LMFDB - Embedded newform 483.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [483,2,Mod(1,483)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(483, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("483.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 483 = 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	483.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:	$3.85677441763$
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	$\chi$	$=$	483.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +4.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +4.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} +8.00000 q^{10} -5.00000 q^{11} +2.00000 q^{12} -2.00000 q^{13} -2.00000 q^{14} +4.00000 q^{15} -4.00000 q^{16} +2.00000 q^{18} -5.00000 q^{19} +8.00000 q^{20} -1.00000 q^{21} -10.0000 q^{22} -1.00000 q^{23} +11.0000 q^{25} -4.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} -2.00000 q^{29} +8.00000 q^{30} +6.00000 q^{31} -8.00000 q^{32} -5.00000 q^{33} -4.00000 q^{35} +2.00000 q^{36} +6.00000 q^{37} -10.0000 q^{38} -2.00000 q^{39} +5.00000 q^{41} -2.00000 q^{42} +8.00000 q^{43} -10.0000 q^{44} +4.00000 q^{45} -2.00000 q^{46} -9.00000 q^{47} -4.00000 q^{48} +1.00000 q^{49} +22.0000 q^{50} -4.00000 q^{52} +9.00000 q^{53} +2.00000 q^{54} -20.0000 q^{55} -5.00000 q^{57} -4.00000 q^{58} +9.00000 q^{59} +8.00000 q^{60} -5.00000 q^{61} +12.0000 q^{62} -1.00000 q^{63} -8.00000 q^{64} -8.00000 q^{65} -10.0000 q^{66} +4.00000 q^{67} -1.00000 q^{69} -8.00000 q^{70} +12.0000 q^{71} +12.0000 q^{74} +11.0000 q^{75} -10.0000 q^{76} +5.00000 q^{77} -4.00000 q^{78} -10.0000 q^{79} -16.0000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -18.0000 q^{83} -2.00000 q^{84} +16.0000 q^{86} -2.00000 q^{87} +10.0000 q^{89} +8.00000 q^{90} +2.00000 q^{91} -2.00000 q^{92} +6.00000 q^{93} -18.0000 q^{94} -20.0000 q^{95} -8.00000 q^{96} -18.0000 q^{97} +2.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.00000	1.00000
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	2.00000	0.816497
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	8.00000	2.52982
$11$	−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
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−2.00000	−0.534522
$15$	4.00000	1.03280
$16$	−4.00000	−1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	2.00000	0.471405
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	8.00000	1.78885
$21$	−1.00000	−0.218218
$22$	−10.0000	−2.13201
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	11.0000	2.20000
$26$	−4.00000	−0.784465
$27$	1.00000	0.192450
$28$	−2.00000	−0.377964
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	8.00000	1.46059
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−8.00000	−1.41421
$33$	−5.00000	−0.870388
$34$	0	0
$35$	−4.00000	−0.676123
$36$	2.00000	0.333333
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−10.0000	−1.62221
$39$	−2.00000	−0.320256
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	−2.00000	−0.308607
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−10.0000	−1.50756
$45$	4.00000	0.596285
$46$	−2.00000	−0.294884
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	−4.00000	−0.577350
$49$	1.00000	0.142857
$50$	22.0000	3.11127
$51$	0	0
$52$	−4.00000	−0.554700
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	2.00000	0.272166
$55$	−20.0000	−2.69680
$56$	0	0
$57$	−5.00000	−0.662266
$58$	−4.00000	−0.525226
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	8.00000	1.03280
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	12.0000	1.52400
$63$	−1.00000	−0.125988
$64$	−8.00000	−1.00000
$65$	−8.00000	−0.992278
$66$	−10.0000	−1.23091
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	−8.00000	−0.956183
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	12.0000	1.39497
$75$	11.0000	1.27017
$76$	−10.0000	−1.14708
$77$	5.00000	0.569803
$78$	−4.00000	−0.452911
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	−16.0000	−1.78885
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	−18.0000	−1.97576	−0.987878	−	0.155230i	$-0.950388\pi$
$83$	−18.0000	−1.97576	−0.987878	+	0.155230i	$0.950388\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	16.0000	1.72532
$87$	−2.00000	−0.214423
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	8.00000	0.843274
$91$	2.00000	0.209657
$92$	−2.00000	−0.208514
$93$	6.00000	0.622171
$94$	−18.0000	−1.85656
$95$	−20.0000	−2.05196
$96$	−8.00000	−0.816497
$97$	−18.0000	−1.82762	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−18.0000	−1.82762	−0.913812	+	0.406138i	$0.866875\pi$
$98$	2.00000	0.202031
$99$	−5.00000	−0.502519
$100$	22.0000	2.20000
$101$	5.00000	0.497519	0.248759	−	0.968565i	$-0.419977\pi$
$101$	5.00000	0.497519	0.248759	+	0.968565i	$0.419977\pi$
$102$	0	0
$103$	−19.0000	−1.87213	−0.936063	−	0.351833i	$-0.885559\pi$
$103$	−19.0000	−1.87213	−0.936063	+	0.351833i	$0.885559\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	18.0000	1.74831
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	2.00000	0.192450
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	−40.0000	−3.81385
$111$	6.00000	0.569495
$112$	4.00000	0.377964
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	−10.0000	−0.936586
$115$	−4.00000	−0.373002
$116$	−4.00000	−0.371391
$117$	−2.00000	−0.184900
$118$	18.0000	1.65703
$119$	0	0
$120$	0	0
$121$	14.0000	1.27273
$122$	−10.0000	−0.905357
$123$	5.00000	0.450835
$124$	12.0000	1.07763
$125$	24.0000	2.14663
$126$	−2.00000	−0.178174
$127$	9.00000	0.798621	0.399310	−	0.916816i	$-0.369250\pi$
$127$	9.00000	0.798621	0.399310	+	0.916816i	$0.369250\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	−16.0000	−1.40329
$131$	−5.00000	−0.436852	−0.218426	−	0.975854i	$-0.570092\pi$
$131$	−5.00000	−0.436852	−0.218426	+	0.975854i	$0.570092\pi$
$132$	−10.0000	−0.870388
$133$	5.00000	0.433555
$134$	8.00000	0.691095
$135$	4.00000	0.344265
$136$	0	0
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	−2.00000	−0.170251
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	−8.00000	−0.676123
$141$	−9.00000	−0.757937
$142$	24.0000	2.01404
$143$	10.0000	0.836242
$144$	−4.00000	−0.333333
$145$	−8.00000	−0.664364
$146$	0	0
$147$	1.00000	0.0824786
$148$	12.0000	0.986394
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	22.0000	1.79629
$151$	−19.0000	−1.54620	−0.773099	−	0.634285i	$-0.781294\pi$
$151$	−19.0000	−1.54620	−0.773099	+	0.634285i	$0.781294\pi$
$152$	0	0
$153$	0	0
$154$	10.0000	0.805823
$155$	24.0000	1.92773
$156$	−4.00000	−0.320256
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	−20.0000	−1.59111
$159$	9.00000	0.713746
$160$	−32.0000	−2.52982
$161$	1.00000	0.0788110
$162$	2.00000	0.157135
$163$	13.0000	1.01824	0.509119	−	0.860696i	$-0.329971\pi$
$163$	13.0000	1.01824	0.509119	+	0.860696i	$0.329971\pi$
$164$	10.0000	0.780869
$165$	−20.0000	−1.55700
$166$	−36.0000	−2.79414
$167$	19.0000	1.47026	0.735132	−	0.677924i	$-0.237120\pi$
$167$	19.0000	1.47026	0.735132	+	0.677924i	$0.237120\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−5.00000	−0.382360
$172$	16.0000	1.21999
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	−4.00000	−0.303239
$175$	−11.0000	−0.831522
$176$	20.0000	1.50756
$177$	9.00000	0.676481
$178$	20.0000	1.49906
$179$	8.00000	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$180$	8.00000	0.596285
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	4.00000	0.296500
$183$	−5.00000	−0.369611
$184$	0	0
$185$	24.0000	1.76452
$186$	12.0000	0.879883
$187$	0	0
$188$	−18.0000	−1.31278
$189$	−1.00000	−0.0727393
$190$	−40.0000	−2.90191
$191$	−23.0000	−1.66422	−0.832111	−	0.554609i	$-0.812868\pi$
$191$	−23.0000	−1.66422	−0.832111	+	0.554609i	$0.812868\pi$
$192$	−8.00000	−0.577350
$193$	−19.0000	−1.36765	−0.683825	−	0.729646i	$-0.739685\pi$
$193$	−19.0000	−1.36765	−0.683825	+	0.729646i	$0.739685\pi$
$194$	−36.0000	−2.58465
$195$	−8.00000	−0.572892
$196$	2.00000	0.142857
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	−10.0000	−0.710669
$199$	3.00000	0.212664	0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000	0.212664	0.106332	+	0.994331i	$0.466089\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	10.0000	0.703598
$203$	2.00000	0.140372
$204$	0	0
$205$	20.0000	1.39686
$206$	−38.0000	−2.64759
$207$	−1.00000	−0.0695048
$208$	8.00000	0.554700
$209$	25.0000	1.72929
$210$	−8.00000	−0.552052
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	18.0000	1.23625
$213$	12.0000	0.822226
$214$	8.00000	0.546869
$215$	32.0000	2.18238
$216$	0	0
$217$	−6.00000	−0.407307
$218$	−24.0000	−1.62549
$219$	0	0
$220$	−40.0000	−2.69680
$221$	0	0
$222$	12.0000	0.805387
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	8.00000	0.534522
$225$	11.0000	0.733333
$226$	−4.00000	−0.266076
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	−10.0000	−0.662266
$229$	−13.0000	−0.859064	−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000	−0.859064	−0.429532	+	0.903052i	$0.641321\pi$
$230$	−8.00000	−0.527504
$231$	5.00000	0.328976
$232$	0	0
$233$	−12.0000	−0.786146	−0.393073	−	0.919507i	$-0.628588\pi$
$233$	−12.0000	−0.786146	−0.393073	+	0.919507i	$0.628588\pi$
$234$	−4.00000	−0.261488
$235$	−36.0000	−2.34838
$236$	18.0000	1.17170
$237$	−10.0000	−0.649570
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	−16.0000	−1.03280
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	28.0000	1.79991
$243$	1.00000	0.0641500
$244$	−10.0000	−0.640184
$245$	4.00000	0.255551
$246$	10.0000	0.637577
$247$	10.0000	0.636285
$248$	0	0
$249$	−18.0000	−1.14070
$250$	48.0000	3.03579
$251$	10.0000	0.631194	0.315597	−	0.948893i	$-0.397795\pi$
$251$	10.0000	0.631194	0.315597	+	0.948893i	$0.397795\pi$
$252$	−2.00000	−0.125988
$253$	5.00000	0.314347
$254$	18.0000	1.12942
$255$	0	0
$256$	16.0000	1.00000
$257$	17.0000	1.06043	0.530215	−	0.847863i	$-0.322111\pi$
$257$	17.0000	1.06043	0.530215	+	0.847863i	$0.322111\pi$
$258$	16.0000	0.996116
$259$	−6.00000	−0.372822
$260$	−16.0000	−0.992278
$261$	−2.00000	−0.123797
$262$	−10.0000	−0.617802
$263$	7.00000	0.431638	0.215819	−	0.976433i	$-0.430758\pi$
$263$	7.00000	0.431638	0.215819	+	0.976433i	$0.430758\pi$
$264$	0	0
$265$	36.0000	2.21146
$266$	10.0000	0.613139
$267$	10.0000	0.611990
$268$	8.00000	0.488678
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	8.00000	0.486864
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	18.0000	1.08742
$275$	−55.0000	−3.31662
$276$	−2.00000	−0.120386
$277$	7.00000	0.420589	0.210295	−	0.977638i	$-0.432558\pi$
$277$	7.00000	0.420589	0.210295	+	0.977638i	$0.432558\pi$
$278$	24.0000	1.43942
$279$	6.00000	0.359211
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	−18.0000	−1.07188
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	24.0000	1.42414
$285$	−20.0000	−1.18470
$286$	20.0000	1.18262
$287$	−5.00000	−0.295141
$288$	−8.00000	−0.471405
$289$	−17.0000	−1.00000
$290$	−16.0000	−0.939552
$291$	−18.0000	−1.05518
$292$	0	0
$293$	−4.00000	−0.233682	−0.116841	−	0.993151i	$-0.537277\pi$
$293$	−4.00000	−0.233682	−0.116841	+	0.993151i	$0.537277\pi$
$294$	2.00000	0.116642
$295$	36.0000	2.09600
$296$	0	0
$297$	−5.00000	−0.290129
$298$	6.00000	0.347571
$299$	2.00000	0.115663
$300$	22.0000	1.27017
$301$	−8.00000	−0.461112
$302$	−38.0000	−2.18665
$303$	5.00000	0.287242
$304$	20.0000	1.14708
$305$	−20.0000	−1.14520
$306$	0	0
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	10.0000	0.569803
$309$	−19.0000	−1.08087
$310$	48.0000	2.72622
$311$	−15.0000	−0.850572	−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000	−0.850572	−0.425286	+	0.905059i	$0.639826\pi$
$312$	0	0
$313$	−25.0000	−1.41308	−0.706542	−	0.707671i	$-0.749746\pi$
$313$	−25.0000	−1.41308	−0.706542	+	0.707671i	$0.749746\pi$
$314$	14.0000	0.790066
$315$	−4.00000	−0.225374
$316$	−20.0000	−1.12509
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	18.0000	1.00939
$319$	10.0000	0.559893
$320$	−32.0000	−1.78885
$321$	4.00000	0.223258
$322$	2.00000	0.111456
$323$	0	0
$324$	2.00000	0.111111
$325$	−22.0000	−1.22034
$326$	26.0000	1.44001
$327$	−12.0000	−0.663602
$328$	0	0
$329$	9.00000	0.496186
$330$	−40.0000	−2.20193
$331$	29.0000	1.59398	0.796992	−	0.603990i	$-0.206423\pi$
$331$	29.0000	1.59398	0.796992	+	0.603990i	$0.206423\pi$
$332$	−36.0000	−1.97576
$333$	6.00000	0.328798
$334$	38.0000	2.07927
$335$	16.0000	0.874173
$336$	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	$0.295482\pi$
$338$	−18.0000	−0.979071
$339$	−2.00000	−0.108625
$340$	0	0
$341$	−30.0000	−1.62459
$342$	−10.0000	−0.540738
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−4.00000	−0.215353
$346$	4.00000	0.215041
$347$	2.00000	0.107366	0.0536828	−	0.998558i	$-0.482904\pi$
$347$	2.00000	0.107366	0.0536828	+	0.998558i	$0.482904\pi$
$348$	−4.00000	−0.214423
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	−22.0000	−1.17595
$351$	−2.00000	−0.106752
$352$	40.0000	2.13201
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	18.0000	0.956689
$355$	48.0000	2.54758
$356$	20.0000	1.06000
$357$	0	0
$358$	16.0000	0.845626
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	−28.0000	−1.47165
$363$	14.0000	0.734809
$364$	4.00000	0.209657
$365$	0	0
$366$	−10.0000	−0.522708
$367$	27.0000	1.40939	0.704694	−	0.709511i	$-0.251084\pi$
$367$	27.0000	1.40939	0.704694	+	0.709511i	$0.251084\pi$
$368$	4.00000	0.208514
$369$	5.00000	0.260290
$370$	48.0000	2.49540
$371$	−9.00000	−0.467257
$372$	12.0000	0.622171
$373$	−34.0000	−1.76045	−0.880227	−	0.474554i	$-0.842610\pi$
$373$	−34.0000	−1.76045	−0.880227	+	0.474554i	$0.842610\pi$
$374$	0	0
$375$	24.0000	1.23935
$376$	0	0
$377$	4.00000	0.206010
$378$	−2.00000	−0.102869
$379$	−30.0000	−1.54100	−0.770498	−	0.637442i	$-0.779993\pi$
$379$	−30.0000	−1.54100	−0.770498	+	0.637442i	$0.779993\pi$
$380$	−40.0000	−2.05196
$381$	9.00000	0.461084
$382$	−46.0000	−2.35356
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	0	0
$385$	20.0000	1.01929
$386$	−38.0000	−1.93415
$387$	8.00000	0.406663
$388$	−36.0000	−1.82762
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−16.0000	−0.810191
$391$	0	0
$392$	0	0
$393$	−5.00000	−0.252217
$394$	−4.00000	−0.201517
$395$	−40.0000	−2.01262
$396$	−10.0000	−0.502519
$397$	30.0000	1.50566	0.752828	−	0.658217i	$-0.228689\pi$
$397$	30.0000	1.50566	0.752828	+	0.658217i	$0.228689\pi$
$398$	6.00000	0.300753
$399$	5.00000	0.250313
$400$	−44.0000	−2.20000
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	8.00000	0.399004
$403$	−12.0000	−0.597763
$404$	10.0000	0.497519
$405$	4.00000	0.198762
$406$	4.00000	0.198517
$407$	−30.0000	−1.48704
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	40.0000	1.97546
$411$	9.00000	0.443937
$412$	−38.0000	−1.87213
$413$	−9.00000	−0.442861
$414$	−2.00000	−0.0982946
$415$	−72.0000	−3.53434
$416$	16.0000	0.784465
$417$	12.0000	0.587643
$418$	50.0000	2.44558
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	−8.00000	−0.390360
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	26.0000	1.26566
$423$	−9.00000	−0.437595
$424$	0	0
$425$	0	0
$426$	24.0000	1.16280
$427$	5.00000	0.241967
$428$	8.00000	0.386695
$429$	10.0000	0.482805
$430$	64.0000	3.08635
$431$	27.0000	1.30054	0.650272	−	0.759701i	$-0.274655\pi$
$431$	27.0000	1.30054	0.650272	+	0.759701i	$0.274655\pi$
$432$	−4.00000	−0.192450
$433$	−11.0000	−0.528626	−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	−12.0000	−0.576018
$435$	−8.00000	−0.383571
$436$	−24.0000	−1.14939
$437$	5.00000	0.239182
$438$	0	0
$439$	−36.0000	−1.71819	−0.859093	−	0.511819i	$-0.828972\pi$
$439$	−36.0000	−1.71819	−0.859093	+	0.511819i	$0.828972\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	12.0000	0.569495
$445$	40.0000	1.89618
$446$	20.0000	0.947027
$447$	3.00000	0.141895
$448$	8.00000	0.377964
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	22.0000	1.03709
$451$	−25.0000	−1.17720
$452$	−4.00000	−0.188144
$453$	−19.0000	−0.892698
$454$	−36.0000	−1.68956
$455$	8.00000	0.375046
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	−26.0000	−1.21490
$459$	0	0
$460$	−8.00000	−0.373002
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	10.0000	0.465242
$463$	−35.0000	−1.62659	−0.813294	−	0.581853i	$-0.802328\pi$
$463$	−35.0000	−1.62659	−0.813294	+	0.581853i	$0.802328\pi$
$464$	8.00000	0.371391
$465$	24.0000	1.11297
$466$	−24.0000	−1.11178
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	−4.00000	−0.184900
$469$	−4.00000	−0.184703
$470$	−72.0000	−3.32111
$471$	7.00000	0.322543
$472$	0	0
$473$	−40.0000	−1.83920
$474$	−20.0000	−0.918630
$475$	−55.0000	−2.52357
$476$	0	0
$477$	9.00000	0.412082
$478$	24.0000	1.09773
$479$	32.0000	1.46212	0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000	1.46212	0.731059	+	0.682315i	$0.239027\pi$
$480$	−32.0000	−1.46059
$481$	−12.0000	−0.547153
$482$	34.0000	1.54866
$483$	1.00000	0.0455016
$484$	28.0000	1.27273
$485$	−72.0000	−3.26935
$486$	2.00000	0.0907218
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	0	0
$489$	13.0000	0.587880
$490$	8.00000	0.361403
$491$	14.0000	0.631811	0.315906	−	0.948791i	$-0.397692\pi$
$491$	14.0000	0.631811	0.315906	+	0.948791i	$0.397692\pi$
$492$	10.0000	0.450835
$493$	0	0
$494$	20.0000	0.899843
$495$	−20.0000	−0.898933
$496$	−24.0000	−1.07763
$497$	−12.0000	−0.538274
$498$	−36.0000	−1.61320
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	48.0000	2.14663
$501$	19.0000	0.848857
$502$	20.0000	0.892644
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	0	0
$505$	20.0000	0.889988
$506$	10.0000	0.444554
$507$	−9.00000	−0.399704
$508$	18.0000	0.798621
$509$	−19.0000	−0.842160	−0.421080	−	0.907023i	$-0.638349\pi$
$509$	−19.0000	−0.842160	−0.421080	+	0.907023i	$0.638349\pi$
$510$	0	0
$511$	0	0
$512$	32.0000	1.41421
$513$	−5.00000	−0.220755
$514$	34.0000	1.49968
$515$	−76.0000	−3.34896
$516$	16.0000	0.704361
$517$	45.0000	1.97910
$518$	−12.0000	−0.527250
$519$	2.00000	0.0877903
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	−4.00000	−0.175075
$523$	19.0000	0.830812	0.415406	−	0.909636i	$-0.363640\pi$
$523$	19.0000	0.830812	0.415406	+	0.909636i	$0.363640\pi$
$524$	−10.0000	−0.436852
$525$	−11.0000	−0.480079
$526$	14.0000	0.610429
$527$	0	0
$528$	20.0000	0.870388
$529$	1.00000	0.0434783
$530$	72.0000	3.12748
$531$	9.00000	0.390567
$532$	10.0000	0.433555
$533$	−10.0000	−0.433148
$534$	20.0000	0.865485
$535$	16.0000	0.691740
$536$	0	0
$537$	8.00000	0.345225
$538$	−36.0000	−1.55207
$539$	−5.00000	−0.215365
$540$	8.00000	0.344265
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	−16.0000	−0.687259
$543$	−14.0000	−0.600798
$544$	0	0
$545$	−48.0000	−2.05609
$546$	4.00000	0.171184
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	18.0000	0.768922
$549$	−5.00000	−0.213395
$550$	−110.000	−4.69042
$551$	10.0000	0.426014
$552$	0	0
$553$	10.0000	0.425243
$554$	14.0000	0.594803
$555$	24.0000	1.01874
$556$	24.0000	1.01783
$557$	46.0000	1.94908	0.974541	−	0.224208i	$-0.0719796\pi$
$557$	46.0000	1.94908	0.974541	+	0.224208i	$0.0719796\pi$
$558$	12.0000	0.508001
$559$	−16.0000	−0.676728
$560$	16.0000	0.676123
$561$	0	0
$562$	4.00000	0.168730
$563$	40.0000	1.68580	0.842900	−	0.538071i	$-0.180847\pi$
$563$	40.0000	1.68580	0.842900	+	0.538071i	$0.180847\pi$
$564$	−18.0000	−0.757937
$565$	−8.00000	−0.336563
$566$	−40.0000	−1.68133
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−17.0000	−0.712677	−0.356339	−	0.934357i	$-0.615975\pi$
$569$	−17.0000	−0.712677	−0.356339	+	0.934357i	$0.615975\pi$
$570$	−40.0000	−1.67542
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	20.0000	0.836242
$573$	−23.0000	−0.960839
$574$	−10.0000	−0.417392
$575$	−11.0000	−0.458732
$576$	−8.00000	−0.333333
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	−34.0000	−1.41421
$579$	−19.0000	−0.789613
$580$	−16.0000	−0.664364
$581$	18.0000	0.746766
$582$	−36.0000	−1.49225
$583$	−45.0000	−1.86371
$584$	0	0
$585$	−8.00000	−0.330759
$586$	−8.00000	−0.330477
$587$	−17.0000	−0.701665	−0.350833	−	0.936438i	$-0.614101\pi$
$587$	−17.0000	−0.701665	−0.350833	+	0.936438i	$0.614101\pi$
$588$	2.00000	0.0824786
$589$	−30.0000	−1.23613
$590$	72.0000	2.96419
$591$	−2.00000	−0.0822690
$592$	−24.0000	−0.986394
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	−10.0000	−0.410305
$595$	0	0
$596$	6.00000	0.245770
$597$	3.00000	0.122782
$598$	4.00000	0.163572
$599$	42.0000	1.71607	0.858037	−	0.513588i	$-0.171684\pi$
$599$	42.0000	1.71607	0.858037	+	0.513588i	$0.171684\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	−16.0000	−0.652111
$603$	4.00000	0.162893
$604$	−38.0000	−1.54620
$605$	56.0000	2.27672
$606$	10.0000	0.406222
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	40.0000	1.62221
$609$	2.00000	0.0810441
$610$	−40.0000	−1.61955
$611$	18.0000	0.728202
$612$	0	0
$613$	34.0000	1.37325	0.686624	−	0.727013i	$-0.259092\pi$
$613$	34.0000	1.37325	0.686624	+	0.727013i	$0.259092\pi$
$614$	−16.0000	−0.645707
$615$	20.0000	0.806478
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	−38.0000	−1.52858
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	48.0000	1.92773
$621$	−1.00000	−0.0401286
$622$	−30.0000	−1.20289
$623$	−10.0000	−0.400642
$624$	8.00000	0.320256
$625$	41.0000	1.64000
$626$	−50.0000	−1.99840
$627$	25.0000	0.998404
$628$	14.0000	0.558661
$629$	0	0
$630$	−8.00000	−0.318728
$631$	−10.0000	−0.398094	−0.199047	−	0.979990i	$-0.563785\pi$
$631$	−10.0000	−0.398094	−0.199047	+	0.979990i	$0.563785\pi$
$632$	0	0
$633$	13.0000	0.516704
$634$	60.0000	2.38290
$635$	36.0000	1.42862
$636$	18.0000	0.713746
$637$	−2.00000	−0.0792429
$638$	20.0000	0.791808
$639$	12.0000	0.474713
$640$	0	0
$641$	−9.00000	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$641$	−9.00000	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$642$	8.00000	0.315735
$643$	−31.0000	−1.22252	−0.611260	−	0.791430i	$-0.709337\pi$
$643$	−31.0000	−1.22252	−0.611260	+	0.791430i	$0.709337\pi$
$644$	2.00000	0.0788110
$645$	32.0000	1.26000
$646$	0	0
$647$	−28.0000	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$648$	0	0
$649$	−45.0000	−1.76640
$650$	−44.0000	−1.72582
$651$	−6.00000	−0.235159
$652$	26.0000	1.01824
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	−24.0000	−0.938474
$655$	−20.0000	−0.781465
$656$	−20.0000	−0.780869
$657$	0	0
$658$	18.0000	0.701713
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	−40.0000	−1.55700
$661$	−5.00000	−0.194477	−0.0972387	−	0.995261i	$-0.531001\pi$
$661$	−5.00000	−0.194477	−0.0972387	+	0.995261i	$0.531001\pi$
$662$	58.0000	2.25423
$663$	0	0
$664$	0	0
$665$	20.0000	0.775567
$666$	12.0000	0.464991
$667$	2.00000	0.0774403
$668$	38.0000	1.47026
$669$	10.0000	0.386622
$670$	32.0000	1.23627
$671$	25.0000	0.965114
$672$	8.00000	0.308607
$673$	49.0000	1.88881	0.944406	−	0.328783i	$-0.106638\pi$
$673$	49.0000	1.88881	0.944406	+	0.328783i	$0.106638\pi$
$674$	44.0000	1.69482
$675$	11.0000	0.423390
$676$	−18.0000	−0.692308
$677$	36.0000	1.38359	0.691796	−	0.722093i	$-0.256820\pi$
$677$	36.0000	1.38359	0.691796	+	0.722093i	$0.256820\pi$
$678$	−4.00000	−0.153619
$679$	18.0000	0.690777
$680$	0	0
$681$	−18.0000	−0.689761
$682$	−60.0000	−2.29752
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	−10.0000	−0.382360
$685$	36.0000	1.37549
$686$	−2.00000	−0.0763604
$687$	−13.0000	−0.495981
$688$	−32.0000	−1.21999
$689$	−18.0000	−0.685745
$690$	−8.00000	−0.304555
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	4.00000	0.152057
$693$	5.00000	0.189934
$694$	4.00000	0.151838
$695$	48.0000	1.82074
$696$	0	0
$697$	0	0
$698$	−4.00000	−0.151402
$699$	−12.0000	−0.453882
$700$	−22.0000	−0.831522
$701$	−15.0000	−0.566542	−0.283271	−	0.959040i	$-0.591420\pi$
$701$	−15.0000	−0.566542	−0.283271	+	0.959040i	$0.591420\pi$
$702$	−4.00000	−0.150970
$703$	−30.0000	−1.13147
$704$	40.0000	1.50756
$705$	−36.0000	−1.35584
$706$	−28.0000	−1.05379
$707$	−5.00000	−0.188044
$708$	18.0000	0.676481
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	96.0000	3.60282
$711$	−10.0000	−0.375029
$712$	0	0
$713$	−6.00000	−0.224702
$714$	0	0
$715$	40.0000	1.49592
$716$	16.0000	0.597948
$717$	12.0000	0.448148
$718$	64.0000	2.38846
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	−16.0000	−0.596285
$721$	19.0000	0.707597
$722$	12.0000	0.446594
$723$	17.0000	0.632237
$724$	−28.0000	−1.04061
$725$	−22.0000	−0.817059
$726$	28.0000	1.03918
$727$	41.0000	1.52061	0.760303	−	0.649569i	$-0.225051\pi$
$727$	41.0000	1.52061	0.760303	+	0.649569i	$0.225051\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	−10.0000	−0.369611
$733$	−6.00000	−0.221615	−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000	−0.221615	−0.110808	+	0.993842i	$0.535344\pi$
$734$	54.0000	1.99318
$735$	4.00000	0.147542
$736$	8.00000	0.294884
$737$	−20.0000	−0.736709
$738$	10.0000	0.368105
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	48.0000	1.76452
$741$	10.0000	0.367359
$742$	−18.0000	−0.660801
$743$	15.0000	0.550297	0.275148	−	0.961402i	$-0.411273\pi$
$743$	15.0000	0.550297	0.275148	+	0.961402i	$0.411273\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	−68.0000	−2.48966
$747$	−18.0000	−0.658586
$748$	0	0
$749$	−4.00000	−0.146157
$750$	48.0000	1.75271
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	36.0000	1.31278
$753$	10.0000	0.364420
$754$	8.00000	0.291343
$755$	−76.0000	−2.76592
$756$	−2.00000	−0.0727393
$757$	8.00000	0.290765	0.145382	−	0.989376i	$-0.453559\pi$
$757$	8.00000	0.290765	0.145382	+	0.989376i	$0.453559\pi$
$758$	−60.0000	−2.17930
$759$	5.00000	0.181489
$760$	0	0
$761$	−21.0000	−0.761249	−0.380625	−	0.924730i	$-0.624291\pi$
$761$	−21.0000	−0.761249	−0.380625	+	0.924730i	$0.624291\pi$
$762$	18.0000	0.652071
$763$	12.0000	0.434429
$764$	−46.0000	−1.66422
$765$	0	0
$766$	12.0000	0.433578
$767$	−18.0000	−0.649942
$768$	16.0000	0.577350
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	40.0000	1.44150
$771$	17.0000	0.612240
$772$	−38.0000	−1.36765
$773$	20.0000	0.719350	0.359675	−	0.933078i	$-0.382888\pi$
$773$	20.0000	0.719350	0.359675	+	0.933078i	$0.382888\pi$
$774$	16.0000	0.575108
$775$	66.0000	2.37079
$776$	0	0
$777$	−6.00000	−0.215249
$778$	12.0000	0.430221
$779$	−25.0000	−0.895718
$780$	−16.0000	−0.572892
$781$	−60.0000	−2.14697
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	−4.00000	−0.142857
$785$	28.0000	0.999363
$786$	−10.0000	−0.356688
$787$	−23.0000	−0.819861	−0.409931	−	0.912117i	$-0.634447\pi$
$787$	−23.0000	−0.819861	−0.409931	+	0.912117i	$0.634447\pi$
$788$	−4.00000	−0.142494
$789$	7.00000	0.249207
$790$	−80.0000	−2.84627
$791$	2.00000	0.0711118
$792$	0	0
$793$	10.0000	0.355110
$794$	60.0000	2.12932
$795$	36.0000	1.27679
$796$	6.00000	0.212664
$797$	46.0000	1.62940	0.814702	−	0.579880i	$-0.196901\pi$
$797$	46.0000	1.62940	0.814702	+	0.579880i	$0.196901\pi$
$798$	10.0000	0.353996
$799$	0	0
$800$	−88.0000	−3.11127
$801$	10.0000	0.353333
$802$	6.00000	0.211867
$803$	0	0
$804$	8.00000	0.282138
$805$	4.00000	0.140981
$806$	−24.0000	−0.845364
$807$	−18.0000	−0.633630
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	8.00000	0.281091
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	4.00000	0.140372
$813$	−8.00000	−0.280572
$814$	−60.0000	−2.10300
$815$	52.0000	1.82148
$816$	0	0
$817$	−40.0000	−1.39942
$818$	28.0000	0.978997
$819$	2.00000	0.0698857
$820$	40.0000	1.39686
$821$	−4.00000	−0.139601	−0.0698005	−	0.997561i	$-0.522236\pi$
$821$	−4.00000	−0.139601	−0.0698005	+	0.997561i	$0.522236\pi$
$822$	18.0000	0.627822
$823$	27.0000	0.941161	0.470580	−	0.882357i	$-0.344045\pi$
$823$	27.0000	0.941161	0.470580	+	0.882357i	$0.344045\pi$
$824$	0	0
$825$	−55.0000	−1.91485
$826$	−18.0000	−0.626300
$827$	−21.0000	−0.730242	−0.365121	−	0.930960i	$-0.618972\pi$
$827$	−21.0000	−0.730242	−0.365121	+	0.930960i	$0.618972\pi$
$828$	−2.00000	−0.0695048
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	−144.000	−4.99831
$831$	7.00000	0.242827
$832$	16.0000	0.554700
$833$	0	0
$834$	24.0000	0.831052
$835$	76.0000	2.63009
$836$	50.0000	1.72929
$837$	6.00000	0.207390
$838$	−36.0000	−1.24360
$839$	46.0000	1.58810	0.794048	−	0.607855i	$-0.207970\pi$
$839$	46.0000	1.58810	0.794048	+	0.607855i	$0.207970\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	16.0000	0.551396
$843$	2.00000	0.0688837
$844$	26.0000	0.894957
$845$	−36.0000	−1.23844
$846$	−18.0000	−0.618853
$847$	−14.0000	−0.481046
$848$	−36.0000	−1.23625
$849$	−20.0000	−0.686398
$850$	0	0
$851$	−6.00000	−0.205677
$852$	24.0000	0.822226
$853$	8.00000	0.273915	0.136957	−	0.990577i	$-0.456268\pi$
$853$	8.00000	0.273915	0.136957	+	0.990577i	$0.456268\pi$
$854$	10.0000	0.342193
$855$	−20.0000	−0.683986
$856$	0	0
$857$	1.00000	0.0341593	0.0170797	−	0.999854i	$-0.494563\pi$
$857$	1.00000	0.0341593	0.0170797	+	0.999854i	$0.494563\pi$
$858$	20.0000	0.682789
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	64.0000	2.18238
$861$	−5.00000	−0.170400
$862$	54.0000	1.83925
$863$	−30.0000	−1.02121	−0.510606	−	0.859815i	$-0.670579\pi$
$863$	−30.0000	−1.02121	−0.510606	+	0.859815i	$0.670579\pi$
$864$	−8.00000	−0.272166
$865$	8.00000	0.272008
$866$	−22.0000	−0.747590
$867$	−17.0000	−0.577350
$868$	−12.0000	−0.407307
$869$	50.0000	1.69613
$870$	−16.0000	−0.542451
$871$	−8.00000	−0.271070
$872$	0	0
$873$	−18.0000	−0.609208
$874$	10.0000	0.338255
$875$	−24.0000	−0.811348
$876$	0	0
$877$	−17.0000	−0.574049	−0.287025	−	0.957923i	$-0.592666\pi$
$877$	−17.0000	−0.574049	−0.287025	+	0.957923i	$0.592666\pi$
$878$	−72.0000	−2.42988
$879$	−4.00000	−0.134917
$880$	80.0000	2.69680
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	2.00000	0.0673435
$883$	32.0000	1.07689	0.538443	−	0.842662i	$-0.319013\pi$
$883$	32.0000	1.07689	0.538443	+	0.842662i	$0.319013\pi$
$884$	0	0
$885$	36.0000	1.21013
$886$	−72.0000	−2.41889
$887$	−16.0000	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$887$	−16.0000	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$888$	0	0
$889$	−9.00000	−0.301850
$890$	80.0000	2.68161
$891$	−5.00000	−0.167506
$892$	20.0000	0.669650
$893$	45.0000	1.50587
$894$	6.00000	0.200670
$895$	32.0000	1.06964
$896$	0	0
$897$	2.00000	0.0667781
$898$	12.0000	0.400445
$899$	−12.0000	−0.400222
$900$	22.0000	0.733333
$901$	0	0
$902$	−50.0000	−1.66482
$903$	−8.00000	−0.266223
$904$	0	0
$905$	−56.0000	−1.86150
$906$	−38.0000	−1.26247
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	−36.0000	−1.19470
$909$	5.00000	0.165840
$910$	16.0000	0.530395
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	20.0000	0.662266
$913$	90.0000	2.97857
$914$	20.0000	0.661541
$915$	−20.0000	−0.661180
$916$	−26.0000	−0.859064
$917$	5.00000	0.165115
$918$	0	0
$919$	2.00000	0.0659739	0.0329870	−	0.999456i	$-0.489498\pi$
$919$	2.00000	0.0659739	0.0329870	+	0.999456i	$0.489498\pi$
$920$	0	0
$921$	−8.00000	−0.263609
$922$	4.00000	0.131733
$923$	−24.0000	−0.789970
$924$	10.0000	0.328976
$925$	66.0000	2.17007
$926$	−70.0000	−2.30034
$927$	−19.0000	−0.624042
$928$	16.0000	0.525226
$929$	26.0000	0.853032	0.426516	−	0.904480i	$-0.359741\pi$
$929$	26.0000	0.853032	0.426516	+	0.904480i	$0.359741\pi$
$930$	48.0000	1.57398
$931$	−5.00000	−0.163868
$932$	−24.0000	−0.786146
$933$	−15.0000	−0.491078
$934$	−12.0000	−0.392652
$935$	0	0
$936$	0	0
$937$	−27.0000	−0.882052	−0.441026	−	0.897494i	$-0.645385\pi$
$937$	−27.0000	−0.882052	−0.441026	+	0.897494i	$0.645385\pi$
$938$	−8.00000	−0.261209
$939$	−25.0000	−0.815844
$940$	−72.0000	−2.34838
$941$	20.0000	0.651981	0.325991	−	0.945373i	$-0.394302\pi$
$941$	20.0000	0.651981	0.325991	+	0.945373i	$0.394302\pi$
$942$	14.0000	0.456145
$943$	−5.00000	−0.162822
$944$	−36.0000	−1.17170
$945$	−4.00000	−0.130120
$946$	−80.0000	−2.60102
$947$	32.0000	1.03986	0.519930	−	0.854209i	$-0.325958\pi$
$947$	32.0000	1.03986	0.519930	+	0.854209i	$0.325958\pi$
$948$	−20.0000	−0.649570
$949$	0	0
$950$	−110.000	−3.56887
$951$	30.0000	0.972817
$952$	0	0
$953$	−55.0000	−1.78162	−0.890812	−	0.454371i	$-0.849864\pi$
$953$	−55.0000	−1.78162	−0.890812	+	0.454371i	$0.849864\pi$
$954$	18.0000	0.582772
$955$	−92.0000	−2.97705
$956$	24.0000	0.776215
$957$	10.0000	0.323254
$958$	64.0000	2.06775
$959$	−9.00000	−0.290625
$960$	−32.0000	−1.03280
$961$	5.00000	0.161290
$962$	−24.0000	−0.773791
$963$	4.00000	0.128898
$964$	34.0000	1.09507
$965$	−76.0000	−2.44653
$966$	2.00000	0.0643489
$967$	28.0000	0.900419	0.450210	−	0.892923i	$-0.351349\pi$
$967$	28.0000	0.900419	0.450210	+	0.892923i	$0.351349\pi$
$968$	0	0
$969$	0	0
$970$	−144.000	−4.62356
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	2.00000	0.0641500
$973$	−12.0000	−0.384702
$974$	−40.0000	−1.28168
$975$	−22.0000	−0.704564
$976$	20.0000	0.640184
$977$	−3.00000	−0.0959785	−0.0479893	−	0.998848i	$-0.515281\pi$
$977$	−3.00000	−0.0959785	−0.0479893	+	0.998848i	$0.515281\pi$
$978$	26.0000	0.831388
$979$	−50.0000	−1.59801
$980$	8.00000	0.255551
$981$	−12.0000	−0.383131
$982$	28.0000	0.893516
$983$	42.0000	1.33959	0.669796	−	0.742545i	$-0.266382\pi$
$983$	42.0000	1.33959	0.669796	+	0.742545i	$0.266382\pi$
$984$	0	0
$985$	−8.00000	−0.254901
$986$	0	0
$987$	9.00000	0.286473
$988$	20.0000	0.636285
$989$	−8.00000	−0.254385
$990$	−40.0000	−1.27128
$991$	5.00000	0.158830	0.0794151	−	0.996842i	$-0.474695\pi$
$991$	5.00000	0.158830	0.0794151	+	0.996842i	$0.474695\pi$
$992$	−48.0000	−1.52400
$993$	29.0000	0.920287
$994$	−24.0000	−0.761234
$995$	12.0000	0.380426
$996$	−36.0000	−1.14070
$997$	−56.0000	−1.77354	−0.886769	−	0.462213i	$-0.847056\pi$
$997$	−56.0000	−1.77354	−0.886769	+	0.462213i	$0.847056\pi$
$998$	−40.0000	−1.26618
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.a.b.1.1	✓	1
3.2	odd	2		1449.2.a.a.1.1		1
4.3	odd	2		7728.2.a.l.1.1		1
7.6	odd	2		3381.2.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.b.1.1	✓	1	1.1	even	1	trivial
1449.2.a.a.1.1		1	3.2	odd	2
3381.2.a.l.1.1		1	7.6	odd	2
7728.2.a.l.1.1		1	4.3	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:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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