LMFDB - Embedded newform 507.2.k.h.488.2

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			488.2
Root			$0.500000 - 1.56488i$ of defining polynomial
Character	$\chi$	$=$	507.488
Dual form			507.2.k.h.80.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.389774 + 1.45466i) q^{2} +(-0.866025 + 1.50000i) q^{3} +(-0.232051 + 0.133975i) q^{4} +(-2.90931 + 2.90931i) q^{5} +(-2.51954 - 0.675108i) q^{6} +(1.84443 + 1.84443i) q^{8} +(-1.50000 - 2.59808i) q^{9} +O(q^{10})$	\(q+(0.389774 + 1.45466i) q^{2} +(-0.866025 + 1.50000i) q^{3} +(-0.232051 + 0.133975i) q^{4} +(-2.90931 + 2.90931i) q^{5} +(-2.51954 - 0.675108i) q^{6} +(1.84443 + 1.84443i) q^{8} +(-1.50000 - 2.59808i) q^{9} +(-5.36603 - 3.09808i) q^{10} +(-1.06488 + 0.285334i) q^{11} -0.464102i q^{12} +(-1.84443 - 6.88351i) q^{15} +(-2.23205 + 3.86603i) q^{16} +(3.19465 - 3.19465i) q^{18} +(0.285334 - 1.06488i) q^{20} +(-0.830127 - 1.43782i) q^{22} +(-4.36397 + 1.16932i) q^{24} -11.9282i q^{25} +5.19615 q^{27} +(9.29423 - 5.36603i) q^{30} +(-1.45466 - 0.389774i) q^{32} +(0.494214 - 1.84443i) q^{33} +(0.696152 + 0.401924i) q^{36} -10.7321 q^{40} +(2.62398 + 9.79282i) q^{41} +(-3.46410 + 2.00000i) q^{43} +(0.208879 - 0.208879i) q^{44} +(11.9226 + 3.19465i) q^{45} +(-6.59817 - 6.59817i) q^{47} +(-3.86603 - 6.69615i) q^{48} +(-6.06218 - 3.50000i) q^{49} +(17.3514 - 4.64930i) q^{50} +(2.02533 + 7.55862i) q^{54} +(2.26795 - 3.92820i) q^{55} +(-3.97420 + 14.8319i) q^{59} +(1.35022 + 1.35022i) q^{60} +(6.92820 + 12.0000i) q^{61} +6.66025i q^{64} +2.87564 q^{66} +(-4.75374 - 1.27376i) q^{71} +(2.02533 - 7.55862i) q^{72} +(17.8923 + 10.3301i) q^{75} +10.3923 q^{79} +(-4.75374 - 17.7412i) q^{80} +(-4.50000 + 7.79423i) q^{81} +(-13.2224 + 7.63397i) q^{82} +(9.29861 - 9.29861i) q^{83} +(-4.25953 - 4.25953i) q^{86} +(-2.49038 - 1.43782i) q^{88} +(-17.7412 + 4.75374i) q^{89} +18.5885i q^{90} +(7.02628 - 12.1699i) q^{94} +(1.84443 - 1.84443i) q^{96} +(2.72842 - 10.1826i) q^{98} +(2.33864 + 2.33864i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$-1$	$e\left(\frac{11}{12}\right)$

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 $

$2$	0.389774	+	1.45466i	0.275612	+	1.02860i	0.955430	+	0.295217i	$0.0953919\pi$
$2$	0.389774	+	1.45466i	0.275612	+	1.02860i	−0.679818	+	0.733380i	$0.737941\pi$
$3$	−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	$0.878184\pi$
$6$	−2.51954	−	0.675108i	−1.02860	−	0.275612i
$7$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$7$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$8$	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
$11$	−1.06488	+	0.285334i	−0.321074	+	0.0860316i	−0.415756	−	0.909476i	$-0.636483\pi$
$11$	−1.06488	+	0.285334i	−0.321074	+	0.0860316i	0.0946823	+	0.995508i	$0.469816\pi$
$12$		−	0.464102i		−	0.133975i
$13$		0			0
$13$		0			0
$14$		0			0
$15$	−1.84443	−	6.88351i	−0.476230	−	1.77731i
$16$	−2.23205	+	3.86603i	−0.558013	+	0.966506i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$	3.19465	−	3.19465i	0.752986	−	0.752986i
$19$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$19$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$20$	0.285334	−	1.06488i	0.0638027	−	0.238115i
$21$		0			0
$22$	−0.830127	−	1.43782i	−0.176984	−	0.306545i
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$	−4.36397	+	1.16932i	−0.890792	+	0.238687i
$25$		−	11.9282i		−	2.38564i
$26$		0			0
$27$	5.19615			1.00000
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$	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	$0.750000\pi$
$32$	−1.45466	−	0.389774i	−0.257149	−	0.0689030i
$33$	0.494214	−	1.84443i	0.0860316	−	0.321074i
$34$		0			0
$35$		0			0
$36$	0.696152	+	0.401924i	0.116025	+	0.0669873i
$37$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$37$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$38$		0			0
$39$		0			0
$40$	−10.7321			−1.69689
$41$	2.62398	+	9.79282i	0.409797	+	1.52938i	0.795034	+	0.606564i	$0.207453\pi$
$41$	2.62398	+	9.79282i	0.409797	+	1.52938i	−0.385238	+	0.922817i	$0.625881\pi$
$42$		0			0
$43$	−3.46410	+	2.00000i	−0.528271	+	0.304997i	−0.740312	−	0.672264i	$-0.765322\pi$
$43$	−3.46410	+	2.00000i	−0.528271	+	0.304997i	0.212041	+	0.977261i	$0.431989\pi$
$44$	0.208879	−	0.208879i	0.0314897	−	0.0314897i
$45$	11.9226	+	3.19465i	1.77731	+	0.476230i
$46$		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
$50$	17.3514	−	4.64930i	2.45386	−	0.657511i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	2.02533	+	7.55862i	0.275612	+	1.02860i
$55$	2.26795	−	3.92820i	0.305810	−	0.529679i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.97420	+	14.8319i	−0.517396	+	1.93095i	−0.229170	+	0.973386i	$0.573601\pi$
$59$	−3.97420	+	14.8319i	−0.517396	+	1.93095i	−0.288226	+	0.957562i	$0.593066\pi$
$60$	1.35022	+	1.35022i	0.174312	+	0.174312i
$61$	6.92820	+	12.0000i	0.887066	+	1.53644i	0.843328	+	0.537400i	$0.180593\pi$
$61$	6.92820	+	12.0000i	0.887066	+	1.53644i	0.0437377	+	0.999043i	$0.486073\pi$
$62$		0			0
$63$		0			0
$64$			6.66025i			0.832532i
$65$		0			0
$66$	2.87564			0.353967
$67$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$67$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4.75374	−	1.27376i	−0.564166	−	0.151168i	−0.0345462	−	0.999403i	$-0.510999\pi$
$71$	−4.75374	−	1.27376i	−0.564166	−	0.151168i	−0.529619	+	0.848235i	$0.677665\pi$
$72$	2.02533	−	7.55862i	0.238687	−	0.890792i
$73$		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	$0.301246\pi$
$80$	−4.75374	−	17.7412i	−0.531485	−	1.98353i
$81$	−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
$89$	−17.7412	+	4.75374i	−1.88056	+	0.503896i	−0.881041	+	0.473040i	$0.843157\pi$
$89$	−17.7412	+	4.75374i	−1.88056	+	0.503896i	−0.999524	+	0.0308556i	$0.990177\pi$
$90$			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		−0.965926	−	0.258819i	$-0.916667\pi$
$97$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$98$	2.72842	−	10.1826i	0.275612	−	1.02860i
$99$	2.33864	+	2.33864i	0.235043	+	0.235043i
$100$	1.59808	+	2.76795i	0.159808	+	0.276795i
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		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	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$	−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	$0.750000\pi$
$110$	6.59817	+	1.76798i	0.629111	+	0.168570i
$111$		0			0
$112$		0			0
$113$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$113$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$114$		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
$123$	−16.9617	−	4.54486i	−1.52938	−	0.409797i
$124$		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	$-0.0543501\pi$
$127$	15.0000	+	8.66025i	1.33103	+	0.768473i	0.345576	+	0.938391i	$0.387683\pi$
$128$	−12.5977	+	3.37554i	−1.11349	+	0.298359i
$129$		−	6.92820i		−	0.609994i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$	0.132424	+	0.494214i	0.0115260	+	0.0430158i
$133$		0			0
$134$		0			0
$135$	−15.1172	+	15.1172i	−1.30108	+	1.30108i
$136$		0			0
$137$	3.76532	−	14.0524i	0.321693	−	1.20057i	−0.595902	−	0.803057i	$-0.703205\pi$
$137$	3.76532	−	14.0524i	0.321693	−	1.20057i	0.917595	−	0.397516i	$-0.130128\pi$
$138$		0			0
$139$	10.0000	+	17.3205i	0.848189	+	1.46911i	0.882823	+	0.469706i	$0.155640\pi$
$139$	10.0000	+	17.3205i	0.848189	+	1.46911i	−0.0346338	+	0.999400i	$0.511026\pi$
$140$		0			0
$141$	15.6114	−	4.18307i	1.31472	−	0.352278i
$142$		−	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
$149$	−0.285334	−	0.0764551i	−0.0233755	−	0.00626345i	0.247112	−	0.968987i	$-0.420518\pi$
$149$	−0.285334	−	0.0764551i	−0.0233755	−	0.00626345i	−0.270488	+	0.962723i	$0.587185\pi$
$150$	−8.05283	+	30.0536i	−0.657511	+	2.45386i
$151$		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	$0.474569\pi$
$158$	4.05065	+	15.1172i	0.322252	+	1.20266i
$159$		0			0
$160$	5.36603	−	3.09808i	0.424222	−	0.244924i
$161$		0			0
$162$	−13.0919	−	3.50797i	−1.02860	−	0.275612i
$163$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$163$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$164$	−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
$167$	11.1430	−	2.98577i	0.862274	−	0.231046i	0.199530	−	0.979892i	$-0.436058\pi$
$167$	11.1430	−	2.98577i	0.862274	−	0.231046i	0.662744	+	0.748846i	$0.269392\pi$
$168$		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	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$		0			0
$176$	1.27376	−	4.75374i	0.0960134	−	0.358327i
$177$	−18.8061	−	18.8061i	−1.41355	−	1.41355i
$178$	−13.8301	−	23.9545i	−1.03661	−	1.79546i
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$	−3.19465	+	0.856003i	−0.238115	+	0.0638027i
$181$		−	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
$188$	2.41510	+	0.647124i	0.176139	+	0.0471964i
$189$		0			0
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	−9.99038	−	5.76795i	−0.720994	−	0.416266i
$193$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$193$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$194$		0			0
$195$		0			0
$196$	1.87564			0.133975
$197$	−6.31284	−	23.5598i	−0.449771	−	1.67857i	−0.703022	−	0.711168i	$-0.748166\pi$
$197$	−6.31284	−	23.5598i	−0.449771	−	1.67857i	0.253251	−	0.967401i	$-0.418500\pi$
$198$	−2.49038	+	4.31347i	−0.176984	+	0.306545i
$199$	21.0000	−	12.1244i	1.48865	−	0.859473i	0.488735	−	0.872433i	$-0.337459\pi$
$199$	21.0000	−	12.1244i	1.48865	−	0.859473i	0.999916	+	0.0129598i	$0.00412534\pi$
$200$	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
$206$	−23.2745	+	6.23638i	−1.62161	+	0.434509i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−1.73205	+	3.00000i	−0.119239	+	0.206529i	−0.919466	−	0.393169i	$-0.871379\pi$
$211$	−1.73205	+	3.00000i	−0.119239	+	0.206529i	0.800227	+	0.599697i	$0.204712\pi$
$212$		0			0
$213$	6.02751	−	6.02751i	0.412998	−	0.412998i
$214$		0			0
$215$	4.25953	−	15.8968i	0.290498	−	1.08415i
$216$	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		−0.258819	−	0.965926i	$-0.583333\pi$
$223$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$224$		0			0
$225$	−30.9904	+	17.8923i	−2.06603	+	1.19282i
$226$		0			0
$227$	28.5989	+	7.66306i	1.89818	+	0.508615i	0.997203	+	0.0747413i	$0.0238131\pi$
$227$	28.5989	+	7.66306i	1.89818	+	0.508615i	0.900974	+	0.433874i	$0.142854\pi$
$228$		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
$236$	−1.06488	−	3.97420i	−0.0693179	−	0.258698i
$237$	−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	$-0.0639945\pi$
$240$	30.7287	+	8.23373i	1.98353	+	0.531485i
$241$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$241$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$242$	−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
$245$	27.8194	−	7.45418i	1.77731	−	0.476230i
$246$		−	26.4449i		−	1.68606i
$247$		0			0
$248$		0			0
$249$	5.89508	+	22.0007i	0.373586	+	1.39424i
$250$	−21.4641	+	37.1769i	−1.35751	+	2.35127i
$251$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$251$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$252$		0			0
$253$		0			0
$254$	−6.75108	+	25.1954i	−0.423601	+	1.58090i
$255$		0			0
$256$	−3.16025	−	5.47372i	−0.197516	−	0.342108i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$	10.0782	−	2.70043i	0.627439	−	0.168122i
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$	4.31347	−	2.49038i	0.265476	−	0.153272i
$265$		0			0
$266$		0			0
$267$	8.23373	−	30.7287i	0.503896	−	1.88056i
$268$		0			0
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$	−27.8827	−	16.0981i	−1.69689	−	0.979698i
$271$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$271$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$272$		0			0
$273$		0			0
$274$	21.9090			1.32357
$275$	3.40353	+	12.7021i	0.205240	+	0.765968i
$276$		0			0
$277$	19.0526	−	11.0000i	1.14476	−	0.660926i	0.197153	−	0.980373i	$-0.436830\pi$
$277$	19.0526	−	11.0000i	1.14476	−	0.660926i	0.947604	+	0.319447i	$0.103497\pi$
$278$	−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	$-0.161200\pi$
$283$	15.0000	+	8.66025i	0.891657	+	0.514799i	0.0171732	+	0.999853i	$0.494533\pi$
$284$	1.27376	−	0.341303i	0.0755839	−	0.0202526i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	1.16932	+	4.36397i	0.0689030	+	0.257149i
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−8.44260	+	31.5082i	−0.493222	+	1.84073i	0.0465452	+	0.998916i	$0.485179\pi$
$293$	−8.44260	+	31.5082i	−0.493222	+	1.84073i	−0.539767	+	0.841814i	$0.681488\pi$
$294$	12.9110	+	12.9110i	0.752986	+	0.752986i
$295$	−31.5885	−	54.7128i	−1.83915	−	3.18550i
$296$		0			0
$297$	−5.53329	+	1.48264i	−0.321074	+	0.0860316i
$298$		−	0.444864i		−	0.0257703i
$299$		0			0
$300$	−5.53590			−0.319615
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−55.0681	−	14.7554i	−3.15319	−	0.844894i
$306$		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	$0.934671\pi$
$314$	0.779548	+	2.90931i	0.0439924	+	0.164182i
$315$		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		−0.965926	−	0.258819i	$-0.916667\pi$
$331$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$332$	−0.911972	+	3.40353i	−0.0500510	+	0.186793i
$333$		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
$344$	−10.0782	−	2.70043i	−0.543378	−	0.145598i
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$349$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$350$		0			0
$351$		0			0
$352$	1.66025			0.0884918
$353$	−0.647124	−	2.41510i	−0.0344429	−	0.128543i	0.946564	−	0.322517i	$-0.104529\pi$
$353$	−0.647124	−	2.41510i	−0.0344429	−	0.128543i	−0.981007	+	0.193974i	$0.937862\pi$
$354$	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
$362$	14.5466	−	3.89774i	0.764550	−	0.204861i
$363$		−	16.9474i		−	0.889510i
$364$		0			0
$365$		0			0
$366$	−9.35458	−	34.9118i	−0.488972	−	1.82487i
$367$	4.00000	−	6.92820i	0.208798	−	0.361649i	−0.742538	−	0.669804i	$-0.766378\pi$
$367$	4.00000	−	6.92820i	0.208798	−	0.361649i	0.951336	+	0.308155i	$0.0997115\pi$
$368$		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	0.987749	−	0.156053i	$-0.0498770\pi$
$373$	6.92820	+	12.0000i	0.358729	+	0.621336i	−0.629020	+	0.777389i	$0.716544\pi$
$374$		0			0
$375$	−47.6903	+	12.7786i	−2.46272	+	0.659884i
$376$		−	24.3397i		−	1.25523i
$377$		0			0
$378$		0			0
$379$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$379$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$380$		0			0
$381$	−25.9808	+	15.0000i	−1.33103	+	0.768473i
$382$		0			0
$383$	16.3910	+	4.39195i	0.837541	+	0.224418i	0.652001	−	0.758218i	$-0.273930\pi$
$383$	16.3910	+	4.39195i	0.837541	+	0.224418i	0.185540	+	0.982637i	$0.440597\pi$
$384$	5.84661	−	21.8198i	0.298359	−	1.11349i
$385$		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
$392$	−4.72576	−	17.6368i	−0.238687	−	0.890792i
$393$		0			0
$394$	31.8109	−	18.3660i	1.60261	−	0.925267i
$395$	−30.2345	+	30.2345i	−1.52126	+	1.52126i
$396$	−0.856003	−	0.229365i	−0.0430158	−	0.0115260i
$397$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$397$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$398$	25.8220	+	25.8220i	1.29434	+	1.29434i
$399$		0			0
$400$	46.1147	+	26.6244i	2.30574	+	1.33122i
$401$	−29.9491	+	8.02485i	−1.49559	+	0.400742i	−0.911620	−	0.411035i	$-0.865168\pi$
$401$	−29.9491	+	8.02485i	−1.49559	+	0.400742i	−0.583969	+	0.811776i	$0.698501\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−9.58394	−	35.7678i	−0.476230	−	1.77731i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$409$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$410$	16.2586	−	60.6778i	0.802954	−	2.99666i
$411$	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	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$421$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$422$	−5.03908	−	1.35022i	−0.245298	−	0.0657275i
$423$	−7.24530	+	27.0398i	−0.352278	+	1.31472i
$424$		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
$431$	10.3635	+	38.6771i	0.499192	+	1.86301i	0.505158	+	0.863027i	$0.331434\pi$
$431$	10.3635	+	38.6771i	0.499192	+	1.86301i	−0.00596647	+	0.999982i	$0.501899\pi$
$432$	−11.5981	+	20.0885i	−0.558013	+	0.966506i
$433$	−18.0000	+	10.3923i	−0.865025	+	0.499422i	−0.865692	−	0.500577i	$-0.833121\pi$
$433$	−18.0000	+	10.3923i	−0.865025	+	0.499422i	0.000666943		1.00000i	$0.499788\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−34.6410	−	20.0000i	−1.65333	−	0.954548i	−0.975691	−	0.219149i	$-0.929672\pi$
$439$	−34.6410	−	20.0000i	−1.65333	−	0.954548i	−0.677634	−	0.735399i	$-0.736995\pi$
$440$	11.4284	−	3.06222i	0.544826	−	0.145986i
$441$			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
$449$	−2.77689	+	10.3635i	−0.131049	+	0.489083i	−0.999983	−	0.00584565i	$-0.998139\pi$
$449$	−2.77689	+	10.3635i	−0.131049	+	0.489083i	0.868933	+	0.494929i	$0.164806\pi$
$450$	−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		−0.258819	−	0.965926i	$-0.583333\pi$
$457$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	24.1305	+	6.46575i	1.12387	+	0.301140i	0.772448	−	0.635077i	$-0.219032\pi$
$461$	24.1305	+	6.46575i	1.12387	+	0.301140i	0.351421	+	0.936217i	$0.385698\pi$
$462$		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
$470$	14.9643	+	55.8476i	0.690253	+	2.57606i
$471$	−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
$474$	−26.1838	−	7.01593i	−1.20266	−	0.322252i
$475$		0			0
$476$		0			0
$477$		0			0
$478$	33.6340	+	19.4186i	1.53838	+	0.888185i
$479$	32.2878	−	8.65148i	1.47527	−	0.395296i	0.570533	−	0.821275i	$-0.306737\pi$
$479$	32.2878	−	8.65148i	1.47527	−	0.395296i	0.904733	+	0.425978i	$0.140070\pi$
$480$			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		−0.965926	−	0.258819i	$-0.916667\pi$
$487$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$488$	−9.35458	+	34.9118i	−0.423462	+	1.58038i
$489$		0			0
$490$	21.6865	+	37.5622i	0.979698	+	1.69689i
$491$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$491$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$492$	4.54486	−	1.21779i	0.204898	−	0.0549023i
$493$		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	$0.750000\pi$
$500$	−7.37772	−	1.97685i	−0.329942	−	0.0884076i
$501$	−5.17150	+	19.3003i	−0.231046	+	0.862274i
$502$		0			0
$503$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−4.64102			−0.205912
$509$	−3.04174	−	11.3519i	−0.134823	−	0.503165i	−0.999999	−	0.00169644i	$-0.999460\pi$
$509$	−3.04174	−	11.3519i	−0.134823	−	0.503165i	0.865176	−	0.501468i	$-0.167207\pi$
$510$		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	−0.244507	+	0.969648i	$0.578626\pi$
$523$	−22.0000	+	38.1051i	−0.961993	+	1.66622i	−0.717486	+	0.696573i	$0.754707\pi$
$524$		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
$531$	44.4957	−	11.9226i	1.93095	−	0.517396i
$532$		0			0
$533$		0			0
$534$	47.9090			2.07322
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	7.45418	+	1.99734i	0.321074	+	0.0860316i
$540$	1.48264	−	5.53329i	0.0638027	−	0.238115i
$541$		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	$0.295725\pi$
$548$	1.00891	+	3.76532i	0.0430986	+	0.160846i
$549$	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
$557$	3.40353	−	0.911972i	0.144212	−	0.0386415i	−0.185991	−	0.982551i	$-0.559549\pi$
$557$	3.40353	−	0.911972i	0.144212	−	0.0386415i	0.330203	+	0.943910i	$0.392883\pi$
$558$		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	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$	−3.06222	+	3.06222i	−0.128943	+	0.128943i
$565$		0			0
$566$	−6.75108	+	25.1954i	−0.283769	+	1.05904i
$567$		0			0
$568$	−6.41858	−	11.1173i	−0.269318	−	0.466472i
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		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	$0.750000\pi$
$578$	24.7292	+	6.62616i	1.02860	+	0.275612i
$579$		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
$587$	−10.7812	−	40.2362i	−0.444990	−	1.66072i	−0.715964	−	0.698138i	$-0.754012\pi$
$587$	−10.7812	−	40.2362i	−0.444990	−	1.66072i	0.270974	−	0.962587i	$-0.412654\pi$
$588$	−1.62436	+	2.81347i	−0.0669873	+	0.116025i
$589$		0			0
$590$	67.2760	−	67.2760i	2.76971	−	2.76971i
$591$	40.8068	+	10.9342i	1.67857	+	0.449771i
$592$		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
$596$	0.0764551	−	0.0204861i	0.00313172	−	0.000839143i
$597$			42.0000i			1.71895i
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$	13.9479	+	52.0543i	0.569421	+	2.12511i
$601$	−24.2487	+	42.0000i	−0.989126	+	1.71322i	−0.367193	+	0.930145i	$0.619681\pi$
$601$	−24.2487	+	42.0000i	−0.989126	+	1.71322i	−0.621932	+	0.783071i	$0.713652\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	10.4195	−	38.8860i	0.423611	−	1.58094i
$606$		0			0
$607$	−16.0000	−	27.7128i	−0.649420	−	1.12483i	−0.983262	−	0.182199i	$-0.941678\pi$
$607$	−16.0000	−	27.7128i	−0.649420	−	1.12483i	0.333842	−	0.942629i	$-0.391655\pi$
$608$		0			0
$609$		0			0
$610$		−	85.8564i		−	3.47622i
$611$		0			0
$612$		0			0
$613$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$613$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$614$		0			0
$615$	62.5692	−	36.1244i	2.52303	−	1.45667i
$616$		0			0
$617$	45.2752	+	12.1315i	1.82271	+	0.488394i	0.997118	−	0.0758689i	$-0.0241730\pi$
$617$	45.2752	+	12.1315i	1.82271	+	0.488394i	0.825595	+	0.564263i	$0.190840\pi$
$618$	10.8017	−	40.3126i	0.434509	−	1.62161i
$619$		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
$626$	−13.5022	−	50.3908i	−0.539655	−	2.01402i
$627$		0			0
$628$	−0.464102	+	0.267949i	−0.0185197	+	0.0106923i
$629$		0			0
$630$		0			0
$631$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$631$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$632$	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
$635$	−68.8351	+	18.4443i	−2.73164	+	0.731940i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	3.82129	+	14.2612i	0.151168	+	0.564166i
$640$	26.8301	−	46.4711i	1.06055	−	1.83693i
$641$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$641$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$642$		0			0
$643$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$643$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$644$		0			0
$645$	20.1563	+	20.1563i	0.793654	+	0.793654i
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$	−22.6758	+	6.07598i	−0.890792	+	0.238687i
$649$		−	16.9282i		−	0.664490i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$	−43.7161	−	11.7137i	−1.70683	−	0.457343i
$657$		0			0
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$	−1.82309	−	1.05256i	−0.0709635	−	0.0409708i
$661$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$661$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$662$		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	$-0.420301\pi$
$673$	−12.1244	−	7.00000i	−0.467360	−	0.269830i	−0.715134	+	0.698988i	$0.753634\pi$
$674$	10.0782	−	2.70043i	0.388196	−	0.104017i
$675$		−	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
$683$	1.69152	−	6.31284i	0.0647242	−	0.241554i	−0.925983	−	0.377565i	$-0.876762\pi$
$683$	1.69152	−	6.31284i	0.0647242	−	0.241554i	0.990707	+	0.136011i	$0.0434282\pi$
$684$		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		−0.258819	−	0.965926i	$-0.583333\pi$
$691$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−79.4839	−	21.2976i	−3.01500	−	0.807866i
$696$		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
$704$	−1.90040	−	7.09239i	−0.0716240	−	0.267304i
$705$	−33.2487	+	57.5885i	−1.25222	+	2.16891i
$706$	3.26091	−	1.88269i	0.122726	−	0.0708558i
$707$		0			0
$708$	6.88351	+	1.84443i	0.258698	+	0.0693179i
$709$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$709$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$710$	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
$717$	11.5608	+	43.1455i	0.431746	+	1.61130i
$718$	−28.4904	+	49.3468i	−1.06325	+	1.84161i
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$	−38.9624	+	38.9624i	−1.45204	+	1.45204i
$721$		0			0
$722$	−7.40571	+	27.6385i	−0.275612	+	1.02860i
$723$		0			0
$724$	1.33975	+	2.32051i	0.0497913	+	0.0862410i
$725$		0			0
$726$	24.6527	−	6.60567i	0.914948	−	0.245159i
$727$		−	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	$0.750000\pi$
$734$	11.6373	+	3.11819i	0.429539	+	0.115095i
$735$	−12.9110	+	48.1846i	−0.476230	+	1.77731i
$736$		0			0
$737$		0			0
$738$	39.6673	+	22.9019i	1.46017	+	0.843031i
$739$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$739$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−14.0524	−	52.4441i	−0.515531	−	1.92399i	−0.344836	−	0.938663i	$-0.612065\pi$
$743$	−14.0524	−	52.4441i	−0.515531	−	1.92399i	−0.170695	−	0.985324i	$-0.554601\pi$
$744$		0			0
$745$	1.05256	−	0.607695i	0.0385628	−	0.0222642i
$746$	−14.7554	+	14.7554i	−0.540235	+	0.540235i
$747$	−38.1064	−	10.2106i	−1.39424	−	0.373586i
$748$		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	$-0.593725\pi$
$751$	−34.6410	−	20.0000i	−1.26407	−	0.729810i	−0.973859	+	0.227153i	$0.927058\pi$
$752$	40.2362	−	10.7812i	1.46726	−	0.393152i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	20.7846	−	36.0000i	0.755429	−	1.30844i	−0.189731	−	0.981836i	$-0.560762\pi$
$757$	20.7846	−	36.0000i	0.755429	−	1.30844i	0.945161	−	0.326606i	$-0.105905\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	7.03642	−	26.2603i	0.255070	−	0.951934i	−0.712982	−	0.701183i	$-0.752656\pi$
$761$	7.03642	−	26.2603i	0.255070	−	0.951934i	0.968052	−	0.250751i	$-0.0806776\pi$
$762$	−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		−0.258819	−	0.965926i	$-0.583333\pi$
$769$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−12.4933	−	3.34756i	−0.449351	−	0.120403i	0.0270446	−	0.999634i	$-0.491390\pi$
$773$	−12.4933	−	3.34756i	−0.449351	−	0.120403i	−0.476396	+	0.879231i	$0.658057\pi$
$774$	−4.67729	+	17.4559i	−0.168122	+	0.627439i
$775$		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		0.258819	−	0.965926i	$-0.416667\pi$
$787$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$788$	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	$-0.166667\pi$
$797$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$798$		0			0
$799$		0			0
$800$	−4.64930	+	17.3514i	−0.164378	+	0.613466i
$801$	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	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$	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
$821$	14.8319	+	55.3534i	0.517637	+	1.93185i	0.260939	+	0.965355i	$0.415968\pi$
$821$	14.8319	+	55.3534i	0.517637	+	1.93185i	0.256697	+	0.966492i	$0.417366\pi$
$822$	−18.9737	+	32.8634i	−0.661785	+	1.14624i
$823$	−48.4974	+	28.0000i	−1.69051	+	0.976019i	−0.736413	+	0.676532i	$0.763482\pi$
$823$	−48.4974	+	28.0000i	−1.69051	+	0.976019i	−0.954100	+	0.299487i	$0.903185\pi$
$824$	−29.5109	+	29.5109i	−1.02806	+	1.02806i
$825$	−22.0007	−	5.89508i	−0.765968	−	0.205240i
$826$		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	$-0.493151\pi$
$829$	−24.0000	−	13.8564i	−0.833554	−	0.481253i	−0.855068	+	0.518516i	$0.826485\pi$
$830$	−78.7044	+	21.0888i	−2.73187	+	0.732002i
$831$			38.1051i			1.32185i
$832$		0			0
$833$		0			0
$834$	−13.5022	−	50.3908i	−0.467542	−	1.74489i
$835$	−23.7321	+	41.1051i	−0.821281	+	1.42250i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	11.5048	−	42.9366i	0.397191	−	1.48234i	−0.420826	−	0.907141i	$-0.638260\pi$
$839$	11.5048	−	42.9366i	0.397191	−	1.48234i	0.818017	−	0.575195i	$-0.195074\pi$
$840$		0			0
$841$	−14.5000	−	25.1147i	−0.500000	−	0.866025i
$842$		0			0
$843$	23.3510	−	6.25687i	0.804250	−	0.215498i
$844$		−	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
$852$	−0.591155	+	2.20622i	−0.0202526	+	0.0755839i
$853$		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	$0.443267\pi$
$860$	1.14134	+	4.25953i	0.0389193	+	0.145249i
$861$		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	$-0.0513559\pi$
$864$	−7.55862	−	2.02533i	−0.257149	−	0.0689030i
$865$		0			0
$866$	−22.1332	−	22.1332i	−0.752116	−	0.752116i
$867$	14.7224	+	25.5000i	0.500000	+	0.866025i
$868$		0			0
$869$	−11.0666	+	2.96528i	−0.375408	+	0.100590i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$877$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$878$	15.5910	−	58.1863i	0.526169	−	1.96369i
$879$	−39.9508	−	39.9508i	−1.34751	−	1.34751i
$880$	10.1244	+	17.5359i	0.341292	+	0.591135i
$881$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$881$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$882$	−30.5478	+	8.18525i	−1.02860	+	0.275612i
$883$		−	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	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$		0			0
$890$	109.927	+	29.4549i	3.68477	+	0.987331i
$891$	2.56801	−	9.58394i	0.0860316	−	0.321074i
$892$		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	$-0.240490\pi$
$907$	15.0000	+	8.66025i	0.498067	+	0.287559i	−0.229848	+	0.973227i	$0.573823\pi$
$908$	−7.66306	+	2.05331i	−0.254307	+	0.0681415i
$909$		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	0.691652	+	0.722231i	$0.256883\pi$
$919$	29.4449	+	51.0000i	0.971296	+	1.68233i	0.279645	+	0.960104i	$0.409783\pi$
$920$		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
$929$	57.4832	+	15.4026i	1.88596	+	0.505342i	0.999058	+	0.0433864i	$0.0138146\pi$
$929$	57.4832	+	15.4026i	1.88596	+	0.505342i	0.886903	+	0.461955i	$0.152852\pi$
$930$		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	$0.137898\pi$
$942$	−5.03908	−	1.35022i	−0.164182	−	0.0439924i
$943$		0			0
$944$	−48.4699	−	48.4699i	−1.57756	−	1.57756i
$945$		0			0
$946$	5.75129	+	3.32051i	0.186991	+	0.107959i
$947$	20.0799	−	5.38038i	0.652508	−	0.174839i	0.0826452	−	0.996579i	$-0.473663\pi$
$947$	20.0799	−	5.38038i	0.652508	−	0.174839i	0.569862	+	0.821740i	$0.306996\pi$
$948$		−	4.82309i		−	0.156647i
$949$		0			0
$950$		0			0
$951$	−7.51015	−	28.0282i	−0.243533	−	0.908878i
$952$		0			0
$953$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$953$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$954$		0			0
$955$		0			0
$956$	−1.78846	+	6.67463i	−0.0578430	+	0.215873i
$957$		0			0
$958$	25.1699	+	43.5955i	0.813202	+	1.40851i
$959$		0			0
$960$	45.8459	−	12.2844i	1.47967	−	0.396477i
$961$		−	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	$0.750000\pi$
$968$	−24.6527	−	6.60567i	−0.792368	−	0.212314i
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$	3.61731	+	2.08846i	0.116025	+	0.0669873i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−61.8564			−1.97998
$977$	9.16618	+	34.2087i	0.293252	+	1.09443i	0.942596	+	0.333937i	$0.108377\pi$
$977$	9.16618	+	34.2087i	0.293252	+	1.09443i	−0.649343	+	0.760495i	$0.724956\pi$
$978$		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
$990$	−5.30393	−	19.7945i	−0.168570	−	0.629111i
$991$	4.00000	−	6.92820i	0.127064	−	0.220082i	−0.795474	−	0.605988i	$-0.792778\pi$
$991$	4.00000	−	6.92820i	0.127064	−	0.220082i	0.922538	+	0.385906i	$0.126111\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−25.8220	+	96.3691i	−0.818613	+	3.05511i
$996$	−4.31550	−	4.31550i	−0.136742	−	0.136742i
$997$	−29.0000	−	50.2295i	−0.918439	−	1.59078i	−0.801786	−	0.597611i	$-0.796117\pi$
$997$	−29.0000	−	50.2295i	−0.918439	−	1.59078i	−0.116653	−	0.993173i	$-0.537216\pi$
$998$		0			0
$999$		0			0

Twists

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

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

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)

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