LMFDB - Embedded newform 7104.2.a.ba.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7104,2,Mod(1,7104)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7104.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7104, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7104 = 2^{6} \cdot 3 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7104.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,1,0,4,0,-1,0,1,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$56.7257255959$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 888)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7104.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} +4.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +3.00000 q^{11} +5.00000 q^{13} +4.00000 q^{15} +7.00000 q^{17} -5.00000 q^{19} -1.00000 q^{21} +1.00000 q^{23} +11.0000 q^{25} +1.00000 q^{27} +8.00000 q^{29} +6.00000 q^{31} +3.00000 q^{33} -4.00000 q^{35} -1.00000 q^{37} +5.00000 q^{39} -2.00000 q^{41} -12.0000 q^{43} +4.00000 q^{45} -2.00000 q^{47} -6.00000 q^{49} +7.00000 q^{51} +1.00000 q^{53} +12.0000 q^{55} -5.00000 q^{57} +4.00000 q^{59} -14.0000 q^{61} -1.00000 q^{63} +20.0000 q^{65} -14.0000 q^{67} +1.00000 q^{69} +4.00000 q^{71} -11.0000 q^{73} +11.0000 q^{75} -3.00000 q^{77} +10.0000 q^{79} +1.00000 q^{81} -5.00000 q^{83} +28.0000 q^{85} +8.00000 q^{87} -9.00000 q^{89} -5.00000 q^{91} +6.00000 q^{93} -20.0000 q^{95} -10.0000 q^{97} +3.00000 q^{99} +O(q^{100})$

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
$3$	1.00000	0.577350
$4$	0	0
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	0	0
$15$	4.00000	1.03280
$16$	0	0
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	0	0
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	−4.00000	−0.676123
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	5.00000	0.800641
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	0	0
$45$	4.00000	0.596285
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	7.00000	0.980196
$52$	0	0
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	0	0
$55$	12.0000	1.61808
$56$	0	0
$57$	−5.00000	−0.662266
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	20.0000	2.48069
$66$	0	0
$67$	−14.0000	−1.71037	−0.855186	−	0.518321i	$-0.826557\pi$
$67$	−14.0000	−1.71037	−0.855186	+	0.518321i	$0.826557\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	0	0
$75$	11.0000	1.27017
$76$	0	0
$77$	−3.00000	−0.341882
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.00000	−0.548821	−0.274411	−	0.961613i	$-0.588483\pi$
$83$	−5.00000	−0.548821	−0.274411	+	0.961613i	$0.588483\pi$
$84$	0	0
$85$	28.0000	3.03703
$86$	0	0
$87$	8.00000	0.857690
$88$	0	0
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	0	0
$91$	−5.00000	−0.524142
$92$	0	0
$93$	6.00000	0.622171
$94$	0	0
$95$	−20.0000	−2.05196
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	3.00000	0.301511
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	−15.0000	−1.45010	−0.725052	−	0.688694i	$-0.758184\pi$
$107$	−15.0000	−1.45010	−0.725052	+	0.688694i	$0.758184\pi$
$108$	0	0
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	0	0
$111$	−1.00000	−0.0949158
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	5.00000	0.462250
$118$	0	0
$119$	−7.00000	−0.641689
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	24.0000	2.14663
$126$	0	0
$127$	13.0000	1.15356	0.576782	−	0.816898i	$-0.304308\pi$
$127$	13.0000	1.15356	0.576782	+	0.816898i	$0.304308\pi$
$128$	0	0
$129$	−12.0000	−1.05654
$130$	0	0
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	5.00000	0.433555
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	0	0
$143$	15.0000	1.25436
$144$	0	0
$145$	32.0000	2.65746
$146$	0	0
$147$	−6.00000	−0.494872
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−19.0000	−1.54620	−0.773099	−	0.634285i	$-0.781294\pi$
$151$	−19.0000	−1.54620	−0.773099	+	0.634285i	$0.781294\pi$
$152$	0	0
$153$	7.00000	0.565916
$154$	0	0
$155$	24.0000	1.92773
$156$	0	0
$157$	−20.0000	−1.59617	−0.798087	−	0.602542i	$-0.794154\pi$
$157$	−20.0000	−1.59617	−0.798087	+	0.602542i	$0.794154\pi$
$158$	0	0
$159$	1.00000	0.0793052
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−7.00000	−0.548282	−0.274141	−	0.961689i	$-0.588394\pi$
$163$	−7.00000	−0.548282	−0.274141	+	0.961689i	$0.588394\pi$
$164$	0	0
$165$	12.0000	0.934199
$166$	0	0
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	−5.00000	−0.382360
$172$	0	0
$173$	−1.00000	−0.0760286	−0.0380143	−	0.999277i	$-0.512103\pi$
$173$	−1.00000	−0.0760286	−0.0380143	+	0.999277i	$0.512103\pi$
$174$	0	0
$175$	−11.0000	−0.831522
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	0	0
$183$	−14.0000	−1.03491
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	21.0000	1.53567
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	21.0000	1.51951	0.759753	−	0.650211i	$-0.225320\pi$
$191$	21.0000	1.51951	0.759753	+	0.650211i	$0.225320\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	20.0000	1.43223
$196$	0	0
$197$	−5.00000	−0.356235	−0.178118	−	0.984009i	$-0.557001\pi$
$197$	−5.00000	−0.356235	−0.178118	+	0.984009i	$0.557001\pi$
$198$	0	0
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	0	0
$201$	−14.0000	−0.987484
$202$	0	0
$203$	−8.00000	−0.561490
$204$	0	0
$205$	−8.00000	−0.558744
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−15.0000	−1.03757
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	4.00000	0.274075
$214$	0	0
$215$	−48.0000	−3.27357
$216$	0	0
$217$	−6.00000	−0.407307
$218$	0	0
$219$	−11.0000	−0.743311
$220$	0	0
$221$	35.0000	2.35435
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	11.0000	0.733333
$226$	0	0
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	0	0
$237$	10.0000	0.649570
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−16.0000	−1.03065	−0.515325	−	0.856995i	$-0.672329\pi$
$241$	−16.0000	−1.03065	−0.515325	+	0.856995i	$0.672329\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−24.0000	−1.53330
$246$	0	0
$247$	−25.0000	−1.59071
$248$	0	0
$249$	−5.00000	−0.316862
$250$	0	0
$251$	10.0000	0.631194	0.315597	−	0.948893i	$-0.397795\pi$
$251$	10.0000	0.631194	0.315597	+	0.948893i	$0.397795\pi$
$252$	0	0
$253$	3.00000	0.188608
$254$	0	0
$255$	28.0000	1.75343
$256$	0	0
$257$	−1.00000	−0.0623783	−0.0311891	−	0.999514i	$-0.509929\pi$
$257$	−1.00000	−0.0623783	−0.0311891	+	0.999514i	$0.509929\pi$
$258$	0	0
$259$	1.00000	0.0621370
$260$	0	0
$261$	8.00000	0.495188
$262$	0	0
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	0	0
$267$	−9.00000	−0.550791
$268$	0	0
$269$	−1.00000	−0.0609711	−0.0304855	−	0.999535i	$-0.509705\pi$
$269$	−1.00000	−0.0609711	−0.0304855	+	0.999535i	$0.509705\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	0	0
$273$	−5.00000	−0.302614
$274$	0	0
$275$	33.0000	1.98997
$276$	0	0
$277$	19.0000	1.14160	0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000	1.14160	0.570800	+	0.821089i	$0.306633\pi$
$278$	0	0
$279$	6.00000	0.359211
$280$	0	0
$281$	19.0000	1.13344	0.566722	−	0.823909i	$-0.308211\pi$
$281$	19.0000	1.13344	0.566722	+	0.823909i	$0.308211\pi$
$282$	0	0
$283$	−11.0000	−0.653882	−0.326941	−	0.945045i	$-0.606018\pi$
$283$	−11.0000	−0.653882	−0.326941	+	0.945045i	$0.606018\pi$
$284$	0	0
$285$	−20.0000	−1.18470
$286$	0	0
$287$	2.00000	0.118056
$288$	0	0
$289$	32.0000	1.88235
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	−7.00000	−0.408944	−0.204472	−	0.978872i	$-0.565548\pi$
$293$	−7.00000	−0.408944	−0.204472	+	0.978872i	$0.565548\pi$
$294$	0	0
$295$	16.0000	0.931556
$296$	0	0
$297$	3.00000	0.174078
$298$	0	0
$299$	5.00000	0.289157
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	−10.0000	−0.574485
$304$	0	0
$305$	−56.0000	−3.20655
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	−10.0000	−0.568880
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	−4.00000	−0.225374
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	24.0000	1.34374
$320$	0	0
$321$	−15.0000	−0.837218
$322$	0	0
$323$	−35.0000	−1.94745
$324$	0	0
$325$	55.0000	3.05085
$326$	0	0
$327$	7.00000	0.387101
$328$	0	0
$329$	2.00000	0.110264
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	0	0
$333$	−1.00000	−0.0547997
$334$	0	0
$335$	−56.0000	−3.05961
$336$	0	0
$337$	5.00000	0.272367	0.136184	−	0.990684i	$-0.456516\pi$
$337$	5.00000	0.272367	0.136184	+	0.990684i	$0.456516\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	18.0000	0.974755
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	4.00000	0.215353
$346$	0	0
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	0	0
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	16.0000	0.849192
$356$	0	0
$357$	−7.00000	−0.370479
$358$	0	0
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	−2.00000	−0.104973
$364$	0	0
$365$	−44.0000	−2.30307
$366$	0	0
$367$	23.0000	1.20059	0.600295	−	0.799779i	$-0.295050\pi$
$367$	23.0000	1.20059	0.600295	+	0.799779i	$0.295050\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	−1.00000	−0.0519174
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	0	0
$375$	24.0000	1.23935
$376$	0	0
$377$	40.0000	2.06010
$378$	0	0
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	0	0
$381$	13.0000	0.666010
$382$	0	0
$383$	−1.00000	−0.0510976	−0.0255488	−	0.999674i	$-0.508133\pi$
$383$	−1.00000	−0.0510976	−0.0255488	+	0.999674i	$0.508133\pi$
$384$	0	0
$385$	−12.0000	−0.611577
$386$	0	0
$387$	−12.0000	−0.609994
$388$	0	0
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	7.00000	0.354005
$392$	0	0
$393$	18.0000	0.907980
$394$	0	0
$395$	40.0000	2.01262
$396$	0	0
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	0	0
$399$	5.00000	0.250313
$400$	0	0
$401$	−25.0000	−1.24844	−0.624220	−	0.781248i	$-0.714583\pi$
$401$	−25.0000	−1.24844	−0.624220	+	0.781248i	$0.714583\pi$
$402$	0	0
$403$	30.0000	1.49441
$404$	0	0
$405$	4.00000	0.198762
$406$	0	0
$407$	−3.00000	−0.148704
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	−20.0000	−0.981761
$416$	0	0
$417$	−2.00000	−0.0979404
$418$	0	0
$419$	−39.0000	−1.90527	−0.952637	−	0.304109i	$-0.901641\pi$
$419$	−39.0000	−1.90527	−0.952637	+	0.304109i	$0.901641\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	0	0
$423$	−2.00000	−0.0972433
$424$	0	0
$425$	77.0000	3.73505
$426$	0	0
$427$	14.0000	0.677507
$428$	0	0
$429$	15.0000	0.724207
$430$	0	0
$431$	3.00000	0.144505	0.0722525	−	0.997386i	$-0.476981\pi$
$431$	3.00000	0.144505	0.0722525	+	0.997386i	$0.476981\pi$
$432$	0	0
$433$	7.00000	0.336399	0.168199	−	0.985753i	$-0.446205\pi$
$433$	7.00000	0.336399	0.168199	+	0.985753i	$0.446205\pi$
$434$	0	0
$435$	32.0000	1.53428
$436$	0	0
$437$	−5.00000	−0.239182
$438$	0	0
$439$	−34.0000	−1.62273	−0.811366	−	0.584539i	$-0.801275\pi$
$439$	−34.0000	−1.62273	−0.811366	+	0.584539i	$0.801275\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	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
$447$	6.00000	0.283790
$448$	0	0
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	0	0
$453$	−19.0000	−0.892698
$454$	0	0
$455$	−20.0000	−0.937614
$456$	0	0
$457$	32.0000	1.49690	0.748448	−	0.663193i	$-0.230799\pi$
$457$	32.0000	1.49690	0.748448	+	0.663193i	$0.230799\pi$
$458$	0	0
$459$	7.00000	0.326732
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	−30.0000	−1.39422	−0.697109	−	0.716965i	$-0.745531\pi$
$463$	−30.0000	−1.39422	−0.697109	+	0.716965i	$0.745531\pi$
$464$	0	0
$465$	24.0000	1.11297
$466$	0	0
$467$	22.0000	1.01804	0.509019	−	0.860755i	$-0.330008\pi$
$467$	22.0000	1.01804	0.509019	+	0.860755i	$0.330008\pi$
$468$	0	0
$469$	14.0000	0.646460
$470$	0	0
$471$	−20.0000	−0.921551
$472$	0	0
$473$	−36.0000	−1.65528
$474$	0	0
$475$	−55.0000	−2.52357
$476$	0	0
$477$	1.00000	0.0457869
$478$	0	0
$479$	−21.0000	−0.959514	−0.479757	−	0.877401i	$-0.659275\pi$
$479$	−21.0000	−0.959514	−0.479757	+	0.877401i	$0.659275\pi$
$480$	0	0
$481$	−5.00000	−0.227980
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	−40.0000	−1.81631
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	0	0
$489$	−7.00000	−0.316551
$490$	0	0
$491$	−41.0000	−1.85030	−0.925152	−	0.379597i	$-0.876063\pi$
$491$	−41.0000	−1.85030	−0.925152	+	0.379597i	$0.876063\pi$
$492$	0	0
$493$	56.0000	2.52211
$494$	0	0
$495$	12.0000	0.539360
$496$	0	0
$497$	−4.00000	−0.179425
$498$	0	0
$499$	−31.0000	−1.38775	−0.693875	−	0.720095i	$-0.744098\pi$
$499$	−31.0000	−1.38775	−0.693875	+	0.720095i	$0.744098\pi$
$500$	0	0
$501$	3.00000	0.134030
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−40.0000	−1.77998
$506$	0	0
$507$	12.0000	0.532939
$508$	0	0
$509$	−11.0000	−0.487566	−0.243783	−	0.969830i	$-0.578389\pi$
$509$	−11.0000	−0.487566	−0.243783	+	0.969830i	$0.578389\pi$
$510$	0	0
$511$	11.0000	0.486611
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	−40.0000	−1.76261
$516$	0	0
$517$	−6.00000	−0.263880
$518$	0	0
$519$	−1.00000	−0.0438951
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	0	0
$525$	−11.0000	−0.480079
$526$	0	0
$527$	42.0000	1.82955
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	−10.0000	−0.433148
$534$	0	0
$535$	−60.0000	−2.59403
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−18.0000	−0.775315
$540$	0	0
$541$	25.0000	1.07483	0.537417	−	0.843317i	$-0.319400\pi$
$541$	25.0000	1.07483	0.537417	+	0.843317i	$0.319400\pi$
$542$	0	0
$543$	−16.0000	−0.686626
$544$	0	0
$545$	28.0000	1.19939
$546$	0	0
$547$	33.0000	1.41098	0.705489	−	0.708721i	$-0.250727\pi$
$547$	33.0000	1.41098	0.705489	+	0.708721i	$0.250727\pi$
$548$	0	0
$549$	−14.0000	−0.597505
$550$	0	0
$551$	−40.0000	−1.70406
$552$	0	0
$553$	−10.0000	−0.425243
$554$	0	0
$555$	−4.00000	−0.169791
$556$	0	0
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	−60.0000	−2.53773
$560$	0	0
$561$	21.0000	0.886621
$562$	0	0
$563$	8.00000	0.337160	0.168580	−	0.985688i	$-0.446082\pi$
$563$	8.00000	0.337160	0.168580	+	0.985688i	$0.446082\pi$
$564$	0	0
$565$	24.0000	1.00969
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	41.0000	1.71881	0.859405	−	0.511296i	$-0.170834\pi$
$569$	41.0000	1.71881	0.859405	+	0.511296i	$0.170834\pi$
$570$	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
$573$	21.0000	0.877288
$574$	0	0
$575$	11.0000	0.458732
$576$	0	0
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	0	0
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	5.00000	0.207435
$582$	0	0
$583$	3.00000	0.124247
$584$	0	0
$585$	20.0000	0.826898
$586$	0	0
$587$	−16.0000	−0.660391	−0.330195	−	0.943913i	$-0.607115\pi$
$587$	−16.0000	−0.660391	−0.330195	+	0.943913i	$0.607115\pi$
$588$	0	0
$589$	−30.0000	−1.23613
$590$	0	0
$591$	−5.00000	−0.205673
$592$	0	0
$593$	12.0000	0.492781	0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000	0.492781	0.246390	+	0.969171i	$0.420755\pi$
$594$	0	0
$595$	−28.0000	−1.14789
$596$	0	0
$597$	−12.0000	−0.491127
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	−49.0000	−1.99875	−0.999376	−	0.0353259i	$-0.988753\pi$
$601$	−49.0000	−1.99875	−0.999376	+	0.0353259i	$0.988753\pi$
$602$	0	0
$603$	−14.0000	−0.570124
$604$	0	0
$605$	−8.00000	−0.325246
$606$	0	0
$607$	42.0000	1.70473	0.852364	−	0.522949i	$-0.175168\pi$
$607$	42.0000	1.70473	0.852364	+	0.522949i	$0.175168\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	0	0
$611$	−10.0000	−0.404557
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	0	0
$615$	−8.00000	−0.322591
$616$	0	0
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	0	0
$619$	46.0000	1.84890	0.924448	−	0.381308i	$-0.124526\pi$
$619$	46.0000	1.84890	0.924448	+	0.381308i	$0.124526\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	9.00000	0.360577
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	−15.0000	−0.599042
$628$	0	0
$629$	−7.00000	−0.279108
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	4.00000	0.158986
$634$	0	0
$635$	52.0000	2.06356
$636$	0	0
$637$	−30.0000	−1.18864
$638$	0	0
$639$	4.00000	0.158238
$640$	0	0
$641$	16.0000	0.631962	0.315981	−	0.948766i	$-0.397666\pi$
$641$	16.0000	0.631962	0.315981	+	0.948766i	$0.397666\pi$
$642$	0	0
$643$	23.0000	0.907031	0.453516	−	0.891248i	$-0.350170\pi$
$643$	23.0000	0.907031	0.453516	+	0.891248i	$0.350170\pi$
$644$	0	0
$645$	−48.0000	−1.89000
$646$	0	0
$647$	−7.00000	−0.275198	−0.137599	−	0.990488i	$-0.543939\pi$
$647$	−7.00000	−0.275198	−0.137599	+	0.990488i	$0.543939\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	−6.00000	−0.235159
$652$	0	0
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	0	0
$655$	72.0000	2.81327
$656$	0	0
$657$	−11.0000	−0.429151
$658$	0	0
$659$	−8.00000	−0.311636	−0.155818	−	0.987786i	$-0.549801\pi$
$659$	−8.00000	−0.311636	−0.155818	+	0.987786i	$0.549801\pi$
$660$	0	0
$661$	21.0000	0.816805	0.408403	−	0.912802i	$-0.366086\pi$
$661$	21.0000	0.816805	0.408403	+	0.912802i	$0.366086\pi$
$662$	0	0
$663$	35.0000	1.35929
$664$	0	0
$665$	20.0000	0.775567
$666$	0	0
$667$	8.00000	0.309761
$668$	0	0
$669$	−8.00000	−0.309298
$670$	0	0
$671$	−42.0000	−1.62139
$672$	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
$675$	11.0000	0.423390
$676$	0	0
$677$	27.0000	1.03769	0.518847	−	0.854867i	$-0.326361\pi$
$677$	27.0000	1.03769	0.518847	+	0.854867i	$0.326361\pi$
$678$	0	0
$679$	10.0000	0.383765
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	−6.00000	−0.229584	−0.114792	−	0.993390i	$-0.536620\pi$
$683$	−6.00000	−0.229584	−0.114792	+	0.993390i	$0.536620\pi$
$684$	0	0
$685$	48.0000	1.83399
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.00000	0.190485
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	0	0
$693$	−3.00000	−0.113961
$694$	0	0
$695$	−8.00000	−0.303457
$696$	0	0
$697$	−14.0000	−0.530288
$698$	0	0
$699$	14.0000	0.529529
$700$	0	0
$701$	−16.0000	−0.604312	−0.302156	−	0.953259i	$-0.597706\pi$
$701$	−16.0000	−0.604312	−0.302156	+	0.953259i	$0.597706\pi$
$702$	0	0
$703$	5.00000	0.188579
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	10.0000	0.376089
$708$	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
$711$	10.0000	0.375029
$712$	0	0
$713$	6.00000	0.224702
$714$	0	0
$715$	60.0000	2.24387
$716$	0	0
$717$	24.0000	0.896296
$718$	0	0
$719$	12.0000	0.447524	0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000	0.447524	0.223762	+	0.974644i	$0.428166\pi$
$720$	0	0
$721$	10.0000	0.372419
$722$	0	0
$723$	−16.0000	−0.595046
$724$	0	0
$725$	88.0000	3.26824
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−84.0000	−3.10685
$732$	0	0
$733$	−18.0000	−0.664845	−0.332423	−	0.943131i	$-0.607866\pi$
$733$	−18.0000	−0.664845	−0.332423	+	0.943131i	$0.607866\pi$
$734$	0	0
$735$	−24.0000	−0.885253
$736$	0	0
$737$	−42.0000	−1.54709
$738$	0	0
$739$	10.0000	0.367856	0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000	0.367856	0.183928	+	0.982940i	$0.441119\pi$
$740$	0	0
$741$	−25.0000	−0.918398
$742$	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
$745$	24.0000	0.879292
$746$	0	0
$747$	−5.00000	−0.182940
$748$	0	0
$749$	15.0000	0.548088
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	10.0000	0.364420
$754$	0	0
$755$	−76.0000	−2.76592
$756$	0	0
$757$	7.00000	0.254419	0.127210	−	0.991876i	$-0.459398\pi$
$757$	7.00000	0.254419	0.127210	+	0.991876i	$0.459398\pi$
$758$	0	0
$759$	3.00000	0.108893
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	−7.00000	−0.253417
$764$	0	0
$765$	28.0000	1.01234
$766$	0	0
$767$	20.0000	0.722158
$768$	0	0
$769$	−24.0000	−0.865462	−0.432731	−	0.901523i	$-0.642450\pi$
$769$	−24.0000	−0.865462	−0.432731	+	0.901523i	$0.642450\pi$
$770$	0	0
$771$	−1.00000	−0.0360141
$772$	0	0
$773$	11.0000	0.395643	0.197821	−	0.980238i	$-0.436613\pi$
$773$	11.0000	0.395643	0.197821	+	0.980238i	$0.436613\pi$
$774$	0	0
$775$	66.0000	2.37079
$776$	0	0
$777$	1.00000	0.0358748
$778$	0	0
$779$	10.0000	0.358287
$780$	0	0
$781$	12.0000	0.429394
$782$	0	0
$783$	8.00000	0.285897
$784$	0	0
$785$	−80.0000	−2.85532
$786$	0	0
$787$	−42.0000	−1.49714	−0.748569	−	0.663057i	$-0.769259\pi$
$787$	−42.0000	−1.49714	−0.748569	+	0.663057i	$0.769259\pi$
$788$	0	0
$789$	6.00000	0.213606
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−70.0000	−2.48577
$794$	0	0
$795$	4.00000	0.141865
$796$	0	0
$797$	40.0000	1.41687	0.708436	−	0.705775i	$-0.249401\pi$
$797$	40.0000	1.41687	0.708436	+	0.705775i	$0.249401\pi$
$798$	0	0
$799$	−14.0000	−0.495284
$800$	0	0
$801$	−9.00000	−0.317999
$802$	0	0
$803$	−33.0000	−1.16454
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	−1.00000	−0.0352017
$808$	0	0
$809$	39.0000	1.37117	0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000	1.37117	0.685583	+	0.727994i	$0.259547\pi$
$810$	0	0
$811$	40.0000	1.40459	0.702295	−	0.711886i	$-0.252159\pi$
$811$	40.0000	1.40459	0.702295	+	0.711886i	$0.252159\pi$
$812$	0	0
$813$	24.0000	0.841717
$814$	0	0
$815$	−28.0000	−0.980797
$816$	0	0
$817$	60.0000	2.09913
$818$	0	0
$819$	−5.00000	−0.174714
$820$	0	0
$821$	45.0000	1.57051	0.785255	−	0.619172i	$-0.212532\pi$
$821$	45.0000	1.57051	0.785255	+	0.619172i	$0.212532\pi$
$822$	0	0
$823$	−23.0000	−0.801730	−0.400865	−	0.916137i	$-0.631290\pi$
$823$	−23.0000	−0.801730	−0.400865	+	0.916137i	$0.631290\pi$
$824$	0	0
$825$	33.0000	1.14891
$826$	0	0
$827$	18.0000	0.625921	0.312961	−	0.949766i	$-0.398679\pi$
$827$	18.0000	0.625921	0.312961	+	0.949766i	$0.398679\pi$
$828$	0	0
$829$	11.0000	0.382046	0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000	0.382046	0.191023	+	0.981586i	$0.438820\pi$
$830$	0	0
$831$	19.0000	0.659103
$832$	0	0
$833$	−42.0000	−1.45521
$834$	0	0
$835$	12.0000	0.415277
$836$	0	0
$837$	6.00000	0.207390
$838$	0	0
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	19.0000	0.654395
$844$	0	0
$845$	48.0000	1.65125
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	−11.0000	−0.377519
$850$	0	0
$851$	−1.00000	−0.0342796
$852$	0	0
$853$	17.0000	0.582069	0.291034	−	0.956713i	$-0.406001\pi$
$853$	17.0000	0.582069	0.291034	+	0.956713i	$0.406001\pi$
$854$	0	0
$855$	−20.0000	−0.683986
$856$	0	0
$857$	33.0000	1.12726	0.563629	−	0.826028i	$-0.309405\pi$
$857$	33.0000	1.12726	0.563629	+	0.826028i	$0.309405\pi$
$858$	0	0
$859$	41.0000	1.39890	0.699451	−	0.714681i	$-0.253428\pi$
$859$	41.0000	1.39890	0.699451	+	0.714681i	$0.253428\pi$
$860$	0	0
$861$	2.00000	0.0681598
$862$	0	0
$863$	34.0000	1.15737	0.578687	−	0.815550i	$-0.303565\pi$
$863$	34.0000	1.15737	0.578687	+	0.815550i	$0.303565\pi$
$864$	0	0
$865$	−4.00000	−0.136004
$866$	0	0
$867$	32.0000	1.08678
$868$	0	0
$869$	30.0000	1.01768
$870$	0	0
$871$	−70.0000	−2.37186
$872$	0	0
$873$	−10.0000	−0.338449
$874$	0	0
$875$	−24.0000	−0.811348
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	0	0
$879$	−7.00000	−0.236104
$880$	0	0
$881$	−48.0000	−1.61716	−0.808581	−	0.588386i	$-0.799764\pi$
$881$	−48.0000	−1.61716	−0.808581	+	0.588386i	$0.799764\pi$
$882$	0	0
$883$	13.0000	0.437485	0.218742	−	0.975783i	$-0.429805\pi$
$883$	13.0000	0.437485	0.218742	+	0.975783i	$0.429805\pi$
$884$	0	0
$885$	16.0000	0.537834
$886$	0	0
$887$	−42.0000	−1.41022	−0.705111	−	0.709097i	$-0.749103\pi$
$887$	−42.0000	−1.41022	−0.705111	+	0.709097i	$0.749103\pi$
$888$	0	0
$889$	−13.0000	−0.436006
$890$	0	0
$891$	3.00000	0.100504
$892$	0	0
$893$	10.0000	0.334637
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.00000	0.166945
$898$	0	0
$899$	48.0000	1.60089
$900$	0	0
$901$	7.00000	0.233204
$902$	0	0
$903$	12.0000	0.399335
$904$	0	0
$905$	−64.0000	−2.12743
$906$	0	0
$907$	−19.0000	−0.630885	−0.315442	−	0.948945i	$-0.602153\pi$
$907$	−19.0000	−0.630885	−0.315442	+	0.948945i	$0.602153\pi$
$908$	0	0
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	−15.0000	−0.496428
$914$	0	0
$915$	−56.0000	−1.85130
$916$	0	0
$917$	−18.0000	−0.594412
$918$	0	0
$919$	22.0000	0.725713	0.362857	−	0.931845i	$-0.381802\pi$
$919$	22.0000	0.725713	0.362857	+	0.931845i	$0.381802\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	20.0000	0.658308
$924$	0	0
$925$	−11.0000	−0.361678
$926$	0	0
$927$	−10.0000	−0.328443
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	30.0000	0.983210
$932$	0	0
$933$	−24.0000	−0.785725
$934$	0	0
$935$	84.0000	2.74709
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	0	0
$941$	−54.0000	−1.76035	−0.880175	−	0.474650i	$-0.842575\pi$
$941$	−54.0000	−1.76035	−0.880175	+	0.474650i	$0.842575\pi$
$942$	0	0
$943$	−2.00000	−0.0651290
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	14.0000	0.454939	0.227469	−	0.973785i	$-0.426955\pi$
$947$	14.0000	0.454939	0.227469	+	0.973785i	$0.426955\pi$
$948$	0	0
$949$	−55.0000	−1.78538
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	−12.0000	−0.388718	−0.194359	−	0.980930i	$-0.562263\pi$
$953$	−12.0000	−0.388718	−0.194359	+	0.980930i	$0.562263\pi$
$954$	0	0
$955$	84.0000	2.71818
$956$	0	0
$957$	24.0000	0.775810
$958$	0	0
$959$	−12.0000	−0.387500
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	−15.0000	−0.483368
$964$	0	0
$965$	−8.00000	−0.257529
$966$	0	0
$967$	52.0000	1.67221	0.836104	−	0.548572i	$-0.184828\pi$
$967$	52.0000	1.67221	0.836104	+	0.548572i	$0.184828\pi$
$968$	0	0
$969$	−35.0000	−1.12436
$970$	0	0
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	0	0
$973$	2.00000	0.0641171
$974$	0	0
$975$	55.0000	1.76141
$976$	0	0
$977$	23.0000	0.735835	0.367918	−	0.929858i	$-0.380071\pi$
$977$	23.0000	0.735835	0.367918	+	0.929858i	$0.380071\pi$
$978$	0	0
$979$	−27.0000	−0.862924
$980$	0	0
$981$	7.00000	0.223493
$982$	0	0
$983$	−30.0000	−0.956851	−0.478426	−	0.878128i	$-0.658792\pi$
$983$	−30.0000	−0.956851	−0.478426	+	0.878128i	$0.658792\pi$
$984$	0	0
$985$	−20.0000	−0.637253
$986$	0	0
$987$	2.00000	0.0636607
$988$	0	0
$989$	−12.0000	−0.381578
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	0	0
$993$	12.0000	0.380808
$994$	0	0
$995$	−48.0000	−1.52170
$996$	0	0
$997$	−33.0000	−1.04512	−0.522560	−	0.852602i	$-0.675023\pi$
$997$	−33.0000	−1.04512	−0.522560	+	0.852602i	$0.675023\pi$
$998$	0	0
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7104.2.a.ba.1.1		1
4.3	odd	2		7104.2.a.n.1.1		1
8.3	odd	2		1776.2.a.g.1.1		1
8.5	even	2		888.2.a.a.1.1	✓	1
24.5	odd	2		2664.2.a.h.1.1		1
24.11	even	2		5328.2.a.w.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
888.2.a.a.1.1	✓	1	8.5	even	2
1776.2.a.g.1.1		1	8.3	odd	2
2664.2.a.h.1.1		1	24.5	odd	2
5328.2.a.w.1.1		1	24.11	even	2
7104.2.a.n.1.1		1	4.3	odd	2
7104.2.a.ba.1.1		1	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7104 = 2^{6} \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7104.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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