LMFDB - Embedded newform 1950.4.a.l.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1950,4,Mod(1,1950)] 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("1950.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1950, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1950.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,2,-3,4,0,-6,8,8,9,0,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$115.053724511$
Analytic rank:	$1$
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	$\chi$	$=$	1950.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{6} +8.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +40.0000 q^{11} -12.0000 q^{12} -13.0000 q^{13} +16.0000 q^{14} +16.0000 q^{16} -130.000 q^{17} +18.0000 q^{18} -20.0000 q^{19} -24.0000 q^{21} +80.0000 q^{22} -24.0000 q^{24} -26.0000 q^{26} -27.0000 q^{27} +32.0000 q^{28} -18.0000 q^{29} -184.000 q^{31} +32.0000 q^{32} -120.000 q^{33} -260.000 q^{34} +36.0000 q^{36} +74.0000 q^{37} -40.0000 q^{38} +39.0000 q^{39} -362.000 q^{41} -48.0000 q^{42} -76.0000 q^{43} +160.000 q^{44} +452.000 q^{47} -48.0000 q^{48} -279.000 q^{49} +390.000 q^{51} -52.0000 q^{52} -382.000 q^{53} -54.0000 q^{54} +64.0000 q^{56} +60.0000 q^{57} -36.0000 q^{58} +464.000 q^{59} +358.000 q^{61} -368.000 q^{62} +72.0000 q^{63} +64.0000 q^{64} -240.000 q^{66} +700.000 q^{67} -520.000 q^{68} -748.000 q^{71} +72.0000 q^{72} -1058.00 q^{73} +148.000 q^{74} -80.0000 q^{76} +320.000 q^{77} +78.0000 q^{78} -976.000 q^{79} +81.0000 q^{81} -724.000 q^{82} +1008.00 q^{83} -96.0000 q^{84} -152.000 q^{86} +54.0000 q^{87} +320.000 q^{88} -386.000 q^{89} -104.000 q^{91} +552.000 q^{93} +904.000 q^{94} -96.0000 q^{96} +614.000 q^{97} -558.000 q^{98} +360.000 q^{99} +O(q^{100})\)

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−6.00000	−0.408248
$7$	8.00000	0.431959	0.215980	−	0.976398i	$-0.430705\pi$
$7$	8.00000	0.431959	0.215980	+	0.976398i	$0.430705\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	0	0
$11$	40.0000	1.09640	0.548202	−	0.836346i	$-0.315312\pi$
$11$	40.0000	1.09640	0.548202	+	0.836346i	$0.315312\pi$
$12$	−12.0000	−0.288675
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	16.0000	0.305441
$15$	0	0
$16$	16.0000	0.250000
$17$	−130.000	−1.85468	−0.927342	−	0.374215i	$-0.877912\pi$
$17$	−130.000	−1.85468	−0.927342	+	0.374215i	$0.877912\pi$
$18$	18.0000	0.235702
$19$	−20.0000	−0.241490	−0.120745	−	0.992684i	$-0.538528\pi$
$19$	−20.0000	−0.241490	−0.120745	+	0.992684i	$0.538528\pi$
$20$	0	0
$21$	−24.0000	−0.249392
$22$	80.0000	0.775275
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−24.0000	−0.204124
$25$	0	0
$26$	−26.0000	−0.196116
$27$	−27.0000	−0.192450
$28$	32.0000	0.215980
$29$	−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	$-0.678944\pi$
$31$	−184.000	−1.06604	−0.533022	+	0.846101i	$0.678944\pi$
$32$	32.0000	0.176777
$33$	−120.000	−0.633010
$34$	−260.000	−1.31146
$35$	0	0
$36$	36.0000	0.166667
$37$	74.0000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	74.0000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−40.0000	−0.170759
$39$	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	$0.742148\pi$
$42$	−48.0000	−0.176347
$43$	−76.0000	−0.269532	−0.134766	−	0.990877i	$-0.543028\pi$
$43$	−76.0000	−0.269532	−0.134766	+	0.990877i	$0.543028\pi$
$44$	160.000	0.548202
$45$	0	0
$46$	0	0
$47$	452.000	1.40279	0.701393	−	0.712774i	$-0.252562\pi$
$47$	452.000	1.40279	0.701393	+	0.712774i	$0.252562\pi$
$48$	−48.0000	−0.144338
$49$	−279.000	−0.813411
$50$	0	0
$51$	390.000	1.07080
$52$	−52.0000	−0.138675
$53$	−382.000	−0.990033	−0.495016	−	0.868884i	$-0.664838\pi$
$53$	−382.000	−0.990033	−0.495016	+	0.868884i	$0.664838\pi$
$54$	−54.0000	−0.136083
$55$	0	0
$56$	64.0000	0.152721
$57$	60.0000	0.139424
$58$	−36.0000	−0.0815005
$59$	464.000	1.02386	0.511929	−	0.859028i	$-0.328931\pi$
$59$	464.000	1.02386	0.511929	+	0.859028i	$0.328931\pi$
$60$	0	0
$61$	358.000	0.751430	0.375715	−	0.926735i	$-0.377397\pi$
$61$	358.000	0.751430	0.375715	+	0.926735i	$0.377397\pi$
$62$	−368.000	−0.753807
$63$	72.0000	0.143986
$64$	64.0000	0.125000
$65$	0	0
$66$	−240.000	−0.447605
$67$	700.000	1.27640	0.638199	−	0.769872i	$-0.279680\pi$
$67$	700.000	1.27640	0.638199	+	0.769872i	$0.279680\pi$
$68$	−520.000	−0.927342
$69$	0	0
$70$	0	0
$71$	−748.000	−1.25030	−0.625150	−	0.780505i	$-0.714962\pi$
$71$	−748.000	−1.25030	−0.625150	+	0.780505i	$0.714962\pi$
$72$	72.0000	0.117851
$73$	−1058.00	−1.69629	−0.848147	−	0.529760i	$-0.822282\pi$
$73$	−1058.00	−1.69629	−0.848147	+	0.529760i	$0.822282\pi$
$74$	148.000	0.232495
$75$	0	0
$76$	−80.0000	−0.120745
$77$	320.000	0.473602
$78$	78.0000	0.113228
$79$	−976.000	−1.38998	−0.694991	−	0.719018i	$-0.744592\pi$
$79$	−976.000	−1.38998	−0.694991	+	0.719018i	$0.744592\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−724.000	−0.975030
$83$	1008.00	1.33304	0.666520	−	0.745487i	$-0.267783\pi$
$83$	1008.00	1.33304	0.666520	+	0.745487i	$0.267783\pi$
$84$	−96.0000	−0.124696
$85$	0	0
$86$	−152.000	−0.190588
$87$	54.0000	0.0665449
$88$	320.000	0.387638
$89$	−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
$94$	904.000	0.991920
$95$	0	0
$96$	−96.0000	−0.102062
$97$	614.000	0.642704	0.321352	−	0.946960i	$-0.395863\pi$
$97$	614.000	0.642704	0.321352	+	0.946960i	$0.395863\pi$
$98$	−558.000	−0.575168
$99$	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	$0.417871\pi$
$102$	780.000	0.757172
$103$	−112.000	−0.107143	−0.0535713	−	0.998564i	$-0.517060\pi$
$103$	−112.000	−0.107143	−0.0535713	+	0.998564i	$0.517060\pi$
$104$	−104.000	−0.0980581
$105$	0	0
$106$	−764.000	−0.700059
$107$	372.000	0.336099	0.168050	−	0.985779i	$-0.446253\pi$
$107$	372.000	0.336099	0.168050	+	0.985779i	$0.446253\pi$
$108$	−108.000	−0.0962250
$109$	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
$112$	128.000	0.107990
$113$	−1914.00	−1.59340	−0.796699	−	0.604376i	$-0.793422\pi$
$113$	−1914.00	−1.59340	−0.796699	+	0.604376i	$0.793422\pi$
$114$	120.000	0.0985880
$115$	0	0
$116$	−72.0000	−0.0576296
$117$	−117.000	−0.0924500
$118$	928.000	0.723977
$119$	−1040.00	−0.801148
$120$	0	0
$121$	269.000	0.202104
$122$	716.000	0.531341
$123$	1086.00	0.796108
$124$	−736.000	−0.533022
$125$	0	0
$126$	144.000	0.101814
$127$	−1296.00	−0.905523	−0.452761	−	0.891632i	$-0.649561\pi$
$127$	−1296.00	−0.905523	−0.452761	+	0.891632i	$0.649561\pi$
$128$	128.000	0.0883883
$129$	228.000	0.155615
$130$	0	0
$131$	−892.000	−0.594919	−0.297460	−	0.954734i	$-0.596139\pi$
$131$	−892.000	−0.594919	−0.297460	+	0.954734i	$0.596139\pi$
$132$	−480.000	−0.316505
$133$	−160.000	−0.104314
$134$	1400.00	0.902549
$135$	0	0
$136$	−1040.00	−0.655730
$137$	−2326.00	−1.45054	−0.725269	−	0.688466i	$-0.758284\pi$
$137$	−2326.00	−1.45054	−0.725269	+	0.688466i	$0.758284\pi$
$138$	0	0
$139$	1932.00	1.17892	0.589461	−	0.807797i	$-0.299340\pi$
$139$	1932.00	1.17892	0.589461	+	0.807797i	$0.299340\pi$
$140$	0	0
$141$	−1356.00	−0.809899
$142$	−1496.00	−0.884095
$143$	−520.000	−0.304088
$144$	144.000	0.0833333
$145$	0	0
$146$	−2116.00	−1.19946
$147$	837.000	0.469623
$148$	296.000	0.164399
$149$	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	$-0.658846\pi$
$151$	−1776.00	−0.957145	−0.478572	+	0.878048i	$0.658846\pi$
$152$	−160.000	−0.0853797
$153$	−1170.00	−0.618228
$154$	640.000	0.334887
$155$	0	0
$156$	156.000	0.0800641
$157$	2410.00	1.22509	0.612544	−	0.790436i	$-0.290146\pi$
$157$	2410.00	1.22509	0.612544	+	0.790436i	$0.290146\pi$
$158$	−1952.00	−0.982866
$159$	1146.00	0.571596
$160$	0	0
$161$	0	0
$162$	162.000	0.0785674
$163$	−3212.00	−1.54346	−0.771728	−	0.635953i	$-0.780607\pi$
$163$	−3212.00	−1.54346	−0.771728	+	0.635953i	$0.780607\pi$
$164$	−1448.00	−0.689450
$165$	0	0
$166$	2016.00	0.942602
$167$	−1668.00	−0.772896	−0.386448	−	0.922311i	$-0.626298\pi$
$167$	−1668.00	−0.772896	−0.386448	+	0.922311i	$0.626298\pi$
$168$	−192.000	−0.0881733
$169$	169.000	0.0769231
$170$	0	0
$171$	−180.000	−0.0804967
$172$	−304.000	−0.134766
$173$	−3598.00	−1.58122	−0.790609	−	0.612321i	$-0.790236\pi$
$173$	−3598.00	−1.58122	−0.790609	+	0.612321i	$0.790236\pi$
$174$	108.000	0.0470544
$175$	0	0
$176$	640.000	0.274101
$177$	−1392.00	−0.591125
$178$	−772.000	−0.325078
$179$	1068.00	0.445956	0.222978	−	0.974824i	$-0.428422\pi$
$179$	1068.00	0.445956	0.222978	+	0.974824i	$0.428422\pi$
$180$	0	0
$181$	−4786.00	−1.96542	−0.982709	−	0.185158i	$-0.940720\pi$
$181$	−4786.00	−1.96542	−0.982709	+	0.185158i	$0.940720\pi$
$182$	−208.000	−0.0847142
$183$	−1074.00	−0.433838
$184$	0	0
$185$	0	0
$186$	1104.00	0.435211
$187$	−5200.00	−2.03348
$188$	1808.00	0.701393
$189$	−216.000	−0.0831306
$190$	0	0
$191$	−1312.00	−0.497031	−0.248516	−	0.968628i	$-0.579943\pi$
$191$	−1312.00	−0.497031	−0.248516	+	0.968628i	$0.579943\pi$
$192$	−192.000	−0.0721688
$193$	350.000	0.130537	0.0652683	−	0.997868i	$-0.479210\pi$
$193$	350.000	0.130537	0.0652683	+	0.997868i	$0.479210\pi$
$194$	1228.00	0.454460
$195$	0	0
$196$	−1116.00	−0.406706
$197$	342.000	0.123688	0.0618439	−	0.998086i	$-0.480302\pi$
$197$	342.000	0.123688	0.0618439	+	0.998086i	$0.480302\pi$
$198$	720.000	0.258425
$199$	−3368.00	−1.19975	−0.599877	−	0.800092i	$-0.704784\pi$
$199$	−3368.00	−1.19975	−0.599877	+	0.800092i	$0.704784\pi$
$200$	0	0
$201$	−2100.00	−0.736928
$202$	1036.00	0.360855
$203$	−144.000	−0.0497873
$204$	1560.00	0.535401
$205$	0	0
$206$	−224.000	−0.0757613
$207$	0	0
$208$	−208.000	−0.0693375
$209$	−800.000	−0.264771
$210$	0	0
$211$	−2004.00	−0.653844	−0.326922	−	0.945051i	$-0.606011\pi$
$211$	−2004.00	−0.653844	−0.326922	+	0.945051i	$0.606011\pi$
$212$	−1528.00	−0.495016
$213$	2244.00	0.721861
$214$	744.000	0.237658
$215$	0	0
$216$	−216.000	−0.0680414
$217$	−1472.00	−0.460488
$218$	1868.00	0.580353
$219$	3174.00	0.979356
$220$	0	0
$221$	1690.00	0.514397
$222$	−444.000	−0.134231
$223$	5608.00	1.68403	0.842017	−	0.539451i	$-0.181368\pi$
$223$	5608.00	1.68403	0.842017	+	0.539451i	$0.181368\pi$
$224$	256.000	0.0763604
$225$	0	0
$226$	−3828.00	−1.12670
$227$	1928.00	0.563726	0.281863	−	0.959455i	$-0.409048\pi$
$227$	1928.00	0.563726	0.281863	+	0.959455i	$0.409048\pi$
$228$	240.000	0.0697122
$229$	−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
$232$	−144.000	−0.0407503
$233$	−2562.00	−0.720353	−0.360176	−	0.932884i	$-0.617283\pi$
$233$	−2562.00	−0.720353	−0.360176	+	0.932884i	$0.617283\pi$
$234$	−234.000	−0.0653720
$235$	0	0
$236$	1856.00	0.511929
$237$	2928.00	0.802506
$238$	−2080.00	−0.566497
$239$	7164.00	1.93891	0.969457	−	0.245260i	$-0.0788733\pi$
$239$	7164.00	1.93891	0.969457	+	0.245260i	$0.0788733\pi$
$240$	0	0
$241$	−6182.00	−1.65236	−0.826178	−	0.563410i	$-0.809489\pi$
$241$	−6182.00	−1.65236	−0.826178	+	0.563410i	$0.809489\pi$
$242$	538.000	0.142909
$243$	−243.000	−0.0641500
$244$	1432.00	0.375715
$245$	0	0
$246$	2172.00	0.562934
$247$	260.000	0.0669773
$248$	−1472.00	−0.376904
$249$	−3024.00	−0.769631
$250$	0	0
$251$	−1396.00	−0.351055	−0.175527	−	0.984475i	$-0.556163\pi$
$251$	−1396.00	−0.351055	−0.175527	+	0.984475i	$0.556163\pi$
$252$	288.000	0.0719932
$253$	0	0
$254$	−2592.00	−0.640301
$255$	0	0
$256$	256.000	0.0625000
$257$	−6906.00	−1.67620	−0.838102	−	0.545514i	$-0.816335\pi$
$257$	−6906.00	−1.67620	−0.838102	+	0.545514i	$0.816335\pi$
$258$	456.000	0.110036
$259$	592.000	0.142027
$260$	0	0
$261$	−162.000	−0.0384197
$262$	−1784.00	−0.420671
$263$	6848.00	1.60557	0.802787	−	0.596266i	$-0.203350\pi$
$263$	6848.00	1.60557	0.802787	+	0.596266i	$0.203350\pi$
$264$	−960.000	−0.223803
$265$	0	0
$266$	−320.000	−0.0737611
$267$	1158.00	0.265425
$268$	2800.00	0.638199
$269$	−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	$-0.317836\pi$
$271$	4832.00	1.08311	0.541556	+	0.840665i	$0.317836\pi$
$272$	−2080.00	−0.463671
$273$	312.000	0.0691689
$274$	−4652.00	−1.02568
$275$	0	0
$276$	0	0
$277$	4082.00	0.885428	0.442714	−	0.896663i	$-0.354016\pi$
$277$	4082.00	0.885428	0.442714	+	0.896663i	$0.354016\pi$
$278$	3864.00	0.833623
$279$	−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	$0.384278\pi$
$282$	−2712.00	−0.572685
$283$	−7796.00	−1.63754	−0.818770	−	0.574121i	$-0.805344\pi$
$283$	−7796.00	−1.63754	−0.818770	+	0.574121i	$0.805344\pi$
$284$	−2992.00	−0.625150
$285$	0	0
$286$	−1040.00	−0.215023
$287$	−2896.00	−0.595629
$288$	288.000	0.0589256
$289$	11987.0	2.43985
$290$	0	0
$291$	−1842.00	−0.371065
$292$	−4232.00	−0.848147
$293$	−3922.00	−0.781999	−0.390999	−	0.920391i	$-0.627871\pi$
$293$	−3922.00	−0.781999	−0.390999	+	0.920391i	$0.627871\pi$
$294$	1674.00	0.332074
$295$	0	0
$296$	592.000	0.116248
$297$	−1080.00	−0.211003
$298$	1764.00	0.342905
$299$	0	0
$300$	0	0
$301$	−608.000	−0.116427
$302$	−3552.00	−0.676803
$303$	−1554.00	−0.294637
$304$	−320.000	−0.0603726
$305$	0	0
$306$	−2340.00	−0.437153
$307$	−5956.00	−1.10725	−0.553627	−	0.832765i	$-0.686757\pi$
$307$	−5956.00	−1.10725	−0.553627	+	0.832765i	$0.686757\pi$
$308$	1280.00	0.236801
$309$	336.000	0.0618588
$310$	0	0
$311$	2352.00	0.428841	0.214421	−	0.976741i	$-0.431214\pi$
$311$	2352.00	0.428841	0.214421	+	0.976741i	$0.431214\pi$
$312$	312.000	0.0566139
$313$	−8442.00	−1.52450	−0.762252	−	0.647280i	$-0.775907\pi$
$313$	−8442.00	−1.52450	−0.762252	+	0.647280i	$0.775907\pi$
$314$	4820.00	0.866269
$315$	0	0
$316$	−3904.00	−0.694991
$317$	5550.00	0.983341	0.491670	−	0.870781i	$-0.336386\pi$
$317$	5550.00	0.983341	0.491670	+	0.870781i	$0.336386\pi$
$318$	2292.00	0.404179
$319$	−720.000	−0.126371
$320$	0	0
$321$	−1116.00	−0.194047
$322$	0	0
$323$	2600.00	0.447888
$324$	324.000	0.0555556
$325$	0	0
$326$	−6424.00	−1.09139
$327$	−2802.00	−0.473856
$328$	−2896.00	−0.487515
$329$	3616.00	0.605947
$330$	0	0
$331$	140.000	0.0232480	0.0116240	−	0.999932i	$-0.496300\pi$
$331$	140.000	0.0232480	0.0116240	+	0.999932i	$0.496300\pi$
$332$	4032.00	0.666520
$333$	666.000	0.109599
$334$	−3336.00	−0.546520
$335$	0	0
$336$	−384.000	−0.0623480
$337$	6174.00	0.997980	0.498990	−	0.866608i	$-0.333704\pi$
$337$	6174.00	0.997980	0.498990	+	0.866608i	$0.333704\pi$
$338$	338.000	0.0543928
$339$	5742.00	0.919949
$340$	0	0
$341$	−7360.00	−1.16882
$342$	−360.000	−0.0569198
$343$	−4976.00	−0.783320
$344$	−608.000	−0.0952941
$345$	0	0
$346$	−7196.00	−1.11809
$347$	2988.00	0.462260	0.231130	−	0.972923i	$-0.425758\pi$
$347$	2988.00	0.462260	0.231130	+	0.972923i	$0.425758\pi$
$348$	216.000	0.0332725
$349$	−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
$352$	1280.00	0.193819
$353$	10754.0	1.62147	0.810733	−	0.585416i	$-0.199069\pi$
$353$	10754.0	1.62147	0.810733	+	0.585416i	$0.199069\pi$
$354$	−2784.00	−0.417989
$355$	0	0
$356$	−1544.00	−0.229865
$357$	3120.00	0.462543
$358$	2136.00	0.315338
$359$	3588.00	0.527486	0.263743	−	0.964593i	$-0.415043\pi$
$359$	3588.00	0.527486	0.263743	+	0.964593i	$0.415043\pi$
$360$	0	0
$361$	−6459.00	−0.941682
$362$	−9572.00	−1.38976
$363$	−807.000	−0.116685
$364$	−416.000	−0.0599020
$365$	0	0
$366$	−2148.00	−0.306770
$367$	−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
$372$	2208.00	0.307741
$373$	10914.0	1.51503	0.757514	−	0.652819i	$-0.226414\pi$
$373$	10914.0	1.51503	0.757514	+	0.652819i	$0.226414\pi$
$374$	−10400.0	−1.43789
$375$	0	0
$376$	3616.00	0.495960
$377$	234.000	0.0319671
$378$	−432.000	−0.0587822
$379$	8100.00	1.09781	0.548904	−	0.835886i	$-0.315045\pi$
$379$	8100.00	1.09781	0.548904	+	0.835886i	$0.315045\pi$
$380$	0	0
$381$	3888.00	0.522804
$382$	−2624.00	−0.351454
$383$	−6180.00	−0.824499	−0.412250	−	0.911071i	$-0.635257\pi$
$383$	−6180.00	−0.824499	−0.412250	+	0.911071i	$0.635257\pi$
$384$	−384.000	−0.0510310
$385$	0	0
$386$	700.000	0.0923033
$387$	−684.000	−0.0898441
$388$	2456.00	0.321352
$389$	−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
$392$	−2232.00	−0.287584
$393$	2676.00	0.343477
$394$	684.000	0.0874605
$395$	0	0
$396$	1440.00	0.182734
$397$	−6078.00	−0.768378	−0.384189	−	0.923254i	$-0.625519\pi$
$397$	−6078.00	−0.768378	−0.384189	+	0.923254i	$0.625519\pi$
$398$	−6736.00	−0.848355
$399$	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	$0.463650\pi$
$402$	−4200.00	−0.521087
$403$	2392.00	0.295668
$404$	2072.00	0.255163
$405$	0	0
$406$	−288.000	−0.0352049
$407$	2960.00	0.360496
$408$	3120.00	0.378586
$409$	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
$412$	−448.000	−0.0535713
$413$	3712.00	0.442265
$414$	0	0
$415$	0	0
$416$	−416.000	−0.0490290
$417$	−5796.00	−0.680651
$418$	−1600.00	−0.187221
$419$	−14188.0	−1.65425	−0.827123	−	0.562021i	$-0.810024\pi$
$419$	−14188.0	−1.65425	−0.827123	+	0.562021i	$0.810024\pi$
$420$	0	0
$421$	8638.00	0.999977	0.499989	−	0.866032i	$-0.333338\pi$
$421$	8638.00	0.999977	0.499989	+	0.866032i	$0.333338\pi$
$422$	−4008.00	−0.462337
$423$	4068.00	0.467596
$424$	−3056.00	−0.350029
$425$	0	0
$426$	4488.00	0.510433
$427$	2864.00	0.324587
$428$	1488.00	0.168050
$429$	1560.00	0.175565
$430$	0	0
$431$	4292.00	0.479671	0.239836	−	0.970813i	$-0.422906\pi$
$431$	4292.00	0.479671	0.239836	+	0.970813i	$0.422906\pi$
$432$	−432.000	−0.0481125
$433$	5982.00	0.663918	0.331959	−	0.943294i	$-0.392290\pi$
$433$	5982.00	0.663918	0.331959	+	0.943294i	$0.392290\pi$
$434$	−2944.00	−0.325614
$435$	0	0
$436$	3736.00	0.410371
$437$	0	0
$438$	6348.00	0.692510
$439$	256.000	0.0278319	0.0139160	−	0.999903i	$-0.495570\pi$
$439$	256.000	0.0278319	0.0139160	+	0.999903i	$0.495570\pi$
$440$	0	0
$441$	−2511.00	−0.271137
$442$	3380.00	0.363733
$443$	−12556.0	−1.34662	−0.673311	−	0.739359i	$-0.735128\pi$
$443$	−12556.0	−1.34662	−0.673311	+	0.739359i	$0.735128\pi$
$444$	−888.000	−0.0949158
$445$	0	0
$446$	11216.0	1.19079
$447$	−2646.00	−0.279981
$448$	512.000	0.0539949
$449$	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
$452$	−7656.00	−0.796699
$453$	5328.00	0.552608
$454$	3856.00	0.398615
$455$	0	0
$456$	480.000	0.0492940
$457$	−1266.00	−0.129586	−0.0647932	−	0.997899i	$-0.520639\pi$
$457$	−1266.00	−0.129586	−0.0647932	+	0.997899i	$0.520639\pi$
$458$	−7876.00	−0.803540
$459$	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	$0.375377\pi$
$462$	−1920.00	−0.193347
$463$	6752.00	0.677737	0.338868	−	0.940834i	$-0.389956\pi$
$463$	6752.00	0.677737	0.338868	+	0.940834i	$0.389956\pi$
$464$	−288.000	−0.0288148
$465$	0	0
$466$	−5124.00	−0.509366
$467$	−7924.00	−0.785180	−0.392590	−	0.919714i	$-0.628421\pi$
$467$	−7924.00	−0.785180	−0.392590	+	0.919714i	$0.628421\pi$
$468$	−468.000	−0.0462250
$469$	5600.00	0.551352
$470$	0	0
$471$	−7230.00	−0.707305
$472$	3712.00	0.361989
$473$	−3040.00	−0.295517
$474$	5856.00	0.567458
$475$	0	0
$476$	−4160.00	−0.400574
$477$	−3438.00	−0.330011
$478$	14328.0	1.37102
$479$	−11084.0	−1.05729	−0.528644	−	0.848844i	$-0.677299\pi$
$479$	−11084.0	−1.05729	−0.528644	+	0.848844i	$0.677299\pi$
$480$	0	0
$481$	−962.000	−0.0911922
$482$	−12364.0	−1.16839
$483$	0	0
$484$	1076.00	0.101052
$485$	0	0
$486$	−486.000	−0.0453609
$487$	−4432.00	−0.412388	−0.206194	−	0.978511i	$-0.566108\pi$
$487$	−4432.00	−0.412388	−0.206194	+	0.978511i	$0.566108\pi$
$488$	2864.00	0.265670
$489$	9636.00	0.891114
$490$	0	0
$491$	−1140.00	−0.104781	−0.0523905	−	0.998627i	$-0.516684\pi$
$491$	−1140.00	−0.104781	−0.0523905	+	0.998627i	$0.516684\pi$
$492$	4344.00	0.398054
$493$	2340.00	0.213769
$494$	520.000	0.0473601
$495$	0	0
$496$	−2944.00	−0.266511
$497$	−5984.00	−0.540079
$498$	−6048.00	−0.544212
$499$	1764.00	0.158251	0.0791257	−	0.996865i	$-0.474787\pi$
$499$	1764.00	0.158251	0.0791257	+	0.996865i	$0.474787\pi$
$500$	0	0
$501$	5004.00	0.446232
$502$	−2792.00	−0.248233
$503$	−16976.0	−1.50482	−0.752408	−	0.658697i	$-0.771108\pi$
$503$	−16976.0	−1.50482	−0.752408	+	0.658697i	$0.771108\pi$
$504$	576.000	0.0509069
$505$	0	0
$506$	0	0
$507$	−507.000	−0.0444116
$508$	−5184.00	−0.452761
$509$	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
$512$	512.000	0.0441942
$513$	540.000	0.0464748
$514$	−13812.0	−1.18526
$515$	0	0
$516$	912.000	0.0778073
$517$	18080.0	1.53802
$518$	1184.00	0.100429
$519$	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	$0.297777\pi$
$522$	−324.000	−0.0271668
$523$	−20284.0	−1.69590	−0.847952	−	0.530074i	$-0.822164\pi$
$523$	−20284.0	−1.69590	−0.847952	+	0.530074i	$0.822164\pi$
$524$	−3568.00	−0.297460
$525$	0	0
$526$	13696.0	1.13531
$527$	23920.0	1.97718
$528$	−1920.00	−0.158252
$529$	−12167.0	−1.00000
$530$	0	0
$531$	4176.00	0.341286
$532$	−640.000	−0.0521570
$533$	4706.00	0.382438
$534$	2316.00	0.187684
$535$	0	0
$536$	5600.00	0.451275
$537$	−3204.00	−0.257473
$538$	−12068.0	−0.967079
$539$	−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	$0.693318\pi$
$542$	9664.00	0.765875
$543$	14358.0	1.13473
$544$	−4160.00	−0.327865
$545$	0	0
$546$	624.000	0.0489098
$547$	20956.0	1.63805	0.819025	−	0.573757i	$-0.194515\pi$
$547$	20956.0	1.63805	0.819025	+	0.573757i	$0.194515\pi$
$548$	−9304.00	−0.725269
$549$	3222.00	0.250477
$550$	0	0
$551$	360.000	0.0278340
$552$	0	0
$553$	−7808.00	−0.600416
$554$	8164.00	0.626092
$555$	0	0
$556$	7728.00	0.589461
$557$	4134.00	0.314476	0.157238	−	0.987561i	$-0.449741\pi$
$557$	4134.00	0.314476	0.157238	+	0.987561i	$0.449741\pi$
$558$	−3312.00	−0.251269
$559$	988.000	0.0747548
$560$	0	0
$561$	15600.0	1.17403
$562$	6700.00	0.502887
$563$	16228.0	1.21479	0.607397	−	0.794399i	$-0.292214\pi$
$563$	16228.0	1.21479	0.607397	+	0.794399i	$0.292214\pi$
$564$	−5424.00	−0.404950
$565$	0	0
$566$	−15592.0	−1.15792
$567$	648.000	0.0479955
$568$	−5984.00	−0.442048
$569$	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	$-0.639910\pi$
$571$	−11612.0	−0.851046	−0.425523	+	0.904948i	$0.639910\pi$
$572$	−2080.00	−0.152044
$573$	3936.00	0.286961
$574$	−5792.00	−0.421173
$575$	0	0
$576$	576.000	0.0416667
$577$	−6354.00	−0.458441	−0.229221	−	0.973375i	$-0.573618\pi$
$577$	−6354.00	−0.458441	−0.229221	+	0.973375i	$0.573618\pi$
$578$	23974.0	1.72524
$579$	−1050.00	−0.0753653
$580$	0	0
$581$	8064.00	0.575819
$582$	−3684.00	−0.262383
$583$	−15280.0	−1.08548
$584$	−8464.00	−0.599731
$585$	0	0
$586$	−7844.00	−0.552957
$587$	13240.0	0.930960	0.465480	−	0.885059i	$-0.345882\pi$
$587$	13240.0	0.930960	0.465480	+	0.885059i	$0.345882\pi$
$588$	3348.00	0.234812
$589$	3680.00	0.257439
$590$	0	0
$591$	−1026.00	−0.0714112
$592$	1184.00	0.0821995
$593$	1146.00	0.0793602	0.0396801	−	0.999212i	$-0.487366\pi$
$593$	1146.00	0.0793602	0.0396801	+	0.999212i	$0.487366\pi$
$594$	−2160.00	−0.149202
$595$	0	0
$596$	3528.00	0.242471
$597$	10104.0	0.692679
$598$	0	0
$599$	10464.0	0.713769	0.356884	−	0.934149i	$-0.383839\pi$
$599$	10464.0	0.713769	0.356884	+	0.934149i	$0.383839\pi$
$600$	0	0
$601$	6650.00	0.451346	0.225673	−	0.974203i	$-0.427542\pi$
$601$	6650.00	0.451346	0.225673	+	0.974203i	$0.427542\pi$
$602$	−1216.00	−0.0823263
$603$	6300.00	0.425466
$604$	−7104.00	−0.478572
$605$	0	0
$606$	−3108.00	−0.208340
$607$	6664.00	0.445607	0.222803	−	0.974863i	$-0.428479\pi$
$607$	6664.00	0.445607	0.222803	+	0.974863i	$0.428479\pi$
$608$	−640.000	−0.0426898
$609$	432.000	0.0287447
$610$	0	0
$611$	−5876.00	−0.389063
$612$	−4680.00	−0.309114
$613$	−2134.00	−0.140606	−0.0703030	−	0.997526i	$-0.522397\pi$
$613$	−2134.00	−0.140606	−0.0703030	+	0.997526i	$0.522397\pi$
$614$	−11912.0	−0.782947
$615$	0	0
$616$	2560.00	0.167444
$617$	714.000	0.0465876	0.0232938	−	0.999729i	$-0.492585\pi$
$617$	714.000	0.0465876	0.0232938	+	0.999729i	$0.492585\pi$
$618$	672.000	0.0437408
$619$	29228.0	1.89786	0.948928	−	0.315494i	$-0.102170\pi$
$619$	29228.0	1.89786	0.948928	+	0.315494i	$0.102170\pi$
$620$	0	0
$621$	0	0
$622$	4704.00	0.303237
$623$	−3088.00	−0.198584
$624$	624.000	0.0400320
$625$	0	0
$626$	−16884.0	−1.07799
$627$	2400.00	0.152866
$628$	9640.00	0.612544
$629$	−9620.00	−0.609816
$630$	0	0
$631$	−13536.0	−0.853977	−0.426989	−	0.904257i	$-0.640426\pi$
$631$	−13536.0	−0.853977	−0.426989	+	0.904257i	$0.640426\pi$
$632$	−7808.00	−0.491433
$633$	6012.00	0.377497
$634$	11100.0	0.695327
$635$	0	0
$636$	4584.00	0.285798
$637$	3627.00	0.225600
$638$	−1440.00	−0.0893576
$639$	−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	$0.322013\pi$
$642$	−2232.00	−0.137212
$643$	−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
$645$	0	0
$646$	5200.00	0.316705
$647$	−25176.0	−1.52978	−0.764892	−	0.644158i	$-0.777208\pi$
$647$	−25176.0	−1.52978	−0.764892	+	0.644158i	$0.777208\pi$
$648$	648.000	0.0392837
$649$	18560.0	1.12256
$650$	0	0
$651$	4416.00	0.265863
$652$	−12848.0	−0.771728
$653$	16034.0	0.960887	0.480443	−	0.877026i	$-0.340476\pi$
$653$	16034.0	0.960887	0.480443	+	0.877026i	$0.340476\pi$
$654$	−5604.00	−0.335067
$655$	0	0
$656$	−5792.00	−0.344725
$657$	−9522.00	−0.565432
$658$	7232.00	0.428469
$659$	25356.0	1.49883	0.749415	−	0.662100i	$-0.230335\pi$
$659$	25356.0	1.49883	0.749415	+	0.662100i	$0.230335\pi$
$660$	0	0
$661$	18310.0	1.07742	0.538711	−	0.842490i	$-0.318911\pi$
$661$	18310.0	1.07742	0.538711	+	0.842490i	$0.318911\pi$
$662$	280.000	0.0164388
$663$	−5070.00	−0.296987
$664$	8064.00	0.471301
$665$	0	0
$666$	1332.00	0.0774984
$667$	0	0
$668$	−6672.00	−0.386448
$669$	−16824.0	−0.972277
$670$	0	0
$671$	14320.0	0.823871
$672$	−768.000	−0.0440867
$673$	−24802.0	−1.42057	−0.710287	−	0.703912i	$-0.751435\pi$
$673$	−24802.0	−1.42057	−0.710287	+	0.703912i	$0.751435\pi$
$674$	12348.0	0.705678
$675$	0	0
$676$	676.000	0.0384615
$677$	22706.0	1.28901	0.644507	−	0.764598i	$-0.277063\pi$
$677$	22706.0	1.28901	0.644507	+	0.764598i	$0.277063\pi$
$678$	11484.0	0.650502
$679$	4912.00	0.277622
$680$	0	0
$681$	−5784.00	−0.325467
$682$	−14720.0	−0.826478
$683$	14792.0	0.828697	0.414349	−	0.910118i	$-0.364009\pi$
$683$	14792.0	0.828697	0.414349	+	0.910118i	$0.364009\pi$
$684$	−720.000	−0.0402484
$685$	0	0
$686$	−9952.00	−0.553891
$687$	11814.0	0.656088
$688$	−1216.00	−0.0673831
$689$	4966.00	0.274586
$690$	0	0
$691$	−1148.00	−0.0632011	−0.0316006	−	0.999501i	$-0.510060\pi$
$691$	−1148.00	−0.0632011	−0.0316006	+	0.999501i	$0.510060\pi$
$692$	−14392.0	−0.790609
$693$	2880.00	0.157867
$694$	5976.00	0.326867
$695$	0	0
$696$	432.000	0.0235272
$697$	47060.0	2.55742
$698$	−324.000	−0.0175696
$699$	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	$0.368804\pi$
$702$	702.000	0.0377426
$703$	−1480.00	−0.0794015
$704$	2560.00	0.137051
$705$	0	0
$706$	21508.0	1.14655
$707$	4144.00	0.220440
$708$	−5568.00	−0.295563
$709$	−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
$712$	−3088.00	−0.162539
$713$	0	0
$714$	6240.00	0.327067
$715$	0	0
$716$	4272.00	0.222978
$717$	−21492.0	−1.11943
$718$	7176.00	0.372989
$719$	9288.00	0.481758	0.240879	−	0.970555i	$-0.422564\pi$
$719$	9288.00	0.481758	0.240879	+	0.970555i	$0.422564\pi$
$720$	0	0
$721$	−896.000	−0.0462813
$722$	−12918.0	−0.665870
$723$	18546.0	0.953988
$724$	−19144.0	−0.982709
$725$	0	0
$726$	−1614.00	−0.0825085
$727$	21544.0	1.09907	0.549534	−	0.835471i	$-0.314805\pi$
$727$	21544.0	1.09907	0.549534	+	0.835471i	$0.314805\pi$
$728$	−832.000	−0.0423571
$729$	729.000	0.0370370
$730$	0	0
$731$	9880.00	0.499897
$732$	−4296.00	−0.216919
$733$	−19990.0	−1.00730	−0.503648	−	0.863909i	$-0.668009\pi$
$733$	−19990.0	−1.00730	−0.503648	+	0.863909i	$0.668009\pi$
$734$	−22544.0	−1.13367
$735$	0	0
$736$	0	0
$737$	28000.0	1.39945
$738$	−6516.00	−0.325010
$739$	532.000	0.0264816	0.0132408	−	0.999912i	$-0.495785\pi$
$739$	532.000	0.0264816	0.0132408	+	0.999912i	$0.495785\pi$
$740$	0	0
$741$	−780.000	−0.0386694
$742$	−6112.00	−0.302397
$743$	25452.0	1.25672	0.628360	−	0.777922i	$-0.283726\pi$
$743$	25452.0	1.25672	0.628360	+	0.777922i	$0.283726\pi$
$744$	4416.00	0.217605
$745$	0	0
$746$	21828.0	1.07129
$747$	9072.00	0.444347
$748$	−20800.0	−1.01674
$749$	2976.00	0.145181
$750$	0	0
$751$	6440.00	0.312915	0.156457	−	0.987685i	$-0.449993\pi$
$751$	6440.00	0.312915	0.156457	+	0.987685i	$0.449993\pi$
$752$	7232.00	0.350697
$753$	4188.00	0.202682
$754$	468.000	0.0226042
$755$	0	0
$756$	−864.000	−0.0415653
$757$	786.000	0.0377380	0.0188690	−	0.999822i	$-0.493993\pi$
$757$	786.000	0.0377380	0.0188690	+	0.999822i	$0.493993\pi$
$758$	16200.0	0.776267
$759$	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	$0.511359\pi$
$762$	7776.00	0.369678
$763$	7472.00	0.354528
$764$	−5248.00	−0.248516
$765$	0	0
$766$	−12360.0	−0.583009
$767$	−6032.00	−0.283967
$768$	−768.000	−0.0360844
$769$	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
$772$	1400.00	0.0652683
$773$	3822.00	0.177837	0.0889184	−	0.996039i	$-0.471659\pi$
$773$	3822.00	0.177837	0.0889184	+	0.996039i	$0.471659\pi$
$774$	−1368.00	−0.0635294
$775$	0	0
$776$	4912.00	0.227230
$777$	−1776.00	−0.0819995
$778$	−15044.0	−0.693256
$779$	7240.00	0.332991
$780$	0	0
$781$	−29920.0	−1.37083
$782$	0	0
$783$	486.000	0.0221816
$784$	−4464.00	−0.203353
$785$	0	0
$786$	5352.00	0.242875
$787$	11900.0	0.538995	0.269498	−	0.963001i	$-0.413142\pi$
$787$	11900.0	0.538995	0.269498	+	0.963001i	$0.413142\pi$
$788$	1368.00	0.0618439
$789$	−20544.0	−0.926978
$790$	0	0
$791$	−15312.0	−0.688283
$792$	2880.00	0.129213
$793$	−4654.00	−0.208409
$794$	−12156.0	−0.543325
$795$	0	0
$796$	−13472.0	−0.599877
$797$	21274.0	0.945500	0.472750	−	0.881197i	$-0.343261\pi$
$797$	21274.0	0.945500	0.472750	+	0.881197i	$0.343261\pi$
$798$	960.000	0.0425860
$799$	−58760.0	−2.60173
$800$	0	0
$801$	−3474.00	−0.153243
$802$	3660.00	0.161146
$803$	−42320.0	−1.85983
$804$	−8400.00	−0.368464
$805$	0	0
$806$	4784.00	0.209069
$807$	18102.0	0.789617
$808$	4144.00	0.180427
$809$	−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	$-0.578270\pi$
$811$	−11244.0	−0.486844	−0.243422	+	0.969921i	$0.578270\pi$
$812$	−576.000	−0.0248936
$813$	−14496.0	−0.625334
$814$	5920.00	0.254909
$815$	0	0
$816$	6240.00	0.267701
$817$	1520.00	0.0650894
$818$	24868.0	1.06295
$819$	−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	$0.406981\pi$
$822$	13956.0	0.592179
$823$	−14384.0	−0.609228	−0.304614	−	0.952476i	$-0.598527\pi$
$823$	−14384.0	−0.609228	−0.304614	+	0.952476i	$0.598527\pi$
$824$	−896.000	−0.0378806
$825$	0	0
$826$	7424.00	0.312729
$827$	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
$832$	−832.000	−0.0346688
$833$	36270.0	1.50862
$834$	−11592.0	−0.481293
$835$	0	0
$836$	−3200.00	−0.132386
$837$	4968.00	0.205160
$838$	−28376.0	−1.16973
$839$	23116.0	0.951195	0.475598	−	0.879663i	$-0.342232\pi$
$839$	23116.0	0.951195	0.475598	+	0.879663i	$0.342232\pi$
$840$	0	0
$841$	−24065.0	−0.986715
$842$	17276.0	0.707091
$843$	−10050.0	−0.410605
$844$	−8016.00	−0.326922
$845$	0	0
$846$	8136.00	0.330640
$847$	2152.00	0.0873006
$848$	−6112.00	−0.247508
$849$	23388.0	0.945435
$850$	0	0
$851$	0	0
$852$	8976.00	0.360930
$853$	−934.000	−0.0374907	−0.0187453	−	0.999824i	$-0.505967\pi$
$853$	−934.000	−0.0374907	−0.0187453	+	0.999824i	$0.505967\pi$
$854$	5728.00	0.229518
$855$	0	0
$856$	2976.00	0.118829
$857$	−12642.0	−0.503900	−0.251950	−	0.967740i	$-0.581072\pi$
$857$	−12642.0	−0.503900	−0.251950	+	0.967740i	$0.581072\pi$
$858$	3120.00	0.124143
$859$	−22796.0	−0.905459	−0.452730	−	0.891648i	$-0.649550\pi$
$859$	−22796.0	−0.905459	−0.452730	+	0.891648i	$0.649550\pi$
$860$	0	0
$861$	8688.00	0.343886
$862$	8584.00	0.339179
$863$	76.0000	0.00299776	0.00149888	−	0.999999i	$-0.499523\pi$
$863$	76.0000	0.00299776	0.00149888	+	0.999999i	$0.499523\pi$
$864$	−864.000	−0.0340207
$865$	0	0
$866$	11964.0	0.469461
$867$	−35961.0	−1.40865
$868$	−5888.00	−0.230244
$869$	−39040.0	−1.52398
$870$	0	0
$871$	−9100.00	−0.354009
$872$	7472.00	0.290176
$873$	5526.00	0.214235
$874$	0	0
$875$	0	0
$876$	12696.0	0.489678
$877$	46130.0	1.77617	0.888084	−	0.459681i	$-0.152036\pi$
$877$	46130.0	1.77617	0.888084	+	0.459681i	$0.152036\pi$
$878$	512.000	0.0196801
$879$	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	$0.459220\pi$
$882$	−5022.00	−0.191723
$883$	−47404.0	−1.80665	−0.903325	−	0.428957i	$-0.858881\pi$
$883$	−47404.0	−1.80665	−0.903325	+	0.428957i	$0.858881\pi$
$884$	6760.00	0.257198
$885$	0	0
$886$	−25112.0	−0.952206
$887$	−33672.0	−1.27463	−0.637314	−	0.770604i	$-0.719955\pi$
$887$	−33672.0	−1.27463	−0.637314	+	0.770604i	$0.719955\pi$
$888$	−1776.00	−0.0671156
$889$	−10368.0	−0.391149
$890$	0	0
$891$	3240.00	0.121823
$892$	22432.0	0.842017
$893$	−9040.00	−0.338759
$894$	−5292.00	−0.197976
$895$	0	0
$896$	1024.00	0.0381802
$897$	0	0
$898$	11148.0	0.414269
$899$	3312.00	0.122871
$900$	0	0
$901$	49660.0	1.83620
$902$	−28960.0	−1.06903
$903$	1824.00	0.0672192
$904$	−15312.0	−0.563351
$905$	0	0
$906$	10656.0	0.390753
$907$	14540.0	0.532296	0.266148	−	0.963932i	$-0.414249\pi$
$907$	14540.0	0.532296	0.266148	+	0.963932i	$0.414249\pi$
$908$	7712.00	0.281863
$909$	4662.00	0.170109
$910$	0	0
$911$	−7840.00	−0.285127	−0.142564	−	0.989786i	$-0.545535\pi$
$911$	−7840.00	−0.285127	−0.142564	+	0.989786i	$0.545535\pi$
$912$	960.000	0.0348561
$913$	40320.0	1.46155
$914$	−2532.00	−0.0916314
$915$	0	0
$916$	−15752.0	−0.568189
$917$	−7136.00	−0.256981
$918$	7020.00	0.252391
$919$	47720.0	1.71288	0.856440	−	0.516246i	$-0.172671\pi$
$919$	47720.0	1.71288	0.856440	+	0.516246i	$0.172671\pi$
$920$	0	0
$921$	17868.0	0.639273
$922$	15108.0	0.539648
$923$	9724.00	0.346771
$924$	−3840.00	−0.136717
$925$	0	0
$926$	13504.0	0.479232
$927$	−1008.00	−0.0357142
$928$	−576.000	−0.0203751
$929$	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
$932$	−10248.0	−0.360176
$933$	−7056.00	−0.247592
$934$	−15848.0	−0.555206
$935$	0	0
$936$	−936.000	−0.0326860
$937$	−22058.0	−0.769054	−0.384527	−	0.923114i	$-0.625635\pi$
$937$	−22058.0	−0.769054	−0.384527	+	0.923114i	$0.625635\pi$
$938$	11200.0	0.389865
$939$	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	$0.367533\pi$
$942$	−14460.0	−0.500140
$943$	0	0
$944$	7424.00	0.255965
$945$	0	0
$946$	−6080.00	−0.208962
$947$	30488.0	1.04617	0.523087	−	0.852279i	$-0.324780\pi$
$947$	30488.0	1.04617	0.523087	+	0.852279i	$0.324780\pi$
$948$	11712.0	0.401253
$949$	13754.0	0.470468
$950$	0	0
$951$	−16650.0	−0.567732
$952$	−8320.00	−0.283249
$953$	−9522.00	−0.323660	−0.161830	−	0.986819i	$-0.551740\pi$
$953$	−9522.00	−0.323660	−0.161830	+	0.986819i	$0.551740\pi$
$954$	−6876.00	−0.233353
$955$	0	0
$956$	28656.0	0.969457
$957$	2160.00	0.0729602
$958$	−22168.0	−0.747615
$959$	−18608.0	−0.626573
$960$	0	0
$961$	4065.00	0.136451
$962$	−1924.00	−0.0644826
$963$	3348.00	0.112033
$964$	−24728.0	−0.826178
$965$	0	0
$966$	0	0
$967$	7616.00	0.253272	0.126636	−	0.991949i	$-0.459582\pi$
$967$	7616.00	0.253272	0.126636	+	0.991949i	$0.459582\pi$
$968$	2152.00	0.0714544
$969$	−7800.00	−0.258588
$970$	0	0
$971$	51316.0	1.69599	0.847996	−	0.530002i	$-0.177809\pi$
$971$	51316.0	1.69599	0.847996	+	0.530002i	$0.177809\pi$
$972$	−972.000	−0.0320750
$973$	15456.0	0.509246
$974$	−8864.00	−0.291603
$975$	0	0
$976$	5728.00	0.187857
$977$	48666.0	1.59362	0.796808	−	0.604232i	$-0.206520\pi$
$977$	48666.0	1.59362	0.796808	+	0.604232i	$0.206520\pi$
$978$	19272.0	0.630113
$979$	−15440.0	−0.504050
$980$	0	0
$981$	8406.00	0.273581
$982$	−2280.00	−0.0740914
$983$	−17388.0	−0.564182	−0.282091	−	0.959388i	$-0.591028\pi$
$983$	−17388.0	−0.564182	−0.282091	+	0.959388i	$0.591028\pi$
$984$	8688.00	0.281467
$985$	0	0
$986$	4680.00	0.151158
$987$	−10848.0	−0.349844
$988$	1040.00	0.0334887
$989$	0	0
$990$	0	0
$991$	11496.0	0.368499	0.184249	−	0.982880i	$-0.441015\pi$
$991$	11496.0	0.368499	0.184249	+	0.982880i	$0.441015\pi$
$992$	−5888.00	−0.188452
$993$	−420.000	−0.0134223
$994$	−11968.0	−0.381893
$995$	0	0
$996$	−12096.0	−0.384816
$997$	−48862.0	−1.55213	−0.776066	−	0.630652i	$-0.782788\pi$
$997$	−48862.0	−1.55213	−0.776066	+	0.630652i	$0.782788\pi$
$998$	3528.00	0.111901
$999$	−1998.00	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.4.a.l.1.1		1
5.4	even	2		78.4.a.c.1.1	✓	1
15.14	odd	2		234.4.a.h.1.1		1
20.19	odd	2		624.4.a.d.1.1		1
40.19	odd	2		2496.4.a.j.1.1		1
40.29	even	2		2496.4.a.a.1.1		1
60.59	even	2		1872.4.a.d.1.1		1
65.34	odd	4		1014.4.b.h.337.1		2
65.44	odd	4		1014.4.b.h.337.2		2
65.64	even	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	5.4	even	2
234.4.a.h.1.1		1	15.14	odd	2
624.4.a.d.1.1		1	20.19	odd	2
1014.4.a.j.1.1		1	65.64	even	2
1014.4.b.h.337.1		2	65.34	odd	4
1014.4.b.h.337.2		2	65.44	odd	4
1872.4.a.d.1.1		1	60.59	even	2
1950.4.a.l.1.1		1	1.1	even	1	trivial
2496.4.a.a.1.1		1	40.29	even	2
2496.4.a.j.1.1		1	40.19	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1950.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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