LMFDB - Embedded newform 4704.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4704,2,Mod(1,4704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4704, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4704.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4704 = 2^{5} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4704.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:	$37.5616291108$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 672)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4704.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -4.00000 q^{5} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -4.00000 q^{5} +1.00000 q^{9} +6.00000 q^{11} +5.00000 q^{13} +4.00000 q^{15} +2.00000 q^{17} +1.00000 q^{19} -6.00000 q^{23} +11.0000 q^{25} -1.00000 q^{27} -3.00000 q^{31} -6.00000 q^{33} +3.00000 q^{37} -5.00000 q^{39} -6.00000 q^{41} +5.00000 q^{43} -4.00000 q^{45} -4.00000 q^{47} -2.00000 q^{51} -6.00000 q^{53} -24.0000 q^{55} -1.00000 q^{57} -6.00000 q^{59} -2.00000 q^{61} -20.0000 q^{65} +7.00000 q^{67} +6.00000 q^{69} +16.0000 q^{71} -3.00000 q^{73} -11.0000 q^{75} +11.0000 q^{79} +1.00000 q^{81} +12.0000 q^{83} -8.00000 q^{85} +4.00000 q^{89} +3.00000 q^{93} -4.00000 q^{95} -6.00000 q^{97} +6.00000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	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
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	0	0
$33$	−6.00000	−1.04447
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	0	0
$39$	−5.00000	−0.800641
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	0	0
$45$	−4.00000	−0.596285
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−24.0000	−3.23616
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−20.0000	−2.48069
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	0	0
$73$	−3.00000	−0.351123	−0.175562	−	0.984468i	$-0.556174\pi$
$73$	−3.00000	−0.351123	−0.175562	+	0.984468i	$0.556174\pi$
$74$	0	0
$75$	−11.0000	−1.27017
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.00000	0.311086
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	−11.0000	−1.08386	−0.541931	−	0.840423i	$-0.682307\pi$
$103$	−11.0000	−1.08386	−0.541931	+	0.840423i	$0.682307\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	0	0
$109$	−15.0000	−1.43674	−0.718370	−	0.695662i	$-0.755111\pi$
$109$	−15.0000	−1.43674	−0.718370	+	0.695662i	$0.755111\pi$
$110$	0	0
$111$	−3.00000	−0.284747
$112$	0	0
$113$	−16.0000	−1.50515	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000	−1.50515	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	24.0000	2.23801
$116$	0	0
$117$	5.00000	0.462250
$118$	0	0
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	−24.0000	−2.14663
$126$	0	0
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	0	0
$129$	−5.00000	−0.440225
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	0	0
$143$	30.0000	2.50873
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	12.0000	0.963863
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	0	0
$165$	24.0000	1.86840
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.00000	0.450988
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	25.0000	1.85824	0.929118	−	0.369784i	$-0.120568\pi$
$181$	25.0000	1.85824	0.929118	+	0.369784i	$0.120568\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	12.0000	0.877527
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−1.00000	−0.0719816	−0.0359908	−	0.999352i	$-0.511459\pi$
$193$	−1.00000	−0.0719816	−0.0359908	+	0.999352i	$0.511459\pi$
$194$	0	0
$195$	20.0000	1.43223
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	0	0
$203$	0	0
$204$	0	0
$205$	24.0000	1.67623
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	6.00000	0.415029
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	0	0
$213$	−16.0000	−1.09630
$214$	0	0
$215$	−20.0000	−1.36399
$216$	0	0
$217$	0	0
$218$	0	0
$219$	3.00000	0.202721
$220$	0	0
$221$	10.0000	0.672673
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	11.0000	0.733333
$226$	0	0
$227$	22.0000	1.46019	0.730096	−	0.683345i	$-0.239475\pi$
$227$	22.0000	1.46019	0.730096	+	0.683345i	$0.239475\pi$
$228$	0	0
$229$	−11.0000	−0.726900	−0.363450	−	0.931614i	$-0.618401\pi$
$229$	−11.0000	−0.726900	−0.363450	+	0.931614i	$0.618401\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.00000	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$234$	0	0
$235$	16.0000	1.04372
$236$	0	0
$237$	−11.0000	−0.714527
$238$	0	0
$239$	2.00000	0.129369	0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000	0.129369	0.0646846	+	0.997906i	$0.479396\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.00000	0.318142
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	−14.0000	−0.883672	−0.441836	−	0.897096i	$-0.645673\pi$
$251$	−14.0000	−0.883672	−0.441836	+	0.897096i	$0.645673\pi$
$252$	0	0
$253$	−36.0000	−2.26330
$254$	0	0
$255$	8.00000	0.500979
$256$	0	0
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	0	0
$267$	−4.00000	−0.244796
$268$	0	0
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	66.0000	3.97995
$276$	0	0
$277$	1.00000	0.0600842	0.0300421	−	0.999549i	$-0.490436\pi$
$277$	1.00000	0.0600842	0.0300421	+	0.999549i	$0.490436\pi$
$278$	0	0
$279$	−3.00000	−0.179605
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	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
$285$	4.00000	0.236940
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	6.00000	0.351726
$292$	0	0
$293$	−12.0000	−0.701047	−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000	−0.701047	−0.350524	+	0.936554i	$0.613996\pi$
$294$	0	0
$295$	24.0000	1.39733
$296$	0	0
$297$	−6.00000	−0.348155
$298$	0	0
$299$	−30.0000	−1.73494
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−2.00000	−0.114897
$304$	0	0
$305$	8.00000	0.458079
$306$	0	0
$307$	11.0000	0.627803	0.313902	−	0.949456i	$-0.398364\pi$
$307$	11.0000	0.627803	0.313902	+	0.949456i	$0.398364\pi$
$308$	0	0
$309$	11.0000	0.625768
$310$	0	0
$311$	−2.00000	−0.113410	−0.0567048	−	0.998391i	$-0.518059\pi$
$311$	−2.00000	−0.113410	−0.0567048	+	0.998391i	$0.518059\pi$
$312$	0	0
$313$	31.0000	1.75222	0.876112	−	0.482108i	$-0.160129\pi$
$313$	31.0000	1.75222	0.876112	+	0.482108i	$0.160129\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−20.0000	−1.12331	−0.561656	−	0.827371i	$-0.689836\pi$
$317$	−20.0000	−1.12331	−0.561656	+	0.827371i	$0.689836\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−10.0000	−0.558146
$322$	0	0
$323$	2.00000	0.111283
$324$	0	0
$325$	55.0000	3.05085
$326$	0	0
$327$	15.0000	0.829502
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−5.00000	−0.274825	−0.137412	−	0.990514i	$-0.543879\pi$
$331$	−5.00000	−0.274825	−0.137412	+	0.990514i	$0.543879\pi$
$332$	0	0
$333$	3.00000	0.164399
$334$	0	0
$335$	−28.0000	−1.52980
$336$	0	0
$337$	1.00000	0.0544735	0.0272367	−	0.999629i	$-0.491329\pi$
$337$	1.00000	0.0544735	0.0272367	+	0.999629i	$0.491329\pi$
$338$	0	0
$339$	16.0000	0.869001
$340$	0	0
$341$	−18.0000	−0.974755
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−24.0000	−1.29212
$346$	0	0
$347$	−22.0000	−1.18102	−0.590511	−	0.807030i	$-0.701074\pi$
$347$	−22.0000	−1.18102	−0.590511	+	0.807030i	$0.701074\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	−4.00000	−0.212899	−0.106449	−	0.994318i	$-0.533948\pi$
$353$	−4.00000	−0.212899	−0.106449	+	0.994318i	$0.533948\pi$
$354$	0	0
$355$	−64.0000	−3.39677
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	−25.0000	−1.31216
$364$	0	0
$365$	12.0000	0.628109
$366$	0	0
$367$	27.0000	1.40939	0.704694	−	0.709511i	$-0.251084\pi$
$367$	27.0000	1.40939	0.704694	+	0.709511i	$0.251084\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−29.0000	−1.50156	−0.750782	−	0.660551i	$-0.770323\pi$
$373$	−29.0000	−1.50156	−0.750782	+	0.660551i	$0.770323\pi$
$374$	0	0
$375$	24.0000	1.23935
$376$	0	0
$377$	0	0
$378$	0	0
$379$	27.0000	1.38690	0.693448	−	0.720506i	$-0.256091\pi$
$379$	27.0000	1.38690	0.693448	+	0.720506i	$0.256091\pi$
$380$	0	0
$381$	7.00000	0.358621
$382$	0	0
$383$	26.0000	1.32854	0.664269	−	0.747494i	$-0.268743\pi$
$383$	26.0000	1.32854	0.664269	+	0.747494i	$0.268743\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.00000	0.254164
$388$	0	0
$389$	−4.00000	−0.202808	−0.101404	−	0.994845i	$-0.532333\pi$
$389$	−4.00000	−0.202808	−0.101404	+	0.994845i	$0.532333\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	−6.00000	−0.302660
$394$	0	0
$395$	−44.0000	−2.21388
$396$	0	0
$397$	−21.0000	−1.05396	−0.526980	−	0.849878i	$-0.676676\pi$
$397$	−21.0000	−1.05396	−0.526980	+	0.849878i	$0.676676\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	−15.0000	−0.747203
$404$	0	0
$405$	−4.00000	−0.198762
$406$	0	0
$407$	18.0000	0.892227
$408$	0	0
$409$	13.0000	0.642809	0.321404	−	0.946942i	$-0.395845\pi$
$409$	13.0000	0.642809	0.321404	+	0.946942i	$0.395845\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−48.0000	−2.35623
$416$	0	0
$417$	−5.00000	−0.244851
$418$	0	0
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	0	0
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	0	0
$423$	−4.00000	−0.194487
$424$	0	0
$425$	22.0000	1.06716
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−30.0000	−1.44841
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−25.0000	−1.20142	−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000	−1.20142	−0.600712	+	0.799466i	$0.705116\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.00000	−0.287019
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−26.0000	−1.23530	−0.617649	−	0.786454i	$-0.711915\pi$
$443$	−26.0000	−1.23530	−0.617649	+	0.786454i	$0.711915\pi$
$444$	0	0
$445$	−16.0000	−0.758473
$446$	0	0
$447$	4.00000	0.189194
$448$	0	0
$449$	36.0000	1.69895	0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000	1.69895	0.849473	+	0.527633i	$0.176920\pi$
$450$	0	0
$451$	−36.0000	−1.69517
$452$	0	0
$453$	8.00000	0.375873
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.0000	1.16945	0.584725	−	0.811231i	$-0.301202\pi$
$457$	25.0000	1.16945	0.584725	+	0.811231i	$0.301202\pi$
$458$	0	0
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−4.00000	−0.186299	−0.0931493	−	0.995652i	$-0.529693\pi$
$461$	−4.00000	−0.186299	−0.0931493	+	0.995652i	$0.529693\pi$
$462$	0	0
$463$	−5.00000	−0.232370	−0.116185	−	0.993228i	$-0.537067\pi$
$463$	−5.00000	−0.232370	−0.116185	+	0.993228i	$0.537067\pi$
$464$	0	0
$465$	−12.0000	−0.556487
$466$	0	0
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−10.0000	−0.460776
$472$	0	0
$473$	30.0000	1.37940
$474$	0	0
$475$	11.0000	0.504715
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	2.00000	0.0913823	0.0456912	−	0.998956i	$-0.485451\pi$
$479$	2.00000	0.0913823	0.0456912	+	0.998956i	$0.485451\pi$
$480$	0	0
$481$	15.0000	0.683941
$482$	0	0
$483$	0	0
$484$	0	0
$485$	24.0000	1.08978
$486$	0	0
$487$	−1.00000	−0.0453143	−0.0226572	−	0.999743i	$-0.507213\pi$
$487$	−1.00000	−0.0453143	−0.0226572	+	0.999743i	$0.507213\pi$
$488$	0	0
$489$	−20.0000	−0.904431
$490$	0	0
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−24.0000	−1.07872
$496$	0	0
$497$	0	0
$498$	0	0
$499$	29.0000	1.29822	0.649109	−	0.760695i	$-0.275142\pi$
$499$	29.0000	1.29822	0.649109	+	0.760695i	$0.275142\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	0	0
$503$	18.0000	0.802580	0.401290	−	0.915951i	$-0.368562\pi$
$503$	18.0000	0.802580	0.401290	+	0.915951i	$0.368562\pi$
$504$	0	0
$505$	−8.00000	−0.355995
$506$	0	0
$507$	−12.0000	−0.532939
$508$	0	0
$509$	−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
$511$	0	0
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	44.0000	1.93887
$516$	0	0
$517$	−24.0000	−1.05552
$518$	0	0
$519$	−22.0000	−0.965693
$520$	0	0
$521$	24.0000	1.05146	0.525730	−	0.850652i	$-0.323792\pi$
$521$	24.0000	1.05146	0.525730	+	0.850652i	$0.323792\pi$
$522$	0	0
$523$	−17.0000	−0.743358	−0.371679	−	0.928361i	$-0.621218\pi$
$523$	−17.0000	−0.743358	−0.371679	+	0.928361i	$0.621218\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.00000	−0.261364
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	−6.00000	−0.260378
$532$	0	0
$533$	−30.0000	−1.29944
$534$	0	0
$535$	−40.0000	−1.72935
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	37.0000	1.59075	0.795377	−	0.606115i	$-0.207273\pi$
$541$	37.0000	1.59075	0.795377	+	0.606115i	$0.207273\pi$
$542$	0	0
$543$	−25.0000	−1.07285
$544$	0	0
$545$	60.0000	2.57012
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	12.0000	0.509372
$556$	0	0
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	0	0
$559$	25.0000	1.05739
$560$	0	0
$561$	−12.0000	−0.506640
$562$	0	0
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	64.0000	2.69250
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	29.0000	1.21361	0.606806	−	0.794850i	$-0.292450\pi$
$571$	29.0000	1.21361	0.606806	+	0.794850i	$0.292450\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−66.0000	−2.75239
$576$	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
$579$	1.00000	0.0415586
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−36.0000	−1.49097
$584$	0	0
$585$	−20.0000	−0.826898
$586$	0	0
$587$	40.0000	1.65098	0.825488	−	0.564419i	$-0.190900\pi$
$587$	40.0000	1.65098	0.825488	+	0.564419i	$0.190900\pi$
$588$	0	0
$589$	−3.00000	−0.123613
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	−26.0000	−1.06769	−0.533846	−	0.845582i	$-0.679254\pi$
$593$	−26.0000	−1.06769	−0.533846	+	0.845582i	$0.679254\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−20.0000	−0.818546
$598$	0	0
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	0	0
$601$	−21.0000	−0.856608	−0.428304	−	0.903635i	$-0.640889\pi$
$601$	−21.0000	−0.856608	−0.428304	+	0.903635i	$0.640889\pi$
$602$	0	0
$603$	7.00000	0.285062
$604$	0	0
$605$	−100.000	−4.06558
$606$	0	0
$607$	−5.00000	−0.202944	−0.101472	−	0.994838i	$-0.532355\pi$
$607$	−5.00000	−0.202944	−0.101472	+	0.994838i	$0.532355\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−20.0000	−0.809113
$612$	0	0
$613$	−46.0000	−1.85792	−0.928961	−	0.370177i	$-0.879297\pi$
$613$	−46.0000	−1.85792	−0.928961	+	0.370177i	$0.879297\pi$
$614$	0	0
$615$	−24.0000	−0.967773
$616$	0	0
$617$	−14.0000	−0.563619	−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000	−0.563619	−0.281809	+	0.959470i	$0.590935\pi$
$618$	0	0
$619$	−25.0000	−1.00483	−0.502417	−	0.864625i	$-0.667556\pi$
$619$	−25.0000	−1.00483	−0.502417	+	0.864625i	$0.667556\pi$
$620$	0	0
$621$	6.00000	0.240772
$622$	0	0
$623$	0	0
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	−6.00000	−0.239617
$628$	0	0
$629$	6.00000	0.239236
$630$	0	0
$631$	28.0000	1.11466	0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000	1.11466	0.557331	+	0.830290i	$0.311825\pi$
$632$	0	0
$633$	−12.0000	−0.476957
$634$	0	0
$635$	28.0000	1.11115
$636$	0	0
$637$	0	0
$638$	0	0
$639$	16.0000	0.632950
$640$	0	0
$641$	38.0000	1.50091	0.750455	−	0.660922i	$-0.229834\pi$
$641$	38.0000	1.50091	0.750455	+	0.660922i	$0.229834\pi$
$642$	0	0
$643$	−23.0000	−0.907031	−0.453516	−	0.891248i	$-0.649830\pi$
$643$	−23.0000	−0.907031	−0.453516	+	0.891248i	$0.649830\pi$
$644$	0	0
$645$	20.0000	0.787499
$646$	0	0
$647$	10.0000	0.393141	0.196570	−	0.980490i	$-0.437020\pi$
$647$	10.0000	0.393141	0.196570	+	0.980490i	$0.437020\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	0	0
$651$	0	0
$652$	0	0
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	0	0
$655$	−24.0000	−0.937758
$656$	0	0
$657$	−3.00000	−0.117041
$658$	0	0
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	0	0
$661$	3.00000	0.116686	0.0583432	−	0.998297i	$-0.481418\pi$
$661$	3.00000	0.116686	0.0583432	+	0.998297i	$0.481418\pi$
$662$	0	0
$663$	−10.0000	−0.388368
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−16.0000	−0.618596
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	−29.0000	−1.11787	−0.558934	−	0.829212i	$-0.688789\pi$
$673$	−29.0000	−1.11787	−0.558934	+	0.829212i	$0.688789\pi$
$674$	0	0
$675$	−11.0000	−0.423390
$676$	0	0
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−22.0000	−0.843042
$682$	0	0
$683$	18.0000	0.688751	0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000	0.688751	0.344375	+	0.938832i	$0.388091\pi$
$684$	0	0
$685$	48.0000	1.83399
$686$	0	0
$687$	11.0000	0.419676
$688$	0	0
$689$	−30.0000	−1.14291
$690$	0	0
$691$	23.0000	0.874961	0.437481	−	0.899228i	$-0.355871\pi$
$691$	23.0000	0.874961	0.437481	+	0.899228i	$0.355871\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−20.0000	−0.758643
$696$	0	0
$697$	−12.0000	−0.454532
$698$	0	0
$699$	8.00000	0.302588
$700$	0	0
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	0	0
$703$	3.00000	0.113147
$704$	0	0
$705$	−16.0000	−0.602595
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	11.0000	0.412532
$712$	0	0
$713$	18.0000	0.674105
$714$	0	0
$715$	−120.000	−4.48775
$716$	0	0
$717$	−2.00000	−0.0746914
$718$	0	0
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−10.0000	−0.371904
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−41.0000	−1.52061	−0.760303	−	0.649569i	$-0.774949\pi$
$727$	−41.0000	−1.52061	−0.760303	+	0.649569i	$0.774949\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	10.0000	0.369863
$732$	0	0
$733$	21.0000	0.775653	0.387826	−	0.921732i	$-0.373226\pi$
$733$	21.0000	0.775653	0.387826	+	0.921732i	$0.373226\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	42.0000	1.54709
$738$	0	0
$739$	−51.0000	−1.87607	−0.938033	−	0.346547i	$-0.887354\pi$
$739$	−51.0000	−1.87607	−0.938033	+	0.346547i	$0.887354\pi$
$740$	0	0
$741$	−5.00000	−0.183680
$742$	0	0
$743$	−18.0000	−0.660356	−0.330178	−	0.943919i	$-0.607109\pi$
$743$	−18.0000	−0.660356	−0.330178	+	0.943919i	$0.607109\pi$
$744$	0	0
$745$	16.0000	0.586195
$746$	0	0
$747$	12.0000	0.439057
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−7.00000	−0.255434	−0.127717	−	0.991811i	$-0.540765\pi$
$751$	−7.00000	−0.255434	−0.127717	+	0.991811i	$0.540765\pi$
$752$	0	0
$753$	14.0000	0.510188
$754$	0	0
$755$	32.0000	1.16460
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	0	0
$759$	36.0000	1.30672
$760$	0	0
$761$	50.0000	1.81250	0.906249	−	0.422744i	$-0.138933\pi$
$761$	50.0000	1.81250	0.906249	+	0.422744i	$0.138933\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−8.00000	−0.289241
$766$	0	0
$767$	−30.0000	−1.08324
$768$	0	0
$769$	31.0000	1.11789	0.558944	−	0.829205i	$-0.311207\pi$
$769$	31.0000	1.11789	0.558944	+	0.829205i	$0.311207\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	0	0
$773$	−22.0000	−0.791285	−0.395643	−	0.918405i	$-0.629478\pi$
$773$	−22.0000	−0.791285	−0.395643	+	0.918405i	$0.629478\pi$
$774$	0	0
$775$	−33.0000	−1.18539
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.00000	−0.214972
$780$	0	0
$781$	96.0000	3.43515
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−40.0000	−1.42766
$786$	0	0
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−10.0000	−0.355110
$794$	0	0
$795$	−24.0000	−0.851192
$796$	0	0
$797$	−46.0000	−1.62940	−0.814702	−	0.579880i	$-0.803099\pi$
$797$	−46.0000	−1.62940	−0.814702	+	0.579880i	$0.803099\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	4.00000	0.141333
$802$	0	0
$803$	−18.0000	−0.635206
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.00000	0.0704033
$808$	0	0
$809$	8.00000	0.281265	0.140633	−	0.990062i	$-0.455086\pi$
$809$	8.00000	0.281265	0.140633	+	0.990062i	$0.455086\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	−16.0000	−0.561144
$814$	0	0
$815$	−80.0000	−2.80228
$816$	0	0
$817$	5.00000	0.174928
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.00000	0.139601	0.0698005	−	0.997561i	$-0.477764\pi$
$821$	4.00000	0.139601	0.0698005	+	0.997561i	$0.477764\pi$
$822$	0	0
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	0	0
$825$	−66.0000	−2.29783
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	5.00000	0.173657	0.0868286	−	0.996223i	$-0.472327\pi$
$829$	5.00000	0.173657	0.0868286	+	0.996223i	$0.472327\pi$
$830$	0	0
$831$	−1.00000	−0.0346896
$832$	0	0
$833$	0	0
$834$	0	0
$835$	32.0000	1.10741
$836$	0	0
$837$	3.00000	0.103695
$838$	0	0
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	−6.00000	−0.206651
$844$	0	0
$845$	−48.0000	−1.65125
$846$	0	0
$847$	0	0
$848$	0	0
$849$	11.0000	0.377519
$850$	0	0
$851$	−18.0000	−0.617032
$852$	0	0
$853$	−33.0000	−1.12990	−0.564949	−	0.825126i	$-0.691104\pi$
$853$	−33.0000	−1.12990	−0.564949	+	0.825126i	$0.691104\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	24.0000	0.819824	0.409912	−	0.912125i	$-0.365559\pi$
$857$	24.0000	0.819824	0.409912	+	0.912125i	$0.365559\pi$
$858$	0	0
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	54.0000	1.83818	0.919091	−	0.394046i	$-0.128925\pi$
$863$	54.0000	1.83818	0.919091	+	0.394046i	$0.128925\pi$
$864$	0	0
$865$	−88.0000	−2.99209
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	66.0000	2.23890
$870$	0	0
$871$	35.0000	1.18593
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	0	0
$876$	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
$879$	12.0000	0.404750
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	0	0
$883$	7.00000	0.235569	0.117784	−	0.993039i	$-0.462421\pi$
$883$	7.00000	0.235569	0.117784	+	0.993039i	$0.462421\pi$
$884$	0	0
$885$	−24.0000	−0.806751
$886$	0	0
$887$	34.0000	1.14161	0.570804	−	0.821086i	$-0.306632\pi$
$887$	34.0000	1.14161	0.570804	+	0.821086i	$0.306632\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	6.00000	0.201008
$892$	0	0
$893$	−4.00000	−0.133855
$894$	0	0
$895$	0	0
$896$	0	0
$897$	30.0000	1.00167
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−100.000	−3.32411
$906$	0	0
$907$	43.0000	1.42779	0.713896	−	0.700252i	$-0.246929\pi$
$907$	43.0000	1.42779	0.713896	+	0.700252i	$0.246929\pi$
$908$	0	0
$909$	2.00000	0.0663358
$910$	0	0
$911$	−26.0000	−0.861418	−0.430709	−	0.902491i	$-0.641737\pi$
$911$	−26.0000	−0.861418	−0.430709	+	0.902491i	$0.641737\pi$
$912$	0	0
$913$	72.0000	2.38285
$914$	0	0
$915$	−8.00000	−0.264472
$916$	0	0
$917$	0	0
$918$	0	0
$919$	43.0000	1.41844	0.709220	−	0.704988i	$-0.249047\pi$
$919$	43.0000	1.41844	0.709220	+	0.704988i	$0.249047\pi$
$920$	0	0
$921$	−11.0000	−0.362462
$922$	0	0
$923$	80.0000	2.63323
$924$	0	0
$925$	33.0000	1.08503
$926$	0	0
$927$	−11.0000	−0.361287
$928$	0	0
$929$	−36.0000	−1.18112	−0.590561	−	0.806993i	$-0.701093\pi$
$929$	−36.0000	−1.18112	−0.590561	+	0.806993i	$0.701093\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	2.00000	0.0654771
$934$	0	0
$935$	−48.0000	−1.56977
$936$	0	0
$937$	51.0000	1.66610	0.833049	−	0.553200i	$-0.186593\pi$
$937$	51.0000	1.66610	0.833049	+	0.553200i	$0.186593\pi$
$938$	0	0
$939$	−31.0000	−1.01165
$940$	0	0
$941$	54.0000	1.76035	0.880175	−	0.474650i	$-0.157425\pi$
$941$	54.0000	1.76035	0.880175	+	0.474650i	$0.157425\pi$
$942$	0	0
$943$	36.0000	1.17232
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30.0000	−0.974869	−0.487435	−	0.873160i	$-0.662067\pi$
$947$	−30.0000	−0.974869	−0.487435	+	0.873160i	$0.662067\pi$
$948$	0	0
$949$	−15.0000	−0.486921
$950$	0	0
$951$	20.0000	0.648544
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	10.0000	0.322245
$964$	0	0
$965$	4.00000	0.128765
$966$	0	0
$967$	−53.0000	−1.70437	−0.852183	−	0.523245i	$-0.824721\pi$
$967$	−53.0000	−1.70437	−0.852183	+	0.523245i	$0.824721\pi$
$968$	0	0
$969$	−2.00000	−0.0642493
$970$	0	0
$971$	−26.0000	−0.834380	−0.417190	−	0.908819i	$-0.636985\pi$
$971$	−26.0000	−0.834380	−0.417190	+	0.908819i	$0.636985\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−55.0000	−1.76141
$976$	0	0
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	−15.0000	−0.478913
$982$	0	0
$983$	26.0000	0.829271	0.414636	−	0.909988i	$-0.363909\pi$
$983$	26.0000	0.829271	0.414636	+	0.909988i	$0.363909\pi$
$984$	0	0
$985$	−72.0000	−2.29411
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−30.0000	−0.953945
$990$	0	0
$991$	25.0000	0.794151	0.397076	−	0.917786i	$-0.370025\pi$
$991$	25.0000	0.794151	0.397076	+	0.917786i	$0.370025\pi$
$992$	0	0
$993$	5.00000	0.158670
$994$	0	0
$995$	−80.0000	−2.53617
$996$	0	0
$997$	−5.00000	−0.158352	−0.0791758	−	0.996861i	$-0.525229\pi$
$997$	−5.00000	−0.158352	−0.0791758	+	0.996861i	$0.525229\pi$
$998$	0	0
$999$	−3.00000	−0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4704.2.a.a.1.1		1
4.3	odd	2		4704.2.a.r.1.1		1
7.2	even	3		672.2.q.j.193.1	yes	2
7.4	even	3		672.2.q.j.289.1	yes	2
7.6	odd	2		4704.2.a.bh.1.1		1
8.3	odd	2		9408.2.a.bp.1.1		1
8.5	even	2		9408.2.a.dd.1.1		1
21.2	odd	6		2016.2.s.a.865.1		2
21.11	odd	6		2016.2.s.a.289.1		2
28.11	odd	6		672.2.q.e.289.1	yes	2
28.23	odd	6		672.2.q.e.193.1	✓	2
28.27	even	2		4704.2.a.p.1.1		1
56.11	odd	6		1344.2.q.l.961.1		2
56.13	odd	2		9408.2.a.a.1.1		1
56.27	even	2		9408.2.a.bs.1.1		1
56.37	even	6		1344.2.q.a.193.1		2
56.51	odd	6		1344.2.q.l.193.1		2
56.53	even	6		1344.2.q.a.961.1		2
84.11	even	6		2016.2.s.b.289.1		2
84.23	even	6		2016.2.s.b.865.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.q.e.193.1	✓	2	28.23	odd	6
672.2.q.e.289.1	yes	2	28.11	odd	6
672.2.q.j.193.1	yes	2	7.2	even	3
672.2.q.j.289.1	yes	2	7.4	even	3
1344.2.q.a.193.1		2	56.37	even	6
1344.2.q.a.961.1		2	56.53	even	6
1344.2.q.l.193.1		2	56.51	odd	6
1344.2.q.l.961.1		2	56.11	odd	6
2016.2.s.a.289.1		2	21.11	odd	6
2016.2.s.a.865.1		2	21.2	odd	6
2016.2.s.b.289.1		2	84.11	even	6
2016.2.s.b.865.1		2	84.23	even	6
4704.2.a.a.1.1		1	1.1	even	1	trivial
4704.2.a.p.1.1		1	28.27	even	2
4704.2.a.r.1.1		1	4.3	odd	2
4704.2.a.bh.1.1		1	7.6	odd	2
9408.2.a.a.1.1		1	56.13	odd	2
9408.2.a.bp.1.1		1	8.3	odd	2
9408.2.a.bs.1.1		1	56.27	even	2
9408.2.a.dd.1.1		1	8.5	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:	\( N \)	\(=\)	\( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4704.a (trivial)

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L-functions

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