LMFDB - Embedded newform 1690.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1690,2,Mod(1,1690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1690, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1690.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1690 = 2 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1690.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:	$13.4947179416$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1690.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.00000 q^{11} -2.00000 q^{12} +1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} -5.00000 q^{19} +1.00000 q^{20} -2.00000 q^{21} -3.00000 q^{22} -2.00000 q^{24} +1.00000 q^{25} +4.00000 q^{27} +1.00000 q^{28} -2.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +6.00000 q^{33} -6.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -11.0000 q^{37} -5.00000 q^{38} +1.00000 q^{40} -6.00000 q^{41} -2.00000 q^{42} +2.00000 q^{43} -3.00000 q^{44} +1.00000 q^{45} +3.00000 q^{47} -2.00000 q^{48} -6.00000 q^{49} +1.00000 q^{50} +12.0000 q^{51} -9.00000 q^{53} +4.00000 q^{54} -3.00000 q^{55} +1.00000 q^{56} +10.0000 q^{57} -2.00000 q^{60} +8.00000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +6.00000 q^{66} +16.0000 q^{67} -6.00000 q^{68} +1.00000 q^{70} -6.00000 q^{71} +1.00000 q^{72} -14.0000 q^{73} -11.0000 q^{74} -2.00000 q^{75} -5.00000 q^{76} -3.00000 q^{77} -16.0000 q^{79} +1.00000 q^{80} -11.0000 q^{81} -6.00000 q^{82} +6.00000 q^{83} -2.00000 q^{84} -6.00000 q^{85} +2.00000 q^{86} -3.00000 q^{88} -9.00000 q^{89} +1.00000 q^{90} -8.00000 q^{93} +3.00000 q^{94} -5.00000 q^{95} -2.00000 q^{96} +10.0000 q^{97} -6.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.00000	−0.816497
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	−2.00000	−0.577350
$13$	0	0
$13$	0	0
$14$	1.00000	0.267261
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	1.00000	0.235702
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	1.00000	0.223607
$21$	−2.00000	−0.436436
$22$	−3.00000	−0.639602
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−2.00000	−0.408248
$25$	1.00000	0.200000
$26$	0	0
$27$	4.00000	0.769800
$28$	1.00000	0.188982
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−2.00000	−0.365148
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	6.00000	1.04447
$34$	−6.00000	−1.02899
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	−11.0000	−1.80839	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−11.0000	−1.80839	−0.904194	+	0.427121i	$0.859528\pi$
$38$	−5.00000	−0.811107
$39$	0	0
$40$	1.00000	0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−2.00000	−0.308607
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	−3.00000	−0.452267
$45$	1.00000	0.149071
$46$	0	0
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	−2.00000	−0.288675
$49$	−6.00000	−0.857143
$50$	1.00000	0.141421
$51$	12.0000	1.68034
$52$	0	0
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	4.00000	0.544331
$55$	−3.00000	−0.404520
$56$	1.00000	0.133631
$57$	10.0000	1.32453
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−2.00000	−0.258199
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	4.00000	0.508001
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	6.00000	0.738549
$67$	16.0000	1.95471	0.977356	−	0.211604i	$-0.0678686\pi$
$67$	16.0000	1.95471	0.977356	+	0.211604i	$0.0678686\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	1.00000	0.119523
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	1.00000	0.117851
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	−11.0000	−1.27872
$75$	−2.00000	−0.230940
$76$	−5.00000	−0.573539
$77$	−3.00000	−0.341882
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	1.00000	0.111803
$81$	−11.0000	−1.22222
$82$	−6.00000	−0.662589
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	−2.00000	−0.218218
$85$	−6.00000	−0.650791
$86$	2.00000	0.215666
$87$	0	0
$88$	−3.00000	−0.319801
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	0	0
$93$	−8.00000	−0.829561
$94$	3.00000	0.309426
$95$	−5.00000	−0.512989
$96$	−2.00000	−0.204124
$97$	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
$99$	−3.00000	−0.301511
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	12.0000	1.18818
$103$	5.00000	0.492665	0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000	0.492665	0.246332	+	0.969185i	$0.420775\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	−9.00000	−0.874157
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	4.00000	0.384900
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−3.00000	−0.286039
$111$	22.0000	2.08815
$112$	1.00000	0.0944911
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	10.0000	0.936586
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.00000	−0.550019
$120$	−2.00000	−0.182574
$121$	−2.00000	−0.181818
$122$	8.00000	0.724286
$123$	12.0000	1.08200
$124$	4.00000	0.359211
$125$	1.00000	0.0894427
$126$	1.00000	0.0890871
$127$	−1.00000	−0.0887357	−0.0443678	−	0.999015i	$-0.514127\pi$
$127$	−1.00000	−0.0887357	−0.0443678	+	0.999015i	$0.514127\pi$
$128$	1.00000	0.0883883
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−9.00000	−0.786334	−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000	−0.786334	−0.393167	+	0.919467i	$0.628621\pi$
$132$	6.00000	0.522233
$133$	−5.00000	−0.433555
$134$	16.0000	1.38219
$135$	4.00000	0.344265
$136$	−6.00000	−0.514496
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−19.0000	−1.61156	−0.805779	−	0.592216i	$-0.798253\pi$
$139$	−19.0000	−1.61156	−0.805779	+	0.592216i	$0.798253\pi$
$140$	1.00000	0.0845154
$141$	−6.00000	−0.505291
$142$	−6.00000	−0.503509
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	−14.0000	−1.15865
$147$	12.0000	0.989743
$148$	−11.0000	−0.904194
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	−2.00000	−0.163299
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	−5.00000	−0.405554
$153$	−6.00000	−0.485071
$154$	−3.00000	−0.241747
$155$	4.00000	0.321288
$156$	0	0
$157$	17.0000	1.35675	0.678374	−	0.734717i	$-0.262685\pi$
$157$	17.0000	1.35675	0.678374	+	0.734717i	$0.262685\pi$
$158$	−16.0000	−1.27289
$159$	18.0000	1.42749
$160$	1.00000	0.0790569
$161$	0	0
$162$	−11.0000	−0.864242
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	−6.00000	−0.468521
$165$	6.00000	0.467099
$166$	6.00000	0.465690
$167$	15.0000	1.16073	0.580367	−	0.814355i	$-0.302909\pi$
$167$	15.0000	1.16073	0.580367	+	0.814355i	$0.302909\pi$
$168$	−2.00000	−0.154303
$169$	0	0
$170$	−6.00000	−0.460179
$171$	−5.00000	−0.382360
$172$	2.00000	0.152499
$173$	15.0000	1.14043	0.570214	−	0.821496i	$-0.306860\pi$
$173$	15.0000	1.14043	0.570214	+	0.821496i	$0.306860\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−3.00000	−0.226134
$177$	0	0
$178$	−9.00000	−0.674579
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	1.00000	0.0745356
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	0	0
$183$	−16.0000	−1.18275
$184$	0	0
$185$	−11.0000	−0.808736
$186$	−8.00000	−0.586588
$187$	18.0000	1.31629
$188$	3.00000	0.218797
$189$	4.00000	0.290957
$190$	−5.00000	−0.362738
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−2.00000	−0.144338
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−6.00000	−0.428571
$197$	27.0000	1.92367	0.961835	−	0.273629i	$-0.0882242\pi$
$197$	27.0000	1.92367	0.961835	+	0.273629i	$0.0882242\pi$
$198$	−3.00000	−0.213201
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	1.00000	0.0707107
$201$	−32.0000	−2.25711
$202$	−6.00000	−0.422159
$203$	0	0
$204$	12.0000	0.840168
$205$	−6.00000	−0.419058
$206$	5.00000	0.348367
$207$	0	0
$208$	0	0
$209$	15.0000	1.03757
$210$	−2.00000	−0.138013
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	−9.00000	−0.618123
$213$	12.0000	0.822226
$214$	−12.0000	−0.820303
$215$	2.00000	0.136399
$216$	4.00000	0.272166
$217$	4.00000	0.271538
$218$	−2.00000	−0.135457
$219$	28.0000	1.89206
$220$	−3.00000	−0.202260
$221$	0	0
$222$	22.0000	1.47654
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	1.00000	0.0668153
$225$	1.00000	0.0666667
$226$	−12.0000	−0.798228
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	10.0000	0.662266
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	3.00000	0.195698
$236$	0	0
$237$	32.0000	2.07862
$238$	−6.00000	−0.388922
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	−2.00000	−0.129099
$241$	−23.0000	−1.48156	−0.740780	−	0.671748i	$-0.765544\pi$
$241$	−23.0000	−1.48156	−0.740780	+	0.671748i	$0.765544\pi$
$242$	−2.00000	−0.128565
$243$	10.0000	0.641500
$244$	8.00000	0.512148
$245$	−6.00000	−0.383326
$246$	12.0000	0.765092
$247$	0	0
$248$	4.00000	0.254000
$249$	−12.0000	−0.760469
$250$	1.00000	0.0632456
$251$	−15.0000	−0.946792	−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000	−0.946792	−0.473396	+	0.880850i	$0.656972\pi$
$252$	1.00000	0.0629941
$253$	0	0
$254$	−1.00000	−0.0627456
$255$	12.0000	0.751469
$256$	1.00000	0.0625000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	−4.00000	−0.249029
$259$	−11.0000	−0.683507
$260$	0	0
$261$	0	0
$262$	−9.00000	−0.556022
$263$	9.00000	0.554964	0.277482	−	0.960731i	$-0.410500\pi$
$263$	9.00000	0.554964	0.277482	+	0.960731i	$0.410500\pi$
$264$	6.00000	0.369274
$265$	−9.00000	−0.552866
$266$	−5.00000	−0.306570
$267$	18.0000	1.10158
$268$	16.0000	0.977356
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	4.00000	0.243432
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	−6.00000	−0.362473
$275$	−3.00000	−0.180907
$276$	0	0
$277$	−1.00000	−0.0600842	−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000	−0.0600842	−0.0300421	+	0.999549i	$0.509564\pi$
$278$	−19.0000	−1.13954
$279$	4.00000	0.239474
$280$	1.00000	0.0597614
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	−6.00000	−0.357295
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	−6.00000	−0.356034
$285$	10.0000	0.592349
$286$	0	0
$287$	−6.00000	−0.354169
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	0	0
$291$	−20.0000	−1.17242
$292$	−14.0000	−0.819288
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	12.0000	0.699854
$295$	0	0
$296$	−11.0000	−0.639362
$297$	−12.0000	−0.696311
$298$	18.0000	1.04271
$299$	0	0
$300$	−2.00000	−0.115470
$301$	2.00000	0.115278
$302$	16.0000	0.920697
$303$	12.0000	0.689382
$304$	−5.00000	−0.286770
$305$	8.00000	0.458079
$306$	−6.00000	−0.342997
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	−3.00000	−0.170941
$309$	−10.0000	−0.568880
$310$	4.00000	0.227185
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	17.0000	0.959366
$315$	1.00000	0.0563436
$316$	−16.0000	−0.900070
$317$	−15.0000	−0.842484	−0.421242	−	0.906948i	$-0.638406\pi$
$317$	−15.0000	−0.842484	−0.421242	+	0.906948i	$0.638406\pi$
$318$	18.0000	1.00939
$319$	0	0
$320$	1.00000	0.0559017
$321$	24.0000	1.33955
$322$	0	0
$323$	30.0000	1.66924
$324$	−11.0000	−0.611111
$325$	0	0
$326$	−2.00000	−0.110770
$327$	4.00000	0.221201
$328$	−6.00000	−0.331295
$329$	3.00000	0.165395
$330$	6.00000	0.330289
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	6.00000	0.329293
$333$	−11.0000	−0.602796
$334$	15.0000	0.820763
$335$	16.0000	0.874173
$336$	−2.00000	−0.109109
$337$	−16.0000	−0.871576	−0.435788	−	0.900049i	$-0.643530\pi$
$337$	−16.0000	−0.871576	−0.435788	+	0.900049i	$0.643530\pi$
$338$	0	0
$339$	24.0000	1.30350
$340$	−6.00000	−0.325396
$341$	−12.0000	−0.649836
$342$	−5.00000	−0.270369
$343$	−13.0000	−0.701934
$344$	2.00000	0.107833
$345$	0	0
$346$	15.0000	0.806405
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	−3.00000	−0.159901
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	−6.00000	−0.318447
$356$	−9.00000	−0.476999
$357$	12.0000	0.635107
$358$	−24.0000	−1.26844
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	1.00000	0.0527046
$361$	6.00000	0.315789
$362$	8.00000	0.420471
$363$	4.00000	0.209946
$364$	0	0
$365$	−14.0000	−0.732793
$366$	−16.0000	−0.836333
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	−11.0000	−0.571863
$371$	−9.00000	−0.467257
$372$	−8.00000	−0.414781
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	18.0000	0.930758
$375$	−2.00000	−0.103280
$376$	3.00000	0.154713
$377$	0	0
$378$	4.00000	0.205738
$379$	19.0000	0.975964	0.487982	−	0.872854i	$-0.337733\pi$
$379$	19.0000	0.975964	0.487982	+	0.872854i	$0.337733\pi$
$380$	−5.00000	−0.256495
$381$	2.00000	0.102463
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	−2.00000	−0.102062
$385$	−3.00000	−0.152894
$386$	4.00000	0.203595
$387$	2.00000	0.101666
$388$	10.0000	0.507673
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	0	0
$392$	−6.00000	−0.303046
$393$	18.0000	0.907980
$394$	27.0000	1.36024
$395$	−16.0000	−0.805047
$396$	−3.00000	−0.150756
$397$	13.0000	0.652451	0.326226	−	0.945292i	$-0.394223\pi$
$397$	13.0000	0.652451	0.326226	+	0.945292i	$0.394223\pi$
$398$	−10.0000	−0.501255
$399$	10.0000	0.500626
$400$	1.00000	0.0500000
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	−32.0000	−1.59601
$403$	0	0
$404$	−6.00000	−0.298511
$405$	−11.0000	−0.546594
$406$	0	0
$407$	33.0000	1.63575
$408$	12.0000	0.594089
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	−6.00000	−0.296319
$411$	12.0000	0.591916
$412$	5.00000	0.246332
$413$	0	0
$414$	0	0
$415$	6.00000	0.294528
$416$	0	0
$417$	38.0000	1.86087
$418$	15.0000	0.733674
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	−2.00000	−0.0975900
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	23.0000	1.11962
$423$	3.00000	0.145865
$424$	−9.00000	−0.437079
$425$	−6.00000	−0.291043
$426$	12.0000	0.581402
$427$	8.00000	0.387147
$428$	−12.0000	−0.580042
$429$	0	0
$430$	2.00000	0.0964486
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	4.00000	0.192450
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	28.0000	1.33789
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	−3.00000	−0.143019
$441$	−6.00000	−0.285714
$442$	0	0
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	22.0000	1.04407
$445$	−9.00000	−0.426641
$446$	19.0000	0.899676
$447$	−36.0000	−1.70274
$448$	1.00000	0.0472456
$449$	−9.00000	−0.424736	−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000	−0.424736	−0.212368	+	0.977190i	$0.568118\pi$
$450$	1.00000	0.0471405
$451$	18.0000	0.847587
$452$	−12.0000	−0.564433
$453$	−32.0000	−1.50349
$454$	24.0000	1.12638
$455$	0	0
$456$	10.0000	0.468293
$457$	4.00000	0.187112	0.0935561	−	0.995614i	$-0.470177\pi$
$457$	4.00000	0.187112	0.0935561	+	0.995614i	$0.470177\pi$
$458$	4.00000	0.186908
$459$	−24.0000	−1.12022
$460$	0	0
$461$	42.0000	1.95614	0.978068	−	0.208288i	$-0.0667892\pi$
$461$	42.0000	1.95614	0.978068	+	0.208288i	$0.0667892\pi$
$462$	6.00000	0.279145
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	−24.0000	−1.11178
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	3.00000	0.138380
$471$	−34.0000	−1.56664
$472$	0	0
$473$	−6.00000	−0.275880
$474$	32.0000	1.46981
$475$	−5.00000	−0.229416
$476$	−6.00000	−0.275010
$477$	−9.00000	−0.412082
$478$	0	0
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	−2.00000	−0.0912871
$481$	0	0
$482$	−23.0000	−1.04762
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	10.0000	0.454077
$486$	10.0000	0.453609
$487$	19.0000	0.860972	0.430486	−	0.902597i	$-0.358342\pi$
$487$	19.0000	0.860972	0.430486	+	0.902597i	$0.358342\pi$
$488$	8.00000	0.362143
$489$	4.00000	0.180886
$490$	−6.00000	−0.271052
$491$	27.0000	1.21849	0.609246	−	0.792981i	$-0.291472\pi$
$491$	27.0000	1.21849	0.609246	+	0.792981i	$0.291472\pi$
$492$	12.0000	0.541002
$493$	0	0
$494$	0	0
$495$	−3.00000	−0.134840
$496$	4.00000	0.179605
$497$	−6.00000	−0.269137
$498$	−12.0000	−0.537733
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	1.00000	0.0447214
$501$	−30.0000	−1.34030
$502$	−15.0000	−0.669483
$503$	9.00000	0.401290	0.200645	−	0.979664i	$-0.435696\pi$
$503$	9.00000	0.401290	0.200645	+	0.979664i	$0.435696\pi$
$504$	1.00000	0.0445435
$505$	−6.00000	−0.266996
$506$	0	0
$507$	0	0
$508$	−1.00000	−0.0443678
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	12.0000	0.531369
$511$	−14.0000	−0.619324
$512$	1.00000	0.0441942
$513$	−20.0000	−0.883022
$514$	12.0000	0.529297
$515$	5.00000	0.220326
$516$	−4.00000	−0.176090
$517$	−9.00000	−0.395820
$518$	−11.0000	−0.483312
$519$	−30.0000	−1.31685
$520$	0	0
$521$	−27.0000	−1.18289	−0.591446	−	0.806345i	$-0.701443\pi$
$521$	−27.0000	−1.18289	−0.591446	+	0.806345i	$0.701443\pi$
$522$	0	0
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	−9.00000	−0.393167
$525$	−2.00000	−0.0872872
$526$	9.00000	0.392419
$527$	−24.0000	−1.04546
$528$	6.00000	0.261116
$529$	−23.0000	−1.00000
$530$	−9.00000	−0.390935
$531$	0	0
$532$	−5.00000	−0.216777
$533$	0	0
$534$	18.0000	0.778936
$535$	−12.0000	−0.518805
$536$	16.0000	0.691095
$537$	48.0000	2.07135
$538$	6.00000	0.258678
$539$	18.0000	0.775315
$540$	4.00000	0.172133
$541$	−20.0000	−0.859867	−0.429934	−	0.902861i	$-0.641463\pi$
$541$	−20.0000	−0.859867	−0.429934	+	0.902861i	$0.641463\pi$
$542$	−20.0000	−0.859074
$543$	−16.0000	−0.686626
$544$	−6.00000	−0.257248
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	−34.0000	−1.45374	−0.726868	−	0.686778i	$-0.759025\pi$
$547$	−34.0000	−1.45374	−0.726868	+	0.686778i	$0.759025\pi$
$548$	−6.00000	−0.256307
$549$	8.00000	0.341432
$550$	−3.00000	−0.127920
$551$	0	0
$552$	0	0
$553$	−16.0000	−0.680389
$554$	−1.00000	−0.0424859
$555$	22.0000	0.933848
$556$	−19.0000	−0.805779
$557$	−21.0000	−0.889799	−0.444899	−	0.895581i	$-0.646761\pi$
$557$	−21.0000	−0.889799	−0.444899	+	0.895581i	$0.646761\pi$
$558$	4.00000	0.169334
$559$	0	0
$560$	1.00000	0.0422577
$561$	−36.0000	−1.51992
$562$	6.00000	0.253095
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	−6.00000	−0.252646
$565$	−12.0000	−0.504844
$566$	14.0000	0.588464
$567$	−11.0000	−0.461957
$568$	−6.00000	−0.251754
$569$	9.00000	0.377300	0.188650	−	0.982044i	$-0.439589\pi$
$569$	9.00000	0.377300	0.188650	+	0.982044i	$0.439589\pi$
$570$	10.0000	0.418854
$571$	17.0000	0.711428	0.355714	−	0.934595i	$-0.384238\pi$
$571$	17.0000	0.711428	0.355714	+	0.934595i	$0.384238\pi$
$572$	0	0
$573$	0	0
$574$	−6.00000	−0.250435
$575$	0	0
$576$	1.00000	0.0416667
$577$	−32.0000	−1.33218	−0.666089	−	0.745873i	$-0.732033\pi$
$577$	−32.0000	−1.33218	−0.666089	+	0.745873i	$0.732033\pi$
$578$	19.0000	0.790296
$579$	−8.00000	−0.332469
$580$	0	0
$581$	6.00000	0.248922
$582$	−20.0000	−0.829027
$583$	27.0000	1.11823
$584$	−14.0000	−0.579324
$585$	0	0
$586$	−9.00000	−0.371787
$587$	6.00000	0.247647	0.123823	−	0.992304i	$-0.460484\pi$
$587$	6.00000	0.247647	0.123823	+	0.992304i	$0.460484\pi$
$588$	12.0000	0.494872
$589$	−20.0000	−0.824086
$590$	0	0
$591$	−54.0000	−2.22126
$592$	−11.0000	−0.452097
$593$	−24.0000	−0.985562	−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000	−0.985562	−0.492781	+	0.870153i	$0.664020\pi$
$594$	−12.0000	−0.492366
$595$	−6.00000	−0.245976
$596$	18.0000	0.737309
$597$	20.0000	0.818546
$598$	0	0
$599$	−18.0000	−0.735460	−0.367730	−	0.929933i	$-0.619865\pi$
$599$	−18.0000	−0.735460	−0.367730	+	0.929933i	$0.619865\pi$
$600$	−2.00000	−0.0816497
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	2.00000	0.0815139
$603$	16.0000	0.651570
$604$	16.0000	0.651031
$605$	−2.00000	−0.0813116
$606$	12.0000	0.487467
$607$	−13.0000	−0.527654	−0.263827	−	0.964570i	$-0.584985\pi$
$607$	−13.0000	−0.527654	−0.263827	+	0.964570i	$0.584985\pi$
$608$	−5.00000	−0.202777
$609$	0	0
$610$	8.00000	0.323911
$611$	0	0
$612$	−6.00000	−0.242536
$613$	13.0000	0.525065	0.262533	−	0.964923i	$-0.415442\pi$
$613$	13.0000	0.525065	0.262533	+	0.964923i	$0.415442\pi$
$614$	−2.00000	−0.0807134
$615$	12.0000	0.483887
$616$	−3.00000	−0.120873
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	−10.0000	−0.402259
$619$	1.00000	0.0401934	0.0200967	−	0.999798i	$-0.493603\pi$
$619$	1.00000	0.0401934	0.0200967	+	0.999798i	$0.493603\pi$
$620$	4.00000	0.160644
$621$	0	0
$622$	30.0000	1.20289
$623$	−9.00000	−0.360577
$624$	0	0
$625$	1.00000	0.0400000
$626$	14.0000	0.559553
$627$	−30.0000	−1.19808
$628$	17.0000	0.678374
$629$	66.0000	2.63159
$630$	1.00000	0.0398410
$631$	28.0000	1.11466	0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000	1.11466	0.557331	+	0.830290i	$0.311825\pi$
$632$	−16.0000	−0.636446
$633$	−46.0000	−1.82834
$634$	−15.0000	−0.595726
$635$	−1.00000	−0.0396838
$636$	18.0000	0.713746
$637$	0	0
$638$	0	0
$639$	−6.00000	−0.237356
$640$	1.00000	0.0395285
$641$	−9.00000	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$641$	−9.00000	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$642$	24.0000	0.947204
$643$	−2.00000	−0.0788723	−0.0394362	−	0.999222i	$-0.512556\pi$
$643$	−2.00000	−0.0788723	−0.0394362	+	0.999222i	$0.512556\pi$
$644$	0	0
$645$	−4.00000	−0.157500
$646$	30.0000	1.18033
$647$	−9.00000	−0.353827	−0.176913	−	0.984226i	$-0.556611\pi$
$647$	−9.00000	−0.353827	−0.176913	+	0.984226i	$0.556611\pi$
$648$	−11.0000	−0.432121
$649$	0	0
$650$	0	0
$651$	−8.00000	−0.313545
$652$	−2.00000	−0.0783260
$653$	−27.0000	−1.05659	−0.528296	−	0.849060i	$-0.677169\pi$
$653$	−27.0000	−1.05659	−0.528296	+	0.849060i	$0.677169\pi$
$654$	4.00000	0.156412
$655$	−9.00000	−0.351659
$656$	−6.00000	−0.234261
$657$	−14.0000	−0.546192
$658$	3.00000	0.116952
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	6.00000	0.233550
$661$	4.00000	0.155582	0.0777910	−	0.996970i	$-0.475213\pi$
$661$	4.00000	0.155582	0.0777910	+	0.996970i	$0.475213\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	6.00000	0.232845
$665$	−5.00000	−0.193892
$666$	−11.0000	−0.426241
$667$	0	0
$668$	15.0000	0.580367
$669$	−38.0000	−1.46916
$670$	16.0000	0.618134
$671$	−24.0000	−0.926510
$672$	−2.00000	−0.0771517
$673$	20.0000	0.770943	0.385472	−	0.922720i	$-0.374039\pi$
$673$	20.0000	0.770943	0.385472	+	0.922720i	$0.374039\pi$
$674$	−16.0000	−0.616297
$675$	4.00000	0.153960
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	24.0000	0.921714
$679$	10.0000	0.383765
$680$	−6.00000	−0.230089
$681$	−48.0000	−1.83936
$682$	−12.0000	−0.459504
$683$	6.00000	0.229584	0.114792	−	0.993390i	$-0.463380\pi$
$683$	6.00000	0.229584	0.114792	+	0.993390i	$0.463380\pi$
$684$	−5.00000	−0.191180
$685$	−6.00000	−0.229248
$686$	−13.0000	−0.496342
$687$	−8.00000	−0.305219
$688$	2.00000	0.0762493
$689$	0	0
$690$	0	0
$691$	13.0000	0.494543	0.247272	−	0.968946i	$-0.420466\pi$
$691$	13.0000	0.494543	0.247272	+	0.968946i	$0.420466\pi$
$692$	15.0000	0.570214
$693$	−3.00000	−0.113961
$694$	6.00000	0.227757
$695$	−19.0000	−0.720711
$696$	0	0
$697$	36.0000	1.36360
$698$	−2.00000	−0.0757011
$699$	48.0000	1.81553
$700$	1.00000	0.0377964
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	55.0000	2.07436
$704$	−3.00000	−0.113067
$705$	−6.00000	−0.225973
$706$	−6.00000	−0.225813
$707$	−6.00000	−0.225653
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	−6.00000	−0.225176
$711$	−16.0000	−0.600047
$712$	−9.00000	−0.337289
$713$	0	0
$714$	12.0000	0.449089
$715$	0	0
$716$	−24.0000	−0.896922
$717$	0	0
$718$	6.00000	0.223918
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	1.00000	0.0372678
$721$	5.00000	0.186210
$722$	6.00000	0.223297
$723$	46.0000	1.71076
$724$	8.00000	0.297318
$725$	0	0
$726$	4.00000	0.148454
$727$	29.0000	1.07555	0.537775	−	0.843088i	$-0.319265\pi$
$727$	29.0000	1.07555	0.537775	+	0.843088i	$0.319265\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	−14.0000	−0.518163
$731$	−12.0000	−0.443836
$732$	−16.0000	−0.591377
$733$	25.0000	0.923396	0.461698	−	0.887037i	$-0.347240\pi$
$733$	25.0000	0.923396	0.461698	+	0.887037i	$0.347240\pi$
$734$	32.0000	1.18114
$735$	12.0000	0.442627
$736$	0	0
$737$	−48.0000	−1.76810
$738$	−6.00000	−0.220863
$739$	7.00000	0.257499	0.128750	−	0.991677i	$-0.458904\pi$
$739$	7.00000	0.257499	0.128750	+	0.991677i	$0.458904\pi$
$740$	−11.0000	−0.404368
$741$	0	0
$742$	−9.00000	−0.330400
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	−8.00000	−0.293294
$745$	18.0000	0.659469
$746$	14.0000	0.512576
$747$	6.00000	0.219529
$748$	18.0000	0.658145
$749$	−12.0000	−0.438470
$750$	−2.00000	−0.0730297
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	3.00000	0.109399
$753$	30.0000	1.09326
$754$	0	0
$755$	16.0000	0.582300
$756$	4.00000	0.145479
$757$	23.0000	0.835949	0.417975	−	0.908459i	$-0.362740\pi$
$757$	23.0000	0.835949	0.417975	+	0.908459i	$0.362740\pi$
$758$	19.0000	0.690111
$759$	0	0
$760$	−5.00000	−0.181369
$761$	3.00000	0.108750	0.0543750	−	0.998521i	$-0.482683\pi$
$761$	3.00000	0.108750	0.0543750	+	0.998521i	$0.482683\pi$
$762$	2.00000	0.0724524
$763$	−2.00000	−0.0724049
$764$	0	0
$765$	−6.00000	−0.216930
$766$	0	0
$767$	0	0
$768$	−2.00000	−0.0721688
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	−3.00000	−0.108112
$771$	−24.0000	−0.864339
$772$	4.00000	0.143963
$773$	−39.0000	−1.40273	−0.701366	−	0.712801i	$-0.747426\pi$
$773$	−39.0000	−1.40273	−0.701366	+	0.712801i	$0.747426\pi$
$774$	2.00000	0.0718885
$775$	4.00000	0.143684
$776$	10.0000	0.358979
$777$	22.0000	0.789246
$778$	−30.0000	−1.07555
$779$	30.0000	1.07486
$780$	0	0
$781$	18.0000	0.644091
$782$	0	0
$783$	0	0
$784$	−6.00000	−0.214286
$785$	17.0000	0.606756
$786$	18.0000	0.642039
$787$	22.0000	0.784215	0.392108	−	0.919919i	$-0.371746\pi$
$787$	22.0000	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$788$	27.0000	0.961835
$789$	−18.0000	−0.640817
$790$	−16.0000	−0.569254
$791$	−12.0000	−0.426671
$792$	−3.00000	−0.106600
$793$	0	0
$794$	13.0000	0.461353
$795$	18.0000	0.638394
$796$	−10.0000	−0.354441
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	10.0000	0.353996
$799$	−18.0000	−0.636794
$800$	1.00000	0.0353553
$801$	−9.00000	−0.317999
$802$	15.0000	0.529668
$803$	42.0000	1.48215
$804$	−32.0000	−1.12855
$805$	0	0
$806$	0	0
$807$	−12.0000	−0.422420
$808$	−6.00000	−0.211079
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	−11.0000	−0.386501
$811$	49.0000	1.72062	0.860311	−	0.509769i	$-0.170269\pi$
$811$	49.0000	1.72062	0.860311	+	0.509769i	$0.170269\pi$
$812$	0	0
$813$	40.0000	1.40286
$814$	33.0000	1.15665
$815$	−2.00000	−0.0700569
$816$	12.0000	0.420084
$817$	−10.0000	−0.349856
$818$	−5.00000	−0.174821
$819$	0	0
$820$	−6.00000	−0.209529
$821$	−36.0000	−1.25641	−0.628204	−	0.778048i	$-0.716210\pi$
$821$	−36.0000	−1.25641	−0.628204	+	0.778048i	$0.716210\pi$
$822$	12.0000	0.418548
$823$	−43.0000	−1.49889	−0.749443	−	0.662069i	$-0.769679\pi$
$823$	−43.0000	−1.49889	−0.749443	+	0.662069i	$0.769679\pi$
$824$	5.00000	0.174183
$825$	6.00000	0.208893
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	0	0
$829$	−40.0000	−1.38926	−0.694629	−	0.719368i	$-0.744431\pi$
$829$	−40.0000	−1.38926	−0.694629	+	0.719368i	$0.744431\pi$
$830$	6.00000	0.208263
$831$	2.00000	0.0693792
$832$	0	0
$833$	36.0000	1.24733
$834$	38.0000	1.31583
$835$	15.0000	0.519096
$836$	15.0000	0.518786
$837$	16.0000	0.553041
$838$	−36.0000	−1.24360
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	−2.00000	−0.0690066
$841$	−29.0000	−1.00000
$842$	10.0000	0.344623
$843$	−12.0000	−0.413302
$844$	23.0000	0.791693
$845$	0	0
$846$	3.00000	0.103142
$847$	−2.00000	−0.0687208
$848$	−9.00000	−0.309061
$849$	−28.0000	−0.960958
$850$	−6.00000	−0.205798
$851$	0	0
$852$	12.0000	0.411113
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	8.00000	0.273754
$855$	−5.00000	−0.170996
$856$	−12.0000	−0.410152
$857$	−12.0000	−0.409912	−0.204956	−	0.978771i	$-0.565705\pi$
$857$	−12.0000	−0.409912	−0.204956	+	0.978771i	$0.565705\pi$
$858$	0	0
$859$	−31.0000	−1.05771	−0.528853	−	0.848713i	$-0.677378\pi$
$859$	−31.0000	−1.05771	−0.528853	+	0.848713i	$0.677378\pi$
$860$	2.00000	0.0681994
$861$	12.0000	0.408959
$862$	−30.0000	−1.02180
$863$	−36.0000	−1.22545	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000	−1.22545	−0.612727	+	0.790295i	$0.709928\pi$
$864$	4.00000	0.136083
$865$	15.0000	0.510015
$866$	−16.0000	−0.543702
$867$	−38.0000	−1.29055
$868$	4.00000	0.135769
$869$	48.0000	1.62829
$870$	0	0
$871$	0	0
$872$	−2.00000	−0.0677285
$873$	10.0000	0.338449
$874$	0	0
$875$	1.00000	0.0338062
$876$	28.0000	0.946032
$877$	10.0000	0.337676	0.168838	−	0.985644i	$-0.445999\pi$
$877$	10.0000	0.337676	0.168838	+	0.985644i	$0.445999\pi$
$878$	20.0000	0.674967
$879$	18.0000	0.607125
$880$	−3.00000	−0.101130
$881$	−15.0000	−0.505363	−0.252681	−	0.967550i	$-0.581312\pi$
$881$	−15.0000	−0.505363	−0.252681	+	0.967550i	$0.581312\pi$
$882$	−6.00000	−0.202031
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	0	0
$885$	0	0
$886$	−18.0000	−0.604722
$887$	−27.0000	−0.906571	−0.453286	−	0.891365i	$-0.649748\pi$
$887$	−27.0000	−0.906571	−0.453286	+	0.891365i	$0.649748\pi$
$888$	22.0000	0.738272
$889$	−1.00000	−0.0335389
$890$	−9.00000	−0.301681
$891$	33.0000	1.10554
$892$	19.0000	0.636167
$893$	−15.0000	−0.501956
$894$	−36.0000	−1.20402
$895$	−24.0000	−0.802232
$896$	1.00000	0.0334077
$897$	0	0
$898$	−9.00000	−0.300334
$899$	0	0
$900$	1.00000	0.0333333
$901$	54.0000	1.79900
$902$	18.0000	0.599334
$903$	−4.00000	−0.133112
$904$	−12.0000	−0.399114
$905$	8.00000	0.265929
$906$	−32.0000	−1.06313
$907$	8.00000	0.265636	0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000	0.265636	0.132818	+	0.991140i	$0.457597\pi$
$908$	24.0000	0.796468
$909$	−6.00000	−0.199007
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	10.0000	0.331133
$913$	−18.0000	−0.595713
$914$	4.00000	0.132308
$915$	−16.0000	−0.528944
$916$	4.00000	0.132164
$917$	−9.00000	−0.297206
$918$	−24.0000	−0.792118
$919$	−34.0000	−1.12156	−0.560778	−	0.827966i	$-0.689498\pi$
$919$	−34.0000	−1.12156	−0.560778	+	0.827966i	$0.689498\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	42.0000	1.38320
$923$	0	0
$924$	6.00000	0.197386
$925$	−11.0000	−0.361678
$926$	−8.00000	−0.262896
$927$	5.00000	0.164222
$928$	0	0
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	−8.00000	−0.262330
$931$	30.0000	0.983210
$932$	−24.0000	−0.786146
$933$	−60.0000	−1.96431
$934$	−12.0000	−0.392652
$935$	18.0000	0.588663
$936$	0	0
$937$	50.0000	1.63343	0.816714	−	0.577042i	$-0.195793\pi$
$937$	50.0000	1.63343	0.816714	+	0.577042i	$0.195793\pi$
$938$	16.0000	0.522419
$939$	−28.0000	−0.913745
$940$	3.00000	0.0978492
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	−34.0000	−1.10778
$943$	0	0
$944$	0	0
$945$	4.00000	0.130120
$946$	−6.00000	−0.195077
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	32.0000	1.03931
$949$	0	0
$950$	−5.00000	−0.162221
$951$	30.0000	0.972817
$952$	−6.00000	−0.194461
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	−9.00000	−0.291386
$955$	0	0
$956$	0	0
$957$	0	0
$958$	30.0000	0.969256
$959$	−6.00000	−0.193750
$960$	−2.00000	−0.0645497
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−12.0000	−0.386695
$964$	−23.0000	−0.740780
$965$	4.00000	0.128765
$966$	0	0
$967$	−5.00000	−0.160789	−0.0803946	−	0.996763i	$-0.525618\pi$
$967$	−5.00000	−0.160789	−0.0803946	+	0.996763i	$0.525618\pi$
$968$	−2.00000	−0.0642824
$969$	−60.0000	−1.92748
$970$	10.0000	0.321081
$971$	33.0000	1.05902	0.529510	−	0.848304i	$-0.322376\pi$
$971$	33.0000	1.05902	0.529510	+	0.848304i	$0.322376\pi$
$972$	10.0000	0.320750
$973$	−19.0000	−0.609112
$974$	19.0000	0.608799
$975$	0	0
$976$	8.00000	0.256074
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	4.00000	0.127906
$979$	27.0000	0.862924
$980$	−6.00000	−0.191663
$981$	−2.00000	−0.0638551
$982$	27.0000	0.861605
$983$	−39.0000	−1.24391	−0.621953	−	0.783054i	$-0.713661\pi$
$983$	−39.0000	−1.24391	−0.621953	+	0.783054i	$0.713661\pi$
$984$	12.0000	0.382546
$985$	27.0000	0.860292
$986$	0	0
$987$	−6.00000	−0.190982
$988$	0	0
$989$	0	0
$990$	−3.00000	−0.0953463
$991$	−22.0000	−0.698853	−0.349427	−	0.936964i	$-0.613624\pi$
$991$	−22.0000	−0.698853	−0.349427	+	0.936964i	$0.613624\pi$
$992$	4.00000	0.127000
$993$	40.0000	1.26936
$994$	−6.00000	−0.190308
$995$	−10.0000	−0.317021
$996$	−12.0000	−0.380235
$997$	41.0000	1.29848	0.649242	−	0.760582i	$-0.275086\pi$
$997$	41.0000	1.29848	0.649242	+	0.760582i	$0.275086\pi$
$998$	4.00000	0.126618
$999$	−44.0000	−1.39210

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1690.2.a.g.1.1		1
5.4	even	2		8450.2.a.k.1.1		1
13.2	odd	12		1690.2.l.i.1161.2		4
13.3	even	3		1690.2.e.e.191.1		2
13.4	even	6		130.2.e.b.81.1	yes	2
13.5	odd	4		1690.2.d.a.1351.1		2
13.6	odd	12		1690.2.l.i.361.1		4
13.7	odd	12		1690.2.l.i.361.2		4
13.8	odd	4		1690.2.d.a.1351.2		2
13.9	even	3		1690.2.e.e.991.1		2
13.10	even	6		130.2.e.b.61.1	✓	2
13.11	odd	12		1690.2.l.i.1161.1		4
13.12	even	2		1690.2.a.a.1.1		1
39.17	odd	6		1170.2.i.f.991.1		2
39.23	odd	6		1170.2.i.f.451.1		2
52.23	odd	6		1040.2.q.c.321.1		2
52.43	odd	6		1040.2.q.c.81.1		2
65.4	even	6		650.2.e.a.601.1		2
65.17	odd	12		650.2.o.b.549.1		4
65.23	odd	12		650.2.o.b.399.1		4
65.43	odd	12		650.2.o.b.549.2		4
65.49	even	6		650.2.e.a.451.1		2
65.62	odd	12		650.2.o.b.399.2		4
65.64	even	2		8450.2.a.w.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.e.b.61.1	✓	2	13.10	even	6
130.2.e.b.81.1	yes	2	13.4	even	6
650.2.e.a.451.1		2	65.49	even	6
650.2.e.a.601.1		2	65.4	even	6
650.2.o.b.399.1		4	65.23	odd	12
650.2.o.b.399.2		4	65.62	odd	12
650.2.o.b.549.1		4	65.17	odd	12
650.2.o.b.549.2		4	65.43	odd	12
1040.2.q.c.81.1		2	52.43	odd	6
1040.2.q.c.321.1		2	52.23	odd	6
1170.2.i.f.451.1		2	39.23	odd	6
1170.2.i.f.991.1		2	39.17	odd	6
1690.2.a.a.1.1		1	13.12	even	2
1690.2.a.g.1.1		1	1.1	even	1	trivial
1690.2.d.a.1351.1		2	13.5	odd	4
1690.2.d.a.1351.2		2	13.8	odd	4
1690.2.e.e.191.1		2	13.3	even	3
1690.2.e.e.991.1		2	13.9	even	3
1690.2.l.i.361.1		4	13.6	odd	12
1690.2.l.i.361.2		4	13.7	odd	12
1690.2.l.i.1161.1		4	13.11	odd	12
1690.2.l.i.1161.2		4	13.2	odd	12
8450.2.a.k.1.1		1	5.4	even	2
8450.2.a.w.1.1		1	65.64	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1690.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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