LMFDB - Embedded newform 3344.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, 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("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.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:	$26.7019744359$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -2.00000 q^{5} +3.00000 q^{7} -2.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} -2.00000 q^{15} -7.00000 q^{17} -1.00000 q^{19} +3.00000 q^{21} +5.00000 q^{23} -1.00000 q^{25} -5.00000 q^{27} +1.00000 q^{29} -10.0000 q^{31} +1.00000 q^{33} -6.00000 q^{35} -6.00000 q^{37} +1.00000 q^{39} +6.00000 q^{41} +4.00000 q^{43} +4.00000 q^{45} +2.00000 q^{49} -7.00000 q^{51} -1.00000 q^{53} -2.00000 q^{55} -1.00000 q^{57} -3.00000 q^{59} -12.0000 q^{61} -6.00000 q^{63} -2.00000 q^{65} -3.00000 q^{67} +5.00000 q^{69} +10.0000 q^{71} +3.00000 q^{73} -1.00000 q^{75} +3.00000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -8.00000 q^{83} +14.0000 q^{85} +1.00000 q^{87} -8.00000 q^{89} +3.00000 q^{91} -10.0000 q^{93} +2.00000 q^{95} +8.00000 q^{97} -2.00000 q^{99} +O(q^{100})$

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	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	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
$15$	−2.00000	−0.516398
$16$	0	0
$17$	−7.00000	−1.69775	−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000	−1.69775	−0.848875	+	0.528594i	$0.822719\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−6.00000	−1.01419
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	4.00000	0.596285
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	−7.00000	−0.980196
$52$	0	0
$53$	−1.00000	−0.137361	−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000	−0.137361	−0.0686803	+	0.997639i	$0.521879\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	0	0
$61$	−12.0000	−1.53644	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000	−1.53644	−0.768221	+	0.640184i	$0.778858\pi$
$62$	0	0
$63$	−6.00000	−0.755929
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	0	0
$69$	5.00000	0.601929
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	0	0
$73$	3.00000	0.351123	0.175562	−	0.984468i	$-0.443826\pi$
$73$	3.00000	0.351123	0.175562	+	0.984468i	$0.443826\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	3.00000	0.341882
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	14.0000	1.51851
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	0	0
$93$	−10.0000	−1.03695
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	0	0
$105$	−6.00000	−0.585540
$106$	0	0
$107$	13.0000	1.25676	0.628379	−	0.777908i	$-0.283719\pi$
$107$	13.0000	1.25676	0.628379	+	0.777908i	$0.283719\pi$
$108$	0	0
$109$	−13.0000	−1.24517	−0.622587	−	0.782551i	$-0.713918\pi$
$109$	−13.0000	−1.24517	−0.622587	+	0.782551i	$0.713918\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	−10.0000	−0.932505
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	−21.0000	−1.92507
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	22.0000	1.92215	0.961074	−	0.276289i	$-0.0891049\pi$
$131$	22.0000	1.92215	0.961074	+	0.276289i	$0.0891049\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	0	0
$135$	10.0000	0.860663
$136$	0	0
$137$	−17.0000	−1.45241	−0.726204	−	0.687479i	$-0.758717\pi$
$137$	−17.0000	−1.45241	−0.726204	+	0.687479i	$0.758717\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	2.00000	0.164957
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	14.0000	1.13183
$154$	0	0
$155$	20.0000	1.60644
$156$	0	0
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	0	0
$159$	−1.00000	−0.0793052
$160$	0	0
$161$	15.0000	1.18217
$162$	0	0
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	0	0
$167$	22.0000	1.70241	0.851206	−	0.524832i	$-0.175872\pi$
$167$	22.0000	1.70241	0.851206	+	0.524832i	$0.175872\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	2.00000	0.152944
$172$	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
$175$	−3.00000	−0.226779
$176$	0	0
$177$	−3.00000	−0.225494
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	−12.0000	−0.887066
$184$	0	0
$185$	12.0000	0.882258
$186$	0	0
$187$	−7.00000	−0.511891
$188$	0	0
$189$	−15.0000	−1.09109
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	0	0
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	−23.0000	−1.63043	−0.815213	−	0.579161i	$-0.803380\pi$
$199$	−23.0000	−1.63043	−0.815213	+	0.579161i	$0.803380\pi$
$200$	0	0
$201$	−3.00000	−0.211604
$202$	0	0
$203$	3.00000	0.210559
$204$	0	0
$205$	−12.0000	−0.838116
$206$	0	0
$207$	−10.0000	−0.695048
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	0	0
$213$	10.0000	0.685189
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	−30.0000	−2.03653
$218$	0	0
$219$	3.00000	0.202721
$220$	0	0
$221$	−7.00000	−0.470871
$222$	0	0
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	0	0
$225$	2.00000	0.133333
$226$	0	0
$227$	15.0000	0.995585	0.497792	−	0.867296i	$-0.334144\pi$
$227$	15.0000	0.995585	0.497792	+	0.867296i	$0.334144\pi$
$228$	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
$231$	3.00000	0.197386
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.00000	−0.519656
$238$	0	0
$239$	−25.0000	−1.61712	−0.808558	−	0.588417i	$-0.799751\pi$
$239$	−25.0000	−1.61712	−0.808558	+	0.588417i	$0.799751\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	−4.00000	−0.255551
$246$	0	0
$247$	−1.00000	−0.0636285
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	0	0
$253$	5.00000	0.314347
$254$	0	0
$255$	14.0000	0.876714
$256$	0	0
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	0	0
$259$	−18.0000	−1.11847
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	−8.00000	−0.489592
$268$	0	0
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	15.0000	0.911185	0.455593	−	0.890188i	$-0.349427\pi$
$271$	15.0000	0.911185	0.455593	+	0.890188i	$0.349427\pi$
$272$	0	0
$273$	3.00000	0.181568
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−16.0000	−0.961347	−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000	−0.961347	−0.480673	+	0.876900i	$0.659608\pi$
$278$	0	0
$279$	20.0000	1.19737
$280$	0	0
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	0	0
$285$	2.00000	0.118470
$286$	0	0
$287$	18.0000	1.06251
$288$	0	0
$289$	32.0000	1.88235
$290$	0	0
$291$	8.00000	0.468968
$292$	0	0
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	0	0
$297$	−5.00000	−0.290129
$298$	0	0
$299$	5.00000	0.289157
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	24.0000	1.37424
$306$	0	0
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	0	0
$309$	−12.0000	−0.682656
$310$	0	0
$311$	−3.00000	−0.170114	−0.0850572	−	0.996376i	$-0.527107\pi$
$311$	−3.00000	−0.170114	−0.0850572	+	0.996376i	$0.527107\pi$
$312$	0	0
$313$	21.0000	1.18699	0.593495	−	0.804838i	$-0.297748\pi$
$313$	21.0000	1.18699	0.593495	+	0.804838i	$0.297748\pi$
$314$	0	0
$315$	12.0000	0.676123
$316$	0	0
$317$	21.0000	1.17948	0.589739	−	0.807594i	$-0.299231\pi$
$317$	21.0000	1.17948	0.589739	+	0.807594i	$0.299231\pi$
$318$	0	0
$319$	1.00000	0.0559893
$320$	0	0
$321$	13.0000	0.725589
$322$	0	0
$323$	7.00000	0.389490
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−13.0000	−0.718902
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−29.0000	−1.59398	−0.796992	−	0.603990i	$-0.793577\pi$
$331$	−29.0000	−1.59398	−0.796992	+	0.603990i	$0.793577\pi$
$332$	0	0
$333$	12.0000	0.657596
$334$	0	0
$335$	6.00000	0.327815
$336$	0	0
$337$	−20.0000	−1.08947	−0.544735	−	0.838608i	$-0.683370\pi$
$337$	−20.0000	−1.08947	−0.544735	+	0.838608i	$0.683370\pi$
$338$	0	0
$339$	−12.0000	−0.651751
$340$	0	0
$341$	−10.0000	−0.541530
$342$	0	0
$343$	−15.0000	−0.809924
$344$	0	0
$345$	−10.0000	−0.538382
$346$	0	0
$347$	26.0000	1.39575	0.697877	−	0.716218i	$-0.254128\pi$
$347$	26.0000	1.39575	0.697877	+	0.716218i	$0.254128\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	1.00000	0.0532246	0.0266123	−	0.999646i	$-0.491528\pi$
$353$	1.00000	0.0532246	0.0266123	+	0.999646i	$0.491528\pi$
$354$	0	0
$355$	−20.0000	−1.06149
$356$	0	0
$357$	−21.0000	−1.11144
$358$	0	0
$359$	17.0000	0.897226	0.448613	−	0.893726i	$-0.351918\pi$
$359$	17.0000	0.897226	0.448613	+	0.893726i	$0.351918\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	0	0
$369$	−12.0000	−0.624695
$370$	0	0
$371$	−3.00000	−0.155752
$372$	0	0
$373$	11.0000	0.569558	0.284779	−	0.958593i	$-0.408080\pi$
$373$	11.0000	0.569558	0.284779	+	0.958593i	$0.408080\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	0	0
$377$	1.00000	0.0515026
$378$	0	0
$379$	5.00000	0.256833	0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000	0.256833	0.128416	+	0.991720i	$0.459011\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	0	0
$383$	−2.00000	−0.102195	−0.0510976	−	0.998694i	$-0.516272\pi$
$383$	−2.00000	−0.102195	−0.0510976	+	0.998694i	$0.516272\pi$
$384$	0	0
$385$	−6.00000	−0.305788
$386$	0	0
$387$	−8.00000	−0.406663
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	−35.0000	−1.77003
$392$	0	0
$393$	22.0000	1.10975
$394$	0	0
$395$	16.0000	0.805047
$396$	0	0
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	0	0
$399$	−3.00000	−0.150188
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	−10.0000	−0.498135
$404$	0	0
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	−6.00000	−0.297409
$408$	0	0
$409$	34.0000	1.68119	0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000	1.68119	0.840596	+	0.541663i	$0.182205\pi$
$410$	0	0
$411$	−17.0000	−0.838548
$412$	0	0
$413$	−9.00000	−0.442861
$414$	0	0
$415$	16.0000	0.785409
$416$	0	0
$417$	14.0000	0.685583
$418$	0	0
$419$	−10.0000	−0.488532	−0.244266	−	0.969708i	$-0.578547\pi$
$419$	−10.0000	−0.488532	−0.244266	+	0.969708i	$0.578547\pi$
$420$	0	0
$421$	15.0000	0.731055	0.365528	−	0.930800i	$-0.380889\pi$
$421$	15.0000	0.731055	0.365528	+	0.930800i	$0.380889\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7.00000	0.339550
$426$	0	0
$427$	−36.0000	−1.74216
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	−34.0000	−1.63772	−0.818861	−	0.573992i	$-0.805394\pi$
$431$	−34.0000	−1.63772	−0.818861	+	0.573992i	$0.805394\pi$
$432$	0	0
$433$	24.0000	1.15337	0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.0000	1.15337	0.576683	+	0.816968i	$0.304347\pi$
$434$	0	0
$435$	−2.00000	−0.0958927
$436$	0	0
$437$	−5.00000	−0.239182
$438$	0	0
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	0	0
$441$	−4.00000	−0.190476
$442$	0	0
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	0	0
$445$	16.0000	0.758473
$446$	0	0
$447$	−18.0000	−0.851371
$448$	0	0
$449$	24.0000	1.13263	0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000	1.13263	0.566315	+	0.824189i	$0.308369\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	0	0
$453$	−12.0000	−0.563809
$454$	0	0
$455$	−6.00000	−0.281284
$456$	0	0
$457$	−37.0000	−1.73079	−0.865393	−	0.501093i	$-0.832931\pi$
$457$	−37.0000	−1.73079	−0.865393	+	0.501093i	$0.832931\pi$
$458$	0	0
$459$	35.0000	1.63366
$460$	0	0
$461$	−8.00000	−0.372597	−0.186299	−	0.982493i	$-0.559649\pi$
$461$	−8.00000	−0.372597	−0.186299	+	0.982493i	$0.559649\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	0	0
$465$	20.0000	0.927478
$466$	0	0
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	0	0
$469$	−9.00000	−0.415581
$470$	0	0
$471$	−22.0000	−1.01371
$472$	0	0
$473$	4.00000	0.183920
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	15.0000	0.682524
$484$	0	0
$485$	−16.0000	−0.726523
$486$	0	0
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	0	0
$489$	2.00000	0.0904431
$490$	0	0
$491$	−10.0000	−0.451294	−0.225647	−	0.974209i	$-0.572450\pi$
$491$	−10.0000	−0.451294	−0.225647	+	0.974209i	$0.572450\pi$
$492$	0	0
$493$	−7.00000	−0.315264
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	30.0000	1.34568
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	22.0000	0.982888
$502$	0	0
$503$	19.0000	0.847168	0.423584	−	0.905857i	$-0.360772\pi$
$503$	19.0000	0.847168	0.423584	+	0.905857i	$0.360772\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	−12.0000	−0.532939
$508$	0	0
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	0	0
$511$	9.00000	0.398137
$512$	0	0
$513$	5.00000	0.220755
$514$	0	0
$515$	24.0000	1.05757
$516$	0	0
$517$	0	0
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	−20.0000	−0.876216	−0.438108	−	0.898922i	$-0.644351\pi$
$521$	−20.0000	−0.876216	−0.438108	+	0.898922i	$0.644351\pi$
$522$	0	0
$523$	21.0000	0.918266	0.459133	−	0.888368i	$-0.348160\pi$
$523$	21.0000	0.918266	0.459133	+	0.888368i	$0.348160\pi$
$524$	0	0
$525$	−3.00000	−0.130931
$526$	0	0
$527$	70.0000	3.04925
$528$	0	0
$529$	2.00000	0.0869565
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	6.00000	0.259889
$534$	0	0
$535$	−26.0000	−1.12408
$536$	0	0
$537$	−4.00000	−0.172613
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	−16.0000	−0.687894	−0.343947	−	0.938989i	$-0.611764\pi$
$541$	−16.0000	−0.687894	−0.343947	+	0.938989i	$0.611764\pi$
$542$	0	0
$543$	−2.00000	−0.0858282
$544$	0	0
$545$	26.0000	1.11372
$546$	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
$549$	24.0000	1.02430
$550$	0	0
$551$	−1.00000	−0.0426014
$552$	0	0
$553$	−24.0000	−1.02058
$554$	0	0
$555$	12.0000	0.509372
$556$	0	0
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	−7.00000	−0.295540
$562$	0	0
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	24.0000	1.00969
$566$	0	0
$567$	3.00000	0.125988
$568$	0	0
$569$	−12.0000	−0.503066	−0.251533	−	0.967849i	$-0.580935\pi$
$569$	−12.0000	−0.503066	−0.251533	+	0.967849i	$0.580935\pi$
$570$	0	0
$571$	14.0000	0.585882	0.292941	−	0.956131i	$-0.405366\pi$
$571$	14.0000	0.585882	0.292941	+	0.956131i	$0.405366\pi$
$572$	0	0
$573$	−3.00000	−0.125327
$574$	0	0
$575$	−5.00000	−0.208514
$576$	0	0
$577$	23.0000	0.957503	0.478751	−	0.877951i	$-0.341090\pi$
$577$	23.0000	0.957503	0.478751	+	0.877951i	$0.341090\pi$
$578$	0	0
$579$	−4.00000	−0.166234
$580$	0	0
$581$	−24.0000	−0.995688
$582$	0	0
$583$	−1.00000	−0.0414158
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	10.0000	0.412043
$590$	0	0
$591$	−12.0000	−0.493614
$592$	0	0
$593$	42.0000	1.72473	0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000	1.72473	0.862367	+	0.506284i	$0.168981\pi$
$594$	0	0
$595$	42.0000	1.72183
$596$	0	0
$597$	−23.0000	−0.941327
$598$	0	0
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	0	0
$603$	6.00000	0.244339
$604$	0	0
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	0	0
$609$	3.00000	0.121566
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−28.0000	−1.13091	−0.565455	−	0.824779i	$-0.691299\pi$
$613$	−28.0000	−1.13091	−0.565455	+	0.824779i	$0.691299\pi$
$614$	0	0
$615$	−12.0000	−0.483887
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	−25.0000	−1.00322
$622$	0	0
$623$	−24.0000	−0.961540
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	−1.00000	−0.0399362
$628$	0	0
$629$	42.0000	1.67465
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	−1.00000	−0.0397464
$634$	0	0
$635$	32.0000	1.26988
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	−20.0000	−0.791188
$640$	0	0
$641$	40.0000	1.57991	0.789953	−	0.613168i	$-0.210105\pi$
$641$	40.0000	1.57991	0.789953	+	0.613168i	$0.210105\pi$
$642$	0	0
$643$	26.0000	1.02534	0.512670	−	0.858586i	$-0.328656\pi$
$643$	26.0000	1.02534	0.512670	+	0.858586i	$0.328656\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	0	0
$647$	−3.00000	−0.117942	−0.0589711	−	0.998260i	$-0.518782\pi$
$647$	−3.00000	−0.117942	−0.0589711	+	0.998260i	$0.518782\pi$
$648$	0	0
$649$	−3.00000	−0.117760
$650$	0	0
$651$	−30.0000	−1.17579
$652$	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
$655$	−44.0000	−1.71922
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	−11.0000	−0.428499	−0.214250	−	0.976779i	$-0.568731\pi$
$659$	−11.0000	−0.428499	−0.214250	+	0.976779i	$0.568731\pi$
$660$	0	0
$661$	29.0000	1.12797	0.563985	−	0.825785i	$-0.309268\pi$
$661$	29.0000	1.12797	0.563985	+	0.825785i	$0.309268\pi$
$662$	0	0
$663$	−7.00000	−0.271857
$664$	0	0
$665$	6.00000	0.232670
$666$	0	0
$667$	5.00000	0.193601
$668$	0	0
$669$	−10.0000	−0.386622
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	−24.0000	−0.925132	−0.462566	−	0.886585i	$-0.653071\pi$
$673$	−24.0000	−0.925132	−0.462566	+	0.886585i	$0.653071\pi$
$674$	0	0
$675$	5.00000	0.192450
$676$	0	0
$677$	−29.0000	−1.11456	−0.557280	−	0.830324i	$-0.688155\pi$
$677$	−29.0000	−1.11456	−0.557280	+	0.830324i	$0.688155\pi$
$678$	0	0
$679$	24.0000	0.921035
$680$	0	0
$681$	15.0000	0.574801
$682$	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
$685$	34.0000	1.29907
$686$	0	0
$687$	20.0000	0.763048
$688$	0	0
$689$	−1.00000	−0.0380970
$690$	0	0
$691$	14.0000	0.532585	0.266293	−	0.963892i	$-0.414201\pi$
$691$	14.0000	0.532585	0.266293	+	0.963892i	$0.414201\pi$
$692$	0	0
$693$	−6.00000	−0.227921
$694$	0	0
$695$	−28.0000	−1.06210
$696$	0	0
$697$	−42.0000	−1.59086
$698$	0	0
$699$	−10.0000	−0.378235
$700$	0	0
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	0	0
$703$	6.00000	0.226294
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18.0000	0.676960
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	0	0
$713$	−50.0000	−1.87251
$714$	0	0
$715$	−2.00000	−0.0747958
$716$	0	0
$717$	−25.0000	−0.933642
$718$	0	0
$719$	33.0000	1.23069	0.615346	−	0.788257i	$-0.289016\pi$
$719$	33.0000	1.23069	0.615346	+	0.788257i	$0.289016\pi$
$720$	0	0
$721$	−36.0000	−1.34071
$722$	0	0
$723$	−4.00000	−0.148762
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−47.0000	−1.74313	−0.871567	−	0.490277i	$-0.836896\pi$
$727$	−47.0000	−1.74313	−0.871567	+	0.490277i	$0.836896\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−28.0000	−1.03562
$732$	0	0
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	0	0
$735$	−4.00000	−0.147542
$736$	0	0
$737$	−3.00000	−0.110506
$738$	0	0
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	0	0
$741$	−1.00000	−0.0367359
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	36.0000	1.31894
$746$	0	0
$747$	16.0000	0.585409
$748$	0	0
$749$	39.0000	1.42503
$750$	0	0
$751$	50.0000	1.82453	0.912263	−	0.409605i	$-0.134333\pi$
$751$	50.0000	1.82453	0.912263	+	0.409605i	$0.134333\pi$
$752$	0	0
$753$	6.00000	0.218652
$754$	0	0
$755$	24.0000	0.873449
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	0	0
$759$	5.00000	0.181489
$760$	0	0
$761$	−11.0000	−0.398750	−0.199375	−	0.979923i	$-0.563891\pi$
$761$	−11.0000	−0.398750	−0.199375	+	0.979923i	$0.563891\pi$
$762$	0	0
$763$	−39.0000	−1.41189
$764$	0	0
$765$	−28.0000	−1.01234
$766$	0	0
$767$	−3.00000	−0.108324
$768$	0	0
$769$	−49.0000	−1.76699	−0.883493	−	0.468445i	$-0.844814\pi$
$769$	−49.0000	−1.76699	−0.883493	+	0.468445i	$0.844814\pi$
$770$	0	0
$771$	−22.0000	−0.792311
$772$	0	0
$773$	−23.0000	−0.827253	−0.413626	−	0.910447i	$-0.635738\pi$
$773$	−23.0000	−0.827253	−0.413626	+	0.910447i	$0.635738\pi$
$774$	0	0
$775$	10.0000	0.359211
$776$	0	0
$777$	−18.0000	−0.645746
$778$	0	0
$779$	−6.00000	−0.214972
$780$	0	0
$781$	10.0000	0.357828
$782$	0	0
$783$	−5.00000	−0.178685
$784$	0	0
$785$	44.0000	1.57043
$786$	0	0
$787$	−49.0000	−1.74666	−0.873331	−	0.487128i	$-0.838045\pi$
$787$	−49.0000	−1.74666	−0.873331	+	0.487128i	$0.838045\pi$
$788$	0	0
$789$	−4.00000	−0.142404
$790$	0	0
$791$	−36.0000	−1.28001
$792$	0	0
$793$	−12.0000	−0.426132
$794$	0	0
$795$	2.00000	0.0709327
$796$	0	0
$797$	19.0000	0.673015	0.336507	−	0.941681i	$-0.390754\pi$
$797$	19.0000	0.673015	0.336507	+	0.941681i	$0.390754\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	16.0000	0.565332
$802$	0	0
$803$	3.00000	0.105868
$804$	0	0
$805$	−30.0000	−1.05736
$806$	0	0
$807$	18.0000	0.633630
$808$	0	0
$809$	−13.0000	−0.457056	−0.228528	−	0.973537i	$-0.573391\pi$
$809$	−13.0000	−0.457056	−0.228528	+	0.973537i	$0.573391\pi$
$810$	0	0
$811$	25.0000	0.877869	0.438934	−	0.898519i	$-0.355356\pi$
$811$	25.0000	0.877869	0.438934	+	0.898519i	$0.355356\pi$
$812$	0	0
$813$	15.0000	0.526073
$814$	0	0
$815$	−4.00000	−0.140114
$816$	0	0
$817$	−4.00000	−0.139942
$818$	0	0
$819$	−6.00000	−0.209657
$820$	0	0
$821$	−16.0000	−0.558404	−0.279202	−	0.960232i	$-0.590070\pi$
$821$	−16.0000	−0.558404	−0.279202	+	0.960232i	$0.590070\pi$
$822$	0	0
$823$	43.0000	1.49889	0.749443	−	0.662069i	$-0.230321\pi$
$823$	43.0000	1.49889	0.749443	+	0.662069i	$0.230321\pi$
$824$	0	0
$825$	−1.00000	−0.0348155
$826$	0	0
$827$	−37.0000	−1.28662	−0.643308	−	0.765607i	$-0.722439\pi$
$827$	−37.0000	−1.28662	−0.643308	+	0.765607i	$0.722439\pi$
$828$	0	0
$829$	25.0000	0.868286	0.434143	−	0.900844i	$-0.357051\pi$
$829$	25.0000	0.868286	0.434143	+	0.900844i	$0.357051\pi$
$830$	0	0
$831$	−16.0000	−0.555034
$832$	0	0
$833$	−14.0000	−0.485071
$834$	0	0
$835$	−44.0000	−1.52268
$836$	0	0
$837$	50.0000	1.72825
$838$	0	0
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	16.0000	0.551069
$844$	0	0
$845$	24.0000	0.825625
$846$	0	0
$847$	3.00000	0.103081
$848$	0	0
$849$	14.0000	0.480479
$850$	0	0
$851$	−30.0000	−1.02839
$852$	0	0
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	−30.0000	−1.02478	−0.512390	−	0.858753i	$-0.671240\pi$
$857$	−30.0000	−1.02478	−0.512390	+	0.858753i	$0.671240\pi$
$858$	0	0
$859$	30.0000	1.02359	0.511793	−	0.859109i	$-0.328981\pi$
$859$	30.0000	1.02359	0.511793	+	0.859109i	$0.328981\pi$
$860$	0	0
$861$	18.0000	0.613438
$862$	0	0
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	0	0
$865$	−28.0000	−0.952029
$866$	0	0
$867$	32.0000	1.08678
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	−3.00000	−0.101651
$872$	0	0
$873$	−16.0000	−0.541518
$874$	0	0
$875$	36.0000	1.21702
$876$	0	0
$877$	27.0000	0.911725	0.455863	−	0.890050i	$-0.349331\pi$
$877$	27.0000	0.911725	0.455863	+	0.890050i	$0.349331\pi$
$878$	0	0
$879$	−9.00000	−0.303562
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	0	0
$885$	6.00000	0.201688
$886$	0	0
$887$	4.00000	0.134307	0.0671534	−	0.997743i	$-0.478608\pi$
$887$	4.00000	0.134307	0.0671534	+	0.997743i	$0.478608\pi$
$888$	0	0
$889$	−48.0000	−1.60987
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	0	0
$894$	0	0
$895$	8.00000	0.267411
$896$	0	0
$897$	5.00000	0.166945
$898$	0	0
$899$	−10.0000	−0.333519
$900$	0	0
$901$	7.00000	0.233204
$902$	0	0
$903$	12.0000	0.399335
$904$	0	0
$905$	4.00000	0.132964
$906$	0	0
$907$	−13.0000	−0.431658	−0.215829	−	0.976431i	$-0.569245\pi$
$907$	−13.0000	−0.431658	−0.215829	+	0.976431i	$0.569245\pi$
$908$	0	0
$909$	−12.0000	−0.398015
$910$	0	0
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	24.0000	0.793416
$916$	0	0
$917$	66.0000	2.17951
$918$	0	0
$919$	−3.00000	−0.0989609	−0.0494804	−	0.998775i	$-0.515757\pi$
$919$	−3.00000	−0.0989609	−0.0494804	+	0.998775i	$0.515757\pi$
$920$	0	0
$921$	16.0000	0.527218
$922$	0	0
$923$	10.0000	0.329154
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	24.0000	0.788263
$928$	0	0
$929$	29.0000	0.951459	0.475730	−	0.879592i	$-0.342184\pi$
$929$	29.0000	0.951459	0.475730	+	0.879592i	$0.342184\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	0	0
$933$	−3.00000	−0.0982156
$934$	0	0
$935$	14.0000	0.457849
$936$	0	0
$937$	−5.00000	−0.163343	−0.0816714	−	0.996659i	$-0.526026\pi$
$937$	−5.00000	−0.163343	−0.0816714	+	0.996659i	$0.526026\pi$
$938$	0	0
$939$	21.0000	0.685309
$940$	0	0
$941$	33.0000	1.07577	0.537885	−	0.843018i	$-0.319224\pi$
$941$	33.0000	1.07577	0.537885	+	0.843018i	$0.319224\pi$
$942$	0	0
$943$	30.0000	0.976934
$944$	0	0
$945$	30.0000	0.975900
$946$	0	0
$947$	48.0000	1.55979	0.779895	−	0.625910i	$-0.215272\pi$
$947$	48.0000	1.55979	0.779895	+	0.625910i	$0.215272\pi$
$948$	0	0
$949$	3.00000	0.0973841
$950$	0	0
$951$	21.0000	0.680972
$952$	0	0
$953$	60.0000	1.94359	0.971795	−	0.235826i	$-0.0757795\pi$
$953$	60.0000	1.94359	0.971795	+	0.235826i	$0.0757795\pi$
$954$	0	0
$955$	6.00000	0.194155
$956$	0	0
$957$	1.00000	0.0323254
$958$	0	0
$959$	−51.0000	−1.64688
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	−26.0000	−0.837838
$964$	0	0
$965$	8.00000	0.257529
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	0	0
$969$	7.00000	0.224872
$970$	0	0
$971$	44.0000	1.41203	0.706014	−	0.708198i	$-0.250492\pi$
$971$	44.0000	1.41203	0.706014	+	0.708198i	$0.250492\pi$
$972$	0	0
$973$	42.0000	1.34646
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	−60.0000	−1.91957	−0.959785	−	0.280736i	$-0.909421\pi$
$977$	−60.0000	−1.91957	−0.959785	+	0.280736i	$0.909421\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	0	0
$981$	26.0000	0.830116
$982$	0	0
$983$	46.0000	1.46717	0.733586	−	0.679597i	$-0.237845\pi$
$983$	46.0000	1.46717	0.733586	+	0.679597i	$0.237845\pi$
$984$	0	0
$985$	24.0000	0.764704
$986$	0	0
$987$	0	0
$988$	0	0
$989$	20.0000	0.635963
$990$	0	0
$991$	−44.0000	−1.39771	−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−44.0000	−1.39771	−0.698853	+	0.715265i	$0.746306\pi$
$992$	0	0
$993$	−29.0000	−0.920287
$994$	0	0
$995$	46.0000	1.45830
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	0	0
$999$	30.0000	0.949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.h.1.1		1
4.3	odd	2		418.2.a.a.1.1	✓	1
12.11	even	2		3762.2.a.g.1.1		1
44.43	even	2		4598.2.a.b.1.1		1
76.75	even	2		7942.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.a.1.1	✓	1	4.3	odd	2
3344.2.a.h.1.1		1	1.1	even	1	trivial
3762.2.a.g.1.1		1	12.11	even	2
4598.2.a.b.1.1		1	44.43	even	2
7942.2.a.i.1.1		1	76.75	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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