LMFDB - Embedded newform 507.2.k.g.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			$0.500000 - 2.19293i$ of defining polynomial
Character	$\chi$	$=$	507.488
Dual form			507.2.k.g.80.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.619657 + 2.31259i) q^{2} +(0.866025 - 1.50000i) q^{3} +(-3.23205 + 1.86603i) q^{4} +(1.23931 - 1.23931i) q^{5} +(4.00552 + 1.07328i) q^{6} +(-2.93225 - 2.93225i) q^{8} +(-1.50000 - 2.59808i) q^{9} +O(q^{10})$	\(q+(0.619657 + 2.31259i) q^{2} +(0.866025 - 1.50000i) q^{3} +(-3.23205 + 1.86603i) q^{4} +(1.23931 - 1.23931i) q^{5} +(4.00552 + 1.07328i) q^{6} +(-2.93225 - 2.93225i) q^{8} +(-1.50000 - 2.59808i) q^{9} +(3.63397 + 2.09808i) q^{10} +(6.31812 - 1.69293i) q^{11} +6.46410i q^{12} +(-0.785693 - 2.93225i) q^{15} +(1.23205 - 2.13397i) q^{16} +(5.07880 - 5.07880i) q^{18} +(-1.69293 + 6.31812i) q^{20} +(7.83013 + 13.5622i) q^{22} +(-6.93777 + 1.85897i) q^{24} +1.92820i q^{25} -5.19615 q^{27} +(6.29423 - 3.63397i) q^{30} +(-2.31259 - 0.619657i) q^{32} +(2.93225 - 10.9433i) q^{33} +(9.69615 + 5.59808i) q^{36} -7.26795 q^{40} +(2.02501 + 7.55743i) q^{41} +(-3.46410 + 2.00000i) q^{43} +(-17.2614 + 17.2614i) q^{44} +(-5.07880 - 1.36086i) q^{45} +(7.10381 + 7.10381i) q^{47} +(-2.13397 - 3.69615i) q^{48} +(-6.06218 - 3.50000i) q^{49} +(-4.45915 + 1.19482i) q^{50} +(-3.21983 - 12.0166i) q^{54} +(5.73205 - 9.92820i) q^{55} +(0.121547 - 0.453620i) q^{59} +(8.01105 + 8.01105i) q^{60} +(-6.92820 - 12.0000i) q^{61} -10.6603i q^{64} +27.1244 q^{66} +(-15.5685 - 4.17156i) q^{71} +(-3.21983 + 12.0166i) q^{72} +(2.89230 + 1.66987i) q^{75} -10.3923 q^{79} +(-1.11777 - 4.17156i) q^{80} +(-4.50000 + 7.79423i) q^{81} +(-16.2224 + 9.36603i) q^{82} +(8.91829 - 8.91829i) q^{83} +(-6.77174 - 6.77174i) q^{86} +(-23.4904 - 13.5622i) q^{88} +(-4.17156 + 1.11777i) q^{89} -12.5885i q^{90} +(-12.0263 + 20.8301i) q^{94} +(-2.93225 + 2.93225i) q^{96} +(4.33760 - 16.1881i) q^{98} +(-13.8755 - 13.8755i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.619657	+	2.31259i	0.438164	+	1.63525i	0.733380	+	0.679818i	$0.237941\pi$
$2$	0.619657	+	2.31259i	0.438164	+	1.63525i	−0.295217	+	0.955430i	$0.595392\pi$
$3$	0.866025	−	1.50000i	0.500000	−	0.866025i
$3$	0.866025	−	1.50000i	0.500000	−	0.866025i
$4$	−3.23205	+	1.86603i	−1.61603	+	0.933013i
$5$	1.23931	−	1.23931i	0.554238	−	0.554238i	−0.373423	−	0.927661i	$-0.621816\pi$
$5$	1.23931	−	1.23931i	0.554238	−	0.554238i	0.927661	+	0.373423i	$0.121816\pi$
$6$	4.00552	+	1.07328i	1.63525	+	0.438164i
$7$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$7$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$8$	−2.93225	−	2.93225i	−1.03671	−	1.03671i
$9$	−1.50000	−	2.59808i	−0.500000	−	0.866025i
$10$	3.63397	+	2.09808i	1.14916	+	0.663470i
$11$	6.31812	−	1.69293i	1.90498	−	0.510439i	0.909476	−	0.415756i	$-0.136483\pi$
$11$	6.31812	−	1.69293i	1.90498	−	0.510439i	0.995508	−	0.0946823i	$-0.0301835\pi$
$12$			6.46410i			1.86603i
$13$		0			0
$13$		0			0
$14$		0			0
$15$	−0.785693	−	2.93225i	−0.202865	−	0.757103i
$16$	1.23205	−	2.13397i	0.308013	−	0.533494i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$	5.07880	−	5.07880i	1.19709	−	1.19709i
$19$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$19$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$20$	−1.69293	+	6.31812i	−0.378552	+	1.41277i
$21$		0			0
$22$	7.83013	+	13.5622i	1.66939	+	2.89147i
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$	−6.93777	+	1.85897i	−1.41617	+	0.379461i
$25$			1.92820i			0.385641i
$26$		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	$0.666667\pi$
$30$	6.29423	−	3.63397i	1.14916	−	0.663470i
$31$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$31$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$32$	−2.31259	−	0.619657i	−0.408812	−	0.109541i
$33$	2.93225	−	10.9433i	0.510439	−	1.90498i
$34$		0			0
$35$		0			0
$36$	9.69615	+	5.59808i	1.61603	+	0.933013i
$37$		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
$40$	−7.26795			−1.14916
$41$	2.02501	+	7.55743i	0.316253	+	1.18027i	0.922817	+	0.385238i	$0.125881\pi$
$41$	2.02501	+	7.55743i	0.316253	+	1.18027i	−0.606564	+	0.795034i	$0.707453\pi$
$42$		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	$0.431989\pi$
$44$	−17.2614	+	17.2614i	−2.60226	+	2.60226i
$45$	−5.07880	−	1.36086i	−0.757103	−	0.202865i
$46$		0			0
$47$	7.10381	+	7.10381i	1.03620	+	1.03620i	0.999320	+	0.0368772i	$0.0117410\pi$
$47$	7.10381	+	7.10381i	1.03620	+	1.03620i	0.0368772	+	0.999320i	$0.488259\pi$
$48$	−2.13397	−	3.69615i	−0.308013	−	0.533494i
$49$	−6.06218	−	3.50000i	−0.866025	−	0.500000i
$50$	−4.45915	+	1.19482i	−0.630618	+	0.168974i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−3.21983	−	12.0166i	−0.438164	−	1.63525i
$55$	5.73205	−	9.92820i	0.772910	−	1.33872i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.121547	−	0.453620i	0.0158241	−	0.0590563i	−0.957562	−	0.288226i	$-0.906934\pi$
$59$	0.121547	−	0.453620i	0.0158241	−	0.0590563i	0.973386	+	0.229170i	$0.0736011\pi$
$60$	8.01105	+	8.01105i	1.03422	+	1.03422i
$61$	−6.92820	−	12.0000i	−0.887066	−	1.53644i	−0.843328	−	0.537400i	$-0.819407\pi$
$61$	−6.92820	−	12.0000i	−0.887066	−	1.53644i	−0.0437377	−	0.999043i	$-0.513927\pi$
$62$		0			0
$63$		0			0
$64$		−	10.6603i		−	1.33253i
$65$		0			0
$66$	27.1244			3.33878
$67$		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
$71$	−15.5685	−	4.17156i	−1.84764	−	0.495073i	−0.848235	−	0.529619i	$-0.822335\pi$
$71$	−15.5685	−	4.17156i	−1.84764	−	0.495073i	−0.999403	+	0.0345462i	$0.989001\pi$
$72$	−3.21983	+	12.0166i	−0.379461	+	1.41617i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		0			0
$75$	2.89230	+	1.66987i	0.333975	+	0.192820i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−10.3923			−1.16923			−0.584613	−	0.811312i	$-0.698754\pi$
$79$	−10.3923			−1.16923			−0.584613	+	0.811312i	$0.698754\pi$
$80$	−1.11777	−	4.17156i	−0.124970	−	0.466395i
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$	−16.2224	+	9.36603i	−1.79147	+	1.03430i
$83$	8.91829	−	8.91829i	0.978910	−	0.978910i	−0.0208726	−	0.999782i	$-0.506644\pi$
$83$	8.91829	−	8.91829i	0.978910	−	0.978910i	0.999782	+	0.0208726i	$0.00664445\pi$
$84$		0			0
$85$		0			0
$86$	−6.77174	−	6.77174i	−0.730215	−	0.730215i
$87$		0			0
$88$	−23.4904	−	13.5622i	−2.50408	−	1.44573i
$89$	−4.17156	+	1.11777i	−0.442185	+	0.118483i	−0.473040	−	0.881041i	$-0.656843\pi$
$89$	−4.17156	+	1.11777i	−0.442185	+	0.118483i	0.0308556	+	0.999524i	$0.490177\pi$
$90$		−	12.5885i		−	1.32694i
$91$		0			0
$92$		0			0
$93$		0			0
$94$	−12.0263	+	20.8301i	−1.24042	+	2.14846i
$95$		0			0
$96$	−2.93225	+	2.93225i	−0.299271	+	0.299271i
$97$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$97$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$98$	4.33760	−	16.1881i	0.438164	−	1.63525i
$99$	−13.8755	−	13.8755i	−1.39454	−	1.39454i
$100$	−3.59808	−	6.23205i	−0.359808	−	0.623205i
$101$		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	$0.666667\pi$
$108$	16.7942	−	9.69615i	1.61603	−	0.933013i
$109$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$109$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$110$	26.5118	+	7.10381i	2.52780	+	0.677322i
$111$		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
$118$	1.12436			0.103505
$119$		0			0
$120$	−6.29423	+	10.9019i	−0.574582	+	0.995205i
$121$	27.5263	−	15.8923i	2.50239	−	1.44475i
$122$	23.4580	−	23.4580i	2.12379	−	2.12379i
$123$	13.0899	+	3.50742i	1.18027	+	0.316253i
$124$		0			0
$125$	8.58622	+	8.58622i	0.767975	+	0.767975i
$126$		0			0
$127$	−15.0000	−	8.66025i	−1.33103	−	0.768473i	−0.345576	−	0.938391i	$-0.612317\pi$
$127$	−15.0000	−	8.66025i	−1.33103	−	0.768473i	−0.985458	+	0.169917i	$0.945650\pi$
$128$	20.0276	−	5.36639i	1.77021	−	0.474326i
$129$			6.92820i			0.609994i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$	10.9433	+	40.8409i	0.952492	+	3.55475i
$133$		0			0
$134$		0			0
$135$	−6.43966	+	6.43966i	−0.554238	+	0.554238i
$136$		0			0
$137$	−4.74673	+	17.7150i	−0.405540	+	1.51350i	0.397516	+	0.917595i	$0.369872\pi$
$137$	−4.74673	+	17.7150i	−0.405540	+	1.51350i	−0.803057	+	0.595902i	$0.796795\pi$
$138$		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
$141$	16.8078	−	4.50363i	1.41547	−	0.379274i
$142$		−	38.5885i		−	3.23827i
$143$		0			0
$144$	−7.39230			−0.616025
$145$		0			0
$146$		0			0
$147$	−10.5000	+	6.06218i	−0.866025	+	0.500000i
$148$		0			0
$149$	23.5795	+	6.31812i	1.93171	+	0.517600i	0.968987	+	0.247112i	$0.0794817\pi$
$149$	23.5795	+	6.31812i	1.93171	+	0.517600i	0.962723	+	0.270488i	$0.0871850\pi$
$150$	−2.06950	+	7.72347i	−0.168974	+	0.630618i
$151$		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	$0.474569\pi$
$158$	−6.43966	−	24.0331i	−0.512312	−	1.91197i
$159$		0			0
$160$	−3.63397	+	2.09808i	−0.287291	+	0.165867i
$161$		0			0
$162$	−20.8133	−	5.57691i	−1.63525	−	0.438164i
$163$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$163$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$164$	−20.6473	−	20.6473i	−1.61228	−	1.61228i
$165$	−9.92820	−	17.1962i	−0.772910	−	1.33872i
$166$	26.1506	+	15.0981i	2.02968	+	1.17184i
$167$	−22.3402	+	5.98604i	−1.72874	+	0.463214i	−0.979892	−	0.199530i	$-0.936058\pi$
$167$	−22.3402	+	5.98604i	−1.72874	+	0.463214i	−0.748846	+	0.662744i	$0.769392\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$		0			0
$172$	7.46410	−	12.9282i	0.569132	−	0.985766i
$173$		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
$176$	4.17156	−	15.5685i	0.314443	−	1.17352i
$177$	−0.575167	−	0.575167i	−0.0432322	−	0.0432322i
$178$	−5.16987	−	8.95448i	−0.387498	−	0.671167i
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$	18.9543	−	5.07880i	1.41277	−	0.378552i
$181$		−	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
$188$	−36.2158	−	9.70398i	−2.64131	−	0.707736i
$189$		0			0
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	−15.9904	−	9.23205i	−1.15401	−	0.666266i
$193$		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
$196$	26.1244			1.86603
$197$	−3.59639	−	13.4219i	−0.256232	−	0.956273i	−0.967401	−	0.253251i	$-0.918500\pi$
$197$	−3.59639	−	13.4219i	−0.256232	−	0.956273i	0.711168	−	0.703022i	$-0.248166\pi$
$198$	23.4904	−	40.6865i	1.66939	−	2.89147i
$199$	−21.0000	+	12.1244i	−1.48865	+	0.859473i	−0.999916	−	0.0129598i	$-0.995875\pi$
$199$	−21.0000	+	12.1244i	−1.48865	+	0.859473i	−0.488735	+	0.872433i	$0.662541\pi$
$200$	5.65397	−	5.65397i	0.399796	−	0.399796i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	11.8756	+	6.85641i	0.829431	+	0.478872i
$206$	−37.0015	+	9.91451i	−2.57801	+	0.690777i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	1.73205	−	3.00000i	0.119239	−	0.206529i	−0.800227	−	0.599697i	$-0.795288\pi$
$211$	1.73205	−	3.00000i	0.119239	−	0.206529i	0.919466	+	0.393169i	$0.128621\pi$
$212$		0			0
$213$	−19.7400	+	19.7400i	−1.35257	+	1.35257i
$214$		0			0
$215$	−1.81448	+	6.77174i	−0.123747	+	0.461829i
$216$	15.2364	+	15.2364i	1.03671	+	1.03671i
$217$		0			0
$218$		0			0
$219$		0			0
$220$			42.7846i			2.88454i
$221$		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
$225$	5.00962	−	2.89230i	0.333975	−	0.192820i
$226$		0			0
$227$	5.41087	+	1.44984i	0.359132	+	0.0962292i	0.433874	−	0.900974i	$-0.357146\pi$
$227$	5.41087	+	1.44984i	0.359132	+	0.0962292i	−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
$235$	17.6077			1.14860
$236$	0.453620	+	1.69293i	0.0295282	+	0.110201i
$237$	−9.00000	+	15.5885i	−0.584613	+	1.01258i
$238$		0			0
$239$	−12.0611	+	12.0611i	−0.780165	+	0.780165i	−0.979858	−	0.199693i	$-0.936005\pi$
$239$	−12.0611	+	12.0611i	−0.780165	+	0.780165i	0.199693	+	0.979858i	$0.436005\pi$
$240$	−7.22536	−	1.93603i	−0.466395	−	0.124970i
$241$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$241$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$242$	53.8092	+	53.8092i	3.45899	+	3.45899i
$243$	7.79423	+	13.5000i	0.500000	+	0.866025i
$244$	44.7846	+	25.8564i	2.86704	+	1.65529i
$245$	−11.8505	+	3.17534i	−0.757103	+	0.202865i
$246$			32.4449i			2.06861i
$247$		0			0
$248$		0			0
$249$	−5.65397	−	21.1009i	−0.358306	−	1.33722i
$250$	−14.5359	+	25.1769i	−0.919331	+	1.59233i
$251$		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
$254$	10.7328	−	40.0552i	0.673434	−	2.51329i
$255$		0			0
$256$	14.1603	+	24.5263i	0.885016	+	1.53289i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$	−16.0221	+	4.29311i	−0.997492	+	0.267277i
$259$		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	$0.666667\pi$
$264$	−40.6865	+	23.4904i	−2.50408	+	1.44573i
$265$		0			0
$266$		0			0
$267$	−1.93603	+	7.22536i	−0.118483	+	0.442185i
$268$		0			0
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$	−18.8827	−	10.9019i	−1.14916	−	0.663470i
$271$		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
$274$	−43.9090			−2.65264
$275$	3.26432	+	12.1826i	0.196846	+	0.734639i
$276$		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	$0.103497\pi$
$278$	−33.8587	+	33.8587i	−2.03071	+	2.03071i
$279$		0			0
$280$		0			0
$281$	−21.5545	−	21.5545i	−1.28583	−	1.28583i	−0.937293	−	0.348542i	$-0.886677\pi$
$281$	−21.5545	−	21.5545i	−1.28583	−	1.28583i	−0.348542	−	0.937293i	$-0.613323\pi$
$282$	20.8301	+	36.0788i	1.24042	+	2.14846i
$283$	−15.0000	−	8.66025i	−0.891657	−	0.514799i	−0.0171732	−	0.999853i	$-0.505467\pi$
$283$	−15.0000	−	8.66025i	−0.891657	−	0.514799i	−0.874484	+	0.485054i	$0.838800\pi$
$284$	58.1024	−	15.5685i	3.44774	−	0.923819i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	1.85897	+	6.93777i	0.109541	+	0.408812i
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.68915	+	10.0361i	−0.157102	+	0.586313i	0.841814	+	0.539767i	$0.181488\pi$
$293$	−2.68915	+	10.0361i	−0.157102	+	0.586313i	−0.998916	+	0.0465452i	$0.985179\pi$
$294$	−20.5257	−	20.5257i	−1.19709	−	1.19709i
$295$	−0.411543	−	0.712813i	−0.0239609	−	0.0415016i
$296$		0			0
$297$	−32.8299	+	8.79674i	−1.90498	+	0.510439i
$298$			58.4449i			3.38562i
$299$		0			0
$300$	−12.4641			−0.719615
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−23.4580	−	6.28555i	−1.34320	−	0.359909i
$306$		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	$-0.0653285\pi$
$313$	34.6410			1.95803			0.979013	+	0.203798i	$0.0653285\pi$
$314$	1.23931	+	4.62518i	0.0699385	+	0.261014i
$315$		0			0
$316$	33.5885	−	19.3923i	1.88950	−	1.09090i
$317$	22.2187	−	22.2187i	1.24792	−	1.24792i	0.291290	−	0.956635i	$-0.405916\pi$
$317$	22.2187	−	22.2187i	1.24792	−	1.24792i	0.956635	−	0.291290i	$-0.0940844\pi$
$318$		0			0
$319$		0			0
$320$	−13.2114	−	13.2114i	−0.738540	−	0.738540i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		−	33.5885i		−	1.86603i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	16.2224	−	28.0981i	0.895734	−	1.55146i
$329$		0			0
$330$	33.6156	−	33.6156i	1.85048	−	1.85048i
$331$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$331$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$332$	−12.1826	+	45.4661i	−0.668608	+	2.49528i
$333$		0			0
$334$	−27.6865	−	47.9545i	−1.51494	−	2.62395i
$335$		0			0
$336$		0			0
$337$			6.92820i			0.377403i	0.982034	+	0.188702i	$0.0604279\pi$
$337$			6.92820i			0.377403i	−0.982034	+	0.188702i	$0.939572\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	16.0221	+	4.29311i	0.863854	+	0.231469i
$345$		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
$352$	−15.6603			−0.834694
$353$	9.70398	+	36.2158i	0.516491	+	1.92757i	0.322517	+	0.946564i	$0.395471\pi$
$353$	9.70398	+	36.2158i	0.516491	+	1.92757i	0.193974	+	0.981007i	$0.437862\pi$
$354$	0.973721	−	1.68653i	0.0517527	−	0.0896382i
$355$	−24.4641	+	14.1244i	−1.29842	+	0.749643i
$356$	11.3969	−	11.3969i	0.604035	−	0.604035i
$357$		0			0
$358$		0			0
$359$	1.48241	+	1.48241i	0.0782385	+	0.0782385i	0.745143	−	0.666905i	$-0.232381\pi$
$359$	1.48241	+	1.48241i	0.0782385	+	0.0782385i	−0.666905	+	0.745143i	$0.732381\pi$
$360$	10.9019	+	18.8827i	0.574582	+	0.995205i
$361$	16.4545	+	9.50000i	0.866025	+	0.500000i
$362$	23.1259	−	6.19657i	1.21547	−	0.325684i
$363$		−	55.0526i		−	2.88951i
$364$		0			0
$365$		0			0
$366$	−14.8718	−	55.5022i	−0.777360	−	2.90115i
$367$	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
$369$	16.5973	−	16.5973i	0.864019	−	0.864019i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−6.92820	−	12.0000i	−0.358729	−	0.621336i	0.629020	−	0.777389i	$-0.283456\pi$
$373$	−6.92820	−	12.0000i	−0.358729	−	0.621336i	−0.987749	+	0.156053i	$0.950123\pi$
$374$		0			0
$375$	20.3152	−	5.44344i	1.04907	−	0.281098i
$376$		−	41.6603i		−	2.14846i
$377$		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
$383$	34.0692	+	9.12882i	1.74086	+	0.466461i	0.982637	−	0.185540i	$-0.0594034\pi$
$383$	34.0692	+	9.12882i	1.74086	+	0.466461i	0.758218	+	0.652001i	$0.226070\pi$
$384$	9.29485	−	34.6889i	0.474326	−	1.77021i
$385$		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
$392$	7.51294	+	28.0387i	0.379461	+	1.41617i
$393$		0			0
$394$	28.8109	−	16.6340i	1.45147	−	0.838008i
$395$	−12.8793	+	12.8793i	−0.648029	+	0.648029i
$396$	70.7386	+	18.9543i	3.55475	+	0.952492i
$397$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$397$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$398$	−41.0515	−	41.0515i	−2.05772	−	2.05772i
$399$		0			0
$400$	4.11474	+	2.37564i	0.205737	+	0.118782i
$401$	24.4868	−	6.56121i	1.22281	−	0.327651i	0.411035	−	0.911620i	$-0.365168\pi$
$401$	24.4868	−	6.56121i	1.22281	−	0.327651i	0.811776	+	0.583969i	$0.198501\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	4.08258	+	15.2364i	0.202865	+	0.757103i
$406$		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	$0.0833333\pi$
$410$	−8.49724	+	31.7121i	−0.419649	+	1.56615i
$411$	22.4618	+	22.4618i	1.10796	+	1.10796i
$412$	−29.8564	−	51.7128i	−1.47092	−	2.54771i
$413$		0			0
$414$		0			0
$415$		−	22.1051i		−	1.08510i
$416$		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	$0.750000\pi$
$422$	8.01105	+	2.14655i	0.389972	+	0.104493i
$423$	7.80052	−	29.1120i	0.379274	−	1.41547i
$424$		0			0
$425$		0			0
$426$	−57.8827	−	33.4186i	−2.80443	−	1.61914i
$427$		0			0
$428$		0			0
$429$		0			0
$430$	−16.7846			−0.809426
$431$	−2.84327	−	10.6112i	−0.136955	−	0.511125i	−0.999982	−	0.00596647i	$-0.998101\pi$
$431$	−2.84327	−	10.6112i	−0.136955	−	0.511125i	0.863027	−	0.505158i	$-0.168566\pi$
$432$	−6.40192	+	11.0885i	−0.308013	+	0.533494i
$433$	18.0000	−	10.3923i	0.865025	−	0.499422i	−0.000666943	−	1.00000i	$-0.500212\pi$
$433$	18.0000	−	10.3923i	0.865025	−	0.499422i	0.865692	+	0.500577i	$0.166879\pi$
$434$		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	$-0.736995\pi$
$440$	−45.9197	+	12.3042i	−2.18914	+	0.586578i
$441$			21.0000i			1.00000i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	−3.78461	+	6.55514i	−0.179408	+	0.310743i
$446$		0			0
$447$	29.8976	−	29.8976i	1.41411	−	1.41411i
$448$		0			0
$449$	10.6112	−	39.6016i	0.500775	−	1.86892i	0.00584565	−	0.999983i	$-0.498139\pi$
$449$	10.6112	−	39.6016i	0.500775	−	1.86892i	0.494929	−	0.868933i	$-0.335194\pi$
$450$	9.79296	+	9.79296i	0.461645	+	0.461645i
$451$	25.5885	+	44.3205i	1.20491	+	2.08697i
$452$		0			0
$453$		0			0
$454$			13.4115i			0.629435i
$455$		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
$461$	−33.7371	−	9.03984i	−1.57129	−	0.421027i	−0.635077	−	0.772448i	$-0.719032\pi$
$461$	−33.7371	−	9.03984i	−1.57129	−	0.421027i	−0.936217	+	0.351421i	$0.885698\pi$
$462$		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
$470$	10.9107	+	40.7194i	0.503274	+	1.87825i
$471$	1.73205	−	3.00000i	0.0798087	−	0.138233i
$472$	−1.68653	+	0.973721i	−0.0776290	+	0.0448191i
$473$	−18.5007	+	18.5007i	−0.850664	+	0.850664i
$474$	−41.6266	−	11.1538i	−1.91197	−	0.512312i
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−35.3660	−	20.4186i	−1.61760	−	0.933924i
$479$	27.2975	−	7.31433i	1.24725	−	0.334200i	0.425978	−	0.904733i	$-0.359930\pi$
$479$	27.2975	−	7.31433i	1.24725	−	0.334200i	0.821275	+	0.570533i	$0.193263\pi$
$480$			7.26795i			0.331735i
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−59.3109	+	102.729i	−2.69595	+	4.66952i
$485$		0			0
$486$	−26.3902	+	26.3902i	−1.19709	+	1.19709i
$487$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$487$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$488$	−14.8718	+	55.5022i	−0.673213	+	2.51247i
$489$		0			0
$490$	−14.6865	−	25.4378i	−0.663470	−	1.14916i
$491$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$491$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$492$	−48.8520	+	13.0899i	−2.20242	+	0.590136i
$493$		0			0
$494$		0			0
$495$	−34.3923			−1.54582
$496$		0			0
$497$		0			0
$498$	45.2942	−	26.1506i	2.02968	−	1.17184i
$499$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$499$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$500$	−43.7732	−	11.7290i	−1.95760	−	0.524536i
$501$	−10.3681	+	38.6944i	−0.463214	+	1.72874i
$502$		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
$508$	64.6410			2.86798
$509$	−11.2754	−	42.0803i	−0.499772	−	1.86517i	−0.501468	−	0.865176i	$-0.667207\pi$
$509$	−11.2754	−	42.0803i	−0.499772	−	1.86517i	0.00169644	−	0.999999i	$-0.499460\pi$
$510$		0			0
$511$		0			0
$512$	−18.6223	+	18.6223i	−0.822996	+	0.822996i
$513$		0			0
$514$		0			0
$515$	19.8290	+	19.8290i	0.873771	+	0.873771i
$516$	−12.9282	−	22.3923i	−0.569132	−	0.985766i
$517$	56.9090	+	32.8564i	2.50285	+	1.44502i
$518$		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
$528$	−19.7400	−	19.7400i	−0.859075	−	0.859075i
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$	−1.36086	+	0.364642i	−0.0590563	+	0.0158241i
$532$		0			0
$533$		0			0
$534$	−17.9090			−0.774997
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−44.2268	−	11.8505i	−1.90498	−	0.510439i
$540$	8.79674	−	32.8299i	0.378552	−	1.41277i
$541$		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	$0.295725\pi$
$548$	−17.7150	−	66.1134i	−0.756749	−	2.82422i
$549$	−20.7846	+	36.0000i	−0.887066	+	1.53644i
$550$	−26.1506	+	15.0981i	−1.11507	+	0.643784i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	37.2445	+	37.2445i	1.58237	+	1.58237i
$555$		0			0
$556$	−64.6410	−	37.3205i	−2.74139	−	1.58274i
$557$	45.4661	−	12.1826i	1.92646	−	0.516194i	0.943910	−	0.330203i	$-0.107117\pi$
$557$	45.4661	−	12.1826i	1.92646	−	0.516194i	0.982551	−	0.185991i	$-0.0595495\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	36.4904	−	63.2032i	1.53925	−	2.66607i
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$	−45.9197	+	45.9197i	−1.93357	+	1.93357i
$565$		0			0
$566$	10.7328	−	40.0552i	0.451132	−	1.68365i
$567$		0			0
$568$	33.4186	+	57.8827i	1.40221	+	2.42870i
$569$		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	$-0.706248\pi$
$571$		−	38.1051i		−	1.59465i	0.603550	−	0.797325i	$-0.293752\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−27.6962	+	15.9904i	−1.15401	+	0.666266i
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	39.3140	+	10.5342i	1.63525	+	0.438164i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−24.8756			−1.02760
$587$	−6.40709	−	23.9116i	−0.264449	−	0.986937i	−0.962587	−	0.270974i	$-0.912654\pi$
$587$	−6.40709	−	23.9116i	−0.264449	−	0.986937i	0.698138	−	0.715964i	$-0.254012\pi$
$588$	22.6244	−	39.1865i	0.933013	−	1.61603i
$589$		0			0
$590$	1.39343	−	1.39343i	0.0573666	−	0.0573666i
$591$	−23.2475	−	6.22914i	−0.956273	−	0.256232i
$592$		0			0
$593$	20.4042	+	20.4042i	0.837900	+	0.837900i	0.988582	−	0.150683i	$-0.0481472\pi$
$593$	20.4042	+	20.4042i	0.837900	+	0.837900i	−0.150683	+	0.988582i	$0.548147\pi$
$594$	−40.6865	−	70.4711i	−1.66939	−	2.89147i
$595$		0			0
$596$	−88.0000	+	23.5795i	−3.60462	+	0.965855i
$597$			42.0000i			1.71895i
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$	−3.58447	−	13.3774i	−0.146335	−	0.546132i
$601$	24.2487	−	42.0000i	0.989126	−	1.71322i	0.367193	−	0.930145i	$-0.380319\pi$
$601$	24.2487	−	42.0000i	0.989126	−	1.71322i	0.621932	−	0.783071i	$-0.286348\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	14.4181	−	53.8092i	0.586181	−	2.18766i
$606$		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
$610$		−	58.1436i		−	2.35417i
$611$		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
$615$	20.5692	−	11.8756i	0.829431	−	0.478872i
$616$		0			0
$617$	15.9006	+	4.26054i	0.640132	+	0.171523i	0.564263	−	0.825595i	$-0.309160\pi$
$617$	15.9006	+	4.26054i	0.640132	+	0.171523i	0.0758689	+	0.997118i	$0.475827\pi$
$618$	−17.1724	+	64.0884i	−0.690777	+	2.57801i
$619$		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
$625$	11.6410			0.465641
$626$	21.4655	+	80.1105i	0.857936	+	3.20186i
$627$		0			0
$628$	−6.46410	+	3.73205i	−0.257946	+	0.148925i
$629$		0			0
$630$		0			0
$631$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$631$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$632$	30.4728	+	30.4728i	1.21214	+	1.21214i
$633$	−3.00000	−	5.19615i	−0.119239	−	0.206529i
$634$	65.1506	+	37.6147i	2.58746	+	1.49387i
$635$	−29.3225	+	7.85693i	−1.16363	+	0.311793i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	12.5147	+	46.7054i	0.495073	+	1.84764i
$640$	18.1699	−	31.4711i	0.718227	−	1.24401i
$641$		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
$645$	8.58622	+	8.58622i	0.338082	+	0.338082i
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$	36.0497	−	9.65949i	1.41617	−	0.379461i
$649$		−	3.07180i		−	0.120579i
$650$		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
$656$	18.6223	+	4.98982i	0.727078	+	0.194820i
$657$		0			0
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$	64.1769	+	37.0526i	2.49808	+	1.44227i
$661$		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
$664$	−52.3013			−2.02968
$665$		0			0
$666$		0			0
$667$		0			0
$668$	61.0346	−	61.0346i	2.36150	−	2.36150i
$669$		0			0
$670$		0			0
$671$	−64.0884	−	64.0884i	−2.47411	−	2.47411i
$672$		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	$0.753634\pi$
$674$	−16.0221	+	4.29311i	−0.617148	+	0.165364i
$675$		−	10.0192i		−	0.385641i
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	6.86071	−	6.86071i	0.262903	−	0.262903i
$682$		0			0
$683$	13.4219	−	50.0913i	0.513576	−	1.91669i	0.136011	−	0.990707i	$-0.456572\pi$
$683$	13.4219	−	50.0913i	0.513576	−	1.91669i	0.377565	−	0.925983i	$-0.376762\pi$
$684$		0			0
$685$	16.0718	+	27.8372i	0.614072	+	1.06360i
$686$		0			0
$687$		0			0
$688$			9.85641i			0.375772i
$689$		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
$695$	33.8587	+	9.07241i	1.28433	+	0.344136i
$696$		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
$704$	−18.0471	−	67.3527i	−0.680176	−	2.53845i
$705$	15.2487	−	26.4115i	0.574300	−	0.994716i
$706$	−77.7391	+	44.8827i	−2.92575	+	1.68918i
$707$		0			0
$708$	2.93225	+	0.785693i	0.110201	+	0.0295282i
$709$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$709$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$710$	−47.8232	−	47.8232i	−1.79477	−	1.79477i
$711$	15.5885	+	27.0000i	0.584613	+	1.01258i
$712$	15.5096	+	8.95448i	0.581248	+	0.335583i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	7.64641	+	28.5368i	0.285560	+	1.06573i
$718$	−2.50962	+	4.34679i	−0.0936581	+	0.162221i
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$	−9.16138	+	9.16138i	−0.341425	+	0.341425i
$721$		0			0
$722$	−11.7735	+	43.9392i	−0.438164	+	1.63525i
$723$		0			0
$724$	18.6603	+	32.3205i	0.693503	+	1.20118i
$725$		0			0
$726$	127.314	−	34.1137i	4.72507	−	1.26608i
$727$			51.9615i			1.92715i	0.267445	+	0.963573i	$0.413821\pi$
$727$			51.9615i			1.92715i	−0.267445	+	0.963573i	$0.586179\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$	77.5692	−	44.7846i	2.86704	−	1.65529i
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$	18.5007	+	4.95725i	0.682874	+	0.182976i
$735$	−5.49985	+	20.5257i	−0.202865	+	0.757103i
$736$		0			0
$737$		0			0
$738$	48.6673	+	28.0981i	1.79147	+	1.03430i
$739$		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
$743$	1.27188	+	4.74673i	0.0466608	+	0.174141i	0.985324	−	0.170695i	$-0.0546013\pi$
$743$	1.27188	+	4.74673i	0.0466608	+	0.174141i	−0.938663	+	0.344836i	$0.887935\pi$
$744$		0			0
$745$	37.0526	−	21.3923i	1.35750	−	0.783753i
$746$	23.4580	−	23.4580i	0.858858	−	0.858858i
$747$	−36.5478	−	9.79296i	−1.33722	−	0.358306i
$748$		0			0
$749$		0			0
$750$	25.1769	+	43.6077i	0.919331	+	1.59233i
$751$	−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	$0.927058\pi$
$752$	23.9116	−	6.40709i	0.871966	−	0.233643i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−20.7846	+	36.0000i	−0.755429	+	1.30844i	0.189731	+	0.981836i	$0.439238\pi$
$757$	−20.7846	+	36.0000i	−0.755429	+	1.30844i	−0.945161	+	0.326606i	$0.894095\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−12.4257	+	46.3734i	−0.450431	+	1.68103i	0.250751	+	0.968052i	$0.419322\pi$
$761$	−12.4257	+	46.3734i	−0.450431	+	1.68103i	−0.701183	+	0.712982i	$0.747344\pi$
$762$	−50.7880	−	50.7880i	−1.83986	−	1.83986i
$763$		0			0
$764$		0			0
$765$		0			0
$766$			84.4449i			3.05112i
$767$		0			0
$768$	49.0526			1.77003
$769$		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
$773$	52.2379	+	13.9971i	1.87887	+	0.503440i	0.999634	+	0.0270446i	$0.00860962\pi$
$773$	52.2379	+	13.9971i	1.87887	+	0.503440i	0.879231	+	0.476396i	$0.158057\pi$
$774$	−7.43588	+	27.7511i	−0.267277	+	0.997492i
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−105.426			−3.77243
$782$		0			0
$783$		0			0
$784$	−14.9378	+	8.62436i	−0.533494	+	0.308013i
$785$	2.47863	−	2.47863i	0.0884660	−	0.0884660i
$786$		0			0
$787$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$787$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$788$	36.6694	+	36.6694i	1.30629	+	1.30629i
$789$		0			0
$790$	−37.7654	−	21.8038i	−1.34363	−	0.775746i
$791$		0			0
$792$			81.3731i			2.89147i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	45.2487	−	78.3731i	1.60380	−	2.77786i
$797$		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
$800$	1.19482	−	4.45915i	0.0422434	−	0.157655i
$801$	9.16138	+	9.16138i	0.323702	+	0.323702i
$802$	30.3468	+	52.5622i	1.07158	+	1.85604i
$803$		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	$0.666667\pi$
$810$	−32.7058	+	18.8827i	−1.14916	+	0.663470i
$811$		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
$820$	−51.1769			−1.78718
$821$	−0.0325685	−	0.121547i	−0.00113665	−	0.00424203i	0.965355	−	0.260939i	$-0.0840323\pi$
$821$	−0.0325685	−	0.121547i	−0.00113665	−	0.00424203i	−0.966492	+	0.256697i	$0.917366\pi$
$822$	−38.0263	+	65.8634i	−1.32632	+	2.29725i
$823$	−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	$0.903185\pi$
$824$	46.9160	−	46.9160i	1.63439	−	1.63439i
$825$	21.1009	+	5.65397i	0.734639	+	0.196846i
$826$		0			0
$827$	−19.4969	−	19.4969i	−0.677975	−	0.677975i	0.281566	−	0.959542i	$-0.409146\pi$
$827$	−19.4969	−	19.4969i	−0.677975	−	0.677975i	−0.959542	+	0.281566i	$0.909146\pi$
$828$		0			0
$829$	24.0000	+	13.8564i	0.833554	+	0.481253i	0.855068	−	0.518516i	$-0.173515\pi$
$829$	24.0000	+	13.8564i	0.833554	+	0.481253i	−0.0215137	+	0.999769i	$0.506849\pi$
$830$	51.1201	−	13.6976i	1.77440	−	0.475450i
$831$		−	38.1051i		−	1.32185i
$832$		0			0
$833$		0			0
$834$	21.4655	+	80.1105i	0.743291	+	2.77400i
$835$	−20.2679	+	35.1051i	−0.701401	+	1.21486i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−9.61500	+	35.8837i	−0.331947	+	1.23884i	0.575195	+	0.818017i	$0.304926\pi$
$839$	−9.61500	+	35.8837i	−0.331947	+	1.23884i	−0.907141	+	0.420826i	$0.861740\pi$
$840$		0			0
$841$	−14.5000	−	25.1147i	−0.500000	−	0.866025i
$842$		0			0
$843$	−50.9985	+	13.6650i	−1.75648	+	0.470648i
$844$			12.9282i			0.445007i
$845$		0			0
$846$	72.1577			2.48083
$847$		0			0
$848$		0			0
$849$	−25.9808	+	15.0000i	−0.891657	+	0.514799i
$850$		0			0
$851$		0			0
$852$	26.9654	−	100.636i	0.923819	−	3.44774i
$853$		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	$-0.556733\pi$
$859$	−10.3923			−0.354581			−0.177290	+	0.984159i	$0.556733\pi$
$860$	−6.77174	−	25.2725i	−0.230914	−	0.861784i
$861$		0			0
$862$	22.7776	−	13.1506i	0.775807	−	0.447912i
$863$	24.2762	−	24.2762i	0.826373	−	0.826373i	−0.160640	−	0.987013i	$-0.551356\pi$
$863$	24.2762	−	24.2762i	0.826373	−	0.826373i	0.987013	+	0.160640i	$0.0513559\pi$
$864$	12.0166	+	3.21983i	0.408812	+	0.109541i
$865$		0			0
$866$	35.1870	+	35.1870i	1.19570	+	1.19570i
$867$	−14.7224	−	25.5000i	−0.500000	−	0.866025i
$868$		0			0
$869$	−65.6598	+	17.5935i	−2.22736	+	0.596818i
$870$		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	$0.0833333\pi$
$878$	24.7863	−	92.5036i	0.836496	−	3.12185i
$879$	12.7252	+	12.7252i	0.429211	+	0.429211i
$880$	−14.1244	−	24.4641i	−0.476132	−	0.824685i
$881$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$881$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$882$	−48.5644	+	13.0128i	−1.63525	+	0.438164i
$883$			51.9615i			1.74864i	0.485346	+	0.874322i	$0.338694\pi$
$883$			51.9615i			1.74864i	−0.485346	+	0.874322i	$0.661306\pi$
$884$		0			0
$885$	−1.42563			−0.0479219
$886$		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
$890$	−17.5045	−	4.69032i	−0.586752	−	0.157220i
$891$	−15.2364	+	56.8630i	−0.510439	+	1.90498i
$892$		0			0
$893$		0			0
$894$	87.6673	+	50.6147i	2.93203	+	1.69281i
$895$		0			0
$896$		0			0
$897$		0			0
$898$	98.1577			3.27556
$899$		0			0
$900$	−10.7942	+	18.6962i	−0.359808	+	0.623205i
$901$		0			0
$902$	−86.6391	+	86.6391i	−2.88477	+	2.88477i
$903$		0			0
$904$		0			0
$905$	−12.3931	−	12.3931i	−0.411962	−	0.411962i
$906$		0			0
$907$	−15.0000	−	8.66025i	−0.498067	−	0.287559i	0.229848	−	0.973227i	$-0.426177\pi$
$907$	−15.0000	−	8.66025i	−0.498067	−	0.287559i	−0.727915	+	0.685668i	$0.759510\pi$
$908$	−20.1937	+	5.41087i	−0.670150	+	0.179566i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	41.2487	−	71.4449i	1.36513	−	2.36448i
$914$		0			0
$915$	−29.7435	+	29.7435i	−0.983291	+	0.983291i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−29.4449	−	51.0000i	−0.971296	−	1.68233i	−0.691652	−	0.722231i	$-0.743117\pi$
$919$	−29.4449	−	51.0000i	−0.971296	−	1.68233i	−0.279645	−	0.960104i	$-0.590217\pi$
$920$		0			0
$921$		0			0
$922$		−	83.6218i		−	2.75394i
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	41.5692	−	24.0000i	1.36531	−	0.788263i
$928$		0			0
$929$	−12.7578	−	3.41844i	−0.418569	−	0.112155i	0.0433864	−	0.999058i	$-0.486185\pi$
$929$	−12.7578	−	3.41844i	−0.418569	−	0.112155i	−0.461955	+	0.886903i	$0.652852\pi$
$930$		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	$-0.308553\pi$
$937$	34.6410			1.13167			0.565836	+	0.824518i	$0.308553\pi$
$938$		0			0
$939$	30.0000	−	51.9615i	0.979013	−	1.69570i
$940$	−56.9090	+	32.8564i	−1.85617	+	1.07166i
$941$	−40.7194	+	40.7194i	−1.32741	+	1.32741i	−0.419796	+	0.907619i	$0.637898\pi$
$941$	−40.7194	+	40.7194i	−1.32741	+	1.32741i	−0.907619	+	0.419796i	$0.862102\pi$
$942$	8.01105	+	2.14655i	0.261014	+	0.0699385i
$943$		0			0
$944$	−0.818262	−	0.818262i	−0.0266322	−	0.0266322i
$945$		0			0
$946$	−54.2487	−	31.3205i	−1.76378	−	1.01832i
$947$	55.9558	−	14.9933i	1.81832	−	0.487217i	0.821740	−	0.569862i	$-0.193004\pi$
$947$	55.9558	−	14.9933i	1.81832	−	0.487217i	0.996579	+	0.0826452i	$0.0263368\pi$
$948$		−	67.1769i		−	2.18180i
$949$		0			0
$950$		0			0
$951$	−14.0861	−	52.5699i	−0.456772	−	1.70470i
$952$		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
$956$	16.4757	−	61.4882i	0.532863	−	1.98867i
$957$		0			0
$958$	33.8301	+	58.5955i	1.09300	+	1.89313i
$959$		0			0
$960$	−31.2585	+	8.37569i	−1.00886	+	0.270324i
$961$		−	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	$0.750000\pi$
$968$	−127.314	−	34.1137i	−4.09203	−	1.09646i
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$	−50.3827	−	29.0885i	−1.61603	−	0.933013i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−34.1436			−1.09291
$977$	−13.3329	−	49.7592i	−0.426559	−	1.59194i	−0.760495	−	0.649343i	$-0.775044\pi$
$977$	−13.3329	−	49.7592i	−0.426559	−	1.59194i	0.333937	−	0.942596i	$-0.391623\pi$
$978$		0			0
$979$	−24.4641	+	14.1244i	−0.781876	+	0.451416i
$980$	32.3763	−	32.3763i	1.03422	−	1.03422i
$981$		0			0
$982$		0			0
$983$	43.4411	+	43.4411i	1.38556	+	1.38556i	0.834406	+	0.551151i	$0.185811\pi$
$983$	43.4411	+	43.4411i	1.38556	+	1.38556i	0.551151	+	0.834406i	$0.314189\pi$
$984$	−28.0981	−	48.6673i	−0.895734	−	1.55146i
$985$	−21.0910	−	12.1769i	−0.672016	−	0.387989i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−21.3114	−	79.5353i	−0.677322	−	2.52780i
$991$	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
$995$	−10.9997	+	41.0515i	−0.348714	+	1.30142i
$996$	57.6487	+	57.6487i	1.82667	+	1.82667i
$997$	−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.g.488.2		8
3.2	odd	2	inner	507.2.k.g.488.1		8
13.2	odd	12	inner	507.2.k.g.80.1		8
13.3	even	3		507.2.k.h.89.1		8
13.4	even	6		507.2.f.d.437.1	yes	8
13.5	odd	4		507.2.k.h.188.2		8
13.6	odd	12		507.2.f.d.239.1	✓	8
13.7	odd	12		507.2.f.d.239.4	yes	8
13.8	odd	4		507.2.k.h.188.1		8
13.9	even	3		507.2.f.d.437.4	yes	8
13.10	even	6		507.2.k.h.89.2		8
13.11	odd	12	inner	507.2.k.g.80.2		8
13.12	even	2	inner	507.2.k.g.488.1		8
39.2	even	12	inner	507.2.k.g.80.2		8
39.5	even	4		507.2.k.h.188.1		8
39.8	even	4		507.2.k.h.188.2		8
39.11	even	12	inner	507.2.k.g.80.1		8
39.17	odd	6		507.2.f.d.437.4	yes	8
39.20	even	12		507.2.f.d.239.1	✓	8
39.23	odd	6		507.2.k.h.89.1		8
39.29	odd	6		507.2.k.h.89.2		8
39.32	even	12		507.2.f.d.239.4	yes	8
39.35	odd	6		507.2.f.d.437.1	yes	8
39.38	odd	2	CM	507.2.k.g.488.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.f.d.239.1	✓	8	13.6	odd	12
507.2.f.d.239.1	✓	8	39.20	even	12
507.2.f.d.239.4	yes	8	13.7	odd	12
507.2.f.d.239.4	yes	8	39.32	even	12
507.2.f.d.437.1	yes	8	13.4	even	6
507.2.f.d.437.1	yes	8	39.35	odd	6
507.2.f.d.437.4	yes	8	13.9	even	3
507.2.f.d.437.4	yes	8	39.17	odd	6
507.2.k.g.80.1		8	13.2	odd	12	inner
507.2.k.g.80.1		8	39.11	even	12	inner
507.2.k.g.80.2		8	13.11	odd	12	inner
507.2.k.g.80.2		8	39.2	even	12	inner
507.2.k.g.488.1		8	3.2	odd	2	inner
507.2.k.g.488.1		8	13.12	even	2	inner
507.2.k.g.488.2		8	1.1	even	1	trivial
507.2.k.g.488.2		8	39.38	odd	2	CM
507.2.k.h.89.1		8	13.3	even	3
507.2.k.h.89.1		8	39.23	odd	6
507.2.k.h.89.2		8	13.10	even	6
507.2.k.h.89.2		8	39.29	odd	6
507.2.k.h.188.1		8	13.8	odd	4
507.2.k.h.188.1		8	39.5	even	4
507.2.k.h.188.2		8	13.5	odd	4
507.2.k.h.188.2		8	39.8	even	4

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)

\(f(q)\)	\(=\)	\(q+(0.619657 + 2.31259i) q^{2} +(0.866025 - 1.50000i) q^{3} +(-3.23205 + 1.86603i) q^{4} +(1.23931 - 1.23931i) q^{5} +(4.00552 + 1.07328i) q^{6} +(-2.93225 - 2.93225i) q^{8} +(-1.50000 - 2.59808i) q^{9} +O(q^{10})\)	\(q+(0.619657 + 2.31259i) q^{2} +(0.866025 - 1.50000i) q^{3} +(-3.23205 + 1.86603i) q^{4} +(1.23931 - 1.23931i) q^{5} +(4.00552 + 1.07328i) q^{6} +(-2.93225 - 2.93225i) q^{8} +(-1.50000 - 2.59808i) q^{9} +(3.63397 + 2.09808i) q^{10} +(6.31812 - 1.69293i) q^{11} +6.46410i q^{12} +(-0.785693 - 2.93225i) q^{15} +(1.23205 - 2.13397i) q^{16} +(5.07880 - 5.07880i) q^{18} +(-1.69293 + 6.31812i) q^{20} +(7.83013 + 13.5622i) q^{22} +(-6.93777 + 1.85897i) q^{24} +1.92820i q^{25} -5.19615 q^{27} +(6.29423 - 3.63397i) q^{30} +(-2.31259 - 0.619657i) q^{32} +(2.93225 - 10.9433i) q^{33} +(9.69615 + 5.59808i) q^{36} -7.26795 q^{40} +(2.02501 + 7.55743i) q^{41} +(-3.46410 + 2.00000i) q^{43} +(-17.2614 + 17.2614i) q^{44} +(-5.07880 - 1.36086i) q^{45} +(7.10381 + 7.10381i) q^{47} +(-2.13397 - 3.69615i) q^{48} +(-6.06218 - 3.50000i) q^{49} +(-4.45915 + 1.19482i) q^{50} +(-3.21983 - 12.0166i) q^{54} +(5.73205 - 9.92820i) q^{55} +(0.121547 - 0.453620i) q^{59} +(8.01105 + 8.01105i) q^{60} +(-6.92820 - 12.0000i) q^{61} -10.6603i q^{64} +27.1244 q^{66} +(-15.5685 - 4.17156i) q^{71} +(-3.21983 + 12.0166i) q^{72} +(2.89230 + 1.66987i) q^{75} -10.3923 q^{79} +(-1.11777 - 4.17156i) q^{80} +(-4.50000 + 7.79423i) q^{81} +(-16.2224 + 9.36603i) q^{82} +(8.91829 - 8.91829i) q^{83} +(-6.77174 - 6.77174i) q^{86} +(-23.4904 - 13.5622i) q^{88} +(-4.17156 + 1.11777i) q^{89} -12.5885i q^{90} +(-12.0263 + 20.8301i) q^{94} +(-2.93225 + 2.93225i) q^{96} +(4.33760 - 16.1881i) q^{98} +(-13.8755 - 13.8755i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{4} - 12 q^{9}+O(q^{10}) \) 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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