7.85641			0.322624			0.161312	+	0.986903i	$0.448427\pi$
$594$		0			0
$595$	6.21166	−	6.21166i	0.254653	−	0.254653i
$596$		0			0
$597$	−35.8564	−	35.8564i	−1.46751	−	1.46751i
$598$		0			0
$599$			29.8744i			1.22063i	0.792158	+	0.610316i	$0.208958\pi$
$599$			29.8744i			1.22063i	−0.792158	+	0.610316i	$0.791042\pi$
$600$		0			0
$601$		−	19.6077i		−	0.799815i	−0.916556	−	0.399907i	$-0.869042\pi$
$601$		−	19.6077i		−	0.799815i	0.916556	−	0.399907i	$-0.130958\pi$
$602$		0			0
$603$	−13.2456	−	13.2456i	−0.539401	−	0.539401i
$604$		0			0
$605$	−16.2679	+	16.2679i	−0.661386	+	0.661386i
$606$		0			0
$607$	−30.9096			−1.25458			−0.627292	−	0.778785i	$-0.715837\pi$
$607$	−30.9096			−1.25458			−0.627292	+	0.778785i	$0.715837\pi$
$608$		0			0
$609$	−6.92820			−0.280745
$610$		0			0
$611$	−36.5665	+	36.5665i	−1.47932	+	1.47932i
$612$		0			0
$613$	23.0526	+	23.0526i	0.931084	+	0.931084i	0.997774	−	0.0666897i	$-0.0212438\pi$
$613$	23.0526	+	23.0526i	0.931084	+	0.931084i	−0.0666897	+	0.997774i	$0.521244\pi$
$614$		0			0
$615$			46.3644i			1.86959i
$616$		0			0
$617$		−	23.3205i		−	0.938848i	−0.882973	−	0.469424i	$-0.844462\pi$
$617$		−	23.3205i		−	0.938848i	0.882973	−	0.469424i	$-0.155538\pi$
$618$		0			0
$619$	3.24453	+	3.24453i	0.130409	+	0.130409i	0.769298	−	0.638890i	$-0.220606\pi$
$619$	3.24453	+	3.24453i	0.130409	+	0.130409i	−0.638890	+	0.769298i	$0.720606\pi$
$620$		0			0
$621$	18.9282	−	18.9282i	0.759563	−	0.759563i
$622$		0			0
$623$	−9.79796			−0.392547
$624$		0			0
$625$	29.0000			1.16000
$626$		0			0
$627$	3.10583	−	3.10583i	0.124035	−	0.124035i
$628$		0			0
$629$	0.928203	+	0.928203i	0.0370099	+	0.0370099i
$630$		0			0
$631$			35.5312i			1.41447i	0.706976	+	0.707237i	$0.250059\pi$
$631$			35.5312i			1.41447i	−0.706976	+	0.707237i	$0.749941\pi$
$632$		0			0
$633$			0.535898i			0.0213000i
$634$		0			0
$635$	−9.79796	−	9.79796i	−0.388820	−	0.388820i
$636$		0			0
$637$	22.1244	−	22.1244i	0.876599	−	0.876599i
$638$		0			0
$639$	−29.8744			−1.18181
$640$		0			0
$641$	34.3923			1.35841			0.679207	−	0.733947i	$-0.262324\pi$
$641$	34.3923			1.35841			0.679207	+	0.733947i	$0.262324\pi$
$642$		0			0
$643$	14.2808	−	14.2808i	0.563181	−	0.563181i	−0.367028	−	0.930210i	$-0.619625\pi$
$643$	14.2808	−	14.2808i	0.563181	−	0.563181i	0.930210	+	0.367028i	$0.119625\pi$
$644$		0			0
$645$	−38.7846	−	38.7846i	−1.52714	−	1.52714i
$646$		0			0
$647$		−	6.69213i		−	0.263095i	−0.991310	−	0.131547i	$-0.958005\pi$
$647$		−	6.69213i		−	0.263095i	0.991310	−	0.131547i	$-0.0419946\pi$
$648$		0			0
$649$		−	13.6077i		−	0.534149i
$650$		0			0
$651$	−11.3137	−	11.3137i	−0.443419	−	0.443419i
$652$		0			0
$653$	−32.6603	+	32.6603i	−1.27809	+	1.27809i	−0.336362	+	0.941733i	$0.609196\pi$
$653$	−32.6603	+	32.6603i	−1.27809	+	1.27809i	−0.941733	+	0.336362i	$0.890804\pi$
$654$		0			0
$655$	43.2586			1.69025
$656$		0			0
$657$	42.2487			1.64828
$658$		0			0
$659$	−13.8004	+	13.8004i	−0.537586	+	0.537586i	−0.922819	−	0.385233i	$-0.874121\pi$
$659$	−13.8004	+	13.8004i	−0.537586	+	0.537586i	0.385233	+	0.922819i	$0.374121\pi$
$660$		0			0
$661$	−21.7321	−	21.7321i	−0.845279	−	0.845279i	0.144261	−	0.989540i	$-0.453920\pi$
$661$	−21.7321	−	21.7321i	−0.845279	−	0.845279i	−0.989540	+	0.144261i	$0.953920\pi$
$662$		0			0
$663$		−	49.9507i		−	1.93993i
$664$		0			0
$665$			3.21539i			0.124687i
$666$		0			0
$667$	11.5911	+	11.5911i	0.448810	+	0.448810i
$668$		0			0
$669$	10.9282	−	10.9282i	0.422509	−	0.422509i
$670$		0			0
$671$	0.480473			0.0185485
$672$		0			0
$673$	6.67949			0.257475			0.128738	−	0.991679i	$-0.458907\pi$
$673$	6.67949			0.257475			0.128738	+	0.991679i	$0.458907\pi$
$674$		0			0
$675$	−2.82843	+	2.82843i	−0.108866	+	0.108866i
$676$		0			0
$677$	31.7321	+	31.7321i	1.21956	+	1.21956i	0.967785	+	0.251776i	$0.0810147\pi$
$677$	31.7321	+	31.7321i	1.21956	+	1.21956i	0.251776	+	0.967785i	$0.418985\pi$
$678$		0			0
$679$			3.58630i			0.137630i
$680$		0			0
$681$			8.53590i			0.327096i
$682$		0			0
$683$	20.9730	+	20.9730i	0.802508	+	0.802508i	0.983487	−	0.180979i	$-0.0579265\pi$
$683$	20.9730	+	20.9730i	0.802508	+	0.802508i	−0.180979	+	0.983487i	$0.557927\pi$
$684$		0			0
$685$	−20.7846	+	20.7846i	−0.794139	+	0.794139i
$686$		0			0
$687$	−4.62158			−0.176324
$688$		0			0
$689$	−31.8564			−1.21363
$690$		0			0
$691$	−23.5983	+	23.5983i	−0.897722	+	0.897722i	−0.995234	−	0.0975119i	$-0.968912\pi$
$691$	−23.5983	+	23.5983i	−0.897722	+	0.897722i	0.0975119	+	0.995234i	$0.468912\pi$
$692$		0			0
$693$	4.14359	+	4.14359i	0.157402	+	0.157402i
$694$		0			0
$695$			0.480473i			0.0182254i
$696$		0			0
$697$		−	24.0000i		−	0.909065i
$698$		0			0
$699$	−18.2832	−	18.2832i	−0.691536	−	0.691536i
$700$		0			0
$701$	26.6603	−	26.6603i	1.00694	−	1.00694i	0.00696818	−	0.999976i	$-0.497782\pi$
$701$	26.6603	−	26.6603i	1.00694	−	1.00694i	0.999976	−	0.00696818i	$-0.00221806\pi$
$702$		0			0
$703$	−0.480473			−0.0181214
$704$		0			0
$705$	−65.5692			−2.46948
$706$		0			0
$707$	−11.5911	+	11.5911i	−0.435929	+	0.435929i
$708$		0			0
$709$	−9.05256	−	9.05256i	−0.339976	−	0.339976i	0.516382	−	0.856358i	$-0.327278\pi$
$709$	−9.05256	−	9.05256i	−0.339976	−	0.339976i	−0.856358	+	0.516382i	$0.827278\pi$
$710$		0			0
$711$		−	68.9919i		−	2.58740i
$712$		0			0
$713$			37.8564i			1.41773i
$714$		0			0
$715$	11.5911	+	11.5911i	0.433483	+	0.433483i
$716$		0			0
$717$	13.8564	−	13.8564i	0.517477	−	0.517477i
$718$		0			0
$719$	46.3644			1.72910			0.864551	−	0.502545i	$-0.167603\pi$
$719$	46.3644			1.72910			0.864551	+	0.502545i	$0.167603\pi$
$720$		0			0
$721$	4.78461			0.178188
$722$		0			0
$723$	−35.5312	+	35.5312i	−1.32142	+	1.32142i
$724$		0			0
$725$	−1.73205	−	1.73205i	−0.0643268	−	0.0643268i
$726$		0			0
$727$		−	20.6312i		−	0.765169i	−0.923921	−	0.382584i	$-0.875034\pi$
$727$		−	20.6312i		−	0.765169i	0.923921	−	0.382584i	$-0.124966\pi$
$728$		0			0
$729$		−	43.7846i		−	1.62165i
$730$		0			0
$731$	20.0764	+	20.0764i	0.742552	+	0.742552i
$732$		0			0
$733$	25.1962	−	25.1962i	0.930641	−	0.930641i	−0.0671048	−	0.997746i	$-0.521376\pi$
$733$	25.1962	−	25.1962i	0.930641	−	0.930641i	0.997746	+	0.0671048i	$0.0213762\pi$
$734$		0			0
$735$	39.6723			1.46334
$736$		0			0
$737$	−5.32051			−0.195983
$738$		0			0
$739$	34.6346	−	34.6346i	1.27406	−	1.27406i	0.330115	−	0.943941i	$-0.392913\pi$
$739$	34.6346	−	34.6346i	1.27406	−	1.27406i	0.943941	−	0.330115i	$-0.107087\pi$
$740$		0			0
$741$	12.9282	+	12.9282i	0.474929	+	0.474929i
$742$		0			0
$743$			10.2784i			0.377079i	0.982066	+	0.188540i	$0.0603754\pi$
$743$			10.2784i			0.377079i	−0.982066	+	0.188540i	$0.939625\pi$
$744$		0			0
$745$		−	35.5692i		−	1.30316i
$746$		0			0
$747$	−25.8719	−	25.8719i	−0.946605	−	0.946605i
$748$		0			0
$749$	9.21539	−	9.21539i	0.336723	−	0.336723i
$750$		0			0
$751$	−15.4548			−0.563954			−0.281977	−	0.959421i	$-0.590990\pi$
$751$	−15.4548			−0.563954			−0.281977	+	0.959421i	$0.590990\pi$
$752$		0			0
$753$	8.53590			0.311065
$754$		0			0
$755$	8.00481	−	8.00481i	0.291325	−	0.291325i
$756$		0			0
$757$	18.2679	+	18.2679i	0.663960	+	0.663960i	0.956311	−	0.292351i	$-0.0944376\pi$
$757$	18.2679	+	18.2679i	0.663960	+	0.663960i	−0.292351	+	0.956311i	$0.594438\pi$
$758$		0			0
$759$		−	23.1822i		−	0.841461i
$760$		0			0
$761$		−	34.6410i		−	1.25574i	−0.778320	−	0.627868i	$-0.783928\pi$
$761$		−	34.6410i		−	1.25574i	0.778320	−	0.627868i	$-0.216072\pi$
$762$		0			0
$763$	2.90276	+	2.90276i	0.105087	+	0.105087i
$764$		0			0
$765$	26.7846	−	26.7846i	0.968400	−	0.968400i
$766$		0			0
$767$	56.6429			2.04526
$768$		0			0
$769$	18.3923			0.663243			0.331622	−	0.943412i	$-0.392404\pi$
$769$	18.3923			0.663243			0.331622	+	0.943412i	$0.392404\pi$
$770$		0			0
$771$	1.79315	−	1.79315i	0.0645788	−	0.0645788i
$772$		0			0
$773$	2.66025	+	2.66025i	0.0956827	+	0.0956827i	0.753328	−	0.657645i	$-0.228447\pi$
$773$	2.66025	+	2.66025i	0.0956827	+	0.0956827i	−0.657645	+	0.753328i	$0.728447\pi$
$774$		0			0
$775$		−	5.65685i		−	0.203200i
$776$		0			0
$777$		−	1.07180i		−	0.0384505i
$778$		0			0
$779$	6.21166	+	6.21166i	0.222556	+	0.222556i
$780$		0			0
$781$	−6.00000	+	6.00000i	−0.214697	+	0.214697i
$782$		0			0
$783$	−9.79796			−0.350150
$784$		0			0
$785$	11.0718			0.395169
$786$		0			0
$787$	12.9682	−	12.9682i	0.462265	−	0.462265i	−0.437132	−	0.899397i	$-0.644006\pi$
$787$	12.9682	−	12.9682i	0.462265	−	0.462265i	0.899397	+	0.437132i	$0.144006\pi$
$788$		0			0
$789$	−24.9282	−	24.9282i	−0.887468	−	0.887468i
$790$		0			0
$791$			0.960947i			0.0341673i
$792$		0			0
$793$			2.00000i			0.0710221i
$794$		0			0
$795$	−28.5617	−	28.5617i	−1.01298	−	1.01298i
$796$		0			0
$797$	−2.41154	+	2.41154i	−0.0854212	+	0.0854212i	−0.748526	−	0.663105i	$-0.769238\pi$
$797$	−2.41154	+	2.41154i	−0.0854212	+	0.0854212i	0.663105	+	0.748526i	$0.269238\pi$
$798$		0			0
$799$	33.9411			1.20075
$800$		0			0
$801$	−42.2487			−1.49278
$802$		0			0
$803$	8.48528	−	8.48528i	0.299439	−	0.299439i
$804$		0			0
$805$	12.0000	+	12.0000i	0.422944	+	0.422944i
$806$		0			0
$807$			20.0764i			0.706722i
$808$		0			0
$809$		−	30.9282i		−	1.08738i	−0.839287	−	0.543689i	$-0.817027\pi$
$809$		−	30.9282i		−	1.08738i	0.839287	−	0.543689i	$-0.182973\pi$
$810$		0			0
$811$	−3.52193	−	3.52193i	−0.123672	−	0.123672i	0.642562	−	0.766234i	$-0.277872\pi$
$811$	−3.52193	−	3.52193i	−0.123672	−	0.123672i	−0.766234	+	0.642562i	$0.777872\pi$
$812$		0			0
$813$	40.7846	−	40.7846i	1.43038	−	1.43038i
$814$		0			0
$815$	−32.4997			−1.13842
$816$		0			0
$817$	−10.3923			−0.363581
$818$		0			0
$819$	−17.2480	+	17.2480i	−0.602693	+	0.602693i
$820$		0			0
$821$	−31.9808	−	31.9808i	−1.11614	−	1.11614i	−0.992303	−	0.123833i	$-0.960481\pi$
$821$	−31.9808	−	31.9808i	−1.11614	−	1.11614i	−0.123833	−	0.992303i	$-0.539519\pi$
$822$		0			0
$823$			18.5606i			0.646983i	0.946231	+	0.323492i	$0.104857\pi$
$823$			18.5606i			0.646983i	−0.946231	+	0.323492i	$0.895143\pi$
$824$		0			0
$825$			3.46410i			0.120605i
$826$		0			0
$827$	−4.48288	−	4.48288i	−0.155885	−	0.155885i	0.624856	−	0.780740i	$-0.285158\pi$
$827$	−4.48288	−	4.48288i	−0.155885	−	0.155885i	−0.780740	+	0.624856i	$0.785158\pi$
$828$		0			0
$829$	−22.1244	+	22.1244i	−0.768411	+	0.768411i	−0.977827	−	0.209416i	$-0.932844\pi$
$829$	−22.1244	+	22.1244i	−0.768411	+	0.768411i	0.209416	+	0.977827i	$0.432844\pi$
$830$		0			0
$831$	27.8038			0.964503
$832$		0			0
$833$	−20.5359			−0.711527
$834$		0			0
$835$	−40.9850	+	40.9850i	−1.41834	+	1.41834i
$836$		0			0
$837$	−16.0000	−	16.0000i	−0.553041	−	0.553041i
$838$		0			0
$839$		−	4.06678i		−	0.140401i	−0.997533	−	0.0702003i	$-0.977636\pi$
$839$		−	4.06678i		−	0.140401i	0.997533	−	0.0702003i	$-0.0223639\pi$
$840$		0			0
$841$			23.0000i			0.793103i
$842$		0			0
$843$	8.48528	+	8.48528i	0.292249	+	0.292249i
$844$		0			0
$845$	−25.7321	+	25.7321i	−0.885209	+	0.885209i
$846$		0			0
$847$	−9.72363			−0.334108
$848$		0			0
$849$	33.3205			1.14356
$850$		0			0
$851$	−1.79315	+	1.79315i	−0.0614684	+	0.0614684i
$852$		0			0
$853$	−34.1244	−	34.1244i	−1.16840	−	1.16840i	−0.982586	−	0.185810i	$-0.940509\pi$
$853$	−34.1244	−	34.1244i	−1.16840	−	1.16840i	−0.185810	−	0.982586i	$-0.559491\pi$
$854$		0			0
$855$			13.8647i			0.474164i
$856$		0			0
$857$			1.85641i			0.0634136i	0.999497	+	0.0317068i	$0.0100943\pi$
$857$			1.85641i			0.0634136i	−0.999497	+	0.0317068i	$0.989906\pi$
$858$		0			0
$859$	27.1846	+	27.1846i	0.927527	+	0.927527i	0.997546	−	0.0700183i	$-0.0223058\pi$
$859$	27.1846	+	27.1846i	0.927527	+	0.927527i	−0.0700183	+	0.997546i	$0.522306\pi$
$860$		0			0
$861$	−13.8564	+	13.8564i	−0.472225	+	0.472225i
$862$		0			0
$863$	−36.5665			−1.24474			−0.622369	−	0.782724i	$-0.713830\pi$
$863$	−36.5665			−1.24474			−0.622369	+	0.782724i	$0.713830\pi$
$864$		0			0
$865$	14.7846			0.502692
$866$		0			0
$867$	9.65926	−	9.65926i	0.328046	−	0.328046i
$868$		0			0
$869$	−13.8564	−	13.8564i	−0.470046	−	0.470046i
$870$		0			0
$871$		−	22.1469i		−	0.750421i
$872$		0			0
$873$			15.4641i			0.523381i
$874$		0			0
$875$	7.17260	+	7.17260i	0.242478	+	0.242478i
$876$		0			0
$877$	8.12436	−	8.12436i	0.274340	−	0.274340i	−0.556505	−	0.830845i	$-0.687858\pi$
$877$	8.12436	−	8.12436i	0.274340	−	0.274340i	0.830845	+	0.556505i	$0.187858\pi$
$878$		0			0
$879$	−46.8449			−1.58004
$880$		0			0
$881$	4.14359			0.139601			0.0698006	−	0.997561i	$-0.477764\pi$
$881$	4.14359			0.139601			0.0698006	+	0.997561i	$0.477764\pi$
$882$		0			0
$883$	−23.2466	+	23.2466i	−0.782310	+	0.782310i	−0.980220	−	0.197910i	$-0.936585\pi$
$883$	−23.2466	+	23.2466i	−0.782310	+	0.782310i	0.197910	+	0.980220i	$0.436585\pi$
$884$		0			0
$885$	50.7846	+	50.7846i	1.70711	+	1.70711i
$886$		0			0
$887$			10.2784i			0.345116i	0.984999	+	0.172558i	$0.0552032\pi$
$887$			10.2784i			0.345116i	−0.984999	+	0.172558i	$0.944797\pi$
$888$		0			0
$889$		−	5.85641i		−	0.196418i
$890$		0			0
$891$	−2.20925	−	2.20925i	−0.0740128	−	0.0740128i
$892$		0			0
$893$	−8.78461	+	8.78461i	−0.293966	+	0.293966i
$894$		0			0
$895$	−20.0764			−0.671080
$896$		0			0
$897$	96.4974			3.22196
$898$		0			0
$899$	9.79796	−	9.79796i	0.326780	−	0.326780i
$900$		0			0
$901$	14.7846	+	14.7846i	0.492547	+	0.492547i
$902$		0			0
$903$		−	23.1822i		−	0.771456i
$904$		0			0
$905$		−	28.1436i		−	0.935525i
$906$		0			0
$907$	−25.3915	−	25.3915i	−0.843110	−	0.843110i	0.146152	−	0.989262i	$-0.453311\pi$
$907$	−25.3915	−	25.3915i	−0.843110	−	0.843110i	−0.989262	+	0.146152i	$0.953311\pi$
$908$		0			0
$909$	−49.9808	+	49.9808i	−1.65776	+	1.65776i
$910$		0			0
$911$	−43.7391			−1.44914			−0.724570	−	0.689201i	$-0.757962\pi$
$911$	−43.7391			−1.44914			−0.724570	+	0.689201i	$0.757962\pi$
$912$		0			0
$913$	−10.3923			−0.343935
$914$		0			0
$915$	−1.79315	+	1.79315i	−0.0592797	+	0.0592797i
$916$		0			0
$917$	12.9282	+	12.9282i	0.426927	+	0.426927i
$918$		0			0
$919$			55.1271i			1.81848i	0.416277	+	0.909238i	$0.363335\pi$
$919$			55.1271i			1.81848i	−0.416277	+	0.909238i	$0.636665\pi$
$920$		0			0
$921$			57.3205i			1.88877i
$922$		0			0
$923$	−24.9754	−	24.9754i	−0.822074	−	0.822074i
$924$		0			0
$925$	0.267949	−	0.267949i	0.00881012	−	0.00881012i
$926$		0			0
$927$	20.6312			0.677617
$928$		0			0
$929$	−3.46410			−0.113653			−0.0568267	−	0.998384i	$-0.518098\pi$
$929$	−3.46410			−0.113653			−0.0568267	+	0.998384i	$0.518098\pi$
$930$		0			0
$931$	5.31508	−	5.31508i	0.174195	−	0.174195i
$932$		0			0
$933$	−19.8564	−	19.8564i	−0.650070	−	0.650070i
$934$		0			0
$935$		−	10.7589i		−	0.351854i
$936$		0			0
$937$			53.1769i			1.73721i	0.495502	+	0.868607i	$0.334984\pi$
$937$			53.1769i			1.73721i	−0.495502	+	0.868607i	$0.665016\pi$
$938$		0			0
$939$	55.6076	+	55.6076i	1.81469	+	1.81469i
$940$		0			0
$941$	29.1962	−	29.1962i	0.951767	−	0.951767i	−0.0471218	−	0.998889i	$-0.515005\pi$
$941$	29.1962	−	29.1962i	0.951767	−	0.951767i	0.998889	+	0.0471218i	$0.0150049\pi$
$942$		0			0
$943$	46.3644			1.50983
$944$		0			0
$945$	−10.1436			−0.329971
$946$		0			0
$947$	−16.0740	+	16.0740i	−0.522334	+	0.522334i	−0.918276	−	0.395941i	$-0.870418\pi$
$947$	−16.0740	+	16.0740i	−0.522334	+	0.522334i	0.395941	+	0.918276i	$0.370418\pi$
$948$		0			0
$949$	35.3205	+	35.3205i	1.14655	+	1.14655i
$950$		0			0
$951$			66.4408i			2.15449i
$952$		0			0
$953$			29.5692i			0.957841i	0.877858	+	0.478920i	$0.158972\pi$
$953$			29.5692i			0.957841i	−0.877858	+	0.478920i	$0.841028\pi$
$954$		0			0
$955$	−29.3939	−	29.3939i	−0.951164	−	0.951164i
$956$		0			0
$957$	−6.00000	+	6.00000i	−0.193952	+	0.193952i
$958$		0			0
$959$	−12.4233			−0.401170
$960$		0			0
$961$	1.00000			0.0322581
$962$		0			0
$963$	39.7367	−	39.7367i	1.28050	−	1.28050i
$964$		0			0
$965$	−4.14359	−	4.14359i	−0.133387	−	0.133387i
$966$		0			0
$967$			7.24693i			0.233046i	0.993188	+	0.116523i	$0.0371748\pi$
$967$			7.24693i			0.233046i	−0.993188	+	0.116523i	$0.962825\pi$
$968$		0			0
$969$		−	12.0000i		−	0.385496i
$970$		0			0
$971$	23.5983	+	23.5983i	0.757306	+	0.757306i	0.975831	−	0.218525i	$-0.0701246\pi$
$971$	23.5983	+	23.5983i	0.757306	+	0.757306i	−0.218525	+	0.975831i	$0.570125\pi$
$972$		0			0
$973$	−0.143594	+	0.143594i	−0.00460340	+	0.00460340i
$974$		0			0
$975$	−14.4195			−0.461795
$976$		0			0
$977$	−51.4641			−1.64648			−0.823241	−	0.567692i	$-0.807837\pi$
$977$	−51.4641			−1.64648			−0.823241	+	0.567692i	$0.807837\pi$
$978$		0			0
$979$	−8.48528	+	8.48528i	−0.271191	+	0.271191i
$980$		0			0
$981$	12.5167	+	12.5167i	0.399626	+	0.399626i
$982$		0			0
$983$		−	57.6038i		−	1.83728i	−0.395100	−	0.918638i	$-0.629290\pi$
$983$		−	57.6038i		−	1.83728i	0.395100	−	0.918638i	$-0.370710\pi$
$984$		0			0
$985$			6.00000i			0.191176i
$986$		0			0
$987$	−19.5959	−	19.5959i	−0.623745	−	0.623745i
$988$		0			0
$989$	−38.7846	+	38.7846i	−1.23328	+	1.23328i
$990$		0			0
$991$	−16.5644			−0.526186			−0.263093	−	0.964770i	$-0.584743\pi$
$991$	−16.5644			−0.526186			−0.263093	+	0.964770i	$0.584743\pi$
$992$		0			0
$993$	−12.5359			−0.397815
$994$		0			0
$995$	32.1480	−	32.1480i	1.01916	−	1.01916i
$996$		0			0
$997$	4.80385	+	4.80385i	0.152139	+	0.152139i	0.779073	−	0.626933i	$-0.215690\pi$
$997$	4.80385	+	4.80385i	0.152139	+	0.152139i	−0.626933	+	0.779073i	$0.715690\pi$
$998$		0			0
$999$		−	1.51575i		−	0.0479562i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.2.e.b.193.1	yes	8
3.2	odd	2		2304.2.k.k.1729.3		8
4.3	odd	2	inner	256.2.e.b.193.4	yes	8
8.3	odd	2		256.2.e.a.193.1	yes	8
8.5	even	2		256.2.e.a.193.4	yes	8
12.11	even	2		2304.2.k.k.1729.4		8
16.3	odd	4	inner	256.2.e.b.65.4	yes	8
16.5	even	4		256.2.e.a.65.4	yes	8
16.11	odd	4		256.2.e.a.65.1	✓	8
16.13	even	4	inner	256.2.e.b.65.1	yes	8
24.5	odd	2		2304.2.k.f.1729.1		8
24.11	even	2		2304.2.k.f.1729.2		8
32.3	odd	8		1024.2.a.j.1.4		4
32.5	even	8		1024.2.b.h.513.1		8
32.11	odd	8		1024.2.b.h.513.2		8
32.13	even	8		1024.2.a.j.1.3		4
32.19	odd	8		1024.2.a.g.1.1		4
32.21	even	8		1024.2.b.h.513.8		8
32.27	odd	8		1024.2.b.h.513.7		8
32.29	even	8		1024.2.a.g.1.2		4
48.5	odd	4		2304.2.k.f.577.2		8
48.11	even	4		2304.2.k.f.577.1		8
48.29	odd	4		2304.2.k.k.577.4		8
48.35	even	4		2304.2.k.k.577.3		8
96.29	odd	8		9216.2.a.bk.1.2		4
96.35	even	8		9216.2.a.bb.1.1		4
96.77	odd	8		9216.2.a.bb.1.4		4
96.83	even	8		9216.2.a.bk.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
256.2.e.a.65.1	✓	8	16.11	odd	4
256.2.e.a.65.4	yes	8	16.5	even	4
256.2.e.a.193.1	yes	8	8.3	odd	2
256.2.e.a.193.4	yes	8	8.5	even	2
256.2.e.b.65.1	yes	8	16.13	even	4	inner
256.2.e.b.65.4	yes	8	16.3	odd	4	inner
256.2.e.b.193.1	yes	8	1.1	even	1	trivial
256.2.e.b.193.4	yes	8	4.3	odd	2	inner
1024.2.a.g.1.1		4	32.19	odd	8
1024.2.a.g.1.2		4	32.29	even	8
1024.2.a.j.1.3		4	32.13	even	8
1024.2.a.j.1.4		4	32.3	odd	8
1024.2.b.h.513.1		8	32.5	even	8
1024.2.b.h.513.2		8	32.11	odd	8
1024.2.b.h.513.7		8	32.27	odd	8
1024.2.b.h.513.8		8	32.21	even	8
2304.2.k.f.577.1		8	48.11	even	4
2304.2.k.f.577.2		8	48.5	odd	4
2304.2.k.f.1729.1		8	24.5	odd	2
2304.2.k.f.1729.2		8	24.11	even	2
2304.2.k.k.577.3		8	48.35	even	4
2304.2.k.k.577.4		8	48.29	odd	4
2304.2.k.k.1729.3		8	3.2	odd	2
2304.2.k.k.1729.4		8	12.11	even	2
9216.2.a.bb.1.1		4	96.35	even	8
9216.2.a.bb.1.4		4	96.77	odd	8
9216.2.a.bk.1.2		4	96.29	odd	8
9216.2.a.bk.1.3		4	96.83	even	8

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:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	256.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.93185 + 1.93185i) q^{3} +(-1.73205 - 1.73205i) q^{5} -1.03528i q^{7} -4.46410i q^{9} +O(q^{10})\)	\(q+(-1.93185 + 1.93185i) q^{3} +(-1.73205 - 1.73205i) q^{5} -1.03528i q^{7} -4.46410i q^{9} +(-0.896575 - 0.896575i) q^{11} +(3.73205 - 3.73205i) q^{13} +6.69213 q^{15} -3.46410 q^{17} +(0.896575 - 0.896575i) q^{19} +(2.00000 + 2.00000i) q^{21} -6.69213i q^{23} +1.00000i q^{25} +(2.82843 + 2.82843i) q^{27} +(-1.73205 + 1.73205i) q^{29} -5.65685 q^{31} +3.46410 q^{33} +(-1.79315 + 1.79315i) q^{35} +(-0.267949 - 0.267949i) q^{37} +14.4195i q^{39} +6.92820i q^{41} +(-5.79555 - 5.79555i) q^{43} +(-7.73205 + 7.73205i) q^{45} -9.79796 q^{47} +5.92820 q^{49} +(6.69213 - 6.69213i) q^{51} +(-4.26795 - 4.26795i) q^{53} +3.10583i q^{55} +3.46410i q^{57} +(7.58871 + 7.58871i) q^{59} +(-0.267949 + 0.267949i) q^{61} -4.62158 q^{63} -12.9282 q^{65} +(2.96713 - 2.96713i) q^{67} +(12.9282 + 12.9282i) q^{69} -6.69213i q^{71} +9.46410i q^{73} +(-1.93185 - 1.93185i) q^{75} +(-0.928203 + 0.928203i) q^{77} +15.4548 q^{79} +2.46410 q^{81} +(5.79555 - 5.79555i) q^{83} +(6.00000 + 6.00000i) q^{85} -6.69213i q^{87} -9.46410i q^{89} +(-3.86370 - 3.86370i) q^{91} +(10.9282 - 10.9282i) q^{93} -3.10583 q^{95} -3.46410 q^{97} +(-4.00240 + 4.00240i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 16 q^{13} + 16 q^{21} - 16 q^{37} - 48 q^{45} - 8 q^{49} - 48 q^{53} - 16 q^{61} - 48 q^{65} + 48 q^{69} + 48 q^{77} - 8 q^{81} + 48 q^{85} + 32 q^{93}+O(q^{100}) \) 8 * q + 16 * q^13 + 16 * q^21 - 16 * q^37 - 48 * q^45 - 8 * q^49 - 48 * q^53 - 16 * q^61 - 48 * q^65 + 48 * q^69 + 48 * q^77 - 8 * q^81 + 48 * q^85 + 32 * q^93

Introduction

L-functions

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