LMFDB - Embedded newform 507.2.k.g.89.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,2,Mod(80,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 1]))

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

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

chi := DirichletCharacter("507.80");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	507.k (of order $12$, degree $4$, not 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:	$4.04841538248$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	8.0.56070144.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 $ x^8 - 4x^7 + 16x^6 - 34x^5 + 63x^4 - 74x^3 + 70x^2 - 38*x + 13
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label			89.1
Root			$0.500000 - 1.56488i$ of defining polynomial
Character	$\chi$	$=$	507.89
Dual form			507.2.k.g.188.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.45466 - 0.389774i) q^{2} +(-0.866025 - 1.50000i) q^{3} +(0.232051 + 0.133975i) q^{4} +(-2.90931 + 2.90931i) q^{5} +(0.675108 + 2.51954i) q^{6} +(1.84443 + 1.84443i) q^{8} +(-1.50000 + 2.59808i) q^{9} +O(q^{10})$	\(q+(-1.45466 - 0.389774i) q^{2} +(-0.866025 - 1.50000i) q^{3} +(0.232051 + 0.133975i) q^{4} +(-2.90931 + 2.90931i) q^{5} +(0.675108 + 2.51954i) q^{6} +(1.84443 + 1.84443i) q^{8} +(-1.50000 + 2.59808i) q^{9} +(5.36603 - 3.09808i) q^{10} +(0.285334 - 1.06488i) q^{11} -0.464102i q^{12} +(6.88351 + 1.84443i) q^{15} +(-2.23205 - 3.86603i) q^{16} +(3.19465 - 3.19465i) q^{18} +(-1.06488 + 0.285334i) q^{20} +(-0.830127 + 1.43782i) q^{22} +(1.16932 - 4.36397i) q^{24} -11.9282i q^{25} +5.19615 q^{27} +(-9.29423 - 5.36603i) q^{30} +(0.389774 + 1.45466i) q^{32} +(-1.84443 + 0.494214i) q^{33} +(-0.696152 + 0.401924i) q^{36} -10.7321 q^{40} +(-9.79282 - 2.62398i) q^{41} +(3.46410 + 2.00000i) q^{43} +(0.208879 - 0.208879i) q^{44} +(-3.19465 - 11.9226i) q^{45} +(-6.59817 - 6.59817i) q^{47} +(-3.86603 + 6.69615i) q^{48} +(6.06218 - 3.50000i) q^{49} +(-4.64930 + 17.3514i) q^{50} +(-7.55862 - 2.02533i) q^{54} +(2.26795 + 3.92820i) q^{55} +(14.8319 - 3.97420i) q^{59} +(1.35022 + 1.35022i) q^{60} +(6.92820 - 12.0000i) q^{61} +6.66025i q^{64} +2.87564 q^{66} +(1.27376 + 4.75374i) q^{71} +(-7.55862 + 2.02533i) q^{72} +(-17.8923 + 10.3301i) q^{75} +10.3923 q^{79} +(17.7412 + 4.75374i) q^{80} +(-4.50000 - 7.79423i) q^{81} +(13.2224 + 7.63397i) q^{82} +(9.29861 - 9.29861i) q^{83} +(-4.25953 - 4.25953i) q^{86} +(2.49038 - 1.43782i) q^{88} +(4.75374 - 17.7412i) q^{89} +18.5885i q^{90} +(7.02628 + 12.1699i) q^{94} +(1.84443 - 1.84443i) q^{96} +(-10.1826 + 2.72842i) q^{98} +(2.33864 + 2.33864i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{4} - 12 q^{9}+O(q^{10}) $ 8 * q - 12 * q^4 - 12 * q^9	$ 8 q - 12 q^{4} - 12 q^{9} + 36 q^{10} - 4 q^{16} + 28 q^{22} - 12 q^{30} + 36 q^{36} - 72 q^{40} - 24 q^{48} + 32 q^{55} + 120 q^{66} - 60 q^{75} - 36 q^{81} - 12 q^{82} - 84 q^{88} - 20 q^{94}+O(q^{100}) $ 8 * q - 12 * q^4 - 12 * q^9 + 36 * q^10 - 4 * q^16 + 28 * q^22 - 12 * q^30 + 36 * q^36 - 72 * q^40 - 24 * q^48 + 32 * q^55 + 120 * q^66 - 60 * q^75 - 36 * q^81 - 12 * q^82 - 84 * q^88 - 20 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$-1$	$e\left(\frac{7}{12}\right)$

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 $

$2$	−1.45466	−	0.389774i	−1.02860	−	0.275612i	−0.295217	−	0.955430i	$-0.595392\pi$
$2$	−1.45466	−	0.389774i	−1.02860	−	0.275612i	−0.733380	+	0.679818i	$0.762059\pi$
$3$	−0.866025	−	1.50000i	−0.500000	−	0.866025i
$3$	−0.866025	−	1.50000i	−0.500000	−	0.866025i
$4$	0.232051	+	0.133975i	0.116025	+	0.0669873i
$5$	−2.90931	+	2.90931i	−1.30108	+	1.30108i	−0.373423	+	0.927661i	$0.621816\pi$
$5$	−2.90931	+	2.90931i	−1.30108	+	1.30108i	−0.927661	+	0.373423i	$0.878184\pi$
$6$	0.675108	+	2.51954i	0.275612	+	1.02860i
$7$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$7$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$8$	1.84443	+	1.84443i	0.652105	+	0.652105i
$9$	−1.50000	+	2.59808i	−0.500000	+	0.866025i
$10$	5.36603	−	3.09808i	1.69689	−	0.979698i
$11$	0.285334	−	1.06488i	0.0860316	−	0.321074i	−0.909476	−	0.415756i	$-0.863517\pi$
$11$	0.285334	−	1.06488i	0.0860316	−	0.321074i	0.995508	+	0.0946823i	$0.0301835\pi$
$12$		−	0.464102i		−	0.133975i
$13$		0			0
$13$		0			0
$14$		0			0
$15$	6.88351	+	1.84443i	1.77731	+	0.476230i
$16$	−2.23205	−	3.86603i	−0.558013	−	0.966506i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$	3.19465	−	3.19465i	0.752986	−	0.752986i
$19$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$19$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$20$	−1.06488	+	0.285334i	−0.238115	+	0.0638027i
$21$		0			0
$22$	−0.830127	+	1.43782i	−0.176984	+	0.306545i
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$	1.16932	−	4.36397i	0.238687	−	0.890792i
$25$		−	11.9282i		−	2.38564i
$26$		0			0
$27$	5.19615			1.00000
$28$		0			0
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$	−9.29423	−	5.36603i	−1.69689	−	0.979698i
$31$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$31$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$32$	0.389774	+	1.45466i	0.0689030	+	0.257149i
$33$	−1.84443	+	0.494214i	−0.321074	+	0.0860316i
$34$		0			0
$35$		0			0
$36$	−0.696152	+	0.401924i	−0.116025	+	0.0669873i
$37$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$37$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$38$		0			0
$39$		0			0
$40$	−10.7321			−1.69689
$41$	−9.79282	−	2.62398i	−1.52938	−	0.409797i	−0.606564	−	0.795034i	$-0.707453\pi$
$41$	−9.79282	−	2.62398i	−1.52938	−	0.409797i	−0.922817	+	0.385238i	$0.874119\pi$
$42$		0			0
$43$	3.46410	+	2.00000i	0.528271	+	0.304997i	0.740312	−	0.672264i	$-0.234678\pi$
$43$	3.46410	+	2.00000i	0.528271	+	0.304997i	−0.212041	+	0.977261i	$0.568011\pi$
$44$	0.208879	−	0.208879i	0.0314897	−	0.0314897i
$45$	−3.19465	−	11.9226i	−0.476230	−	1.77731i
$46$		0			0
$47$	−6.59817	−	6.59817i	−0.962443	−	0.962443i	0.0368772	−	0.999320i	$-0.488259\pi$
$47$	−6.59817	−	6.59817i	−0.962443	−	0.962443i	−0.999320	+	0.0368772i	$0.988259\pi$
$48$	−3.86603	+	6.69615i	−0.558013	+	0.966506i
$49$	6.06218	−	3.50000i	0.866025	−	0.500000i
$50$	−4.64930	+	17.3514i	−0.657511	+	2.45386i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−7.55862	−	2.02533i	−1.02860	−	0.275612i
$55$	2.26795	+	3.92820i	0.305810	+	0.529679i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	14.8319	−	3.97420i	1.93095	−	0.517396i	0.957562	−	0.288226i	$-0.0930655\pi$
$59$	14.8319	−	3.97420i	1.93095	−	0.517396i	0.973386	−	0.229170i	$-0.0736011\pi$
$60$	1.35022	+	1.35022i	0.174312	+	0.174312i
$61$	6.92820	−	12.0000i	0.887066	−	1.53644i	0.0437377	−	0.999043i	$-0.486073\pi$
$61$	6.92820	−	12.0000i	0.887066	−	1.53644i	0.843328	−	0.537400i	$-0.180593\pi$
$62$		0			0
$63$		0			0
$64$			6.66025i			0.832532i
$65$		0			0
$66$	2.87564			0.353967
$67$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$67$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.27376	+	4.75374i	0.151168	+	0.564166i	0.999403	+	0.0345462i	$0.0109986\pi$
$71$	1.27376	+	4.75374i	0.151168	+	0.564166i	−0.848235	+	0.529619i	$0.822335\pi$
$72$	−7.55862	+	2.02533i	−0.890792	+	0.238687i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		0			0
$75$	−17.8923	+	10.3301i	−2.06603	+	1.19282i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	10.3923			1.16923			0.584613	−	0.811312i	$-0.301246\pi$
$79$	10.3923			1.16923			0.584613	+	0.811312i	$0.301246\pi$
$80$	17.7412	+	4.75374i	1.98353	+	0.531485i
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$	13.2224	+	7.63397i	1.46017	+	0.843031i
$83$	9.29861	−	9.29861i	1.02065	−	1.02065i	0.0208726	−	0.999782i	$-0.493356\pi$
$83$	9.29861	−	9.29861i	1.02065	−	1.02065i	0.999782	−	0.0208726i	$-0.00664445\pi$
$84$		0			0
$85$		0			0
$86$	−4.25953	−	4.25953i	−0.459317	−	0.459317i
$87$		0			0
$88$	2.49038	−	1.43782i	0.265476	−	0.153272i
$89$	4.75374	−	17.7412i	0.503896	−	1.88056i	0.0308556	−	0.999524i	$-0.490177\pi$
$89$	4.75374	−	17.7412i	0.503896	−	1.88056i	0.473040	−	0.881041i	$-0.343157\pi$
$90$			18.5885i			1.95940i
$91$		0			0
$92$		0			0
$93$		0			0
$94$	7.02628	+	12.1699i	0.724705	+	1.25523i
$95$		0			0
$96$	1.84443	−	1.84443i	0.188246	−	0.188246i
$97$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$97$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$98$	−10.1826	+	2.72842i	−1.02860	+	0.275612i
$99$	2.33864	+	2.33864i	0.235043	+	0.235043i
$100$	1.59808	−	2.76795i	0.159808	−	0.276795i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$			16.0000i			1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$			16.0000i			1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$	1.20577	+	0.696152i	0.116025	+	0.0669873i
$109$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$109$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$110$	−1.76798	−	6.59817i	−0.168570	−	0.629111i
$111$		0			0
$112$		0			0
$113$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$113$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−23.1244			−2.12877
$119$		0			0
$120$	9.29423	+	16.0981i	0.848443	+	1.46955i
$121$	8.47372	+	4.89230i	0.770338	+	0.444755i
$122$	−14.7554	+	14.7554i	−1.33590	+	1.33590i
$123$	4.54486	+	16.9617i	0.409797	+	1.52938i
$124$		0			0
$125$	20.1563	+	20.1563i	1.80284	+	1.80284i
$126$		0			0
$127$	−15.0000	+	8.66025i	−1.33103	+	0.768473i	−0.985458	−	0.169917i	$-0.945650\pi$
$127$	−15.0000	+	8.66025i	−1.33103	+	0.768473i	−0.345576	+	0.938391i	$0.612317\pi$
$128$	3.37554	−	12.5977i	0.298359	−	1.11349i
$129$		−	6.92820i		−	0.609994i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$	−0.494214	−	0.132424i	−0.0430158	−	0.0115260i
$133$		0			0
$134$		0			0
$135$	−15.1172	+	15.1172i	−1.30108	+	1.30108i
$136$		0			0
$137$	−14.0524	+	3.76532i	−1.20057	+	0.321693i	−0.803057	−	0.595902i	$-0.796795\pi$
$137$	−14.0524	+	3.76532i	−1.20057	+	0.321693i	−0.397516	+	0.917595i	$0.630128\pi$
$138$		0			0
$139$	10.0000	−	17.3205i	0.848189	−	1.46911i	−0.0346338	−	0.999400i	$-0.511026\pi$
$139$	10.0000	−	17.3205i	0.848189	−	1.46911i	0.882823	−	0.469706i	$-0.155640\pi$
$140$		0			0
$141$	−4.18307	+	15.6114i	−0.352278	+	1.31472i
$142$		−	7.41154i		−	0.621963i
$143$		0			0
$144$	13.3923			1.11603
$145$		0			0
$146$		0			0
$147$	−10.5000	−	6.06218i	−0.866025	−	0.500000i
$148$		0			0
$149$	0.0764551	+	0.285334i	0.00626345	+	0.0233755i	0.968987	−	0.247112i	$-0.0794817\pi$
$149$	0.0764551	+	0.285334i	0.00626345	+	0.0233755i	−0.962723	+	0.270488i	$0.912815\pi$
$150$	30.0536	−	8.05283i	2.45386	−	0.657511i
$151$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$151$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$	−15.1172	−	4.05065i	−1.20266	−	0.322252i
$159$		0			0
$160$	−5.36603	−	3.09808i	−0.424222	−	0.244924i
$161$		0			0
$162$	3.50797	+	13.0919i	0.275612	+	1.02860i
$163$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$163$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$164$	−1.92089	−	1.92089i	−0.149996	−	0.149996i
$165$	3.92820	−	6.80385i	0.305810	−	0.529679i
$166$	−17.1506	+	9.90192i	−1.33115	+	0.768538i
$167$	−2.98577	+	11.1430i	−0.231046	+	0.862274i	0.748846	+	0.662744i	$0.230608\pi$
$167$	−2.98577	+	11.1430i	−0.231046	+	0.862274i	−0.979892	+	0.199530i	$0.936058\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$		0			0
$172$	0.535898	+	0.928203i	0.0408619	+	0.0707748i
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$		0			0
$176$	−4.75374	+	1.27376i	−0.358327	+	0.0960134i
$177$	−18.8061	−	18.8061i	−1.41355	−	1.41355i
$178$	−13.8301	+	23.9545i	−1.03661	+	1.79546i
$179$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$179$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$180$	0.856003	−	3.19465i	0.0638027	−	0.238115i
$181$		−	10.0000i		−	0.743294i	−0.928374	−	0.371647i	$-0.878793\pi$
$181$		−	10.0000i		−	0.743294i	0.928374	−	0.371647i	$-0.121207\pi$
$182$		0			0
$183$	−24.0000			−1.77413
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−0.647124	−	2.41510i	−0.0471964	−	0.176139i
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	9.99038	−	5.76795i	0.720994	−	0.416266i
$193$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$193$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$194$		0			0
$195$		0			0
$196$	1.87564			0.133975
$197$	23.5598	+	6.31284i	1.67857	+	0.449771i	0.967401	−	0.253251i	$-0.0814997\pi$
$197$	23.5598	+	6.31284i	1.67857	+	0.449771i	0.711168	+	0.703022i	$0.248166\pi$
$198$	−2.49038	−	4.31347i	−0.176984	−	0.306545i
$199$	−21.0000	−	12.1244i	−1.48865	−	0.859473i	−0.488735	−	0.872433i	$-0.662541\pi$
$199$	−21.0000	−	12.1244i	−1.48865	−	0.859473i	−0.999916	+	0.0129598i	$0.995875\pi$
$200$	22.0007	−	22.0007i	1.55569	−	1.55569i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	36.1244	−	20.8564i	2.52303	−	1.45667i
$206$	6.23638	−	23.2745i	0.434509	−	1.62161i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−1.73205	−	3.00000i	−0.119239	−	0.206529i	0.800227	−	0.599697i	$-0.204712\pi$
$211$	−1.73205	−	3.00000i	−0.119239	−	0.206529i	−0.919466	+	0.393169i	$0.871379\pi$
$212$		0			0
$213$	6.02751	−	6.02751i	0.412998	−	0.412998i
$214$		0			0
$215$	−15.8968	+	4.25953i	−1.08415	+	0.290498i
$216$	9.58394	+	9.58394i	0.652105	+	0.652105i
$217$		0			0
$218$		0			0
$219$		0			0
$220$			1.21539i			0.0819416i
$221$		0			0
$222$		0			0
$223$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$223$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$224$		0			0
$225$	30.9904	+	17.8923i	2.06603	+	1.19282i
$226$		0			0
$227$	−7.66306	−	28.5989i	−0.508615	−	1.89818i	−0.433874	−	0.900974i	$-0.642854\pi$
$227$	−7.66306	−	28.5989i	−0.508615	−	1.89818i	−0.0747413	−	0.997203i	$-0.523813\pi$
$228$		0			0
$229$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$229$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$	38.3923			2.50444
$236$	3.97420	+	1.06488i	0.258698	+	0.0693179i
$237$	−9.00000	−	15.5885i	−0.584613	−	1.01258i
$238$		0			0
$239$	18.2354	−	18.2354i	1.17955	−	1.17955i	0.199693	−	0.979858i	$-0.436005\pi$
$239$	18.2354	−	18.2354i	1.17955	−	1.17955i	0.979858	−	0.199693i	$-0.0639945\pi$
$240$	−8.23373	−	30.7287i	−0.531485	−	1.98353i
$241$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$241$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$242$	−10.4195	−	10.4195i	−0.669788	−	0.669788i
$243$	−7.79423	+	13.5000i	−0.500000	+	0.866025i
$244$	3.21539	−	1.85641i	0.205844	−	0.118844i
$245$	−7.45418	+	27.8194i	−0.476230	+	1.77731i
$246$		−	26.4449i		−	1.68606i
$247$		0			0
$248$		0			0
$249$	−22.0007	−	5.89508i	−1.39424	−	0.373586i
$250$	−21.4641	−	37.1769i	−1.35751	−	2.35127i
$251$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$251$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$252$		0			0
$253$		0			0
$254$	25.1954	−	6.75108i	1.58090	−	0.423601i
$255$		0			0
$256$	−3.16025	+	5.47372i	−0.197516	+	0.342108i
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$	−2.70043	+	10.0782i	−0.168122	+	0.627439i
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$	−4.31347	−	2.49038i	−0.265476	−	0.153272i
$265$		0			0
$266$		0			0
$267$	−30.7287	+	8.23373i	−1.88056	+	0.503896i
$268$		0			0
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$	27.8827	−	16.0981i	1.69689	−	0.979698i
$271$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$271$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$272$		0			0
$273$		0			0
$274$	21.9090			1.32357
$275$	−12.7021	−	3.40353i	−0.765968	−	0.205240i
$276$		0			0
$277$	−19.0526	−	11.0000i	−1.14476	−	0.660926i	−0.197153	−	0.980373i	$-0.563170\pi$
$277$	−19.0526	−	11.0000i	−1.14476	−	0.660926i	−0.947604	+	0.319447i	$0.896503\pi$
$278$	−21.2976	+	21.2976i	−1.27735	+	1.27735i
$279$		0			0
$280$		0			0
$281$	−9.86928	−	9.86928i	−0.588752	−	0.588752i	0.348542	−	0.937293i	$-0.386677\pi$
$281$	−9.86928	−	9.86928i	−0.588752	−	0.588752i	−0.937293	+	0.348542i	$0.886677\pi$
$282$	12.1699	−	21.0788i	0.724705	−	1.25523i
$283$	−15.0000	+	8.66025i	−0.891657	+	0.514799i	−0.874484	−	0.485054i	$-0.838800\pi$
$283$	−15.0000	+	8.66025i	−0.891657	+	0.514799i	−0.0171732	+	0.999853i	$0.505467\pi$
$284$	−0.341303	+	1.27376i	−0.0202526	+	0.0755839i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−4.36397	−	1.16932i	−0.257149	−	0.0689030i
$289$	8.50000	+	14.7224i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	31.5082	−	8.44260i	1.84073	−	0.493222i	0.841814	−	0.539767i	$-0.181488\pi$
$293$	31.5082	−	8.44260i	1.84073	−	0.493222i	0.998916	+	0.0465452i	$0.0148212\pi$
$294$	12.9110	+	12.9110i	0.752986	+	0.752986i
$295$	−31.5885	+	54.7128i	−1.83915	+	3.18550i
$296$		0			0
$297$	1.48264	−	5.53329i	0.0860316	−	0.321074i
$298$		−	0.444864i		−	0.0257703i
$299$		0			0
$300$	−5.53590			−0.319615
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	14.7554	+	55.0681i	0.844894	+	3.15319i
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$	24.0000	−	13.8564i	1.36531	−	0.788263i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−34.6410			−1.95803			−0.979013	−	0.203798i	$-0.934671\pi$
$313$	−34.6410			−1.95803			−0.979013	+	0.203798i	$0.934671\pi$
$314$	−2.90931	−	0.779548i	−0.164182	−	0.0439924i
$315$		0			0
$316$	2.41154	+	1.39230i	0.135660	+	0.0783233i
$317$	−11.8461	+	11.8461i	−0.665345	+	0.665345i	−0.956635	−	0.291290i	$-0.905916\pi$
$317$	−11.8461	+	11.8461i	−0.665345	+	0.665345i	0.291290	+	0.956635i	$0.405916\pi$
$318$		0			0
$319$		0			0
$320$	−19.3768	−	19.3768i	−1.08319	−	1.08319i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		−	2.41154i		−	0.133975i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	−13.2224	−	22.9019i	−0.730087	−	1.26455i
$329$		0			0
$330$	−8.36615	+	8.36615i	−0.460541	+	0.460541i
$331$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$331$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$332$	3.40353	−	0.911972i	0.186793	−	0.0500510i
$333$		0			0
$334$	8.68653	−	15.0455i	0.475306	−	0.823254i
$335$		0			0
$336$		0			0
$337$		−	6.92820i		−	0.377403i	−0.982034	−	0.188702i	$-0.939572\pi$
$337$		−	6.92820i		−	0.377403i	0.982034	−	0.188702i	$-0.0604279\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	2.70043	+	10.0782i	0.145598	+	0.543378i
$345$		0			0
$346$		0			0
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$349$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$350$		0			0
$351$		0			0
$352$	1.66025			0.0884918
$353$	2.41510	+	0.647124i	0.128543	+	0.0344429i	0.322517	−	0.946564i	$-0.395471\pi$
$353$	2.41510	+	0.647124i	0.128543	+	0.0344429i	−0.193974	+	0.981007i	$0.562138\pi$
$354$	20.0263	+	34.6865i	1.06438	+	1.84357i
$355$	−17.5359	−	10.1244i	−0.930709	−	0.537345i
$356$	3.47998	−	3.47998i	0.184439	−	0.184439i
$357$		0			0
$358$		0			0
$359$	26.7545	+	26.7545i	1.41205	+	1.41205i	0.745143	+	0.666905i	$0.232381\pi$
$359$	26.7545	+	26.7545i	1.41205	+	1.41205i	0.666905	+	0.745143i	$0.267619\pi$
$360$	16.0981	−	27.8827i	0.848443	−	1.46955i
$361$	−16.4545	+	9.50000i	−0.866025	+	0.500000i
$362$	−3.89774	+	14.5466i	−0.204861	+	0.764550i
$363$		−	16.9474i		−	0.889510i
$364$		0			0
$365$		0			0
$366$	34.9118	+	9.35458i	1.82487	+	0.488972i
$367$	4.00000	+	6.92820i	0.208798	+	0.361649i	0.951336	−	0.308155i	$-0.0997115\pi$
$367$	4.00000	+	6.92820i	0.208798	+	0.361649i	−0.742538	+	0.669804i	$0.766378\pi$
$368$		0			0
$369$	21.5065	−	21.5065i	1.11959	−	1.11959i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	6.92820	−	12.0000i	0.358729	−	0.621336i	−0.629020	−	0.777389i	$-0.716544\pi$
$373$	6.92820	−	12.0000i	0.358729	−	0.621336i	0.987749	+	0.156053i	$0.0498770\pi$
$374$		0			0
$375$	12.7786	−	47.6903i	0.659884	−	2.46272i
$376$		−	24.3397i		−	1.25523i
$377$		0			0
$378$		0			0
$379$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$379$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$380$		0			0
$381$	25.9808	+	15.0000i	1.33103	+	0.768473i
$382$		0			0
$383$	−4.39195	−	16.3910i	−0.224418	−	0.837541i	−0.982637	−	0.185540i	$-0.940597\pi$
$383$	−4.39195	−	16.3910i	−0.224418	−	0.837541i	0.758218	−	0.652001i	$-0.226070\pi$
$384$	−21.8198	+	5.84661i	−1.11349	+	0.298359i
$385$		0			0
$386$		0			0
$387$	−10.3923	+	6.00000i	−0.528271	+	0.304997i
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$	17.6368	+	4.72576i	0.890792	+	0.238687i
$393$		0			0
$394$	−31.8109	−	18.3660i	−1.60261	−	0.925267i
$395$	−30.2345	+	30.2345i	−1.52126	+	1.52126i
$396$	0.229365	+	0.856003i	0.0115260	+	0.0430158i
$397$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$397$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$398$	25.8220	+	25.8220i	1.29434	+	1.29434i
$399$		0			0
$400$	−46.1147	+	26.6244i	−2.30574	+	1.33122i
$401$	8.02485	−	29.9491i	0.400742	−	1.49559i	−0.411035	−	0.911620i	$-0.634832\pi$
$401$	8.02485	−	29.9491i	0.400742	−	1.49559i	0.811776	−	0.583969i	$-0.198501\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	35.7678	+	9.58394i	1.77731	+	0.476230i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$409$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$410$	−60.6778	+	16.2586i	−2.99666	+	0.802954i
$411$	17.8177	+	17.8177i	0.878881	+	0.878881i
$412$	−2.14359	+	3.71281i	−0.105607	+	0.182917i
$413$		0			0
$414$		0			0
$415$			54.1051i			2.65592i
$416$		0			0
$417$	−34.6410			−1.69638
$418$		0			0
$419$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$419$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$420$		0			0
$421$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$421$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$422$	1.35022	+	5.03908i	0.0657275	+	0.245298i
$423$	27.0398	−	7.24530i	1.31472	−	0.352278i
$424$		0			0
$425$		0			0
$426$	−11.1173	+	6.41858i	−0.538636	+	0.310981i
$427$		0			0
$428$		0			0
$429$		0			0
$430$	24.7846			1.19522
$431$	−38.6771	−	10.3635i	−1.86301	−	0.499192i	−0.863027	−	0.505158i	$-0.831434\pi$
$431$	−38.6771	−	10.3635i	−1.86301	−	0.499192i	−0.999982	+	0.00596647i	$0.998101\pi$
$432$	−11.5981	−	20.0885i	−0.558013	−	0.966506i
$433$	18.0000	+	10.3923i	0.865025	+	0.499422i	0.865692	−	0.500577i	$-0.166879\pi$
$433$	18.0000	+	10.3923i	0.865025	+	0.499422i	−0.000666943		1.00000i	$0.500212\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	34.6410	−	20.0000i	1.65333	−	0.954548i	0.677634	−	0.735399i	$-0.263005\pi$
$439$	34.6410	−	20.0000i	1.65333	−	0.954548i	0.975691	−	0.219149i	$-0.0703280\pi$
$440$	−3.06222	+	11.4284i	−0.145986	+	0.544826i
$441$			21.0000i			1.00000i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	37.7846	+	65.4449i	1.79116	+	3.10238i
$446$		0			0
$447$	0.361790	−	0.361790i	0.0171121	−	0.0171121i
$448$		0			0
$449$	10.3635	−	2.77689i	0.489083	−	0.131049i	−0.00584565	−	0.999983i	$-0.501861\pi$
$449$	10.3635	−	2.77689i	0.489083	−	0.131049i	0.494929	+	0.868933i	$0.335194\pi$
$450$	−38.1064	−	38.1064i	−1.79635	−	1.79635i
$451$	−5.58846	+	9.67949i	−0.263150	+	0.455789i
$452$		0			0
$453$		0			0
$454$			44.5885i			2.09264i
$455$		0			0
$456$		0			0
$457$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$457$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6.46575	−	24.1305i	−0.301140	−	1.12387i	−0.936217	−	0.351421i	$-0.885698\pi$
$461$	−6.46575	−	24.1305i	−0.301140	−	1.12387i	0.635077	−	0.772448i	$-0.280968\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$		0			0
$470$	−55.8476	−	14.9643i	−2.57606	−	0.690253i
$471$	−1.73205	−	3.00000i	−0.0798087	−	0.138233i
$472$	34.6865	+	20.0263i	1.59658	+	0.921784i
$473$	3.11819	−	3.11819i	0.143375	−	0.143375i
$474$	7.01593	+	26.1838i	0.322252	+	1.20266i
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−33.6340	+	19.4186i	−1.53838	+	0.888185i
$479$	−8.65148	+	32.2878i	−0.395296	+	1.47527i	0.425978	+	0.904733i	$0.359930\pi$
$479$	−8.65148	+	32.2878i	−0.395296	+	1.47527i	−0.821275	+	0.570533i	$0.806737\pi$
$480$			10.7321i			0.489849i
$481$		0			0
$482$		0			0
$483$		0			0
$484$	1.31089	+	2.27053i	0.0595859	+	0.103206i
$485$		0			0
$486$	16.5999	−	16.5999i	0.752986	−	0.752986i
$487$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$487$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$488$	34.9118	−	9.35458i	1.58038	−	0.423462i
$489$		0			0
$490$	21.6865	−	37.5622i	0.979698	−	1.69689i
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$	−1.21779	+	4.54486i	−0.0549023	+	0.204898i
$493$		0			0
$494$		0			0
$495$	−13.6077			−0.611620
$496$		0			0
$497$		0			0
$498$	29.7058	+	17.1506i	1.33115	+	0.768538i
$499$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$499$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$500$	1.97685	+	7.37772i	0.0884076	+	0.329942i
$501$	19.3003	−	5.17150i	0.862274	−	0.231046i
$502$		0			0
$503$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$503$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−4.64102			−0.205912
$509$	11.3519	+	3.04174i	0.503165	+	0.134823i	0.501468	−	0.865176i	$-0.332793\pi$
$509$	11.3519	+	3.04174i	0.503165	+	0.134823i	0.00169644	+	0.999999i	$0.499460\pi$
$510$		0			0
$511$		0			0
$512$	−11.7137	+	11.7137i	−0.517678	+	0.517678i
$513$		0			0
$514$		0			0
$515$	−46.5490	−	46.5490i	−2.05119	−	2.05119i
$516$	0.928203	−	1.60770i	0.0408619	−	0.0707748i
$517$	−8.90897	+	5.14359i	−0.391816	+	0.226215i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−22.0000	−	38.1051i	−0.961993	−	1.66622i	−0.717486	−	0.696573i	$-0.754707\pi$
$523$	−22.0000	−	38.1051i	−0.961993	−	1.66622i	−0.244507	−	0.969648i	$-0.578626\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	6.02751	+	6.02751i	0.262313	+	0.262313i
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$	−11.9226	+	44.4957i	−0.517396	+	1.93095i
$532$		0			0
$533$		0			0
$534$	47.9090			2.07322
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.99734	−	7.45418i	−0.0860316	−	0.321074i
$540$	−5.53329	+	1.48264i	−0.238115	+	0.0638027i
$541$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$541$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$542$		0			0
$543$	−15.0000	+	8.66025i	−0.643712	+	0.371647i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	28.0000			1.19719			0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000			1.19719			0.598597	+	0.801050i	$0.295725\pi$
$548$	−3.76532	−	1.00891i	−0.160846	−	0.0430986i
$549$	20.7846	+	36.0000i	0.887066	+	1.53644i
$550$	17.1506	+	9.90192i	0.731306	+	0.422219i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	23.4274	+	23.4274i	0.995335	+	0.995335i
$555$		0			0
$556$	4.64102	−	2.67949i	0.196823	−	0.113636i
$557$	−0.911972	+	3.40353i	−0.0386415	+	0.144212i	−0.982551	−	0.185991i	$-0.940451\pi$
$557$	−0.911972	+	3.40353i	−0.0386415	+	0.144212i	0.943910	+	0.330203i	$0.107117\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	10.5096	+	18.2032i	0.443322	+	0.767855i
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$	−3.06222	+	3.06222i	−0.128943	+	0.128943i
$565$		0			0
$566$	25.1954	−	6.75108i	1.05904	−	0.283769i
$567$		0			0
$568$	−6.41858	+	11.1173i	−0.269318	+	0.466472i
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$			38.1051i			1.59465i	0.603550	+	0.797325i	$0.293752\pi$
$571$			38.1051i			1.59465i	−0.603550	+	0.797325i	$0.706248\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−17.3038	−	9.99038i	−0.720994	−	0.416266i
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	−6.62616	−	24.7292i	−0.275612	−	1.02860i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−49.1244			−2.02931
$587$	40.2362	+	10.7812i	1.66072	+	0.444990i	0.962587	−	0.270974i	$-0.0873457\pi$
$587$	40.2362	+	10.7812i	1.66072	+	0.444990i	0.698138	+	0.715964i	$0.254012\pi$
$588$	−1.62436	−	2.81347i	−0.0669873	−	0.116025i
$589$		0			0
$590$	67.2760	−	67.2760i	2.76971	−	2.76971i
$591$	−10.9342	−	40.8068i	−0.449771	−	1.67857i
$592$		0			0
$593$	−27.7429	−	27.7429i	−1.13926	−	1.13926i	−0.988582	−	0.150683i	$-0.951853\pi$
$593$	−27.7429	−	27.7429i	−1.13926	−	1.13926i	−0.150683	−	0.988582i	$-0.548147\pi$
$594$	−4.31347	+	7.47114i	−0.176984	+	0.306545i
$595$		0			0
$596$	−0.0204861	+	0.0764551i	−0.000839143	+	0.00313172i
$597$			42.0000i			1.71895i
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$	−52.0543	−	13.9479i	−2.12511	−	0.569421i
$601$	−24.2487	−	42.0000i	−0.989126	−	1.71322i	−0.621932	−	0.783071i	$-0.713652\pi$
$601$	−24.2487	−	42.0000i	−0.989126	−	1.71322i	−0.367193	−	0.930145i	$-0.619681\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−38.8860	+	10.4195i	−1.58094	+	0.423611i
$606$		0			0
$607$	−16.0000	+	27.7128i	−0.649420	+	1.12483i	0.333842	+	0.942629i	$0.391655\pi$
$607$	−16.0000	+	27.7128i	−0.649420	+	1.12483i	−0.983262	+	0.182199i	$0.941678\pi$
$608$		0			0
$609$		0			0
$610$		−	85.8564i		−	3.47622i
$611$		0			0
$612$		0			0
$613$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$613$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$614$		0			0
$615$	−62.5692	−	36.1244i	−2.52303	−	1.45667i
$616$		0			0
$617$	−12.1315	−	45.2752i	−0.488394	−	1.82271i	−0.564263	−	0.825595i	$-0.690840\pi$
$617$	−12.1315	−	45.2752i	−0.488394	−	1.82271i	0.0758689	−	0.997118i	$-0.475827\pi$
$618$	−40.3126	+	10.8017i	−1.62161	+	0.434509i
$619$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$619$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−57.6410			−2.30564
$626$	50.3908	+	13.5022i	2.01402	+	0.539655i
$627$		0			0
$628$	0.464102	+	0.267949i	0.0185197	+	0.0106923i
$629$		0			0
$630$		0			0
$631$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$631$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$632$	19.1679	+	19.1679i	0.762457	+	0.762457i
$633$	−3.00000	+	5.19615i	−0.119239	+	0.206529i
$634$	21.8494	−	12.6147i	0.867749	−	0.500995i
$635$	18.4443	−	68.8351i	0.731940	−	2.73164i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−14.2612	−	3.82129i	−0.564166	−	0.151168i
$640$	26.8301	+	46.4711i	1.06055	+	1.83693i
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$643$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$644$		0			0
$645$	20.1563	+	20.1563i	0.793654	+	0.793654i
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$	6.07598	−	22.6758i	0.238687	−	0.890792i
$649$		−	16.9282i		−	0.664490i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$	11.7137	+	43.7161i	0.457343	+	1.70683i
$657$		0			0
$658$		0			0
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$	1.82309	−	1.05256i	0.0709635	−	0.0409708i
$661$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$661$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$662$		0			0
$663$		0			0
$664$	34.3013			1.33115
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−2.18573	+	2.18573i	−0.0845686	+	0.0845686i
$669$		0			0
$670$		0			0
$671$	−10.8017	−	10.8017i	−0.416996	−	0.416996i
$672$		0			0
$673$	12.1244	−	7.00000i	0.467360	−	0.269830i	−0.247774	−	0.968818i	$-0.579699\pi$
$673$	12.1244	−	7.00000i	0.467360	−	0.269830i	0.715134	+	0.698988i	$0.246366\pi$
$674$	−2.70043	+	10.0782i	−0.104017	+	0.388196i
$675$		−	61.9808i		−	2.38564i
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−36.2620	+	36.2620i	−1.38956	+	1.38956i
$682$		0			0
$683$	−6.31284	+	1.69152i	−0.241554	+	0.0647242i	−0.377565	−	0.925983i	$-0.623238\pi$
$683$	−6.31284	+	1.69152i	−0.241554	+	0.0647242i	0.136011	+	0.990707i	$0.456572\pi$
$684$		0			0
$685$	29.9282	−	51.8372i	1.14350	−	1.98060i
$686$		0			0
$687$		0			0
$688$		−	17.8564i		−	0.680769i
$689$		0			0
$690$		0			0
$691$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$691$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	21.2976	+	79.4839i	0.807866	+	3.01500i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	7.09239	+	1.90040i	0.267304	+	0.0716240i
$705$	−33.2487	−	57.5885i	−1.25222	−	2.16891i
$706$	−3.26091	−	1.88269i	−0.122726	−	0.0708558i
$707$		0			0
$708$	−1.84443	−	6.88351i	−0.0693179	−	0.258698i
$709$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$709$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$710$	21.5625	+	21.5625i	0.809226	+	0.809226i
$711$	−15.5885	+	27.0000i	−0.584613	+	1.01258i
$712$	41.4904	−	23.9545i	1.55492	−	0.897732i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−43.1455	−	11.5608i	−1.61130	−	0.431746i
$718$	−28.4904	−	49.3468i	−1.06325	−	1.84161i
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$	−38.9624	+	38.9624i	−1.45204	+	1.45204i
$721$		0			0
$722$	27.6385	−	7.40571i	1.02860	−	0.275612i
$723$		0			0
$724$	1.33975	−	2.32051i	0.0497913	−	0.0862410i
$725$		0			0
$726$	−6.60567	+	24.6527i	−0.245159	+	0.914948i
$727$		−	51.9615i		−	1.92715i	−0.267445	−	0.963573i	$-0.586179\pi$
$727$		−	51.9615i		−	1.92715i	0.267445	−	0.963573i	$-0.413821\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$	−5.56922	−	3.21539i	−0.205844	−	0.118844i
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$	−3.11819	−	11.6373i	−0.115095	−	0.429539i
$735$	48.1846	−	12.9110i	1.77731	−	0.476230i
$736$		0			0
$737$		0			0
$738$	−39.6673	+	22.9019i	−1.46017	+	0.843031i
$739$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$739$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	52.4441	+	14.0524i	1.92399	+	0.515531i	0.985324	+	0.170695i	$0.0546013\pi$
$743$	52.4441	+	14.0524i	1.92399	+	0.515531i	0.938663	+	0.344836i	$0.112065\pi$
$744$		0			0
$745$	−1.05256	−	0.607695i	−0.0385628	−	0.0222642i
$746$	−14.7554	+	14.7554i	−0.540235	+	0.540235i
$747$	10.2106	+	38.1064i	0.373586	+	1.39424i
$748$		0			0
$749$		0			0
$750$	−37.1769	+	64.3923i	−1.35751	+	2.35127i
$751$	34.6410	−	20.0000i	1.26407	−	0.729810i	0.290209	−	0.956963i	$-0.406275\pi$
$751$	34.6410	−	20.0000i	1.26407	−	0.729810i	0.973859	+	0.227153i	$0.0729417\pi$
$752$	−10.7812	+	40.2362i	−0.393152	+	1.46726i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	20.7846	+	36.0000i	0.755429	+	1.30844i	0.945161	+	0.326606i	$0.105905\pi$
$757$	20.7846	+	36.0000i	0.755429	+	1.30844i	−0.189731	+	0.981836i	$0.560762\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−26.2603	+	7.03642i	−0.951934	+	0.255070i	−0.701183	−	0.712982i	$-0.747344\pi$
$761$	−26.2603	+	7.03642i	−0.951934	+	0.255070i	−0.250751	+	0.968052i	$0.580678\pi$
$762$	−31.9465	−	31.9465i	−1.15730	−	1.15730i
$763$		0			0
$764$		0			0
$765$		0			0
$766$			25.5551i			0.923345i
$767$		0			0
$768$	10.9474			0.395032
$769$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$769$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	3.34756	+	12.4933i	0.120403	+	0.449351i	0.999634	−	0.0270446i	$-0.00860962\pi$
$773$	3.34756	+	12.4933i	0.120403	+	0.449351i	−0.879231	+	0.476396i	$0.841943\pi$
$774$	17.4559	−	4.67729i	0.627439	−	0.168122i
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	5.42563			0.194144
$782$		0			0
$783$		0			0
$784$	−27.0622	−	15.6244i	−0.966506	−	0.558013i
$785$	−5.81863	+	5.81863i	−0.207676	+	0.207676i
$786$		0			0
$787$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$787$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$788$	4.62132	+	4.62132i	0.164628	+	0.164628i
$789$		0			0
$790$	55.7654	−	32.1962i	1.98404	−	1.14549i
$791$		0			0
$792$			8.62693i			0.306545i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−3.24871	−	5.62693i	−0.115148	−	0.199441i
$797$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$797$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$798$		0			0
$799$		0			0
$800$	17.3514	−	4.64930i	0.613466	−	0.164378i
$801$	38.9624	+	38.9624i	1.37667	+	1.37667i
$802$	−23.3468	+	40.4378i	−0.824404	+	1.42791i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$	−48.2942	−	27.8827i	−1.69689	−	0.979698i
$811$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$811$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	11.1769			0.390315
$821$	−55.3534	−	14.8319i	−1.93185	−	0.517637i	−0.966492	−	0.256697i	$-0.917366\pi$
$821$	−55.3534	−	14.8319i	−1.93185	−	0.517637i	−0.965355	−	0.260939i	$-0.915968\pi$
$822$	−18.9737	−	32.8634i	−0.661785	−	1.14624i
$823$	48.4974	+	28.0000i	1.69051	+	0.976019i	0.954100	+	0.299487i	$0.0968155\pi$
$823$	48.4974	+	28.0000i	1.69051	+	0.976019i	0.736413	+	0.676532i	$0.236518\pi$
$824$	−29.5109	+	29.5109i	−1.02806	+	1.02806i
$825$	5.89508	+	22.0007i	0.205240	+	0.765968i
$826$		0			0
$827$	35.6913	+	35.6913i	1.24111	+	1.24111i	0.959542	+	0.281566i	$0.0908540\pi$
$827$	35.6913	+	35.6913i	1.24111	+	1.24111i	0.281566	+	0.959542i	$0.409146\pi$
$828$		0			0
$829$	24.0000	−	13.8564i	0.833554	−	0.481253i	−0.0215137	−	0.999769i	$-0.506849\pi$
$829$	24.0000	−	13.8564i	0.833554	−	0.481253i	0.855068	+	0.518516i	$0.173515\pi$
$830$	21.0888	−	78.7044i	0.732002	−	2.73187i
$831$			38.1051i			1.32185i
$832$		0			0
$833$		0			0
$834$	50.3908	+	13.5022i	1.74489	+	0.467542i
$835$	−23.7321	−	41.1051i	−0.821281	−	1.42250i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−42.9366	+	11.5048i	−1.48234	+	0.397191i	−0.907141	−	0.420826i	$-0.861740\pi$
$839$	−42.9366	+	11.5048i	−1.48234	+	0.397191i	−0.575195	+	0.818017i	$0.695074\pi$
$840$		0			0
$841$	−14.5000	+	25.1147i	−0.500000	+	0.866025i
$842$		0			0
$843$	−6.25687	+	23.3510i	−0.215498	+	0.804250i
$844$		−	0.928203i		−	0.0319501i
$845$		0			0
$846$	−42.1577			−1.44941
$847$		0			0
$848$		0			0
$849$	25.9808	+	15.0000i	0.891657	+	0.514799i
$850$		0			0
$851$		0			0
$852$	2.20622	−	0.591155i	0.0755839	−	0.0202526i
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	10.3923			0.354581			0.177290	−	0.984159i	$-0.443267\pi$
$859$	10.3923			0.354581			0.177290	+	0.984159i	$0.443267\pi$
$860$	−4.25953	−	1.14134i	−0.145249	−	0.0389193i
$861$		0			0
$862$	52.2224	+	30.1506i	1.77870	+	1.02693i
$863$	33.7144	−	33.7144i	1.14765	−	1.14765i	0.160640	−	0.987013i	$-0.448644\pi$
$863$	33.7144	−	33.7144i	1.14765	−	1.14765i	0.987013	−	0.160640i	$-0.0513559\pi$
$864$	2.02533	+	7.55862i	0.0689030	+	0.257149i
$865$		0			0
$866$	−22.1332	−	22.1332i	−0.752116	−	0.752116i
$867$	14.7224	−	25.5000i	0.500000	−	0.866025i
$868$		0			0
$869$	2.96528	−	11.0666i	0.100590	−	0.375408i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$877$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$878$	−58.1863	+	15.5910i	−1.96369	+	0.526169i
$879$	−39.9508	−	39.9508i	−1.34751	−	1.34751i
$880$	10.1244	−	17.5359i	0.341292	−	0.591135i
$881$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$881$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$882$	8.18525	−	30.5478i	0.275612	−	1.02860i
$883$		−	51.9615i		−	1.74864i	−0.485346	−	0.874322i	$-0.661306\pi$
$883$		−	51.9615i		−	1.74864i	0.485346	−	0.874322i	$-0.338694\pi$
$884$		0			0
$885$	109.426			3.67830
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$		0			0
$890$	−29.4549	−	109.927i	−0.987331	−	3.68477i
$891$	−9.58394	+	2.56801i	−0.321074	+	0.0860316i
$892$		0			0
$893$		0			0
$894$	−0.667296	+	0.385263i	−0.0223177	+	0.0128851i
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−16.1577			−0.539189
$899$		0			0
$900$	4.79423	+	8.30385i	0.159808	+	0.276795i
$901$		0			0
$902$	11.9021	−	11.9021i	0.396297	−	0.396297i
$903$		0			0
$904$		0			0
$905$	29.0931	+	29.0931i	0.967088	+	0.967088i
$906$		0			0
$907$	−15.0000	+	8.66025i	−0.498067	+	0.287559i	−0.727915	−	0.685668i	$-0.759510\pi$
$907$	−15.0000	+	8.66025i	−0.498067	+	0.287559i	0.229848	+	0.973227i	$0.426177\pi$
$908$	2.05331	−	7.66306i	0.0681415	−	0.254307i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	−7.24871	−	12.5551i	−0.239897	−	0.415514i
$914$		0			0
$915$	69.8235	−	69.8235i	2.30829	−	2.30829i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	29.4449	−	51.0000i	0.971296	−	1.68233i	0.279645	−	0.960104i	$-0.409783\pi$
$919$	29.4449	−	51.0000i	0.971296	−	1.68233i	0.691652	−	0.722231i	$-0.256883\pi$
$920$		0			0
$921$		0			0
$922$			37.6218i			1.23901i
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−41.5692	−	24.0000i	−1.36531	−	0.788263i
$928$		0			0
$929$	−15.4026	−	57.4832i	−0.505342	−	1.88596i	−0.461955	−	0.886903i	$-0.652852\pi$
$929$	−15.4026	−	57.4832i	−0.505342	−	1.88596i	−0.0433864	−	0.999058i	$-0.513815\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−34.6410			−1.13167			−0.565836	−	0.824518i	$-0.691447\pi$
$937$	−34.6410			−1.13167			−0.565836	+	0.824518i	$0.691447\pi$
$938$		0			0
$939$	30.0000	+	51.9615i	0.979013	+	1.69570i
$940$	8.90897	+	5.14359i	0.290578	+	0.167766i
$941$	14.9643	−	14.9643i	0.487823	−	0.487823i	−0.419796	−	0.907619i	$-0.637898\pi$
$941$	14.9643	−	14.9643i	0.487823	−	0.487823i	0.907619	+	0.419796i	$0.137898\pi$
$942$	1.35022	+	5.03908i	0.0439924	+	0.164182i
$943$		0			0
$944$	−48.4699	−	48.4699i	−1.57756	−	1.57756i
$945$		0			0
$946$	−5.75129	+	3.32051i	−0.186991	+	0.107959i
$947$	−5.38038	+	20.0799i	−0.174839	+	0.652508i	0.821740	+	0.569862i	$0.193004\pi$
$947$	−5.38038	+	20.0799i	−0.174839	+	0.652508i	−0.996579	+	0.0826452i	$0.973663\pi$
$948$		−	4.82309i		−	0.156647i
$949$		0			0
$950$		0			0
$951$	28.0282	+	7.51015i	0.908878	+	0.243533i
$952$		0			0
$953$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$953$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$954$		0			0
$955$		0			0
$956$	6.67463	−	1.78846i	0.215873	−	0.0578430i
$957$		0			0
$958$	25.1699	−	43.5955i	0.813202	−	1.40851i
$959$		0			0
$960$	−12.2844	+	45.8459i	−0.396477	+	1.47967i
$961$		−	31.0000i		−	1.00000i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	6.60567	+	24.6527i	0.212314	+	0.792368i
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$	−3.61731	+	2.08846i	−0.116025	+	0.0669873i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−61.8564			−1.97998
$977$	−34.2087	−	9.16618i	−1.09443	−	0.293252i	−0.333937	−	0.942596i	$-0.608377\pi$
$977$	−34.2087	−	9.16618i	−1.09443	−	0.293252i	−0.760495	+	0.649343i	$0.775044\pi$
$978$		0			0
$979$	−17.5359	−	10.1244i	−0.560450	−	0.323576i
$980$	−5.45684	+	5.45684i	−0.174312	+	0.174312i
$981$		0			0
$982$		0			0
$983$	8.88085	+	8.88085i	0.283255	+	0.283255i	0.834406	−	0.551151i	$-0.185811\pi$
$983$	8.88085	+	8.88085i	0.283255	+	0.283255i	−0.551151	+	0.834406i	$0.685811\pi$
$984$	−22.9019	+	39.6673i	−0.730087	+	1.26455i
$985$	−86.9090	+	50.1769i	−2.76915	+	1.59877i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	19.7945	+	5.30393i	0.629111	+	0.168570i
$991$	4.00000	+	6.92820i	0.127064	+	0.220082i	0.922538	−	0.385906i	$-0.126111\pi$
$991$	4.00000	+	6.92820i	0.127064	+	0.220082i	−0.795474	+	0.605988i	$0.792778\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	96.3691	−	25.8220i	3.05511	−	0.818613i
$996$	−4.31550	−	4.31550i	−0.136742	−	0.136742i
$997$	−29.0000	+	50.2295i	−0.918439	+	1.59078i	−0.116653	+	0.993173i	$0.537216\pi$
$997$	−29.0000	+	50.2295i	−0.918439	+	1.59078i	−0.801786	+	0.597611i	$0.796117\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.k.g.89.1		8
3.2	odd	2	inner	507.2.k.g.89.2		8
13.2	odd	12		507.2.f.d.239.2	✓	8
13.3	even	3		507.2.f.d.437.3	yes	8
13.4	even	6		507.2.k.h.488.1		8
13.5	odd	4		507.2.k.h.80.1		8
13.6	odd	12	inner	507.2.k.g.188.2		8
13.7	odd	12	inner	507.2.k.g.188.1		8
13.8	odd	4		507.2.k.h.80.2		8
13.9	even	3		507.2.k.h.488.2		8
13.10	even	6		507.2.f.d.437.2	yes	8
13.11	odd	12		507.2.f.d.239.3	yes	8
13.12	even	2	inner	507.2.k.g.89.2		8
39.2	even	12		507.2.f.d.239.3	yes	8
39.5	even	4		507.2.k.h.80.2		8
39.8	even	4		507.2.k.h.80.1		8
39.11	even	12		507.2.f.d.239.2	✓	8
39.17	odd	6		507.2.k.h.488.2		8
39.20	even	12	inner	507.2.k.g.188.2		8
39.23	odd	6		507.2.f.d.437.3	yes	8
39.29	odd	6		507.2.f.d.437.2	yes	8
39.32	even	12	inner	507.2.k.g.188.1		8
39.35	odd	6		507.2.k.h.488.1		8
39.38	odd	2	CM	507.2.k.g.89.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.f.d.239.2	✓	8	13.2	odd	12
507.2.f.d.239.2	✓	8	39.11	even	12
507.2.f.d.239.3	yes	8	13.11	odd	12
507.2.f.d.239.3	yes	8	39.2	even	12
507.2.f.d.437.2	yes	8	13.10	even	6
507.2.f.d.437.2	yes	8	39.29	odd	6
507.2.f.d.437.3	yes	8	13.3	even	3
507.2.f.d.437.3	yes	8	39.23	odd	6
507.2.k.g.89.1		8	1.1	even	1	trivial
507.2.k.g.89.1		8	39.38	odd	2	CM
507.2.k.g.89.2		8	3.2	odd	2	inner
507.2.k.g.89.2		8	13.12	even	2	inner
507.2.k.g.188.1		8	13.7	odd	12	inner
507.2.k.g.188.1		8	39.32	even	12	inner
507.2.k.g.188.2		8	13.6	odd	12	inner
507.2.k.g.188.2		8	39.20	even	12	inner
507.2.k.h.80.1		8	13.5	odd	4
507.2.k.h.80.1		8	39.8	even	4
507.2.k.h.80.2		8	13.8	odd	4
507.2.k.h.80.2		8	39.5	even	4
507.2.k.h.488.1		8	13.4	even	6
507.2.k.h.488.1		8	39.35	odd	6
507.2.k.h.488.2		8	13.9	even	3
507.2.k.h.488.2		8	39.17	odd	6

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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.k (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.45466 - 0.389774i) q^{2} +(-0.866025 - 1.50000i) q^{3} +(0.232051 + 0.133975i) q^{4} +(-2.90931 + 2.90931i) q^{5} +(0.675108 + 2.51954i) q^{6} +(1.84443 + 1.84443i) q^{8} +(-1.50000 + 2.59808i) q^{9} +O(q^{10})\)	\(q+(-1.45466 - 0.389774i) q^{2} +(-0.866025 - 1.50000i) q^{3} +(0.232051 + 0.133975i) q^{4} +(-2.90931 + 2.90931i) q^{5} +(0.675108 + 2.51954i) q^{6} +(1.84443 + 1.84443i) q^{8} +(-1.50000 + 2.59808i) q^{9} +(5.36603 - 3.09808i) q^{10} +(0.285334 - 1.06488i) q^{11} -0.464102i q^{12} +(6.88351 + 1.84443i) q^{15} +(-2.23205 - 3.86603i) q^{16} +(3.19465 - 3.19465i) q^{18} +(-1.06488 + 0.285334i) q^{20} +(-0.830127 + 1.43782i) q^{22} +(1.16932 - 4.36397i) q^{24} -11.9282i q^{25} +5.19615 q^{27} +(-9.29423 - 5.36603i) q^{30} +(0.389774 + 1.45466i) q^{32} +(-1.84443 + 0.494214i) q^{33} +(-0.696152 + 0.401924i) q^{36} -10.7321 q^{40} +(-9.79282 - 2.62398i) q^{41} +(3.46410 + 2.00000i) q^{43} +(0.208879 - 0.208879i) q^{44} +(-3.19465 - 11.9226i) q^{45} +(-6.59817 - 6.59817i) q^{47} +(-3.86603 + 6.69615i) q^{48} +(6.06218 - 3.50000i) q^{49} +(-4.64930 + 17.3514i) q^{50} +(-7.55862 - 2.02533i) q^{54} +(2.26795 + 3.92820i) q^{55} +(14.8319 - 3.97420i) q^{59} +(1.35022 + 1.35022i) q^{60} +(6.92820 - 12.0000i) q^{61} +6.66025i q^{64} +2.87564 q^{66} +(1.27376 + 4.75374i) q^{71} +(-7.55862 + 2.02533i) q^{72} +(-17.8923 + 10.3301i) q^{75} +10.3923 q^{79} +(17.7412 + 4.75374i) q^{80} +(-4.50000 - 7.79423i) q^{81} +(13.2224 + 7.63397i) q^{82} +(9.29861 - 9.29861i) q^{83} +(-4.25953 - 4.25953i) q^{86} +(2.49038 - 1.43782i) q^{88} +(4.75374 - 17.7412i) q^{89} +18.5885i q^{90} +(7.02628 + 12.1699i) q^{94} +(1.84443 - 1.84443i) q^{96} +(-10.1826 + 2.72842i) q^{98} +(2.33864 + 2.33864i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{4} - 12 q^{9}+O(q^{10}) \) 8 * q - 12 * q^4 - 12 * q^9	\( 8 q - 12 q^{4} - 12 q^{9} + 36 q^{10} - 4 q^{16} + 28 q^{22} - 12 q^{30} + 36 q^{36} - 72 q^{40} - 24 q^{48} + 32 q^{55} + 120 q^{66} - 60 q^{75} - 36 q^{81} - 12 q^{82} - 84 q^{88} - 20 q^{94}+O(q^{100}) \) 8 * q - 12 * q^4 - 12 * q^9 + 36 * q^10 - 4 * q^16 + 28 * q^22 - 12 * q^30 + 36 * q^36 - 72 * q^40 - 24 * q^48 + 32 * q^55 + 120 * q^66 - 60 * q^75 - 36 * q^81 - 12 * q^82 - 84 * q^88 - 20 * q^94

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