LMFDB - Embedded newform 5712.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5712,2,Mod(1,5712)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5712, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5712.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5712.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:	$45.6105496346$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2856)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5712.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q-1.00000 q^{3} +3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -5.00000 q^{11} -3.00000 q^{13} -3.00000 q^{15} +1.00000 q^{17} +5.00000 q^{19} -1.00000 q^{21} -5.00000 q^{23} +4.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} -8.00000 q^{31} +5.00000 q^{33} +3.00000 q^{35} -6.00000 q^{37} +3.00000 q^{39} +9.00000 q^{41} +1.00000 q^{43} +3.00000 q^{45} +1.00000 q^{49} -1.00000 q^{51} +4.00000 q^{53} -15.0000 q^{55} -5.00000 q^{57} -12.0000 q^{59} -6.00000 q^{61} +1.00000 q^{63} -9.00000 q^{65} +16.0000 q^{67} +5.00000 q^{69} -8.00000 q^{71} -10.0000 q^{73} -4.00000 q^{75} -5.00000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -14.0000 q^{83} +3.00000 q^{85} -2.00000 q^{87} -8.00000 q^{89} -3.00000 q^{91} +8.00000 q^{93} +15.0000 q^{95} -8.00000 q^{97} -5.00000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	5.00000	0.870388
$34$	0	0
$35$	3.00000	0.507093
$36$	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
$39$	3.00000	0.480384
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	3.00000	0.447214
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.00000	−0.140028
$52$	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
$55$	−15.0000	−2.02260
$56$	0	0
$57$	−5.00000	−0.662266
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−9.00000	−1.11631
$66$	0	0
$67$	16.0000	1.95471	0.977356	−	0.211604i	$-0.0678686\pi$
$67$	16.0000	1.95471	0.977356	+	0.211604i	$0.0678686\pi$
$68$	0	0
$69$	5.00000	0.601929
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	−5.00000	−0.569803
$78$	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
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	0	0
$87$	−2.00000	−0.214423
$88$	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
$93$	8.00000	0.829561
$94$	0	0
$95$	15.0000	1.53897
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	−5.00000	−0.502519
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	0	0
$107$	15.0000	1.45010	0.725052	−	0.688694i	$-0.241816\pi$
$107$	15.0000	1.45010	0.725052	+	0.688694i	$0.241816\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	0	0
$115$	−15.0000	−1.39876
$116$	0	0
$117$	−3.00000	−0.277350
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	−9.00000	−0.811503
$124$	0	0
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−1.00000	−0.0887357	−0.0443678	−	0.999015i	$-0.514127\pi$
$127$	−1.00000	−0.0887357	−0.0443678	+	0.999015i	$0.514127\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	0	0
$133$	5.00000	0.433555
$134$	0	0
$135$	−3.00000	−0.258199
$136$	0	0
$137$	4.00000	0.341743	0.170872	−	0.985293i	$-0.445342\pi$
$137$	4.00000	0.341743	0.170872	+	0.985293i	$0.445342\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	15.0000	1.25436
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	−24.0000	−1.92773
$156$	0	0
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	0	0
$159$	−4.00000	−0.317221
$160$	0	0
$161$	−5.00000	−0.394055
$162$	0	0
$163$	−18.0000	−1.40987	−0.704934	−	0.709273i	$-0.749024\pi$
$163$	−18.0000	−1.40987	−0.704934	+	0.709273i	$0.749024\pi$
$164$	0	0
$165$	15.0000	1.16775
$166$	0	0
$167$	−15.0000	−1.16073	−0.580367	−	0.814355i	$-0.697091\pi$
$167$	−15.0000	−1.16073	−0.580367	+	0.814355i	$0.697091\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	5.00000	0.382360
$172$	0	0
$173$	7.00000	0.532200	0.266100	−	0.963945i	$-0.414265\pi$
$173$	7.00000	0.532200	0.266100	+	0.963945i	$0.414265\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	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
$183$	6.00000	0.443533
$184$	0	0
$185$	−18.0000	−1.32339
$186$	0	0
$187$	−5.00000	−0.365636
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	0	0
$195$	9.00000	0.644503
$196$	0	0
$197$	−27.0000	−1.92367	−0.961835	−	0.273629i	$-0.911776\pi$
$197$	−27.0000	−1.92367	−0.961835	+	0.273629i	$0.911776\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	−16.0000	−1.12855
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	27.0000	1.88576
$206$	0	0
$207$	−5.00000	−0.347524
$208$	0	0
$209$	−25.0000	−1.72929
$210$	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
$213$	8.00000	0.548151
$214$	0	0
$215$	3.00000	0.204598
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	−3.00000	−0.201802
$222$	0	0
$223$	25.0000	1.67412	0.837062	−	0.547108i	$-0.184271\pi$
$223$	25.0000	1.67412	0.837062	+	0.547108i	$0.184271\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	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
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	5.00000	0.328976
$232$	0	0
$233$	−9.00000	−0.589610	−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000	−0.589610	−0.294805	+	0.955557i	$0.595255\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	−22.0000	−1.42306	−0.711531	−	0.702655i	$-0.751998\pi$
$239$	−22.0000	−1.42306	−0.711531	+	0.702655i	$0.751998\pi$
$240$	0	0
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.00000	0.191663
$246$	0	0
$247$	−15.0000	−0.954427
$248$	0	0
$249$	14.0000	0.887214
$250$	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
$253$	25.0000	1.57174
$254$	0	0
$255$	−3.00000	−0.187867
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	0	0
$267$	8.00000	0.489592
$268$	0	0
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	0	0
$271$	29.0000	1.76162	0.880812	−	0.473466i	$-0.156997\pi$
$271$	29.0000	1.76162	0.880812	+	0.473466i	$0.156997\pi$
$272$	0	0
$273$	3.00000	0.181568
$274$	0	0
$275$	−20.0000	−1.20605
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	−15.0000	−0.888523
$286$	0	0
$287$	9.00000	0.531253
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	8.00000	0.468968
$292$	0	0
$293$	20.0000	1.16841	0.584206	−	0.811605i	$-0.301406\pi$
$293$	20.0000	1.16841	0.584206	+	0.811605i	$0.301406\pi$
$294$	0	0
$295$	−36.0000	−2.09600
$296$	0	0
$297$	5.00000	0.290129
$298$	0	0
$299$	15.0000	0.867472
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	−10.0000	−0.574485
$304$	0	0
$305$	−18.0000	−1.03068
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	13.0000	0.739544
$310$	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
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	3.00000	0.169031
$316$	0	0
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	−10.0000	−0.559893
$320$	0	0
$321$	−15.0000	−0.837218
$322$	0	0
$323$	5.00000	0.278207
$324$	0	0
$325$	−12.0000	−0.665640
$326$	0	0
$327$	6.00000	0.331801
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−17.0000	−0.934405	−0.467202	−	0.884150i	$-0.654738\pi$
$331$	−17.0000	−0.934405	−0.467202	+	0.884150i	$0.654738\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	0	0
$335$	48.0000	2.62252
$336$	0	0
$337$	−8.00000	−0.435788	−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000	−0.435788	−0.217894	+	0.975972i	$0.569919\pi$
$338$	0	0
$339$	15.0000	0.814688
$340$	0	0
$341$	40.0000	2.16612
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	15.0000	0.807573
$346$	0	0
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	−33.0000	−1.76645	−0.883225	−	0.468950i	$-0.844632\pi$
$349$	−33.0000	−1.76645	−0.883225	+	0.468950i	$0.844632\pi$
$350$	0	0
$351$	3.00000	0.160128
$352$	0	0
$353$	4.00000	0.212899	0.106449	−	0.994318i	$-0.466052\pi$
$353$	4.00000	0.212899	0.106449	+	0.994318i	$0.466052\pi$
$354$	0	0
$355$	−24.0000	−1.27379
$356$	0	0
$357$	−1.00000	−0.0529256
$358$	0	0
$359$	−10.0000	−0.527780	−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000	−0.527780	−0.263890	+	0.964553i	$0.585006\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	−14.0000	−0.734809
$364$	0	0
$365$	−30.0000	−1.57027
$366$	0	0
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	0	0
$369$	9.00000	0.468521
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0	0
$375$	3.00000	0.154919
$376$	0	0
$377$	−6.00000	−0.309016
$378$	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
$381$	1.00000	0.0512316
$382$	0	0
$383$	−38.0000	−1.94171	−0.970855	−	0.239669i	$-0.922961\pi$
$383$	−38.0000	−1.94171	−0.970855	+	0.239669i	$0.922961\pi$
$384$	0	0
$385$	−15.0000	−0.764471
$386$	0	0
$387$	1.00000	0.0508329
$388$	0	0
$389$	8.00000	0.405616	0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000	0.405616	0.202808	+	0.979219i	$0.434993\pi$
$390$	0	0
$391$	−5.00000	−0.252861
$392$	0	0
$393$	−15.0000	−0.756650
$394$	0	0
$395$	−24.0000	−1.20757
$396$	0	0
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	0	0
$399$	−5.00000	−0.250313
$400$	0	0
$401$	−21.0000	−1.04869	−0.524345	−	0.851506i	$-0.675690\pi$
$401$	−21.0000	−1.04869	−0.524345	+	0.851506i	$0.675690\pi$
$402$	0	0
$403$	24.0000	1.19553
$404$	0	0
$405$	3.00000	0.149071
$406$	0	0
$407$	30.0000	1.48704
$408$	0	0
$409$	3.00000	0.148340	0.0741702	−	0.997246i	$-0.476369\pi$
$409$	3.00000	0.148340	0.0741702	+	0.997246i	$0.476369\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	0	0
$413$	−12.0000	−0.590481
$414$	0	0
$415$	−42.0000	−2.06170
$416$	0	0
$417$	10.0000	0.489702
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	7.00000	0.341159	0.170580	−	0.985344i	$-0.445436\pi$
$421$	7.00000	0.341159	0.170580	+	0.985344i	$0.445436\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	−6.00000	−0.290360
$428$	0	0
$429$	−15.0000	−0.724207
$430$	0	0
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	0	0
$433$	−39.0000	−1.87422	−0.937110	−	0.349034i	$-0.886510\pi$
$433$	−39.0000	−1.87422	−0.937110	+	0.349034i	$0.886510\pi$
$434$	0	0
$435$	−6.00000	−0.287678
$436$	0	0
$437$	−25.0000	−1.19591
$438$	0	0
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	8.00000	0.380091	0.190046	−	0.981775i	$-0.439136\pi$
$443$	8.00000	0.380091	0.190046	+	0.981775i	$0.439136\pi$
$444$	0	0
$445$	−24.0000	−1.13771
$446$	0	0
$447$	−10.0000	−0.472984
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	−45.0000	−2.11897
$452$	0	0
$453$	−16.0000	−0.751746
$454$	0	0
$455$	−9.00000	−0.421927
$456$	0	0
$457$	−31.0000	−1.45012	−0.725059	−	0.688686i	$-0.758188\pi$
$457$	−31.0000	−1.45012	−0.725059	+	0.688686i	$0.758188\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	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
$469$	16.0000	0.738811
$470$	0	0
$471$	−5.00000	−0.230388
$472$	0	0
$473$	−5.00000	−0.229900
$474$	0	0
$475$	20.0000	0.917663
$476$	0	0
$477$	4.00000	0.183147
$478$	0	0
$479$	9.00000	0.411220	0.205610	−	0.978634i	$-0.434082\pi$
$479$	9.00000	0.411220	0.205610	+	0.978634i	$0.434082\pi$
$480$	0	0
$481$	18.0000	0.820729
$482$	0	0
$483$	5.00000	0.227508
$484$	0	0
$485$	−24.0000	−1.08978
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	0	0
$489$	18.0000	0.813988
$490$	0	0
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	0	0
$493$	2.00000	0.0900755
$494$	0	0
$495$	−15.0000	−0.674200
$496$	0	0
$497$	−8.00000	−0.358849
$498$	0	0
$499$	40.0000	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$500$	0	0
$501$	15.0000	0.670151
$502$	0	0
$503$	39.0000	1.73892	0.869462	−	0.494000i	$-0.164466\pi$
$503$	39.0000	1.73892	0.869462	+	0.494000i	$0.164466\pi$
$504$	0	0
$505$	30.0000	1.33498
$506$	0	0
$507$	4.00000	0.177646
$508$	0	0
$509$	−8.00000	−0.354594	−0.177297	−	0.984157i	$-0.556735\pi$
$509$	−8.00000	−0.354594	−0.177297	+	0.984157i	$0.556735\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	−39.0000	−1.71855
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−7.00000	−0.307266
$520$	0	0
$521$	45.0000	1.97149	0.985743	−	0.168259i	$-0.0538144\pi$
$521$	45.0000	1.97149	0.985743	+	0.168259i	$0.0538144\pi$
$522$	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
$525$	−4.00000	−0.174574
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	2.00000	0.0869565
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	−27.0000	−1.16950
$534$	0	0
$535$	45.0000	1.94552
$536$	0	0
$537$	6.00000	0.258919
$538$	0	0
$539$	−5.00000	−0.215365
$540$	0	0
$541$	−20.0000	−0.859867	−0.429934	−	0.902861i	$-0.641463\pi$
$541$	−20.0000	−0.859867	−0.429934	+	0.902861i	$0.641463\pi$
$542$	0	0
$543$	2.00000	0.0858282
$544$	0	0
$545$	−18.0000	−0.771035
$546$	0	0
$547$	14.0000	0.598597	0.299298	−	0.954160i	$-0.403247\pi$
$547$	14.0000	0.598597	0.299298	+	0.954160i	$0.403247\pi$
$548$	0	0
$549$	−6.00000	−0.256074
$550$	0	0
$551$	10.0000	0.426014
$552$	0	0
$553$	−8.00000	−0.340195
$554$	0	0
$555$	18.0000	0.764057
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	−3.00000	−0.126886
$560$	0	0
$561$	5.00000	0.211100
$562$	0	0
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	0	0
$565$	−45.0000	−1.89316
$566$	0	0
$567$	1.00000	0.0419961
$568$	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
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	−10.0000	−0.417756
$574$	0	0
$575$	−20.0000	−0.834058
$576$	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
$581$	−14.0000	−0.580818
$582$	0	0
$583$	−20.0000	−0.828315
$584$	0	0
$585$	−9.00000	−0.372104
$586$	0	0
$587$	34.0000	1.40333	0.701665	−	0.712507i	$-0.252440\pi$
$587$	34.0000	1.40333	0.701665	+	0.712507i	$0.252440\pi$
$588$	0	0
$589$	−40.0000	−1.64817
$590$	0	0
$591$	27.0000	1.11063
$592$	0	0
$593$	−24.0000	−0.985562	−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000	−0.985562	−0.492781	+	0.870153i	$0.664020\pi$
$594$	0	0
$595$	3.00000	0.122988
$596$	0	0
$597$	2.00000	0.0818546
$598$	0	0
$599$	26.0000	1.06233	0.531166	−	0.847268i	$-0.321754\pi$
$599$	26.0000	1.06233	0.531166	+	0.847268i	$0.321754\pi$
$600$	0	0
$601$	−20.0000	−0.815817	−0.407909	−	0.913023i	$-0.633742\pi$
$601$	−20.0000	−0.815817	−0.407909	+	0.913023i	$0.633742\pi$
$602$	0	0
$603$	16.0000	0.651570
$604$	0	0
$605$	42.0000	1.70754
$606$	0	0
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	0	0
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−25.0000	−1.00974	−0.504870	−	0.863195i	$-0.668460\pi$
$613$	−25.0000	−1.00974	−0.504870	+	0.863195i	$0.668460\pi$
$614$	0	0
$615$	−27.0000	−1.08875
$616$	0	0
$617$	−10.0000	−0.402585	−0.201292	−	0.979531i	$-0.564514\pi$
$617$	−10.0000	−0.402585	−0.201292	+	0.979531i	$0.564514\pi$
$618$	0	0
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	0	0
$621$	5.00000	0.200643
$622$	0	0
$623$	−8.00000	−0.320513
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	25.0000	0.998404
$628$	0	0
$629$	−6.00000	−0.239236
$630$	0	0
$631$	−13.0000	−0.517522	−0.258761	−	0.965941i	$-0.583314\pi$
$631$	−13.0000	−0.517522	−0.258761	+	0.965941i	$0.583314\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	−3.00000	−0.119051
$636$	0	0
$637$	−3.00000	−0.118864
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	37.0000	1.46141	0.730706	−	0.682692i	$-0.239191\pi$
$641$	37.0000	1.46141	0.730706	+	0.682692i	$0.239191\pi$
$642$	0	0
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	0	0
$645$	−3.00000	−0.118125
$646$	0	0
$647$	−22.0000	−0.864909	−0.432455	−	0.901656i	$-0.642352\pi$
$647$	−22.0000	−0.864909	−0.432455	+	0.901656i	$0.642352\pi$
$648$	0	0
$649$	60.0000	2.35521
$650$	0	0
$651$	8.00000	0.313545
$652$	0	0
$653$	−15.0000	−0.586995	−0.293498	−	0.955960i	$-0.594819\pi$
$653$	−15.0000	−0.586995	−0.293498	+	0.955960i	$0.594819\pi$
$654$	0	0
$655$	45.0000	1.75830
$656$	0	0
$657$	−10.0000	−0.390137
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	−9.00000	−0.350059	−0.175030	−	0.984563i	$-0.556002\pi$
$661$	−9.00000	−0.350059	−0.175030	+	0.984563i	$0.556002\pi$
$662$	0	0
$663$	3.00000	0.116510
$664$	0	0
$665$	15.0000	0.581675
$666$	0	0
$667$	−10.0000	−0.387202
$668$	0	0
$669$	−25.0000	−0.966556
$670$	0	0
$671$	30.0000	1.15814
$672$	0	0
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	49.0000	1.88322	0.941611	−	0.336701i	$-0.109311\pi$
$677$	49.0000	1.88322	0.941611	+	0.336701i	$0.109311\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	0	0
$681$	−15.0000	−0.574801
$682$	0	0
$683$	−19.0000	−0.727015	−0.363507	−	0.931591i	$-0.618421\pi$
$683$	−19.0000	−0.727015	−0.363507	+	0.931591i	$0.618421\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	14.0000	0.534133
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	0	0
$693$	−5.00000	−0.189934
$694$	0	0
$695$	−30.0000	−1.13796
$696$	0	0
$697$	9.00000	0.340899
$698$	0	0
$699$	9.00000	0.340411
$700$	0	0
$701$	20.0000	0.755390	0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000	0.755390	0.377695	+	0.925930i	$0.376717\pi$
$702$	0	0
$703$	−30.0000	−1.13147
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.0000	0.376089
$708$	0	0
$709$	18.0000	0.676004	0.338002	−	0.941145i	$-0.390249\pi$
$709$	18.0000	0.676004	0.338002	+	0.941145i	$0.390249\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	40.0000	1.49801
$714$	0	0
$715$	45.0000	1.68290
$716$	0	0
$717$	22.0000	0.821605
$718$	0	0
$719$	23.0000	0.857755	0.428878	−	0.903363i	$-0.358909\pi$
$719$	23.0000	0.857755	0.428878	+	0.903363i	$0.358909\pi$
$720$	0	0
$721$	−13.0000	−0.484145
$722$	0	0
$723$	−12.0000	−0.446285
$724$	0	0
$725$	8.00000	0.297113
$726$	0	0
$727$	36.0000	1.33517	0.667583	−	0.744535i	$-0.267329\pi$
$727$	36.0000	1.33517	0.667583	+	0.744535i	$0.267329\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.00000	0.0369863
$732$	0	0
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	0	0
$735$	−3.00000	−0.110657
$736$	0	0
$737$	−80.0000	−2.94684
$738$	0	0
$739$	−1.00000	−0.0367856	−0.0183928	−	0.999831i	$-0.505855\pi$
$739$	−1.00000	−0.0367856	−0.0183928	+	0.999831i	$0.505855\pi$
$740$	0	0
$741$	15.0000	0.551039
$742$	0	0
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	0	0
$745$	30.0000	1.09911
$746$	0	0
$747$	−14.0000	−0.512233
$748$	0	0
$749$	15.0000	0.548088
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	0	0
$753$	−6.00000	−0.218652
$754$	0	0
$755$	48.0000	1.74690
$756$	0	0
$757$	21.0000	0.763258	0.381629	−	0.924316i	$-0.375363\pi$
$757$	21.0000	0.763258	0.381629	+	0.924316i	$0.375363\pi$
$758$	0	0
$759$	−25.0000	−0.907443
$760$	0	0
$761$	26.0000	0.942499	0.471250	−	0.882000i	$-0.343803\pi$
$761$	26.0000	0.942499	0.471250	+	0.882000i	$0.343803\pi$
$762$	0	0
$763$	−6.00000	−0.217215
$764$	0	0
$765$	3.00000	0.108465
$766$	0	0
$767$	36.0000	1.29988
$768$	0	0
$769$	15.0000	0.540914	0.270457	−	0.962732i	$-0.412825\pi$
$769$	15.0000	0.540914	0.270457	+	0.962732i	$0.412825\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	0	0
$773$	−26.0000	−0.935155	−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000	−0.935155	−0.467578	+	0.883952i	$0.654873\pi$
$774$	0	0
$775$	−32.0000	−1.14947
$776$	0	0
$777$	6.00000	0.215249
$778$	0	0
$779$	45.0000	1.61229
$780$	0	0
$781$	40.0000	1.43131
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	15.0000	0.535373
$786$	0	0
$787$	26.0000	0.926800	0.463400	−	0.886149i	$-0.346629\pi$
$787$	26.0000	0.926800	0.463400	+	0.886149i	$0.346629\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−15.0000	−0.533339
$792$	0	0
$793$	18.0000	0.639199
$794$	0	0
$795$	−12.0000	−0.425596
$796$	0	0
$797$	44.0000	1.55856	0.779280	−	0.626676i	$-0.215585\pi$
$797$	44.0000	1.55856	0.779280	+	0.626676i	$0.215585\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−8.00000	−0.282666
$802$	0	0
$803$	50.0000	1.76446
$804$	0	0
$805$	−15.0000	−0.528681
$806$	0	0
$807$	−9.00000	−0.316815
$808$	0	0
$809$	−21.0000	−0.738321	−0.369160	−	0.929366i	$-0.620355\pi$
$809$	−21.0000	−0.738321	−0.369160	+	0.929366i	$0.620355\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	0	0
$813$	−29.0000	−1.01707
$814$	0	0
$815$	−54.0000	−1.89154
$816$	0	0
$817$	5.00000	0.174928
$818$	0	0
$819$	−3.00000	−0.104828
$820$	0	0
$821$	9.00000	0.314102	0.157051	−	0.987590i	$-0.449801\pi$
$821$	9.00000	0.314102	0.157051	+	0.987590i	$0.449801\pi$
$822$	0	0
$823$	46.0000	1.60346	0.801730	−	0.597687i	$-0.203913\pi$
$823$	46.0000	1.60346	0.801730	+	0.597687i	$0.203913\pi$
$824$	0	0
$825$	20.0000	0.696311
$826$	0	0
$827$	33.0000	1.14752	0.573761	−	0.819023i	$-0.305484\pi$
$827$	33.0000	1.14752	0.573761	+	0.819023i	$0.305484\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	−45.0000	−1.55729
$836$	0	0
$837$	8.00000	0.276520
$838$	0	0
$839$	−37.0000	−1.27738	−0.638691	−	0.769463i	$-0.720524\pi$
$839$	−37.0000	−1.27738	−0.638691	+	0.769463i	$0.720524\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12.0000	−0.412813
$846$	0	0
$847$	14.0000	0.481046
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	30.0000	1.02839
$852$	0	0
$853$	−16.0000	−0.547830	−0.273915	−	0.961754i	$-0.588319\pi$
$853$	−16.0000	−0.547830	−0.273915	+	0.961754i	$0.588319\pi$
$854$	0	0
$855$	15.0000	0.512989
$856$	0	0
$857$	30.0000	1.02478	0.512390	−	0.858753i	$-0.328760\pi$
$857$	30.0000	1.02478	0.512390	+	0.858753i	$0.328760\pi$
$858$	0	0
$859$	56.0000	1.91070	0.955348	−	0.295484i	$-0.0954809\pi$
$859$	56.0000	1.91070	0.955348	+	0.295484i	$0.0954809\pi$
$860$	0	0
$861$	−9.00000	−0.306719
$862$	0	0
$863$	−54.0000	−1.83818	−0.919091	−	0.394046i	$-0.871075\pi$
$863$	−54.0000	−1.83818	−0.919091	+	0.394046i	$0.871075\pi$
$864$	0	0
$865$	21.0000	0.714021
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	40.0000	1.35691
$870$	0	0
$871$	−48.0000	−1.62642
$872$	0	0
$873$	−8.00000	−0.270759
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	12.0000	0.405211	0.202606	−	0.979260i	$-0.435059\pi$
$877$	12.0000	0.405211	0.202606	+	0.979260i	$0.435059\pi$
$878$	0	0
$879$	−20.0000	−0.674583
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	−29.0000	−0.975928	−0.487964	−	0.872864i	$-0.662260\pi$
$883$	−29.0000	−0.975928	−0.487964	+	0.872864i	$0.662260\pi$
$884$	0	0
$885$	36.0000	1.21013
$886$	0	0
$887$	−9.00000	−0.302190	−0.151095	−	0.988519i	$-0.548280\pi$
$887$	−9.00000	−0.302190	−0.151095	+	0.988519i	$0.548280\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	0	0
$891$	−5.00000	−0.167506
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−18.0000	−0.601674
$896$	0	0
$897$	−15.0000	−0.500835
$898$	0	0
$899$	−16.0000	−0.533630
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	−1.00000	−0.0332779
$904$	0	0
$905$	−6.00000	−0.199447
$906$	0	0
$907$	−18.0000	−0.597680	−0.298840	−	0.954303i	$-0.596600\pi$
$907$	−18.0000	−0.597680	−0.298840	+	0.954303i	$0.596600\pi$
$908$	0	0
$909$	10.0000	0.331679
$910$	0	0
$911$	−53.0000	−1.75597	−0.877984	−	0.478690i	$-0.841112\pi$
$911$	−53.0000	−1.75597	−0.877984	+	0.478690i	$0.841112\pi$
$912$	0	0
$913$	70.0000	2.31666
$914$	0	0
$915$	18.0000	0.595062
$916$	0	0
$917$	15.0000	0.495344
$918$	0	0
$919$	−49.0000	−1.61636	−0.808180	−	0.588935i	$-0.799547\pi$
$919$	−49.0000	−1.61636	−0.808180	+	0.588935i	$0.799547\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	0	0
$923$	24.0000	0.789970
$924$	0	0
$925$	−24.0000	−0.789115
$926$	0	0
$927$	−13.0000	−0.426976
$928$	0	0
$929$	35.0000	1.14831	0.574156	−	0.818746i	$-0.305330\pi$
$929$	35.0000	1.14831	0.574156	+	0.818746i	$0.305330\pi$
$930$	0	0
$931$	5.00000	0.163868
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	−15.0000	−0.490552
$936$	0	0
$937$	−18.0000	−0.588034	−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000	−0.588034	−0.294017	+	0.955800i	$0.594992\pi$
$938$	0	0
$939$	−14.0000	−0.456873
$940$	0	0
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	0	0
$943$	−45.0000	−1.46540
$944$	0	0
$945$	−3.00000	−0.0975900
$946$	0	0
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	0	0
$949$	30.0000	0.973841
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	54.0000	1.74923	0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000	1.74923	0.874616	+	0.484817i	$0.161114\pi$
$954$	0	0
$955$	30.0000	0.970777
$956$	0	0
$957$	10.0000	0.323254
$958$	0	0
$959$	4.00000	0.129167
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	15.0000	0.483368
$964$	0	0
$965$	12.0000	0.386294
$966$	0	0
$967$	49.0000	1.57573	0.787867	−	0.615846i	$-0.211185\pi$
$967$	49.0000	1.57573	0.787867	+	0.615846i	$0.211185\pi$
$968$	0	0
$969$	−5.00000	−0.160623
$970$	0	0
$971$	38.0000	1.21948	0.609739	−	0.792602i	$-0.291274\pi$
$971$	38.0000	1.21948	0.609739	+	0.792602i	$0.291274\pi$
$972$	0	0
$973$	−10.0000	−0.320585
$974$	0	0
$975$	12.0000	0.384308
$976$	0	0
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	0	0
$979$	40.0000	1.27841
$980$	0	0
$981$	−6.00000	−0.191565
$982$	0	0
$983$	−25.0000	−0.797376	−0.398688	−	0.917087i	$-0.630534\pi$
$983$	−25.0000	−0.797376	−0.398688	+	0.917087i	$0.630534\pi$
$984$	0	0
$985$	−81.0000	−2.58087
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5.00000	−0.158991
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	0	0
$993$	17.0000	0.539479
$994$	0	0
$995$	−6.00000	−0.190213
$996$	0	0
$997$	−38.0000	−1.20347	−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000	−1.20347	−0.601736	+	0.798695i	$0.705524\pi$
$998$	0	0
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5712.2.a.m.1.1		1
4.3	odd	2		2856.2.a.j.1.1	✓	1
12.11	even	2		8568.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2856.2.a.j.1.1	✓	1	4.3	odd	2
5712.2.a.m.1.1		1	1.1	even	1	trivial
8568.2.a.a.1.1		1	12.11	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:	\( N \)	\(=\)	\( 5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5712.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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