LMFDB - Embedded newform 624.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(1,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, 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("624.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.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:	$36.8171918436$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} -12.0000 q^{5} -2.00000 q^{7} +9.00000 q^{9} +36.0000 q^{11} +13.0000 q^{13} -36.0000 q^{15} -78.0000 q^{17} -74.0000 q^{19} -6.00000 q^{21} +96.0000 q^{23} +19.0000 q^{25} +27.0000 q^{27} +18.0000 q^{29} +214.000 q^{31} +108.000 q^{33} +24.0000 q^{35} -286.000 q^{37} +39.0000 q^{39} -384.000 q^{41} -524.000 q^{43} -108.000 q^{45} -300.000 q^{47} -339.000 q^{49} -234.000 q^{51} +558.000 q^{53} -432.000 q^{55} -222.000 q^{57} -576.000 q^{59} +74.0000 q^{61} -18.0000 q^{63} -156.000 q^{65} -38.0000 q^{67} +288.000 q^{69} +456.000 q^{71} -682.000 q^{73} +57.0000 q^{75} -72.0000 q^{77} -704.000 q^{79} +81.0000 q^{81} +888.000 q^{83} +936.000 q^{85} +54.0000 q^{87} -1020.00 q^{89} -26.0000 q^{91} +642.000 q^{93} +888.000 q^{95} +110.000 q^{97} +324.000 q^{99} +O(q^{100})$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−12.0000	−1.07331	−0.536656	−	0.843801i	$-0.680313\pi$
$5$	−12.0000	−1.07331	−0.536656	+	0.843801i	$0.680313\pi$
$6$	0	0
$7$	−2.00000	−0.107990	−0.0539949	−	0.998541i	$-0.517195\pi$
$7$	−2.00000	−0.107990	−0.0539949	+	0.998541i	$0.517195\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	36.0000	0.986764	0.493382	−	0.869813i	$-0.335760\pi$
$11$	36.0000	0.986764	0.493382	+	0.869813i	$0.335760\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−36.0000	−0.619677
$16$	0	0
$17$	−78.0000	−1.11281	−0.556405	−	0.830911i	$-0.687820\pi$
$17$	−78.0000	−1.11281	−0.556405	+	0.830911i	$0.687820\pi$
$18$	0	0
$19$	−74.0000	−0.893514	−0.446757	−	0.894655i	$-0.647421\pi$
$19$	−74.0000	−0.893514	−0.446757	+	0.894655i	$0.647421\pi$
$20$	0	0
$21$	−6.00000	−0.0623480
$22$	0	0
$23$	96.0000	0.870321	0.435161	−	0.900353i	$-0.356692\pi$
$23$	96.0000	0.870321	0.435161	+	0.900353i	$0.356692\pi$
$24$	0	0
$25$	19.0000	0.152000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	18.0000	0.115259	0.0576296	−	0.998338i	$-0.481646\pi$
$29$	18.0000	0.115259	0.0576296	+	0.998338i	$0.481646\pi$
$30$	0	0
$31$	214.000	1.23986	0.619928	−	0.784659i	$-0.287162\pi$
$31$	214.000	1.23986	0.619928	+	0.784659i	$0.287162\pi$
$32$	0	0
$33$	108.000	0.569709
$34$	0	0
$35$	24.0000	0.115907
$36$	0	0
$37$	−286.000	−1.27076	−0.635380	−	0.772200i	$-0.719156\pi$
$37$	−286.000	−1.27076	−0.635380	+	0.772200i	$0.719156\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	−384.000	−1.46270	−0.731350	−	0.682002i	$-0.761110\pi$
$41$	−384.000	−1.46270	−0.731350	+	0.682002i	$0.761110\pi$
$42$	0	0
$43$	−524.000	−1.85835	−0.929177	−	0.369634i	$-0.879483\pi$
$43$	−524.000	−1.85835	−0.929177	+	0.369634i	$0.879483\pi$
$44$	0	0
$45$	−108.000	−0.357771
$46$	0	0
$47$	−300.000	−0.931053	−0.465527	−	0.885034i	$-0.654135\pi$
$47$	−300.000	−0.931053	−0.465527	+	0.885034i	$0.654135\pi$
$48$	0	0
$49$	−339.000	−0.988338
$50$	0	0
$51$	−234.000	−0.642481
$52$	0	0
$53$	558.000	1.44617	0.723087	−	0.690757i	$-0.242723\pi$
$53$	558.000	1.44617	0.723087	+	0.690757i	$0.242723\pi$
$54$	0	0
$55$	−432.000	−1.05911
$56$	0	0
$57$	−222.000	−0.515870
$58$	0	0
$59$	−576.000	−1.27100	−0.635498	−	0.772102i	$-0.719205\pi$
$59$	−576.000	−1.27100	−0.635498	+	0.772102i	$0.719205\pi$
$60$	0	0
$61$	74.0000	0.155323	0.0776617	−	0.996980i	$-0.475255\pi$
$61$	74.0000	0.155323	0.0776617	+	0.996980i	$0.475255\pi$
$62$	0	0
$63$	−18.0000	−0.0359966
$64$	0	0
$65$	−156.000	−0.297683
$66$	0	0
$67$	−38.0000	−0.0692901	−0.0346451	−	0.999400i	$-0.511030\pi$
$67$	−38.0000	−0.0692901	−0.0346451	+	0.999400i	$0.511030\pi$
$68$	0	0
$69$	288.000	0.502480
$70$	0	0
$71$	456.000	0.762215	0.381107	−	0.924531i	$-0.375543\pi$
$71$	456.000	0.762215	0.381107	+	0.924531i	$0.375543\pi$
$72$	0	0
$73$	−682.000	−1.09345	−0.546726	−	0.837311i	$-0.684126\pi$
$73$	−682.000	−1.09345	−0.546726	+	0.837311i	$0.684126\pi$
$74$	0	0
$75$	57.0000	0.0877572
$76$	0	0
$77$	−72.0000	−0.106561
$78$	0	0
$79$	−704.000	−1.00261	−0.501305	−	0.865271i	$-0.667147\pi$
$79$	−704.000	−1.00261	−0.501305	+	0.865271i	$0.667147\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	888.000	1.17435	0.587173	−	0.809462i	$-0.300241\pi$
$83$	888.000	1.17435	0.587173	+	0.809462i	$0.300241\pi$
$84$	0	0
$85$	936.000	1.19439
$86$	0	0
$87$	54.0000	0.0665449
$88$	0	0
$89$	−1020.00	−1.21483	−0.607415	−	0.794385i	$-0.707793\pi$
$89$	−1020.00	−1.21483	−0.607415	+	0.794385i	$0.707793\pi$
$90$	0	0
$91$	−26.0000	−0.0299510
$92$	0	0
$93$	642.000	0.715831
$94$	0	0
$95$	888.000	0.959020
$96$	0	0
$97$	110.000	0.115142	0.0575712	−	0.998341i	$-0.481664\pi$
$97$	110.000	0.115142	0.0575712	+	0.998341i	$0.481664\pi$
$98$	0	0
$99$	324.000	0.328921
$100$	0	0
$101$	−990.000	−0.975333	−0.487667	−	0.873030i	$-0.662152\pi$
$101$	−990.000	−0.975333	−0.487667	+	0.873030i	$0.662152\pi$
$102$	0	0
$103$	−1208.00	−1.15561	−0.577805	−	0.816175i	$-0.696090\pi$
$103$	−1208.00	−1.15561	−0.577805	+	0.816175i	$0.696090\pi$
$104$	0	0
$105$	72.0000	0.0669189
$106$	0	0
$107$	−996.000	−0.899878	−0.449939	−	0.893059i	$-0.648554\pi$
$107$	−996.000	−0.899878	−0.449939	+	0.893059i	$0.648554\pi$
$108$	0	0
$109$	−1402.00	−1.23199	−0.615997	−	0.787749i	$-0.711246\pi$
$109$	−1402.00	−1.23199	−0.615997	+	0.787749i	$0.711246\pi$
$110$	0	0
$111$	−858.000	−0.733673
$112$	0	0
$113$	1926.00	1.60339	0.801694	−	0.597735i	$-0.203932\pi$
$113$	1926.00	1.60339	0.801694	+	0.597735i	$0.203932\pi$
$114$	0	0
$115$	−1152.00	−0.934127
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	156.000	0.120172
$120$	0	0
$121$	−35.0000	−0.0262960
$122$	0	0
$123$	−1152.00	−0.844491
$124$	0	0
$125$	1272.00	0.910169
$126$	0	0
$127$	988.000	0.690321	0.345161	−	0.938544i	$-0.387824\pi$
$127$	988.000	0.690321	0.345161	+	0.938544i	$0.387824\pi$
$128$	0	0
$129$	−1572.00	−1.07292
$130$	0	0
$131$	2100.00	1.40059	0.700297	−	0.713851i	$-0.253051\pi$
$131$	2100.00	1.40059	0.700297	+	0.713851i	$0.253051\pi$
$132$	0	0
$133$	148.000	0.0964904
$134$	0	0
$135$	−324.000	−0.206559
$136$	0	0
$137$	−2496.00	−1.55655	−0.778276	−	0.627922i	$-0.783906\pi$
$137$	−2496.00	−1.55655	−0.778276	+	0.627922i	$0.783906\pi$
$138$	0	0
$139$	2464.00	1.50355	0.751776	−	0.659418i	$-0.229197\pi$
$139$	2464.00	1.50355	0.751776	+	0.659418i	$0.229197\pi$
$140$	0	0
$141$	−900.000	−0.537544
$142$	0	0
$143$	468.000	0.273679
$144$	0	0
$145$	−216.000	−0.123709
$146$	0	0
$147$	−1017.00	−0.570617
$148$	0	0
$149$	216.000	0.118761	0.0593806	−	0.998235i	$-0.481087\pi$
$149$	216.000	0.118761	0.0593806	+	0.998235i	$0.481087\pi$
$150$	0	0
$151$	898.000	0.483962	0.241981	−	0.970281i	$-0.422203\pi$
$151$	898.000	0.483962	0.241981	+	0.970281i	$0.422203\pi$
$152$	0	0
$153$	−702.000	−0.370937
$154$	0	0
$155$	−2568.00	−1.33075
$156$	0	0
$157$	−1510.00	−0.767587	−0.383793	−	0.923419i	$-0.625383\pi$
$157$	−1510.00	−0.767587	−0.383793	+	0.923419i	$0.625383\pi$
$158$	0	0
$159$	1674.00	0.834949
$160$	0	0
$161$	−192.000	−0.0939858
$162$	0	0
$163$	394.000	0.189328	0.0946640	−	0.995509i	$-0.469822\pi$
$163$	394.000	0.189328	0.0946640	+	0.995509i	$0.469822\pi$
$164$	0	0
$165$	−1296.00	−0.611476
$166$	0	0
$167$	−84.0000	−0.0389228	−0.0194614	−	0.999811i	$-0.506195\pi$
$167$	−84.0000	−0.0389228	−0.0194614	+	0.999811i	$0.506195\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−666.000	−0.297838
$172$	0	0
$173$	1194.00	0.524729	0.262365	−	0.964969i	$-0.415498\pi$
$173$	1194.00	0.524729	0.262365	+	0.964969i	$0.415498\pi$
$174$	0	0
$175$	−38.0000	−0.0164145
$176$	0	0
$177$	−1728.00	−0.733810
$178$	0	0
$179$	−3156.00	−1.31782	−0.658912	−	0.752220i	$-0.728983\pi$
$179$	−3156.00	−1.31782	−0.658912	+	0.752220i	$0.728983\pi$
$180$	0	0
$181$	−1078.00	−0.442691	−0.221346	−	0.975195i	$-0.571045\pi$
$181$	−1078.00	−0.442691	−0.221346	+	0.975195i	$0.571045\pi$
$182$	0	0
$183$	222.000	0.0896760
$184$	0	0
$185$	3432.00	1.36392
$186$	0	0
$187$	−2808.00	−1.09808
$188$	0	0
$189$	−54.0000	−0.0207827
$190$	0	0
$191$	−3192.00	−1.20924	−0.604620	−	0.796514i	$-0.706675\pi$
$191$	−3192.00	−1.20924	−0.604620	+	0.796514i	$0.706675\pi$
$192$	0	0
$193$	722.000	0.269278	0.134639	−	0.990895i	$-0.457012\pi$
$193$	722.000	0.269278	0.134639	+	0.990895i	$0.457012\pi$
$194$	0	0
$195$	−468.000	−0.171868
$196$	0	0
$197$	2796.00	1.01120	0.505601	−	0.862767i	$-0.331271\pi$
$197$	2796.00	1.01120	0.505601	+	0.862767i	$0.331271\pi$
$198$	0	0
$199$	340.000	0.121115	0.0605577	−	0.998165i	$-0.480712\pi$
$199$	340.000	0.121115	0.0605577	+	0.998165i	$0.480712\pi$
$200$	0	0
$201$	−114.000	−0.0400047
$202$	0	0
$203$	−36.0000	−0.0124468
$204$	0	0
$205$	4608.00	1.56994
$206$	0	0
$207$	864.000	0.290107
$208$	0	0
$209$	−2664.00	−0.881688
$210$	0	0
$211$	1924.00	0.627742	0.313871	−	0.949466i	$-0.398374\pi$
$211$	1924.00	0.627742	0.313871	+	0.949466i	$0.398374\pi$
$212$	0	0
$213$	1368.00	0.440065
$214$	0	0
$215$	6288.00	1.99460
$216$	0	0
$217$	−428.000	−0.133892
$218$	0	0
$219$	−2046.00	−0.631305
$220$	0	0
$221$	−1014.00	−0.308638
$222$	0	0
$223$	−5042.00	−1.51407	−0.757034	−	0.653375i	$-0.773352\pi$
$223$	−5042.00	−1.51407	−0.757034	+	0.653375i	$0.773352\pi$
$224$	0	0
$225$	171.000	0.0506667
$226$	0	0
$227$	2676.00	0.782433	0.391217	−	0.920299i	$-0.372054\pi$
$227$	2676.00	0.782433	0.391217	+	0.920299i	$0.372054\pi$
$228$	0	0
$229$	−2410.00	−0.695447	−0.347723	−	0.937597i	$-0.613045\pi$
$229$	−2410.00	−0.695447	−0.347723	+	0.937597i	$0.613045\pi$
$230$	0	0
$231$	−216.000	−0.0615228
$232$	0	0
$233$	3726.00	1.04763	0.523816	−	0.851831i	$-0.324508\pi$
$233$	3726.00	1.04763	0.523816	+	0.851831i	$0.324508\pi$
$234$	0	0
$235$	3600.00	0.999311
$236$	0	0
$237$	−2112.00	−0.578857
$238$	0	0
$239$	−1248.00	−0.337767	−0.168884	−	0.985636i	$-0.554016\pi$
$239$	−1248.00	−0.337767	−0.168884	+	0.985636i	$0.554016\pi$
$240$	0	0
$241$	−4210.00	−1.12527	−0.562635	−	0.826706i	$-0.690212\pi$
$241$	−4210.00	−1.12527	−0.562635	+	0.826706i	$0.690212\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	4068.00	1.06080
$246$	0	0
$247$	−962.000	−0.247816
$248$	0	0
$249$	2664.00	0.678009
$250$	0	0
$251$	7692.00	1.93432	0.967161	−	0.254165i	$-0.0818007\pi$
$251$	7692.00	1.93432	0.967161	+	0.254165i	$0.0818007\pi$
$252$	0	0
$253$	3456.00	0.858802
$254$	0	0
$255$	2808.00	0.689583
$256$	0	0
$257$	1326.00	0.321843	0.160921	−	0.986967i	$-0.448553\pi$
$257$	1326.00	0.321843	0.160921	+	0.986967i	$0.448553\pi$
$258$	0	0
$259$	572.000	0.137229
$260$	0	0
$261$	162.000	0.0384197
$262$	0	0
$263$	6048.00	1.41801	0.709003	−	0.705205i	$-0.249145\pi$
$263$	6048.00	1.41801	0.709003	+	0.705205i	$0.249145\pi$
$264$	0	0
$265$	−6696.00	−1.55220
$266$	0	0
$267$	−3060.00	−0.701382
$268$	0	0
$269$	6474.00	1.46739	0.733693	−	0.679481i	$-0.237795\pi$
$269$	6474.00	1.46739	0.733693	+	0.679481i	$0.237795\pi$
$270$	0	0
$271$	−5978.00	−1.33999	−0.669996	−	0.742365i	$-0.733704\pi$
$271$	−5978.00	−1.33999	−0.669996	+	0.742365i	$0.733704\pi$
$272$	0	0
$273$	−78.0000	−0.0172922
$274$	0	0
$275$	684.000	0.149988
$276$	0	0
$277$	8750.00	1.89797	0.948983	−	0.315327i	$-0.102114\pi$
$277$	8750.00	1.89797	0.948983	+	0.315327i	$0.102114\pi$
$278$	0	0
$279$	1926.00	0.413285
$280$	0	0
$281$	8976.00	1.90556	0.952782	−	0.303656i	$-0.0982075\pi$
$281$	8976.00	1.90556	0.952782	+	0.303656i	$0.0982075\pi$
$282$	0	0
$283$	592.000	0.124349	0.0621745	−	0.998065i	$-0.480196\pi$
$283$	592.000	0.124349	0.0621745	+	0.998065i	$0.480196\pi$
$284$	0	0
$285$	2664.00	0.553690
$286$	0	0
$287$	768.000	0.157957
$288$	0	0
$289$	1171.00	0.238347
$290$	0	0
$291$	330.000	0.0664775
$292$	0	0
$293$	−4608.00	−0.918779	−0.459389	−	0.888235i	$-0.651932\pi$
$293$	−4608.00	−0.918779	−0.459389	+	0.888235i	$0.651932\pi$
$294$	0	0
$295$	6912.00	1.36418
$296$	0	0
$297$	972.000	0.189903
$298$	0	0
$299$	1248.00	0.241384
$300$	0	0
$301$	1048.00	0.200683
$302$	0	0
$303$	−2970.00	−0.563109
$304$	0	0
$305$	−888.000	−0.166711
$306$	0	0
$307$	3166.00	0.588577	0.294289	−	0.955717i	$-0.404917\pi$
$307$	3166.00	0.588577	0.294289	+	0.955717i	$0.404917\pi$
$308$	0	0
$309$	−3624.00	−0.667191
$310$	0	0
$311$	−2472.00	−0.450721	−0.225361	−	0.974275i	$-0.572356\pi$
$311$	−2472.00	−0.450721	−0.225361	+	0.974275i	$0.572356\pi$
$312$	0	0
$313$	−3094.00	−0.558732	−0.279366	−	0.960185i	$-0.590124\pi$
$313$	−3094.00	−0.558732	−0.279366	+	0.960185i	$0.590124\pi$
$314$	0	0
$315$	216.000	0.0386356
$316$	0	0
$317$	2316.00	0.410345	0.205173	−	0.978726i	$-0.434224\pi$
$317$	2316.00	0.410345	0.205173	+	0.978726i	$0.434224\pi$
$318$	0	0
$319$	648.000	0.113734
$320$	0	0
$321$	−2988.00	−0.519545
$322$	0	0
$323$	5772.00	0.994312
$324$	0	0
$325$	247.000	0.0421572
$326$	0	0
$327$	−4206.00	−0.711292
$328$	0	0
$329$	600.000	0.100544
$330$	0	0
$331$	4426.00	0.734970	0.367485	−	0.930030i	$-0.380219\pi$
$331$	4426.00	0.734970	0.367485	+	0.930030i	$0.380219\pi$
$332$	0	0
$333$	−2574.00	−0.423587
$334$	0	0
$335$	456.000	0.0743700
$336$	0	0
$337$	866.000	0.139982	0.0699911	−	0.997548i	$-0.477703\pi$
$337$	866.000	0.139982	0.0699911	+	0.997548i	$0.477703\pi$
$338$	0	0
$339$	5778.00	0.925716
$340$	0	0
$341$	7704.00	1.22345
$342$	0	0
$343$	1364.00	0.214720
$344$	0	0
$345$	−3456.00	−0.539318
$346$	0	0
$347$	2556.00	0.395427	0.197714	−	0.980260i	$-0.436648\pi$
$347$	2556.00	0.395427	0.197714	+	0.980260i	$0.436648\pi$
$348$	0	0
$349$	−11014.0	−1.68930	−0.844650	−	0.535318i	$-0.820192\pi$
$349$	−11014.0	−1.68930	−0.844650	+	0.535318i	$0.820192\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	−9720.00	−1.46556	−0.732781	−	0.680465i	$-0.761778\pi$
$353$	−9720.00	−1.46556	−0.732781	+	0.680465i	$0.761778\pi$
$354$	0	0
$355$	−5472.00	−0.818095
$356$	0	0
$357$	468.000	0.0693815
$358$	0	0
$359$	2988.00	0.439277	0.219639	−	0.975581i	$-0.429512\pi$
$359$	2988.00	0.439277	0.219639	+	0.975581i	$0.429512\pi$
$360$	0	0
$361$	−1383.00	−0.201633
$362$	0	0
$363$	−105.000	−0.0151820
$364$	0	0
$365$	8184.00	1.17362
$366$	0	0
$367$	2068.00	0.294138	0.147069	−	0.989126i	$-0.453016\pi$
$367$	2068.00	0.294138	0.147069	+	0.989126i	$0.453016\pi$
$368$	0	0
$369$	−3456.00	−0.487567
$370$	0	0
$371$	−1116.00	−0.156172
$372$	0	0
$373$	902.000	0.125211	0.0626056	−	0.998038i	$-0.480059\pi$
$373$	902.000	0.125211	0.0626056	+	0.998038i	$0.480059\pi$
$374$	0	0
$375$	3816.00	0.525486
$376$	0	0
$377$	234.000	0.0319671
$378$	0	0
$379$	−12818.0	−1.73725	−0.868623	−	0.495473i	$-0.834995\pi$
$379$	−12818.0	−1.73725	−0.868623	+	0.495473i	$0.834995\pi$
$380$	0	0
$381$	2964.00	0.398557
$382$	0	0
$383$	1332.00	0.177708	0.0888538	−	0.996045i	$-0.471680\pi$
$383$	1332.00	0.177708	0.0888538	+	0.996045i	$0.471680\pi$
$384$	0	0
$385$	864.000	0.114373
$386$	0	0
$387$	−4716.00	−0.619452
$388$	0	0
$389$	3054.00	0.398056	0.199028	−	0.979994i	$-0.436221\pi$
$389$	3054.00	0.398056	0.199028	+	0.979994i	$0.436221\pi$
$390$	0	0
$391$	−7488.00	−0.968502
$392$	0	0
$393$	6300.00	0.808633
$394$	0	0
$395$	8448.00	1.07611
$396$	0	0
$397$	11162.0	1.41110	0.705548	−	0.708663i	$-0.250701\pi$
$397$	11162.0	1.41110	0.705548	+	0.708663i	$0.250701\pi$
$398$	0	0
$399$	444.000	0.0557088
$400$	0	0
$401$	−14820.0	−1.84557	−0.922787	−	0.385310i	$-0.874095\pi$
$401$	−14820.0	−1.84557	−0.922787	+	0.385310i	$0.874095\pi$
$402$	0	0
$403$	2782.00	0.343874
$404$	0	0
$405$	−972.000	−0.119257
$406$	0	0
$407$	−10296.0	−1.25394
$408$	0	0
$409$	−9682.00	−1.17052	−0.585262	−	0.810844i	$-0.699008\pi$
$409$	−9682.00	−1.17052	−0.585262	+	0.810844i	$0.699008\pi$
$410$	0	0
$411$	−7488.00	−0.898676
$412$	0	0
$413$	1152.00	0.137255
$414$	0	0
$415$	−10656.0	−1.26044
$416$	0	0
$417$	7392.00	0.868076
$418$	0	0
$419$	348.000	0.0405750	0.0202875	−	0.999794i	$-0.493542\pi$
$419$	348.000	0.0405750	0.0202875	+	0.999794i	$0.493542\pi$
$420$	0	0
$421$	2486.00	0.287792	0.143896	−	0.989593i	$-0.454037\pi$
$421$	2486.00	0.287792	0.143896	+	0.989593i	$0.454037\pi$
$422$	0	0
$423$	−2700.00	−0.310351
$424$	0	0
$425$	−1482.00	−0.169147
$426$	0	0
$427$	−148.000	−0.0167734
$428$	0	0
$429$	1404.00	0.158009
$430$	0	0
$431$	1812.00	0.202508	0.101254	−	0.994861i	$-0.467715\pi$
$431$	1812.00	0.202508	0.101254	+	0.994861i	$0.467715\pi$
$432$	0	0
$433$	−6226.00	−0.690999	−0.345499	−	0.938419i	$-0.612290\pi$
$433$	−6226.00	−0.690999	−0.345499	+	0.938419i	$0.612290\pi$
$434$	0	0
$435$	−648.000	−0.0714235
$436$	0	0
$437$	−7104.00	−0.777644
$438$	0	0
$439$	12544.0	1.36376	0.681882	−	0.731462i	$-0.261162\pi$
$439$	12544.0	1.36376	0.681882	+	0.731462i	$0.261162\pi$
$440$	0	0
$441$	−3051.00	−0.329446
$442$	0	0
$443$	8556.00	0.917625	0.458812	−	0.888533i	$-0.348275\pi$
$443$	8556.00	0.917625	0.458812	+	0.888533i	$0.348275\pi$
$444$	0	0
$445$	12240.0	1.30389
$446$	0	0
$447$	648.000	0.0685668
$448$	0	0
$449$	4116.00	0.432619	0.216310	−	0.976325i	$-0.430598\pi$
$449$	4116.00	0.432619	0.216310	+	0.976325i	$0.430598\pi$
$450$	0	0
$451$	−13824.0	−1.44334
$452$	0	0
$453$	2694.00	0.279415
$454$	0	0
$455$	312.000	0.0321468
$456$	0	0
$457$	−6514.00	−0.666766	−0.333383	−	0.942791i	$-0.608190\pi$
$457$	−6514.00	−0.666766	−0.333383	+	0.942791i	$0.608190\pi$
$458$	0	0
$459$	−2106.00	−0.214160
$460$	0	0
$461$	10500.0	1.06081	0.530405	−	0.847744i	$-0.322040\pi$
$461$	10500.0	1.06081	0.530405	+	0.847744i	$0.322040\pi$
$462$	0	0
$463$	5542.00	0.556282	0.278141	−	0.960540i	$-0.410282\pi$
$463$	5542.00	0.556282	0.278141	+	0.960540i	$0.410282\pi$
$464$	0	0
$465$	−7704.00	−0.768311
$466$	0	0
$467$	5220.00	0.517244	0.258622	−	0.965979i	$-0.416732\pi$
$467$	5220.00	0.517244	0.258622	+	0.965979i	$0.416732\pi$
$468$	0	0
$469$	76.0000	0.00748263
$470$	0	0
$471$	−4530.00	−0.443166
$472$	0	0
$473$	−18864.0	−1.83376
$474$	0	0
$475$	−1406.00	−0.135814
$476$	0	0
$477$	5022.00	0.482058
$478$	0	0
$479$	−11592.0	−1.10575	−0.552873	−	0.833266i	$-0.686468\pi$
$479$	−11592.0	−1.10575	−0.552873	+	0.833266i	$0.686468\pi$
$480$	0	0
$481$	−3718.00	−0.352445
$482$	0	0
$483$	−576.000	−0.0542627
$484$	0	0
$485$	−1320.00	−0.123584
$486$	0	0
$487$	−12170.0	−1.13239	−0.566196	−	0.824270i	$-0.691586\pi$
$487$	−12170.0	−1.13239	−0.566196	+	0.824270i	$0.691586\pi$
$488$	0	0
$489$	1182.00	0.109309
$490$	0	0
$491$	−1812.00	−0.166547	−0.0832733	−	0.996527i	$-0.526537\pi$
$491$	−1812.00	−0.166547	−0.0832733	+	0.996527i	$0.526537\pi$
$492$	0	0
$493$	−1404.00	−0.128262
$494$	0	0
$495$	−3888.00	−0.353036
$496$	0	0
$497$	−912.000	−0.0823115
$498$	0	0
$499$	1330.00	0.119317	0.0596583	−	0.998219i	$-0.480999\pi$
$499$	1330.00	0.119317	0.0596583	+	0.998219i	$0.480999\pi$
$500$	0	0
$501$	−252.000	−0.0224721
$502$	0	0
$503$	2688.00	0.238274	0.119137	−	0.992878i	$-0.461987\pi$
$503$	2688.00	0.238274	0.119137	+	0.992878i	$0.461987\pi$
$504$	0	0
$505$	11880.0	1.04684
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	5124.00	0.446203	0.223101	−	0.974795i	$-0.428382\pi$
$509$	5124.00	0.446203	0.223101	+	0.974795i	$0.428382\pi$
$510$	0	0
$511$	1364.00	0.118082
$512$	0	0
$513$	−1998.00	−0.171957
$514$	0	0
$515$	14496.0	1.24033
$516$	0	0
$517$	−10800.0	−0.918730
$518$	0	0
$519$	3582.00	0.302953
$520$	0	0
$521$	−882.000	−0.0741672	−0.0370836	−	0.999312i	$-0.511807\pi$
$521$	−882.000	−0.0741672	−0.0370836	+	0.999312i	$0.511807\pi$
$522$	0	0
$523$	2320.00	0.193970	0.0969852	−	0.995286i	$-0.469080\pi$
$523$	2320.00	0.193970	0.0969852	+	0.995286i	$0.469080\pi$
$524$	0	0
$525$	−114.000	−0.00947689
$526$	0	0
$527$	−16692.0	−1.37972
$528$	0	0
$529$	−2951.00	−0.242541
$530$	0	0
$531$	−5184.00	−0.423666
$532$	0	0
$533$	−4992.00	−0.405680
$534$	0	0
$535$	11952.0	0.965851
$536$	0	0
$537$	−9468.00	−0.760846
$538$	0	0
$539$	−12204.0	−0.975257
$540$	0	0
$541$	21422.0	1.70241	0.851205	−	0.524833i	$-0.175872\pi$
$541$	21422.0	1.70241	0.851205	+	0.524833i	$0.175872\pi$
$542$	0	0
$543$	−3234.00	−0.255588
$544$	0	0
$545$	16824.0	1.32231
$546$	0	0
$547$	−7040.00	−0.550290	−0.275145	−	0.961403i	$-0.588726\pi$
$547$	−7040.00	−0.550290	−0.275145	+	0.961403i	$0.588726\pi$
$548$	0	0
$549$	666.000	0.0517745
$550$	0	0
$551$	−1332.00	−0.102986
$552$	0	0
$553$	1408.00	0.108272
$554$	0	0
$555$	10296.0	0.787461
$556$	0	0
$557$	−8400.00	−0.638994	−0.319497	−	0.947587i	$-0.603514\pi$
$557$	−8400.00	−0.638994	−0.319497	+	0.947587i	$0.603514\pi$
$558$	0	0
$559$	−6812.00	−0.515415
$560$	0	0
$561$	−8424.00	−0.633978
$562$	0	0
$563$	−19044.0	−1.42559	−0.712797	−	0.701371i	$-0.752572\pi$
$563$	−19044.0	−1.42559	−0.712797	+	0.701371i	$0.752572\pi$
$564$	0	0
$565$	−23112.0	−1.72094
$566$	0	0
$567$	−162.000	−0.0119989
$568$	0	0
$569$	−4698.00	−0.346134	−0.173067	−	0.984910i	$-0.555368\pi$
$569$	−4698.00	−0.346134	−0.173067	+	0.984910i	$0.555368\pi$
$570$	0	0
$571$	8728.00	0.639677	0.319838	−	0.947472i	$-0.396371\pi$
$571$	8728.00	0.639677	0.319838	+	0.947472i	$0.396371\pi$
$572$	0	0
$573$	−9576.00	−0.698156
$574$	0	0
$575$	1824.00	0.132289
$576$	0	0
$577$	2018.00	0.145599	0.0727993	−	0.997347i	$-0.476807\pi$
$577$	2018.00	0.145599	0.0727993	+	0.997347i	$0.476807\pi$
$578$	0	0
$579$	2166.00	0.155468
$580$	0	0
$581$	−1776.00	−0.126817
$582$	0	0
$583$	20088.0	1.42703
$584$	0	0
$585$	−1404.00	−0.0992278
$586$	0	0
$587$	−11376.0	−0.799894	−0.399947	−	0.916538i	$-0.630971\pi$
$587$	−11376.0	−0.799894	−0.399947	+	0.916538i	$0.630971\pi$
$588$	0	0
$589$	−15836.0	−1.10783
$590$	0	0
$591$	8388.00	0.583818
$592$	0	0
$593$	−25596.0	−1.77252	−0.886258	−	0.463192i	$-0.846704\pi$
$593$	−25596.0	−1.77252	−0.886258	+	0.463192i	$0.846704\pi$
$594$	0	0
$595$	−1872.00	−0.128982
$596$	0	0
$597$	1020.00	0.0699260
$598$	0	0
$599$	−3480.00	−0.237377	−0.118689	−	0.992932i	$-0.537869\pi$
$599$	−3480.00	−0.237377	−0.118689	+	0.992932i	$0.537869\pi$
$600$	0	0
$601$	10010.0	0.679395	0.339698	−	0.940535i	$-0.389675\pi$
$601$	10010.0	0.679395	0.339698	+	0.940535i	$0.389675\pi$
$602$	0	0
$603$	−342.000	−0.0230967
$604$	0	0
$605$	420.000	0.0282238
$606$	0	0
$607$	−3764.00	−0.251690	−0.125845	−	0.992050i	$-0.540164\pi$
$607$	−3764.00	−0.251690	−0.125845	+	0.992050i	$0.540164\pi$
$608$	0	0
$609$	−108.000	−0.00718618
$610$	0	0
$611$	−3900.00	−0.258228
$612$	0	0
$613$	13610.0	0.896742	0.448371	−	0.893848i	$-0.352004\pi$
$613$	13610.0	0.896742	0.448371	+	0.893848i	$0.352004\pi$
$614$	0	0
$615$	13824.0	0.906402
$616$	0	0
$617$	6408.00	0.418114	0.209057	−	0.977903i	$-0.432961\pi$
$617$	6408.00	0.418114	0.209057	+	0.977903i	$0.432961\pi$
$618$	0	0
$619$	6694.00	0.434660	0.217330	−	0.976098i	$-0.430265\pi$
$619$	6694.00	0.434660	0.217330	+	0.976098i	$0.430265\pi$
$620$	0	0
$621$	2592.00	0.167493
$622$	0	0
$623$	2040.00	0.131189
$624$	0	0
$625$	−17639.0	−1.12890
$626$	0	0
$627$	−7992.00	−0.509043
$628$	0	0
$629$	22308.0	1.41411
$630$	0	0
$631$	27250.0	1.71918	0.859592	−	0.510981i	$-0.170718\pi$
$631$	27250.0	1.71918	0.859592	+	0.510981i	$0.170718\pi$
$632$	0	0
$633$	5772.00	0.362427
$634$	0	0
$635$	−11856.0	−0.740931
$636$	0	0
$637$	−4407.00	−0.274116
$638$	0	0
$639$	4104.00	0.254072
$640$	0	0
$641$	−12630.0	−0.778245	−0.389122	−	0.921186i	$-0.627222\pi$
$641$	−12630.0	−0.778245	−0.389122	+	0.921186i	$0.627222\pi$
$642$	0	0
$643$	−14798.0	−0.907583	−0.453792	−	0.891108i	$-0.649929\pi$
$643$	−14798.0	−0.907583	−0.453792	+	0.891108i	$0.649929\pi$
$644$	0	0
$645$	18864.0	1.15158
$646$	0	0
$647$	−26232.0	−1.59395	−0.796976	−	0.604012i	$-0.793568\pi$
$647$	−26232.0	−1.59395	−0.796976	+	0.604012i	$0.793568\pi$
$648$	0	0
$649$	−20736.0	−1.25417
$650$	0	0
$651$	−1284.00	−0.0773025
$652$	0	0
$653$	−30390.0	−1.82121	−0.910607	−	0.413274i	$-0.864385\pi$
$653$	−30390.0	−1.82121	−0.910607	+	0.413274i	$0.864385\pi$
$654$	0	0
$655$	−25200.0	−1.50328
$656$	0	0
$657$	−6138.00	−0.364484
$658$	0	0
$659$	28740.0	1.69886	0.849432	−	0.527698i	$-0.176945\pi$
$659$	28740.0	1.69886	0.849432	+	0.527698i	$0.176945\pi$
$660$	0	0
$661$	−9214.00	−0.542183	−0.271092	−	0.962554i	$-0.587385\pi$
$661$	−9214.00	−0.542183	−0.271092	+	0.962554i	$0.587385\pi$
$662$	0	0
$663$	−3042.00	−0.178192
$664$	0	0
$665$	−1776.00	−0.103564
$666$	0	0
$667$	1728.00	0.100312
$668$	0	0
$669$	−15126.0	−0.874148
$670$	0	0
$671$	2664.00	0.153268
$672$	0	0
$673$	16598.0	0.950677	0.475339	−	0.879803i	$-0.342326\pi$
$673$	16598.0	0.950677	0.475339	+	0.879803i	$0.342326\pi$
$674$	0	0
$675$	513.000	0.0292524
$676$	0	0
$677$	−8610.00	−0.488788	−0.244394	−	0.969676i	$-0.578589\pi$
$677$	−8610.00	−0.488788	−0.244394	+	0.969676i	$0.578589\pi$
$678$	0	0
$679$	−220.000	−0.0124342
$680$	0	0
$681$	8028.00	0.451738
$682$	0	0
$683$	−804.000	−0.0450428	−0.0225214	−	0.999746i	$-0.507169\pi$
$683$	−804.000	−0.0450428	−0.0225214	+	0.999746i	$0.507169\pi$
$684$	0	0
$685$	29952.0	1.67067
$686$	0	0
$687$	−7230.00	−0.401516
$688$	0	0
$689$	7254.00	0.401096
$690$	0	0
$691$	−2270.00	−0.124971	−0.0624854	−	0.998046i	$-0.519903\pi$
$691$	−2270.00	−0.124971	−0.0624854	+	0.998046i	$0.519903\pi$
$692$	0	0
$693$	−648.000	−0.0355202
$694$	0	0
$695$	−29568.0	−1.61378
$696$	0	0
$697$	29952.0	1.62771
$698$	0	0
$699$	11178.0	0.604851
$700$	0	0
$701$	1782.00	0.0960131	0.0480066	−	0.998847i	$-0.484713\pi$
$701$	1782.00	0.0960131	0.0480066	+	0.998847i	$0.484713\pi$
$702$	0	0
$703$	21164.0	1.13544
$704$	0	0
$705$	10800.0	0.576953
$706$	0	0
$707$	1980.00	0.105326
$708$	0	0
$709$	−10690.0	−0.566250	−0.283125	−	0.959083i	$-0.591371\pi$
$709$	−10690.0	−0.566250	−0.283125	+	0.959083i	$0.591371\pi$
$710$	0	0
$711$	−6336.00	−0.334203
$712$	0	0
$713$	20544.0	1.07907
$714$	0	0
$715$	−5616.00	−0.293743
$716$	0	0
$717$	−3744.00	−0.195010
$718$	0	0
$719$	−11568.0	−0.600019	−0.300009	−	0.953936i	$-0.596990\pi$
$719$	−11568.0	−0.600019	−0.300009	+	0.953936i	$0.596990\pi$
$720$	0	0
$721$	2416.00	0.124794
$722$	0	0
$723$	−12630.0	−0.649675
$724$	0	0
$725$	342.000	0.0175194
$726$	0	0
$727$	11644.0	0.594019	0.297010	−	0.954874i	$-0.404011\pi$
$727$	11644.0	0.594019	0.297010	+	0.954874i	$0.404011\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	40872.0	2.06800
$732$	0	0
$733$	−15010.0	−0.756353	−0.378177	−	0.925733i	$-0.623449\pi$
$733$	−15010.0	−0.756353	−0.378177	+	0.925733i	$0.623449\pi$
$734$	0	0
$735$	12204.0	0.612451
$736$	0	0
$737$	−1368.00	−0.0683730
$738$	0	0
$739$	−33410.0	−1.66307	−0.831534	−	0.555474i	$-0.812537\pi$
$739$	−33410.0	−1.66307	−0.831534	+	0.555474i	$0.812537\pi$
$740$	0	0
$741$	−2886.00	−0.143077
$742$	0	0
$743$	6504.00	0.321142	0.160571	−	0.987024i	$-0.448666\pi$
$743$	6504.00	0.321142	0.160571	+	0.987024i	$0.448666\pi$
$744$	0	0
$745$	−2592.00	−0.127468
$746$	0	0
$747$	7992.00	0.391448
$748$	0	0
$749$	1992.00	0.0971777
$750$	0	0
$751$	13912.0	0.675973	0.337987	−	0.941151i	$-0.390254\pi$
$751$	13912.0	0.675973	0.337987	+	0.941151i	$0.390254\pi$
$752$	0	0
$753$	23076.0	1.11678
$754$	0	0
$755$	−10776.0	−0.519442
$756$	0	0
$757$	−23974.0	−1.15106	−0.575528	−	0.817782i	$-0.695204\pi$
$757$	−23974.0	−1.15106	−0.575528	+	0.817782i	$0.695204\pi$
$758$	0	0
$759$	10368.0	0.495829
$760$	0	0
$761$	−288.000	−0.0137188	−0.00685939	−	0.999976i	$-0.502183\pi$
$761$	−288.000	−0.0137188	−0.00685939	+	0.999976i	$0.502183\pi$
$762$	0	0
$763$	2804.00	0.133043
$764$	0	0
$765$	8424.00	0.398131
$766$	0	0
$767$	−7488.00	−0.352511
$768$	0	0
$769$	1514.00	0.0709964	0.0354982	−	0.999370i	$-0.488698\pi$
$769$	1514.00	0.0709964	0.0354982	+	0.999370i	$0.488698\pi$
$770$	0	0
$771$	3978.00	0.185816
$772$	0	0
$773$	15816.0	0.735915	0.367957	−	0.929843i	$-0.380057\pi$
$773$	15816.0	0.735915	0.367957	+	0.929843i	$0.380057\pi$
$774$	0	0
$775$	4066.00	0.188458
$776$	0	0
$777$	1716.00	0.0792293
$778$	0	0
$779$	28416.0	1.30694
$780$	0	0
$781$	16416.0	0.752126
$782$	0	0
$783$	486.000	0.0221816
$784$	0	0
$785$	18120.0	0.823861
$786$	0	0
$787$	−10154.0	−0.459912	−0.229956	−	0.973201i	$-0.573858\pi$
$787$	−10154.0	−0.459912	−0.229956	+	0.973201i	$0.573858\pi$
$788$	0	0
$789$	18144.0	0.818686
$790$	0	0
$791$	−3852.00	−0.173150
$792$	0	0
$793$	962.000	0.0430790
$794$	0	0
$795$	−20088.0	−0.896161
$796$	0	0
$797$	−17442.0	−0.775191	−0.387596	−	0.921830i	$-0.626694\pi$
$797$	−17442.0	−0.775191	−0.387596	+	0.921830i	$0.626694\pi$
$798$	0	0
$799$	23400.0	1.03609
$800$	0	0
$801$	−9180.00	−0.404943
$802$	0	0
$803$	−24552.0	−1.07898
$804$	0	0
$805$	2304.00	0.100876
$806$	0	0
$807$	19422.0	0.847196
$808$	0	0
$809$	−8778.00	−0.381481	−0.190740	−	0.981641i	$-0.561089\pi$
$809$	−8778.00	−0.381481	−0.190740	+	0.981641i	$0.561089\pi$
$810$	0	0
$811$	430.000	0.0186182	0.00930909	−	0.999957i	$-0.497037\pi$
$811$	430.000	0.0186182	0.00930909	+	0.999957i	$0.497037\pi$
$812$	0	0
$813$	−17934.0	−0.773644
$814$	0	0
$815$	−4728.00	−0.203208
$816$	0	0
$817$	38776.0	1.66047
$818$	0	0
$819$	−234.000	−0.00998367
$820$	0	0
$821$	−32976.0	−1.40179	−0.700895	−	0.713264i	$-0.747216\pi$
$821$	−32976.0	−1.40179	−0.700895	+	0.713264i	$0.747216\pi$
$822$	0	0
$823$	1168.00	0.0494701	0.0247351	−	0.999694i	$-0.492126\pi$
$823$	1168.00	0.0494701	0.0247351	+	0.999694i	$0.492126\pi$
$824$	0	0
$825$	2052.00	0.0865957
$826$	0	0
$827$	17172.0	0.722042	0.361021	−	0.932558i	$-0.382428\pi$
$827$	17172.0	0.722042	0.361021	+	0.932558i	$0.382428\pi$
$828$	0	0
$829$	27146.0	1.13730	0.568649	−	0.822580i	$-0.307466\pi$
$829$	27146.0	1.13730	0.568649	+	0.822580i	$0.307466\pi$
$830$	0	0
$831$	26250.0	1.09579
$832$	0	0
$833$	26442.0	1.09983
$834$	0	0
$835$	1008.00	0.0417764
$836$	0	0
$837$	5778.00	0.238610
$838$	0	0
$839$	−30696.0	−1.26310	−0.631552	−	0.775334i	$-0.717582\pi$
$839$	−30696.0	−1.26310	−0.631552	+	0.775334i	$0.717582\pi$
$840$	0	0
$841$	−24065.0	−0.986715
$842$	0	0
$843$	26928.0	1.10018
$844$	0	0
$845$	−2028.00	−0.0825625
$846$	0	0
$847$	70.0000	0.00283970
$848$	0	0
$849$	1776.00	0.0717929
$850$	0	0
$851$	−27456.0	−1.10597
$852$	0	0
$853$	24842.0	0.997156	0.498578	−	0.866845i	$-0.333856\pi$
$853$	24842.0	0.997156	0.498578	+	0.866845i	$0.333856\pi$
$854$	0	0
$855$	7992.00	0.319673
$856$	0	0
$857$	11406.0	0.454634	0.227317	−	0.973821i	$-0.427005\pi$
$857$	11406.0	0.454634	0.227317	+	0.973821i	$0.427005\pi$
$858$	0	0
$859$	−20540.0	−0.815851	−0.407925	−	0.913015i	$-0.633748\pi$
$859$	−20540.0	−0.815851	−0.407925	+	0.913015i	$0.633748\pi$
$860$	0	0
$861$	2304.00	0.0911964
$862$	0	0
$863$	−9108.00	−0.359258	−0.179629	−	0.983734i	$-0.557490\pi$
$863$	−9108.00	−0.359258	−0.179629	+	0.983734i	$0.557490\pi$
$864$	0	0
$865$	−14328.0	−0.563198
$866$	0	0
$867$	3513.00	0.137610
$868$	0	0
$869$	−25344.0	−0.989340
$870$	0	0
$871$	−494.000	−0.0192176
$872$	0	0
$873$	990.000	0.0383808
$874$	0	0
$875$	−2544.00	−0.0982890
$876$	0	0
$877$	−24046.0	−0.925856	−0.462928	−	0.886396i	$-0.653201\pi$
$877$	−24046.0	−0.925856	−0.462928	+	0.886396i	$0.653201\pi$
$878$	0	0
$879$	−13824.0	−0.530457
$880$	0	0
$881$	7998.00	0.305856	0.152928	−	0.988237i	$-0.451130\pi$
$881$	7998.00	0.305856	0.152928	+	0.988237i	$0.451130\pi$
$882$	0	0
$883$	−24032.0	−0.915902	−0.457951	−	0.888978i	$-0.651416\pi$
$883$	−24032.0	−0.915902	−0.457951	+	0.888978i	$0.651416\pi$
$884$	0	0
$885$	20736.0	0.787608
$886$	0	0
$887$	15648.0	0.592343	0.296172	−	0.955135i	$-0.404290\pi$
$887$	15648.0	0.592343	0.296172	+	0.955135i	$0.404290\pi$
$888$	0	0
$889$	−1976.00	−0.0745477
$890$	0	0
$891$	2916.00	0.109640
$892$	0	0
$893$	22200.0	0.831909
$894$	0	0
$895$	37872.0	1.41444
$896$	0	0
$897$	3744.00	0.139363
$898$	0	0
$899$	3852.00	0.142905
$900$	0	0
$901$	−43524.0	−1.60932
$902$	0	0
$903$	3144.00	0.115865
$904$	0	0
$905$	12936.0	0.475146
$906$	0	0
$907$	808.000	0.0295802	0.0147901	−	0.999891i	$-0.495292\pi$
$907$	808.000	0.0295802	0.0147901	+	0.999891i	$0.495292\pi$
$908$	0	0
$909$	−8910.00	−0.325111
$910$	0	0
$911$	−39144.0	−1.42360	−0.711799	−	0.702383i	$-0.752120\pi$
$911$	−39144.0	−1.42360	−0.711799	+	0.702383i	$0.752120\pi$
$912$	0	0
$913$	31968.0	1.15880
$914$	0	0
$915$	−2664.00	−0.0962504
$916$	0	0
$917$	−4200.00	−0.151250
$918$	0	0
$919$	38248.0	1.37289	0.686445	−	0.727182i	$-0.259170\pi$
$919$	38248.0	1.37289	0.686445	+	0.727182i	$0.259170\pi$
$920$	0	0
$921$	9498.00	0.339815
$922$	0	0
$923$	5928.00	0.211400
$924$	0	0
$925$	−5434.00	−0.193155
$926$	0	0
$927$	−10872.0	−0.385203
$928$	0	0
$929$	−54264.0	−1.91641	−0.958205	−	0.286084i	$-0.907646\pi$
$929$	−54264.0	−1.91641	−0.958205	+	0.286084i	$0.907646\pi$
$930$	0	0
$931$	25086.0	0.883094
$932$	0	0
$933$	−7416.00	−0.260224
$934$	0	0
$935$	33696.0	1.17859
$936$	0	0
$937$	12206.0	0.425563	0.212782	−	0.977100i	$-0.431748\pi$
$937$	12206.0	0.425563	0.212782	+	0.977100i	$0.431748\pi$
$938$	0	0
$939$	−9282.00	−0.322584
$940$	0	0
$941$	17664.0	0.611934	0.305967	−	0.952042i	$-0.401020\pi$
$941$	17664.0	0.611934	0.305967	+	0.952042i	$0.401020\pi$
$942$	0	0
$943$	−36864.0	−1.27302
$944$	0	0
$945$	648.000	0.0223063
$946$	0	0
$947$	51984.0	1.78379	0.891897	−	0.452238i	$-0.149374\pi$
$947$	51984.0	1.78379	0.891897	+	0.452238i	$0.149374\pi$
$948$	0	0
$949$	−8866.00	−0.303269
$950$	0	0
$951$	6948.00	0.236913
$952$	0	0
$953$	13782.0	0.468460	0.234230	−	0.972181i	$-0.424743\pi$
$953$	13782.0	0.468460	0.234230	+	0.972181i	$0.424743\pi$
$954$	0	0
$955$	38304.0	1.29789
$956$	0	0
$957$	1944.00	0.0656642
$958$	0	0
$959$	4992.00	0.168092
$960$	0	0
$961$	16005.0	0.537243
$962$	0	0
$963$	−8964.00	−0.299959
$964$	0	0
$965$	−8664.00	−0.289020
$966$	0	0
$967$	−14618.0	−0.486125	−0.243063	−	0.970011i	$-0.578152\pi$
$967$	−14618.0	−0.486125	−0.243063	+	0.970011i	$0.578152\pi$
$968$	0	0
$969$	17316.0	0.574066
$970$	0	0
$971$	18708.0	0.618299	0.309149	−	0.951013i	$-0.399956\pi$
$971$	18708.0	0.618299	0.309149	+	0.951013i	$0.399956\pi$
$972$	0	0
$973$	−4928.00	−0.162368
$974$	0	0
$975$	741.000	0.0243395
$976$	0	0
$977$	−48804.0	−1.59814	−0.799068	−	0.601241i	$-0.794673\pi$
$977$	−48804.0	−1.59814	−0.799068	+	0.601241i	$0.794673\pi$
$978$	0	0
$979$	−36720.0	−1.19875
$980$	0	0
$981$	−12618.0	−0.410664
$982$	0	0
$983$	44736.0	1.45153	0.725766	−	0.687941i	$-0.241485\pi$
$983$	44736.0	1.45153	0.725766	+	0.687941i	$0.241485\pi$
$984$	0	0
$985$	−33552.0	−1.08534
$986$	0	0
$987$	1800.00	0.0580493
$988$	0	0
$989$	−50304.0	−1.61737
$990$	0	0
$991$	21004.0	0.673274	0.336637	−	0.941635i	$-0.390711\pi$
$991$	21004.0	0.673274	0.336637	+	0.941635i	$0.390711\pi$
$992$	0	0
$993$	13278.0	0.424335
$994$	0	0
$995$	−4080.00	−0.129995
$996$	0	0
$997$	9038.00	0.287098	0.143549	−	0.989643i	$-0.454149\pi$
$997$	9038.00	0.287098	0.143549	+	0.989643i	$0.454149\pi$
$998$	0	0
$999$	−7722.00	−0.244558

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.a.g.1.1		1
3.2	odd	2		1872.4.a.m.1.1		1
4.3	odd	2		39.4.a.a.1.1	✓	1
8.3	odd	2		2496.4.a.o.1.1		1
8.5	even	2		2496.4.a.f.1.1		1
12.11	even	2		117.4.a.a.1.1		1
20.19	odd	2		975.4.a.e.1.1		1
28.27	even	2		1911.4.a.f.1.1		1
52.31	even	4		507.4.b.b.337.2		2
52.47	even	4		507.4.b.b.337.1		2
52.51	odd	2		507.4.a.c.1.1		1
156.155	even	2		1521.4.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.a.1.1	✓	1	4.3	odd	2
117.4.a.a.1.1		1	12.11	even	2
507.4.a.c.1.1		1	52.51	odd	2
507.4.b.b.337.1		2	52.47	even	4
507.4.b.b.337.2		2	52.31	even	4
624.4.a.g.1.1		1	1.1	even	1	trivial
975.4.a.e.1.1		1	20.19	odd	2
1521.4.a.f.1.1		1	156.155	even	2
1872.4.a.m.1.1		1	3.2	odd	2
1911.4.a.f.1.1		1	28.27	even	2
2496.4.a.f.1.1		1	8.5	even	2
2496.4.a.o.1.1		1	8.3	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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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