LMFDB - Embedded newform 624.4.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [624,4,Mod(1,624)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-3,0,10,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$36.8171918436$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 78)
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)

$f(q)$

$=$

$q-3.00000 q^{3} +10.0000 q^{5} +8.00000 q^{7} +9.00000 q^{9} -40.0000 q^{11} +13.0000 q^{13} -30.0000 q^{15} +130.000 q^{17} +20.0000 q^{19} -24.0000 q^{21} -25.0000 q^{25} -27.0000 q^{27} -18.0000 q^{29} +184.000 q^{31} +120.000 q^{33} +80.0000 q^{35} -74.0000 q^{37} -39.0000 q^{39} -362.000 q^{41} -76.0000 q^{43} +90.0000 q^{45} +452.000 q^{47} -279.000 q^{49} -390.000 q^{51} +382.000 q^{53} -400.000 q^{55} -60.0000 q^{57} -464.000 q^{59} +358.000 q^{61} +72.0000 q^{63} +130.000 q^{65} +700.000 q^{67} +748.000 q^{71} +1058.00 q^{73} +75.0000 q^{75} -320.000 q^{77} +976.000 q^{79} +81.0000 q^{81} +1008.00 q^{83} +1300.00 q^{85} +54.0000 q^{87} -386.000 q^{89} +104.000 q^{91} -552.000 q^{93} +200.000 q^{95} -614.000 q^{97} -360.000 q^{99} +O(q^{100})$

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
$5$	10.0000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	10.0000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	8.00000	0.431959	0.215980	−	0.976398i	$-0.430705\pi$
$7$	8.00000	0.431959	0.215980	+	0.976398i	$0.430705\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−40.0000	−1.09640	−0.548202	−	0.836346i	$-0.684688\pi$
$11$	−40.0000	−1.09640	−0.548202	+	0.836346i	$0.684688\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−30.0000	−0.516398
$16$	0	0
$17$	130.000	1.85468	0.927342	−	0.374215i	$-0.122088\pi$
$17$	130.000	1.85468	0.927342	+	0.374215i	$0.122088\pi$
$18$	0	0
$19$	20.0000	0.241490	0.120745	−	0.992684i	$-0.461472\pi$
$19$	20.0000	0.241490	0.120745	+	0.992684i	$0.461472\pi$
$20$	0	0
$21$	−24.0000	−0.249392
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−18.0000	−0.115259	−0.0576296	−	0.998338i	$-0.518354\pi$
$29$	−18.0000	−0.115259	−0.0576296	+	0.998338i	$0.518354\pi$
$30$	0	0
$31$	184.000	1.06604	0.533022	−	0.846101i	$-0.321056\pi$
$31$	184.000	1.06604	0.533022	+	0.846101i	$0.321056\pi$
$32$	0	0
$33$	120.000	0.633010
$34$	0	0
$35$	80.0000	0.386356
$36$	0	0
$37$	−74.0000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−74.0000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	−362.000	−1.37890	−0.689450	−	0.724333i	$-0.742148\pi$
$41$	−362.000	−1.37890	−0.689450	+	0.724333i	$0.742148\pi$
$42$	0	0
$43$	−76.0000	−0.269532	−0.134766	−	0.990877i	$-0.543028\pi$
$43$	−76.0000	−0.269532	−0.134766	+	0.990877i	$0.543028\pi$
$44$	0	0
$45$	90.0000	0.298142
$46$	0	0
$47$	452.000	1.40279	0.701393	−	0.712774i	$-0.252562\pi$
$47$	452.000	1.40279	0.701393	+	0.712774i	$0.252562\pi$
$48$	0	0
$49$	−279.000	−0.813411
$50$	0	0
$51$	−390.000	−1.07080
$52$	0	0
$53$	382.000	0.990033	0.495016	−	0.868884i	$-0.335162\pi$
$53$	382.000	0.990033	0.495016	+	0.868884i	$0.335162\pi$
$54$	0	0
$55$	−400.000	−0.980654
$56$	0	0
$57$	−60.0000	−0.139424
$58$	0	0
$59$	−464.000	−1.02386	−0.511929	−	0.859028i	$-0.671069\pi$
$59$	−464.000	−1.02386	−0.511929	+	0.859028i	$0.671069\pi$
$60$	0	0
$61$	358.000	0.751430	0.375715	−	0.926735i	$-0.377397\pi$
$61$	358.000	0.751430	0.375715	+	0.926735i	$0.377397\pi$
$62$	0	0
$63$	72.0000	0.143986
$64$	0	0
$65$	130.000	0.248069
$66$	0	0
$67$	700.000	1.27640	0.638199	−	0.769872i	$-0.279680\pi$
$67$	700.000	1.27640	0.638199	+	0.769872i	$0.279680\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	748.000	1.25030	0.625150	−	0.780505i	$-0.285038\pi$
$71$	748.000	1.25030	0.625150	+	0.780505i	$0.285038\pi$
$72$	0	0
$73$	1058.00	1.69629	0.848147	−	0.529760i	$-0.177718\pi$
$73$	1058.00	1.69629	0.848147	+	0.529760i	$0.177718\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	−320.000	−0.473602
$78$	0	0
$79$	976.000	1.38998	0.694991	−	0.719018i	$-0.255408\pi$
$79$	976.000	1.38998	0.694991	+	0.719018i	$0.255408\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1008.00	1.33304	0.666520	−	0.745487i	$-0.267783\pi$
$83$	1008.00	1.33304	0.666520	+	0.745487i	$0.267783\pi$
$84$	0	0
$85$	1300.00	1.65888
$86$	0	0
$87$	54.0000	0.0665449
$88$	0	0
$89$	−386.000	−0.459729	−0.229865	−	0.973223i	$-0.573828\pi$
$89$	−386.000	−0.459729	−0.229865	+	0.973223i	$0.573828\pi$
$90$	0	0
$91$	104.000	0.119804
$92$	0	0
$93$	−552.000	−0.615481
$94$	0	0
$95$	200.000	0.215995
$96$	0	0
$97$	−614.000	−0.642704	−0.321352	−	0.946960i	$-0.604137\pi$
$97$	−614.000	−0.642704	−0.321352	+	0.946960i	$0.604137\pi$
$98$	0	0
$99$	−360.000	−0.365468
$100$	0	0
$101$	518.000	0.510326	0.255163	−	0.966898i	$-0.417871\pi$
$101$	518.000	0.510326	0.255163	+	0.966898i	$0.417871\pi$
$102$	0	0
$103$	−112.000	−0.107143	−0.0535713	−	0.998564i	$-0.517060\pi$
$103$	−112.000	−0.107143	−0.0535713	+	0.998564i	$0.517060\pi$
$104$	0	0
$105$	−240.000	−0.223063
$106$	0	0
$107$	372.000	0.336099	0.168050	−	0.985779i	$-0.446253\pi$
$107$	372.000	0.336099	0.168050	+	0.985779i	$0.446253\pi$
$108$	0	0
$109$	934.000	0.820743	0.410371	−	0.911918i	$-0.365399\pi$
$109$	934.000	0.820743	0.410371	+	0.911918i	$0.365399\pi$
$110$	0	0
$111$	222.000	0.189832
$112$	0	0
$113$	1914.00	1.59340	0.796699	−	0.604376i	$-0.206578\pi$
$113$	1914.00	1.59340	0.796699	+	0.604376i	$0.206578\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	1040.00	0.801148
$120$	0	0
$121$	269.000	0.202104
$122$	0	0
$123$	1086.00	0.796108
$124$	0	0
$125$	−1500.00	−1.07331
$126$	0	0
$127$	−1296.00	−0.905523	−0.452761	−	0.891632i	$-0.649561\pi$
$127$	−1296.00	−0.905523	−0.452761	+	0.891632i	$0.649561\pi$
$128$	0	0
$129$	228.000	0.155615
$130$	0	0
$131$	892.000	0.594919	0.297460	−	0.954734i	$-0.403861\pi$
$131$	892.000	0.594919	0.297460	+	0.954734i	$0.403861\pi$
$132$	0	0
$133$	160.000	0.104314
$134$	0	0
$135$	−270.000	−0.172133
$136$	0	0
$137$	2326.00	1.45054	0.725269	−	0.688466i	$-0.241716\pi$
$137$	2326.00	1.45054	0.725269	+	0.688466i	$0.241716\pi$
$138$	0	0
$139$	−1932.00	−1.17892	−0.589461	−	0.807797i	$-0.700660\pi$
$139$	−1932.00	−1.17892	−0.589461	+	0.807797i	$0.700660\pi$
$140$	0	0
$141$	−1356.00	−0.809899
$142$	0	0
$143$	−520.000	−0.304088
$144$	0	0
$145$	−180.000	−0.103091
$146$	0	0
$147$	837.000	0.469623
$148$	0	0
$149$	882.000	0.484941	0.242471	−	0.970159i	$-0.422042\pi$
$149$	882.000	0.484941	0.242471	+	0.970159i	$0.422042\pi$
$150$	0	0
$151$	1776.00	0.957145	0.478572	−	0.878048i	$-0.341154\pi$
$151$	1776.00	0.957145	0.478572	+	0.878048i	$0.341154\pi$
$152$	0	0
$153$	1170.00	0.618228
$154$	0	0
$155$	1840.00	0.953499
$156$	0	0
$157$	−2410.00	−1.22509	−0.612544	−	0.790436i	$-0.709854\pi$
$157$	−2410.00	−1.22509	−0.612544	+	0.790436i	$0.709854\pi$
$158$	0	0
$159$	−1146.00	−0.571596
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−3212.00	−1.54346	−0.771728	−	0.635953i	$-0.780607\pi$
$163$	−3212.00	−1.54346	−0.771728	+	0.635953i	$0.780607\pi$
$164$	0	0
$165$	1200.00	0.566181
$166$	0	0
$167$	−1668.00	−0.772896	−0.386448	−	0.922311i	$-0.626298\pi$
$167$	−1668.00	−0.772896	−0.386448	+	0.922311i	$0.626298\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	180.000	0.0804967
$172$	0	0
$173$	3598.00	1.58122	0.790609	−	0.612321i	$-0.209764\pi$
$173$	3598.00	1.58122	0.790609	+	0.612321i	$0.209764\pi$
$174$	0	0
$175$	−200.000	−0.0863919
$176$	0	0
$177$	1392.00	0.591125
$178$	0	0
$179$	−1068.00	−0.445956	−0.222978	−	0.974824i	$-0.571578\pi$
$179$	−1068.00	−0.445956	−0.222978	+	0.974824i	$0.571578\pi$
$180$	0	0
$181$	−4786.00	−1.96542	−0.982709	−	0.185158i	$-0.940720\pi$
$181$	−4786.00	−1.96542	−0.982709	+	0.185158i	$0.940720\pi$
$182$	0	0
$183$	−1074.00	−0.433838
$184$	0	0
$185$	−740.000	−0.294086
$186$	0	0
$187$	−5200.00	−2.03348
$188$	0	0
$189$	−216.000	−0.0831306
$190$	0	0
$191$	1312.00	0.497031	0.248516	−	0.968628i	$-0.420057\pi$
$191$	1312.00	0.497031	0.248516	+	0.968628i	$0.420057\pi$
$192$	0	0
$193$	−350.000	−0.130537	−0.0652683	−	0.997868i	$-0.520790\pi$
$193$	−350.000	−0.130537	−0.0652683	+	0.997868i	$0.520790\pi$
$194$	0	0
$195$	−390.000	−0.143223
$196$	0	0
$197$	−342.000	−0.123688	−0.0618439	−	0.998086i	$-0.519698\pi$
$197$	−342.000	−0.123688	−0.0618439	+	0.998086i	$0.519698\pi$
$198$	0	0
$199$	3368.00	1.19975	0.599877	−	0.800092i	$-0.295216\pi$
$199$	3368.00	1.19975	0.599877	+	0.800092i	$0.295216\pi$
$200$	0	0
$201$	−2100.00	−0.736928
$202$	0	0
$203$	−144.000	−0.0497873
$204$	0	0
$205$	−3620.00	−1.23333
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−800.000	−0.264771
$210$	0	0
$211$	2004.00	0.653844	0.326922	−	0.945051i	$-0.393989\pi$
$211$	2004.00	0.653844	0.326922	+	0.945051i	$0.393989\pi$
$212$	0	0
$213$	−2244.00	−0.721861
$214$	0	0
$215$	−760.000	−0.241077
$216$	0	0
$217$	1472.00	0.460488
$218$	0	0
$219$	−3174.00	−0.979356
$220$	0	0
$221$	1690.00	0.514397
$222$	0	0
$223$	5608.00	1.68403	0.842017	−	0.539451i	$-0.181368\pi$
$223$	5608.00	1.68403	0.842017	+	0.539451i	$0.181368\pi$
$224$	0	0
$225$	−225.000	−0.0666667
$226$	0	0
$227$	1928.00	0.563726	0.281863	−	0.959455i	$-0.409048\pi$
$227$	1928.00	0.563726	0.281863	+	0.959455i	$0.409048\pi$
$228$	0	0
$229$	−3938.00	−1.13638	−0.568189	−	0.822898i	$-0.692356\pi$
$229$	−3938.00	−1.13638	−0.568189	+	0.822898i	$0.692356\pi$
$230$	0	0
$231$	960.000	0.273434
$232$	0	0
$233$	2562.00	0.720353	0.360176	−	0.932884i	$-0.382717\pi$
$233$	2562.00	0.720353	0.360176	+	0.932884i	$0.382717\pi$
$234$	0	0
$235$	4520.00	1.25469
$236$	0	0
$237$	−2928.00	−0.802506
$238$	0	0
$239$	−7164.00	−1.93891	−0.969457	−	0.245260i	$-0.921127\pi$
$239$	−7164.00	−1.93891	−0.969457	+	0.245260i	$0.921127\pi$
$240$	0	0
$241$	−6182.00	−1.65236	−0.826178	−	0.563410i	$-0.809489\pi$
$241$	−6182.00	−1.65236	−0.826178	+	0.563410i	$0.809489\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−2790.00	−0.727537
$246$	0	0
$247$	260.000	0.0669773
$248$	0	0
$249$	−3024.00	−0.769631
$250$	0	0
$251$	1396.00	0.351055	0.175527	−	0.984475i	$-0.443837\pi$
$251$	1396.00	0.351055	0.175527	+	0.984475i	$0.443837\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−3900.00	−0.957755
$256$	0	0
$257$	6906.00	1.67620	0.838102	−	0.545514i	$-0.183665\pi$
$257$	6906.00	1.67620	0.838102	+	0.545514i	$0.183665\pi$
$258$	0	0
$259$	−592.000	−0.142027
$260$	0	0
$261$	−162.000	−0.0384197
$262$	0	0
$263$	6848.00	1.60557	0.802787	−	0.596266i	$-0.203350\pi$
$263$	6848.00	1.60557	0.802787	+	0.596266i	$0.203350\pi$
$264$	0	0
$265$	3820.00	0.885512
$266$	0	0
$267$	1158.00	0.265425
$268$	0	0
$269$	−6034.00	−1.36766	−0.683828	−	0.729643i	$-0.739686\pi$
$269$	−6034.00	−1.36766	−0.683828	+	0.729643i	$0.739686\pi$
$270$	0	0
$271$	−4832.00	−1.08311	−0.541556	−	0.840665i	$-0.682164\pi$
$271$	−4832.00	−1.08311	−0.541556	+	0.840665i	$0.682164\pi$
$272$	0	0
$273$	−312.000	−0.0691689
$274$	0	0
$275$	1000.00	0.219281
$276$	0	0
$277$	−4082.00	−0.885428	−0.442714	−	0.896663i	$-0.645984\pi$
$277$	−4082.00	−0.885428	−0.442714	+	0.896663i	$0.645984\pi$
$278$	0	0
$279$	1656.00	0.355348
$280$	0	0
$281$	3350.00	0.711189	0.355595	−	0.934640i	$-0.384278\pi$
$281$	3350.00	0.711189	0.355595	+	0.934640i	$0.384278\pi$
$282$	0	0
$283$	−7796.00	−1.63754	−0.818770	−	0.574121i	$-0.805344\pi$
$283$	−7796.00	−1.63754	−0.818770	+	0.574121i	$0.805344\pi$
$284$	0	0
$285$	−600.000	−0.124705
$286$	0	0
$287$	−2896.00	−0.595629
$288$	0	0
$289$	11987.0	2.43985
$290$	0	0
$291$	1842.00	0.371065
$292$	0	0
$293$	3922.00	0.781999	0.390999	−	0.920391i	$-0.372129\pi$
$293$	3922.00	0.781999	0.390999	+	0.920391i	$0.372129\pi$
$294$	0	0
$295$	−4640.00	−0.915767
$296$	0	0
$297$	1080.00	0.211003
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−608.000	−0.116427
$302$	0	0
$303$	−1554.00	−0.294637
$304$	0	0
$305$	3580.00	0.672099
$306$	0	0
$307$	−5956.00	−1.10725	−0.553627	−	0.832765i	$-0.686757\pi$
$307$	−5956.00	−1.10725	−0.553627	+	0.832765i	$0.686757\pi$
$308$	0	0
$309$	336.000	0.0618588
$310$	0	0
$311$	−2352.00	−0.428841	−0.214421	−	0.976741i	$-0.568786\pi$
$311$	−2352.00	−0.428841	−0.214421	+	0.976741i	$0.568786\pi$
$312$	0	0
$313$	8442.00	1.52450	0.762252	−	0.647280i	$-0.224093\pi$
$313$	8442.00	1.52450	0.762252	+	0.647280i	$0.224093\pi$
$314$	0	0
$315$	720.000	0.128785
$316$	0	0
$317$	−5550.00	−0.983341	−0.491670	−	0.870781i	$-0.663614\pi$
$317$	−5550.00	−0.983341	−0.491670	+	0.870781i	$0.663614\pi$
$318$	0	0
$319$	720.000	0.126371
$320$	0	0
$321$	−1116.00	−0.194047
$322$	0	0
$323$	2600.00	0.447888
$324$	0	0
$325$	−325.000	−0.0554700
$326$	0	0
$327$	−2802.00	−0.473856
$328$	0	0
$329$	3616.00	0.605947
$330$	0	0
$331$	−140.000	−0.0232480	−0.0116240	−	0.999932i	$-0.503700\pi$
$331$	−140.000	−0.0232480	−0.0116240	+	0.999932i	$0.503700\pi$
$332$	0	0
$333$	−666.000	−0.109599
$334$	0	0
$335$	7000.00	1.14164
$336$	0	0
$337$	−6174.00	−0.997980	−0.498990	−	0.866608i	$-0.666296\pi$
$337$	−6174.00	−0.997980	−0.498990	+	0.866608i	$0.666296\pi$
$338$	0	0
$339$	−5742.00	−0.919949
$340$	0	0
$341$	−7360.00	−1.16882
$342$	0	0
$343$	−4976.00	−0.783320
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2988.00	0.462260	0.231130	−	0.972923i	$-0.425758\pi$
$347$	2988.00	0.462260	0.231130	+	0.972923i	$0.425758\pi$
$348$	0	0
$349$	−162.000	−0.0248472	−0.0124236	−	0.999923i	$-0.503955\pi$
$349$	−162.000	−0.0248472	−0.0124236	+	0.999923i	$0.503955\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	−10754.0	−1.62147	−0.810733	−	0.585416i	$-0.800931\pi$
$353$	−10754.0	−1.62147	−0.810733	+	0.585416i	$0.800931\pi$
$354$	0	0
$355$	7480.00	1.11830
$356$	0	0
$357$	−3120.00	−0.462543
$358$	0	0
$359$	−3588.00	−0.527486	−0.263743	−	0.964593i	$-0.584957\pi$
$359$	−3588.00	−0.527486	−0.263743	+	0.964593i	$0.584957\pi$
$360$	0	0
$361$	−6459.00	−0.941682
$362$	0	0
$363$	−807.000	−0.116685
$364$	0	0
$365$	10580.0	1.51721
$366$	0	0
$367$	−11272.0	−1.60325	−0.801626	−	0.597826i	$-0.796032\pi$
$367$	−11272.0	−1.60325	−0.801626	+	0.597826i	$0.796032\pi$
$368$	0	0
$369$	−3258.00	−0.459633
$370$	0	0
$371$	3056.00	0.427654
$372$	0	0
$373$	−10914.0	−1.51503	−0.757514	−	0.652819i	$-0.773586\pi$
$373$	−10914.0	−1.51503	−0.757514	+	0.652819i	$0.773586\pi$
$374$	0	0
$375$	4500.00	0.619677
$376$	0	0
$377$	−234.000	−0.0319671
$378$	0	0
$379$	−8100.00	−1.09781	−0.548904	−	0.835886i	$-0.684955\pi$
$379$	−8100.00	−1.09781	−0.548904	+	0.835886i	$0.684955\pi$
$380$	0	0
$381$	3888.00	0.522804
$382$	0	0
$383$	−6180.00	−0.824499	−0.412250	−	0.911071i	$-0.635257\pi$
$383$	−6180.00	−0.824499	−0.412250	+	0.911071i	$0.635257\pi$
$384$	0	0
$385$	−3200.00	−0.423603
$386$	0	0
$387$	−684.000	−0.0898441
$388$	0	0
$389$	−7522.00	−0.980413	−0.490206	−	0.871606i	$-0.663079\pi$
$389$	−7522.00	−0.980413	−0.490206	+	0.871606i	$0.663079\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−2676.00	−0.343477
$394$	0	0
$395$	9760.00	1.24324
$396$	0	0
$397$	6078.00	0.768378	0.384189	−	0.923254i	$-0.374481\pi$
$397$	6078.00	0.768378	0.384189	+	0.923254i	$0.374481\pi$
$398$	0	0
$399$	−480.000	−0.0602257
$400$	0	0
$401$	1830.00	0.227895	0.113947	−	0.993487i	$-0.463650\pi$
$401$	1830.00	0.227895	0.113947	+	0.993487i	$0.463650\pi$
$402$	0	0
$403$	2392.00	0.295668
$404$	0	0
$405$	810.000	0.0993808
$406$	0	0
$407$	2960.00	0.360496
$408$	0	0
$409$	12434.0	1.50323	0.751616	−	0.659601i	$-0.229275\pi$
$409$	12434.0	1.50323	0.751616	+	0.659601i	$0.229275\pi$
$410$	0	0
$411$	−6978.00	−0.837468
$412$	0	0
$413$	−3712.00	−0.442265
$414$	0	0
$415$	10080.0	1.19231
$416$	0	0
$417$	5796.00	0.680651
$418$	0	0
$419$	14188.0	1.65425	0.827123	−	0.562021i	$-0.189976\pi$
$419$	14188.0	1.65425	0.827123	+	0.562021i	$0.189976\pi$
$420$	0	0
$421$	8638.00	0.999977	0.499989	−	0.866032i	$-0.333338\pi$
$421$	8638.00	0.999977	0.499989	+	0.866032i	$0.333338\pi$
$422$	0	0
$423$	4068.00	0.467596
$424$	0	0
$425$	−3250.00	−0.370937
$426$	0	0
$427$	2864.00	0.324587
$428$	0	0
$429$	1560.00	0.175565
$430$	0	0
$431$	−4292.00	−0.479671	−0.239836	−	0.970813i	$-0.577094\pi$
$431$	−4292.00	−0.479671	−0.239836	+	0.970813i	$0.577094\pi$
$432$	0	0
$433$	−5982.00	−0.663918	−0.331959	−	0.943294i	$-0.607710\pi$
$433$	−5982.00	−0.663918	−0.331959	+	0.943294i	$0.607710\pi$
$434$	0	0
$435$	540.000	0.0595196
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−256.000	−0.0278319	−0.0139160	−	0.999903i	$-0.504430\pi$
$439$	−256.000	−0.0278319	−0.0139160	+	0.999903i	$0.504430\pi$
$440$	0	0
$441$	−2511.00	−0.271137
$442$	0	0
$443$	−12556.0	−1.34662	−0.673311	−	0.739359i	$-0.735128\pi$
$443$	−12556.0	−1.34662	−0.673311	+	0.739359i	$0.735128\pi$
$444$	0	0
$445$	−3860.00	−0.411194
$446$	0	0
$447$	−2646.00	−0.279981
$448$	0	0
$449$	5574.00	0.585865	0.292932	−	0.956133i	$-0.405369\pi$
$449$	5574.00	0.585865	0.292932	+	0.956133i	$0.405369\pi$
$450$	0	0
$451$	14480.0	1.51183
$452$	0	0
$453$	−5328.00	−0.552608
$454$	0	0
$455$	1040.00	0.107156
$456$	0	0
$457$	1266.00	0.129586	0.0647932	−	0.997899i	$-0.479361\pi$
$457$	1266.00	0.129586	0.0647932	+	0.997899i	$0.479361\pi$
$458$	0	0
$459$	−3510.00	−0.356934
$460$	0	0
$461$	7554.00	0.763178	0.381589	−	0.924332i	$-0.375377\pi$
$461$	7554.00	0.763178	0.381589	+	0.924332i	$0.375377\pi$
$462$	0	0
$463$	6752.00	0.677737	0.338868	−	0.940834i	$-0.389956\pi$
$463$	6752.00	0.677737	0.338868	+	0.940834i	$0.389956\pi$
$464$	0	0
$465$	−5520.00	−0.550503
$466$	0	0
$467$	−7924.00	−0.785180	−0.392590	−	0.919714i	$-0.628421\pi$
$467$	−7924.00	−0.785180	−0.392590	+	0.919714i	$0.628421\pi$
$468$	0	0
$469$	5600.00	0.551352
$470$	0	0
$471$	7230.00	0.707305
$472$	0	0
$473$	3040.00	0.295517
$474$	0	0
$475$	−500.000	−0.0482980
$476$	0	0
$477$	3438.00	0.330011
$478$	0	0
$479$	11084.0	1.05729	0.528644	−	0.848844i	$-0.322701\pi$
$479$	11084.0	1.05729	0.528644	+	0.848844i	$0.322701\pi$
$480$	0	0
$481$	−962.000	−0.0911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6140.00	−0.574852
$486$	0	0
$487$	−4432.00	−0.412388	−0.206194	−	0.978511i	$-0.566108\pi$
$487$	−4432.00	−0.412388	−0.206194	+	0.978511i	$0.566108\pi$
$488$	0	0
$489$	9636.00	0.891114
$490$	0	0
$491$	1140.00	0.104781	0.0523905	−	0.998627i	$-0.483316\pi$
$491$	1140.00	0.104781	0.0523905	+	0.998627i	$0.483316\pi$
$492$	0	0
$493$	−2340.00	−0.213769
$494$	0	0
$495$	−3600.00	−0.326885
$496$	0	0
$497$	5984.00	0.540079
$498$	0	0
$499$	−1764.00	−0.158251	−0.0791257	−	0.996865i	$-0.525213\pi$
$499$	−1764.00	−0.158251	−0.0791257	+	0.996865i	$0.525213\pi$
$500$	0	0
$501$	5004.00	0.446232
$502$	0	0
$503$	−16976.0	−1.50482	−0.752408	−	0.658697i	$-0.771108\pi$
$503$	−16976.0	−1.50482	−0.752408	+	0.658697i	$0.771108\pi$
$504$	0	0
$505$	5180.00	0.456449
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	9474.00	0.825005	0.412503	−	0.910956i	$-0.364655\pi$
$509$	9474.00	0.825005	0.412503	+	0.910956i	$0.364655\pi$
$510$	0	0
$511$	8464.00	0.732731
$512$	0	0
$513$	−540.000	−0.0464748
$514$	0	0
$515$	−1120.00	−0.0958313
$516$	0	0
$517$	−18080.0	−1.53802
$518$	0	0
$519$	−10794.0	−0.912917
$520$	0	0
$521$	14114.0	1.18684	0.593422	−	0.804892i	$-0.297777\pi$
$521$	14114.0	1.18684	0.593422	+	0.804892i	$0.297777\pi$
$522$	0	0
$523$	−20284.0	−1.69590	−0.847952	−	0.530074i	$-0.822164\pi$
$523$	−20284.0	−1.69590	−0.847952	+	0.530074i	$0.822164\pi$
$524$	0	0
$525$	600.000	0.0498784
$526$	0	0
$527$	23920.0	1.97718
$528$	0	0
$529$	−12167.0	−1.00000
$530$	0	0
$531$	−4176.00	−0.341286
$532$	0	0
$533$	−4706.00	−0.382438
$534$	0	0
$535$	3720.00	0.300616
$536$	0	0
$537$	3204.00	0.257473
$538$	0	0
$539$	11160.0	0.891828
$540$	0	0
$541$	−14362.0	−1.14135	−0.570675	−	0.821176i	$-0.693318\pi$
$541$	−14362.0	−1.14135	−0.570675	+	0.821176i	$0.693318\pi$
$542$	0	0
$543$	14358.0	1.13473
$544$	0	0
$545$	9340.00	0.734095
$546$	0	0
$547$	20956.0	1.63805	0.819025	−	0.573757i	$-0.194515\pi$
$547$	20956.0	1.63805	0.819025	+	0.573757i	$0.194515\pi$
$548$	0	0
$549$	3222.00	0.250477
$550$	0	0
$551$	−360.000	−0.0278340
$552$	0	0
$553$	7808.00	0.600416
$554$	0	0
$555$	2220.00	0.169791
$556$	0	0
$557$	−4134.00	−0.314476	−0.157238	−	0.987561i	$-0.550259\pi$
$557$	−4134.00	−0.314476	−0.157238	+	0.987561i	$0.550259\pi$
$558$	0	0
$559$	−988.000	−0.0747548
$560$	0	0
$561$	15600.0	1.17403
$562$	0	0
$563$	16228.0	1.21479	0.607397	−	0.794399i	$-0.292214\pi$
$563$	16228.0	1.21479	0.607397	+	0.794399i	$0.292214\pi$
$564$	0	0
$565$	19140.0	1.42518
$566$	0	0
$567$	648.000	0.0479955
$568$	0	0
$569$	2514.00	0.185224	0.0926119	−	0.995702i	$-0.470478\pi$
$569$	2514.00	0.185224	0.0926119	+	0.995702i	$0.470478\pi$
$570$	0	0
$571$	11612.0	0.851046	0.425523	−	0.904948i	$-0.360090\pi$
$571$	11612.0	0.851046	0.425523	+	0.904948i	$0.360090\pi$
$572$	0	0
$573$	−3936.00	−0.286961
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6354.00	0.458441	0.229221	−	0.973375i	$-0.426382\pi$
$577$	6354.00	0.458441	0.229221	+	0.973375i	$0.426382\pi$
$578$	0	0
$579$	1050.00	0.0753653
$580$	0	0
$581$	8064.00	0.575819
$582$	0	0
$583$	−15280.0	−1.08548
$584$	0	0
$585$	1170.00	0.0826898
$586$	0	0
$587$	13240.0	0.930960	0.465480	−	0.885059i	$-0.345882\pi$
$587$	13240.0	0.930960	0.465480	+	0.885059i	$0.345882\pi$
$588$	0	0
$589$	3680.00	0.257439
$590$	0	0
$591$	1026.00	0.0714112
$592$	0	0
$593$	−1146.00	−0.0793602	−0.0396801	−	0.999212i	$-0.512634\pi$
$593$	−1146.00	−0.0793602	−0.0396801	+	0.999212i	$0.512634\pi$
$594$	0	0
$595$	10400.0	0.716569
$596$	0	0
$597$	−10104.0	−0.692679
$598$	0	0
$599$	−10464.0	−0.713769	−0.356884	−	0.934149i	$-0.616161\pi$
$599$	−10464.0	−0.713769	−0.356884	+	0.934149i	$0.616161\pi$
$600$	0	0
$601$	6650.00	0.451346	0.225673	−	0.974203i	$-0.427542\pi$
$601$	6650.00	0.451346	0.225673	+	0.974203i	$0.427542\pi$
$602$	0	0
$603$	6300.00	0.425466
$604$	0	0
$605$	2690.00	0.180767
$606$	0	0
$607$	6664.00	0.445607	0.222803	−	0.974863i	$-0.428479\pi$
$607$	6664.00	0.445607	0.222803	+	0.974863i	$0.428479\pi$
$608$	0	0
$609$	432.000	0.0287447
$610$	0	0
$611$	5876.00	0.389063
$612$	0	0
$613$	2134.00	0.140606	0.0703030	−	0.997526i	$-0.477603\pi$
$613$	2134.00	0.140606	0.0703030	+	0.997526i	$0.477603\pi$
$614$	0	0
$615$	10860.0	0.712061
$616$	0	0
$617$	−714.000	−0.0465876	−0.0232938	−	0.999729i	$-0.507415\pi$
$617$	−714.000	−0.0465876	−0.0232938	+	0.999729i	$0.507415\pi$
$618$	0	0
$619$	−29228.0	−1.89786	−0.948928	−	0.315494i	$-0.897830\pi$
$619$	−29228.0	−1.89786	−0.948928	+	0.315494i	$0.897830\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3088.00	−0.198584
$624$	0	0
$625$	−11875.0	−0.760000
$626$	0	0
$627$	2400.00	0.152866
$628$	0	0
$629$	−9620.00	−0.609816
$630$	0	0
$631$	13536.0	0.853977	0.426989	−	0.904257i	$-0.359574\pi$
$631$	13536.0	0.853977	0.426989	+	0.904257i	$0.359574\pi$
$632$	0	0
$633$	−6012.00	−0.377497
$634$	0	0
$635$	−12960.0	−0.809924
$636$	0	0
$637$	−3627.00	−0.225600
$638$	0	0
$639$	6732.00	0.416767
$640$	0	0
$641$	17218.0	1.06095	0.530476	−	0.847700i	$-0.322013\pi$
$641$	17218.0	1.06095	0.530476	+	0.847700i	$0.322013\pi$
$642$	0	0
$643$	−15044.0	−0.922671	−0.461335	−	0.887226i	$-0.652630\pi$
$643$	−15044.0	−0.922671	−0.461335	+	0.887226i	$0.652630\pi$
$644$	0	0
$645$	2280.00	0.139186
$646$	0	0
$647$	−25176.0	−1.52978	−0.764892	−	0.644158i	$-0.777208\pi$
$647$	−25176.0	−1.52978	−0.764892	+	0.644158i	$0.777208\pi$
$648$	0	0
$649$	18560.0	1.12256
$650$	0	0
$651$	−4416.00	−0.265863
$652$	0	0
$653$	−16034.0	−0.960887	−0.480443	−	0.877026i	$-0.659524\pi$
$653$	−16034.0	−0.960887	−0.480443	+	0.877026i	$0.659524\pi$
$654$	0	0
$655$	8920.00	0.532112
$656$	0	0
$657$	9522.00	0.565432
$658$	0	0
$659$	−25356.0	−1.49883	−0.749415	−	0.662100i	$-0.769665\pi$
$659$	−25356.0	−1.49883	−0.749415	+	0.662100i	$0.769665\pi$
$660$	0	0
$661$	18310.0	1.07742	0.538711	−	0.842490i	$-0.318911\pi$
$661$	18310.0	1.07742	0.538711	+	0.842490i	$0.318911\pi$
$662$	0	0
$663$	−5070.00	−0.296987
$664$	0	0
$665$	1600.00	0.0933013
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−16824.0	−0.972277
$670$	0	0
$671$	−14320.0	−0.823871
$672$	0	0
$673$	24802.0	1.42057	0.710287	−	0.703912i	$-0.248565\pi$
$673$	24802.0	1.42057	0.710287	+	0.703912i	$0.248565\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	−22706.0	−1.28901	−0.644507	−	0.764598i	$-0.722937\pi$
$677$	−22706.0	−1.28901	−0.644507	+	0.764598i	$0.722937\pi$
$678$	0	0
$679$	−4912.00	−0.277622
$680$	0	0
$681$	−5784.00	−0.325467
$682$	0	0
$683$	14792.0	0.828697	0.414349	−	0.910118i	$-0.364009\pi$
$683$	14792.0	0.828697	0.414349	+	0.910118i	$0.364009\pi$
$684$	0	0
$685$	23260.0	1.29740
$686$	0	0
$687$	11814.0	0.656088
$688$	0	0
$689$	4966.00	0.274586
$690$	0	0
$691$	1148.00	0.0632011	0.0316006	−	0.999501i	$-0.489940\pi$
$691$	1148.00	0.0632011	0.0316006	+	0.999501i	$0.489940\pi$
$692$	0	0
$693$	−2880.00	−0.157867
$694$	0	0
$695$	−19320.0	−1.05446
$696$	0	0
$697$	−47060.0	−2.55742
$698$	0	0
$699$	−7686.00	−0.415896
$700$	0	0
$701$	14870.0	0.801187	0.400594	−	0.916256i	$-0.368804\pi$
$701$	14870.0	0.801187	0.400594	+	0.916256i	$0.368804\pi$
$702$	0	0
$703$	−1480.00	−0.0794015
$704$	0	0
$705$	−13560.0	−0.724396
$706$	0	0
$707$	4144.00	0.220440
$708$	0	0
$709$	−6354.00	−0.336572	−0.168286	−	0.985738i	$-0.553823\pi$
$709$	−6354.00	−0.336572	−0.168286	+	0.985738i	$0.553823\pi$
$710$	0	0
$711$	8784.00	0.463327
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−5200.00	−0.271985
$716$	0	0
$717$	21492.0	1.11943
$718$	0	0
$719$	−9288.00	−0.481758	−0.240879	−	0.970555i	$-0.577436\pi$
$719$	−9288.00	−0.481758	−0.240879	+	0.970555i	$0.577436\pi$
$720$	0	0
$721$	−896.000	−0.0462813
$722$	0	0
$723$	18546.0	0.953988
$724$	0	0
$725$	450.000	0.0230518
$726$	0	0
$727$	21544.0	1.09907	0.549534	−	0.835471i	$-0.314805\pi$
$727$	21544.0	1.09907	0.549534	+	0.835471i	$0.314805\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−9880.00	−0.499897
$732$	0	0
$733$	19990.0	1.00730	0.503648	−	0.863909i	$-0.331991\pi$
$733$	19990.0	1.00730	0.503648	+	0.863909i	$0.331991\pi$
$734$	0	0
$735$	8370.00	0.420044
$736$	0	0
$737$	−28000.0	−1.39945
$738$	0	0
$739$	−532.000	−0.0264816	−0.0132408	−	0.999912i	$-0.504215\pi$
$739$	−532.000	−0.0264816	−0.0132408	+	0.999912i	$0.504215\pi$
$740$	0	0
$741$	−780.000	−0.0386694
$742$	0	0
$743$	25452.0	1.25672	0.628360	−	0.777922i	$-0.283726\pi$
$743$	25452.0	1.25672	0.628360	+	0.777922i	$0.283726\pi$
$744$	0	0
$745$	8820.00	0.433745
$746$	0	0
$747$	9072.00	0.444347
$748$	0	0
$749$	2976.00	0.145181
$750$	0	0
$751$	−6440.00	−0.312915	−0.156457	−	0.987685i	$-0.550007\pi$
$751$	−6440.00	−0.312915	−0.156457	+	0.987685i	$0.550007\pi$
$752$	0	0
$753$	−4188.00	−0.202682
$754$	0	0
$755$	17760.0	0.856096
$756$	0	0
$757$	−786.000	−0.0377380	−0.0188690	−	0.999822i	$-0.506007\pi$
$757$	−786.000	−0.0377380	−0.0188690	+	0.999822i	$0.506007\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1498.00	−0.0713567	−0.0356784	−	0.999363i	$-0.511359\pi$
$761$	−1498.00	−0.0713567	−0.0356784	+	0.999363i	$0.511359\pi$
$762$	0	0
$763$	7472.00	0.354528
$764$	0	0
$765$	11700.0	0.552960
$766$	0	0
$767$	−6032.00	−0.283967
$768$	0	0
$769$	14738.0	0.691113	0.345556	−	0.938398i	$-0.387690\pi$
$769$	14738.0	0.691113	0.345556	+	0.938398i	$0.387690\pi$
$770$	0	0
$771$	−20718.0	−0.967757
$772$	0	0
$773$	−3822.00	−0.177837	−0.0889184	−	0.996039i	$-0.528341\pi$
$773$	−3822.00	−0.177837	−0.0889184	+	0.996039i	$0.528341\pi$
$774$	0	0
$775$	−4600.00	−0.213209
$776$	0	0
$777$	1776.00	0.0819995
$778$	0	0
$779$	−7240.00	−0.332991
$780$	0	0
$781$	−29920.0	−1.37083
$782$	0	0
$783$	486.000	0.0221816
$784$	0	0
$785$	−24100.0	−1.09575
$786$	0	0
$787$	11900.0	0.538995	0.269498	−	0.963001i	$-0.413142\pi$
$787$	11900.0	0.538995	0.269498	+	0.963001i	$0.413142\pi$
$788$	0	0
$789$	−20544.0	−0.926978
$790$	0	0
$791$	15312.0	0.688283
$792$	0	0
$793$	4654.00	0.208409
$794$	0	0
$795$	−11460.0	−0.511251
$796$	0	0
$797$	−21274.0	−0.945500	−0.472750	−	0.881197i	$-0.656739\pi$
$797$	−21274.0	−0.945500	−0.472750	+	0.881197i	$0.656739\pi$
$798$	0	0
$799$	58760.0	2.60173
$800$	0	0
$801$	−3474.00	−0.153243
$802$	0	0
$803$	−42320.0	−1.85983
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18102.0	0.789617
$808$	0	0
$809$	−27566.0	−1.19798	−0.598992	−	0.800755i	$-0.704432\pi$
$809$	−27566.0	−1.19798	−0.598992	+	0.800755i	$0.704432\pi$
$810$	0	0
$811$	11244.0	0.486844	0.243422	−	0.969921i	$-0.421730\pi$
$811$	11244.0	0.486844	0.243422	+	0.969921i	$0.421730\pi$
$812$	0	0
$813$	14496.0	0.625334
$814$	0	0
$815$	−32120.0	−1.38051
$816$	0	0
$817$	−1520.00	−0.0650894
$818$	0	0
$819$	936.000	0.0399347
$820$	0	0
$821$	13554.0	0.576173	0.288086	−	0.957604i	$-0.406981\pi$
$821$	13554.0	0.576173	0.288086	+	0.957604i	$0.406981\pi$
$822$	0	0
$823$	−14384.0	−0.609228	−0.304614	−	0.952476i	$-0.598527\pi$
$823$	−14384.0	−0.609228	−0.304614	+	0.952476i	$0.598527\pi$
$824$	0	0
$825$	−3000.00	−0.126602
$826$	0	0
$827$	2488.00	0.104615	0.0523073	−	0.998631i	$-0.483342\pi$
$827$	2488.00	0.104615	0.0523073	+	0.998631i	$0.483342\pi$
$828$	0	0
$829$	−20858.0	−0.873858	−0.436929	−	0.899496i	$-0.643934\pi$
$829$	−20858.0	−0.873858	−0.436929	+	0.899496i	$0.643934\pi$
$830$	0	0
$831$	12246.0	0.511202
$832$	0	0
$833$	−36270.0	−1.50862
$834$	0	0
$835$	−16680.0	−0.691300
$836$	0	0
$837$	−4968.00	−0.205160
$838$	0	0
$839$	−23116.0	−0.951195	−0.475598	−	0.879663i	$-0.657768\pi$
$839$	−23116.0	−0.951195	−0.475598	+	0.879663i	$0.657768\pi$
$840$	0	0
$841$	−24065.0	−0.986715
$842$	0	0
$843$	−10050.0	−0.410605
$844$	0	0
$845$	1690.00	0.0688021
$846$	0	0
$847$	2152.00	0.0873006
$848$	0	0
$849$	23388.0	0.945435
$850$	0	0
$851$	0	0
$852$	0	0
$853$	934.000	0.0374907	0.0187453	−	0.999824i	$-0.494033\pi$
$853$	934.000	0.0374907	0.0187453	+	0.999824i	$0.494033\pi$
$854$	0	0
$855$	1800.00	0.0719985
$856$	0	0
$857$	12642.0	0.503900	0.251950	−	0.967740i	$-0.418928\pi$
$857$	12642.0	0.503900	0.251950	+	0.967740i	$0.418928\pi$
$858$	0	0
$859$	22796.0	0.905459	0.452730	−	0.891648i	$-0.350450\pi$
$859$	22796.0	0.905459	0.452730	+	0.891648i	$0.350450\pi$
$860$	0	0
$861$	8688.00	0.343886
$862$	0	0
$863$	76.0000	0.00299776	0.00149888	−	0.999999i	$-0.499523\pi$
$863$	76.0000	0.00299776	0.00149888	+	0.999999i	$0.499523\pi$
$864$	0	0
$865$	35980.0	1.41429
$866$	0	0
$867$	−35961.0	−1.40865
$868$	0	0
$869$	−39040.0	−1.52398
$870$	0	0
$871$	9100.00	0.354009
$872$	0	0
$873$	−5526.00	−0.214235
$874$	0	0
$875$	−12000.0	−0.463627
$876$	0	0
$877$	−46130.0	−1.77617	−0.888084	−	0.459681i	$-0.847964\pi$
$877$	−46130.0	−1.77617	−0.888084	+	0.459681i	$0.847964\pi$
$878$	0	0
$879$	−11766.0	−0.451487
$880$	0	0
$881$	6682.00	0.255530	0.127765	−	0.991804i	$-0.459220\pi$
$881$	6682.00	0.255530	0.127765	+	0.991804i	$0.459220\pi$
$882$	0	0
$883$	−47404.0	−1.80665	−0.903325	−	0.428957i	$-0.858881\pi$
$883$	−47404.0	−1.80665	−0.903325	+	0.428957i	$0.858881\pi$
$884$	0	0
$885$	13920.0	0.528718
$886$	0	0
$887$	−33672.0	−1.27463	−0.637314	−	0.770604i	$-0.719955\pi$
$887$	−33672.0	−1.27463	−0.637314	+	0.770604i	$0.719955\pi$
$888$	0	0
$889$	−10368.0	−0.391149
$890$	0	0
$891$	−3240.00	−0.121823
$892$	0	0
$893$	9040.00	0.338759
$894$	0	0
$895$	−10680.0	−0.398875
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3312.00	−0.122871
$900$	0	0
$901$	49660.0	1.83620
$902$	0	0
$903$	1824.00	0.0672192
$904$	0	0
$905$	−47860.0	−1.75792
$906$	0	0
$907$	14540.0	0.532296	0.266148	−	0.963932i	$-0.414249\pi$
$907$	14540.0	0.532296	0.266148	+	0.963932i	$0.414249\pi$
$908$	0	0
$909$	4662.00	0.170109
$910$	0	0
$911$	7840.00	0.285127	0.142564	−	0.989786i	$-0.454465\pi$
$911$	7840.00	0.285127	0.142564	+	0.989786i	$0.454465\pi$
$912$	0	0
$913$	−40320.0	−1.46155
$914$	0	0
$915$	−10740.0	−0.388037
$916$	0	0
$917$	7136.00	0.256981
$918$	0	0
$919$	−47720.0	−1.71288	−0.856440	−	0.516246i	$-0.827329\pi$
$919$	−47720.0	−1.71288	−0.856440	+	0.516246i	$0.827329\pi$
$920$	0	0
$921$	17868.0	0.639273
$922$	0	0
$923$	9724.00	0.346771
$924$	0	0
$925$	1850.00	0.0657596
$926$	0	0
$927$	−1008.00	−0.0357142
$928$	0	0
$929$	7502.00	0.264944	0.132472	−	0.991187i	$-0.457709\pi$
$929$	7502.00	0.264944	0.132472	+	0.991187i	$0.457709\pi$
$930$	0	0
$931$	−5580.00	−0.196431
$932$	0	0
$933$	7056.00	0.247592
$934$	0	0
$935$	−52000.0	−1.81880
$936$	0	0
$937$	22058.0	0.769054	0.384527	−	0.923114i	$-0.374365\pi$
$937$	22058.0	0.769054	0.384527	+	0.923114i	$0.374365\pi$
$938$	0	0
$939$	−25326.0	−0.880173
$940$	0	0
$941$	23338.0	0.808498	0.404249	−	0.914649i	$-0.367533\pi$
$941$	23338.0	0.808498	0.404249	+	0.914649i	$0.367533\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−2160.00	−0.0743543
$946$	0	0
$947$	30488.0	1.04617	0.523087	−	0.852279i	$-0.324780\pi$
$947$	30488.0	1.04617	0.523087	+	0.852279i	$0.324780\pi$
$948$	0	0
$949$	13754.0	0.470468
$950$	0	0
$951$	16650.0	0.567732
$952$	0	0
$953$	9522.00	0.323660	0.161830	−	0.986819i	$-0.448260\pi$
$953$	9522.00	0.323660	0.161830	+	0.986819i	$0.448260\pi$
$954$	0	0
$955$	13120.0	0.444558
$956$	0	0
$957$	−2160.00	−0.0729602
$958$	0	0
$959$	18608.0	0.626573
$960$	0	0
$961$	4065.00	0.136451
$962$	0	0
$963$	3348.00	0.112033
$964$	0	0
$965$	−3500.00	−0.116755
$966$	0	0
$967$	7616.00	0.253272	0.126636	−	0.991949i	$-0.459582\pi$
$967$	7616.00	0.253272	0.126636	+	0.991949i	$0.459582\pi$
$968$	0	0
$969$	−7800.00	−0.258588
$970$	0	0
$971$	−51316.0	−1.69599	−0.847996	−	0.530002i	$-0.822191\pi$
$971$	−51316.0	−1.69599	−0.847996	+	0.530002i	$0.822191\pi$
$972$	0	0
$973$	−15456.0	−0.509246
$974$	0	0
$975$	975.000	0.0320256
$976$	0	0
$977$	−48666.0	−1.59362	−0.796808	−	0.604232i	$-0.793480\pi$
$977$	−48666.0	−1.59362	−0.796808	+	0.604232i	$0.793480\pi$
$978$	0	0
$979$	15440.0	0.504050
$980$	0	0
$981$	8406.00	0.273581
$982$	0	0
$983$	−17388.0	−0.564182	−0.282091	−	0.959388i	$-0.591028\pi$
$983$	−17388.0	−0.564182	−0.282091	+	0.959388i	$0.591028\pi$
$984$	0	0
$985$	−3420.00	−0.110630
$986$	0	0
$987$	−10848.0	−0.349844
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−11496.0	−0.368499	−0.184249	−	0.982880i	$-0.558985\pi$
$991$	−11496.0	−0.368499	−0.184249	+	0.982880i	$0.558985\pi$
$992$	0	0
$993$	420.000	0.0134223
$994$	0	0
$995$	33680.0	1.07309
$996$	0	0
$997$	48862.0	1.55213	0.776066	−	0.630652i	$-0.217212\pi$
$997$	48862.0	1.55213	0.776066	+	0.630652i	$0.217212\pi$
$998$	0	0
$999$	1998.00	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.a.d.1.1		1
3.2	odd	2		1872.4.a.d.1.1		1
4.3	odd	2		78.4.a.c.1.1	✓	1
8.3	odd	2		2496.4.a.a.1.1		1
8.5	even	2		2496.4.a.j.1.1		1
12.11	even	2		234.4.a.h.1.1		1
20.19	odd	2		1950.4.a.l.1.1		1
52.31	even	4		1014.4.b.h.337.2		2
52.47	even	4		1014.4.b.h.337.1		2
52.51	odd	2		1014.4.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.4.a.c.1.1	✓	1	4.3	odd	2
234.4.a.h.1.1		1	12.11	even	2
624.4.a.d.1.1		1	1.1	even	1	trivial
1014.4.a.j.1.1		1	52.51	odd	2
1014.4.b.h.337.1		2	52.47	even	4
1014.4.b.h.337.2		2	52.31	even	4
1872.4.a.d.1.1		1	3.2	odd	2
1950.4.a.l.1.1		1	20.19	odd	2
2496.4.a.a.1.1		1	8.3	odd	2
2496.4.a.j.1.1		1	8.5	even	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 $\F_{q}$
Belyi maps

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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