LMFDB - Embedded newform 507.2.k.h.188.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("507.80");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.04841538248$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	8.0.56070144.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 $ x^8 - 4x^7 + 16x^6 - 34x^5 + 63x^4 - 74x^3 + 70x^2 - 38*x + 13
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label			188.1
Root			$0.500000 + 2.19293i$ of defining polynomial
Character	$\chi$	$=$	507.188
Dual form			507.2.k.h.89.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.31259 + 0.619657i) q^{2} +(0.866025 - 1.50000i) q^{3} +(3.23205 - 1.86603i) q^{4} +(1.23931 + 1.23931i) q^{5} +(-1.07328 + 4.00552i) q^{6} +(-2.93225 + 2.93225i) q^{8} +(-1.50000 - 2.59808i) q^{9} +O(q^{10})$	\(q+(-2.31259 + 0.619657i) q^{2} +(0.866025 - 1.50000i) q^{3} +(3.23205 - 1.86603i) q^{4} +(1.23931 + 1.23931i) q^{5} +(-1.07328 + 4.00552i) q^{6} +(-2.93225 + 2.93225i) q^{8} +(-1.50000 - 2.59808i) q^{9} +(-3.63397 - 2.09808i) q^{10} +(-1.69293 - 6.31812i) q^{11} -6.46410i q^{12} +(2.93225 - 0.785693i) q^{15} +(1.23205 - 2.13397i) q^{16} +(5.07880 + 5.07880i) q^{18} +(6.31812 + 1.69293i) q^{20} +(7.83013 + 13.5622i) q^{22} +(1.85897 + 6.93777i) q^{24} -1.92820i q^{25} -5.19615 q^{27} +(-6.29423 + 3.63397i) q^{30} +(0.619657 - 2.31259i) q^{32} +(-10.9433 - 2.93225i) q^{33} +(-9.69615 - 5.59808i) q^{36} -7.26795 q^{40} +(-7.55743 + 2.02501i) q^{41} +(3.46410 - 2.00000i) q^{43} +(-17.2614 - 17.2614i) q^{44} +(1.36086 - 5.07880i) q^{45} +(7.10381 - 7.10381i) q^{47} +(-2.13397 - 3.69615i) q^{48} +(6.06218 + 3.50000i) q^{49} +(1.19482 + 4.45915i) q^{50} +(12.0166 - 3.21983i) q^{54} +(5.73205 - 9.92820i) q^{55} +(-0.453620 - 0.121547i) q^{59} +(8.01105 - 8.01105i) q^{60} +(-6.92820 - 12.0000i) q^{61} +10.6603i q^{64} +27.1244 q^{66} +(4.17156 - 15.5685i) q^{71} +(12.0166 + 3.21983i) q^{72} +(-2.89230 - 1.66987i) q^{75} -10.3923 q^{79} +(4.17156 - 1.11777i) q^{80} +(-4.50000 + 7.79423i) q^{81} +(16.2224 - 9.36603i) q^{82} +(8.91829 + 8.91829i) q^{83} +(-6.77174 + 6.77174i) q^{86} +(23.4904 + 13.5622i) q^{88} +(1.11777 + 4.17156i) q^{89} +12.5885i q^{90} +(-12.0263 + 20.8301i) q^{94} +(-2.93225 - 2.93225i) q^{96} +(-16.1881 - 4.33760i) q^{98} +(-13.8755 + 13.8755i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$-1$	$e\left(\frac{5}{12}\right)$

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 $

$2$	−2.31259	+	0.619657i	−1.63525	+	0.438164i	−0.955430	−	0.295217i	$-0.904608\pi$
$2$	−2.31259	+	0.619657i	−1.63525	+	0.438164i	−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
$4$	3.23205	−	1.86603i	1.61603	−	0.933013i
$5$	1.23931	+	1.23931i	0.554238	+	0.554238i	0.927661	−	0.373423i	$-0.121816\pi$
$5$	1.23931	+	1.23931i	0.554238	+	0.554238i	−0.373423	+	0.927661i	$0.621816\pi$
$6$	−1.07328	+	4.00552i	−0.438164	+	1.63525i
$7$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$7$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$8$	−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
$11$	−1.69293	−	6.31812i	−0.510439	−	1.90498i	−0.415756	−	0.909476i	$-0.636483\pi$
$11$	−1.69293	−	6.31812i	−0.510439	−	1.90498i	−0.0946823	−	0.995508i	$-0.530184\pi$
$12$		−	6.46410i		−	1.86603i
$13$		0			0
$13$		0			0
$14$		0			0
$15$	2.93225	−	0.785693i	0.757103	−	0.202865i
$16$	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		0.258819	−	0.965926i	$-0.416667\pi$
$19$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$20$	6.31812	+	1.69293i	1.41277	+	0.378552i
$21$		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	$0.166667\pi$
$24$	1.85897	+	6.93777i	0.379461	+	1.41617i
$25$		−	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	$-0.750000\pi$
$31$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$32$	0.619657	−	2.31259i	0.109541	−	0.408812i
$33$	−10.9433	−	2.93225i	−1.90498	−	0.510439i
$34$		0			0
$35$		0			0
$36$	−9.69615	−	5.59808i	−1.61603	−	0.933013i
$37$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$37$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$38$		0			0
$39$		0			0
$40$	−7.26795			−1.14916
$41$	−7.55743	+	2.02501i	−1.18027	+	0.316253i	−0.795034	−	0.606564i	$-0.792547\pi$
$41$	−7.55743	+	2.02501i	−1.18027	+	0.316253i	−0.385238	+	0.922817i	$0.625881\pi$
$42$		0			0
$43$	3.46410	−	2.00000i	0.528271	−	0.304997i	−0.212041	−	0.977261i	$-0.568011\pi$
$43$	3.46410	−	2.00000i	0.528271	−	0.304997i	0.740312	+	0.672264i	$0.234678\pi$
$44$	−17.2614	−	17.2614i	−2.60226	−	2.60226i
$45$	1.36086	−	5.07880i	0.202865	−	0.757103i
$46$		0			0
$47$	7.10381	−	7.10381i	1.03620	−	1.03620i	0.0368772	−	0.999320i	$-0.488259\pi$
$47$	7.10381	−	7.10381i	1.03620	−	1.03620i	0.999320	−	0.0368772i	$-0.0117410\pi$
$48$	−2.13397	−	3.69615i	−0.308013	−	0.533494i
$49$	6.06218	+	3.50000i	0.866025	+	0.500000i
$50$	1.19482	+	4.45915i	0.168974	+	0.630618i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	12.0166	−	3.21983i	1.63525	−	0.438164i
$55$	5.73205	−	9.92820i	0.772910	−	1.33872i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.453620	−	0.121547i	−0.0590563	−	0.0158241i	0.229170	−	0.973386i	$-0.426399\pi$
$59$	−0.453620	−	0.121547i	−0.0590563	−	0.0158241i	−0.288226	+	0.957562i	$0.593066\pi$
$60$	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		0.965926	−	0.258819i	$-0.0833333\pi$
$67$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.17156	−	15.5685i	0.495073	−	1.84764i	−0.0345462	−	0.999403i	$-0.510999\pi$
$71$	4.17156	−	15.5685i	0.495073	−	1.84764i	0.529619	−	0.848235i	$-0.322335\pi$
$72$	12.0166	+	3.21983i	1.41617	+	0.379461i
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$		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	$0.698754\pi$
$80$	4.17156	−	1.11777i	0.466395	−	0.124970i
$81$	−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	0.999782	−	0.0208726i	$-0.00664445\pi$
$83$	8.91829	+	8.91829i	0.978910	+	0.978910i	−0.0208726	+	0.999782i	$0.506644\pi$
$84$		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
$89$	1.11777	+	4.17156i	0.118483	+	0.442185i	0.999524	−	0.0308556i	$-0.00982320\pi$
$89$	1.11777	+	4.17156i	0.118483	+	0.442185i	−0.881041	+	0.473040i	$0.843157\pi$
$90$			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		0.258819	−	0.965926i	$-0.416667\pi$
$97$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$98$	−16.1881	−	4.33760i	−1.63525	−	0.438164i
$99$	−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	$-0.710980\pi$
$103$		−	16.0000i		−	1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$		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	$-0.750000\pi$
$109$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$110$	−7.10381	+	26.5118i	−0.677322	+	2.52780i
$111$		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
$123$	−3.50742	+	13.0899i	−0.316253	+	1.18027i
$124$		0			0
$125$	8.58622	−	8.58622i	0.767975	−	0.767975i
$126$		0			0
$127$	15.0000	+	8.66025i	1.33103	+	0.768473i	0.985458	−	0.169917i	$-0.0543501\pi$
$127$	15.0000	+	8.66025i	1.33103	+	0.768473i	0.345576	+	0.938391i	$0.387683\pi$
$128$	−5.36639	−	20.0276i	−0.474326	−	1.77021i
$129$		−	6.92820i		−	0.609994i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$	−40.8409	+	10.9433i	−3.55475	+	0.952492i
$133$		0			0
$134$		0			0
$135$	−6.43966	−	6.43966i	−0.554238	−	0.554238i
$136$		0			0
$137$	17.7150	+	4.74673i	1.51350	+	0.405540i	0.917595	−	0.397516i	$-0.130128\pi$
$137$	17.7150	+	4.74673i	1.51350	+	0.405540i	0.595902	+	0.803057i	$0.296795\pi$
$138$		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
$141$	−4.50363	−	16.8078i	−0.379274	−	1.41547i
$142$			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
$149$	−6.31812	+	23.5795i	−0.517600	+	1.93171i	−0.247112	+	0.968987i	$0.579482\pi$
$149$	−6.31812	+	23.5795i	−0.517600	+	1.93171i	−0.270488	+	0.962723i	$0.587185\pi$
$150$	7.72347	+	2.06950i	0.630618	+	0.168974i
$151$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$151$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$152$		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	$0.474569\pi$
$158$	24.0331	−	6.43966i	1.91197	−	0.512312i
$159$		0			0
$160$	3.63397	−	2.09808i	0.287291	−	0.165867i
$161$		0			0
$162$	5.57691	−	20.8133i	0.438164	−	1.63525i
$163$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$163$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$164$	−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
$167$	5.98604	+	22.3402i	0.463214	+	1.72874i	0.662744	+	0.748846i	$0.269392\pi$
$167$	5.98604	+	22.3402i	0.463214	+	1.72874i	−0.199530	+	0.979892i	$0.563942\pi$
$168$		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
$176$	−15.5685	−	4.17156i	−1.17352	−	0.314443i
$177$	−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	$0.166667\pi$
$180$	−5.07880	−	18.9543i	−0.378552	−	1.41277i
$181$			10.0000i			0.743294i	0.928374	+	0.371647i	$0.121207\pi$
$181$			10.0000i			0.743294i	−0.928374	+	0.371647i	$0.878793\pi$
$182$		0			0
$183$	−24.0000			−1.77413
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	9.70398	−	36.2158i	0.707736	−	2.64131i
$189$		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		−0.258819	−	0.965926i	$-0.583333\pi$
$193$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$194$		0			0
$195$		0			0
$196$	26.1244			1.86603
$197$	13.4219	−	3.59639i	0.956273	−	0.256232i	0.253251	−	0.967401i	$-0.418500\pi$
$197$	13.4219	−	3.59639i	0.956273	−	0.256232i	0.703022	+	0.711168i	$0.251834\pi$
$198$	23.4904	−	40.6865i	1.66939	−	2.89147i
$199$	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	$0.00412534\pi$
$200$	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
$206$	9.91451	+	37.0015i	0.690777	+	2.57801i
$207$		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
$215$	6.77174	+	1.81448i	0.461829	+	0.123747i
$216$	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		0.965926	−	0.258819i	$-0.0833333\pi$
$223$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$224$		0			0
$225$	−5.00962	+	2.89230i	−0.333975	+	0.192820i
$226$		0			0
$227$	−1.44984	+	5.41087i	−0.0962292	+	0.359132i	−0.997203	−	0.0747413i	$-0.976187\pi$
$227$	−1.44984	+	5.41087i	−0.0962292	+	0.359132i	0.900974	+	0.433874i	$0.142854\pi$
$228$		0			0
$229$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$229$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$230$		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
$236$	−1.69293	+	0.453620i	−0.110201	+	0.0295282i
$237$	−9.00000	+	15.5885i	−0.584613	+	1.01258i
$238$		0			0
$239$	−12.0611	−	12.0611i	−0.780165	−	0.780165i	0.199693	−	0.979858i	$-0.436005\pi$
$239$	−12.0611	−	12.0611i	−0.780165	−	0.780165i	−0.979858	+	0.199693i	$0.936005\pi$
$240$	1.93603	−	7.22536i	0.124970	−	0.466395i
$241$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$241$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$242$	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
$245$	3.17534	+	11.8505i	0.202865	+	0.757103i
$246$		−	32.4449i		−	2.06861i
$247$		0			0
$248$		0			0
$249$	21.1009	−	5.65397i	1.33722	−	0.358306i
$250$	−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
$254$	−40.0552	−	10.7328i	−2.51329	−	0.673434i
$255$		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	$0.166667\pi$
$258$	4.29311	+	16.0221i	0.267277	+	0.997492i
$259$		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
$267$	7.22536	+	1.93603i	0.442185	+	0.118483i
$268$		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		−0.258819	−	0.965926i	$-0.583333\pi$
$271$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$272$		0			0
$273$		0			0
$274$	−43.9090			−2.65264
$275$	−12.1826	+	3.26432i	−0.734639	+	0.196846i
$276$		0			0
$277$	−19.0526	+	11.0000i	−1.14476	+	0.660926i	−0.947604	−	0.319447i	$-0.896503\pi$
$277$	−19.0526	+	11.0000i	−1.14476	+	0.660926i	−0.197153	+	0.980373i	$0.563170\pi$
$278$	−33.8587	−	33.8587i	−2.03071	−	2.03071i
$279$		0			0
$280$		0			0
$281$	−21.5545	+	21.5545i	−1.28583	+	1.28583i	−0.348542	+	0.937293i	$0.613323\pi$
$281$	−21.5545	+	21.5545i	−1.28583	+	1.28583i	−0.937293	+	0.348542i	$0.886677\pi$
$282$	20.8301	+	36.0788i	1.24042	+	2.14846i
$283$	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	$0.494533\pi$
$284$	−15.5685	−	58.1024i	−0.923819	−	3.44774i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−6.93777	+	1.85897i	−0.408812	+	0.109541i
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.0361	+	2.68915i	0.586313	+	0.157102i	0.539767	−	0.841814i	$-0.318512\pi$
$293$	10.0361	+	2.68915i	0.586313	+	0.157102i	0.0465452	+	0.998916i	$0.485179\pi$
$294$	−20.5257	+	20.5257i	−1.19709	+	1.19709i
$295$	−0.411543	−	0.712813i	−0.0239609	−	0.0415016i
$296$		0			0
$297$	8.79674	+	32.8299i	0.510439	+	1.90498i
$298$		−	58.4449i		−	3.38562i
$299$		0			0
$300$	−12.4641			−0.719615
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	6.28555	−	23.4580i	0.359909	−	1.34320i
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		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	$0.0653285\pi$
$314$	−4.62518	+	1.23931i	−0.261014	+	0.0699385i
$315$		0			0
$316$	−33.5885	+	19.3923i	−1.88950	+	1.09090i
$317$	22.2187	+	22.2187i	1.24792	+	1.24792i	0.956635	+	0.291290i	$0.0940844\pi$
$317$	22.2187	+	22.2187i	1.24792	+	1.24792i	0.291290	+	0.956635i	$0.405916\pi$
$318$		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		0.258819	−	0.965926i	$-0.416667\pi$
$331$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$332$	45.4661	+	12.1826i	2.49528	+	0.668608i
$333$		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	$-0.939572\pi$
$337$		−	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
$344$	−4.29311	+	16.0221i	−0.231469	+	0.863854i
$345$		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		−0.258819	−	0.965926i	$-0.583333\pi$
$349$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$350$		0			0
$351$		0			0
$352$	−15.6603			−0.834694
$353$	−36.2158	+	9.70398i	−1.92757	+	0.516491i	−0.946564	+	0.322517i	$0.895471\pi$
$353$	−36.2158	+	9.70398i	−1.92757	+	0.516491i	−0.981007	+	0.193974i	$0.937862\pi$
$354$	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	−0.666905	−	0.745143i	$-0.732381\pi$
$359$	1.48241	−	1.48241i	0.0782385	−	0.0782385i	0.745143	+	0.666905i	$0.232381\pi$
$360$	10.9019	+	18.8827i	0.574582	+	0.995205i
$361$	−16.4545	−	9.50000i	−0.866025	−	0.500000i
$362$	−6.19657	−	23.1259i	−0.325684	−	1.21547i
$363$			55.0526i			2.88951i
$364$		0			0
$365$		0			0
$366$	55.5022	−	14.8718i	2.90115	−	0.777360i
$367$	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
$375$	−5.44344	−	20.3152i	−0.281098	−	1.04907i
$376$			41.6603i			2.14846i
$377$		0			0
$378$		0			0
$379$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$379$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$380$		0			0
$381$	25.9808	−	15.0000i	1.33103	−	0.768473i
$382$		0			0
$383$	−9.12882	+	34.0692i	−0.466461	+	1.74086i	0.185540	+	0.982637i	$0.440597\pi$
$383$	−9.12882	+	34.0692i	−0.466461	+	1.74086i	−0.652001	+	0.758218i	$0.726070\pi$
$384$	−34.6889	−	9.29485i	−1.77021	−	0.474326i
$385$		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
$392$	−28.0387	+	7.51294i	−1.41617	+	0.379461i
$393$		0			0
$394$	−28.8109	+	16.6340i	−1.45147	+	0.838008i
$395$	−12.8793	−	12.8793i	−0.648029	−	0.648029i
$396$	−18.9543	+	70.7386i	−0.952492	+	3.55475i
$397$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$397$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$398$	−41.0515	+	41.0515i	−2.05772	+	2.05772i
$399$		0			0
$400$	−4.11474	−	2.37564i	−0.205737	−	0.118782i
$401$	−6.56121	−	24.4868i	−0.327651	−	1.22281i	−0.911620	−	0.411035i	$-0.865168\pi$
$401$	−6.56121	−	24.4868i	−0.327651	−	1.22281i	0.583969	−	0.811776i	$-0.301499\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−15.2364	+	4.08258i	−0.757103	+	0.202865i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$409$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$410$	31.7121	+	8.49724i	1.56615	+	0.419649i
$411$	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	$-0.750000\pi$
$421$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$422$	−2.14655	+	8.01105i	−0.104493	+	0.389972i
$423$	−29.1120	−	7.80052i	−1.41547	−	0.379274i
$424$		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
$431$	10.6112	−	2.84327i	0.511125	−	0.136955i	0.00596647	−	0.999982i	$-0.498101\pi$
$431$	10.6112	−	2.84327i	0.511125	−	0.136955i	0.505158	+	0.863027i	$0.331434\pi$
$432$	−6.40192	+	11.0885i	−0.308013	+	0.533494i
$433$	−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	$0.0703280\pi$
$439$	34.6410	+	20.0000i	1.65333	+	0.954548i	0.677634	+	0.735399i	$0.263005\pi$
$440$	12.3042	+	45.9197i	0.586578	+	2.18914i
$441$		−	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
$449$	−39.6016	−	10.6112i	−1.86892	−	0.500775i	−0.999983	−	0.00584565i	$-0.998139\pi$
$449$	−39.6016	−	10.6112i	−1.86892	−	0.500775i	−0.868933	−	0.494929i	$-0.835194\pi$
$450$	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		0.965926	−	0.258819i	$-0.0833333\pi$
$457$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9.03984	−	33.7371i	0.421027	−	1.57129i	−0.351421	−	0.936217i	$-0.614302\pi$
$461$	9.03984	−	33.7371i	0.421027	−	1.57129i	0.772448	−	0.635077i	$-0.219032\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		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
$470$	−40.7194	+	10.9107i	−1.87825	+	0.503274i
$471$	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
$474$	11.1538	−	41.6266i	0.512312	−	1.91197i
$475$		0			0
$476$		0			0
$477$		0			0
$478$	35.3660	+	20.4186i	1.61760	+	0.933924i
$479$	−7.31433	−	27.2975i	−0.334200	−	1.24725i	−0.904733	−	0.425978i	$-0.859930\pi$
$479$	−7.31433	−	27.2975i	−0.334200	−	1.24725i	0.570533	−	0.821275i	$-0.306737\pi$
$480$		−	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		0.258819	−	0.965926i	$-0.416667\pi$
$487$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$488$	55.5022	+	14.8718i	2.51247	+	0.673213i
$489$		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	$0.166667\pi$
$492$	13.0899	+	48.8520i	0.590136	+	2.20242i
$493$		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	$-0.750000\pi$
$499$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$500$	11.7290	−	43.7732i	0.524536	−	1.95760i
$501$	38.6944	+	10.3681i	1.72874	+	0.463214i
$502$		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
$509$	42.0803	−	11.2754i	1.86517	−	0.499772i	0.865176	−	0.501468i	$-0.167207\pi$
$509$	42.0803	−	11.2754i	1.86517	−	0.499772i	0.999999	+	0.00169644i	$0.000539995\pi$
$510$		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
$531$	0.364642	+	1.36086i	0.0158241	+	0.0590563i
$532$		0			0
$533$		0			0
$534$	−17.9090			−0.774997
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	11.8505	−	44.2268i	0.510439	−	1.90498i
$540$	−32.8299	−	8.79674i	−1.41277	−	0.378552i
$541$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$541$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$542$		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	$0.295725\pi$
$548$	66.1134	−	17.7150i	2.82422	−	0.756749i
$549$	−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
$557$	−12.1826	−	45.4661i	−0.516194	−	1.92646i	−0.330203	−	0.943910i	$-0.607117\pi$
$557$	−12.1826	−	45.4661i	−0.516194	−	1.92646i	−0.185991	−	0.982551i	$-0.559549\pi$
$558$		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
$566$	−40.0552	−	10.7328i	−1.68365	−	0.451132i
$567$		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	$0.293752\pi$
$571$			38.1051i			1.59465i	−0.603550	+	0.797325i	$0.706248\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	27.6962	−	15.9904i	1.15401	−	0.666266i
$577$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$577$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$578$	−10.5342	+	39.3140i	−0.438164	+	1.63525i
$579$		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
$587$	23.9116	−	6.40709i	0.986937	−	0.264449i	0.270974	−	0.962587i	$-0.412654\pi$
$587$	23.9116	−	6.40709i	0.986937	−	0.264449i	0.715964	+	0.698138i	$0.245988\pi$
$588$	22.6244	−	39.1865i	0.933013	−	1.61603i
$589$		0			0
$590$	1.39343	+	1.39343i	0.0573666	+	0.0573666i
$591$	6.22914	−	23.2475i	0.256232	−	0.956273i
$592$		0			0
$593$	20.4042	−	20.4042i	0.837900	−	0.837900i	−0.150683	−	0.988582i	$-0.548147\pi$
$593$	20.4042	−	20.4042i	0.837900	−	0.837900i	0.988582	+	0.150683i	$0.0481472\pi$
$594$	−40.6865	−	70.4711i	−1.66939	−	2.89147i
$595$		0			0
$596$	23.5795	+	88.0000i	0.965855	+	3.60462i
$597$		−	42.0000i		−	1.71895i
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$	13.3774	−	3.58447i	0.546132	−	0.146335i
$601$	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
$605$	−53.8092	−	14.4181i	−2.18766	−	0.586181i
$606$		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		0.965926	−	0.258819i	$-0.0833333\pi$
$613$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$614$		0			0
$615$	−20.5692	+	11.8756i	−0.829431	+	0.478872i
$616$		0			0
$617$	−4.26054	+	15.9006i	−0.171523	+	0.640132i	0.825595	+	0.564263i	$0.190840\pi$
$617$	−4.26054	+	15.9006i	−0.171523	+	0.640132i	−0.997118	+	0.0758689i	$0.975827\pi$
$618$	64.0884	+	17.1724i	2.57801	+	0.690777i
$619$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$619$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	11.6410			0.465641
$626$	−80.1105	+	21.4655i	−3.20186	+	0.857936i
$627$		0			0
$628$	6.46410	−	3.73205i	0.257946	−	0.148925i
$629$		0			0
$630$		0			0
$631$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$631$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$632$	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
$635$	7.85693	+	29.3225i	0.311793	+	1.16363i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−46.7054	+	12.5147i	−1.84764	+	0.495073i
$640$	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		0.258819	−	0.965926i	$-0.416667\pi$
$643$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$644$		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	$0.166667\pi$
$648$	−9.65949	−	36.0497i	−0.379461	−	1.41617i
$649$			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
$656$	−4.98982	+	18.6223i	−0.194820	+	0.727078i
$657$		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		−0.258819	−	0.965926i	$-0.583333\pi$
$661$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$662$		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	0.715134	−	0.698988i	$-0.246366\pi$
$673$	12.1244	+	7.00000i	0.467360	+	0.269830i	−0.247774	+	0.968818i	$0.579699\pi$
$674$	4.29311	+	16.0221i	0.165364	+	0.617148i
$675$			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
$683$	−50.0913	−	13.4219i	−1.91669	−	0.513576i	−0.990707	−	0.136011i	$-0.956572\pi$
$683$	−50.0913	−	13.4219i	−1.91669	−	0.513576i	−0.925983	−	0.377565i	$-0.876762\pi$
$684$		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		0.965926	−	0.258819i	$-0.0833333\pi$
$691$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−9.07241	+	33.8587i	−0.344136	+	1.28433i
$696$		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
$704$	67.3527	−	18.0471i	2.53845	−	0.680176i
$705$	15.2487	−	26.4115i	0.574300	−	0.994716i
$706$	77.7391	−	44.8827i	2.92575	−	1.68918i
$707$		0			0
$708$	−0.785693	+	2.93225i	−0.0295282	+	0.110201i
$709$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$709$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$710$	−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
$717$	−28.5368	+	7.64641i	−1.06573	+	0.285560i
$718$	−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
$722$	43.9392	+	11.7735i	1.63525	+	0.438164i
$723$		0			0
$724$	18.6603	+	32.3205i	0.693503	+	1.20118i
$725$		0			0
$726$	−34.1137	−	127.314i	−1.26608	−	4.72507i
$727$		−	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
$732$	−77.5692	+	44.7846i	−2.86704	+	1.65529i
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$	−4.95725	+	18.5007i	−0.182976	+	0.682874i
$735$	20.5257	+	5.49985i	0.757103	+	0.202865i
$736$		0			0
$737$		0			0
$738$	−48.6673	−	28.0981i	−1.79147	−	1.03430i
$739$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$739$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−4.74673	+	1.27188i	−0.174141	+	0.0466608i	−0.344836	−	0.938663i	$-0.612065\pi$
$743$	−4.74673	+	1.27188i	−0.174141	+	0.0466608i	0.170695	+	0.985324i	$0.445399\pi$
$744$		0			0
$745$	−37.0526	+	21.3923i	−1.35750	+	0.783753i
$746$	23.4580	+	23.4580i	0.858858	+	0.858858i
$747$	9.79296	−	36.5478i	0.358306	−	1.33722i
$748$		0			0
$749$		0			0
$750$	25.1769	+	43.6077i	0.919331	+	1.59233i
$751$	34.6410	+	20.0000i	1.26407	+	0.729810i	0.973859	−	0.227153i	$-0.0729417\pi$
$751$	34.6410	+	20.0000i	1.26407	+	0.729810i	0.290209	+	0.956963i	$0.406275\pi$
$752$	−6.40709	−	23.9116i	−0.233643	−	0.871966i
$753$		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
$761$	46.3734	+	12.4257i	1.68103	+	0.450431i	0.968052	−	0.250751i	$-0.0806776\pi$
$761$	46.3734	+	12.4257i	1.68103	+	0.450431i	0.712982	+	0.701183i	$0.247344\pi$
$762$	−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		0.965926	−	0.258819i	$-0.0833333\pi$
$769$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−13.9971	+	52.2379i	−0.503440	+	1.87887i	−0.0270446	+	0.999634i	$0.508610\pi$
$773$	−13.9971	+	52.2379i	−0.503440	+	1.87887i	−0.476396	+	0.879231i	$0.658057\pi$
$774$	27.7511	+	7.43588i	0.997492	+	0.267277i
$775$		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		−0.965926	−	0.258819i	$-0.916667\pi$
$787$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$788$	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
$800$	−4.45915	−	1.19482i	−0.157655	−	0.0422434i
$801$	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	$-0.750000\pi$
$811$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$812$		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
$821$	0.121547	−	0.0325685i	0.00424203	−	0.00113665i	−0.256697	−	0.966492i	$-0.582634\pi$
$821$	0.121547	−	0.0325685i	0.00424203	−	0.00113665i	0.260939	+	0.965355i	$0.415968\pi$
$822$	−38.0263	+	65.8634i	−1.32632	+	2.29725i
$823$	48.4974	−	28.0000i	1.69051	−	0.976019i	0.736413	−	0.676532i	$-0.236518\pi$
$823$	48.4974	−	28.0000i	1.69051	−	0.976019i	0.954100	−	0.299487i	$-0.0968155\pi$
$824$	46.9160	+	46.9160i	1.63439	+	1.63439i
$825$	−5.65397	+	21.1009i	−0.196846	+	0.734639i
$826$		0			0
$827$	−19.4969	+	19.4969i	−0.677975	+	0.677975i	−0.959542	−	0.281566i	$-0.909146\pi$
$827$	−19.4969	+	19.4969i	−0.677975	+	0.677975i	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	$0.826485\pi$
$830$	−13.6976	−	51.1201i	−0.475450	−	1.77440i
$831$			38.1051i			1.32185i
$832$		0			0
$833$		0			0
$834$	−80.1105	+	21.4655i	−2.77400	+	0.743291i
$835$	−20.2679	+	35.1051i	−0.701401	+	1.21486i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	35.8837	+	9.61500i	1.23884	+	0.331947i	0.818017	−	0.575195i	$-0.195074\pi$
$839$	35.8837	+	9.61500i	1.23884	+	0.331947i	0.420826	+	0.907141i	$0.361740\pi$
$840$		0			0
$841$	−14.5000	−	25.1147i	−0.500000	−	0.866025i
$842$		0			0
$843$	13.6650	+	50.9985i	0.470648	+	1.75648i
$844$		−	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
$852$	−100.636	−	26.9654i	−3.44774	−	0.923819i
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$		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	$0.556733\pi$
$860$	25.2725	−	6.77174i	0.861784	−	0.230914i
$861$		0			0
$862$	−22.7776	+	13.1506i	−0.775807	+	0.447912i
$863$	24.2762	+	24.2762i	0.826373	+	0.826373i	0.987013	−	0.160640i	$-0.0513559\pi$
$863$	24.2762	+	24.2762i	0.826373	+	0.826373i	−0.160640	+	0.987013i	$0.551356\pi$
$864$	−3.21983	+	12.0166i	−0.109541	+	0.408812i
$865$		0			0
$866$	35.1870	−	35.1870i	1.19570	−	1.19570i
$867$	−14.7224	−	25.5000i	−0.500000	−	0.866025i
$868$		0			0
$869$	17.5935	+	65.6598i	0.596818	+	2.22736i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$877$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$878$	−92.5036	−	24.7863i	−3.12185	−	0.836496i
$879$	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	$0.166667\pi$
$882$	13.0128	+	48.5644i	0.438164	+	1.63525i
$883$		−	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
$885$	−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
$890$	4.69032	−	17.5045i	0.157220	−	0.586752i
$891$	56.8630	+	15.2364i	1.90498	+	0.510439i
$892$		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	0.727915	−	0.685668i	$-0.240490\pi$
$907$	15.0000	+	8.66025i	0.498067	+	0.287559i	−0.229848	+	0.973227i	$0.573823\pi$
$908$	5.41087	+	20.1937i	0.179566	+	0.670150i
$909$		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
$929$	3.41844	−	12.7578i	0.112155	−	0.418569i	−0.886903	−	0.461955i	$-0.847148\pi$
$929$	3.41844	−	12.7578i	0.112155	−	0.418569i	0.999058	+	0.0433864i	$0.0138146\pi$
$930$		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	−0.907619	−	0.419796i	$-0.862102\pi$
$941$	−40.7194	−	40.7194i	−1.32741	−	1.32741i	−0.419796	−	0.907619i	$-0.637898\pi$
$942$	−2.14655	+	8.01105i	−0.0699385	+	0.261014i
$943$		0			0
$944$	−0.818262	+	0.818262i	−0.0266322	+	0.0266322i
$945$		0			0
$946$	54.2487	+	31.3205i	1.76378	+	1.01832i
$947$	−14.9933	−	55.9558i	−0.487217	−	1.81832i	−0.569862	−	0.821740i	$-0.693004\pi$
$947$	−14.9933	−	55.9558i	−0.487217	−	1.81832i	0.0826452	−	0.996579i	$-0.473663\pi$
$948$			67.1769i			2.18180i
$949$		0			0
$950$		0			0
$951$	52.5699	−	14.0861i	1.70470	−	0.456772i
$952$		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
$956$	−61.4882	−	16.4757i	−1.98867	−	0.532863i
$957$		0			0
$958$	33.8301	+	58.5955i	1.09300	+	1.89313i
$959$		0			0
$960$	8.37569	+	31.2585i	0.270324	+	1.00886i
$961$			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	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	34.1137	−	127.314i	1.09646	−	4.09203i
$969$		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
$977$	49.7592	−	13.3329i	1.59194	−	0.426559i	0.649343	−	0.760495i	$-0.275044\pi$
$977$	49.7592	−	13.3329i	1.59194	−	0.426559i	0.942596	+	0.333937i	$0.108377\pi$
$978$		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	0.551151	−	0.834406i	$-0.314189\pi$
$983$	43.4411	−	43.4411i	1.38556	−	1.38556i	0.834406	−	0.551151i	$-0.185811\pi$
$984$	−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
$990$	79.5353	−	21.3114i	2.52780	−	0.677322i
$991$	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
$995$	41.0515	+	10.9997i	1.30142	+	0.348714i
$996$	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.h.188.1		8
3.2	odd	2	inner	507.2.k.h.188.2		8
13.2	odd	12	inner	507.2.k.h.89.1		8
13.3	even	3		507.2.k.g.80.2		8
13.4	even	6		507.2.f.d.239.1	✓	8
13.5	odd	4		507.2.k.g.488.2		8
13.6	odd	12		507.2.f.d.437.4	yes	8
13.7	odd	12		507.2.f.d.437.1	yes	8
13.8	odd	4		507.2.k.g.488.1		8
13.9	even	3		507.2.f.d.239.4	yes	8
13.10	even	6		507.2.k.g.80.1		8
13.11	odd	12	inner	507.2.k.h.89.2		8
13.12	even	2	inner	507.2.k.h.188.2		8
39.2	even	12	inner	507.2.k.h.89.2		8
39.5	even	4		507.2.k.g.488.1		8
39.8	even	4		507.2.k.g.488.2		8
39.11	even	12	inner	507.2.k.h.89.1		8
39.17	odd	6		507.2.f.d.239.4	yes	8
39.20	even	12		507.2.f.d.437.4	yes	8
39.23	odd	6		507.2.k.g.80.2		8
39.29	odd	6		507.2.k.g.80.1		8
39.32	even	12		507.2.f.d.437.1	yes	8
39.35	odd	6		507.2.f.d.239.1	✓	8
39.38	odd	2	CM	507.2.k.h.188.1		8

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

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