LMFDB - Embedded newform 6960.2.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6960,2,Mod(1,6960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6960, 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("6960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6960 = 2^{4} \cdot 3 \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6960.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:	$55.5758798068$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1740)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+1.00000 q^{3} +1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} -2.00000 q^{13} +1.00000 q^{15} -4.00000 q^{17} -6.00000 q^{19} +2.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +1.00000 q^{29} -4.00000 q^{31} -3.00000 q^{33} +2.00000 q^{35} +3.00000 q^{37} -2.00000 q^{39} +3.00000 q^{41} -1.00000 q^{43} +1.00000 q^{45} -2.00000 q^{47} -3.00000 q^{49} -4.00000 q^{51} -1.00000 q^{53} -3.00000 q^{55} -6.00000 q^{57} -4.00000 q^{59} +4.00000 q^{61} +2.00000 q^{63} -2.00000 q^{65} -10.0000 q^{67} +1.00000 q^{69} -6.00000 q^{71} +1.00000 q^{73} +1.00000 q^{75} -6.00000 q^{77} +4.00000 q^{79} +1.00000 q^{81} -17.0000 q^{83} -4.00000 q^{85} +1.00000 q^{87} +6.00000 q^{89} -4.00000 q^{91} -4.00000 q^{93} -6.00000 q^{95} -13.0000 q^{97} -3.00000 q^{99} +O(q^{100})$

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	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	−3.00000	−0.522233
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−1.00000	−0.137361	−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000	−0.137361	−0.0686803	+	0.997639i	$0.521879\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	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
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−6.00000	−0.683763
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−17.0000	−1.86599	−0.932996	−	0.359886i	$-0.882816\pi$
$83$	−17.0000	−1.86599	−0.932996	+	0.359886i	$0.882816\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	−13.0000	−1.31995	−0.659975	−	0.751288i	$-0.729433\pi$
$97$	−13.0000	−1.31995	−0.659975	+	0.751288i	$0.729433\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	0	0
$101$	5.00000	0.497519	0.248759	−	0.968565i	$-0.419977\pi$
$101$	5.00000	0.497519	0.248759	+	0.968565i	$0.419977\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	−9.00000	−0.862044	−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000	−0.862044	−0.431022	+	0.902342i	$0.641847\pi$
$110$	0	0
$111$	3.00000	0.284747
$112$	0	0
$113$	16.0000	1.50515	0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000	1.50515	0.752577	+	0.658505i	$0.228811\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	−8.00000	−0.733359
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	3.00000	0.270501
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	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
$133$	−12.0000	−1.04053
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	0	0
$143$	6.00000	0.501745
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	9.00000	0.732410	0.366205	−	0.930534i	$-0.380657\pi$
$151$	9.00000	0.732410	0.366205	+	0.930534i	$0.380657\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	−1.00000	−0.0793052
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	0	0
$165$	−3.00000	−0.233550
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−6.00000	−0.458831
$172$	0	0
$173$	7.00000	0.532200	0.266100	−	0.963945i	$-0.414265\pi$
$173$	7.00000	0.532200	0.266100	+	0.963945i	$0.414265\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	0	0
$183$	4.00000	0.295689
$184$	0	0
$185$	3.00000	0.220564
$186$	0	0
$187$	12.0000	0.877527
$188$	0	0
$189$	2.00000	0.145479
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	0	0
$197$	−1.00000	−0.0712470	−0.0356235	−	0.999365i	$-0.511342\pi$
$197$	−1.00000	−0.0712470	−0.0356235	+	0.999365i	$0.511342\pi$
$198$	0	0
$199$	−13.0000	−0.921546	−0.460773	−	0.887518i	$-0.652428\pi$
$199$	−13.0000	−0.921546	−0.460773	+	0.887518i	$0.652428\pi$
$200$	0	0
$201$	−10.0000	−0.705346
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	3.00000	0.209529
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	18.0000	1.24509
$210$	0	0
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	0	0
$213$	−6.00000	−0.411113
$214$	0	0
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	1.00000	0.0675737
$220$	0	0
$221$	8.00000	0.538138
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−3.00000	−0.199117	−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000	−0.199117	−0.0995585	+	0.995032i	$0.531743\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	−6.00000	−0.394771
$232$	0	0
$233$	1.00000	0.0655122	0.0327561	−	0.999463i	$-0.489572\pi$
$233$	1.00000	0.0655122	0.0327561	+	0.999463i	$0.489572\pi$
$234$	0	0
$235$	−2.00000	−0.130466
$236$	0	0
$237$	4.00000	0.259828
$238$	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
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	12.0000	0.763542
$248$	0	0
$249$	−17.0000	−1.07733
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	−3.00000	−0.188608
$254$	0	0
$255$	−4.00000	−0.250490
$256$	0	0
$257$	13.0000	0.810918	0.405459	−	0.914113i	$-0.367112\pi$
$257$	13.0000	0.810918	0.405459	+	0.914113i	$0.367112\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	−1.00000	−0.0614295
$266$	0	0
$267$	6.00000	0.367194
$268$	0	0
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	−10.0000	−0.607457	−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000	−0.607457	−0.303728	+	0.952759i	$0.598232\pi$
$272$	0	0
$273$	−4.00000	−0.242091
$274$	0	0
$275$	−3.00000	−0.180907
$276$	0	0
$277$	4.00000	0.240337	0.120168	−	0.992754i	$-0.461657\pi$
$277$	4.00000	0.240337	0.120168	+	0.992754i	$0.461657\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−16.0000	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$282$	0	0
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	0	0
$285$	−6.00000	−0.355409
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−13.0000	−0.762073
$292$	0	0
$293$	−12.0000	−0.701047	−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000	−0.701047	−0.350524	+	0.936554i	$0.613996\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	−3.00000	−0.174078
$298$	0	0
$299$	−2.00000	−0.115663
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	5.00000	0.287242
$304$	0	0
$305$	4.00000	0.229039
$306$	0	0
$307$	33.0000	1.88341	0.941705	−	0.336440i	$-0.109223\pi$
$307$	33.0000	1.88341	0.941705	+	0.336440i	$0.109223\pi$
$308$	0	0
$309$	−14.0000	−0.796432
$310$	0	0
$311$	9.00000	0.510343	0.255172	−	0.966896i	$-0.417868\pi$
$311$	9.00000	0.510343	0.255172	+	0.966896i	$0.417868\pi$
$312$	0	0
$313$	12.0000	0.678280	0.339140	−	0.940736i	$-0.389864\pi$
$313$	12.0000	0.678280	0.339140	+	0.940736i	$0.389864\pi$
$314$	0	0
$315$	2.00000	0.112687
$316$	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
$319$	−3.00000	−0.167968
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	24.0000	1.33540
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	−9.00000	−0.497701
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	36.0000	1.97874	0.989369	−	0.145424i	$-0.0464545\pi$
$331$	36.0000	1.97874	0.989369	+	0.145424i	$0.0464545\pi$
$332$	0	0
$333$	3.00000	0.164399
$334$	0	0
$335$	−10.0000	−0.546358
$336$	0	0
$337$	−30.0000	−1.63420	−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000	−1.63420	−0.817102	+	0.576493i	$0.804421\pi$
$338$	0	0
$339$	16.0000	0.869001
$340$	0	0
$341$	12.0000	0.649836
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	5.00000	0.268414	0.134207	−	0.990953i	$-0.457151\pi$
$347$	5.00000	0.268414	0.134207	+	0.990953i	$0.457151\pi$
$348$	0	0
$349$	−29.0000	−1.55233	−0.776167	−	0.630527i	$-0.782839\pi$
$349$	−29.0000	−1.55233	−0.776167	+	0.630527i	$0.782839\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	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
$355$	−6.00000	−0.318447
$356$	0	0
$357$	−8.00000	−0.423405
$358$	0	0
$359$	25.0000	1.31945	0.659725	−	0.751507i	$-0.270673\pi$
$359$	25.0000	1.31945	0.659725	+	0.751507i	$0.270673\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−2.00000	−0.104973
$364$	0	0
$365$	1.00000	0.0523424
$366$	0	0
$367$	17.0000	0.887393	0.443696	−	0.896177i	$-0.353667\pi$
$367$	17.0000	0.887393	0.443696	+	0.896177i	$0.353667\pi$
$368$	0	0
$369$	3.00000	0.156174
$370$	0	0
$371$	−2.00000	−0.103835
$372$	0	0
$373$	−12.0000	−0.621336	−0.310668	−	0.950518i	$-0.600553\pi$
$373$	−12.0000	−0.621336	−0.310668	+	0.950518i	$0.600553\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	32.0000	1.64373	0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000	1.64373	0.821865	+	0.569683i	$0.192934\pi$
$380$	0	0
$381$	−7.00000	−0.358621
$382$	0	0
$383$	−15.0000	−0.766464	−0.383232	−	0.923652i	$-0.625189\pi$
$383$	−15.0000	−0.766464	−0.383232	+	0.923652i	$0.625189\pi$
$384$	0	0
$385$	−6.00000	−0.305788
$386$	0	0
$387$	−1.00000	−0.0508329
$388$	0	0
$389$	11.0000	0.557722	0.278861	−	0.960331i	$-0.410043\pi$
$389$	11.0000	0.557722	0.278861	+	0.960331i	$0.410043\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	−20.0000	−1.00887
$394$	0	0
$395$	4.00000	0.201262
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	−12.0000	−0.600751
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−9.00000	−0.446113
$408$	0	0
$409$	−24.0000	−1.18672	−0.593362	−	0.804936i	$-0.702200\pi$
$409$	−24.0000	−1.18672	−0.593362	+	0.804936i	$0.702200\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	0	0
$413$	−8.00000	−0.393654
$414$	0	0
$415$	−17.0000	−0.834497
$416$	0	0
$417$	5.00000	0.244851
$418$	0	0
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	−2.00000	−0.0972433
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	8.00000	0.387147
$428$	0	0
$429$	6.00000	0.289683
$430$	0	0
$431$	−34.0000	−1.63772	−0.818861	−	0.573992i	$-0.805394\pi$
$431$	−34.0000	−1.63772	−0.818861	+	0.573992i	$0.805394\pi$
$432$	0	0
$433$	29.0000	1.39365	0.696826	−	0.717241i	$-0.254595\pi$
$433$	29.0000	1.39365	0.696826	+	0.717241i	$0.254595\pi$
$434$	0	0
$435$	1.00000	0.0479463
$436$	0	0
$437$	−6.00000	−0.287019
$438$	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
$441$	−3.00000	−0.142857
$442$	0	0
$443$	16.0000	0.760183	0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000	0.760183	0.380091	+	0.924949i	$0.375893\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	0	0
$447$	2.00000	0.0945968
$448$	0	0
$449$	21.0000	0.991051	0.495526	−	0.868593i	$-0.334975\pi$
$449$	21.0000	0.991051	0.495526	+	0.868593i	$0.334975\pi$
$450$	0	0
$451$	−9.00000	−0.423793
$452$	0	0
$453$	9.00000	0.422857
$454$	0	0
$455$	−4.00000	−0.187523
$456$	0	0
$457$	−12.0000	−0.561336	−0.280668	−	0.959805i	$-0.590556\pi$
$457$	−12.0000	−0.561336	−0.280668	+	0.959805i	$0.590556\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	27.0000	1.25752	0.628758	−	0.777601i	$-0.283564\pi$
$461$	27.0000	1.25752	0.628758	+	0.777601i	$0.283564\pi$
$462$	0	0
$463$	−14.0000	−0.650635	−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000	−0.650635	−0.325318	+	0.945605i	$0.605471\pi$
$464$	0	0
$465$	−4.00000	−0.185496
$466$	0	0
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	0	0
$469$	−20.0000	−0.923514
$470$	0	0
$471$	−14.0000	−0.645086
$472$	0	0
$473$	3.00000	0.137940
$474$	0	0
$475$	−6.00000	−0.275299
$476$	0	0
$477$	−1.00000	−0.0457869
$478$	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
$483$	2.00000	0.0910032
$484$	0	0
$485$	−13.0000	−0.590300
$486$	0	0
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	0	0
$489$	−11.0000	−0.497437
$490$	0	0
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	0	0
$495$	−3.00000	−0.134840
$496$	0	0
$497$	−12.0000	−0.538274
$498$	0	0
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	0	0
$501$	16.0000	0.714827
$502$	0	0
$503$	10.0000	0.445878	0.222939	−	0.974832i	$-0.428435\pi$
$503$	10.0000	0.445878	0.222939	+	0.974832i	$0.428435\pi$
$504$	0	0
$505$	5.00000	0.222497
$506$	0	0
$507$	−9.00000	−0.399704
$508$	0	0
$509$	−42.0000	−1.86162	−0.930809	−	0.365507i	$-0.880896\pi$
$509$	−42.0000	−1.86162	−0.930809	+	0.365507i	$0.880896\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	−6.00000	−0.264906
$514$	0	0
$515$	−14.0000	−0.616914
$516$	0	0
$517$	6.00000	0.263880
$518$	0	0
$519$	7.00000	0.307266
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	8.00000	0.349816	0.174908	−	0.984585i	$-0.444037\pi$
$523$	8.00000	0.349816	0.174908	+	0.984585i	$0.444037\pi$
$524$	0	0
$525$	2.00000	0.0872872
$526$	0	0
$527$	16.0000	0.696971
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−6.00000	−0.259889
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	6.00000	0.258919
$538$	0	0
$539$	9.00000	0.387657
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	5.00000	0.214571
$544$	0	0
$545$	−9.00000	−0.385518
$546$	0	0
$547$	−32.0000	−1.36822	−0.684111	−	0.729378i	$-0.739809\pi$
$547$	−32.0000	−1.36822	−0.684111	+	0.729378i	$0.739809\pi$
$548$	0	0
$549$	4.00000	0.170716
$550$	0	0
$551$	−6.00000	−0.255609
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	3.00000	0.127343
$556$	0	0
$557$	23.0000	0.974541	0.487271	−	0.873251i	$-0.337993\pi$
$557$	23.0000	0.974541	0.487271	+	0.873251i	$0.337993\pi$
$558$	0	0
$559$	2.00000	0.0845910
$560$	0	0
$561$	12.0000	0.506640
$562$	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
$565$	16.0000	0.673125
$566$	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
$571$	21.0000	0.878823	0.439411	−	0.898286i	$-0.355187\pi$
$571$	21.0000	0.878823	0.439411	+	0.898286i	$0.355187\pi$
$572$	0	0
$573$	−3.00000	−0.125327
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	0	0
$579$	6.00000	0.249351
$580$	0	0
$581$	−34.0000	−1.41056
$582$	0	0
$583$	3.00000	0.124247
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	0	0
$591$	−1.00000	−0.0411345
$592$	0	0
$593$	−2.00000	−0.0821302	−0.0410651	−	0.999156i	$-0.513075\pi$
$593$	−2.00000	−0.0821302	−0.0410651	+	0.999156i	$0.513075\pi$
$594$	0	0
$595$	−8.00000	−0.327968
$596$	0	0
$597$	−13.0000	−0.532055
$598$	0	0
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	−10.0000	−0.407231
$604$	0	0
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	0	0
$611$	4.00000	0.161823
$612$	0	0
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	0	0
$615$	3.00000	0.120972
$616$	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
$619$	−6.00000	−0.241160	−0.120580	−	0.992704i	$-0.538475\pi$
$619$	−6.00000	−0.241160	−0.120580	+	0.992704i	$0.538475\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	12.0000	0.480770
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	18.0000	0.718851
$628$	0	0
$629$	−12.0000	−0.478471
$630$	0	0
$631$	−24.0000	−0.955425	−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000	−0.955425	−0.477712	+	0.878516i	$0.658534\pi$
$632$	0	0
$633$	10.0000	0.397464
$634$	0	0
$635$	−7.00000	−0.277787
$636$	0	0
$637$	6.00000	0.237729
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	13.0000	0.513469	0.256735	−	0.966482i	$-0.417353\pi$
$641$	13.0000	0.513469	0.256735	+	0.966482i	$0.417353\pi$
$642$	0	0
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	0	0
$645$	−1.00000	−0.0393750
$646$	0	0
$647$	7.00000	0.275198	0.137599	−	0.990488i	$-0.456061\pi$
$647$	7.00000	0.275198	0.137599	+	0.990488i	$0.456061\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	−8.00000	−0.313545
$652$	0	0
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	−20.0000	−0.781465
$656$	0	0
$657$	1.00000	0.0390137
$658$	0	0
$659$	13.0000	0.506408	0.253204	−	0.967413i	$-0.418516\pi$
$659$	13.0000	0.506408	0.253204	+	0.967413i	$0.418516\pi$
$660$	0	0
$661$	3.00000	0.116686	0.0583432	−	0.998297i	$-0.481418\pi$
$661$	3.00000	0.116686	0.0583432	+	0.998297i	$0.481418\pi$
$662$	0	0
$663$	8.00000	0.310694
$664$	0	0
$665$	−12.0000	−0.465340
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	4.00000	0.154649
$670$	0	0
$671$	−12.0000	−0.463255
$672$	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
$677$	−8.00000	−0.307465	−0.153732	−	0.988113i	$-0.549129\pi$
$677$	−8.00000	−0.307465	−0.153732	+	0.988113i	$0.549129\pi$
$678$	0	0
$679$	−26.0000	−0.997788
$680$	0	0
$681$	−3.00000	−0.114960
$682$	0	0
$683$	33.0000	1.26271	0.631355	−	0.775494i	$-0.282499\pi$
$683$	33.0000	1.26271	0.631355	+	0.775494i	$0.282499\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	−2.00000	−0.0763048
$688$	0	0
$689$	2.00000	0.0761939
$690$	0	0
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	0	0
$693$	−6.00000	−0.227921
$694$	0	0
$695$	5.00000	0.189661
$696$	0	0
$697$	−12.0000	−0.454532
$698$	0	0
$699$	1.00000	0.0378235
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	−18.0000	−0.678883
$704$	0	0
$705$	−2.00000	−0.0753244
$706$	0	0
$707$	10.0000	0.376089
$708$	0	0
$709$	−21.0000	−0.788672	−0.394336	−	0.918966i	$-0.629025\pi$
$709$	−21.0000	−0.788672	−0.394336	+	0.918966i	$0.629025\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	−4.00000	−0.149801
$714$	0	0
$715$	6.00000	0.224387
$716$	0	0
$717$	6.00000	0.224074
$718$	0	0
$719$	−40.0000	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$720$	0	0
$721$	−28.0000	−1.04277
$722$	0	0
$723$	17.0000	0.632237
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	20.0000	0.741759	0.370879	−	0.928681i	$-0.379056\pi$
$727$	20.0000	0.741759	0.370879	+	0.928681i	$0.379056\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.00000	0.147945
$732$	0	0
$733$	−46.0000	−1.69905	−0.849524	−	0.527549i	$-0.823111\pi$
$733$	−46.0000	−1.69905	−0.849524	+	0.527549i	$0.823111\pi$
$734$	0	0
$735$	−3.00000	−0.110657
$736$	0	0
$737$	30.0000	1.10506
$738$	0	0
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	0	0
$741$	12.0000	0.440831
$742$	0	0
$743$	54.0000	1.98107	0.990534	−	0.137268i	$-0.0438322\pi$
$743$	54.0000	1.98107	0.990534	+	0.137268i	$0.0438322\pi$
$744$	0	0
$745$	2.00000	0.0732743
$746$	0	0
$747$	−17.0000	−0.621997
$748$	0	0
$749$	−8.00000	−0.292314
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	0	0
$755$	9.00000	0.327544
$756$	0	0
$757$	33.0000	1.19941	0.599703	−	0.800223i	$-0.295286\pi$
$757$	33.0000	1.19941	0.599703	+	0.800223i	$0.295286\pi$
$758$	0	0
$759$	−3.00000	−0.108893
$760$	0	0
$761$	36.0000	1.30500	0.652499	−	0.757789i	$-0.273720\pi$
$761$	36.0000	1.30500	0.652499	+	0.757789i	$0.273720\pi$
$762$	0	0
$763$	−18.0000	−0.651644
$764$	0	0
$765$	−4.00000	−0.144620
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	−20.0000	−0.721218	−0.360609	−	0.932717i	$-0.617431\pi$
$769$	−20.0000	−0.721218	−0.360609	+	0.932717i	$0.617431\pi$
$770$	0	0
$771$	13.0000	0.468184
$772$	0	0
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	6.00000	0.215249
$778$	0	0
$779$	−18.0000	−0.644917
$780$	0	0
$781$	18.0000	0.644091
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−14.0000	−0.499681
$786$	0	0
$787$	−8.00000	−0.285169	−0.142585	−	0.989783i	$-0.545541\pi$
$787$	−8.00000	−0.285169	−0.142585	+	0.989783i	$0.545541\pi$
$788$	0	0
$789$	12.0000	0.427211
$790$	0	0
$791$	32.0000	1.13779
$792$	0	0
$793$	−8.00000	−0.284088
$794$	0	0
$795$	−1.00000	−0.0354663
$796$	0	0
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	−3.00000	−0.105868
$804$	0	0
$805$	2.00000	0.0704907
$806$	0	0
$807$	2.00000	0.0704033
$808$	0	0
$809$	−21.0000	−0.738321	−0.369160	−	0.929366i	$-0.620355\pi$
$809$	−21.0000	−0.738321	−0.369160	+	0.929366i	$0.620355\pi$
$810$	0	0
$811$	−33.0000	−1.15879	−0.579393	−	0.815048i	$-0.696710\pi$
$811$	−33.0000	−1.15879	−0.579393	+	0.815048i	$0.696710\pi$
$812$	0	0
$813$	−10.0000	−0.350715
$814$	0	0
$815$	−11.0000	−0.385313
$816$	0	0
$817$	6.00000	0.209913
$818$	0	0
$819$	−4.00000	−0.139771
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	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
$825$	−3.00000	−0.104447
$826$	0	0
$827$	−18.0000	−0.625921	−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000	−0.625921	−0.312961	+	0.949766i	$0.601321\pi$
$828$	0	0
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	0	0
$831$	4.00000	0.138758
$832$	0	0
$833$	12.0000	0.415775
$834$	0	0
$835$	16.0000	0.553703
$836$	0	0
$837$	−4.00000	−0.138260
$838$	0	0
$839$	8.00000	0.276191	0.138095	−	0.990419i	$-0.455902\pi$
$839$	8.00000	0.276191	0.138095	+	0.990419i	$0.455902\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−16.0000	−0.551069
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	−4.00000	−0.137442
$848$	0	0
$849$	−6.00000	−0.205919
$850$	0	0
$851$	3.00000	0.102839
$852$	0	0
$853$	−27.0000	−0.924462	−0.462231	−	0.886759i	$-0.652951\pi$
$853$	−27.0000	−0.924462	−0.462231	+	0.886759i	$0.652951\pi$
$854$	0	0
$855$	−6.00000	−0.205196
$856$	0	0
$857$	−9.00000	−0.307434	−0.153717	−	0.988115i	$-0.549124\pi$
$857$	−9.00000	−0.307434	−0.153717	+	0.988115i	$0.549124\pi$
$858$	0	0
$859$	−22.0000	−0.750630	−0.375315	−	0.926897i	$-0.622466\pi$
$859$	−22.0000	−0.750630	−0.375315	+	0.926897i	$0.622466\pi$
$860$	0	0
$861$	6.00000	0.204479
$862$	0	0
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	0	0
$865$	7.00000	0.238007
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−12.0000	−0.407072
$870$	0	0
$871$	20.0000	0.677674
$872$	0	0
$873$	−13.0000	−0.439983
$874$	0	0
$875$	2.00000	0.0676123
$876$	0	0
$877$	4.00000	0.135070	0.0675352	−	0.997717i	$-0.478487\pi$
$877$	4.00000	0.135070	0.0675352	+	0.997717i	$0.478487\pi$
$878$	0	0
$879$	−12.0000	−0.404750
$880$	0	0
$881$	45.0000	1.51609	0.758044	−	0.652203i	$-0.226155\pi$
$881$	45.0000	1.51609	0.758044	+	0.652203i	$0.226155\pi$
$882$	0	0
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	0	0
$885$	−4.00000	−0.134459
$886$	0	0
$887$	10.0000	0.335767	0.167884	−	0.985807i	$-0.446307\pi$
$887$	10.0000	0.335767	0.167884	+	0.985807i	$0.446307\pi$
$888$	0	0
$889$	−14.0000	−0.469545
$890$	0	0
$891$	−3.00000	−0.100504
$892$	0	0
$893$	12.0000	0.401565
$894$	0	0
$895$	6.00000	0.200558
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	0	0
$899$	−4.00000	−0.133407
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	−2.00000	−0.0665558
$904$	0	0
$905$	5.00000	0.166206
$906$	0	0
$907$	−3.00000	−0.0996134	−0.0498067	−	0.998759i	$-0.515861\pi$
$907$	−3.00000	−0.0996134	−0.0498067	+	0.998759i	$0.515861\pi$
$908$	0	0
$909$	5.00000	0.165840
$910$	0	0
$911$	27.0000	0.894550	0.447275	−	0.894397i	$-0.352395\pi$
$911$	27.0000	0.894550	0.447275	+	0.894397i	$0.352395\pi$
$912$	0	0
$913$	51.0000	1.68785
$914$	0	0
$915$	4.00000	0.132236
$916$	0	0
$917$	−40.0000	−1.32092
$918$	0	0
$919$	−44.0000	−1.45143	−0.725713	−	0.687998i	$-0.758490\pi$
$919$	−44.0000	−1.45143	−0.725713	+	0.687998i	$0.758490\pi$
$920$	0	0
$921$	33.0000	1.08739
$922$	0	0
$923$	12.0000	0.394985
$924$	0	0
$925$	3.00000	0.0986394
$926$	0	0
$927$	−14.0000	−0.459820
$928$	0	0
$929$	−8.00000	−0.262471	−0.131236	−	0.991351i	$-0.541894\pi$
$929$	−8.00000	−0.262471	−0.131236	+	0.991351i	$0.541894\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	0	0
$933$	9.00000	0.294647
$934$	0	0
$935$	12.0000	0.392442
$936$	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
$939$	12.0000	0.391605
$940$	0	0
$941$	50.0000	1.62995	0.814977	−	0.579494i	$-0.196750\pi$
$941$	50.0000	1.62995	0.814977	+	0.579494i	$0.196750\pi$
$942$	0	0
$943$	3.00000	0.0976934
$944$	0	0
$945$	2.00000	0.0650600
$946$	0	0
$947$	14.0000	0.454939	0.227469	−	0.973785i	$-0.426955\pi$
$947$	14.0000	0.454939	0.227469	+	0.973785i	$0.426955\pi$
$948$	0	0
$949$	−2.00000	−0.0649227
$950$	0	0
$951$	−32.0000	−1.03767
$952$	0	0
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	0	0
$955$	−3.00000	−0.0970777
$956$	0	0
$957$	−3.00000	−0.0969762
$958$	0	0
$959$	24.0000	0.775000
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−4.00000	−0.128898
$964$	0	0
$965$	6.00000	0.193147
$966$	0	0
$967$	29.0000	0.932577	0.466289	−	0.884633i	$-0.345591\pi$
$967$	29.0000	0.932577	0.466289	+	0.884633i	$0.345591\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	9.00000	0.288824	0.144412	−	0.989518i	$-0.453871\pi$
$971$	9.00000	0.288824	0.144412	+	0.989518i	$0.453871\pi$
$972$	0	0
$973$	10.0000	0.320585
$974$	0	0
$975$	−2.00000	−0.0640513
$976$	0	0
$977$	23.0000	0.735835	0.367918	−	0.929858i	$-0.380071\pi$
$977$	23.0000	0.735835	0.367918	+	0.929858i	$0.380071\pi$
$978$	0	0
$979$	−18.0000	−0.575282
$980$	0	0
$981$	−9.00000	−0.287348
$982$	0	0
$983$	−10.0000	−0.318950	−0.159475	−	0.987202i	$-0.550980\pi$
$983$	−10.0000	−0.318950	−0.159475	+	0.987202i	$0.550980\pi$
$984$	0	0
$985$	−1.00000	−0.0318626
$986$	0	0
$987$	−4.00000	−0.127321
$988$	0	0
$989$	−1.00000	−0.0317982
$990$	0	0
$991$	51.0000	1.62007	0.810034	−	0.586383i	$-0.199448\pi$
$991$	51.0000	1.62007	0.810034	+	0.586383i	$0.199448\pi$
$992$	0	0
$993$	36.0000	1.14243
$994$	0	0
$995$	−13.0000	−0.412128
$996$	0	0
$997$	7.00000	0.221692	0.110846	−	0.993838i	$-0.464644\pi$
$997$	7.00000	0.221692	0.110846	+	0.993838i	$0.464644\pi$
$998$	0	0
$999$	3.00000	0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6960.2.a.bl.1.1		1
4.3	odd	2		1740.2.a.b.1.1	✓	1
12.11	even	2		5220.2.a.c.1.1		1
20.3	even	4		8700.2.g.k.349.1		2
20.7	even	4		8700.2.g.k.349.2		2
20.19	odd	2		8700.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1740.2.a.b.1.1	✓	1	4.3	odd	2
5220.2.a.c.1.1		1	12.11	even	2
6960.2.a.bl.1.1		1	1.1	even	1	trivial
8700.2.a.r.1.1		1	20.19	odd	2
8700.2.g.k.349.1		2	20.3	even	4
8700.2.g.k.349.2		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:	\( N \)	\(=\)	\( 6960 = 2^{4} \cdot 3 \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6960.a (trivial)

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