LMFDB - Embedded newform 5915.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5915,2,Mod(1,5915)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5915, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

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

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

chi := DirichletCharacter("5915.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5915 = 5 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5915.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$47.2315127956$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 455)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5915.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} -1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} -1.00000 q^{7} -2.00000 q^{9} -2.00000 q^{10} -5.00000 q^{11} +2.00000 q^{12} +2.00000 q^{14} +1.00000 q^{15} -4.00000 q^{16} +2.00000 q^{17} +4.00000 q^{18} -6.00000 q^{19} +2.00000 q^{20} -1.00000 q^{21} +10.0000 q^{22} -6.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} -2.00000 q^{28} +2.00000 q^{29} -2.00000 q^{30} -8.00000 q^{31} +8.00000 q^{32} -5.00000 q^{33} -4.00000 q^{34} -1.00000 q^{35} -4.00000 q^{36} -10.0000 q^{37} +12.0000 q^{38} +2.00000 q^{42} +10.0000 q^{43} -10.0000 q^{44} -2.00000 q^{45} +12.0000 q^{46} +8.00000 q^{47} -4.00000 q^{48} +1.00000 q^{49} -2.00000 q^{50} +2.00000 q^{51} +2.00000 q^{53} +10.0000 q^{54} -5.00000 q^{55} -6.00000 q^{57} -4.00000 q^{58} +6.00000 q^{59} +2.00000 q^{60} +2.00000 q^{61} +16.0000 q^{62} +2.00000 q^{63} -8.00000 q^{64} +10.0000 q^{66} -2.00000 q^{67} +4.00000 q^{68} -6.00000 q^{69} +2.00000 q^{70} -15.0000 q^{71} +9.00000 q^{73} +20.0000 q^{74} +1.00000 q^{75} -12.0000 q^{76} +5.00000 q^{77} -3.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +1.00000 q^{83} -2.00000 q^{84} +2.00000 q^{85} -20.0000 q^{86} +2.00000 q^{87} -2.00000 q^{89} +4.00000 q^{90} -12.0000 q^{92} -8.00000 q^{93} -16.0000 q^{94} -6.00000 q^{95} +8.00000 q^{96} -13.0000 q^{97} -2.00000 q^{98} +10.0000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	2.00000	1.00000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.00000	−0.816497
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.00000	−0.666667
$10$	−2.00000	−0.632456
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	2.00000	0.577350
$13$	0	0
$13$	0	0
$14$	2.00000	0.534522
$15$	1.00000	0.258199
$16$	−4.00000	−1.00000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	4.00000	0.942809
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	2.00000	0.447214
$21$	−1.00000	−0.218218
$22$	10.0000	2.13201
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	−2.00000	−0.377964
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	−2.00000	−0.365148
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	8.00000	1.41421
$33$	−5.00000	−0.870388
$34$	−4.00000	−0.685994
$35$	−1.00000	−0.169031
$36$	−4.00000	−0.666667
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	12.0000	1.94666
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	2.00000	0.308607
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	−10.0000	−1.50756
$45$	−2.00000	−0.298142
$46$	12.0000	1.76930
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−4.00000	−0.577350
$49$	1.00000	0.142857
$50$	−2.00000	−0.282843
$51$	2.00000	0.280056
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	10.0000	1.36083
$55$	−5.00000	−0.674200
$56$	0	0
$57$	−6.00000	−0.794719
$58$	−4.00000	−0.525226
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	2.00000	0.258199
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	16.0000	2.03200
$63$	2.00000	0.251976
$64$	−8.00000	−1.00000
$65$	0	0
$66$	10.0000	1.23091
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	4.00000	0.485071
$69$	−6.00000	−0.722315
$70$	2.00000	0.239046
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	20.0000	2.32495
$75$	1.00000	0.115470
$76$	−12.0000	−1.37649
$77$	5.00000	0.569803
$78$	0	0
$79$	−3.00000	−0.337526	−0.168763	−	0.985657i	$-0.553977\pi$
$79$	−3.00000	−0.337526	−0.168763	+	0.985657i	$0.553977\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	0	0
$83$	1.00000	0.109764	0.0548821	−	0.998493i	$-0.482522\pi$
$83$	1.00000	0.109764	0.0548821	+	0.998493i	$0.482522\pi$
$84$	−2.00000	−0.218218
$85$	2.00000	0.216930
$86$	−20.0000	−2.15666
$87$	2.00000	0.214423
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	4.00000	0.421637
$91$	0	0
$92$	−12.0000	−1.25109
$93$	−8.00000	−0.829561
$94$	−16.0000	−1.65027
$95$	−6.00000	−0.615587
$96$	8.00000	0.816497
$97$	−13.0000	−1.31995	−0.659975	−	0.751288i	$-0.729433\pi$
$97$	−13.0000	−1.31995	−0.659975	+	0.751288i	$0.729433\pi$
$98$	−2.00000	−0.202031
$99$	10.0000	1.00504
$100$	2.00000	0.200000
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	−4.00000	−0.396059
$103$	17.0000	1.67506	0.837530	−	0.546392i	$-0.183999\pi$
$103$	17.0000	1.67506	0.837530	+	0.546392i	$0.183999\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	−4.00000	−0.388514
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	−10.0000	−0.962250
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	10.0000	0.953463
$111$	−10.0000	−0.949158
$112$	4.00000	0.377964
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	12.0000	1.12390
$115$	−6.00000	−0.559503
$116$	4.00000	0.371391
$117$	0	0
$118$	−12.0000	−1.10469
$119$	−2.00000	−0.183340
$120$	0	0
$121$	14.0000	1.27273
$122$	−4.00000	−0.362143
$123$	0	0
$124$	−16.0000	−1.43684
$125$	1.00000	0.0894427
$126$	−4.00000	−0.356348
$127$	22.0000	1.95218	0.976092	−	0.217357i	$-0.0697436\pi$
$127$	22.0000	1.95218	0.976092	+	0.217357i	$0.0697436\pi$
$128$	0	0
$129$	10.0000	0.880451
$130$	0	0
$131$	−22.0000	−1.92215	−0.961074	−	0.276289i	$-0.910895\pi$
$131$	−22.0000	−1.92215	−0.961074	+	0.276289i	$0.910895\pi$
$132$	−10.0000	−0.870388
$133$	6.00000	0.520266
$134$	4.00000	0.345547
$135$	−5.00000	−0.430331
$136$	0	0
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	12.0000	1.02151
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	−2.00000	−0.169031
$141$	8.00000	0.673722
$142$	30.0000	2.51754
$143$	0	0
$144$	8.00000	0.666667
$145$	2.00000	0.166091
$146$	−18.0000	−1.48969
$147$	1.00000	0.0824786
$148$	−20.0000	−1.64399
$149$	21.0000	1.72039	0.860194	−	0.509968i	$-0.170343\pi$
$149$	21.0000	1.72039	0.860194	+	0.509968i	$0.170343\pi$
$150$	−2.00000	−0.163299
$151$	17.0000	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	−10.0000	−0.805823
$155$	−8.00000	−0.642575
$156$	0	0
$157$	3.00000	0.239426	0.119713	−	0.992809i	$-0.461803\pi$
$157$	3.00000	0.239426	0.119713	+	0.992809i	$0.461803\pi$
$158$	6.00000	0.477334
$159$	2.00000	0.158610
$160$	8.00000	0.632456
$161$	6.00000	0.472866
$162$	−2.00000	−0.157135
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	−5.00000	−0.389249
$166$	−2.00000	−0.155230
$167$	15.0000	1.16073	0.580367	−	0.814355i	$-0.302909\pi$
$167$	15.0000	1.16073	0.580367	+	0.814355i	$0.302909\pi$
$168$	0	0
$169$	0	0
$170$	−4.00000	−0.306786
$171$	12.0000	0.917663
$172$	20.0000	1.52499
$173$	−3.00000	−0.228086	−0.114043	−	0.993476i	$-0.536380\pi$
$173$	−3.00000	−0.228086	−0.114043	+	0.993476i	$0.536380\pi$
$174$	−4.00000	−0.303239
$175$	−1.00000	−0.0755929
$176$	20.0000	1.50756
$177$	6.00000	0.450988
$178$	4.00000	0.299813
$179$	−15.0000	−1.12115	−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000	−1.12115	−0.560576	+	0.828103i	$0.689420\pi$
$180$	−4.00000	−0.298142
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	−10.0000	−0.735215
$186$	16.0000	1.17318
$187$	−10.0000	−0.731272
$188$	16.0000	1.16692
$189$	5.00000	0.363696
$190$	12.0000	0.870572
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	−8.00000	−0.577350
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	26.0000	1.86669
$195$	0	0
$196$	2.00000	0.142857
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	−20.0000	−1.42134
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	−24.0000	−1.68863
$203$	−2.00000	−0.140372
$204$	4.00000	0.280056
$205$	0	0
$206$	−34.0000	−2.36889
$207$	12.0000	0.834058
$208$	0	0
$209$	30.0000	2.07514
$210$	2.00000	0.138013
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	4.00000	0.274721
$213$	−15.0000	−1.02778
$214$	−20.0000	−1.36717
$215$	10.0000	0.681994
$216$	0	0
$217$	8.00000	0.543075
$218$	−28.0000	−1.89640
$219$	9.00000	0.608164
$220$	−10.0000	−0.674200
$221$	0	0
$222$	20.0000	1.34231
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	−8.00000	−0.534522
$225$	−2.00000	−0.133333
$226$	4.00000	0.266076
$227$	−7.00000	−0.464606	−0.232303	−	0.972643i	$-0.574626\pi$
$227$	−7.00000	−0.464606	−0.232303	+	0.972643i	$0.574626\pi$
$228$	−12.0000	−0.794719
$229$	−8.00000	−0.528655	−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000	−0.528655	−0.264327	+	0.964433i	$0.585150\pi$
$230$	12.0000	0.791257
$231$	5.00000	0.328976
$232$	0	0
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	12.0000	0.781133
$237$	−3.00000	−0.194871
$238$	4.00000	0.259281
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	−4.00000	−0.258199
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	−28.0000	−1.79991
$243$	16.0000	1.02640
$244$	4.00000	0.256074
$245$	1.00000	0.0638877
$246$	0	0
$247$	0	0
$248$	0	0
$249$	1.00000	0.0633724
$250$	−2.00000	−0.126491
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	4.00000	0.251976
$253$	30.0000	1.88608
$254$	−44.0000	−2.76081
$255$	2.00000	0.125245
$256$	16.0000	1.00000
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	−20.0000	−1.24515
$259$	10.0000	0.621370
$260$	0	0
$261$	−4.00000	−0.247594
$262$	44.0000	2.71833
$263$	−22.0000	−1.35658	−0.678289	−	0.734795i	$-0.737278\pi$
$263$	−22.0000	−1.35658	−0.678289	+	0.734795i	$0.737278\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	−12.0000	−0.735767
$267$	−2.00000	−0.122398
$268$	−4.00000	−0.244339
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	10.0000	0.608581
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	−8.00000	−0.485071
$273$	0	0
$274$	−16.0000	−0.966595
$275$	−5.00000	−0.301511
$276$	−12.0000	−0.722315
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	−24.0000	−1.43942
$279$	16.0000	0.957895
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	−16.0000	−0.952786
$283$	−19.0000	−1.12943	−0.564716	−	0.825285i	$-0.691014\pi$
$283$	−19.0000	−1.12943	−0.564716	+	0.825285i	$0.691014\pi$
$284$	−30.0000	−1.78017
$285$	−6.00000	−0.355409
$286$	0	0
$287$	0	0
$288$	−16.0000	−0.942809
$289$	−13.0000	−0.764706
$290$	−4.00000	−0.234888
$291$	−13.0000	−0.762073
$292$	18.0000	1.05337
$293$	−13.0000	−0.759468	−0.379734	−	0.925096i	$-0.623985\pi$
$293$	−13.0000	−0.759468	−0.379734	+	0.925096i	$0.623985\pi$
$294$	−2.00000	−0.116642
$295$	6.00000	0.349334
$296$	0	0
$297$	25.0000	1.45065
$298$	−42.0000	−2.43299
$299$	0	0
$300$	2.00000	0.115470
$301$	−10.0000	−0.576390
$302$	−34.0000	−1.95648
$303$	12.0000	0.689382
$304$	24.0000	1.37649
$305$	2.00000	0.114520
$306$	8.00000	0.457330
$307$	−17.0000	−0.970241	−0.485121	−	0.874447i	$-0.661224\pi$
$307$	−17.0000	−0.970241	−0.485121	+	0.874447i	$0.661224\pi$
$308$	10.0000	0.569803
$309$	17.0000	0.967096
$310$	16.0000	0.908739
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	0	0
$313$	11.0000	0.621757	0.310878	−	0.950450i	$-0.399377\pi$
$313$	11.0000	0.621757	0.310878	+	0.950450i	$0.399377\pi$
$314$	−6.00000	−0.338600
$315$	2.00000	0.112687
$316$	−6.00000	−0.337526
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	−4.00000	−0.224309
$319$	−10.0000	−0.559893
$320$	−8.00000	−0.447214
$321$	10.0000	0.558146
$322$	−12.0000	−0.668734
$323$	−12.0000	−0.667698
$324$	2.00000	0.111111
$325$	0	0
$326$	0	0
$327$	14.0000	0.774202
$328$	0	0
$329$	−8.00000	−0.441054
$330$	10.0000	0.550482
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	2.00000	0.109764
$333$	20.0000	1.09599
$334$	−30.0000	−1.64153
$335$	−2.00000	−0.109272
$336$	4.00000	0.218218
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	0	0
$339$	−2.00000	−0.108625
$340$	4.00000	0.216930
$341$	40.0000	2.16612
$342$	−24.0000	−1.29777
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−6.00000	−0.323029
$346$	6.00000	0.322562
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	4.00000	0.214423
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	−40.0000	−2.13201
$353$	−9.00000	−0.479022	−0.239511	−	0.970894i	$-0.576987\pi$
$353$	−9.00000	−0.479022	−0.239511	+	0.970894i	$0.576987\pi$
$354$	−12.0000	−0.637793
$355$	−15.0000	−0.796117
$356$	−4.00000	−0.212000
$357$	−2.00000	−0.105851
$358$	30.0000	1.58555
$359$	−11.0000	−0.580558	−0.290279	−	0.956942i	$-0.593748\pi$
$359$	−11.0000	−0.580558	−0.290279	+	0.956942i	$0.593748\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−28.0000	−1.47165
$363$	14.0000	0.734809
$364$	0	0
$365$	9.00000	0.471082
$366$	−4.00000	−0.209083
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	24.0000	1.25109
$369$	0	0
$370$	20.0000	1.03975
$371$	−2.00000	−0.103835
$372$	−16.0000	−0.829561
$373$	−38.0000	−1.96757	−0.983783	−	0.179364i	$-0.942596\pi$
$373$	−38.0000	−1.96757	−0.983783	+	0.179364i	$0.942596\pi$
$374$	20.0000	1.03418
$375$	1.00000	0.0516398
$376$	0	0
$377$	0	0
$378$	−10.0000	−0.514344
$379$	11.0000	0.565032	0.282516	−	0.959263i	$-0.408831\pi$
$379$	11.0000	0.565032	0.282516	+	0.959263i	$0.408831\pi$
$380$	−12.0000	−0.615587
$381$	22.0000	1.12709
$382$	24.0000	1.22795
$383$	−21.0000	−1.07305	−0.536525	−	0.843884i	$-0.680263\pi$
$383$	−21.0000	−1.07305	−0.536525	+	0.843884i	$0.680263\pi$
$384$	0	0
$385$	5.00000	0.254824
$386$	−4.00000	−0.203595
$387$	−20.0000	−1.01666
$388$	−26.0000	−1.31995
$389$	−19.0000	−0.963338	−0.481669	−	0.876353i	$-0.659969\pi$
$389$	−19.0000	−0.963338	−0.481669	+	0.876353i	$0.659969\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	−22.0000	−1.10975
$394$	24.0000	1.20910
$395$	−3.00000	−0.150946
$396$	20.0000	1.00504
$397$	3.00000	0.150566	0.0752828	−	0.997162i	$-0.476014\pi$
$397$	3.00000	0.150566	0.0752828	+	0.997162i	$0.476014\pi$
$398$	8.00000	0.401004
$399$	6.00000	0.300376
$400$	−4.00000	−0.200000
$401$	23.0000	1.14857	0.574283	−	0.818657i	$-0.305281\pi$
$401$	23.0000	1.14857	0.574283	+	0.818657i	$0.305281\pi$
$402$	4.00000	0.199502
$403$	0	0
$404$	24.0000	1.19404
$405$	1.00000	0.0496904
$406$	4.00000	0.198517
$407$	50.0000	2.47841
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	8.00000	0.394611
$412$	34.0000	1.67506
$413$	−6.00000	−0.295241
$414$	−24.0000	−1.17954
$415$	1.00000	0.0490881
$416$	0	0
$417$	12.0000	0.587643
$418$	−60.0000	−2.93470
$419$	26.0000	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$420$	−2.00000	−0.0975900
$421$	11.0000	0.536107	0.268054	−	0.963404i	$-0.413620\pi$
$421$	11.0000	0.536107	0.268054	+	0.963404i	$0.413620\pi$
$422$	24.0000	1.16830
$423$	−16.0000	−0.777947
$424$	0	0
$425$	2.00000	0.0970143
$426$	30.0000	1.45350
$427$	−2.00000	−0.0967868
$428$	20.0000	0.966736
$429$	0	0
$430$	−20.0000	−0.964486
$431$	25.0000	1.20421	0.602104	−	0.798418i	$-0.294329\pi$
$431$	25.0000	1.20421	0.602104	+	0.798418i	$0.294329\pi$
$432$	20.0000	0.962250
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	−16.0000	−0.768025
$435$	2.00000	0.0958927
$436$	28.0000	1.34096
$437$	36.0000	1.72211
$438$	−18.0000	−0.860073
$439$	−18.0000	−0.859093	−0.429547	−	0.903045i	$-0.641327\pi$
$439$	−18.0000	−0.859093	−0.429547	+	0.903045i	$0.641327\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	0	0
$443$	−28.0000	−1.33032	−0.665160	−	0.746701i	$-0.731637\pi$
$443$	−28.0000	−1.33032	−0.665160	+	0.746701i	$0.731637\pi$
$444$	−20.0000	−0.949158
$445$	−2.00000	−0.0948091
$446$	32.0000	1.51524
$447$	21.0000	0.993266
$448$	8.00000	0.377964
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	4.00000	0.188562
$451$	0	0
$452$	−4.00000	−0.188144
$453$	17.0000	0.798730
$454$	14.0000	0.657053
$455$	0	0
$456$	0	0
$457$	16.0000	0.748448	0.374224	−	0.927338i	$-0.377909\pi$
$457$	16.0000	0.748448	0.374224	+	0.927338i	$0.377909\pi$
$458$	16.0000	0.747631
$459$	−10.0000	−0.466760
$460$	−12.0000	−0.559503
$461$	8.00000	0.372597	0.186299	−	0.982493i	$-0.440351\pi$
$461$	8.00000	0.372597	0.186299	+	0.982493i	$0.440351\pi$
$462$	−10.0000	−0.465242
$463$	−18.0000	−0.836531	−0.418265	−	0.908325i	$-0.637362\pi$
$463$	−18.0000	−0.836531	−0.418265	+	0.908325i	$0.637362\pi$
$464$	−8.00000	−0.371391
$465$	−8.00000	−0.370991
$466$	28.0000	1.29707
$467$	−7.00000	−0.323921	−0.161961	−	0.986797i	$-0.551782\pi$
$467$	−7.00000	−0.323921	−0.161961	+	0.986797i	$0.551782\pi$
$468$	0	0
$469$	2.00000	0.0923514
$470$	−16.0000	−0.738025
$471$	3.00000	0.138233
$472$	0	0
$473$	−50.0000	−2.29900
$474$	6.00000	0.275589
$475$	−6.00000	−0.275299
$476$	−4.00000	−0.183340
$477$	−4.00000	−0.183147
$478$	0	0
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	8.00000	0.365148
$481$	0	0
$482$	−40.0000	−1.82195
$483$	6.00000	0.273009
$484$	28.0000	1.27273
$485$	−13.0000	−0.590300
$486$	−32.0000	−1.45155
$487$	4.00000	0.181257	0.0906287	−	0.995885i	$-0.471112\pi$
$487$	4.00000	0.181257	0.0906287	+	0.995885i	$0.471112\pi$
$488$	0	0
$489$	0	0
$490$	−2.00000	−0.0903508
$491$	−15.0000	−0.676941	−0.338470	−	0.940977i	$-0.609909\pi$
$491$	−15.0000	−0.676941	−0.338470	+	0.940977i	$0.609909\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	0	0
$495$	10.0000	0.449467
$496$	32.0000	1.43684
$497$	15.0000	0.672842
$498$	−2.00000	−0.0896221
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	2.00000	0.0894427
$501$	15.0000	0.670151
$502$	−48.0000	−2.14234
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	−60.0000	−2.66733
$507$	0	0
$508$	44.0000	1.95218
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	−4.00000	−0.177123
$511$	−9.00000	−0.398137
$512$	−32.0000	−1.41421
$513$	30.0000	1.32453
$514$	−28.0000	−1.23503
$515$	17.0000	0.749110
$516$	20.0000	0.880451
$517$	−40.0000	−1.75920
$518$	−20.0000	−0.878750
$519$	−3.00000	−0.131685
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	8.00000	0.350150
$523$	33.0000	1.44299	0.721495	−	0.692420i	$-0.243455\pi$
$523$	33.0000	1.44299	0.721495	+	0.692420i	$0.243455\pi$
$524$	−44.0000	−1.92215
$525$	−1.00000	−0.0436436
$526$	44.0000	1.91849
$527$	−16.0000	−0.696971
$528$	20.0000	0.870388
$529$	13.0000	0.565217
$530$	−4.00000	−0.173749
$531$	−12.0000	−0.520756
$532$	12.0000	0.520266
$533$	0	0
$534$	4.00000	0.173097
$535$	10.0000	0.432338
$536$	0	0
$537$	−15.0000	−0.647298
$538$	0	0
$539$	−5.00000	−0.215365
$540$	−10.0000	−0.430331
$541$	−23.0000	−0.988847	−0.494424	−	0.869221i	$-0.664621\pi$
$541$	−23.0000	−0.988847	−0.494424	+	0.869221i	$0.664621\pi$
$542$	4.00000	0.171815
$543$	14.0000	0.600798
$544$	16.0000	0.685994
$545$	14.0000	0.599694
$546$	0	0
$547$	30.0000	1.28271	0.641354	−	0.767245i	$-0.278373\pi$
$547$	30.0000	1.28271	0.641354	+	0.767245i	$0.278373\pi$
$548$	16.0000	0.683486
$549$	−4.00000	−0.170716
$550$	10.0000	0.426401
$551$	−12.0000	−0.511217
$552$	0	0
$553$	3.00000	0.127573
$554$	16.0000	0.679775
$555$	−10.0000	−0.424476
$556$	24.0000	1.01783
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	−32.0000	−1.35467
$559$	0	0
$560$	4.00000	0.169031
$561$	−10.0000	−0.422200
$562$	−36.0000	−1.51857
$563$	−19.0000	−0.800755	−0.400377	−	0.916350i	$-0.631121\pi$
$563$	−19.0000	−0.800755	−0.400377	+	0.916350i	$0.631121\pi$
$564$	16.0000	0.673722
$565$	−2.00000	−0.0841406
$566$	38.0000	1.59726
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−11.0000	−0.461144	−0.230572	−	0.973055i	$-0.574060\pi$
$569$	−11.0000	−0.461144	−0.230572	+	0.973055i	$0.574060\pi$
$570$	12.0000	0.502625
$571$	−15.0000	−0.627730	−0.313865	−	0.949468i	$-0.601624\pi$
$571$	−15.0000	−0.627730	−0.313865	+	0.949468i	$0.601624\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	0	0
$575$	−6.00000	−0.250217
$576$	16.0000	0.666667
$577$	25.0000	1.04076	0.520382	−	0.853934i	$-0.325790\pi$
$577$	25.0000	1.04076	0.520382	+	0.853934i	$0.325790\pi$
$578$	26.0000	1.08146
$579$	2.00000	0.0831172
$580$	4.00000	0.166091
$581$	−1.00000	−0.0414870
$582$	26.0000	1.07773
$583$	−10.0000	−0.414158
$584$	0	0
$585$	0	0
$586$	26.0000	1.07405
$587$	11.0000	0.454019	0.227009	−	0.973893i	$-0.427105\pi$
$587$	11.0000	0.454019	0.227009	+	0.973893i	$0.427105\pi$
$588$	2.00000	0.0824786
$589$	48.0000	1.97781
$590$	−12.0000	−0.494032
$591$	−12.0000	−0.493614
$592$	40.0000	1.64399
$593$	25.0000	1.02663	0.513313	−	0.858201i	$-0.328418\pi$
$593$	25.0000	1.02663	0.513313	+	0.858201i	$0.328418\pi$
$594$	−50.0000	−2.05152
$595$	−2.00000	−0.0819920
$596$	42.0000	1.72039
$597$	−4.00000	−0.163709
$598$	0	0
$599$	−19.0000	−0.776319	−0.388159	−	0.921592i	$-0.626889\pi$
$599$	−19.0000	−0.776319	−0.388159	+	0.921592i	$0.626889\pi$
$600$	0	0
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	20.0000	0.815139
$603$	4.00000	0.162893
$604$	34.0000	1.38344
$605$	14.0000	0.569181
$606$	−24.0000	−0.974933
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	−48.0000	−1.94666
$609$	−2.00000	−0.0810441
$610$	−4.00000	−0.161955
$611$	0	0
$612$	−8.00000	−0.323381
$613$	18.0000	0.727013	0.363507	−	0.931592i	$-0.381579\pi$
$613$	18.0000	0.727013	0.363507	+	0.931592i	$0.381579\pi$
$614$	34.0000	1.37213
$615$	0	0
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	−34.0000	−1.36768
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	−16.0000	−0.642575
$621$	30.0000	1.20386
$622$	8.00000	0.320771
$623$	2.00000	0.0801283
$624$	0	0
$625$	1.00000	0.0400000
$626$	−22.0000	−0.879297
$627$	30.0000	1.19808
$628$	6.00000	0.239426
$629$	−20.0000	−0.797452
$630$	−4.00000	−0.159364
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	−12.0000	−0.476957
$634$	12.0000	0.476581
$635$	22.0000	0.873043
$636$	4.00000	0.158610
$637$	0	0
$638$	20.0000	0.791808
$639$	30.0000	1.18678
$640$	0	0
$641$	11.0000	0.434474	0.217237	−	0.976119i	$-0.430296\pi$
$641$	11.0000	0.434474	0.217237	+	0.976119i	$0.430296\pi$
$642$	−20.0000	−0.789337
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	12.0000	0.472866
$645$	10.0000	0.393750
$646$	24.0000	0.944267
$647$	29.0000	1.14011	0.570054	−	0.821607i	$-0.306922\pi$
$647$	29.0000	1.14011	0.570054	+	0.821607i	$0.306922\pi$
$648$	0	0
$649$	−30.0000	−1.17760
$650$	0	0
$651$	8.00000	0.313545
$652$	0	0
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	−28.0000	−1.09489
$655$	−22.0000	−0.859611
$656$	0	0
$657$	−18.0000	−0.702247
$658$	16.0000	0.623745
$659$	15.0000	0.584317	0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000	0.584317	0.292159	+	0.956370i	$0.405627\pi$
$660$	−10.0000	−0.389249
$661$	4.00000	0.155582	0.0777910	−	0.996970i	$-0.475213\pi$
$661$	4.00000	0.155582	0.0777910	+	0.996970i	$0.475213\pi$
$662$	24.0000	0.932786
$663$	0	0
$664$	0	0
$665$	6.00000	0.232670
$666$	−40.0000	−1.54997
$667$	−12.0000	−0.464642
$668$	30.0000	1.16073
$669$	−16.0000	−0.618596
$670$	4.00000	0.154533
$671$	−10.0000	−0.386046
$672$	−8.00000	−0.308607
$673$	−24.0000	−0.925132	−0.462566	−	0.886585i	$-0.653071\pi$
$673$	−24.0000	−0.925132	−0.462566	+	0.886585i	$0.653071\pi$
$674$	−16.0000	−0.616297
$675$	−5.00000	−0.192450
$676$	0	0
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	4.00000	0.153619
$679$	13.0000	0.498894
$680$	0	0
$681$	−7.00000	−0.268241
$682$	−80.0000	−3.06336
$683$	−16.0000	−0.612223	−0.306111	−	0.951996i	$-0.599028\pi$
$683$	−16.0000	−0.612223	−0.306111	+	0.951996i	$0.599028\pi$
$684$	24.0000	0.917663
$685$	8.00000	0.305664
$686$	2.00000	0.0763604
$687$	−8.00000	−0.305219
$688$	−40.0000	−1.52499
$689$	0	0
$690$	12.0000	0.456832
$691$	26.0000	0.989087	0.494543	−	0.869153i	$-0.335335\pi$
$691$	26.0000	0.989087	0.494543	+	0.869153i	$0.335335\pi$
$692$	−6.00000	−0.228086
$693$	−10.0000	−0.379869
$694$	56.0000	2.12573
$695$	12.0000	0.455186
$696$	0	0
$697$	0	0
$698$	−36.0000	−1.36262
$699$	−14.0000	−0.529529
$700$	−2.00000	−0.0755929
$701$	−14.0000	−0.528773	−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000	−0.528773	−0.264386	+	0.964417i	$0.585169\pi$
$702$	0	0
$703$	60.0000	2.26294
$704$	40.0000	1.50756
$705$	8.00000	0.301297
$706$	18.0000	0.677439
$707$	−12.0000	−0.451306
$708$	12.0000	0.450988
$709$	5.00000	0.187779	0.0938895	−	0.995583i	$-0.470070\pi$
$709$	5.00000	0.187779	0.0938895	+	0.995583i	$0.470070\pi$
$710$	30.0000	1.12588
$711$	6.00000	0.225018
$712$	0	0
$713$	48.0000	1.79761
$714$	4.00000	0.149696
$715$	0	0
$716$	−30.0000	−1.12115
$717$	0	0
$718$	22.0000	0.821033
$719$	34.0000	1.26799	0.633993	−	0.773339i	$-0.281415\pi$
$719$	34.0000	1.26799	0.633993	+	0.773339i	$0.281415\pi$
$720$	8.00000	0.298142
$721$	−17.0000	−0.633113
$722$	−34.0000	−1.26535
$723$	20.0000	0.743808
$724$	28.0000	1.04061
$725$	2.00000	0.0742781
$726$	−28.0000	−1.03918
$727$	15.0000	0.556319	0.278160	−	0.960535i	$-0.410276\pi$
$727$	15.0000	0.556319	0.278160	+	0.960535i	$0.410276\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	−18.0000	−0.666210
$731$	20.0000	0.739727
$732$	4.00000	0.147844
$733$	38.0000	1.40356	0.701781	−	0.712393i	$-0.252388\pi$
$733$	38.0000	1.40356	0.701781	+	0.712393i	$0.252388\pi$
$734$	−32.0000	−1.18114
$735$	1.00000	0.0368856
$736$	−48.0000	−1.76930
$737$	10.0000	0.368355
$738$	0	0
$739$	9.00000	0.331070	0.165535	−	0.986204i	$-0.447065\pi$
$739$	9.00000	0.331070	0.165535	+	0.986204i	$0.447065\pi$
$740$	−20.0000	−0.735215
$741$	0	0
$742$	4.00000	0.146845
$743$	−30.0000	−1.10059	−0.550297	−	0.834969i	$-0.685485\pi$
$743$	−30.0000	−1.10059	−0.550297	+	0.834969i	$0.685485\pi$
$744$	0	0
$745$	21.0000	0.769380
$746$	76.0000	2.78256
$747$	−2.00000	−0.0731762
$748$	−20.0000	−0.731272
$749$	−10.0000	−0.365392
$750$	−2.00000	−0.0730297
$751$	23.0000	0.839282	0.419641	−	0.907690i	$-0.362156\pi$
$751$	23.0000	0.839282	0.419641	+	0.907690i	$0.362156\pi$
$752$	−32.0000	−1.16692
$753$	24.0000	0.874609
$754$	0	0
$755$	17.0000	0.618693
$756$	10.0000	0.363696
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	−22.0000	−0.799076
$759$	30.0000	1.08893
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	−44.0000	−1.59395
$763$	−14.0000	−0.506834
$764$	−24.0000	−0.868290
$765$	−4.00000	−0.144620
$766$	42.0000	1.51752
$767$	0	0
$768$	16.0000	0.577350
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	−10.0000	−0.360375
$771$	14.0000	0.504198
$772$	4.00000	0.143963
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	40.0000	1.43777
$775$	−8.00000	−0.287368
$776$	0	0
$777$	10.0000	0.358748
$778$	38.0000	1.36237
$779$	0	0
$780$	0	0
$781$	75.0000	2.68371
$782$	24.0000	0.858238
$783$	−10.0000	−0.357371
$784$	−4.00000	−0.142857
$785$	3.00000	0.107075
$786$	44.0000	1.56943
$787$	43.0000	1.53278	0.766392	−	0.642373i	$-0.222050\pi$
$787$	43.0000	1.53278	0.766392	+	0.642373i	$0.222050\pi$
$788$	−24.0000	−0.854965
$789$	−22.0000	−0.783221
$790$	6.00000	0.213470
$791$	2.00000	0.0711118
$792$	0	0
$793$	0	0
$794$	−6.00000	−0.212932
$795$	2.00000	0.0709327
$796$	−8.00000	−0.283552
$797$	9.00000	0.318796	0.159398	−	0.987214i	$-0.449045\pi$
$797$	9.00000	0.318796	0.159398	+	0.987214i	$0.449045\pi$
$798$	−12.0000	−0.424795
$799$	16.0000	0.566039
$800$	8.00000	0.282843
$801$	4.00000	0.141333
$802$	−46.0000	−1.62432
$803$	−45.0000	−1.58802
$804$	−4.00000	−0.141069
$805$	6.00000	0.211472
$806$	0	0
$807$	0	0
$808$	0	0
$809$	49.0000	1.72275	0.861374	−	0.507971i	$-0.169604\pi$
$809$	49.0000	1.72275	0.861374	+	0.507971i	$0.169604\pi$
$810$	−2.00000	−0.0702728
$811$	24.0000	0.842754	0.421377	−	0.906886i	$-0.361547\pi$
$811$	24.0000	0.842754	0.421377	+	0.906886i	$0.361547\pi$
$812$	−4.00000	−0.140372
$813$	−2.00000	−0.0701431
$814$	−100.000	−3.50500
$815$	0	0
$816$	−8.00000	−0.280056
$817$	−60.0000	−2.09913
$818$	−36.0000	−1.25871
$819$	0	0
$820$	0	0
$821$	−53.0000	−1.84971	−0.924856	−	0.380317i	$-0.875815\pi$
$821$	−53.0000	−1.84971	−0.924856	+	0.380317i	$0.875815\pi$
$822$	−16.0000	−0.558064
$823$	−22.0000	−0.766872	−0.383436	−	0.923567i	$-0.625259\pi$
$823$	−22.0000	−0.766872	−0.383436	+	0.923567i	$0.625259\pi$
$824$	0	0
$825$	−5.00000	−0.174078
$826$	12.0000	0.417533
$827$	−16.0000	−0.556375	−0.278187	−	0.960527i	$-0.589734\pi$
$827$	−16.0000	−0.556375	−0.278187	+	0.960527i	$0.589734\pi$
$828$	24.0000	0.834058
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	−2.00000	−0.0694210
$831$	−8.00000	−0.277517
$832$	0	0
$833$	2.00000	0.0692959
$834$	−24.0000	−0.831052
$835$	15.0000	0.519096
$836$	60.0000	2.07514
$837$	40.0000	1.38260
$838$	−52.0000	−1.79631
$839$	−20.0000	−0.690477	−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000	−0.690477	−0.345238	+	0.938515i	$0.612202\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−22.0000	−0.758170
$843$	18.0000	0.619953
$844$	−24.0000	−0.826114
$845$	0	0
$846$	32.0000	1.10018
$847$	−14.0000	−0.481046
$848$	−8.00000	−0.274721
$849$	−19.0000	−0.652078
$850$	−4.00000	−0.137199
$851$	60.0000	2.05677
$852$	−30.0000	−1.02778
$853$	27.0000	0.924462	0.462231	−	0.886759i	$-0.347049\pi$
$853$	27.0000	0.924462	0.462231	+	0.886759i	$0.347049\pi$
$854$	4.00000	0.136877
$855$	12.0000	0.410391
$856$	0	0
$857$	15.0000	0.512390	0.256195	−	0.966625i	$-0.417531\pi$
$857$	15.0000	0.512390	0.256195	+	0.966625i	$0.417531\pi$
$858$	0	0
$859$	16.0000	0.545913	0.272956	−	0.962026i	$-0.411998\pi$
$859$	16.0000	0.545913	0.272956	+	0.962026i	$0.411998\pi$
$860$	20.0000	0.681994
$861$	0	0
$862$	−50.0000	−1.70301
$863$	−30.0000	−1.02121	−0.510606	−	0.859815i	$-0.670579\pi$
$863$	−30.0000	−1.02121	−0.510606	+	0.859815i	$0.670579\pi$
$864$	−40.0000	−1.36083
$865$	−3.00000	−0.102003
$866$	52.0000	1.76703
$867$	−13.0000	−0.441503
$868$	16.0000	0.543075
$869$	15.0000	0.508840
$870$	−4.00000	−0.135613
$871$	0	0
$872$	0	0
$873$	26.0000	0.879967
$874$	−72.0000	−2.43544
$875$	−1.00000	−0.0338062
$876$	18.0000	0.608164
$877$	30.0000	1.01303	0.506514	−	0.862232i	$-0.330934\pi$
$877$	30.0000	1.01303	0.506514	+	0.862232i	$0.330934\pi$
$878$	36.0000	1.21494
$879$	−13.0000	−0.438479
$880$	20.0000	0.674200
$881$	20.0000	0.673817	0.336909	−	0.941537i	$-0.390619\pi$
$881$	20.0000	0.673817	0.336909	+	0.941537i	$0.390619\pi$
$882$	4.00000	0.134687
$883$	26.0000	0.874970	0.437485	−	0.899226i	$-0.355869\pi$
$883$	26.0000	0.874970	0.437485	+	0.899226i	$0.355869\pi$
$884$	0	0
$885$	6.00000	0.201688
$886$	56.0000	1.88136
$887$	16.0000	0.537227	0.268614	−	0.963248i	$-0.413434\pi$
$887$	16.0000	0.537227	0.268614	+	0.963248i	$0.413434\pi$
$888$	0	0
$889$	−22.0000	−0.737856
$890$	4.00000	0.134080
$891$	−5.00000	−0.167506
$892$	−32.0000	−1.07144
$893$	−48.0000	−1.60626
$894$	−42.0000	−1.40469
$895$	−15.0000	−0.501395
$896$	0	0
$897$	0	0
$898$	−36.0000	−1.20134
$899$	−16.0000	−0.533630
$900$	−4.00000	−0.133333
$901$	4.00000	0.133259
$902$	0	0
$903$	−10.0000	−0.332779
$904$	0	0
$905$	14.0000	0.465376
$906$	−34.0000	−1.12957
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	−14.0000	−0.464606
$909$	−24.0000	−0.796030
$910$	0	0
$911$	−15.0000	−0.496972	−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000	−0.496972	−0.248486	+	0.968635i	$0.579933\pi$
$912$	24.0000	0.794719
$913$	−5.00000	−0.165476
$914$	−32.0000	−1.05847
$915$	2.00000	0.0661180
$916$	−16.0000	−0.528655
$917$	22.0000	0.726504
$918$	20.0000	0.660098
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	−17.0000	−0.560169
$922$	−16.0000	−0.526932
$923$	0	0
$924$	10.0000	0.328976
$925$	−10.0000	−0.328798
$926$	36.0000	1.18303
$927$	−34.0000	−1.11671
$928$	16.0000	0.525226
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	16.0000	0.524661
$931$	−6.00000	−0.196642
$932$	−28.0000	−0.917170
$933$	−4.00000	−0.130954
$934$	14.0000	0.458094
$935$	−10.0000	−0.327035
$936$	0	0
$937$	−30.0000	−0.980057	−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000	−0.980057	−0.490029	+	0.871706i	$0.663014\pi$
$938$	−4.00000	−0.130605
$939$	11.0000	0.358971
$940$	16.0000	0.521862
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	−6.00000	−0.195491
$943$	0	0
$944$	−24.0000	−0.781133
$945$	5.00000	0.162650
$946$	100.000	3.25128
$947$	42.0000	1.36482	0.682408	−	0.730971i	$-0.260933\pi$
$947$	42.0000	1.36482	0.682408	+	0.730971i	$0.260933\pi$
$948$	−6.00000	−0.194871
$949$	0	0
$950$	12.0000	0.389331
$951$	−6.00000	−0.194563
$952$	0	0
$953$	−12.0000	−0.388718	−0.194359	−	0.980930i	$-0.562263\pi$
$953$	−12.0000	−0.388718	−0.194359	+	0.980930i	$0.562263\pi$
$954$	8.00000	0.259010
$955$	−12.0000	−0.388311
$956$	0	0
$957$	−10.0000	−0.323254
$958$	72.0000	2.32621
$959$	−8.00000	−0.258333
$960$	−8.00000	−0.258199
$961$	33.0000	1.06452
$962$	0	0
$963$	−20.0000	−0.644491
$964$	40.0000	1.28831
$965$	2.00000	0.0643823
$966$	−12.0000	−0.386094
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	0	0
$969$	−12.0000	−0.385496
$970$	26.0000	0.834810
$971$	−52.0000	−1.66876	−0.834380	−	0.551190i	$-0.814174\pi$
$971$	−52.0000	−1.66876	−0.834380	+	0.551190i	$0.814174\pi$
$972$	32.0000	1.02640
$973$	−12.0000	−0.384702
$974$	−8.00000	−0.256337
$975$	0	0
$976$	−8.00000	−0.256074
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	10.0000	0.319601
$980$	2.00000	0.0638877
$981$	−28.0000	−0.893971
$982$	30.0000	0.957338
$983$	31.0000	0.988746	0.494373	−	0.869250i	$-0.335398\pi$
$983$	31.0000	0.988746	0.494373	+	0.869250i	$0.335398\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	−8.00000	−0.254772
$987$	−8.00000	−0.254643
$988$	0	0
$989$	−60.0000	−1.90789
$990$	−20.0000	−0.635642
$991$	48.0000	1.52477	0.762385	−	0.647124i	$-0.224028\pi$
$991$	48.0000	1.52477	0.762385	+	0.647124i	$0.224028\pi$
$992$	−64.0000	−2.03200
$993$	−12.0000	−0.380808
$994$	−30.0000	−0.951542
$995$	−4.00000	−0.126809
$996$	2.00000	0.0633724
$997$	41.0000	1.29848	0.649242	−	0.760582i	$-0.275086\pi$
$997$	41.0000	1.29848	0.649242	+	0.760582i	$0.275086\pi$
$998$	−40.0000	−1.26618
$999$	50.0000	1.58193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5915.2.a.b.1.1		1
13.4	even	6		455.2.i.b.211.1	✓	2
13.10	even	6		455.2.i.b.386.1	yes	2
13.12	even	2		5915.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
455.2.i.b.211.1	✓	2	13.4	even	6
455.2.i.b.386.1	yes	2	13.10	even	6
5915.2.a.b.1.1		1	1.1	even	1	trivial
5915.2.a.j.1.1		1	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5915 = 5 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5915.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more