LMFDB - Embedded newform 920.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [920,2,Mod(1,920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	920.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:	$7.34623698596$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} +1.00000 q^{5} -2.00000 q^{7} +6.00000 q^{9} +O(q^{10})$

$q-3.00000 q^{3} +1.00000 q^{5} -2.00000 q^{7} +6.00000 q^{9} +1.00000 q^{13} -3.00000 q^{15} +6.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -9.00000 q^{27} -3.00000 q^{29} +3.00000 q^{31} -2.00000 q^{35} -8.00000 q^{37} -3.00000 q^{39} +3.00000 q^{41} -2.00000 q^{43} +6.00000 q^{45} -11.0000 q^{47} -3.00000 q^{49} -14.0000 q^{53} -8.00000 q^{59} -4.00000 q^{61} -12.0000 q^{63} +1.00000 q^{65} -4.00000 q^{67} -3.00000 q^{69} +7.00000 q^{71} -9.00000 q^{73} -3.00000 q^{75} +9.00000 q^{81} +4.00000 q^{83} +9.00000 q^{87} -2.00000 q^{89} -2.00000 q^{91} -9.00000 q^{93} +18.0000 q^{97} +O(q^{100})$

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
$3$	−3.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	6.00000	2.00000
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	6.00000	1.30931
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−9.00000	−1.73205
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	−3.00000	−0.480384
$40$	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
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	6.00000	0.894427
$46$	0	0
$47$	−11.0000	−1.60451	−0.802257	−	0.596978i	$-0.796368\pi$
$47$	−11.0000	−1.60451	−0.802257	+	0.596978i	$0.796368\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−14.0000	−1.92305	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−14.0000	−1.92305	−0.961524	+	0.274721i	$0.911414\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	0	0
$63$	−12.0000	−1.51186
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	7.00000	0.830747	0.415374	−	0.909651i	$-0.363651\pi$
$71$	7.00000	0.830747	0.415374	+	0.909651i	$0.363651\pi$
$72$	0	0
$73$	−9.00000	−1.05337	−0.526685	−	0.850060i	$-0.676565\pi$
$73$	−9.00000	−1.05337	−0.526685	+	0.850060i	$0.676565\pi$
$74$	0	0
$75$	−3.00000	−0.346410
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.00000	0.964901
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	−9.00000	−0.933257
$94$	0	0
$95$	0	0
$96$	0	0
$97$	18.0000	1.82762	0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	6.00000	0.585540
$106$	0	0
$107$	−16.0000	−1.54678	−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000	−1.54678	−0.773389	+	0.633932i	$0.781440\pi$
$108$	0	0
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	0	0
$111$	24.0000	2.27798
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	6.00000	0.554700
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	−9.00000	−0.811503
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	11.0000	0.976092	0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000	0.976092	0.488046	+	0.872818i	$0.337710\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	0	0
$131$	−9.00000	−0.786334	−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000	−0.786334	−0.393167	+	0.919467i	$0.628621\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−9.00000	−0.774597
$136$	0	0
$137$	4.00000	0.341743	0.170872	−	0.985293i	$-0.445342\pi$
$137$	4.00000	0.341743	0.170872	+	0.985293i	$0.445342\pi$
$138$	0	0
$139$	−11.0000	−0.933008	−0.466504	−	0.884519i	$-0.654487\pi$
$139$	−11.0000	−0.933008	−0.466504	+	0.884519i	$0.654487\pi$
$140$	0	0
$141$	33.0000	2.77910
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−3.00000	−0.249136
$146$	0	0
$147$	9.00000	0.742307
$148$	0	0
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	7.00000	0.569652	0.284826	−	0.958579i	$-0.408064\pi$
$151$	7.00000	0.569652	0.284826	+	0.958579i	$0.408064\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.00000	0.240966
$156$	0	0
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	0	0
$159$	42.0000	3.33082
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	7.00000	0.548282	0.274141	−	0.961689i	$-0.411606\pi$
$163$	7.00000	0.548282	0.274141	+	0.961689i	$0.411606\pi$
$164$	0	0
$165$	0	0
$166$	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
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	24.0000	1.80395
$178$	0	0
$179$	21.0000	1.56961	0.784807	−	0.619740i	$-0.212762\pi$
$179$	21.0000	1.56961	0.784807	+	0.619740i	$0.212762\pi$
$180$	0	0
$181$	12.0000	0.891953	0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000	0.891953	0.445976	+	0.895045i	$0.352856\pi$
$182$	0	0
$183$	12.0000	0.887066
$184$	0	0
$185$	−8.00000	−0.588172
$186$	0	0
$187$	0	0
$188$	0	0
$189$	18.0000	1.30931
$190$	0	0
$191$	2.00000	0.144715	0.0723575	−	0.997379i	$-0.476948\pi$
$191$	2.00000	0.144715	0.0723575	+	0.997379i	$0.476948\pi$
$192$	0	0
$193$	−1.00000	−0.0719816	−0.0359908	−	0.999352i	$-0.511459\pi$
$193$	−1.00000	−0.0719816	−0.0359908	+	0.999352i	$0.511459\pi$
$194$	0	0
$195$	−3.00000	−0.214834
$196$	0	0
$197$	3.00000	0.213741	0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000	0.213741	0.106871	+	0.994273i	$0.465917\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	6.00000	0.421117
$204$	0	0
$205$	3.00000	0.209529
$206$	0	0
$207$	6.00000	0.417029
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	0	0
$213$	−21.0000	−1.43890
$214$	0	0
$215$	−2.00000	−0.136399
$216$	0	0
$217$	−6.00000	−0.407307
$218$	0	0
$219$	27.0000	1.82449
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	6.00000	0.400000
$226$	0	0
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	0	0
$232$	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
$235$	−11.0000	−0.717561
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.00000	−0.0646846	−0.0323423	−	0.999477i	$-0.510297\pi$
$239$	−1.00000	−0.0646846	−0.0323423	+	0.999477i	$0.510297\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.00000	0.311891	0.155946	−	0.987766i	$-0.450158\pi$
$257$	5.00000	0.311891	0.155946	+	0.987766i	$0.450158\pi$
$258$	0	0
$259$	16.0000	0.994192
$260$	0	0
$261$	−18.0000	−1.11417
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	−14.0000	−0.860013
$266$	0	0
$267$	6.00000	0.367194
$268$	0	0
$269$	17.0000	1.03651	0.518254	−	0.855227i	$-0.326582\pi$
$269$	17.0000	1.03651	0.518254	+	0.855227i	$0.326582\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	0	0
$273$	6.00000	0.363137
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−29.0000	−1.74244	−0.871221	−	0.490892i	$-0.836671\pi$
$277$	−29.0000	−1.74244	−0.871221	+	0.490892i	$0.836671\pi$
$278$	0	0
$279$	18.0000	1.07763
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	10.0000	0.594438	0.297219	−	0.954809i	$-0.403941\pi$
$283$	10.0000	0.594438	0.297219	+	0.954809i	$0.403941\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.00000	−0.354169
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−54.0000	−3.16554
$292$	0	0
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.00000	0.0578315
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	−54.0000	−3.10222
$304$	0	0
$305$	−4.00000	−0.229039
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	12.0000	0.682656
$310$	0	0
$311$	−29.0000	−1.64444	−0.822220	−	0.569170i	$-0.807264\pi$
$311$	−29.0000	−1.64444	−0.822220	+	0.569170i	$0.807264\pi$
$312$	0	0
$313$	−20.0000	−1.13047	−0.565233	−	0.824931i	$-0.691214\pi$
$313$	−20.0000	−1.13047	−0.565233	+	0.824931i	$0.691214\pi$
$314$	0	0
$315$	−12.0000	−0.676123
$316$	0	0
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	48.0000	2.67910
$322$	0	0
$323$	0	0
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	54.0000	2.98621
$328$	0	0
$329$	22.0000	1.21290
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	0	0
$333$	−48.0000	−2.63038
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	0	0
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	−3.00000	−0.161515
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	−7.00000	−0.374701	−0.187351	−	0.982293i	$-0.559990\pi$
$349$	−7.00000	−0.374701	−0.187351	+	0.982293i	$0.559990\pi$
$350$	0	0
$351$	−9.00000	−0.480384
$352$	0	0
$353$	19.0000	1.01127	0.505634	−	0.862748i	$-0.331259\pi$
$353$	19.0000	1.01127	0.505634	+	0.862748i	$0.331259\pi$
$354$	0	0
$355$	7.00000	0.371521
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	33.0000	1.73205
$364$	0	0
$365$	−9.00000	−0.471082
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	18.0000	0.937043
$370$	0	0
$371$	28.0000	1.45369
$372$	0	0
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	0	0
$377$	−3.00000	−0.154508
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	−33.0000	−1.69064
$382$	0	0
$383$	30.0000	1.53293	0.766464	−	0.642287i	$-0.222014\pi$
$383$	30.0000	1.53293	0.766464	+	0.642287i	$0.222014\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−12.0000	−0.609994
$388$	0	0
$389$	−16.0000	−0.811232	−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000	−0.811232	−0.405616	+	0.914044i	$0.632943\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	27.0000	1.36197
$394$	0	0
$395$	0	0
$396$	0	0
$397$	25.0000	1.25471	0.627357	−	0.778732i	$-0.284137\pi$
$397$	25.0000	1.25471	0.627357	+	0.778732i	$0.284137\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	3.00000	0.149441
$404$	0	0
$405$	9.00000	0.447214
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−11.0000	−0.543915	−0.271957	−	0.962309i	$-0.587671\pi$
$409$	−11.0000	−0.543915	−0.271957	+	0.962309i	$0.587671\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	0	0
$413$	16.0000	0.787309
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	33.0000	1.61602
$418$	0	0
$419$	22.0000	1.07477	0.537385	−	0.843337i	$-0.319412\pi$
$419$	22.0000	1.07477	0.537385	+	0.843337i	$0.319412\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	0	0
$423$	−66.0000	−3.20903
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.00000	0.387147
$428$	0	0
$429$	0	0
$430$	0	0
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	9.00000	0.431517
$436$	0	0
$437$	0	0
$438$	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
$441$	−18.0000	−0.857143
$442$	0	0
$443$	33.0000	1.56788	0.783939	−	0.620838i	$-0.213208\pi$
$443$	33.0000	1.56788	0.783939	+	0.620838i	$0.213208\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	0	0
$447$	66.0000	3.12169
$448$	0	0
$449$	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
$451$	0	0
$452$	0	0
$453$	−21.0000	−0.986666
$454$	0	0
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	−4.00000	−0.187112	−0.0935561	−	0.995614i	$-0.529823\pi$
$457$	−4.00000	−0.187112	−0.0935561	+	0.995614i	$0.529823\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	13.0000	0.605470	0.302735	−	0.953075i	$-0.402100\pi$
$461$	13.0000	0.605470	0.302735	+	0.953075i	$0.402100\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	0	0
$465$	−9.00000	−0.417365
$466$	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
$469$	8.00000	0.369406
$470$	0	0
$471$	18.0000	0.829396
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−84.0000	−3.84610
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	6.00000	0.273009
$484$	0	0
$485$	18.0000	0.817338
$486$	0	0
$487$	−25.0000	−1.13286	−0.566429	−	0.824110i	$-0.691675\pi$
$487$	−25.0000	−1.13286	−0.566429	+	0.824110i	$0.691675\pi$
$488$	0	0
$489$	−21.0000	−0.949653
$490$	0	0
$491$	−31.0000	−1.39901	−0.699505	−	0.714628i	$-0.746596\pi$
$491$	−31.0000	−1.39901	−0.699505	+	0.714628i	$0.746596\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−14.0000	−0.627986
$498$	0	0
$499$	−25.0000	−1.11915	−0.559577	−	0.828778i	$-0.689036\pi$
$499$	−25.0000	−1.11915	−0.559577	+	0.828778i	$0.689036\pi$
$500$	0	0
$501$	−48.0000	−2.14448
$502$	0	0
$503$	−14.0000	−0.624229	−0.312115	−	0.950044i	$-0.601037\pi$
$503$	−14.0000	−0.624229	−0.312115	+	0.950044i	$0.601037\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	0	0
$507$	36.0000	1.59882
$508$	0	0
$509$	21.0000	0.930809	0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000	0.930809	0.465404	+	0.885098i	$0.345909\pi$
$510$	0	0
$511$	18.0000	0.796273
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.00000	−0.176261
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−42.0000	−1.84360
$520$	0	0
$521$	−4.00000	−0.175243	−0.0876216	−	0.996154i	$-0.527927\pi$
$521$	−4.00000	−0.175243	−0.0876216	+	0.996154i	$0.527927\pi$
$522$	0	0
$523$	42.0000	1.83653	0.918266	−	0.395964i	$-0.129590\pi$
$523$	42.0000	1.83653	0.918266	+	0.395964i	$0.129590\pi$
$524$	0	0
$525$	6.00000	0.261861
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−48.0000	−2.08302
$532$	0	0
$533$	3.00000	0.129944
$534$	0	0
$535$	−16.0000	−0.691740
$536$	0	0
$537$	−63.0000	−2.71865
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−7.00000	−0.300954	−0.150477	−	0.988614i	$-0.548081\pi$
$541$	−7.00000	−0.300954	−0.150477	+	0.988614i	$0.548081\pi$
$542$	0	0
$543$	−36.0000	−1.54491
$544$	0	0
$545$	−18.0000	−0.771035
$546$	0	0
$547$	35.0000	1.49649	0.748246	−	0.663421i	$-0.230896\pi$
$547$	35.0000	1.49649	0.748246	+	0.663421i	$0.230896\pi$
$548$	0	0
$549$	−24.0000	−1.02430
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	24.0000	1.01874
$556$	0	0
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−16.0000	−0.674320	−0.337160	−	0.941447i	$-0.609466\pi$
$563$	−16.0000	−0.674320	−0.337160	+	0.941447i	$0.609466\pi$
$564$	0	0
$565$	2.00000	0.0841406
$566$	0	0
$567$	−18.0000	−0.755929
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	0	0
$573$	−6.00000	−0.250654
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−9.00000	−0.374675	−0.187337	−	0.982296i	$-0.559986\pi$
$577$	−9.00000	−0.374675	−0.187337	+	0.982296i	$0.559986\pi$
$578$	0	0
$579$	3.00000	0.124676
$580$	0	0
$581$	−8.00000	−0.331896
$582$	0	0
$583$	0	0
$584$	0	0
$585$	6.00000	0.248069
$586$	0	0
$587$	33.0000	1.36206	0.681028	−	0.732257i	$-0.261533\pi$
$587$	33.0000	1.36206	0.681028	+	0.732257i	$0.261533\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−9.00000	−0.370211
$592$	0	0
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	12.0000	0.491127
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	37.0000	1.50926	0.754631	−	0.656150i	$-0.227816\pi$
$601$	37.0000	1.50926	0.754631	+	0.656150i	$0.227816\pi$
$602$	0	0
$603$	−24.0000	−0.977356
$604$	0	0
$605$	−11.0000	−0.447214
$606$	0	0
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	0	0
$609$	−18.0000	−0.729397
$610$	0	0
$611$	−11.0000	−0.445012
$612$	0	0
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	0	0
$615$	−9.00000	−0.362915
$616$	0	0
$617$	−48.0000	−1.93241	−0.966204	−	0.257780i	$-0.917009\pi$
$617$	−48.0000	−1.93241	−0.966204	+	0.257780i	$0.917009\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	−9.00000	−0.361158
$622$	0	0
$623$	4.00000	0.160257
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	0	0
$633$	48.0000	1.90783
$634$	0	0
$635$	11.0000	0.436522
$636$	0	0
$637$	−3.00000	−0.118864
$638$	0	0
$639$	42.0000	1.66149
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	0	0
$643$	34.0000	1.34083	0.670415	−	0.741987i	$-0.266116\pi$
$643$	34.0000	1.34083	0.670415	+	0.741987i	$0.266116\pi$
$644$	0	0
$645$	6.00000	0.236250
$646$	0	0
$647$	−39.0000	−1.53325	−0.766624	−	0.642096i	$-0.778065\pi$
$647$	−39.0000	−1.53325	−0.766624	+	0.642096i	$0.778065\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	18.0000	0.705476
$652$	0	0
$653$	−3.00000	−0.117399	−0.0586995	−	0.998276i	$-0.518695\pi$
$653$	−3.00000	−0.117399	−0.0586995	+	0.998276i	$0.518695\pi$
$654$	0	0
$655$	−9.00000	−0.351659
$656$	0	0
$657$	−54.0000	−2.10674
$658$	0	0
$659$	8.00000	0.311636	0.155818	−	0.987786i	$-0.450199\pi$
$659$	8.00000	0.311636	0.155818	+	0.987786i	$0.450199\pi$
$660$	0	0
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.00000	−0.116160
$668$	0	0
$669$	48.0000	1.85579
$670$	0	0
$671$	0	0
$672$	0	0
$673$	13.0000	0.501113	0.250557	−	0.968102i	$-0.419386\pi$
$673$	13.0000	0.501113	0.250557	+	0.968102i	$0.419386\pi$
$674$	0	0
$675$	−9.00000	−0.346410
$676$	0	0
$677$	−46.0000	−1.76792	−0.883962	−	0.467559i	$-0.845134\pi$
$677$	−46.0000	−1.76792	−0.883962	+	0.467559i	$0.845134\pi$
$678$	0	0
$679$	−36.0000	−1.38155
$680$	0	0
$681$	−6.00000	−0.229920
$682$	0	0
$683$	35.0000	1.33924	0.669619	−	0.742705i	$-0.266457\pi$
$683$	35.0000	1.33924	0.669619	+	0.742705i	$0.266457\pi$
$684$	0	0
$685$	4.00000	0.152832
$686$	0	0
$687$	6.00000	0.228914
$688$	0	0
$689$	−14.0000	−0.533358
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.0000	−0.417254
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−63.0000	−2.38288
$700$	0	0
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	33.0000	1.24285
$706$	0	0
$707$	−36.0000	−1.35392
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.00000	0.112351
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.00000	0.112037
$718$	0	0
$719$	28.0000	1.04422	0.522112	−	0.852877i	$-0.325144\pi$
$719$	28.0000	1.04422	0.522112	+	0.852877i	$0.325144\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	−6.00000	−0.223142
$724$	0	0
$725$	−3.00000	−0.111417
$726$	0	0
$727$	−6.00000	−0.222528	−0.111264	−	0.993791i	$-0.535490\pi$
$727$	−6.00000	−0.222528	−0.111264	+	0.993791i	$0.535490\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−8.00000	−0.295487	−0.147743	−	0.989026i	$-0.547201\pi$
$733$	−8.00000	−0.295487	−0.147743	+	0.989026i	$0.547201\pi$
$734$	0	0
$735$	9.00000	0.331970
$736$	0	0
$737$	0	0
$738$	0	0
$739$	17.0000	0.625355	0.312678	−	0.949859i	$-0.398774\pi$
$739$	17.0000	0.625355	0.312678	+	0.949859i	$0.398774\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	0	0
$745$	−22.0000	−0.806018
$746$	0	0
$747$	24.0000	0.878114
$748$	0	0
$749$	32.0000	1.16925
$750$	0	0
$751$	−50.0000	−1.82453	−0.912263	−	0.409605i	$-0.865667\pi$
$751$	−50.0000	−1.82453	−0.912263	+	0.409605i	$0.865667\pi$
$752$	0	0
$753$	−48.0000	−1.74922
$754$	0	0
$755$	7.00000	0.254756
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	29.0000	1.05125	0.525625	−	0.850717i	$-0.323832\pi$
$761$	29.0000	1.05125	0.525625	+	0.850717i	$0.323832\pi$
$762$	0	0
$763$	36.0000	1.30329
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	−15.0000	−0.540212
$772$	0	0
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	3.00000	0.107763
$776$	0	0
$777$	−48.0000	−1.72199
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	27.0000	0.964901
$784$	0	0
$785$	−6.00000	−0.214149
$786$	0	0
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	0	0
$789$	36.0000	1.28163
$790$	0	0
$791$	−4.00000	−0.142224
$792$	0	0
$793$	−4.00000	−0.142044
$794$	0	0
$795$	42.0000	1.48959
$796$	0	0
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−12.0000	−0.423999
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−2.00000	−0.0704907
$806$	0	0
$807$	−51.0000	−1.79529
$808$	0	0
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	0	0
$811$	−5.00000	−0.175574	−0.0877869	−	0.996139i	$-0.527979\pi$
$811$	−5.00000	−0.175574	−0.0877869	+	0.996139i	$0.527979\pi$
$812$	0	0
$813$	60.0000	2.10429
$814$	0	0
$815$	7.00000	0.245199
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−12.0000	−0.419314
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	33.0000	1.15031	0.575154	−	0.818045i	$-0.304942\pi$
$823$	33.0000	1.15031	0.575154	+	0.818045i	$0.304942\pi$
$824$	0	0
$825$	0	0
$826$	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
$829$	6.00000	0.208389	0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000	0.208389	0.104194	+	0.994557i	$0.466774\pi$
$830$	0	0
$831$	87.0000	3.01800
$832$	0	0
$833$	0	0
$834$	0	0
$835$	16.0000	0.553703
$836$	0	0
$837$	−27.0000	−0.933257
$838$	0	0
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	−18.0000	−0.619953
$844$	0	0
$845$	−12.0000	−0.412813
$846$	0	0
$847$	22.0000	0.755929
$848$	0	0
$849$	−30.0000	−1.02960
$850$	0	0
$851$	−8.00000	−0.274236
$852$	0	0
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	41.0000	1.40053	0.700267	−	0.713881i	$-0.253064\pi$
$857$	41.0000	1.40053	0.700267	+	0.713881i	$0.253064\pi$
$858$	0	0
$859$	−17.0000	−0.580033	−0.290016	−	0.957022i	$-0.593661\pi$
$859$	−17.0000	−0.580033	−0.290016	+	0.957022i	$0.593661\pi$
$860$	0	0
$861$	18.0000	0.613438
$862$	0	0
$863$	−17.0000	−0.578687	−0.289343	−	0.957225i	$-0.593437\pi$
$863$	−17.0000	−0.578687	−0.289343	+	0.957225i	$0.593437\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	51.0000	1.73205
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−4.00000	−0.135535
$872$	0	0
$873$	108.000	3.65525
$874$	0	0
$875$	−2.00000	−0.0676123
$876$	0	0
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	0	0
$879$	72.0000	2.42850
$880$	0	0
$881$	36.0000	1.21287	0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000	1.21287	0.606435	+	0.795133i	$0.292599\pi$
$882$	0	0
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	0	0
$885$	24.0000	0.806751
$886$	0	0
$887$	−15.0000	−0.503651	−0.251825	−	0.967773i	$-0.581031\pi$
$887$	−15.0000	−0.503651	−0.251825	+	0.967773i	$0.581031\pi$
$888$	0	0
$889$	−22.0000	−0.737856
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	21.0000	0.701953
$896$	0	0
$897$	−3.00000	−0.100167
$898$	0	0
$899$	−9.00000	−0.300167
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−12.0000	−0.399335
$904$	0	0
$905$	12.0000	0.398893
$906$	0	0
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	0	0
$909$	108.000	3.58213
$910$	0	0
$911$	−44.0000	−1.45779	−0.728893	−	0.684628i	$-0.759965\pi$
$911$	−44.0000	−1.45779	−0.728893	+	0.684628i	$0.759965\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	12.0000	0.396708
$916$	0	0
$917$	18.0000	0.594412
$918$	0	0
$919$	−50.0000	−1.64935	−0.824674	−	0.565608i	$-0.808641\pi$
$919$	−50.0000	−1.64935	−0.824674	+	0.565608i	$0.808641\pi$
$920$	0	0
$921$	60.0000	1.97707
$922$	0	0
$923$	7.00000	0.230408
$924$	0	0
$925$	−8.00000	−0.263038
$926$	0	0
$927$	−24.0000	−0.788263
$928$	0	0
$929$	19.0000	0.623370	0.311685	−	0.950186i	$-0.399107\pi$
$929$	19.0000	0.623370	0.311685	+	0.950186i	$0.399107\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	87.0000	2.84825
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−44.0000	−1.43742	−0.718709	−	0.695311i	$-0.755266\pi$
$937$	−44.0000	−1.43742	−0.718709	+	0.695311i	$0.755266\pi$
$938$	0	0
$939$	60.0000	1.95803
$940$	0	0
$941$	12.0000	0.391189	0.195594	−	0.980685i	$-0.437336\pi$
$941$	12.0000	0.391189	0.195594	+	0.980685i	$0.437336\pi$
$942$	0	0
$943$	3.00000	0.0976934
$944$	0	0
$945$	18.0000	0.585540
$946$	0	0
$947$	−47.0000	−1.52729	−0.763647	−	0.645634i	$-0.776593\pi$
$947$	−47.0000	−1.52729	−0.763647	+	0.645634i	$0.776593\pi$
$948$	0	0
$949$	−9.00000	−0.292152
$950$	0	0
$951$	42.0000	1.36194
$952$	0	0
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	0	0
$955$	2.00000	0.0647185
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.00000	−0.258333
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	−96.0000	−3.09356
$964$	0	0
$965$	−1.00000	−0.0321911
$966$	0	0
$967$	43.0000	1.38279	0.691393	−	0.722478i	$-0.256997\pi$
$967$	43.0000	1.38279	0.691393	+	0.722478i	$0.256997\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−14.0000	−0.449281	−0.224641	−	0.974442i	$-0.572121\pi$
$971$	−14.0000	−0.449281	−0.224641	+	0.974442i	$0.572121\pi$
$972$	0	0
$973$	22.0000	0.705288
$974$	0	0
$975$	−3.00000	−0.0960769
$976$	0	0
$977$	48.0000	1.53566	0.767828	−	0.640656i	$-0.221338\pi$
$977$	48.0000	1.53566	0.767828	+	0.640656i	$0.221338\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−108.000	−3.44817
$982$	0	0
$983$	−14.0000	−0.446531	−0.223265	−	0.974758i	$-0.571672\pi$
$983$	−14.0000	−0.446531	−0.223265	+	0.974758i	$0.571672\pi$
$984$	0	0
$985$	3.00000	0.0955879
$986$	0	0
$987$	−66.0000	−2.10080
$988$	0	0
$989$	−2.00000	−0.0635963
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	21.0000	0.666415
$994$	0	0
$995$	−4.00000	−0.126809
$996$	0	0
$997$	58.0000	1.83688	0.918439	−	0.395562i	$-0.129450\pi$
$997$	58.0000	1.83688	0.918439	+	0.395562i	$0.129450\pi$
$998$	0	0
$999$	72.0000	2.27798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.2.a.a.1.1	✓	1
3.2	odd	2		8280.2.a.d.1.1		1
4.3	odd	2		1840.2.a.i.1.1		1
5.2	odd	4		4600.2.e.b.4049.2		2
5.3	odd	4		4600.2.e.b.4049.1		2
5.4	even	2		4600.2.a.p.1.1		1
8.3	odd	2		7360.2.a.a.1.1		1
8.5	even	2		7360.2.a.ba.1.1		1
20.19	odd	2		9200.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.a.1.1	✓	1	1.1	even	1	trivial
1840.2.a.i.1.1		1	4.3	odd	2
4600.2.a.p.1.1		1	5.4	even	2
4600.2.e.b.4049.1		2	5.3	odd	4
4600.2.e.b.4049.2		2	5.2	odd	4
7360.2.a.a.1.1		1	8.3	odd	2
7360.2.a.ba.1.1		1	8.5	even	2
8280.2.a.d.1.1		1	3.2	odd	2
9200.2.a.c.1.1		1	20.19	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:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	920.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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