LMFDB - Embedded newform 8303.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8303,2,Mod(1,8303)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8303, 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("8303.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8303 = 19^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8303.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:	$66.2997887983$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 437)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8303.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -5.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-2.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -5.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +4.00000 q^{12} +2.00000 q^{15} +4.00000 q^{16} -7.00000 q^{17} +2.00000 q^{20} +10.0000 q^{21} +1.00000 q^{23} -4.00000 q^{25} +4.00000 q^{27} +10.0000 q^{28} -6.00000 q^{29} -4.00000 q^{31} +2.00000 q^{33} +5.00000 q^{35} -2.00000 q^{36} -2.00000 q^{37} +2.00000 q^{41} -5.00000 q^{43} +2.00000 q^{44} -1.00000 q^{45} -3.00000 q^{47} -8.00000 q^{48} +18.0000 q^{49} +14.0000 q^{51} +4.00000 q^{53} +1.00000 q^{55} -6.00000 q^{59} -4.00000 q^{60} +11.0000 q^{61} -5.00000 q^{63} -8.00000 q^{64} +16.0000 q^{67} +14.0000 q^{68} -2.00000 q^{69} +10.0000 q^{71} -7.00000 q^{73} +8.00000 q^{75} +5.00000 q^{77} -4.00000 q^{79} -4.00000 q^{80} -11.0000 q^{81} +4.00000 q^{83} -20.0000 q^{84} +7.00000 q^{85} +12.0000 q^{87} +16.0000 q^{89} -2.00000 q^{92} +8.00000 q^{93} +4.00000 q^{97} -1.00000 q^{99} +O(q^{100})$

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	−2.00000	−1.00000
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−5.00000	−1.88982	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−5.00000	−1.88982	−0.944911	+	0.327327i	$0.893852\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	4.00000	1.15470
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	4.00000	1.00000
$17$	−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
$19$	0	0
$19$	0	0
$20$	2.00000	0.447214
$21$	10.0000	2.18218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	4.00000	0.769800
$28$	10.0000	1.88982
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	5.00000	0.845154
$36$	−2.00000	−0.333333
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	2.00000	0.301511
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	−8.00000	−1.15470
$49$	18.0000	2.57143
$50$	0	0
$51$	14.0000	1.96039
$52$	0	0
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	−4.00000	−0.516398
$61$	11.0000	1.40841	0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000	1.40841	0.704203	+	0.709999i	$0.251305\pi$
$62$	0	0
$63$	−5.00000	−0.629941
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	16.0000	1.95471	0.977356	−	0.211604i	$-0.0678686\pi$
$67$	16.0000	1.95471	0.977356	+	0.211604i	$0.0678686\pi$
$68$	14.0000	1.69775
$69$	−2.00000	−0.240772
$70$	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
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	0	0
$75$	8.00000	0.923760
$76$	0	0
$77$	5.00000	0.569803
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−4.00000	−0.447214
$81$	−11.0000	−1.22222
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	−20.0000	−2.18218
$85$	7.00000	0.759257
$86$	0	0
$87$	12.0000	1.28654
$88$	0	0
$89$	16.0000	1.69600	0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000	1.69600	0.847998	+	0.529999i	$0.177808\pi$
$90$	0	0
$91$	0	0
$92$	−2.00000	−0.208514
$93$	8.00000	0.829561
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.00000	0.406138	0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000	0.406138	0.203069	+	0.979164i	$0.434908\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	8.00000	0.800000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	−10.0000	−0.975900
$106$	0	0
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	−8.00000	−0.769800
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	−20.0000	−1.88982
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	12.0000	1.11417
$117$	0	0
$118$	0	0
$119$	35.0000	3.20844
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	−4.00000	−0.360668
$124$	8.00000	0.718421
$125$	9.00000	0.804984
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	10.0000	0.880451
$130$	0	0
$131$	9.00000	0.786334	0.393167	−	0.919467i	$-0.371379\pi$
$131$	9.00000	0.786334	0.393167	+	0.919467i	$0.371379\pi$
$132$	−4.00000	−0.348155
$133$	0	0
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	0	0
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	−10.0000	−0.845154
$141$	6.00000	0.505291
$142$	0	0
$143$	0	0
$144$	4.00000	0.333333
$145$	6.00000	0.498273
$146$	0	0
$147$	−36.0000	−2.96923
$148$	4.00000	0.328798
$149$	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
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	−7.00000	−0.565916
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	0	0
$159$	−8.00000	−0.634441
$160$	0	0
$161$	−5.00000	−0.394055
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−4.00000	−0.312348
$165$	−2.00000	−0.155700
$166$	0	0
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	10.0000	0.762493
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	0	0
$175$	20.0000	1.51186
$176$	−4.00000	−0.301511
$177$	12.0000	0.901975
$178$	0	0
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	2.00000	0.149071
$181$	−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
$183$	−22.0000	−1.62629
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	7.00000	0.511891
$188$	6.00000	0.437595
$189$	−20.0000	−1.45479
$190$	0	0
$191$	−1.00000	−0.0723575	−0.0361787	−	0.999345i	$-0.511519\pi$
$191$	−1.00000	−0.0723575	−0.0361787	+	0.999345i	$0.511519\pi$
$192$	16.0000	1.15470
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	0	0
$195$	0	0
$196$	−36.0000	−2.57143
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−25.0000	−1.77220	−0.886102	−	0.463491i	$-0.846597\pi$
$199$	−25.0000	−1.77220	−0.886102	+	0.463491i	$0.846597\pi$
$200$	0	0
$201$	−32.0000	−2.25711
$202$	0	0
$203$	30.0000	2.10559
$204$	−28.0000	−1.96039
$205$	−2.00000	−0.139686
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	0	0
$210$	0	0
$211$	22.0000	1.51454	0.757271	−	0.653101i	$-0.226532\pi$
$211$	22.0000	1.51454	0.757271	+	0.653101i	$0.226532\pi$
$212$	−8.00000	−0.549442
$213$	−20.0000	−1.37038
$214$	0	0
$215$	5.00000	0.340997
$216$	0	0
$217$	20.0000	1.35769
$218$	0	0
$219$	14.0000	0.946032
$220$	−2.00000	−0.134840
$221$	0	0
$222$	0	0
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	0	0
$229$	−15.0000	−0.991228	−0.495614	−	0.868543i	$-0.665057\pi$
$229$	−15.0000	−0.991228	−0.495614	+	0.868543i	$0.665057\pi$
$230$	0	0
$231$	−10.0000	−0.657952
$232$	0	0
$233$	−13.0000	−0.851658	−0.425829	−	0.904804i	$-0.640018\pi$
$233$	−13.0000	−0.851658	−0.425829	+	0.904804i	$0.640018\pi$
$234$	0	0
$235$	3.00000	0.195698
$236$	12.0000	0.781133
$237$	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	$0.799751\pi$
$240$	8.00000	0.516398
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	−22.0000	−1.40841
$245$	−18.0000	−1.14998
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	−15.0000	−0.946792	−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000	−0.946792	−0.473396	+	0.880850i	$0.656972\pi$
$252$	10.0000	0.629941
$253$	−1.00000	−0.0628695
$254$	0	0
$255$	−14.0000	−0.876714
$256$	16.0000	1.00000
$257$	20.0000	1.24757	0.623783	−	0.781598i	$-0.285595\pi$
$257$	20.0000	1.24757	0.623783	+	0.781598i	$0.285595\pi$
$258$	0	0
$259$	10.0000	0.621370
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	5.00000	0.308313	0.154157	−	0.988046i	$-0.450734\pi$
$263$	5.00000	0.308313	0.154157	+	0.988046i	$0.450734\pi$
$264$	0	0
$265$	−4.00000	−0.245718
$266$	0	0
$267$	−32.0000	−1.95837
$268$	−32.0000	−1.95471
$269$	−32.0000	−1.95107	−0.975537	−	0.219834i	$-0.929448\pi$
$269$	−32.0000	−1.95107	−0.975537	+	0.219834i	$0.929448\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	−28.0000	−1.69775
$273$	0	0
$274$	0	0
$275$	4.00000	0.241209
$276$	4.00000	0.240772
$277$	13.0000	0.781094	0.390547	−	0.920583i	$-0.372286\pi$
$277$	13.0000	0.781094	0.390547	+	0.920583i	$0.372286\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	−17.0000	−1.01055	−0.505273	−	0.862960i	$-0.668608\pi$
$283$	−17.0000	−1.01055	−0.505273	+	0.862960i	$0.668608\pi$
$284$	−20.0000	−1.18678
$285$	0	0
$286$	0	0
$287$	−10.0000	−0.590281
$288$	0	0
$289$	32.0000	1.88235
$290$	0	0
$291$	−8.00000	−0.468968
$292$	14.0000	0.819288
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	0	0
$300$	−16.0000	−0.923760
$301$	25.0000	1.44098
$302$	0	0
$303$	−28.0000	−1.60856
$304$	0	0
$305$	−11.0000	−0.629858
$306$	0	0
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	−10.0000	−0.569803
$309$	−28.0000	−1.59286
$310$	0	0
$311$	13.0000	0.737162	0.368581	−	0.929596i	$-0.379844\pi$
$311$	13.0000	0.737162	0.368581	+	0.929596i	$0.379844\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	5.00000	0.281718
$316$	8.00000	0.450035
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	8.00000	0.447214
$321$	36.0000	2.00932
$322$	0	0
$323$	0	0
$324$	22.0000	1.22222
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	15.0000	0.826977
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	−8.00000	−0.439057
$333$	−2.00000	−0.109599
$334$	0	0
$335$	−16.0000	−0.874173
$336$	40.0000	2.18218
$337$	−28.0000	−1.52526	−0.762629	−	0.646837i	$-0.776092\pi$
$337$	−28.0000	−1.52526	−0.762629	+	0.646837i	$0.776092\pi$
$338$	0	0
$339$	4.00000	0.217250
$340$	−14.0000	−0.759257
$341$	4.00000	0.216612
$342$	0	0
$343$	−55.0000	−2.96972
$344$	0	0
$345$	2.00000	0.107676
$346$	0	0
$347$	−19.0000	−1.01997	−0.509987	−	0.860182i	$-0.670350\pi$
$347$	−19.0000	−1.01997	−0.509987	+	0.860182i	$0.670350\pi$
$348$	−24.0000	−1.28654
$349$	9.00000	0.481759	0.240879	−	0.970555i	$-0.422564\pi$
$349$	9.00000	0.481759	0.240879	+	0.970555i	$0.422564\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	0	0
$355$	−10.0000	−0.530745
$356$	−32.0000	−1.69600
$357$	−70.0000	−3.70479
$358$	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
$361$	0	0
$362$	0	0
$363$	20.0000	1.04973
$364$	0	0
$365$	7.00000	0.366397
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	4.00000	0.208514
$369$	2.00000	0.104116
$370$	0	0
$371$	−20.0000	−1.03835
$372$	−16.0000	−0.829561
$373$	−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
$375$	−18.0000	−0.929516
$376$	0	0
$377$	0	0
$378$	0	0
$379$	34.0000	1.74646	0.873231	−	0.487306i	$-0.162020\pi$
$379$	34.0000	1.74646	0.873231	+	0.487306i	$0.162020\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	20.0000	1.02195	0.510976	−	0.859595i	$-0.329284\pi$
$383$	20.0000	1.02195	0.510976	+	0.859595i	$0.329284\pi$
$384$	0	0
$385$	−5.00000	−0.254824
$386$	0	0
$387$	−5.00000	−0.254164
$388$	−8.00000	−0.406138
$389$	27.0000	1.36895	0.684477	−	0.729034i	$-0.260031\pi$
$389$	27.0000	1.36895	0.684477	+	0.729034i	$0.260031\pi$
$390$	0	0
$391$	−7.00000	−0.354005
$392$	0	0
$393$	−18.0000	−0.907980
$394$	0	0
$395$	4.00000	0.201262
$396$	2.00000	0.100504
$397$	−23.0000	−1.15434	−0.577168	−	0.816625i	$-0.695842\pi$
$397$	−23.0000	−1.15434	−0.577168	+	0.816625i	$0.695842\pi$
$398$	0	0
$399$	0	0
$400$	−16.0000	−0.800000
$401$	20.0000	0.998752	0.499376	−	0.866385i	$-0.333563\pi$
$401$	20.0000	0.998752	0.499376	+	0.866385i	$0.333563\pi$
$402$	0	0
$403$	0	0
$404$	−28.0000	−1.39305
$405$	11.0000	0.546594
$406$	0	0
$407$	2.00000	0.0991363
$408$	0	0
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	−28.0000	−1.37946
$413$	30.0000	1.47620
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	10.0000	0.489702
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	20.0000	0.975900
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	0	0
$423$	−3.00000	−0.145865
$424$	0	0
$425$	28.0000	1.35820
$426$	0	0
$427$	−55.0000	−2.66164
$428$	36.0000	1.74013
$429$	0	0
$430$	0	0
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	16.0000	0.769800
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	−12.0000	−0.575356
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−18.0000	−0.859093	−0.429547	−	0.903045i	$-0.641327\pi$
$439$	−18.0000	−0.859093	−0.429547	+	0.903045i	$0.641327\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	0	0
$443$	21.0000	0.997740	0.498870	−	0.866677i	$-0.333748\pi$
$443$	21.0000	0.997740	0.498870	+	0.866677i	$0.333748\pi$
$444$	−8.00000	−0.379663
$445$	−16.0000	−0.758473
$446$	0	0
$447$	−18.0000	−0.851371
$448$	40.0000	1.88982
$449$	−28.0000	−1.32140	−0.660701	−	0.750649i	$-0.729741\pi$
$449$	−28.0000	−1.32140	−0.660701	+	0.750649i	$0.729741\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	4.00000	0.188144
$453$	−4.00000	−0.187936
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15.0000	0.701670	0.350835	−	0.936437i	$-0.385898\pi$
$457$	15.0000	0.701670	0.350835	+	0.936437i	$0.385898\pi$
$458$	0	0
$459$	−28.0000	−1.30693
$460$	2.00000	0.0932505
$461$	−15.0000	−0.698620	−0.349310	−	0.937007i	$-0.613584\pi$
$461$	−15.0000	−0.698620	−0.349310	+	0.937007i	$0.613584\pi$
$462$	0	0
$463$	−23.0000	−1.06890	−0.534450	−	0.845200i	$-0.679481\pi$
$463$	−23.0000	−1.06890	−0.534450	+	0.845200i	$0.679481\pi$
$464$	−24.0000	−1.11417
$465$	−8.00000	−0.370991
$466$	0	0
$467$	17.0000	0.786666	0.393333	−	0.919396i	$-0.371322\pi$
$467$	17.0000	0.786666	0.393333	+	0.919396i	$0.371322\pi$
$468$	0	0
$469$	−80.0000	−3.69406
$470$	0	0
$471$	20.0000	0.921551
$472$	0	0
$473$	5.00000	0.229900
$474$	0	0
$475$	0	0
$476$	−70.0000	−3.20844
$477$	4.00000	0.183147
$478$	0	0
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	10.0000	0.455016
$484$	20.0000	0.909091
$485$	−4.00000	−0.181631
$486$	0	0
$487$	26.0000	1.17817	0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000	1.17817	0.589086	+	0.808070i	$0.299488\pi$
$488$	0	0
$489$	−8.00000	−0.361773
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	8.00000	0.360668
$493$	42.0000	1.89158
$494$	0	0
$495$	1.00000	0.0449467
$496$	−16.0000	−0.718421
$497$	−50.0000	−2.24281
$498$	0	0
$499$	5.00000	0.223831	0.111915	−	0.993718i	$-0.464301\pi$
$499$	5.00000	0.223831	0.111915	+	0.993718i	$0.464301\pi$
$500$	−18.0000	−0.804984
$501$	−36.0000	−1.60836
$502$	0	0
$503$	−28.0000	−1.24846	−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000	−1.24846	−0.624229	+	0.781241i	$0.714587\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	0	0
$507$	26.0000	1.15470
$508$	−4.00000	−0.177471
$509$	−24.0000	−1.06378	−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000	−1.06378	−0.531891	+	0.846813i	$0.678518\pi$
$510$	0	0
$511$	35.0000	1.54831
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−14.0000	−0.616914
$516$	−20.0000	−0.880451
$517$	3.00000	0.131940
$518$	0	0
$519$	20.0000	0.877903
$520$	0	0
$521$	36.0000	1.57719	0.788594	−	0.614914i	$-0.210809\pi$
$521$	36.0000	1.57719	0.788594	+	0.614914i	$0.210809\pi$
$522$	0	0
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	−18.0000	−0.786334
$525$	−40.0000	−1.74574
$526$	0	0
$527$	28.0000	1.21970
$528$	8.00000	0.348155
$529$	1.00000	0.0434783
$530$	0	0
$531$	−6.00000	−0.260378
$532$	0	0
$533$	0	0
$534$	0	0
$535$	18.0000	0.778208
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	−18.0000	−0.775315
$540$	8.00000	0.344265
$541$	23.0000	0.988847	0.494424	−	0.869221i	$-0.335379\pi$
$541$	23.0000	0.988847	0.494424	+	0.869221i	$0.335379\pi$
$542$	0	0
$543$	4.00000	0.171656
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	−18.0000	−0.768922
$549$	11.0000	0.469469
$550$	0	0
$551$	0	0
$552$	0	0
$553$	20.0000	0.850487
$554$	0	0
$555$	−4.00000	−0.169791
$556$	10.0000	0.424094
$557$	−31.0000	−1.31351	−0.656756	−	0.754103i	$-0.728072\pi$
$557$	−31.0000	−1.31351	−0.656756	+	0.754103i	$0.728072\pi$
$558$	0	0
$559$	0	0
$560$	20.0000	0.845154
$561$	−14.0000	−0.591080
$562$	0	0
$563$	−42.0000	−1.77009	−0.885044	−	0.465506i	$-0.845872\pi$
$563$	−42.0000	−1.77009	−0.885044	+	0.465506i	$0.845872\pi$
$564$	−12.0000	−0.505291
$565$	2.00000	0.0841406
$566$	0	0
$567$	55.0000	2.30978
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	0	0
$573$	2.00000	0.0835512
$574$	0	0
$575$	−4.00000	−0.166812
$576$	−8.00000	−0.333333
$577$	−5.00000	−0.208153	−0.104076	−	0.994569i	$-0.533189\pi$
$577$	−5.00000	−0.208153	−0.104076	+	0.994569i	$0.533189\pi$
$578$	0	0
$579$	32.0000	1.32987
$580$	−12.0000	−0.498273
$581$	−20.0000	−0.829740
$582$	0	0
$583$	−4.00000	−0.165663
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.00000	0.206372	0.103186	−	0.994662i	$-0.467096\pi$
$587$	5.00000	0.206372	0.103186	+	0.994662i	$0.467096\pi$
$588$	72.0000	2.96923
$589$	0	0
$590$	0	0
$591$	12.0000	0.493614
$592$	−8.00000	−0.328798
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	−35.0000	−1.43486
$596$	−18.0000	−0.737309
$597$	50.0000	2.04636
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	16.0000	0.651570
$604$	−4.00000	−0.162758
$605$	10.0000	0.406558
$606$	0	0
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	0	0
$609$	−60.0000	−2.43132
$610$	0	0
$611$	0	0
$612$	14.0000	0.565916
$613$	9.00000	0.363507	0.181753	−	0.983344i	$-0.441823\pi$
$613$	9.00000	0.363507	0.181753	+	0.983344i	$0.441823\pi$
$614$	0	0
$615$	4.00000	0.161296
$616$	0	0
$617$	−27.0000	−1.08698	−0.543490	−	0.839416i	$-0.682897\pi$
$617$	−27.0000	−1.08698	−0.543490	+	0.839416i	$0.682897\pi$
$618$	0	0
$619$	−12.0000	−0.482321	−0.241160	−	0.970485i	$-0.577528\pi$
$619$	−12.0000	−0.482321	−0.241160	+	0.970485i	$0.577528\pi$
$620$	−8.00000	−0.321288
$621$	4.00000	0.160514
$622$	0	0
$623$	−80.0000	−3.20513
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	20.0000	0.798087
$629$	14.0000	0.558217
$630$	0	0
$631$	7.00000	0.278666	0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000	0.278666	0.139333	+	0.990246i	$0.455504\pi$
$632$	0	0
$633$	−44.0000	−1.74884
$634$	0	0
$635$	−2.00000	−0.0793676
$636$	16.0000	0.634441
$637$	0	0
$638$	0	0
$639$	10.0000	0.395594
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	−9.00000	−0.354925	−0.177463	−	0.984128i	$-0.556789\pi$
$643$	−9.00000	−0.354925	−0.177463	+	0.984128i	$0.556789\pi$
$644$	10.0000	0.394055
$645$	−10.0000	−0.393750
$646$	0	0
$647$	35.0000	1.37599	0.687996	−	0.725714i	$-0.258491\pi$
$647$	35.0000	1.37599	0.687996	+	0.725714i	$0.258491\pi$
$648$	0	0
$649$	6.00000	0.235521
$650$	0	0
$651$	−40.0000	−1.56772
$652$	−8.00000	−0.313304
$653$	9.00000	0.352197	0.176099	−	0.984373i	$-0.443652\pi$
$653$	9.00000	0.352197	0.176099	+	0.984373i	$0.443652\pi$
$654$	0	0
$655$	−9.00000	−0.351659
$656$	8.00000	0.312348
$657$	−7.00000	−0.273096
$658$	0	0
$659$	34.0000	1.32445	0.662226	−	0.749304i	$-0.269612\pi$
$659$	34.0000	1.32445	0.662226	+	0.749304i	$0.269612\pi$
$660$	4.00000	0.155700
$661$	−32.0000	−1.24466	−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000	−1.24466	−0.622328	+	0.782757i	$0.713813\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.00000	−0.232321
$668$	−36.0000	−1.39288
$669$	−20.0000	−0.773245
$670$	0	0
$671$	−11.0000	−0.424650
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	0	0
$675$	−16.0000	−0.615840
$676$	26.0000	1.00000
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	−20.0000	−0.767530
$680$	0	0
$681$	48.0000	1.83936
$682$	0	0
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	0	0
$685$	−9.00000	−0.343872
$686$	0	0
$687$	30.0000	1.14457
$688$	−20.0000	−0.762493
$689$	0	0
$690$	0	0
$691$	1.00000	0.0380418	0.0190209	−	0.999819i	$-0.493945\pi$
$691$	1.00000	0.0380418	0.0190209	+	0.999819i	$0.493945\pi$
$692$	20.0000	0.760286
$693$	5.00000	0.189934
$694$	0	0
$695$	5.00000	0.189661
$696$	0	0
$697$	−14.0000	−0.530288
$698$	0	0
$699$	26.0000	0.983410
$700$	−40.0000	−1.51186
$701$	14.0000	0.528773	0.264386	−	0.964417i	$-0.414831\pi$
$701$	14.0000	0.528773	0.264386	+	0.964417i	$0.414831\pi$
$702$	0	0
$703$	0	0
$704$	8.00000	0.301511
$705$	−6.00000	−0.225973
$706$	0	0
$707$	−70.0000	−2.63262
$708$	−24.0000	−0.901975
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	−4.00000	−0.149801
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	50.0000	1.86728
$718$	0	0
$719$	23.0000	0.857755	0.428878	−	0.903363i	$-0.358909\pi$
$719$	23.0000	0.857755	0.428878	+	0.903363i	$0.358909\pi$
$720$	−4.00000	−0.149071
$721$	−70.0000	−2.60694
$722$	0	0
$723$	−28.0000	−1.04133
$724$	4.00000	0.148659
$725$	24.0000	0.891338
$726$	0	0
$727$	1.00000	0.0370879	0.0185440	−	0.999828i	$-0.494097\pi$
$727$	1.00000	0.0370879	0.0185440	+	0.999828i	$0.494097\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	35.0000	1.29452
$732$	44.0000	1.62629
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	0	0
$735$	36.0000	1.32788
$736$	0	0
$737$	−16.0000	−0.589368
$738$	0	0
$739$	−45.0000	−1.65535	−0.827676	−	0.561206i	$-0.810337\pi$
$739$	−45.0000	−1.65535	−0.827676	+	0.561206i	$0.810337\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	0	0
$743$	20.0000	0.733729	0.366864	−	0.930274i	$-0.380431\pi$
$743$	20.0000	0.733729	0.366864	+	0.930274i	$0.380431\pi$
$744$	0	0
$745$	−9.00000	−0.329734
$746$	0	0
$747$	4.00000	0.146352
$748$	−14.0000	−0.511891
$749$	90.0000	3.28853
$750$	0	0
$751$	52.0000	1.89751	0.948753	−	0.316017i	$-0.102346\pi$
$751$	52.0000	1.89751	0.948753	+	0.316017i	$0.102346\pi$
$752$	−12.0000	−0.437595
$753$	30.0000	1.09326
$754$	0	0
$755$	−2.00000	−0.0727875
$756$	40.0000	1.45479
$757$	35.0000	1.27210	0.636048	−	0.771649i	$-0.280568\pi$
$757$	35.0000	1.27210	0.636048	+	0.771649i	$0.280568\pi$
$758$	0	0
$759$	2.00000	0.0725954
$760$	0	0
$761$	25.0000	0.906249	0.453125	−	0.891447i	$-0.350309\pi$
$761$	25.0000	0.906249	0.453125	+	0.891447i	$0.350309\pi$
$762$	0	0
$763$	0	0
$764$	2.00000	0.0723575
$765$	7.00000	0.253086
$766$	0	0
$767$	0	0
$768$	−32.0000	−1.15470
$769$	3.00000	0.108183	0.0540914	−	0.998536i	$-0.482774\pi$
$769$	3.00000	0.108183	0.0540914	+	0.998536i	$0.482774\pi$
$770$	0	0
$771$	−40.0000	−1.44056
$772$	32.0000	1.15171
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	0	0
$775$	16.0000	0.574737
$776$	0	0
$777$	−20.0000	−0.717496
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−10.0000	−0.357828
$782$	0	0
$783$	−24.0000	−0.857690
$784$	72.0000	2.57143
$785$	10.0000	0.356915
$786$	0	0
$787$	−40.0000	−1.42585	−0.712923	−	0.701242i	$-0.752629\pi$
$787$	−40.0000	−1.42585	−0.712923	+	0.701242i	$0.752629\pi$
$788$	12.0000	0.427482
$789$	−10.0000	−0.356009
$790$	0	0
$791$	10.0000	0.355559
$792$	0	0
$793$	0	0
$794$	0	0
$795$	8.00000	0.283731
$796$	50.0000	1.77220
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	21.0000	0.742927
$800$	0	0
$801$	16.0000	0.565332
$802$	0	0
$803$	7.00000	0.247025
$804$	64.0000	2.25711
$805$	5.00000	0.176227
$806$	0	0
$807$	64.0000	2.25291
$808$	0	0
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	−60.0000	−2.10559
$813$	−48.0000	−1.68343
$814$	0	0
$815$	−4.00000	−0.140114
$816$	56.0000	1.96039
$817$	0	0
$818$	0	0
$819$	0	0
$820$	4.00000	0.139686
$821$	−31.0000	−1.08191	−0.540954	−	0.841052i	$-0.681937\pi$
$821$	−31.0000	−1.08191	−0.540954	+	0.841052i	$0.681937\pi$
$822$	0	0
$823$	31.0000	1.08059	0.540296	−	0.841475i	$-0.318312\pi$
$823$	31.0000	1.08059	0.540296	+	0.841475i	$0.318312\pi$
$824$	0	0
$825$	−8.00000	−0.278524
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	−2.00000	−0.0695048
$829$	−36.0000	−1.25033	−0.625166	−	0.780492i	$-0.714969\pi$
$829$	−36.0000	−1.25033	−0.625166	+	0.780492i	$0.714969\pi$
$830$	0	0
$831$	−26.0000	−0.901930
$832$	0	0
$833$	−126.000	−4.36564
$834$	0	0
$835$	−18.0000	−0.622916
$836$	0	0
$837$	−16.0000	−0.553041
$838$	0	0
$839$	−54.0000	−1.86429	−0.932144	−	0.362089i	$-0.882064\pi$
$839$	−54.0000	−1.86429	−0.932144	+	0.362089i	$0.882064\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−36.0000	−1.23991
$844$	−44.0000	−1.51454
$845$	13.0000	0.447214
$846$	0	0
$847$	50.0000	1.71802
$848$	16.0000	0.549442
$849$	34.0000	1.16688
$850$	0	0
$851$	−2.00000	−0.0685591
$852$	40.0000	1.37038
$853$	−38.0000	−1.30110	−0.650548	−	0.759465i	$-0.725461\pi$
$853$	−38.0000	−1.30110	−0.650548	+	0.759465i	$0.725461\pi$
$854$	0	0
$855$	0	0
$856$	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
$859$	15.0000	0.511793	0.255897	−	0.966704i	$-0.417629\pi$
$859$	15.0000	0.511793	0.255897	+	0.966704i	$0.417629\pi$
$860$	−10.0000	−0.340997
$861$	20.0000	0.681598
$862$	0	0
$863$	−50.0000	−1.70202	−0.851010	−	0.525150i	$-0.824009\pi$
$863$	−50.0000	−1.70202	−0.851010	+	0.525150i	$0.824009\pi$
$864$	0	0
$865$	10.0000	0.340010
$866$	0	0
$867$	−64.0000	−2.17355
$868$	−40.0000	−1.35769
$869$	4.00000	0.135691
$870$	0	0
$871$	0	0
$872$	0	0
$873$	4.00000	0.135379
$874$	0	0
$875$	−45.0000	−1.52128
$876$	−28.0000	−0.946032
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	0	0
$879$	48.0000	1.61900
$880$	4.00000	0.134840
$881$	−15.0000	−0.505363	−0.252681	−	0.967550i	$-0.581312\pi$
$881$	−15.0000	−0.505363	−0.252681	+	0.967550i	$0.581312\pi$
$882$	0	0
$883$	−25.0000	−0.841317	−0.420658	−	0.907219i	$-0.638201\pi$
$883$	−25.0000	−0.841317	−0.420658	+	0.907219i	$0.638201\pi$
$884$	0	0
$885$	−12.0000	−0.403376
$886$	0	0
$887$	26.0000	0.872995	0.436497	−	0.899706i	$-0.356219\pi$
$887$	26.0000	0.872995	0.436497	+	0.899706i	$0.356219\pi$
$888$	0	0
$889$	−10.0000	−0.335389
$890$	0	0
$891$	11.0000	0.368514
$892$	−20.0000	−0.669650
$893$	0	0
$894$	0	0
$895$	−6.00000	−0.200558
$896$	0	0
$897$	0	0
$898$	0	0
$899$	24.0000	0.800445
$900$	8.00000	0.266667
$901$	−28.0000	−0.932815
$902$	0	0
$903$	−50.0000	−1.66390
$904$	0	0
$905$	2.00000	0.0664822
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	48.0000	1.59294
$909$	14.0000	0.464351
$910$	0	0
$911$	14.0000	0.463841	0.231920	−	0.972735i	$-0.425499\pi$
$911$	14.0000	0.463841	0.231920	+	0.972735i	$0.425499\pi$
$912$	0	0
$913$	−4.00000	−0.132381
$914$	0	0
$915$	22.0000	0.727298
$916$	30.0000	0.991228
$917$	−45.0000	−1.48603
$918$	0	0
$919$	44.0000	1.45143	0.725713	−	0.687998i	$-0.241510\pi$
$919$	44.0000	1.45143	0.725713	+	0.687998i	$0.241510\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	0	0
$923$	0	0
$924$	20.0000	0.657952
$925$	8.00000	0.263038
$926$	0	0
$927$	14.0000	0.459820
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	0	0
$932$	26.0000	0.851658
$933$	−26.0000	−0.851202
$934$	0	0
$935$	−7.00000	−0.228924
$936$	0	0
$937$	−43.0000	−1.40475	−0.702374	−	0.711808i	$-0.747877\pi$
$937$	−43.0000	−1.40475	−0.702374	+	0.711808i	$0.747877\pi$
$938$	0	0
$939$	−12.0000	−0.391605
$940$	−6.00000	−0.195698
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	0	0
$943$	2.00000	0.0651290
$944$	−24.0000	−0.781133
$945$	20.0000	0.650600
$946$	0	0
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	−16.0000	−0.519656
$949$	0	0
$950$	0	0
$951$	28.0000	0.907962
$952$	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
$955$	1.00000	0.0323592
$956$	50.0000	1.61712
$957$	−12.0000	−0.387905
$958$	0	0
$959$	−45.0000	−1.45313
$960$	−16.0000	−0.516398
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−18.0000	−0.580042
$964$	−28.0000	−0.901819
$965$	16.0000	0.515058
$966$	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
$969$	0	0
$970$	0	0
$971$	4.00000	0.128366	0.0641831	−	0.997938i	$-0.479556\pi$
$971$	4.00000	0.128366	0.0641831	+	0.997938i	$0.479556\pi$
$972$	−20.0000	−0.641500
$973$	25.0000	0.801463
$974$	0	0
$975$	0	0
$976$	44.0000	1.40841
$977$	−32.0000	−1.02377	−0.511885	−	0.859054i	$-0.671053\pi$
$977$	−32.0000	−1.02377	−0.511885	+	0.859054i	$0.671053\pi$
$978$	0	0
$979$	−16.0000	−0.511362
$980$	36.0000	1.14998
$981$	0	0
$982$	0	0
$983$	32.0000	1.02064	0.510321	−	0.859984i	$-0.329527\pi$
$983$	32.0000	1.02064	0.510321	+	0.859984i	$0.329527\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	−30.0000	−0.954911
$988$	0	0
$989$	−5.00000	−0.158991
$990$	0	0
$991$	38.0000	1.20711	0.603555	−	0.797321i	$-0.293750\pi$
$991$	38.0000	1.20711	0.603555	+	0.797321i	$0.293750\pi$
$992$	0	0
$993$	−16.0000	−0.507745
$994$	0	0
$995$	25.0000	0.792553
$996$	16.0000	0.506979
$997$	−7.00000	−0.221692	−0.110846	−	0.993838i	$-0.535356\pi$
$997$	−7.00000	−0.221692	−0.110846	+	0.993838i	$0.535356\pi$
$998$	0	0
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8303.2.a.b.1.1		1
19.18	odd	2		437.2.a.a.1.1	✓	1
57.56	even	2		3933.2.a.c.1.1		1
76.75	even	2		6992.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
437.2.a.a.1.1	✓	1	19.18	odd	2
3933.2.a.c.1.1		1	57.56	even	2
6992.2.a.c.1.1		1	76.75	even	2
8303.2.a.b.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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8303 = 19^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8303.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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