LMFDB - Embedded newform 2352.4.a.bc.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 168)
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)

$f(q)$

$=$

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

$q+3.00000 q^{3} +2.00000 q^{5} +9.00000 q^{9} +18.0000 q^{11} +33.0000 q^{13} +6.00000 q^{15} -68.0000 q^{17} -25.0000 q^{19} -92.0000 q^{23} -121.000 q^{25} +27.0000 q^{27} +92.0000 q^{29} -25.0000 q^{31} +54.0000 q^{33} -213.000 q^{37} +99.0000 q^{39} +94.0000 q^{41} +67.0000 q^{43} +18.0000 q^{45} -278.000 q^{47} -204.000 q^{51} -400.000 q^{53} +36.0000 q^{55} -75.0000 q^{57} -744.000 q^{59} -734.000 q^{61} +66.0000 q^{65} -555.000 q^{67} -276.000 q^{69} +642.000 q^{71} +973.000 q^{73} -363.000 q^{75} +785.000 q^{79} +81.0000 q^{81} +822.000 q^{83} -136.000 q^{85} +276.000 q^{87} +424.000 q^{89} -75.0000 q^{93} -50.0000 q^{95} -734.000 q^{97} +162.000 q^{99} +O(q^{100})$

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
$5$	2.00000	0.178885	0.0894427	−	0.995992i	$-0.471491\pi$
$5$	2.00000	0.178885	0.0894427	+	0.995992i	$0.471491\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	18.0000	0.493382	0.246691	−	0.969094i	$-0.420657\pi$
$11$	18.0000	0.493382	0.246691	+	0.969094i	$0.420657\pi$
$12$	0	0
$13$	33.0000	0.704043	0.352021	−	0.935992i	$-0.385494\pi$
$13$	33.0000	0.704043	0.352021	+	0.935992i	$0.385494\pi$
$14$	0	0
$15$	6.00000	0.103280
$16$	0	0
$17$	−68.0000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−68.0000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	−25.0000	−0.301863	−0.150931	−	0.988544i	$-0.548227\pi$
$19$	−25.0000	−0.301863	−0.150931	+	0.988544i	$0.548227\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−92.0000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−92.0000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−121.000	−0.968000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	92.0000	0.589102	0.294551	−	0.955636i	$-0.404830\pi$
$29$	92.0000	0.589102	0.294551	+	0.955636i	$0.404830\pi$
$30$	0	0
$31$	−25.0000	−0.144843	−0.0724215	−	0.997374i	$-0.523073\pi$
$31$	−25.0000	−0.144843	−0.0724215	+	0.997374i	$0.523073\pi$
$32$	0	0
$33$	54.0000	0.284854
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−213.000	−0.946405	−0.473202	−	0.880954i	$-0.656902\pi$
$37$	−213.000	−0.946405	−0.473202	+	0.880954i	$0.656902\pi$
$38$	0	0
$39$	99.0000	0.406479
$40$	0	0
$41$	94.0000	0.358057	0.179028	−	0.983844i	$-0.442705\pi$
$41$	94.0000	0.358057	0.179028	+	0.983844i	$0.442705\pi$
$42$	0	0
$43$	67.0000	0.237614	0.118807	−	0.992917i	$-0.462093\pi$
$43$	67.0000	0.237614	0.118807	+	0.992917i	$0.462093\pi$
$44$	0	0
$45$	18.0000	0.0596285
$46$	0	0
$47$	−278.000	−0.862776	−0.431388	−	0.902167i	$-0.641976\pi$
$47$	−278.000	−0.862776	−0.431388	+	0.902167i	$0.641976\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−204.000	−0.560112
$52$	0	0
$53$	−400.000	−1.03668	−0.518342	−	0.855174i	$-0.673450\pi$
$53$	−400.000	−1.03668	−0.518342	+	0.855174i	$0.673450\pi$
$54$	0	0
$55$	36.0000	0.0882589
$56$	0	0
$57$	−75.0000	−0.174281
$58$	0	0
$59$	−744.000	−1.64170	−0.820852	−	0.571141i	$-0.806501\pi$
$59$	−744.000	−1.64170	−0.820852	+	0.571141i	$0.806501\pi$
$60$	0	0
$61$	−734.000	−1.54064	−0.770320	−	0.637657i	$-0.779904\pi$
$61$	−734.000	−1.54064	−0.770320	+	0.637657i	$0.779904\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	66.0000	0.125943
$66$	0	0
$67$	−555.000	−1.01200	−0.506000	−	0.862533i	$-0.668877\pi$
$67$	−555.000	−1.01200	−0.506000	+	0.862533i	$0.668877\pi$
$68$	0	0
$69$	−276.000	−0.481543
$70$	0	0
$71$	642.000	1.07312	0.536559	−	0.843863i	$-0.319724\pi$
$71$	642.000	1.07312	0.536559	+	0.843863i	$0.319724\pi$
$72$	0	0
$73$	973.000	1.56001	0.780007	−	0.625771i	$-0.215215\pi$
$73$	973.000	1.56001	0.780007	+	0.625771i	$0.215215\pi$
$74$	0	0
$75$	−363.000	−0.558875
$76$	0	0
$77$	0	0
$78$	0	0
$79$	785.000	1.11797	0.558984	−	0.829179i	$-0.311191\pi$
$79$	785.000	1.11797	0.558984	+	0.829179i	$0.311191\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	822.000	1.08706	0.543531	−	0.839389i	$-0.317087\pi$
$83$	822.000	1.08706	0.543531	+	0.839389i	$0.317087\pi$
$84$	0	0
$85$	−136.000	−0.173544
$86$	0	0
$87$	276.000	0.340118
$88$	0	0
$89$	424.000	0.504988	0.252494	−	0.967598i	$-0.418749\pi$
$89$	424.000	0.504988	0.252494	+	0.967598i	$0.418749\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−75.0000	−0.0836251
$94$	0	0
$95$	−50.0000	−0.0539989
$96$	0	0
$97$	−734.000	−0.768313	−0.384157	−	0.923268i	$-0.625508\pi$
$97$	−734.000	−0.768313	−0.384157	+	0.923268i	$0.625508\pi$
$98$	0	0
$99$	162.000	0.164461
$100$	0	0
$101$	−270.000	−0.266000	−0.133000	−	0.991116i	$-0.542461\pi$
$101$	−270.000	−0.266000	−0.133000	+	0.991116i	$0.542461\pi$
$102$	0	0
$103$	−417.000	−0.398915	−0.199457	−	0.979906i	$-0.563918\pi$
$103$	−417.000	−0.398915	−0.199457	+	0.979906i	$0.563918\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1644.00	1.48534	0.742670	−	0.669657i	$-0.233559\pi$
$107$	1644.00	1.48534	0.742670	+	0.669657i	$0.233559\pi$
$108$	0	0
$109$	−551.000	−0.484186	−0.242093	−	0.970253i	$-0.577834\pi$
$109$	−551.000	−0.484186	−0.242093	+	0.970253i	$0.577834\pi$
$110$	0	0
$111$	−639.000	−0.546407
$112$	0	0
$113$	−402.000	−0.334664	−0.167332	−	0.985901i	$-0.553515\pi$
$113$	−402.000	−0.334664	−0.167332	+	0.985901i	$0.553515\pi$
$114$	0	0
$115$	−184.000	−0.149201
$116$	0	0
$117$	297.000	0.234681
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1007.00	−0.756574
$122$	0	0
$123$	282.000	0.206724
$124$	0	0
$125$	−492.000	−0.352047
$126$	0	0
$127$	−1889.00	−1.31986	−0.659928	−	0.751329i	$-0.729413\pi$
$127$	−1889.00	−1.31986	−0.659928	+	0.751329i	$0.729413\pi$
$128$	0	0
$129$	201.000	0.137187
$130$	0	0
$131$	−554.000	−0.369490	−0.184745	−	0.982786i	$-0.559146\pi$
$131$	−554.000	−0.369490	−0.184745	+	0.982786i	$0.559146\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	54.0000	0.0344265
$136$	0	0
$137$	−2956.00	−1.84342	−0.921708	−	0.387883i	$-0.873206\pi$
$137$	−2956.00	−1.84342	−0.921708	+	0.387883i	$0.873206\pi$
$138$	0	0
$139$	795.000	0.485115	0.242558	−	0.970137i	$-0.422014\pi$
$139$	795.000	0.485115	0.242558	+	0.970137i	$0.422014\pi$
$140$	0	0
$141$	−834.000	−0.498124
$142$	0	0
$143$	594.000	0.347362
$144$	0	0
$145$	184.000	0.105382
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−908.000	−0.499237	−0.249618	−	0.968344i	$-0.580305\pi$
$149$	−908.000	−0.499237	−0.249618	+	0.968344i	$0.580305\pi$
$150$	0	0
$151$	3272.00	1.76339	0.881694	−	0.471822i	$-0.156403\pi$
$151$	3272.00	1.76339	0.881694	+	0.471822i	$0.156403\pi$
$152$	0	0
$153$	−612.000	−0.323381
$154$	0	0
$155$	−50.0000	−0.0259103
$156$	0	0
$157$	234.000	0.118951	0.0594753	−	0.998230i	$-0.481057\pi$
$157$	234.000	0.118951	0.0594753	+	0.998230i	$0.481057\pi$
$158$	0	0
$159$	−1200.00	−0.598529
$160$	0	0
$161$	0	0
$162$	0	0
$163$	836.000	0.401721	0.200861	−	0.979620i	$-0.435626\pi$
$163$	836.000	0.401721	0.200861	+	0.979620i	$0.435626\pi$
$164$	0	0
$165$	108.000	0.0509563
$166$	0	0
$167$	−6.00000	−0.00278020	−0.00139010	−	0.999999i	$-0.500442\pi$
$167$	−6.00000	−0.00278020	−0.00139010	+	0.999999i	$0.500442\pi$
$168$	0	0
$169$	−1108.00	−0.504324
$170$	0	0
$171$	−225.000	−0.100621
$172$	0	0
$173$	−532.000	−0.233799	−0.116899	−	0.993144i	$-0.537296\pi$
$173$	−532.000	−0.233799	−0.116899	+	0.993144i	$0.537296\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2232.00	−0.947838
$178$	0	0
$179$	−4070.00	−1.69948	−0.849738	−	0.527206i	$-0.823240\pi$
$179$	−4070.00	−1.69948	−0.849738	+	0.527206i	$0.823240\pi$
$180$	0	0
$181$	−4747.00	−1.94940	−0.974701	−	0.223513i	$-0.928247\pi$
$181$	−4747.00	−1.94940	−0.974701	+	0.223513i	$0.928247\pi$
$182$	0	0
$183$	−2202.00	−0.889489
$184$	0	0
$185$	−426.000	−0.169298
$186$	0	0
$187$	−1224.00	−0.478651
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2402.00	0.909961	0.454981	−	0.890501i	$-0.349646\pi$
$191$	2402.00	0.909961	0.454981	+	0.890501i	$0.349646\pi$
$192$	0	0
$193$	−2925.00	−1.09091	−0.545456	−	0.838139i	$-0.683644\pi$
$193$	−2925.00	−1.09091	−0.545456	+	0.838139i	$0.683644\pi$
$194$	0	0
$195$	198.000	0.0727132
$196$	0	0
$197$	60.0000	0.0216996	0.0108498	−	0.999941i	$-0.496546\pi$
$197$	60.0000	0.0216996	0.0108498	+	0.999941i	$0.496546\pi$
$198$	0	0
$199$	3184.00	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	3184.00	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−1665.00	−0.584279
$202$	0	0
$203$	0	0
$204$	0	0
$205$	188.000	0.0640512
$206$	0	0
$207$	−828.000	−0.278019
$208$	0	0
$209$	−450.000	−0.148934
$210$	0	0
$211$	1612.00	0.525946	0.262973	−	0.964803i	$-0.415297\pi$
$211$	1612.00	0.525946	0.262973	+	0.964803i	$0.415297\pi$
$212$	0	0
$213$	1926.00	0.619565
$214$	0	0
$215$	134.000	0.0425057
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2919.00	0.900675
$220$	0	0
$221$	−2244.00	−0.683022
$222$	0	0
$223$	5736.00	1.72247	0.861235	−	0.508206i	$-0.169691\pi$
$223$	5736.00	1.72247	0.861235	+	0.508206i	$0.169691\pi$
$224$	0	0
$225$	−1089.00	−0.322667
$226$	0	0
$227$	−4970.00	−1.45317	−0.726587	−	0.687075i	$-0.758895\pi$
$227$	−4970.00	−1.45317	−0.726587	+	0.687075i	$0.758895\pi$
$228$	0	0
$229$	−763.000	−0.220177	−0.110088	−	0.993922i	$-0.535113\pi$
$229$	−763.000	−0.220177	−0.110088	+	0.993922i	$0.535113\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5374.00	−1.51100	−0.755499	−	0.655150i	$-0.772605\pi$
$233$	−5374.00	−1.51100	−0.755499	+	0.655150i	$0.772605\pi$
$234$	0	0
$235$	−556.000	−0.154338
$236$	0	0
$237$	2355.00	0.645459
$238$	0	0
$239$	−878.000	−0.237628	−0.118814	−	0.992917i	$-0.537909\pi$
$239$	−878.000	−0.237628	−0.118814	+	0.992917i	$0.537909\pi$
$240$	0	0
$241$	670.000	0.179081	0.0895404	−	0.995983i	$-0.471460\pi$
$241$	670.000	0.179081	0.0895404	+	0.995983i	$0.471460\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−825.000	−0.212524
$248$	0	0
$249$	2466.00	0.627616
$250$	0	0
$251$	4700.00	1.18192	0.590959	−	0.806702i	$-0.298750\pi$
$251$	4700.00	1.18192	0.590959	+	0.806702i	$0.298750\pi$
$252$	0	0
$253$	−1656.00	−0.411509
$254$	0	0
$255$	−408.000	−0.100196
$256$	0	0
$257$	1734.00	0.420871	0.210436	−	0.977608i	$-0.432512\pi$
$257$	1734.00	0.420871	0.210436	+	0.977608i	$0.432512\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	828.000	0.196367
$262$	0	0
$263$	−3036.00	−0.711817	−0.355908	−	0.934521i	$-0.615828\pi$
$263$	−3036.00	−0.711817	−0.355908	+	0.934521i	$0.615828\pi$
$264$	0	0
$265$	−800.000	−0.185448
$266$	0	0
$267$	1272.00	0.291555
$268$	0	0
$269$	7110.00	1.61154	0.805770	−	0.592228i	$-0.201752\pi$
$269$	7110.00	1.61154	0.805770	+	0.592228i	$0.201752\pi$
$270$	0	0
$271$	−4808.00	−1.07773	−0.538866	−	0.842392i	$-0.681147\pi$
$271$	−4808.00	−1.07773	−0.538866	+	0.842392i	$0.681147\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2178.00	−0.477594
$276$	0	0
$277$	−7091.00	−1.53811	−0.769056	−	0.639182i	$-0.779273\pi$
$277$	−7091.00	−1.53811	−0.769056	+	0.639182i	$0.779273\pi$
$278$	0	0
$279$	−225.000	−0.0482810
$280$	0	0
$281$	−1992.00	−0.422892	−0.211446	−	0.977390i	$-0.567817\pi$
$281$	−1992.00	−0.422892	−0.211446	+	0.977390i	$0.567817\pi$
$282$	0	0
$283$	−1517.00	−0.318644	−0.159322	−	0.987227i	$-0.550931\pi$
$283$	−1517.00	−0.318644	−0.159322	+	0.987227i	$0.550931\pi$
$284$	0	0
$285$	−150.000	−0.0311763
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−289.000	−0.0588235
$290$	0	0
$291$	−2202.00	−0.443586
$292$	0	0
$293$	8016.00	1.59829	0.799146	−	0.601137i	$-0.205285\pi$
$293$	8016.00	1.59829	0.799146	+	0.601137i	$0.205285\pi$
$294$	0	0
$295$	−1488.00	−0.293677
$296$	0	0
$297$	486.000	0.0949514
$298$	0	0
$299$	−3036.00	−0.587212
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−810.000	−0.153575
$304$	0	0
$305$	−1468.00	−0.275598
$306$	0	0
$307$	4169.00	0.775040	0.387520	−	0.921861i	$-0.373332\pi$
$307$	4169.00	0.775040	0.387520	+	0.921861i	$0.373332\pi$
$308$	0	0
$309$	−1251.00	−0.230314
$310$	0	0
$311$	9102.00	1.65957	0.829786	−	0.558081i	$-0.188462\pi$
$311$	9102.00	1.65957	0.829786	+	0.558081i	$0.188462\pi$
$312$	0	0
$313$	−7993.00	−1.44342	−0.721711	−	0.692195i	$-0.756644\pi$
$313$	−7993.00	−1.44342	−0.721711	+	0.692195i	$0.756644\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6288.00	−1.11410	−0.557049	−	0.830479i	$-0.688067\pi$
$317$	−6288.00	−1.11410	−0.557049	+	0.830479i	$0.688067\pi$
$318$	0	0
$319$	1656.00	0.290653
$320$	0	0
$321$	4932.00	0.857562
$322$	0	0
$323$	1700.00	0.292850
$324$	0	0
$325$	−3993.00	−0.681513
$326$	0	0
$327$	−1653.00	−0.279545
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−8015.00	−1.33095	−0.665475	−	0.746420i	$-0.731771\pi$
$331$	−8015.00	−1.33095	−0.665475	+	0.746420i	$0.731771\pi$
$332$	0	0
$333$	−1917.00	−0.315468
$334$	0	0
$335$	−1110.00	−0.181032
$336$	0	0
$337$	2093.00	0.338317	0.169159	−	0.985589i	$-0.445895\pi$
$337$	2093.00	0.338317	0.169159	+	0.985589i	$0.445895\pi$
$338$	0	0
$339$	−1206.00	−0.193218
$340$	0	0
$341$	−450.000	−0.0714630
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−552.000	−0.0861411
$346$	0	0
$347$	996.000	0.154087	0.0770433	−	0.997028i	$-0.475452\pi$
$347$	996.000	0.154087	0.0770433	+	0.997028i	$0.475452\pi$
$348$	0	0
$349$	5522.00	0.846951	0.423475	−	0.905908i	$-0.360810\pi$
$349$	5522.00	0.846951	0.423475	+	0.905908i	$0.360810\pi$
$350$	0	0
$351$	891.000	0.135493
$352$	0	0
$353$	510.000	0.0768968	0.0384484	−	0.999261i	$-0.487758\pi$
$353$	510.000	0.0768968	0.0384484	+	0.999261i	$0.487758\pi$
$354$	0	0
$355$	1284.00	0.191965
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2364.00	−0.347541	−0.173770	−	0.984786i	$-0.555595\pi$
$359$	−2364.00	−0.347541	−0.173770	+	0.984786i	$0.555595\pi$
$360$	0	0
$361$	−6234.00	−0.908879
$362$	0	0
$363$	−3021.00	−0.436808
$364$	0	0
$365$	1946.00	0.279064
$366$	0	0
$367$	−2423.00	−0.344631	−0.172315	−	0.985042i	$-0.555125\pi$
$367$	−2423.00	−0.344631	−0.172315	+	0.985042i	$0.555125\pi$
$368$	0	0
$369$	846.000	0.119352
$370$	0	0
$371$	0	0
$372$	0	0
$373$	8375.00	1.16258	0.581288	−	0.813698i	$-0.302549\pi$
$373$	8375.00	1.16258	0.581288	+	0.813698i	$0.302549\pi$
$374$	0	0
$375$	−1476.00	−0.203254
$376$	0	0
$377$	3036.00	0.414753
$378$	0	0
$379$	−9651.00	−1.30802	−0.654009	−	0.756487i	$-0.726914\pi$
$379$	−9651.00	−1.30802	−0.654009	+	0.756487i	$0.726914\pi$
$380$	0	0
$381$	−5667.00	−0.762019
$382$	0	0
$383$	−7080.00	−0.944572	−0.472286	−	0.881445i	$-0.656571\pi$
$383$	−7080.00	−0.944572	−0.472286	+	0.881445i	$0.656571\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	603.000	0.0792047
$388$	0	0
$389$	5158.00	0.672290	0.336145	−	0.941810i	$-0.390877\pi$
$389$	5158.00	0.672290	0.336145	+	0.941810i	$0.390877\pi$
$390$	0	0
$391$	6256.00	0.809155
$392$	0	0
$393$	−1662.00	−0.213325
$394$	0	0
$395$	1570.00	0.199988
$396$	0	0
$397$	−13385.0	−1.69213	−0.846063	−	0.533083i	$-0.821033\pi$
$397$	−13385.0	−1.69213	−0.846063	+	0.533083i	$0.821033\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14328.0	1.78430	0.892152	−	0.451735i	$-0.149195\pi$
$401$	14328.0	1.78430	0.892152	+	0.451735i	$0.149195\pi$
$402$	0	0
$403$	−825.000	−0.101976
$404$	0	0
$405$	162.000	0.0198762
$406$	0	0
$407$	−3834.00	−0.466939
$408$	0	0
$409$	5349.00	0.646677	0.323339	−	0.946283i	$-0.395195\pi$
$409$	5349.00	0.646677	0.323339	+	0.946283i	$0.395195\pi$
$410$	0	0
$411$	−8868.00	−1.06430
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1644.00	0.194460
$416$	0	0
$417$	2385.00	0.280081
$418$	0	0
$419$	−12734.0	−1.48472	−0.742359	−	0.670003i	$-0.766293\pi$
$419$	−12734.0	−1.48472	−0.742359	+	0.670003i	$0.766293\pi$
$420$	0	0
$421$	−3287.00	−0.380519	−0.190260	−	0.981734i	$-0.560933\pi$
$421$	−3287.00	−0.380519	−0.190260	+	0.981734i	$0.560933\pi$
$422$	0	0
$423$	−2502.00	−0.287592
$424$	0	0
$425$	8228.00	0.939098
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1782.00	0.200550
$430$	0	0
$431$	−11946.0	−1.33508	−0.667539	−	0.744575i	$-0.732652\pi$
$431$	−11946.0	−1.33508	−0.667539	+	0.744575i	$0.732652\pi$
$432$	0	0
$433$	−4433.00	−0.492001	−0.246000	−	0.969270i	$-0.579116\pi$
$433$	−4433.00	−0.492001	−0.246000	+	0.969270i	$0.579116\pi$
$434$	0	0
$435$	552.000	0.0608422
$436$	0	0
$437$	2300.00	0.251771
$438$	0	0
$439$	12360.0	1.34376	0.671880	−	0.740660i	$-0.265487\pi$
$439$	12360.0	1.34376	0.671880	+	0.740660i	$0.265487\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−10376.0	−1.11282	−0.556409	−	0.830908i	$-0.687821\pi$
$443$	−10376.0	−1.11282	−0.556409	+	0.830908i	$0.687821\pi$
$444$	0	0
$445$	848.000	0.0903350
$446$	0	0
$447$	−2724.00	−0.288234
$448$	0	0
$449$	−16382.0	−1.72186	−0.860929	−	0.508725i	$-0.830117\pi$
$449$	−16382.0	−1.72186	−0.860929	+	0.508725i	$0.830117\pi$
$450$	0	0
$451$	1692.00	0.176659
$452$	0	0
$453$	9816.00	1.01809
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−9659.00	−0.988685	−0.494342	−	0.869267i	$-0.664591\pi$
$457$	−9659.00	−0.988685	−0.494342	+	0.869267i	$0.664591\pi$
$458$	0	0
$459$	−1836.00	−0.186704
$460$	0	0
$461$	7524.00	0.760147	0.380073	−	0.924956i	$-0.375899\pi$
$461$	7524.00	0.760147	0.380073	+	0.924956i	$0.375899\pi$
$462$	0	0
$463$	−13151.0	−1.32004	−0.660020	−	0.751248i	$-0.729452\pi$
$463$	−13151.0	−1.32004	−0.660020	+	0.751248i	$0.729452\pi$
$464$	0	0
$465$	−150.000	−0.0149593
$466$	0	0
$467$	9150.00	0.906663	0.453331	−	0.891342i	$-0.350235\pi$
$467$	9150.00	0.906663	0.453331	+	0.891342i	$0.350235\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	702.000	0.0686761
$472$	0	0
$473$	1206.00	0.117235
$474$	0	0
$475$	3025.00	0.292203
$476$	0	0
$477$	−3600.00	−0.345561
$478$	0	0
$479$	11816.0	1.12711	0.563556	−	0.826078i	$-0.309433\pi$
$479$	11816.0	1.12711	0.563556	+	0.826078i	$0.309433\pi$
$480$	0	0
$481$	−7029.00	−0.666309
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1468.00	−0.137440
$486$	0	0
$487$	−13087.0	−1.21772	−0.608859	−	0.793279i	$-0.708372\pi$
$487$	−13087.0	−1.21772	−0.608859	+	0.793279i	$0.708372\pi$
$488$	0	0
$489$	2508.00	0.231934
$490$	0	0
$491$	−3612.00	−0.331990	−0.165995	−	0.986127i	$-0.553084\pi$
$491$	−3612.00	−0.331990	−0.165995	+	0.986127i	$0.553084\pi$
$492$	0	0
$493$	−6256.00	−0.571513
$494$	0	0
$495$	324.000	0.0294196
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−3493.00	−0.313363	−0.156682	−	0.987649i	$-0.550080\pi$
$499$	−3493.00	−0.313363	−0.156682	+	0.987649i	$0.550080\pi$
$500$	0	0
$501$	−18.0000	−0.00160515
$502$	0	0
$503$	2354.00	0.208667	0.104334	−	0.994542i	$-0.466729\pi$
$503$	2354.00	0.208667	0.104334	+	0.994542i	$0.466729\pi$
$504$	0	0
$505$	−540.000	−0.0475835
$506$	0	0
$507$	−3324.00	−0.291172
$508$	0	0
$509$	−13146.0	−1.14477	−0.572383	−	0.819986i	$-0.693981\pi$
$509$	−13146.0	−1.14477	−0.572383	+	0.819986i	$0.693981\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−675.000	−0.0580935
$514$	0	0
$515$	−834.000	−0.0713601
$516$	0	0
$517$	−5004.00	−0.425678
$518$	0	0
$519$	−1596.00	−0.134984
$520$	0	0
$521$	13380.0	1.12512	0.562561	−	0.826756i	$-0.309816\pi$
$521$	13380.0	1.12512	0.562561	+	0.826756i	$0.309816\pi$
$522$	0	0
$523$	5353.00	0.447553	0.223777	−	0.974640i	$-0.428161\pi$
$523$	5353.00	0.447553	0.223777	+	0.974640i	$0.428161\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1700.00	0.140518
$528$	0	0
$529$	−3703.00	−0.304348
$530$	0	0
$531$	−6696.00	−0.547235
$532$	0	0
$533$	3102.00	0.252087
$534$	0	0
$535$	3288.00	0.265706
$536$	0	0
$537$	−12210.0	−0.981193
$538$	0	0
$539$	0	0
$540$	0	0
$541$	21685.0	1.72331	0.861655	−	0.507494i	$-0.169428\pi$
$541$	21685.0	1.72331	0.861655	+	0.507494i	$0.169428\pi$
$542$	0	0
$543$	−14241.0	−1.12549
$544$	0	0
$545$	−1102.00	−0.0866137
$546$	0	0
$547$	−3724.00	−0.291091	−0.145545	−	0.989352i	$-0.546494\pi$
$547$	−3724.00	−0.291091	−0.145545	+	0.989352i	$0.546494\pi$
$548$	0	0
$549$	−6606.00	−0.513547
$550$	0	0
$551$	−2300.00	−0.177828
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1278.00	−0.0977443
$556$	0	0
$557$	24166.0	1.83832	0.919162	−	0.393880i	$-0.128867\pi$
$557$	24166.0	1.83832	0.919162	+	0.393880i	$0.128867\pi$
$558$	0	0
$559$	2211.00	0.167290
$560$	0	0
$561$	−3672.00	−0.276349
$562$	0	0
$563$	25118.0	1.88028	0.940140	−	0.340789i	$-0.110694\pi$
$563$	25118.0	1.88028	0.940140	+	0.340789i	$0.110694\pi$
$564$	0	0
$565$	−804.000	−0.0598664
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6162.00	−0.453997	−0.226999	−	0.973895i	$-0.572891\pi$
$569$	−6162.00	−0.453997	−0.226999	+	0.973895i	$0.572891\pi$
$570$	0	0
$571$	12619.0	0.924849	0.462424	−	0.886659i	$-0.346980\pi$
$571$	12619.0	0.924849	0.462424	+	0.886659i	$0.346980\pi$
$572$	0	0
$573$	7206.00	0.525366
$574$	0	0
$575$	11132.0	0.807368
$576$	0	0
$577$	−8937.00	−0.644804	−0.322402	−	0.946603i	$-0.604490\pi$
$577$	−8937.00	−0.644804	−0.322402	+	0.946603i	$0.604490\pi$
$578$	0	0
$579$	−8775.00	−0.629839
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−7200.00	−0.511481
$584$	0	0
$585$	594.000	0.0419810
$586$	0	0
$587$	−12784.0	−0.898896	−0.449448	−	0.893306i	$-0.648379\pi$
$587$	−12784.0	−0.898896	−0.449448	+	0.893306i	$0.648379\pi$
$588$	0	0
$589$	625.000	0.0437227
$590$	0	0
$591$	180.000	0.0125283
$592$	0	0
$593$	13746.0	0.951907	0.475953	−	0.879471i	$-0.342103\pi$
$593$	13746.0	0.951907	0.475953	+	0.879471i	$0.342103\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9552.00	0.654836
$598$	0	0
$599$	−6076.00	−0.414455	−0.207228	−	0.978293i	$-0.566444\pi$
$599$	−6076.00	−0.414455	−0.207228	+	0.978293i	$0.566444\pi$
$600$	0	0
$601$	−3977.00	−0.269925	−0.134963	−	0.990851i	$-0.543091\pi$
$601$	−3977.00	−0.269925	−0.134963	+	0.990851i	$0.543091\pi$
$602$	0	0
$603$	−4995.00	−0.337334
$604$	0	0
$605$	−2014.00	−0.135340
$606$	0	0
$607$	5873.00	0.392715	0.196357	−	0.980532i	$-0.437089\pi$
$607$	5873.00	0.392715	0.196357	+	0.980532i	$0.437089\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9174.00	−0.607431
$612$	0	0
$613$	−11438.0	−0.753632	−0.376816	−	0.926288i	$-0.622981\pi$
$613$	−11438.0	−0.753632	−0.376816	+	0.926288i	$0.622981\pi$
$614$	0	0
$615$	564.000	0.0369800
$616$	0	0
$617$	25866.0	1.68772	0.843862	−	0.536560i	$-0.180276\pi$
$617$	25866.0	1.68772	0.843862	+	0.536560i	$0.180276\pi$
$618$	0	0
$619$	12069.0	0.783674	0.391837	−	0.920035i	$-0.371840\pi$
$619$	12069.0	0.783674	0.391837	+	0.920035i	$0.371840\pi$
$620$	0	0
$621$	−2484.00	−0.160514
$622$	0	0
$623$	0	0
$624$	0	0
$625$	14141.0	0.905024
$626$	0	0
$627$	−1350.00	−0.0859869
$628$	0	0
$629$	14484.0	0.918148
$630$	0	0
$631$	−392.000	−0.0247310	−0.0123655	−	0.999924i	$-0.503936\pi$
$631$	−392.000	−0.0247310	−0.0123655	+	0.999924i	$0.503936\pi$
$632$	0	0
$633$	4836.00	0.303655
$634$	0	0
$635$	−3778.00	−0.236103
$636$	0	0
$637$	0	0
$638$	0	0
$639$	5778.00	0.357706
$640$	0	0
$641$	−27884.0	−1.71818	−0.859089	−	0.511827i	$-0.828969\pi$
$641$	−27884.0	−1.71818	−0.859089	+	0.511827i	$0.828969\pi$
$642$	0	0
$643$	−4069.00	−0.249558	−0.124779	−	0.992185i	$-0.539822\pi$
$643$	−4069.00	−0.249558	−0.124779	+	0.992185i	$0.539822\pi$
$644$	0	0
$645$	402.000	0.0245407
$646$	0	0
$647$	−23586.0	−1.43317	−0.716585	−	0.697499i	$-0.754296\pi$
$647$	−23586.0	−1.43317	−0.716585	+	0.697499i	$0.754296\pi$
$648$	0	0
$649$	−13392.0	−0.809988
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7822.00	−0.468757	−0.234379	−	0.972145i	$-0.575306\pi$
$653$	−7822.00	−0.468757	−0.234379	+	0.972145i	$0.575306\pi$
$654$	0	0
$655$	−1108.00	−0.0660964
$656$	0	0
$657$	8757.00	0.520005
$658$	0	0
$659$	4096.00	0.242121	0.121060	−	0.992645i	$-0.461371\pi$
$659$	4096.00	0.242121	0.121060	+	0.992645i	$0.461371\pi$
$660$	0	0
$661$	−24601.0	−1.44761	−0.723803	−	0.690006i	$-0.757608\pi$
$661$	−24601.0	−1.44761	−0.723803	+	0.690006i	$0.757608\pi$
$662$	0	0
$663$	−6732.00	−0.394343
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8464.00	−0.491345
$668$	0	0
$669$	17208.0	0.994469
$670$	0	0
$671$	−13212.0	−0.760125
$672$	0	0
$673$	16231.0	0.929657	0.464828	−	0.885401i	$-0.346116\pi$
$673$	16231.0	0.929657	0.464828	+	0.885401i	$0.346116\pi$
$674$	0	0
$675$	−3267.00	−0.186292
$676$	0	0
$677$	1404.00	0.0797047	0.0398524	−	0.999206i	$-0.487311\pi$
$677$	1404.00	0.0797047	0.0398524	+	0.999206i	$0.487311\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−14910.0	−0.838990
$682$	0	0
$683$	−22248.0	−1.24641	−0.623204	−	0.782060i	$-0.714169\pi$
$683$	−22248.0	−1.24641	−0.623204	+	0.782060i	$0.714169\pi$
$684$	0	0
$685$	−5912.00	−0.329760
$686$	0	0
$687$	−2289.00	−0.127119
$688$	0	0
$689$	−13200.0	−0.729869
$690$	0	0
$691$	−32483.0	−1.78829	−0.894147	−	0.447773i	$-0.852217\pi$
$691$	−32483.0	−1.78829	−0.894147	+	0.447773i	$0.852217\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1590.00	0.0867801
$696$	0	0
$697$	−6392.00	−0.347366
$698$	0	0
$699$	−16122.0	−0.872375
$700$	0	0
$701$	16180.0	0.871769	0.435885	−	0.900003i	$-0.356436\pi$
$701$	16180.0	0.871769	0.435885	+	0.900003i	$0.356436\pi$
$702$	0	0
$703$	5325.00	0.285684
$704$	0	0
$705$	−1668.00	−0.0891071
$706$	0	0
$707$	0	0
$708$	0	0
$709$	24358.0	1.29025	0.645123	−	0.764079i	$-0.276806\pi$
$709$	24358.0	1.29025	0.645123	+	0.764079i	$0.276806\pi$
$710$	0	0
$711$	7065.00	0.372656
$712$	0	0
$713$	2300.00	0.120807
$714$	0	0
$715$	1188.00	0.0621380
$716$	0	0
$717$	−2634.00	−0.137195
$718$	0	0
$719$	26658.0	1.38272	0.691360	−	0.722510i	$-0.257012\pi$
$719$	26658.0	1.38272	0.691360	+	0.722510i	$0.257012\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	2010.00	0.103392
$724$	0	0
$725$	−11132.0	−0.570251
$726$	0	0
$727$	20701.0	1.05606	0.528031	−	0.849225i	$-0.322930\pi$
$727$	20701.0	1.05606	0.528031	+	0.849225i	$0.322930\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−4556.00	−0.230519
$732$	0	0
$733$	−12839.0	−0.646957	−0.323478	−	0.946236i	$-0.604852\pi$
$733$	−12839.0	−0.646957	−0.323478	+	0.946236i	$0.604852\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9990.00	−0.499303
$738$	0	0
$739$	21815.0	1.08590	0.542948	−	0.839766i	$-0.317308\pi$
$739$	21815.0	1.08590	0.542948	+	0.839766i	$0.317308\pi$
$740$	0	0
$741$	−2475.00	−0.122701
$742$	0	0
$743$	24286.0	1.19915	0.599574	−	0.800319i	$-0.295337\pi$
$743$	24286.0	1.19915	0.599574	+	0.800319i	$0.295337\pi$
$744$	0	0
$745$	−1816.00	−0.0893062
$746$	0	0
$747$	7398.00	0.362354
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−15173.0	−0.737245	−0.368622	−	0.929579i	$-0.620170\pi$
$751$	−15173.0	−0.737245	−0.368622	+	0.929579i	$0.620170\pi$
$752$	0	0
$753$	14100.0	0.682381
$754$	0	0
$755$	6544.00	0.315444
$756$	0	0
$757$	−13742.0	−0.659791	−0.329895	−	0.944018i	$-0.607013\pi$
$757$	−13742.0	−0.659791	−0.329895	+	0.944018i	$0.607013\pi$
$758$	0	0
$759$	−4968.00	−0.237585
$760$	0	0
$761$	34764.0	1.65597	0.827986	−	0.560749i	$-0.189487\pi$
$761$	34764.0	1.65597	0.827986	+	0.560749i	$0.189487\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1224.00	−0.0578481
$766$	0	0
$767$	−24552.0	−1.15583
$768$	0	0
$769$	−31241.0	−1.46499	−0.732496	−	0.680771i	$-0.761645\pi$
$769$	−31241.0	−1.46499	−0.732496	+	0.680771i	$0.761645\pi$
$770$	0	0
$771$	5202.00	0.242990
$772$	0	0
$773$	27622.0	1.28524	0.642622	−	0.766183i	$-0.277846\pi$
$773$	27622.0	1.28524	0.642622	+	0.766183i	$0.277846\pi$
$774$	0	0
$775$	3025.00	0.140208
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2350.00	−0.108084
$780$	0	0
$781$	11556.0	0.529457
$782$	0	0
$783$	2484.00	0.113373
$784$	0	0
$785$	468.000	0.0212785
$786$	0	0
$787$	−30544.0	−1.38345	−0.691726	−	0.722160i	$-0.743149\pi$
$787$	−30544.0	−1.38345	−0.691726	+	0.722160i	$0.743149\pi$
$788$	0	0
$789$	−9108.00	−0.410968
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−24222.0	−1.08468
$794$	0	0
$795$	−2400.00	−0.107068
$796$	0	0
$797$	31668.0	1.40745	0.703725	−	0.710472i	$-0.251519\pi$
$797$	31668.0	1.40745	0.703725	+	0.710472i	$0.251519\pi$
$798$	0	0
$799$	18904.0	0.837016
$800$	0	0
$801$	3816.00	0.168329
$802$	0	0
$803$	17514.0	0.769683
$804$	0	0
$805$	0	0
$806$	0	0
$807$	21330.0	0.930423
$808$	0	0
$809$	11562.0	0.502470	0.251235	−	0.967926i	$-0.419163\pi$
$809$	11562.0	0.502470	0.251235	+	0.967926i	$0.419163\pi$
$810$	0	0
$811$	3144.00	0.136129	0.0680646	−	0.997681i	$-0.478318\pi$
$811$	3144.00	0.136129	0.0680646	+	0.997681i	$0.478318\pi$
$812$	0	0
$813$	−14424.0	−0.622228
$814$	0	0
$815$	1672.00	0.0718621
$816$	0	0
$817$	−1675.00	−0.0717268
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10926.0	0.464458	0.232229	−	0.972661i	$-0.425398\pi$
$821$	10926.0	0.464458	0.232229	+	0.972661i	$0.425398\pi$
$822$	0	0
$823$	35224.0	1.49190	0.745949	−	0.666003i	$-0.231996\pi$
$823$	35224.0	1.49190	0.745949	+	0.666003i	$0.231996\pi$
$824$	0	0
$825$	−6534.00	−0.275739
$826$	0	0
$827$	38322.0	1.61135	0.805675	−	0.592358i	$-0.201803\pi$
$827$	38322.0	1.61135	0.805675	+	0.592358i	$0.201803\pi$
$828$	0	0
$829$	18089.0	0.757849	0.378925	−	0.925428i	$-0.376294\pi$
$829$	18089.0	0.757849	0.378925	+	0.925428i	$0.376294\pi$
$830$	0	0
$831$	−21273.0	−0.888029
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−12.0000	−0.000497338 0
$836$	0	0
$837$	−675.000	−0.0278750
$838$	0	0
$839$	2412.00	0.0992509	0.0496254	−	0.998768i	$-0.484197\pi$
$839$	2412.00	0.0992509	0.0496254	+	0.998768i	$0.484197\pi$
$840$	0	0
$841$	−15925.0	−0.652958
$842$	0	0
$843$	−5976.00	−0.244157
$844$	0	0
$845$	−2216.00	−0.0902162
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−4551.00	−0.183969
$850$	0	0
$851$	19596.0	0.789356
$852$	0	0
$853$	8611.00	0.345645	0.172822	−	0.984953i	$-0.444711\pi$
$853$	8611.00	0.345645	0.172822	+	0.984953i	$0.444711\pi$
$854$	0	0
$855$	−450.000	−0.0179996
$856$	0	0
$857$	10560.0	0.420913	0.210457	−	0.977603i	$-0.432505\pi$
$857$	10560.0	0.420913	0.210457	+	0.977603i	$0.432505\pi$
$858$	0	0
$859$	7808.00	0.310134	0.155067	−	0.987904i	$-0.450441\pi$
$859$	7808.00	0.310134	0.155067	+	0.987904i	$0.450441\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18910.0	0.745891	0.372945	−	0.927853i	$-0.378348\pi$
$863$	18910.0	0.745891	0.372945	+	0.927853i	$0.378348\pi$
$864$	0	0
$865$	−1064.00	−0.0418232
$866$	0	0
$867$	−867.000	−0.0339618
$868$	0	0
$869$	14130.0	0.551585
$870$	0	0
$871$	−18315.0	−0.712492
$872$	0	0
$873$	−6606.00	−0.256104
$874$	0	0
$875$	0	0
$876$	0	0
$877$	9290.00	0.357698	0.178849	−	0.983877i	$-0.442763\pi$
$877$	9290.00	0.357698	0.178849	+	0.983877i	$0.442763\pi$
$878$	0	0
$879$	24048.0	0.922775
$880$	0	0
$881$	−39428.0	−1.50779	−0.753895	−	0.656995i	$-0.771827\pi$
$881$	−39428.0	−1.50779	−0.753895	+	0.656995i	$0.771827\pi$
$882$	0	0
$883$	−37051.0	−1.41208	−0.706039	−	0.708173i	$-0.749520\pi$
$883$	−37051.0	−1.41208	−0.706039	+	0.708173i	$0.749520\pi$
$884$	0	0
$885$	−4464.00	−0.169554
$886$	0	0
$887$	21450.0	0.811974	0.405987	−	0.913879i	$-0.366928\pi$
$887$	21450.0	0.811974	0.405987	+	0.913879i	$0.366928\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1458.00	0.0548202
$892$	0	0
$893$	6950.00	0.260440
$894$	0	0
$895$	−8140.00	−0.304011
$896$	0	0
$897$	−9108.00	−0.339027
$898$	0	0
$899$	−2300.00	−0.0853274
$900$	0	0
$901$	27200.0	1.00573
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9494.00	−0.348720
$906$	0	0
$907$	−1523.00	−0.0557557	−0.0278778	−	0.999611i	$-0.508875\pi$
$907$	−1523.00	−0.0557557	−0.0278778	+	0.999611i	$0.508875\pi$
$908$	0	0
$909$	−2430.00	−0.0886667
$910$	0	0
$911$	−1324.00	−0.0481516	−0.0240758	−	0.999710i	$-0.507664\pi$
$911$	−1324.00	−0.0481516	−0.0240758	+	0.999710i	$0.507664\pi$
$912$	0	0
$913$	14796.0	0.536337
$914$	0	0
$915$	−4404.00	−0.159117
$916$	0	0
$917$	0	0
$918$	0	0
$919$	22457.0	0.806081	0.403040	−	0.915182i	$-0.367953\pi$
$919$	22457.0	0.806081	0.403040	+	0.915182i	$0.367953\pi$
$920$	0	0
$921$	12507.0	0.447470
$922$	0	0
$923$	21186.0	0.755521
$924$	0	0
$925$	25773.0	0.916120
$926$	0	0
$927$	−3753.00	−0.132972
$928$	0	0
$929$	50146.0	1.77098	0.885488	−	0.464662i	$-0.153824\pi$
$929$	50146.0	1.77098	0.885488	+	0.464662i	$0.153824\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	27306.0	0.958155
$934$	0	0
$935$	−2448.00	−0.0856237
$936$	0	0
$937$	−32841.0	−1.14500	−0.572502	−	0.819903i	$-0.694027\pi$
$937$	−32841.0	−1.14500	−0.572502	+	0.819903i	$0.694027\pi$
$938$	0	0
$939$	−23979.0	−0.833360
$940$	0	0
$941$	−13176.0	−0.456456	−0.228228	−	0.973608i	$-0.573293\pi$
$941$	−13176.0	−0.456456	−0.228228	+	0.973608i	$0.573293\pi$
$942$	0	0
$943$	−8648.00	−0.298640
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20594.0	0.706669	0.353334	−	0.935497i	$-0.385048\pi$
$947$	20594.0	0.706669	0.353334	+	0.935497i	$0.385048\pi$
$948$	0	0
$949$	32109.0	1.09832
$950$	0	0
$951$	−18864.0	−0.643225
$952$	0	0
$953$	−992.000	−0.0337188	−0.0168594	−	0.999858i	$-0.505367\pi$
$953$	−992.000	−0.0337188	−0.0168594	+	0.999858i	$0.505367\pi$
$954$	0	0
$955$	4804.00	0.162779
$956$	0	0
$957$	4968.00	0.167808
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29166.0	−0.979021
$962$	0	0
$963$	14796.0	0.495114
$964$	0	0
$965$	−5850.00	−0.195148
$966$	0	0
$967$	−21459.0	−0.713625	−0.356812	−	0.934176i	$-0.616136\pi$
$967$	−21459.0	−0.713625	−0.356812	+	0.934176i	$0.616136\pi$
$968$	0	0
$969$	5100.00	0.169077
$970$	0	0
$971$	27344.0	0.903719	0.451859	−	0.892089i	$-0.350761\pi$
$971$	27344.0	0.903719	0.451859	+	0.892089i	$0.350761\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−11979.0	−0.393472
$976$	0	0
$977$	−50226.0	−1.64470	−0.822350	−	0.568982i	$-0.807337\pi$
$977$	−50226.0	−1.64470	−0.822350	+	0.568982i	$0.807337\pi$
$978$	0	0
$979$	7632.00	0.249152
$980$	0	0
$981$	−4959.00	−0.161395
$982$	0	0
$983$	4152.00	0.134718	0.0673592	−	0.997729i	$-0.478543\pi$
$983$	4152.00	0.134718	0.0673592	+	0.997729i	$0.478543\pi$
$984$	0	0
$985$	120.000	0.00388174
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6164.00	−0.198184
$990$	0	0
$991$	53727.0	1.72219	0.861097	−	0.508441i	$-0.169778\pi$
$991$	53727.0	1.72219	0.861097	+	0.508441i	$0.169778\pi$
$992$	0	0
$993$	−24045.0	−0.768424
$994$	0	0
$995$	6368.00	0.202894
$996$	0	0
$997$	−29741.0	−0.944741	−0.472371	−	0.881400i	$-0.656602\pi$
$997$	−29741.0	−0.944741	−0.472371	+	0.881400i	$0.656602\pi$
$998$	0	0
$999$	−5751.00	−0.182136

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.bc.1.1		1
4.3	odd	2		1176.4.a.d.1.1		1
7.2	even	3		336.4.q.b.193.1		2
7.4	even	3		336.4.q.b.289.1		2
7.6	odd	2		2352.4.a.j.1.1		1
28.11	odd	6		168.4.q.b.121.1	yes	2
28.23	odd	6		168.4.q.b.25.1	✓	2
28.27	even	2		1176.4.a.k.1.1		1
84.11	even	6		504.4.s.d.289.1		2
84.23	even	6		504.4.s.d.361.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.q.b.25.1	✓	2	28.23	odd	6
168.4.q.b.121.1	yes	2	28.11	odd	6
336.4.q.b.193.1		2	7.2	even	3
336.4.q.b.289.1		2	7.4	even	3
504.4.s.d.289.1		2	84.11	even	6
504.4.s.d.361.1		2	84.23	even	6
1176.4.a.d.1.1		1	4.3	odd	2
1176.4.a.k.1.1		1	28.27	even	2
2352.4.a.j.1.1		1	7.6	odd	2
2352.4.a.bc.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)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more