LMFDB - Embedded newform 507.2.b.d.337.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [507,2,Mod(337,507)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("507.337"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(507, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	507.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,4,-10,0,0,0,0,4,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

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

Self dual:	no
Analytic conductor:	$4.04841538248$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.3
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.2.b.d.337.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.56155i q^{2} +1.00000 q^{3} -0.438447 q^{4} +3.56155i q^{5} +1.56155i q^{6} +0.561553i q^{7} +2.43845i q^{8} +1.00000 q^{9} -5.56155 q^{10} -2.00000i q^{11} -0.438447 q^{12} -0.876894 q^{14} +3.56155i q^{15} -4.68466 q^{16} +1.56155 q^{17} +1.56155i q^{18} -7.12311i q^{19} -1.56155i q^{20} +0.561553i q^{21} +3.12311 q^{22} -2.00000 q^{23} +2.43845i q^{24} -7.68466 q^{25} +1.00000 q^{27} -0.246211i q^{28} +6.68466 q^{29} -5.56155 q^{30} -2.56155i q^{31} -2.43845i q^{32} -2.00000i q^{33} +2.43845i q^{34} -2.00000 q^{35} -0.438447 q^{36} +7.56155i q^{37} +11.1231 q^{38} -8.68466 q^{40} +1.56155i q^{41} -0.876894 q^{42} -4.56155 q^{43} +0.876894i q^{44} +3.56155i q^{45} -3.12311i q^{46} +8.24621i q^{47} -4.68466 q^{48} +6.68466 q^{49} -12.0000i q^{50} +1.56155 q^{51} -0.684658 q^{53} +1.56155i q^{54} +7.12311 q^{55} -1.36932 q^{56} -7.12311i q^{57} +10.4384i q^{58} -2.87689i q^{59} -1.56155i q^{60} +3.87689 q^{61} +4.00000 q^{62} +0.561553i q^{63} -5.56155 q^{64} +3.12311 q^{66} -4.56155i q^{67} -0.684658 q^{68} -2.00000 q^{69} -3.12311i q^{70} -14.0000i q^{71} +2.43845i q^{72} -10.1231i q^{73} -11.8078 q^{74} -7.68466 q^{75} +3.12311i q^{76} +1.12311 q^{77} +5.43845 q^{79} -16.6847i q^{80} +1.00000 q^{81} -2.43845 q^{82} +0.876894i q^{83} -0.246211i q^{84} +5.56155i q^{85} -7.12311i q^{86} +6.68466 q^{87} +4.87689 q^{88} +4.87689i q^{89} -5.56155 q^{90} +0.876894 q^{92} -2.56155i q^{93} -12.8769 q^{94} +25.3693 q^{95} -2.43845i q^{96} +8.56155i q^{97} +10.4384i q^{98} -2.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 10 q^{4} + 4 q^{9} - 14 q^{10} - 10 q^{12} - 20 q^{14} + 6 q^{16} - 2 q^{17} - 4 q^{22} - 8 q^{23} - 6 q^{25} + 4 q^{27} + 2 q^{29} - 14 q^{30} - 8 q^{35} - 10 q^{36} + 28 q^{38} - 10 q^{40}+ \cdots + 52 q^{95}+O(q^{100}) $ 4 * q + 4 * q^3 - 10 * q^4 + 4 * q^9 - 14 * q^10 - 10 * q^12 - 20 * q^14 + 6 * q^16 - 2 * q^17 - 4 * q^22 - 8 * q^23 - 6 * q^25 + 4 * q^27 + 2 * q^29 - 14 * q^30 - 8 * q^35 - 10 * q^36 + 28 * q^38 - 10 * q^40 - 20 * q^42 - 10 * q^43 + 6 * q^48 + 2 * q^49 - 2 * q^51 + 22 * q^53 + 12 * q^55 + 44 * q^56 + 32 * q^61 + 16 * q^62 - 14 * q^64 - 4 * q^66 + 22 * q^68 - 8 * q^69 - 6 * q^74 - 6 * q^75 - 12 * q^77 + 30 * q^79 + 4 * q^81 - 18 * q^82 + 2 * q^87 + 36 * q^88 - 14 * q^90 + 20 * q^92 - 68 * q^94 + 52 * q^95

Character values

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

$n$	$170$	$340$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.56155i	1.10418i	0.833783	+	0.552092i	$0.186170\pi$
$2$	1.56155i	1.10418i	−0.833783	+	0.552092i	$0.813830\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.438447	−0.219224
$5$	3.56155i	1.59277i	0.604787	+	0.796387i	$0.293258\pi$
$5$	3.56155i	1.59277i	−0.604787	+	0.796387i	$0.706742\pi$
$6$	1.56155i	0.637501i
$7$	0.561553i	0.212247i	0.994353	+	0.106124i	$0.0338439\pi$
$7$	0.561553i	0.212247i	−0.994353	+	0.106124i	$0.966156\pi$
$8$	2.43845i	0.862121i
$9$	1.00000	0.333333
$10$	−5.56155	−1.75872
$11$	− 2.00000i	− 0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	− 2.00000i	− 0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	−0.438447	−0.126569
$13$	0	0
$13$	0	0
$14$	−0.876894	−0.234360
$15$	3.56155i	0.919589i
$16$	−4.68466	−1.17116
$17$	1.56155	0.378732	0.189366	−	0.981907i	$-0.439357\pi$
$17$	1.56155	0.378732	0.189366	+	0.981907i	$0.439357\pi$
$18$	1.56155i	0.368062i
$19$	− 7.12311i	− 1.63415i	−0.576530	−	0.817076i	$-0.695593\pi$
$19$	− 7.12311i	− 1.63415i	0.576530	−	0.817076i	$-0.304407\pi$
$20$	− 1.56155i	− 0.349174i
$21$	0.561553i	0.122541i
$22$	3.12311	0.665848
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	2.43845i	0.497746i
$25$	−7.68466	−1.53693
$26$	0	0
$27$	1.00000	0.192450
$28$	− 0.246211i	− 0.0465296i
$29$	6.68466	1.24131	0.620655	−	0.784084i	$-0.286867\pi$
$29$	6.68466	1.24131	0.620655	+	0.784084i	$0.286867\pi$
$30$	−5.56155	−1.01540
$31$	− 2.56155i	− 0.460068i	−0.973183	−	0.230034i	$-0.926116\pi$
$31$	− 2.56155i	− 0.460068i	0.973183	−	0.230034i	$-0.0738838\pi$
$32$	− 2.43845i	− 0.431061i
$33$	− 2.00000i	− 0.348155i
$34$	2.43845i	0.418190i
$35$	−2.00000	−0.338062
$36$	−0.438447	−0.0730745
$37$	7.56155i	1.24311i	0.783370	+	0.621556i	$0.213499\pi$
$37$	7.56155i	1.24311i	−0.783370	+	0.621556i	$0.786501\pi$
$38$	11.1231	1.80441
$39$	0	0
$40$	−8.68466	−1.37317
$41$	1.56155i	0.243874i	0.992538	+	0.121937i	$0.0389105\pi$
$41$	1.56155i	0.243874i	−0.992538	+	0.121937i	$0.961089\pi$
$42$	−0.876894	−0.135308
$43$	−4.56155	−0.695630	−0.347815	−	0.937563i	$-0.613076\pi$
$43$	−4.56155	−0.695630	−0.347815	+	0.937563i	$0.613076\pi$
$44$	0.876894i	0.132197i
$45$	3.56155i	0.530925i
$46$	− 3.12311i	− 0.460477i
$47$	8.24621i	1.20283i	0.798935	+	0.601417i	$0.205397\pi$
$47$	8.24621i	1.20283i	−0.798935	+	0.601417i	$0.794603\pi$
$48$	−4.68466	−0.676172
$49$	6.68466	0.954951
$50$	− 12.0000i	− 1.69706i
$51$	1.56155	0.218661
$52$	0	0
$53$	−0.684658	−0.0940451	−0.0470225	−	0.998894i	$-0.514973\pi$
$53$	−0.684658	−0.0940451	−0.0470225	+	0.998894i	$0.514973\pi$
$54$	1.56155i	0.212500i
$55$	7.12311	0.960479
$56$	−1.36932	−0.182983
$57$	− 7.12311i	− 0.943478i
$58$	10.4384i	1.37064i
$59$	− 2.87689i	− 0.374540i	−0.982309	−	0.187270i	$-0.940036\pi$
$59$	− 2.87689i	− 0.374540i	0.982309	−	0.187270i	$-0.0599639\pi$
$60$	− 1.56155i	− 0.201596i
$61$	3.87689	0.496385	0.248193	−	0.968711i	$-0.420163\pi$
$61$	3.87689	0.496385	0.248193	+	0.968711i	$0.420163\pi$
$62$	4.00000	0.508001
$63$	0.561553i	0.0707490i
$64$	−5.56155	−0.695194
$65$	0	0
$66$	3.12311	0.384428
$67$	− 4.56155i	− 0.557282i	−0.960395	−	0.278641i	$-0.910116\pi$
$67$	− 4.56155i	− 0.557282i	0.960395	−	0.278641i	$-0.0898840\pi$
$68$	−0.684658	−0.0830270
$69$	−2.00000	−0.240772
$70$	− 3.12311i	− 0.373283i
$71$	− 14.0000i	− 1.66149i	−0.556650	−	0.830747i	$-0.687914\pi$
$71$	− 14.0000i	− 1.66149i	0.556650	−	0.830747i	$-0.312086\pi$
$72$	2.43845i	0.287374i
$73$	− 10.1231i	− 1.18482i	−0.805637	−	0.592410i	$-0.798177\pi$
$73$	− 10.1231i	− 1.18482i	0.805637	−	0.592410i	$-0.201823\pi$
$74$	−11.8078	−1.37262
$75$	−7.68466	−0.887348
$76$	3.12311i	0.358245i
$77$	1.12311	0.127990
$78$	0	0
$79$	5.43845	0.611873	0.305937	−	0.952052i	$-0.401030\pi$
$79$	5.43845	0.611873	0.305937	+	0.952052i	$0.401030\pi$
$80$	− 16.6847i	− 1.86540i
$81$	1.00000	0.111111
$82$	−2.43845	−0.269281
$83$	0.876894i	0.0962517i	0.998841	+	0.0481258i	$0.0153248\pi$
$83$	0.876894i	0.0962517i	−0.998841	+	0.0481258i	$0.984675\pi$
$84$	− 0.246211i	− 0.0268638i
$85$	5.56155i	0.603235i
$86$	− 7.12311i	− 0.768104i
$87$	6.68466	0.716671
$88$	4.87689	0.519879
$89$	4.87689i	0.516950i	0.966018	+	0.258475i	$0.0832199\pi$
$89$	4.87689i	0.516950i	−0.966018	+	0.258475i	$0.916780\pi$
$90$	−5.56155	−0.586239
$91$	0	0
$92$	0.876894	0.0914226
$93$	− 2.56155i	− 0.265621i
$94$	−12.8769	−1.32815
$95$	25.3693	2.60284
$96$	− 2.43845i	− 0.248873i
$97$	8.56155i	0.869294i	0.900601	+	0.434647i	$0.143127\pi$
$97$	8.56155i	0.869294i	−0.900601	+	0.434647i	$0.856873\pi$
$98$	10.4384i	1.05444i
$99$	− 2.00000i	− 0.201008i
$100$	3.36932	0.336932
$101$	7.56155	0.752403	0.376201	−	0.926538i	$-0.377230\pi$
$101$	7.56155	0.752403	0.376201	+	0.926538i	$0.377230\pi$
$102$	2.43845i	0.241442i
$103$	3.43845	0.338800	0.169400	−	0.985547i	$-0.445817\pi$
$103$	3.43845	0.338800	0.169400	+	0.985547i	$0.445817\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	− 1.06913i	− 0.103843i
$107$	−8.24621	−0.797191	−0.398596	−	0.917127i	$-0.630502\pi$
$107$	−8.24621	−0.797191	−0.398596	+	0.917127i	$0.630502\pi$
$108$	−0.438447	−0.0421896
$109$	2.80776i	0.268935i	0.990918	+	0.134468i	$0.0429324\pi$
$109$	2.80776i	0.268935i	−0.990918	+	0.134468i	$0.957068\pi$
$110$	11.1231i	1.06055i
$111$	7.56155i	0.717711i
$112$	− 2.63068i	− 0.248576i
$113$	5.80776	0.546348	0.273174	−	0.961965i	$-0.411926\pi$
$113$	5.80776	0.546348	0.273174	+	0.961965i	$0.411926\pi$
$114$	11.1231	1.04177
$115$	− 7.12311i	− 0.664233i
$116$	−2.93087	−0.272124
$117$	0	0
$118$	4.49242	0.413561
$119$	0.876894i	0.0803848i
$120$	−8.68466	−0.792797
$121$	7.00000	0.636364
$122$	6.05398i	0.548101i
$123$	1.56155i	0.140800i
$124$	1.12311i	0.100858i
$125$	− 9.56155i	− 0.855211i
$126$	−0.876894	−0.0781200
$127$	−5.43845	−0.482584	−0.241292	−	0.970453i	$-0.577571\pi$
$127$	−5.43845	−0.482584	−0.241292	+	0.970453i	$0.577571\pi$
$128$	− 13.5616i	− 1.19868i
$129$	−4.56155	−0.401622
$130$	0	0
$131$	7.36932	0.643860	0.321930	−	0.946763i	$-0.395668\pi$
$131$	7.36932	0.643860	0.321930	+	0.946763i	$0.395668\pi$
$132$	0.876894i	0.0763239i
$133$	4.00000	0.346844
$134$	7.12311	0.615343
$135$	3.56155i	0.306530i
$136$	3.80776i	0.326513i
$137$	− 5.56155i	− 0.475156i	−0.971369	−	0.237578i	$-0.923647\pi$
$137$	− 5.56155i	− 0.475156i	0.971369	−	0.237578i	$-0.0763535\pi$
$138$	− 3.12311i	− 0.265856i
$139$	−17.9309	−1.52088	−0.760438	−	0.649410i	$-0.775016\pi$
$139$	−17.9309	−1.52088	−0.760438	+	0.649410i	$0.775016\pi$
$140$	0.876894	0.0741111
$141$	8.24621i	0.694456i
$142$	21.8617	1.83460
$143$	0	0
$144$	−4.68466	−0.390388
$145$	23.8078i	1.97713i
$146$	15.8078	1.30826
$147$	6.68466	0.551341
$148$	− 3.31534i	− 0.272519i
$149$	2.43845i	0.199765i	0.994999	+	0.0998827i	$0.0318468\pi$
$149$	2.43845i	0.199765i	−0.994999	+	0.0998827i	$0.968153\pi$
$150$	− 12.0000i	− 0.979796i
$151$	− 9.36932i	− 0.762464i	−0.924479	−	0.381232i	$-0.875500\pi$
$151$	− 9.36932i	− 0.762464i	0.924479	−	0.381232i	$-0.124500\pi$
$152$	17.3693	1.40884
$153$	1.56155	0.126244
$154$	1.75379i	0.141324i
$155$	9.12311	0.732785
$156$	0	0
$157$	20.3693	1.62565	0.812824	−	0.582509i	$-0.197929\pi$
$157$	20.3693	1.62565	0.812824	+	0.582509i	$0.197929\pi$
$158$	8.49242i	0.675621i
$159$	−0.684658	−0.0542969
$160$	8.68466	0.686583
$161$	− 1.12311i	− 0.0885131i
$162$	1.56155i	0.122687i
$163$	− 4.80776i	− 0.376573i	−0.982114	−	0.188287i	$-0.939707\pi$
$163$	− 4.80776i	− 0.376573i	0.982114	−	0.188287i	$-0.0602934\pi$
$164$	− 0.684658i	− 0.0534628i
$165$	7.12311	0.554533
$166$	−1.36932	−0.106280
$167$	10.2462i	0.792876i	0.918062	+	0.396438i	$0.129754\pi$
$167$	10.2462i	0.792876i	−0.918062	+	0.396438i	$0.870246\pi$
$168$	−1.36932	−0.105645
$169$	0	0
$170$	−8.68466	−0.666083
$171$	− 7.12311i	− 0.544718i
$172$	2.00000	0.152499
$173$	−20.2462	−1.53929	−0.769645	−	0.638471i	$-0.779567\pi$
$173$	−20.2462	−1.53929	−0.769645	+	0.638471i	$0.779567\pi$
$174$	10.4384i	0.791337i
$175$	− 4.31534i	− 0.326209i
$176$	9.36932i	0.706239i
$177$	− 2.87689i	− 0.216241i
$178$	−7.61553	−0.570808
$179$	4.87689	0.364516	0.182258	−	0.983251i	$-0.441659\pi$
$179$	4.87689	0.364516	0.182258	+	0.983251i	$0.441659\pi$
$180$	− 1.56155i	− 0.116391i
$181$	2.68466	0.199549	0.0997745	−	0.995010i	$-0.468188\pi$
$181$	2.68466	0.199549	0.0997745	+	0.995010i	$0.468188\pi$
$182$	0	0
$183$	3.87689	0.286588
$184$	− 4.87689i	− 0.359529i
$185$	−26.9309	−1.98000
$186$	4.00000	0.293294
$187$	− 3.12311i	− 0.228384i
$188$	− 3.61553i	− 0.263689i
$189$	0.561553i	0.0408470i
$190$	39.6155i	2.87401i
$191$	−9.12311	−0.660125	−0.330062	−	0.943959i	$-0.607070\pi$
$191$	−9.12311	−0.660125	−0.330062	+	0.943959i	$0.607070\pi$
$192$	−5.56155	−0.401371
$193$	− 13.4924i	− 0.971206i	−0.874180	−	0.485603i	$-0.838600\pi$
$193$	− 13.4924i	− 0.971206i	0.874180	−	0.485603i	$-0.161400\pi$
$194$	−13.3693	−0.959861
$195$	0	0
$196$	−2.93087	−0.209348
$197$	− 13.3693i	− 0.952524i	−0.879303	−	0.476262i	$-0.841991\pi$
$197$	− 13.3693i	− 0.952524i	0.879303	−	0.476262i	$-0.158009\pi$
$198$	3.12311	0.221949
$199$	−22.1771	−1.57209	−0.786046	−	0.618168i	$-0.787875\pi$
$199$	−22.1771	−1.57209	−0.786046	+	0.618168i	$0.787875\pi$
$200$	− 18.7386i	− 1.32502i
$201$	− 4.56155i	− 0.321747i
$202$	11.8078i	0.830791i
$203$	3.75379i	0.263464i
$204$	−0.684658	−0.0479357
$205$	−5.56155	−0.388436
$206$	5.36932i	0.374098i
$207$	−2.00000	−0.139010
$208$	0	0
$209$	−14.2462	−0.985431
$210$	− 3.12311i	− 0.215515i
$211$	19.6847	1.35515	0.677574	−	0.735455i	$-0.263031\pi$
$211$	19.6847	1.35515	0.677574	+	0.735455i	$0.263031\pi$
$212$	0.300187	0.0206169
$213$	− 14.0000i	− 0.959264i
$214$	− 12.8769i	− 0.880246i
$215$	− 16.2462i	− 1.10798i
$216$	2.43845i	0.165915i
$217$	1.43845	0.0976482
$218$	−4.38447	−0.296954
$219$	− 10.1231i	− 0.684056i
$220$	−3.12311	−0.210560
$221$	0	0
$222$	−11.8078	−0.792485
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	1.36932	0.0914913
$225$	−7.68466	−0.512311
$226$	9.06913i	0.603270i
$227$	− 7.12311i	− 0.472777i	−0.971659	−	0.236389i	$-0.924036\pi$
$227$	− 7.12311i	− 0.472777i	0.971659	−	0.236389i	$-0.0759638\pi$
$228$	3.12311i	0.206833i
$229$	− 16.2462i	− 1.07358i	−0.843716	−	0.536790i	$-0.819637\pi$
$229$	− 16.2462i	− 1.07358i	0.843716	−	0.536790i	$-0.180363\pi$
$230$	11.1231	0.733436
$231$	1.12311	0.0738949
$232$	16.3002i	1.07016i
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	0	0
$235$	−29.3693	−1.91584
$236$	1.26137i	0.0821079i
$237$	5.43845	0.353265
$238$	−1.36932	−0.0887596
$239$	25.3693i	1.64100i	0.571643	+	0.820502i	$0.306306\pi$
$239$	25.3693i	1.64100i	−0.571643	+	0.820502i	$0.693694\pi$
$240$	− 16.6847i	− 1.07699i
$241$	− 17.8078i	− 1.14710i	−0.819171	−	0.573549i	$-0.805566\pi$
$241$	− 17.8078i	− 1.14710i	0.819171	−	0.573549i	$-0.194434\pi$
$242$	10.9309i	0.702663i
$243$	1.00000	0.0641500
$244$	−1.69981	−0.108819
$245$	23.8078i	1.52102i
$246$	−2.43845	−0.155470
$247$	0	0
$248$	6.24621	0.396635
$249$	0.876894i	0.0555709i
$250$	14.9309	0.944311
$251$	−18.7386	−1.18277	−0.591386	−	0.806389i	$-0.701419\pi$
$251$	−18.7386	−1.18277	−0.591386	+	0.806389i	$0.701419\pi$
$252$	− 0.246211i	− 0.0155099i
$253$	4.00000i	0.251478i
$254$	− 8.49242i	− 0.532862i
$255$	5.56155i	0.348278i
$256$	10.0540	0.628373
$257$	−29.1771	−1.82002	−0.910008	−	0.414590i	$-0.863925\pi$
$257$	−29.1771	−1.82002	−0.910008	+	0.414590i	$0.863925\pi$
$258$	− 7.12311i	− 0.443465i
$259$	−4.24621	−0.263847
$260$	0	0
$261$	6.68466	0.413770
$262$	11.5076i	0.710941i
$263$	9.36932	0.577737	0.288868	−	0.957369i	$-0.406721\pi$
$263$	9.36932	0.577737	0.288868	+	0.957369i	$0.406721\pi$
$264$	4.87689	0.300152
$265$	− 2.43845i	− 0.149793i
$266$	6.24621i	0.382980i
$267$	4.87689i	0.298461i
$268$	2.00000i	0.122169i
$269$	−21.3693	−1.30291	−0.651455	−	0.758687i	$-0.725841\pi$
$269$	−21.3693	−1.30291	−0.651455	+	0.758687i	$0.725841\pi$
$270$	−5.56155	−0.338465
$271$	− 29.9309i	− 1.81817i	−0.416610	−	0.909085i	$-0.636782\pi$
$271$	− 29.9309i	− 1.81817i	0.416610	−	0.909085i	$-0.363218\pi$
$272$	−7.31534	−0.443558
$273$	0	0
$274$	8.68466	0.524659
$275$	15.3693i	0.926805i
$276$	0.876894	0.0527828
$277$	−5.31534	−0.319368	−0.159684	−	0.987168i	$-0.551048\pi$
$277$	−5.31534	−0.319368	−0.159684	+	0.987168i	$0.551048\pi$
$278$	− 28.0000i	− 1.67933i
$279$	− 2.56155i	− 0.153356i
$280$	− 4.87689i	− 0.291450i
$281$	17.8078i	1.06232i	0.847271	+	0.531161i	$0.178244\pi$
$281$	17.8078i	1.06232i	−0.847271	+	0.531161i	$0.821756\pi$
$282$	−12.8769	−0.766808
$283$	13.6847	0.813469	0.406734	−	0.913547i	$-0.366667\pi$
$283$	13.6847	0.813469	0.406734	+	0.913547i	$0.366667\pi$
$284$	6.13826i	0.364239i
$285$	25.3693	1.50275
$286$	0	0
$287$	−0.876894	−0.0517614
$288$	− 2.43845i	− 0.143687i
$289$	−14.5616	−0.856562
$290$	−37.1771	−2.18311
$291$	8.56155i	0.501887i
$292$	4.43845i	0.259740i
$293$	20.4384i	1.19403i	0.802231	+	0.597013i	$0.203646\pi$
$293$	20.4384i	1.19403i	−0.802231	+	0.597013i	$0.796354\pi$
$294$	10.4384i	0.608783i
$295$	10.2462	0.596557
$296$	−18.4384	−1.07171
$297$	− 2.00000i	− 0.116052i
$298$	−3.80776	−0.220578
$299$	0	0
$300$	3.36932	0.194528
$301$	− 2.56155i	− 0.147645i
$302$	14.6307	0.841901
$303$	7.56155	0.434400
$304$	33.3693i	1.91386i
$305$	13.8078i	0.790630i
$306$	2.43845i	0.139397i
$307$	30.8078i	1.75829i	0.476553	+	0.879146i	$0.341886\pi$
$307$	30.8078i	1.75829i	−0.476553	+	0.879146i	$0.658114\pi$
$308$	−0.492423	−0.0280584
$309$	3.43845	0.195606
$310$	14.2462i	0.809130i
$311$	19.1231	1.08437	0.542186	−	0.840259i	$-0.317597\pi$
$311$	19.1231	1.08437	0.542186	+	0.840259i	$0.317597\pi$
$312$	0	0
$313$	−13.6847	−0.773503	−0.386751	−	0.922184i	$-0.626403\pi$
$313$	−13.6847	−0.773503	−0.386751	+	0.922184i	$0.626403\pi$
$314$	31.8078i	1.79502i
$315$	−2.00000	−0.112687
$316$	−2.38447	−0.134137
$317$	− 14.0540i	− 0.789350i	−0.918821	−	0.394675i	$-0.870857\pi$
$317$	− 14.0540i	− 0.789350i	0.918821	−	0.394675i	$-0.129143\pi$
$318$	− 1.06913i	− 0.0599539i
$319$	− 13.3693i	− 0.748538i
$320$	− 19.8078i	− 1.10729i
$321$	−8.24621	−0.460259
$322$	1.75379	0.0977348
$323$	− 11.1231i	− 0.618906i
$324$	−0.438447	−0.0243582
$325$	0	0
$326$	7.50758	0.415806
$327$	2.80776i	0.155270i
$328$	−3.80776	−0.210249
$329$	−4.63068	−0.255298
$330$	11.1231i	0.612307i
$331$	3.19224i	0.175461i	0.996144	+	0.0877306i	$0.0279615\pi$
$331$	3.19224i	0.175461i	−0.996144	+	0.0877306i	$0.972039\pi$
$332$	− 0.384472i	− 0.0211006i
$333$	7.56155i	0.414371i
$334$	−16.0000	−0.875481
$335$	16.2462	0.887625
$336$	− 2.63068i	− 0.143516i
$337$	6.12311	0.333547	0.166773	−	0.985995i	$-0.446665\pi$
$337$	6.12311	0.333547	0.166773	+	0.985995i	$0.446665\pi$
$338$	0	0
$339$	5.80776	0.315434
$340$	− 2.43845i	− 0.132243i
$341$	−5.12311	−0.277432
$342$	11.1231	0.601469
$343$	7.68466i	0.414933i
$344$	− 11.1231i	− 0.599718i
$345$	− 7.12311i	− 0.383495i
$346$	− 31.6155i	− 1.69966i
$347$	27.6155	1.48248	0.741240	−	0.671241i	$-0.234238\pi$
$347$	27.6155	1.48248	0.741240	+	0.671241i	$0.234238\pi$
$348$	−2.93087	−0.157111
$349$	6.80776i	0.364411i	0.983260	+	0.182206i	$0.0583237\pi$
$349$	6.80776i	0.364411i	−0.983260	+	0.182206i	$0.941676\pi$
$350$	6.73863	0.360195
$351$	0	0
$352$	−4.87689	−0.259939
$353$	− 5.31534i	− 0.282907i	−0.989945	−	0.141454i	$-0.954822\pi$
$353$	− 5.31534i	− 0.282907i	0.989945	−	0.141454i	$-0.0451776\pi$
$354$	4.49242	0.238770
$355$	49.8617	2.64639
$356$	− 2.13826i	− 0.113328i
$357$	0.876894i	0.0464102i
$358$	7.61553i	0.402493i
$359$	− 9.36932i	− 0.494494i	−0.968953	−	0.247247i	$-0.920474\pi$
$359$	− 9.36932i	− 0.494494i	0.968953	−	0.247247i	$-0.0795258\pi$
$360$	−8.68466	−0.457722
$361$	−31.7386	−1.67045
$362$	4.19224i	0.220339i
$363$	7.00000	0.367405
$364$	0	0
$365$	36.0540	1.88715
$366$	6.05398i	0.316446i
$367$	−17.0540	−0.890210	−0.445105	−	0.895478i	$-0.646834\pi$
$367$	−17.0540	−0.890210	−0.445105	+	0.895478i	$0.646834\pi$
$368$	9.36932	0.488409
$369$	1.56155i	0.0812912i
$370$	− 42.0540i	− 2.18628i
$371$	− 0.384472i	− 0.0199608i
$372$	1.12311i	0.0582303i
$373$	28.3693	1.46891	0.734454	−	0.678659i	$-0.237438\pi$
$373$	28.3693	1.46891	0.734454	+	0.678659i	$0.237438\pi$
$374$	4.87689	0.252178
$375$	− 9.56155i	− 0.493756i
$376$	−20.1080	−1.03699
$377$	0	0
$378$	−0.876894	−0.0451026
$379$	− 23.6847i	− 1.21660i	−0.793708	−	0.608300i	$-0.791852\pi$
$379$	− 23.6847i	− 1.21660i	0.793708	−	0.608300i	$-0.208148\pi$
$380$	−11.1231	−0.570603
$381$	−5.43845	−0.278620
$382$	− 14.2462i	− 0.728900i
$383$	22.7386i	1.16189i	0.813943	+	0.580945i	$0.197317\pi$
$383$	22.7386i	1.16189i	−0.813943	+	0.580945i	$0.802683\pi$
$384$	− 13.5616i	− 0.692060i
$385$	4.00000i	0.203859i
$386$	21.0691	1.07239
$387$	−4.56155	−0.231877
$388$	− 3.75379i	− 0.190570i
$389$	−34.0540	−1.72661	−0.863303	−	0.504687i	$-0.831608\pi$
$389$	−34.0540	−1.72661	−0.863303	+	0.504687i	$0.831608\pi$
$390$	0	0
$391$	−3.12311	−0.157942
$392$	16.3002i	0.823284i
$393$	7.36932	0.371733
$394$	20.8769	1.05176
$395$	19.3693i	0.974576i
$396$	0.876894i	0.0440656i
$397$	25.0540i	1.25742i	0.777639	+	0.628711i	$0.216417\pi$
$397$	25.0540i	1.25742i	−0.777639	+	0.628711i	$0.783583\pi$
$398$	− 34.6307i	− 1.73588i
$399$	4.00000	0.200250
$400$	36.0000	1.80000
$401$	14.4384i	0.721022i	0.932755	+	0.360511i	$0.117398\pi$
$401$	14.4384i	0.721022i	−0.932755	+	0.360511i	$0.882602\pi$
$402$	7.12311	0.355268
$403$	0	0
$404$	−3.31534	−0.164944
$405$	3.56155i	0.176975i
$406$	−5.86174	−0.290913
$407$	15.1231	0.749625
$408$	3.80776i	0.188512i
$409$	− 6.36932i	− 0.314942i	−0.987524	−	0.157471i	$-0.949666\pi$
$409$	− 6.36932i	− 0.314942i	0.987524	−	0.157471i	$-0.0503342\pi$
$410$	− 8.68466i	− 0.428905i
$411$	− 5.56155i	− 0.274331i
$412$	−1.50758	−0.0742730
$413$	1.61553	0.0794949
$414$	− 3.12311i	− 0.153492i
$415$	−3.12311	−0.153307
$416$	0	0
$417$	−17.9309	−0.878078
$418$	− 22.2462i	− 1.08810i
$419$	−34.2462	−1.67304	−0.836518	−	0.547939i	$-0.815413\pi$
$419$	−34.2462	−1.67304	−0.836518	+	0.547939i	$0.815413\pi$
$420$	0.876894	0.0427881
$421$	− 31.2462i	− 1.52285i	−0.648255	−	0.761424i	$-0.724501\pi$
$421$	− 31.2462i	− 1.52285i	0.648255	−	0.761424i	$-0.275499\pi$
$422$	30.7386i	1.49633i
$423$	8.24621i	0.400945i
$424$	− 1.66950i	− 0.0810783i
$425$	−12.0000	−0.582086
$426$	21.8617	1.05920
$427$	2.17708i	0.105356i
$428$	3.61553	0.174763
$429$	0	0
$430$	25.3693	1.22342
$431$	11.1231i	0.535781i	0.963449	+	0.267891i	$0.0863266\pi$
$431$	11.1231i	0.535781i	−0.963449	+	0.267891i	$0.913673\pi$
$432$	−4.68466	−0.225391
$433$	−8.75379	−0.420680	−0.210340	−	0.977628i	$-0.567457\pi$
$433$	−8.75379	−0.420680	−0.210340	+	0.977628i	$0.567457\pi$
$434$	2.24621i	0.107822i
$435$	23.8078i	1.14149i
$436$	− 1.23106i	− 0.0589569i
$437$	14.2462i	0.681489i
$438$	15.8078	0.755324
$439$	13.6847	0.653133	0.326567	−	0.945174i	$-0.394108\pi$
$439$	13.6847	0.653133	0.326567	+	0.945174i	$0.394108\pi$
$440$	17.3693i	0.828050i
$441$	6.68466	0.318317
$442$	0	0
$443$	−34.7386	−1.65048	−0.825241	−	0.564781i	$-0.808961\pi$
$443$	−34.7386	−1.65048	−0.825241	+	0.564781i	$0.808961\pi$
$444$	− 3.31534i	− 0.157339i
$445$	−17.3693	−0.823385
$446$	12.4924	0.591533
$447$	2.43845i	0.115335i
$448$	− 3.12311i	− 0.147553i
$449$	− 8.24621i	− 0.389163i	−0.980886	−	0.194581i	$-0.937665\pi$
$449$	− 8.24621i	− 0.389163i	0.980886	−	0.194581i	$-0.0623348\pi$
$450$	− 12.0000i	− 0.565685i
$451$	3.12311	0.147061
$452$	−2.54640	−0.119772
$453$	− 9.36932i	− 0.440209i
$454$	11.1231	0.522033
$455$	0	0
$456$	17.3693	0.813393
$457$	− 12.6155i	− 0.590130i	−0.955477	−	0.295065i	$-0.904659\pi$
$457$	− 12.6155i	− 0.590130i	0.955477	−	0.295065i	$-0.0953412\pi$
$458$	25.3693	1.18543
$459$	1.56155	0.0728870
$460$	3.12311i	0.145616i
$461$	16.1922i	0.754148i	0.926183	+	0.377074i	$0.123070\pi$
$461$	16.1922i	0.754148i	−0.926183	+	0.377074i	$0.876930\pi$
$462$	1.75379i	0.0815936i
$463$	− 14.3153i	− 0.665290i	−0.943052	−	0.332645i	$-0.892059\pi$
$463$	− 14.3153i	− 0.665290i	0.943052	−	0.332645i	$-0.107941\pi$
$464$	−31.3153	−1.45378
$465$	9.12311	0.423074
$466$	− 40.6004i	− 1.88078i
$467$	26.0000	1.20314	0.601568	−	0.798821i	$-0.294543\pi$
$467$	26.0000	1.20314	0.601568	+	0.798821i	$0.294543\pi$
$468$	0	0
$469$	2.56155	0.118282
$470$	− 45.8617i	− 2.11544i
$471$	20.3693	0.938569
$472$	7.01515	0.322899
$473$	9.12311i	0.419481i
$474$	8.49242i	0.390070i
$475$	54.7386i	2.51158i
$476$	− 0.384472i	− 0.0176222i
$477$	−0.684658	−0.0313484
$478$	−39.6155	−1.81197
$479$	10.2462i	0.468161i	0.972217	+	0.234081i	$0.0752080\pi$
$479$	10.2462i	0.468161i	−0.972217	+	0.234081i	$0.924792\pi$
$480$	8.68466	0.396399
$481$	0	0
$482$	27.8078	1.26661
$483$	− 1.12311i	− 0.0511031i
$484$	−3.06913	−0.139506
$485$	−30.4924	−1.38459
$486$	1.56155i	0.0708335i
$487$	− 7.12311i	− 0.322779i	−0.986891	−	0.161389i	$-0.948403\pi$
$487$	− 7.12311i	− 0.322779i	0.986891	−	0.161389i	$-0.0515975\pi$
$488$	9.45360i	0.427944i
$489$	− 4.80776i	− 0.217415i
$490$	−37.1771	−1.67949
$491$	36.2462	1.63577	0.817884	−	0.575383i	$-0.195147\pi$
$491$	36.2462	1.63577	0.817884	+	0.575383i	$0.195147\pi$
$492$	− 0.684658i	− 0.0308668i
$493$	10.4384	0.470124
$494$	0	0
$495$	7.12311	0.320160
$496$	12.0000i	0.538816i
$497$	7.86174	0.352647
$498$	−1.36932	−0.0613606
$499$	− 4.49242i	− 0.201108i	−0.994932	−	0.100554i	$-0.967938\pi$
$499$	− 4.49242i	− 0.201108i	0.994932	−	0.100554i	$-0.0320616\pi$
$500$	4.19224i	0.187482i
$501$	10.2462i	0.457767i
$502$	− 29.2614i	− 1.30600i
$503$	−28.2462	−1.25944	−0.629718	−	0.776824i	$-0.716830\pi$
$503$	−28.2462	−1.25944	−0.629718	+	0.776824i	$0.716830\pi$
$504$	−1.36932	−0.0609942
$505$	26.9309i	1.19841i
$506$	−6.24621	−0.277678
$507$	0	0
$508$	2.38447	0.105794
$509$	13.8078i	0.612018i	0.952029	+	0.306009i	$0.0989938\pi$
$509$	13.8078i	0.612018i	−0.952029	+	0.306009i	$0.901006\pi$
$510$	−8.68466	−0.384563
$511$	5.68466	0.251474
$512$	− 11.4233i	− 0.504843i
$513$	− 7.12311i	− 0.314493i
$514$	− 45.5616i	− 2.00963i
$515$	12.2462i	0.539633i
$516$	2.00000	0.0880451
$517$	16.4924	0.725336
$518$	− 6.63068i	− 0.291335i
$519$	−20.2462	−0.888710
$520$	0	0
$521$	−9.06913	−0.397326	−0.198663	−	0.980068i	$-0.563660\pi$
$521$	−9.06913	−0.397326	−0.198663	+	0.980068i	$0.563660\pi$
$522$	10.4384i	0.456878i
$523$	33.8617	1.48067	0.740335	−	0.672238i	$-0.234667\pi$
$523$	33.8617	1.48067	0.740335	+	0.672238i	$0.234667\pi$
$524$	−3.23106	−0.141149
$525$	− 4.31534i	− 0.188337i
$526$	14.6307i	0.637928i
$527$	− 4.00000i	− 0.174243i
$528$	9.36932i	0.407747i
$529$	−19.0000	−0.826087
$530$	3.80776	0.165399
$531$	− 2.87689i	− 0.124847i
$532$	−1.75379	−0.0760364
$533$	0	0
$534$	−7.61553	−0.329556
$535$	− 29.3693i	− 1.26975i
$536$	11.1231	0.480445
$537$	4.87689	0.210454
$538$	− 33.3693i	− 1.43865i
$539$	− 13.3693i	− 0.575857i
$540$	− 1.56155i	− 0.0671985i
$541$	19.7386i	0.848630i	0.905515	+	0.424315i	$0.139485\pi$
$541$	19.7386i	0.848630i	−0.905515	+	0.424315i	$0.860515\pi$
$542$	46.7386	2.00760
$543$	2.68466	0.115210
$544$	− 3.80776i	− 0.163257i
$545$	−10.0000	−0.428353
$546$	0	0
$547$	3.93087	0.168072	0.0840359	−	0.996463i	$-0.473219\pi$
$547$	3.93087	0.168072	0.0840359	+	0.996463i	$0.473219\pi$
$548$	2.43845i	0.104165i
$549$	3.87689	0.165462
$550$	−24.0000	−1.02336
$551$	− 47.6155i	− 2.02849i
$552$	− 4.87689i	− 0.207574i
$553$	3.05398i	0.129868i
$554$	− 8.30019i	− 0.352641i
$555$	−26.9309	−1.14315
$556$	7.86174	0.333412
$557$	− 42.9309i	− 1.81904i	−0.415661	−	0.909520i	$-0.636450\pi$
$557$	− 42.9309i	− 1.81904i	0.415661	−	0.909520i	$-0.363550\pi$
$558$	4.00000	0.169334
$559$	0	0
$560$	9.36932	0.395926
$561$	− 3.12311i	− 0.131858i
$562$	−27.8078	−1.17300
$563$	23.3693	0.984899	0.492450	−	0.870341i	$-0.336102\pi$
$563$	23.3693	0.984899	0.492450	+	0.870341i	$0.336102\pi$
$564$	− 3.61553i	− 0.152241i
$565$	20.6847i	0.870210i
$566$	21.3693i	0.898219i
$567$	0.561553i	0.0235830i
$568$	34.1383	1.43241
$569$	8.73863	0.366343	0.183171	−	0.983081i	$-0.441364\pi$
$569$	8.73863	0.366343	0.183171	+	0.983081i	$0.441364\pi$
$570$	39.6155i	1.65931i
$571$	5.36932	0.224699	0.112349	−	0.993669i	$-0.464162\pi$
$571$	5.36932	0.224699	0.112349	+	0.993669i	$0.464162\pi$
$572$	0	0
$573$	−9.12311	−0.381123
$574$	− 1.36932i	− 0.0571542i
$575$	15.3693	0.640945
$576$	−5.56155	−0.231731
$577$	17.3153i	0.720847i	0.932789	+	0.360424i	$0.117368\pi$
$577$	17.3153i	0.720847i	−0.932789	+	0.360424i	$0.882632\pi$
$578$	− 22.7386i	− 0.945802i
$579$	− 13.4924i	− 0.560726i
$580$	− 10.4384i	− 0.433433i
$581$	−0.492423	−0.0204291
$582$	−13.3693	−0.554176
$583$	1.36932i	0.0567113i
$584$	24.6847	1.02146
$585$	0	0
$586$	−31.9157	−1.31843
$587$	− 39.3693i	− 1.62495i	−0.582999	−	0.812473i	$-0.698121\pi$
$587$	− 39.3693i	− 1.62495i	0.582999	−	0.812473i	$-0.301879\pi$
$588$	−2.93087	−0.120867
$589$	−18.2462	−0.751822
$590$	16.0000i	0.658710i
$591$	− 13.3693i	− 0.549940i
$592$	− 35.4233i	− 1.45589i
$593$	− 17.4233i	− 0.715489i	−0.933820	−	0.357744i	$-0.883546\pi$
$593$	− 17.4233i	− 0.715489i	0.933820	−	0.357744i	$-0.116454\pi$
$594$	3.12311	0.128143
$595$	−3.12311	−0.128035
$596$	− 1.06913i	− 0.0437933i
$597$	−22.1771	−0.907648
$598$	0	0
$599$	−41.6155	−1.70036	−0.850182	−	0.526489i	$-0.823508\pi$
$599$	−41.6155	−1.70036	−0.850182	+	0.526489i	$0.823508\pi$
$600$	− 18.7386i	− 0.765002i
$601$	−7.06913	−0.288356	−0.144178	−	0.989552i	$-0.546054\pi$
$601$	−7.06913	−0.288356	−0.144178	+	0.989552i	$0.546054\pi$
$602$	4.00000	0.163028
$603$	− 4.56155i	− 0.185761i
$604$	4.10795i	0.167150i
$605$	24.9309i	1.01358i
$606$	11.8078i	0.479658i
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	−17.3693	−0.704419
$609$	3.75379i	0.152111i
$610$	−21.5616	−0.873002
$611$	0	0
$612$	−0.684658	−0.0276757
$613$	34.8617i	1.40805i	0.710174	+	0.704026i	$0.248616\pi$
$613$	34.8617i	1.40805i	−0.710174	+	0.704026i	$0.751384\pi$
$614$	−48.1080	−1.94148
$615$	−5.56155	−0.224263
$616$	2.73863i	0.110343i
$617$	− 9.80776i	− 0.394846i	−0.980318	−	0.197423i	$-0.936743\pi$
$617$	− 9.80776i	− 0.394846i	0.980318	−	0.197423i	$-0.0632572\pi$
$618$	5.36932i	0.215986i
$619$	29.3002i	1.17767i	0.808252	+	0.588837i	$0.200414\pi$
$619$	29.3002i	1.17767i	−0.808252	+	0.588837i	$0.799586\pi$
$620$	−4.00000	−0.160644
$621$	−2.00000	−0.0802572
$622$	29.8617i	1.19735i
$623$	−2.73863	−0.109721
$624$	0	0
$625$	−4.36932	−0.174773
$626$	− 21.3693i	− 0.854090i
$627$	−14.2462	−0.568939
$628$	−8.93087	−0.356380
$629$	11.8078i	0.470806i
$630$	− 3.12311i	− 0.124428i
$631$	− 18.5616i	− 0.738924i	−0.929246	−	0.369462i	$-0.879542\pi$
$631$	− 18.5616i	− 0.738924i	0.929246	−	0.369462i	$-0.120458\pi$
$632$	13.2614i	0.527509i
$633$	19.6847	0.782395
$634$	21.9460	0.871588
$635$	− 19.3693i	− 0.768648i
$636$	0.300187	0.0119032
$637$	0	0
$638$	20.8769	0.826524
$639$	− 14.0000i	− 0.553831i
$640$	48.3002	1.90923
$641$	19.1771	0.757449	0.378725	−	0.925509i	$-0.376363\pi$
$641$	19.1771	0.757449	0.378725	+	0.925509i	$0.376363\pi$
$642$	− 12.8769i	− 0.508210i
$643$	31.5464i	1.24407i	0.782990	+	0.622034i	$0.213694\pi$
$643$	31.5464i	1.24407i	−0.782990	+	0.622034i	$0.786306\pi$
$644$	0.492423i	0.0194042i
$645$	− 16.2462i	− 0.639694i
$646$	17.3693	0.683387
$647$	6.38447	0.250999	0.125500	−	0.992094i	$-0.459947\pi$
$647$	6.38447	0.250999	0.125500	+	0.992094i	$0.459947\pi$
$648$	2.43845i	0.0957913i
$649$	−5.75379	−0.225856
$650$	0	0
$651$	1.43845	0.0563772
$652$	2.10795i	0.0825537i
$653$	23.1231	0.904877	0.452439	−	0.891796i	$-0.350554\pi$
$653$	23.1231	0.904877	0.452439	+	0.891796i	$0.350554\pi$
$654$	−4.38447	−0.171446
$655$	26.2462i	1.02552i
$656$	− 7.31534i	− 0.285616i
$657$	− 10.1231i	− 0.394940i
$658$	− 7.23106i	− 0.281896i
$659$	−2.24621	−0.0875000	−0.0437500	−	0.999043i	$-0.513930\pi$
$659$	−2.24621	−0.0875000	−0.0437500	+	0.999043i	$0.513930\pi$
$660$	−3.12311	−0.121567
$661$	5.63068i	0.219008i	0.993986	+	0.109504i	$0.0349263\pi$
$661$	5.63068i	0.219008i	−0.993986	+	0.109504i	$0.965074\pi$
$662$	−4.98485	−0.193742
$663$	0	0
$664$	−2.13826	−0.0829806
$665$	14.2462i	0.552444i
$666$	−11.8078	−0.457542
$667$	−13.3693	−0.517662
$668$	− 4.49242i	− 0.173817i
$669$	− 8.00000i	− 0.309298i
$670$	25.3693i	0.980102i
$671$	− 7.75379i	− 0.299332i
$672$	1.36932	0.0528225
$673$	23.2462	0.896076	0.448038	−	0.894015i	$-0.352123\pi$
$673$	23.2462	0.896076	0.448038	+	0.894015i	$0.352123\pi$
$674$	9.56155i	0.368297i
$675$	−7.68466	−0.295783
$676$	0	0
$677$	−15.6155	−0.600153	−0.300077	−	0.953915i	$-0.597012\pi$
$677$	−15.6155	−0.600153	−0.300077	+	0.953915i	$0.597012\pi$
$678$	9.06913i	0.348298i
$679$	−4.80776	−0.184505
$680$	−13.5616	−0.520062
$681$	− 7.12311i	− 0.272958i
$682$	− 8.00000i	− 0.306336i
$683$	− 38.1080i	− 1.45816i	−0.684428	−	0.729080i	$-0.739948\pi$
$683$	− 38.1080i	− 1.45816i	0.684428	−	0.729080i	$-0.260052\pi$
$684$	3.12311i	0.119415i
$685$	19.8078	0.756816
$686$	−12.0000	−0.458162
$687$	− 16.2462i	− 0.619832i
$688$	21.3693	0.814698
$689$	0	0
$690$	11.1231	0.423449
$691$	51.3002i	1.95155i	0.218774	+	0.975776i	$0.429794\pi$
$691$	51.3002i	1.95155i	−0.218774	+	0.975776i	$0.570206\pi$
$692$	8.87689	0.337449
$693$	1.12311	0.0426633
$694$	43.1231i	1.63693i
$695$	− 63.8617i	− 2.42241i
$696$	16.3002i	0.617857i
$697$	2.43845i	0.0923628i
$698$	−10.6307	−0.402377
$699$	−26.0000	−0.983410
$700$	1.89205i	0.0715127i
$701$	5.36932	0.202796	0.101398	−	0.994846i	$-0.467668\pi$
$701$	5.36932	0.202796	0.101398	+	0.994846i	$0.467668\pi$
$702$	0	0
$703$	53.8617	2.03143
$704$	11.1231i	0.419218i
$705$	−29.3693	−1.10611
$706$	8.30019	0.312382
$707$	4.24621i	0.159695i
$708$	1.26137i	0.0474050i
$709$	− 7.49242i	− 0.281384i	−0.990053	−	0.140692i	$-0.955067\pi$
$709$	− 7.49242i	− 0.281384i	0.990053	−	0.140692i	$-0.0449327\pi$
$710$	77.8617i	2.92210i
$711$	5.43845	0.203958
$712$	−11.8920	−0.445673
$713$	5.12311i	0.191862i
$714$	−1.36932	−0.0512454
$715$	0	0
$716$	−2.13826	−0.0799106
$717$	25.3693i	0.947435i
$718$	14.6307	0.546012
$719$	23.3693	0.871528	0.435764	−	0.900061i	$-0.356478\pi$
$719$	23.3693	0.871528	0.435764	+	0.900061i	$0.356478\pi$
$720$	− 16.6847i	− 0.621801i
$721$	1.93087i	0.0719093i
$722$	− 49.5616i	− 1.84449i
$723$	− 17.8078i	− 0.662278i
$724$	−1.17708	−0.0437459
$725$	−51.3693	−1.90781
$726$	10.9309i	0.405683i
$727$	38.6695	1.43417	0.717086	−	0.696984i	$-0.245475\pi$
$727$	38.6695	1.43417	0.717086	+	0.696984i	$0.245475\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	56.3002i	2.08376i
$731$	−7.12311	−0.263458
$732$	−1.69981	−0.0628269
$733$	− 20.5076i	− 0.757465i	−0.925506	−	0.378732i	$-0.876360\pi$
$733$	− 20.5076i	− 0.757465i	0.925506	−	0.378732i	$-0.123640\pi$
$734$	− 26.6307i	− 0.982956i
$735$	23.8078i	0.878163i
$736$	4.87689i	0.179765i
$737$	−9.12311	−0.336054
$738$	−2.43845	−0.0897605
$739$	10.2462i	0.376913i	0.982082	+	0.188456i	$0.0603484\pi$
$739$	10.2462i	0.376913i	−0.982082	+	0.188456i	$0.939652\pi$
$740$	11.8078	0.434062
$741$	0	0
$742$	0.600373	0.0220404
$743$	12.6307i	0.463375i	0.972790	+	0.231687i	$0.0744247\pi$
$743$	12.6307i	0.463375i	−0.972790	+	0.231687i	$0.925575\pi$
$744$	6.24621	0.228997
$745$	−8.68466	−0.318181
$746$	44.3002i	1.62195i
$747$	0.876894i	0.0320839i
$748$	1.36932i	0.0500672i
$749$	− 4.63068i	− 0.169201i
$750$	14.9309	0.545198
$751$	−44.1080	−1.60952	−0.804761	−	0.593599i	$-0.797707\pi$
$751$	−44.1080	−1.60952	−0.804761	+	0.593599i	$0.797707\pi$
$752$	− 38.6307i	− 1.40872i
$753$	−18.7386	−0.682874
$754$	0	0
$755$	33.3693	1.21443
$756$	− 0.246211i	− 0.00895462i
$757$	30.0000	1.09037	0.545184	−	0.838316i	$-0.316460\pi$
$757$	30.0000	1.09037	0.545184	+	0.838316i	$0.316460\pi$
$758$	36.9848	1.34335
$759$	4.00000i	0.145191i
$760$	61.8617i	2.24396i
$761$	9.36932i	0.339637i	0.985475	+	0.169819i	$0.0543182\pi$
$761$	9.36932i	0.339637i	−0.985475	+	0.169819i	$0.945682\pi$
$762$	− 8.49242i	− 0.307648i
$763$	−1.57671	−0.0570807
$764$	4.00000	0.144715
$765$	5.56155i	0.201078i
$766$	−35.5076	−1.28294
$767$	0	0
$768$	10.0540	0.362792
$769$	18.0000i	0.649097i	0.945869	+	0.324548i	$0.105212\pi$
$769$	18.0000i	0.649097i	−0.945869	+	0.324548i	$0.894788\pi$
$770$	−6.24621	−0.225098
$771$	−29.1771	−1.05079
$772$	5.91571i	0.212911i
$773$	24.2462i	0.872076i	0.899928	+	0.436038i	$0.143619\pi$
$773$	24.2462i	0.872076i	−0.899928	+	0.436038i	$0.856381\pi$
$774$	− 7.12311i	− 0.256035i
$775$	19.6847i	0.707094i
$776$	−20.8769	−0.749437
$777$	−4.24621	−0.152332
$778$	− 53.1771i	− 1.90649i
$779$	11.1231	0.398527
$780$	0	0
$781$	−28.0000	−1.00192
$782$	− 4.87689i	− 0.174397i
$783$	6.68466	0.238890
$784$	−31.3153	−1.11841
$785$	72.5464i	2.58929i
$786$	11.5076i	0.410462i
$787$	44.1771i	1.57474i	0.616479	+	0.787371i	$0.288559\pi$
$787$	44.1771i	1.57474i	−0.616479	+	0.787371i	$0.711441\pi$
$788$	5.86174i	0.208816i
$789$	9.36932	0.333557
$790$	−30.2462	−1.07611
$791$	3.26137i	0.115961i
$792$	4.87689	0.173293
$793$	0	0
$794$	−39.1231	−1.38843
$795$	− 2.43845i	− 0.0864828i
$796$	9.72348	0.344640
$797$	0.384472	0.0136187	0.00680935	−	0.999977i	$-0.497833\pi$
$797$	0.384472	0.0136187	0.00680935	+	0.999977i	$0.497833\pi$
$798$	6.24621i	0.221113i
$799$	12.8769i	0.455552i
$800$	18.7386i	0.662511i
$801$	4.87689i	0.172317i
$802$	−22.5464	−0.796141
$803$	−20.2462	−0.714473
$804$	2.00000i	0.0705346i
$805$	4.00000	0.140981
$806$	0	0
$807$	−21.3693	−0.752236
$808$	18.4384i	0.648662i
$809$	−16.3002	−0.573084	−0.286542	−	0.958068i	$-0.592506\pi$
$809$	−16.3002	−0.573084	−0.286542	+	0.958068i	$0.592506\pi$
$810$	−5.56155	−0.195413
$811$	− 2.56155i	− 0.0899483i	−0.998988	−	0.0449741i	$-0.985679\pi$
$811$	− 2.56155i	− 0.0899483i	0.998988	−	0.0449741i	$-0.0143205\pi$
$812$	− 1.64584i	− 0.0577576i
$813$	− 29.9309i	− 1.04972i
$814$	23.6155i	0.827724i
$815$	17.1231	0.599796
$816$	−7.31534	−0.256088
$817$	32.4924i	1.13677i
$818$	9.94602	0.347755
$819$	0	0
$820$	2.43845	0.0851543
$821$	− 6.49242i	− 0.226587i	−0.993562	−	0.113294i	$-0.963860\pi$
$821$	− 6.49242i	− 0.226587i	0.993562	−	0.113294i	$-0.0361401\pi$
$822$	8.68466	0.302912
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	8.38447i	0.292087i
$825$	15.3693i	0.535091i
$826$	2.52273i	0.0877771i
$827$	− 14.7386i	− 0.512513i	−0.966609	−	0.256256i	$-0.917511\pi$
$827$	− 14.7386i	− 0.512513i	0.966609	−	0.256256i	$-0.0824891\pi$
$828$	0.876894	0.0304742
$829$	−13.4924	−0.468611	−0.234306	−	0.972163i	$-0.575282\pi$
$829$	−13.4924	−0.468611	−0.234306	+	0.972163i	$0.575282\pi$
$830$	− 4.87689i	− 0.169279i
$831$	−5.31534	−0.184387
$832$	0	0
$833$	10.4384	0.361671
$834$	− 28.0000i	− 0.969561i
$835$	−36.4924	−1.26287
$836$	6.24621	0.216030
$837$	− 2.56155i	− 0.0885402i
$838$	− 53.4773i	− 1.84734i
$839$	21.6155i	0.746251i	0.927781	+	0.373125i	$0.121714\pi$
$839$	21.6155i	0.746251i	−0.927781	+	0.373125i	$0.878286\pi$
$840$	− 4.87689i	− 0.168269i
$841$	15.6847	0.540850
$842$	48.7926	1.68150
$843$	17.8078i	0.613332i
$844$	−8.63068	−0.297080
$845$	0	0
$846$	−12.8769	−0.442717
$847$	3.93087i	0.135066i
$848$	3.20739	0.110142
$849$	13.6847	0.469656
$850$	− 18.7386i	− 0.642730i
$851$	− 15.1231i	− 0.518413i
$852$	6.13826i	0.210293i
$853$	2.12311i	0.0726938i	0.999339	+	0.0363469i	$0.0115721\pi$
$853$	2.12311i	0.0726938i	−0.999339	+	0.0363469i	$0.988428\pi$
$854$	−3.39963	−0.116333
$855$	25.3693	0.867612
$856$	− 20.1080i	− 0.687276i
$857$	−35.5616	−1.21476	−0.607380	−	0.794412i	$-0.707779\pi$
$857$	−35.5616	−1.21476	−0.607380	+	0.794412i	$0.707779\pi$
$858$	0	0
$859$	24.5616	0.838029	0.419015	−	0.907979i	$-0.362376\pi$
$859$	24.5616	0.838029	0.419015	+	0.907979i	$0.362376\pi$
$860$	7.12311i	0.242896i
$861$	−0.876894	−0.0298845
$862$	−17.3693	−0.591601
$863$	− 30.4924i	− 1.03797i	−0.854782	−	0.518987i	$-0.826309\pi$
$863$	− 30.4924i	− 1.03797i	0.854782	−	0.518987i	$-0.173691\pi$
$864$	− 2.43845i	− 0.0829577i
$865$	− 72.1080i	− 2.45174i
$866$	− 13.6695i	− 0.464509i
$867$	−14.5616	−0.494536
$868$	−0.630683	−0.0214068
$869$	− 10.8769i	− 0.368973i
$870$	−37.1771	−1.26042
$871$	0	0
$872$	−6.84658	−0.231855
$873$	8.56155i	0.289765i
$874$	−22.2462	−0.752489
$875$	5.36932	0.181516
$876$	4.43845i	0.149961i
$877$	− 23.5616i	− 0.795617i	−0.917468	−	0.397809i	$-0.869771\pi$
$877$	− 23.5616i	− 0.795617i	0.917468	−	0.397809i	$-0.130229\pi$
$878$	21.3693i	0.721180i
$879$	20.4384i	0.689372i
$880$	−33.3693	−1.12488
$881$	9.06913	0.305547	0.152773	−	0.988261i	$-0.451180\pi$
$881$	9.06913	0.305547	0.152773	+	0.988261i	$0.451180\pi$
$882$	10.4384i	0.351481i
$883$	−8.80776	−0.296405	−0.148202	−	0.988957i	$-0.547349\pi$
$883$	−8.80776	−0.296405	−0.148202	+	0.988957i	$0.547349\pi$
$884$	0	0
$885$	10.2462	0.344423
$886$	− 54.2462i	− 1.82244i
$887$	24.6307	0.827017	0.413509	−	0.910500i	$-0.364303\pi$
$887$	24.6307	0.827017	0.413509	+	0.910500i	$0.364303\pi$
$888$	−18.4384	−0.618754
$889$	− 3.05398i	− 0.102427i
$890$	− 27.1231i	− 0.909169i
$891$	− 2.00000i	− 0.0670025i
$892$	3.50758i	0.117442i
$893$	58.7386	1.96561
$894$	−3.80776	−0.127351
$895$	17.3693i	0.580592i
$896$	7.61553	0.254417
$897$	0	0
$898$	12.8769	0.429708
$899$	− 17.1231i	− 0.571088i
$900$	3.36932	0.112311
$901$	−1.06913	−0.0356179
$902$	4.87689i	0.162383i
$903$	− 2.56155i	− 0.0852431i
$904$	14.1619i	0.471019i
$905$	9.56155i	0.317837i
$906$	14.6307	0.486072
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	3.12311i	0.103644i
$909$	7.56155	0.250801
$910$	0	0
$911$	38.7386	1.28347	0.641734	−	0.766927i	$-0.278215\pi$
$911$	38.7386	1.28347	0.641734	+	0.766927i	$0.278215\pi$
$912$	33.3693i	1.10497i
$913$	1.75379	0.0580419
$914$	19.6998	0.651612
$915$	13.8078i	0.456471i
$916$	7.12311i	0.235354i
$917$	4.13826i	0.136657i
$918$	2.43845i	0.0804807i
$919$	11.5076	0.379600	0.189800	−	0.981823i	$-0.439216\pi$
$919$	11.5076	0.379600	0.189800	+	0.981823i	$0.439216\pi$
$920$	17.3693	0.572649
$921$	30.8078i	1.01515i
$922$	−25.2850	−0.832718
$923$	0	0
$924$	−0.492423	−0.0161995
$925$	− 58.1080i	− 1.91058i
$926$	22.3542	0.734603
$927$	3.43845	0.112933
$928$	− 16.3002i	− 0.535080i
$929$	7.80776i	0.256164i	0.991764	+	0.128082i	$0.0408821\pi$
$929$	7.80776i	0.256164i	−0.991764	+	0.128082i	$0.959118\pi$
$930$	14.2462i	0.467152i
$931$	− 47.6155i	− 1.56054i
$932$	11.3996	0.373407
$933$	19.1231	0.626062
$934$	40.6004i	1.32848i
$935$	11.1231	0.363764
$936$	0	0
$937$	−7.56155	−0.247025	−0.123513	−	0.992343i	$-0.539416\pi$
$937$	−7.56155	−0.247025	−0.123513	+	0.992343i	$0.539416\pi$
$938$	4.00000i	0.130605i
$939$	−13.6847	−0.446582
$940$	12.8769	0.419998
$941$	− 30.4924i	− 0.994025i	−0.867744	−	0.497012i	$-0.834430\pi$
$941$	− 30.4924i	− 0.994025i	0.867744	−	0.497012i	$-0.165570\pi$
$942$	31.8078i	1.03635i
$943$	− 3.12311i	− 0.101702i
$944$	13.4773i	0.438648i
$945$	−2.00000	−0.0650600
$946$	−14.2462	−0.463184
$947$	− 38.7386i	− 1.25884i	−0.777067	−	0.629418i	$-0.783293\pi$
$947$	− 38.7386i	− 1.25884i	0.777067	−	0.629418i	$-0.216707\pi$
$948$	−2.38447	−0.0774440
$949$	0	0
$950$	−85.4773	−2.77325
$951$	− 14.0540i	− 0.455731i
$952$	−2.13826	−0.0693014
$953$	−30.9848	−1.00370	−0.501849	−	0.864955i	$-0.667347\pi$
$953$	−30.9848	−1.00370	−0.501849	+	0.864955i	$0.667347\pi$
$954$	− 1.06913i	− 0.0346144i
$955$	− 32.4924i	− 1.05143i
$956$	− 11.1231i	− 0.359747i
$957$	− 13.3693i	− 0.432169i
$958$	−16.0000	−0.516937
$959$	3.12311	0.100850
$960$	− 19.8078i	− 0.639293i
$961$	24.4384	0.788337
$962$	0	0
$963$	−8.24621	−0.265730
$964$	7.80776i	0.251471i
$965$	48.0540	1.54691
$966$	1.75379	0.0564272
$967$	0.876894i	0.0281990i	0.999901	+	0.0140995i	$0.00448816\pi$
$967$	0.876894i	0.0281990i	−0.999901	+	0.0140995i	$0.995512\pi$
$968$	17.0691i	0.548623i
$969$	− 11.1231i	− 0.357326i
$970$	− 47.6155i	− 1.52884i
$971$	−12.9848	−0.416704	−0.208352	−	0.978054i	$-0.566810\pi$
$971$	−12.9848	−0.416704	−0.208352	+	0.978054i	$0.566810\pi$
$972$	−0.438447	−0.0140632
$973$	− 10.0691i	− 0.322801i
$974$	11.1231	0.356407
$975$	0	0
$976$	−18.1619	−0.581349
$977$	− 61.1771i	− 1.95723i	−0.205705	−	0.978614i	$-0.565949\pi$
$977$	− 61.1771i	− 1.95723i	0.205705	−	0.978614i	$-0.434051\pi$
$978$	7.50758	0.240066
$979$	9.75379	0.311732
$980$	− 10.4384i	− 0.333444i
$981$	2.80776i	0.0896450i
$982$	56.6004i	1.80619i
$983$	13.6155i	0.434268i	0.976142	+	0.217134i	$0.0696708\pi$
$983$	13.6155i	0.434268i	−0.976142	+	0.217134i	$0.930329\pi$
$984$	−3.80776	−0.121387
$985$	47.6155	1.51716
$986$	16.3002i	0.519104i
$987$	−4.63068	−0.147396
$988$	0	0
$989$	9.12311	0.290098
$990$	11.1231i	0.353516i
$991$	−50.3542	−1.59955	−0.799776	−	0.600298i	$-0.795049\pi$
$991$	−50.3542	−1.59955	−0.799776	+	0.600298i	$0.795049\pi$
$992$	−6.24621	−0.198317
$993$	3.19224i	0.101303i
$994$	12.2765i	0.389388i
$995$	− 78.9848i	− 2.50399i
$996$	− 0.384472i	− 0.0121825i
$997$	−20.6155	−0.652900	−0.326450	−	0.945214i	$-0.605853\pi$
$997$	−20.6155	−0.652900	−0.326450	+	0.945214i	$0.605853\pi$
$998$	7.01515	0.222061
$999$	7.56155i	0.239237i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.b.d.337.3		4
3.2	odd	2		1521.2.b.h.1351.2		4
13.2	odd	12		507.2.e.g.22.1		4
13.3	even	3		507.2.j.g.316.2		8
13.4	even	6		507.2.j.g.361.2		8
13.5	odd	4		507.2.a.d.1.2		2
13.6	odd	12		507.2.e.g.484.1		4
13.7	odd	12		39.2.e.b.16.2	✓	4
13.8	odd	4		507.2.a.g.1.1		2
13.9	even	3		507.2.j.g.361.3		8
13.10	even	6		507.2.j.g.316.3		8
13.11	odd	12		39.2.e.b.22.2	yes	4
13.12	even	2	inner	507.2.b.d.337.2		4
39.5	even	4		1521.2.a.m.1.1		2
39.8	even	4		1521.2.a.g.1.2		2
39.11	even	12		117.2.g.c.100.1		4
39.20	even	12		117.2.g.c.55.1		4
39.38	odd	2		1521.2.b.h.1351.3		4
52.7	even	12		624.2.q.h.289.1		4
52.11	even	12		624.2.q.h.529.1		4
52.31	even	4		8112.2.a.bo.1.2		2
52.47	even	4		8112.2.a.bk.1.1		2
65.7	even	12		975.2.bb.i.874.2		8
65.24	odd	12		975.2.i.k.451.1		4
65.33	even	12		975.2.bb.i.874.3		8
65.37	even	12		975.2.bb.i.724.3		8
65.59	odd	12		975.2.i.k.601.1		4
65.63	even	12		975.2.bb.i.724.2		8
156.11	odd	12		1872.2.t.r.1153.2		4
156.59	odd	12		1872.2.t.r.289.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.e.b.16.2	✓	4	13.7	odd	12
39.2.e.b.22.2	yes	4	13.11	odd	12
117.2.g.c.55.1		4	39.20	even	12
117.2.g.c.100.1		4	39.11	even	12
507.2.a.d.1.2		2	13.5	odd	4
507.2.a.g.1.1		2	13.8	odd	4
507.2.b.d.337.2		4	13.12	even	2	inner
507.2.b.d.337.3		4	1.1	even	1	trivial
507.2.e.g.22.1		4	13.2	odd	12
507.2.e.g.484.1		4	13.6	odd	12
507.2.j.g.316.2		8	13.3	even	3
507.2.j.g.316.3		8	13.10	even	6
507.2.j.g.361.2		8	13.4	even	6
507.2.j.g.361.3		8	13.9	even	3
624.2.q.h.289.1		4	52.7	even	12
624.2.q.h.529.1		4	52.11	even	12
975.2.i.k.451.1		4	65.24	odd	12
975.2.i.k.601.1		4	65.59	odd	12
975.2.bb.i.724.2		8	65.63	even	12
975.2.bb.i.724.3		8	65.37	even	12
975.2.bb.i.874.2		8	65.7	even	12
975.2.bb.i.874.3		8	65.33	even	12
1521.2.a.g.1.2		2	39.8	even	4
1521.2.a.m.1.1		2	39.5	even	4
1521.2.b.h.1351.2		4	3.2	odd	2
1521.2.b.h.1351.3		4	39.38	odd	2
1872.2.t.r.289.2		4	156.59	odd	12
1872.2.t.r.1153.2		4	156.11	odd	12
8112.2.a.bk.1.1		2	52.47	even	4
8112.2.a.bo.1.2		2	52.31	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.56155i q^{2} +1.00000 q^{3} -0.438447 q^{4} +3.56155i q^{5} +1.56155i q^{6} +0.561553i q^{7} +2.43845i q^{8} +1.00000 q^{9} -5.56155 q^{10} -2.00000i q^{11} -0.438447 q^{12} -0.876894 q^{14} +3.56155i q^{15} -4.68466 q^{16} +1.56155 q^{17} +1.56155i q^{18} -7.12311i q^{19} -1.56155i q^{20} +0.561553i q^{21} +3.12311 q^{22} -2.00000 q^{23} +2.43845i q^{24} -7.68466 q^{25} +1.00000 q^{27} -0.246211i q^{28} +6.68466 q^{29} -5.56155 q^{30} -2.56155i q^{31} -2.43845i q^{32} -2.00000i q^{33} +2.43845i q^{34} -2.00000 q^{35} -0.438447 q^{36} +7.56155i q^{37} +11.1231 q^{38} -8.68466 q^{40} +1.56155i q^{41} -0.876894 q^{42} -4.56155 q^{43} +0.876894i q^{44} +3.56155i q^{45} -3.12311i q^{46} +8.24621i q^{47} -4.68466 q^{48} +6.68466 q^{49} -12.0000i q^{50} +1.56155 q^{51} -0.684658 q^{53} +1.56155i q^{54} +7.12311 q^{55} -1.36932 q^{56} -7.12311i q^{57} +10.4384i q^{58} -2.87689i q^{59} -1.56155i q^{60} +3.87689 q^{61} +4.00000 q^{62} +0.561553i q^{63} -5.56155 q^{64} +3.12311 q^{66} -4.56155i q^{67} -0.684658 q^{68} -2.00000 q^{69} -3.12311i q^{70} -14.0000i q^{71} +2.43845i q^{72} -10.1231i q^{73} -11.8078 q^{74} -7.68466 q^{75} +3.12311i q^{76} +1.12311 q^{77} +5.43845 q^{79} -16.6847i q^{80} +1.00000 q^{81} -2.43845 q^{82} +0.876894i q^{83} -0.246211i q^{84} +5.56155i q^{85} -7.12311i q^{86} +6.68466 q^{87} +4.87689 q^{88} +4.87689i q^{89} -5.56155 q^{90} +0.876894 q^{92} -2.56155i q^{93} -12.8769 q^{94} +25.3693 q^{95} -2.43845i q^{96} +8.56155i q^{97} +10.4384i q^{98} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 10 q^{4} + 4 q^{9} - 14 q^{10} - 10 q^{12} - 20 q^{14} + 6 q^{16} - 2 q^{17} - 4 q^{22} - 8 q^{23} - 6 q^{25} + 4 q^{27} + 2 q^{29} - 14 q^{30} - 8 q^{35} - 10 q^{36} + 28 q^{38} - 10 q^{40}+ \cdots + 52 q^{95}+O(q^{100}) \) 4 * q + 4 * q^3 - 10 * q^4 + 4 * q^9 - 14 * q^10 - 10 * q^12 - 20 * q^14 + 6 * q^16 - 2 * q^17 - 4 * q^22 - 8 * q^23 - 6 * q^25 + 4 * q^27 + 2 * q^29 - 14 * q^30 - 8 * q^35 - 10 * q^36 + 28 * q^38 - 10 * q^40 - 20 * q^42 - 10 * q^43 + 6 * q^48 + 2 * q^49 - 2 * q^51 + 22 * q^53 + 12 * q^55 + 44 * q^56 + 32 * q^61 + 16 * q^62 - 14 * q^64 - 4 * q^66 + 22 * q^68 - 8 * q^69 - 6 * q^74 - 6 * q^75 - 12 * q^77 + 30 * q^79 + 4 * q^81 - 18 * q^82 + 2 * q^87 + 36 * q^88 - 14 * q^90 + 20 * q^92 - 68 * q^94 + 52 * q^95

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