LMFDB - Embedded newform 124.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [124,2,Mod(1,124)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(124, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("124.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 124 = 2^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	124.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:	$0.990144985064$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	124.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{3} -3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-2.00000 q^{3} -3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -6.00000 q^{11} +2.00000 q^{13} +6.00000 q^{15} +6.00000 q^{17} -1.00000 q^{19} +2.00000 q^{21} -6.00000 q^{23} +4.00000 q^{25} +4.00000 q^{27} +1.00000 q^{31} +12.0000 q^{33} +3.00000 q^{35} -10.0000 q^{37} -4.00000 q^{39} -9.00000 q^{41} +8.00000 q^{43} -3.00000 q^{45} -6.00000 q^{49} -12.0000 q^{51} +18.0000 q^{55} +2.00000 q^{57} -3.00000 q^{59} -10.0000 q^{61} -1.00000 q^{63} -6.00000 q^{65} -4.00000 q^{67} +12.0000 q^{69} -15.0000 q^{71} +14.0000 q^{73} -8.00000 q^{75} +6.00000 q^{77} +8.00000 q^{79} -11.0000 q^{81} +6.00000 q^{83} -18.0000 q^{85} +12.0000 q^{89} -2.00000 q^{91} -2.00000 q^{93} +3.00000 q^{95} -7.00000 q^{97} -6.00000 q^{99} +O(q^{100})$

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	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	0	0
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	6.00000	1.54919
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	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
$25$	4.00000	0.800000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	12.0000	2.08893
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	−3.00000	−0.447214
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−12.0000	−1.68034
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	18.0000	2.42712
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	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
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−6.00000	−0.744208
$66$	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
$69$	12.0000	1.44463
$70$	0	0
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	0	0
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	0	0
$75$	−8.00000	−0.923760
$76$	0	0
$77$	6.00000	0.683763
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−18.0000	−1.95237
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	3.00000	0.307794
$96$	0	0
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	0	0
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	0	0
$105$	−6.00000	−0.585540
$106$	0	0
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	20.0000	1.89832
$112$	0	0
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	0	0
$115$	18.0000	1.67851
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−6.00000	−0.550019
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	18.0000	1.62301
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	−16.0000	−1.40872
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−12.0000	−1.03280
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−12.0000	−1.00349
$144$	0	0
$145$	0	0
$146$	0	0
$147$	12.0000	0.989743
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	−3.00000	−0.240966
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	0	0
$165$	−36.0000	−2.80260
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	6.00000	0.450988
$178$	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
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	20.0000	1.47844
$184$	0	0
$185$	30.0000	2.20564
$186$	0	0
$187$	−36.0000	−2.63258
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	−9.00000	−0.651217	−0.325609	−	0.945505i	$-0.605569\pi$
$191$	−9.00000	−0.651217	−0.325609	+	0.945505i	$0.605569\pi$
$192$	0	0
$193$	−19.0000	−1.36765	−0.683825	−	0.729646i	$-0.739685\pi$
$193$	−19.0000	−1.36765	−0.683825	+	0.729646i	$0.739685\pi$
$194$	0	0
$195$	12.0000	0.859338
$196$	0	0
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	0	0
$204$	0	0
$205$	27.0000	1.88576
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	6.00000	0.415029
$210$	0	0
$211$	−19.0000	−1.30801	−0.654007	−	0.756489i	$-0.726913\pi$
$211$	−19.0000	−1.30801	−0.654007	+	0.756489i	$0.726913\pi$
$212$	0	0
$213$	30.0000	2.05557
$214$	0	0
$215$	−24.0000	−1.63679
$216$	0	0
$217$	−1.00000	−0.0678844
$218$	0	0
$219$	−28.0000	−1.89206
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	−28.0000	−1.87502	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−28.0000	−1.87502	−0.937509	+	0.347960i	$0.886874\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	−12.0000	−0.789542
$232$	0	0
$233$	21.0000	1.37576	0.687878	−	0.725826i	$-0.258542\pi$
$233$	21.0000	1.37576	0.687878	+	0.725826i	$0.258542\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−16.0000	−1.03931
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	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
$243$	10.0000	0.641500
$244$	0	0
$245$	18.0000	1.14998
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	36.0000	2.26330
$254$	0	0
$255$	36.0000	2.25441
$256$	0	0
$257$	−15.0000	−0.935674	−0.467837	−	0.883815i	$-0.654967\pi$
$257$	−15.0000	−0.935674	−0.467837	+	0.883815i	$0.654967\pi$
$258$	0	0
$259$	10.0000	0.621370
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−24.0000	−1.46878
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	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
$273$	4.00000	0.242091
$274$	0	0
$275$	−24.0000	−1.44725
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	9.00000	0.536895	0.268447	−	0.963294i	$-0.413489\pi$
$281$	9.00000	0.536895	0.268447	+	0.963294i	$0.413489\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	−6.00000	−0.355409
$286$	0	0
$287$	9.00000	0.531253
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	14.0000	0.820695
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	9.00000	0.524000
$296$	0	0
$297$	−24.0000	−1.39262
$298$	0	0
$299$	−12.0000	−0.693978
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	30.0000	1.71780
$306$	0	0
$307$	11.0000	0.627803	0.313902	−	0.949456i	$-0.398364\pi$
$307$	11.0000	0.627803	0.313902	+	0.949456i	$0.398364\pi$
$308$	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
$313$	−34.0000	−1.92179	−0.960897	−	0.276907i	$-0.910691\pi$
$313$	−34.0000	−1.92179	−0.960897	+	0.276907i	$0.910691\pi$
$314$	0	0
$315$	3.00000	0.169031
$316$	0	0
$317$	9.00000	0.505490	0.252745	−	0.967533i	$-0.418667\pi$
$317$	9.00000	0.505490	0.252745	+	0.967533i	$0.418667\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−6.00000	−0.334887
$322$	0	0
$323$	−6.00000	−0.333849
$324$	0	0
$325$	8.00000	0.443760
$326$	0	0
$327$	−22.0000	−1.21660
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	0	0
$333$	−10.0000	−0.547997
$334$	0	0
$335$	12.0000	0.655630
$336$	0	0
$337$	−4.00000	−0.217894	−0.108947	−	0.994048i	$-0.534748\pi$
$337$	−4.00000	−0.217894	−0.108947	+	0.994048i	$0.534748\pi$
$338$	0	0
$339$	−18.0000	−0.977626
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	−36.0000	−1.93817
$346$	0	0
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	45.0000	2.38835
$356$	0	0
$357$	12.0000	0.635107
$358$	0	0
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	−50.0000	−2.62432
$364$	0	0
$365$	−42.0000	−2.19838
$366$	0	0
$367$	2.00000	0.104399	0.0521996	−	0.998637i	$-0.483377\pi$
$367$	2.00000	0.104399	0.0521996	+	0.998637i	$0.483377\pi$
$368$	0	0
$369$	−9.00000	−0.468521
$370$	0	0
$371$	0	0
$372$	0	0
$373$	11.0000	0.569558	0.284779	−	0.958593i	$-0.408080\pi$
$373$	11.0000	0.569558	0.284779	+	0.958593i	$0.408080\pi$
$374$	0	0
$375$	−6.00000	−0.309839
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	0	0
$385$	−18.0000	−0.917365
$386$	0	0
$387$	8.00000	0.406663
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	−36.0000	−1.82060
$392$	0	0
$393$	−24.0000	−1.21064
$394$	0	0
$395$	−24.0000	−1.20757
$396$	0	0
$397$	−7.00000	−0.351320	−0.175660	−	0.984451i	$-0.556206\pi$
$397$	−7.00000	−0.351320	−0.175660	+	0.984451i	$0.556206\pi$
$398$	0	0
$399$	−2.00000	−0.100125
$400$	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
$403$	2.00000	0.0996271
$404$	0	0
$405$	33.0000	1.63978
$406$	0	0
$407$	60.0000	2.97409
$408$	0	0
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	0	0
$411$	36.0000	1.77575
$412$	0	0
$413$	3.00000	0.147620
$414$	0	0
$415$	−18.0000	−0.883585
$416$	0	0
$417$	−28.0000	−1.37117
$418$	0	0
$419$	−15.0000	−0.732798	−0.366399	−	0.930458i	$-0.619409\pi$
$419$	−15.0000	−0.732798	−0.366399	+	0.930458i	$0.619409\pi$
$420$	0	0
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	24.0000	1.16417
$426$	0	0
$427$	10.0000	0.483934
$428$	0	0
$429$	24.0000	1.15873
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.00000	0.287019
$438$	0	0
$439$	17.0000	0.811366	0.405683	−	0.914014i	$-0.367034\pi$
$439$	17.0000	0.811366	0.405683	+	0.914014i	$0.367034\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	27.0000	1.28281	0.641404	−	0.767203i	$-0.278352\pi$
$443$	27.0000	1.28281	0.641404	+	0.767203i	$0.278352\pi$
$444$	0	0
$445$	−36.0000	−1.70656
$446$	0	0
$447$	36.0000	1.70274
$448$	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
$451$	54.0000	2.54276
$452$	0	0
$453$	32.0000	1.50349
$454$	0	0
$455$	6.00000	0.281284
$456$	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
$459$	24.0000	1.12022
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	−10.0000	−0.464739	−0.232370	−	0.972628i	$-0.574648\pi$
$463$	−10.0000	−0.464739	−0.232370	+	0.972628i	$0.574648\pi$
$464$	0	0
$465$	6.00000	0.278243
$466$	0	0
$467$	−27.0000	−1.24941	−0.624705	−	0.780860i	$-0.714781\pi$
$467$	−27.0000	−1.24941	−0.624705	+	0.780860i	$0.714781\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	26.0000	1.19802
$472$	0	0
$473$	−48.0000	−2.20704
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−33.0000	−1.50781	−0.753904	−	0.656984i	$-0.771832\pi$
$479$	−33.0000	−1.50781	−0.753904	+	0.656984i	$0.771832\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	0	0
$483$	−12.0000	−0.546019
$484$	0	0
$485$	21.0000	0.953561
$486$	0	0
$487$	14.0000	0.634401	0.317200	−	0.948359i	$-0.397257\pi$
$487$	14.0000	0.634401	0.317200	+	0.948359i	$0.397257\pi$
$488$	0	0
$489$	2.00000	0.0904431
$490$	0	0
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	18.0000	0.809040
$496$	0	0
$497$	15.0000	0.672842
$498$	0	0
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	0	0
$503$	−27.0000	−1.20387	−0.601935	−	0.798545i	$-0.705603\pi$
$503$	−27.0000	−1.20387	−0.601935	+	0.798545i	$0.705603\pi$
$504$	0	0
$505$	9.00000	0.400495
$506$	0	0
$507$	18.0000	0.799408
$508$	0	0
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	0	0
$511$	−14.0000	−0.619324
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	21.0000	0.925371
$516$	0	0
$517$	0	0
$518$	0	0
$519$	12.0000	0.526742
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	2.00000	0.0874539	0.0437269	−	0.999044i	$-0.486077\pi$
$523$	2.00000	0.0874539	0.0437269	+	0.999044i	$0.486077\pi$
$524$	0	0
$525$	8.00000	0.349149
$526$	0	0
$527$	6.00000	0.261364
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	−3.00000	−0.130189
$532$	0	0
$533$	−18.0000	−0.779667
$534$	0	0
$535$	−9.00000	−0.389104
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	36.0000	1.55063
$540$	0	0
$541$	23.0000	0.988847	0.494424	−	0.869221i	$-0.335379\pi$
$541$	23.0000	0.988847	0.494424	+	0.869221i	$0.335379\pi$
$542$	0	0
$543$	20.0000	0.858282
$544$	0	0
$545$	−33.0000	−1.41356
$546$	0	0
$547$	23.0000	0.983409	0.491704	−	0.870762i	$-0.336374\pi$
$547$	23.0000	0.983409	0.491704	+	0.870762i	$0.336374\pi$
$548$	0	0
$549$	−10.0000	−0.426790
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−8.00000	−0.340195
$554$	0	0
$555$	−60.0000	−2.54686
$556$	0	0
$557$	−36.0000	−1.52537	−0.762684	−	0.646771i	$-0.776119\pi$
$557$	−36.0000	−1.52537	−0.762684	+	0.646771i	$0.776119\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	72.0000	3.03984
$562$	0	0
$563$	−15.0000	−0.632175	−0.316087	−	0.948730i	$-0.602369\pi$
$563$	−15.0000	−0.632175	−0.316087	+	0.948730i	$0.602369\pi$
$564$	0	0
$565$	−27.0000	−1.13590
$566$	0	0
$567$	11.0000	0.461957
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	−34.0000	−1.42286	−0.711428	−	0.702759i	$-0.751951\pi$
$571$	−34.0000	−1.42286	−0.711428	+	0.702759i	$0.751951\pi$
$572$	0	0
$573$	18.0000	0.751961
$574$	0	0
$575$	−24.0000	−1.00087
$576$	0	0
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	0	0
$579$	38.0000	1.57923
$580$	0	0
$581$	−6.00000	−0.248922
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−6.00000	−0.248069
$586$	0	0
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	0	0
$589$	−1.00000	−0.0412043
$590$	0	0
$591$	−24.0000	−0.987228
$592$	0	0
$593$	3.00000	0.123195	0.0615976	−	0.998101i	$-0.480380\pi$
$593$	3.00000	0.123195	0.0615976	+	0.998101i	$0.480380\pi$
$594$	0	0
$595$	18.0000	0.737928
$596$	0	0
$597$	−4.00000	−0.163709
$598$	0	0
$599$	−9.00000	−0.367730	−0.183865	−	0.982952i	$-0.558861\pi$
$599$	−9.00000	−0.367730	−0.183865	+	0.982952i	$0.558861\pi$
$600$	0	0
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	−75.0000	−3.04918
$606$	0	0
$607$	44.0000	1.78590	0.892952	−	0.450151i	$-0.148630\pi$
$607$	44.0000	1.78590	0.892952	+	0.450151i	$0.148630\pi$
$608$	0	0
$609$	0	0
$610$	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
$615$	−54.0000	−2.17749
$616$	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
$619$	−40.0000	−1.60774	−0.803868	−	0.594808i	$-0.797228\pi$
$619$	−40.0000	−1.60774	−0.803868	+	0.594808i	$0.797228\pi$
$620$	0	0
$621$	−24.0000	−0.963087
$622$	0	0
$623$	−12.0000	−0.480770
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	−12.0000	−0.479234
$628$	0	0
$629$	−60.0000	−2.39236
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	0	0
$633$	38.0000	1.51036
$634$	0	0
$635$	−6.00000	−0.238103
$636$	0	0
$637$	−12.0000	−0.475457
$638$	0	0
$639$	−15.0000	−0.593391
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	0	0
$645$	48.0000	1.89000
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	18.0000	0.706562
$650$	0	0
$651$	2.00000	0.0783862
$652$	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
$655$	−36.0000	−1.40664
$656$	0	0
$657$	14.0000	0.546192
$658$	0	0
$659$	−39.0000	−1.51922	−0.759612	−	0.650376i	$-0.774611\pi$
$659$	−39.0000	−1.51922	−0.759612	+	0.650376i	$0.774611\pi$
$660$	0	0
$661$	47.0000	1.82809	0.914044	−	0.405615i	$-0.132943\pi$
$661$	47.0000	1.82809	0.914044	+	0.405615i	$0.132943\pi$
$662$	0	0
$663$	−24.0000	−0.932083
$664$	0	0
$665$	−3.00000	−0.116335
$666$	0	0
$667$	0	0
$668$	0	0
$669$	56.0000	2.16509
$670$	0	0
$671$	60.0000	2.31627
$672$	0	0
$673$	−28.0000	−1.07932	−0.539660	−	0.841883i	$-0.681447\pi$
$673$	−28.0000	−1.07932	−0.539660	+	0.841883i	$0.681447\pi$
$674$	0	0
$675$	16.0000	0.615840
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	7.00000	0.268635
$680$	0	0
$681$	−24.0000	−0.919682
$682$	0	0
$683$	39.0000	1.49229	0.746147	−	0.665782i	$-0.231902\pi$
$683$	39.0000	1.49229	0.746147	+	0.665782i	$0.231902\pi$
$684$	0	0
$685$	54.0000	2.06323
$686$	0	0
$687$	44.0000	1.67870
$688$	0	0
$689$	0	0
$690$	0	0
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	0	0
$693$	6.00000	0.227921
$694$	0	0
$695$	−42.0000	−1.59315
$696$	0	0
$697$	−54.0000	−2.04540
$698$	0	0
$699$	−42.0000	−1.58859
$700$	0	0
$701$	−33.0000	−1.24639	−0.623196	−	0.782065i	$-0.714166\pi$
$701$	−33.0000	−1.24639	−0.623196	+	0.782065i	$0.714166\pi$
$702$	0	0
$703$	10.0000	0.377157
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.00000	0.112827
$708$	0	0
$709$	20.0000	0.751116	0.375558	−	0.926799i	$-0.377451\pi$
$709$	20.0000	0.751116	0.375558	+	0.926799i	$0.377451\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	−6.00000	−0.224702
$714$	0	0
$715$	36.0000	1.34632
$716$	0	0
$717$	−48.0000	−1.79259
$718$	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
$721$	7.00000	0.260694
$722$	0	0
$723$	−4.00000	−0.148762
$724$	0	0
$725$	0	0
$726$	0	0
$727$	17.0000	0.630495	0.315248	−	0.949009i	$-0.397912\pi$
$727$	17.0000	0.630495	0.315248	+	0.949009i	$0.397912\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	48.0000	1.77534
$732$	0	0
$733$	5.00000	0.184679	0.0923396	−	0.995728i	$-0.470565\pi$
$733$	5.00000	0.184679	0.0923396	+	0.995728i	$0.470565\pi$
$734$	0	0
$735$	−36.0000	−1.32788
$736$	0	0
$737$	24.0000	0.884051
$738$	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
$741$	4.00000	0.146944
$742$	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
$745$	54.0000	1.97841
$746$	0	0
$747$	6.00000	0.219529
$748$	0	0
$749$	−3.00000	−0.109618
$750$	0	0
$751$	17.0000	0.620339	0.310169	−	0.950681i	$-0.399614\pi$
$751$	17.0000	0.620339	0.310169	+	0.950681i	$0.399614\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	48.0000	1.74690
$756$	0	0
$757$	20.0000	0.726912	0.363456	−	0.931611i	$-0.381597\pi$
$757$	20.0000	0.726912	0.363456	+	0.931611i	$0.381597\pi$
$758$	0	0
$759$	−72.0000	−2.61343
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	−11.0000	−0.398227
$764$	0	0
$765$	−18.0000	−0.650791
$766$	0	0
$767$	−6.00000	−0.216647
$768$	0	0
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	0	0
$773$	−12.0000	−0.431610	−0.215805	−	0.976436i	$-0.569238\pi$
$773$	−12.0000	−0.431610	−0.215805	+	0.976436i	$0.569238\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	−20.0000	−0.717496
$778$	0	0
$779$	9.00000	0.322458
$780$	0	0
$781$	90.0000	3.22045
$782$	0	0
$783$	0	0
$784$	0	0
$785$	39.0000	1.39197
$786$	0	0
$787$	−16.0000	−0.570338	−0.285169	−	0.958477i	$-0.592050\pi$
$787$	−16.0000	−0.570338	−0.285169	+	0.958477i	$0.592050\pi$
$788$	0	0
$789$	12.0000	0.427211
$790$	0	0
$791$	−9.00000	−0.320003
$792$	0	0
$793$	−20.0000	−0.710221
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24.0000	0.850124	0.425062	−	0.905164i	$-0.360252\pi$
$797$	24.0000	0.850124	0.425062	+	0.905164i	$0.360252\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	12.0000	0.423999
$802$	0	0
$803$	−84.0000	−2.96430
$804$	0	0
$805$	−18.0000	−0.634417
$806$	0	0
$807$	0	0
$808$	0	0
$809$	48.0000	1.68759	0.843795	−	0.536666i	$-0.180316\pi$
$809$	48.0000	1.68759	0.843795	+	0.536666i	$0.180316\pi$
$810$	0	0
$811$	−52.0000	−1.82597	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000	−1.82597	−0.912983	+	0.407997i	$0.866228\pi$
$812$	0	0
$813$	−16.0000	−0.561144
$814$	0	0
$815$	3.00000	0.105085
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	0	0
$825$	48.0000	1.67115
$826$	0	0
$827$	18.0000	0.625921	0.312961	−	0.949766i	$-0.398679\pi$
$827$	18.0000	0.625921	0.312961	+	0.949766i	$0.398679\pi$
$828$	0	0
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	0	0
$831$	−16.0000	−0.555034
$832$	0	0
$833$	−36.0000	−1.24733
$834$	0	0
$835$	−36.0000	−1.24583
$836$	0	0
$837$	4.00000	0.138260
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	−18.0000	−0.619953
$844$	0	0
$845$	27.0000	0.928828
$846$	0	0
$847$	−25.0000	−0.859010
$848$	0	0
$849$	8.00000	0.274559
$850$	0	0
$851$	60.0000	2.05677
$852$	0	0
$853$	−34.0000	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$854$	0	0
$855$	3.00000	0.102598
$856$	0	0
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	−18.0000	−0.613438
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	18.0000	0.612018
$866$	0	0
$867$	−38.0000	−1.29055
$868$	0	0
$869$	−48.0000	−1.62829
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	−7.00000	−0.236914
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	−19.0000	−0.641584	−0.320792	−	0.947150i	$-0.603949\pi$
$877$	−19.0000	−0.641584	−0.320792	+	0.947150i	$0.603949\pi$
$878$	0	0
$879$	−12.0000	−0.404750
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	0	0
$885$	−18.0000	−0.605063
$886$	0	0
$887$	27.0000	0.906571	0.453286	−	0.891365i	$-0.350252\pi$
$887$	27.0000	0.906571	0.453286	+	0.891365i	$0.350252\pi$
$888$	0	0
$889$	−2.00000	−0.0670778
$890$	0	0
$891$	66.0000	2.21108
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−18.0000	−0.601674
$896$	0	0
$897$	24.0000	0.801337
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	16.0000	0.532447
$904$	0	0
$905$	30.0000	0.997234
$906$	0	0
$907$	−19.0000	−0.630885	−0.315442	−	0.948945i	$-0.602153\pi$
$907$	−19.0000	−0.630885	−0.315442	+	0.948945i	$0.602153\pi$
$908$	0	0
$909$	−3.00000	−0.0995037
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	0	0
$913$	−36.0000	−1.19143
$914$	0	0
$915$	−60.0000	−1.98354
$916$	0	0
$917$	−12.0000	−0.396275
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	0	0
$921$	−22.0000	−0.724925
$922$	0	0
$923$	−30.0000	−0.987462
$924$	0	0
$925$	−40.0000	−1.31519
$926$	0	0
$927$	−7.00000	−0.229910
$928$	0	0
$929$	−12.0000	−0.393707	−0.196854	−	0.980433i	$-0.563072\pi$
$929$	−12.0000	−0.393707	−0.196854	+	0.980433i	$0.563072\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	0	0
$933$	−18.0000	−0.589294
$934$	0	0
$935$	108.000	3.53198
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	68.0000	2.21910
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	54.0000	1.75848
$944$	0	0
$945$	12.0000	0.390360
$946$	0	0
$947$	−18.0000	−0.584921	−0.292461	−	0.956278i	$-0.594474\pi$
$947$	−18.0000	−0.584921	−0.292461	+	0.956278i	$0.594474\pi$
$948$	0	0
$949$	28.0000	0.908918
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	0	0
$955$	27.0000	0.873699
$956$	0	0
$957$	0	0
$958$	0	0
$959$	18.0000	0.581250
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	3.00000	0.0966736
$964$	0	0
$965$	57.0000	1.83489
$966$	0	0
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	0	0
$969$	12.0000	0.385496
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	−14.0000	−0.448819
$974$	0	0
$975$	−16.0000	−0.512410
$976$	0	0
$977$	51.0000	1.63163	0.815817	−	0.578310i	$-0.196287\pi$
$977$	51.0000	1.63163	0.815817	+	0.578310i	$0.196287\pi$
$978$	0	0
$979$	−72.0000	−2.30113
$980$	0	0
$981$	11.0000	0.351203
$982$	0	0
$983$	42.0000	1.33959	0.669796	−	0.742545i	$-0.266382\pi$
$983$	42.0000	1.33959	0.669796	+	0.742545i	$0.266382\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−48.0000	−1.52631
$990$	0	0
$991$	50.0000	1.58830	0.794151	−	0.607720i	$-0.207916\pi$
$991$	50.0000	1.58830	0.794151	+	0.607720i	$0.207916\pi$
$992$	0	0
$993$	−4.00000	−0.126936
$994$	0	0
$995$	−6.00000	−0.190213
$996$	0	0
$997$	−1.00000	−0.0316703	−0.0158352	−	0.999875i	$-0.505041\pi$
$997$	−1.00000	−0.0316703	−0.0158352	+	0.999875i	$0.505041\pi$
$998$	0	0
$999$	−40.0000	−1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.2.a.a.1.1	✓	1
3.2	odd	2		1116.2.a.f.1.1		1
4.3	odd	2		496.2.a.d.1.1		1
5.2	odd	4		3100.2.c.b.249.2		2
5.3	odd	4		3100.2.c.b.249.1		2
5.4	even	2		3100.2.a.f.1.1		1
7.6	odd	2		6076.2.a.b.1.1		1
8.3	odd	2		1984.2.a.c.1.1		1
8.5	even	2		1984.2.a.l.1.1		1
12.11	even	2		4464.2.a.w.1.1		1
31.30	odd	2		3844.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.2.a.a.1.1	✓	1	1.1	even	1	trivial
496.2.a.d.1.1		1	4.3	odd	2
1116.2.a.f.1.1		1	3.2	odd	2
1984.2.a.c.1.1		1	8.3	odd	2
1984.2.a.l.1.1		1	8.5	even	2
3100.2.a.f.1.1		1	5.4	even	2
3100.2.c.b.249.1		2	5.3	odd	4
3100.2.c.b.249.2		2	5.2	odd	4
3844.2.a.c.1.1		1	31.30	odd	2
4464.2.a.w.1.1		1	12.11	even	2
6076.2.a.b.1.1		1	7.6	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:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	124.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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