LMFDB - Embedded newform 9920.2.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9920,2,Mod(1,9920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9920, 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("9920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9920 = 2^{6} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9920.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:	$79.2115988051$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 155)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	9920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+1.00000 q^{3} +1.00000 q^{5} -2.00000 q^{9} +4.00000 q^{11} +6.00000 q^{13} +1.00000 q^{15} +5.00000 q^{17} +1.00000 q^{19} +8.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} +10.0000 q^{29} -1.00000 q^{31} +4.00000 q^{33} -1.00000 q^{37} +6.00000 q^{39} -3.00000 q^{41} +7.00000 q^{43} -2.00000 q^{45} -6.00000 q^{47} -7.00000 q^{49} +5.00000 q^{51} -5.00000 q^{53} +4.00000 q^{55} +1.00000 q^{57} -11.0000 q^{59} +12.0000 q^{61} +6.00000 q^{65} +2.00000 q^{67} +8.00000 q^{69} +9.00000 q^{71} -9.00000 q^{73} +1.00000 q^{75} -10.0000 q^{79} +1.00000 q^{81} -9.00000 q^{83} +5.00000 q^{85} +10.0000 q^{87} -1.00000 q^{93} +1.00000 q^{95} -14.0000 q^{97} -8.00000 q^{99} +O(q^{100})$

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	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	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
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	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	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	4.00000	0.696311
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	0	0
$39$	6.00000	0.960769
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	5.00000	0.700140
$52$	0	0
$53$	−5.00000	−0.686803	−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000	−0.686803	−0.343401	+	0.939189i	$0.611579\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−11.0000	−1.43208	−0.716039	−	0.698060i	$-0.754047\pi$
$59$	−11.0000	−1.43208	−0.716039	+	0.698060i	$0.754047\pi$
$60$	0	0
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.00000	0.744208
$66$	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
$69$	8.00000	0.963087
$70$	0	0
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	0	0
$73$	−9.00000	−1.05337	−0.526685	−	0.850060i	$-0.676565\pi$
$73$	−9.00000	−1.05337	−0.526685	+	0.850060i	$0.676565\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	0	0
$85$	5.00000	0.542326
$86$	0	0
$87$	10.0000	1.07211
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	−8.00000	−0.804030
$100$	0	0
$101$	7.00000	0.696526	0.348263	−	0.937397i	$-0.386772\pi$
$101$	7.00000	0.696526	0.348263	+	0.937397i	$0.386772\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	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
$111$	−1.00000	−0.0949158
$112$	0	0
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	0	0
$115$	8.00000	0.746004
$116$	0	0
$117$	−12.0000	−1.10940
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−3.00000	−0.270501
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	7.00000	0.616316
$130$	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
$133$	0	0
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	24.0000	2.00698
$144$	0	0
$145$	10.0000	0.830455
$146$	0	0
$147$	−7.00000	−0.577350
$148$	0	0
$149$	−1.00000	−0.0819232	−0.0409616	−	0.999161i	$-0.513042\pi$
$149$	−1.00000	−0.0819232	−0.0409616	+	0.999161i	$0.513042\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	0	0
$153$	−10.0000	−0.808452
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	−5.00000	−0.396526
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−24.0000	−1.87983	−0.939913	−	0.341415i	$-0.889094\pi$
$163$	−24.0000	−1.87983	−0.939913	+	0.341415i	$0.889094\pi$
$164$	0	0
$165$	4.00000	0.311400
$166$	0	0
$167$	−15.0000	−1.16073	−0.580367	−	0.814355i	$-0.697091\pi$
$167$	−15.0000	−1.16073	−0.580367	+	0.814355i	$0.697091\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	−4.00000	−0.304114	−0.152057	−	0.988372i	$-0.548590\pi$
$173$	−4.00000	−0.304114	−0.152057	+	0.988372i	$0.548590\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−11.0000	−0.826811
$178$	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
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	0	0
$183$	12.0000	0.887066
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	20.0000	1.46254
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	6.00000	0.429669
$196$	0	0
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.00000	−0.209529
$206$	0	0
$207$	−16.0000	−1.11208
$208$	0	0
$209$	4.00000	0.276686
$210$	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
$213$	9.00000	0.616670
$214$	0	0
$215$	7.00000	0.477396
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−9.00000	−0.608164
$220$	0	0
$221$	30.0000	2.01802
$222$	0	0
$223$	−17.0000	−1.13840	−0.569202	−	0.822198i	$-0.692748\pi$
$223$	−17.0000	−1.13840	−0.569202	+	0.822198i	$0.692748\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	0	0
$227$	10.0000	0.663723	0.331862	−	0.943328i	$-0.392323\pi$
$227$	10.0000	0.663723	0.331862	+	0.943328i	$0.392323\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.0000	1.04819	0.524097	−	0.851658i	$-0.324403\pi$
$233$	16.0000	1.04819	0.524097	+	0.851658i	$0.324403\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	0	0
$237$	−10.0000	−0.649570
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	−7.00000	−0.447214
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	−9.00000	−0.570352
$250$	0	0
$251$	22.0000	1.38863	0.694314	−	0.719672i	$-0.255708\pi$
$251$	22.0000	1.38863	0.694314	+	0.719672i	$0.255708\pi$
$252$	0	0
$253$	32.0000	2.01182
$254$	0	0
$255$	5.00000	0.313112
$256$	0	0
$257$	10.0000	0.623783	0.311891	−	0.950118i	$-0.399037\pi$
$257$	10.0000	0.623783	0.311891	+	0.950118i	$0.399037\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−20.0000	−1.23797
$262$	0	0
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	0	0
$265$	−5.00000	−0.307148
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	−14.0000	−0.850439	−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000	−0.850439	−0.425220	+	0.905090i	$0.639803\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	−9.00000	−0.540758	−0.270379	−	0.962754i	$-0.587149\pi$
$277$	−9.00000	−0.540758	−0.270379	+	0.962754i	$0.587149\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	0	0
$281$	7.00000	0.417585	0.208792	−	0.977960i	$-0.433047\pi$
$281$	7.00000	0.417585	0.208792	+	0.977960i	$0.433047\pi$
$282$	0	0
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	0	0
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	−14.0000	−0.820695
$292$	0	0
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−11.0000	−0.640445
$296$	0	0
$297$	−20.0000	−1.16052
$298$	0	0
$299$	48.0000	2.77591
$300$	0	0
$301$	0	0
$302$	0	0
$303$	7.00000	0.402139
$304$	0	0
$305$	12.0000	0.687118
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−7.00000	−0.396934	−0.198467	−	0.980108i	$-0.563596\pi$
$311$	−7.00000	−0.396934	−0.198467	+	0.980108i	$0.563596\pi$
$312$	0	0
$313$	29.0000	1.63918	0.819588	−	0.572953i	$-0.194202\pi$
$313$	29.0000	1.63918	0.819588	+	0.572953i	$0.194202\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−32.0000	−1.79730	−0.898650	−	0.438667i	$-0.855451\pi$
$317$	−32.0000	−1.79730	−0.898650	+	0.438667i	$0.855451\pi$
$318$	0	0
$319$	40.0000	2.23957
$320$	0	0
$321$	2.00000	0.111629
$322$	0	0
$323$	5.00000	0.278207
$324$	0	0
$325$	6.00000	0.332820
$326$	0	0
$327$	−15.0000	−0.829502
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	2.00000	0.109272
$336$	0	0
$337$	−19.0000	−1.03500	−0.517498	−	0.855684i	$-0.673136\pi$
$337$	−19.0000	−1.03500	−0.517498	+	0.855684i	$0.673136\pi$
$338$	0	0
$339$	−4.00000	−0.217250
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	0	0
$344$	0	0
$345$	8.00000	0.430706
$346$	0	0
$347$	−8.00000	−0.429463	−0.214731	−	0.976673i	$-0.568888\pi$
$347$	−8.00000	−0.429463	−0.214731	+	0.976673i	$0.568888\pi$
$348$	0	0
$349$	29.0000	1.55233	0.776167	−	0.630527i	$-0.217161\pi$
$349$	29.0000	1.55233	0.776167	+	0.630527i	$0.217161\pi$
$350$	0	0
$351$	−30.0000	−1.60128
$352$	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
$355$	9.00000	0.477670
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	−9.00000	−0.471082
$366$	0	0
$367$	25.0000	1.30499	0.652495	−	0.757793i	$-0.273722\pi$
$367$	25.0000	1.30499	0.652495	+	0.757793i	$0.273722\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	0	0
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	60.0000	3.09016
$378$	0	0
$379$	5.00000	0.256833	0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000	0.256833	0.128416	+	0.991720i	$0.459011\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	0	0
$383$	1.00000	0.0510976	0.0255488	−	0.999674i	$-0.491867\pi$
$383$	1.00000	0.0510976	0.0255488	+	0.999674i	$0.491867\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−14.0000	−0.711660
$388$	0	0
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	40.0000	2.02289
$392$	0	0
$393$	1.00000	0.0504433
$394$	0	0
$395$	−10.0000	−0.503155
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−16.0000	−0.799002	−0.399501	−	0.916733i	$-0.630817\pi$
$401$	−16.0000	−0.799002	−0.399501	+	0.916733i	$0.630817\pi$
$402$	0	0
$403$	−6.00000	−0.298881
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−4.00000	−0.198273
$408$	0	0
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	0	0
$411$	3.00000	0.147979
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−9.00000	−0.441793
$416$	0	0
$417$	−2.00000	−0.0979404
$418$	0	0
$419$	15.0000	0.732798	0.366399	−	0.930458i	$-0.380591\pi$
$419$	15.0000	0.732798	0.366399	+	0.930458i	$0.380591\pi$
$420$	0	0
$421$	1.00000	0.0487370	0.0243685	−	0.999703i	$-0.492242\pi$
$421$	1.00000	0.0487370	0.0243685	+	0.999703i	$0.492242\pi$
$422$	0	0
$423$	12.0000	0.583460
$424$	0	0
$425$	5.00000	0.242536
$426$	0	0
$427$	0	0
$428$	0	0
$429$	24.0000	1.15873
$430$	0	0
$431$	−33.0000	−1.58955	−0.794777	−	0.606902i	$-0.792412\pi$
$431$	−33.0000	−1.58955	−0.794777	+	0.606902i	$0.792412\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	10.0000	0.479463
$436$	0	0
$437$	8.00000	0.382692
$438$	0	0
$439$	−35.0000	−1.67046	−0.835229	−	0.549902i	$-0.814665\pi$
$439$	−35.0000	−1.67046	−0.835229	+	0.549902i	$0.814665\pi$
$440$	0	0
$441$	14.0000	0.666667
$442$	0	0
$443$	34.0000	1.61539	0.807694	−	0.589601i	$-0.200715\pi$
$443$	34.0000	1.61539	0.807694	+	0.589601i	$0.200715\pi$
$444$	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
$451$	−12.0000	−0.565058
$452$	0	0
$453$	10.0000	0.469841
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	0	0
$459$	−25.0000	−1.16690
$460$	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
$463$	−12.0000	−0.557687	−0.278844	−	0.960337i	$-0.589951\pi$
$463$	−12.0000	−0.557687	−0.278844	+	0.960337i	$0.589951\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	28.0000	1.28744
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	10.0000	0.457869
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−14.0000	−0.635707
$486$	0	0
$487$	−19.0000	−0.860972	−0.430486	−	0.902597i	$-0.641658\pi$
$487$	−19.0000	−0.860972	−0.430486	+	0.902597i	$0.641658\pi$
$488$	0	0
$489$	−24.0000	−1.08532
$490$	0	0
$491$	4.00000	0.180517	0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000	0.180517	0.0902587	+	0.995918i	$0.471231\pi$
$492$	0	0
$493$	50.0000	2.25189
$494$	0	0
$495$	−8.00000	−0.359573
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−34.0000	−1.52205	−0.761025	−	0.648723i	$-0.775303\pi$
$499$	−34.0000	−1.52205	−0.761025	+	0.648723i	$0.775303\pi$
$500$	0	0
$501$	−15.0000	−0.670151
$502$	0	0
$503$	−22.0000	−0.980932	−0.490466	−	0.871460i	$-0.663173\pi$
$503$	−22.0000	−0.980932	−0.490466	+	0.871460i	$0.663173\pi$
$504$	0	0
$505$	7.00000	0.311496
$506$	0	0
$507$	23.0000	1.02147
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	−24.0000	−1.05552
$518$	0	0
$519$	−4.00000	−0.175581
$520$	0	0
$521$	19.0000	0.832405	0.416203	−	0.909272i	$-0.363361\pi$
$521$	19.0000	0.832405	0.416203	+	0.909272i	$0.363361\pi$
$522$	0	0
$523$	−11.0000	−0.480996	−0.240498	−	0.970650i	$-0.577311\pi$
$523$	−11.0000	−0.480996	−0.240498	+	0.970650i	$0.577311\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.00000	−0.217803
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	22.0000	0.954719
$532$	0	0
$533$	−18.0000	−0.779667
$534$	0	0
$535$	2.00000	0.0864675
$536$	0	0
$537$	4.00000	0.172613
$538$	0	0
$539$	−28.0000	−1.20605
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	0	0
$543$	18.0000	0.772454
$544$	0	0
$545$	−15.0000	−0.642529
$546$	0	0
$547$	−40.0000	−1.71028	−0.855138	−	0.518400i	$-0.826528\pi$
$547$	−40.0000	−1.71028	−0.855138	+	0.518400i	$0.826528\pi$
$548$	0	0
$549$	−24.0000	−1.02430
$550$	0	0
$551$	10.0000	0.426014
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.00000	−0.0424476
$556$	0	0
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	42.0000	1.77641
$560$	0	0
$561$	20.0000	0.844401
$562$	0	0
$563$	6.00000	0.252870	0.126435	−	0.991975i	$-0.459647\pi$
$563$	6.00000	0.252870	0.126435	+	0.991975i	$0.459647\pi$
$564$	0	0
$565$	−4.00000	−0.168281
$566$	0	0
$567$	0	0
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	26.0000	1.08807	0.544033	−	0.839064i	$-0.316897\pi$
$571$	26.0000	1.08807	0.544033	+	0.839064i	$0.316897\pi$
$572$	0	0
$573$	4.00000	0.167102
$574$	0	0
$575$	8.00000	0.333623
$576$	0	0
$577$	6.00000	0.249783	0.124892	−	0.992170i	$-0.460142\pi$
$577$	6.00000	0.249783	0.124892	+	0.992170i	$0.460142\pi$
$578$	0	0
$579$	14.0000	0.581820
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−20.0000	−0.828315
$584$	0	0
$585$	−12.0000	−0.496139
$586$	0	0
$587$	8.00000	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$587$	8.00000	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$588$	0	0
$589$	−1.00000	−0.0412043
$590$	0	0
$591$	22.0000	0.904959
$592$	0	0
$593$	−4.00000	−0.164260	−0.0821302	−	0.996622i	$-0.526172\pi$
$593$	−4.00000	−0.164260	−0.0821302	+	0.996622i	$0.526172\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.00000	−0.327418
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	5.00000	0.203279
$606$	0	0
$607$	−42.0000	−1.70473	−0.852364	−	0.522949i	$-0.824832\pi$
$607$	−42.0000	−1.70473	−0.852364	+	0.522949i	$0.824832\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−36.0000	−1.45640
$612$	0	0
$613$	5.00000	0.201948	0.100974	−	0.994889i	$-0.467804\pi$
$613$	5.00000	0.201948	0.100974	+	0.994889i	$0.467804\pi$
$614$	0	0
$615$	−3.00000	−0.120972
$616$	0	0
$617$	16.0000	0.644136	0.322068	−	0.946717i	$-0.395622\pi$
$617$	16.0000	0.644136	0.322068	+	0.946717i	$0.395622\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	−40.0000	−1.60514
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.00000	0.159745
$628$	0	0
$629$	−5.00000	−0.199363
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	−20.0000	−0.794929
$634$	0	0
$635$	−8.00000	−0.317470
$636$	0	0
$637$	−42.0000	−1.66410
$638$	0	0
$639$	−18.0000	−0.712069
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	−7.00000	−0.276053	−0.138027	−	0.990429i	$-0.544076\pi$
$643$	−7.00000	−0.276053	−0.138027	+	0.990429i	$0.544076\pi$
$644$	0	0
$645$	7.00000	0.275625
$646$	0	0
$647$	−21.0000	−0.825595	−0.412798	−	0.910823i	$-0.635448\pi$
$647$	−21.0000	−0.825595	−0.412798	+	0.910823i	$0.635448\pi$
$648$	0	0
$649$	−44.0000	−1.72715
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	1.00000	0.0390732
$656$	0	0
$657$	18.0000	0.702247
$658$	0	0
$659$	−39.0000	−1.51922	−0.759612	−	0.650376i	$-0.774611\pi$
$659$	−39.0000	−1.51922	−0.759612	+	0.650376i	$0.774611\pi$
$660$	0	0
$661$	−29.0000	−1.12797	−0.563985	−	0.825785i	$-0.690732\pi$
$661$	−29.0000	−1.12797	−0.563985	+	0.825785i	$0.690732\pi$
$662$	0	0
$663$	30.0000	1.16510
$664$	0	0
$665$	0	0
$666$	0	0
$667$	80.0000	3.09761
$668$	0	0
$669$	−17.0000	−0.657258
$670$	0	0
$671$	48.0000	1.85302
$672$	0	0
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	−27.0000	−1.03769	−0.518847	−	0.854867i	$-0.673639\pi$
$677$	−27.0000	−1.03769	−0.518847	+	0.854867i	$0.673639\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	10.0000	0.383201
$682$	0	0
$683$	−48.0000	−1.83667	−0.918334	−	0.395805i	$-0.870466\pi$
$683$	−48.0000	−1.83667	−0.918334	+	0.395805i	$0.870466\pi$
$684$	0	0
$685$	3.00000	0.114624
$686$	0	0
$687$	14.0000	0.534133
$688$	0	0
$689$	−30.0000	−1.14291
$690$	0	0
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.00000	−0.0758643
$696$	0	0
$697$	−15.0000	−0.568166
$698$	0	0
$699$	16.0000	0.605176
$700$	0	0
$701$	26.0000	0.982006	0.491003	−	0.871158i	$-0.336630\pi$
$701$	26.0000	0.982006	0.491003	+	0.871158i	$0.336630\pi$
$702$	0	0
$703$	−1.00000	−0.0377157
$704$	0	0
$705$	−6.00000	−0.225973
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	20.0000	0.750059
$712$	0	0
$713$	−8.00000	−0.299602
$714$	0	0
$715$	24.0000	0.897549
$716$	0	0
$717$	6.00000	0.224074
$718$	0	0
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	4.00000	0.148762
$724$	0	0
$725$	10.0000	0.371391
$726$	0	0
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	35.0000	1.29452
$732$	0	0
$733$	40.0000	1.47743	0.738717	−	0.674016i	$-0.235432\pi$
$733$	40.0000	1.47743	0.738717	+	0.674016i	$0.235432\pi$
$734$	0	0
$735$	−7.00000	−0.258199
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	0	0
$741$	6.00000	0.220416
$742$	0	0
$743$	−15.0000	−0.550297	−0.275148	−	0.961402i	$-0.588727\pi$
$743$	−15.0000	−0.550297	−0.275148	+	0.961402i	$0.588727\pi$
$744$	0	0
$745$	−1.00000	−0.0366372
$746$	0	0
$747$	18.0000	0.658586
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	0	0
$753$	22.0000	0.801725
$754$	0	0
$755$	10.0000	0.363937
$756$	0	0
$757$	45.0000	1.63555	0.817776	−	0.575536i	$-0.195207\pi$
$757$	45.0000	1.63555	0.817776	+	0.575536i	$0.195207\pi$
$758$	0	0
$759$	32.0000	1.16153
$760$	0	0
$761$	−44.0000	−1.59500	−0.797499	−	0.603320i	$-0.793844\pi$
$761$	−44.0000	−1.59500	−0.797499	+	0.603320i	$0.793844\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−10.0000	−0.361551
$766$	0	0
$767$	−66.0000	−2.38312
$768$	0	0
$769$	−29.0000	−1.04577	−0.522883	−	0.852404i	$-0.675144\pi$
$769$	−29.0000	−1.04577	−0.522883	+	0.852404i	$0.675144\pi$
$770$	0	0
$771$	10.0000	0.360141
$772$	0	0
$773$	−2.00000	−0.0719350	−0.0359675	−	0.999353i	$-0.511451\pi$
$773$	−2.00000	−0.0719350	−0.0359675	+	0.999353i	$0.511451\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.00000	−0.107486
$780$	0	0
$781$	36.0000	1.28818
$782$	0	0
$783$	−50.0000	−1.78685
$784$	0	0
$785$	0	0
$786$	0	0
$787$	20.0000	0.712923	0.356462	−	0.934310i	$-0.383983\pi$
$787$	20.0000	0.712923	0.356462	+	0.934310i	$0.383983\pi$
$788$	0	0
$789$	−9.00000	−0.320408
$790$	0	0
$791$	0	0
$792$	0	0
$793$	72.0000	2.55679
$794$	0	0
$795$	−5.00000	−0.177332
$796$	0	0
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	−30.0000	−1.06132
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−36.0000	−1.27041
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−6.00000	−0.211210
$808$	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
$811$	3.00000	0.105344	0.0526721	−	0.998612i	$-0.483226\pi$
$811$	3.00000	0.105344	0.0526721	+	0.998612i	$0.483226\pi$
$812$	0	0
$813$	−14.0000	−0.491001
$814$	0	0
$815$	−24.0000	−0.840683
$816$	0	0
$817$	7.00000	0.244899
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−56.0000	−1.95441	−0.977207	−	0.212290i	$-0.931908\pi$
$821$	−56.0000	−1.95441	−0.977207	+	0.212290i	$0.931908\pi$
$822$	0	0
$823$	−23.0000	−0.801730	−0.400865	−	0.916137i	$-0.631290\pi$
$823$	−23.0000	−0.801730	−0.400865	+	0.916137i	$0.631290\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	−37.0000	−1.28662	−0.643308	−	0.765607i	$-0.722439\pi$
$827$	−37.0000	−1.28662	−0.643308	+	0.765607i	$0.722439\pi$
$828$	0	0
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	0	0
$831$	−9.00000	−0.312207
$832$	0	0
$833$	−35.0000	−1.21268
$834$	0	0
$835$	−15.0000	−0.519096
$836$	0	0
$837$	5.00000	0.172825
$838$	0	0
$839$	−35.0000	−1.20833	−0.604167	−	0.796858i	$-0.706494\pi$
$839$	−35.0000	−1.20833	−0.604167	+	0.796858i	$0.706494\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	0	0
$843$	7.00000	0.241093
$844$	0	0
$845$	23.0000	0.791224
$846$	0	0
$847$	0	0
$848$	0	0
$849$	6.00000	0.205919
$850$	0	0
$851$	−8.00000	−0.274236
$852$	0	0
$853$	32.0000	1.09566	0.547830	−	0.836590i	$-0.315454\pi$
$853$	32.0000	1.09566	0.547830	+	0.836590i	$0.315454\pi$
$854$	0	0
$855$	−2.00000	−0.0683986
$856$	0	0
$857$	36.0000	1.22974	0.614868	−	0.788630i	$-0.289209\pi$
$857$	36.0000	1.22974	0.614868	+	0.788630i	$0.289209\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−45.0000	−1.53182	−0.765909	−	0.642949i	$-0.777711\pi$
$863$	−45.0000	−1.53182	−0.765909	+	0.642949i	$0.777711\pi$
$864$	0	0
$865$	−4.00000	−0.136004
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	−40.0000	−1.35691
$870$	0	0
$871$	12.0000	0.406604
$872$	0	0
$873$	28.0000	0.947656
$874$	0	0
$875$	0	0
$876$	0	0
$877$	44.0000	1.48577	0.742887	−	0.669417i	$-0.233456\pi$
$877$	44.0000	1.48577	0.742887	+	0.669417i	$0.233456\pi$
$878$	0	0
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	11.0000	0.370179	0.185090	−	0.982722i	$-0.440742\pi$
$883$	11.0000	0.370179	0.185090	+	0.982722i	$0.440742\pi$
$884$	0	0
$885$	−11.0000	−0.369761
$886$	0	0
$887$	18.0000	0.604381	0.302190	−	0.953248i	$-0.402282\pi$
$887$	18.0000	0.604381	0.302190	+	0.953248i	$0.402282\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	−6.00000	−0.200782
$894$	0	0
$895$	4.00000	0.133705
$896$	0	0
$897$	48.0000	1.60267
$898$	0	0
$899$	−10.0000	−0.333519
$900$	0	0
$901$	−25.0000	−0.832871
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18.0000	0.598340
$906$	0	0
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	0	0
$909$	−14.0000	−0.464351
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	−36.0000	−1.19143
$914$	0	0
$915$	12.0000	0.396708
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−17.0000	−0.560778	−0.280389	−	0.959886i	$-0.590464\pi$
$919$	−17.0000	−0.560778	−0.280389	+	0.959886i	$0.590464\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	54.0000	1.77743
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	48.0000	1.57483	0.787414	−	0.616424i	$-0.211419\pi$
$929$	48.0000	1.57483	0.787414	+	0.616424i	$0.211419\pi$
$930$	0	0
$931$	−7.00000	−0.229416
$932$	0	0
$933$	−7.00000	−0.229170
$934$	0	0
$935$	20.0000	0.654070
$936$	0	0
$937$	28.0000	0.914720	0.457360	−	0.889282i	$-0.348795\pi$
$937$	28.0000	0.914720	0.457360	+	0.889282i	$0.348795\pi$
$938$	0	0
$939$	29.0000	0.946379
$940$	0	0
$941$	−54.0000	−1.76035	−0.880175	−	0.474650i	$-0.842575\pi$
$941$	−54.0000	−1.76035	−0.880175	+	0.474650i	$0.842575\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−55.0000	−1.78726	−0.893630	−	0.448805i	$-0.851850\pi$
$947$	−55.0000	−1.78726	−0.893630	+	0.448805i	$0.851850\pi$
$948$	0	0
$949$	−54.0000	−1.75291
$950$	0	0
$951$	−32.0000	−1.03767
$952$	0	0
$953$	−9.00000	−0.291539	−0.145769	−	0.989319i	$-0.546566\pi$
$953$	−9.00000	−0.291539	−0.145769	+	0.989319i	$0.546566\pi$
$954$	0	0
$955$	4.00000	0.129437
$956$	0	0
$957$	40.0000	1.29302
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−4.00000	−0.128898
$964$	0	0
$965$	14.0000	0.450676
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	0	0
$969$	5.00000	0.160623
$970$	0	0
$971$	−33.0000	−1.05902	−0.529510	−	0.848304i	$-0.677624\pi$
$971$	−33.0000	−1.05902	−0.529510	+	0.848304i	$0.677624\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	6.00000	0.192154
$976$	0	0
$977$	−44.0000	−1.40768	−0.703842	−	0.710356i	$-0.748534\pi$
$977$	−44.0000	−1.40768	−0.703842	+	0.710356i	$0.748534\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	30.0000	0.957826
$982$	0	0
$983$	56.0000	1.78612	0.893061	−	0.449935i	$-0.148553\pi$
$983$	56.0000	1.78612	0.893061	+	0.449935i	$0.148553\pi$
$984$	0	0
$985$	22.0000	0.700978
$986$	0	0
$987$	0	0
$988$	0	0
$989$	56.0000	1.78070
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	0	0
$993$	−12.0000	−0.380808
$994$	0	0
$995$	−8.00000	−0.253617
$996$	0	0
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	0	0
$999$	5.00000	0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9920.2.a.ba.1.1		1
4.3	odd	2		9920.2.a.l.1.1		1
8.3	odd	2		2480.2.a.k.1.1		1
8.5	even	2		155.2.a.c.1.1	✓	1
24.5	odd	2		1395.2.a.b.1.1		1
40.13	odd	4		775.2.b.c.249.1		2
40.29	even	2		775.2.a.a.1.1		1
40.37	odd	4		775.2.b.c.249.2		2
56.13	odd	2		7595.2.a.h.1.1		1
120.29	odd	2		6975.2.a.l.1.1		1
248.61	odd	2		4805.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
155.2.a.c.1.1	✓	1	8.5	even	2
775.2.a.a.1.1		1	40.29	even	2
775.2.b.c.249.1		2	40.13	odd	4
775.2.b.c.249.2		2	40.37	odd	4
1395.2.a.b.1.1		1	24.5	odd	2
2480.2.a.k.1.1		1	8.3	odd	2
4805.2.a.e.1.1		1	248.61	odd	2
6975.2.a.l.1.1		1	120.29	odd	2
7595.2.a.h.1.1		1	56.13	odd	2
9920.2.a.l.1.1		1	4.3	odd	2
9920.2.a.ba.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9920 = 2^{6} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9920.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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