LMFDB - Embedded newform 5520.2.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1,5520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.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:	$44.0774219157$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1380)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -1.00000 q^{5} +5.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} -1.00000 q^{5} +5.00000 q^{7} +1.00000 q^{9} +4.00000 q^{13} -1.00000 q^{15} -3.00000 q^{17} +4.00000 q^{19} +5.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +1.00000 q^{29} -1.00000 q^{31} -5.00000 q^{35} -1.00000 q^{37} +4.00000 q^{39} +11.0000 q^{41} -4.00000 q^{43} -1.00000 q^{45} -6.00000 q^{47} +18.0000 q^{49} -3.00000 q^{51} +1.00000 q^{53} +4.00000 q^{57} +1.00000 q^{59} +6.00000 q^{61} +5.00000 q^{63} -4.00000 q^{65} +9.00000 q^{67} +1.00000 q^{69} -13.0000 q^{71} +1.00000 q^{75} -16.0000 q^{79} +1.00000 q^{81} -9.00000 q^{83} +3.00000 q^{85} +1.00000 q^{87} +8.00000 q^{89} +20.0000 q^{91} -1.00000 q^{93} -4.00000 q^{95} +2.00000 q^{97} +O(q^{100})$

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	5.00000	1.88982	0.944911	−	0.327327i	$-0.106148\pi$
$7$	5.00000	1.88982	0.944911	+	0.327327i	$0.106148\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	5.00000	1.09109
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−5.00000	−0.845154
$36$	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
$39$	4.00000	0.640513
$40$	0	0
$41$	11.0000	1.71791	0.858956	−	0.512050i	$-0.171114\pi$
$41$	11.0000	1.71791	0.858956	+	0.512050i	$0.171114\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	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
$49$	18.0000	2.57143
$50$	0	0
$51$	−3.00000	−0.420084
$52$	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
$55$	0	0
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	1.00000	0.130189	0.0650945	−	0.997879i	$-0.479265\pi$
$59$	1.00000	0.130189	0.0650945	+	0.997879i	$0.479265\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	5.00000	0.629941
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	9.00000	1.09952	0.549762	−	0.835321i	$-0.314718\pi$
$67$	9.00000	1.09952	0.549762	+	0.835321i	$0.314718\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−13.0000	−1.54282	−0.771408	−	0.636341i	$-0.780447\pi$
$71$	−13.0000	−1.54282	−0.771408	+	0.636341i	$0.780447\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	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
$85$	3.00000	0.325396
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	20.0000	2.09657
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.00000	−0.696526	−0.348263	−	0.937397i	$-0.613228\pi$
$101$	−7.00000	−0.696526	−0.348263	+	0.937397i	$0.613228\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	−5.00000	−0.487950
$106$	0	0
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	0	0
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	0	0
$111$	−1.00000	−0.0949158
$112$	0	0
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	4.00000	0.369800
$118$	0	0
$119$	−15.0000	−1.37505
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	11.0000	0.991837
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	18.0000	1.59724	0.798621	−	0.601834i	$-0.205563\pi$
$127$	18.0000	1.59724	0.798621	+	0.601834i	$0.205563\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	0	0
$133$	20.0000	1.73422
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	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
$139$	−11.0000	−0.933008	−0.466504	−	0.884519i	$-0.654487\pi$
$139$	−11.0000	−0.933008	−0.466504	+	0.884519i	$0.654487\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	18.0000	1.48461
$148$	0	0
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	0	0
$159$	1.00000	0.0793052
$160$	0	0
$161$	5.00000	0.394055
$162$	0	0
$163$	−18.0000	−1.40987	−0.704934	−	0.709273i	$-0.749024\pi$
$163$	−18.0000	−1.40987	−0.704934	+	0.709273i	$0.749024\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	4.00000	0.305888
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	5.00000	0.377964
$176$	0	0
$177$	1.00000	0.0751646
$178$	0	0
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	0	0
$188$	0	0
$189$	5.00000	0.363696
$190$	0	0
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	0	0
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	0	0
$195$	−4.00000	−0.286446
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	9.00000	0.634811
$202$	0	0
$203$	5.00000	0.350931
$204$	0	0
$205$	−11.0000	−0.768273
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	0	0
$210$	0	0
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	0	0
$213$	−13.0000	−0.890745
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	−5.00000	−0.339422
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−28.0000	−1.85843	−0.929213	−	0.369546i	$-0.879513\pi$
$227$	−28.0000	−1.85843	−0.929213	+	0.369546i	$0.879513\pi$
$228$	0	0
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.00000	0.524097	0.262049	−	0.965055i	$-0.415602\pi$
$233$	8.00000	0.524097	0.262049	+	0.965055i	$0.415602\pi$
$234$	0	0
$235$	6.00000	0.391397
$236$	0	0
$237$	−16.0000	−1.03931
$238$	0	0
$239$	21.0000	1.35838	0.679189	−	0.733964i	$-0.262332\pi$
$239$	21.0000	1.35838	0.679189	+	0.733964i	$0.262332\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−18.0000	−1.14998
$246$	0	0
$247$	16.0000	1.01806
$248$	0	0
$249$	−9.00000	−0.570352
$250$	0	0
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	3.00000	0.187867
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−5.00000	−0.310685
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	29.0000	1.78822	0.894108	−	0.447851i	$-0.147810\pi$
$263$	29.0000	1.78822	0.894108	+	0.447851i	$0.147810\pi$
$264$	0	0
$265$	−1.00000	−0.0614295
$266$	0	0
$267$	8.00000	0.489592
$268$	0	0
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	0	0
$271$	13.0000	0.789694	0.394847	−	0.918747i	$-0.370798\pi$
$271$	13.0000	0.789694	0.394847	+	0.918747i	$0.370798\pi$
$272$	0	0
$273$	20.0000	1.21046
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	20.0000	1.19310	0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000	1.19310	0.596550	+	0.802576i	$0.296538\pi$
$282$	0	0
$283$	−1.00000	−0.0594438	−0.0297219	−	0.999558i	$-0.509462\pi$
$283$	−1.00000	−0.0594438	−0.0297219	+	0.999558i	$0.509462\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	0	0
$287$	55.0000	3.24655
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	2.00000	0.117242
$292$	0	0
$293$	7.00000	0.408944	0.204472	−	0.978872i	$-0.434452\pi$
$293$	7.00000	0.408944	0.204472	+	0.978872i	$0.434452\pi$
$294$	0	0
$295$	−1.00000	−0.0582223
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.00000	0.231326
$300$	0	0
$301$	−20.0000	−1.15278
$302$	0	0
$303$	−7.00000	−0.402139
$304$	0	0
$305$	−6.00000	−0.343559
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	0	0
$311$	32.0000	1.81455	0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000	1.81455	0.907277	+	0.420534i	$0.138157\pi$
$312$	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
$315$	−5.00000	−0.281718
$316$	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
$319$	0	0
$320$	0	0
$321$	3.00000	0.167444
$322$	0	0
$323$	−12.0000	−0.667698
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	−12.0000	−0.663602
$328$	0	0
$329$	−30.0000	−1.65395
$330$	0	0
$331$	−25.0000	−1.37412	−0.687062	−	0.726599i	$-0.741100\pi$
$331$	−25.0000	−1.37412	−0.687062	+	0.726599i	$0.741100\pi$
$332$	0	0
$333$	−1.00000	−0.0547997
$334$	0	0
$335$	−9.00000	−0.491723
$336$	0	0
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	0	0
$339$	1.00000	0.0543125
$340$	0	0
$341$	0	0
$342$	0	0
$343$	55.0000	2.96972
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	8.00000	0.429463	0.214731	−	0.976673i	$-0.431112\pi$
$347$	8.00000	0.429463	0.214731	+	0.976673i	$0.431112\pi$
$348$	0	0
$349$	19.0000	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$349$	19.0000	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	13.0000	0.689968
$356$	0	0
$357$	−15.0000	−0.793884
$358$	0	0
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	−11.0000	−0.577350
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7.00000	0.365397	0.182699	−	0.983169i	$-0.441517\pi$
$367$	7.00000	0.365397	0.182699	+	0.983169i	$0.441517\pi$
$368$	0	0
$369$	11.0000	0.572637
$370$	0	0
$371$	5.00000	0.259587
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	24.0000	1.23280	0.616399	−	0.787434i	$-0.288591\pi$
$379$	24.0000	1.23280	0.616399	+	0.787434i	$0.288591\pi$
$380$	0	0
$381$	18.0000	0.922168
$382$	0	0
$383$	−27.0000	−1.37964	−0.689818	−	0.723983i	$-0.742309\pi$
$383$	−27.0000	−1.37964	−0.689818	+	0.723983i	$0.742309\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	−3.00000	−0.151717
$392$	0	0
$393$	−20.0000	−1.00887
$394$	0	0
$395$	16.0000	0.805047
$396$	0	0
$397$	32.0000	1.60603	0.803017	−	0.595956i	$-0.203227\pi$
$397$	32.0000	1.60603	0.803017	+	0.595956i	$0.203227\pi$
$398$	0	0
$399$	20.0000	1.00125
$400$	0	0
$401$	38.0000	1.89763	0.948815	−	0.315833i	$-0.102284\pi$
$401$	38.0000	1.89763	0.948815	+	0.315833i	$0.102284\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−7.00000	−0.346128	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000	−0.346128	−0.173064	+	0.984911i	$0.555367\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	0	0
$413$	5.00000	0.246034
$414$	0	0
$415$	9.00000	0.441793
$416$	0	0
$417$	−11.0000	−0.538672
$418$	0	0
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	−3.00000	−0.145521
$426$	0	0
$427$	30.0000	1.45180
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	−17.0000	−0.816968	−0.408484	−	0.912766i	$-0.633942\pi$
$433$	−17.0000	−0.816968	−0.408484	+	0.912766i	$0.633942\pi$
$434$	0	0
$435$	−1.00000	−0.0479463
$436$	0	0
$437$	4.00000	0.191346
$438$	0	0
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	0	0
$443$	2.00000	0.0950229	0.0475114	−	0.998871i	$-0.484871\pi$
$443$	2.00000	0.0950229	0.0475114	+	0.998871i	$0.484871\pi$
$444$	0	0
$445$	−8.00000	−0.379236
$446$	0	0
$447$	22.0000	1.04056
$448$	0	0
$449$	1.00000	0.0471929	0.0235965	−	0.999722i	$-0.492488\pi$
$449$	1.00000	0.0471929	0.0235965	+	0.999722i	$0.492488\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	16.0000	0.751746
$454$	0	0
$455$	−20.0000	−0.937614
$456$	0	0
$457$	9.00000	0.421002	0.210501	−	0.977594i	$-0.432490\pi$
$457$	9.00000	0.421002	0.210501	+	0.977594i	$0.432490\pi$
$458$	0	0
$459$	−3.00000	−0.140028
$460$	0	0
$461$	34.0000	1.58354	0.791769	−	0.610821i	$-0.209160\pi$
$461$	34.0000	1.58354	0.791769	+	0.610821i	$0.209160\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	3.00000	0.138823	0.0694117	−	0.997588i	$-0.477888\pi$
$467$	3.00000	0.138823	0.0694117	+	0.997588i	$0.477888\pi$
$468$	0	0
$469$	45.0000	2.07791
$470$	0	0
$471$	5.00000	0.230388
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	1.00000	0.0457869
$478$	0	0
$479$	−38.0000	−1.73626	−0.868132	−	0.496333i	$-0.834679\pi$
$479$	−38.0000	−1.73626	−0.868132	+	0.496333i	$0.834679\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	5.00000	0.227508
$484$	0	0
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	14.0000	0.634401	0.317200	−	0.948359i	$-0.397257\pi$
$487$	14.0000	0.634401	0.317200	+	0.948359i	$0.397257\pi$
$488$	0	0
$489$	−18.0000	−0.813988
$490$	0	0
$491$	−1.00000	−0.0451294	−0.0225647	−	0.999745i	$-0.507183\pi$
$491$	−1.00000	−0.0451294	−0.0225647	+	0.999745i	$0.507183\pi$
$492$	0	0
$493$	−3.00000	−0.135113
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−65.0000	−2.91565
$498$	0	0
$499$	−3.00000	−0.134298	−0.0671492	−	0.997743i	$-0.521390\pi$
$499$	−3.00000	−0.134298	−0.0671492	+	0.997743i	$0.521390\pi$
$500$	0	0
$501$	6.00000	0.268060
$502$	0	0
$503$	−31.0000	−1.38222	−0.691111	−	0.722749i	$-0.742878\pi$
$503$	−31.0000	−1.38222	−0.691111	+	0.722749i	$0.742878\pi$
$504$	0	0
$505$	7.00000	0.311496
$506$	0	0
$507$	3.00000	0.133235
$508$	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
$511$	0	0
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	−16.0000	−0.705044
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	32.0000	1.40195	0.700973	−	0.713188i	$-0.252749\pi$
$521$	32.0000	1.40195	0.700973	+	0.713188i	$0.252749\pi$
$522$	0	0
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	0	0
$525$	5.00000	0.218218
$526$	0	0
$527$	3.00000	0.130682
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	1.00000	0.0433963
$532$	0	0
$533$	44.0000	1.90585
$534$	0	0
$535$	−3.00000	−0.129701
$536$	0	0
$537$	−24.0000	−1.03568
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	−8.00000	−0.343313
$544$	0	0
$545$	12.0000	0.514024
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	4.00000	0.170406
$552$	0	0
$553$	−80.0000	−3.40195
$554$	0	0
$555$	1.00000	0.0424476
$556$	0	0
$557$	−23.0000	−0.974541	−0.487271	−	0.873251i	$-0.662007\pi$
$557$	−23.0000	−0.974541	−0.487271	+	0.873251i	$0.662007\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−27.0000	−1.13791	−0.568957	−	0.822367i	$-0.692653\pi$
$563$	−27.0000	−1.13791	−0.568957	+	0.822367i	$0.692653\pi$
$564$	0	0
$565$	−1.00000	−0.0420703
$566$	0	0
$567$	5.00000	0.209980
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	−10.0000	−0.418487	−0.209243	−	0.977864i	$-0.567100\pi$
$571$	−10.0000	−0.418487	−0.209243	+	0.977864i	$0.567100\pi$
$572$	0	0
$573$	10.0000	0.417756
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	4.00000	0.166522	0.0832611	−	0.996528i	$-0.473466\pi$
$577$	4.00000	0.166522	0.0832611	+	0.996528i	$0.473466\pi$
$578$	0	0
$579$	12.0000	0.498703
$580$	0	0
$581$	−45.0000	−1.86691
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	6.00000	0.247647	0.123823	−	0.992304i	$-0.460484\pi$
$587$	6.00000	0.247647	0.123823	+	0.992304i	$0.460484\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	2.00000	0.0822690
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	15.0000	0.614940
$596$	0	0
$597$	−2.00000	−0.0818546
$598$	0	0
$599$	44.0000	1.79779	0.898896	−	0.438163i	$-0.144371\pi$
$599$	44.0000	1.79779	0.898896	+	0.438163i	$0.144371\pi$
$600$	0	0
$601$	7.00000	0.285536	0.142768	−	0.989756i	$-0.454400\pi$
$601$	7.00000	0.285536	0.142768	+	0.989756i	$0.454400\pi$
$602$	0	0
$603$	9.00000	0.366508
$604$	0	0
$605$	11.0000	0.447214
$606$	0	0
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	0	0
$609$	5.00000	0.202610
$610$	0	0
$611$	−24.0000	−0.970936
$612$	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
$615$	−11.0000	−0.443563
$616$	0	0
$617$	−1.00000	−0.0402585	−0.0201292	−	0.999797i	$-0.506408\pi$
$617$	−1.00000	−0.0402585	−0.0201292	+	0.999797i	$0.506408\pi$
$618$	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
$623$	40.0000	1.60257
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.00000	0.119618
$630$	0	0
$631$	−38.0000	−1.51276	−0.756378	−	0.654135i	$-0.773033\pi$
$631$	−38.0000	−1.51276	−0.756378	+	0.654135i	$0.773033\pi$
$632$	0	0
$633$	23.0000	0.914168
$634$	0	0
$635$	−18.0000	−0.714308
$636$	0	0
$637$	72.0000	2.85274
$638$	0	0
$639$	−13.0000	−0.514272
$640$	0	0
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	0	0
$643$	1.00000	0.0394362	0.0197181	−	0.999806i	$-0.493723\pi$
$643$	1.00000	0.0394362	0.0197181	+	0.999806i	$0.493723\pi$
$644$	0	0
$645$	4.00000	0.157500
$646$	0	0
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−5.00000	−0.195965
$652$	0	0
$653$	−44.0000	−1.72185	−0.860927	−	0.508729i	$-0.830115\pi$
$653$	−44.0000	−1.72185	−0.860927	+	0.508729i	$0.830115\pi$
$654$	0	0
$655$	20.0000	0.781465
$656$	0	0
$657$	0	0
$658$	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
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	0	0
$663$	−12.0000	−0.466041
$664$	0	0
$665$	−20.0000	−0.775567
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	−14.0000	−0.541271
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	39.0000	1.49889	0.749446	−	0.662066i	$-0.230320\pi$
$677$	39.0000	1.49889	0.749446	+	0.662066i	$0.230320\pi$
$678$	0	0
$679$	10.0000	0.383765
$680$	0	0
$681$	−28.0000	−1.07296
$682$	0	0
$683$	−34.0000	−1.30097	−0.650487	−	0.759517i	$-0.725435\pi$
$683$	−34.0000	−1.30097	−0.650487	+	0.759517i	$0.725435\pi$
$684$	0	0
$685$	−2.00000	−0.0764161
$686$	0	0
$687$	−20.0000	−0.763048
$688$	0	0
$689$	4.00000	0.152388
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11.0000	0.417254
$696$	0	0
$697$	−33.0000	−1.24996
$698$	0	0
$699$	8.00000	0.302588
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	−4.00000	−0.150863
$704$	0	0
$705$	6.00000	0.225973
$706$	0	0
$707$	−35.0000	−1.31631
$708$	0	0
$709$	18.0000	0.676004	0.338002	−	0.941145i	$-0.390249\pi$
$709$	18.0000	0.676004	0.338002	+	0.941145i	$0.390249\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	0	0
$713$	−1.00000	−0.0374503
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21.0000	0.784259
$718$	0	0
$719$	25.0000	0.932343	0.466171	−	0.884694i	$-0.345633\pi$
$719$	25.0000	0.932343	0.466171	+	0.884694i	$0.345633\pi$
$720$	0	0
$721$	80.0000	2.97936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	−13.0000	−0.482143	−0.241072	−	0.970507i	$-0.577499\pi$
$727$	−13.0000	−0.482143	−0.241072	+	0.970507i	$0.577499\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.0000	0.443836
$732$	0	0
$733$	−21.0000	−0.775653	−0.387826	−	0.921732i	$-0.626774\pi$
$733$	−21.0000	−0.775653	−0.387826	+	0.921732i	$0.626774\pi$
$734$	0	0
$735$	−18.0000	−0.663940
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−45.0000	−1.65535	−0.827676	−	0.561206i	$-0.810337\pi$
$739$	−45.0000	−1.65535	−0.827676	+	0.561206i	$0.810337\pi$
$740$	0	0
$741$	16.0000	0.587775
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	−22.0000	−0.806018
$746$	0	0
$747$	−9.00000	−0.329293
$748$	0	0
$749$	15.0000	0.548088
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	0	0
$753$	−6.00000	−0.218652
$754$	0	0
$755$	−16.0000	−0.582300
$756$	0	0
$757$	29.0000	1.05402	0.527011	−	0.849858i	$-0.323312\pi$
$757$	29.0000	1.05402	0.527011	+	0.849858i	$0.323312\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−45.0000	−1.63125	−0.815624	−	0.578582i	$-0.803606\pi$
$761$	−45.0000	−1.63125	−0.815624	+	0.578582i	$0.803606\pi$
$762$	0	0
$763$	−60.0000	−2.17215
$764$	0	0
$765$	3.00000	0.108465
$766$	0	0
$767$	4.00000	0.144432
$768$	0	0
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	0	0
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	−5.00000	−0.179374
$778$	0	0
$779$	44.0000	1.57646
$780$	0	0
$781$	0	0
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−5.00000	−0.178458
$786$	0	0
$787$	−23.0000	−0.819861	−0.409931	−	0.912117i	$-0.634447\pi$
$787$	−23.0000	−0.819861	−0.409931	+	0.912117i	$0.634447\pi$
$788$	0	0
$789$	29.0000	1.03243
$790$	0	0
$791$	5.00000	0.177780
$792$	0	0
$793$	24.0000	0.852265
$794$	0	0
$795$	−1.00000	−0.0354663
$796$	0	0
$797$	15.0000	0.531327	0.265664	−	0.964066i	$-0.414409\pi$
$797$	15.0000	0.531327	0.265664	+	0.964066i	$0.414409\pi$
$798$	0	0
$799$	18.0000	0.636794
$800$	0	0
$801$	8.00000	0.282666
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−5.00000	−0.176227
$806$	0	0
$807$	15.0000	0.528025
$808$	0	0
$809$	13.0000	0.457056	0.228528	−	0.973537i	$-0.426609\pi$
$809$	13.0000	0.457056	0.228528	+	0.973537i	$0.426609\pi$
$810$	0	0
$811$	39.0000	1.36948	0.684738	−	0.728790i	$-0.259917\pi$
$811$	39.0000	1.36948	0.684738	+	0.728790i	$0.259917\pi$
$812$	0	0
$813$	13.0000	0.455930
$814$	0	0
$815$	18.0000	0.630512
$816$	0	0
$817$	−16.0000	−0.559769
$818$	0	0
$819$	20.0000	0.698857
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	0	0
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−27.0000	−0.938882	−0.469441	−	0.882964i	$-0.655545\pi$
$827$	−27.0000	−0.938882	−0.469441	+	0.882964i	$0.655545\pi$
$828$	0	0
$829$	17.0000	0.590434	0.295217	−	0.955430i	$-0.404608\pi$
$829$	17.0000	0.590434	0.295217	+	0.955430i	$0.404608\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	−54.0000	−1.87099
$834$	0	0
$835$	−6.00000	−0.207639
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	26.0000	0.897620	0.448810	−	0.893627i	$-0.351848\pi$
$839$	26.0000	0.897620	0.448810	+	0.893627i	$0.351848\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	20.0000	0.688837
$844$	0	0
$845$	−3.00000	−0.103203
$846$	0	0
$847$	−55.0000	−1.88982
$848$	0	0
$849$	−1.00000	−0.0343199
$850$	0	0
$851$	−1.00000	−0.0342796
$852$	0	0
$853$	−54.0000	−1.84892	−0.924462	−	0.381273i	$-0.875486\pi$
$853$	−54.0000	−1.84892	−0.924462	+	0.381273i	$0.875486\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	−2.00000	−0.0683187	−0.0341593	−	0.999416i	$-0.510875\pi$
$857$	−2.00000	−0.0683187	−0.0341593	+	0.999416i	$0.510875\pi$
$858$	0	0
$859$	27.0000	0.921228	0.460614	−	0.887601i	$-0.347629\pi$
$859$	27.0000	0.921228	0.460614	+	0.887601i	$0.347629\pi$
$860$	0	0
$861$	55.0000	1.87439
$862$	0	0
$863$	−42.0000	−1.42970	−0.714848	−	0.699280i	$-0.753504\pi$
$863$	−42.0000	−1.42970	−0.714848	+	0.699280i	$0.753504\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	0	0
$870$	0	0
$871$	36.0000	1.21981
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	−5.00000	−0.169031
$876$	0	0
$877$	42.0000	1.41824	0.709120	−	0.705088i	$-0.249093\pi$
$877$	42.0000	1.41824	0.709120	+	0.705088i	$0.249093\pi$
$878$	0	0
$879$	7.00000	0.236104
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	0	0
$883$	−46.0000	−1.54802	−0.774012	−	0.633171i	$-0.781753\pi$
$883$	−46.0000	−1.54802	−0.774012	+	0.633171i	$0.781753\pi$
$884$	0	0
$885$	−1.00000	−0.0336146
$886$	0	0
$887$	−38.0000	−1.27592	−0.637958	−	0.770072i	$-0.720220\pi$
$887$	−38.0000	−1.27592	−0.637958	+	0.770072i	$0.720220\pi$
$888$	0	0
$889$	90.0000	3.01850
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−24.0000	−0.803129
$894$	0	0
$895$	24.0000	0.802232
$896$	0	0
$897$	4.00000	0.133556
$898$	0	0
$899$	−1.00000	−0.0333519
$900$	0	0
$901$	−3.00000	−0.0999445
$902$	0	0
$903$	−20.0000	−0.665558
$904$	0	0
$905$	8.00000	0.265929
$906$	0	0
$907$	17.0000	0.564476	0.282238	−	0.959344i	$-0.408923\pi$
$907$	17.0000	0.564476	0.282238	+	0.959344i	$0.408923\pi$
$908$	0	0
$909$	−7.00000	−0.232175
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−6.00000	−0.198354
$916$	0	0
$917$	−100.000	−3.30229
$918$	0	0
$919$	−48.0000	−1.58337	−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−48.0000	−1.58337	−0.791687	+	0.610927i	$0.790797\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	0	0
$923$	−52.0000	−1.71160
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	16.0000	0.525509
$928$	0	0
$929$	5.00000	0.164045	0.0820223	−	0.996630i	$-0.473862\pi$
$929$	5.00000	0.164045	0.0820223	+	0.996630i	$0.473862\pi$
$930$	0	0
$931$	72.0000	2.35970
$932$	0	0
$933$	32.0000	1.04763
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	21.0000	0.685309
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	0	0
$943$	11.0000	0.358209
$944$	0	0
$945$	−5.00000	−0.162650
$946$	0	0
$947$	8.00000	0.259965	0.129983	−	0.991516i	$-0.458508\pi$
$947$	8.00000	0.259965	0.129983	+	0.991516i	$0.458508\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	−10.0000	−0.323592
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10.0000	0.322917
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	3.00000	0.0966736
$964$	0	0
$965$	−12.0000	−0.386294
$966$	0	0
$967$	−2.00000	−0.0643157	−0.0321578	−	0.999483i	$-0.510238\pi$
$967$	−2.00000	−0.0643157	−0.0321578	+	0.999483i	$0.510238\pi$
$968$	0	0
$969$	−12.0000	−0.385496
$970$	0	0
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	0	0
$973$	−55.0000	−1.76322
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	31.0000	0.991778	0.495889	−	0.868386i	$-0.334842\pi$
$977$	31.0000	0.991778	0.495889	+	0.868386i	$0.334842\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−12.0000	−0.383131
$982$	0	0
$983$	11.0000	0.350846	0.175423	−	0.984493i	$-0.443871\pi$
$983$	11.0000	0.350846	0.175423	+	0.984493i	$0.443871\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	0	0
$987$	−30.0000	−0.954911
$988$	0	0
$989$	−4.00000	−0.127193
$990$	0	0
$991$	35.0000	1.11181	0.555906	−	0.831245i	$-0.312372\pi$
$991$	35.0000	1.11181	0.555906	+	0.831245i	$0.312372\pi$
$992$	0	0
$993$	−25.0000	−0.793351
$994$	0	0
$995$	2.00000	0.0634043
$996$	0	0
$997$	40.0000	1.26681	0.633406	−	0.773819i	$-0.281656\pi$
$997$	40.0000	1.26681	0.633406	+	0.773819i	$0.281656\pi$
$998$	0	0
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.ba.1.1		1
4.3	odd	2		1380.2.a.a.1.1	✓	1
12.11	even	2		4140.2.a.e.1.1		1
20.3	even	4		6900.2.f.f.6349.1		2
20.7	even	4		6900.2.f.f.6349.2		2
20.19	odd	2		6900.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.a.a.1.1	✓	1	4.3	odd	2
4140.2.a.e.1.1		1	12.11	even	2
5520.2.a.ba.1.1		1	1.1	even	1	trivial
6900.2.a.i.1.1		1	20.19	odd	2
6900.2.f.f.6349.1		2	20.3	even	4
6900.2.f.f.6349.2		2	20.7	even	4

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:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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