LMFDB - Embedded newform 6045.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.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:	$48.2695680219$
Analytic rank:	$1$
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	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +2.00000 q^{7} +1.00000 q^{9} -2.00000 q^{10} -5.00000 q^{11} -2.00000 q^{12} -1.00000 q^{13} -4.00000 q^{14} -1.00000 q^{15} -4.00000 q^{16} +4.00000 q^{17} -2.00000 q^{18} -2.00000 q^{19} +2.00000 q^{20} -2.00000 q^{21} +10.0000 q^{22} -2.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} +2.00000 q^{30} +1.00000 q^{31} +8.00000 q^{32} +5.00000 q^{33} -8.00000 q^{34} +2.00000 q^{35} +2.00000 q^{36} +1.00000 q^{37} +4.00000 q^{38} +1.00000 q^{39} +10.0000 q^{41} +4.00000 q^{42} -10.0000 q^{43} -10.0000 q^{44} +1.00000 q^{45} +4.00000 q^{46} +4.00000 q^{47} +4.00000 q^{48} -3.00000 q^{49} -2.00000 q^{50} -4.00000 q^{51} -2.00000 q^{52} +12.0000 q^{53} +2.00000 q^{54} -5.00000 q^{55} +2.00000 q^{57} -10.0000 q^{59} -2.00000 q^{60} +14.0000 q^{61} -2.00000 q^{62} +2.00000 q^{63} -8.00000 q^{64} -1.00000 q^{65} -10.0000 q^{66} -12.0000 q^{67} +8.00000 q^{68} +2.00000 q^{69} -4.00000 q^{70} -4.00000 q^{71} -7.00000 q^{73} -2.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -10.0000 q^{77} -2.00000 q^{78} -8.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -20.0000 q^{82} -4.00000 q^{83} -4.00000 q^{84} +4.00000 q^{85} +20.0000 q^{86} +15.0000 q^{89} -2.00000 q^{90} -2.00000 q^{91} -4.00000 q^{92} -1.00000 q^{93} -8.00000 q^{94} -2.00000 q^{95} -8.00000 q^{96} +10.0000 q^{97} +6.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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 $

$2$	−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
$3$	−1.00000	−0.577350
$4$	2.00000	1.00000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	2.00000	0.816497
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	−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
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−4.00000	−1.06904
$15$	−1.00000	−0.258199
$16$	−4.00000	−1.00000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−2.00000	−0.471405
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	2.00000	0.447214
$21$	−2.00000	−0.436436
$22$	10.0000	2.13201
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	4.00000	0.755929
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	2.00000	0.365148
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	8.00000	1.41421
$33$	5.00000	0.870388
$34$	−8.00000	−1.37199
$35$	2.00000	0.338062
$36$	2.00000	0.333333
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	4.00000	0.648886
$39$	1.00000	0.160128
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	4.00000	0.617213
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	−10.0000	−1.50756
$45$	1.00000	0.149071
$46$	4.00000	0.589768
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	4.00000	0.577350
$49$	−3.00000	−0.428571
$50$	−2.00000	−0.282843
$51$	−4.00000	−0.560112
$52$	−2.00000	−0.277350
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	2.00000	0.272166
$55$	−5.00000	−0.674200
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	−2.00000	−0.258199
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	−2.00000	−0.254000
$63$	2.00000	0.251976
$64$	−8.00000	−1.00000
$65$	−1.00000	−0.124035
$66$	−10.0000	−1.23091
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	8.00000	0.970143
$69$	2.00000	0.240772
$70$	−4.00000	−0.478091
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	−2.00000	−0.232495
$75$	−1.00000	−0.115470
$76$	−4.00000	−0.458831
$77$	−10.0000	−1.13961
$78$	−2.00000	−0.226455
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	−20.0000	−2.20863
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	−4.00000	−0.436436
$85$	4.00000	0.433861
$86$	20.0000	2.15666
$87$	0	0
$88$	0	0
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	−2.00000	−0.210819
$91$	−2.00000	−0.209657
$92$	−4.00000	−0.417029
$93$	−1.00000	−0.103695
$94$	−8.00000	−0.825137
$95$	−2.00000	−0.205196
$96$	−8.00000	−0.816497
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	6.00000	0.606092
$99$	−5.00000	−0.502519
$100$	2.00000	0.200000
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	8.00000	0.792118
$103$	−5.00000	−0.492665	−0.246332	−	0.969185i	$-0.579225\pi$
$103$	−5.00000	−0.492665	−0.246332	+	0.969185i	$0.579225\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	−24.0000	−2.33109
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	−2.00000	−0.192450
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	10.0000	0.953463
$111$	−1.00000	−0.0949158
$112$	−8.00000	−0.755929
$113$	−3.00000	−0.282216	−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000	−0.282216	−0.141108	+	0.989994i	$0.545067\pi$
$114$	−4.00000	−0.374634
$115$	−2.00000	−0.186501
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	20.0000	1.84115
$119$	8.00000	0.733359
$120$	0	0
$121$	14.0000	1.27273
$122$	−28.0000	−2.53500
$123$	−10.0000	−0.901670
$124$	2.00000	0.179605
$125$	1.00000	0.0894427
$126$	−4.00000	−0.356348
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	0	0
$129$	10.0000	0.880451
$130$	2.00000	0.175412
$131$	−21.0000	−1.83478	−0.917389	−	0.397991i	$-0.869707\pi$
$131$	−21.0000	−1.83478	−0.917389	+	0.397991i	$0.869707\pi$
$132$	10.0000	0.870388
$133$	−4.00000	−0.346844
$134$	24.0000	2.07328
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	−4.00000	−0.340503
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	4.00000	0.338062
$141$	−4.00000	−0.336861
$142$	8.00000	0.671345
$143$	5.00000	0.418121
$144$	−4.00000	−0.333333
$145$	0	0
$146$	14.0000	1.15865
$147$	3.00000	0.247436
$148$	2.00000	0.164399
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	2.00000	0.163299
$151$	13.0000	1.05792	0.528962	−	0.848645i	$-0.322581\pi$
$151$	13.0000	1.05792	0.528962	+	0.848645i	$0.322581\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	20.0000	1.61165
$155$	1.00000	0.0803219
$156$	2.00000	0.160128
$157$	−19.0000	−1.51637	−0.758183	−	0.652042i	$-0.773912\pi$
$157$	−19.0000	−1.51637	−0.758183	+	0.652042i	$0.773912\pi$
$158$	16.0000	1.27289
$159$	−12.0000	−0.951662
$160$	8.00000	0.632456
$161$	−4.00000	−0.315244
$162$	−2.00000	−0.157135
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	20.0000	1.56174
$165$	5.00000	0.389249
$166$	8.00000	0.620920
$167$	−3.00000	−0.232147	−0.116073	−	0.993241i	$-0.537031\pi$
$167$	−3.00000	−0.232147	−0.116073	+	0.993241i	$0.537031\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−8.00000	−0.613572
$171$	−2.00000	−0.152944
$172$	−20.0000	−1.52499
$173$	3.00000	0.228086	0.114043	−	0.993476i	$-0.463620\pi$
$173$	3.00000	0.228086	0.114043	+	0.993476i	$0.463620\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	20.0000	1.50756
$177$	10.0000	0.751646
$178$	−30.0000	−2.24860
$179$	8.00000	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$180$	2.00000	0.149071
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	4.00000	0.296500
$183$	−14.0000	−1.03491
$184$	0	0
$185$	1.00000	0.0735215
$186$	2.00000	0.146647
$187$	−20.0000	−1.46254
$188$	8.00000	0.583460
$189$	−2.00000	−0.145479
$190$	4.00000	0.290191
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	8.00000	0.577350
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	−20.0000	−1.43592
$195$	1.00000	0.0716115
$196$	−6.00000	−0.428571
$197$	21.0000	1.49619	0.748094	−	0.663593i	$-0.230969\pi$
$197$	21.0000	1.49619	0.748094	+	0.663593i	$0.230969\pi$
$198$	10.0000	0.710669
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	6.00000	0.422159
$203$	0	0
$204$	−8.00000	−0.560112
$205$	10.0000	0.698430
$206$	10.0000	0.696733
$207$	−2.00000	−0.139010
$208$	4.00000	0.277350
$209$	10.0000	0.691714
$210$	4.00000	0.276026
$211$	3.00000	0.206529	0.103264	−	0.994654i	$-0.467071\pi$
$211$	3.00000	0.206529	0.103264	+	0.994654i	$0.467071\pi$
$212$	24.0000	1.64833
$213$	4.00000	0.274075
$214$	−6.00000	−0.410152
$215$	−10.0000	−0.681994
$216$	0	0
$217$	2.00000	0.135769
$218$	12.0000	0.812743
$219$	7.00000	0.473016
$220$	−10.0000	−0.674200
$221$	−4.00000	−0.269069
$222$	2.00000	0.134231
$223$	−13.0000	−0.870544	−0.435272	−	0.900299i	$-0.643348\pi$
$223$	−13.0000	−0.870544	−0.435272	+	0.900299i	$0.643348\pi$
$224$	16.0000	1.06904
$225$	1.00000	0.0666667
$226$	6.00000	0.399114
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	4.00000	0.264906
$229$	−13.0000	−0.859064	−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000	−0.859064	−0.429532	+	0.903052i	$0.641321\pi$
$230$	4.00000	0.263752
$231$	10.0000	0.657952
$232$	0	0
$233$	5.00000	0.327561	0.163780	−	0.986497i	$-0.447631\pi$
$233$	5.00000	0.327561	0.163780	+	0.986497i	$0.447631\pi$
$234$	2.00000	0.130744
$235$	4.00000	0.260931
$236$	−20.0000	−1.30189
$237$	8.00000	0.519656
$238$	−16.0000	−1.03713
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	4.00000	0.258199
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−28.0000	−1.79991
$243$	−1.00000	−0.0641500
$244$	28.0000	1.79252
$245$	−3.00000	−0.191663
$246$	20.0000	1.27515
$247$	2.00000	0.127257
$248$	0	0
$249$	4.00000	0.253490
$250$	−2.00000	−0.126491
$251$	22.0000	1.38863	0.694314	−	0.719672i	$-0.255708\pi$
$251$	22.0000	1.38863	0.694314	+	0.719672i	$0.255708\pi$
$252$	4.00000	0.251976
$253$	10.0000	0.628695
$254$	40.0000	2.50982
$255$	−4.00000	−0.250490
$256$	16.0000	1.00000
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	−20.0000	−1.24515
$259$	2.00000	0.124274
$260$	−2.00000	−0.124035
$261$	0	0
$262$	42.0000	2.59477
$263$	−10.0000	−0.616626	−0.308313	−	0.951285i	$-0.599764\pi$
$263$	−10.0000	−0.616626	−0.308313	+	0.951285i	$0.599764\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	8.00000	0.490511
$267$	−15.0000	−0.917985
$268$	−24.0000	−1.46603
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	2.00000	0.121716
$271$	13.0000	0.789694	0.394847	−	0.918747i	$-0.370798\pi$
$271$	13.0000	0.789694	0.394847	+	0.918747i	$0.370798\pi$
$272$	−16.0000	−0.970143
$273$	2.00000	0.121046
$274$	36.0000	2.17484
$275$	−5.00000	−0.301511
$276$	4.00000	0.240772
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−24.0000	−1.43942
$279$	1.00000	0.0598684
$280$	0	0
$281$	−28.0000	−1.67034	−0.835170	−	0.549992i	$-0.814631\pi$
$281$	−28.0000	−1.67034	−0.835170	+	0.549992i	$0.814631\pi$
$282$	8.00000	0.476393
$283$	−5.00000	−0.297219	−0.148610	−	0.988896i	$-0.547480\pi$
$283$	−5.00000	−0.297219	−0.148610	+	0.988896i	$0.547480\pi$
$284$	−8.00000	−0.474713
$285$	2.00000	0.118470
$286$	−10.0000	−0.591312
$287$	20.0000	1.18056
$288$	8.00000	0.471405
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−10.0000	−0.586210
$292$	−14.0000	−0.819288
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	−6.00000	−0.349927
$295$	−10.0000	−0.582223
$296$	0	0
$297$	5.00000	0.290129
$298$	−8.00000	−0.463428
$299$	2.00000	0.115663
$300$	−2.00000	−0.115470
$301$	−20.0000	−1.15278
$302$	−26.0000	−1.49613
$303$	3.00000	0.172345
$304$	8.00000	0.458831
$305$	14.0000	0.801638
$306$	−8.00000	−0.457330
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	−20.0000	−1.13961
$309$	5.00000	0.284440
$310$	−2.00000	−0.113592
$311$	31.0000	1.75785	0.878924	−	0.476961i	$-0.158262\pi$
$311$	31.0000	1.75785	0.878924	+	0.476961i	$0.158262\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	38.0000	2.14446
$315$	2.00000	0.112687
$316$	−16.0000	−0.900070
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	24.0000	1.34585
$319$	0	0
$320$	−8.00000	−0.447214
$321$	−3.00000	−0.167444
$322$	8.00000	0.445823
$323$	−8.00000	−0.445132
$324$	2.00000	0.111111
$325$	−1.00000	−0.0554700
$326$	−32.0000	−1.77232
$327$	6.00000	0.331801
$328$	0	0
$329$	8.00000	0.441054
$330$	−10.0000	−0.550482
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	−8.00000	−0.439057
$333$	1.00000	0.0547997
$334$	6.00000	0.328305
$335$	−12.0000	−0.655630
$336$	8.00000	0.436436
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	−2.00000	−0.108786
$339$	3.00000	0.162938
$340$	8.00000	0.433861
$341$	−5.00000	−0.270765
$342$	4.00000	0.216295
$343$	−20.0000	−1.07990
$344$	0	0
$345$	2.00000	0.107676
$346$	−6.00000	−0.322562
$347$	−2.00000	−0.107366	−0.0536828	−	0.998558i	$-0.517096\pi$
$347$	−2.00000	−0.107366	−0.0536828	+	0.998558i	$0.517096\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	−4.00000	−0.213809
$351$	1.00000	0.0533761
$352$	−40.0000	−2.13201
$353$	3.00000	0.159674	0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000	0.159674	0.0798369	+	0.996808i	$0.474560\pi$
$354$	−20.0000	−1.06299
$355$	−4.00000	−0.212298
$356$	30.0000	1.59000
$357$	−8.00000	−0.423405
$358$	−16.0000	−0.845626
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	24.0000	1.26141
$363$	−14.0000	−0.734809
$364$	−4.00000	−0.209657
$365$	−7.00000	−0.366397
$366$	28.0000	1.46358
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	8.00000	0.417029
$369$	10.0000	0.520579
$370$	−2.00000	−0.103975
$371$	24.0000	1.24602
$372$	−2.00000	−0.103695
$373$	−31.0000	−1.60512	−0.802560	−	0.596572i	$-0.796529\pi$
$373$	−31.0000	−1.60512	−0.802560	+	0.596572i	$0.796529\pi$
$374$	40.0000	2.06835
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	0	0
$378$	4.00000	0.205738
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	−4.00000	−0.205196
$381$	20.0000	1.02463
$382$	6.00000	0.306987
$383$	−27.0000	−1.37964	−0.689818	−	0.723983i	$-0.742309\pi$
$383$	−27.0000	−1.37964	−0.689818	+	0.723983i	$0.742309\pi$
$384$	0	0
$385$	−10.0000	−0.509647
$386$	32.0000	1.62876
$387$	−10.0000	−0.508329
$388$	20.0000	1.01535
$389$	−36.0000	−1.82527	−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000	−1.82527	−0.912636	+	0.408773i	$0.865957\pi$
$390$	−2.00000	−0.101274
$391$	−8.00000	−0.404577
$392$	0	0
$393$	21.0000	1.05931
$394$	−42.0000	−2.11593
$395$	−8.00000	−0.402524
$396$	−10.0000	−0.502519
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	−28.0000	−1.40351
$399$	4.00000	0.200250
$400$	−4.00000	−0.200000
$401$	−5.00000	−0.249688	−0.124844	−	0.992176i	$-0.539843\pi$
$401$	−5.00000	−0.249688	−0.124844	+	0.992176i	$0.539843\pi$
$402$	−24.0000	−1.19701
$403$	−1.00000	−0.0498135
$404$	−6.00000	−0.298511
$405$	1.00000	0.0496904
$406$	0	0
$407$	−5.00000	−0.247841
$408$	0	0
$409$	7.00000	0.346128	0.173064	−	0.984911i	$-0.444633\pi$
$409$	7.00000	0.346128	0.173064	+	0.984911i	$0.444633\pi$
$410$	−20.0000	−0.987730
$411$	18.0000	0.887875
$412$	−10.0000	−0.492665
$413$	−20.0000	−0.984136
$414$	4.00000	0.196589
$415$	−4.00000	−0.196352
$416$	−8.00000	−0.392232
$417$	−12.0000	−0.587643
$418$	−20.0000	−0.978232
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	−4.00000	−0.195180
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	−6.00000	−0.292075
$423$	4.00000	0.194487
$424$	0	0
$425$	4.00000	0.194029
$426$	−8.00000	−0.387601
$427$	28.0000	1.35501
$428$	6.00000	0.290021
$429$	−5.00000	−0.241402
$430$	20.0000	0.964486
$431$	−10.0000	−0.481683	−0.240842	−	0.970564i	$-0.577423\pi$
$431$	−10.0000	−0.481683	−0.240842	+	0.970564i	$0.577423\pi$
$432$	4.00000	0.192450
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−12.0000	−0.574696
$437$	4.00000	0.191346
$438$	−14.0000	−0.668946
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	8.00000	0.380521
$443$	−31.0000	−1.47285	−0.736427	−	0.676517i	$-0.763489\pi$
$443$	−31.0000	−1.47285	−0.736427	+	0.676517i	$0.763489\pi$
$444$	−2.00000	−0.0949158
$445$	15.0000	0.711068
$446$	26.0000	1.23114
$447$	−4.00000	−0.189194
$448$	−16.0000	−0.755929
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	−2.00000	−0.0942809
$451$	−50.0000	−2.35441
$452$	−6.00000	−0.282216
$453$	−13.0000	−0.610793
$454$	−56.0000	−2.62821
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	−5.00000	−0.233890	−0.116945	−	0.993138i	$-0.537310\pi$
$457$	−5.00000	−0.233890	−0.116945	+	0.993138i	$0.537310\pi$
$458$	26.0000	1.21490
$459$	−4.00000	−0.186704
$460$	−4.00000	−0.186501
$461$	13.0000	0.605470	0.302735	−	0.953075i	$-0.402100\pi$
$461$	13.0000	0.605470	0.302735	+	0.953075i	$0.402100\pi$
$462$	−20.0000	−0.930484
$463$	−31.0000	−1.44069	−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−31.0000	−1.44069	−0.720346	+	0.693615i	$0.756017\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	−10.0000	−0.463241
$467$	40.0000	1.85098	0.925490	−	0.378773i	$-0.123654\pi$
$467$	40.0000	1.85098	0.925490	+	0.378773i	$0.123654\pi$
$468$	−2.00000	−0.0924500
$469$	−24.0000	−1.10822
$470$	−8.00000	−0.369012
$471$	19.0000	0.875474
$472$	0	0
$473$	50.0000	2.29900
$474$	−16.0000	−0.734904
$475$	−2.00000	−0.0917663
$476$	16.0000	0.733359
$477$	12.0000	0.549442
$478$	−16.0000	−0.731823
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	−8.00000	−0.365148
$481$	−1.00000	−0.0455961
$482$	−20.0000	−0.910975
$483$	4.00000	0.182006
$484$	28.0000	1.27273
$485$	10.0000	0.454077
$486$	2.00000	0.0907218
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	0	0
$489$	−16.0000	−0.723545
$490$	6.00000	0.271052
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	−20.0000	−0.901670
$493$	0	0
$494$	−4.00000	−0.179969
$495$	−5.00000	−0.224733
$496$	−4.00000	−0.179605
$497$	−8.00000	−0.358849
$498$	−8.00000	−0.358489
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	2.00000	0.0894427
$501$	3.00000	0.134030
$502$	−44.0000	−1.96382
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	−3.00000	−0.133498
$506$	−20.0000	−0.889108
$507$	−1.00000	−0.0444116
$508$	−40.0000	−1.77471
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	8.00000	0.354246
$511$	−14.0000	−0.619324
$512$	−32.0000	−1.41421
$513$	2.00000	0.0883022
$514$	6.00000	0.264649
$515$	−5.00000	−0.220326
$516$	20.0000	0.880451
$517$	−20.0000	−0.879599
$518$	−4.00000	−0.175750
$519$	−3.00000	−0.131685
$520$	0	0
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	0	0
$523$	−10.0000	−0.437269	−0.218635	−	0.975807i	$-0.570160\pi$
$523$	−10.0000	−0.437269	−0.218635	+	0.975807i	$0.570160\pi$
$524$	−42.0000	−1.83478
$525$	−2.00000	−0.0872872
$526$	20.0000	0.872041
$527$	4.00000	0.174243
$528$	−20.0000	−0.870388
$529$	−19.0000	−0.826087
$530$	−24.0000	−1.04249
$531$	−10.0000	−0.433963
$532$	−8.00000	−0.346844
$533$	−10.0000	−0.433148
$534$	30.0000	1.29823
$535$	3.00000	0.129701
$536$	0	0
$537$	−8.00000	−0.345225
$538$	0	0
$539$	15.0000	0.646096
$540$	−2.00000	−0.0860663
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	−26.0000	−1.11680
$543$	12.0000	0.514969
$544$	32.0000	1.37199
$545$	−6.00000	−0.257012
$546$	−4.00000	−0.171184
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	−36.0000	−1.53784
$549$	14.0000	0.597505
$550$	10.0000	0.426401
$551$	0	0
$552$	0	0
$553$	−16.0000	−0.680389
$554$	−4.00000	−0.169944
$555$	−1.00000	−0.0424476
$556$	24.0000	1.01783
$557$	−35.0000	−1.48300	−0.741499	−	0.670954i	$-0.765885\pi$
$557$	−35.0000	−1.48300	−0.741499	+	0.670954i	$0.765885\pi$
$558$	−2.00000	−0.0846668
$559$	10.0000	0.422955
$560$	−8.00000	−0.338062
$561$	20.0000	0.844401
$562$	56.0000	2.36222
$563$	−31.0000	−1.30649	−0.653247	−	0.757145i	$-0.726594\pi$
$563$	−31.0000	−1.30649	−0.653247	+	0.757145i	$0.726594\pi$
$564$	−8.00000	−0.336861
$565$	−3.00000	−0.126211
$566$	10.0000	0.420331
$567$	2.00000	0.0839921
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	−4.00000	−0.167542
$571$	10.0000	0.418487	0.209243	−	0.977864i	$-0.432900\pi$
$571$	10.0000	0.418487	0.209243	+	0.977864i	$0.432900\pi$
$572$	10.0000	0.418121
$573$	3.00000	0.125327
$574$	−40.0000	−1.66957
$575$	−2.00000	−0.0834058
$576$	−8.00000	−0.333333
$577$	26.0000	1.08239	0.541197	−	0.840896i	$-0.317971\pi$
$577$	26.0000	1.08239	0.541197	+	0.840896i	$0.317971\pi$
$578$	2.00000	0.0831890
$579$	16.0000	0.664937
$580$	0	0
$581$	−8.00000	−0.331896
$582$	20.0000	0.829027
$583$	−60.0000	−2.48495
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	36.0000	1.48715
$587$	37.0000	1.52715	0.763577	−	0.645717i	$-0.223441\pi$
$587$	37.0000	1.52715	0.763577	+	0.645717i	$0.223441\pi$
$588$	6.00000	0.247436
$589$	−2.00000	−0.0824086
$590$	20.0000	0.823387
$591$	−21.0000	−0.863825
$592$	−4.00000	−0.164399
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	−10.0000	−0.410305
$595$	8.00000	0.327968
$596$	8.00000	0.327693
$597$	−14.0000	−0.572982
$598$	−4.00000	−0.163572
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	28.0000	1.14214	0.571072	−	0.820900i	$-0.306528\pi$
$601$	28.0000	1.14214	0.571072	+	0.820900i	$0.306528\pi$
$602$	40.0000	1.63028
$603$	−12.0000	−0.488678
$604$	26.0000	1.05792
$605$	14.0000	0.569181
$606$	−6.00000	−0.243733
$607$	−23.0000	−0.933541	−0.466771	−	0.884378i	$-0.654583\pi$
$607$	−23.0000	−0.933541	−0.466771	+	0.884378i	$0.654583\pi$
$608$	−16.0000	−0.648886
$609$	0	0
$610$	−28.0000	−1.13369
$611$	−4.00000	−0.161823
$612$	8.00000	0.323381
$613$	−18.0000	−0.727013	−0.363507	−	0.931592i	$-0.618421\pi$
$613$	−18.0000	−0.727013	−0.363507	+	0.931592i	$0.618421\pi$
$614$	64.0000	2.58283
$615$	−10.0000	−0.403239
$616$	0	0
$617$	−38.0000	−1.52982	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−38.0000	−1.52982	−0.764911	+	0.644136i	$0.777217\pi$
$618$	−10.0000	−0.402259
$619$	11.0000	0.442127	0.221064	−	0.975259i	$-0.429047\pi$
$619$	11.0000	0.442127	0.221064	+	0.975259i	$0.429047\pi$
$620$	2.00000	0.0803219
$621$	2.00000	0.0802572
$622$	−62.0000	−2.48597
$623$	30.0000	1.20192
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	12.0000	0.479616
$627$	−10.0000	−0.399362
$628$	−38.0000	−1.51637
$629$	4.00000	0.159490
$630$	−4.00000	−0.159364
$631$	1.00000	0.0398094	0.0199047	−	0.999802i	$-0.493664\pi$
$631$	1.00000	0.0398094	0.0199047	+	0.999802i	$0.493664\pi$
$632$	0	0
$633$	−3.00000	−0.119239
$634$	12.0000	0.476581
$635$	−20.0000	−0.793676
$636$	−24.0000	−0.951662
$637$	3.00000	0.118864
$638$	0	0
$639$	−4.00000	−0.158238
$640$	0	0
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	6.00000	0.236801
$643$	23.0000	0.907031	0.453516	−	0.891248i	$-0.350170\pi$
$643$	23.0000	0.907031	0.453516	+	0.891248i	$0.350170\pi$
$644$	−8.00000	−0.315244
$645$	10.0000	0.393750
$646$	16.0000	0.629512
$647$	−30.0000	−1.17942	−0.589711	−	0.807614i	$-0.700758\pi$
$647$	−30.0000	−1.17942	−0.589711	+	0.807614i	$0.700758\pi$
$648$	0	0
$649$	50.0000	1.96267
$650$	2.00000	0.0784465
$651$	−2.00000	−0.0783862
$652$	32.0000	1.25322
$653$	−15.0000	−0.586995	−0.293498	−	0.955960i	$-0.594819\pi$
$653$	−15.0000	−0.586995	−0.293498	+	0.955960i	$0.594819\pi$
$654$	−12.0000	−0.469237
$655$	−21.0000	−0.820538
$656$	−40.0000	−1.56174
$657$	−7.00000	−0.273096
$658$	−16.0000	−0.623745
$659$	−44.0000	−1.71400	−0.856998	−	0.515319i	$-0.827673\pi$
$659$	−44.0000	−1.71400	−0.856998	+	0.515319i	$0.827673\pi$
$660$	10.0000	0.389249
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	0	0
$663$	4.00000	0.155347
$664$	0	0
$665$	−4.00000	−0.155113
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	−6.00000	−0.232147
$669$	13.0000	0.502609
$670$	24.0000	0.927201
$671$	−70.0000	−2.70232
$672$	−16.0000	−0.617213
$673$	28.0000	1.07932	0.539660	−	0.841883i	$-0.318553\pi$
$673$	28.0000	1.07932	0.539660	+	0.841883i	$0.318553\pi$
$674$	20.0000	0.770371
$675$	−1.00000	−0.0384900
$676$	2.00000	0.0769231
$677$	−16.0000	−0.614930	−0.307465	−	0.951559i	$-0.599481\pi$
$677$	−16.0000	−0.614930	−0.307465	+	0.951559i	$0.599481\pi$
$678$	−6.00000	−0.230429
$679$	20.0000	0.767530
$680$	0	0
$681$	−28.0000	−1.07296
$682$	10.0000	0.382920
$683$	6.00000	0.229584	0.114792	−	0.993390i	$-0.463380\pi$
$683$	6.00000	0.229584	0.114792	+	0.993390i	$0.463380\pi$
$684$	−4.00000	−0.152944
$685$	−18.0000	−0.687745
$686$	40.0000	1.52721
$687$	13.0000	0.495981
$688$	40.0000	1.52499
$689$	−12.0000	−0.457164
$690$	−4.00000	−0.152277
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	6.00000	0.228086
$693$	−10.0000	−0.379869
$694$	4.00000	0.151838
$695$	12.0000	0.455186
$696$	0	0
$697$	40.0000	1.51511
$698$	20.0000	0.757011
$699$	−5.00000	−0.189117
$700$	4.00000	0.151186
$701$	−19.0000	−0.717620	−0.358810	−	0.933411i	$-0.616817\pi$
$701$	−19.0000	−0.717620	−0.358810	+	0.933411i	$0.616817\pi$
$702$	−2.00000	−0.0754851
$703$	−2.00000	−0.0754314
$704$	40.0000	1.50756
$705$	−4.00000	−0.150649
$706$	−6.00000	−0.225813
$707$	−6.00000	−0.225653
$708$	20.0000	0.751646
$709$	13.0000	0.488225	0.244113	−	0.969747i	$-0.421503\pi$
$709$	13.0000	0.488225	0.244113	+	0.969747i	$0.421503\pi$
$710$	8.00000	0.300235
$711$	−8.00000	−0.300023
$712$	0	0
$713$	−2.00000	−0.0749006
$714$	16.0000	0.598785
$715$	5.00000	0.186989
$716$	16.0000	0.597948
$717$	−8.00000	−0.298765
$718$	−36.0000	−1.34351
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	−4.00000	−0.149071
$721$	−10.0000	−0.372419
$722$	30.0000	1.11648
$723$	−10.0000	−0.371904
$724$	−24.0000	−0.891953
$725$	0	0
$726$	28.0000	1.03918
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	14.0000	0.518163
$731$	−40.0000	−1.47945
$732$	−28.0000	−1.03491
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	−56.0000	−2.06700
$735$	3.00000	0.110657
$736$	−16.0000	−0.589768
$737$	60.0000	2.21013
$738$	−20.0000	−0.736210
$739$	−47.0000	−1.72892	−0.864461	−	0.502699i	$-0.832340\pi$
$739$	−47.0000	−1.72892	−0.864461	+	0.502699i	$0.832340\pi$
$740$	2.00000	0.0735215
$741$	−2.00000	−0.0734718
$742$	−48.0000	−1.76214
$743$	19.0000	0.697042	0.348521	−	0.937301i	$-0.386684\pi$
$743$	19.0000	0.697042	0.348521	+	0.937301i	$0.386684\pi$
$744$	0	0
$745$	4.00000	0.146549
$746$	62.0000	2.26998
$747$	−4.00000	−0.146352
$748$	−40.0000	−1.46254
$749$	6.00000	0.219235
$750$	2.00000	0.0730297
$751$	−27.0000	−0.985244	−0.492622	−	0.870243i	$-0.663961\pi$
$751$	−27.0000	−0.985244	−0.492622	+	0.870243i	$0.663961\pi$
$752$	−16.0000	−0.583460
$753$	−22.0000	−0.801725
$754$	0	0
$755$	13.0000	0.473118
$756$	−4.00000	−0.145479
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	68.0000	2.46987
$759$	−10.0000	−0.362977
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	−40.0000	−1.44905
$763$	−12.0000	−0.434429
$764$	−6.00000	−0.217072
$765$	4.00000	0.144620
$766$	54.0000	1.95110
$767$	10.0000	0.361079
$768$	−16.0000	−0.577350
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	20.0000	0.720750
$771$	3.00000	0.108042
$772$	−32.0000	−1.15171
$773$	−11.0000	−0.395643	−0.197821	−	0.980238i	$-0.563387\pi$
$773$	−11.0000	−0.395643	−0.197821	+	0.980238i	$0.563387\pi$
$774$	20.0000	0.718885
$775$	1.00000	0.0359211
$776$	0	0
$777$	−2.00000	−0.0717496
$778$	72.0000	2.58133
$779$	−20.0000	−0.716574
$780$	2.00000	0.0716115
$781$	20.0000	0.715656
$782$	16.0000	0.572159
$783$	0	0
$784$	12.0000	0.428571
$785$	−19.0000	−0.678139
$786$	−42.0000	−1.49809
$787$	−5.00000	−0.178231	−0.0891154	−	0.996021i	$-0.528404\pi$
$787$	−5.00000	−0.178231	−0.0891154	+	0.996021i	$0.528404\pi$
$788$	42.0000	1.49619
$789$	10.0000	0.356009
$790$	16.0000	0.569254
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−14.0000	−0.497155
$794$	−28.0000	−0.993683
$795$	−12.0000	−0.425596
$796$	28.0000	0.992434
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	−8.00000	−0.283197
$799$	16.0000	0.566039
$800$	8.00000	0.282843
$801$	15.0000	0.529999
$802$	10.0000	0.353112
$803$	35.0000	1.23512
$804$	24.0000	0.846415
$805$	−4.00000	−0.140981
$806$	2.00000	0.0704470
$807$	0	0
$808$	0	0
$809$	−22.0000	−0.773479	−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000	−0.773479	−0.386739	+	0.922189i	$0.626399\pi$
$810$	−2.00000	−0.0702728
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	−13.0000	−0.455930
$814$	10.0000	0.350500
$815$	16.0000	0.560456
$816$	16.0000	0.560112
$817$	20.0000	0.699711
$818$	−14.0000	−0.489499
$819$	−2.00000	−0.0698857
$820$	20.0000	0.698430
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	−36.0000	−1.25564
$823$	−54.0000	−1.88232	−0.941161	−	0.337959i	$-0.890263\pi$
$823$	−54.0000	−1.88232	−0.941161	+	0.337959i	$0.890263\pi$
$824$	0	0
$825$	5.00000	0.174078
$826$	40.0000	1.39178
$827$	24.0000	0.834562	0.417281	−	0.908778i	$-0.362983\pi$
$827$	24.0000	0.834562	0.417281	+	0.908778i	$0.362983\pi$
$828$	−4.00000	−0.139010
$829$	−44.0000	−1.52818	−0.764092	−	0.645108i	$-0.776812\pi$
$829$	−44.0000	−1.52818	−0.764092	+	0.645108i	$0.776812\pi$
$830$	8.00000	0.277684
$831$	−2.00000	−0.0693792
$832$	8.00000	0.277350
$833$	−12.0000	−0.415775
$834$	24.0000	0.831052
$835$	−3.00000	−0.103819
$836$	20.0000	0.691714
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−4.00000	−0.138095	−0.0690477	−	0.997613i	$-0.521996\pi$
$839$	−4.00000	−0.138095	−0.0690477	+	0.997613i	$0.521996\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	44.0000	1.51634
$843$	28.0000	0.964371
$844$	6.00000	0.206529
$845$	1.00000	0.0344010
$846$	−8.00000	−0.275046
$847$	28.0000	0.962091
$848$	−48.0000	−1.64833
$849$	5.00000	0.171600
$850$	−8.00000	−0.274398
$851$	−2.00000	−0.0685591
$852$	8.00000	0.274075
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	−56.0000	−1.91628
$855$	−2.00000	−0.0683986
$856$	0	0
$857$	−39.0000	−1.33221	−0.666107	−	0.745856i	$-0.732041\pi$
$857$	−39.0000	−1.33221	−0.666107	+	0.745856i	$0.732041\pi$
$858$	10.0000	0.341394
$859$	26.0000	0.887109	0.443554	−	0.896248i	$-0.353717\pi$
$859$	26.0000	0.887109	0.443554	+	0.896248i	$0.353717\pi$
$860$	−20.0000	−0.681994
$861$	−20.0000	−0.681598
$862$	20.0000	0.681203
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	−8.00000	−0.272166
$865$	3.00000	0.102003
$866$	8.00000	0.271851
$867$	1.00000	0.0339618
$868$	4.00000	0.135769
$869$	40.0000	1.35691
$870$	0	0
$871$	12.0000	0.406604
$872$	0	0
$873$	10.0000	0.338449
$874$	−8.00000	−0.270604
$875$	2.00000	0.0676123
$876$	14.0000	0.473016
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	−16.0000	−0.539974
$879$	18.0000	0.607125
$880$	20.0000	0.674200
$881$	−12.0000	−0.404290	−0.202145	−	0.979356i	$-0.564791\pi$
$881$	−12.0000	−0.404290	−0.202145	+	0.979356i	$0.564791\pi$
$882$	6.00000	0.202031
$883$	−56.0000	−1.88455	−0.942275	−	0.334840i	$-0.891318\pi$
$883$	−56.0000	−1.88455	−0.942275	+	0.334840i	$0.891318\pi$
$884$	−8.00000	−0.269069
$885$	10.0000	0.336146
$886$	62.0000	2.08293
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−40.0000	−1.34156
$890$	−30.0000	−1.00560
$891$	−5.00000	−0.167506
$892$	−26.0000	−0.870544
$893$	−8.00000	−0.267710
$894$	8.00000	0.267560
$895$	8.00000	0.267411
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	−60.0000	−2.00223
$899$	0	0
$900$	2.00000	0.0666667
$901$	48.0000	1.59911
$902$	100.000	3.32964
$903$	20.0000	0.665558
$904$	0	0
$905$	−12.0000	−0.398893
$906$	26.0000	0.863792
$907$	−37.0000	−1.22856	−0.614282	−	0.789086i	$-0.710554\pi$
$907$	−37.0000	−1.22856	−0.614282	+	0.789086i	$0.710554\pi$
$908$	56.0000	1.85843
$909$	−3.00000	−0.0995037
$910$	4.00000	0.132599
$911$	−56.0000	−1.85536	−0.927681	−	0.373373i	$-0.878201\pi$
$911$	−56.0000	−1.85536	−0.927681	+	0.373373i	$0.878201\pi$
$912$	−8.00000	−0.264906
$913$	20.0000	0.661903
$914$	10.0000	0.330771
$915$	−14.0000	−0.462826
$916$	−26.0000	−0.859064
$917$	−42.0000	−1.38696
$918$	8.00000	0.264039
$919$	31.0000	1.02260	0.511298	−	0.859404i	$-0.329165\pi$
$919$	31.0000	1.02260	0.511298	+	0.859404i	$0.329165\pi$
$920$	0	0
$921$	32.0000	1.05444
$922$	−26.0000	−0.856264
$923$	4.00000	0.131662
$924$	20.0000	0.657952
$925$	1.00000	0.0328798
$926$	62.0000	2.03745
$927$	−5.00000	−0.164222
$928$	0	0
$929$	3.00000	0.0984268	0.0492134	−	0.998788i	$-0.484329\pi$
$929$	3.00000	0.0984268	0.0492134	+	0.998788i	$0.484329\pi$
$930$	2.00000	0.0655826
$931$	6.00000	0.196642
$932$	10.0000	0.327561
$933$	−31.0000	−1.01489
$934$	−80.0000	−2.61768
$935$	−20.0000	−0.654070
$936$	0	0
$937$	37.0000	1.20874	0.604369	−	0.796705i	$-0.293425\pi$
$937$	37.0000	1.20874	0.604369	+	0.796705i	$0.293425\pi$
$938$	48.0000	1.56726
$939$	6.00000	0.195803
$940$	8.00000	0.260931
$941$	45.0000	1.46696	0.733479	−	0.679712i	$-0.237895\pi$
$941$	45.0000	1.46696	0.733479	+	0.679712i	$0.237895\pi$
$942$	−38.0000	−1.23811
$943$	−20.0000	−0.651290
$944$	40.0000	1.30189
$945$	−2.00000	−0.0650600
$946$	−100.000	−3.25128
$947$	45.0000	1.46230	0.731152	−	0.682215i	$-0.238983\pi$
$947$	45.0000	1.46230	0.731152	+	0.682215i	$0.238983\pi$
$948$	16.0000	0.519656
$949$	7.00000	0.227230
$950$	4.00000	0.129777
$951$	6.00000	0.194563
$952$	0	0
$953$	−16.0000	−0.518291	−0.259145	−	0.965838i	$-0.583441\pi$
$953$	−16.0000	−0.518291	−0.259145	+	0.965838i	$0.583441\pi$
$954$	−24.0000	−0.777029
$955$	−3.00000	−0.0970777
$956$	16.0000	0.517477
$957$	0	0
$958$	−48.0000	−1.55081
$959$	−36.0000	−1.16250
$960$	8.00000	0.258199
$961$	1.00000	0.0322581
$962$	2.00000	0.0644826
$963$	3.00000	0.0966736
$964$	20.0000	0.644157
$965$	−16.0000	−0.515058
$966$	−8.00000	−0.257396
$967$	−21.0000	−0.675314	−0.337657	−	0.941269i	$-0.609634\pi$
$967$	−21.0000	−0.675314	−0.337657	+	0.941269i	$0.609634\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	−20.0000	−0.642161
$971$	−27.0000	−0.866471	−0.433236	−	0.901281i	$-0.642628\pi$
$971$	−27.0000	−0.866471	−0.433236	+	0.901281i	$0.642628\pi$
$972$	−2.00000	−0.0641500
$973$	24.0000	0.769405
$974$	−16.0000	−0.512673
$975$	1.00000	0.0320256
$976$	−56.0000	−1.79252
$977$	−54.0000	−1.72761	−0.863807	−	0.503824i	$-0.831926\pi$
$977$	−54.0000	−1.72761	−0.863807	+	0.503824i	$0.831926\pi$
$978$	32.0000	1.02325
$979$	−75.0000	−2.39701
$980$	−6.00000	−0.191663
$981$	−6.00000	−0.191565
$982$	60.0000	1.91468
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	0	0
$985$	21.0000	0.669116
$986$	0	0
$987$	−8.00000	−0.254643
$988$	4.00000	0.127257
$989$	20.0000	0.635963
$990$	10.0000	0.317821
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	8.00000	0.254000
$993$	0	0
$994$	16.0000	0.507489
$995$	14.0000	0.443830
$996$	8.00000	0.253490
$997$	23.0000	0.728417	0.364209	−	0.931317i	$-0.381339\pi$
$997$	23.0000	0.728417	0.364209	+	0.931317i	$0.381339\pi$
$998$	−8.00000	−0.253236
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.b.1.1	✓	1	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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