LMFDB - Embedded newform 8619.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8619,2,Mod(1,8619)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8619, 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("8619.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8619 = 3 \cdot 13^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8619.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:	$68.8230615021$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8619.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -2.00000 q^{4} -3.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} -2.00000 q^{4} -3.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} +3.00000 q^{11} -2.00000 q^{12} -3.00000 q^{15} +4.00000 q^{16} -1.00000 q^{17} +1.00000 q^{19} +6.00000 q^{20} +4.00000 q^{21} +9.00000 q^{23} +4.00000 q^{25} +1.00000 q^{27} -8.00000 q^{28} +6.00000 q^{29} -2.00000 q^{31} +3.00000 q^{33} -12.0000 q^{35} -2.00000 q^{36} +4.00000 q^{37} +3.00000 q^{41} -7.00000 q^{43} -6.00000 q^{44} -3.00000 q^{45} +6.00000 q^{47} +4.00000 q^{48} +9.00000 q^{49} -1.00000 q^{51} -6.00000 q^{53} -9.00000 q^{55} +1.00000 q^{57} -6.00000 q^{59} +6.00000 q^{60} +8.00000 q^{61} +4.00000 q^{63} -8.00000 q^{64} +4.00000 q^{67} +2.00000 q^{68} +9.00000 q^{69} -12.0000 q^{71} -2.00000 q^{73} +4.00000 q^{75} -2.00000 q^{76} +12.0000 q^{77} -10.0000 q^{79} -12.0000 q^{80} +1.00000 q^{81} +6.00000 q^{83} -8.00000 q^{84} +3.00000 q^{85} +6.00000 q^{87} -18.0000 q^{92} -2.00000 q^{93} -3.00000 q^{95} +16.0000 q^{97} +3.00000 q^{99} +O(q^{100})$

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 $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−2.00000	−1.00000
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−2.00000	−0.577350
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−3.00000	−0.774597
$16$	4.00000	1.00000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	6.00000	1.34164
$21$	4.00000	0.872872
$22$	0	0
$23$	9.00000	1.87663	0.938315	−	0.345782i	$-0.112386\pi$
$23$	9.00000	1.87663	0.938315	+	0.345782i	$0.112386\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	−8.00000	−1.51186
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	−12.0000	−2.02837
$36$	−2.00000	−0.333333
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	−7.00000	−1.06749	−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000	−1.06749	−0.533745	+	0.845645i	$0.679216\pi$
$44$	−6.00000	−0.904534
$45$	−3.00000	−0.447214
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	4.00000	0.577350
$49$	9.00000	1.28571
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−9.00000	−1.21356
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	6.00000	0.774597
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	0	0
$63$	4.00000	0.503953
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	2.00000	0.242536
$69$	9.00000	1.08347
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	−2.00000	−0.229416
$77$	12.0000	1.36753
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	−12.0000	−1.34164
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	−8.00000	−0.872872
$85$	3.00000	0.325396
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	−18.0000	−1.87663
$93$	−2.00000	−0.207390
$94$	0	0
$95$	−3.00000	−0.307794
$96$	0	0
$97$	16.0000	1.62455	0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000	1.62455	0.812277	+	0.583272i	$0.198228\pi$
$98$	0	0
$99$	3.00000	0.301511
$100$	−8.00000	−0.800000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	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
$105$	−12.0000	−1.17108
$106$	0	0
$107$	9.00000	0.870063	0.435031	−	0.900415i	$-0.356737\pi$
$107$	9.00000	0.870063	0.435031	+	0.900415i	$0.356737\pi$
$108$	−2.00000	−0.192450
$109$	−20.0000	−1.91565	−0.957826	−	0.287348i	$-0.907226\pi$
$109$	−20.0000	−1.91565	−0.957826	+	0.287348i	$0.907226\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	16.0000	1.51186
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	0	0
$115$	−27.0000	−2.51776
$116$	−12.0000	−1.11417
$117$	0	0
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	3.00000	0.270501
$124$	4.00000	0.359211
$125$	3.00000	0.268328
$126$	0	0
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	0	0
$129$	−7.00000	−0.616316
$130$	0	0
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	−6.00000	−0.522233
$133$	4.00000	0.346844
$134$	0	0
$135$	−3.00000	−0.258199
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	24.0000	2.02837
$141$	6.00000	0.505291
$142$	0	0
$143$	0	0
$144$	4.00000	0.333333
$145$	−18.0000	−1.49482
$146$	0	0
$147$	9.00000	0.742307
$148$	−8.00000	−0.657596
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	11.0000	0.877896	0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000	0.877896	0.438948	+	0.898513i	$0.355351\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	36.0000	2.83720
$162$	0	0
$163$	−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
$165$	−9.00000	−0.700649
$166$	0	0
$167$	−21.0000	−1.62503	−0.812514	−	0.582941i	$-0.801902\pi$
$167$	−21.0000	−1.62503	−0.812514	+	0.582941i	$0.801902\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	1.00000	0.0764719
$172$	14.0000	1.06749
$173$	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
$175$	16.0000	1.20949
$176$	12.0000	0.904534
$177$	−6.00000	−0.450988
$178$	0	0
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	6.00000	0.447214
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	8.00000	0.591377
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	−3.00000	−0.219382
$188$	−12.0000	−0.875190
$189$	4.00000	0.290957
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	−8.00000	−0.577350
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	0	0
$195$	0	0
$196$	−18.0000	−1.28571
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	24.0000	1.68447
$204$	2.00000	0.140028
$205$	−9.00000	−0.628587
$206$	0	0
$207$	9.00000	0.625543
$208$	0	0
$209$	3.00000	0.207514
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	12.0000	0.824163
$213$	−12.0000	−0.822226
$214$	0	0
$215$	21.0000	1.43219
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	−2.00000	−0.135147
$220$	18.0000	1.21356
$221$	0	0
$222$	0	0
$223$	1.00000	0.0669650	0.0334825	−	0.999439i	$-0.489340\pi$
$223$	1.00000	0.0669650	0.0334825	+	0.999439i	$0.489340\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	−3.00000	−0.199117	−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000	−0.199117	−0.0995585	+	0.995032i	$0.531743\pi$
$228$	−2.00000	−0.132453
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	0	0
$233$	21.0000	1.37576	0.687878	−	0.725826i	$-0.258542\pi$
$233$	21.0000	1.37576	0.687878	+	0.725826i	$0.258542\pi$
$234$	0	0
$235$	−18.0000	−1.17419
$236$	12.0000	0.781133
$237$	−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	$0.373129\pi$
$240$	−12.0000	−0.774597
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	−16.0000	−1.02430
$245$	−27.0000	−1.72497
$246$	0	0
$247$	0	0
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	−8.00000	−0.503953
$253$	27.0000	1.69748
$254$	0	0
$255$	3.00000	0.187867
$256$	16.0000	1.00000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	0	0
$259$	16.0000	0.994192
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	18.0000	1.10573
$266$	0	0
$267$	0	0
$268$	−8.00000	−0.488678
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	0	0
$271$	−11.0000	−0.668202	−0.334101	−	0.942537i	$-0.608433\pi$
$271$	−11.0000	−0.668202	−0.334101	+	0.942537i	$0.608433\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	0	0
$275$	12.0000	0.723627
$276$	−18.0000	−1.08347
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	0	0
$279$	−2.00000	−0.119737
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	−10.0000	−0.594438	−0.297219	−	0.954809i	$-0.596059\pi$
$283$	−10.0000	−0.594438	−0.297219	+	0.954809i	$0.596059\pi$
$284$	24.0000	1.42414
$285$	−3.00000	−0.177705
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	16.0000	0.937937
$292$	4.00000	0.234082
$293$	24.0000	1.40209	0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000	1.40209	0.701047	+	0.713115i	$0.252716\pi$
$294$	0	0
$295$	18.0000	1.04800
$296$	0	0
$297$	3.00000	0.174078
$298$	0	0
$299$	0	0
$300$	−8.00000	−0.461880
$301$	−28.0000	−1.61389
$302$	0	0
$303$	0	0
$304$	4.00000	0.229416
$305$	−24.0000	−1.37424
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	−24.0000	−1.36753
$309$	5.00000	0.284440
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	0	0
$315$	−12.0000	−0.676123
$316$	20.0000	1.12509
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	18.0000	1.00781
$320$	24.0000	1.34164
$321$	9.00000	0.502331
$322$	0	0
$323$	−1.00000	−0.0556415
$324$	−2.00000	−0.111111
$325$	0	0
$326$	0	0
$327$	−20.0000	−1.10600
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	13.0000	0.714545	0.357272	−	0.934000i	$-0.383707\pi$
$331$	13.0000	0.714545	0.357272	+	0.934000i	$0.383707\pi$
$332$	−12.0000	−0.658586
$333$	4.00000	0.219199
$334$	0	0
$335$	−12.0000	−0.655630
$336$	16.0000	0.872872
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	−9.00000	−0.488813
$340$	−6.00000	−0.325396
$341$	−6.00000	−0.324918
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	−27.0000	−1.45363
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−12.0000	−0.643268
$349$	19.0000	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$349$	19.0000	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−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
$355$	36.0000	1.91068
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	−2.00000	−0.104973
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	36.0000	1.87663
$369$	3.00000	0.156174
$370$	0	0
$371$	−24.0000	−1.24602
$372$	4.00000	0.207390
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	3.00000	0.154919
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−32.0000	−1.64373	−0.821865	−	0.569683i	$-0.807066\pi$
$379$	−32.0000	−1.64373	−0.821865	+	0.569683i	$0.807066\pi$
$380$	6.00000	0.307794
$381$	−13.0000	−0.666010
$382$	0	0
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	−36.0000	−1.83473
$386$	0	0
$387$	−7.00000	−0.355830
$388$	−32.0000	−1.62455
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	0	0
$391$	−9.00000	−0.455150
$392$	0	0
$393$	3.00000	0.151330
$394$	0	0
$395$	30.0000	1.50946
$396$	−6.00000	−0.301511
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	16.0000	0.800000
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−3.00000	−0.149071
$406$	0	0
$407$	12.0000	0.594818
$408$	0	0
$409$	19.0000	0.939490	0.469745	−	0.882802i	$-0.344346\pi$
$409$	19.0000	0.939490	0.469745	+	0.882802i	$0.344346\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	−10.0000	−0.492665
$413$	−24.0000	−1.18096
$414$	0	0
$415$	−18.0000	−0.883585
$416$	0	0
$417$	2.00000	0.0979404
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	24.0000	1.17108
$421$	25.0000	1.21843	0.609213	−	0.793007i	$-0.291486\pi$
$421$	25.0000	1.21843	0.609213	+	0.793007i	$0.291486\pi$
$422$	0	0
$423$	6.00000	0.291730
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	32.0000	1.54859
$428$	−18.0000	−0.870063
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	4.00000	0.192450
$433$	−1.00000	−0.0480569	−0.0240285	−	0.999711i	$-0.507649\pi$
$433$	−1.00000	−0.0480569	−0.0240285	+	0.999711i	$0.507649\pi$
$434$	0	0
$435$	−18.0000	−0.863034
$436$	40.0000	1.91565
$437$	9.00000	0.430528
$438$	0	0
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	0	0
$447$	18.0000	0.851371
$448$	−32.0000	−1.51186
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	18.0000	0.846649
$453$	−8.00000	−0.375873
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.0000	0.888783	0.444391	−	0.895833i	$-0.353420\pi$
$457$	19.0000	0.888783	0.444391	+	0.895833i	$0.353420\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	54.0000	2.51776
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	24.0000	1.11417
$465$	6.00000	0.278243
$466$	0	0
$467$	42.0000	1.94353	0.971764	−	0.235954i	$-0.0758216\pi$
$467$	42.0000	1.94353	0.971764	+	0.235954i	$0.0758216\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	11.0000	0.506853
$472$	0	0
$473$	−21.0000	−0.965581
$474$	0	0
$475$	4.00000	0.183533
$476$	8.00000	0.366679
$477$	−6.00000	−0.274721
$478$	0	0
$479$	27.0000	1.23366	0.616831	−	0.787096i	$-0.288416\pi$
$479$	27.0000	1.23366	0.616831	+	0.787096i	$0.288416\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	36.0000	1.63806
$484$	4.00000	0.181818
$485$	−48.0000	−2.17957
$486$	0	0
$487$	22.0000	0.996915	0.498458	−	0.866914i	$-0.333900\pi$
$487$	22.0000	0.996915	0.498458	+	0.866914i	$0.333900\pi$
$488$	0	0
$489$	−2.00000	−0.0904431
$490$	0	0
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	−6.00000	−0.270501
$493$	−6.00000	−0.270226
$494$	0	0
$495$	−9.00000	−0.404520
$496$	−8.00000	−0.359211
$497$	−48.0000	−2.15309
$498$	0	0
$499$	22.0000	0.984855	0.492428	−	0.870353i	$-0.336110\pi$
$499$	22.0000	0.984855	0.492428	+	0.870353i	$0.336110\pi$
$500$	−6.00000	−0.268328
$501$	−21.0000	−0.938211
$502$	0	0
$503$	15.0000	0.668817	0.334408	−	0.942428i	$-0.391463\pi$
$503$	15.0000	0.668817	0.334408	+	0.942428i	$0.391463\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	26.0000	1.15356
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−15.0000	−0.660979
$516$	14.0000	0.616316
$517$	18.0000	0.791639
$518$	0	0
$519$	15.0000	0.658427
$520$	0	0
$521$	21.0000	0.920027	0.460013	−	0.887912i	$-0.347845\pi$
$521$	21.0000	0.920027	0.460013	+	0.887912i	$0.347845\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−6.00000	−0.262111
$525$	16.0000	0.698297
$526$	0	0
$527$	2.00000	0.0871214
$528$	12.0000	0.522233
$529$	58.0000	2.52174
$530$	0	0
$531$	−6.00000	−0.260378
$532$	−8.00000	−0.346844
$533$	0	0
$534$	0	0
$535$	−27.0000	−1.16731
$536$	0	0
$537$	−6.00000	−0.258919
$538$	0	0
$539$	27.0000	1.16297
$540$	6.00000	0.258199
$541$	16.0000	0.687894	0.343947	−	0.938989i	$-0.388236\pi$
$541$	16.0000	0.687894	0.343947	+	0.938989i	$0.388236\pi$
$542$	0	0
$543$	14.0000	0.600798
$544$	0	0
$545$	60.0000	2.57012
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−12.0000	−0.512615
$549$	8.00000	0.341432
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	−40.0000	−1.70097
$554$	0	0
$555$	−12.0000	−0.509372
$556$	−4.00000	−0.169638
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	0	0
$560$	−48.0000	−2.02837
$561$	−3.00000	−0.126660
$562$	0	0
$563$	−30.0000	−1.26435	−0.632175	−	0.774826i	$-0.717837\pi$
$563$	−30.0000	−1.26435	−0.632175	+	0.774826i	$0.717837\pi$
$564$	−12.0000	−0.505291
$565$	27.0000	1.13590
$566$	0	0
$567$	4.00000	0.167984
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	0	0
$573$	18.0000	0.751961
$574$	0	0
$575$	36.0000	1.50130
$576$	−8.00000	−0.333333
$577$	7.00000	0.291414	0.145707	−	0.989328i	$-0.453454\pi$
$577$	7.00000	0.291414	0.145707	+	0.989328i	$0.453454\pi$
$578$	0	0
$579$	22.0000	0.914289
$580$	36.0000	1.49482
$581$	24.0000	0.995688
$582$	0	0
$583$	−18.0000	−0.745484
$584$	0	0
$585$	0	0
$586$	0	0
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	−18.0000	−0.742307
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	−3.00000	−0.123404
$592$	16.0000	0.657596
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	12.0000	0.491952
$596$	−36.0000	−1.47462
$597$	−16.0000	−0.654836
$598$	0	0
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	0	0
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	16.0000	0.651031
$605$	6.00000	0.243935
$606$	0	0
$607$	38.0000	1.54237	0.771186	−	0.636610i	$-0.219664\pi$
$607$	38.0000	1.54237	0.771186	+	0.636610i	$0.219664\pi$
$608$	0	0
$609$	24.0000	0.972529
$610$	0	0
$611$	0	0
$612$	2.00000	0.0808452
$613$	−11.0000	−0.444286	−0.222143	−	0.975014i	$-0.571305\pi$
$613$	−11.0000	−0.444286	−0.222143	+	0.975014i	$0.571305\pi$
$614$	0	0
$615$	−9.00000	−0.362915
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	−12.0000	−0.481932
$621$	9.00000	0.361158
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	3.00000	0.119808
$628$	−22.0000	−0.877896
$629$	−4.00000	−0.159490
$630$	0	0
$631$	37.0000	1.47295	0.736473	−	0.676467i	$-0.236490\pi$
$631$	37.0000	1.47295	0.736473	+	0.676467i	$0.236490\pi$
$632$	0	0
$633$	2.00000	0.0794929
$634$	0	0
$635$	39.0000	1.54767
$636$	12.0000	0.475831
$637$	0	0
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	−33.0000	−1.30342	−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000	−1.30342	−0.651711	+	0.758468i	$0.725948\pi$
$642$	0	0
$643$	−32.0000	−1.26196	−0.630978	−	0.775800i	$-0.717346\pi$
$643$	−32.0000	−1.26196	−0.630978	+	0.775800i	$0.717346\pi$
$644$	−72.0000	−2.83720
$645$	21.0000	0.826874
$646$	0	0
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	0	0
$649$	−18.0000	−0.706562
$650$	0	0
$651$	−8.00000	−0.313545
$652$	4.00000	0.156652
$653$	27.0000	1.05659	0.528296	−	0.849060i	$-0.322831\pi$
$653$	27.0000	1.05659	0.528296	+	0.849060i	$0.322831\pi$
$654$	0	0
$655$	−9.00000	−0.351659
$656$	12.0000	0.468521
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	18.0000	0.700649
$661$	31.0000	1.20576	0.602880	−	0.797832i	$-0.294020\pi$
$661$	31.0000	1.20576	0.602880	+	0.797832i	$0.294020\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−12.0000	−0.465340
$666$	0	0
$667$	54.0000	2.09089
$668$	42.0000	1.62503
$669$	1.00000	0.0386622
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	−27.0000	−1.03769	−0.518847	−	0.854867i	$-0.673639\pi$
$677$	−27.0000	−1.03769	−0.518847	+	0.854867i	$0.673639\pi$
$678$	0	0
$679$	64.0000	2.45609
$680$	0	0
$681$	−3.00000	−0.114960
$682$	0	0
$683$	−3.00000	−0.114792	−0.0573959	−	0.998351i	$-0.518280\pi$
$683$	−3.00000	−0.114792	−0.0573959	+	0.998351i	$0.518280\pi$
$684$	−2.00000	−0.0764719
$685$	−18.0000	−0.687745
$686$	0	0
$687$	−14.0000	−0.534133
$688$	−28.0000	−1.06749
$689$	0	0
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−30.0000	−1.14043
$693$	12.0000	0.455842
$694$	0	0
$695$	−6.00000	−0.227593
$696$	0	0
$697$	−3.00000	−0.113633
$698$	0	0
$699$	21.0000	0.794293
$700$	−32.0000	−1.20949
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	−24.0000	−0.904534
$705$	−18.0000	−0.677919
$706$	0	0
$707$	0	0
$708$	12.0000	0.450988
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	0	0
$713$	−18.0000	−0.674105
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	12.0000	0.448148
$718$	0	0
$719$	3.00000	0.111881	0.0559406	−	0.998434i	$-0.482184\pi$
$719$	3.00000	0.111881	0.0559406	+	0.998434i	$0.482184\pi$
$720$	−12.0000	−0.447214
$721$	20.0000	0.744839
$722$	0	0
$723$	−8.00000	−0.297523
$724$	−28.0000	−1.04061
$725$	24.0000	0.891338
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	7.00000	0.258904
$732$	−16.0000	−0.591377
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	0	0
$735$	−27.0000	−0.995910
$736$	0	0
$737$	12.0000	0.442026
$738$	0	0
$739$	1.00000	0.0367856	0.0183928	−	0.999831i	$-0.494145\pi$
$739$	1.00000	0.0367856	0.0183928	+	0.999831i	$0.494145\pi$
$740$	24.0000	0.882258
$741$	0	0
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	−54.0000	−1.97841
$746$	0	0
$747$	6.00000	0.219529
$748$	6.00000	0.219382
$749$	36.0000	1.31541
$750$	0	0
$751$	−46.0000	−1.67856	−0.839282	−	0.543696i	$-0.817024\pi$
$751$	−46.0000	−1.67856	−0.839282	+	0.543696i	$0.817024\pi$
$752$	24.0000	0.875190
$753$	24.0000	0.874609
$754$	0	0
$755$	24.0000	0.873449
$756$	−8.00000	−0.290957
$757$	−43.0000	−1.56286	−0.781431	−	0.623992i	$-0.785510\pi$
$757$	−43.0000	−1.56286	−0.781431	+	0.623992i	$0.785510\pi$
$758$	0	0
$759$	27.0000	0.980038
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	−80.0000	−2.89619
$764$	−36.0000	−1.30243
$765$	3.00000	0.108465
$766$	0	0
$767$	0	0
$768$	16.0000	0.577350
$769$	−41.0000	−1.47850	−0.739249	−	0.673432i	$-0.764819\pi$
$769$	−41.0000	−1.47850	−0.739249	+	0.673432i	$0.764819\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	−44.0000	−1.58359
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	16.0000	0.573997
$778$	0	0
$779$	3.00000	0.107486
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	6.00000	0.214423
$784$	36.0000	1.28571
$785$	−33.0000	−1.17782
$786$	0	0
$787$	40.0000	1.42585	0.712923	−	0.701242i	$-0.247371\pi$
$787$	40.0000	1.42585	0.712923	+	0.701242i	$0.247371\pi$
$788$	6.00000	0.213741
$789$	12.0000	0.427211
$790$	0	0
$791$	−36.0000	−1.28001
$792$	0	0
$793$	0	0
$794$	0	0
$795$	18.0000	0.638394
$796$	32.0000	1.13421
$797$	−48.0000	−1.70025	−0.850124	−	0.526583i	$-0.823473\pi$
$797$	−48.0000	−1.70025	−0.850124	+	0.526583i	$0.823473\pi$
$798$	0	0
$799$	−6.00000	−0.212265
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6.00000	−0.211735
$804$	−8.00000	−0.282138
$805$	−108.000	−3.80650
$806$	0	0
$807$	−15.0000	−0.528025
$808$	0	0
$809$	−51.0000	−1.79306	−0.896532	−	0.442978i	$-0.853922\pi$
$809$	−51.0000	−1.79306	−0.896532	+	0.442978i	$0.853922\pi$
$810$	0	0
$811$	10.0000	0.351147	0.175574	−	0.984466i	$-0.443822\pi$
$811$	10.0000	0.351147	0.175574	+	0.984466i	$0.443822\pi$
$812$	−48.0000	−1.68447
$813$	−11.0000	−0.385787
$814$	0	0
$815$	6.00000	0.210171
$816$	−4.00000	−0.140028
$817$	−7.00000	−0.244899
$818$	0	0
$819$	0	0
$820$	18.0000	0.628587
$821$	21.0000	0.732905	0.366453	−	0.930437i	$-0.380572\pi$
$821$	21.0000	0.732905	0.366453	+	0.930437i	$0.380572\pi$
$822$	0	0
$823$	−28.0000	−0.976019	−0.488009	−	0.872838i	$-0.662277\pi$
$823$	−28.0000	−0.976019	−0.488009	+	0.872838i	$0.662277\pi$
$824$	0	0
$825$	12.0000	0.417786
$826$	0	0
$827$	21.0000	0.730242	0.365121	−	0.930960i	$-0.381028\pi$
$827$	21.0000	0.730242	0.365121	+	0.930960i	$0.381028\pi$
$828$	−18.0000	−0.625543
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	0	0
$833$	−9.00000	−0.311832
$834$	0	0
$835$	63.0000	2.18020
$836$	−6.00000	−0.207514
$837$	−2.00000	−0.0691301
$838$	0	0
$839$	57.0000	1.96786	0.983929	−	0.178559i	$-0.0571434\pi$
$839$	57.0000	1.96786	0.983929	+	0.178559i	$0.0571434\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−12.0000	−0.413302
$844$	−4.00000	−0.137686
$845$	0	0
$846$	0	0
$847$	−8.00000	−0.274883
$848$	−24.0000	−0.824163
$849$	−10.0000	−0.343199
$850$	0	0
$851$	36.0000	1.23406
$852$	24.0000	0.822226
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	−3.00000	−0.102598
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	−42.0000	−1.43219
$861$	12.0000	0.408959
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	−45.0000	−1.53005
$866$	0	0
$867$	1.00000	0.0339618
$868$	16.0000	0.543075
$869$	−30.0000	−1.01768
$870$	0	0
$871$	0	0
$872$	0	0
$873$	16.0000	0.541518
$874$	0	0
$875$	12.0000	0.405674
$876$	4.00000	0.135147
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	0	0
$879$	24.0000	0.809500
$880$	−36.0000	−1.21356
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	11.0000	0.370179	0.185090	−	0.982722i	$-0.440742\pi$
$883$	11.0000	0.370179	0.185090	+	0.982722i	$0.440742\pi$
$884$	0	0
$885$	18.0000	0.605063
$886$	0	0
$887$	39.0000	1.30949	0.654746	−	0.755849i	$-0.272776\pi$
$887$	39.0000	1.30949	0.654746	+	0.755849i	$0.272776\pi$
$888$	0	0
$889$	−52.0000	−1.74402
$890$	0	0
$891$	3.00000	0.100504
$892$	−2.00000	−0.0669650
$893$	6.00000	0.200782
$894$	0	0
$895$	18.0000	0.601674
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−12.0000	−0.400222
$900$	−8.00000	−0.266667
$901$	6.00000	0.199889
$902$	0	0
$903$	−28.0000	−0.931782
$904$	0	0
$905$	−42.0000	−1.39613
$906$	0	0
$907$	2.00000	0.0664089	0.0332045	−	0.999449i	$-0.489429\pi$
$907$	2.00000	0.0664089	0.0332045	+	0.999449i	$0.489429\pi$
$908$	6.00000	0.199117
$909$	0	0
$910$	0	0
$911$	−27.0000	−0.894550	−0.447275	−	0.894397i	$-0.647605\pi$
$911$	−27.0000	−0.894550	−0.447275	+	0.894397i	$0.647605\pi$
$912$	4.00000	0.132453
$913$	18.0000	0.595713
$914$	0	0
$915$	−24.0000	−0.793416
$916$	28.0000	0.925146
$917$	12.0000	0.396275
$918$	0	0
$919$	11.0000	0.362857	0.181428	−	0.983404i	$-0.441928\pi$
$919$	11.0000	0.362857	0.181428	+	0.983404i	$0.441928\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	0	0
$924$	−24.0000	−0.789542
$925$	16.0000	0.526077
$926$	0	0
$927$	5.00000	0.164222
$928$	0	0
$929$	−15.0000	−0.492134	−0.246067	−	0.969253i	$-0.579138\pi$
$929$	−15.0000	−0.492134	−0.246067	+	0.969253i	$0.579138\pi$
$930$	0	0
$931$	9.00000	0.294963
$932$	−42.0000	−1.37576
$933$	−24.0000	−0.785725
$934$	0	0
$935$	9.00000	0.294331
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	−16.0000	−0.522140
$940$	36.0000	1.17419
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	27.0000	0.879241
$944$	−24.0000	−0.781133
$945$	−12.0000	−0.390360
$946$	0	0
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	20.0000	0.649570
$949$	0	0
$950$	0	0
$951$	6.00000	0.194563
$952$	0	0
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	0	0
$955$	−54.0000	−1.74740
$956$	−24.0000	−0.776215
$957$	18.0000	0.581857
$958$	0	0
$959$	24.0000	0.775000
$960$	24.0000	0.774597
$961$	−27.0000	−0.870968
$962$	0	0
$963$	9.00000	0.290021
$964$	16.0000	0.515325
$965$	−66.0000	−2.12462
$966$	0	0
$967$	−41.0000	−1.31847	−0.659236	−	0.751936i	$-0.729120\pi$
$967$	−41.0000	−1.31847	−0.659236	+	0.751936i	$0.729120\pi$
$968$	0	0
$969$	−1.00000	−0.0321246
$970$	0	0
$971$	6.00000	0.192549	0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000	0.192549	0.0962746	+	0.995355i	$0.469307\pi$
$972$	−2.00000	−0.0641500
$973$	8.00000	0.256468
$974$	0	0
$975$	0	0
$976$	32.0000	1.02430
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0	0
$979$	0	0
$980$	54.0000	1.72497
$981$	−20.0000	−0.638551
$982$	0	0
$983$	9.00000	0.287055	0.143528	−	0.989646i	$-0.454155\pi$
$983$	9.00000	0.287055	0.143528	+	0.989646i	$0.454155\pi$
$984$	0	0
$985$	9.00000	0.286764
$986$	0	0
$987$	24.0000	0.763928
$988$	0	0
$989$	−63.0000	−2.00328
$990$	0	0
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	0	0
$993$	13.0000	0.412543
$994$	0	0
$995$	48.0000	1.52170
$996$	−12.0000	−0.380235
$997$	62.0000	1.96356	0.981780	−	0.190022i	$-0.0608559\pi$
$997$	62.0000	1.96356	0.981780	+	0.190022i	$0.0608559\pi$
$998$	0	0
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8619.2.a.g.1.1		1
13.12	even	2		51.2.a.a.1.1	✓	1
39.38	odd	2		153.2.a.b.1.1		1
52.51	odd	2		816.2.a.g.1.1		1
65.12	odd	4		1275.2.b.b.1174.1		2
65.38	odd	4		1275.2.b.b.1174.2		2
65.64	even	2		1275.2.a.d.1.1		1
91.90	odd	2		2499.2.a.d.1.1		1
104.51	odd	2		3264.2.a.r.1.1		1
104.77	even	2		3264.2.a.a.1.1		1
143.142	odd	2		6171.2.a.e.1.1		1
156.155	even	2		2448.2.a.c.1.1		1
195.194	odd	2		3825.2.a.i.1.1		1
221.12	odd	16		867.2.h.c.688.2		8
221.25	even	8		867.2.e.e.829.1		4
221.38	even	4		867.2.d.a.577.1		2
221.64	even	4		867.2.d.a.577.2		2
221.77	even	8		867.2.e.e.829.2		4
221.90	odd	16		867.2.h.c.688.1		8
221.116	odd	16		867.2.h.c.757.1		8
221.129	odd	16		867.2.h.c.712.2		8
221.142	odd	16		867.2.h.c.733.2		8
221.155	even	8		867.2.e.e.616.2		4
221.168	even	8		867.2.e.e.616.1		4
221.181	odd	16		867.2.h.c.733.1		8
221.194	odd	16		867.2.h.c.712.1		8
221.207	odd	16		867.2.h.c.757.2		8
221.220	even	2		867.2.a.c.1.1		1
273.272	even	2		7497.2.a.j.1.1		1
312.77	odd	2		9792.2.a.by.1.1		1
312.155	even	2		9792.2.a.cd.1.1		1
663.662	odd	2		2601.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.a.a.1.1	✓	1	13.12	even	2
153.2.a.b.1.1		1	39.38	odd	2
816.2.a.g.1.1		1	52.51	odd	2
867.2.a.c.1.1		1	221.220	even	2
867.2.d.a.577.1		2	221.38	even	4
867.2.d.a.577.2		2	221.64	even	4
867.2.e.e.616.1		4	221.168	even	8
867.2.e.e.616.2		4	221.155	even	8
867.2.e.e.829.1		4	221.25	even	8
867.2.e.e.829.2		4	221.77	even	8
867.2.h.c.688.1		8	221.90	odd	16
867.2.h.c.688.2		8	221.12	odd	16
867.2.h.c.712.1		8	221.194	odd	16
867.2.h.c.712.2		8	221.129	odd	16
867.2.h.c.733.1		8	221.181	odd	16
867.2.h.c.733.2		8	221.142	odd	16
867.2.h.c.757.1		8	221.116	odd	16
867.2.h.c.757.2		8	221.207	odd	16
1275.2.a.d.1.1		1	65.64	even	2
1275.2.b.b.1174.1		2	65.12	odd	4
1275.2.b.b.1174.2		2	65.38	odd	4
2448.2.a.c.1.1		1	156.155	even	2
2499.2.a.d.1.1		1	91.90	odd	2
2601.2.a.f.1.1		1	663.662	odd	2
3264.2.a.a.1.1		1	104.77	even	2
3264.2.a.r.1.1		1	104.51	odd	2
3825.2.a.i.1.1		1	195.194	odd	2
6171.2.a.e.1.1		1	143.142	odd	2
7497.2.a.j.1.1		1	273.272	even	2
8619.2.a.g.1.1		1	1.1	even	1	trivial
9792.2.a.by.1.1		1	312.77	odd	2
9792.2.a.cd.1.1		1	312.155	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:	\( N \)	\(=\)	\( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8619.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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