LMFDB - Embedded newform 1932.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1932,2,Mod(1,1932)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1932, 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("1932.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1932.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:	$15.4270976705$
Analytic rank:	$0$
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	$\chi$	$=$	1932.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

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

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
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	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
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−5.00000	−1.00000
$26$	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
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	0	0
$33$	5.00000	0.870388
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	6.00000	0.960769
$40$	0	0
$41$	−7.00000	−1.09322	−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000	−1.09322	−0.546608	+	0.837389i	$0.684081\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	11.0000	1.51097	0.755483	−	0.655168i	$-0.227402\pi$
$53$	11.0000	1.51097	0.755483	+	0.655168i	$0.227402\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−7.00000	−0.927173
$58$	0	0
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	−11.0000	−1.40841	−0.704203	−	0.709999i	$-0.748695\pi$
$61$	−11.0000	−1.40841	−0.704203	+	0.709999i	$0.748695\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	0	0
$75$	5.00000	0.577350
$76$	0	0
$77$	5.00000	0.569803
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−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
$85$	0	0
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	0	0
$93$	−10.0000	−1.03695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	−5.00000	−0.502519
$100$	0	0
$101$	−19.0000	−1.89057	−0.945285	−	0.326245i	$-0.894217\pi$
$101$	−19.0000	−1.89057	−0.945285	+	0.326245i	$0.894217\pi$
$102$	0	0
$103$	−3.00000	−0.295599	−0.147799	−	0.989017i	$-0.547219\pi$
$103$	−3.00000	−0.295599	−0.147799	+	0.989017i	$0.547219\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−6.00000	−0.554700
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	7.00000	0.631169
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.0000	0.976092	0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000	0.976092	0.488046	+	0.872818i	$0.337710\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	1.00000	0.0873704	0.0436852	−	0.999045i	$-0.486090\pi$
$131$	1.00000	0.0873704	0.0436852	+	0.999045i	$0.486090\pi$
$132$	0	0
$133$	−7.00000	−0.606977
$134$	0	0
$135$	0	0
$136$	0	0
$137$	23.0000	1.96502	0.982511	−	0.186203i	$-0.0596182\pi$
$137$	23.0000	1.96502	0.982511	+	0.186203i	$0.0596182\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	−9.00000	−0.757937
$142$	0	0
$143$	30.0000	2.50873
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	1.00000	0.0819232	0.0409616	−	0.999161i	$-0.486958\pi$
$149$	1.00000	0.0819232	0.0409616	+	0.999161i	$0.486958\pi$
$150$	0	0
$151$	−17.0000	−1.38344	−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000	−1.38344	−0.691720	+	0.722166i	$0.743147\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.00000	0.0798087	0.0399043	−	0.999204i	$-0.487295\pi$
$157$	1.00000	0.0798087	0.0399043	+	0.999204i	$0.487295\pi$
$158$	0	0
$159$	−11.0000	−0.872357
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	19.0000	1.48819	0.744097	−	0.668071i	$-0.232880\pi$
$163$	19.0000	1.48819	0.744097	+	0.668071i	$0.232880\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.00000	−0.232147	−0.116073	−	0.993241i	$-0.537031\pi$
$167$	−3.00000	−0.232147	−0.116073	+	0.993241i	$0.537031\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	7.00000	0.535303
$172$	0	0
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	5.00000	0.377964
$176$	0	0
$177$	−3.00000	−0.225494
$178$	0	0
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	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
$183$	11.0000	0.813143
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−20.0000	−1.46254
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	5.00000	0.361787	0.180894	−	0.983503i	$-0.442101\pi$
$191$	5.00000	0.361787	0.180894	+	0.983503i	$0.442101\pi$
$192$	0	0
$193$	13.0000	0.935760	0.467880	−	0.883792i	$-0.345018\pi$
$193$	13.0000	0.935760	0.467880	+	0.883792i	$0.345018\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	11.0000	0.779769	0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000	0.779769	0.389885	+	0.920864i	$0.372515\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−35.0000	−2.42100
$210$	0	0
$211$	−21.0000	−1.44570	−0.722850	−	0.691005i	$-0.757168\pi$
$211$	−21.0000	−1.44570	−0.722850	+	0.691005i	$0.757168\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−10.0000	−0.678844
$218$	0	0
$219$	−8.00000	−0.540590
$220$	0	0
$221$	−24.0000	−1.61441
$222$	0	0
$223$	18.0000	1.20537	0.602685	−	0.797980i	$-0.294098\pi$
$223$	18.0000	1.20537	0.602685	+	0.797980i	$0.294098\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	0	0
$227$	−22.0000	−1.46019	−0.730096	−	0.683345i	$-0.760525\pi$
$227$	−22.0000	−1.46019	−0.730096	+	0.683345i	$0.760525\pi$
$228$	0	0
$229$	−19.0000	−1.25556	−0.627778	−	0.778393i	$-0.716035\pi$
$229$	−19.0000	−1.25556	−0.627778	+	0.778393i	$0.716035\pi$
$230$	0	0
$231$	−5.00000	−0.328976
$232$	0	0
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−10.0000	−0.649570
$238$	0	0
$239$	−4.00000	−0.258738	−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000	−0.258738	−0.129369	+	0.991596i	$0.541295\pi$
$240$	0	0
$241$	11.0000	0.708572	0.354286	−	0.935137i	$-0.384724\pi$
$241$	11.0000	0.708572	0.354286	+	0.935137i	$0.384724\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−42.0000	−2.67240
$248$	0	0
$249$	14.0000	0.887214
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	5.00000	0.314347
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	−21.0000	−1.29492	−0.647458	−	0.762101i	$-0.724168\pi$
$263$	−21.0000	−1.29492	−0.647458	+	0.762101i	$0.724168\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	10.0000	0.611990
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	−6.00000	−0.363137
$274$	0	0
$275$	25.0000	1.50756
$276$	0	0
$277$	11.0000	0.660926	0.330463	−	0.943819i	$-0.392795\pi$
$277$	11.0000	0.660926	0.330463	+	0.943819i	$0.392795\pi$
$278$	0	0
$279$	10.0000	0.598684
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.00000	0.413197
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	32.0000	1.86946	0.934730	−	0.355359i	$-0.115641\pi$
$293$	32.0000	1.86946	0.934730	+	0.355359i	$0.115641\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.00000	0.290129
$298$	0	0
$299$	6.00000	0.346989
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	19.0000	1.09152
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	3.00000	0.170664
$310$	0	0
$311$	15.0000	0.850572	0.425286	−	0.905059i	$-0.360174\pi$
$311$	15.0000	0.850572	0.425286	+	0.905059i	$0.360174\pi$
$312$	0	0
$313$	21.0000	1.18699	0.593495	−	0.804838i	$-0.297748\pi$
$313$	21.0000	1.18699	0.593495	+	0.804838i	$0.297748\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	−10.0000	−0.559893
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	28.0000	1.55796
$324$	0	0
$325$	30.0000	1.66410
$326$	0	0
$327$	−4.00000	−0.221201
$328$	0	0
$329$	−9.00000	−0.496186
$330$	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
$333$	−2.00000	−0.109599
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	−18.0000	−0.977626
$340$	0	0
$341$	−50.0000	−2.70765
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	0	0
$353$	−22.0000	−1.17094	−0.585471	−	0.810693i	$-0.699090\pi$
$353$	−22.0000	−1.17094	−0.585471	+	0.810693i	$0.699090\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	0	0
$363$	−14.0000	−0.734809
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11.0000	0.574195	0.287098	−	0.957901i	$-0.407310\pi$
$367$	11.0000	0.574195	0.287098	+	0.957901i	$0.407310\pi$
$368$	0	0
$369$	−7.00000	−0.364405
$370$	0	0
$371$	−11.0000	−0.571092
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	2.00000	0.102733	0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000	0.102733	0.0513665	+	0.998680i	$0.483642\pi$
$380$	0	0
$381$	−11.0000	−0.563547
$382$	0	0
$383$	−26.0000	−1.32854	−0.664269	−	0.747494i	$-0.731257\pi$
$383$	−26.0000	−1.32854	−0.664269	+	0.747494i	$0.731257\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.00000	0.406663
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	−1.00000	−0.0504433
$394$	0	0
$395$	0	0
$396$	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
$399$	7.00000	0.350438
$400$	0	0
$401$	−11.0000	−0.549314	−0.274657	−	0.961542i	$-0.588564\pi$
$401$	−11.0000	−0.549314	−0.274657	+	0.961542i	$0.588564\pi$
$402$	0	0
$403$	−60.0000	−2.98881
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.0000	0.495682
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	−23.0000	−1.13451
$412$	0	0
$413$	−3.00000	−0.147620
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−16.0000	−0.783523
$418$	0	0
$419$	−14.0000	−0.683945	−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000	−0.683945	−0.341972	+	0.939710i	$0.611095\pi$
$420$	0	0
$421$	−8.00000	−0.389896	−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000	−0.389896	−0.194948	+	0.980814i	$0.562454\pi$
$422$	0	0
$423$	9.00000	0.437595
$424$	0	0
$425$	−20.0000	−0.970143
$426$	0	0
$427$	11.0000	0.532327
$428$	0	0
$429$	−30.0000	−1.44841
$430$	0	0
$431$	7.00000	0.337178	0.168589	−	0.985686i	$-0.446079\pi$
$431$	7.00000	0.337178	0.168589	+	0.985686i	$0.446079\pi$
$432$	0	0
$433$	−41.0000	−1.97033	−0.985167	−	0.171598i	$-0.945107\pi$
$433$	−41.0000	−1.97033	−0.985167	+	0.171598i	$0.945107\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.00000	−0.334855
$438$	0	0
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1.00000	−0.0472984
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	35.0000	1.64809
$452$	0	0
$453$	17.0000	0.798730
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	−1.00000	−0.0464739	−0.0232370	−	0.999730i	$-0.507397\pi$
$463$	−1.00000	−0.0464739	−0.0232370	+	0.999730i	$0.507397\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.0000	0.462745	0.231372	−	0.972865i	$-0.425678\pi$
$467$	10.0000	0.462745	0.231372	+	0.972865i	$0.425678\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	−1.00000	−0.0460776
$472$	0	0
$473$	−40.0000	−1.83920
$474$	0	0
$475$	−35.0000	−1.60591
$476$	0	0
$477$	11.0000	0.503655
$478$	0	0
$479$	−28.0000	−1.27935	−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000	−1.27935	−0.639676	+	0.768644i	$0.720932\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	0	0
$486$	0	0
$487$	28.0000	1.26880	0.634401	−	0.773004i	$-0.281247\pi$
$487$	28.0000	1.26880	0.634401	+	0.773004i	$0.281247\pi$
$488$	0	0
$489$	−19.0000	−0.859210
$490$	0	0
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	0	0
$493$	8.00000	0.360302
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	3.00000	0.134030
$502$	0	0
$503$	−4.00000	−0.178351	−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000	−0.178351	−0.0891756	+	0.996016i	$0.528423\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−23.0000	−1.02147
$508$	0	0
$509$	−3.00000	−0.132973	−0.0664863	−	0.997787i	$-0.521179\pi$
$509$	−3.00000	−0.132973	−0.0664863	+	0.997787i	$0.521179\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	0	0
$513$	−7.00000	−0.309058
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−45.0000	−1.97910
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	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
$525$	−5.00000	−0.218218
$526$	0	0
$527$	40.0000	1.74243
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	3.00000	0.130189
$532$	0	0
$533$	42.0000	1.81922
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−16.0000	−0.690451
$538$	0	0
$539$	−5.00000	−0.215365
$540$	0	0
$541$	−17.0000	−0.730887	−0.365444	−	0.930834i	$-0.619083\pi$
$541$	−17.0000	−0.730887	−0.365444	+	0.930834i	$0.619083\pi$
$542$	0	0
$543$	−14.0000	−0.600798
$544$	0	0
$545$	0	0
$546$	0	0
$547$	24.0000	1.02617	0.513083	−	0.858339i	$-0.328503\pi$
$547$	24.0000	1.02617	0.513083	+	0.858339i	$0.328503\pi$
$548$	0	0
$549$	−11.0000	−0.469469
$550$	0	0
$551$	14.0000	0.596420
$552$	0	0
$553$	−10.0000	−0.425243
$554$	0	0
$555$	0	0
$556$	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
$559$	−48.0000	−2.03018
$560$	0	0
$561$	20.0000	0.844401
$562$	0	0
$563$	8.00000	0.337160	0.168580	−	0.985688i	$-0.446082\pi$
$563$	8.00000	0.337160	0.168580	+	0.985688i	$0.446082\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	9.00000	0.377300	0.188650	−	0.982044i	$-0.439589\pi$
$569$	9.00000	0.377300	0.188650	+	0.982044i	$0.439589\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	0	0
$573$	−5.00000	−0.208878
$574$	0	0
$575$	5.00000	0.208514
$576$	0	0
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	0	0
$579$	−13.0000	−0.540262
$580$	0	0
$581$	14.0000	0.580818
$582$	0	0
$583$	−55.0000	−2.27787
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21.0000	0.866763	0.433381	−	0.901211i	$-0.357320\pi$
$587$	21.0000	0.866763	0.433381	+	0.901211i	$0.357320\pi$
$588$	0	0
$589$	70.0000	2.88430
$590$	0	0
$591$	−10.0000	−0.411345
$592$	0	0
$593$	38.0000	1.56047	0.780236	−	0.625485i	$-0.215099\pi$
$593$	38.0000	1.56047	0.780236	+	0.625485i	$0.215099\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−11.0000	−0.450200
$598$	0	0
$599$	−26.0000	−1.06233	−0.531166	−	0.847268i	$-0.678246\pi$
$599$	−26.0000	−1.06233	−0.531166	+	0.847268i	$0.678246\pi$
$600$	0	0
$601$	16.0000	0.652654	0.326327	−	0.945257i	$-0.394189\pi$
$601$	16.0000	0.652654	0.326327	+	0.945257i	$0.394189\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	0	0
$611$	−54.0000	−2.18461
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	10.0000	0.400642
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	35.0000	1.39777
$628$	0	0
$629$	−8.00000	−0.318981
$630$	0	0
$631$	−30.0000	−1.19428	−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000	−1.19428	−0.597141	+	0.802137i	$0.703697\pi$
$632$	0	0
$633$	21.0000	0.834675
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−6.00000	−0.237729
$638$	0	0
$639$	0	0
$640$	0	0
$641$	49.0000	1.93538	0.967692	−	0.252136i	$-0.0811330\pi$
$641$	49.0000	1.93538	0.967692	+	0.252136i	$0.0811330\pi$
$642$	0	0
$643$	−19.0000	−0.749287	−0.374643	−	0.927169i	$-0.622235\pi$
$643$	−19.0000	−0.749287	−0.374643	+	0.927169i	$0.622235\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	28.0000	1.10079	0.550397	−	0.834903i	$-0.314476\pi$
$647$	28.0000	1.10079	0.550397	+	0.834903i	$0.314476\pi$
$648$	0	0
$649$	−15.0000	−0.588802
$650$	0	0
$651$	10.0000	0.391931
$652$	0	0
$653$	−28.0000	−1.09572	−0.547862	−	0.836569i	$-0.684558\pi$
$653$	−28.0000	−1.09572	−0.547862	+	0.836569i	$0.684558\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8.00000	0.312110
$658$	0	0
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	0	0
$661$	−11.0000	−0.427850	−0.213925	−	0.976850i	$-0.568625\pi$
$661$	−11.0000	−0.427850	−0.213925	+	0.976850i	$0.568625\pi$
$662$	0	0
$663$	24.0000	0.932083
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.00000	−0.0774403
$668$	0	0
$669$	−18.0000	−0.695920
$670$	0	0
$671$	55.0000	2.12325
$672$	0	0
$673$	17.0000	0.655302	0.327651	−	0.944799i	$-0.393743\pi$
$673$	17.0000	0.655302	0.327651	+	0.944799i	$0.393743\pi$
$674$	0	0
$675$	5.00000	0.192450
$676$	0	0
$677$	28.0000	1.07613	0.538064	−	0.842904i	$-0.319156\pi$
$677$	28.0000	1.07613	0.538064	+	0.842904i	$0.319156\pi$
$678$	0	0
$679$	−10.0000	−0.383765
$680$	0	0
$681$	22.0000	0.843042
$682$	0	0
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	19.0000	0.724895
$688$	0	0
$689$	−66.0000	−2.51440
$690$	0	0
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	0	0
$693$	5.00000	0.189934
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−28.0000	−1.06058
$698$	0	0
$699$	24.0000	0.907763
$700$	0	0
$701$	−5.00000	−0.188847	−0.0944237	−	0.995532i	$-0.530101\pi$
$701$	−5.00000	−0.188847	−0.0944237	+	0.995532i	$0.530101\pi$
$702$	0	0
$703$	−14.0000	−0.528020
$704$	0	0
$705$	0	0
$706$	0	0
$707$	19.0000	0.714569
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	0	0
$713$	−10.0000	−0.374503
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.00000	0.149383
$718$	0	0
$719$	−40.0000	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$720$	0	0
$721$	3.00000	0.111726
$722$	0	0
$723$	−11.0000	−0.409094
$724$	0	0
$725$	−10.0000	−0.371391
$726$	0	0
$727$	−23.0000	−0.853023	−0.426511	−	0.904482i	$-0.640258\pi$
$727$	−23.0000	−0.853023	−0.426511	+	0.904482i	$0.640258\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	32.0000	1.18356
$732$	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
$735$	0	0
$736$	0	0
$737$	−40.0000	−1.47342
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	42.0000	1.54291
$742$	0	0
$743$	51.0000	1.87101	0.935504	−	0.353315i	$-0.114946\pi$
$743$	51.0000	1.87101	0.935504	+	0.353315i	$0.114946\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−14.0000	−0.512233
$748$	0	0
$749$	−4.00000	−0.146157
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	0	0
$753$	18.0000	0.655956
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−20.0000	−0.726912	−0.363456	−	0.931611i	$-0.618403\pi$
$757$	−20.0000	−0.726912	−0.363456	+	0.931611i	$0.618403\pi$
$758$	0	0
$759$	−5.00000	−0.181489
$760$	0	0
$761$	15.0000	0.543750	0.271875	−	0.962333i	$-0.412356\pi$
$761$	15.0000	0.543750	0.271875	+	0.962333i	$0.412356\pi$
$762$	0	0
$763$	−4.00000	−0.144810
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−18.0000	−0.649942
$768$	0	0
$769$	−38.0000	−1.37032	−0.685158	−	0.728395i	$-0.740267\pi$
$769$	−38.0000	−1.37032	−0.685158	+	0.728395i	$0.740267\pi$
$770$	0	0
$771$	3.00000	0.108042
$772$	0	0
$773$	4.00000	0.143870	0.0719350	−	0.997409i	$-0.477083\pi$
$773$	4.00000	0.143870	0.0719350	+	0.997409i	$0.477083\pi$
$774$	0	0
$775$	−50.0000	−1.79605
$776$	0	0
$777$	−2.00000	−0.0717496
$778$	0	0
$779$	−49.0000	−1.75561
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	0	0
$786$	0	0
$787$	45.0000	1.60408	0.802038	−	0.597272i	$-0.203749\pi$
$787$	45.0000	1.60408	0.802038	+	0.597272i	$0.203749\pi$
$788$	0	0
$789$	21.0000	0.747620
$790$	0	0
$791$	−18.0000	−0.640006
$792$	0	0
$793$	66.0000	2.34373
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	0	0
$799$	36.0000	1.27359
$800$	0	0
$801$	−10.0000	−0.353333
$802$	0	0
$803$	−40.0000	−1.41157
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.0000	0.352017
$808$	0	0
$809$	50.0000	1.75791	0.878953	−	0.476908i	$-0.158243\pi$
$809$	50.0000	1.75791	0.878953	+	0.476908i	$0.158243\pi$
$810$	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
$813$	−8.00000	−0.280572
$814$	0	0
$815$	0	0
$816$	0	0
$817$	56.0000	1.95919
$818$	0	0
$819$	6.00000	0.209657
$820$	0	0
$821$	−36.0000	−1.25641	−0.628204	−	0.778048i	$-0.716210\pi$
$821$	−36.0000	−1.25641	−0.628204	+	0.778048i	$0.716210\pi$
$822$	0	0
$823$	41.0000	1.42917	0.714585	−	0.699549i	$-0.246616\pi$
$823$	41.0000	1.42917	0.714585	+	0.699549i	$0.246616\pi$
$824$	0	0
$825$	−25.0000	−0.870388
$826$	0	0
$827$	19.0000	0.660695	0.330347	−	0.943859i	$-0.392834\pi$
$827$	19.0000	0.660695	0.330347	+	0.943859i	$0.392834\pi$
$828$	0	0
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	0	0
$831$	−11.0000	−0.381586
$832$	0	0
$833$	4.00000	0.138592
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−10.0000	−0.345651
$838$	0	0
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	10.0000	0.344418
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−14.0000	−0.481046
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	2.00000	0.0685591
$852$	0	0
$853$	−28.0000	−0.958702	−0.479351	−	0.877623i	$-0.659128\pi$
$853$	−28.0000	−0.958702	−0.479351	+	0.877623i	$0.659128\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	37.0000	1.26390	0.631948	−	0.775011i	$-0.282256\pi$
$857$	37.0000	1.26390	0.631948	+	0.775011i	$0.282256\pi$
$858$	0	0
$859$	−24.0000	−0.818869	−0.409435	−	0.912339i	$-0.634274\pi$
$859$	−24.0000	−0.818869	−0.409435	+	0.912339i	$0.634274\pi$
$860$	0	0
$861$	−7.00000	−0.238559
$862$	0	0
$863$	26.0000	0.885050	0.442525	−	0.896756i	$-0.354083\pi$
$863$	26.0000	0.885050	0.442525	+	0.896756i	$0.354083\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	−50.0000	−1.69613
$870$	0	0
$871$	−48.0000	−1.62642
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−5.00000	−0.168838	−0.0844190	−	0.996430i	$-0.526903\pi$
$877$	−5.00000	−0.168838	−0.0844190	+	0.996430i	$0.526903\pi$
$878$	0	0
$879$	−32.0000	−1.07933
$880$	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
$883$	48.0000	1.61533	0.807664	−	0.589643i	$-0.200731\pi$
$883$	48.0000	1.61533	0.807664	+	0.589643i	$0.200731\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−56.0000	−1.88030	−0.940148	−	0.340766i	$-0.889313\pi$
$887$	−56.0000	−1.88030	−0.940148	+	0.340766i	$0.889313\pi$
$888$	0	0
$889$	−11.0000	−0.368928
$890$	0	0
$891$	−5.00000	−0.167506
$892$	0	0
$893$	63.0000	2.10821
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−6.00000	−0.200334
$898$	0	0
$899$	20.0000	0.667037
$900$	0	0
$901$	44.0000	1.46585
$902$	0	0
$903$	8.00000	0.266223
$904$	0	0
$905$	0	0
$906$	0	0
$907$	40.0000	1.32818	0.664089	−	0.747653i	$-0.268820\pi$
$907$	40.0000	1.32818	0.664089	+	0.747653i	$0.268820\pi$
$908$	0	0
$909$	−19.0000	−0.630190
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	0	0
$913$	70.0000	2.31666
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.00000	−0.0330229
$918$	0	0
$919$	−42.0000	−1.38545	−0.692726	−	0.721201i	$-0.743591\pi$
$919$	−42.0000	−1.38545	−0.692726	+	0.721201i	$0.743591\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	10.0000	0.328798
$926$	0	0
$927$	−3.00000	−0.0985329
$928$	0	0
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	0	0
$931$	7.00000	0.229416
$932$	0	0
$933$	−15.0000	−0.491078
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.00000	0.228680	0.114340	−	0.993442i	$-0.463525\pi$
$937$	7.00000	0.228680	0.114340	+	0.993442i	$0.463525\pi$
$938$	0	0
$939$	−21.0000	−0.685309
$940$	0	0
$941$	40.0000	1.30396	0.651981	−	0.758235i	$-0.273938\pi$
$941$	40.0000	1.30396	0.651981	+	0.758235i	$0.273938\pi$
$942$	0	0
$943$	7.00000	0.227951
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	−48.0000	−1.55815
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	39.0000	1.26333	0.631667	−	0.775240i	$-0.282371\pi$
$953$	39.0000	1.26333	0.631667	+	0.775240i	$0.282371\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	10.0000	0.323254
$958$	0	0
$959$	−23.0000	−0.742709
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	0	0
$966$	0	0
$967$	12.0000	0.385894	0.192947	−	0.981209i	$-0.438195\pi$
$967$	12.0000	0.385894	0.192947	+	0.981209i	$0.438195\pi$
$968$	0	0
$969$	−28.0000	−0.899490
$970$	0	0
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	0	0
$975$	−30.0000	−0.960769
$976$	0	0
$977$	−37.0000	−1.18373	−0.591867	−	0.806035i	$-0.701609\pi$
$977$	−37.0000	−1.18373	−0.591867	+	0.806035i	$0.701609\pi$
$978$	0	0
$979$	50.0000	1.59801
$980$	0	0
$981$	4.00000	0.127710
$982$	0	0
$983$	−18.0000	−0.574111	−0.287055	−	0.957914i	$-0.592676\pi$
$983$	−18.0000	−0.574111	−0.287055	+	0.957914i	$0.592676\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	9.00000	0.286473
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	−41.0000	−1.30241	−0.651204	−	0.758903i	$-0.725736\pi$
$991$	−41.0000	−1.30241	−0.651204	+	0.758903i	$0.725736\pi$
$992$	0	0
$993$	5.00000	0.158670
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−8.00000	−0.253363	−0.126681	−	0.991943i	$-0.540433\pi$
$997$	−8.00000	−0.253363	−0.126681	+	0.991943i	$0.540433\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1932.2.a.a.1.1	✓	1
3.2	odd	2		5796.2.a.c.1.1		1
4.3	odd	2		7728.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.a.a.1.1	✓	1	1.1	even	1	trivial
5796.2.a.c.1.1		1	3.2	odd	2
7728.2.a.r.1.1		1	4.3	odd	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:	\( N \)	\(=\)	\( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1932.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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