LMFDB - Embedded newform 2352.4.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.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:	$138.772492334$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} -2.00000 q^{5} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} -2.00000 q^{5} +9.00000 q^{9} +8.00000 q^{11} +42.0000 q^{13} -6.00000 q^{15} +2.00000 q^{17} -124.000 q^{19} -76.0000 q^{23} -121.000 q^{25} +27.0000 q^{27} +254.000 q^{29} -72.0000 q^{31} +24.0000 q^{33} +398.000 q^{37} +126.000 q^{39} -462.000 q^{41} -212.000 q^{43} -18.0000 q^{45} -264.000 q^{47} +6.00000 q^{51} -162.000 q^{53} -16.0000 q^{55} -372.000 q^{57} -772.000 q^{59} -30.0000 q^{61} -84.0000 q^{65} +764.000 q^{67} -228.000 q^{69} +236.000 q^{71} -418.000 q^{73} -363.000 q^{75} -552.000 q^{79} +81.0000 q^{81} +1036.00 q^{83} -4.00000 q^{85} +762.000 q^{87} -30.0000 q^{89} -216.000 q^{93} +248.000 q^{95} +1190.00 q^{97} +72.0000 q^{99} +O(q^{100})$

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−2.00000	−0.178885	−0.0894427	−	0.995992i	$-0.528509\pi$
$5$	−2.00000	−0.178885	−0.0894427	+	0.995992i	$0.528509\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	8.00000	0.219281	0.109640	−	0.993971i	$-0.465030\pi$
$11$	8.00000	0.219281	0.109640	+	0.993971i	$0.465030\pi$
$12$	0	0
$13$	42.0000	0.896054	0.448027	−	0.894020i	$-0.352127\pi$
$13$	42.0000	0.896054	0.448027	+	0.894020i	$0.352127\pi$
$14$	0	0
$15$	−6.00000	−0.103280
$16$	0	0
$17$	2.00000	0.0285336	0.0142668	−	0.999898i	$-0.495459\pi$
$17$	2.00000	0.0285336	0.0142668	+	0.999898i	$0.495459\pi$
$18$	0	0
$19$	−124.000	−1.49724	−0.748620	−	0.663000i	$-0.769283\pi$
$19$	−124.000	−1.49724	−0.748620	+	0.663000i	$0.769283\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−76.0000	−0.689004	−0.344502	−	0.938786i	$-0.611952\pi$
$23$	−76.0000	−0.689004	−0.344502	+	0.938786i	$0.611952\pi$
$24$	0	0
$25$	−121.000	−0.968000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	254.000	1.62644	0.813218	−	0.581960i	$-0.197714\pi$
$29$	254.000	1.62644	0.813218	+	0.581960i	$0.197714\pi$
$30$	0	0
$31$	−72.0000	−0.417148	−0.208574	−	0.978007i	$-0.566882\pi$
$31$	−72.0000	−0.417148	−0.208574	+	0.978007i	$0.566882\pi$
$32$	0	0
$33$	24.0000	0.126602
$34$	0	0
$35$	0	0
$36$	0	0
$37$	398.000	1.76840	0.884200	−	0.467109i	$-0.154704\pi$
$37$	398.000	1.76840	0.884200	+	0.467109i	$0.154704\pi$
$38$	0	0
$39$	126.000	0.517337
$40$	0	0
$41$	−462.000	−1.75981	−0.879906	−	0.475148i	$-0.842394\pi$
$41$	−462.000	−1.75981	−0.879906	+	0.475148i	$0.842394\pi$
$42$	0	0
$43$	−212.000	−0.751853	−0.375927	−	0.926649i	$-0.622676\pi$
$43$	−212.000	−0.751853	−0.375927	+	0.926649i	$0.622676\pi$
$44$	0	0
$45$	−18.0000	−0.0596285
$46$	0	0
$47$	−264.000	−0.819327	−0.409663	−	0.912237i	$-0.634354\pi$
$47$	−264.000	−0.819327	−0.409663	+	0.912237i	$0.634354\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	6.00000	0.0164739
$52$	0	0
$53$	−162.000	−0.419857	−0.209928	−	0.977717i	$-0.567323\pi$
$53$	−162.000	−0.419857	−0.209928	+	0.977717i	$0.567323\pi$
$54$	0	0
$55$	−16.0000	−0.0392262
$56$	0	0
$57$	−372.000	−0.864432
$58$	0	0
$59$	−772.000	−1.70349	−0.851744	−	0.523958i	$-0.824455\pi$
$59$	−772.000	−1.70349	−0.851744	+	0.523958i	$0.824455\pi$
$60$	0	0
$61$	−30.0000	−0.0629690	−0.0314845	−	0.999504i	$-0.510023\pi$
$61$	−30.0000	−0.0629690	−0.0314845	+	0.999504i	$0.510023\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−84.0000	−0.160291
$66$	0	0
$67$	764.000	1.39310	0.696548	−	0.717510i	$-0.254718\pi$
$67$	764.000	1.39310	0.696548	+	0.717510i	$0.254718\pi$
$68$	0	0
$69$	−228.000	−0.397797
$70$	0	0
$71$	236.000	0.394480	0.197240	−	0.980355i	$-0.436802\pi$
$71$	236.000	0.394480	0.197240	+	0.980355i	$0.436802\pi$
$72$	0	0
$73$	−418.000	−0.670181	−0.335090	−	0.942186i	$-0.608767\pi$
$73$	−418.000	−0.670181	−0.335090	+	0.942186i	$0.608767\pi$
$74$	0	0
$75$	−363.000	−0.558875
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−552.000	−0.786137	−0.393069	−	0.919509i	$-0.628587\pi$
$79$	−552.000	−0.786137	−0.393069	+	0.919509i	$0.628587\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1036.00	1.37007	0.685035	−	0.728510i	$-0.259787\pi$
$83$	1036.00	1.37007	0.685035	+	0.728510i	$0.259787\pi$
$84$	0	0
$85$	−4.00000	−0.00510425
$86$	0	0
$87$	762.000	0.939023
$88$	0	0
$89$	−30.0000	−0.0357303	−0.0178651	−	0.999840i	$-0.505687\pi$
$89$	−30.0000	−0.0357303	−0.0178651	+	0.999840i	$0.505687\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−216.000	−0.240840
$94$	0	0
$95$	248.000	0.267834
$96$	0	0
$97$	1190.00	1.24563	0.622815	−	0.782369i	$-0.285989\pi$
$97$	1190.00	1.24563	0.622815	+	0.782369i	$0.285989\pi$
$98$	0	0
$99$	72.0000	0.0730937
$100$	0	0
$101$	−1370.00	−1.34970	−0.674852	−	0.737953i	$-0.735793\pi$
$101$	−1370.00	−1.34970	−0.674852	+	0.737953i	$0.735793\pi$
$102$	0	0
$103$	464.000	0.443876	0.221938	−	0.975061i	$-0.428762\pi$
$103$	464.000	0.443876	0.221938	+	0.975061i	$0.428762\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2136.00	1.92986	0.964930	−	0.262509i	$-0.0845500\pi$
$107$	2136.00	1.92986	0.964930	+	0.262509i	$0.0845500\pi$
$108$	0	0
$109$	−1226.00	−1.07733	−0.538667	−	0.842518i	$-0.681072\pi$
$109$	−1226.00	−1.07733	−0.538667	+	0.842518i	$0.681072\pi$
$110$	0	0
$111$	1194.00	1.02099
$112$	0	0
$113$	338.000	0.281384	0.140692	−	0.990053i	$-0.455067\pi$
$113$	338.000	0.281384	0.140692	+	0.990053i	$0.455067\pi$
$114$	0	0
$115$	152.000	0.123253
$116$	0	0
$117$	378.000	0.298685
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1267.00	−0.951916
$122$	0	0
$123$	−1386.00	−1.01603
$124$	0	0
$125$	492.000	0.352047
$126$	0	0
$127$	−2088.00	−1.45890	−0.729449	−	0.684035i	$-0.760223\pi$
$127$	−2088.00	−1.45890	−0.729449	+	0.684035i	$0.760223\pi$
$128$	0	0
$129$	−636.000	−0.434083
$130$	0	0
$131$	−292.000	−0.194749	−0.0973747	−	0.995248i	$-0.531045\pi$
$131$	−292.000	−0.194749	−0.0973747	+	0.995248i	$0.531045\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−54.0000	−0.0344265
$136$	0	0
$137$	818.000	0.510120	0.255060	−	0.966925i	$-0.417905\pi$
$137$	818.000	0.510120	0.255060	+	0.966925i	$0.417905\pi$
$138$	0	0
$139$	−2156.00	−1.31561	−0.657804	−	0.753189i	$-0.728515\pi$
$139$	−2156.00	−1.31561	−0.657804	+	0.753189i	$0.728515\pi$
$140$	0	0
$141$	−792.000	−0.473039
$142$	0	0
$143$	336.000	0.196488
$144$	0	0
$145$	−508.000	−0.290946
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2850.00	−1.56699	−0.783494	−	0.621400i	$-0.786564\pi$
$149$	−2850.00	−1.56699	−0.783494	+	0.621400i	$0.786564\pi$
$150$	0	0
$151$	−1672.00	−0.901096	−0.450548	−	0.892752i	$-0.648771\pi$
$151$	−1672.00	−0.901096	−0.450548	+	0.892752i	$0.648771\pi$
$152$	0	0
$153$	18.0000	0.00951120
$154$	0	0
$155$	144.000	0.0746217
$156$	0	0
$157$	−446.000	−0.226718	−0.113359	−	0.993554i	$-0.536161\pi$
$157$	−446.000	−0.226718	−0.113359	+	0.993554i	$0.536161\pi$
$158$	0	0
$159$	−486.000	−0.242404
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2708.00	−1.30127	−0.650635	−	0.759391i	$-0.725497\pi$
$163$	−2708.00	−1.30127	−0.650635	+	0.759391i	$0.725497\pi$
$164$	0	0
$165$	−48.0000	−0.0226472
$166$	0	0
$167$	896.000	0.415177	0.207589	−	0.978216i	$-0.433439\pi$
$167$	896.000	0.415177	0.207589	+	0.978216i	$0.433439\pi$
$168$	0	0
$169$	−433.000	−0.197087
$170$	0	0
$171$	−1116.00	−0.499080
$172$	0	0
$173$	−4034.00	−1.77283	−0.886414	−	0.462893i	$-0.846811\pi$
$173$	−4034.00	−1.77283	−0.886414	+	0.462893i	$0.846811\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2316.00	−0.983510
$178$	0	0
$179$	3480.00	1.45311	0.726557	−	0.687106i	$-0.241119\pi$
$179$	3480.00	1.45311	0.726557	+	0.687106i	$0.241119\pi$
$180$	0	0
$181$	2898.00	1.19009	0.595046	−	0.803692i	$-0.297134\pi$
$181$	2898.00	1.19009	0.595046	+	0.803692i	$0.297134\pi$
$182$	0	0
$183$	−90.0000	−0.0363551
$184$	0	0
$185$	−796.000	−0.316341
$186$	0	0
$187$	16.0000	0.00625688
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2652.00	−1.00467	−0.502335	−	0.864673i	$-0.667526\pi$
$191$	−2652.00	−1.00467	−0.502335	+	0.864673i	$0.667526\pi$
$192$	0	0
$193$	146.000	0.0544524	0.0272262	−	0.999629i	$-0.491333\pi$
$193$	146.000	0.0544524	0.0272262	+	0.999629i	$0.491333\pi$
$194$	0	0
$195$	−252.000	−0.0925441
$196$	0	0
$197$	−2546.00	−0.920787	−0.460393	−	0.887715i	$-0.652292\pi$
$197$	−2546.00	−0.920787	−0.460393	+	0.887715i	$0.652292\pi$
$198$	0	0
$199$	−2536.00	−0.903378	−0.451689	−	0.892175i	$-0.649178\pi$
$199$	−2536.00	−0.903378	−0.451689	+	0.892175i	$0.649178\pi$
$200$	0	0
$201$	2292.00	0.804305
$202$	0	0
$203$	0	0
$204$	0	0
$205$	924.000	0.314805
$206$	0	0
$207$	−684.000	−0.229668
$208$	0	0
$209$	−992.000	−0.328316
$210$	0	0
$211$	1300.00	0.424150	0.212075	−	0.977253i	$-0.431978\pi$
$211$	1300.00	0.424150	0.212075	+	0.977253i	$0.431978\pi$
$212$	0	0
$213$	708.000	0.227753
$214$	0	0
$215$	424.000	0.134496
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1254.00	−0.386929
$220$	0	0
$221$	84.0000	0.0255677
$222$	0	0
$223$	2576.00	0.773550	0.386775	−	0.922174i	$-0.373589\pi$
$223$	2576.00	0.773550	0.386775	+	0.922174i	$0.373589\pi$
$224$	0	0
$225$	−1089.00	−0.322667
$226$	0	0
$227$	−1836.00	−0.536826	−0.268413	−	0.963304i	$-0.586499\pi$
$227$	−1836.00	−0.536826	−0.268413	+	0.963304i	$0.586499\pi$
$228$	0	0
$229$	1874.00	0.540775	0.270387	−	0.962752i	$-0.412848\pi$
$229$	1874.00	0.540775	0.270387	+	0.962752i	$0.412848\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3730.00	1.04876	0.524379	−	0.851485i	$-0.324298\pi$
$233$	3730.00	1.04876	0.524379	+	0.851485i	$0.324298\pi$
$234$	0	0
$235$	528.000	0.146566
$236$	0	0
$237$	−1656.00	−0.453877
$238$	0	0
$239$	−2004.00	−0.542377	−0.271188	−	0.962526i	$-0.587417\pi$
$239$	−2004.00	−0.542377	−0.271188	+	0.962526i	$0.587417\pi$
$240$	0	0
$241$	646.000	0.172666	0.0863330	−	0.996266i	$-0.472485\pi$
$241$	646.000	0.172666	0.0863330	+	0.996266i	$0.472485\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5208.00	−1.34161
$248$	0	0
$249$	3108.00	0.791010
$250$	0	0
$251$	1260.00	0.316855	0.158427	−	0.987371i	$-0.449358\pi$
$251$	1260.00	0.316855	0.158427	+	0.987371i	$0.449358\pi$
$252$	0	0
$253$	−608.000	−0.151086
$254$	0	0
$255$	−12.0000	−0.00294694
$256$	0	0
$257$	−5910.00	−1.43446	−0.717229	−	0.696838i	$-0.754590\pi$
$257$	−5910.00	−1.43446	−0.717229	+	0.696838i	$0.754590\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2286.00	0.542145
$262$	0	0
$263$	−2988.00	−0.700563	−0.350281	−	0.936645i	$-0.613914\pi$
$263$	−2988.00	−0.700563	−0.350281	+	0.936645i	$0.613914\pi$
$264$	0	0
$265$	324.000	0.0751063
$266$	0	0
$267$	−90.0000	−0.0206289
$268$	0	0
$269$	1318.00	0.298736	0.149368	−	0.988782i	$-0.452276\pi$
$269$	1318.00	0.298736	0.149368	+	0.988782i	$0.452276\pi$
$270$	0	0
$271$	−5640.00	−1.26423	−0.632114	−	0.774876i	$-0.717812\pi$
$271$	−5640.00	−1.26423	−0.632114	+	0.774876i	$0.717812\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−968.000	−0.212264
$276$	0	0
$277$	6446.00	1.39820	0.699102	−	0.715022i	$-0.253583\pi$
$277$	6446.00	1.39820	0.699102	+	0.715022i	$0.253583\pi$
$278$	0	0
$279$	−648.000	−0.139049
$280$	0	0
$281$	4930.00	1.04662	0.523308	−	0.852144i	$-0.324698\pi$
$281$	4930.00	1.04662	0.523308	+	0.852144i	$0.324698\pi$
$282$	0	0
$283$	−6260.00	−1.31491	−0.657453	−	0.753496i	$-0.728366\pi$
$283$	−6260.00	−1.31491	−0.657453	+	0.753496i	$0.728366\pi$
$284$	0	0
$285$	744.000	0.154634
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4909.00	−0.999186
$290$	0	0
$291$	3570.00	0.719165
$292$	0	0
$293$	2310.00	0.460586	0.230293	−	0.973121i	$-0.426032\pi$
$293$	2310.00	0.460586	0.230293	+	0.973121i	$0.426032\pi$
$294$	0	0
$295$	1544.00	0.304729
$296$	0	0
$297$	216.000	0.0422006
$298$	0	0
$299$	−3192.00	−0.617385
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−4110.00	−0.779252
$304$	0	0
$305$	60.0000	0.0112642
$306$	0	0
$307$	196.000	0.0364375	0.0182187	−	0.999834i	$-0.494200\pi$
$307$	196.000	0.0364375	0.0182187	+	0.999834i	$0.494200\pi$
$308$	0	0
$309$	1392.00	0.256272
$310$	0	0
$311$	−6736.00	−1.22818	−0.614089	−	0.789237i	$-0.710477\pi$
$311$	−6736.00	−1.22818	−0.614089	+	0.789237i	$0.710477\pi$
$312$	0	0
$313$	−394.000	−0.0711508	−0.0355754	−	0.999367i	$-0.511326\pi$
$313$	−394.000	−0.0711508	−0.0355754	+	0.999367i	$0.511326\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6714.00	−1.18958	−0.594788	−	0.803882i	$-0.702764\pi$
$317$	−6714.00	−1.18958	−0.594788	+	0.803882i	$0.702764\pi$
$318$	0	0
$319$	2032.00	0.356646
$320$	0	0
$321$	6408.00	1.11420
$322$	0	0
$323$	−248.000	−0.0427216
$324$	0	0
$325$	−5082.00	−0.867380
$326$	0	0
$327$	−3678.00	−0.622000
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−692.000	−0.114912	−0.0574558	−	0.998348i	$-0.518299\pi$
$331$	−692.000	−0.114912	−0.0574558	+	0.998348i	$0.518299\pi$
$332$	0	0
$333$	3582.00	0.589467
$334$	0	0
$335$	−1528.00	−0.249205
$336$	0	0
$337$	−1566.00	−0.253132	−0.126566	−	0.991958i	$-0.540396\pi$
$337$	−1566.00	−0.253132	−0.126566	+	0.991958i	$0.540396\pi$
$338$	0	0
$339$	1014.00	0.162457
$340$	0	0
$341$	−576.000	−0.0914726
$342$	0	0
$343$	0	0
$344$	0	0
$345$	456.000	0.0711600
$346$	0	0
$347$	5328.00	0.824271	0.412135	−	0.911123i	$-0.364783\pi$
$347$	5328.00	0.824271	0.412135	+	0.911123i	$0.364783\pi$
$348$	0	0
$349$	−11326.0	−1.73715	−0.868577	−	0.495554i	$-0.834965\pi$
$349$	−11326.0	−1.73715	−0.868577	+	0.495554i	$0.834965\pi$
$350$	0	0
$351$	1134.00	0.172446
$352$	0	0
$353$	2130.00	0.321157	0.160579	−	0.987023i	$-0.448664\pi$
$353$	2130.00	0.321157	0.160579	+	0.987023i	$0.448664\pi$
$354$	0	0
$355$	−472.000	−0.0705666
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3044.00	−0.447510	−0.223755	−	0.974645i	$-0.571832\pi$
$359$	−3044.00	−0.447510	−0.223755	+	0.974645i	$0.571832\pi$
$360$	0	0
$361$	8517.00	1.24173
$362$	0	0
$363$	−3801.00	−0.549589
$364$	0	0
$365$	836.000	0.119886
$366$	0	0
$367$	12416.0	1.76597	0.882984	−	0.469404i	$-0.155531\pi$
$367$	12416.0	1.76597	0.882984	+	0.469404i	$0.155531\pi$
$368$	0	0
$369$	−4158.00	−0.586604
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−7442.00	−1.03306	−0.516531	−	0.856268i	$-0.672777\pi$
$373$	−7442.00	−1.03306	−0.516531	+	0.856268i	$0.672777\pi$
$374$	0	0
$375$	1476.00	0.203254
$376$	0	0
$377$	10668.0	1.45737
$378$	0	0
$379$	−100.000	−0.0135532	−0.00677659	−	0.999977i	$-0.502157\pi$
$379$	−100.000	−0.0135532	−0.00677659	+	0.999977i	$0.502157\pi$
$380$	0	0
$381$	−6264.00	−0.842295
$382$	0	0
$383$	8080.00	1.07799	0.538993	−	0.842310i	$-0.318805\pi$
$383$	8080.00	1.07799	0.538993	+	0.842310i	$0.318805\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1908.00	−0.250618
$388$	0	0
$389$	−5482.00	−0.714520	−0.357260	−	0.934005i	$-0.616289\pi$
$389$	−5482.00	−0.714520	−0.357260	+	0.934005i	$0.616289\pi$
$390$	0	0
$391$	−152.000	−0.0196598
$392$	0	0
$393$	−876.000	−0.112439
$394$	0	0
$395$	1104.00	0.140629
$396$	0	0
$397$	−10446.0	−1.32058	−0.660289	−	0.751011i	$-0.729566\pi$
$397$	−10446.0	−1.32058	−0.660289	+	0.751011i	$0.729566\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11334.0	−1.41145	−0.705727	−	0.708484i	$-0.749379\pi$
$401$	−11334.0	−1.41145	−0.705727	+	0.708484i	$0.749379\pi$
$402$	0	0
$403$	−3024.00	−0.373787
$404$	0	0
$405$	−162.000	−0.0198762
$406$	0	0
$407$	3184.00	0.387776
$408$	0	0
$409$	−8594.00	−1.03899	−0.519494	−	0.854474i	$-0.673879\pi$
$409$	−8594.00	−1.03899	−0.519494	+	0.854474i	$0.673879\pi$
$410$	0	0
$411$	2454.00	0.294518
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−2072.00	−0.245085
$416$	0	0
$417$	−6468.00	−0.759567
$418$	0	0
$419$	10500.0	1.22424	0.612122	−	0.790763i	$-0.290316\pi$
$419$	10500.0	1.22424	0.612122	+	0.790763i	$0.290316\pi$
$420$	0	0
$421$	−12066.0	−1.39682	−0.698410	−	0.715698i	$-0.746109\pi$
$421$	−12066.0	−1.39682	−0.698410	+	0.715698i	$0.746109\pi$
$422$	0	0
$423$	−2376.00	−0.273109
$424$	0	0
$425$	−242.000	−0.0276205
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1008.00	0.113442
$430$	0	0
$431$	−4332.00	−0.484142	−0.242071	−	0.970259i	$-0.577827\pi$
$431$	−4332.00	−0.484142	−0.242071	+	0.970259i	$0.577827\pi$
$432$	0	0
$433$	1918.00	0.212871	0.106436	−	0.994320i	$-0.466056\pi$
$433$	1918.00	0.212871	0.106436	+	0.994320i	$0.466056\pi$
$434$	0	0
$435$	−1524.00	−0.167977
$436$	0	0
$437$	9424.00	1.03160
$438$	0	0
$439$	−7992.00	−0.868878	−0.434439	−	0.900701i	$-0.643053\pi$
$439$	−7992.00	−0.868878	−0.434439	+	0.900701i	$0.643053\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3184.00	−0.341482	−0.170741	−	0.985316i	$-0.554616\pi$
$443$	−3184.00	−0.341482	−0.170741	+	0.985316i	$0.554616\pi$
$444$	0	0
$445$	60.0000	0.00639162
$446$	0	0
$447$	−8550.00	−0.904700
$448$	0	0
$449$	11426.0	1.20095	0.600475	−	0.799644i	$-0.294978\pi$
$449$	11426.0	1.20095	0.600475	+	0.799644i	$0.294978\pi$
$450$	0	0
$451$	−3696.00	−0.385893
$452$	0	0
$453$	−5016.00	−0.520248
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16934.0	−1.73335	−0.866673	−	0.498877i	$-0.833746\pi$
$457$	−16934.0	−1.73335	−0.866673	+	0.498877i	$0.833746\pi$
$458$	0	0
$459$	54.0000	0.00549129
$460$	0	0
$461$	17038.0	1.72134	0.860671	−	0.509161i	$-0.170044\pi$
$461$	17038.0	1.72134	0.860671	+	0.509161i	$0.170044\pi$
$462$	0	0
$463$	13592.0	1.36431	0.682153	−	0.731209i	$-0.261044\pi$
$463$	13592.0	1.36431	0.682153	+	0.731209i	$0.261044\pi$
$464$	0	0
$465$	432.000	0.0430828
$466$	0	0
$467$	8612.00	0.853353	0.426676	−	0.904404i	$-0.359684\pi$
$467$	8612.00	0.853353	0.426676	+	0.904404i	$0.359684\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1338.00	−0.130896
$472$	0	0
$473$	−1696.00	−0.164867
$474$	0	0
$475$	15004.0	1.44933
$476$	0	0
$477$	−1458.00	−0.139952
$478$	0	0
$479$	7432.00	0.708928	0.354464	−	0.935070i	$-0.384663\pi$
$479$	7432.00	0.708928	0.354464	+	0.935070i	$0.384663\pi$
$480$	0	0
$481$	16716.0	1.58458
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2380.00	−0.222825
$486$	0	0
$487$	6616.00	0.615605	0.307802	−	0.951450i	$-0.400406\pi$
$487$	6616.00	0.615605	0.307802	+	0.951450i	$0.400406\pi$
$488$	0	0
$489$	−8124.00	−0.751288
$490$	0	0
$491$	−17040.0	−1.56620	−0.783100	−	0.621896i	$-0.786363\pi$
$491$	−17040.0	−1.56620	−0.783100	+	0.621896i	$0.786363\pi$
$492$	0	0
$493$	508.000	0.0464081
$494$	0	0
$495$	−144.000	−0.0130754
$496$	0	0
$497$	0	0
$498$	0	0
$499$	2948.00	0.264470	0.132235	−	0.991218i	$-0.457785\pi$
$499$	2948.00	0.264470	0.132235	+	0.991218i	$0.457785\pi$
$500$	0	0
$501$	2688.00	0.239703
$502$	0	0
$503$	17304.0	1.53389	0.766946	−	0.641712i	$-0.221776\pi$
$503$	17304.0	1.53389	0.766946	+	0.641712i	$0.221776\pi$
$504$	0	0
$505$	2740.00	0.241442
$506$	0	0
$507$	−1299.00	−0.113788
$508$	0	0
$509$	−4650.00	−0.404927	−0.202463	−	0.979290i	$-0.564895\pi$
$509$	−4650.00	−0.404927	−0.202463	+	0.979290i	$0.564895\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−3348.00	−0.288144
$514$	0	0
$515$	−928.000	−0.0794030
$516$	0	0
$517$	−2112.00	−0.179663
$518$	0	0
$519$	−12102.0	−1.02354
$520$	0	0
$521$	−16854.0	−1.41725	−0.708625	−	0.705585i	$-0.750684\pi$
$521$	−16854.0	−1.41725	−0.708625	+	0.705585i	$0.750684\pi$
$522$	0	0
$523$	−124.000	−0.0103674	−0.00518369	−	0.999987i	$-0.501650\pi$
$523$	−124.000	−0.0103674	−0.00518369	+	0.999987i	$0.501650\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−144.000	−0.0119027
$528$	0	0
$529$	−6391.00	−0.525273
$530$	0	0
$531$	−6948.00	−0.567830
$532$	0	0
$533$	−19404.0	−1.57689
$534$	0	0
$535$	−4272.00	−0.345224
$536$	0	0
$537$	10440.0	0.838956
$538$	0	0
$539$	0	0
$540$	0	0
$541$	5382.00	0.427708	0.213854	−	0.976866i	$-0.431398\pi$
$541$	5382.00	0.427708	0.213854	+	0.976866i	$0.431398\pi$
$542$	0	0
$543$	8694.00	0.687100
$544$	0	0
$545$	2452.00	0.192720
$546$	0	0
$547$	−17460.0	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$547$	−17460.0	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$548$	0	0
$549$	−270.000	−0.0209897
$550$	0	0
$551$	−31496.0	−2.43516
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−2388.00	−0.182640
$556$	0	0
$557$	−9514.00	−0.723736	−0.361868	−	0.932229i	$-0.617861\pi$
$557$	−9514.00	−0.723736	−0.361868	+	0.932229i	$0.617861\pi$
$558$	0	0
$559$	−8904.00	−0.673701
$560$	0	0
$561$	48.0000	0.00361241
$562$	0	0
$563$	3988.00	0.298533	0.149267	−	0.988797i	$-0.452309\pi$
$563$	3988.00	0.298533	0.149267	+	0.988797i	$0.452309\pi$
$564$	0	0
$565$	−676.000	−0.0503355
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11346.0	0.835939	0.417969	−	0.908461i	$-0.362742\pi$
$569$	11346.0	0.835939	0.417969	+	0.908461i	$0.362742\pi$
$570$	0	0
$571$	8436.00	0.618276	0.309138	−	0.951017i	$-0.399959\pi$
$571$	8436.00	0.618276	0.309138	+	0.951017i	$0.399959\pi$
$572$	0	0
$573$	−7956.00	−0.580047
$574$	0	0
$575$	9196.00	0.666956
$576$	0	0
$577$	−2098.00	−0.151371	−0.0756853	−	0.997132i	$-0.524114\pi$
$577$	−2098.00	−0.151371	−0.0756853	+	0.997132i	$0.524114\pi$
$578$	0	0
$579$	438.000	0.0314381
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−1296.00	−0.0920666
$584$	0	0
$585$	−756.000	−0.0534303
$586$	0	0
$587$	9436.00	0.663484	0.331742	−	0.943370i	$-0.392364\pi$
$587$	9436.00	0.663484	0.331742	+	0.943370i	$0.392364\pi$
$588$	0	0
$589$	8928.00	0.624570
$590$	0	0
$591$	−7638.00	−0.531616
$592$	0	0
$593$	1314.00	0.0909941	0.0454971	−	0.998964i	$-0.485513\pi$
$593$	1314.00	0.0909941	0.0454971	+	0.998964i	$0.485513\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−7608.00	−0.521566
$598$	0	0
$599$	8940.00	0.609814	0.304907	−	0.952382i	$-0.401375\pi$
$599$	8940.00	0.609814	0.304907	+	0.952382i	$0.401375\pi$
$600$	0	0
$601$	−16058.0	−1.08988	−0.544941	−	0.838474i	$-0.683448\pi$
$601$	−16058.0	−1.08988	−0.544941	+	0.838474i	$0.683448\pi$
$602$	0	0
$603$	6876.00	0.464365
$604$	0	0
$605$	2534.00	0.170284
$606$	0	0
$607$	3936.00	0.263192	0.131596	−	0.991303i	$-0.457990\pi$
$607$	3936.00	0.263192	0.131596	+	0.991303i	$0.457990\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−11088.0	−0.734161
$612$	0	0
$613$	174.000	0.0114646	0.00573230	−	0.999984i	$-0.498175\pi$
$613$	174.000	0.0114646	0.00573230	+	0.999984i	$0.498175\pi$
$614$	0	0
$615$	2772.00	0.181753
$616$	0	0
$617$	16018.0	1.04515	0.522577	−	0.852592i	$-0.324971\pi$
$617$	16018.0	1.04515	0.522577	+	0.852592i	$0.324971\pi$
$618$	0	0
$619$	−3068.00	−0.199214	−0.0996069	−	0.995027i	$-0.531759\pi$
$619$	−3068.00	−0.199214	−0.0996069	+	0.995027i	$0.531759\pi$
$620$	0	0
$621$	−2052.00	−0.132599
$622$	0	0
$623$	0	0
$624$	0	0
$625$	14141.0	0.905024
$626$	0	0
$627$	−2976.00	−0.189553
$628$	0	0
$629$	796.000	0.0504588
$630$	0	0
$631$	−24656.0	−1.55553	−0.777765	−	0.628555i	$-0.783647\pi$
$631$	−24656.0	−1.55553	−0.777765	+	0.628555i	$0.783647\pi$
$632$	0	0
$633$	3900.00	0.244883
$634$	0	0
$635$	4176.00	0.260976
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2124.00	0.131493
$640$	0	0
$641$	7594.00	0.467933	0.233966	−	0.972245i	$-0.424829\pi$
$641$	7594.00	0.467933	0.233966	+	0.972245i	$0.424829\pi$
$642$	0	0
$643$	−3724.00	−0.228398	−0.114199	−	0.993458i	$-0.536430\pi$
$643$	−3724.00	−0.228398	−0.114199	+	0.993458i	$0.536430\pi$
$644$	0	0
$645$	1272.00	0.0776511
$646$	0	0
$647$	3792.00	0.230416	0.115208	−	0.993341i	$-0.463247\pi$
$647$	3792.00	0.230416	0.115208	+	0.993341i	$0.463247\pi$
$648$	0	0
$649$	−6176.00	−0.373543
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24702.0	1.48034	0.740171	−	0.672418i	$-0.234744\pi$
$653$	24702.0	1.48034	0.740171	+	0.672418i	$0.234744\pi$
$654$	0	0
$655$	584.000	0.0348378
$656$	0	0
$657$	−3762.00	−0.223394
$658$	0	0
$659$	20144.0	1.19074	0.595371	−	0.803451i	$-0.297005\pi$
$659$	20144.0	1.19074	0.595371	+	0.803451i	$0.297005\pi$
$660$	0	0
$661$	2522.00	0.148403	0.0742015	−	0.997243i	$-0.476359\pi$
$661$	2522.00	0.148403	0.0742015	+	0.997243i	$0.476359\pi$
$662$	0	0
$663$	252.000	0.0147615
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−19304.0	−1.12062
$668$	0	0
$669$	7728.00	0.446609
$670$	0	0
$671$	−240.000	−0.0138079
$672$	0	0
$673$	−10414.0	−0.596479	−0.298239	−	0.954491i	$-0.596399\pi$
$673$	−10414.0	−0.596479	−0.298239	+	0.954491i	$0.596399\pi$
$674$	0	0
$675$	−3267.00	−0.186292
$676$	0	0
$677$	22230.0	1.26199	0.630996	−	0.775786i	$-0.282647\pi$
$677$	22230.0	1.26199	0.630996	+	0.775786i	$0.282647\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−5508.00	−0.309937
$682$	0	0
$683$	−18192.0	−1.01918	−0.509588	−	0.860418i	$-0.670202\pi$
$683$	−18192.0	−1.01918	−0.509588	+	0.860418i	$0.670202\pi$
$684$	0	0
$685$	−1636.00	−0.0912531
$686$	0	0
$687$	5622.00	0.312216
$688$	0	0
$689$	−6804.00	−0.376214
$690$	0	0
$691$	8108.00	0.446372	0.223186	−	0.974776i	$-0.428354\pi$
$691$	8108.00	0.446372	0.223186	+	0.974776i	$0.428354\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4312.00	0.235343
$696$	0	0
$697$	−924.000	−0.0502138
$698$	0	0
$699$	11190.0	0.605500
$700$	0	0
$701$	−5794.00	−0.312177	−0.156089	−	0.987743i	$-0.549889\pi$
$701$	−5794.00	−0.312177	−0.156089	+	0.987743i	$0.549889\pi$
$702$	0	0
$703$	−49352.0	−2.64772
$704$	0	0
$705$	1584.00	0.0846197
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1954.00	−0.103504	−0.0517518	−	0.998660i	$-0.516480\pi$
$709$	−1954.00	−0.103504	−0.0517518	+	0.998660i	$0.516480\pi$
$710$	0	0
$711$	−4968.00	−0.262046
$712$	0	0
$713$	5472.00	0.287417
$714$	0	0
$715$	−672.000	−0.0351488
$716$	0	0
$717$	−6012.00	−0.313141
$718$	0	0
$719$	−32016.0	−1.66063	−0.830317	−	0.557292i	$-0.811840\pi$
$719$	−32016.0	−1.66063	−0.830317	+	0.557292i	$0.811840\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1938.00	0.0996888
$724$	0	0
$725$	−30734.0	−1.57439
$726$	0	0
$727$	−23072.0	−1.17702	−0.588510	−	0.808490i	$-0.700285\pi$
$727$	−23072.0	−1.17702	−0.588510	+	0.808490i	$0.700285\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−424.000	−0.0214531
$732$	0	0
$733$	−31782.0	−1.60149	−0.800747	−	0.599003i	$-0.795564\pi$
$733$	−31782.0	−1.60149	−0.800747	+	0.599003i	$0.795564\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6112.00	0.305480
$738$	0	0
$739$	24396.0	1.21437	0.607186	−	0.794559i	$-0.292298\pi$
$739$	24396.0	1.21437	0.607186	+	0.794559i	$0.292298\pi$
$740$	0	0
$741$	−15624.0	−0.774578
$742$	0	0
$743$	32604.0	1.60986	0.804929	−	0.593371i	$-0.202203\pi$
$743$	32604.0	1.60986	0.804929	+	0.593371i	$0.202203\pi$
$744$	0	0
$745$	5700.00	0.280311
$746$	0	0
$747$	9324.00	0.456690
$748$	0	0
$749$	0	0
$750$	0	0
$751$	7680.00	0.373165	0.186583	−	0.982439i	$-0.440259\pi$
$751$	7680.00	0.373165	0.186583	+	0.982439i	$0.440259\pi$
$752$	0	0
$753$	3780.00	0.182936
$754$	0	0
$755$	3344.00	0.161193
$756$	0	0
$757$	366.000	0.0175727	0.00878633	−	0.999961i	$-0.497203\pi$
$757$	366.000	0.0175727	0.00878633	+	0.999961i	$0.497203\pi$
$758$	0	0
$759$	−1824.00	−0.0872293
$760$	0	0
$761$	−29374.0	−1.39922	−0.699610	−	0.714525i	$-0.746643\pi$
$761$	−29374.0	−1.39922	−0.699610	+	0.714525i	$0.746643\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−36.0000	−0.00170142
$766$	0	0
$767$	−32424.0	−1.52642
$768$	0	0
$769$	38990.0	1.82837	0.914184	−	0.405299i	$-0.132833\pi$
$769$	38990.0	1.82837	0.914184	+	0.405299i	$0.132833\pi$
$770$	0	0
$771$	−17730.0	−0.828185
$772$	0	0
$773$	20470.0	0.952464	0.476232	−	0.879320i	$-0.342002\pi$
$773$	20470.0	0.952464	0.476232	+	0.879320i	$0.342002\pi$
$774$	0	0
$775$	8712.00	0.403799
$776$	0	0
$777$	0	0
$778$	0	0
$779$	57288.0	2.63486
$780$	0	0
$781$	1888.00	0.0865019
$782$	0	0
$783$	6858.00	0.313008
$784$	0	0
$785$	892.000	0.0405565
$786$	0	0
$787$	29916.0	1.35501	0.677503	−	0.735520i	$-0.263062\pi$
$787$	29916.0	1.35501	0.677503	+	0.735520i	$0.263062\pi$
$788$	0	0
$789$	−8964.00	−0.404470
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−1260.00	−0.0564236
$794$	0	0
$795$	972.000	0.0433626
$796$	0	0
$797$	−4914.00	−0.218398	−0.109199	−	0.994020i	$-0.534828\pi$
$797$	−4914.00	−0.218398	−0.109199	+	0.994020i	$0.534828\pi$
$798$	0	0
$799$	−528.000	−0.0233783
$800$	0	0
$801$	−270.000	−0.0119101
$802$	0	0
$803$	−3344.00	−0.146958
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3954.00	0.172475
$808$	0	0
$809$	34250.0	1.48846	0.744231	−	0.667922i	$-0.232816\pi$
$809$	34250.0	1.48846	0.744231	+	0.667922i	$0.232816\pi$
$810$	0	0
$811$	−41804.0	−1.81003	−0.905017	−	0.425376i	$-0.860142\pi$
$811$	−41804.0	−1.81003	−0.905017	+	0.425376i	$0.860142\pi$
$812$	0	0
$813$	−16920.0	−0.729902
$814$	0	0
$815$	5416.00	0.232778
$816$	0	0
$817$	26288.0	1.12570
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30862.0	1.31193	0.655963	−	0.754793i	$-0.272263\pi$
$821$	30862.0	1.31193	0.655963	+	0.754793i	$0.272263\pi$
$822$	0	0
$823$	−10576.0	−0.447942	−0.223971	−	0.974596i	$-0.571902\pi$
$823$	−10576.0	−0.447942	−0.223971	+	0.974596i	$0.571902\pi$
$824$	0	0
$825$	−2904.00	−0.122551
$826$	0	0
$827$	10680.0	0.449069	0.224534	−	0.974466i	$-0.427914\pi$
$827$	10680.0	0.449069	0.224534	+	0.974466i	$0.427914\pi$
$828$	0	0
$829$	1178.00	0.0493530	0.0246765	−	0.999695i	$-0.492144\pi$
$829$	1178.00	0.0493530	0.0246765	+	0.999695i	$0.492144\pi$
$830$	0	0
$831$	19338.0	0.807254
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1792.00	−0.0742691
$836$	0	0
$837$	−1944.00	−0.0802801
$838$	0	0
$839$	−5600.00	−0.230433	−0.115217	−	0.993340i	$-0.536756\pi$
$839$	−5600.00	−0.230433	−0.115217	+	0.993340i	$0.536756\pi$
$840$	0	0
$841$	40127.0	1.64529
$842$	0	0
$843$	14790.0	0.604264
$844$	0	0
$845$	866.000	0.0352560
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−18780.0	−0.759161
$850$	0	0
$851$	−30248.0	−1.21843
$852$	0	0
$853$	826.000	0.0331556	0.0165778	−	0.999863i	$-0.494723\pi$
$853$	826.000	0.0331556	0.0165778	+	0.999863i	$0.494723\pi$
$854$	0	0
$855$	2232.00	0.0892781
$856$	0	0
$857$	−45918.0	−1.83026	−0.915128	−	0.403164i	$-0.867910\pi$
$857$	−45918.0	−1.83026	−0.915128	+	0.403164i	$0.867910\pi$
$858$	0	0
$859$	42380.0	1.68334	0.841669	−	0.539994i	$-0.181574\pi$
$859$	42380.0	1.68334	0.841669	+	0.539994i	$0.181574\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26524.0	1.04622	0.523110	−	0.852265i	$-0.324772\pi$
$863$	26524.0	1.04622	0.523110	+	0.852265i	$0.324772\pi$
$864$	0	0
$865$	8068.00	0.317133
$866$	0	0
$867$	−14727.0	−0.576880
$868$	0	0
$869$	−4416.00	−0.172385
$870$	0	0
$871$	32088.0	1.24829
$872$	0	0
$873$	10710.0	0.415210
$874$	0	0
$875$	0	0
$876$	0	0
$877$	20614.0	0.793712	0.396856	−	0.917881i	$-0.370101\pi$
$877$	20614.0	0.793712	0.396856	+	0.917881i	$0.370101\pi$
$878$	0	0
$879$	6930.00	0.265919
$880$	0	0
$881$	23730.0	0.907473	0.453737	−	0.891136i	$-0.350091\pi$
$881$	23730.0	0.907473	0.453737	+	0.891136i	$0.350091\pi$
$882$	0	0
$883$	9028.00	0.344073	0.172036	−	0.985091i	$-0.444965\pi$
$883$	9028.00	0.344073	0.172036	+	0.985091i	$0.444965\pi$
$884$	0	0
$885$	4632.00	0.175936
$886$	0	0
$887$	37200.0	1.40818	0.704089	−	0.710112i	$-0.251356\pi$
$887$	37200.0	1.40818	0.704089	+	0.710112i	$0.251356\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	648.000	0.0243646
$892$	0	0
$893$	32736.0	1.22673
$894$	0	0
$895$	−6960.00	−0.259941
$896$	0	0
$897$	−9576.00	−0.356447
$898$	0	0
$899$	−18288.0	−0.678464
$900$	0	0
$901$	−324.000	−0.0119800
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5796.00	−0.212890
$906$	0	0
$907$	−23988.0	−0.878179	−0.439090	−	0.898443i	$-0.644699\pi$
$907$	−23988.0	−0.878179	−0.439090	+	0.898443i	$0.644699\pi$
$908$	0	0
$909$	−12330.0	−0.449901
$910$	0	0
$911$	−15276.0	−0.555561	−0.277781	−	0.960645i	$-0.589599\pi$
$911$	−15276.0	−0.555561	−0.277781	+	0.960645i	$0.589599\pi$
$912$	0	0
$913$	8288.00	0.300430
$914$	0	0
$915$	180.000	0.00650341
$916$	0	0
$917$	0	0
$918$	0	0
$919$	10760.0	0.386224	0.193112	−	0.981177i	$-0.438142\pi$
$919$	10760.0	0.386224	0.193112	+	0.981177i	$0.438142\pi$
$920$	0	0
$921$	588.000	0.0210372
$922$	0	0
$923$	9912.00	0.353475
$924$	0	0
$925$	−48158.0	−1.71181
$926$	0	0
$927$	4176.00	0.147959
$928$	0	0
$929$	52890.0	1.86788	0.933942	−	0.357424i	$-0.116345\pi$
$929$	52890.0	1.86788	0.933942	+	0.357424i	$0.116345\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−20208.0	−0.709089
$934$	0	0
$935$	−32.0000	−0.00111926
$936$	0	0
$937$	6118.00	0.213305	0.106652	−	0.994296i	$-0.465987\pi$
$937$	6118.00	0.213305	0.106652	+	0.994296i	$0.465987\pi$
$938$	0	0
$939$	−1182.00	−0.0410789
$940$	0	0
$941$	32230.0	1.11654	0.558272	−	0.829658i	$-0.311465\pi$
$941$	32230.0	1.11654	0.558272	+	0.829658i	$0.311465\pi$
$942$	0	0
$943$	35112.0	1.21252
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18544.0	0.636324	0.318162	−	0.948036i	$-0.396934\pi$
$947$	18544.0	0.636324	0.318162	+	0.948036i	$0.396934\pi$
$948$	0	0
$949$	−17556.0	−0.600518
$950$	0	0
$951$	−20142.0	−0.686802
$952$	0	0
$953$	25930.0	0.881380	0.440690	−	0.897659i	$-0.354734\pi$
$953$	25930.0	0.881380	0.440690	+	0.897659i	$0.354734\pi$
$954$	0	0
$955$	5304.00	0.179721
$956$	0	0
$957$	6096.00	0.205910
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−24607.0	−0.825988
$962$	0	0
$963$	19224.0	0.643286
$964$	0	0
$965$	−292.000	−0.00974074
$966$	0	0
$967$	−8192.00	−0.272427	−0.136214	−	0.990680i	$-0.543493\pi$
$967$	−8192.00	−0.272427	−0.136214	+	0.990680i	$0.543493\pi$
$968$	0	0
$969$	−744.000	−0.0246653
$970$	0	0
$971$	−54444.0	−1.79937	−0.899686	−	0.436537i	$-0.856205\pi$
$971$	−54444.0	−1.79937	−0.899686	+	0.436537i	$0.856205\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−15246.0	−0.500782
$976$	0	0
$977$	−25446.0	−0.833255	−0.416627	−	0.909077i	$-0.636788\pi$
$977$	−25446.0	−0.833255	−0.416627	+	0.909077i	$0.636788\pi$
$978$	0	0
$979$	−240.000	−0.00783497
$980$	0	0
$981$	−11034.0	−0.359112
$982$	0	0
$983$	33192.0	1.07697	0.538484	−	0.842635i	$-0.318997\pi$
$983$	33192.0	1.07697	0.538484	+	0.842635i	$0.318997\pi$
$984$	0	0
$985$	5092.00	0.164715
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16112.0	0.518030
$990$	0	0
$991$	−11024.0	−0.353369	−0.176685	−	0.984268i	$-0.556537\pi$
$991$	−11024.0	−0.353369	−0.176685	+	0.984268i	$0.556537\pi$
$992$	0	0
$993$	−2076.00	−0.0663443
$994$	0	0
$995$	5072.00	0.161601
$996$	0	0
$997$	40714.0	1.29331	0.646653	−	0.762785i	$-0.276168\pi$
$997$	40714.0	1.29331	0.646653	+	0.762785i	$0.276168\pi$
$998$	0	0
$999$	10746.0	0.340329

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.ba.1.1		1
4.3	odd	2		294.4.a.h.1.1		1
7.6	odd	2		336.4.a.d.1.1		1
12.11	even	2		882.4.a.d.1.1		1
21.20	even	2		1008.4.a.j.1.1		1
28.3	even	6		294.4.e.a.79.1		2
28.11	odd	6		294.4.e.d.79.1		2
28.19	even	6		294.4.e.a.67.1		2
28.23	odd	6		294.4.e.d.67.1		2
28.27	even	2		42.4.a.b.1.1	✓	1
56.13	odd	2		1344.4.a.t.1.1		1
56.27	even	2		1344.4.a.f.1.1		1
84.11	even	6		882.4.g.r.667.1		2
84.23	even	6		882.4.g.r.361.1		2
84.47	odd	6		882.4.g.s.361.1		2
84.59	odd	6		882.4.g.s.667.1		2
84.83	odd	2		126.4.a.c.1.1		1
140.27	odd	4		1050.4.g.n.799.2		2
140.83	odd	4		1050.4.g.n.799.1		2
140.139	even	2		1050.4.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.a.b.1.1	✓	1	28.27	even	2
126.4.a.c.1.1		1	84.83	odd	2
294.4.a.h.1.1		1	4.3	odd	2
294.4.e.a.67.1		2	28.19	even	6
294.4.e.a.79.1		2	28.3	even	6
294.4.e.d.67.1		2	28.23	odd	6
294.4.e.d.79.1		2	28.11	odd	6
336.4.a.d.1.1		1	7.6	odd	2
882.4.a.d.1.1		1	12.11	even	2
882.4.g.r.361.1		2	84.23	even	6
882.4.g.r.667.1		2	84.11	even	6
882.4.g.s.361.1		2	84.47	odd	6
882.4.g.s.667.1		2	84.59	odd	6
1008.4.a.j.1.1		1	21.20	even	2
1050.4.a.d.1.1		1	140.139	even	2
1050.4.g.n.799.1		2	140.83	odd	4
1050.4.g.n.799.2		2	140.27	odd	4
1344.4.a.f.1.1		1	56.27	even	2
1344.4.a.t.1.1		1	56.13	odd	2
2352.4.a.ba.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

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