LMFDB - Embedded newform 3120.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3120,2,Mod(1,3120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3120, 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("3120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3120.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:	$24.9133254306$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3120.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{13} -1.00000 q^{15} +8.00000 q^{17} +6.00000 q^{19} -2.00000 q^{21} -6.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.00000 q^{29} -4.00000 q^{33} +2.00000 q^{35} -2.00000 q^{37} -1.00000 q^{39} -2.00000 q^{41} +4.00000 q^{43} -1.00000 q^{45} -3.00000 q^{49} +8.00000 q^{51} -10.0000 q^{53} +4.00000 q^{55} +6.00000 q^{57} -4.00000 q^{59} -10.0000 q^{61} -2.00000 q^{63} +1.00000 q^{65} -12.0000 q^{67} -6.00000 q^{69} +8.00000 q^{71} -8.00000 q^{73} +1.00000 q^{75} +8.00000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -12.0000 q^{83} -8.00000 q^{85} -4.00000 q^{87} -14.0000 q^{89} +2.00000 q^{91} -6.00000 q^{95} -16.0000 q^{97} -4.00000 q^{99} +O(q^{100})$

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	8.00000	1.94029	0.970143	−	0.242536i	$-0.0779791\pi$
$17$	8.00000	1.94029	0.970143	+	0.242536i	$0.0779791\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	2.00000	0.338062
$36$	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
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	8.00000	1.12022
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	8.00000	0.911685
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	0	0
$87$	−4.00000	−0.428845
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−16.0000	−1.59206	−0.796030	−	0.605257i	$-0.793070\pi$
$101$	−16.0000	−1.59206	−0.796030	+	0.605257i	$0.793070\pi$
$102$	0	0
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	20.0000	1.88144	0.940721	−	0.339182i	$-0.110150\pi$
$113$	20.0000	1.88144	0.940721	+	0.339182i	$0.110150\pi$
$114$	0	0
$115$	6.00000	0.559503
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−16.0000	−1.46672
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	8.00000	0.646762
$154$	0	0
$155$	0	0
$156$	0	0
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	0	0
$165$	4.00000	0.311400
$166$	0	0
$167$	4.00000	0.309529	0.154765	−	0.987951i	$-0.450538\pi$
$167$	4.00000	0.309529	0.154765	+	0.987951i	$0.450538\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.00000	0.458831
$172$	0	0
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	−32.0000	−2.34007
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	0	0
$203$	8.00000	0.561490
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−24.0000	−1.66011
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	8.00000	0.548151
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−8.00000	−0.540590
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.00000	−0.519656
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.00000	0.191663
$246$	0	0
$247$	−6.00000	−0.381771
$248$	0	0
$249$	−12.0000	−0.760469
$250$	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
$253$	24.0000	1.50887
$254$	0	0
$255$	−8.00000	−0.500979
$256$	0	0
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	−4.00000	−0.247594
$262$	0	0
$263$	−2.00000	−0.123325	−0.0616626	−	0.998097i	$-0.519640\pi$
$263$	−2.00000	−0.123325	−0.0616626	+	0.998097i	$0.519640\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	0	0
$267$	−14.0000	−0.856786
$268$	0	0
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	−6.00000	−0.355409
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	47.0000	2.76471
$290$	0	0
$291$	−16.0000	−0.937937
$292$	0	0
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	6.00000	0.346989
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	−16.0000	−0.919176
$304$	0	0
$305$	10.0000	0.572598
$306$	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
$309$	12.0000	0.682656
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	2.00000	0.112687
$316$	0	0
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	0	0
$319$	16.0000	0.895828
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	48.0000	2.67079
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	12.0000	0.663602
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	12.0000	0.655630
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	20.0000	1.08625
$340$	0	0
$341$	0	0
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	6.00000	0.323029
$346$	0	0
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	0	0
$349$	−28.0000	−1.49881	−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000	−1.49881	−0.749403	+	0.662114i	$0.769659\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	−16.0000	−0.846810
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	8.00000	0.418739
$366$	0	0
$367$	36.0000	1.87918	0.939592	−	0.342296i	$-0.111204\pi$
$367$	36.0000	1.87918	0.939592	+	0.342296i	$0.111204\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	20.0000	1.03835
$372$	0	0
$373$	−2.00000	−0.103556	−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000	−0.103556	−0.0517780	+	0.998659i	$0.516489\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	18.0000	0.924598	0.462299	−	0.886724i	$-0.347025\pi$
$379$	18.0000	0.924598	0.462299	+	0.886724i	$0.347025\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	−8.00000	−0.407718
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	8.00000	0.405616	0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000	0.405616	0.202808	+	0.979219i	$0.434993\pi$
$390$	0	0
$391$	−48.0000	−2.42746
$392$	0	0
$393$	−10.0000	−0.504433
$394$	0	0
$395$	8.00000	0.402524
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	0	0
$399$	−12.0000	−0.600751
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	8.00000	0.391762
$418$	0	0
$419$	−10.0000	−0.488532	−0.244266	−	0.969708i	$-0.578547\pi$
$419$	−10.0000	−0.488532	−0.244266	+	0.969708i	$0.578547\pi$
$420$	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
$423$	0	0
$424$	0	0
$425$	8.00000	0.388057
$426$	0	0
$427$	20.0000	0.967868
$428$	0	0
$429$	4.00000	0.193122
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	4.00000	0.191785
$436$	0	0
$437$	−36.0000	−1.72211
$438$	0	0
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−16.0000	−0.760183	−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000	−0.760183	−0.380091	+	0.924949i	$0.624107\pi$
$444$	0	0
$445$	14.0000	0.663664
$446$	0	0
$447$	−10.0000	−0.472984
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	0	0
$459$	8.00000	0.373408
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	26.0000	1.20832	0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000	1.20832	0.604161	+	0.796862i	$0.293508\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	0	0
$469$	24.0000	1.10822
$470$	0	0
$471$	22.0000	1.01371
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	6.00000	0.275299
$476$	0	0
$477$	−10.0000	−0.457869
$478$	0	0
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	0	0
$483$	12.0000	0.546019
$484$	0	0
$485$	16.0000	0.726523
$486$	0	0
$487$	−26.0000	−1.17817	−0.589086	−	0.808070i	$-0.700512\pi$
$487$	−26.0000	−1.17817	−0.589086	+	0.808070i	$0.700512\pi$
$488$	0	0
$489$	16.0000	0.723545
$490$	0	0
$491$	−42.0000	−1.89543	−0.947717	−	0.319113i	$-0.896615\pi$
$491$	−42.0000	−1.89543	−0.947717	+	0.319113i	$0.896615\pi$
$492$	0	0
$493$	−32.0000	−1.44121
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	−16.0000	−0.717698
$498$	0	0
$499$	−38.0000	−1.70111	−0.850557	−	0.525883i	$-0.823735\pi$
$499$	−38.0000	−1.70111	−0.850557	+	0.525883i	$0.823735\pi$
$500$	0	0
$501$	4.00000	0.178707
$502$	0	0
$503$	−10.0000	−0.445878	−0.222939	−	0.974832i	$-0.571565\pi$
$503$	−10.0000	−0.445878	−0.222939	+	0.974832i	$0.571565\pi$
$504$	0	0
$505$	16.0000	0.711991
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	0	0
$513$	6.00000	0.264906
$514$	0	0
$515$	−12.0000	−0.528783
$516$	0	0
$517$	0	0
$518$	0	0
$519$	22.0000	0.965693
$520$	0	0
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	0	0
$523$	−36.0000	−1.57417	−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000	−1.57417	−0.787085	+	0.616844i	$0.788411\pi$
$524$	0	0
$525$	−2.00000	−0.0872872
$526$	0	0
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	2.00000	0.0866296
$534$	0	0
$535$	12.0000	0.518805
$536$	0	0
$537$	10.0000	0.431532
$538$	0	0
$539$	12.0000	0.516877
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	0	0
$543$	10.0000	0.429141
$544$	0	0
$545$	−12.0000	−0.514024
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	0	0
$549$	−10.0000	−0.426790
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	2.00000	0.0848953
$556$	0	0
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	−32.0000	−1.35104
$562$	0	0
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	−20.0000	−0.841406
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.00000	−0.250217
$576$	0	0
$577$	12.0000	0.499567	0.249783	−	0.968302i	$-0.419641\pi$
$577$	12.0000	0.499567	0.249783	+	0.968302i	$0.419641\pi$
$578$	0	0
$579$	−4.00000	−0.166234
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	40.0000	1.65663
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	22.0000	0.903432	0.451716	−	0.892162i	$-0.350812\pi$
$593$	22.0000	0.903432	0.451716	+	0.892162i	$0.350812\pi$
$594$	0	0
$595$	16.0000	0.655936
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	0	0
$605$	−5.00000	−0.203279
$606$	0	0
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	0	0
$609$	8.00000	0.324176
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	0	0
$615$	2.00000	0.0806478
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	0	0
$623$	28.0000	1.12180
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−24.0000	−0.958468
$628$	0	0
$629$	−16.0000	−0.637962
$630$	0	0
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	0	0
$633$	20.0000	0.794929
$634$	0	0
$635$	4.00000	0.158735
$636$	0	0
$637$	3.00000	0.118864
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	−44.0000	−1.73519	−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000	−1.73519	−0.867595	+	0.497271i	$0.834335\pi$
$644$	0	0
$645$	−4.00000	−0.157500
$646$	0	0
$647$	30.0000	1.17942	0.589711	−	0.807614i	$-0.299242\pi$
$647$	30.0000	1.17942	0.589711	+	0.807614i	$0.299242\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	0	0
$655$	10.0000	0.390732
$656$	0	0
$657$	−8.00000	−0.312110
$658$	0	0
$659$	−2.00000	−0.0779089	−0.0389545	−	0.999241i	$-0.512403\pi$
$659$	−2.00000	−0.0779089	−0.0389545	+	0.999241i	$0.512403\pi$
$660$	0	0
$661$	48.0000	1.86698	0.933492	−	0.358599i	$-0.116745\pi$
$661$	48.0000	1.86698	0.933492	+	0.358599i	$0.116745\pi$
$662$	0	0
$663$	−8.00000	−0.310694
$664$	0	0
$665$	12.0000	0.465340
$666$	0	0
$667$	24.0000	0.929284
$668$	0	0
$669$	2.00000	0.0773245
$670$	0	0
$671$	40.0000	1.54418
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	0	0
$679$	32.0000	1.22805
$680$	0	0
$681$	4.00000	0.153280
$682$	0	0
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	0	0
$687$	4.00000	0.152610
$688$	0	0
$689$	10.0000	0.380970
$690$	0	0
$691$	14.0000	0.532585	0.266293	−	0.963892i	$-0.414201\pi$
$691$	14.0000	0.532585	0.266293	+	0.963892i	$0.414201\pi$
$692$	0	0
$693$	8.00000	0.303895
$694$	0	0
$695$	−8.00000	−0.303457
$696$	0	0
$697$	−16.0000	−0.606043
$698$	0	0
$699$	24.0000	0.907763
$700$	0	0
$701$	32.0000	1.20862	0.604312	−	0.796748i	$-0.293448\pi$
$701$	32.0000	1.20862	0.604312	+	0.796748i	$0.293448\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	0	0
$705$	0	0
$706$	0	0
$707$	32.0000	1.20348
$708$	0	0
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−4.00000	−0.149592
$716$	0	0
$717$	−16.0000	−0.597531
$718$	0	0
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	0	0
$723$	2.00000	0.0743808
$724$	0	0
$725$	−4.00000	−0.148556
$726$	0	0
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	32.0000	1.18356
$732$	0	0
$733$	38.0000	1.40356	0.701781	−	0.712393i	$-0.252388\pi$
$733$	38.0000	1.40356	0.701781	+	0.712393i	$0.252388\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	48.0000	1.76810
$738$	0	0
$739$	−30.0000	−1.10357	−0.551784	−	0.833987i	$-0.686053\pi$
$739$	−30.0000	−1.10357	−0.551784	+	0.833987i	$0.686053\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	0	0
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	0	0
$745$	10.0000	0.366372
$746$	0	0
$747$	−12.0000	−0.439057
$748$	0	0
$749$	24.0000	0.876941
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	0	0
$753$	−6.00000	−0.218652
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−6.00000	−0.218074	−0.109037	−	0.994038i	$-0.534777\pi$
$757$	−6.00000	−0.218074	−0.109037	+	0.994038i	$0.534777\pi$
$758$	0	0
$759$	24.0000	0.871145
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	−24.0000	−0.868858
$764$	0	0
$765$	−8.00000	−0.289241
$766$	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
$771$	12.0000	0.432169
$772$	0	0
$773$	38.0000	1.36677	0.683383	−	0.730061i	$-0.260508\pi$
$773$	38.0000	1.36677	0.683383	+	0.730061i	$0.260508\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.00000	0.143499
$778$	0	0
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−32.0000	−1.14505
$782$	0	0
$783$	−4.00000	−0.142948
$784$	0	0
$785$	−22.0000	−0.785214
$786$	0	0
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	0	0
$789$	−2.00000	−0.0712019
$790$	0	0
$791$	−40.0000	−1.42224
$792$	0	0
$793$	10.0000	0.355110
$794$	0	0
$795$	10.0000	0.354663
$796$	0	0
$797$	−10.0000	−0.354218	−0.177109	−	0.984191i	$-0.556675\pi$
$797$	−10.0000	−0.354218	−0.177109	+	0.984191i	$0.556675\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−14.0000	−0.494666
$802$	0	0
$803$	32.0000	1.12926
$804$	0	0
$805$	−12.0000	−0.422944
$806$	0	0
$807$	−24.0000	−0.844840
$808$	0	0
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	0	0
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	0	0
$813$	−16.0000	−0.561144
$814$	0	0
$815$	−16.0000	−0.560456
$816$	0	0
$817$	24.0000	0.839654
$818$	0	0
$819$	2.00000	0.0698857
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	0	0
$823$	52.0000	1.81261	0.906303	−	0.422628i	$-0.138892\pi$
$823$	52.0000	1.81261	0.906303	+	0.422628i	$0.138892\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	0	0
$833$	−24.0000	−0.831551
$834$	0	0
$835$	−4.00000	−0.138426
$836$	0	0
$837$	0	0
$838$	0	0
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	−6.00000	−0.206651
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−10.0000	−0.343604
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	0	0
$855$	−6.00000	−0.205196
$856$	0	0
$857$	4.00000	0.136637	0.0683187	−	0.997664i	$-0.478237\pi$
$857$	4.00000	0.136637	0.0683187	+	0.997664i	$0.478237\pi$
$858$	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
$861$	4.00000	0.136320
$862$	0	0
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	0	0
$865$	−22.0000	−0.748022
$866$	0	0
$867$	47.0000	1.59620
$868$	0	0
$869$	32.0000	1.08553
$870$	0	0
$871$	12.0000	0.406604
$872$	0	0
$873$	−16.0000	−0.541518
$874$	0	0
$875$	2.00000	0.0676123
$876$	0	0
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	0	0
$879$	−26.0000	−0.876958
$880$	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
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	0	0
$885$	4.00000	0.134459
$886$	0	0
$887$	2.00000	0.0671534	0.0335767	−	0.999436i	$-0.489310\pi$
$887$	2.00000	0.0671534	0.0335767	+	0.999436i	$0.489310\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	−4.00000	−0.134005
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−10.0000	−0.334263
$896$	0	0
$897$	6.00000	0.200334
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−80.0000	−2.66519
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	−10.0000	−0.332411
$906$	0	0
$907$	−52.0000	−1.72663	−0.863316	−	0.504664i	$-0.831616\pi$
$907$	−52.0000	−1.72663	−0.863316	+	0.504664i	$0.831616\pi$
$908$	0	0
$909$	−16.0000	−0.530687
$910$	0	0
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	0	0
$913$	48.0000	1.58857
$914$	0	0
$915$	10.0000	0.330590
$916$	0	0
$917$	20.0000	0.660458
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	0	0
$923$	−8.00000	−0.263323
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	12.0000	0.394132
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	0	0
$933$	−24.0000	−0.785725
$934$	0	0
$935$	32.0000	1.04651
$936$	0	0
$937$	18.0000	0.588034	0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000	0.588034	0.294017	+	0.955800i	$0.405008\pi$
$938$	0	0
$939$	6.00000	0.195803
$940$	0	0
$941$	−34.0000	−1.10837	−0.554184	−	0.832394i	$-0.686970\pi$
$941$	−34.0000	−1.10837	−0.554184	+	0.832394i	$0.686970\pi$
$942$	0	0
$943$	12.0000	0.390774
$944$	0	0
$945$	2.00000	0.0650600
$946$	0	0
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	0	0
$951$	−30.0000	−0.972817
$952$	0	0
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	16.0000	0.517207
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−12.0000	−0.386695
$964$	0	0
$965$	4.00000	0.128765
$966$	0	0
$967$	14.0000	0.450210	0.225105	−	0.974335i	$-0.427728\pi$
$967$	14.0000	0.450210	0.225105	+	0.974335i	$0.427728\pi$
$968$	0	0
$969$	48.0000	1.54198
$970$	0	0
$971$	−38.0000	−1.21948	−0.609739	−	0.792602i	$-0.708726\pi$
$971$	−38.0000	−1.21948	−0.609739	+	0.792602i	$0.708726\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0	0
$979$	56.0000	1.78977
$980$	0	0
$981$	12.0000	0.383131
$982$	0	0
$983$	52.0000	1.65854	0.829271	−	0.558846i	$-0.188756\pi$
$983$	52.0000	1.65854	0.829271	+	0.558846i	$0.188756\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−24.0000	−0.763156
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	−10.0000	−0.317340
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	0	0
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.a.o.1.1		1
3.2	odd	2		9360.2.a.bh.1.1		1
4.3	odd	2		390.2.a.e.1.1	✓	1
12.11	even	2		1170.2.a.e.1.1		1
20.3	even	4		1950.2.e.f.1249.1		2
20.7	even	4		1950.2.e.f.1249.2		2
20.19	odd	2		1950.2.a.h.1.1		1
52.31	even	4		5070.2.b.e.1351.1		2
52.47	even	4		5070.2.b.e.1351.2		2
52.51	odd	2		5070.2.a.e.1.1		1
60.23	odd	4		5850.2.e.i.5149.2		2
60.47	odd	4		5850.2.e.i.5149.1		2
60.59	even	2		5850.2.a.bi.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.e.1.1	✓	1	4.3	odd	2
1170.2.a.e.1.1		1	12.11	even	2
1950.2.a.h.1.1		1	20.19	odd	2
1950.2.e.f.1249.1		2	20.3	even	4
1950.2.e.f.1249.2		2	20.7	even	4
3120.2.a.o.1.1		1	1.1	even	1	trivial
5070.2.a.e.1.1		1	52.51	odd	2
5070.2.b.e.1351.1		2	52.31	even	4
5070.2.b.e.1351.2		2	52.47	even	4
5850.2.a.bi.1.1		1	60.59	even	2
5850.2.e.i.5149.1		2	60.47	odd	4
5850.2.e.i.5149.2		2	60.23	odd	4
9360.2.a.bh.1.1		1	3.2	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:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.a (trivial)

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L-functions

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