LMFDB - Embedded newform 336.4.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,4,Mod(1,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, 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("336.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	336.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:	$19.8246417619$
Analytic rank:	$0$
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	$\chi$	$=$	336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} +4.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} +4.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +26.0000 q^{11} +2.00000 q^{13} +12.0000 q^{15} -36.0000 q^{17} +76.0000 q^{19} +21.0000 q^{21} +114.000 q^{23} -109.000 q^{25} +27.0000 q^{27} +6.00000 q^{29} +256.000 q^{31} +78.0000 q^{33} +28.0000 q^{35} -86.0000 q^{37} +6.00000 q^{39} +160.000 q^{41} +220.000 q^{43} +36.0000 q^{45} -308.000 q^{47} +49.0000 q^{49} -108.000 q^{51} +258.000 q^{53} +104.000 q^{55} +228.000 q^{57} -264.000 q^{59} +606.000 q^{61} +63.0000 q^{63} +8.00000 q^{65} +520.000 q^{67} +342.000 q^{69} +286.000 q^{71} -530.000 q^{73} -327.000 q^{75} +182.000 q^{77} +44.0000 q^{79} +81.0000 q^{81} -1012.00 q^{83} -144.000 q^{85} +18.0000 q^{87} +768.000 q^{89} +14.0000 q^{91} +768.000 q^{93} +304.000 q^{95} +222.000 q^{97} +234.000 q^{99} +O(q^{100})$

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
$5$	4.00000	0.357771	0.178885	−	0.983870i	$-0.442751\pi$
$5$	4.00000	0.357771	0.178885	+	0.983870i	$0.442751\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	26.0000	0.712663	0.356332	−	0.934360i	$-0.384027\pi$
$11$	26.0000	0.712663	0.356332	+	0.934360i	$0.384027\pi$
$12$	0	0
$13$	2.00000	0.0426692	0.0213346	−	0.999772i	$-0.493208\pi$
$13$	2.00000	0.0426692	0.0213346	+	0.999772i	$0.493208\pi$
$14$	0	0
$15$	12.0000	0.206559
$16$	0	0
$17$	−36.0000	−0.513605	−0.256802	−	0.966464i	$-0.582669\pi$
$17$	−36.0000	−0.513605	−0.256802	+	0.966464i	$0.582669\pi$
$18$	0	0
$19$	76.0000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	76.0000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	114.000	1.03351	0.516753	−	0.856134i	$-0.327141\pi$
$23$	114.000	1.03351	0.516753	+	0.856134i	$0.327141\pi$
$24$	0	0
$25$	−109.000	−0.872000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	6.00000	0.0384197	0.0192099	−	0.999815i	$-0.493885\pi$
$29$	6.00000	0.0384197	0.0192099	+	0.999815i	$0.493885\pi$
$30$	0	0
$31$	256.000	1.48319	0.741596	−	0.670847i	$-0.234069\pi$
$31$	256.000	1.48319	0.741596	+	0.670847i	$0.234069\pi$
$32$	0	0
$33$	78.0000	0.411456
$34$	0	0
$35$	28.0000	0.135225
$36$	0	0
$37$	−86.0000	−0.382117	−0.191058	−	0.981579i	$-0.561192\pi$
$37$	−86.0000	−0.382117	−0.191058	+	0.981579i	$0.561192\pi$
$38$	0	0
$39$	6.00000	0.0246351
$40$	0	0
$41$	160.000	0.609459	0.304729	−	0.952439i	$-0.401434\pi$
$41$	160.000	0.609459	0.304729	+	0.952439i	$0.401434\pi$
$42$	0	0
$43$	220.000	0.780225	0.390113	−	0.920767i	$-0.372436\pi$
$43$	220.000	0.780225	0.390113	+	0.920767i	$0.372436\pi$
$44$	0	0
$45$	36.0000	0.119257
$46$	0	0
$47$	−308.000	−0.955881	−0.477941	−	0.878392i	$-0.658617\pi$
$47$	−308.000	−0.955881	−0.477941	+	0.878392i	$0.658617\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−108.000	−0.296530
$52$	0	0
$53$	258.000	0.668661	0.334330	−	0.942456i	$-0.391490\pi$
$53$	258.000	0.668661	0.334330	+	0.942456i	$0.391490\pi$
$54$	0	0
$55$	104.000	0.254970
$56$	0	0
$57$	228.000	0.529813
$58$	0	0
$59$	−264.000	−0.582540	−0.291270	−	0.956641i	$-0.594078\pi$
$59$	−264.000	−0.582540	−0.291270	+	0.956641i	$0.594078\pi$
$60$	0	0
$61$	606.000	1.27197	0.635986	−	0.771700i	$-0.280593\pi$
$61$	606.000	1.27197	0.635986	+	0.771700i	$0.280593\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	8.00000	0.0152658
$66$	0	0
$67$	520.000	0.948181	0.474090	−	0.880476i	$-0.342777\pi$
$67$	520.000	0.948181	0.474090	+	0.880476i	$0.342777\pi$
$68$	0	0
$69$	342.000	0.596695
$70$	0	0
$71$	286.000	0.478056	0.239028	−	0.971013i	$-0.423171\pi$
$71$	286.000	0.478056	0.239028	+	0.971013i	$0.423171\pi$
$72$	0	0
$73$	−530.000	−0.849751	−0.424875	−	0.905252i	$-0.639682\pi$
$73$	−530.000	−0.849751	−0.424875	+	0.905252i	$0.639682\pi$
$74$	0	0
$75$	−327.000	−0.503449
$76$	0	0
$77$	182.000	0.269361
$78$	0	0
$79$	44.0000	0.0626631	0.0313316	−	0.999509i	$-0.490025\pi$
$79$	44.0000	0.0626631	0.0313316	+	0.999509i	$0.490025\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1012.00	−1.33833	−0.669165	−	0.743114i	$-0.733348\pi$
$83$	−1012.00	−1.33833	−0.669165	+	0.743114i	$0.733348\pi$
$84$	0	0
$85$	−144.000	−0.183753
$86$	0	0
$87$	18.0000	0.0221816
$88$	0	0
$89$	768.000	0.914695	0.457347	−	0.889288i	$-0.348800\pi$
$89$	768.000	0.914695	0.457347	+	0.889288i	$0.348800\pi$
$90$	0	0
$91$	14.0000	0.0161275
$92$	0	0
$93$	768.000	0.856321
$94$	0	0
$95$	304.000	0.328313
$96$	0	0
$97$	222.000	0.232378	0.116189	−	0.993227i	$-0.462932\pi$
$97$	222.000	0.232378	0.116189	+	0.993227i	$0.462932\pi$
$98$	0	0
$99$	234.000	0.237554
$100$	0	0
$101$	−320.000	−0.315259	−0.157630	−	0.987498i	$-0.550385\pi$
$101$	−320.000	−0.315259	−0.157630	+	0.987498i	$0.550385\pi$
$102$	0	0
$103$	592.000	0.566325	0.283163	−	0.959072i	$-0.408616\pi$
$103$	592.000	0.566325	0.283163	+	0.959072i	$0.408616\pi$
$104$	0	0
$105$	84.0000	0.0780720
$106$	0	0
$107$	−1782.00	−1.61002	−0.805011	−	0.593259i	$-0.797841\pi$
$107$	−1782.00	−1.61002	−0.805011	+	0.593259i	$0.797841\pi$
$108$	0	0
$109$	230.000	0.202110	0.101055	−	0.994881i	$-0.467778\pi$
$109$	230.000	0.202110	0.101055	+	0.994881i	$0.467778\pi$
$110$	0	0
$111$	−258.000	−0.220615
$112$	0	0
$113$	−1718.00	−1.43023	−0.715114	−	0.699007i	$-0.753625\pi$
$113$	−1718.00	−1.43023	−0.715114	+	0.699007i	$0.753625\pi$
$114$	0	0
$115$	456.000	0.369758
$116$	0	0
$117$	18.0000	0.0142231
$118$	0	0
$119$	−252.000	−0.194124
$120$	0	0
$121$	−655.000	−0.492111
$122$	0	0
$123$	480.000	0.351871
$124$	0	0
$125$	−936.000	−0.669747
$126$	0	0
$127$	2444.00	1.70764	0.853819	−	0.520571i	$-0.174281\pi$
$127$	2444.00	1.70764	0.853819	+	0.520571i	$0.174281\pi$
$128$	0	0
$129$	660.000	0.450463
$130$	0	0
$131$	−1996.00	−1.33123	−0.665616	−	0.746295i	$-0.731831\pi$
$131$	−1996.00	−1.33123	−0.665616	+	0.746295i	$0.731831\pi$
$132$	0	0
$133$	532.000	0.346844
$134$	0	0
$135$	108.000	0.0688530
$136$	0	0
$137$	−1746.00	−1.08884	−0.544419	−	0.838813i	$-0.683250\pi$
$137$	−1746.00	−1.08884	−0.544419	+	0.838813i	$0.683250\pi$
$138$	0	0
$139$	−1804.00	−1.10081	−0.550407	−	0.834896i	$-0.685528\pi$
$139$	−1804.00	−1.10081	−0.550407	+	0.834896i	$0.685528\pi$
$140$	0	0
$141$	−924.000	−0.551878
$142$	0	0
$143$	52.0000	0.0304088
$144$	0	0
$145$	24.0000	0.0137455
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	−2814.00	−1.54719	−0.773597	−	0.633678i	$-0.781544\pi$
$149$	−2814.00	−1.54719	−0.773597	+	0.633678i	$0.781544\pi$
$150$	0	0
$151$	792.000	0.426835	0.213417	−	0.976961i	$-0.431541\pi$
$151$	792.000	0.426835	0.213417	+	0.976961i	$0.431541\pi$
$152$	0	0
$153$	−324.000	−0.171202
$154$	0	0
$155$	1024.00	0.530643
$156$	0	0
$157$	−2778.00	−1.41216	−0.706078	−	0.708134i	$-0.749537\pi$
$157$	−2778.00	−1.41216	−0.706078	+	0.708134i	$0.749537\pi$
$158$	0	0
$159$	774.000	0.386052
$160$	0	0
$161$	798.000	0.390629
$162$	0	0
$163$	−2880.00	−1.38392	−0.691960	−	0.721936i	$-0.743253\pi$
$163$	−2880.00	−1.38392	−0.691960	+	0.721936i	$0.743253\pi$
$164$	0	0
$165$	312.000	0.147207
$166$	0	0
$167$	1060.00	0.491169	0.245585	−	0.969375i	$-0.421020\pi$
$167$	1060.00	0.491169	0.245585	+	0.969375i	$0.421020\pi$
$168$	0	0
$169$	−2193.00	−0.998179
$170$	0	0
$171$	684.000	0.305888
$172$	0	0
$173$	−1440.00	−0.632839	−0.316420	−	0.948619i	$-0.602481\pi$
$173$	−1440.00	−0.632839	−0.316420	+	0.948619i	$0.602481\pi$
$174$	0	0
$175$	−763.000	−0.329585
$176$	0	0
$177$	−792.000	−0.336330
$178$	0	0
$179$	−1210.00	−0.505249	−0.252625	−	0.967564i	$-0.581294\pi$
$179$	−1210.00	−0.505249	−0.252625	+	0.967564i	$0.581294\pi$
$180$	0	0
$181$	1618.00	0.664447	0.332224	−	0.943201i	$-0.392201\pi$
$181$	1618.00	0.664447	0.332224	+	0.943201i	$0.392201\pi$
$182$	0	0
$183$	1818.00	0.734374
$184$	0	0
$185$	−344.000	−0.136710
$186$	0	0
$187$	−936.000	−0.366027
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	−4018.00	−1.52216	−0.761079	−	0.648659i	$-0.775330\pi$
$191$	−4018.00	−1.52216	−0.761079	+	0.648659i	$0.775330\pi$
$192$	0	0
$193$	3382.00	1.26136	0.630678	−	0.776045i	$-0.282777\pi$
$193$	3382.00	1.26136	0.630678	+	0.776045i	$0.282777\pi$
$194$	0	0
$195$	24.0000	0.00881372
$196$	0	0
$197$	−4302.00	−1.55586	−0.777931	−	0.628350i	$-0.783731\pi$
$197$	−4302.00	−1.55586	−0.777931	+	0.628350i	$0.783731\pi$
$198$	0	0
$199$	2640.00	0.940425	0.470213	−	0.882553i	$-0.344177\pi$
$199$	2640.00	0.940425	0.470213	+	0.882553i	$0.344177\pi$
$200$	0	0
$201$	1560.00	0.547432
$202$	0	0
$203$	42.0000	0.0145213
$204$	0	0
$205$	640.000	0.218047
$206$	0	0
$207$	1026.00	0.344502
$208$	0	0
$209$	1976.00	0.653985
$210$	0	0
$211$	−3396.00	−1.10801	−0.554005	−	0.832513i	$-0.686901\pi$
$211$	−3396.00	−1.10801	−0.554005	+	0.832513i	$0.686901\pi$
$212$	0	0
$213$	858.000	0.276006
$214$	0	0
$215$	880.000	0.279142
$216$	0	0
$217$	1792.00	0.560594
$218$	0	0
$219$	−1590.00	−0.490604
$220$	0	0
$221$	−72.0000	−0.0219151
$222$	0	0
$223$	−3480.00	−1.04501	−0.522507	−	0.852635i	$-0.675003\pi$
$223$	−3480.00	−1.04501	−0.522507	+	0.852635i	$0.675003\pi$
$224$	0	0
$225$	−981.000	−0.290667
$226$	0	0
$227$	1504.00	0.439753	0.219877	−	0.975528i	$-0.429435\pi$
$227$	1504.00	0.439753	0.219877	+	0.975528i	$0.429435\pi$
$228$	0	0
$229$	5122.00	1.47804	0.739020	−	0.673683i	$-0.235289\pi$
$229$	5122.00	1.47804	0.739020	+	0.673683i	$0.235289\pi$
$230$	0	0
$231$	546.000	0.155516
$232$	0	0
$233$	1630.00	0.458304	0.229152	−	0.973391i	$-0.426405\pi$
$233$	1630.00	0.458304	0.229152	+	0.973391i	$0.426405\pi$
$234$	0	0
$235$	−1232.00	−0.341986
$236$	0	0
$237$	132.000	0.0361786
$238$	0	0
$239$	2522.00	0.682572	0.341286	−	0.939960i	$-0.389138\pi$
$239$	2522.00	0.682572	0.341286	+	0.939960i	$0.389138\pi$
$240$	0	0
$241$	3022.00	0.807735	0.403867	−	0.914817i	$-0.367666\pi$
$241$	3022.00	0.807735	0.403867	+	0.914817i	$0.367666\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	196.000	0.0511101
$246$	0	0
$247$	152.000	0.0391560
$248$	0	0
$249$	−3036.00	−0.772685
$250$	0	0
$251$	−3480.00	−0.875122	−0.437561	−	0.899189i	$-0.644158\pi$
$251$	−3480.00	−0.875122	−0.437561	+	0.899189i	$0.644158\pi$
$252$	0	0
$253$	2964.00	0.736542
$254$	0	0
$255$	−432.000	−0.106090
$256$	0	0
$257$	3496.00	0.848539	0.424269	−	0.905536i	$-0.360531\pi$
$257$	3496.00	0.848539	0.424269	+	0.905536i	$0.360531\pi$
$258$	0	0
$259$	−602.000	−0.144426
$260$	0	0
$261$	54.0000	0.0128066
$262$	0	0
$263$	5058.00	1.18589	0.592946	−	0.805242i	$-0.297965\pi$
$263$	5058.00	1.18589	0.592946	+	0.805242i	$0.297965\pi$
$264$	0	0
$265$	1032.00	0.239227
$266$	0	0
$267$	2304.00	0.528099
$268$	0	0
$269$	3596.00	0.815063	0.407532	−	0.913191i	$-0.366390\pi$
$269$	3596.00	0.815063	0.407532	+	0.913191i	$0.366390\pi$
$270$	0	0
$271$	2424.00	0.543349	0.271674	−	0.962389i	$-0.412423\pi$
$271$	2424.00	0.543349	0.271674	+	0.962389i	$0.412423\pi$
$272$	0	0
$273$	42.0000	0.00931119
$274$	0	0
$275$	−2834.00	−0.621442
$276$	0	0
$277$	3634.00	0.788252	0.394126	−	0.919056i	$-0.371047\pi$
$277$	3634.00	0.788252	0.394126	+	0.919056i	$0.371047\pi$
$278$	0	0
$279$	2304.00	0.494397
$280$	0	0
$281$	−2970.00	−0.630517	−0.315259	−	0.949006i	$-0.602091\pi$
$281$	−2970.00	−0.630517	−0.315259	+	0.949006i	$0.602091\pi$
$282$	0	0
$283$	7028.00	1.47622	0.738112	−	0.674679i	$-0.235718\pi$
$283$	7028.00	1.47622	0.738112	+	0.674679i	$0.235718\pi$
$284$	0	0
$285$	912.000	0.189552
$286$	0	0
$287$	1120.00	0.230354
$288$	0	0
$289$	−3617.00	−0.736210
$290$	0	0
$291$	666.000	0.134164
$292$	0	0
$293$	−528.000	−0.105277	−0.0526384	−	0.998614i	$-0.516763\pi$
$293$	−528.000	−0.105277	−0.0526384	+	0.998614i	$0.516763\pi$
$294$	0	0
$295$	−1056.00	−0.208416
$296$	0	0
$297$	702.000	0.137152
$298$	0	0
$299$	228.000	0.0440989
$300$	0	0
$301$	1540.00	0.294897
$302$	0	0
$303$	−960.000	−0.182015
$304$	0	0
$305$	2424.00	0.455075
$306$	0	0
$307$	−4300.00	−0.799394	−0.399697	−	0.916647i	$-0.630885\pi$
$307$	−4300.00	−0.799394	−0.399697	+	0.916647i	$0.630885\pi$
$308$	0	0
$309$	1776.00	0.326968
$310$	0	0
$311$	4580.00	0.835074	0.417537	−	0.908660i	$-0.362893\pi$
$311$	4580.00	0.835074	0.417537	+	0.908660i	$0.362893\pi$
$312$	0	0
$313$	2266.00	0.409207	0.204604	−	0.978845i	$-0.434409\pi$
$313$	2266.00	0.409207	0.204604	+	0.978845i	$0.434409\pi$
$314$	0	0
$315$	252.000	0.0450749
$316$	0	0
$317$	−7926.00	−1.40432	−0.702159	−	0.712021i	$-0.747780\pi$
$317$	−7926.00	−1.40432	−0.702159	+	0.712021i	$0.747780\pi$
$318$	0	0
$319$	156.000	0.0273803
$320$	0	0
$321$	−5346.00	−0.929547
$322$	0	0
$323$	−2736.00	−0.471316
$324$	0	0
$325$	−218.000	−0.0372076
$326$	0	0
$327$	690.000	0.116688
$328$	0	0
$329$	−2156.00	−0.361289
$330$	0	0
$331$	4132.00	0.686149	0.343074	−	0.939308i	$-0.388532\pi$
$331$	4132.00	0.686149	0.343074	+	0.939308i	$0.388532\pi$
$332$	0	0
$333$	−774.000	−0.127372
$334$	0	0
$335$	2080.00	0.339231
$336$	0	0
$337$	−3622.00	−0.585469	−0.292734	−	0.956194i	$-0.594565\pi$
$337$	−3622.00	−0.585469	−0.292734	+	0.956194i	$0.594565\pi$
$338$	0	0
$339$	−5154.00	−0.825743
$340$	0	0
$341$	6656.00	1.05702
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	1368.00	0.213480
$346$	0	0
$347$	10254.0	1.58635	0.793175	−	0.608994i	$-0.208426\pi$
$347$	10254.0	1.58635	0.793175	+	0.608994i	$0.208426\pi$
$348$	0	0
$349$	−8178.00	−1.25432	−0.627161	−	0.778890i	$-0.715783\pi$
$349$	−8178.00	−1.25432	−0.627161	+	0.778890i	$0.715783\pi$
$350$	0	0
$351$	54.0000	0.00821170
$352$	0	0
$353$	9932.00	1.49753	0.748763	−	0.662837i	$-0.230648\pi$
$353$	9932.00	1.49753	0.748763	+	0.662837i	$0.230648\pi$
$354$	0	0
$355$	1144.00	0.171034
$356$	0	0
$357$	−756.000	−0.112078
$358$	0	0
$359$	−6546.00	−0.962353	−0.481176	−	0.876624i	$-0.659790\pi$
$359$	−6546.00	−0.962353	−0.481176	+	0.876624i	$0.659790\pi$
$360$	0	0
$361$	−1083.00	−0.157895
$362$	0	0
$363$	−1965.00	−0.284121
$364$	0	0
$365$	−2120.00	−0.304016
$366$	0	0
$367$	12176.0	1.73183	0.865916	−	0.500190i	$-0.166737\pi$
$367$	12176.0	1.73183	0.865916	+	0.500190i	$0.166737\pi$
$368$	0	0
$369$	1440.00	0.203153
$370$	0	0
$371$	1806.00	0.252730
$372$	0	0
$373$	−8210.00	−1.13967	−0.569836	−	0.821758i	$-0.692993\pi$
$373$	−8210.00	−1.13967	−0.569836	+	0.821758i	$0.692993\pi$
$374$	0	0
$375$	−2808.00	−0.386679
$376$	0	0
$377$	12.0000	0.00163934
$378$	0	0
$379$	−6908.00	−0.936254	−0.468127	−	0.883661i	$-0.655071\pi$
$379$	−6908.00	−0.936254	−0.468127	+	0.883661i	$0.655071\pi$
$380$	0	0
$381$	7332.00	0.985905
$382$	0	0
$383$	10248.0	1.36723	0.683614	−	0.729844i	$-0.260407\pi$
$383$	10248.0	1.36723	0.683614	+	0.729844i	$0.260407\pi$
$384$	0	0
$385$	728.000	0.0963697
$386$	0	0
$387$	1980.00	0.260075
$388$	0	0
$389$	−274.000	−0.0357130	−0.0178565	−	0.999841i	$-0.505684\pi$
$389$	−274.000	−0.0357130	−0.0178565	+	0.999841i	$0.505684\pi$
$390$	0	0
$391$	−4104.00	−0.530814
$392$	0	0
$393$	−5988.00	−0.768587
$394$	0	0
$395$	176.000	0.0224190
$396$	0	0
$397$	−10010.0	−1.26546	−0.632730	−	0.774373i	$-0.718066\pi$
$397$	−10010.0	−1.26546	−0.632730	+	0.774373i	$0.718066\pi$
$398$	0	0
$399$	1596.00	0.200250
$400$	0	0
$401$	1998.00	0.248816	0.124408	−	0.992231i	$-0.460297\pi$
$401$	1998.00	0.248816	0.124408	+	0.992231i	$0.460297\pi$
$402$	0	0
$403$	512.000	0.0632867
$404$	0	0
$405$	324.000	0.0397523
$406$	0	0
$407$	−2236.00	−0.272320
$408$	0	0
$409$	−12842.0	−1.55256	−0.776279	−	0.630390i	$-0.782895\pi$
$409$	−12842.0	−1.55256	−0.776279	+	0.630390i	$0.782895\pi$
$410$	0	0
$411$	−5238.00	−0.628641
$412$	0	0
$413$	−1848.00	−0.220180
$414$	0	0
$415$	−4048.00	−0.478816
$416$	0	0
$417$	−5412.00	−0.635556
$418$	0	0
$419$	10400.0	1.21259	0.606293	−	0.795242i	$-0.292656\pi$
$419$	10400.0	1.21259	0.606293	+	0.795242i	$0.292656\pi$
$420$	0	0
$421$	15586.0	1.80431	0.902156	−	0.431410i	$-0.141984\pi$
$421$	15586.0	1.80431	0.902156	+	0.431410i	$0.141984\pi$
$422$	0	0
$423$	−2772.00	−0.318627
$424$	0	0
$425$	3924.00	0.447863
$426$	0	0
$427$	4242.00	0.480761
$428$	0	0
$429$	156.000	0.0175565
$430$	0	0
$431$	−8066.00	−0.901451	−0.450726	−	0.892663i	$-0.648835\pi$
$431$	−8066.00	−0.901451	−0.450726	+	0.892663i	$0.648835\pi$
$432$	0	0
$433$	−5222.00	−0.579569	−0.289784	−	0.957092i	$-0.593584\pi$
$433$	−5222.00	−0.579569	−0.289784	+	0.957092i	$0.593584\pi$
$434$	0	0
$435$	72.0000	0.00793594
$436$	0	0
$437$	8664.00	0.948410
$438$	0	0
$439$	10920.0	1.18721	0.593603	−	0.804758i	$-0.297705\pi$
$439$	10920.0	1.18721	0.593603	+	0.804758i	$0.297705\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−1650.00	−0.176961	−0.0884807	−	0.996078i	$-0.528201\pi$
$443$	−1650.00	−0.176961	−0.0884807	+	0.996078i	$0.528201\pi$
$444$	0	0
$445$	3072.00	0.327251
$446$	0	0
$447$	−8442.00	−0.893273
$448$	0	0
$449$	11858.0	1.24636	0.623178	−	0.782080i	$-0.285841\pi$
$449$	11858.0	1.24636	0.623178	+	0.782080i	$0.285841\pi$
$450$	0	0
$451$	4160.00	0.434339
$452$	0	0
$453$	2376.00	0.246433
$454$	0	0
$455$	56.0000	0.00576994
$456$	0	0
$457$	−17894.0	−1.83161	−0.915805	−	0.401623i	$-0.868446\pi$
$457$	−17894.0	−1.83161	−0.915805	+	0.401623i	$0.868446\pi$
$458$	0	0
$459$	−972.000	−0.0988433
$460$	0	0
$461$	−2088.00	−0.210950	−0.105475	−	0.994422i	$-0.533636\pi$
$461$	−2088.00	−0.210950	−0.105475	+	0.994422i	$0.533636\pi$
$462$	0	0
$463$	−13532.0	−1.35828	−0.679142	−	0.734007i	$-0.737648\pi$
$463$	−13532.0	−1.35828	−0.679142	+	0.734007i	$0.737648\pi$
$464$	0	0
$465$	3072.00	0.306367
$466$	0	0
$467$	−6344.00	−0.628620	−0.314310	−	0.949320i	$-0.601773\pi$
$467$	−6344.00	−0.628620	−0.314310	+	0.949320i	$0.601773\pi$
$468$	0	0
$469$	3640.00	0.358379
$470$	0	0
$471$	−8334.00	−0.815309
$472$	0	0
$473$	5720.00	0.556038
$474$	0	0
$475$	−8284.00	−0.800202
$476$	0	0
$477$	2322.00	0.222887
$478$	0	0
$479$	−9948.00	−0.948926	−0.474463	−	0.880275i	$-0.657358\pi$
$479$	−9948.00	−0.948926	−0.474463	+	0.880275i	$0.657358\pi$
$480$	0	0
$481$	−172.000	−0.0163046
$482$	0	0
$483$	2394.00	0.225530
$484$	0	0
$485$	888.000	0.0831382
$486$	0	0
$487$	−20144.0	−1.87436	−0.937178	−	0.348850i	$-0.886572\pi$
$487$	−20144.0	−1.87436	−0.937178	+	0.348850i	$0.886572\pi$
$488$	0	0
$489$	−8640.00	−0.799007
$490$	0	0
$491$	−10530.0	−0.967846	−0.483923	−	0.875111i	$-0.660788\pi$
$491$	−10530.0	−0.967846	−0.483923	+	0.875111i	$0.660788\pi$
$492$	0	0
$493$	−216.000	−0.0197326
$494$	0	0
$495$	936.000	0.0849900
$496$	0	0
$497$	2002.00	0.180688
$498$	0	0
$499$	5548.00	0.497721	0.248860	−	0.968539i	$-0.419944\pi$
$499$	5548.00	0.497721	0.248860	+	0.968539i	$0.419944\pi$
$500$	0	0
$501$	3180.00	0.283577
$502$	0	0
$503$	2152.00	0.190761	0.0953807	−	0.995441i	$-0.469593\pi$
$503$	2152.00	0.190761	0.0953807	+	0.995441i	$0.469593\pi$
$504$	0	0
$505$	−1280.00	−0.112791
$506$	0	0
$507$	−6579.00	−0.576299
$508$	0	0
$509$	−19780.0	−1.72246	−0.861231	−	0.508214i	$-0.830306\pi$
$509$	−19780.0	−1.72246	−0.861231	+	0.508214i	$0.830306\pi$
$510$	0	0
$511$	−3710.00	−0.321176
$512$	0	0
$513$	2052.00	0.176604
$514$	0	0
$515$	2368.00	0.202615
$516$	0	0
$517$	−8008.00	−0.681221
$518$	0	0
$519$	−4320.00	−0.365370
$520$	0	0
$521$	23052.0	1.93844	0.969219	−	0.246199i	$-0.0791817\pi$
$521$	23052.0	1.93844	0.969219	+	0.246199i	$0.0791817\pi$
$522$	0	0
$523$	13364.0	1.11734	0.558668	−	0.829391i	$-0.311313\pi$
$523$	13364.0	1.11734	0.558668	+	0.829391i	$0.311313\pi$
$524$	0	0
$525$	−2289.00	−0.190286
$526$	0	0
$527$	−9216.00	−0.761775
$528$	0	0
$529$	829.000	0.0681351
$530$	0	0
$531$	−2376.00	−0.194180
$532$	0	0
$533$	320.000	0.0260051
$534$	0	0
$535$	−7128.00	−0.576019
$536$	0	0
$537$	−3630.00	−0.291706
$538$	0	0
$539$	1274.00	0.101809
$540$	0	0
$541$	−9766.00	−0.776106	−0.388053	−	0.921637i	$-0.626852\pi$
$541$	−9766.00	−0.776106	−0.388053	+	0.921637i	$0.626852\pi$
$542$	0	0
$543$	4854.00	0.383619
$544$	0	0
$545$	920.000	0.0723091
$546$	0	0
$547$	−3768.00	−0.294530	−0.147265	−	0.989097i	$-0.547047\pi$
$547$	−3768.00	−0.294530	−0.147265	+	0.989097i	$0.547047\pi$
$548$	0	0
$549$	5454.00	0.423991
$550$	0	0
$551$	456.000	0.0352564
$552$	0	0
$553$	308.000	0.0236844
$554$	0	0
$555$	−1032.00	−0.0789297
$556$	0	0
$557$	1618.00	0.123082	0.0615412	−	0.998105i	$-0.480398\pi$
$557$	1618.00	0.123082	0.0615412	+	0.998105i	$0.480398\pi$
$558$	0	0
$559$	440.000	0.0332916
$560$	0	0
$561$	−2808.00	−0.211326
$562$	0	0
$563$	5768.00	0.431780	0.215890	−	0.976418i	$-0.430735\pi$
$563$	5768.00	0.431780	0.215890	+	0.976418i	$0.430735\pi$
$564$	0	0
$565$	−6872.00	−0.511694
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	−8114.00	−0.597815	−0.298907	−	0.954282i	$-0.596622\pi$
$569$	−8114.00	−0.597815	−0.298907	+	0.954282i	$0.596622\pi$
$570$	0	0
$571$	−7816.00	−0.572836	−0.286418	−	0.958105i	$-0.592465\pi$
$571$	−7816.00	−0.572836	−0.286418	+	0.958105i	$0.592465\pi$
$572$	0	0
$573$	−12054.0	−0.878819
$574$	0	0
$575$	−12426.0	−0.901217
$576$	0	0
$577$	−18278.0	−1.31876	−0.659379	−	0.751811i	$-0.729181\pi$
$577$	−18278.0	−1.31876	−0.659379	+	0.751811i	$0.729181\pi$
$578$	0	0
$579$	10146.0	0.728244
$580$	0	0
$581$	−7084.00	−0.505841
$582$	0	0
$583$	6708.00	0.476530
$584$	0	0
$585$	72.0000	0.00508860
$586$	0	0
$587$	24960.0	1.75504	0.877521	−	0.479539i	$-0.159196\pi$
$587$	24960.0	1.75504	0.877521	+	0.479539i	$0.159196\pi$
$588$	0	0
$589$	19456.0	1.36107
$590$	0	0
$591$	−12906.0	−0.898277
$592$	0	0
$593$	10892.0	0.754268	0.377134	−	0.926159i	$-0.376910\pi$
$593$	10892.0	0.754268	0.377134	+	0.926159i	$0.376910\pi$
$594$	0	0
$595$	−1008.00	−0.0694521
$596$	0	0
$597$	7920.00	0.542955
$598$	0	0
$599$	20022.0	1.36574	0.682869	−	0.730541i	$-0.260732\pi$
$599$	20022.0	1.36574	0.682869	+	0.730541i	$0.260732\pi$
$600$	0	0
$601$	10370.0	0.703829	0.351914	−	0.936032i	$-0.385531\pi$
$601$	10370.0	0.703829	0.351914	+	0.936032i	$0.385531\pi$
$602$	0	0
$603$	4680.00	0.316060
$604$	0	0
$605$	−2620.00	−0.176063
$606$	0	0
$607$	8992.00	0.601275	0.300638	−	0.953738i	$-0.402801\pi$
$607$	8992.00	0.601275	0.300638	+	0.953738i	$0.402801\pi$
$608$	0	0
$609$	126.000	0.00838387
$610$	0	0
$611$	−616.000	−0.0407867
$612$	0	0
$613$	12318.0	0.811614	0.405807	−	0.913959i	$-0.366991\pi$
$613$	12318.0	0.811614	0.405807	+	0.913959i	$0.366991\pi$
$614$	0	0
$615$	1920.00	0.125889
$616$	0	0
$617$	−26026.0	−1.69816	−0.849082	−	0.528261i	$-0.822844\pi$
$617$	−26026.0	−1.69816	−0.849082	+	0.528261i	$0.822844\pi$
$618$	0	0
$619$	18332.0	1.19035	0.595174	−	0.803597i	$-0.297083\pi$
$619$	18332.0	1.19035	0.595174	+	0.803597i	$0.297083\pi$
$620$	0	0
$621$	3078.00	0.198898
$622$	0	0
$623$	5376.00	0.345722
$624$	0	0
$625$	9881.00	0.632384
$626$	0	0
$627$	5928.00	0.377578
$628$	0	0
$629$	3096.00	0.196257
$630$	0	0
$631$	−10572.0	−0.666980	−0.333490	−	0.942754i	$-0.608226\pi$
$631$	−10572.0	−0.666980	−0.333490	+	0.942754i	$0.608226\pi$
$632$	0	0
$633$	−10188.0	−0.639710
$634$	0	0
$635$	9776.00	0.610943
$636$	0	0
$637$	98.0000	0.00609561
$638$	0	0
$639$	2574.00	0.159352
$640$	0	0
$641$	24822.0	1.52950	0.764750	−	0.644327i	$-0.222862\pi$
$641$	24822.0	1.52950	0.764750	+	0.644327i	$0.222862\pi$
$642$	0	0
$643$	6620.00	0.406014	0.203007	−	0.979177i	$-0.434929\pi$
$643$	6620.00	0.406014	0.203007	+	0.979177i	$0.434929\pi$
$644$	0	0
$645$	2640.00	0.161163
$646$	0	0
$647$	−15956.0	−0.969544	−0.484772	−	0.874641i	$-0.661097\pi$
$647$	−15956.0	−0.969544	−0.484772	+	0.874641i	$0.661097\pi$
$648$	0	0
$649$	−6864.00	−0.415155
$650$	0	0
$651$	5376.00	0.323659
$652$	0	0
$653$	−30802.0	−1.84590	−0.922952	−	0.384915i	$-0.874231\pi$
$653$	−30802.0	−1.84590	−0.922952	+	0.384915i	$0.874231\pi$
$654$	0	0
$655$	−7984.00	−0.476276
$656$	0	0
$657$	−4770.00	−0.283250
$658$	0	0
$659$	12714.0	0.751543	0.375772	−	0.926712i	$-0.377378\pi$
$659$	12714.0	0.751543	0.375772	+	0.926712i	$0.377378\pi$
$660$	0	0
$661$	5822.00	0.342586	0.171293	−	0.985220i	$-0.445205\pi$
$661$	5822.00	0.342586	0.171293	+	0.985220i	$0.445205\pi$
$662$	0	0
$663$	−216.000	−0.0126527
$664$	0	0
$665$	2128.00	0.124091
$666$	0	0
$667$	684.000	0.0397070
$668$	0	0
$669$	−10440.0	−0.603339
$670$	0	0
$671$	15756.0	0.906488
$672$	0	0
$673$	28510.0	1.63296	0.816478	−	0.577376i	$-0.195923\pi$
$673$	28510.0	1.63296	0.816478	+	0.577376i	$0.195923\pi$
$674$	0	0
$675$	−2943.00	−0.167816
$676$	0	0
$677$	−21864.0	−1.24121	−0.620607	−	0.784122i	$-0.713114\pi$
$677$	−21864.0	−1.24121	−0.620607	+	0.784122i	$0.713114\pi$
$678$	0	0
$679$	1554.00	0.0878307
$680$	0	0
$681$	4512.00	0.253892
$682$	0	0
$683$	17862.0	1.00069	0.500344	−	0.865826i	$-0.333207\pi$
$683$	17862.0	1.00069	0.500344	+	0.865826i	$0.333207\pi$
$684$	0	0
$685$	−6984.00	−0.389555
$686$	0	0
$687$	15366.0	0.853347
$688$	0	0
$689$	516.000	0.0285313
$690$	0	0
$691$	6068.00	0.334063	0.167032	−	0.985952i	$-0.446582\pi$
$691$	6068.00	0.334063	0.167032	+	0.985952i	$0.446582\pi$
$692$	0	0
$693$	1638.00	0.0897871
$694$	0	0
$695$	−7216.00	−0.393840
$696$	0	0
$697$	−5760.00	−0.313021
$698$	0	0
$699$	4890.00	0.264602
$700$	0	0
$701$	7734.00	0.416703	0.208352	−	0.978054i	$-0.433190\pi$
$701$	7734.00	0.416703	0.208352	+	0.978054i	$0.433190\pi$
$702$	0	0
$703$	−6536.00	−0.350654
$704$	0	0
$705$	−3696.00	−0.197446
$706$	0	0
$707$	−2240.00	−0.119157
$708$	0	0
$709$	19902.0	1.05421	0.527105	−	0.849800i	$-0.323277\pi$
$709$	19902.0	1.05421	0.527105	+	0.849800i	$0.323277\pi$
$710$	0	0
$711$	396.000	0.0208877
$712$	0	0
$713$	29184.0	1.53289
$714$	0	0
$715$	208.000	0.0108794
$716$	0	0
$717$	7566.00	0.394083
$718$	0	0
$719$	−12160.0	−0.630725	−0.315363	−	0.948971i	$-0.602126\pi$
$719$	−12160.0	−0.630725	−0.315363	+	0.948971i	$0.602126\pi$
$720$	0	0
$721$	4144.00	0.214051
$722$	0	0
$723$	9066.00	0.466346
$724$	0	0
$725$	−654.000	−0.0335020
$726$	0	0
$727$	−3088.00	−0.157534	−0.0787672	−	0.996893i	$-0.525098\pi$
$727$	−3088.00	−0.157534	−0.0787672	+	0.996893i	$0.525098\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−7920.00	−0.400727
$732$	0	0
$733$	10178.0	0.512869	0.256435	−	0.966562i	$-0.417452\pi$
$733$	10178.0	0.512869	0.256435	+	0.966562i	$0.417452\pi$
$734$	0	0
$735$	588.000	0.0295084
$736$	0	0
$737$	13520.0	0.675734
$738$	0	0
$739$	5840.00	0.290701	0.145350	−	0.989380i	$-0.453569\pi$
$739$	5840.00	0.290701	0.145350	+	0.989380i	$0.453569\pi$
$740$	0	0
$741$	456.000	0.0226067
$742$	0	0
$743$	14414.0	0.711707	0.355854	−	0.934542i	$-0.384190\pi$
$743$	14414.0	0.711707	0.355854	+	0.934542i	$0.384190\pi$
$744$	0	0
$745$	−11256.0	−0.553541
$746$	0	0
$747$	−9108.00	−0.446110
$748$	0	0
$749$	−12474.0	−0.608531
$750$	0	0
$751$	12608.0	0.612613	0.306307	−	0.951933i	$-0.400907\pi$
$751$	12608.0	0.612613	0.306307	+	0.951933i	$0.400907\pi$
$752$	0	0
$753$	−10440.0	−0.505252
$754$	0	0
$755$	3168.00	0.152709
$756$	0	0
$757$	22606.0	1.08538	0.542688	−	0.839935i	$-0.317407\pi$
$757$	22606.0	1.08538	0.542688	+	0.839935i	$0.317407\pi$
$758$	0	0
$759$	8892.00	0.425243
$760$	0	0
$761$	5952.00	0.283521	0.141761	−	0.989901i	$-0.454724\pi$
$761$	5952.00	0.283521	0.141761	+	0.989901i	$0.454724\pi$
$762$	0	0
$763$	1610.00	0.0763905
$764$	0	0
$765$	−1296.00	−0.0612510
$766$	0	0
$767$	−528.000	−0.0248566
$768$	0	0
$769$	14138.0	0.662977	0.331489	−	0.943459i	$-0.392449\pi$
$769$	14138.0	0.662977	0.331489	+	0.943459i	$0.392449\pi$
$770$	0	0
$771$	10488.0	0.489904
$772$	0	0
$773$	12616.0	0.587019	0.293510	−	0.955956i	$-0.405177\pi$
$773$	12616.0	0.587019	0.293510	+	0.955956i	$0.405177\pi$
$774$	0	0
$775$	−27904.0	−1.29334
$776$	0	0
$777$	−1806.00	−0.0833847
$778$	0	0
$779$	12160.0	0.559278
$780$	0	0
$781$	7436.00	0.340693
$782$	0	0
$783$	162.000	0.00739388
$784$	0	0
$785$	−11112.0	−0.505228
$786$	0	0
$787$	12292.0	0.556750	0.278375	−	0.960472i	$-0.410204\pi$
$787$	12292.0	0.556750	0.278375	+	0.960472i	$0.410204\pi$
$788$	0	0
$789$	15174.0	0.684675
$790$	0	0
$791$	−12026.0	−0.540576
$792$	0	0
$793$	1212.00	0.0542741
$794$	0	0
$795$	3096.00	0.138118
$796$	0	0
$797$	−20784.0	−0.923723	−0.461861	−	0.886952i	$-0.652818\pi$
$797$	−20784.0	−0.923723	−0.461861	+	0.886952i	$0.652818\pi$
$798$	0	0
$799$	11088.0	0.490945
$800$	0	0
$801$	6912.00	0.304898
$802$	0	0
$803$	−13780.0	−0.605586
$804$	0	0
$805$	3192.00	0.139756
$806$	0	0
$807$	10788.0	0.470577
$808$	0	0
$809$	−13574.0	−0.589909	−0.294955	−	0.955511i	$-0.595305\pi$
$809$	−13574.0	−0.589909	−0.294955	+	0.955511i	$0.595305\pi$
$810$	0	0
$811$	−32308.0	−1.39887	−0.699437	−	0.714694i	$-0.746566\pi$
$811$	−32308.0	−1.39887	−0.699437	+	0.714694i	$0.746566\pi$
$812$	0	0
$813$	7272.00	0.313703
$814$	0	0
$815$	−11520.0	−0.495126
$816$	0	0
$817$	16720.0	0.715984
$818$	0	0
$819$	126.000	0.00537582
$820$	0	0
$821$	−2390.00	−0.101598	−0.0507988	−	0.998709i	$-0.516177\pi$
$821$	−2390.00	−0.101598	−0.0507988	+	0.998709i	$0.516177\pi$
$822$	0	0
$823$	−28020.0	−1.18677	−0.593387	−	0.804917i	$-0.702210\pi$
$823$	−28020.0	−1.18677	−0.593387	+	0.804917i	$0.702210\pi$
$824$	0	0
$825$	−8502.00	−0.358790
$826$	0	0
$827$	−34006.0	−1.42987	−0.714936	−	0.699190i	$-0.753544\pi$
$827$	−34006.0	−1.42987	−0.714936	+	0.699190i	$0.753544\pi$
$828$	0	0
$829$	−6310.00	−0.264361	−0.132181	−	0.991226i	$-0.542198\pi$
$829$	−6310.00	−0.264361	−0.132181	+	0.991226i	$0.542198\pi$
$830$	0	0
$831$	10902.0	0.455098
$832$	0	0
$833$	−1764.00	−0.0733721
$834$	0	0
$835$	4240.00	0.175726
$836$	0	0
$837$	6912.00	0.285440
$838$	0	0
$839$	16524.0	0.679943	0.339971	−	0.940436i	$-0.389583\pi$
$839$	16524.0	0.679943	0.339971	+	0.940436i	$0.389583\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	0	0
$843$	−8910.00	−0.364029
$844$	0	0
$845$	−8772.00	−0.357119
$846$	0	0
$847$	−4585.00	−0.186001
$848$	0	0
$849$	21084.0	0.852298
$850$	0	0
$851$	−9804.00	−0.394920
$852$	0	0
$853$	36662.0	1.47161	0.735805	−	0.677194i	$-0.236804\pi$
$853$	36662.0	1.47161	0.735805	+	0.677194i	$0.236804\pi$
$854$	0	0
$855$	2736.00	0.109438
$856$	0	0
$857$	−42000.0	−1.67409	−0.837044	−	0.547136i	$-0.815718\pi$
$857$	−42000.0	−1.67409	−0.837044	+	0.547136i	$0.815718\pi$
$858$	0	0
$859$	−28388.0	−1.12757	−0.563787	−	0.825920i	$-0.690656\pi$
$859$	−28388.0	−1.12757	−0.563787	+	0.825920i	$0.690656\pi$
$860$	0	0
$861$	3360.00	0.132995
$862$	0	0
$863$	−23918.0	−0.943428	−0.471714	−	0.881752i	$-0.656364\pi$
$863$	−23918.0	−0.943428	−0.471714	+	0.881752i	$0.656364\pi$
$864$	0	0
$865$	−5760.00	−0.226411
$866$	0	0
$867$	−10851.0	−0.425051
$868$	0	0
$869$	1144.00	0.0446577
$870$	0	0
$871$	1040.00	0.0404582
$872$	0	0
$873$	1998.00	0.0774594
$874$	0	0
$875$	−6552.00	−0.253141
$876$	0	0
$877$	−2346.00	−0.0903293	−0.0451646	−	0.998980i	$-0.514381\pi$
$877$	−2346.00	−0.0903293	−0.0451646	+	0.998980i	$0.514381\pi$
$878$	0	0
$879$	−1584.00	−0.0607816
$880$	0	0
$881$	51876.0	1.98382	0.991911	−	0.126937i	$-0.0405145\pi$
$881$	51876.0	1.98382	0.991911	+	0.126937i	$0.0405145\pi$
$882$	0	0
$883$	11372.0	0.433407	0.216703	−	0.976237i	$-0.430470\pi$
$883$	11372.0	0.433407	0.216703	+	0.976237i	$0.430470\pi$
$884$	0	0
$885$	−3168.00	−0.120329
$886$	0	0
$887$	−38764.0	−1.46738	−0.733691	−	0.679483i	$-0.762204\pi$
$887$	−38764.0	−1.46738	−0.733691	+	0.679483i	$0.762204\pi$
$888$	0	0
$889$	17108.0	0.645426
$890$	0	0
$891$	2106.00	0.0791848
$892$	0	0
$893$	−23408.0	−0.877177
$894$	0	0
$895$	−4840.00	−0.180764
$896$	0	0
$897$	684.000	0.0254605
$898$	0	0
$899$	1536.00	0.0569838
$900$	0	0
$901$	−9288.00	−0.343427
$902$	0	0
$903$	4620.00	0.170259
$904$	0	0
$905$	6472.00	0.237720
$906$	0	0
$907$	11212.0	0.410461	0.205231	−	0.978714i	$-0.434206\pi$
$907$	11212.0	0.410461	0.205231	+	0.978714i	$0.434206\pi$
$908$	0	0
$909$	−2880.00	−0.105086
$910$	0	0
$911$	−1458.00	−0.0530249	−0.0265125	−	0.999648i	$-0.508440\pi$
$911$	−1458.00	−0.0530249	−0.0265125	+	0.999648i	$0.508440\pi$
$912$	0	0
$913$	−26312.0	−0.953779
$914$	0	0
$915$	7272.00	0.262738
$916$	0	0
$917$	−13972.0	−0.503158
$918$	0	0
$919$	38608.0	1.38581	0.692906	−	0.721028i	$-0.256330\pi$
$919$	38608.0	1.38581	0.692906	+	0.721028i	$0.256330\pi$
$920$	0	0
$921$	−12900.0	−0.461530
$922$	0	0
$923$	572.000	0.0203983
$924$	0	0
$925$	9374.00	0.333206
$926$	0	0
$927$	5328.00	0.188775
$928$	0	0
$929$	7376.00	0.260494	0.130247	−	0.991482i	$-0.458423\pi$
$929$	7376.00	0.260494	0.130247	+	0.991482i	$0.458423\pi$
$930$	0	0
$931$	3724.00	0.131095
$932$	0	0
$933$	13740.0	0.482130
$934$	0	0
$935$	−3744.00	−0.130954
$936$	0	0
$937$	−24878.0	−0.867373	−0.433687	−	0.901064i	$-0.642788\pi$
$937$	−24878.0	−0.867373	−0.433687	+	0.901064i	$0.642788\pi$
$938$	0	0
$939$	6798.00	0.236256
$940$	0	0
$941$	−26148.0	−0.905845	−0.452923	−	0.891550i	$-0.649619\pi$
$941$	−26148.0	−0.905845	−0.452923	+	0.891550i	$0.649619\pi$
$942$	0	0
$943$	18240.0	0.629879
$944$	0	0
$945$	756.000	0.0260240
$946$	0	0
$947$	−18350.0	−0.629667	−0.314834	−	0.949147i	$-0.601949\pi$
$947$	−18350.0	−0.629667	−0.314834	+	0.949147i	$0.601949\pi$
$948$	0	0
$949$	−1060.00	−0.0362582
$950$	0	0
$951$	−23778.0	−0.810783
$952$	0	0
$953$	−57510.0	−1.95481	−0.977404	−	0.211381i	$-0.932204\pi$
$953$	−57510.0	−1.95481	−0.977404	+	0.211381i	$0.932204\pi$
$954$	0	0
$955$	−16072.0	−0.544584
$956$	0	0
$957$	468.000	0.0158080
$958$	0	0
$959$	−12222.0	−0.411542
$960$	0	0
$961$	35745.0	1.19986
$962$	0	0
$963$	−16038.0	−0.536674
$964$	0	0
$965$	13528.0	0.451276
$966$	0	0
$967$	33364.0	1.10953	0.554764	−	0.832008i	$-0.312808\pi$
$967$	33364.0	1.10953	0.554764	+	0.832008i	$0.312808\pi$
$968$	0	0
$969$	−8208.00	−0.272114
$970$	0	0
$971$	−26892.0	−0.888780	−0.444390	−	0.895833i	$-0.646580\pi$
$971$	−26892.0	−0.888780	−0.444390	+	0.895833i	$0.646580\pi$
$972$	0	0
$973$	−12628.0	−0.416069
$974$	0	0
$975$	−654.000	−0.0214818
$976$	0	0
$977$	−31970.0	−1.04689	−0.523445	−	0.852060i	$-0.675353\pi$
$977$	−31970.0	−1.04689	−0.523445	+	0.852060i	$0.675353\pi$
$978$	0	0
$979$	19968.0	0.651869
$980$	0	0
$981$	2070.00	0.0673700
$982$	0	0
$983$	19728.0	0.640107	0.320054	−	0.947399i	$-0.396299\pi$
$983$	19728.0	0.640107	0.320054	+	0.947399i	$0.396299\pi$
$984$	0	0
$985$	−17208.0	−0.556642
$986$	0	0
$987$	−6468.00	−0.208590
$988$	0	0
$989$	25080.0	0.806368
$990$	0	0
$991$	21400.0	0.685967	0.342984	−	0.939341i	$-0.388562\pi$
$991$	21400.0	0.685967	0.342984	+	0.939341i	$0.388562\pi$
$992$	0	0
$993$	12396.0	0.396148
$994$	0	0
$995$	10560.0	0.336457
$996$	0	0
$997$	−4754.00	−0.151014	−0.0755069	−	0.997145i	$-0.524057\pi$
$997$	−4754.00	−0.151014	−0.0755069	+	0.997145i	$0.524057\pi$
$998$	0	0
$999$	−2322.00	−0.0735384

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.4.a.j.1.1		1
3.2	odd	2		1008.4.a.i.1.1		1
4.3	odd	2		168.4.a.c.1.1	✓	1
7.6	odd	2		2352.4.a.h.1.1		1
8.3	odd	2		1344.4.a.s.1.1		1
8.5	even	2		1344.4.a.e.1.1		1
12.11	even	2		504.4.a.b.1.1		1
28.27	even	2		1176.4.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.a.c.1.1	✓	1	4.3	odd	2
336.4.a.j.1.1		1	1.1	even	1	trivial
504.4.a.b.1.1		1	12.11	even	2
1008.4.a.i.1.1		1	3.2	odd	2
1176.4.a.j.1.1		1	28.27	even	2
1344.4.a.e.1.1		1	8.5	even	2
1344.4.a.s.1.1		1	8.3	odd	2
2352.4.a.h.1.1		1	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	336.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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