LMFDB - Embedded newform 2352.4.a.s.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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 147)
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)

$f(q)$

$=$

$q-3.00000 q^{3} +18.0000 q^{5} +9.00000 q^{9} +O(q^{10})$

$q-3.00000 q^{3} +18.0000 q^{5} +9.00000 q^{9} +50.0000 q^{11} -36.0000 q^{13} -54.0000 q^{15} +126.000 q^{17} +72.0000 q^{19} -14.0000 q^{23} +199.000 q^{25} -27.0000 q^{27} +158.000 q^{29} +36.0000 q^{31} -150.000 q^{33} -162.000 q^{37} +108.000 q^{39} -270.000 q^{41} +324.000 q^{43} +162.000 q^{45} +72.0000 q^{47} -378.000 q^{51} -22.0000 q^{53} +900.000 q^{55} -216.000 q^{57} -468.000 q^{59} +792.000 q^{61} -648.000 q^{65} -232.000 q^{67} +42.0000 q^{69} +734.000 q^{71} +180.000 q^{73} -597.000 q^{75} -236.000 q^{79} +81.0000 q^{81} -36.0000 q^{83} +2268.00 q^{85} -474.000 q^{87} +234.000 q^{89} -108.000 q^{93} +1296.00 q^{95} +468.000 q^{97} +450.000 q^{99} +O(q^{100})$

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
$5$	18.0000	1.60997	0.804984	−	0.593296i	$-0.202174\pi$
$5$	18.0000	1.60997	0.804984	+	0.593296i	$0.202174\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	50.0000	1.37051	0.685253	−	0.728305i	$-0.259692\pi$
$11$	50.0000	1.37051	0.685253	+	0.728305i	$0.259692\pi$
$12$	0	0
$13$	−36.0000	−0.768046	−0.384023	−	0.923323i	$-0.625462\pi$
$13$	−36.0000	−0.768046	−0.384023	+	0.923323i	$0.625462\pi$
$14$	0	0
$15$	−54.0000	−0.929516
$16$	0	0
$17$	126.000	1.79762	0.898808	−	0.438342i	$-0.144434\pi$
$17$	126.000	1.79762	0.898808	+	0.438342i	$0.144434\pi$
$18$	0	0
$19$	72.0000	0.869365	0.434682	−	0.900584i	$-0.356861\pi$
$19$	72.0000	0.869365	0.434682	+	0.900584i	$0.356861\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−14.0000	−0.126922	−0.0634609	−	0.997984i	$-0.520214\pi$
$23$	−14.0000	−0.126922	−0.0634609	+	0.997984i	$0.520214\pi$
$24$	0	0
$25$	199.000	1.59200
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	158.000	1.01172	0.505860	−	0.862616i	$-0.331175\pi$
$29$	158.000	1.01172	0.505860	+	0.862616i	$0.331175\pi$
$30$	0	0
$31$	36.0000	0.208574	0.104287	−	0.994547i	$-0.466744\pi$
$31$	36.0000	0.208574	0.104287	+	0.994547i	$0.466744\pi$
$32$	0	0
$33$	−150.000	−0.791262
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−162.000	−0.719801	−0.359900	−	0.932991i	$-0.617189\pi$
$37$	−162.000	−0.719801	−0.359900	+	0.932991i	$0.617189\pi$
$38$	0	0
$39$	108.000	0.443432
$40$	0	0
$41$	−270.000	−1.02846	−0.514231	−	0.857652i	$-0.671922\pi$
$41$	−270.000	−1.02846	−0.514231	+	0.857652i	$0.671922\pi$
$42$	0	0
$43$	324.000	1.14906	0.574529	−	0.818484i	$-0.305185\pi$
$43$	324.000	1.14906	0.574529	+	0.818484i	$0.305185\pi$
$44$	0	0
$45$	162.000	0.536656
$46$	0	0
$47$	72.0000	0.223453	0.111726	−	0.993739i	$-0.464362\pi$
$47$	72.0000	0.223453	0.111726	+	0.993739i	$0.464362\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−378.000	−1.03785
$52$	0	0
$53$	−22.0000	−0.0570176	−0.0285088	−	0.999594i	$-0.509076\pi$
$53$	−22.0000	−0.0570176	−0.0285088	+	0.999594i	$0.509076\pi$
$54$	0	0
$55$	900.000	2.20647
$56$	0	0
$57$	−216.000	−0.501928
$58$	0	0
$59$	−468.000	−1.03268	−0.516342	−	0.856382i	$-0.672707\pi$
$59$	−468.000	−1.03268	−0.516342	+	0.856382i	$0.672707\pi$
$60$	0	0
$61$	792.000	1.66238	0.831190	−	0.555988i	$-0.187660\pi$
$61$	792.000	1.66238	0.831190	+	0.555988i	$0.187660\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−648.000	−1.23653
$66$	0	0
$67$	−232.000	−0.423034	−0.211517	−	0.977374i	$-0.567840\pi$
$67$	−232.000	−0.423034	−0.211517	+	0.977374i	$0.567840\pi$
$68$	0	0
$69$	42.0000	0.0732783
$70$	0	0
$71$	734.000	1.22690	0.613449	−	0.789734i	$-0.289782\pi$
$71$	734.000	1.22690	0.613449	+	0.789734i	$0.289782\pi$
$72$	0	0
$73$	180.000	0.288595	0.144297	−	0.989534i	$-0.453908\pi$
$73$	180.000	0.288595	0.144297	+	0.989534i	$0.453908\pi$
$74$	0	0
$75$	−597.000	−0.919142
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−236.000	−0.336102	−0.168051	−	0.985778i	$-0.553747\pi$
$79$	−236.000	−0.336102	−0.168051	+	0.985778i	$0.553747\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−36.0000	−0.0476086	−0.0238043	−	0.999717i	$-0.507578\pi$
$83$	−36.0000	−0.0476086	−0.0238043	+	0.999717i	$0.507578\pi$
$84$	0	0
$85$	2268.00	2.89411
$86$	0	0
$87$	−474.000	−0.584116
$88$	0	0
$89$	234.000	0.278696	0.139348	−	0.990243i	$-0.455499\pi$
$89$	234.000	0.278696	0.139348	+	0.990243i	$0.455499\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−108.000	−0.120420
$94$	0	0
$95$	1296.00	1.39965
$96$	0	0
$97$	468.000	0.489878	0.244939	−	0.969538i	$-0.421232\pi$
$97$	468.000	0.489878	0.244939	+	0.969538i	$0.421232\pi$
$98$	0	0
$99$	450.000	0.456835
$100$	0	0
$101$	−666.000	−0.656133	−0.328067	−	0.944655i	$-0.606397\pi$
$101$	−666.000	−0.656133	−0.328067	+	0.944655i	$0.606397\pi$
$102$	0	0
$103$	252.000	0.241071	0.120535	−	0.992709i	$-0.461539\pi$
$103$	252.000	0.241071	0.120535	+	0.992709i	$0.461539\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−670.000	−0.605340	−0.302670	−	0.953095i	$-0.597878\pi$
$107$	−670.000	−0.605340	−0.302670	+	0.953095i	$0.597878\pi$
$108$	0	0
$109$	162.000	0.142356	0.0711779	−	0.997464i	$-0.477324\pi$
$109$	162.000	0.142356	0.0711779	+	0.997464i	$0.477324\pi$
$110$	0	0
$111$	486.000	0.415577
$112$	0	0
$113$	−1390.00	−1.15717	−0.578585	−	0.815622i	$-0.696395\pi$
$113$	−1390.00	−1.15717	−0.578585	+	0.815622i	$0.696395\pi$
$114$	0	0
$115$	−252.000	−0.204340
$116$	0	0
$117$	−324.000	−0.256015
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1169.00	0.878287
$122$	0	0
$123$	810.000	0.593782
$124$	0	0
$125$	1332.00	0.953102
$126$	0	0
$127$	−916.000	−0.640015	−0.320007	−	0.947415i	$-0.603685\pi$
$127$	−916.000	−0.640015	−0.320007	+	0.947415i	$0.603685\pi$
$128$	0	0
$129$	−972.000	−0.663410
$130$	0	0
$131$	−2268.00	−1.51264	−0.756321	−	0.654201i	$-0.773005\pi$
$131$	−2268.00	−1.51264	−0.756321	+	0.654201i	$0.773005\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−486.000	−0.309839
$136$	0	0
$137$	806.000	0.502637	0.251318	−	0.967904i	$-0.419136\pi$
$137$	806.000	0.502637	0.251318	+	0.967904i	$0.419136\pi$
$138$	0	0
$139$	−2628.00	−1.60363	−0.801813	−	0.597575i	$-0.796131\pi$
$139$	−2628.00	−1.60363	−0.801813	+	0.597575i	$0.796131\pi$
$140$	0	0
$141$	−216.000	−0.129011
$142$	0	0
$143$	−1800.00	−1.05261
$144$	0	0
$145$	2844.00	1.62884
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2390.00	−1.31407	−0.657035	−	0.753860i	$-0.728190\pi$
$149$	−2390.00	−1.31407	−0.657035	+	0.753860i	$0.728190\pi$
$150$	0	0
$151$	−3240.00	−1.74614	−0.873071	−	0.487593i	$-0.837875\pi$
$151$	−3240.00	−1.74614	−0.873071	+	0.487593i	$0.837875\pi$
$152$	0	0
$153$	1134.00	0.599206
$154$	0	0
$155$	648.000	0.335798
$156$	0	0
$157$	3024.00	1.53721	0.768603	−	0.639726i	$-0.220952\pi$
$157$	3024.00	1.53721	0.768603	+	0.639726i	$0.220952\pi$
$158$	0	0
$159$	66.0000	0.0329191
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1784.00	0.857262	0.428631	−	0.903480i	$-0.358996\pi$
$163$	1784.00	0.857262	0.428631	+	0.903480i	$0.358996\pi$
$164$	0	0
$165$	−2700.00	−1.27391
$166$	0	0
$167$	3024.00	1.40122	0.700611	−	0.713543i	$-0.252911\pi$
$167$	3024.00	1.40122	0.700611	+	0.713543i	$0.252911\pi$
$168$	0	0
$169$	−901.000	−0.410105
$170$	0	0
$171$	648.000	0.289788
$172$	0	0
$173$	1566.00	0.688213	0.344106	−	0.938931i	$-0.388182\pi$
$173$	1566.00	0.688213	0.344106	+	0.938931i	$0.388182\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1404.00	0.596221
$178$	0	0
$179$	−3802.00	−1.58757	−0.793784	−	0.608199i	$-0.791892\pi$
$179$	−3802.00	−1.58757	−0.793784	+	0.608199i	$0.791892\pi$
$180$	0	0
$181$	−468.000	−0.192189	−0.0960944	−	0.995372i	$-0.530635\pi$
$181$	−468.000	−0.192189	−0.0960944	+	0.995372i	$0.530635\pi$
$182$	0	0
$183$	−2376.00	−0.959776
$184$	0	0
$185$	−2916.00	−1.15886
$186$	0	0
$187$	6300.00	2.46365
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−482.000	−0.182598	−0.0912992	−	0.995824i	$-0.529102\pi$
$191$	−482.000	−0.182598	−0.0912992	+	0.995824i	$0.529102\pi$
$192$	0	0
$193$	−810.000	−0.302099	−0.151049	−	0.988526i	$-0.548265\pi$
$193$	−810.000	−0.302099	−0.151049	+	0.988526i	$0.548265\pi$
$194$	0	0
$195$	1944.00	0.713911
$196$	0	0
$197$	−2462.00	−0.890407	−0.445204	−	0.895429i	$-0.646869\pi$
$197$	−2462.00	−0.890407	−0.445204	+	0.895429i	$0.646869\pi$
$198$	0	0
$199$	4536.00	1.61582	0.807911	−	0.589305i	$-0.200598\pi$
$199$	4536.00	1.61582	0.807911	+	0.589305i	$0.200598\pi$
$200$	0	0
$201$	696.000	0.244239
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−4860.00	−1.65579
$206$	0	0
$207$	−126.000	−0.0423073
$208$	0	0
$209$	3600.00	1.19147
$210$	0	0
$211$	−2916.00	−0.951402	−0.475701	−	0.879607i	$-0.657805\pi$
$211$	−2916.00	−0.951402	−0.475701	+	0.879607i	$0.657805\pi$
$212$	0	0
$213$	−2202.00	−0.708350
$214$	0	0
$215$	5832.00	1.84995
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−540.000	−0.166620
$220$	0	0
$221$	−4536.00	−1.38065
$222$	0	0
$223$	1080.00	0.324315	0.162157	−	0.986765i	$-0.448155\pi$
$223$	1080.00	0.324315	0.162157	+	0.986765i	$0.448155\pi$
$224$	0	0
$225$	1791.00	0.530667
$226$	0	0
$227$	1332.00	0.389462	0.194731	−	0.980857i	$-0.437617\pi$
$227$	1332.00	0.389462	0.194731	+	0.980857i	$0.437617\pi$
$228$	0	0
$229$	1620.00	0.467479	0.233739	−	0.972299i	$-0.424904\pi$
$229$	1620.00	0.467479	0.233739	+	0.972299i	$0.424904\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6718.00	1.88889	0.944444	−	0.328673i	$-0.106601\pi$
$233$	6718.00	1.88889	0.944444	+	0.328673i	$0.106601\pi$
$234$	0	0
$235$	1296.00	0.359752
$236$	0	0
$237$	708.000	0.194049
$238$	0	0
$239$	3578.00	0.968375	0.484187	−	0.874964i	$-0.339115\pi$
$239$	3578.00	0.968375	0.484187	+	0.874964i	$0.339115\pi$
$240$	0	0
$241$	−756.000	−0.202067	−0.101034	−	0.994883i	$-0.532215\pi$
$241$	−756.000	−0.202067	−0.101034	+	0.994883i	$0.532215\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2592.00	−0.667713
$248$	0	0
$249$	108.000	0.0274868
$250$	0	0
$251$	6516.00	1.63859	0.819295	−	0.573372i	$-0.194365\pi$
$251$	6516.00	1.63859	0.819295	+	0.573372i	$0.194365\pi$
$252$	0	0
$253$	−700.000	−0.173947
$254$	0	0
$255$	−6804.00	−1.67091
$256$	0	0
$257$	−6030.00	−1.46358	−0.731792	−	0.681528i	$-0.761316\pi$
$257$	−6030.00	−1.46358	−0.731792	+	0.681528i	$0.761316\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1422.00	0.337240
$262$	0	0
$263$	−590.000	−0.138331	−0.0691653	−	0.997605i	$-0.522034\pi$
$263$	−590.000	−0.138331	−0.0691653	+	0.997605i	$0.522034\pi$
$264$	0	0
$265$	−396.000	−0.0917966
$266$	0	0
$267$	−702.000	−0.160905
$268$	0	0
$269$	−990.000	−0.224392	−0.112196	−	0.993686i	$-0.535788\pi$
$269$	−990.000	−0.224392	−0.112196	+	0.993686i	$0.535788\pi$
$270$	0	0
$271$	3420.00	0.766606	0.383303	−	0.923623i	$-0.374787\pi$
$271$	3420.00	0.766606	0.383303	+	0.923623i	$0.374787\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9950.00	2.18185
$276$	0	0
$277$	−2734.00	−0.593033	−0.296516	−	0.955028i	$-0.595825\pi$
$277$	−2734.00	−0.593033	−0.296516	+	0.955028i	$0.595825\pi$
$278$	0	0
$279$	324.000	0.0695246
$280$	0	0
$281$	598.000	0.126953	0.0634763	−	0.997983i	$-0.479781\pi$
$281$	598.000	0.126953	0.0634763	+	0.997983i	$0.479781\pi$
$282$	0	0
$283$	−3600.00	−0.756176	−0.378088	−	0.925770i	$-0.623418\pi$
$283$	−3600.00	−0.756176	−0.378088	+	0.925770i	$0.623418\pi$
$284$	0	0
$285$	−3888.00	−0.808089
$286$	0	0
$287$	0	0
$288$	0	0
$289$	10963.0	2.23143
$290$	0	0
$291$	−1404.00	−0.282831
$292$	0	0
$293$	7902.00	1.57556	0.787781	−	0.615955i	$-0.211230\pi$
$293$	7902.00	1.57556	0.787781	+	0.615955i	$0.211230\pi$
$294$	0	0
$295$	−8424.00	−1.66259
$296$	0	0
$297$	−1350.00	−0.263754
$298$	0	0
$299$	504.000	0.0974818
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1998.00	0.378819
$304$	0	0
$305$	14256.0	2.67638
$306$	0	0
$307$	10224.0	1.90070	0.950349	−	0.311185i	$-0.100726\pi$
$307$	10224.0	1.90070	0.950349	+	0.311185i	$0.100726\pi$
$308$	0	0
$309$	−756.000	−0.139182
$310$	0	0
$311$	−3888.00	−0.708901	−0.354451	−	0.935075i	$-0.615332\pi$
$311$	−3888.00	−0.708901	−0.354451	+	0.935075i	$0.615332\pi$
$312$	0	0
$313$	5112.00	0.923154	0.461577	−	0.887100i	$-0.347284\pi$
$313$	5112.00	0.923154	0.461577	+	0.887100i	$0.347284\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10102.0	−1.78986	−0.894929	−	0.446209i	$-0.852774\pi$
$317$	−10102.0	−1.78986	−0.894929	+	0.446209i	$0.852774\pi$
$318$	0	0
$319$	7900.00	1.38657
$320$	0	0
$321$	2010.00	0.349493
$322$	0	0
$323$	9072.00	1.56279
$324$	0	0
$325$	−7164.00	−1.22273
$326$	0	0
$327$	−486.000	−0.0821892
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−5508.00	−0.914644	−0.457322	−	0.889301i	$-0.651191\pi$
$331$	−5508.00	−0.914644	−0.457322	+	0.889301i	$0.651191\pi$
$332$	0	0
$333$	−1458.00	−0.239934
$334$	0	0
$335$	−4176.00	−0.681072
$336$	0	0
$337$	−9234.00	−1.49261	−0.746303	−	0.665607i	$-0.768173\pi$
$337$	−9234.00	−1.49261	−0.746303	+	0.665607i	$0.768173\pi$
$338$	0	0
$339$	4170.00	0.668092
$340$	0	0
$341$	1800.00	0.285852
$342$	0	0
$343$	0	0
$344$	0	0
$345$	756.000	0.117976
$346$	0	0
$347$	6494.00	1.00466	0.502329	−	0.864677i	$-0.332477\pi$
$347$	6494.00	1.00466	0.502329	+	0.864677i	$0.332477\pi$
$348$	0	0
$349$	10080.0	1.54605	0.773023	−	0.634378i	$-0.218744\pi$
$349$	10080.0	1.54605	0.773023	+	0.634378i	$0.218744\pi$
$350$	0	0
$351$	972.000	0.147811
$352$	0	0
$353$	−738.000	−0.111274	−0.0556371	−	0.998451i	$-0.517719\pi$
$353$	−738.000	−0.111274	−0.0556371	+	0.998451i	$0.517719\pi$
$354$	0	0
$355$	13212.0	1.97527
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−194.000	−0.0285207	−0.0142603	−	0.999898i	$-0.504539\pi$
$359$	−194.000	−0.0285207	−0.0142603	+	0.999898i	$0.504539\pi$
$360$	0	0
$361$	−1675.00	−0.244205
$362$	0	0
$363$	−3507.00	−0.507079
$364$	0	0
$365$	3240.00	0.464628
$366$	0	0
$367$	4752.00	0.675892	0.337946	−	0.941165i	$-0.390268\pi$
$367$	4752.00	0.675892	0.337946	+	0.941165i	$0.390268\pi$
$368$	0	0
$369$	−2430.00	−0.342820
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−2306.00	−0.320108	−0.160054	−	0.987108i	$-0.551167\pi$
$373$	−2306.00	−0.320108	−0.160054	+	0.987108i	$0.551167\pi$
$374$	0	0
$375$	−3996.00	−0.550273
$376$	0	0
$377$	−5688.00	−0.777047
$378$	0	0
$379$	7452.00	1.00998	0.504991	−	0.863124i	$-0.331496\pi$
$379$	7452.00	1.00998	0.504991	+	0.863124i	$0.331496\pi$
$380$	0	0
$381$	2748.00	0.369513
$382$	0	0
$383$	1152.00	0.153693	0.0768465	−	0.997043i	$-0.475515\pi$
$383$	1152.00	0.153693	0.0768465	+	0.997043i	$0.475515\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2916.00	0.383020
$388$	0	0
$389$	1894.00	0.246863	0.123431	−	0.992353i	$-0.460610\pi$
$389$	1894.00	0.246863	0.123431	+	0.992353i	$0.460610\pi$
$390$	0	0
$391$	−1764.00	−0.228157
$392$	0	0
$393$	6804.00	0.873324
$394$	0	0
$395$	−4248.00	−0.541114
$396$	0	0
$397$	9216.00	1.16508	0.582541	−	0.812801i	$-0.302058\pi$
$397$	9216.00	1.16508	0.582541	+	0.812801i	$0.302058\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11650.0	−1.45081	−0.725403	−	0.688324i	$-0.758347\pi$
$401$	−11650.0	−1.45081	−0.725403	+	0.688324i	$0.758347\pi$
$402$	0	0
$403$	−1296.00	−0.160194
$404$	0	0
$405$	1458.00	0.178885
$406$	0	0
$407$	−8100.00	−0.986492
$408$	0	0
$409$	−7524.00	−0.909628	−0.454814	−	0.890586i	$-0.650294\pi$
$409$	−7524.00	−0.909628	−0.454814	+	0.890586i	$0.650294\pi$
$410$	0	0
$411$	−2418.00	−0.290197
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−648.000	−0.0766484
$416$	0	0
$417$	7884.00	0.925854
$418$	0	0
$419$	3852.00	0.449123	0.224561	−	0.974460i	$-0.427905\pi$
$419$	3852.00	0.449123	0.224561	+	0.974460i	$0.427905\pi$
$420$	0	0
$421$	10402.0	1.20419	0.602093	−	0.798426i	$-0.294334\pi$
$421$	10402.0	1.20419	0.602093	+	0.798426i	$0.294334\pi$
$422$	0	0
$423$	648.000	0.0744843
$424$	0	0
$425$	25074.0	2.86181
$426$	0	0
$427$	0	0
$428$	0	0
$429$	5400.00	0.607726
$430$	0	0
$431$	10390.0	1.16118	0.580590	−	0.814196i	$-0.302822\pi$
$431$	10390.0	1.16118	0.580590	+	0.814196i	$0.302822\pi$
$432$	0	0
$433$	11232.0	1.24659	0.623297	−	0.781985i	$-0.285793\pi$
$433$	11232.0	1.24659	0.623297	+	0.781985i	$0.285793\pi$
$434$	0	0
$435$	−8532.00	−0.940409
$436$	0	0
$437$	−1008.00	−0.110341
$438$	0	0
$439$	−14616.0	−1.58903	−0.794514	−	0.607245i	$-0.792275\pi$
$439$	−14616.0	−1.58903	−0.794514	+	0.607245i	$0.792275\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−11938.0	−1.28034	−0.640171	−	0.768232i	$-0.721137\pi$
$443$	−11938.0	−1.28034	−0.640171	+	0.768232i	$0.721137\pi$
$444$	0	0
$445$	4212.00	0.448692
$446$	0	0
$447$	7170.00	0.758679
$448$	0	0
$449$	8186.00	0.860404	0.430202	−	0.902733i	$-0.358442\pi$
$449$	8186.00	0.860404	0.430202	+	0.902733i	$0.358442\pi$
$450$	0	0
$451$	−13500.0	−1.40951
$452$	0	0
$453$	9720.00	1.00814
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2106.00	0.215568	0.107784	−	0.994174i	$-0.465625\pi$
$457$	2106.00	0.215568	0.107784	+	0.994174i	$0.465625\pi$
$458$	0	0
$459$	−3402.00	−0.345952
$460$	0	0
$461$	9486.00	0.958367	0.479183	−	0.877715i	$-0.340933\pi$
$461$	9486.00	0.958367	0.479183	+	0.877715i	$0.340933\pi$
$462$	0	0
$463$	12652.0	1.26995	0.634977	−	0.772531i	$-0.281010\pi$
$463$	12652.0	1.26995	0.634977	+	0.772531i	$0.281010\pi$
$464$	0	0
$465$	−1944.00	−0.193873
$466$	0	0
$467$	3708.00	0.367421	0.183711	−	0.982980i	$-0.441189\pi$
$467$	3708.00	0.367421	0.183711	+	0.982980i	$0.441189\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−9072.00	−0.887507
$472$	0	0
$473$	16200.0	1.57479
$474$	0	0
$475$	14328.0	1.38403
$476$	0	0
$477$	−198.000	−0.0190059
$478$	0	0
$479$	8064.00	0.769214	0.384607	−	0.923080i	$-0.374337\pi$
$479$	8064.00	0.769214	0.384607	+	0.923080i	$0.374337\pi$
$480$	0	0
$481$	5832.00	0.552841
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8424.00	0.788689
$486$	0	0
$487$	11664.0	1.08531	0.542655	−	0.839955i	$-0.317419\pi$
$487$	11664.0	1.08531	0.542655	+	0.839955i	$0.317419\pi$
$488$	0	0
$489$	−5352.00	−0.494940
$490$	0	0
$491$	9814.00	0.902036	0.451018	−	0.892515i	$-0.351061\pi$
$491$	9814.00	0.902036	0.451018	+	0.892515i	$0.351061\pi$
$492$	0	0
$493$	19908.0	1.81868
$494$	0	0
$495$	8100.00	0.735491
$496$	0	0
$497$	0	0
$498$	0	0
$499$	15228.0	1.36613	0.683065	−	0.730358i	$-0.260647\pi$
$499$	15228.0	1.36613	0.683065	+	0.730358i	$0.260647\pi$
$500$	0	0
$501$	−9072.00	−0.808996
$502$	0	0
$503$	11088.0	0.982882	0.491441	−	0.870911i	$-0.336470\pi$
$503$	11088.0	0.982882	0.491441	+	0.870911i	$0.336470\pi$
$504$	0	0
$505$	−11988.0	−1.05635
$506$	0	0
$507$	2703.00	0.236774
$508$	0	0
$509$	−5814.00	−0.506289	−0.253144	−	0.967429i	$-0.581465\pi$
$509$	−5814.00	−0.506289	−0.253144	+	0.967429i	$0.581465\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1944.00	−0.167309
$514$	0	0
$515$	4536.00	0.388117
$516$	0	0
$517$	3600.00	0.306243
$518$	0	0
$519$	−4698.00	−0.397340
$520$	0	0
$521$	−11682.0	−0.982337	−0.491169	−	0.871065i	$-0.663430\pi$
$521$	−11682.0	−0.982337	−0.491169	+	0.871065i	$0.663430\pi$
$522$	0	0
$523$	2988.00	0.249820	0.124910	−	0.992168i	$-0.460136\pi$
$523$	2988.00	0.249820	0.124910	+	0.992168i	$0.460136\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4536.00	0.374936
$528$	0	0
$529$	−11971.0	−0.983891
$530$	0	0
$531$	−4212.00	−0.344228
$532$	0	0
$533$	9720.00	0.789906
$534$	0	0
$535$	−12060.0	−0.974578
$536$	0	0
$537$	11406.0	0.916583
$538$	0	0
$539$	0	0
$540$	0	0
$541$	7130.00	0.566622	0.283311	−	0.959028i	$-0.408567\pi$
$541$	7130.00	0.566622	0.283311	+	0.959028i	$0.408567\pi$
$542$	0	0
$543$	1404.00	0.110960
$544$	0	0
$545$	2916.00	0.229188
$546$	0	0
$547$	5488.00	0.428976	0.214488	−	0.976727i	$-0.431192\pi$
$547$	5488.00	0.428976	0.214488	+	0.976727i	$0.431192\pi$
$548$	0	0
$549$	7128.00	0.554127
$550$	0	0
$551$	11376.0	0.879553
$552$	0	0
$553$	0	0
$554$	0	0
$555$	8748.00	0.669067
$556$	0	0
$557$	5746.00	0.437102	0.218551	−	0.975826i	$-0.429867\pi$
$557$	5746.00	0.437102	0.218551	+	0.975826i	$0.429867\pi$
$558$	0	0
$559$	−11664.0	−0.882531
$560$	0	0
$561$	−18900.0	−1.42239
$562$	0	0
$563$	−13068.0	−0.978243	−0.489121	−	0.872216i	$-0.662682\pi$
$563$	−13068.0	−0.978243	−0.489121	+	0.872216i	$0.662682\pi$
$564$	0	0
$565$	−25020.0	−1.86301
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1130.00	−0.0832549	−0.0416275	−	0.999133i	$-0.513254\pi$
$569$	−1130.00	−0.0832549	−0.0416275	+	0.999133i	$0.513254\pi$
$570$	0	0
$571$	−16864.0	−1.23597	−0.617983	−	0.786192i	$-0.712050\pi$
$571$	−16864.0	−1.23597	−0.617983	+	0.786192i	$0.712050\pi$
$572$	0	0
$573$	1446.00	0.105423
$574$	0	0
$575$	−2786.00	−0.202060
$576$	0	0
$577$	−2088.00	−0.150649	−0.0753246	−	0.997159i	$-0.523999\pi$
$577$	−2088.00	−0.150649	−0.0753246	+	0.997159i	$0.523999\pi$
$578$	0	0
$579$	2430.00	0.174417
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−1100.00	−0.0781430
$584$	0	0
$585$	−5832.00	−0.412177
$586$	0	0
$587$	−10260.0	−0.721423	−0.360712	−	0.932677i	$-0.617466\pi$
$587$	−10260.0	−0.721423	−0.360712	+	0.932677i	$0.617466\pi$
$588$	0	0
$589$	2592.00	0.181327
$590$	0	0
$591$	7386.00	0.514077
$592$	0	0
$593$	3582.00	0.248052	0.124026	−	0.992279i	$-0.460419\pi$
$593$	3582.00	0.248052	0.124026	+	0.992279i	$0.460419\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−13608.0	−0.932895
$598$	0	0
$599$	−7034.00	−0.479802	−0.239901	−	0.970797i	$-0.577115\pi$
$599$	−7034.00	−0.479802	−0.239901	+	0.970797i	$0.577115\pi$
$600$	0	0
$601$	−18072.0	−1.22658	−0.613288	−	0.789859i	$-0.710154\pi$
$601$	−18072.0	−1.22658	−0.613288	+	0.789859i	$0.710154\pi$
$602$	0	0
$603$	−2088.00	−0.141011
$604$	0	0
$605$	21042.0	1.41401
$606$	0	0
$607$	−28584.0	−1.91135	−0.955674	−	0.294425i	$-0.904872\pi$
$607$	−28584.0	−1.91135	−0.955674	+	0.294425i	$0.904872\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2592.00	−0.171622
$612$	0	0
$613$	−10910.0	−0.718843	−0.359421	−	0.933175i	$-0.617026\pi$
$613$	−10910.0	−0.718843	−0.359421	+	0.933175i	$0.617026\pi$
$614$	0	0
$615$	14580.0	0.955971
$616$	0	0
$617$	−5522.00	−0.360304	−0.180152	−	0.983639i	$-0.557659\pi$
$617$	−5522.00	−0.360304	−0.180152	+	0.983639i	$0.557659\pi$
$618$	0	0
$619$	−2412.00	−0.156618	−0.0783089	−	0.996929i	$-0.524952\pi$
$619$	−2412.00	−0.156618	−0.0783089	+	0.996929i	$0.524952\pi$
$620$	0	0
$621$	378.000	0.0244261
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−899.000	−0.0575360
$626$	0	0
$627$	−10800.0	−0.687895
$628$	0	0
$629$	−20412.0	−1.29393
$630$	0	0
$631$	−24676.0	−1.55679	−0.778396	−	0.627773i	$-0.783966\pi$
$631$	−24676.0	−1.55679	−0.778396	+	0.627773i	$0.783966\pi$
$632$	0	0
$633$	8748.00	0.549292
$634$	0	0
$635$	−16488.0	−1.03040
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6606.00	0.408966
$640$	0	0
$641$	−27482.0	−1.69341	−0.846703	−	0.532065i	$-0.821416\pi$
$641$	−27482.0	−1.69341	−0.846703	+	0.532065i	$0.821416\pi$
$642$	0	0
$643$	22752.0	1.39541	0.697707	−	0.716383i	$-0.254204\pi$
$643$	22752.0	1.39541	0.697707	+	0.716383i	$0.254204\pi$
$644$	0	0
$645$	−17496.0	−1.06807
$646$	0	0
$647$	−14832.0	−0.901246	−0.450623	−	0.892714i	$-0.648798\pi$
$647$	−14832.0	−0.901246	−0.450623	+	0.892714i	$0.648798\pi$
$648$	0	0
$649$	−23400.0	−1.41530
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2822.00	0.169117	0.0845585	−	0.996419i	$-0.473052\pi$
$653$	2822.00	0.169117	0.0845585	+	0.996419i	$0.473052\pi$
$654$	0	0
$655$	−40824.0	−2.43531
$656$	0	0
$657$	1620.00	0.0961982
$658$	0	0
$659$	15826.0	0.935498	0.467749	−	0.883861i	$-0.345065\pi$
$659$	15826.0	0.935498	0.467749	+	0.883861i	$0.345065\pi$
$660$	0	0
$661$	−23832.0	−1.40236	−0.701178	−	0.712986i	$-0.747342\pi$
$661$	−23832.0	−1.40236	−0.701178	+	0.712986i	$0.747342\pi$
$662$	0	0
$663$	13608.0	0.797121
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2212.00	−0.128409
$668$	0	0
$669$	−3240.00	−0.187243
$670$	0	0
$671$	39600.0	2.27830
$672$	0	0
$673$	13770.0	0.788699	0.394350	−	0.918961i	$-0.370970\pi$
$673$	13770.0	0.788699	0.394350	+	0.918961i	$0.370970\pi$
$674$	0	0
$675$	−5373.00	−0.306381
$676$	0	0
$677$	8334.00	0.473119	0.236560	−	0.971617i	$-0.423980\pi$
$677$	8334.00	0.473119	0.236560	+	0.971617i	$0.423980\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−3996.00	−0.224856
$682$	0	0
$683$	18598.0	1.04192	0.520961	−	0.853580i	$-0.325574\pi$
$683$	18598.0	1.04192	0.520961	+	0.853580i	$0.325574\pi$
$684$	0	0
$685$	14508.0	0.809229
$686$	0	0
$687$	−4860.00	−0.269899
$688$	0	0
$689$	792.000	0.0437922
$690$	0	0
$691$	−8964.00	−0.493497	−0.246749	−	0.969080i	$-0.579362\pi$
$691$	−8964.00	−0.493497	−0.246749	+	0.969080i	$0.579362\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−47304.0	−2.58179
$696$	0	0
$697$	−34020.0	−1.84878
$698$	0	0
$699$	−20154.0	−1.09055
$700$	0	0
$701$	3542.00	0.190841	0.0954205	−	0.995437i	$-0.469580\pi$
$701$	3542.00	0.190841	0.0954205	+	0.995437i	$0.469580\pi$
$702$	0	0
$703$	−11664.0	−0.625770
$704$	0	0
$705$	−3888.00	−0.207703
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−486.000	−0.0257435	−0.0128717	−	0.999917i	$-0.504097\pi$
$709$	−486.000	−0.0257435	−0.0128717	+	0.999917i	$0.504097\pi$
$710$	0	0
$711$	−2124.00	−0.112034
$712$	0	0
$713$	−504.000	−0.0264726
$714$	0	0
$715$	−32400.0	−1.69467
$716$	0	0
$717$	−10734.0	−0.559091
$718$	0	0
$719$	26928.0	1.39672	0.698362	−	0.715744i	$-0.253912\pi$
$719$	26928.0	1.39672	0.698362	+	0.715744i	$0.253912\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	2268.00	0.116664
$724$	0	0
$725$	31442.0	1.61066
$726$	0	0
$727$	−20628.0	−1.05234	−0.526169	−	0.850380i	$-0.676372\pi$
$727$	−20628.0	−1.05234	−0.526169	+	0.850380i	$0.676372\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	40824.0	2.06557
$732$	0	0
$733$	−9756.00	−0.491604	−0.245802	−	0.969320i	$-0.579051\pi$
$733$	−9756.00	−0.491604	−0.245802	+	0.969320i	$0.579051\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−11600.0	−0.579771
$738$	0	0
$739$	−19064.0	−0.948959	−0.474479	−	0.880267i	$-0.657364\pi$
$739$	−19064.0	−0.948959	−0.474479	+	0.880267i	$0.657364\pi$
$740$	0	0
$741$	7776.00	0.385504
$742$	0	0
$743$	3766.00	0.185950	0.0929752	−	0.995668i	$-0.470362\pi$
$743$	3766.00	0.185950	0.0929752	+	0.995668i	$0.470362\pi$
$744$	0	0
$745$	−43020.0	−2.11561
$746$	0	0
$747$	−324.000	−0.0158695
$748$	0	0
$749$	0	0
$750$	0	0
$751$	11664.0	0.566745	0.283372	−	0.959010i	$-0.408547\pi$
$751$	11664.0	0.566745	0.283372	+	0.959010i	$0.408547\pi$
$752$	0	0
$753$	−19548.0	−0.946041
$754$	0	0
$755$	−58320.0	−2.81123
$756$	0	0
$757$	−34182.0	−1.64117	−0.820585	−	0.571524i	$-0.806352\pi$
$757$	−34182.0	−1.64117	−0.820585	+	0.571524i	$0.806352\pi$
$758$	0	0
$759$	2100.00	0.100428
$760$	0	0
$761$	−4734.00	−0.225502	−0.112751	−	0.993623i	$-0.535966\pi$
$761$	−4734.00	−0.225502	−0.112751	+	0.993623i	$0.535966\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	20412.0	0.964703
$766$	0	0
$767$	16848.0	0.793150
$768$	0	0
$769$	−30240.0	−1.41805	−0.709026	−	0.705182i	$-0.750865\pi$
$769$	−30240.0	−1.41805	−0.709026	+	0.705182i	$0.750865\pi$
$770$	0	0
$771$	18090.0	0.845001
$772$	0	0
$773$	27702.0	1.28897	0.644484	−	0.764618i	$-0.277072\pi$
$773$	27702.0	1.28897	0.644484	+	0.764618i	$0.277072\pi$
$774$	0	0
$775$	7164.00	0.332050
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−19440.0	−0.894108
$780$	0	0
$781$	36700.0	1.68147
$782$	0	0
$783$	−4266.00	−0.194705
$784$	0	0
$785$	54432.0	2.47486
$786$	0	0
$787$	22644.0	1.02563	0.512815	−	0.858499i	$-0.328603\pi$
$787$	22644.0	1.02563	0.512815	+	0.858499i	$0.328603\pi$
$788$	0	0
$789$	1770.00	0.0798652
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−28512.0	−1.27679
$794$	0	0
$795$	1188.00	0.0529988
$796$	0	0
$797$	30150.0	1.33998	0.669992	−	0.742368i	$-0.266297\pi$
$797$	30150.0	1.33998	0.669992	+	0.742368i	$0.266297\pi$
$798$	0	0
$799$	9072.00	0.401682
$800$	0	0
$801$	2106.00	0.0928987
$802$	0	0
$803$	9000.00	0.395521
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2970.00	0.129553
$808$	0	0
$809$	−11318.0	−0.491866	−0.245933	−	0.969287i	$-0.579094\pi$
$809$	−11318.0	−0.491866	−0.245933	+	0.969287i	$0.579094\pi$
$810$	0	0
$811$	29628.0	1.28284	0.641418	−	0.767192i	$-0.278347\pi$
$811$	29628.0	1.28284	0.641418	+	0.767192i	$0.278347\pi$
$812$	0	0
$813$	−10260.0	−0.442600
$814$	0	0
$815$	32112.0	1.38016
$816$	0	0
$817$	23328.0	0.998952
$818$	0	0
$819$	0	0
$820$	0	0
$821$	17770.0	0.755393	0.377696	−	0.925930i	$-0.376716\pi$
$821$	17770.0	0.755393	0.377696	+	0.925930i	$0.376716\pi$
$822$	0	0
$823$	−7868.00	−0.333246	−0.166623	−	0.986021i	$-0.553286\pi$
$823$	−7868.00	−0.333246	−0.166623	+	0.986021i	$0.553286\pi$
$824$	0	0
$825$	−29850.0	−1.25969
$826$	0	0
$827$	−35726.0	−1.50219	−0.751097	−	0.660192i	$-0.770475\pi$
$827$	−35726.0	−1.50219	−0.751097	+	0.660192i	$0.770475\pi$
$828$	0	0
$829$	27108.0	1.13571	0.567853	−	0.823130i	$-0.307774\pi$
$829$	27108.0	1.13571	0.567853	+	0.823130i	$0.307774\pi$
$830$	0	0
$831$	8202.00	0.342388
$832$	0	0
$833$	0	0
$834$	0	0
$835$	54432.0	2.25592
$836$	0	0
$837$	−972.000	−0.0401401
$838$	0	0
$839$	−23256.0	−0.956956	−0.478478	−	0.878099i	$-0.658811\pi$
$839$	−23256.0	−0.956956	−0.478478	+	0.878099i	$0.658811\pi$
$840$	0	0
$841$	575.000	0.0235762
$842$	0	0
$843$	−1794.00	−0.0732961
$844$	0	0
$845$	−16218.0	−0.660256
$846$	0	0
$847$	0	0
$848$	0	0
$849$	10800.0	0.436578
$850$	0	0
$851$	2268.00	0.0913584
$852$	0	0
$853$	35280.0	1.41614	0.708068	−	0.706144i	$-0.249567\pi$
$853$	35280.0	1.41614	0.708068	+	0.706144i	$0.249567\pi$
$854$	0	0
$855$	11664.0	0.466550
$856$	0	0
$857$	−19710.0	−0.785625	−0.392813	−	0.919619i	$-0.628498\pi$
$857$	−19710.0	−0.785625	−0.392813	+	0.919619i	$0.628498\pi$
$858$	0	0
$859$	−3888.00	−0.154432	−0.0772159	−	0.997014i	$-0.524603\pi$
$859$	−3888.00	−0.154432	−0.0772159	+	0.997014i	$0.524603\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	36634.0	1.44500	0.722500	−	0.691370i	$-0.242993\pi$
$863$	36634.0	1.44500	0.722500	+	0.691370i	$0.242993\pi$
$864$	0	0
$865$	28188.0	1.10800
$866$	0	0
$867$	−32889.0	−1.28831
$868$	0	0
$869$	−11800.0	−0.460630
$870$	0	0
$871$	8352.00	0.324910
$872$	0	0
$873$	4212.00	0.163293
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1226.00	0.0472053	0.0236027	−	0.999721i	$-0.492486\pi$
$877$	1226.00	0.0472053	0.0236027	+	0.999721i	$0.492486\pi$
$878$	0	0
$879$	−23706.0	−0.909651
$880$	0	0
$881$	−38538.0	−1.47376	−0.736878	−	0.676026i	$-0.763701\pi$
$881$	−38538.0	−1.47376	−0.736878	+	0.676026i	$0.763701\pi$
$882$	0	0
$883$	37260.0	1.42004	0.710022	−	0.704180i	$-0.248685\pi$
$883$	37260.0	1.42004	0.710022	+	0.704180i	$0.248685\pi$
$884$	0	0
$885$	25272.0	0.959897
$886$	0	0
$887$	−26640.0	−1.00844	−0.504219	−	0.863576i	$-0.668219\pi$
$887$	−26640.0	−1.00844	−0.504219	+	0.863576i	$0.668219\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4050.00	0.152278
$892$	0	0
$893$	5184.00	0.194262
$894$	0	0
$895$	−68436.0	−2.55594
$896$	0	0
$897$	−1512.00	−0.0562812
$898$	0	0
$899$	5688.00	0.211018
$900$	0	0
$901$	−2772.00	−0.102496
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8424.00	−0.309418
$906$	0	0
$907$	12636.0	0.462593	0.231296	−	0.972883i	$-0.425703\pi$
$907$	12636.0	0.462593	0.231296	+	0.972883i	$0.425703\pi$
$908$	0	0
$909$	−5994.00	−0.218711
$910$	0	0
$911$	33638.0	1.22336	0.611678	−	0.791107i	$-0.290495\pi$
$911$	33638.0	1.22336	0.611678	+	0.791107i	$0.290495\pi$
$912$	0	0
$913$	−1800.00	−0.0652479
$914$	0	0
$915$	−42768.0	−1.54521
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−36936.0	−1.32580	−0.662898	−	0.748710i	$-0.730674\pi$
$919$	−36936.0	−1.32580	−0.662898	+	0.748710i	$0.730674\pi$
$920$	0	0
$921$	−30672.0	−1.09737
$922$	0	0
$923$	−26424.0	−0.942315
$924$	0	0
$925$	−32238.0	−1.14592
$926$	0	0
$927$	2268.00	0.0803570
$928$	0	0
$929$	−22302.0	−0.787626	−0.393813	−	0.919191i	$-0.628844\pi$
$929$	−22302.0	−0.787626	−0.393813	+	0.919191i	$0.628844\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	11664.0	0.409284
$934$	0	0
$935$	113400.	3.96639
$936$	0	0
$937$	13824.0	0.481975	0.240987	−	0.970528i	$-0.422529\pi$
$937$	13824.0	0.481975	0.240987	+	0.970528i	$0.422529\pi$
$938$	0	0
$939$	−15336.0	−0.532983
$940$	0	0
$941$	13554.0	0.469551	0.234776	−	0.972050i	$-0.424564\pi$
$941$	13554.0	0.469551	0.234776	+	0.972050i	$0.424564\pi$
$942$	0	0
$943$	3780.00	0.130534
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−44878.0	−1.53996	−0.769978	−	0.638070i	$-0.779733\pi$
$947$	−44878.0	−1.53996	−0.769978	+	0.638070i	$0.779733\pi$
$948$	0	0
$949$	−6480.00	−0.221654
$950$	0	0
$951$	30306.0	1.03337
$952$	0	0
$953$	38362.0	1.30395	0.651976	−	0.758239i	$-0.273940\pi$
$953$	38362.0	1.30395	0.651976	+	0.758239i	$0.273940\pi$
$954$	0	0
$955$	−8676.00	−0.293978
$956$	0	0
$957$	−23700.0	−0.800535
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28495.0	−0.956497
$962$	0	0
$963$	−6030.00	−0.201780
$964$	0	0
$965$	−14580.0	−0.486370
$966$	0	0
$967$	−26444.0	−0.879402	−0.439701	−	0.898144i	$-0.644916\pi$
$967$	−26444.0	−0.879402	−0.439701	+	0.898144i	$0.644916\pi$
$968$	0	0
$969$	−27216.0	−0.902274
$970$	0	0
$971$	−17820.0	−0.588951	−0.294475	−	0.955659i	$-0.595145\pi$
$971$	−17820.0	−0.588951	−0.294475	+	0.955659i	$0.595145\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	21492.0	0.705943
$976$	0	0
$977$	34438.0	1.12771	0.563853	−	0.825875i	$-0.309318\pi$
$977$	34438.0	1.12771	0.563853	+	0.825875i	$0.309318\pi$
$978$	0	0
$979$	11700.0	0.381955
$980$	0	0
$981$	1458.00	0.0474519
$982$	0	0
$983$	−26064.0	−0.845689	−0.422845	−	0.906202i	$-0.638968\pi$
$983$	−26064.0	−0.845689	−0.422845	+	0.906202i	$0.638968\pi$
$984$	0	0
$985$	−44316.0	−1.43353
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4536.00	−0.145841
$990$	0	0
$991$	−33696.0	−1.08011	−0.540055	−	0.841630i	$-0.681597\pi$
$991$	−33696.0	−1.08011	−0.540055	+	0.841630i	$0.681597\pi$
$992$	0	0
$993$	16524.0	0.528070
$994$	0	0
$995$	81648.0	2.60142
$996$	0	0
$997$	36072.0	1.14585	0.572925	−	0.819608i	$-0.305809\pi$
$997$	36072.0	1.14585	0.572925	+	0.819608i	$0.305809\pi$
$998$	0	0
$999$	4374.00	0.138526

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.s.1.1		1
4.3	odd	2		147.4.a.h.1.1	yes	1
7.6	odd	2		2352.4.a.t.1.1		1
12.11	even	2		441.4.a.a.1.1		1
28.3	even	6		147.4.e.d.79.1		2
28.11	odd	6		147.4.e.a.79.1		2
28.19	even	6		147.4.e.d.67.1		2
28.23	odd	6		147.4.e.a.67.1		2
28.27	even	2		147.4.a.f.1.1	✓	1
84.11	even	6		441.4.e.o.226.1		2
84.23	even	6		441.4.e.o.361.1		2
84.47	odd	6		441.4.e.l.361.1		2
84.59	odd	6		441.4.e.l.226.1		2
84.83	odd	2		441.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.a.f.1.1	✓	1	28.27	even	2
147.4.a.h.1.1	yes	1	4.3	odd	2
147.4.e.a.67.1		2	28.23	odd	6
147.4.e.a.79.1		2	28.11	odd	6
147.4.e.d.67.1		2	28.19	even	6
147.4.e.d.79.1		2	28.3	even	6
441.4.a.a.1.1		1	12.11	even	2
441.4.a.c.1.1		1	84.83	odd	2
441.4.e.l.226.1		2	84.59	odd	6
441.4.e.l.361.1		2	84.47	odd	6
441.4.e.o.226.1		2	84.11	even	6
441.4.e.o.361.1		2	84.23	even	6
2352.4.a.s.1.1		1	1.1	even	1	trivial
2352.4.a.t.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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

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