LMFDB - Embedded newform 1960.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1960,2,Mod(1,1960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1960 = 2^{3} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1960.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$15.6506787962$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+2.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} -1.00000 q^{11} +3.00000 q^{13} +2.00000 q^{15} +2.00000 q^{17} +5.00000 q^{19} +7.00000 q^{23} +1.00000 q^{25} -4.00000 q^{27} -6.00000 q^{29} -4.00000 q^{31} -2.00000 q^{33} -5.00000 q^{37} +6.00000 q^{39} +5.00000 q^{41} +6.00000 q^{43} +1.00000 q^{45} +9.00000 q^{47} +4.00000 q^{51} +11.0000 q^{53} -1.00000 q^{55} +10.0000 q^{57} -8.00000 q^{59} +12.0000 q^{61} +3.00000 q^{65} -4.00000 q^{67} +14.0000 q^{69} -4.00000 q^{71} -12.0000 q^{73} +2.00000 q^{75} +14.0000 q^{79} -11.0000 q^{81} +4.00000 q^{83} +2.00000 q^{85} -12.0000 q^{87} -6.00000 q^{89} -8.00000 q^{93} +5.00000 q^{95} -6.00000 q^{97} -1.00000 q^{99} +O(q^{100})$

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
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.00000	1.45960	0.729800	−	0.683660i	$-0.239613\pi$
$23$	7.00000	1.45960	0.729800	+	0.683660i	$0.239613\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	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
$33$	−2.00000	−0.348155
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	0	0
$39$	6.00000	0.960769
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	11.0000	1.51097	0.755483	−	0.655168i	$-0.227402\pi$
$53$	11.0000	1.51097	0.755483	+	0.655168i	$0.227402\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	10.0000	1.32453
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.00000	0.372104
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	14.0000	1.68540
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	0	0
$75$	2.00000	0.230940
$76$	0	0
$77$	0	0
$78$	0	0
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	−12.0000	−1.28654
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	5.00000	0.512989
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	−20.0000	−1.97066	−0.985329	−	0.170664i	$-0.945409\pi$
$103$	−20.0000	−1.97066	−0.985329	+	0.170664i	$0.945409\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	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
$115$	7.00000	0.652753
$116$	0	0
$117$	3.00000	0.277350
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	10.0000	0.901670
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	17.0000	1.50851	0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000	1.50851	0.754253	+	0.656584i	$0.227999\pi$
$128$	0	0
$129$	12.0000	1.05654
$130$	0	0
$131$	−7.00000	−0.611593	−0.305796	−	0.952097i	$-0.598923\pi$
$131$	−7.00000	−0.611593	−0.305796	+	0.952097i	$0.598923\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	18.0000	1.51587
$142$	0	0
$143$	−3.00000	−0.250873
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−5.00000	−0.399043	−0.199522	−	0.979893i	$-0.563939\pi$
$157$	−5.00000	−0.399043	−0.199522	+	0.979893i	$0.563939\pi$
$158$	0	0
$159$	22.0000	1.74471
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	0	0
$167$	−5.00000	−0.386912	−0.193456	−	0.981109i	$-0.561970\pi$
$167$	−5.00000	−0.386912	−0.193456	+	0.981109i	$0.561970\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	5.00000	0.382360
$172$	0	0
$173$	−19.0000	−1.44454	−0.722272	−	0.691609i	$-0.756902\pi$
$173$	−19.0000	−1.44454	−0.722272	+	0.691609i	$0.756902\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−16.0000	−1.20263
$178$	0	0
$179$	9.00000	0.672692	0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000	0.672692	0.336346	+	0.941739i	$0.390809\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	24.0000	1.77413
$184$	0	0
$185$	−5.00000	−0.367607
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	−20.0000	−1.43963	−0.719816	−	0.694165i	$-0.755774\pi$
$193$	−20.0000	−1.43963	−0.719816	+	0.694165i	$0.755774\pi$
$194$	0	0
$195$	6.00000	0.429669
$196$	0	0
$197$	−27.0000	−1.92367	−0.961835	−	0.273629i	$-0.911776\pi$
$197$	−27.0000	−1.92367	−0.961835	+	0.273629i	$0.911776\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	0	0
$204$	0	0
$205$	5.00000	0.349215
$206$	0	0
$207$	7.00000	0.486534
$208$	0	0
$209$	−5.00000	−0.345857
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	0	0
$213$	−8.00000	−0.548151
$214$	0	0
$215$	6.00000	0.409197
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−24.0000	−1.62177
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	0	0
$229$	28.0000	1.85029	0.925146	−	0.379611i	$-0.123942\pi$
$229$	28.0000	1.85029	0.925146	+	0.379611i	$0.123942\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.00000	0.131024	0.0655122	−	0.997852i	$-0.479132\pi$
$233$	2.00000	0.131024	0.0655122	+	0.997852i	$0.479132\pi$
$234$	0	0
$235$	9.00000	0.587095
$236$	0	0
$237$	28.0000	1.81880
$238$	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
$241$	−23.0000	−1.48156	−0.740780	−	0.671748i	$-0.765544\pi$
$241$	−23.0000	−1.48156	−0.740780	+	0.671748i	$0.765544\pi$
$242$	0	0
$243$	−10.0000	−0.641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	15.0000	0.954427
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	−29.0000	−1.83046	−0.915232	−	0.402928i	$-0.867993\pi$
$251$	−29.0000	−1.83046	−0.915232	+	0.402928i	$0.867993\pi$
$252$	0	0
$253$	−7.00000	−0.440086
$254$	0	0
$255$	4.00000	0.250490
$256$	0	0
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	11.0000	0.675725
$266$	0	0
$267$	−12.0000	−0.734388
$268$	0	0
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−3.00000	−0.178965	−0.0894825	−	0.995988i	$-0.528521\pi$
$281$	−3.00000	−0.178965	−0.0894825	+	0.995988i	$0.528521\pi$
$282$	0	0
$283$	22.0000	1.30776	0.653882	−	0.756596i	$-0.273139\pi$
$283$	22.0000	1.30776	0.653882	+	0.756596i	$0.273139\pi$
$284$	0	0
$285$	10.0000	0.592349
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−12.0000	−0.703452
$292$	0	0
$293$	−21.0000	−1.22683	−0.613417	−	0.789760i	$-0.710205\pi$
$293$	−21.0000	−1.22683	−0.613417	+	0.789760i	$0.710205\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	21.0000	1.21446
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−24.0000	−1.37876
$304$	0	0
$305$	12.0000	0.687118
$306$	0	0
$307$	6.00000	0.342438	0.171219	−	0.985233i	$-0.445229\pi$
$307$	6.00000	0.342438	0.171219	+	0.985233i	$0.445229\pi$
$308$	0	0
$309$	−40.0000	−2.27552
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	24.0000	1.33955
$322$	0	0
$323$	10.0000	0.556415
$324$	0	0
$325$	3.00000	0.166410
$326$	0	0
$327$	−8.00000	−0.442401
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−27.0000	−1.48405	−0.742027	−	0.670370i	$-0.766135\pi$
$331$	−27.0000	−1.48405	−0.742027	+	0.670370i	$0.766135\pi$
$332$	0	0
$333$	−5.00000	−0.273998
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	0	0
$339$	40.0000	2.17250
$340$	0	0
$341$	4.00000	0.216612
$342$	0	0
$343$	0	0
$344$	0	0
$345$	14.0000	0.753735
$346$	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
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	−12.0000	−0.640513
$352$	0	0
$353$	−20.0000	−1.06449	−0.532246	−	0.846590i	$-0.678652\pi$
$353$	−20.0000	−1.06449	−0.532246	+	0.846590i	$0.678652\pi$
$354$	0	0
$355$	−4.00000	−0.212298
$356$	0	0
$357$	0	0
$358$	0	0
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	−20.0000	−1.04973
$364$	0	0
$365$	−12.0000	−0.628109
$366$	0	0
$367$	−19.0000	−0.991792	−0.495896	−	0.868382i	$-0.665160\pi$
$367$	−19.0000	−0.991792	−0.495896	+	0.868382i	$0.665160\pi$
$368$	0	0
$369$	5.00000	0.260290
$370$	0	0
$371$	0	0
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	2.00000	0.103280
$376$	0	0
$377$	−18.0000	−0.927047
$378$	0	0
$379$	−21.0000	−1.07870	−0.539349	−	0.842082i	$-0.681330\pi$
$379$	−21.0000	−1.07870	−0.539349	+	0.842082i	$0.681330\pi$
$380$	0	0
$381$	34.0000	1.74187
$382$	0	0
$383$	21.0000	1.07305	0.536525	−	0.843884i	$-0.319737\pi$
$383$	21.0000	1.07305	0.536525	+	0.843884i	$0.319737\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.00000	0.304997
$388$	0	0
$389$	−16.0000	−0.811232	−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000	−0.811232	−0.405616	+	0.914044i	$0.632943\pi$
$390$	0	0
$391$	14.0000	0.708010
$392$	0	0
$393$	−14.0000	−0.706207
$394$	0	0
$395$	14.0000	0.704416
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.0000	0.649189	0.324595	−	0.945853i	$-0.394772\pi$
$401$	13.0000	0.649189	0.324595	+	0.945853i	$0.394772\pi$
$402$	0	0
$403$	−12.0000	−0.597763
$404$	0	0
$405$	−11.0000	−0.546594
$406$	0	0
$407$	5.00000	0.247841
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	−24.0000	−1.18383
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	−8.00000	−0.391762
$418$	0	0
$419$	5.00000	0.244266	0.122133	−	0.992514i	$-0.461027\pi$
$419$	5.00000	0.244266	0.122133	+	0.992514i	$0.461027\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	0	0
$423$	9.00000	0.437595
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−6.00000	−0.289683
$430$	0	0
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	0	0
$433$	−24.0000	−1.15337	−0.576683	−	0.816968i	$-0.695653\pi$
$433$	−24.0000	−1.15337	−0.576683	+	0.816968i	$0.695653\pi$
$434$	0	0
$435$	−12.0000	−0.575356
$436$	0	0
$437$	35.0000	1.67428
$438$	0	0
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−32.0000	−1.52037	−0.760183	−	0.649709i	$-0.774891\pi$
$443$	−32.0000	−1.52037	−0.760183	+	0.649709i	$0.774891\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	0	0
$447$	20.0000	0.945968
$448$	0	0
$449$	13.0000	0.613508	0.306754	−	0.951789i	$-0.400757\pi$
$449$	13.0000	0.613508	0.306754	+	0.951789i	$0.400757\pi$
$450$	0	0
$451$	−5.00000	−0.235441
$452$	0	0
$453$	−20.0000	−0.939682
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	0	0
$459$	−8.00000	−0.373408
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	9.00000	0.418265	0.209133	−	0.977887i	$-0.432936\pi$
$463$	9.00000	0.418265	0.209133	+	0.977887i	$0.432936\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	10.0000	0.462745	0.231372	−	0.972865i	$-0.425678\pi$
$467$	10.0000	0.462745	0.231372	+	0.972865i	$0.425678\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−10.0000	−0.460776
$472$	0	0
$473$	−6.00000	−0.275880
$474$	0	0
$475$	5.00000	0.229416
$476$	0	0
$477$	11.0000	0.503655
$478$	0	0
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	−15.0000	−0.683941
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.00000	−0.272446
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	0	0
$489$	8.00000	0.361773
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	−12.0000	−0.540453
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	0	0
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	−10.0000	−0.446767
$502$	0	0
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	−8.00000	−0.355292
$508$	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
$511$	0	0
$512$	0	0
$513$	−20.0000	−0.883022
$514$	0	0
$515$	−20.0000	−0.881305
$516$	0	0
$517$	−9.00000	−0.395820
$518$	0	0
$519$	−38.0000	−1.66801
$520$	0	0
$521$	15.0000	0.657162	0.328581	−	0.944476i	$-0.393430\pi$
$521$	15.0000	0.657162	0.328581	+	0.944476i	$0.393430\pi$
$522$	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
$525$	0	0
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	26.0000	1.13043
$530$	0	0
$531$	−8.00000	−0.347170
$532$	0	0
$533$	15.0000	0.649722
$534$	0	0
$535$	12.0000	0.518805
$536$	0	0
$537$	18.0000	0.776757
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−40.0000	−1.71973	−0.859867	−	0.510518i	$-0.829454\pi$
$541$	−40.0000	−1.71973	−0.859867	+	0.510518i	$0.829454\pi$
$542$	0	0
$543$	−4.00000	−0.171656
$544$	0	0
$545$	−4.00000	−0.171341
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	12.0000	0.512148
$550$	0	0
$551$	−30.0000	−1.27804
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−10.0000	−0.424476
$556$	0	0
$557$	21.0000	0.889799	0.444899	−	0.895581i	$-0.353239\pi$
$557$	21.0000	0.889799	0.444899	+	0.895581i	$0.353239\pi$
$558$	0	0
$559$	18.0000	0.761319
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	0	0
$565$	20.0000	0.841406
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11.0000	−0.461144	−0.230572	−	0.973055i	$-0.574060\pi$
$569$	−11.0000	−0.461144	−0.230572	+	0.973055i	$0.574060\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	0	0
$573$	24.0000	1.00261
$574$	0	0
$575$	7.00000	0.291920
$576$	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
$579$	−40.0000	−1.66234
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−11.0000	−0.455573
$584$	0	0
$585$	3.00000	0.124035
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	−20.0000	−0.824086
$590$	0	0
$591$	−54.0000	−2.22126
$592$	0	0
$593$	12.0000	0.492781	0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000	0.492781	0.246390	+	0.969171i	$0.420755\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.00000	0.327418
$598$	0	0
$599$	34.0000	1.38920	0.694601	−	0.719395i	$-0.255581\pi$
$599$	34.0000	1.38920	0.694601	+	0.719395i	$0.255581\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	−10.0000	−0.406558
$606$	0	0
$607$	−33.0000	−1.33943	−0.669714	−	0.742619i	$-0.733583\pi$
$607$	−33.0000	−1.33943	−0.669714	+	0.742619i	$0.733583\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	27.0000	1.09230
$612$	0	0
$613$	−41.0000	−1.65597	−0.827987	−	0.560747i	$-0.810514\pi$
$613$	−41.0000	−1.65597	−0.827987	+	0.560747i	$0.810514\pi$
$614$	0	0
$615$	10.0000	0.403239
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	0	0
$619$	−35.0000	−1.40677	−0.703384	−	0.710810i	$-0.748329\pi$
$619$	−35.0000	−1.40677	−0.703384	+	0.710810i	$0.748329\pi$
$620$	0	0
$621$	−28.0000	−1.12360
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−10.0000	−0.399362
$628$	0	0
$629$	−10.0000	−0.398726
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	0	0
$633$	−26.0000	−1.03341
$634$	0	0
$635$	17.0000	0.674624
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−4.00000	−0.158238
$640$	0	0
$641$	35.0000	1.38242	0.691208	−	0.722655i	$-0.257079\pi$
$641$	35.0000	1.38242	0.691208	+	0.722655i	$0.257079\pi$
$642$	0	0
$643$	34.0000	1.34083	0.670415	−	0.741987i	$-0.266116\pi$
$643$	34.0000	1.34083	0.670415	+	0.741987i	$0.266116\pi$
$644$	0	0
$645$	12.0000	0.472500
$646$	0	0
$647$	−23.0000	−0.904223	−0.452112	−	0.891961i	$-0.649329\pi$
$647$	−23.0000	−0.904223	−0.452112	+	0.891961i	$0.649329\pi$
$648$	0	0
$649$	8.00000	0.314027
$650$	0	0
$651$	0	0
$652$	0	0
$653$	45.0000	1.76099	0.880493	−	0.474059i	$-0.157212\pi$
$653$	45.0000	1.76099	0.880493	+	0.474059i	$0.157212\pi$
$654$	0	0
$655$	−7.00000	−0.273513
$656$	0	0
$657$	−12.0000	−0.468165
$658$	0	0
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	0	0
$661$	36.0000	1.40024	0.700119	−	0.714026i	$-0.253130\pi$
$661$	36.0000	1.40024	0.700119	+	0.714026i	$0.253130\pi$
$662$	0	0
$663$	12.0000	0.466041
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−42.0000	−1.62625
$668$	0	0
$669$	32.0000	1.23719
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	−42.0000	−1.61898	−0.809491	−	0.587133i	$-0.800257\pi$
$673$	−42.0000	−1.61898	−0.809491	+	0.587133i	$0.800257\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−21.0000	−0.807096	−0.403548	−	0.914959i	$-0.632223\pi$
$677$	−21.0000	−0.807096	−0.403548	+	0.914959i	$0.632223\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	16.0000	0.613121
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	0	0
$687$	56.0000	2.13653
$688$	0	0
$689$	33.0000	1.25720
$690$	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
$695$	−4.00000	−0.151729
$696$	0	0
$697$	10.0000	0.378777
$698$	0	0
$699$	4.00000	0.151294
$700$	0	0
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	−25.0000	−0.942893
$704$	0	0
$705$	18.0000	0.677919
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	0	0
$711$	14.0000	0.525041
$712$	0	0
$713$	−28.0000	−1.04861
$714$	0	0
$715$	−3.00000	−0.112194
$716$	0	0
$717$	12.0000	0.448148
$718$	0	0
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−46.0000	−1.71076
$724$	0	0
$725$	−6.00000	−0.222834
$726$	0	0
$727$	−41.0000	−1.52061	−0.760303	−	0.649569i	$-0.774949\pi$
$727$	−41.0000	−1.52061	−0.760303	+	0.649569i	$0.774949\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	12.0000	0.443836
$732$	0	0
$733$	−31.0000	−1.14501	−0.572506	−	0.819901i	$-0.694029\pi$
$733$	−31.0000	−1.14501	−0.572506	+	0.819901i	$0.694029\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.00000	0.147342
$738$	0	0
$739$	−41.0000	−1.50821	−0.754105	−	0.656754i	$-0.771929\pi$
$739$	−41.0000	−1.50821	−0.754105	+	0.656754i	$0.771929\pi$
$740$	0	0
$741$	30.0000	1.10208
$742$	0	0
$743$	−3.00000	−0.110059	−0.0550297	−	0.998485i	$-0.517525\pi$
$743$	−3.00000	−0.110059	−0.0550297	+	0.998485i	$0.517525\pi$
$744$	0	0
$745$	10.0000	0.366372
$746$	0	0
$747$	4.00000	0.146352
$748$	0	0
$749$	0	0
$750$	0	0
$751$	18.0000	0.656829	0.328415	−	0.944534i	$-0.393486\pi$
$751$	18.0000	0.656829	0.328415	+	0.944534i	$0.393486\pi$
$752$	0	0
$753$	−58.0000	−2.11364
$754$	0	0
$755$	−10.0000	−0.363937
$756$	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
$759$	−14.0000	−0.508168
$760$	0	0
$761$	3.00000	0.108750	0.0543750	−	0.998521i	$-0.482683\pi$
$761$	3.00000	0.108750	0.0543750	+	0.998521i	$0.482683\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.00000	0.0723102
$766$	0	0
$767$	−24.0000	−0.866590
$768$	0	0
$769$	41.0000	1.47850	0.739249	−	0.673432i	$-0.235181\pi$
$769$	41.0000	1.47850	0.739249	+	0.673432i	$0.235181\pi$
$770$	0	0
$771$	−24.0000	−0.864339
$772$	0	0
$773$	−15.0000	−0.539513	−0.269756	−	0.962929i	$-0.586943\pi$
$773$	−15.0000	−0.539513	−0.269756	+	0.962929i	$0.586943\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	0	0
$778$	0	0
$779$	25.0000	0.895718
$780$	0	0
$781$	4.00000	0.143131
$782$	0	0
$783$	24.0000	0.857690
$784$	0	0
$785$	−5.00000	−0.178458
$786$	0	0
$787$	18.0000	0.641631	0.320815	−	0.947142i	$-0.396043\pi$
$787$	18.0000	0.641631	0.320815	+	0.947142i	$0.396043\pi$
$788$	0	0
$789$	16.0000	0.569615
$790$	0	0
$791$	0	0
$792$	0	0
$793$	36.0000	1.27840
$794$	0	0
$795$	22.0000	0.780260
$796$	0	0
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	18.0000	0.636794
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	12.0000	0.423471
$804$	0	0
$805$	0	0
$806$	0	0
$807$	24.0000	0.844840
$808$	0	0
$809$	55.0000	1.93370	0.966849	−	0.255351i	$-0.0821909\pi$
$809$	55.0000	1.93370	0.966849	+	0.255351i	$0.0821909\pi$
$810$	0	0
$811$	5.00000	0.175574	0.0877869	−	0.996139i	$-0.472021\pi$
$811$	5.00000	0.175574	0.0877869	+	0.996139i	$0.472021\pi$
$812$	0	0
$813$	16.0000	0.561144
$814$	0	0
$815$	4.00000	0.140114
$816$	0	0
$817$	30.0000	1.04957
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	0	0
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	−14.0000	−0.486828	−0.243414	−	0.969923i	$-0.578267\pi$
$827$	−14.0000	−0.486828	−0.243414	+	0.969923i	$0.578267\pi$
$828$	0	0
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	0	0
$831$	4.00000	0.138758
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−5.00000	−0.173032
$836$	0	0
$837$	16.0000	0.553041
$838$	0	0
$839$	14.0000	0.483334	0.241667	−	0.970359i	$-0.422306\pi$
$839$	14.0000	0.483334	0.241667	+	0.970359i	$0.422306\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−6.00000	−0.206651
$844$	0	0
$845$	−4.00000	−0.137604
$846$	0	0
$847$	0	0
$848$	0	0
$849$	44.0000	1.51008
$850$	0	0
$851$	−35.0000	−1.19978
$852$	0	0
$853$	−23.0000	−0.787505	−0.393753	−	0.919216i	$-0.628823\pi$
$853$	−23.0000	−0.787505	−0.393753	+	0.919216i	$0.628823\pi$
$854$	0	0
$855$	5.00000	0.170996
$856$	0	0
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−13.0000	−0.442525	−0.221263	−	0.975214i	$-0.571018\pi$
$863$	−13.0000	−0.442525	−0.221263	+	0.975214i	$0.571018\pi$
$864$	0	0
$865$	−19.0000	−0.646019
$866$	0	0
$867$	−26.0000	−0.883006
$868$	0	0
$869$	−14.0000	−0.474917
$870$	0	0
$871$	−12.0000	−0.406604
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	0	0
$876$	0	0
$877$	9.00000	0.303908	0.151954	−	0.988388i	$-0.451443\pi$
$877$	9.00000	0.303908	0.151954	+	0.988388i	$0.451443\pi$
$878$	0	0
$879$	−42.0000	−1.41662
$880$	0	0
$881$	7.00000	0.235836	0.117918	−	0.993023i	$-0.462378\pi$
$881$	7.00000	0.235836	0.117918	+	0.993023i	$0.462378\pi$
$882$	0	0
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	0	0
$885$	−16.0000	−0.537834
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	11.0000	0.368514
$892$	0	0
$893$	45.0000	1.50587
$894$	0	0
$895$	9.00000	0.300837
$896$	0	0
$897$	42.0000	1.40234
$898$	0	0
$899$	24.0000	0.800445
$900$	0	0
$901$	22.0000	0.732926
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	22.0000	0.730498	0.365249	−	0.930910i	$-0.380984\pi$
$907$	22.0000	0.730498	0.365249	+	0.930910i	$0.380984\pi$
$908$	0	0
$909$	−12.0000	−0.398015
$910$	0	0
$911$	42.0000	1.39152	0.695761	−	0.718273i	$-0.255067\pi$
$911$	42.0000	1.39152	0.695761	+	0.718273i	$0.255067\pi$
$912$	0	0
$913$	−4.00000	−0.132381
$914$	0	0
$915$	24.0000	0.793416
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−18.0000	−0.593765	−0.296883	−	0.954914i	$-0.595947\pi$
$919$	−18.0000	−0.593765	−0.296883	+	0.954914i	$0.595947\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	0	0
$923$	−12.0000	−0.394985
$924$	0	0
$925$	−5.00000	−0.164399
$926$	0	0
$927$	−20.0000	−0.656886
$928$	0	0
$929$	31.0000	1.01708	0.508539	−	0.861039i	$-0.330186\pi$
$929$	31.0000	1.01708	0.508539	+	0.861039i	$0.330186\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	8.00000	0.261908
$934$	0	0
$935$	−2.00000	−0.0654070
$936$	0	0
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	0	0
$939$	−32.0000	−1.04428
$940$	0	0
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	0	0
$943$	35.0000	1.13976
$944$	0	0
$945$	0	0
$946$	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
$949$	−36.0000	−1.16861
$950$	0	0
$951$	4.00000	0.129709
$952$	0	0
$953$	−20.0000	−0.647864	−0.323932	−	0.946080i	$-0.605005\pi$
$953$	−20.0000	−0.647864	−0.323932	+	0.946080i	$0.605005\pi$
$954$	0	0
$955$	12.0000	0.388311
$956$	0	0
$957$	12.0000	0.387905
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	12.0000	0.386695
$964$	0	0
$965$	−20.0000	−0.643823
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	0	0
$969$	20.0000	0.642493
$970$	0	0
$971$	−7.00000	−0.224641	−0.112320	−	0.993672i	$-0.535828\pi$
$971$	−7.00000	−0.224641	−0.112320	+	0.993672i	$0.535828\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	6.00000	0.192154
$976$	0	0
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	6.00000	0.191761
$980$	0	0
$981$	−4.00000	−0.127710
$982$	0	0
$983$	3.00000	0.0956851	0.0478426	−	0.998855i	$-0.484765\pi$
$983$	3.00000	0.0956851	0.0478426	+	0.998855i	$0.484765\pi$
$984$	0	0
$985$	−27.0000	−0.860292
$986$	0	0
$987$	0	0
$988$	0	0
$989$	42.0000	1.33552
$990$	0	0
$991$	44.0000	1.39771	0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000	1.39771	0.698853	+	0.715265i	$0.253694\pi$
$992$	0	0
$993$	−54.0000	−1.71364
$994$	0	0
$995$	4.00000	0.126809
$996$	0	0
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	0	0
$999$	20.0000	0.632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1960.2.a.m.1.1		1
4.3	odd	2		3920.2.a.i.1.1		1
5.4	even	2		9800.2.a.g.1.1		1
7.2	even	3		1960.2.q.c.361.1		2
7.3	odd	6		280.2.q.c.121.1	yes	2
7.4	even	3		1960.2.q.c.961.1		2
7.5	odd	6		280.2.q.c.81.1	✓	2
7.6	odd	2		1960.2.a.a.1.1		1
21.5	even	6		2520.2.bi.e.361.1		2
21.17	even	6		2520.2.bi.e.1801.1		2
28.3	even	6		560.2.q.c.401.1		2
28.19	even	6		560.2.q.c.81.1		2
28.27	even	2		3920.2.a.bf.1.1		1
35.3	even	12		1400.2.bh.e.849.1		4
35.12	even	12		1400.2.bh.e.249.1		4
35.17	even	12		1400.2.bh.e.849.2		4
35.19	odd	6		1400.2.q.a.1201.1		2
35.24	odd	6		1400.2.q.a.401.1		2
35.33	even	12		1400.2.bh.e.249.2		4
35.34	odd	2		9800.2.a.bi.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.q.c.81.1	✓	2	7.5	odd	6
280.2.q.c.121.1	yes	2	7.3	odd	6
560.2.q.c.81.1		2	28.19	even	6
560.2.q.c.401.1		2	28.3	even	6
1400.2.q.a.401.1		2	35.24	odd	6
1400.2.q.a.1201.1		2	35.19	odd	6
1400.2.bh.e.249.1		4	35.12	even	12
1400.2.bh.e.249.2		4	35.33	even	12
1400.2.bh.e.849.1		4	35.3	even	12
1400.2.bh.e.849.2		4	35.17	even	12
1960.2.a.a.1.1		1	7.6	odd	2
1960.2.a.m.1.1		1	1.1	even	1	trivial
1960.2.q.c.361.1		2	7.2	even	3
1960.2.q.c.961.1		2	7.4	even	3
2520.2.bi.e.361.1		2	21.5	even	6
2520.2.bi.e.1801.1		2	21.17	even	6
3920.2.a.i.1.1		1	4.3	odd	2
3920.2.a.bf.1.1		1	28.27	even	2
9800.2.a.g.1.1		1	5.4	even	2
9800.2.a.bi.1.1		1	35.34	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:	\( N \)	\(=\)	\( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1960.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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