LMFDB - Embedded newform 428.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [428,2,Mod(1,428)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(428, 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("428.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 428 = 2^{2} \cdot 107 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	428.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:	$3.41759720651$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	428.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} +2.00000 q^{5} -4.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} +2.00000 q^{5} -4.00000 q^{7} -2.00000 q^{9} -5.00000 q^{11} +1.00000 q^{13} -2.00000 q^{15} +2.00000 q^{17} -1.00000 q^{19} +4.00000 q^{21} -3.00000 q^{23} -1.00000 q^{25} +5.00000 q^{27} -10.0000 q^{29} +4.00000 q^{31} +5.00000 q^{33} -8.00000 q^{35} -7.00000 q^{37} -1.00000 q^{39} +3.00000 q^{41} -6.00000 q^{43} -4.00000 q^{45} +9.00000 q^{49} -2.00000 q^{51} +1.00000 q^{53} -10.0000 q^{55} +1.00000 q^{57} +8.00000 q^{59} +7.00000 q^{61} +8.00000 q^{63} +2.00000 q^{65} +2.00000 q^{67} +3.00000 q^{69} -6.00000 q^{71} -8.00000 q^{73} +1.00000 q^{75} +20.0000 q^{77} +13.0000 q^{79} +1.00000 q^{81} +12.0000 q^{83} +4.00000 q^{85} +10.0000 q^{87} -3.00000 q^{89} -4.00000 q^{91} -4.00000 q^{93} -2.00000 q^{95} -12.0000 q^{97} +10.0000 q^{99} +O(q^{100})$

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	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	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
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	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	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	5.00000	0.870388
$34$	0	0
$35$	−8.00000	−1.35225
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	−4.00000	−0.596285
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	0	0
$55$	−10.0000	−1.34840
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	0	0
$63$	8.00000	1.00791
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	0	0
$69$	3.00000	0.361158
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	20.0000	2.27921
$78$	0	0
$79$	13.0000	1.46261	0.731307	−	0.682048i	$-0.238911\pi$
$79$	13.0000	1.46261	0.731307	+	0.682048i	$0.238911\pi$
$80$	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
$85$	4.00000	0.433861
$86$	0	0
$87$	10.0000	1.07211
$88$	0	0
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	−2.00000	−0.205196
$96$	0	0
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	0	0
$99$	10.0000	1.00504
$100$	0	0
$101$	19.0000	1.89057	0.945285	−	0.326245i	$-0.105783\pi$
$101$	19.0000	1.89057	0.945285	+	0.326245i	$0.105783\pi$
$102$	0	0
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	0	0
$105$	8.00000	0.780720
$106$	0	0
$107$	1.00000	0.0966736
$107$	1.00000	0.0966736
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	7.00000	0.664411
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	−8.00000	−0.733359
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	−3.00000	−0.270501
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	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
$133$	4.00000	0.346844
$134$	0	0
$135$	10.0000	0.860663
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	−18.0000	−1.52674	−0.763370	−	0.645961i	$-0.776457\pi$
$139$	−18.0000	−1.52674	−0.763370	+	0.645961i	$0.776457\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.00000	−0.418121
$144$	0	0
$145$	−20.0000	−1.66091
$146$	0	0
$147$	−9.00000	−0.742307
$148$	0	0
$149$	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
$151$	1.00000	0.0813788	0.0406894	−	0.999172i	$-0.487045\pi$
$151$	1.00000	0.0813788	0.0406894	+	0.999172i	$0.487045\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	−16.0000	−1.27694	−0.638470	−	0.769647i	$-0.720432\pi$
$157$	−16.0000	−1.27694	−0.638470	+	0.769647i	$0.720432\pi$
$158$	0	0
$159$	−1.00000	−0.0793052
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	0	0
$165$	10.0000	0.778499
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	2.00000	0.152944
$172$	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
$175$	4.00000	0.302372
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	−7.00000	−0.517455
$184$	0	0
$185$	−14.0000	−1.02930
$186$	0	0
$187$	−10.0000	−0.731272
$188$	0	0
$189$	−20.0000	−1.45479
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	0	0
$193$	17.0000	1.22369	0.611843	−	0.790979i	$-0.290428\pi$
$193$	17.0000	1.22369	0.611843	+	0.790979i	$0.290428\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	0	0
$197$	−13.0000	−0.926212	−0.463106	−	0.886303i	$-0.653265\pi$
$197$	−13.0000	−0.926212	−0.463106	+	0.886303i	$0.653265\pi$
$198$	0	0
$199$	−5.00000	−0.354441	−0.177220	−	0.984171i	$-0.556711\pi$
$199$	−5.00000	−0.354441	−0.177220	+	0.984171i	$0.556711\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	0	0
$203$	40.0000	2.80745
$204$	0	0
$205$	6.00000	0.419058
$206$	0	0
$207$	6.00000	0.417029
$208$	0	0
$209$	5.00000	0.345857
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	0	0
$213$	6.00000	0.411113
$214$	0	0
$215$	−12.0000	−0.818393
$216$	0	0
$217$	−16.0000	−1.08615
$218$	0	0
$219$	8.00000	0.540590
$220$	0	0
$221$	2.00000	0.134535
$222$	0	0
$223$	−19.0000	−1.27233	−0.636167	−	0.771551i	$-0.719481\pi$
$223$	−19.0000	−1.27233	−0.636167	+	0.771551i	$0.719481\pi$
$224$	0	0
$225$	2.00000	0.133333
$226$	0	0
$227$	−11.0000	−0.730096	−0.365048	−	0.930989i	$-0.618947\pi$
$227$	−11.0000	−0.730096	−0.365048	+	0.930989i	$0.618947\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	−20.0000	−1.31590
$232$	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
$237$	−13.0000	−0.844441
$238$	0	0
$239$	−27.0000	−1.74648	−0.873242	−	0.487286i	$-0.837987\pi$
$239$	−27.0000	−1.74648	−0.873242	+	0.487286i	$0.837987\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	18.0000	1.14998
$246$	0	0
$247$	−1.00000	−0.0636285
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	8.00000	0.504956	0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000	0.504956	0.252478	+	0.967603i	$0.418755\pi$
$252$	0	0
$253$	15.0000	0.943042
$254$	0	0
$255$	−4.00000	−0.250490
$256$	0	0
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	0	0
$259$	28.0000	1.73984
$260$	0	0
$261$	20.0000	1.23797
$262$	0	0
$263$	−11.0000	−0.678289	−0.339145	−	0.940734i	$-0.610138\pi$
$263$	−11.0000	−0.678289	−0.339145	+	0.940734i	$0.610138\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	3.00000	0.183597
$268$	0	0
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	5.00000	0.301511
$276$	0	0
$277$	−32.0000	−1.92269	−0.961347	−	0.275340i	$-0.911209\pi$
$277$	−32.0000	−1.92269	−0.961347	+	0.275340i	$0.911209\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	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
$285$	2.00000	0.118470
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	12.0000	0.703452
$292$	0	0
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	0	0
$295$	16.0000	0.931556
$296$	0	0
$297$	−25.0000	−1.45065
$298$	0	0
$299$	−3.00000	−0.173494
$300$	0	0
$301$	24.0000	1.38334
$302$	0	0
$303$	−19.0000	−1.09152
$304$	0	0
$305$	14.0000	0.801638
$306$	0	0
$307$	26.0000	1.48390	0.741949	−	0.670456i	$-0.233902\pi$
$307$	26.0000	1.48390	0.741949	+	0.670456i	$0.233902\pi$
$308$	0	0
$309$	12.0000	0.682656
$310$	0	0
$311$	20.0000	1.13410	0.567048	−	0.823685i	$-0.308085\pi$
$311$	20.0000	1.13410	0.567048	+	0.823685i	$0.308085\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	16.0000	0.901498
$316$	0	0
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	50.0000	2.79946
$320$	0	0
$321$	−1.00000	−0.0558146
$322$	0	0
$323$	−2.00000	−0.111283
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−6.00000	−0.331801
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−36.0000	−1.97874	−0.989369	−	0.145424i	$-0.953545\pi$
$331$	−36.0000	−1.97874	−0.989369	+	0.145424i	$0.953545\pi$
$332$	0	0
$333$	14.0000	0.767195
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	−15.0000	−0.817102	−0.408551	−	0.912735i	$-0.633966\pi$
$337$	−15.0000	−0.817102	−0.408551	+	0.912735i	$0.633966\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	0	0
$341$	−20.0000	−1.08306
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	6.00000	0.323029
$346$	0	0
$347$	10.0000	0.536828	0.268414	−	0.963304i	$-0.413500\pi$
$347$	10.0000	0.536828	0.268414	+	0.963304i	$0.413500\pi$
$348$	0	0
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	0	0
$357$	8.00000	0.423405
$358$	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
$361$	−18.0000	−0.947368
$362$	0	0
$363$	−14.0000	−0.734809
$364$	0	0
$365$	−16.0000	−0.837478
$366$	0	0
$367$	−20.0000	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$367$	−20.0000	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−4.00000	−0.207670
$372$	0	0
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	0	0
$377$	−10.0000	−0.515026
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	20.0000	1.02463
$382$	0	0
$383$	29.0000	1.48183	0.740915	−	0.671598i	$-0.234392\pi$
$383$	29.0000	1.48183	0.740915	+	0.671598i	$0.234392\pi$
$384$	0	0
$385$	40.0000	2.03859
$386$	0	0
$387$	12.0000	0.609994
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	0	0
$393$	−6.00000	−0.302660
$394$	0	0
$395$	26.0000	1.30820
$396$	0	0
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	2.00000	0.0993808
$406$	0	0
$407$	35.0000	1.73489
$408$	0	0
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	0	0
$413$	−32.0000	−1.57462
$414$	0	0
$415$	24.0000	1.17811
$416$	0	0
$417$	18.0000	0.881464
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	−7.00000	−0.341159	−0.170580	−	0.985344i	$-0.554564\pi$
$421$	−7.00000	−0.341159	−0.170580	+	0.985344i	$0.554564\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	−28.0000	−1.35501
$428$	0	0
$429$	5.00000	0.241402
$430$	0	0
$431$	20.0000	0.963366	0.481683	−	0.876346i	$-0.340026\pi$
$431$	20.0000	0.963366	0.481683	+	0.876346i	$0.340026\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	20.0000	0.958927
$436$	0	0
$437$	3.00000	0.143509
$438$	0	0
$439$	−13.0000	−0.620456	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	−13.0000	−0.620456	−0.310228	+	0.950662i	$0.600405\pi$
$440$	0	0
$441$	−18.0000	−0.857143
$442$	0	0
$443$	10.0000	0.475114	0.237557	−	0.971374i	$-0.423653\pi$
$443$	10.0000	0.475114	0.237557	+	0.971374i	$0.423653\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	0	0
$447$	−9.00000	−0.425685
$448$	0	0
$449$	−4.00000	−0.188772	−0.0943858	−	0.995536i	$-0.530089\pi$
$449$	−4.00000	−0.188772	−0.0943858	+	0.995536i	$0.530089\pi$
$450$	0	0
$451$	−15.0000	−0.706322
$452$	0	0
$453$	−1.00000	−0.0469841
$454$	0	0
$455$	−8.00000	−0.375046
$456$	0	0
$457$	−17.0000	−0.795226	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000	−0.795226	−0.397613	+	0.917553i	$0.630161\pi$
$458$	0	0
$459$	10.0000	0.466760
$460$	0	0
$461$	25.0000	1.16437	0.582183	−	0.813058i	$-0.302199\pi$
$461$	25.0000	1.16437	0.582183	+	0.813058i	$0.302199\pi$
$462$	0	0
$463$	35.0000	1.62659	0.813294	−	0.581853i	$-0.197672\pi$
$463$	35.0000	1.62659	0.813294	+	0.581853i	$0.197672\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	−5.00000	−0.231372	−0.115686	−	0.993286i	$-0.536907\pi$
$467$	−5.00000	−0.231372	−0.115686	+	0.993286i	$0.536907\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	16.0000	0.737241
$472$	0	0
$473$	30.0000	1.37940
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−7.00000	−0.319173
$482$	0	0
$483$	−12.0000	−0.546019
$484$	0	0
$485$	−24.0000	−1.08978
$486$	0	0
$487$	6.00000	0.271886	0.135943	−	0.990717i	$-0.456594\pi$
$487$	6.00000	0.271886	0.135943	+	0.990717i	$0.456594\pi$
$488$	0	0
$489$	11.0000	0.497437
$490$	0	0
$491$	16.0000	0.722070	0.361035	−	0.932552i	$-0.382424\pi$
$491$	16.0000	0.722070	0.361035	+	0.932552i	$0.382424\pi$
$492$	0	0
$493$	−20.0000	−0.900755
$494$	0	0
$495$	20.0000	0.898933
$496$	0	0
$497$	24.0000	1.07655
$498$	0	0
$499$	22.0000	0.984855	0.492428	−	0.870353i	$-0.336110\pi$
$499$	22.0000	0.984855	0.492428	+	0.870353i	$0.336110\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	0	0
$503$	−9.00000	−0.401290	−0.200645	−	0.979664i	$-0.564304\pi$
$503$	−9.00000	−0.401290	−0.200645	+	0.979664i	$0.564304\pi$
$504$	0	0
$505$	38.0000	1.69098
$506$	0	0
$507$	12.0000	0.532939
$508$	0	0
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	32.0000	1.41560
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	−24.0000	−1.05757
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−22.0000	−0.965693
$520$	0	0
$521$	−28.0000	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$522$	0	0
$523$	−10.0000	−0.437269	−0.218635	−	0.975807i	$-0.570160\pi$
$523$	−10.0000	−0.437269	−0.218635	+	0.975807i	$0.570160\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	8.00000	0.348485
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−16.0000	−0.694341
$532$	0	0
$533$	3.00000	0.129944
$534$	0	0
$535$	2.00000	0.0864675
$536$	0	0
$537$	−4.00000	−0.172613
$538$	0	0
$539$	−45.0000	−1.93829
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	0	0
$543$	−14.0000	−0.600798
$544$	0	0
$545$	12.0000	0.514024
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	−14.0000	−0.597505
$550$	0	0
$551$	10.0000	0.426014
$552$	0	0
$553$	−52.0000	−2.21126
$554$	0	0
$555$	14.0000	0.594267
$556$	0	0
$557$	42.0000	1.77960	0.889799	−	0.456354i	$-0.150845\pi$
$557$	42.0000	1.77960	0.889799	+	0.456354i	$0.150845\pi$
$558$	0	0
$559$	−6.00000	−0.253773
$560$	0	0
$561$	10.0000	0.422200
$562$	0	0
$563$	42.0000	1.77009	0.885044	−	0.465506i	$-0.154128\pi$
$563$	42.0000	1.77009	0.885044	+	0.465506i	$0.154128\pi$
$564$	0	0
$565$	−24.0000	−1.00969
$566$	0	0
$567$	−4.00000	−0.167984
$568$	0	0
$569$	−13.0000	−0.544988	−0.272494	−	0.962157i	$-0.587849\pi$
$569$	−13.0000	−0.544988	−0.272494	+	0.962157i	$0.587849\pi$
$570$	0	0
$571$	−27.0000	−1.12991	−0.564957	−	0.825120i	$-0.691107\pi$
$571$	−27.0000	−1.12991	−0.564957	+	0.825120i	$0.691107\pi$
$572$	0	0
$573$	−24.0000	−1.00261
$574$	0	0
$575$	3.00000	0.125109
$576$	0	0
$577$	−21.0000	−0.874241	−0.437121	−	0.899403i	$-0.644002\pi$
$577$	−21.0000	−0.874241	−0.437121	+	0.899403i	$0.644002\pi$
$578$	0	0
$579$	−17.0000	−0.706496
$580$	0	0
$581$	−48.0000	−1.99138
$582$	0	0
$583$	−5.00000	−0.207079
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	13.0000	0.536567	0.268284	−	0.963340i	$-0.413544\pi$
$587$	13.0000	0.536567	0.268284	+	0.963340i	$0.413544\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	13.0000	0.534749
$592$	0	0
$593$	−10.0000	−0.410651	−0.205325	−	0.978694i	$-0.565825\pi$
$593$	−10.0000	−0.410651	−0.205325	+	0.978694i	$0.565825\pi$
$594$	0	0
$595$	−16.0000	−0.655936
$596$	0	0
$597$	5.00000	0.204636
$598$	0	0
$599$	15.0000	0.612883	0.306442	−	0.951889i	$-0.400862\pi$
$599$	15.0000	0.612883	0.306442	+	0.951889i	$0.400862\pi$
$600$	0	0
$601$	−4.00000	−0.163163	−0.0815817	−	0.996667i	$-0.525997\pi$
$601$	−4.00000	−0.163163	−0.0815817	+	0.996667i	$0.525997\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	28.0000	1.13836
$606$	0	0
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	0	0
$609$	−40.0000	−1.62088
$610$	0	0
$611$	0	0
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	0	0
$617$	−28.0000	−1.12724	−0.563619	−	0.826035i	$-0.690591\pi$
$617$	−28.0000	−1.12724	−0.563619	+	0.826035i	$0.690591\pi$
$618$	0	0
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	−15.0000	−0.601929
$622$	0	0
$623$	12.0000	0.480770
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	−5.00000	−0.199681
$628$	0	0
$629$	−14.0000	−0.558217
$630$	0	0
$631$	4.00000	0.159237	0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000	0.159237	0.0796187	+	0.996825i	$0.474630\pi$
$632$	0	0
$633$	20.0000	0.794929
$634$	0	0
$635$	−40.0000	−1.58735
$636$	0	0
$637$	9.00000	0.356593
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	50.0000	1.97488	0.987441	−	0.157991i	$-0.0505015\pi$
$641$	50.0000	1.97488	0.987441	+	0.157991i	$0.0505015\pi$
$642$	0	0
$643$	9.00000	0.354925	0.177463	−	0.984128i	$-0.443211\pi$
$643$	9.00000	0.354925	0.177463	+	0.984128i	$0.443211\pi$
$644$	0	0
$645$	12.0000	0.472500
$646$	0	0
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	0	0
$649$	−40.0000	−1.57014
$650$	0	0
$651$	16.0000	0.627089
$652$	0	0
$653$	−26.0000	−1.01746	−0.508729	−	0.860927i	$-0.669885\pi$
$653$	−26.0000	−1.01746	−0.508729	+	0.860927i	$0.669885\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	0	0
$657$	16.0000	0.624219
$658$	0	0
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	0	0
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	0	0
$663$	−2.00000	−0.0776736
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	30.0000	1.16160
$668$	0	0
$669$	19.0000	0.734582
$670$	0	0
$671$	−35.0000	−1.35116
$672$	0	0
$673$	16.0000	0.616755	0.308377	−	0.951264i	$-0.400214\pi$
$673$	16.0000	0.616755	0.308377	+	0.951264i	$0.400214\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	−38.0000	−1.46046	−0.730229	−	0.683202i	$-0.760587\pi$
$677$	−38.0000	−1.46046	−0.730229	+	0.683202i	$0.760587\pi$
$678$	0	0
$679$	48.0000	1.84207
$680$	0	0
$681$	11.0000	0.421521
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	4.00000	0.152832
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.00000	0.0380970
$690$	0	0
$691$	−45.0000	−1.71188	−0.855940	−	0.517075i	$-0.827021\pi$
$691$	−45.0000	−1.71188	−0.855940	+	0.517075i	$0.827021\pi$
$692$	0	0
$693$	−40.0000	−1.51947
$694$	0	0
$695$	−36.0000	−1.36556
$696$	0	0
$697$	6.00000	0.227266
$698$	0	0
$699$	9.00000	0.340411
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	7.00000	0.264010
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−76.0000	−2.85827
$708$	0	0
$709$	44.0000	1.65245	0.826227	−	0.563337i	$-0.190483\pi$
$709$	44.0000	1.65245	0.826227	+	0.563337i	$0.190483\pi$
$710$	0	0
$711$	−26.0000	−0.975076
$712$	0	0
$713$	−12.0000	−0.449404
$714$	0	0
$715$	−10.0000	−0.373979
$716$	0	0
$717$	27.0000	1.00833
$718$	0	0
$719$	4.00000	0.149175	0.0745874	−	0.997214i	$-0.476236\pi$
$719$	4.00000	0.149175	0.0745874	+	0.997214i	$0.476236\pi$
$720$	0	0
$721$	48.0000	1.78761
$722$	0	0
$723$	26.0000	0.966950
$724$	0	0
$725$	10.0000	0.371391
$726$	0	0
$727$	−7.00000	−0.259616	−0.129808	−	0.991539i	$-0.541436\pi$
$727$	−7.00000	−0.259616	−0.129808	+	0.991539i	$0.541436\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−12.0000	−0.443836
$732$	0	0
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	0	0
$735$	−18.0000	−0.663940
$736$	0	0
$737$	−10.0000	−0.368355
$738$	0	0
$739$	−22.0000	−0.809283	−0.404642	−	0.914475i	$-0.632604\pi$
$739$	−22.0000	−0.809283	−0.404642	+	0.914475i	$0.632604\pi$
$740$	0	0
$741$	1.00000	0.0367359
$742$	0	0
$743$	−41.0000	−1.50414	−0.752072	−	0.659081i	$-0.770945\pi$
$743$	−41.0000	−1.50414	−0.752072	+	0.659081i	$0.770945\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	0	0
$747$	−24.0000	−0.878114
$748$	0	0
$749$	−4.00000	−0.146157
$750$	0	0
$751$	38.0000	1.38664	0.693320	−	0.720630i	$-0.256147\pi$
$751$	38.0000	1.38664	0.693320	+	0.720630i	$0.256147\pi$
$752$	0	0
$753$	−8.00000	−0.291536
$754$	0	0
$755$	2.00000	0.0727875
$756$	0	0
$757$	18.0000	0.654221	0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000	0.654221	0.327111	+	0.944986i	$0.393925\pi$
$758$	0	0
$759$	−15.0000	−0.544466
$760$	0	0
$761$	47.0000	1.70375	0.851874	−	0.523746i	$-0.175466\pi$
$761$	47.0000	1.70375	0.851874	+	0.523746i	$0.175466\pi$
$762$	0	0
$763$	−24.0000	−0.868858
$764$	0	0
$765$	−8.00000	−0.289241
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	−12.0000	−0.432731	−0.216366	−	0.976312i	$-0.569420\pi$
$769$	−12.0000	−0.432731	−0.216366	+	0.976312i	$0.569420\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	0	0
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	−28.0000	−1.00449
$778$	0	0
$779$	−3.00000	−0.107486
$780$	0	0
$781$	30.0000	1.07348
$782$	0	0
$783$	−50.0000	−1.78685
$784$	0	0
$785$	−32.0000	−1.14213
$786$	0	0
$787$	6.00000	0.213877	0.106938	−	0.994266i	$-0.465895\pi$
$787$	6.00000	0.213877	0.106938	+	0.994266i	$0.465895\pi$
$788$	0	0
$789$	11.0000	0.391610
$790$	0	0
$791$	48.0000	1.70668
$792$	0	0
$793$	7.00000	0.248577
$794$	0	0
$795$	−2.00000	−0.0709327
$796$	0	0
$797$	−17.0000	−0.602171	−0.301085	−	0.953597i	$-0.597349\pi$
$797$	−17.0000	−0.602171	−0.301085	+	0.953597i	$0.597349\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	40.0000	1.41157
$804$	0	0
$805$	24.0000	0.845889
$806$	0	0
$807$	12.0000	0.422420
$808$	0	0
$809$	−22.0000	−0.773479	−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000	−0.773479	−0.386739	+	0.922189i	$0.626399\pi$
$810$	0	0
$811$	−7.00000	−0.245803	−0.122902	−	0.992419i	$-0.539220\pi$
$811$	−7.00000	−0.245803	−0.122902	+	0.992419i	$0.539220\pi$
$812$	0	0
$813$	−24.0000	−0.841717
$814$	0	0
$815$	−22.0000	−0.770626
$816$	0	0
$817$	6.00000	0.209913
$818$	0	0
$819$	8.00000	0.279543
$820$	0	0
$821$	−32.0000	−1.11681	−0.558404	−	0.829569i	$-0.688586\pi$
$821$	−32.0000	−1.11681	−0.558404	+	0.829569i	$0.688586\pi$
$822$	0	0
$823$	−52.0000	−1.81261	−0.906303	−	0.422628i	$-0.861108\pi$
$823$	−52.0000	−1.81261	−0.906303	+	0.422628i	$0.861108\pi$
$824$	0	0
$825$	−5.00000	−0.174078
$826$	0	0
$827$	18.0000	0.625921	0.312961	−	0.949766i	$-0.398679\pi$
$827$	18.0000	0.625921	0.312961	+	0.949766i	$0.398679\pi$
$828$	0	0
$829$	52.0000	1.80603	0.903017	−	0.429604i	$-0.141347\pi$
$829$	52.0000	1.80603	0.903017	+	0.429604i	$0.141347\pi$
$830$	0	0
$831$	32.0000	1.11007
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	16.0000	0.553703
$836$	0	0
$837$	20.0000	0.691301
$838$	0	0
$839$	−8.00000	−0.276191	−0.138095	−	0.990419i	$-0.544098\pi$
$839$	−8.00000	−0.276191	−0.138095	+	0.990419i	$0.544098\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	0	0
$843$	18.0000	0.619953
$844$	0	0
$845$	−24.0000	−0.825625
$846$	0	0
$847$	−56.0000	−1.92418
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	21.0000	0.719871
$852$	0	0
$853$	−20.0000	−0.684787	−0.342393	−	0.939557i	$-0.611238\pi$
$853$	−20.0000	−0.684787	−0.342393	+	0.939557i	$0.611238\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	57.0000	1.94708	0.973541	−	0.228510i	$-0.0733855\pi$
$857$	57.0000	1.94708	0.973541	+	0.228510i	$0.0733855\pi$
$858$	0	0
$859$	9.00000	0.307076	0.153538	−	0.988143i	$-0.450933\pi$
$859$	9.00000	0.307076	0.153538	+	0.988143i	$0.450933\pi$
$860$	0	0
$861$	12.0000	0.408959
$862$	0	0
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	0	0
$865$	44.0000	1.49604
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	−65.0000	−2.20497
$870$	0	0
$871$	2.00000	0.0677674
$872$	0	0
$873$	24.0000	0.812277
$874$	0	0
$875$	48.0000	1.62270
$876$	0	0
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	0	0
$879$	9.00000	0.303562
$880$	0	0
$881$	27.0000	0.909653	0.454827	−	0.890580i	$-0.349701\pi$
$881$	27.0000	0.909653	0.454827	+	0.890580i	$0.349701\pi$
$882$	0	0
$883$	−31.0000	−1.04323	−0.521617	−	0.853180i	$-0.674671\pi$
$883$	−31.0000	−1.04323	−0.521617	+	0.853180i	$0.674671\pi$
$884$	0	0
$885$	−16.0000	−0.537834
$886$	0	0
$887$	22.0000	0.738688	0.369344	−	0.929293i	$-0.379582\pi$
$887$	22.0000	0.738688	0.369344	+	0.929293i	$0.379582\pi$
$888$	0	0
$889$	80.0000	2.68311
$890$	0	0
$891$	−5.00000	−0.167506
$892$	0	0
$893$	0	0
$894$	0	0
$895$	8.00000	0.267411
$896$	0	0
$897$	3.00000	0.100167
$898$	0	0
$899$	−40.0000	−1.33407
$900$	0	0
$901$	2.00000	0.0666297
$902$	0	0
$903$	−24.0000	−0.798670
$904$	0	0
$905$	28.0000	0.930751
$906$	0	0
$907$	46.0000	1.52740	0.763702	−	0.645568i	$-0.223379\pi$
$907$	46.0000	1.52740	0.763702	+	0.645568i	$0.223379\pi$
$908$	0	0
$909$	−38.0000	−1.26038
$910$	0	0
$911$	2.00000	0.0662630	0.0331315	−	0.999451i	$-0.489452\pi$
$911$	2.00000	0.0662630	0.0331315	+	0.999451i	$0.489452\pi$
$912$	0	0
$913$	−60.0000	−1.98571
$914$	0	0
$915$	−14.0000	−0.462826
$916$	0	0
$917$	−24.0000	−0.792550
$918$	0	0
$919$	−28.0000	−0.923635	−0.461817	−	0.886975i	$-0.652802\pi$
$919$	−28.0000	−0.923635	−0.461817	+	0.886975i	$0.652802\pi$
$920$	0	0
$921$	−26.0000	−0.856729
$922$	0	0
$923$	−6.00000	−0.197492
$924$	0	0
$925$	7.00000	0.230159
$926$	0	0
$927$	24.0000	0.788263
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−9.00000	−0.294963
$932$	0	0
$933$	−20.0000	−0.654771
$934$	0	0
$935$	−20.0000	−0.654070
$936$	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
$939$	−10.0000	−0.326338
$940$	0	0
$941$	26.0000	0.847576	0.423788	−	0.905761i	$-0.360700\pi$
$941$	26.0000	0.847576	0.423788	+	0.905761i	$0.360700\pi$
$942$	0	0
$943$	−9.00000	−0.293080
$944$	0	0
$945$	−40.0000	−1.30120
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	−8.00000	−0.259691
$950$	0	0
$951$	6.00000	0.194563
$952$	0	0
$953$	−8.00000	−0.259145	−0.129573	−	0.991570i	$-0.541361\pi$
$953$	−8.00000	−0.259145	−0.129573	+	0.991570i	$0.541361\pi$
$954$	0	0
$955$	48.0000	1.55324
$956$	0	0
$957$	−50.0000	−1.61627
$958$	0	0
$959$	−8.00000	−0.258333
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−2.00000	−0.0644491
$964$	0	0
$965$	34.0000	1.09450
$966$	0	0
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	0	0
$969$	2.00000	0.0642493
$970$	0	0
$971$	30.0000	0.962746	0.481373	−	0.876516i	$-0.340138\pi$
$971$	30.0000	0.962746	0.481373	+	0.876516i	$0.340138\pi$
$972$	0	0
$973$	72.0000	2.30821
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	−23.0000	−0.735835	−0.367918	−	0.929858i	$-0.619929\pi$
$977$	−23.0000	−0.735835	−0.367918	+	0.929858i	$0.619929\pi$
$978$	0	0
$979$	15.0000	0.479402
$980$	0	0
$981$	−12.0000	−0.383131
$982$	0	0
$983$	12.0000	0.382741	0.191370	−	0.981518i	$-0.438707\pi$
$983$	12.0000	0.382741	0.191370	+	0.981518i	$0.438707\pi$
$984$	0	0
$985$	−26.0000	−0.828429
$986$	0	0
$987$	0	0
$988$	0	0
$989$	18.0000	0.572367
$990$	0	0
$991$	−46.0000	−1.46124	−0.730619	−	0.682785i	$-0.760768\pi$
$991$	−46.0000	−1.46124	−0.730619	+	0.682785i	$0.760768\pi$
$992$	0	0
$993$	36.0000	1.14243
$994$	0	0
$995$	−10.0000	−0.317021
$996$	0	0
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	0	0
$999$	−35.0000	−1.10735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	428.2.a.a.1.1	✓	1
3.2	odd	2		3852.2.a.c.1.1		1
4.3	odd	2		1712.2.a.h.1.1		1
8.3	odd	2		6848.2.a.e.1.1		1
8.5	even	2		6848.2.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
428.2.a.a.1.1	✓	1	1.1	even	1	trivial
1712.2.a.h.1.1		1	4.3	odd	2
3852.2.a.c.1.1		1	3.2	odd	2
6848.2.a.e.1.1		1	8.3	odd	2
6848.2.a.o.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:	\( N \)	\(=\)	\( 428 = 2^{2} \cdot 107 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	428.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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