LMFDB - Embedded newform 370.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [370,2,Mod(1,370)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(370, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("370.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 370 = 2 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	370.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:	$2.95446487479$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	370.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +3.00000 q^{11} -2.00000 q^{12} -4.00000 q^{13} +1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} -3.00000 q^{22} +6.00000 q^{23} +2.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} +4.00000 q^{27} -1.00000 q^{28} +3.00000 q^{29} -2.00000 q^{30} +5.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} -3.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} -2.00000 q^{38} +8.00000 q^{39} +1.00000 q^{40} +3.00000 q^{41} -2.00000 q^{42} -1.00000 q^{43} +3.00000 q^{44} -1.00000 q^{45} -6.00000 q^{46} +12.0000 q^{47} -2.00000 q^{48} -6.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} -4.00000 q^{52} +3.00000 q^{53} -4.00000 q^{54} -3.00000 q^{55} +1.00000 q^{56} -4.00000 q^{57} -3.00000 q^{58} +2.00000 q^{60} -1.00000 q^{61} -5.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} +6.00000 q^{66} -4.00000 q^{67} +3.00000 q^{68} -12.0000 q^{69} -1.00000 q^{70} +6.00000 q^{71} -1.00000 q^{72} -16.0000 q^{73} -1.00000 q^{74} -2.00000 q^{75} +2.00000 q^{76} -3.00000 q^{77} -8.00000 q^{78} +8.00000 q^{79} -1.00000 q^{80} -11.0000 q^{81} -3.00000 q^{82} -12.0000 q^{83} +2.00000 q^{84} -3.00000 q^{85} +1.00000 q^{86} -6.00000 q^{87} -3.00000 q^{88} -6.00000 q^{89} +1.00000 q^{90} +4.00000 q^{91} +6.00000 q^{92} -10.0000 q^{93} -12.0000 q^{94} -2.00000 q^{95} +2.00000 q^{96} +17.0000 q^{97} +6.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.00000	0.816497
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−2.00000	−0.577350
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	1.00000	0.267261
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	−1.00000	−0.235702
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−1.00000	−0.223607
$21$	2.00000	0.436436
$22$	−3.00000	−0.639602
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	2.00000	0.408248
$25$	1.00000	0.200000
$26$	4.00000	0.784465
$27$	4.00000	0.769800
$28$	−1.00000	−0.188982
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	−2.00000	−0.365148
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	−1.00000	−0.176777
$33$	−6.00000	−1.04447
$34$	−3.00000	−0.514496
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−2.00000	−0.324443
$39$	8.00000	1.28103
$40$	1.00000	0.158114
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	−2.00000	−0.308607
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	3.00000	0.452267
$45$	−1.00000	−0.149071
$46$	−6.00000	−0.884652
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	−2.00000	−0.288675
$49$	−6.00000	−0.857143
$50$	−1.00000	−0.141421
$51$	−6.00000	−0.840168
$52$	−4.00000	−0.554700
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	−4.00000	−0.544331
$55$	−3.00000	−0.404520
$56$	1.00000	0.133631
$57$	−4.00000	−0.529813
$58$	−3.00000	−0.393919
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	2.00000	0.258199
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	−5.00000	−0.635001
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	6.00000	0.738549
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	3.00000	0.363803
$69$	−12.0000	−1.44463
$70$	−1.00000	−0.119523
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	−1.00000	−0.117851
$73$	−16.0000	−1.87266	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−16.0000	−1.87266	−0.936329	+	0.351123i	$0.885800\pi$
$74$	−1.00000	−0.116248
$75$	−2.00000	−0.230940
$76$	2.00000	0.229416
$77$	−3.00000	−0.341882
$78$	−8.00000	−0.905822
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−1.00000	−0.111803
$81$	−11.0000	−1.22222
$82$	−3.00000	−0.331295
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	2.00000	0.218218
$85$	−3.00000	−0.325396
$86$	1.00000	0.107833
$87$	−6.00000	−0.643268
$88$	−3.00000	−0.319801
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	1.00000	0.105409
$91$	4.00000	0.419314
$92$	6.00000	0.625543
$93$	−10.0000	−1.03695
$94$	−12.0000	−1.23771
$95$	−2.00000	−0.205196
$96$	2.00000	0.204124
$97$	17.0000	1.72609	0.863044	−	0.505128i	$-0.168555\pi$
$97$	17.0000	1.72609	0.863044	+	0.505128i	$0.168555\pi$
$98$	6.00000	0.606092
$99$	3.00000	0.301511
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	6.00000	0.594089
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	4.00000	0.392232
$105$	−2.00000	−0.195180
$106$	−3.00000	−0.291386
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	4.00000	0.384900
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	3.00000	0.286039
$111$	−2.00000	−0.189832
$112$	−1.00000	−0.0944911
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	4.00000	0.374634
$115$	−6.00000	−0.559503
$116$	3.00000	0.278543
$117$	−4.00000	−0.369800
$118$	0	0
$119$	−3.00000	−0.275010
$120$	−2.00000	−0.182574
$121$	−2.00000	−0.181818
$122$	1.00000	0.0905357
$123$	−6.00000	−0.541002
$124$	5.00000	0.449013
$125$	−1.00000	−0.0894427
$126$	1.00000	0.0890871
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	−1.00000	−0.0883883
$129$	2.00000	0.176090
$130$	−4.00000	−0.350823
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−6.00000	−0.522233
$133$	−2.00000	−0.173422
$134$	4.00000	0.345547
$135$	−4.00000	−0.344265
$136$	−3.00000	−0.257248
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	12.0000	1.02151
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	1.00000	0.0845154
$141$	−24.0000	−2.02116
$142$	−6.00000	−0.503509
$143$	−12.0000	−1.00349
$144$	1.00000	0.0833333
$145$	−3.00000	−0.249136
$146$	16.0000	1.32417
$147$	12.0000	0.989743
$148$	1.00000	0.0821995
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	2.00000	0.163299
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−2.00000	−0.162221
$153$	3.00000	0.242536
$154$	3.00000	0.241747
$155$	−5.00000	−0.401610
$156$	8.00000	0.640513
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	−8.00000	−0.636446
$159$	−6.00000	−0.475831
$160$	1.00000	0.0790569
$161$	−6.00000	−0.472866
$162$	11.0000	0.864242
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	3.00000	0.234261
$165$	6.00000	0.467099
$166$	12.0000	0.931381
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	−2.00000	−0.154303
$169$	3.00000	0.230769
$170$	3.00000	0.230089
$171$	2.00000	0.152944
$172$	−1.00000	−0.0762493
$173$	15.0000	1.14043	0.570214	−	0.821496i	$-0.306860\pi$
$173$	15.0000	1.14043	0.570214	+	0.821496i	$0.306860\pi$
$174$	6.00000	0.454859
$175$	−1.00000	−0.0755929
$176$	3.00000	0.226134
$177$	0	0
$178$	6.00000	0.449719
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	−1.00000	−0.0745356
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−4.00000	−0.296500
$183$	2.00000	0.147844
$184$	−6.00000	−0.442326
$185$	−1.00000	−0.0735215
$186$	10.0000	0.733236
$187$	9.00000	0.658145
$188$	12.0000	0.875190
$189$	−4.00000	−0.290957
$190$	2.00000	0.145095
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	−2.00000	−0.144338
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−17.0000	−1.22053
$195$	−8.00000	−0.572892
$196$	−6.00000	−0.428571
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−3.00000	−0.213201
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	−1.00000	−0.0707107
$201$	8.00000	0.564276
$202$	6.00000	0.422159
$203$	−3.00000	−0.210559
$204$	−6.00000	−0.420084
$205$	−3.00000	−0.209529
$206$	4.00000	0.278693
$207$	6.00000	0.417029
$208$	−4.00000	−0.277350
$209$	6.00000	0.415029
$210$	2.00000	0.138013
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	3.00000	0.206041
$213$	−12.0000	−0.822226
$214$	−6.00000	−0.410152
$215$	1.00000	0.0681994
$216$	−4.00000	−0.272166
$217$	−5.00000	−0.339422
$218$	−11.0000	−0.745014
$219$	32.0000	2.16236
$220$	−3.00000	−0.202260
$221$	−12.0000	−0.807207
$222$	2.00000	0.134231
$223$	17.0000	1.13840	0.569202	−	0.822198i	$-0.307252\pi$
$223$	17.0000	1.13840	0.569202	+	0.822198i	$0.307252\pi$
$224$	1.00000	0.0668153
$225$	1.00000	0.0666667
$226$	−9.00000	−0.598671
$227$	−27.0000	−1.79205	−0.896026	−	0.444001i	$-0.853559\pi$
$227$	−27.0000	−1.79205	−0.896026	+	0.444001i	$0.853559\pi$
$228$	−4.00000	−0.264906
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	6.00000	0.395628
$231$	6.00000	0.394771
$232$	−3.00000	−0.196960
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	4.00000	0.261488
$235$	−12.0000	−0.782794
$236$	0	0
$237$	−16.0000	−1.03931
$238$	3.00000	0.194461
$239$	−9.00000	−0.582162	−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000	−0.582162	−0.291081	+	0.956698i	$0.594015\pi$
$240$	2.00000	0.129099
$241$	−28.0000	−1.80364	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000	−1.80364	−0.901819	+	0.432113i	$0.857768\pi$
$242$	2.00000	0.128565
$243$	10.0000	0.641500
$244$	−1.00000	−0.0640184
$245$	6.00000	0.383326
$246$	6.00000	0.382546
$247$	−8.00000	−0.509028
$248$	−5.00000	−0.317500
$249$	24.0000	1.52094
$250$	1.00000	0.0632456
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	−1.00000	−0.0629941
$253$	18.0000	1.13165
$254$	−20.0000	−1.25491
$255$	6.00000	0.375735
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	−2.00000	−0.124515
$259$	−1.00000	−0.0621370
$260$	4.00000	0.248069
$261$	3.00000	0.185695
$262$	0	0
$263$	9.00000	0.554964	0.277482	−	0.960731i	$-0.410500\pi$
$263$	9.00000	0.554964	0.277482	+	0.960731i	$0.410500\pi$
$264$	6.00000	0.369274
$265$	−3.00000	−0.184289
$266$	2.00000	0.122628
$267$	12.0000	0.734388
$268$	−4.00000	−0.244339
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	4.00000	0.243432
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	3.00000	0.181902
$273$	−8.00000	−0.484182
$274$	−12.0000	−0.724947
$275$	3.00000	0.180907
$276$	−12.0000	−0.722315
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	13.0000	0.779688
$279$	5.00000	0.299342
$280$	−1.00000	−0.0597614
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	24.0000	1.42918
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	6.00000	0.356034
$285$	4.00000	0.236940
$286$	12.0000	0.709575
$287$	−3.00000	−0.177084
$288$	−1.00000	−0.0589256
$289$	−8.00000	−0.470588
$290$	3.00000	0.176166
$291$	−34.0000	−1.99312
$292$	−16.0000	−0.936329
$293$	21.0000	1.22683	0.613417	−	0.789760i	$-0.289795\pi$
$293$	21.0000	1.22683	0.613417	+	0.789760i	$0.289795\pi$
$294$	−12.0000	−0.699854
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	12.0000	0.696311
$298$	−6.00000	−0.347571
$299$	−24.0000	−1.38796
$300$	−2.00000	−0.115470
$301$	1.00000	0.0576390
$302$	−8.00000	−0.460348
$303$	12.0000	0.689382
$304$	2.00000	0.114708
$305$	1.00000	0.0572598
$306$	−3.00000	−0.171499
$307$	−34.0000	−1.94048	−0.970241	−	0.242140i	$-0.922151\pi$
$307$	−34.0000	−1.94048	−0.970241	+	0.242140i	$0.922151\pi$
$308$	−3.00000	−0.170941
$309$	8.00000	0.455104
$310$	5.00000	0.283981
$311$	−9.00000	−0.510343	−0.255172	−	0.966896i	$-0.582132\pi$
$311$	−9.00000	−0.510343	−0.255172	+	0.966896i	$0.582132\pi$
$312$	−8.00000	−0.452911
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	13.0000	0.733632
$315$	1.00000	0.0563436
$316$	8.00000	0.450035
$317$	−21.0000	−1.17948	−0.589739	−	0.807594i	$-0.700769\pi$
$317$	−21.0000	−1.17948	−0.589739	+	0.807594i	$0.700769\pi$
$318$	6.00000	0.336463
$319$	9.00000	0.503903
$320$	−1.00000	−0.0559017
$321$	−12.0000	−0.669775
$322$	6.00000	0.334367
$323$	6.00000	0.333849
$324$	−11.0000	−0.611111
$325$	−4.00000	−0.221880
$326$	−11.0000	−0.609234
$327$	−22.0000	−1.21660
$328$	−3.00000	−0.165647
$329$	−12.0000	−0.661581
$330$	−6.00000	−0.330289
$331$	26.0000	1.42909	0.714545	−	0.699590i	$-0.246634\pi$
$331$	26.0000	1.42909	0.714545	+	0.699590i	$0.246634\pi$
$332$	−12.0000	−0.658586
$333$	1.00000	0.0547997
$334$	−12.0000	−0.656611
$335$	4.00000	0.218543
$336$	2.00000	0.109109
$337$	−4.00000	−0.217894	−0.108947	−	0.994048i	$-0.534748\pi$
$337$	−4.00000	−0.217894	−0.108947	+	0.994048i	$0.534748\pi$
$338$	−3.00000	−0.163178
$339$	−18.0000	−0.977626
$340$	−3.00000	−0.162698
$341$	15.0000	0.812296
$342$	−2.00000	−0.108148
$343$	13.0000	0.701934
$344$	1.00000	0.0539164
$345$	12.0000	0.646058
$346$	−15.0000	−0.806405
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−6.00000	−0.321634
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	1.00000	0.0534522
$351$	−16.0000	−0.854017
$352$	−3.00000	−0.159901
$353$	−21.0000	−1.11772	−0.558859	−	0.829263i	$-0.688761\pi$
$353$	−21.0000	−1.11772	−0.558859	+	0.829263i	$0.688761\pi$
$354$	0	0
$355$	−6.00000	−0.318447
$356$	−6.00000	−0.317999
$357$	6.00000	0.317554
$358$	0	0
$359$	−36.0000	−1.90001	−0.950004	−	0.312239i	$-0.898921\pi$
$359$	−36.0000	−1.90001	−0.950004	+	0.312239i	$0.898921\pi$
$360$	1.00000	0.0527046
$361$	−15.0000	−0.789474
$362$	−2.00000	−0.105118
$363$	4.00000	0.209946
$364$	4.00000	0.209657
$365$	16.0000	0.837478
$366$	−2.00000	−0.104542
$367$	35.0000	1.82699	0.913493	−	0.406855i	$-0.133375\pi$
$367$	35.0000	1.82699	0.913493	+	0.406855i	$0.133375\pi$
$368$	6.00000	0.312772
$369$	3.00000	0.156174
$370$	1.00000	0.0519875
$371$	−3.00000	−0.155752
$372$	−10.0000	−0.518476
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	−9.00000	−0.465379
$375$	2.00000	0.103280
$376$	−12.0000	−0.618853
$377$	−12.0000	−0.618031
$378$	4.00000	0.205738
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	−2.00000	−0.102598
$381$	−40.0000	−2.04926
$382$	−3.00000	−0.153493
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	2.00000	0.102062
$385$	3.00000	0.152894
$386$	−14.0000	−0.712581
$387$	−1.00000	−0.0508329
$388$	17.0000	0.863044
$389$	9.00000	0.456318	0.228159	−	0.973624i	$-0.426729\pi$
$389$	9.00000	0.456318	0.228159	+	0.973624i	$0.426729\pi$
$390$	8.00000	0.405096
$391$	18.0000	0.910299
$392$	6.00000	0.303046
$393$	0	0
$394$	6.00000	0.302276
$395$	−8.00000	−0.402524
$396$	3.00000	0.150756
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	16.0000	0.802008
$399$	4.00000	0.200250
$400$	1.00000	0.0500000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	−8.00000	−0.399004
$403$	−20.0000	−0.996271
$404$	−6.00000	−0.298511
$405$	11.0000	0.546594
$406$	3.00000	0.148888
$407$	3.00000	0.148704
$408$	6.00000	0.297044
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	3.00000	0.148159
$411$	−24.0000	−1.18383
$412$	−4.00000	−0.197066
$413$	0	0
$414$	−6.00000	−0.294884
$415$	12.0000	0.589057
$416$	4.00000	0.196116
$417$	26.0000	1.27323
$418$	−6.00000	−0.293470
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	−2.00000	−0.0975900
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−5.00000	−0.243396
$423$	12.0000	0.583460
$424$	−3.00000	−0.145693
$425$	3.00000	0.145521
$426$	12.0000	0.581402
$427$	1.00000	0.0483934
$428$	6.00000	0.290021
$429$	24.0000	1.15873
$430$	−1.00000	−0.0482243
$431$	15.0000	0.722525	0.361262	−	0.932464i	$-0.382346\pi$
$431$	15.0000	0.722525	0.361262	+	0.932464i	$0.382346\pi$
$432$	4.00000	0.192450
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	5.00000	0.240008
$435$	6.00000	0.287678
$436$	11.0000	0.526804
$437$	12.0000	0.574038
$438$	−32.0000	−1.52902
$439$	35.0000	1.67046	0.835229	−	0.549902i	$-0.185335\pi$
$439$	35.0000	1.67046	0.835229	+	0.549902i	$0.185335\pi$
$440$	3.00000	0.143019
$441$	−6.00000	−0.285714
$442$	12.0000	0.570782
$443$	30.0000	1.42534	0.712672	−	0.701498i	$-0.247485\pi$
$443$	30.0000	1.42534	0.712672	+	0.701498i	$0.247485\pi$
$444$	−2.00000	−0.0949158
$445$	6.00000	0.284427
$446$	−17.0000	−0.804973
$447$	−12.0000	−0.567581
$448$	−1.00000	−0.0472456
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	−1.00000	−0.0471405
$451$	9.00000	0.423793
$452$	9.00000	0.423324
$453$	−16.0000	−0.751746
$454$	27.0000	1.26717
$455$	−4.00000	−0.187523
$456$	4.00000	0.187317
$457$	17.0000	0.795226	0.397613	−	0.917553i	$-0.369839\pi$
$457$	17.0000	0.795226	0.397613	+	0.917553i	$0.369839\pi$
$458$	−14.0000	−0.654177
$459$	12.0000	0.560112
$460$	−6.00000	−0.279751
$461$	33.0000	1.53696	0.768482	−	0.639872i	$-0.221013\pi$
$461$	33.0000	1.53696	0.768482	+	0.639872i	$0.221013\pi$
$462$	−6.00000	−0.279145
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	3.00000	0.139272
$465$	10.0000	0.463739
$466$	0	0
$467$	27.0000	1.24941	0.624705	−	0.780860i	$-0.285219\pi$
$467$	27.0000	1.24941	0.624705	+	0.780860i	$0.285219\pi$
$468$	−4.00000	−0.184900
$469$	4.00000	0.184703
$470$	12.0000	0.553519
$471$	26.0000	1.19802
$472$	0	0
$473$	−3.00000	−0.137940
$474$	16.0000	0.734904
$475$	2.00000	0.0917663
$476$	−3.00000	−0.137505
$477$	3.00000	0.137361
$478$	9.00000	0.411650
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	−2.00000	−0.0912871
$481$	−4.00000	−0.182384
$482$	28.0000	1.27537
$483$	12.0000	0.546019
$484$	−2.00000	−0.0909091
$485$	−17.0000	−0.771930
$486$	−10.0000	−0.453609
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	1.00000	0.0452679
$489$	−22.0000	−0.994874
$490$	−6.00000	−0.271052
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	−6.00000	−0.270501
$493$	9.00000	0.405340
$494$	8.00000	0.359937
$495$	−3.00000	−0.134840
$496$	5.00000	0.224507
$497$	−6.00000	−0.269137
$498$	−24.0000	−1.07547
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	−1.00000	−0.0447214
$501$	−24.0000	−1.07224
$502$	−18.0000	−0.803379
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	1.00000	0.0445435
$505$	6.00000	0.266996
$506$	−18.0000	−0.800198
$507$	−6.00000	−0.266469
$508$	20.0000	0.887357
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	−6.00000	−0.265684
$511$	16.0000	0.707798
$512$	−1.00000	−0.0441942
$513$	8.00000	0.353209
$514$	−18.0000	−0.793946
$515$	4.00000	0.176261
$516$	2.00000	0.0880451
$517$	36.0000	1.58328
$518$	1.00000	0.0439375
$519$	−30.0000	−1.31685
$520$	−4.00000	−0.175412
$521$	−39.0000	−1.70862	−0.854311	−	0.519763i	$-0.826020\pi$
$521$	−39.0000	−1.70862	−0.854311	+	0.519763i	$0.826020\pi$
$522$	−3.00000	−0.131306
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	2.00000	0.0872872
$526$	−9.00000	−0.392419
$527$	15.0000	0.653410
$528$	−6.00000	−0.261116
$529$	13.0000	0.565217
$530$	3.00000	0.130312
$531$	0	0
$532$	−2.00000	−0.0867110
$533$	−12.0000	−0.519778
$534$	−12.0000	−0.519291
$535$	−6.00000	−0.259403
$536$	4.00000	0.172774
$537$	0	0
$538$	18.0000	0.776035
$539$	−18.0000	−0.775315
$540$	−4.00000	−0.172133
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	−2.00000	−0.0859074
$543$	−4.00000	−0.171656
$544$	−3.00000	−0.128624
$545$	−11.0000	−0.471188
$546$	8.00000	0.342368
$547$	−1.00000	−0.0427569	−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−1.00000	−0.0427569	−0.0213785	+	0.999771i	$0.506805\pi$
$548$	12.0000	0.512615
$549$	−1.00000	−0.0426790
$550$	−3.00000	−0.127920
$551$	6.00000	0.255609
$552$	12.0000	0.510754
$553$	−8.00000	−0.340195
$554$	28.0000	1.18961
$555$	2.00000	0.0848953
$556$	−13.0000	−0.551323
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	−5.00000	−0.211667
$559$	4.00000	0.169182
$560$	1.00000	0.0422577
$561$	−18.0000	−0.759961
$562$	−24.0000	−1.01238
$563$	−3.00000	−0.126435	−0.0632175	−	0.998000i	$-0.520136\pi$
$563$	−3.00000	−0.126435	−0.0632175	+	0.998000i	$0.520136\pi$
$564$	−24.0000	−1.01058
$565$	−9.00000	−0.378633
$566$	4.00000	0.168133
$567$	11.0000	0.461957
$568$	−6.00000	−0.251754
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	−4.00000	−0.167542
$571$	−31.0000	−1.29731	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−31.0000	−1.29731	−0.648655	+	0.761083i	$0.724668\pi$
$572$	−12.0000	−0.501745
$573$	−6.00000	−0.250654
$574$	3.00000	0.125218
$575$	6.00000	0.250217
$576$	1.00000	0.0416667
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	8.00000	0.332756
$579$	−28.0000	−1.16364
$580$	−3.00000	−0.124568
$581$	12.0000	0.497844
$582$	34.0000	1.40935
$583$	9.00000	0.372742
$584$	16.0000	0.662085
$585$	4.00000	0.165380
$586$	−21.0000	−0.867502
$587$	27.0000	1.11441	0.557205	−	0.830375i	$-0.311874\pi$
$587$	27.0000	1.11441	0.557205	+	0.830375i	$0.311874\pi$
$588$	12.0000	0.494872
$589$	10.0000	0.412043
$590$	0	0
$591$	12.0000	0.493614
$592$	1.00000	0.0410997
$593$	−36.0000	−1.47834	−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$594$	−12.0000	−0.492366
$595$	3.00000	0.122988
$596$	6.00000	0.245770
$597$	32.0000	1.30967
$598$	24.0000	0.981433
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	2.00000	0.0816497
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	−1.00000	−0.0407570
$603$	−4.00000	−0.162893
$604$	8.00000	0.325515
$605$	2.00000	0.0813116
$606$	−12.0000	−0.487467
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	−2.00000	−0.0811107
$609$	6.00000	0.243132
$610$	−1.00000	−0.0404888
$611$	−48.0000	−1.94187
$612$	3.00000	0.121268
$613$	29.0000	1.17130	0.585649	−	0.810564i	$-0.300840\pi$
$613$	29.0000	1.17130	0.585649	+	0.810564i	$0.300840\pi$
$614$	34.0000	1.37213
$615$	6.00000	0.241943
$616$	3.00000	0.120873
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	−8.00000	−0.321807
$619$	35.0000	1.40677	0.703384	−	0.710810i	$-0.251671\pi$
$619$	35.0000	1.40677	0.703384	+	0.710810i	$0.251671\pi$
$620$	−5.00000	−0.200805
$621$	24.0000	0.963087
$622$	9.00000	0.360867
$623$	6.00000	0.240385
$624$	8.00000	0.320256
$625$	1.00000	0.0400000
$626$	−26.0000	−1.03917
$627$	−12.0000	−0.479234
$628$	−13.0000	−0.518756
$629$	3.00000	0.119618
$630$	−1.00000	−0.0398410
$631$	−7.00000	−0.278666	−0.139333	−	0.990246i	$-0.544496\pi$
$631$	−7.00000	−0.278666	−0.139333	+	0.990246i	$0.544496\pi$
$632$	−8.00000	−0.318223
$633$	−10.0000	−0.397464
$634$	21.0000	0.834017
$635$	−20.0000	−0.793676
$636$	−6.00000	−0.237915
$637$	24.0000	0.950915
$638$	−9.00000	−0.356313
$639$	6.00000	0.237356
$640$	1.00000	0.0395285
$641$	15.0000	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$642$	12.0000	0.473602
$643$	−13.0000	−0.512670	−0.256335	−	0.966588i	$-0.582515\pi$
$643$	−13.0000	−0.512670	−0.256335	+	0.966588i	$0.582515\pi$
$644$	−6.00000	−0.236433
$645$	−2.00000	−0.0787499
$646$	−6.00000	−0.236067
$647$	−48.0000	−1.88707	−0.943537	−	0.331266i	$-0.892524\pi$
$647$	−48.0000	−1.88707	−0.943537	+	0.331266i	$0.892524\pi$
$648$	11.0000	0.432121
$649$	0	0
$650$	4.00000	0.156893
$651$	10.0000	0.391931
$652$	11.0000	0.430793
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	22.0000	0.860268
$655$	0	0
$656$	3.00000	0.117130
$657$	−16.0000	−0.624219
$658$	12.0000	0.467809
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	6.00000	0.233550
$661$	5.00000	0.194477	0.0972387	−	0.995261i	$-0.468999\pi$
$661$	5.00000	0.194477	0.0972387	+	0.995261i	$0.468999\pi$
$662$	−26.0000	−1.01052
$663$	24.0000	0.932083
$664$	12.0000	0.465690
$665$	2.00000	0.0775567
$666$	−1.00000	−0.0387492
$667$	18.0000	0.696963
$668$	12.0000	0.464294
$669$	−34.0000	−1.31452
$670$	−4.00000	−0.154533
$671$	−3.00000	−0.115814
$672$	−2.00000	−0.0771517
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	4.00000	0.154074
$675$	4.00000	0.153960
$676$	3.00000	0.115385
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	18.0000	0.691286
$679$	−17.0000	−0.652400
$680$	3.00000	0.115045
$681$	54.0000	2.06928
$682$	−15.0000	−0.574380
$683$	−15.0000	−0.573959	−0.286980	−	0.957937i	$-0.592651\pi$
$683$	−15.0000	−0.573959	−0.286980	+	0.957937i	$0.592651\pi$
$684$	2.00000	0.0764719
$685$	−12.0000	−0.458496
$686$	−13.0000	−0.496342
$687$	−28.0000	−1.06827
$688$	−1.00000	−0.0381246
$689$	−12.0000	−0.457164
$690$	−12.0000	−0.456832
$691$	−19.0000	−0.722794	−0.361397	−	0.932412i	$-0.617700\pi$
$691$	−19.0000	−0.722794	−0.361397	+	0.932412i	$0.617700\pi$
$692$	15.0000	0.570214
$693$	−3.00000	−0.113961
$694$	12.0000	0.455514
$695$	13.0000	0.493118
$696$	6.00000	0.227429
$697$	9.00000	0.340899
$698$	−26.0000	−0.984115
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	16.0000	0.603881
$703$	2.00000	0.0754314
$704$	3.00000	0.113067
$705$	24.0000	0.903892
$706$	21.0000	0.790345
$707$	6.00000	0.225653
$708$	0	0
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	6.00000	0.225176
$711$	8.00000	0.300023
$712$	6.00000	0.224860
$713$	30.0000	1.12351
$714$	−6.00000	−0.224544
$715$	12.0000	0.448775
$716$	0	0
$717$	18.0000	0.672222
$718$	36.0000	1.34351
$719$	−42.0000	−1.56634	−0.783168	−	0.621810i	$-0.786397\pi$
$719$	−42.0000	−1.56634	−0.783168	+	0.621810i	$0.786397\pi$
$720$	−1.00000	−0.0372678
$721$	4.00000	0.148968
$722$	15.0000	0.558242
$723$	56.0000	2.08266
$724$	2.00000	0.0743294
$725$	3.00000	0.111417
$726$	−4.00000	−0.148454
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	−4.00000	−0.148250
$729$	13.0000	0.481481
$730$	−16.0000	−0.592187
$731$	−3.00000	−0.110959
$732$	2.00000	0.0739221
$733$	−31.0000	−1.14501	−0.572506	−	0.819901i	$-0.694029\pi$
$733$	−31.0000	−1.14501	−0.572506	+	0.819901i	$0.694029\pi$
$734$	−35.0000	−1.29187
$735$	−12.0000	−0.442627
$736$	−6.00000	−0.221163
$737$	−12.0000	−0.442026
$738$	−3.00000	−0.110432
$739$	11.0000	0.404642	0.202321	−	0.979319i	$-0.435152\pi$
$739$	11.0000	0.404642	0.202321	+	0.979319i	$0.435152\pi$
$740$	−1.00000	−0.0367607
$741$	16.0000	0.587775
$742$	3.00000	0.110133
$743$	−51.0000	−1.87101	−0.935504	−	0.353315i	$-0.885054\pi$
$743$	−51.0000	−1.87101	−0.935504	+	0.353315i	$0.885054\pi$
$744$	10.0000	0.366618
$745$	−6.00000	−0.219823
$746$	22.0000	0.805477
$747$	−12.0000	−0.439057
$748$	9.00000	0.329073
$749$	−6.00000	−0.219235
$750$	−2.00000	−0.0730297
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	12.0000	0.437595
$753$	−36.0000	−1.31191
$754$	12.0000	0.437014
$755$	−8.00000	−0.291150
$756$	−4.00000	−0.145479
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	16.0000	0.581146
$759$	−36.0000	−1.30672
$760$	2.00000	0.0725476
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	40.0000	1.44905
$763$	−11.0000	−0.398227
$764$	3.00000	0.108536
$765$	−3.00000	−0.108465
$766$	24.0000	0.867155
$767$	0	0
$768$	−2.00000	−0.0721688
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	−3.00000	−0.108112
$771$	−36.0000	−1.29651
$772$	14.0000	0.503871
$773$	−39.0000	−1.40273	−0.701366	−	0.712801i	$-0.747426\pi$
$773$	−39.0000	−1.40273	−0.701366	+	0.712801i	$0.747426\pi$
$774$	1.00000	0.0359443
$775$	5.00000	0.179605
$776$	−17.0000	−0.610264
$777$	2.00000	0.0717496
$778$	−9.00000	−0.322666
$779$	6.00000	0.214972
$780$	−8.00000	−0.286446
$781$	18.0000	0.644091
$782$	−18.0000	−0.643679
$783$	12.0000	0.428845
$784$	−6.00000	−0.214286
$785$	13.0000	0.463990
$786$	0	0
$787$	−22.0000	−0.784215	−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000	−0.784215	−0.392108	+	0.919919i	$0.628254\pi$
$788$	−6.00000	−0.213741
$789$	−18.0000	−0.640817
$790$	8.00000	0.284627
$791$	−9.00000	−0.320003
$792$	−3.00000	−0.106600
$793$	4.00000	0.142044
$794$	34.0000	1.20661
$795$	6.00000	0.212798
$796$	−16.0000	−0.567105
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	−4.00000	−0.141598
$799$	36.0000	1.27359
$800$	−1.00000	−0.0353553
$801$	−6.00000	−0.212000
$802$	−6.00000	−0.211867
$803$	−48.0000	−1.69388
$804$	8.00000	0.282138
$805$	6.00000	0.211472
$806$	20.0000	0.704470
$807$	36.0000	1.26726
$808$	6.00000	0.211079
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	−11.0000	−0.386501
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	−3.00000	−0.105279
$813$	−4.00000	−0.140286
$814$	−3.00000	−0.105150
$815$	−11.0000	−0.385313
$816$	−6.00000	−0.210042
$817$	−2.00000	−0.0699711
$818$	−32.0000	−1.11885
$819$	4.00000	0.139771
$820$	−3.00000	−0.104765
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	24.0000	0.837096
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	4.00000	0.139347
$825$	−6.00000	−0.208893
$826$	0	0
$827$	−3.00000	−0.104320	−0.0521601	−	0.998639i	$-0.516611\pi$
$827$	−3.00000	−0.104320	−0.0521601	+	0.998639i	$0.516611\pi$
$828$	6.00000	0.208514
$829$	47.0000	1.63238	0.816189	−	0.577785i	$-0.196083\pi$
$829$	47.0000	1.63238	0.816189	+	0.577785i	$0.196083\pi$
$830$	−12.0000	−0.416526
$831$	56.0000	1.94262
$832$	−4.00000	−0.138675
$833$	−18.0000	−0.623663
$834$	−26.0000	−0.900306
$835$	−12.0000	−0.415277
$836$	6.00000	0.207514
$837$	20.0000	0.691301
$838$	36.0000	1.24360
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	2.00000	0.0690066
$841$	−20.0000	−0.689655
$842$	10.0000	0.344623
$843$	−48.0000	−1.65321
$844$	5.00000	0.172107
$845$	−3.00000	−0.103203
$846$	−12.0000	−0.412568
$847$	2.00000	0.0687208
$848$	3.00000	0.103020
$849$	8.00000	0.274559
$850$	−3.00000	−0.102899
$851$	6.00000	0.205677
$852$	−12.0000	−0.411113
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	−1.00000	−0.0342193
$855$	−2.00000	−0.0683986
$856$	−6.00000	−0.205076
$857$	−15.0000	−0.512390	−0.256195	−	0.966625i	$-0.582469\pi$
$857$	−15.0000	−0.512390	−0.256195	+	0.966625i	$0.582469\pi$
$858$	−24.0000	−0.819346
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	1.00000	0.0340997
$861$	6.00000	0.204479
$862$	−15.0000	−0.510902
$863$	39.0000	1.32758	0.663788	−	0.747921i	$-0.268948\pi$
$863$	39.0000	1.32758	0.663788	+	0.747921i	$0.268948\pi$
$864$	−4.00000	−0.136083
$865$	−15.0000	−0.510015
$866$	−2.00000	−0.0679628
$867$	16.0000	0.543388
$868$	−5.00000	−0.169711
$869$	24.0000	0.814144
$870$	−6.00000	−0.203419
$871$	16.0000	0.542139
$872$	−11.0000	−0.372507
$873$	17.0000	0.575363
$874$	−12.0000	−0.405906
$875$	1.00000	0.0338062
$876$	32.0000	1.08118
$877$	−13.0000	−0.438979	−0.219489	−	0.975615i	$-0.570439\pi$
$877$	−13.0000	−0.438979	−0.219489	+	0.975615i	$0.570439\pi$
$878$	−35.0000	−1.18119
$879$	−42.0000	−1.41662
$880$	−3.00000	−0.101130
$881$	15.0000	0.505363	0.252681	−	0.967550i	$-0.418688\pi$
$881$	15.0000	0.505363	0.252681	+	0.967550i	$0.418688\pi$
$882$	6.00000	0.202031
$883$	29.0000	0.975928	0.487964	−	0.872864i	$-0.337740\pi$
$883$	29.0000	0.975928	0.487964	+	0.872864i	$0.337740\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	−30.0000	−1.00787
$887$	33.0000	1.10803	0.554016	−	0.832506i	$-0.313095\pi$
$887$	33.0000	1.10803	0.554016	+	0.832506i	$0.313095\pi$
$888$	2.00000	0.0671156
$889$	−20.0000	−0.670778
$890$	−6.00000	−0.201120
$891$	−33.0000	−1.10554
$892$	17.0000	0.569202
$893$	24.0000	0.803129
$894$	12.0000	0.401340
$895$	0	0
$896$	1.00000	0.0334077
$897$	48.0000	1.60267
$898$	−6.00000	−0.200223
$899$	15.0000	0.500278
$900$	1.00000	0.0333333
$901$	9.00000	0.299833
$902$	−9.00000	−0.299667
$903$	−2.00000	−0.0665558
$904$	−9.00000	−0.299336
$905$	−2.00000	−0.0664822
$906$	16.0000	0.531564
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−27.0000	−0.896026
$909$	−6.00000	−0.199007
$910$	4.00000	0.132599
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	−4.00000	−0.132453
$913$	−36.0000	−1.19143
$914$	−17.0000	−0.562310
$915$	−2.00000	−0.0661180
$916$	14.0000	0.462573
$917$	0	0
$918$	−12.0000	−0.396059
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	6.00000	0.197814
$921$	68.0000	2.24068
$922$	−33.0000	−1.08680
$923$	−24.0000	−0.789970
$924$	6.00000	0.197386
$925$	1.00000	0.0328798
$926$	4.00000	0.131448
$927$	−4.00000	−0.131377
$928$	−3.00000	−0.0984798
$929$	45.0000	1.47640	0.738201	−	0.674581i	$-0.235676\pi$
$929$	45.0000	1.47640	0.738201	+	0.674581i	$0.235676\pi$
$930$	−10.0000	−0.327913
$931$	−12.0000	−0.393284
$932$	0	0
$933$	18.0000	0.589294
$934$	−27.0000	−0.883467
$935$	−9.00000	−0.294331
$936$	4.00000	0.130744
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	−4.00000	−0.130605
$939$	−52.0000	−1.69696
$940$	−12.0000	−0.391397
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	−26.0000	−0.847126
$943$	18.0000	0.586161
$944$	0	0
$945$	4.00000	0.130120
$946$	3.00000	0.0975384
$947$	33.0000	1.07236	0.536178	−	0.844105i	$-0.319868\pi$
$947$	33.0000	1.07236	0.536178	+	0.844105i	$0.319868\pi$
$948$	−16.0000	−0.519656
$949$	64.0000	2.07753
$950$	−2.00000	−0.0648886
$951$	42.0000	1.36194
$952$	3.00000	0.0972306
$953$	−60.0000	−1.94359	−0.971795	−	0.235826i	$-0.924220\pi$
$953$	−60.0000	−1.94359	−0.971795	+	0.235826i	$0.924220\pi$
$954$	−3.00000	−0.0971286
$955$	−3.00000	−0.0970777
$956$	−9.00000	−0.291081
$957$	−18.0000	−0.581857
$958$	−24.0000	−0.775405
$959$	−12.0000	−0.387500
$960$	2.00000	0.0645497
$961$	−6.00000	−0.193548
$962$	4.00000	0.128965
$963$	6.00000	0.193347
$964$	−28.0000	−0.901819
$965$	−14.0000	−0.450676
$966$	−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	$0.722376\pi$
$968$	2.00000	0.0642824
$969$	−12.0000	−0.385496
$970$	17.0000	0.545837
$971$	57.0000	1.82922	0.914609	−	0.404341i	$-0.132499\pi$
$971$	57.0000	1.82922	0.914609	+	0.404341i	$0.132499\pi$
$972$	10.0000	0.320750
$973$	13.0000	0.416761
$974$	−2.00000	−0.0640841
$975$	8.00000	0.256205
$976$	−1.00000	−0.0320092
$977$	−45.0000	−1.43968	−0.719839	−	0.694141i	$-0.755784\pi$
$977$	−45.0000	−1.43968	−0.719839	+	0.694141i	$0.755784\pi$
$978$	22.0000	0.703482
$979$	−18.0000	−0.575282
$980$	6.00000	0.191663
$981$	11.0000	0.351203
$982$	12.0000	0.382935
$983$	51.0000	1.62665	0.813324	−	0.581811i	$-0.197656\pi$
$983$	51.0000	1.62665	0.813324	+	0.581811i	$0.197656\pi$
$984$	6.00000	0.191273
$985$	6.00000	0.191176
$986$	−9.00000	−0.286618
$987$	24.0000	0.763928
$988$	−8.00000	−0.254514
$989$	−6.00000	−0.190789
$990$	3.00000	0.0953463
$991$	47.0000	1.49300	0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000	1.49300	0.746502	+	0.665383i	$0.231732\pi$
$992$	−5.00000	−0.158750
$993$	−52.0000	−1.65017
$994$	6.00000	0.190308
$995$	16.0000	0.507234
$996$	24.0000	0.760469
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	−14.0000	−0.443162
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	370.2.a.a.1.1	✓	1
3.2	odd	2		3330.2.a.v.1.1		1
4.3	odd	2		2960.2.a.j.1.1		1
5.2	odd	4		1850.2.b.g.149.1		2
5.3	odd	4		1850.2.b.g.149.2		2
5.4	even	2		1850.2.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.a.1.1	✓	1	1.1	even	1	trivial
1850.2.a.o.1.1		1	5.4	even	2
1850.2.b.g.149.1		2	5.2	odd	4
1850.2.b.g.149.2		2	5.3	odd	4
2960.2.a.j.1.1		1	4.3	odd	2
3330.2.a.v.1.1		1	3.2	odd	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:	\( N \)	\(=\)	\( 370 = 2 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	370.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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