LMFDB - Embedded newform 5610.2.a.bk.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5610,2,Mod(1,5610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5610, 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("5610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5610.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:	$44.7960755339$
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	$\chi$	$=$	5610.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -4.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} -4.00000 q^{21} +1.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} -2.00000 q^{29} +1.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} -1.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} -2.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} +4.00000 q^{41} -4.00000 q^{42} +8.00000 q^{43} +1.00000 q^{44} +1.00000 q^{45} -6.00000 q^{46} +4.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -1.00000 q^{51} -2.00000 q^{52} -12.0000 q^{53} +1.00000 q^{54} +1.00000 q^{55} -4.00000 q^{56} -2.00000 q^{57} -2.00000 q^{58} -12.0000 q^{59} +1.00000 q^{60} -6.00000 q^{61} -8.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +1.00000 q^{66} +6.00000 q^{67} -1.00000 q^{68} -6.00000 q^{69} -4.00000 q^{70} -4.00000 q^{71} +1.00000 q^{72} -2.00000 q^{73} -6.00000 q^{74} +1.00000 q^{75} -2.00000 q^{76} -4.00000 q^{77} -2.00000 q^{78} -10.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +4.00000 q^{82} +8.00000 q^{83} -4.00000 q^{84} -1.00000 q^{85} +8.00000 q^{86} -2.00000 q^{87} +1.00000 q^{88} -10.0000 q^{89} +1.00000 q^{90} +8.00000 q^{91} -6.00000 q^{92} -8.00000 q^{93} +4.00000 q^{94} -2.00000 q^{95} +1.00000 q^{96} -12.0000 q^{97} +9.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.00000	0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−4.00000	−1.06904
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.00000	0.235702
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	1.00000	0.223607
$21$	−4.00000	−0.872872
$22$	1.00000	0.213201
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	−4.00000	−0.755929
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	1.00000	0.182574
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	−1.00000	−0.171499
$35$	−4.00000	−0.676123
$36$	1.00000	0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−2.00000	−0.324443
$39$	−2.00000	−0.320256
$40$	1.00000	0.158114
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	−4.00000	−0.617213
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	1.00000	0.150756
$45$	1.00000	0.149071
$46$	−6.00000	−0.884652
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	1.00000	0.141421
$51$	−1.00000	−0.140028
$52$	−2.00000	−0.277350
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	1.00000	0.136083
$55$	1.00000	0.134840
$56$	−4.00000	−0.534522
$57$	−2.00000	−0.264906
$58$	−2.00000	−0.262613
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	1.00000	0.129099
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−8.00000	−1.01600
$63$	−4.00000	−0.503953
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	1.00000	0.123091
$67$	6.00000	0.733017	0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000	0.733017	0.366508	+	0.930415i	$0.380553\pi$
$68$	−1.00000	−0.121268
$69$	−6.00000	−0.722315
$70$	−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	$0.576281\pi$
$72$	1.00000	0.117851
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−6.00000	−0.697486
$75$	1.00000	0.115470
$76$	−2.00000	−0.229416
$77$	−4.00000	−0.455842
$78$	−2.00000	−0.226455
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	4.00000	0.441726
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	−4.00000	−0.436436
$85$	−1.00000	−0.108465
$86$	8.00000	0.862662
$87$	−2.00000	−0.214423
$88$	1.00000	0.106600
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	1.00000	0.105409
$91$	8.00000	0.838628
$92$	−6.00000	−0.625543
$93$	−8.00000	−0.829561
$94$	4.00000	0.412568
$95$	−2.00000	−0.205196
$96$	1.00000	0.102062
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	9.00000	0.909137
$99$	1.00000	0.100504
$100$	1.00000	0.100000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	−1.00000	−0.0990148
$103$	20.0000	1.97066	0.985329	−	0.170664i	$-0.0545913\pi$
$103$	20.0000	1.97066	0.985329	+	0.170664i	$0.0545913\pi$
$104$	−2.00000	−0.196116
$105$	−4.00000	−0.390360
$106$	−12.0000	−1.16554
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	1.00000	0.0962250
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	1.00000	0.0953463
$111$	−6.00000	−0.569495
$112$	−4.00000	−0.377964
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	−2.00000	−0.187317
$115$	−6.00000	−0.559503
$116$	−2.00000	−0.185695
$117$	−2.00000	−0.184900
$118$	−12.0000	−1.10469
$119$	4.00000	0.366679
$120$	1.00000	0.0912871
$121$	1.00000	0.0909091
$122$	−6.00000	−0.543214
$123$	4.00000	0.360668
$124$	−8.00000	−0.718421
$125$	1.00000	0.0894427
$126$	−4.00000	−0.356348
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	1.00000	0.0883883
$129$	8.00000	0.704361
$130$	−2.00000	−0.175412
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	1.00000	0.0870388
$133$	8.00000	0.693688
$134$	6.00000	0.518321
$135$	1.00000	0.0860663
$136$	−1.00000	−0.0857493
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	−6.00000	−0.510754
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	−4.00000	−0.338062
$141$	4.00000	0.336861
$142$	−4.00000	−0.335673
$143$	−2.00000	−0.167248
$144$	1.00000	0.0833333
$145$	−2.00000	−0.166091
$146$	−2.00000	−0.165521
$147$	9.00000	0.742307
$148$	−6.00000	−0.493197
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	1.00000	0.0816497
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−2.00000	−0.162221
$153$	−1.00000	−0.0808452
$154$	−4.00000	−0.322329
$155$	−8.00000	−0.642575
$156$	−2.00000	−0.160128
$157$	−8.00000	−0.638470	−0.319235	−	0.947676i	$-0.603426\pi$
$157$	−8.00000	−0.638470	−0.319235	+	0.947676i	$0.603426\pi$
$158$	−10.0000	−0.795557
$159$	−12.0000	−0.951662
$160$	1.00000	0.0790569
$161$	24.0000	1.89146
$162$	1.00000	0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	4.00000	0.312348
$165$	1.00000	0.0778499
$166$	8.00000	0.620920
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	−4.00000	−0.308607
$169$	−9.00000	−0.692308
$170$	−1.00000	−0.0766965
$171$	−2.00000	−0.152944
$172$	8.00000	0.609994
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	−2.00000	−0.151620
$175$	−4.00000	−0.302372
$176$	1.00000	0.0753778
$177$	−12.0000	−0.901975
$178$	−10.0000	−0.749532
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	1.00000	0.0745356
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	8.00000	0.592999
$183$	−6.00000	−0.443533
$184$	−6.00000	−0.442326
$185$	−6.00000	−0.441129
$186$	−8.00000	−0.586588
$187$	−1.00000	−0.0731272
$188$	4.00000	0.291730
$189$	−4.00000	−0.290957
$190$	−2.00000	−0.145095
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	1.00000	0.0721688
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	−12.0000	−0.861550
$195$	−2.00000	−0.143223
$196$	9.00000	0.642857
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	1.00000	0.0710669
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	1.00000	0.0707107
$201$	6.00000	0.423207
$202$	0	0
$203$	8.00000	0.561490
$204$	−1.00000	−0.0700140
$205$	4.00000	0.279372
$206$	20.0000	1.39347
$207$	−6.00000	−0.417029
$208$	−2.00000	−0.138675
$209$	−2.00000	−0.138343
$210$	−4.00000	−0.276026
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−12.0000	−0.824163
$213$	−4.00000	−0.274075
$214$	12.0000	0.820303
$215$	8.00000	0.545595
$216$	1.00000	0.0680414
$217$	32.0000	2.17230
$218$	6.00000	0.406371
$219$	−2.00000	−0.135147
$220$	1.00000	0.0674200
$221$	2.00000	0.134535
$222$	−6.00000	−0.402694
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	−4.00000	−0.267261
$225$	1.00000	0.0666667
$226$	−12.0000	−0.798228
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	−2.00000	−0.132453
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	−6.00000	−0.395628
$231$	−4.00000	−0.263181
$232$	−2.00000	−0.131306
$233$	2.00000	0.131024	0.0655122	−	0.997852i	$-0.479132\pi$
$233$	2.00000	0.131024	0.0655122	+	0.997852i	$0.479132\pi$
$234$	−2.00000	−0.130744
$235$	4.00000	0.260931
$236$	−12.0000	−0.781133
$237$	−10.0000	−0.649570
$238$	4.00000	0.259281
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	1.00000	0.0645497
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	1.00000	0.0642824
$243$	1.00000	0.0641500
$244$	−6.00000	−0.384111
$245$	9.00000	0.574989
$246$	4.00000	0.255031
$247$	4.00000	0.254514
$248$	−8.00000	−0.508001
$249$	8.00000	0.506979
$250$	1.00000	0.0632456
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	−4.00000	−0.251976
$253$	−6.00000	−0.377217
$254$	16.0000	1.00393
$255$	−1.00000	−0.0626224
$256$	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	$0.310261\pi$
$258$	8.00000	0.498058
$259$	24.0000	1.49129
$260$	−2.00000	−0.124035
$261$	−2.00000	−0.123797
$262$	8.00000	0.494242
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	1.00000	0.0615457
$265$	−12.0000	−0.737154
$266$	8.00000	0.490511
$267$	−10.0000	−0.611990
$268$	6.00000	0.366508
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	1.00000	0.0608581
$271$	28.0000	1.70088	0.850439	−	0.526073i	$-0.176336\pi$
$271$	28.0000	1.70088	0.850439	+	0.526073i	$0.176336\pi$
$272$	−1.00000	−0.0606339
$273$	8.00000	0.484182
$274$	10.0000	0.604122
$275$	1.00000	0.0603023
$276$	−6.00000	−0.361158
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	−4.00000	−0.239046
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	4.00000	0.238197
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	−4.00000	−0.237356
$285$	−2.00000	−0.118470
$286$	−2.00000	−0.118262
$287$	−16.0000	−0.944450
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	−2.00000	−0.117444
$291$	−12.0000	−0.703452
$292$	−2.00000	−0.117041
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	9.00000	0.524891
$295$	−12.0000	−0.698667
$296$	−6.00000	−0.348743
$297$	1.00000	0.0580259
$298$	−12.0000	−0.695141
$299$	12.0000	0.693978
$300$	1.00000	0.0577350
$301$	−32.0000	−1.84445
$302$	−8.00000	−0.460348
$303$	0	0
$304$	−2.00000	−0.114708
$305$	−6.00000	−0.343559
$306$	−1.00000	−0.0571662
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	−4.00000	−0.227921
$309$	20.0000	1.13776
$310$	−8.00000	−0.454369
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−2.00000	−0.113228
$313$	28.0000	1.58265	0.791327	−	0.611393i	$-0.209391\pi$
$313$	28.0000	1.58265	0.791327	+	0.611393i	$0.209391\pi$
$314$	−8.00000	−0.451466
$315$	−4.00000	−0.225374
$316$	−10.0000	−0.562544
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	−12.0000	−0.672927
$319$	−2.00000	−0.111979
$320$	1.00000	0.0559017
$321$	12.0000	0.669775
$322$	24.0000	1.33747
$323$	2.00000	0.111283
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	12.0000	0.664619
$327$	6.00000	0.331801
$328$	4.00000	0.220863
$329$	−16.0000	−0.882109
$330$	1.00000	0.0550482
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	8.00000	0.439057
$333$	−6.00000	−0.328798
$334$	0	0
$335$	6.00000	0.327815
$336$	−4.00000	−0.218218
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	−9.00000	−0.489535
$339$	−12.0000	−0.651751
$340$	−1.00000	−0.0542326
$341$	−8.00000	−0.433224
$342$	−2.00000	−0.108148
$343$	−8.00000	−0.431959
$344$	8.00000	0.431331
$345$	−6.00000	−0.323029
$346$	−14.0000	−0.752645
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	−2.00000	−0.107211
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	−4.00000	−0.213809
$351$	−2.00000	−0.106752
$352$	1.00000	0.0533002
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	−12.0000	−0.637793
$355$	−4.00000	−0.212298
$356$	−10.0000	−0.529999
$357$	4.00000	0.211702
$358$	−12.0000	−0.634220
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	1.00000	0.0527046
$361$	−15.0000	−0.789474
$362$	10.0000	0.525588
$363$	1.00000	0.0524864
$364$	8.00000	0.419314
$365$	−2.00000	−0.104685
$366$	−6.00000	−0.313625
$367$	−2.00000	−0.104399	−0.0521996	−	0.998637i	$-0.516623\pi$
$367$	−2.00000	−0.104399	−0.0521996	+	0.998637i	$0.516623\pi$
$368$	−6.00000	−0.312772
$369$	4.00000	0.208232
$370$	−6.00000	−0.311925
$371$	48.0000	2.49204
$372$	−8.00000	−0.414781
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	−1.00000	−0.0517088
$375$	1.00000	0.0516398
$376$	4.00000	0.206284
$377$	4.00000	0.206010
$378$	−4.00000	−0.205738
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	−2.00000	−0.102598
$381$	16.0000	0.819705
$382$	−24.0000	−1.22795
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	1.00000	0.0510310
$385$	−4.00000	−0.203859
$386$	−10.0000	−0.508987
$387$	8.00000	0.406663
$388$	−12.0000	−0.609208
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	−2.00000	−0.101274
$391$	6.00000	0.303433
$392$	9.00000	0.454569
$393$	8.00000	0.403547
$394$	−18.0000	−0.906827
$395$	−10.0000	−0.503155
$396$	1.00000	0.0502519
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	−24.0000	−1.20301
$399$	8.00000	0.400501
$400$	1.00000	0.0500000
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	6.00000	0.299253
$403$	16.0000	0.797017
$404$	0	0
$405$	1.00000	0.0496904
$406$	8.00000	0.397033
$407$	−6.00000	−0.297409
$408$	−1.00000	−0.0495074
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	4.00000	0.197546
$411$	10.0000	0.493264
$412$	20.0000	0.985329
$413$	48.0000	2.36193
$414$	−6.00000	−0.294884
$415$	8.00000	0.392705
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	−2.00000	−0.0978232
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	−4.00000	−0.195180
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	−4.00000	−0.194717
$423$	4.00000	0.194487
$424$	−12.0000	−0.582772
$425$	−1.00000	−0.0485071
$426$	−4.00000	−0.193801
$427$	24.0000	1.16144
$428$	12.0000	0.580042
$429$	−2.00000	−0.0965609
$430$	8.00000	0.385794
$431$	2.00000	0.0963366	0.0481683	−	0.998839i	$-0.484662\pi$
$431$	2.00000	0.0963366	0.0481683	+	0.998839i	$0.484662\pi$
$432$	1.00000	0.0481125
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	32.0000	1.53605
$435$	−2.00000	−0.0958927
$436$	6.00000	0.287348
$437$	12.0000	0.574038
$438$	−2.00000	−0.0955637
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	1.00000	0.0476731
$441$	9.00000	0.428571
$442$	2.00000	0.0951303
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	−6.00000	−0.284747
$445$	−10.0000	−0.474045
$446$	24.0000	1.13643
$447$	−12.0000	−0.567581
$448$	−4.00000	−0.188982
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	1.00000	0.0471405
$451$	4.00000	0.188353
$452$	−12.0000	−0.564433
$453$	−8.00000	−0.375873
$454$	4.00000	0.187729
$455$	8.00000	0.375046
$456$	−2.00000	−0.0936586
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	−22.0000	−1.02799
$459$	−1.00000	−0.0466760
$460$	−6.00000	−0.279751
$461$	16.0000	0.745194	0.372597	−	0.927993i	$-0.378467\pi$
$461$	16.0000	0.745194	0.372597	+	0.927993i	$0.378467\pi$
$462$	−4.00000	−0.186097
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	−2.00000	−0.0928477
$465$	−8.00000	−0.370991
$466$	2.00000	0.0926482
$467$	−2.00000	−0.0925490	−0.0462745	−	0.998929i	$-0.514735\pi$
$467$	−2.00000	−0.0925490	−0.0462745	+	0.998929i	$0.514735\pi$
$468$	−2.00000	−0.0924500
$469$	−24.0000	−1.10822
$470$	4.00000	0.184506
$471$	−8.00000	−0.368621
$472$	−12.0000	−0.552345
$473$	8.00000	0.367840
$474$	−10.0000	−0.459315
$475$	−2.00000	−0.0917663
$476$	4.00000	0.183340
$477$	−12.0000	−0.549442
$478$	−12.0000	−0.548867
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	1.00000	0.0456435
$481$	12.0000	0.547153
$482$	4.00000	0.182195
$483$	24.0000	1.09204
$484$	1.00000	0.0454545
$485$	−12.0000	−0.544892
$486$	1.00000	0.0453609
$487$	−22.0000	−0.996915	−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000	−0.996915	−0.498458	+	0.866914i	$0.666100\pi$
$488$	−6.00000	−0.271607
$489$	12.0000	0.542659
$490$	9.00000	0.406579
$491$	34.0000	1.53440	0.767199	−	0.641409i	$-0.221650\pi$
$491$	34.0000	1.53440	0.767199	+	0.641409i	$0.221650\pi$
$492$	4.00000	0.180334
$493$	2.00000	0.0900755
$494$	4.00000	0.179969
$495$	1.00000	0.0449467
$496$	−8.00000	−0.359211
$497$	16.0000	0.717698
$498$	8.00000	0.358489
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	16.0000	0.714115
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	−4.00000	−0.178174
$505$	0	0
$506$	−6.00000	−0.266733
$507$	−9.00000	−0.399704
$508$	16.0000	0.709885
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	−1.00000	−0.0442807
$511$	8.00000	0.353899
$512$	1.00000	0.0441942
$513$	−2.00000	−0.0883022
$514$	18.0000	0.793946
$515$	20.0000	0.881305
$516$	8.00000	0.352180
$517$	4.00000	0.175920
$518$	24.0000	1.05450
$519$	−14.0000	−0.614532
$520$	−2.00000	−0.0877058
$521$	38.0000	1.66481	0.832405	−	0.554168i	$-0.186963\pi$
$521$	38.0000	1.66481	0.832405	+	0.554168i	$0.186963\pi$
$522$	−2.00000	−0.0875376
$523$	32.0000	1.39926	0.699631	−	0.714504i	$-0.253348\pi$
$523$	32.0000	1.39926	0.699631	+	0.714504i	$0.253348\pi$
$524$	8.00000	0.349482
$525$	−4.00000	−0.174574
$526$	0	0
$527$	8.00000	0.348485
$528$	1.00000	0.0435194
$529$	13.0000	0.565217
$530$	−12.0000	−0.521247
$531$	−12.0000	−0.520756
$532$	8.00000	0.346844
$533$	−8.00000	−0.346518
$534$	−10.0000	−0.432742
$535$	12.0000	0.518805
$536$	6.00000	0.259161
$537$	−12.0000	−0.517838
$538$	−6.00000	−0.258678
$539$	9.00000	0.387657
$540$	1.00000	0.0430331
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	28.0000	1.20270
$543$	10.0000	0.429141
$544$	−1.00000	−0.0428746
$545$	6.00000	0.257012
$546$	8.00000	0.342368
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	10.0000	0.427179
$549$	−6.00000	−0.256074
$550$	1.00000	0.0426401
$551$	4.00000	0.170406
$552$	−6.00000	−0.255377
$553$	40.0000	1.70097
$554$	22.0000	0.934690
$555$	−6.00000	−0.254686
$556$	0	0
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	−8.00000	−0.338667
$559$	−16.0000	−0.676728
$560$	−4.00000	−0.169031
$561$	−1.00000	−0.0422200
$562$	−18.0000	−0.759284
$563$	−32.0000	−1.34864	−0.674320	−	0.738440i	$-0.735563\pi$
$563$	−32.0000	−1.34864	−0.674320	+	0.738440i	$0.735563\pi$
$564$	4.00000	0.168430
$565$	−12.0000	−0.504844
$566$	−20.0000	−0.840663
$567$	−4.00000	−0.167984
$568$	−4.00000	−0.167836
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	−2.00000	−0.0837708
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	−2.00000	−0.0836242
$573$	−24.0000	−1.00261
$574$	−16.0000	−0.667827
$575$	−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	$0.750276\pi$
$578$	1.00000	0.0415945
$579$	−10.0000	−0.415586
$580$	−2.00000	−0.0830455
$581$	−32.0000	−1.32758
$582$	−12.0000	−0.497416
$583$	−12.0000	−0.496989
$584$	−2.00000	−0.0827606
$585$	−2.00000	−0.0826898
$586$	30.0000	1.23929
$587$	−22.0000	−0.908037	−0.454019	−	0.890992i	$-0.650010\pi$
$587$	−22.0000	−0.908037	−0.454019	+	0.890992i	$0.650010\pi$
$588$	9.00000	0.371154
$589$	16.0000	0.659269
$590$	−12.0000	−0.494032
$591$	−18.0000	−0.740421
$592$	−6.00000	−0.246598
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	1.00000	0.0410305
$595$	4.00000	0.163984
$596$	−12.0000	−0.491539
$597$	−24.0000	−0.982255
$598$	12.0000	0.490716
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	1.00000	0.0408248
$601$	−36.0000	−1.46847	−0.734235	−	0.678895i	$-0.762459\pi$
$601$	−36.0000	−1.46847	−0.734235	+	0.678895i	$0.762459\pi$
$602$	−32.0000	−1.30422
$603$	6.00000	0.244339
$604$	−8.00000	−0.325515
$605$	1.00000	0.0406558
$606$	0	0
$607$	12.0000	0.487065	0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000	0.487065	0.243532	+	0.969893i	$0.421694\pi$
$608$	−2.00000	−0.0811107
$609$	8.00000	0.324176
$610$	−6.00000	−0.242933
$611$	−8.00000	−0.323645
$612$	−1.00000	−0.0404226
$613$	18.0000	0.727013	0.363507	−	0.931592i	$-0.381579\pi$
$613$	18.0000	0.727013	0.363507	+	0.931592i	$0.381579\pi$
$614$	4.00000	0.161427
$615$	4.00000	0.161296
$616$	−4.00000	−0.161165
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	20.0000	0.804518
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	−8.00000	−0.321288
$621$	−6.00000	−0.240772
$622$	0	0
$623$	40.0000	1.60257
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	28.0000	1.11911
$627$	−2.00000	−0.0798723
$628$	−8.00000	−0.319235
$629$	6.00000	0.239236
$630$	−4.00000	−0.159364
$631$	48.0000	1.91085	0.955425	−	0.295234i	$-0.0953977\pi$
$631$	48.0000	1.91085	0.955425	+	0.295234i	$0.0953977\pi$
$632$	−10.0000	−0.397779
$633$	−4.00000	−0.158986
$634$	18.0000	0.714871
$635$	16.0000	0.634941
$636$	−12.0000	−0.475831
$637$	−18.0000	−0.713186
$638$	−2.00000	−0.0791808
$639$	−4.00000	−0.158238
$640$	1.00000	0.0395285
$641$	38.0000	1.50091	0.750455	−	0.660922i	$-0.229834\pi$
$641$	38.0000	1.50091	0.750455	+	0.660922i	$0.229834\pi$
$642$	12.0000	0.473602
$643$	24.0000	0.946468	0.473234	−	0.880937i	$-0.343087\pi$
$643$	24.0000	0.946468	0.473234	+	0.880937i	$0.343087\pi$
$644$	24.0000	0.945732
$645$	8.00000	0.315000
$646$	2.00000	0.0786889
$647$	−16.0000	−0.629025	−0.314512	−	0.949253i	$-0.601841\pi$
$647$	−16.0000	−0.629025	−0.314512	+	0.949253i	$0.601841\pi$
$648$	1.00000	0.0392837
$649$	−12.0000	−0.471041
$650$	−2.00000	−0.0784465
$651$	32.0000	1.25418
$652$	12.0000	0.469956
$653$	−34.0000	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$654$	6.00000	0.234619
$655$	8.00000	0.312586
$656$	4.00000	0.156174
$657$	−2.00000	−0.0780274
$658$	−16.0000	−0.623745
$659$	26.0000	1.01282	0.506408	−	0.862294i	$-0.330973\pi$
$659$	26.0000	1.01282	0.506408	+	0.862294i	$0.330973\pi$
$660$	1.00000	0.0389249
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	−8.00000	−0.310929
$663$	2.00000	0.0776736
$664$	8.00000	0.310460
$665$	8.00000	0.310227
$666$	−6.00000	−0.232495
$667$	12.0000	0.464642
$668$	0	0
$669$	24.0000	0.927894
$670$	6.00000	0.231800
$671$	−6.00000	−0.231627
$672$	−4.00000	−0.154303
$673$	−50.0000	−1.92736	−0.963679	−	0.267063i	$-0.913947\pi$
$673$	−50.0000	−1.92736	−0.963679	+	0.267063i	$0.913947\pi$
$674$	22.0000	0.847408
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	−12.0000	−0.460857
$679$	48.0000	1.84207
$680$	−1.00000	−0.0383482
$681$	4.00000	0.153280
$682$	−8.00000	−0.306336
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	−2.00000	−0.0764719
$685$	10.0000	0.382080
$686$	−8.00000	−0.305441
$687$	−22.0000	−0.839352
$688$	8.00000	0.304997
$689$	24.0000	0.914327
$690$	−6.00000	−0.228416
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	−14.0000	−0.532200
$693$	−4.00000	−0.151947
$694$	4.00000	0.151838
$695$	0	0
$696$	−2.00000	−0.0758098
$697$	−4.00000	−0.151511
$698$	−20.0000	−0.757011
$699$	2.00000	0.0756469
$700$	−4.00000	−0.151186
$701$	−24.0000	−0.906467	−0.453234	−	0.891392i	$-0.649730\pi$
$701$	−24.0000	−0.906467	−0.453234	+	0.891392i	$0.649730\pi$
$702$	−2.00000	−0.0754851
$703$	12.0000	0.452589
$704$	1.00000	0.0376889
$705$	4.00000	0.150649
$706$	−14.0000	−0.526897
$707$	0	0
$708$	−12.0000	−0.450988
$709$	42.0000	1.57734	0.788672	−	0.614815i	$-0.210769\pi$
$709$	42.0000	1.57734	0.788672	+	0.614815i	$0.210769\pi$
$710$	−4.00000	−0.150117
$711$	−10.0000	−0.375029
$712$	−10.0000	−0.374766
$713$	48.0000	1.79761
$714$	4.00000	0.149696
$715$	−2.00000	−0.0747958
$716$	−12.0000	−0.448461
$717$	−12.0000	−0.448148
$718$	32.0000	1.19423
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	1.00000	0.0372678
$721$	−80.0000	−2.97936
$722$	−15.0000	−0.558242
$723$	4.00000	0.148762
$724$	10.0000	0.371647
$725$	−2.00000	−0.0742781
$726$	1.00000	0.0371135
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	8.00000	0.296500
$729$	1.00000	0.0370370
$730$	−2.00000	−0.0740233
$731$	−8.00000	−0.295891
$732$	−6.00000	−0.221766
$733$	18.0000	0.664845	0.332423	−	0.943131i	$-0.392134\pi$
$733$	18.0000	0.664845	0.332423	+	0.943131i	$0.392134\pi$
$734$	−2.00000	−0.0738213
$735$	9.00000	0.331970
$736$	−6.00000	−0.221163
$737$	6.00000	0.221013
$738$	4.00000	0.147242
$739$	−38.0000	−1.39785	−0.698926	−	0.715194i	$-0.746338\pi$
$739$	−38.0000	−1.39785	−0.698926	+	0.715194i	$0.746338\pi$
$740$	−6.00000	−0.220564
$741$	4.00000	0.146944
$742$	48.0000	1.76214
$743$	12.0000	0.440237	0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000	0.440237	0.220119	+	0.975473i	$0.429356\pi$
$744$	−8.00000	−0.293294
$745$	−12.0000	−0.439646
$746$	10.0000	0.366126
$747$	8.00000	0.292705
$748$	−1.00000	−0.0365636
$749$	−48.0000	−1.75388
$750$	1.00000	0.0365148
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	4.00000	0.145865
$753$	16.0000	0.583072
$754$	4.00000	0.145671
$755$	−8.00000	−0.291150
$756$	−4.00000	−0.145479
$757$	−52.0000	−1.88997	−0.944986	−	0.327111i	$-0.893925\pi$
$757$	−52.0000	−1.88997	−0.944986	+	0.327111i	$0.893925\pi$
$758$	20.0000	0.726433
$759$	−6.00000	−0.217786
$760$	−2.00000	−0.0725476
$761$	2.00000	0.0724999	0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000	0.0724999	0.0362500	+	0.999343i	$0.488459\pi$
$762$	16.0000	0.579619
$763$	−24.0000	−0.868858
$764$	−24.0000	−0.868290
$765$	−1.00000	−0.0361551
$766$	−20.0000	−0.722629
$767$	24.0000	0.866590
$768$	1.00000	0.0360844
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	−4.00000	−0.144150
$771$	18.0000	0.648254
$772$	−10.0000	−0.359908
$773$	32.0000	1.15096	0.575480	−	0.817816i	$-0.304815\pi$
$773$	32.0000	1.15096	0.575480	+	0.817816i	$0.304815\pi$
$774$	8.00000	0.287554
$775$	−8.00000	−0.287368
$776$	−12.0000	−0.430775
$777$	24.0000	0.860995
$778$	−6.00000	−0.215110
$779$	−8.00000	−0.286630
$780$	−2.00000	−0.0716115
$781$	−4.00000	−0.143131
$782$	6.00000	0.214560
$783$	−2.00000	−0.0714742
$784$	9.00000	0.321429
$785$	−8.00000	−0.285532
$786$	8.00000	0.285351
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	−18.0000	−0.641223
$789$	0	0
$790$	−10.0000	−0.355784
$791$	48.0000	1.70668
$792$	1.00000	0.0355335
$793$	12.0000	0.426132
$794$	−22.0000	−0.780751
$795$	−12.0000	−0.425596
$796$	−24.0000	−0.850657
$797$	−32.0000	−1.13350	−0.566749	−	0.823890i	$-0.691799\pi$
$797$	−32.0000	−1.13350	−0.566749	+	0.823890i	$0.691799\pi$
$798$	8.00000	0.283197
$799$	−4.00000	−0.141510
$800$	1.00000	0.0353553
$801$	−10.0000	−0.353333
$802$	−2.00000	−0.0706225
$803$	−2.00000	−0.0705785
$804$	6.00000	0.211604
$805$	24.0000	0.845889
$806$	16.0000	0.563576
$807$	−6.00000	−0.211210
$808$	0	0
$809$	−16.0000	−0.562530	−0.281265	−	0.959630i	$-0.590754\pi$
$809$	−16.0000	−0.562530	−0.281265	+	0.959630i	$0.590754\pi$
$810$	1.00000	0.0351364
$811$	−52.0000	−1.82597	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000	−1.82597	−0.912983	+	0.407997i	$0.866228\pi$
$812$	8.00000	0.280745
$813$	28.0000	0.982003
$814$	−6.00000	−0.210300
$815$	12.0000	0.420342
$816$	−1.00000	−0.0350070
$817$	−16.0000	−0.559769
$818$	−18.0000	−0.629355
$819$	8.00000	0.279543
$820$	4.00000	0.139686
$821$	50.0000	1.74501	0.872506	−	0.488603i	$-0.162493\pi$
$821$	50.0000	1.74501	0.872506	+	0.488603i	$0.162493\pi$
$822$	10.0000	0.348790
$823$	26.0000	0.906303	0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000	0.906303	0.453152	+	0.891434i	$0.350300\pi$
$824$	20.0000	0.696733
$825$	1.00000	0.0348155
$826$	48.0000	1.67013
$827$	−28.0000	−0.973655	−0.486828	−	0.873498i	$-0.661846\pi$
$827$	−28.0000	−0.973655	−0.486828	+	0.873498i	$0.661846\pi$
$828$	−6.00000	−0.208514
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	8.00000	0.277684
$831$	22.0000	0.763172
$832$	−2.00000	−0.0693375
$833$	−9.00000	−0.311832
$834$	0	0
$835$	0	0
$836$	−2.00000	−0.0691714
$837$	−8.00000	−0.276520
$838$	36.0000	1.24360
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	−4.00000	−0.138013
$841$	−25.0000	−0.862069
$842$	−34.0000	−1.17172
$843$	−18.0000	−0.619953
$844$	−4.00000	−0.137686
$845$	−9.00000	−0.309609
$846$	4.00000	0.137523
$847$	−4.00000	−0.137442
$848$	−12.0000	−0.412082
$849$	−20.0000	−0.686398
$850$	−1.00000	−0.0342997
$851$	36.0000	1.23406
$852$	−4.00000	−0.137038
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	24.0000	0.821263
$855$	−2.00000	−0.0683986
$856$	12.0000	0.410152
$857$	54.0000	1.84460	0.922302	−	0.386469i	$-0.126305\pi$
$857$	54.0000	1.84460	0.922302	+	0.386469i	$0.126305\pi$
$858$	−2.00000	−0.0682789
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	8.00000	0.272798
$861$	−16.0000	−0.545279
$862$	2.00000	0.0681203
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	1.00000	0.0340207
$865$	−14.0000	−0.476014
$866$	−34.0000	−1.15537
$867$	1.00000	0.0339618
$868$	32.0000	1.08615
$869$	−10.0000	−0.339227
$870$	−2.00000	−0.0678064
$871$	−12.0000	−0.406604
$872$	6.00000	0.203186
$873$	−12.0000	−0.406138
$874$	12.0000	0.405906
$875$	−4.00000	−0.135225
$876$	−2.00000	−0.0675737
$877$	−46.0000	−1.55331	−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000	−1.55331	−0.776655	+	0.629926i	$0.783085\pi$
$878$	−10.0000	−0.337484
$879$	30.0000	1.01187
$880$	1.00000	0.0337100
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	9.00000	0.303046
$883$	−50.0000	−1.68263	−0.841317	−	0.540542i	$-0.818219\pi$
$883$	−50.0000	−1.68263	−0.841317	+	0.540542i	$0.818219\pi$
$884$	2.00000	0.0672673
$885$	−12.0000	−0.403376
$886$	18.0000	0.604722
$887$	−28.0000	−0.940148	−0.470074	−	0.882627i	$-0.655773\pi$
$887$	−28.0000	−0.940148	−0.470074	+	0.882627i	$0.655773\pi$
$888$	−6.00000	−0.201347
$889$	−64.0000	−2.14649
$890$	−10.0000	−0.335201
$891$	1.00000	0.0335013
$892$	24.0000	0.803579
$893$	−8.00000	−0.267710
$894$	−12.0000	−0.401340
$895$	−12.0000	−0.401116
$896$	−4.00000	−0.133631
$897$	12.0000	0.400668
$898$	−6.00000	−0.200223
$899$	16.0000	0.533630
$900$	1.00000	0.0333333
$901$	12.0000	0.399778
$902$	4.00000	0.133185
$903$	−32.0000	−1.06489
$904$	−12.0000	−0.399114
$905$	10.0000	0.332411
$906$	−8.00000	−0.265782
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	4.00000	0.132745
$909$	0	0
$910$	8.00000	0.265197
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	−2.00000	−0.0662266
$913$	8.00000	0.264761
$914$	22.0000	0.727695
$915$	−6.00000	−0.198354
$916$	−22.0000	−0.726900
$917$	−32.0000	−1.05673
$918$	−1.00000	−0.0330049
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	−6.00000	−0.197814
$921$	4.00000	0.131804
$922$	16.0000	0.526932
$923$	8.00000	0.263323
$924$	−4.00000	−0.131590
$925$	−6.00000	−0.197279
$926$	16.0000	0.525793
$927$	20.0000	0.656886
$928$	−2.00000	−0.0656532
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	−8.00000	−0.262330
$931$	−18.0000	−0.589926
$932$	2.00000	0.0655122
$933$	0	0
$934$	−2.00000	−0.0654420
$935$	−1.00000	−0.0327035
$936$	−2.00000	−0.0653720
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	−24.0000	−0.783628
$939$	28.0000	0.913745
$940$	4.00000	0.130466
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	−8.00000	−0.260654
$943$	−24.0000	−0.781548
$944$	−12.0000	−0.390567
$945$	−4.00000	−0.130120
$946$	8.00000	0.260102
$947$	−52.0000	−1.68977	−0.844886	−	0.534946i	$-0.820332\pi$
$947$	−52.0000	−1.68977	−0.844886	+	0.534946i	$0.820332\pi$
$948$	−10.0000	−0.324785
$949$	4.00000	0.129845
$950$	−2.00000	−0.0648886
$951$	18.0000	0.583690
$952$	4.00000	0.129641
$953$	42.0000	1.36051	0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000	1.36051	0.680257	+	0.732974i	$0.261868\pi$
$954$	−12.0000	−0.388514
$955$	−24.0000	−0.776622
$956$	−12.0000	−0.388108
$957$	−2.00000	−0.0646508
$958$	10.0000	0.323085
$959$	−40.0000	−1.29167
$960$	1.00000	0.0322749
$961$	33.0000	1.06452
$962$	12.0000	0.386896
$963$	12.0000	0.386695
$964$	4.00000	0.128831
$965$	−10.0000	−0.321911
$966$	24.0000	0.772187
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	1.00000	0.0321412
$969$	2.00000	0.0642493
$970$	−12.0000	−0.385297
$971$	40.0000	1.28366	0.641831	−	0.766846i	$-0.278175\pi$
$971$	40.0000	1.28366	0.641831	+	0.766846i	$0.278175\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−22.0000	−0.704925
$975$	−2.00000	−0.0640513
$976$	−6.00000	−0.192055
$977$	2.00000	0.0639857	0.0319928	−	0.999488i	$-0.489815\pi$
$977$	2.00000	0.0639857	0.0319928	+	0.999488i	$0.489815\pi$
$978$	12.0000	0.383718
$979$	−10.0000	−0.319601
$980$	9.00000	0.287494
$981$	6.00000	0.191565
$982$	34.0000	1.08498
$983$	38.0000	1.21201	0.606006	−	0.795460i	$-0.292771\pi$
$983$	38.0000	1.21201	0.606006	+	0.795460i	$0.292771\pi$
$984$	4.00000	0.127515
$985$	−18.0000	−0.573528
$986$	2.00000	0.0636930
$987$	−16.0000	−0.509286
$988$	4.00000	0.127257
$989$	−48.0000	−1.52631
$990$	1.00000	0.0317821
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	−8.00000	−0.254000
$993$	−8.00000	−0.253872
$994$	16.0000	0.507489
$995$	−24.0000	−0.760851
$996$	8.00000	0.253490
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	−20.0000	−0.633089
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5610.2.a.bk.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5610.2.a.bk.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:	\( N \)	\(=\)	\( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5610.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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