LMFDB - Embedded newform 8036.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$2$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q-1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{9} -5.00000 q^{11} -2.00000 q^{13} +1.00000 q^{15} -3.00000 q^{17} -3.00000 q^{19} -1.00000 q^{23} -4.00000 q^{25} +5.00000 q^{27} -10.0000 q^{29} -11.0000 q^{31} +5.00000 q^{33} +7.00000 q^{37} +2.00000 q^{39} -1.00000 q^{41} +8.00000 q^{43} +2.00000 q^{45} -7.00000 q^{47} +3.00000 q^{51} -11.0000 q^{53} +5.00000 q^{55} +3.00000 q^{57} -7.00000 q^{59} -1.00000 q^{61} +2.00000 q^{65} +7.00000 q^{67} +1.00000 q^{69} -12.0000 q^{71} -5.00000 q^{73} +4.00000 q^{75} +9.00000 q^{79} +1.00000 q^{81} +3.00000 q^{85} +10.0000 q^{87} -3.00000 q^{89} +11.0000 q^{93} +3.00000 q^{95} +2.00000 q^{97} +10.0000 q^{99} +O(q^{100})$

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
$2$	0	0
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	−11.0000	−1.97566	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−11.0000	−1.97566	−0.987829	+	0.155543i	$0.950287\pi$
$32$	0	0
$33$	5.00000	0.870388
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	−7.00000	−1.02105	−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000	−1.02105	−0.510527	+	0.859861i	$0.670550\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	−11.0000	−1.51097	−0.755483	−	0.655168i	$-0.772598\pi$
$53$	−11.0000	−1.51097	−0.755483	+	0.655168i	$0.772598\pi$
$54$	0	0
$55$	5.00000	0.674200
$56$	0	0
$57$	3.00000	0.397360
$58$	0	0
$59$	−7.00000	−0.911322	−0.455661	−	0.890153i	$-0.650597\pi$
$59$	−7.00000	−0.911322	−0.455661	+	0.890153i	$0.650597\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	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
$73$	−5.00000	−0.585206	−0.292603	−	0.956234i	$-0.594521\pi$
$73$	−5.00000	−0.585206	−0.292603	+	0.956234i	$0.594521\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	0	0
$78$	0	0
$79$	9.00000	1.01258	0.506290	−	0.862364i	$-0.331017\pi$
$79$	9.00000	1.01258	0.506290	+	0.862364i	$0.331017\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	0	0
$87$	10.0000	1.07211
$88$	0	0
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	11.0000	1.14065
$94$	0	0
$95$	3.00000	0.307794
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	10.0000	1.00504
$100$	0	0
$101$	−11.0000	−1.09454	−0.547270	−	0.836956i	$-0.684333\pi$
$101$	−11.0000	−1.09454	−0.547270	+	0.836956i	$0.684333\pi$
$102$	0	0
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.0000	−1.25676	−0.628379	−	0.777908i	$-0.716281\pi$
$107$	−13.0000	−1.25676	−0.628379	+	0.777908i	$0.716281\pi$
$108$	0	0
$109$	17.0000	1.62830	0.814152	−	0.580651i	$-0.197202\pi$
$109$	17.0000	1.62830	0.814152	+	0.580651i	$0.197202\pi$
$110$	0	0
$111$	−7.00000	−0.664411
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	4.00000	0.369800
$118$	0	0
$119$	0	0
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	1.00000	0.0901670
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	7.00000	0.589506
$142$	0	0
$143$	10.0000	0.836242
$144$	0	0
$145$	10.0000	0.830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.00000	−0.573462	−0.286731	−	0.958011i	$-0.592569\pi$
$149$	−7.00000	−0.573462	−0.286731	+	0.958011i	$0.592569\pi$
$150$	0	0
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	11.0000	0.883541
$156$	0	0
$157$	−3.00000	−0.239426	−0.119713	−	0.992809i	$-0.538197\pi$
$157$	−3.00000	−0.239426	−0.119713	+	0.992809i	$0.538197\pi$
$158$	0	0
$159$	11.0000	0.872357
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−17.0000	−1.33154	−0.665771	−	0.746156i	$-0.731897\pi$
$163$	−17.0000	−1.33154	−0.665771	+	0.746156i	$0.731897\pi$
$164$	0	0
$165$	−5.00000	−0.389249
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	6.00000	0.458831
$172$	0	0
$173$	−21.0000	−1.59660	−0.798300	−	0.602260i	$-0.794267\pi$
$173$	−21.0000	−1.59660	−0.798300	+	0.602260i	$0.794267\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.00000	0.526152
$178$	0	0
$179$	7.00000	0.523205	0.261602	−	0.965176i	$-0.415749\pi$
$179$	7.00000	0.523205	0.261602	+	0.965176i	$0.415749\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	1.00000	0.0739221
$184$	0	0
$185$	−7.00000	−0.514650
$186$	0	0
$187$	15.0000	1.09691
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.00000	0.0723575	0.0361787	−	0.999345i	$-0.488481\pi$
$191$	1.00000	0.0723575	0.0361787	+	0.999345i	$0.488481\pi$
$192$	0	0
$193$	17.0000	1.22369	0.611843	−	0.790979i	$-0.290428\pi$
$193$	17.0000	1.22369	0.611843	+	0.790979i	$0.290428\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	0	0
$197$	26.0000	1.85242	0.926212	−	0.377004i	$-0.123046\pi$
$197$	26.0000	1.85242	0.926212	+	0.377004i	$0.123046\pi$
$198$	0	0
$199$	−21.0000	−1.48865	−0.744325	−	0.667817i	$-0.767229\pi$
$199$	−21.0000	−1.48865	−0.744325	+	0.667817i	$0.767229\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1.00000	0.0698430
$206$	0	0
$207$	2.00000	0.139010
$208$	0	0
$209$	15.0000	1.03757
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	12.0000	0.822226
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	0	0
$218$	0	0
$219$	5.00000	0.337869
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	8.00000	0.533333
$226$	0	0
$227$	−9.00000	−0.597351	−0.298675	−	0.954355i	$-0.596545\pi$
$227$	−9.00000	−0.597351	−0.298675	+	0.954355i	$0.596545\pi$
$228$	0	0
$229$	−27.0000	−1.78421	−0.892105	−	0.451828i	$-0.850772\pi$
$229$	−27.0000	−1.78421	−0.892105	+	0.451828i	$0.850772\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.0000	0.851658	0.425829	−	0.904804i	$-0.359982\pi$
$233$	13.0000	0.851658	0.425829	+	0.904804i	$0.359982\pi$
$234$	0	0
$235$	7.00000	0.456630
$236$	0	0
$237$	−9.00000	−0.584613
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	19.0000	1.22390	0.611949	−	0.790897i	$-0.290386\pi$
$241$	19.0000	1.22390	0.611949	+	0.790897i	$0.290386\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	5.00000	0.314347
$254$	0	0
$255$	−3.00000	−0.187867
$256$	0	0
$257$	−7.00000	−0.436648	−0.218324	−	0.975876i	$-0.570059\pi$
$257$	−7.00000	−0.436648	−0.218324	+	0.975876i	$0.570059\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	20.0000	1.23797
$262$	0	0
$263$	19.0000	1.17159	0.585795	−	0.810459i	$-0.300782\pi$
$263$	19.0000	1.17159	0.585795	+	0.810459i	$0.300782\pi$
$264$	0	0
$265$	11.0000	0.675725
$266$	0	0
$267$	3.00000	0.183597
$268$	0	0
$269$	11.0000	0.670682	0.335341	−	0.942097i	$-0.391148\pi$
$269$	11.0000	0.670682	0.335341	+	0.942097i	$0.391148\pi$
$270$	0	0
$271$	−25.0000	−1.51864	−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000	−1.51864	−0.759321	+	0.650716i	$0.774469\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	20.0000	1.20605
$276$	0	0
$277$	23.0000	1.38194	0.690968	−	0.722885i	$-0.257185\pi$
$277$	23.0000	1.38194	0.690968	+	0.722885i	$0.257185\pi$
$278$	0	0
$279$	22.0000	1.31711
$280$	0	0
$281$	−14.0000	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$282$	0	0
$283$	17.0000	1.01055	0.505273	−	0.862960i	$-0.331392\pi$
$283$	17.0000	1.01055	0.505273	+	0.862960i	$0.331392\pi$
$284$	0	0
$285$	−3.00000	−0.177705
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−2.00000	−0.117242
$292$	0	0
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	7.00000	0.407556
$296$	0	0
$297$	−25.0000	−1.45065
$298$	0	0
$299$	2.00000	0.115663
$300$	0	0
$301$	0	0
$302$	0	0
$303$	11.0000	0.631933
$304$	0	0
$305$	1.00000	0.0572598
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	1.00000	0.0568880
$310$	0	0
$311$	−1.00000	−0.0567048	−0.0283524	−	0.999598i	$-0.509026\pi$
$311$	−1.00000	−0.0567048	−0.0283524	+	0.999598i	$0.509026\pi$
$312$	0	0
$313$	−11.0000	−0.621757	−0.310878	−	0.950450i	$-0.600623\pi$
$313$	−11.0000	−0.621757	−0.310878	+	0.950450i	$0.600623\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.0000	0.954815	0.477408	−	0.878682i	$-0.341577\pi$
$317$	17.0000	0.954815	0.477408	+	0.878682i	$0.341577\pi$
$318$	0	0
$319$	50.0000	2.79946
$320$	0	0
$321$	13.0000	0.725589
$322$	0	0
$323$	9.00000	0.500773
$324$	0	0
$325$	8.00000	0.443760
$326$	0	0
$327$	−17.0000	−0.940102
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−3.00000	−0.164895	−0.0824475	−	0.996595i	$-0.526274\pi$
$331$	−3.00000	−0.164895	−0.0824475	+	0.996595i	$0.526274\pi$
$332$	0	0
$333$	−14.0000	−0.767195
$334$	0	0
$335$	−7.00000	−0.382451
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	55.0000	2.97842
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	−9.00000	−0.483145	−0.241573	−	0.970383i	$-0.577663\pi$
$347$	−9.00000	−0.483145	−0.241573	+	0.970383i	$0.577663\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	−10.0000	−0.533761
$352$	0	0
$353$	−9.00000	−0.479022	−0.239511	−	0.970894i	$-0.576987\pi$
$353$	−9.00000	−0.479022	−0.239511	+	0.970894i	$0.576987\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−29.0000	−1.53056	−0.765281	−	0.643697i	$-0.777400\pi$
$359$	−29.0000	−1.53056	−0.765281	+	0.643697i	$0.777400\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	0	0
$363$	−14.0000	−0.734809
$364$	0	0
$365$	5.00000	0.261712
$366$	0	0
$367$	−15.0000	−0.782994	−0.391497	−	0.920179i	$-0.628043\pi$
$367$	−15.0000	−0.782994	−0.391497	+	0.920179i	$0.628043\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	0	0
$372$	0	0
$373$	19.0000	0.983783	0.491891	−	0.870657i	$-0.336306\pi$
$373$	19.0000	0.983783	0.491891	+	0.870657i	$0.336306\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	20.0000	1.03005
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	−27.0000	−1.37964	−0.689818	−	0.723983i	$-0.742309\pi$
$383$	−27.0000	−1.37964	−0.689818	+	0.723983i	$0.742309\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−16.0000	−0.813326
$388$	0	0
$389$	3.00000	0.152106	0.0760530	−	0.997104i	$-0.475768\pi$
$389$	3.00000	0.152106	0.0760530	+	0.997104i	$0.475768\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	0	0
$393$	−3.00000	−0.151330
$394$	0	0
$395$	−9.00000	−0.452839
$396$	0	0
$397$	29.0000	1.45547	0.727734	−	0.685859i	$-0.240573\pi$
$397$	29.0000	1.45547	0.727734	+	0.685859i	$0.240573\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−13.0000	−0.649189	−0.324595	−	0.945853i	$-0.605228\pi$
$401$	−13.0000	−0.649189	−0.324595	+	0.945853i	$0.605228\pi$
$402$	0	0
$403$	22.0000	1.09590
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−35.0000	−1.73489
$408$	0	0
$409$	35.0000	1.73064	0.865319	−	0.501221i	$-0.167116\pi$
$409$	35.0000	1.73064	0.865319	+	0.501221i	$0.167116\pi$
$410$	0	0
$411$	−9.00000	−0.443937
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	20.0000	0.979404
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	0	0
$423$	14.0000	0.680703
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−10.0000	−0.482805
$430$	0	0
$431$	−35.0000	−1.68589	−0.842945	−	0.537999i	$-0.819180\pi$
$431$	−35.0000	−1.68589	−0.842945	+	0.537999i	$0.819180\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	−10.0000	−0.479463
$436$	0	0
$437$	3.00000	0.143509
$438$	0	0
$439$	−7.00000	−0.334092	−0.167046	−	0.985949i	$-0.553423\pi$
$439$	−7.00000	−0.334092	−0.167046	+	0.985949i	$0.553423\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	39.0000	1.85295	0.926473	−	0.376361i	$-0.122825\pi$
$443$	39.0000	1.85295	0.926473	+	0.376361i	$0.122825\pi$
$444$	0	0
$445$	3.00000	0.142214
$446$	0	0
$447$	7.00000	0.331089
$448$	0	0
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	0	0
$451$	5.00000	0.235441
$452$	0	0
$453$	5.00000	0.234920
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.00000	0.233890	0.116945	−	0.993138i	$-0.462690\pi$
$457$	5.00000	0.233890	0.116945	+	0.993138i	$0.462690\pi$
$458$	0	0
$459$	−15.0000	−0.700140
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	−11.0000	−0.510113
$466$	0	0
$467$	−33.0000	−1.52706	−0.763529	−	0.645774i	$-0.776535\pi$
$467$	−33.0000	−1.52706	−0.763529	+	0.645774i	$0.776535\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	3.00000	0.138233
$472$	0	0
$473$	−40.0000	−1.83920
$474$	0	0
$475$	12.0000	0.550598
$476$	0	0
$477$	22.0000	1.00731
$478$	0	0
$479$	−17.0000	−0.776750	−0.388375	−	0.921501i	$-0.626963\pi$
$479$	−17.0000	−0.776750	−0.388375	+	0.921501i	$0.626963\pi$
$480$	0	0
$481$	−14.0000	−0.638345
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	−35.0000	−1.58600	−0.793001	−	0.609221i	$-0.791482\pi$
$487$	−35.0000	−1.58600	−0.793001	+	0.609221i	$0.791482\pi$
$488$	0	0
$489$	17.0000	0.768767
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	30.0000	1.35113
$494$	0	0
$495$	−10.0000	−0.449467
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−3.00000	−0.134298	−0.0671492	−	0.997743i	$-0.521390\pi$
$499$	−3.00000	−0.134298	−0.0671492	+	0.997743i	$0.521390\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	0	0
$503$	40.0000	1.78351	0.891756	−	0.452517i	$-0.149474\pi$
$503$	40.0000	1.78351	0.891756	+	0.452517i	$0.149474\pi$
$504$	0	0
$505$	11.0000	0.489494
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	5.00000	0.221621	0.110811	−	0.993842i	$-0.464655\pi$
$509$	5.00000	0.221621	0.110811	+	0.993842i	$0.464655\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−15.0000	−0.662266
$514$	0	0
$515$	1.00000	0.0440653
$516$	0	0
$517$	35.0000	1.53930
$518$	0	0
$519$	21.0000	0.921798
$520$	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
$523$	39.0000	1.70535	0.852675	−	0.522441i	$-0.174978\pi$
$523$	39.0000	1.70535	0.852675	+	0.522441i	$0.174978\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	33.0000	1.43750
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	14.0000	0.607548
$532$	0	0
$533$	2.00000	0.0866296
$534$	0	0
$535$	13.0000	0.562039
$536$	0	0
$537$	−7.00000	−0.302072
$538$	0	0
$539$	0	0
$540$	0	0
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	0	0
$543$	22.0000	0.944110
$544$	0	0
$545$	−17.0000	−0.728200
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	30.0000	1.27804
$552$	0	0
$553$	0	0
$554$	0	0
$555$	7.00000	0.297133
$556$	0	0
$557$	33.0000	1.39825	0.699127	−	0.714997i	$-0.253572\pi$
$557$	33.0000	1.39825	0.699127	+	0.714997i	$0.253572\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	−15.0000	−0.633300
$562$	0	0
$563$	−25.0000	−1.05362	−0.526812	−	0.849982i	$-0.676613\pi$
$563$	−25.0000	−1.05362	−0.526812	+	0.849982i	$0.676613\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	0	0
$571$	15.0000	0.627730	0.313865	−	0.949468i	$-0.398376\pi$
$571$	15.0000	0.627730	0.313865	+	0.949468i	$0.398376\pi$
$572$	0	0
$573$	−1.00000	−0.0417756
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	−39.0000	−1.62359	−0.811796	−	0.583942i	$-0.801510\pi$
$577$	−39.0000	−1.62359	−0.811796	+	0.583942i	$0.801510\pi$
$578$	0	0
$579$	−17.0000	−0.706496
$580$	0	0
$581$	0	0
$582$	0	0
$583$	55.0000	2.27787
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	0	0
$589$	33.0000	1.35974
$590$	0	0
$591$	−26.0000	−1.06950
$592$	0	0
$593$	5.00000	0.205325	0.102663	−	0.994716i	$-0.467264\pi$
$593$	5.00000	0.205325	0.102663	+	0.994716i	$0.467264\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	21.0000	0.859473
$598$	0	0
$599$	9.00000	0.367730	0.183865	−	0.982952i	$-0.441139\pi$
$599$	9.00000	0.367730	0.183865	+	0.982952i	$0.441139\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	−14.0000	−0.570124
$604$	0	0
$605$	−14.0000	−0.569181
$606$	0	0
$607$	35.0000	1.42061	0.710303	−	0.703896i	$-0.248558\pi$
$607$	35.0000	1.42061	0.710303	+	0.703896i	$0.248558\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	14.0000	0.566379
$612$	0	0
$613$	27.0000	1.09052	0.545260	−	0.838267i	$-0.316431\pi$
$613$	27.0000	1.09052	0.545260	+	0.838267i	$0.316431\pi$
$614$	0	0
$615$	−1.00000	−0.0403239
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	−11.0000	−0.442127	−0.221064	−	0.975259i	$-0.570953\pi$
$619$	−11.0000	−0.442127	−0.221064	+	0.975259i	$0.570953\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	0	0
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−15.0000	−0.599042
$628$	0	0
$629$	−21.0000	−0.837325
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	12.0000	0.476957
$634$	0	0
$635$	−4.00000	−0.158735
$636$	0	0
$637$	0	0
$638$	0	0
$639$	24.0000	0.949425
$640$	0	0
$641$	−15.0000	−0.592464	−0.296232	−	0.955116i	$-0.595730\pi$
$641$	−15.0000	−0.592464	−0.296232	+	0.955116i	$0.595730\pi$
$642$	0	0
$643$	−36.0000	−1.41970	−0.709851	−	0.704352i	$-0.751238\pi$
$643$	−36.0000	−1.41970	−0.709851	+	0.704352i	$0.751238\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	−19.0000	−0.746967	−0.373484	−	0.927637i	$-0.621837\pi$
$647$	−19.0000	−0.746967	−0.373484	+	0.927637i	$0.621837\pi$
$648$	0	0
$649$	35.0000	1.37387
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.00000	0.352197	0.176099	−	0.984373i	$-0.443652\pi$
$653$	9.00000	0.352197	0.176099	+	0.984373i	$0.443652\pi$
$654$	0	0
$655$	−3.00000	−0.117220
$656$	0	0
$657$	10.0000	0.390137
$658$	0	0
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	51.0000	1.98367	0.991835	−	0.127527i	$-0.0407041\pi$
$661$	51.0000	1.98367	0.991835	+	0.127527i	$0.0407041\pi$
$662$	0	0
$663$	−6.00000	−0.233021
$664$	0	0
$665$	0	0
$666$	0	0
$667$	10.0000	0.387202
$668$	0	0
$669$	−8.00000	−0.309298
$670$	0	0
$671$	5.00000	0.193023
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	0	0
$675$	−20.0000	−0.769800
$676$	0	0
$677$	15.0000	0.576497	0.288248	−	0.957556i	$-0.406927\pi$
$677$	15.0000	0.576497	0.288248	+	0.957556i	$0.406927\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	9.00000	0.344881
$682$	0	0
$683$	31.0000	1.18618	0.593091	−	0.805135i	$-0.297907\pi$
$683$	31.0000	1.18618	0.593091	+	0.805135i	$0.297907\pi$
$684$	0	0
$685$	−9.00000	−0.343872
$686$	0	0
$687$	27.0000	1.03011
$688$	0	0
$689$	22.0000	0.838133
$690$	0	0
$691$	9.00000	0.342376	0.171188	−	0.985238i	$-0.445239\pi$
$691$	9.00000	0.342376	0.171188	+	0.985238i	$0.445239\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	20.0000	0.758643
$696$	0	0
$697$	3.00000	0.113633
$698$	0	0
$699$	−13.0000	−0.491705
$700$	0	0
$701$	50.0000	1.88847	0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	0	0
$703$	−21.0000	−0.792030
$704$	0	0
$705$	−7.00000	−0.263635
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−19.0000	−0.713560	−0.356780	−	0.934188i	$-0.616125\pi$
$709$	−19.0000	−0.713560	−0.356780	+	0.934188i	$0.616125\pi$
$710$	0	0
$711$	−18.0000	−0.675053
$712$	0	0
$713$	11.0000	0.411953
$714$	0	0
$715$	−10.0000	−0.373979
$716$	0	0
$717$	16.0000	0.597531
$718$	0	0
$719$	−11.0000	−0.410231	−0.205115	−	0.978738i	$-0.565757\pi$
$719$	−11.0000	−0.410231	−0.205115	+	0.978738i	$0.565757\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−19.0000	−0.706618
$724$	0	0
$725$	40.0000	1.48556
$726$	0	0
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	3.00000	0.110808	0.0554038	−	0.998464i	$-0.482355\pi$
$733$	3.00000	0.110808	0.0554038	+	0.998464i	$0.482355\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−35.0000	−1.28924
$738$	0	0
$739$	−15.0000	−0.551784	−0.275892	−	0.961189i	$-0.588973\pi$
$739$	−15.0000	−0.551784	−0.275892	+	0.961189i	$0.588973\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	0	0
$743$	−48.0000	−1.76095	−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000	−1.76095	−0.880475	+	0.474093i	$0.842776\pi$
$744$	0	0
$745$	7.00000	0.256460
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−7.00000	−0.255434	−0.127717	−	0.991811i	$-0.540765\pi$
$751$	−7.00000	−0.255434	−0.127717	+	0.991811i	$0.540765\pi$
$752$	0	0
$753$	−24.0000	−0.874609
$754$	0	0
$755$	5.00000	0.181969
$756$	0	0
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	0	0
$759$	−5.00000	−0.181489
$760$	0	0
$761$	−49.0000	−1.77625	−0.888124	−	0.459603i	$-0.847992\pi$
$761$	−49.0000	−1.77625	−0.888124	+	0.459603i	$0.847992\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−6.00000	−0.216930
$766$	0	0
$767$	14.0000	0.505511
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	7.00000	0.252099
$772$	0	0
$773$	−19.0000	−0.683383	−0.341691	−	0.939812i	$-0.611000\pi$
$773$	−19.0000	−0.683383	−0.341691	+	0.939812i	$0.611000\pi$
$774$	0	0
$775$	44.0000	1.58053
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.00000	0.107486
$780$	0	0
$781$	60.0000	2.14697
$782$	0	0
$783$	−50.0000	−1.78685
$784$	0	0
$785$	3.00000	0.107075
$786$	0	0
$787$	−47.0000	−1.67537	−0.837685	−	0.546154i	$-0.816091\pi$
$787$	−47.0000	−1.67537	−0.837685	+	0.546154i	$0.816091\pi$
$788$	0	0
$789$	−19.0000	−0.676418
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	−11.0000	−0.390130
$796$	0	0
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	0	0
$799$	21.0000	0.742927
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	25.0000	0.882231
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−11.0000	−0.387218
$808$	0	0
$809$	−31.0000	−1.08990	−0.544951	−	0.838468i	$-0.683452\pi$
$809$	−31.0000	−1.08990	−0.544951	+	0.838468i	$0.683452\pi$
$810$	0	0
$811$	−12.0000	−0.421377	−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000	−0.421377	−0.210688	+	0.977553i	$0.567571\pi$
$812$	0	0
$813$	25.0000	0.876788
$814$	0	0
$815$	17.0000	0.595484
$816$	0	0
$817$	−24.0000	−0.839654
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−17.0000	−0.593304	−0.296652	−	0.954986i	$-0.595870\pi$
$821$	−17.0000	−0.593304	−0.296652	+	0.954986i	$0.595870\pi$
$822$	0	0
$823$	−5.00000	−0.174289	−0.0871445	−	0.996196i	$-0.527774\pi$
$823$	−5.00000	−0.174289	−0.0871445	+	0.996196i	$0.527774\pi$
$824$	0	0
$825$	−20.0000	−0.696311
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	51.0000	1.77130	0.885652	−	0.464350i	$-0.153712\pi$
$829$	51.0000	1.77130	0.885652	+	0.464350i	$0.153712\pi$
$830$	0	0
$831$	−23.0000	−0.797861
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−16.0000	−0.553703
$836$	0	0
$837$	−55.0000	−1.90108
$838$	0	0
$839$	8.00000	0.276191	0.138095	−	0.990419i	$-0.455902\pi$
$839$	8.00000	0.276191	0.138095	+	0.990419i	$0.455902\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	0	0
$843$	14.0000	0.482186
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−17.0000	−0.583438
$850$	0	0
$851$	−7.00000	−0.239957
$852$	0	0
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	0	0
$855$	−6.00000	−0.205196
$856$	0	0
$857$	−13.0000	−0.444072	−0.222036	−	0.975039i	$-0.571270\pi$
$857$	−13.0000	−0.444072	−0.222036	+	0.975039i	$0.571270\pi$
$858$	0	0
$859$	11.0000	0.375315	0.187658	−	0.982235i	$-0.439910\pi$
$859$	11.0000	0.375315	0.187658	+	0.982235i	$0.439910\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−45.0000	−1.53182	−0.765909	−	0.642949i	$-0.777711\pi$
$863$	−45.0000	−1.53182	−0.765909	+	0.642949i	$0.777711\pi$
$864$	0	0
$865$	21.0000	0.714021
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	−45.0000	−1.52652
$870$	0	0
$871$	−14.0000	−0.474372
$872$	0	0
$873$	−4.00000	−0.135379
$874$	0	0
$875$	0	0
$876$	0	0
$877$	11.0000	0.371444	0.185722	−	0.982602i	$-0.440538\pi$
$877$	11.0000	0.371444	0.185722	+	0.982602i	$0.440538\pi$
$878$	0	0
$879$	18.0000	0.607125
$880$	0	0
$881$	−50.0000	−1.68454	−0.842271	−	0.539054i	$-0.818782\pi$
$881$	−50.0000	−1.68454	−0.842271	+	0.539054i	$0.818782\pi$
$882$	0	0
$883$	−32.0000	−1.07689	−0.538443	−	0.842662i	$-0.680987\pi$
$883$	−32.0000	−1.07689	−0.538443	+	0.842662i	$0.680987\pi$
$884$	0	0
$885$	−7.00000	−0.235302
$886$	0	0
$887$	−43.0000	−1.44380	−0.721899	−	0.691998i	$-0.756731\pi$
$887$	−43.0000	−1.44380	−0.721899	+	0.691998i	$0.756731\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−5.00000	−0.167506
$892$	0	0
$893$	21.0000	0.702738
$894$	0	0
$895$	−7.00000	−0.233984
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	0	0
$899$	110.000	3.66871
$900$	0	0
$901$	33.0000	1.09939
$902$	0	0
$903$	0	0
$904$	0	0
$905$	22.0000	0.731305
$906$	0	0
$907$	17.0000	0.564476	0.282238	−	0.959344i	$-0.408923\pi$
$907$	17.0000	0.564476	0.282238	+	0.959344i	$0.408923\pi$
$908$	0	0
$909$	22.0000	0.729694
$910$	0	0
$911$	−28.0000	−0.927681	−0.463841	−	0.885919i	$-0.653529\pi$
$911$	−28.0000	−0.927681	−0.463841	+	0.885919i	$0.653529\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−1.00000	−0.0330590
$916$	0	0
$917$	0	0
$918$	0	0
$919$	5.00000	0.164935	0.0824674	−	0.996594i	$-0.473720\pi$
$919$	5.00000	0.164935	0.0824674	+	0.996594i	$0.473720\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	0	0
$923$	24.0000	0.789970
$924$	0	0
$925$	−28.0000	−0.920634
$926$	0	0
$927$	2.00000	0.0656886
$928$	0	0
$929$	17.0000	0.557752	0.278876	−	0.960327i	$-0.410038\pi$
$929$	17.0000	0.557752	0.278876	+	0.960327i	$0.410038\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1.00000	0.0327385
$934$	0	0
$935$	−15.0000	−0.490552
$936$	0	0
$937$	6.00000	0.196011	0.0980057	−	0.995186i	$-0.468754\pi$
$937$	6.00000	0.196011	0.0980057	+	0.995186i	$0.468754\pi$
$938$	0	0
$939$	11.0000	0.358971
$940$	0	0
$941$	27.0000	0.880175	0.440087	−	0.897955i	$-0.354947\pi$
$941$	27.0000	0.880175	0.440087	+	0.897955i	$0.354947\pi$
$942$	0	0
$943$	1.00000	0.0325645
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15.0000	0.487435	0.243717	−	0.969846i	$-0.421633\pi$
$947$	15.0000	0.487435	0.243717	+	0.969846i	$0.421633\pi$
$948$	0	0
$949$	10.0000	0.324614
$950$	0	0
$951$	−17.0000	−0.551263
$952$	0	0
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	0	0
$955$	−1.00000	−0.0323592
$956$	0	0
$957$	−50.0000	−1.61627
$958$	0	0
$959$	0	0
$960$	0	0
$961$	90.0000	2.90323
$962$	0	0
$963$	26.0000	0.837838
$964$	0	0
$965$	−17.0000	−0.547249
$966$	0	0
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	0	0
$969$	−9.00000	−0.289122
$970$	0	0
$971$	−11.0000	−0.353007	−0.176503	−	0.984300i	$-0.556479\pi$
$971$	−11.0000	−0.353007	−0.176503	+	0.984300i	$0.556479\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−8.00000	−0.256205
$976$	0	0
$977$	−7.00000	−0.223950	−0.111975	−	0.993711i	$-0.535718\pi$
$977$	−7.00000	−0.223950	−0.111975	+	0.993711i	$0.535718\pi$
$978$	0	0
$979$	15.0000	0.479402
$980$	0	0
$981$	−34.0000	−1.08554
$982$	0	0
$983$	−11.0000	−0.350846	−0.175423	−	0.984493i	$-0.556129\pi$
$983$	−11.0000	−0.350846	−0.175423	+	0.984493i	$0.556129\pi$
$984$	0	0
$985$	−26.0000	−0.828429
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	−13.0000	−0.412959	−0.206479	−	0.978451i	$-0.566201\pi$
$991$	−13.0000	−0.412959	−0.206479	+	0.978451i	$0.566201\pi$
$992$	0	0
$993$	3.00000	0.0952021
$994$	0	0
$995$	21.0000	0.665745
$996$	0	0
$997$	37.0000	1.17180	0.585901	−	0.810383i	$-0.300741\pi$
$997$	37.0000	1.17180	0.585901	+	0.810383i	$0.300741\pi$
$998$	0	0
$999$	35.0000	1.10735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.a.1.1		1
7.2	even	3		1148.2.i.b.165.1	✓	2
7.4	even	3		1148.2.i.b.821.1	yes	2
7.6	odd	2		8036.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.i.b.165.1	✓	2	7.2	even	3
1148.2.i.b.821.1	yes	2	7.4	even	3
8036.2.a.a.1.1		1	1.1	even	1	trivial
8036.2.a.e.1.1		1	7.6	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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