LMFDB - Embedded newform 924.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [924,2,Mod(1,924)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(924, 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("924.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 924 = 2^{2} \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	924.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:	$7.37817714677$
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	$\chi$	$=$	924.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +3.00000 q^{13} -3.00000 q^{15} -2.00000 q^{17} -3.00000 q^{19} -1.00000 q^{21} -4.00000 q^{23} +4.00000 q^{25} +1.00000 q^{27} -9.00000 q^{29} -2.00000 q^{31} -1.00000 q^{33} +3.00000 q^{35} -11.0000 q^{37} +3.00000 q^{39} -4.00000 q^{41} -4.00000 q^{43} -3.00000 q^{45} -3.00000 q^{47} +1.00000 q^{49} -2.00000 q^{51} -4.00000 q^{53} +3.00000 q^{55} -3.00000 q^{57} -3.00000 q^{59} +10.0000 q^{61} -1.00000 q^{63} -9.00000 q^{65} +11.0000 q^{67} -4.00000 q^{69} +4.00000 q^{71} +9.00000 q^{73} +4.00000 q^{75} +1.00000 q^{77} -4.00000 q^{79} +1.00000 q^{81} +10.0000 q^{83} +6.00000 q^{85} -9.00000 q^{87} +6.00000 q^{89} -3.00000 q^{91} -2.00000 q^{93} +9.00000 q^{95} +12.0000 q^{97} -1.00000 q^{99} +O(q^{100})$

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
$5$	−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
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	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
$15$	−3.00000	−0.774597
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	−11.0000	−1.80839	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−11.0000	−1.80839	−0.904194	+	0.427121i	$0.859528\pi$
$38$	0	0
$39$	3.00000	0.480384
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	−3.00000	−0.447214
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	0	0
$57$	−3.00000	−0.397360
$58$	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	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−9.00000	−1.11631
$66$	0	0
$67$	11.0000	1.34386	0.671932	−	0.740613i	$-0.265465\pi$
$67$	11.0000	1.34386	0.671932	+	0.740613i	$0.265465\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.0000	1.09764	0.548821	−	0.835940i	$-0.315077\pi$
$83$	10.0000	1.09764	0.548821	+	0.835940i	$0.315077\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	0	0
$87$	−9.00000	−0.964901
$88$	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
$91$	−3.00000	−0.314485
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	9.00000	0.923381
$96$	0	0
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	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
$105$	3.00000	0.292770
$106$	0	0
$107$	−19.0000	−1.83680	−0.918400	−	0.395654i	$-0.870518\pi$
$107$	−19.0000	−1.83680	−0.918400	+	0.395654i	$0.870518\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−11.0000	−1.04407
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	12.0000	1.11901
$116$	0	0
$117$	3.00000	0.277350
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.00000	−0.360668
$124$	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
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	3.00000	0.260133
$134$	0	0
$135$	−3.00000	−0.258199
$136$	0	0
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	0	0
$143$	−3.00000	−0.250873
$144$	0	0
$145$	27.0000	2.24223
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	0	0
$151$	18.0000	1.46482	0.732410	−	0.680864i	$-0.238396\pi$
$151$	18.0000	1.46482	0.732410	+	0.680864i	$0.238396\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	−4.00000	−0.317221
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	−13.0000	−1.01824	−0.509119	−	0.860696i	$-0.670029\pi$
$163$	−13.0000	−1.01824	−0.509119	+	0.860696i	$0.670029\pi$
$164$	0	0
$165$	3.00000	0.233550
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	−3.00000	−0.229416
$172$	0	0
$173$	−12.0000	−0.912343	−0.456172	−	0.889892i	$-0.650780\pi$
$173$	−12.0000	−0.912343	−0.456172	+	0.889892i	$0.650780\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	−3.00000	−0.225494
$178$	0	0
$179$	−8.00000	−0.597948	−0.298974	−	0.954261i	$-0.596644\pi$
$179$	−8.00000	−0.597948	−0.298974	+	0.954261i	$0.596644\pi$
$180$	0	0
$181$	−4.00000	−0.297318	−0.148659	−	0.988889i	$-0.547496\pi$
$181$	−4.00000	−0.297318	−0.148659	+	0.988889i	$0.547496\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	0	0
$185$	33.0000	2.42621
$186$	0	0
$187$	2.00000	0.146254
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	24.0000	1.72756	0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000	1.72756	0.863779	+	0.503871i	$0.168091\pi$
$194$	0	0
$195$	−9.00000	−0.644503
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	11.0000	0.775880
$202$	0	0
$203$	9.00000	0.631676
$204$	0	0
$205$	12.0000	0.838116
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	3.00000	0.207514
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	0	0
$213$	4.00000	0.274075
$214$	0	0
$215$	12.0000	0.818393
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	9.00000	0.608164
$220$	0	0
$221$	−6.00000	−0.403604
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	8.00000	0.528655	0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000	0.528655	0.264327	+	0.964433i	$0.414850\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	0	0
$235$	9.00000	0.587095
$236$	0	0
$237$	−4.00000	−0.259828
$238$	0	0
$239$	11.0000	0.711531	0.355765	−	0.934575i	$-0.384220\pi$
$239$	11.0000	0.711531	0.355765	+	0.934575i	$0.384220\pi$
$240$	0	0
$241$	19.0000	1.22390	0.611949	−	0.790897i	$-0.290386\pi$
$241$	19.0000	1.22390	0.611949	+	0.790897i	$0.290386\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−9.00000	−0.572656
$248$	0	0
$249$	10.0000	0.633724
$250$	0	0
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	0	0
$255$	6.00000	0.375735
$256$	0	0
$257$	25.0000	1.55946	0.779729	−	0.626118i	$-0.215357\pi$
$257$	25.0000	1.55946	0.779729	+	0.626118i	$0.215357\pi$
$258$	0	0
$259$	11.0000	0.683507
$260$	0	0
$261$	−9.00000	−0.557086
$262$	0	0
$263$	−3.00000	−0.184988	−0.0924940	−	0.995713i	$-0.529484\pi$
$263$	−3.00000	−0.184988	−0.0924940	+	0.995713i	$0.529484\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	0	0
$267$	6.00000	0.367194
$268$	0	0
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	−3.00000	−0.182237	−0.0911185	−	0.995840i	$-0.529044\pi$
$271$	−3.00000	−0.182237	−0.0911185	+	0.995840i	$0.529044\pi$
$272$	0	0
$273$	−3.00000	−0.181568
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	12.0000	0.721010	0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000	0.721010	0.360505	+	0.932757i	$0.382604\pi$
$278$	0	0
$279$	−2.00000	−0.119737
$280$	0	0
$281$	23.0000	1.37206	0.686032	−	0.727571i	$-0.259351\pi$
$281$	23.0000	1.37206	0.686032	+	0.727571i	$0.259351\pi$
$282$	0	0
$283$	21.0000	1.24832	0.624160	−	0.781296i	$-0.285441\pi$
$283$	21.0000	1.24832	0.624160	+	0.781296i	$0.285441\pi$
$284$	0	0
$285$	9.00000	0.533114
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	12.0000	0.703452
$292$	0	0
$293$	−16.0000	−0.934730	−0.467365	−	0.884064i	$-0.654797\pi$
$293$	−16.0000	−0.934730	−0.467365	+	0.884064i	$0.654797\pi$
$294$	0	0
$295$	9.00000	0.524000
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−12.0000	−0.693978
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	10.0000	0.574485
$304$	0	0
$305$	−30.0000	−1.71780
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	−14.0000	−0.796432
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	18.0000	1.01742	0.508710	−	0.860938i	$-0.330123\pi$
$313$	18.0000	1.01742	0.508710	+	0.860938i	$0.330123\pi$
$314$	0	0
$315$	3.00000	0.169031
$316$	0	0
$317$	24.0000	1.34797	0.673987	−	0.738743i	$-0.264580\pi$
$317$	24.0000	1.34797	0.673987	+	0.738743i	$0.264580\pi$
$318$	0	0
$319$	9.00000	0.503903
$320$	0	0
$321$	−19.0000	−1.06048
$322$	0	0
$323$	6.00000	0.333849
$324$	0	0
$325$	12.0000	0.665640
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	3.00000	0.165395
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	0	0
$333$	−11.0000	−0.602796
$334$	0	0
$335$	−33.0000	−1.80298
$336$	0	0
$337$	−20.0000	−1.08947	−0.544735	−	0.838608i	$-0.683370\pi$
$337$	−20.0000	−1.08947	−0.544735	+	0.838608i	$0.683370\pi$
$338$	0	0
$339$	10.0000	0.543125
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	12.0000	0.646058
$346$	0	0
$347$	−8.00000	−0.429463	−0.214731	−	0.976673i	$-0.568888\pi$
$347$	−8.00000	−0.429463	−0.214731	+	0.976673i	$0.568888\pi$
$348$	0	0
$349$	−1.00000	−0.0535288	−0.0267644	−	0.999642i	$-0.508520\pi$
$349$	−1.00000	−0.0535288	−0.0267644	+	0.999642i	$0.508520\pi$
$350$	0	0
$351$	3.00000	0.160128
$352$	0	0
$353$	3.00000	0.159674	0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000	0.159674	0.0798369	+	0.996808i	$0.474560\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	0	0
$357$	2.00000	0.105851
$358$	0	0
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−27.0000	−1.41324
$366$	0	0
$367$	−38.0000	−1.98358	−0.991792	−	0.127862i	$-0.959188\pi$
$367$	−38.0000	−1.98358	−0.991792	+	0.127862i	$0.959188\pi$
$368$	0	0
$369$	−4.00000	−0.208232
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	3.00000	0.154919
$376$	0	0
$377$	−27.0000	−1.39057
$378$	0	0
$379$	−35.0000	−1.79783	−0.898915	−	0.438124i	$-0.855643\pi$
$379$	−35.0000	−1.79783	−0.898915	+	0.438124i	$0.855643\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	0	0
$383$	32.0000	1.63512	0.817562	−	0.575841i	$-0.195325\pi$
$383$	32.0000	1.63512	0.817562	+	0.575841i	$0.195325\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	−4.00000	−0.201773
$394$	0	0
$395$	12.0000	0.603786
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	3.00000	0.150188
$400$	0	0
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	0	0
$403$	−6.00000	−0.298881
$404$	0	0
$405$	−3.00000	−0.149071
$406$	0	0
$407$	11.0000	0.545250
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	−14.0000	−0.690569
$412$	0	0
$413$	3.00000	0.147620
$414$	0	0
$415$	−30.0000	−1.47264
$416$	0	0
$417$	−8.00000	−0.391762
$418$	0	0
$419$	−5.00000	−0.244266	−0.122133	−	0.992514i	$-0.538973\pi$
$419$	−5.00000	−0.244266	−0.122133	+	0.992514i	$0.538973\pi$
$420$	0	0
$421$	−37.0000	−1.80327	−0.901635	−	0.432498i	$-0.857632\pi$
$421$	−37.0000	−1.80327	−0.901635	+	0.432498i	$0.857632\pi$
$422$	0	0
$423$	−3.00000	−0.145865
$424$	0	0
$425$	−8.00000	−0.388057
$426$	0	0
$427$	−10.0000	−0.483934
$428$	0	0
$429$	−3.00000	−0.144841
$430$	0	0
$431$	−11.0000	−0.529851	−0.264926	−	0.964269i	$-0.585347\pi$
$431$	−11.0000	−0.529851	−0.264926	+	0.964269i	$0.585347\pi$
$432$	0	0
$433$	8.00000	0.384455	0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000	0.384455	0.192228	+	0.981350i	$0.438429\pi$
$434$	0	0
$435$	27.0000	1.29455
$436$	0	0
$437$	12.0000	0.574038
$438$	0	0
$439$	37.0000	1.76591	0.882957	−	0.469454i	$-0.155549\pi$
$439$	37.0000	1.76591	0.882957	+	0.469454i	$0.155549\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−16.0000	−0.760183	−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000	−0.760183	−0.380091	+	0.924949i	$0.624107\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	0	0
$447$	3.00000	0.141895
$448$	0	0
$449$	32.0000	1.51017	0.755087	−	0.655625i	$-0.227595\pi$
$449$	32.0000	1.51017	0.755087	+	0.655625i	$0.227595\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	0	0
$453$	18.0000	0.845714
$454$	0	0
$455$	9.00000	0.421927
$456$	0	0
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	0	0
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	8.00000	0.372597	0.186299	−	0.982493i	$-0.440351\pi$
$461$	8.00000	0.372597	0.186299	+	0.982493i	$0.440351\pi$
$462$	0	0
$463$	7.00000	0.325318	0.162659	−	0.986682i	$-0.447993\pi$
$463$	7.00000	0.325318	0.162659	+	0.986682i	$0.447993\pi$
$464$	0	0
$465$	6.00000	0.278243
$466$	0	0
$467$	−23.0000	−1.06431	−0.532157	−	0.846646i	$-0.678618\pi$
$467$	−23.0000	−1.06431	−0.532157	+	0.846646i	$0.678618\pi$
$468$	0	0
$469$	−11.0000	−0.507933
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.00000	0.183920
$474$	0	0
$475$	−12.0000	−0.550598
$476$	0	0
$477$	−4.00000	−0.183147
$478$	0	0
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	0	0
$481$	−33.0000	−1.50467
$482$	0	0
$483$	4.00000	0.182006
$484$	0	0
$485$	−36.0000	−1.63468
$486$	0	0
$487$	−40.0000	−1.81257	−0.906287	−	0.422664i	$-0.861095\pi$
$487$	−40.0000	−1.81257	−0.906287	+	0.422664i	$0.861095\pi$
$488$	0	0
$489$	−13.0000	−0.587880
$490$	0	0
$491$	−3.00000	−0.135388	−0.0676941	−	0.997706i	$-0.521564\pi$
$491$	−3.00000	−0.135388	−0.0676941	+	0.997706i	$0.521564\pi$
$492$	0	0
$493$	18.0000	0.810679
$494$	0	0
$495$	3.00000	0.134840
$496$	0	0
$497$	−4.00000	−0.179425
$498$	0	0
$499$	−27.0000	−1.20869	−0.604343	−	0.796724i	$-0.706564\pi$
$499$	−27.0000	−1.20869	−0.604343	+	0.796724i	$0.706564\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−30.0000	−1.33498
$506$	0	0
$507$	−4.00000	−0.177646
$508$	0	0
$509$	26.0000	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$510$	0	0
$511$	−9.00000	−0.398137
$512$	0	0
$513$	−3.00000	−0.132453
$514$	0	0
$515$	42.0000	1.85074
$516$	0	0
$517$	3.00000	0.131940
$518$	0	0
$519$	−12.0000	−0.526742
$520$	0	0
$521$	−31.0000	−1.35813	−0.679067	−	0.734076i	$-0.737616\pi$
$521$	−31.0000	−1.35813	−0.679067	+	0.734076i	$0.737616\pi$
$522$	0	0
$523$	−21.0000	−0.918266	−0.459133	−	0.888368i	$-0.651840\pi$
$523$	−21.0000	−0.918266	−0.459133	+	0.888368i	$0.651840\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	4.00000	0.174243
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−3.00000	−0.130189
$532$	0	0
$533$	−12.0000	−0.519778
$534$	0	0
$535$	57.0000	2.46432
$536$	0	0
$537$	−8.00000	−0.345225
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	−4.00000	−0.171656
$544$	0	0
$545$	6.00000	0.257012
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	27.0000	1.15024
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	33.0000	1.40077
$556$	0	0
$557$	−5.00000	−0.211857	−0.105928	−	0.994374i	$-0.533781\pi$
$557$	−5.00000	−0.211857	−0.105928	+	0.994374i	$0.533781\pi$
$558$	0	0
$559$	−12.0000	−0.507546
$560$	0	0
$561$	2.00000	0.0844401
$562$	0	0
$563$	32.0000	1.34864	0.674320	−	0.738440i	$-0.264437\pi$
$563$	32.0000	1.34864	0.674320	+	0.738440i	$0.264437\pi$
$564$	0	0
$565$	−30.0000	−1.26211
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	−18.0000	−0.751961
$574$	0	0
$575$	−16.0000	−0.667246
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	24.0000	0.997406
$580$	0	0
$581$	−10.0000	−0.414870
$582$	0	0
$583$	4.00000	0.165663
$584$	0	0
$585$	−9.00000	−0.372104
$586$	0	0
$587$	15.0000	0.619116	0.309558	−	0.950881i	$-0.399819\pi$
$587$	15.0000	0.619116	0.309558	+	0.950881i	$0.399819\pi$
$588$	0	0
$589$	6.00000	0.247226
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	20.0000	0.821302	0.410651	−	0.911793i	$-0.365302\pi$
$593$	20.0000	0.821302	0.410651	+	0.911793i	$0.365302\pi$
$594$	0	0
$595$	−6.00000	−0.245976
$596$	0	0
$597$	−20.0000	−0.818546
$598$	0	0
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	37.0000	1.50926	0.754631	−	0.656150i	$-0.227816\pi$
$601$	37.0000	1.50926	0.754631	+	0.656150i	$0.227816\pi$
$602$	0	0
$603$	11.0000	0.447955
$604$	0	0
$605$	−3.00000	−0.121967
$606$	0	0
$607$	−27.0000	−1.09590	−0.547948	−	0.836512i	$-0.684591\pi$
$607$	−27.0000	−1.09590	−0.547948	+	0.836512i	$0.684591\pi$
$608$	0	0
$609$	9.00000	0.364698
$610$	0	0
$611$	−9.00000	−0.364101
$612$	0	0
$613$	34.0000	1.37325	0.686624	−	0.727013i	$-0.259092\pi$
$613$	34.0000	1.37325	0.686624	+	0.727013i	$0.259092\pi$
$614$	0	0
$615$	12.0000	0.483887
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	3.00000	0.119808
$628$	0	0
$629$	22.0000	0.877197
$630$	0	0
$631$	4.00000	0.159237	0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000	0.159237	0.0796187	+	0.996825i	$0.474630\pi$
$632$	0	0
$633$	−10.0000	−0.397464
$634$	0	0
$635$	−6.00000	−0.238103
$636$	0	0
$637$	3.00000	0.118864
$638$	0	0
$639$	4.00000	0.158238
$640$	0	0
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	0	0
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	0	0
$645$	12.0000	0.472500
$646$	0	0
$647$	23.0000	0.904223	0.452112	−	0.891961i	$-0.350671\pi$
$647$	23.0000	0.904223	0.452112	+	0.891961i	$0.350671\pi$
$648$	0	0
$649$	3.00000	0.117760
$650$	0	0
$651$	2.00000	0.0783862
$652$	0	0
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	0	0
$657$	9.00000	0.351123
$658$	0	0
$659$	−23.0000	−0.895953	−0.447976	−	0.894045i	$-0.647855\pi$
$659$	−23.0000	−0.895953	−0.447976	+	0.894045i	$0.647855\pi$
$660$	0	0
$661$	−32.0000	−1.24466	−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000	−1.24466	−0.622328	+	0.782757i	$0.713813\pi$
$662$	0	0
$663$	−6.00000	−0.233021
$664$	0	0
$665$	−9.00000	−0.349005
$666$	0	0
$667$	36.0000	1.39393
$668$	0	0
$669$	−14.0000	−0.541271
$670$	0	0
$671$	−10.0000	−0.386046
$672$	0	0
$673$	−32.0000	−1.23351	−0.616755	−	0.787155i	$-0.711553\pi$
$673$	−32.0000	−1.23351	−0.616755	+	0.787155i	$0.711553\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	−12.0000	−0.460518
$680$	0	0
$681$	18.0000	0.689761
$682$	0	0
$683$	30.0000	1.14792	0.573959	−	0.818884i	$-0.305407\pi$
$683$	30.0000	1.14792	0.573959	+	0.818884i	$0.305407\pi$
$684$	0	0
$685$	42.0000	1.60474
$686$	0	0
$687$	8.00000	0.305219
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−42.0000	−1.59776	−0.798878	−	0.601494i	$-0.794573\pi$
$691$	−42.0000	−1.59776	−0.798878	+	0.601494i	$0.794573\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	24.0000	0.910372
$696$	0	0
$697$	8.00000	0.303022
$698$	0	0
$699$	22.0000	0.832116
$700$	0	0
$701$	−10.0000	−0.377695	−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000	−0.377695	−0.188847	+	0.982006i	$0.560475\pi$
$702$	0	0
$703$	33.0000	1.24462
$704$	0	0
$705$	9.00000	0.338960
$706$	0	0
$707$	−10.0000	−0.376089
$708$	0	0
$709$	−25.0000	−0.938895	−0.469447	−	0.882960i	$-0.655547\pi$
$709$	−25.0000	−0.938895	−0.469447	+	0.882960i	$0.655547\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	8.00000	0.299602
$714$	0	0
$715$	9.00000	0.336581
$716$	0	0
$717$	11.0000	0.410803
$718$	0	0
$719$	−29.0000	−1.08152	−0.540759	−	0.841178i	$-0.681863\pi$
$719$	−29.0000	−1.08152	−0.540759	+	0.841178i	$0.681863\pi$
$720$	0	0
$721$	14.0000	0.521387
$722$	0	0
$723$	19.0000	0.706618
$724$	0	0
$725$	−36.0000	−1.33701
$726$	0	0
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	0	0
$735$	−3.00000	−0.110657
$736$	0	0
$737$	−11.0000	−0.405190
$738$	0	0
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	0	0
$741$	−9.00000	−0.330623
$742$	0	0
$743$	13.0000	0.476924	0.238462	−	0.971152i	$-0.423357\pi$
$743$	13.0000	0.476924	0.238462	+	0.971152i	$0.423357\pi$
$744$	0	0
$745$	−9.00000	−0.329734
$746$	0	0
$747$	10.0000	0.365881
$748$	0	0
$749$	19.0000	0.694245
$750$	0	0
$751$	−41.0000	−1.49611	−0.748056	−	0.663636i	$-0.769012\pi$
$751$	−41.0000	−1.49611	−0.748056	+	0.663636i	$0.769012\pi$
$752$	0	0
$753$	−21.0000	−0.765283
$754$	0	0
$755$	−54.0000	−1.96526
$756$	0	0
$757$	13.0000	0.472493	0.236247	−	0.971693i	$-0.424083\pi$
$757$	13.0000	0.472493	0.236247	+	0.971693i	$0.424083\pi$
$758$	0	0
$759$	4.00000	0.145191
$760$	0	0
$761$	20.0000	0.724999	0.362500	−	0.931984i	$-0.381923\pi$
$761$	20.0000	0.724999	0.362500	+	0.931984i	$0.381923\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	0	0
$765$	6.00000	0.216930
$766$	0	0
$767$	−9.00000	−0.324971
$768$	0	0
$769$	−5.00000	−0.180305	−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000	−0.180305	−0.0901523	+	0.995928i	$0.528735\pi$
$770$	0	0
$771$	25.0000	0.900353
$772$	0	0
$773$	−31.0000	−1.11499	−0.557496	−	0.830179i	$-0.688238\pi$
$773$	−31.0000	−1.11499	−0.557496	+	0.830179i	$0.688238\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	11.0000	0.394623
$778$	0	0
$779$	12.0000	0.429945
$780$	0	0
$781$	−4.00000	−0.143131
$782$	0	0
$783$	−9.00000	−0.321634
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−5.00000	−0.178231	−0.0891154	−	0.996021i	$-0.528404\pi$
$787$	−5.00000	−0.178231	−0.0891154	+	0.996021i	$0.528404\pi$
$788$	0	0
$789$	−3.00000	−0.106803
$790$	0	0
$791$	−10.0000	−0.355559
$792$	0	0
$793$	30.0000	1.06533
$794$	0	0
$795$	12.0000	0.425596
$796$	0	0
$797$	7.00000	0.247953	0.123976	−	0.992285i	$-0.460435\pi$
$797$	7.00000	0.247953	0.123976	+	0.992285i	$0.460435\pi$
$798$	0	0
$799$	6.00000	0.212265
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	−9.00000	−0.317603
$804$	0	0
$805$	−12.0000	−0.422944
$806$	0	0
$807$	−30.0000	−1.05605
$808$	0	0
$809$	27.0000	0.949269	0.474635	−	0.880183i	$-0.342580\pi$
$809$	27.0000	0.949269	0.474635	+	0.880183i	$0.342580\pi$
$810$	0	0
$811$	−21.0000	−0.737410	−0.368705	−	0.929547i	$-0.620199\pi$
$811$	−21.0000	−0.737410	−0.368705	+	0.929547i	$0.620199\pi$
$812$	0	0
$813$	−3.00000	−0.105215
$814$	0	0
$815$	39.0000	1.36611
$816$	0	0
$817$	12.0000	0.419827
$818$	0	0
$819$	−3.00000	−0.104828
$820$	0	0
$821$	13.0000	0.453703	0.226852	−	0.973929i	$-0.427157\pi$
$821$	13.0000	0.453703	0.226852	+	0.973929i	$0.427157\pi$
$822$	0	0
$823$	55.0000	1.91718	0.958590	−	0.284791i	$-0.0919241\pi$
$823$	55.0000	1.91718	0.958590	+	0.284791i	$0.0919241\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	−3.00000	−0.104320	−0.0521601	−	0.998639i	$-0.516611\pi$
$827$	−3.00000	−0.104320	−0.0521601	+	0.998639i	$0.516611\pi$
$828$	0	0
$829$	−24.0000	−0.833554	−0.416777	−	0.909009i	$-0.636840\pi$
$829$	−24.0000	−0.833554	−0.416777	+	0.909009i	$0.636840\pi$
$830$	0	0
$831$	12.0000	0.416275
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	0	0
$839$	−27.0000	−0.932144	−0.466072	−	0.884747i	$-0.654331\pi$
$839$	−27.0000	−0.932144	−0.466072	+	0.884747i	$0.654331\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	23.0000	0.792162
$844$	0	0
$845$	12.0000	0.412813
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	21.0000	0.720718
$850$	0	0
$851$	44.0000	1.50830
$852$	0	0
$853$	54.0000	1.84892	0.924462	−	0.381273i	$-0.124514\pi$
$853$	54.0000	1.84892	0.924462	+	0.381273i	$0.124514\pi$
$854$	0	0
$855$	9.00000	0.307794
$856$	0	0
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	4.00000	0.136320
$862$	0	0
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	0	0
$865$	36.0000	1.22404
$866$	0	0
$867$	−13.0000	−0.441503
$868$	0	0
$869$	4.00000	0.135691
$870$	0	0
$871$	33.0000	1.11816
$872$	0	0
$873$	12.0000	0.406138
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	−24.0000	−0.810422	−0.405211	−	0.914223i	$-0.632802\pi$
$877$	−24.0000	−0.810422	−0.405211	+	0.914223i	$0.632802\pi$
$878$	0	0
$879$	−16.0000	−0.539667
$880$	0	0
$881$	25.0000	0.842271	0.421136	−	0.906998i	$-0.361632\pi$
$881$	25.0000	0.842271	0.421136	+	0.906998i	$0.361632\pi$
$882$	0	0
$883$	29.0000	0.975928	0.487964	−	0.872864i	$-0.337740\pi$
$883$	29.0000	0.975928	0.487964	+	0.872864i	$0.337740\pi$
$884$	0	0
$885$	9.00000	0.302532
$886$	0	0
$887$	18.0000	0.604381	0.302190	−	0.953248i	$-0.402282\pi$
$887$	18.0000	0.604381	0.302190	+	0.953248i	$0.402282\pi$
$888$	0	0
$889$	−2.00000	−0.0670778
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	9.00000	0.301174
$894$	0	0
$895$	24.0000	0.802232
$896$	0	0
$897$	−12.0000	−0.400668
$898$	0	0
$899$	18.0000	0.600334
$900$	0	0
$901$	8.00000	0.266519
$902$	0	0
$903$	4.00000	0.133112
$904$	0	0
$905$	12.0000	0.398893
$906$	0	0
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	0	0
$909$	10.0000	0.331679
$910$	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
$913$	−10.0000	−0.330952
$914$	0	0
$915$	−30.0000	−0.991769
$916$	0	0
$917$	4.00000	0.132092
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	0	0
$923$	12.0000	0.394985
$924$	0	0
$925$	−44.0000	−1.44671
$926$	0	0
$927$	−14.0000	−0.459820
$928$	0	0
$929$	−29.0000	−0.951459	−0.475730	−	0.879592i	$-0.657816\pi$
$929$	−29.0000	−0.951459	−0.475730	+	0.879592i	$0.657816\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−6.00000	−0.196221
$936$	0	0
$937$	30.0000	0.980057	0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000	0.980057	0.490029	+	0.871706i	$0.336986\pi$
$938$	0	0
$939$	18.0000	0.587408
$940$	0	0
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	0	0
$943$	16.0000	0.521032
$944$	0	0
$945$	3.00000	0.0975900
$946$	0	0
$947$	18.0000	0.584921	0.292461	−	0.956278i	$-0.405526\pi$
$947$	18.0000	0.584921	0.292461	+	0.956278i	$0.405526\pi$
$948$	0	0
$949$	27.0000	0.876457
$950$	0	0
$951$	24.0000	0.778253
$952$	0	0
$953$	−45.0000	−1.45769	−0.728846	−	0.684677i	$-0.759943\pi$
$953$	−45.0000	−1.45769	−0.728846	+	0.684677i	$0.759943\pi$
$954$	0	0
$955$	54.0000	1.74740
$956$	0	0
$957$	9.00000	0.290929
$958$	0	0
$959$	14.0000	0.452084
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−19.0000	−0.612266
$964$	0	0
$965$	−72.0000	−2.31776
$966$	0	0
$967$	18.0000	0.578841	0.289420	−	0.957202i	$-0.406537\pi$
$967$	18.0000	0.578841	0.289420	+	0.957202i	$0.406537\pi$
$968$	0	0
$969$	6.00000	0.192748
$970$	0	0
$971$	23.0000	0.738105	0.369053	−	0.929409i	$-0.379682\pi$
$971$	23.0000	0.738105	0.369053	+	0.929409i	$0.379682\pi$
$972$	0	0
$973$	8.00000	0.256468
$974$	0	0
$975$	12.0000	0.384308
$976$	0	0
$977$	8.00000	0.255943	0.127971	−	0.991778i	$-0.459153\pi$
$977$	8.00000	0.255943	0.127971	+	0.991778i	$0.459153\pi$
$978$	0	0
$979$	−6.00000	−0.191761
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	−56.0000	−1.78612	−0.893061	−	0.449935i	$-0.851447\pi$
$983$	−56.0000	−1.78612	−0.893061	+	0.449935i	$0.851447\pi$
$984$	0	0
$985$	54.0000	1.72058
$986$	0	0
$987$	3.00000	0.0954911
$988$	0	0
$989$	16.0000	0.508770
$990$	0	0
$991$	−17.0000	−0.540023	−0.270011	−	0.962857i	$-0.587027\pi$
$991$	−17.0000	−0.540023	−0.270011	+	0.962857i	$0.587027\pi$
$992$	0	0
$993$	8.00000	0.253872
$994$	0	0
$995$	60.0000	1.90213
$996$	0	0
$997$	−58.0000	−1.83688	−0.918439	−	0.395562i	$-0.870550\pi$
$997$	−58.0000	−1.83688	−0.918439	+	0.395562i	$0.870550\pi$
$998$	0	0
$999$	−11.0000	−0.348025

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	924.2.a.e.1.1	✓	1
3.2	odd	2		2772.2.a.j.1.1		1
4.3	odd	2		3696.2.a.d.1.1		1
7.6	odd	2		6468.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
924.2.a.e.1.1	✓	1	1.1	even	1	trivial
2772.2.a.j.1.1		1	3.2	odd	2
3696.2.a.d.1.1		1	4.3	odd	2
6468.2.a.h.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:	\( N \)	\(=\)	\( 924 = 2^{2} \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	924.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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