LMFDB - Embedded newform 116.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [116,2,Mod(1,116)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(116, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("116.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 116 = 2^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	116.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:	$0.926264663447$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	116.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} +3.00000 q^{5} +4.00000 q^{7} +6.00000 q^{9} +O(q^{10})$

$q-3.00000 q^{3} +3.00000 q^{5} +4.00000 q^{7} +6.00000 q^{9} -1.00000 q^{11} -3.00000 q^{13} -9.00000 q^{15} +2.00000 q^{17} +4.00000 q^{19} -12.0000 q^{21} -6.00000 q^{23} +4.00000 q^{25} -9.00000 q^{27} -1.00000 q^{29} +9.00000 q^{31} +3.00000 q^{33} +12.0000 q^{35} -8.00000 q^{37} +9.00000 q^{39} -8.00000 q^{41} -5.00000 q^{43} +18.0000 q^{45} -7.00000 q^{47} +9.00000 q^{49} -6.00000 q^{51} -5.00000 q^{53} -3.00000 q^{55} -12.0000 q^{57} -10.0000 q^{59} +10.0000 q^{61} +24.0000 q^{63} -9.00000 q^{65} +8.00000 q^{67} +18.0000 q^{69} -2.00000 q^{71} -12.0000 q^{75} -4.00000 q^{77} -1.00000 q^{79} +9.00000 q^{81} +6.00000 q^{83} +6.00000 q^{85} +3.00000 q^{87} +12.0000 q^{89} -12.0000 q^{91} -27.0000 q^{93} +12.0000 q^{95} -6.00000 q^{99} +O(q^{100})$

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
$2$	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
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	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
$9$	6.00000	2.00000
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	0	0
$15$	−9.00000	−2.32379
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	−12.0000	−2.61861
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	−9.00000	−1.73205
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	9.00000	1.61645	0.808224	−	0.588875i	$-0.200429\pi$
$31$	9.00000	1.61645	0.808224	+	0.588875i	$0.200429\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	12.0000	2.02837
$36$	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
$39$	9.00000	1.44115
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	0	0
$45$	18.0000	2.68328
$46$	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
$49$	9.00000	1.28571
$50$	0	0
$51$	−6.00000	−0.840168
$52$	0	0
$53$	−5.00000	−0.686803	−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000	−0.686803	−0.343401	+	0.939189i	$0.611579\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	0	0
$57$	−12.0000	−1.58944
$58$	0	0
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	24.0000	3.02372
$64$	0	0
$65$	−9.00000	−1.11631
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	18.0000	2.16695
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	−12.0000	−1.38564
$76$	0	0
$77$	−4.00000	−0.455842
$78$	0	0
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	0	0
$87$	3.00000	0.321634
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	−12.0000	−1.25794
$92$	0	0
$93$	−27.0000	−2.79977
$94$	0	0
$95$	12.0000	1.23117
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	−36.0000	−3.51324
$106$	0	0
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	0	0
$111$	24.0000	2.27798
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−18.0000	−1.67851
$116$	0	0
$117$	−18.0000	−1.66410
$118$	0	0
$119$	8.00000	0.733359
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	24.0000	2.16401
$124$	0	0
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	15.0000	1.32068
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	16.0000	1.38738
$134$	0	0
$135$	−27.0000	−2.32379
$136$	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
$139$	−18.0000	−1.52674	−0.763370	−	0.645961i	$-0.776457\pi$
$139$	−18.0000	−1.52674	−0.763370	+	0.645961i	$0.776457\pi$
$140$	0	0
$141$	21.0000	1.76852
$142$	0	0
$143$	3.00000	0.250873
$144$	0	0
$145$	−3.00000	−0.249136
$146$	0	0
$147$	−27.0000	−2.22692
$148$	0	0
$149$	9.00000	0.737309	0.368654	−	0.929567i	$-0.379819\pi$
$149$	9.00000	0.737309	0.368654	+	0.929567i	$0.379819\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	12.0000	0.970143
$154$	0	0
$155$	27.0000	2.16869
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	0	0
$159$	15.0000	1.18958
$160$	0	0
$161$	−24.0000	−1.89146
$162$	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
$165$	9.00000	0.700649
$166$	0	0
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	24.0000	1.83533
$172$	0	0
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	16.0000	1.20949
$176$	0	0
$177$	30.0000	2.25494
$178$	0	0
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	17.0000	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$182$	0	0
$183$	−30.0000	−2.21766
$184$	0	0
$185$	−24.0000	−1.76452
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	−36.0000	−2.61861
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−8.00000	−0.575853	−0.287926	−	0.957653i	$-0.592966\pi$
$193$	−8.00000	−0.575853	−0.287926	+	0.957653i	$0.592966\pi$
$194$	0	0
$195$	27.0000	1.93351
$196$	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
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	−24.0000	−1.69283
$202$	0	0
$203$	−4.00000	−0.280745
$204$	0	0
$205$	−24.0000	−1.67623
$206$	0	0
$207$	−36.0000	−2.50217
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	0	0
$213$	6.00000	0.411113
$214$	0	0
$215$	−15.0000	−1.02299
$216$	0	0
$217$	36.0000	2.44384
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6.00000	−0.403604
$222$	0	0
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	0	0
$225$	24.0000	1.60000
$226$	0	0
$227$	30.0000	1.99117	0.995585	−	0.0938647i	$-0.0299221\pi$
$227$	30.0000	1.99117	0.995585	+	0.0938647i	$0.0299221\pi$
$228$	0	0
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	0	0
$233$	23.0000	1.50678	0.753390	−	0.657574i	$-0.228417\pi$
$233$	23.0000	1.50678	0.753390	+	0.657574i	$0.228417\pi$
$234$	0	0
$235$	−21.0000	−1.36989
$236$	0	0
$237$	3.00000	0.194871
$238$	0	0
$239$	−22.0000	−1.42306	−0.711531	−	0.702655i	$-0.751998\pi$
$239$	−22.0000	−1.42306	−0.711531	+	0.702655i	$0.751998\pi$
$240$	0	0
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	27.0000	1.72497
$246$	0	0
$247$	−12.0000	−0.763542
$248$	0	0
$249$	−18.0000	−1.14070
$250$	0	0
$251$	−31.0000	−1.95670	−0.978351	−	0.206951i	$-0.933646\pi$
$251$	−31.0000	−1.95670	−0.978351	+	0.206951i	$0.933646\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	0	0
$255$	−18.0000	−1.12720
$256$	0	0
$257$	−27.0000	−1.68421	−0.842107	−	0.539311i	$-0.818685\pi$
$257$	−27.0000	−1.68421	−0.842107	+	0.539311i	$0.818685\pi$
$258$	0	0
$259$	−32.0000	−1.98838
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	13.0000	0.801614	0.400807	−	0.916162i	$-0.368730\pi$
$263$	13.0000	0.801614	0.400807	+	0.916162i	$0.368730\pi$
$264$	0	0
$265$	−15.0000	−0.921443
$266$	0	0
$267$	−36.0000	−2.20316
$268$	0	0
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	−5.00000	−0.303728	−0.151864	−	0.988401i	$-0.548528\pi$
$271$	−5.00000	−0.303728	−0.151864	+	0.988401i	$0.548528\pi$
$272$	0	0
$273$	36.0000	2.17882
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	54.0000	3.23290
$280$	0	0
$281$	−1.00000	−0.0596550	−0.0298275	−	0.999555i	$-0.509496\pi$
$281$	−1.00000	−0.0596550	−0.0298275	+	0.999555i	$0.509496\pi$
$282$	0	0
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	0	0
$285$	−36.0000	−2.13246
$286$	0	0
$287$	−32.0000	−1.88890
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	0	0
$295$	−30.0000	−1.74667
$296$	0	0
$297$	9.00000	0.522233
$298$	0	0
$299$	18.0000	1.04097
$300$	0	0
$301$	−20.0000	−1.15278
$302$	0	0
$303$	12.0000	0.689382
$304$	0	0
$305$	30.0000	1.71780
$306$	0	0
$307$	7.00000	0.399511	0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000	0.399511	0.199756	+	0.979846i	$0.435985\pi$
$308$	0	0
$309$	−18.0000	−1.02398
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	0	0
$315$	72.0000	4.05674
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	1.00000	0.0559893
$320$	0	0
$321$	24.0000	1.33955
$322$	0	0
$323$	8.00000	0.445132
$324$	0	0
$325$	−12.0000	−0.665640
$326$	0	0
$327$	15.0000	0.829502
$328$	0	0
$329$	−28.0000	−1.54369
$330$	0	0
$331$	3.00000	0.164895	0.0824475	−	0.996595i	$-0.473726\pi$
$331$	3.00000	0.164895	0.0824475	+	0.996595i	$0.473726\pi$
$332$	0	0
$333$	−48.0000	−2.63038
$334$	0	0
$335$	24.0000	1.31126
$336$	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
$339$	18.0000	0.977626
$340$	0	0
$341$	−9.00000	−0.487377
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	54.0000	2.90726
$346$	0	0
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	0	0
$349$	35.0000	1.87351	0.936754	−	0.349990i	$-0.113815\pi$
$349$	35.0000	1.87351	0.936754	+	0.349990i	$0.113815\pi$
$350$	0	0
$351$	27.0000	1.44115
$352$	0	0
$353$	22.0000	1.17094	0.585471	−	0.810693i	$-0.300910\pi$
$353$	22.0000	1.17094	0.585471	+	0.810693i	$0.300910\pi$
$354$	0	0
$355$	−6.00000	−0.318447
$356$	0	0
$357$	−24.0000	−1.27021
$358$	0	0
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	30.0000	1.57459
$364$	0	0
$365$	0	0
$366$	0	0
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	0	0
$369$	−48.0000	−2.49878
$370$	0	0
$371$	−20.0000	−1.03835
$372$	0	0
$373$	−19.0000	−0.983783	−0.491891	−	0.870657i	$-0.663694\pi$
$373$	−19.0000	−0.983783	−0.491891	+	0.870657i	$0.663694\pi$
$374$	0	0
$375$	9.00000	0.464758
$376$	0	0
$377$	3.00000	0.154508
$378$	0	0
$379$	32.0000	1.64373	0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000	1.64373	0.821865	+	0.569683i	$0.192934\pi$
$380$	0	0
$381$	24.0000	1.22956
$382$	0	0
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	0	0
$385$	−12.0000	−0.611577
$386$	0	0
$387$	−30.0000	−1.52499
$388$	0	0
$389$	4.00000	0.202808	0.101404	−	0.994845i	$-0.467667\pi$
$389$	4.00000	0.202808	0.101404	+	0.994845i	$0.467667\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	−36.0000	−1.81596
$394$	0	0
$395$	−3.00000	−0.150946
$396$	0	0
$397$	13.0000	0.652451	0.326226	−	0.945292i	$-0.394223\pi$
$397$	13.0000	0.652451	0.326226	+	0.945292i	$0.394223\pi$
$398$	0	0
$399$	−48.0000	−2.40301
$400$	0	0
$401$	−5.00000	−0.249688	−0.124844	−	0.992176i	$-0.539843\pi$
$401$	−5.00000	−0.249688	−0.124844	+	0.992176i	$0.539843\pi$
$402$	0	0
$403$	−27.0000	−1.34497
$404$	0	0
$405$	27.0000	1.34164
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	0	0
$413$	−40.0000	−1.96827
$414$	0	0
$415$	18.0000	0.883585
$416$	0	0
$417$	54.0000	2.64439
$418$	0	0
$419$	10.0000	0.488532	0.244266	−	0.969708i	$-0.421453\pi$
$419$	10.0000	0.488532	0.244266	+	0.969708i	$0.421453\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	−42.0000	−2.04211
$424$	0	0
$425$	8.00000	0.388057
$426$	0	0
$427$	40.0000	1.93574
$428$	0	0
$429$	−9.00000	−0.434524
$430$	0	0
$431$	−14.0000	−0.674356	−0.337178	−	0.941441i	$-0.609472\pi$
$431$	−14.0000	−0.674356	−0.337178	+	0.941441i	$0.609472\pi$
$432$	0	0
$433$	−28.0000	−1.34559	−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000	−1.34559	−0.672797	+	0.739827i	$0.734907\pi$
$434$	0	0
$435$	9.00000	0.431517
$436$	0	0
$437$	−24.0000	−1.14808
$438$	0	0
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	0	0
$441$	54.0000	2.57143
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	36.0000	1.70656
$446$	0	0
$447$	−27.0000	−1.27706
$448$	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
$451$	8.00000	0.376705
$452$	0	0
$453$	−12.0000	−0.563809
$454$	0	0
$455$	−36.0000	−1.68771
$456$	0	0
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	0	0
$459$	−18.0000	−0.840168
$460$	0	0
$461$	−42.0000	−1.95614	−0.978068	−	0.208288i	$-0.933211\pi$
$461$	−42.0000	−1.95614	−0.978068	+	0.208288i	$0.933211\pi$
$462$	0	0
$463$	−10.0000	−0.464739	−0.232370	−	0.972628i	$-0.574648\pi$
$463$	−10.0000	−0.464739	−0.232370	+	0.972628i	$0.574648\pi$
$464$	0	0
$465$	−81.0000	−3.75629
$466$	0	0
$467$	−13.0000	−0.601568	−0.300784	−	0.953692i	$-0.597248\pi$
$467$	−13.0000	−0.601568	−0.300784	+	0.953692i	$0.597248\pi$
$468$	0	0
$469$	32.0000	1.47762
$470$	0	0
$471$	−12.0000	−0.552931
$472$	0	0
$473$	5.00000	0.229900
$474$	0	0
$475$	16.0000	0.734130
$476$	0	0
$477$	−30.0000	−1.37361
$478$	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
$481$	24.0000	1.09431
$482$	0	0
$483$	72.0000	3.27611
$484$	0	0
$485$	0	0
$486$	0	0
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	0	0
$489$	−21.0000	−0.949653
$490$	0	0
$491$	17.0000	0.767199	0.383600	−	0.923499i	$-0.374684\pi$
$491$	17.0000	0.767199	0.383600	+	0.923499i	$0.374684\pi$
$492$	0	0
$493$	−2.00000	−0.0900755
$494$	0	0
$495$	−18.0000	−0.809040
$496$	0	0
$497$	−8.00000	−0.358849
$498$	0	0
$499$	10.0000	0.447661	0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000	0.447661	0.223831	+	0.974628i	$0.428144\pi$
$500$	0	0
$501$	−6.00000	−0.268060
$502$	0	0
$503$	23.0000	1.02552	0.512760	−	0.858532i	$-0.328623\pi$
$503$	23.0000	1.02552	0.512760	+	0.858532i	$0.328623\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	12.0000	0.532939
$508$	0	0
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−36.0000	−1.58944
$514$	0	0
$515$	18.0000	0.793175
$516$	0	0
$517$	7.00000	0.307860
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	39.0000	1.70862	0.854311	−	0.519763i	$-0.173980\pi$
$521$	39.0000	1.70862	0.854311	+	0.519763i	$0.173980\pi$
$522$	0	0
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	0	0
$525$	−48.0000	−2.09489
$526$	0	0
$527$	18.0000	0.784092
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	−60.0000	−2.60378
$532$	0	0
$533$	24.0000	1.03956
$534$	0	0
$535$	−24.0000	−1.03761
$536$	0	0
$537$	−30.0000	−1.29460
$538$	0	0
$539$	−9.00000	−0.387657
$540$	0	0
$541$	40.0000	1.71973	0.859867	−	0.510518i	$-0.170546\pi$
$541$	40.0000	1.71973	0.859867	+	0.510518i	$0.170546\pi$
$542$	0	0
$543$	−51.0000	−2.18862
$544$	0	0
$545$	−15.0000	−0.642529
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	0	0
$549$	60.0000	2.56074
$550$	0	0
$551$	−4.00000	−0.170406
$552$	0	0
$553$	−4.00000	−0.170097
$554$	0	0
$555$	72.0000	3.05623
$556$	0	0
$557$	42.0000	1.77960	0.889799	−	0.456354i	$-0.150845\pi$
$557$	42.0000	1.77960	0.889799	+	0.456354i	$0.150845\pi$
$558$	0	0
$559$	15.0000	0.634432
$560$	0	0
$561$	6.00000	0.253320
$562$	0	0
$563$	−33.0000	−1.39078	−0.695392	−	0.718631i	$-0.744769\pi$
$563$	−33.0000	−1.39078	−0.695392	+	0.718631i	$0.744769\pi$
$564$	0	0
$565$	−18.0000	−0.757266
$566$	0	0
$567$	36.0000	1.51186
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−24.0000	−1.00087
$576$	0	0
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	0	0
$579$	24.0000	0.997406
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	5.00000	0.207079
$584$	0	0
$585$	−54.0000	−2.23263
$586$	0	0
$587$	42.0000	1.73353	0.866763	−	0.498721i	$-0.166197\pi$
$587$	42.0000	1.73353	0.866763	+	0.498721i	$0.166197\pi$
$588$	0	0
$589$	36.0000	1.48335
$590$	0	0
$591$	−78.0000	−3.20849
$592$	0	0
$593$	−29.0000	−1.19089	−0.595444	−	0.803397i	$-0.703024\pi$
$593$	−29.0000	−1.19089	−0.595444	+	0.803397i	$0.703024\pi$
$594$	0	0
$595$	24.0000	0.983904
$596$	0	0
$597$	60.0000	2.45564
$598$	0	0
$599$	7.00000	0.286012	0.143006	−	0.989722i	$-0.454323\pi$
$599$	7.00000	0.286012	0.143006	+	0.989722i	$0.454323\pi$
$600$	0	0
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	0	0
$603$	48.0000	1.95471
$604$	0	0
$605$	−30.0000	−1.21967
$606$	0	0
$607$	47.0000	1.90767	0.953836	−	0.300329i	$-0.0970966\pi$
$607$	47.0000	1.90767	0.953836	+	0.300329i	$0.0970966\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	21.0000	0.849569
$612$	0	0
$613$	−21.0000	−0.848182	−0.424091	−	0.905620i	$-0.639406\pi$
$613$	−21.0000	−0.848182	−0.424091	+	0.905620i	$0.639406\pi$
$614$	0	0
$615$	72.0000	2.90332
$616$	0	0
$617$	−36.0000	−1.44931	−0.724653	−	0.689114i	$-0.758000\pi$
$617$	−36.0000	−1.44931	−0.724653	+	0.689114i	$0.758000\pi$
$618$	0	0
$619$	−41.0000	−1.64793	−0.823965	−	0.566641i	$-0.808243\pi$
$619$	−41.0000	−1.64793	−0.823965	+	0.566641i	$0.808243\pi$
$620$	0	0
$621$	54.0000	2.16695
$622$	0	0
$623$	48.0000	1.92308
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	12.0000	0.479234
$628$	0	0
$629$	−16.0000	−0.637962
$630$	0	0
$631$	−14.0000	−0.557331	−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000	−0.557331	−0.278666	+	0.960388i	$0.589892\pi$
$632$	0	0
$633$	39.0000	1.55011
$634$	0	0
$635$	−24.0000	−0.952411
$636$	0	0
$637$	−27.0000	−1.06978
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	0	0
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	0	0
$645$	45.0000	1.77187
$646$	0	0
$647$	22.0000	0.864909	0.432455	−	0.901656i	$-0.357648\pi$
$647$	22.0000	0.864909	0.432455	+	0.901656i	$0.357648\pi$
$648$	0	0
$649$	10.0000	0.392534
$650$	0	0
$651$	−108.000	−4.23285
$652$	0	0
$653$	4.00000	0.156532	0.0782660	−	0.996933i	$-0.475062\pi$
$653$	4.00000	0.156532	0.0782660	+	0.996933i	$0.475062\pi$
$654$	0	0
$655$	36.0000	1.40664
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	0	0
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	0	0
$663$	18.0000	0.699062
$664$	0	0
$665$	48.0000	1.86136
$666$	0	0
$667$	6.00000	0.232321
$668$	0	0
$669$	−42.0000	−1.62381
$670$	0	0
$671$	−10.0000	−0.386046
$672$	0	0
$673$	25.0000	0.963679	0.481840	−	0.876259i	$-0.339969\pi$
$673$	25.0000	0.963679	0.481840	+	0.876259i	$0.339969\pi$
$674$	0	0
$675$	−36.0000	−1.38564
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−90.0000	−3.44881
$682$	0	0
$683$	8.00000	0.306111	0.153056	−	0.988218i	$-0.451089\pi$
$683$	8.00000	0.306111	0.153056	+	0.988218i	$0.451089\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	−60.0000	−2.28914
$688$	0	0
$689$	15.0000	0.571454
$690$	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
$693$	−24.0000	−0.911685
$694$	0	0
$695$	−54.0000	−2.04834
$696$	0	0
$697$	−16.0000	−0.606043
$698$	0	0
$699$	−69.0000	−2.60982
$700$	0	0
$701$	−35.0000	−1.32193	−0.660966	−	0.750416i	$-0.729853\pi$
$701$	−35.0000	−1.32193	−0.660966	+	0.750416i	$0.729853\pi$
$702$	0	0
$703$	−32.0000	−1.20690
$704$	0	0
$705$	63.0000	2.37272
$706$	0	0
$707$	−16.0000	−0.601742
$708$	0	0
$709$	29.0000	1.08912	0.544559	−	0.838723i	$-0.316697\pi$
$709$	29.0000	1.08912	0.544559	+	0.838723i	$0.316697\pi$
$710$	0	0
$711$	−6.00000	−0.225018
$712$	0	0
$713$	−54.0000	−2.02232
$714$	0	0
$715$	9.00000	0.336581
$716$	0	0
$717$	66.0000	2.46482
$718$	0	0
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	24.0000	0.893807
$722$	0	0
$723$	−51.0000	−1.89671
$724$	0	0
$725$	−4.00000	−0.148556
$726$	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
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−10.0000	−0.369863
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	−81.0000	−2.98773
$736$	0	0
$737$	−8.00000	−0.294684
$738$	0	0
$739$	1.00000	0.0367856	0.0183928	−	0.999831i	$-0.494145\pi$
$739$	1.00000	0.0367856	0.0183928	+	0.999831i	$0.494145\pi$
$740$	0	0
$741$	36.0000	1.32249
$742$	0	0
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	0	0
$745$	27.0000	0.989203
$746$	0	0
$747$	36.0000	1.31717
$748$	0	0
$749$	−32.0000	−1.16925
$750$	0	0
$751$	44.0000	1.60558	0.802791	−	0.596260i	$-0.203347\pi$
$751$	44.0000	1.60558	0.802791	+	0.596260i	$0.203347\pi$
$752$	0	0
$753$	93.0000	3.38911
$754$	0	0
$755$	12.0000	0.436725
$756$	0	0
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	0	0
$759$	−18.0000	−0.653359
$760$	0	0
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	0	0
$763$	−20.0000	−0.724049
$764$	0	0
$765$	36.0000	1.30158
$766$	0	0
$767$	30.0000	1.08324
$768$	0	0
$769$	6.00000	0.216366	0.108183	−	0.994131i	$-0.465497\pi$
$769$	6.00000	0.216366	0.108183	+	0.994131i	$0.465497\pi$
$770$	0	0
$771$	81.0000	2.91714
$772$	0	0
$773$	36.0000	1.29483	0.647415	−	0.762138i	$-0.275850\pi$
$773$	36.0000	1.29483	0.647415	+	0.762138i	$0.275850\pi$
$774$	0	0
$775$	36.0000	1.29316
$776$	0	0
$777$	96.0000	3.44398
$778$	0	0
$779$	−32.0000	−1.14652
$780$	0	0
$781$	2.00000	0.0715656
$782$	0	0
$783$	9.00000	0.321634
$784$	0	0
$785$	12.0000	0.428298
$786$	0	0
$787$	−14.0000	−0.499046	−0.249523	−	0.968369i	$-0.580274\pi$
$787$	−14.0000	−0.499046	−0.249523	+	0.968369i	$0.580274\pi$
$788$	0	0
$789$	−39.0000	−1.38844
$790$	0	0
$791$	−24.0000	−0.853342
$792$	0	0
$793$	−30.0000	−1.06533
$794$	0	0
$795$	45.0000	1.59599
$796$	0	0
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	0	0
$799$	−14.0000	−0.495284
$800$	0	0
$801$	72.0000	2.54399
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−72.0000	−2.53767
$806$	0	0
$807$	−42.0000	−1.47847
$808$	0	0
$809$	−36.0000	−1.26569	−0.632846	−	0.774277i	$-0.718114\pi$
$809$	−36.0000	−1.26569	−0.632846	+	0.774277i	$0.718114\pi$
$810$	0	0
$811$	−24.0000	−0.842754	−0.421377	−	0.906886i	$-0.638453\pi$
$811$	−24.0000	−0.842754	−0.421377	+	0.906886i	$0.638453\pi$
$812$	0	0
$813$	15.0000	0.526073
$814$	0	0
$815$	21.0000	0.735598
$816$	0	0
$817$	−20.0000	−0.699711
$818$	0	0
$819$	−72.0000	−2.51588
$820$	0	0
$821$	−43.0000	−1.50071	−0.750355	−	0.661035i	$-0.770118\pi$
$821$	−43.0000	−1.50071	−0.750355	+	0.661035i	$0.770118\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	0	0
$825$	12.0000	0.417786
$826$	0	0
$827$	−13.0000	−0.452054	−0.226027	−	0.974121i	$-0.572574\pi$
$827$	−13.0000	−0.452054	−0.226027	+	0.974121i	$0.572574\pi$
$828$	0	0
$829$	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
$831$	30.0000	1.04069
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	6.00000	0.207639
$836$	0	0
$837$	−81.0000	−2.79977
$838$	0	0
$839$	9.00000	0.310715	0.155357	−	0.987858i	$-0.450347\pi$
$839$	9.00000	0.310715	0.155357	+	0.987858i	$0.450347\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	3.00000	0.103325
$844$	0	0
$845$	−12.0000	−0.412813
$846$	0	0
$847$	−40.0000	−1.37442
$848$	0	0
$849$	18.0000	0.617758
$850$	0	0
$851$	48.0000	1.64542
$852$	0	0
$853$	−42.0000	−1.43805	−0.719026	−	0.694983i	$-0.755412\pi$
$853$	−42.0000	−1.43805	−0.719026	+	0.694983i	$0.755412\pi$
$854$	0	0
$855$	72.0000	2.46235
$856$	0	0
$857$	49.0000	1.67381	0.836904	−	0.547350i	$-0.184363\pi$
$857$	49.0000	1.67381	0.836904	+	0.547350i	$0.184363\pi$
$858$	0	0
$859$	9.00000	0.307076	0.153538	−	0.988143i	$-0.450933\pi$
$859$	9.00000	0.307076	0.153538	+	0.988143i	$0.450933\pi$
$860$	0	0
$861$	96.0000	3.27167
$862$	0	0
$863$	54.0000	1.83818	0.919091	−	0.394046i	$-0.128925\pi$
$863$	54.0000	1.83818	0.919091	+	0.394046i	$0.128925\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	39.0000	1.32451
$868$	0	0
$869$	1.00000	0.0339227
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−12.0000	−0.405674
$876$	0	0
$877$	−29.0000	−0.979260	−0.489630	−	0.871930i	$-0.662868\pi$
$877$	−29.0000	−0.979260	−0.489630	+	0.871930i	$0.662868\pi$
$878$	0	0
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	0	0
$883$	−24.0000	−0.807664	−0.403832	−	0.914833i	$-0.632322\pi$
$883$	−24.0000	−0.807664	−0.403832	+	0.914833i	$0.632322\pi$
$884$	0	0
$885$	90.0000	3.02532
$886$	0	0
$887$	−55.0000	−1.84672	−0.923360	−	0.383936i	$-0.874568\pi$
$887$	−55.0000	−1.84672	−0.923360	+	0.383936i	$0.874568\pi$
$888$	0	0
$889$	−32.0000	−1.07325
$890$	0	0
$891$	−9.00000	−0.301511
$892$	0	0
$893$	−28.0000	−0.936984
$894$	0	0
$895$	30.0000	1.00279
$896$	0	0
$897$	−54.0000	−1.80301
$898$	0	0
$899$	−9.00000	−0.300167
$900$	0	0
$901$	−10.0000	−0.333148
$902$	0	0
$903$	60.0000	1.99667
$904$	0	0
$905$	51.0000	1.69530
$906$	0	0
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	0	0
$909$	−24.0000	−0.796030
$910$	0	0
$911$	51.0000	1.68971	0.844853	−	0.534999i	$-0.179688\pi$
$911$	51.0000	1.68971	0.844853	+	0.534999i	$0.179688\pi$
$912$	0	0
$913$	−6.00000	−0.198571
$914$	0	0
$915$	−90.0000	−2.97531
$916$	0	0
$917$	48.0000	1.58510
$918$	0	0
$919$	−18.0000	−0.593765	−0.296883	−	0.954914i	$-0.595947\pi$
$919$	−18.0000	−0.593765	−0.296883	+	0.954914i	$0.595947\pi$
$920$	0	0
$921$	−21.0000	−0.691974
$922$	0	0
$923$	6.00000	0.197492
$924$	0	0
$925$	−32.0000	−1.05215
$926$	0	0
$927$	36.0000	1.18240
$928$	0	0
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−6.00000	−0.196221
$936$	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
$939$	−3.00000	−0.0979013
$940$	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
$943$	48.0000	1.56310
$944$	0	0
$945$	−108.000	−3.51324
$946$	0	0
$947$	27.0000	0.877382	0.438691	−	0.898638i	$-0.355442\pi$
$947$	27.0000	0.877382	0.438691	+	0.898638i	$0.355442\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−54.0000	−1.75107
$952$	0	0
$953$	3.00000	0.0971795	0.0485898	−	0.998819i	$-0.484527\pi$
$953$	3.00000	0.0971795	0.0485898	+	0.998819i	$0.484527\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−3.00000	−0.0969762
$958$	0	0
$959$	16.0000	0.516667
$960$	0	0
$961$	50.0000	1.61290
$962$	0	0
$963$	−48.0000	−1.54678
$964$	0	0
$965$	−24.0000	−0.772587
$966$	0	0
$967$	17.0000	0.546683	0.273342	−	0.961917i	$-0.411871\pi$
$967$	17.0000	0.546683	0.273342	+	0.961917i	$0.411871\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	0	0
$973$	−72.0000	−2.30821
$974$	0	0
$975$	36.0000	1.15292
$976$	0	0
$977$	25.0000	0.799821	0.399910	−	0.916554i	$-0.369041\pi$
$977$	25.0000	0.799821	0.399910	+	0.916554i	$0.369041\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	0	0
$981$	−30.0000	−0.957826
$982$	0	0
$983$	45.0000	1.43528	0.717639	−	0.696416i	$-0.245223\pi$
$983$	45.0000	1.43528	0.717639	+	0.696416i	$0.245223\pi$
$984$	0	0
$985$	78.0000	2.48529
$986$	0	0
$987$	84.0000	2.67375
$988$	0	0
$989$	30.0000	0.953945
$990$	0	0
$991$	−34.0000	−1.08005	−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000	−1.08005	−0.540023	+	0.841650i	$0.681584\pi$
$992$	0	0
$993$	−9.00000	−0.285606
$994$	0	0
$995$	−60.0000	−1.90213
$996$	0	0
$997$	−8.00000	−0.253363	−0.126681	−	0.991943i	$-0.540433\pi$
$997$	−8.00000	−0.253363	−0.126681	+	0.991943i	$0.540433\pi$
$998$	0	0
$999$	72.0000	2.27798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.a.a.1.1	✓	1
3.2	odd	2		1044.2.a.c.1.1		1
4.3	odd	2		464.2.a.g.1.1		1
5.2	odd	4		2900.2.c.a.349.2		2
5.3	odd	4		2900.2.c.a.349.1		2
5.4	even	2		2900.2.a.e.1.1		1
7.6	odd	2		5684.2.a.k.1.1		1
8.3	odd	2		1856.2.a.a.1.1		1
8.5	even	2		1856.2.a.o.1.1		1
12.11	even	2		4176.2.a.c.1.1		1
29.12	odd	4		3364.2.c.b.1681.1		2
29.17	odd	4		3364.2.c.b.1681.2		2
29.28	even	2		3364.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.a.a.1.1	✓	1	1.1	even	1	trivial
464.2.a.g.1.1		1	4.3	odd	2
1044.2.a.c.1.1		1	3.2	odd	2
1856.2.a.a.1.1		1	8.3	odd	2
1856.2.a.o.1.1		1	8.5	even	2
2900.2.a.e.1.1		1	5.4	even	2
2900.2.c.a.349.1		2	5.3	odd	4
2900.2.c.a.349.2		2	5.2	odd	4
3364.2.a.c.1.1		1	29.28	even	2
3364.2.c.b.1681.1		2	29.12	odd	4
3364.2.c.b.1681.2		2	29.17	odd	4
4176.2.a.c.1.1		1	12.11	even	2
5684.2.a.k.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:	\( N \)	\(=\)	\( 116 = 2^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	116.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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