LMFDB - Embedded newform 7935.2.a.t.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7935,2,Mod(1,7935)] 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("7935.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7935, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7935 = 3 \cdot 5 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7935.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,8,2,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$63.3612940039$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 345)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	7935.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.44949 q^{2} +1.00000 q^{3} +4.00000 q^{4} +1.00000 q^{5} -2.44949 q^{6} +1.00000 q^{7} -4.89898 q^{8} +1.00000 q^{9} -2.44949 q^{10} -2.44949 q^{11} +4.00000 q^{12} +4.44949 q^{13} -2.44949 q^{14} +1.00000 q^{15} +4.00000 q^{16} +5.44949 q^{17} -2.44949 q^{18} -4.44949 q^{19} +4.00000 q^{20} +1.00000 q^{21} +6.00000 q^{22} -4.89898 q^{24} +1.00000 q^{25} -10.8990 q^{26} +1.00000 q^{27} +4.00000 q^{28} +10.3485 q^{29} -2.44949 q^{30} +0.101021 q^{31} -2.44949 q^{33} -13.3485 q^{34} +1.00000 q^{35} +4.00000 q^{36} -3.89898 q^{37} +10.8990 q^{38} +4.44949 q^{39} -4.89898 q^{40} +5.44949 q^{41} -2.44949 q^{42} -2.00000 q^{43} -9.79796 q^{44} +1.00000 q^{45} +8.44949 q^{47} +4.00000 q^{48} -6.00000 q^{49} -2.44949 q^{50} +5.44949 q^{51} +17.7980 q^{52} -0.550510 q^{53} -2.44949 q^{54} -2.44949 q^{55} -4.89898 q^{56} -4.44949 q^{57} -25.3485 q^{58} +10.3485 q^{59} +4.00000 q^{60} -0.651531 q^{61} -0.247449 q^{62} +1.00000 q^{63} -8.00000 q^{64} +4.44949 q^{65} +6.00000 q^{66} +7.00000 q^{67} +21.7980 q^{68} -2.44949 q^{70} -4.34847 q^{71} -4.89898 q^{72} -5.34847 q^{73} +9.55051 q^{74} +1.00000 q^{75} -17.7980 q^{76} -2.44949 q^{77} -10.8990 q^{78} +4.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -13.3485 q^{82} -15.2474 q^{83} +4.00000 q^{84} +5.44949 q^{85} +4.89898 q^{86} +10.3485 q^{87} +12.0000 q^{88} +16.8990 q^{89} -2.44949 q^{90} +4.44949 q^{91} +0.101021 q^{93} -20.6969 q^{94} -4.44949 q^{95} -3.10102 q^{97} +14.6969 q^{98} -2.44949 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 8 q^{4} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 8 q^{12} + 4 q^{13} + 2 q^{15} + 8 q^{16} + 6 q^{17} - 4 q^{19} + 8 q^{20} + 2 q^{21} + 12 q^{22} + 2 q^{25} - 12 q^{26} + 2 q^{27} + 8 q^{28} + 6 q^{29}+ \cdots - 16 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 + 8 * q^4 + 2 * q^5 + 2 * q^7 + 2 * q^9 + 8 * q^12 + 4 * q^13 + 2 * q^15 + 8 * q^16 + 6 * q^17 - 4 * q^19 + 8 * q^20 + 2 * q^21 + 12 * q^22 + 2 * q^25 - 12 * q^26 + 2 * q^27 + 8 * q^28 + 6 * q^29 + 10 * q^31 - 12 * q^34 + 2 * q^35 + 8 * q^36 + 2 * q^37 + 12 * q^38 + 4 * q^39 + 6 * q^41 - 4 * q^43 + 2 * q^45 + 12 * q^47 + 8 * q^48 - 12 * q^49 + 6 * q^51 + 16 * q^52 - 6 * q^53 - 4 * q^57 - 36 * q^58 + 6 * q^59 + 8 * q^60 - 16 * q^61 + 24 * q^62 + 2 * q^63 - 16 * q^64 + 4 * q^65 + 12 * q^66 + 14 * q^67 + 24 * q^68 + 6 * q^71 + 4 * q^73 + 24 * q^74 + 2 * q^75 - 16 * q^76 - 12 * q^78 + 8 * q^79 + 8 * q^80 + 2 * q^81 - 12 * q^82 - 6 * q^83 + 8 * q^84 + 6 * q^85 + 6 * q^87 + 24 * q^88 + 24 * q^89 + 4 * q^91 + 10 * q^93 - 12 * q^94 - 4 * q^95 - 16 * q^97

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 $

$2$	−2.44949	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$2$	−2.44949	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.00000	2.00000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.44949	−1.00000
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	−4.89898	−1.73205
$9$	1.00000	0.333333
$10$	−2.44949	−0.774597
$11$	−2.44949	−0.738549	−0.369274	−	0.929320i	$-0.620394\pi$
$11$	−2.44949	−0.738549	−0.369274	+	0.929320i	$0.620394\pi$
$12$	4.00000	1.15470
$13$	4.44949	1.23407	0.617033	−	0.786937i	$-0.288334\pi$
$13$	4.44949	1.23407	0.617033	+	0.786937i	$0.288334\pi$
$14$	−2.44949	−0.654654
$15$	1.00000	0.258199
$16$	4.00000	1.00000
$17$	5.44949	1.32170	0.660848	−	0.750520i	$-0.270197\pi$
$17$	5.44949	1.32170	0.660848	+	0.750520i	$0.270197\pi$
$18$	−2.44949	−0.577350
$19$	−4.44949	−1.02078	−0.510391	−	0.859942i	$-0.670499\pi$
$19$	−4.44949	−1.02078	−0.510391	+	0.859942i	$0.670499\pi$
$20$	4.00000	0.894427
$21$	1.00000	0.218218
$22$	6.00000	1.27920
$23$	0	0
$23$	0	0
$24$	−4.89898	−1.00000
$25$	1.00000	0.200000
$26$	−10.8990	−2.13747
$27$	1.00000	0.192450
$28$	4.00000	0.755929
$29$	10.3485	1.92166	0.960831	−	0.277134i	$-0.0893846\pi$
$29$	10.3485	1.92166	0.960831	+	0.277134i	$0.0893846\pi$
$30$	−2.44949	−0.447214
$31$	0.101021	0.0181438	0.00907191	−	0.999959i	$-0.497112\pi$
$31$	0.101021	0.0181438	0.00907191	+	0.999959i	$0.497112\pi$
$32$	0	0
$33$	−2.44949	−0.426401
$34$	−13.3485	−2.28924
$35$	1.00000	0.169031
$36$	4.00000	0.666667
$37$	−3.89898	−0.640988	−0.320494	−	0.947250i	$-0.603849\pi$
$37$	−3.89898	−0.640988	−0.320494	+	0.947250i	$0.603849\pi$
$38$	10.8990	1.76805
$39$	4.44949	0.712489
$40$	−4.89898	−0.774597
$41$	5.44949	0.851067	0.425534	−	0.904943i	$-0.360086\pi$
$41$	5.44949	0.851067	0.425534	+	0.904943i	$0.360086\pi$
$42$	−2.44949	−0.377964
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	−9.79796	−1.47710
$45$	1.00000	0.149071
$46$	0	0
$47$	8.44949	1.23248	0.616242	−	0.787557i	$-0.288654\pi$
$47$	8.44949	1.23248	0.616242	+	0.787557i	$0.288654\pi$
$48$	4.00000	0.577350
$49$	−6.00000	−0.857143
$50$	−2.44949	−0.346410
$51$	5.44949	0.763081
$52$	17.7980	2.46813
$53$	−0.550510	−0.0756184	−0.0378092	−	0.999285i	$-0.512038\pi$
$53$	−0.550510	−0.0756184	−0.0378092	+	0.999285i	$0.512038\pi$
$54$	−2.44949	−0.333333
$55$	−2.44949	−0.330289
$56$	−4.89898	−0.654654
$57$	−4.44949	−0.589349
$58$	−25.3485	−3.32842
$59$	10.3485	1.34726	0.673628	−	0.739071i	$-0.264735\pi$
$59$	10.3485	1.34726	0.673628	+	0.739071i	$0.264735\pi$
$60$	4.00000	0.516398
$61$	−0.651531	−0.0834200	−0.0417100	−	0.999130i	$-0.513281\pi$
$61$	−0.651531	−0.0834200	−0.0417100	+	0.999130i	$0.513281\pi$
$62$	−0.247449	−0.0314260
$63$	1.00000	0.125988
$64$	−8.00000	−1.00000
$65$	4.44949	0.551891
$66$	6.00000	0.738549
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	21.7980	2.64339
$69$	0	0
$70$	−2.44949	−0.292770
$71$	−4.34847	−0.516068	−0.258034	−	0.966136i	$-0.583075\pi$
$71$	−4.34847	−0.516068	−0.258034	+	0.966136i	$0.583075\pi$
$72$	−4.89898	−0.577350
$73$	−5.34847	−0.625991	−0.312995	−	0.949755i	$-0.601332\pi$
$73$	−5.34847	−0.625991	−0.312995	+	0.949755i	$0.601332\pi$
$74$	9.55051	1.11022
$75$	1.00000	0.115470
$76$	−17.7980	−2.04157
$77$	−2.44949	−0.279145
$78$	−10.8990	−1.23407
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	−13.3485	−1.47409
$83$	−15.2474	−1.67362	−0.836812	−	0.547490i	$-0.815584\pi$
$83$	−15.2474	−1.67362	−0.836812	+	0.547490i	$0.815584\pi$
$84$	4.00000	0.436436
$85$	5.44949	0.591080
$86$	4.89898	0.528271
$87$	10.3485	1.10947
$88$	12.0000	1.27920
$89$	16.8990	1.79129	0.895644	−	0.444771i	$-0.146715\pi$
$89$	16.8990	1.79129	0.895644	+	0.444771i	$0.146715\pi$
$90$	−2.44949	−0.258199
$91$	4.44949	0.466433
$92$	0	0
$93$	0.101021	0.0104753
$94$	−20.6969	−2.13473
$95$	−4.44949	−0.456508
$96$	0	0
$97$	−3.10102	−0.314861	−0.157430	−	0.987530i	$-0.550321\pi$
$97$	−3.10102	−0.314861	−0.157430	+	0.987530i	$0.550321\pi$
$98$	14.6969	1.48461
$99$	−2.44949	−0.246183
$100$	4.00000	0.400000
$101$	−12.5505	−1.24882	−0.624411	−	0.781096i	$-0.714661\pi$
$101$	−12.5505	−1.24882	−0.624411	+	0.781096i	$0.714661\pi$
$102$	−13.3485	−1.32170
$103$	12.6969	1.25107	0.625533	−	0.780198i	$-0.284881\pi$
$103$	12.6969	1.25107	0.625533	+	0.780198i	$0.284881\pi$
$104$	−21.7980	−2.13747
$105$	1.00000	0.0975900
$106$	1.34847	0.130975
$107$	5.44949	0.526822	0.263411	−	0.964684i	$-0.415152\pi$
$107$	5.44949	0.526822	0.263411	+	0.964684i	$0.415152\pi$
$108$	4.00000	0.384900
$109$	−11.5505	−1.10634	−0.553169	−	0.833069i	$-0.686582\pi$
$109$	−11.5505	−1.10634	−0.553169	+	0.833069i	$0.686582\pi$
$110$	6.00000	0.572078
$111$	−3.89898	−0.370075
$112$	4.00000	0.377964
$113$	1.65153	0.155363	0.0776815	−	0.996978i	$-0.475248\pi$
$113$	1.65153	0.155363	0.0776815	+	0.996978i	$0.475248\pi$
$114$	10.8990	1.02078
$115$	0	0
$116$	41.3939	3.84332
$117$	4.44949	0.411355
$118$	−25.3485	−2.33352
$119$	5.44949	0.499554
$120$	−4.89898	−0.447214
$121$	−5.00000	−0.454545
$122$	1.59592	0.144488
$123$	5.44949	0.491364
$124$	0.404082	0.0362876
$125$	1.00000	0.0894427
$126$	−2.44949	−0.218218
$127$	−1.55051	−0.137586	−0.0687928	−	0.997631i	$-0.521915\pi$
$127$	−1.55051	−0.137586	−0.0687928	+	0.997631i	$0.521915\pi$
$128$	19.5959	1.73205
$129$	−2.00000	−0.176090
$130$	−10.8990	−0.955904
$131$	−19.5959	−1.71210	−0.856052	−	0.516890i	$-0.827090\pi$
$131$	−19.5959	−1.71210	−0.856052	+	0.516890i	$0.827090\pi$
$132$	−9.79796	−0.852803
$133$	−4.44949	−0.385820
$134$	−17.1464	−1.48123
$135$	1.00000	0.0860663
$136$	−26.6969	−2.28924
$137$	16.8990	1.44378	0.721889	−	0.692009i	$-0.243274\pi$
$137$	16.8990	1.44378	0.721889	+	0.692009i	$0.243274\pi$
$138$	0	0
$139$	−4.79796	−0.406958	−0.203479	−	0.979079i	$-0.565225\pi$
$139$	−4.79796	−0.406958	−0.203479	+	0.979079i	$0.565225\pi$
$140$	4.00000	0.338062
$141$	8.44949	0.711575
$142$	10.6515	0.893857
$143$	−10.8990	−0.911418
$144$	4.00000	0.333333
$145$	10.3485	0.859394
$146$	13.1010	1.08425
$147$	−6.00000	−0.494872
$148$	−15.5959	−1.28198
$149$	1.34847	0.110471	0.0552355	−	0.998473i	$-0.482409\pi$
$149$	1.34847	0.110471	0.0552355	+	0.998473i	$0.482409\pi$
$150$	−2.44949	−0.200000
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	21.7980	1.76805
$153$	5.44949	0.440565
$154$	6.00000	0.483494
$155$	0.101021	0.00811416
$156$	17.7980	1.42498
$157$	−18.5959	−1.48412	−0.742058	−	0.670336i	$-0.766150\pi$
$157$	−18.5959	−1.48412	−0.742058	+	0.670336i	$0.766150\pi$
$158$	−9.79796	−0.779484
$159$	−0.550510	−0.0436583
$160$	0	0
$161$	0	0
$162$	−2.44949	−0.192450
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	21.7980	1.70213
$165$	−2.44949	−0.190693
$166$	37.3485	2.89880
$167$	13.3485	1.03294	0.516468	−	0.856307i	$-0.327247\pi$
$167$	13.3485	1.03294	0.516468	+	0.856307i	$0.327247\pi$
$168$	−4.89898	−0.377964
$169$	6.79796	0.522920
$170$	−13.3485	−1.02378
$171$	−4.44949	−0.340261
$172$	−8.00000	−0.609994
$173$	19.5959	1.48985	0.744925	−	0.667148i	$-0.232485\pi$
$173$	19.5959	1.48985	0.744925	+	0.667148i	$0.232485\pi$
$174$	−25.3485	−1.92166
$175$	1.00000	0.0755929
$176$	−9.79796	−0.738549
$177$	10.3485	0.777839
$178$	−41.3939	−3.10260
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	4.00000	0.298142
$181$	8.89898	0.661456	0.330728	−	0.943726i	$-0.392706\pi$
$181$	8.89898	0.661456	0.330728	+	0.943726i	$0.392706\pi$
$182$	−10.8990	−0.807886
$183$	−0.651531	−0.0481625
$184$	0	0
$185$	−3.89898	−0.286659
$186$	−0.247449	−0.0181438
$187$	−13.3485	−0.976137
$188$	33.7980	2.46497
$189$	1.00000	0.0727393
$190$	10.8990	0.790695
$191$	18.2474	1.32034	0.660170	−	0.751117i	$-0.270484\pi$
$191$	18.2474	1.32034	0.660170	+	0.751117i	$0.270484\pi$
$192$	−8.00000	−0.577350
$193$	−6.69694	−0.482056	−0.241028	−	0.970518i	$-0.577485\pi$
$193$	−6.69694	−0.482056	−0.241028	+	0.970518i	$0.577485\pi$
$194$	7.59592	0.545355
$195$	4.44949	0.318635
$196$	−24.0000	−1.71429
$197$	−22.8990	−1.63148	−0.815742	−	0.578415i	$-0.803671\pi$
$197$	−22.8990	−1.63148	−0.815742	+	0.578415i	$0.803671\pi$
$198$	6.00000	0.426401
$199$	2.89898	0.205503	0.102752	−	0.994707i	$-0.467235\pi$
$199$	2.89898	0.205503	0.102752	+	0.994707i	$0.467235\pi$
$200$	−4.89898	−0.346410
$201$	7.00000	0.493742
$202$	30.7423	2.16302
$203$	10.3485	0.726320
$204$	21.7980	1.52616
$205$	5.44949	0.380609
$206$	−31.1010	−2.16691
$207$	0	0
$208$	17.7980	1.23407
$209$	10.8990	0.753898
$210$	−2.44949	−0.169031
$211$	11.0000	0.757271	0.378636	−	0.925546i	$-0.376393\pi$
$211$	11.0000	0.757271	0.378636	+	0.925546i	$0.376393\pi$
$212$	−2.20204	−0.151237
$213$	−4.34847	−0.297952
$214$	−13.3485	−0.912483
$215$	−2.00000	−0.136399
$216$	−4.89898	−0.333333
$217$	0.101021	0.00685772
$218$	28.2929	1.91623
$219$	−5.34847	−0.361416
$220$	−9.79796	−0.660578
$221$	24.2474	1.63106
$222$	9.55051	0.640988
$223$	21.5959	1.44617	0.723085	−	0.690759i	$-0.242724\pi$
$223$	21.5959	1.44617	0.723085	+	0.690759i	$0.242724\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	−4.04541	−0.269097
$227$	−21.7980	−1.44678	−0.723391	−	0.690439i	$-0.757417\pi$
$227$	−21.7980	−1.44678	−0.723391	+	0.690439i	$0.757417\pi$
$228$	−17.7980	−1.17870
$229$	28.4949	1.88300	0.941498	−	0.337019i	$-0.109419\pi$
$229$	28.4949	1.88300	0.941498	+	0.337019i	$0.109419\pi$
$230$	0	0
$231$	−2.44949	−0.161165
$232$	−50.6969	−3.32842
$233$	21.7980	1.42803	0.714016	−	0.700129i	$-0.246874\pi$
$233$	21.7980	1.42803	0.714016	+	0.700129i	$0.246874\pi$
$234$	−10.8990	−0.712489
$235$	8.44949	0.551184
$236$	41.3939	2.69451
$237$	4.00000	0.259828
$238$	−13.3485	−0.865253
$239$	6.55051	0.423717	0.211859	−	0.977300i	$-0.432048\pi$
$239$	6.55051	0.423717	0.211859	+	0.977300i	$0.432048\pi$
$240$	4.00000	0.258199
$241$	29.3485	1.89050	0.945251	−	0.326346i	$-0.105817\pi$
$241$	29.3485	1.89050	0.945251	+	0.326346i	$0.105817\pi$
$242$	12.2474	0.787296
$243$	1.00000	0.0641500
$244$	−2.60612	−0.166840
$245$	−6.00000	−0.383326
$246$	−13.3485	−0.851067
$247$	−19.7980	−1.25971
$248$	−0.494897	−0.0314260
$249$	−15.2474	−0.966268
$250$	−2.44949	−0.154919
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	4.00000	0.251976
$253$	0	0
$254$	3.79796	0.238305
$255$	5.44949	0.341260
$256$	−32.0000	−2.00000
$257$	−25.3485	−1.58119	−0.790597	−	0.612337i	$-0.790230\pi$
$257$	−25.3485	−1.58119	−0.790597	+	0.612337i	$0.790230\pi$
$258$	4.89898	0.304997
$259$	−3.89898	−0.242271
$260$	17.7980	1.10378
$261$	10.3485	0.640554
$262$	48.0000	2.96545
$263$	−27.2474	−1.68015	−0.840075	−	0.542471i	$-0.817489\pi$
$263$	−27.2474	−1.68015	−0.840075	+	0.542471i	$0.817489\pi$
$264$	12.0000	0.738549
$265$	−0.550510	−0.0338176
$266$	10.8990	0.668259
$267$	16.8990	1.03420
$268$	28.0000	1.71037
$269$	−5.44949	−0.332261	−0.166131	−	0.986104i	$-0.553127\pi$
$269$	−5.44949	−0.332261	−0.166131	+	0.986104i	$0.553127\pi$
$270$	−2.44949	−0.149071
$271$	31.6969	1.92545	0.962726	−	0.270479i	$-0.0871820\pi$
$271$	31.6969	1.92545	0.962726	+	0.270479i	$0.0871820\pi$
$272$	21.7980	1.32170
$273$	4.44949	0.269295
$274$	−41.3939	−2.50070
$275$	−2.44949	−0.147710
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	11.7526	0.704871
$279$	0.101021	0.00604794
$280$	−4.89898	−0.292770
$281$	19.3485	1.15423	0.577116	−	0.816662i	$-0.304178\pi$
$281$	19.3485	1.15423	0.577116	+	0.816662i	$0.304178\pi$
$282$	−20.6969	−1.23248
$283$	2.10102	0.124893	0.0624464	−	0.998048i	$-0.480110\pi$
$283$	2.10102	0.124893	0.0624464	+	0.998048i	$0.480110\pi$
$284$	−17.3939	−1.03214
$285$	−4.44949	−0.263565
$286$	26.6969	1.57862
$287$	5.44949	0.321673
$288$	0	0
$289$	12.6969	0.746879
$290$	−25.3485	−1.48851
$291$	−3.10102	−0.181785
$292$	−21.3939	−1.25198
$293$	21.2474	1.24129	0.620645	−	0.784092i	$-0.286871\pi$
$293$	21.2474	1.24129	0.620645	+	0.784092i	$0.286871\pi$
$294$	14.6969	0.857143
$295$	10.3485	0.602511
$296$	19.1010	1.11022
$297$	−2.44949	−0.142134
$298$	−3.30306	−0.191341
$299$	0	0
$300$	4.00000	0.230940
$301$	−2.00000	−0.115278
$302$	−34.2929	−1.97333
$303$	−12.5505	−0.721008
$304$	−17.7980	−1.02078
$305$	−0.651531	−0.0373065
$306$	−13.3485	−0.763081
$307$	−2.65153	−0.151331	−0.0756654	−	0.997133i	$-0.524108\pi$
$307$	−2.65153	−0.151331	−0.0756654	+	0.997133i	$0.524108\pi$
$308$	−9.79796	−0.558291
$309$	12.6969	0.722304
$310$	−0.247449	−0.0140541
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	−21.7980	−1.23407
$313$	−19.6969	−1.11334	−0.556668	−	0.830735i	$-0.687921\pi$
$313$	−19.6969	−1.11334	−0.556668	+	0.830735i	$0.687921\pi$
$314$	45.5505	2.57056
$315$	1.00000	0.0563436
$316$	16.0000	0.900070
$317$	6.24745	0.350892	0.175446	−	0.984489i	$-0.443863\pi$
$317$	6.24745	0.350892	0.175446	+	0.984489i	$0.443863\pi$
$318$	1.34847	0.0756184
$319$	−25.3485	−1.41924
$320$	−8.00000	−0.447214
$321$	5.44949	0.304161
$322$	0	0
$323$	−24.2474	−1.34916
$324$	4.00000	0.222222
$325$	4.44949	0.246813
$326$	24.4949	1.35665
$327$	−11.5505	−0.638745
$328$	−26.6969	−1.47409
$329$	8.44949	0.465835
$330$	6.00000	0.330289
$331$	24.5959	1.35191	0.675957	−	0.736941i	$-0.263731\pi$
$331$	24.5959	1.35191	0.675957	+	0.736941i	$0.263731\pi$
$332$	−60.9898	−3.34725
$333$	−3.89898	−0.213663
$334$	−32.6969	−1.78910
$335$	7.00000	0.382451
$336$	4.00000	0.218218
$337$	19.7980	1.07846	0.539232	−	0.842157i	$-0.318715\pi$
$337$	19.7980	1.07846	0.539232	+	0.842157i	$0.318715\pi$
$338$	−16.6515	−0.905724
$339$	1.65153	0.0896988
$340$	21.7980	1.18216
$341$	−0.247449	−0.0134001
$342$	10.8990	0.589349
$343$	−13.0000	−0.701934
$344$	9.79796	0.528271
$345$	0	0
$346$	−48.0000	−2.58050
$347$	−28.8990	−1.55138	−0.775689	−	0.631115i	$-0.782598\pi$
$347$	−28.8990	−1.55138	−0.775689	+	0.631115i	$0.782598\pi$
$348$	41.3939	2.21894
$349$	23.4949	1.25765	0.628827	−	0.777546i	$-0.283536\pi$
$349$	23.4949	1.25765	0.628827	+	0.777546i	$0.283536\pi$
$350$	−2.44949	−0.130931
$351$	4.44949	0.237496
$352$	0	0
$353$	−15.5505	−0.827670	−0.413835	−	0.910352i	$-0.635811\pi$
$353$	−15.5505	−0.827670	−0.413835	+	0.910352i	$0.635811\pi$
$354$	−25.3485	−1.34726
$355$	−4.34847	−0.230793
$356$	67.5959	3.58258
$357$	5.44949	0.288418
$358$	14.6969	0.776757
$359$	3.55051	0.187389	0.0936944	−	0.995601i	$-0.470132\pi$
$359$	3.55051	0.187389	0.0936944	+	0.995601i	$0.470132\pi$
$360$	−4.89898	−0.258199
$361$	0.797959	0.0419978
$362$	−21.7980	−1.14568
$363$	−5.00000	−0.262432
$364$	17.7980	0.932867
$365$	−5.34847	−0.279952
$366$	1.59592	0.0834200
$367$	−13.6969	−0.714974	−0.357487	−	0.933918i	$-0.616366\pi$
$367$	−13.6969	−0.714974	−0.357487	+	0.933918i	$0.616366\pi$
$368$	0	0
$369$	5.44949	0.283689
$370$	9.55051	0.496507
$371$	−0.550510	−0.0285811
$372$	0.404082	0.0209507
$373$	−27.5959	−1.42886	−0.714431	−	0.699706i	$-0.753314\pi$
$373$	−27.5959	−1.42886	−0.714431	+	0.699706i	$0.753314\pi$
$374$	32.6969	1.69072
$375$	1.00000	0.0516398
$376$	−41.3939	−2.13473
$377$	46.0454	2.37146
$378$	−2.44949	−0.125988
$379$	−25.3939	−1.30440	−0.652198	−	0.758049i	$-0.726153\pi$
$379$	−25.3939	−1.30440	−0.652198	+	0.758049i	$0.726153\pi$
$380$	−17.7980	−0.913016
$381$	−1.55051	−0.0794350
$382$	−44.6969	−2.28689
$383$	−1.65153	−0.0843893	−0.0421946	−	0.999109i	$-0.513435\pi$
$383$	−1.65153	−0.0843893	−0.0421946	+	0.999109i	$0.513435\pi$
$384$	19.5959	1.00000
$385$	−2.44949	−0.124838
$386$	16.4041	0.834946
$387$	−2.00000	−0.101666
$388$	−12.4041	−0.629722
$389$	3.30306	0.167472	0.0837359	−	0.996488i	$-0.473315\pi$
$389$	3.30306	0.167472	0.0837359	+	0.996488i	$0.473315\pi$
$390$	−10.8990	−0.551891
$391$	0	0
$392$	29.3939	1.48461
$393$	−19.5959	−0.988483
$394$	56.0908	2.82581
$395$	4.00000	0.201262
$396$	−9.79796	−0.492366
$397$	−13.7980	−0.692500	−0.346250	−	0.938142i	$-0.612545\pi$
$397$	−13.7980	−0.692500	−0.346250	+	0.938142i	$0.612545\pi$
$398$	−7.10102	−0.355942
$399$	−4.44949	−0.222753
$400$	4.00000	0.200000
$401$	13.1010	0.654234	0.327117	−	0.944984i	$-0.393923\pi$
$401$	13.1010	0.654234	0.327117	+	0.944984i	$0.393923\pi$
$402$	−17.1464	−0.855186
$403$	0.449490	0.0223907
$404$	−50.2020	−2.49764
$405$	1.00000	0.0496904
$406$	−25.3485	−1.25802
$407$	9.55051	0.473401
$408$	−26.6969	−1.32170
$409$	21.8990	1.08283	0.541417	−	0.840754i	$-0.317888\pi$
$409$	21.8990	1.08283	0.541417	+	0.840754i	$0.317888\pi$
$410$	−13.3485	−0.659234
$411$	16.8990	0.833565
$412$	50.7878	2.50213
$413$	10.3485	0.509215
$414$	0	0
$415$	−15.2474	−0.748468
$416$	0	0
$417$	−4.79796	−0.234957
$418$	−26.6969	−1.30579
$419$	−33.5505	−1.63905	−0.819525	−	0.573044i	$-0.805763\pi$
$419$	−33.5505	−1.63905	−0.819525	+	0.573044i	$0.805763\pi$
$420$	4.00000	0.195180
$421$	22.2474	1.08427	0.542137	−	0.840290i	$-0.317615\pi$
$421$	22.2474	1.08427	0.542137	+	0.840290i	$0.317615\pi$
$422$	−26.9444	−1.31163
$423$	8.44949	0.410828
$424$	2.69694	0.130975
$425$	5.44949	0.264339
$426$	10.6515	0.516068
$427$	−0.651531	−0.0315298
$428$	21.7980	1.05364
$429$	−10.8990	−0.526208
$430$	4.89898	0.236250
$431$	24.4949	1.17988	0.589939	−	0.807448i	$-0.299152\pi$
$431$	24.4949	1.17988	0.589939	+	0.807448i	$0.299152\pi$
$432$	4.00000	0.192450
$433$	−2.79796	−0.134461	−0.0672307	−	0.997737i	$-0.521416\pi$
$433$	−2.79796	−0.134461	−0.0672307	+	0.997737i	$0.521416\pi$
$434$	−0.247449	−0.0118779
$435$	10.3485	0.496171
$436$	−46.2020	−2.21268
$437$	0	0
$438$	13.1010	0.625991
$439$	−36.6969	−1.75145	−0.875725	−	0.482811i	$-0.839616\pi$
$439$	−36.6969	−1.75145	−0.875725	+	0.482811i	$0.839616\pi$
$440$	12.0000	0.572078
$441$	−6.00000	−0.285714
$442$	−59.3939	−2.82508
$443$	−20.4495	−0.971585	−0.485792	−	0.874074i	$-0.661469\pi$
$443$	−20.4495	−0.971585	−0.485792	+	0.874074i	$0.661469\pi$
$444$	−15.5959	−0.740150
$445$	16.8990	0.801088
$446$	−52.8990	−2.50484
$447$	1.34847	0.0637804
$448$	−8.00000	−0.377964
$449$	−21.2474	−1.00273	−0.501365	−	0.865236i	$-0.667168\pi$
$449$	−21.2474	−1.00273	−0.501365	+	0.865236i	$0.667168\pi$
$450$	−2.44949	−0.115470
$451$	−13.3485	−0.628555
$452$	6.60612	0.310726
$453$	14.0000	0.657777
$454$	53.3939	2.50590
$455$	4.44949	0.208595
$456$	21.7980	1.02078
$457$	−0.101021	−0.00472554	−0.00236277	−	0.999997i	$-0.500752\pi$
$457$	−0.101021	−0.00472554	−0.00236277	+	0.999997i	$0.500752\pi$
$458$	−69.7980	−3.26144
$459$	5.44949	0.254360
$460$	0	0
$461$	−18.4949	−0.861393	−0.430697	−	0.902497i	$-0.641732\pi$
$461$	−18.4949	−0.861393	−0.430697	+	0.902497i	$0.641732\pi$
$462$	6.00000	0.279145
$463$	−24.9444	−1.15926	−0.579632	−	0.814878i	$-0.696804\pi$
$463$	−24.9444	−1.15926	−0.579632	+	0.814878i	$0.696804\pi$
$464$	41.3939	1.92166
$465$	0.101021	0.00468471
$466$	−53.3939	−2.47342
$467$	34.8434	1.61236	0.806179	−	0.591671i	$-0.201532\pi$
$467$	34.8434	1.61236	0.806179	+	0.591671i	$0.201532\pi$
$468$	17.7980	0.822711
$469$	7.00000	0.323230
$470$	−20.6969	−0.954679
$471$	−18.5959	−0.856855
$472$	−50.6969	−2.33352
$473$	4.89898	0.225255
$474$	−9.79796	−0.450035
$475$	−4.44949	−0.204157
$476$	21.7980	0.999108
$477$	−0.550510	−0.0252061
$478$	−16.0454	−0.733900
$479$	−32.9444	−1.50527	−0.752634	−	0.658439i	$-0.771217\pi$
$479$	−32.9444	−1.50527	−0.752634	+	0.658439i	$0.771217\pi$
$480$	0	0
$481$	−17.3485	−0.791022
$482$	−71.8888	−3.27444
$483$	0	0
$484$	−20.0000	−0.909091
$485$	−3.10102	−0.140810
$486$	−2.44949	−0.111111
$487$	−18.9444	−0.858452	−0.429226	−	0.903197i	$-0.641214\pi$
$487$	−18.9444	−0.858452	−0.429226	+	0.903197i	$0.641214\pi$
$488$	3.19184	0.144488
$489$	−10.0000	−0.452216
$490$	14.6969	0.663940
$491$	12.5505	0.566397	0.283198	−	0.959061i	$-0.408605\pi$
$491$	12.5505	0.566397	0.283198	+	0.959061i	$0.408605\pi$
$492$	21.7980	0.982728
$493$	56.3939	2.53985
$494$	48.4949	2.18189
$495$	−2.44949	−0.110096
$496$	0.404082	0.0181438
$497$	−4.34847	−0.195056
$498$	37.3485	1.67362
$499$	−1.00000	−0.0447661	−0.0223831	−	0.999749i	$-0.507125\pi$
$499$	−1.00000	−0.0447661	−0.0223831	+	0.999749i	$0.507125\pi$
$500$	4.00000	0.178885
$501$	13.3485	0.596366
$502$	44.0908	1.96787
$503$	−7.04541	−0.314139	−0.157070	−	0.987588i	$-0.550205\pi$
$503$	−7.04541	−0.314139	−0.157070	+	0.987588i	$0.550205\pi$
$504$	−4.89898	−0.218218
$505$	−12.5505	−0.558490
$506$	0	0
$507$	6.79796	0.301908
$508$	−6.20204	−0.275171
$509$	28.8990	1.28092	0.640462	−	0.767990i	$-0.278743\pi$
$509$	28.8990	1.28092	0.640462	+	0.767990i	$0.278743\pi$
$510$	−13.3485	−0.591080
$511$	−5.34847	−0.236602
$512$	39.1918	1.73205
$513$	−4.44949	−0.196450
$514$	62.0908	2.73871
$515$	12.6969	0.559494
$516$	−8.00000	−0.352180
$517$	−20.6969	−0.910250
$518$	9.55051	0.419625
$519$	19.5959	0.860165
$520$	−21.7980	−0.955904
$521$	43.8434	1.92081	0.960406	−	0.278603i	$-0.0898713\pi$
$521$	43.8434	1.92081	0.960406	+	0.278603i	$0.0898713\pi$
$522$	−25.3485	−1.10947
$523$	33.3939	1.46021	0.730106	−	0.683334i	$-0.239471\pi$
$523$	33.3939	1.46021	0.730106	+	0.683334i	$0.239471\pi$
$524$	−78.3837	−3.42421
$525$	1.00000	0.0436436
$526$	66.7423	2.91010
$527$	0.550510	0.0239806
$528$	−9.79796	−0.426401
$529$	0	0
$530$	1.34847	0.0585738
$531$	10.3485	0.449085
$532$	−17.7980	−0.771639
$533$	24.2474	1.05027
$534$	−41.3939	−1.79129
$535$	5.44949	0.235602
$536$	−34.2929	−1.48123
$537$	−6.00000	−0.258919
$538$	13.3485	0.575493
$539$	14.6969	0.633042
$540$	4.00000	0.172133
$541$	12.8990	0.554570	0.277285	−	0.960788i	$-0.410565\pi$
$541$	12.8990	0.554570	0.277285	+	0.960788i	$0.410565\pi$
$542$	−77.6413	−3.33498
$543$	8.89898	0.381892
$544$	0	0
$545$	−11.5505	−0.494769
$546$	−10.8990	−0.466433
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	67.5959	2.88755
$549$	−0.651531	−0.0278067
$550$	6.00000	0.255841
$551$	−46.0454	−1.96160
$552$	0	0
$553$	4.00000	0.170097
$554$	24.4949	1.04069
$555$	−3.89898	−0.165502
$556$	−19.1918	−0.813915
$557$	19.0454	0.806980	0.403490	−	0.914984i	$-0.367797\pi$
$557$	19.0454	0.806980	0.403490	+	0.914984i	$0.367797\pi$
$558$	−0.247449	−0.0104753
$559$	−8.89898	−0.376387
$560$	4.00000	0.169031
$561$	−13.3485	−0.563573
$562$	−47.3939	−1.99919
$563$	−5.44949	−0.229669	−0.114834	−	0.993385i	$-0.536634\pi$
$563$	−5.44949	−0.229669	−0.114834	+	0.993385i	$0.536634\pi$
$564$	33.7980	1.42315
$565$	1.65153	0.0694804
$566$	−5.14643	−0.216321
$567$	1.00000	0.0419961
$568$	21.3031	0.893857
$569$	28.2929	1.18610	0.593049	−	0.805166i	$-0.297924\pi$
$569$	28.2929	1.18610	0.593049	+	0.805166i	$0.297924\pi$
$570$	10.8990	0.456508
$571$	−13.1464	−0.550161	−0.275080	−	0.961421i	$-0.588704\pi$
$571$	−13.1464	−0.550161	−0.275080	+	0.961421i	$0.588704\pi$
$572$	−43.5959	−1.82284
$573$	18.2474	0.762298
$574$	−13.3485	−0.557154
$575$	0	0
$576$	−8.00000	−0.333333
$577$	−28.0000	−1.16566	−0.582828	−	0.812596i	$-0.698054\pi$
$577$	−28.0000	−1.16566	−0.582828	+	0.812596i	$0.698054\pi$
$578$	−31.1010	−1.29363
$579$	−6.69694	−0.278315
$580$	41.3939	1.71879
$581$	−15.2474	−0.632571
$582$	7.59592	0.314861
$583$	1.34847	0.0558479
$584$	26.2020	1.08425
$585$	4.44949	0.183964
$586$	−52.0454	−2.14998
$587$	−6.00000	−0.247647	−0.123823	−	0.992304i	$-0.539516\pi$
$587$	−6.00000	−0.247647	−0.123823	+	0.992304i	$0.539516\pi$
$588$	−24.0000	−0.989743
$589$	−0.449490	−0.0185209
$590$	−25.3485	−1.04358
$591$	−22.8990	−0.941938
$592$	−15.5959	−0.640988
$593$	27.5505	1.13136	0.565682	−	0.824624i	$-0.308613\pi$
$593$	27.5505	1.13136	0.565682	+	0.824624i	$0.308613\pi$
$594$	6.00000	0.246183
$595$	5.44949	0.223407
$596$	5.39388	0.220942
$597$	2.89898	0.118647
$598$	0	0
$599$	−2.69694	−0.110194	−0.0550970	−	0.998481i	$-0.517547\pi$
$599$	−2.69694	−0.110194	−0.0550970	+	0.998481i	$0.517547\pi$
$600$	−4.89898	−0.200000
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	4.89898	0.199667
$603$	7.00000	0.285062
$604$	56.0000	2.27861
$605$	−5.00000	−0.203279
$606$	30.7423	1.24882
$607$	−16.2474	−0.659464	−0.329732	−	0.944075i	$-0.606958\pi$
$607$	−16.2474	−0.659464	−0.329732	+	0.944075i	$0.606958\pi$
$608$	0	0
$609$	10.3485	0.419341
$610$	1.59592	0.0646168
$611$	37.5959	1.52097
$612$	21.7980	0.881130
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	6.49490	0.262113
$615$	5.44949	0.219745
$616$	12.0000	0.483494
$617$	−12.5505	−0.505265	−0.252632	−	0.967562i	$-0.581296\pi$
$617$	−12.5505	−0.505265	−0.252632	+	0.967562i	$0.581296\pi$
$618$	−31.1010	−1.25107
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0.404082	0.0162283
$621$	0	0
$622$	−29.3939	−1.17859
$623$	16.8990	0.677043
$624$	17.7980	0.712489
$625$	1.00000	0.0400000
$626$	48.2474	1.92836
$627$	10.8990	0.435263
$628$	−74.3837	−2.96823
$629$	−21.2474	−0.847191
$630$	−2.44949	−0.0975900
$631$	−29.5505	−1.17639	−0.588194	−	0.808720i	$-0.700161\pi$
$631$	−29.5505	−1.17639	−0.588194	+	0.808720i	$0.700161\pi$
$632$	−19.5959	−0.779484
$633$	11.0000	0.437211
$634$	−15.3031	−0.607762
$635$	−1.55051	−0.0615301
$636$	−2.20204	−0.0873166
$637$	−26.6969	−1.05777
$638$	62.0908	2.45820
$639$	−4.34847	−0.172023
$640$	19.5959	0.774597
$641$	−26.9444	−1.06424	−0.532120	−	0.846669i	$-0.678604\pi$
$641$	−26.9444	−1.06424	−0.532120	+	0.846669i	$0.678604\pi$
$642$	−13.3485	−0.526822
$643$	9.69694	0.382410	0.191205	−	0.981550i	$-0.438760\pi$
$643$	9.69694	0.382410	0.191205	+	0.981550i	$0.438760\pi$
$644$	0	0
$645$	−2.00000	−0.0787499
$646$	59.3939	2.33682
$647$	1.95459	0.0768430	0.0384215	−	0.999262i	$-0.487767\pi$
$647$	1.95459	0.0768430	0.0384215	+	0.999262i	$0.487767\pi$
$648$	−4.89898	−0.192450
$649$	−25.3485	−0.995014
$650$	−10.8990	−0.427493
$651$	0.101021	0.00395931
$652$	−40.0000	−1.56652
$653$	43.8434	1.71572	0.857862	−	0.513881i	$-0.171793\pi$
$653$	43.8434	1.71572	0.857862	+	0.513881i	$0.171793\pi$
$654$	28.2929	1.10634
$655$	−19.5959	−0.765676
$656$	21.7980	0.851067
$657$	−5.34847	−0.208664
$658$	−20.6969	−0.806851
$659$	5.14643	0.200476	0.100238	−	0.994963i	$-0.468040\pi$
$659$	5.14643	0.200476	0.100238	+	0.994963i	$0.468040\pi$
$660$	−9.79796	−0.381385
$661$	−46.0908	−1.79272	−0.896362	−	0.443322i	$-0.853800\pi$
$661$	−46.0908	−1.79272	−0.896362	+	0.443322i	$0.853800\pi$
$662$	−60.2474	−2.34158
$663$	24.2474	0.941693
$664$	74.6969	2.89880
$665$	−4.44949	−0.172544
$666$	9.55051	0.370075
$667$	0	0
$668$	53.3939	2.06587
$669$	21.5959	0.834946
$670$	−17.1464	−0.662424
$671$	1.59592	0.0616097
$672$	0	0
$673$	22.4495	0.865364	0.432682	−	0.901547i	$-0.357567\pi$
$673$	22.4495	0.865364	0.432682	+	0.901547i	$0.357567\pi$
$674$	−48.4949	−1.86795
$675$	1.00000	0.0384900
$676$	27.1918	1.04584
$677$	12.5505	0.482355	0.241178	−	0.970481i	$-0.422466\pi$
$677$	12.5505	0.482355	0.241178	+	0.970481i	$0.422466\pi$
$678$	−4.04541	−0.155363
$679$	−3.10102	−0.119006
$680$	−26.6969	−1.02378
$681$	−21.7980	−0.835300
$682$	0.606123	0.0232097
$683$	18.2474	0.698219	0.349110	−	0.937082i	$-0.386484\pi$
$683$	18.2474	0.698219	0.349110	+	0.937082i	$0.386484\pi$
$684$	−17.7980	−0.680522
$685$	16.8990	0.645677
$686$	31.8434	1.21579
$687$	28.4949	1.08715
$688$	−8.00000	−0.304997
$689$	−2.44949	−0.0933181
$690$	0	0
$691$	16.2020	0.616355	0.308177	−	0.951329i	$-0.400281\pi$
$691$	16.2020	0.616355	0.308177	+	0.951329i	$0.400281\pi$
$692$	78.3837	2.97970
$693$	−2.44949	−0.0930484
$694$	70.7878	2.68707
$695$	−4.79796	−0.181997
$696$	−50.6969	−1.92166
$697$	29.6969	1.12485
$698$	−57.5505	−2.17832
$699$	21.7980	0.824475
$700$	4.00000	0.151186
$701$	52.0454	1.96573	0.982864	−	0.184332i	$-0.0590123\pi$
$701$	52.0454	1.96573	0.982864	+	0.184332i	$0.0590123\pi$
$702$	−10.8990	−0.411355
$703$	17.3485	0.654310
$704$	19.5959	0.738549
$705$	8.44949	0.318226
$706$	38.0908	1.43357
$707$	−12.5505	−0.472011
$708$	41.3939	1.55568
$709$	28.7423	1.07944	0.539721	−	0.841844i	$-0.318530\pi$
$709$	28.7423	1.07944	0.539721	+	0.841844i	$0.318530\pi$
$710$	10.6515	0.399745
$711$	4.00000	0.150012
$712$	−82.7878	−3.10260
$713$	0	0
$714$	−13.3485	−0.499554
$715$	−10.8990	−0.407599
$716$	−24.0000	−0.896922
$717$	6.55051	0.244633
$718$	−8.69694	−0.324567
$719$	−11.9444	−0.445450	−0.222725	−	0.974881i	$-0.571495\pi$
$719$	−11.9444	−0.445450	−0.222725	+	0.974881i	$0.571495\pi$
$720$	4.00000	0.149071
$721$	12.6969	0.472859
$722$	−1.95459	−0.0727424
$723$	29.3485	1.09148
$724$	35.5959	1.32291
$725$	10.3485	0.384332
$726$	12.2474	0.454545
$727$	16.3031	0.604647	0.302324	−	0.953205i	$-0.402238\pi$
$727$	16.3031	0.604647	0.302324	+	0.953205i	$0.402238\pi$
$728$	−21.7980	−0.807886
$729$	1.00000	0.0370370
$730$	13.1010	0.484891
$731$	−10.8990	−0.403113
$732$	−2.60612	−0.0963251
$733$	8.59592	0.317497	0.158749	−	0.987319i	$-0.449254\pi$
$733$	8.59592	0.317497	0.158749	+	0.987319i	$0.449254\pi$
$734$	33.5505	1.23837
$735$	−6.00000	−0.221313
$736$	0	0
$737$	−17.1464	−0.631597
$738$	−13.3485	−0.491364
$739$	−9.20204	−0.338503	−0.169251	−	0.985573i	$-0.554135\pi$
$739$	−9.20204	−0.338503	−0.169251	+	0.985573i	$0.554135\pi$
$740$	−15.5959	−0.573317
$741$	−19.7980	−0.727296
$742$	1.34847	0.0495039
$743$	21.7980	0.799690	0.399845	−	0.916583i	$-0.369064\pi$
$743$	21.7980	0.799690	0.399845	+	0.916583i	$0.369064\pi$
$744$	−0.494897	−0.0181438
$745$	1.34847	0.0494041
$746$	67.5959	2.47486
$747$	−15.2474	−0.557875
$748$	−53.3939	−1.95227
$749$	5.44949	0.199120
$750$	−2.44949	−0.0894427
$751$	−30.6515	−1.11849	−0.559245	−	0.829002i	$-0.688909\pi$
$751$	−30.6515	−1.11849	−0.559245	+	0.829002i	$0.688909\pi$
$752$	33.7980	1.23248
$753$	−18.0000	−0.655956
$754$	−112.788	−4.10749
$755$	14.0000	0.509512
$756$	4.00000	0.145479
$757$	3.69694	0.134368	0.0671838	−	0.997741i	$-0.478599\pi$
$757$	3.69694	0.134368	0.0671838	+	0.997741i	$0.478599\pi$
$758$	62.2020	2.25928
$759$	0	0
$760$	21.7980	0.790695
$761$	14.1464	0.512808	0.256404	−	0.966570i	$-0.417462\pi$
$761$	14.1464	0.512808	0.256404	+	0.966570i	$0.417462\pi$
$762$	3.79796	0.137586
$763$	−11.5505	−0.418157
$764$	72.9898	2.64068
$765$	5.44949	0.197027
$766$	4.04541	0.146167
$767$	46.0454	1.66260
$768$	−32.0000	−1.15470
$769$	−10.4495	−0.376818	−0.188409	−	0.982091i	$-0.560333\pi$
$769$	−10.4495	−0.376818	−0.188409	+	0.982091i	$0.560333\pi$
$770$	6.00000	0.216225
$771$	−25.3485	−0.912903
$772$	−26.7878	−0.964112
$773$	16.2929	0.586013	0.293007	−	0.956110i	$-0.405344\pi$
$773$	16.2929	0.586013	0.293007	+	0.956110i	$0.405344\pi$
$774$	4.89898	0.176090
$775$	0.101021	0.00362876
$776$	15.1918	0.545355
$777$	−3.89898	−0.139875
$778$	−8.09082	−0.290070
$779$	−24.2474	−0.868755
$780$	17.7980	0.637269
$781$	10.6515	0.381142
$782$	0	0
$783$	10.3485	0.369824
$784$	−24.0000	−0.857143
$785$	−18.5959	−0.663717
$786$	48.0000	1.71210
$787$	21.6969	0.773412	0.386706	−	0.922203i	$-0.373613\pi$
$787$	21.6969	0.773412	0.386706	+	0.922203i	$0.373613\pi$
$788$	−91.5959	−3.26297
$789$	−27.2474	−0.970035
$790$	−9.79796	−0.348596
$791$	1.65153	0.0587217
$792$	12.0000	0.426401
$793$	−2.89898	−0.102946
$794$	33.7980	1.19944
$795$	−0.550510	−0.0195246
$796$	11.5959	0.411006
$797$	−9.24745	−0.327561	−0.163781	−	0.986497i	$-0.552369\pi$
$797$	−9.24745	−0.327561	−0.163781	+	0.986497i	$0.552369\pi$
$798$	10.8990	0.385820
$799$	46.0454	1.62897
$800$	0	0
$801$	16.8990	0.597096
$802$	−32.0908	−1.13317
$803$	13.1010	0.462325
$804$	28.0000	0.987484
$805$	0	0
$806$	−1.10102	−0.0387818
$807$	−5.44949	−0.191831
$808$	61.4847	2.16302
$809$	6.55051	0.230304	0.115152	−	0.993348i	$-0.463265\pi$
$809$	6.55051	0.230304	0.115152	+	0.993348i	$0.463265\pi$
$810$	−2.44949	−0.0860663
$811$	−42.3939	−1.48865	−0.744325	−	0.667817i	$-0.767229\pi$
$811$	−42.3939	−1.48865	−0.744325	+	0.667817i	$0.767229\pi$
$812$	41.3939	1.45264
$813$	31.6969	1.11166
$814$	−23.3939	−0.819955
$815$	−10.0000	−0.350285
$816$	21.7980	0.763081
$817$	8.89898	0.311336
$818$	−53.6413	−1.87552
$819$	4.44949	0.155478
$820$	21.7980	0.761218
$821$	21.3031	0.743482	0.371741	−	0.928336i	$-0.378761\pi$
$821$	21.3031	0.743482	0.371741	+	0.928336i	$0.378761\pi$
$822$	−41.3939	−1.44378
$823$	3.59592	0.125346	0.0626729	−	0.998034i	$-0.480038\pi$
$823$	3.59592	0.125346	0.0626729	+	0.998034i	$0.480038\pi$
$824$	−62.2020	−2.16691
$825$	−2.44949	−0.0852803
$826$	−25.3485	−0.881986
$827$	−32.1464	−1.11784	−0.558920	−	0.829221i	$-0.688784\pi$
$827$	−32.1464	−1.11784	−0.558920	+	0.829221i	$0.688784\pi$
$828$	0	0
$829$	−27.6969	−0.961954	−0.480977	−	0.876733i	$-0.659718\pi$
$829$	−27.6969	−0.961954	−0.480977	+	0.876733i	$0.659718\pi$
$830$	37.3485	1.29638
$831$	−10.0000	−0.346896
$832$	−35.5959	−1.23407
$833$	−32.6969	−1.13288
$834$	11.7526	0.406958
$835$	13.3485	0.461943
$836$	43.5959	1.50780
$837$	0.101021	0.00349178
$838$	82.1816	2.83892
$839$	45.1918	1.56020	0.780098	−	0.625658i	$-0.215169\pi$
$839$	45.1918	1.56020	0.780098	+	0.625658i	$0.215169\pi$
$840$	−4.89898	−0.169031
$841$	78.0908	2.69279
$842$	−54.4949	−1.87802
$843$	19.3485	0.666397
$844$	44.0000	1.51454
$845$	6.79796	0.233857
$846$	−20.6969	−0.711575
$847$	−5.00000	−0.171802
$848$	−2.20204	−0.0756184
$849$	2.10102	0.0721068
$850$	−13.3485	−0.457849
$851$	0	0
$852$	−17.3939	−0.595904
$853$	46.0908	1.57812	0.789060	−	0.614316i	$-0.210568\pi$
$853$	46.0908	1.57812	0.789060	+	0.614316i	$0.210568\pi$
$854$	1.59592	0.0546112
$855$	−4.44949	−0.152169
$856$	−26.6969	−0.912483
$857$	−13.5959	−0.464428	−0.232214	−	0.972665i	$-0.574597\pi$
$857$	−13.5959	−0.464428	−0.232214	+	0.972665i	$0.574597\pi$
$858$	26.6969	0.911418
$859$	38.1918	1.30309	0.651544	−	0.758611i	$-0.274121\pi$
$859$	38.1918	1.30309	0.651544	+	0.758611i	$0.274121\pi$
$860$	−8.00000	−0.272798
$861$	5.44949	0.185718
$862$	−60.0000	−2.04361
$863$	6.49490	0.221089	0.110544	−	0.993871i	$-0.464741\pi$
$863$	6.49490	0.221089	0.110544	+	0.993871i	$0.464741\pi$
$864$	0	0
$865$	19.5959	0.666281
$866$	6.85357	0.232894
$867$	12.6969	0.431211
$868$	0.404082	0.0137154
$869$	−9.79796	−0.332373
$870$	−25.3485	−0.859394
$871$	31.1464	1.05536
$872$	56.5857	1.91623
$873$	−3.10102	−0.104954
$874$	0	0
$875$	1.00000	0.0338062
$876$	−21.3939	−0.722832
$877$	−10.4949	−0.354388	−0.177194	−	0.984176i	$-0.556702\pi$
$877$	−10.4949	−0.354388	−0.177194	+	0.984176i	$0.556702\pi$
$878$	89.8888	3.03360
$879$	21.2474	0.716659
$880$	−9.79796	−0.330289
$881$	−8.44949	−0.284671	−0.142335	−	0.989819i	$-0.545461\pi$
$881$	−8.44949	−0.284671	−0.142335	+	0.989819i	$0.545461\pi$
$882$	14.6969	0.494872
$883$	−37.5505	−1.26368	−0.631838	−	0.775101i	$-0.717699\pi$
$883$	−37.5505	−1.26368	−0.631838	+	0.775101i	$0.717699\pi$
$884$	96.9898	3.26212
$885$	10.3485	0.347860
$886$	50.0908	1.68283
$887$	−6.49490	−0.218077	−0.109039	−	0.994038i	$-0.534777\pi$
$887$	−6.49490	−0.218077	−0.109039	+	0.994038i	$0.534777\pi$
$888$	19.1010	0.640988
$889$	−1.55051	−0.0520024
$890$	−41.3939	−1.38753
$891$	−2.44949	−0.0820610
$892$	86.3837	2.89234
$893$	−37.5959	−1.25810
$894$	−3.30306	−0.110471
$895$	−6.00000	−0.200558
$896$	19.5959	0.654654
$897$	0	0
$898$	52.0454	1.73678
$899$	1.04541	0.0348663
$900$	4.00000	0.133333
$901$	−3.00000	−0.0999445
$902$	32.6969	1.08869
$903$	−2.00000	−0.0665558
$904$	−8.09082	−0.269097
$905$	8.89898	0.295812
$906$	−34.2929	−1.13930
$907$	−38.7980	−1.28827	−0.644133	−	0.764914i	$-0.722781\pi$
$907$	−38.7980	−1.28827	−0.644133	+	0.764914i	$0.722781\pi$
$908$	−87.1918	−2.89356
$909$	−12.5505	−0.416274
$910$	−10.8990	−0.361298
$911$	14.2020	0.470535	0.235267	−	0.971931i	$-0.424403\pi$
$911$	14.2020	0.470535	0.235267	+	0.971931i	$0.424403\pi$
$912$	−17.7980	−0.589349
$913$	37.3485	1.23605
$914$	0.247449	0.00818488
$915$	−0.651531	−0.0215389
$916$	113.980	3.76599
$917$	−19.5959	−0.647114
$918$	−13.3485	−0.440565
$919$	45.3939	1.49741	0.748703	−	0.662906i	$-0.230677\pi$
$919$	45.3939	1.49741	0.748703	+	0.662906i	$0.230677\pi$
$920$	0	0
$921$	−2.65153	−0.0873709
$922$	45.3031	1.49198
$923$	−19.3485	−0.636863
$924$	−9.79796	−0.322329
$925$	−3.89898	−0.128198
$926$	61.1010	2.00790
$927$	12.6969	0.417022
$928$	0	0
$929$	22.8434	0.749467	0.374733	−	0.927133i	$-0.377734\pi$
$929$	22.8434	0.749467	0.374733	+	0.927133i	$0.377734\pi$
$930$	−0.247449	−0.00811416
$931$	26.6969	0.874957
$932$	87.1918	2.85606
$933$	12.0000	0.392862
$934$	−85.3485	−2.79269
$935$	−13.3485	−0.436542
$936$	−21.7980	−0.712489
$937$	8.89898	0.290717	0.145358	−	0.989379i	$-0.453566\pi$
$937$	8.89898	0.290717	0.145358	+	0.989379i	$0.453566\pi$
$938$	−17.1464	−0.559851
$939$	−19.6969	−0.642785
$940$	33.7980	1.10237
$941$	24.8536	0.810203	0.405102	−	0.914272i	$-0.367236\pi$
$941$	24.8536	0.810203	0.405102	+	0.914272i	$0.367236\pi$
$942$	45.5505	1.48412
$943$	0	0
$944$	41.3939	1.34726
$945$	1.00000	0.0325300
$946$	−12.0000	−0.390154
$947$	−0.494897	−0.0160820	−0.00804100	−	0.999968i	$-0.502560\pi$
$947$	−0.494897	−0.0160820	−0.00804100	+	0.999968i	$0.502560\pi$
$948$	16.0000	0.519656
$949$	−23.7980	−0.772514
$950$	10.8990	0.353610
$951$	6.24745	0.202587
$952$	−26.6969	−0.865253
$953$	−10.4041	−0.337021	−0.168511	−	0.985700i	$-0.553896\pi$
$953$	−10.4041	−0.337021	−0.168511	+	0.985700i	$0.553896\pi$
$954$	1.34847	0.0436583
$955$	18.2474	0.590474
$956$	26.2020	0.847435
$957$	−25.3485	−0.819400
$958$	80.6969	2.60720
$959$	16.8990	0.545697
$960$	−8.00000	−0.258199
$961$	−30.9898	−0.999671
$962$	42.4949	1.37009
$963$	5.44949	0.175607
$964$	117.394	3.78100
$965$	−6.69694	−0.215582
$966$	0	0
$967$	54.0454	1.73798	0.868992	−	0.494827i	$-0.164769\pi$
$967$	54.0454	1.73798	0.868992	+	0.494827i	$0.164769\pi$
$968$	24.4949	0.787296
$969$	−24.2474	−0.778940
$970$	7.59592	0.243890
$971$	−22.2929	−0.715412	−0.357706	−	0.933834i	$-0.616441\pi$
$971$	−22.2929	−0.715412	−0.357706	+	0.933834i	$0.616441\pi$
$972$	4.00000	0.128300
$973$	−4.79796	−0.153816
$974$	46.4041	1.48688
$975$	4.44949	0.142498
$976$	−2.60612	−0.0834200
$977$	−19.6515	−0.628708	−0.314354	−	0.949306i	$-0.601788\pi$
$977$	−19.6515	−0.628708	−0.314354	+	0.949306i	$0.601788\pi$
$978$	24.4949	0.783260
$979$	−41.3939	−1.32295
$980$	−24.0000	−0.766652
$981$	−11.5505	−0.368779
$982$	−30.7423	−0.981028
$983$	33.7423	1.07621	0.538107	−	0.842877i	$-0.319140\pi$
$983$	33.7423	1.07621	0.538107	+	0.842877i	$0.319140\pi$
$984$	−26.6969	−0.851067
$985$	−22.8990	−0.729622
$986$	−138.136	−4.39915
$987$	8.44949	0.268950
$988$	−79.1918	−2.51943
$989$	0	0
$990$	6.00000	0.190693
$991$	57.7878	1.83569	0.917844	−	0.396941i	$-0.129928\pi$
$991$	57.7878	1.83569	0.917844	+	0.396941i	$0.129928\pi$
$992$	0	0
$993$	24.5959	0.780528
$994$	10.6515	0.337846
$995$	2.89898	0.0919038
$996$	−60.9898	−1.93254
$997$	40.0908	1.26969	0.634844	−	0.772640i	$-0.281064\pi$
$997$	40.0908	1.26969	0.634844	+	0.772640i	$0.281064\pi$
$998$	2.44949	0.0775372
$999$	−3.89898	−0.123358

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7935.2.a.t.1.1		2
23.22	odd	2		345.2.a.i.1.1	✓	2
69.68	even	2		1035.2.a.k.1.2		2
92.91	even	2		5520.2.a.bi.1.1		2
115.22	even	4		1725.2.b.m.1174.1		4
115.68	even	4		1725.2.b.m.1174.4		4
115.114	odd	2		1725.2.a.y.1.2		2
345.344	even	2		5175.2.a.bl.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.2.a.i.1.1	✓	2	23.22	odd	2
1035.2.a.k.1.2		2	69.68	even	2
1725.2.a.y.1.2		2	115.114	odd	2
1725.2.b.m.1174.1		4	115.22	even	4
1725.2.b.m.1174.4		4	115.68	even	4
5175.2.a.bl.1.1		2	345.344	even	2
5520.2.a.bi.1.1		2	92.91	even	2
7935.2.a.t.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7935 = 3 \cdot 5 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7935.a (trivial)

\(f(q)\)	\(=\)	\(q-2.44949 q^{2} +1.00000 q^{3} +4.00000 q^{4} +1.00000 q^{5} -2.44949 q^{6} +1.00000 q^{7} -4.89898 q^{8} +1.00000 q^{9} -2.44949 q^{10} -2.44949 q^{11} +4.00000 q^{12} +4.44949 q^{13} -2.44949 q^{14} +1.00000 q^{15} +4.00000 q^{16} +5.44949 q^{17} -2.44949 q^{18} -4.44949 q^{19} +4.00000 q^{20} +1.00000 q^{21} +6.00000 q^{22} -4.89898 q^{24} +1.00000 q^{25} -10.8990 q^{26} +1.00000 q^{27} +4.00000 q^{28} +10.3485 q^{29} -2.44949 q^{30} +0.101021 q^{31} -2.44949 q^{33} -13.3485 q^{34} +1.00000 q^{35} +4.00000 q^{36} -3.89898 q^{37} +10.8990 q^{38} +4.44949 q^{39} -4.89898 q^{40} +5.44949 q^{41} -2.44949 q^{42} -2.00000 q^{43} -9.79796 q^{44} +1.00000 q^{45} +8.44949 q^{47} +4.00000 q^{48} -6.00000 q^{49} -2.44949 q^{50} +5.44949 q^{51} +17.7980 q^{52} -0.550510 q^{53} -2.44949 q^{54} -2.44949 q^{55} -4.89898 q^{56} -4.44949 q^{57} -25.3485 q^{58} +10.3485 q^{59} +4.00000 q^{60} -0.651531 q^{61} -0.247449 q^{62} +1.00000 q^{63} -8.00000 q^{64} +4.44949 q^{65} +6.00000 q^{66} +7.00000 q^{67} +21.7980 q^{68} -2.44949 q^{70} -4.34847 q^{71} -4.89898 q^{72} -5.34847 q^{73} +9.55051 q^{74} +1.00000 q^{75} -17.7980 q^{76} -2.44949 q^{77} -10.8990 q^{78} +4.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -13.3485 q^{82} -15.2474 q^{83} +4.00000 q^{84} +5.44949 q^{85} +4.89898 q^{86} +10.3485 q^{87} +12.0000 q^{88} +16.8990 q^{89} -2.44949 q^{90} +4.44949 q^{91} +0.101021 q^{93} -20.6969 q^{94} -4.44949 q^{95} -3.10102 q^{97} +14.6969 q^{98} -2.44949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 8 q^{4} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 8 q^{12} + 4 q^{13} + 2 q^{15} + 8 q^{16} + 6 q^{17} - 4 q^{19} + 8 q^{20} + 2 q^{21} + 12 q^{22} + 2 q^{25} - 12 q^{26} + 2 q^{27} + 8 q^{28} + 6 q^{29}+ \cdots - 16 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 + 8 * q^4 + 2 * q^5 + 2 * q^7 + 2 * q^9 + 8 * q^12 + 4 * q^13 + 2 * q^15 + 8 * q^16 + 6 * q^17 - 4 * q^19 + 8 * q^20 + 2 * q^21 + 12 * q^22 + 2 * q^25 - 12 * q^26 + 2 * q^27 + 8 * q^28 + 6 * q^29 + 10 * q^31 - 12 * q^34 + 2 * q^35 + 8 * q^36 + 2 * q^37 + 12 * q^38 + 4 * q^39 + 6 * q^41 - 4 * q^43 + 2 * q^45 + 12 * q^47 + 8 * q^48 - 12 * q^49 + 6 * q^51 + 16 * q^52 - 6 * q^53 - 4 * q^57 - 36 * q^58 + 6 * q^59 + 8 * q^60 - 16 * q^61 + 24 * q^62 + 2 * q^63 - 16 * q^64 + 4 * q^65 + 12 * q^66 + 14 * q^67 + 24 * q^68 + 6 * q^71 + 4 * q^73 + 24 * q^74 + 2 * q^75 - 16 * q^76 - 12 * q^78 + 8 * q^79 + 8 * q^80 + 2 * q^81 - 12 * q^82 - 6 * q^83 + 8 * q^84 + 6 * q^85 + 6 * q^87 + 24 * q^88 + 24 * q^89 + 4 * q^91 + 10 * q^93 - 12 * q^94 - 4 * q^95 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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