LMFDB - Embedded newform 1014.4.a.j.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1014,4,Mod(1,1014)] 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("1014.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1014, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1014 = 2 \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1014.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,2,3,4,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$59.8279367458$
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	$\chi$	$=$	1014.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -10.0000 q^{5} +6.00000 q^{6} +8.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -20.0000 q^{10} -40.0000 q^{11} +12.0000 q^{12} +16.0000 q^{14} -30.0000 q^{15} +16.0000 q^{16} +130.000 q^{17} +18.0000 q^{18} +20.0000 q^{19} -40.0000 q^{20} +24.0000 q^{21} -80.0000 q^{22} +24.0000 q^{24} -25.0000 q^{25} +27.0000 q^{27} +32.0000 q^{28} -18.0000 q^{29} -60.0000 q^{30} +184.000 q^{31} +32.0000 q^{32} -120.000 q^{33} +260.000 q^{34} -80.0000 q^{35} +36.0000 q^{36} +74.0000 q^{37} +40.0000 q^{38} -80.0000 q^{40} +362.000 q^{41} +48.0000 q^{42} +76.0000 q^{43} -160.000 q^{44} -90.0000 q^{45} +452.000 q^{47} +48.0000 q^{48} -279.000 q^{49} -50.0000 q^{50} +390.000 q^{51} +382.000 q^{53} +54.0000 q^{54} +400.000 q^{55} +64.0000 q^{56} +60.0000 q^{57} -36.0000 q^{58} -464.000 q^{59} -120.000 q^{60} +358.000 q^{61} +368.000 q^{62} +72.0000 q^{63} +64.0000 q^{64} -240.000 q^{66} +700.000 q^{67} +520.000 q^{68} -160.000 q^{70} +748.000 q^{71} +72.0000 q^{72} -1058.00 q^{73} +148.000 q^{74} -75.0000 q^{75} +80.0000 q^{76} -320.000 q^{77} -976.000 q^{79} -160.000 q^{80} +81.0000 q^{81} +724.000 q^{82} +1008.00 q^{83} +96.0000 q^{84} -1300.00 q^{85} +152.000 q^{86} -54.0000 q^{87} -320.000 q^{88} +386.000 q^{89} -180.000 q^{90} +552.000 q^{93} +904.000 q^{94} -200.000 q^{95} +96.0000 q^{96} +614.000 q^{97} -558.000 q^{98} -360.000 q^{99} +O(q^{100})\)

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
$5$	−10.0000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−10.0000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	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
$10$	−20.0000	−0.632456
$11$	−40.0000	−1.09640	−0.548202	−	0.836346i	$-0.684688\pi$
$11$	−40.0000	−1.09640	−0.548202	+	0.836346i	$0.684688\pi$
$12$	12.0000	0.288675
$13$	0	0
$13$	0	0
$14$	16.0000	0.305441
$15$	−30.0000	−0.516398
$16$	16.0000	0.250000
$17$	130.000	1.85468	0.927342	−	0.374215i	$-0.122088\pi$
$17$	130.000	1.85468	0.927342	+	0.374215i	$0.122088\pi$
$18$	18.0000	0.235702
$19$	20.0000	0.241490	0.120745	−	0.992684i	$-0.461472\pi$
$19$	20.0000	0.241490	0.120745	+	0.992684i	$0.461472\pi$
$20$	−40.0000	−0.447214
$21$	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
$25$	−25.0000	−0.200000
$26$	0	0
$27$	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	$0.518354\pi$
$30$	−60.0000	−0.365148
$31$	184.000	1.06604	0.533022	−	0.846101i	$-0.321056\pi$
$31$	184.000	1.06604	0.533022	+	0.846101i	$0.321056\pi$
$32$	32.0000	0.176777
$33$	−120.000	−0.633010
$34$	260.000	1.31146
$35$	−80.0000	−0.386356
$36$	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
$39$	0	0
$40$	−80.0000	−0.316228
$41$	362.000	1.37890	0.689450	−	0.724333i	$-0.257852\pi$
$41$	362.000	1.37890	0.689450	+	0.724333i	$0.257852\pi$
$42$	48.0000	0.176347
$43$	76.0000	0.269532	0.134766	−	0.990877i	$-0.456972\pi$
$43$	76.0000	0.269532	0.134766	+	0.990877i	$0.456972\pi$
$44$	−160.000	−0.548202
$45$	−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	$0.252562\pi$
$48$	48.0000	0.144338
$49$	−279.000	−0.813411
$50$	−50.0000	−0.141421
$51$	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	$0.335162\pi$
$54$	54.0000	0.136083
$55$	400.000	0.980654
$56$	64.0000	0.152721
$57$	60.0000	0.139424
$58$	−36.0000	−0.0815005
$59$	−464.000	−1.02386	−0.511929	−	0.859028i	$-0.671069\pi$
$59$	−464.000	−1.02386	−0.511929	+	0.859028i	$0.671069\pi$
$60$	−120.000	−0.258199
$61$	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
$70$	−160.000	−0.273195
$71$	748.000	1.25030	0.625150	−	0.780505i	$-0.285038\pi$
$71$	748.000	1.25030	0.625150	+	0.780505i	$0.285038\pi$
$72$	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
$75$	−75.0000	−0.115470
$76$	80.0000	0.120745
$77$	−320.000	−0.473602
$78$	0	0
$79$	−976.000	−1.38998	−0.694991	−	0.719018i	$-0.744592\pi$
$79$	−976.000	−1.38998	−0.694991	+	0.719018i	$0.744592\pi$
$80$	−160.000	−0.223607
$81$	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
$85$	−1300.00	−1.65888
$86$	152.000	0.190588
$87$	−54.0000	−0.0665449
$88$	−320.000	−0.387638
$89$	386.000	0.459729	0.229865	−	0.973223i	$-0.426172\pi$
$89$	386.000	0.459729	0.229865	+	0.973223i	$0.426172\pi$
$90$	−180.000	−0.210819
$91$	0	0
$92$	0	0
$93$	552.000	0.615481
$94$	904.000	0.991920
$95$	−200.000	−0.215995
$96$	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
$100$	−100.000	−0.100000
$101$	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	$-0.482940\pi$
$103$	112.000	0.107143	0.0535713	+	0.998564i	$0.482940\pi$
$104$	0	0
$105$	−240.000	−0.223063
$106$	764.000	0.700059
$107$	−372.000	−0.336099	−0.168050	−	0.985779i	$-0.553747\pi$
$107$	−372.000	−0.336099	−0.168050	+	0.985779i	$0.553747\pi$
$108$	108.000	0.0962250
$109$	−934.000	−0.820743	−0.410371	−	0.911918i	$-0.634601\pi$
$109$	−934.000	−0.820743	−0.410371	+	0.911918i	$0.634601\pi$
$110$	800.000	0.693427
$111$	222.000	0.189832
$112$	128.000	0.107990
$113$	1914.00	1.59340	0.796699	−	0.604376i	$-0.206578\pi$
$113$	1914.00	1.59340	0.796699	+	0.604376i	$0.206578\pi$
$114$	120.000	0.0985880
$115$	0	0
$116$	−72.0000	−0.0576296
$117$	0	0
$118$	−928.000	−0.723977
$119$	1040.00	0.801148
$120$	−240.000	−0.182574
$121$	269.000	0.202104
$122$	716.000	0.531341
$123$	1086.00	0.796108
$124$	736.000	0.533022
$125$	1500.00	1.07331
$126$	144.000	0.101814
$127$	1296.00	0.905523	0.452761	−	0.891632i	$-0.350439\pi$
$127$	1296.00	0.905523	0.452761	+	0.891632i	$0.350439\pi$
$128$	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
$135$	−270.000	−0.172133
$136$	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	$0.299340\pi$
$140$	−320.000	−0.193178
$141$	1356.00	0.809899
$142$	1496.00	0.884095
$143$	0	0
$144$	144.000	0.0833333
$145$	180.000	0.103091
$146$	−2116.00	−1.19946
$147$	−837.000	−0.469623
$148$	296.000	0.164399
$149$	−882.000	−0.484941	−0.242471	−	0.970159i	$-0.577958\pi$
$149$	−882.000	−0.484941	−0.242471	+	0.970159i	$0.577958\pi$
$150$	−150.000	−0.0816497
$151$	1776.00	0.957145	0.478572	−	0.878048i	$-0.341154\pi$
$151$	1776.00	0.957145	0.478572	+	0.878048i	$0.341154\pi$
$152$	160.000	0.0853797
$153$	1170.00	0.618228
$154$	−640.000	−0.334887
$155$	−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	$0.709854\pi$
$158$	−1952.00	−0.982866
$159$	1146.00	0.571596
$160$	−320.000	−0.158114
$161$	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
$165$	1200.00	0.566181
$166$	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
$169$	0	0
$170$	−2600.00	−1.17301
$171$	180.000	0.0804967
$172$	304.000	0.134766
$173$	3598.00	1.58122	0.790609	−	0.612321i	$-0.209764\pi$
$173$	3598.00	1.58122	0.790609	+	0.612321i	$0.209764\pi$
$174$	−108.000	−0.0470544
$175$	−200.000	−0.0863919
$176$	−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	$0.428422\pi$
$180$	−360.000	−0.149071
$181$	−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
$186$	1104.00	0.435211
$187$	−5200.00	−2.03348
$188$	1808.00	0.701393
$189$	216.000	0.0831306
$190$	−400.000	−0.152732
$191$	−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	$0.704784\pi$
$200$	−200.000	−0.0707107
$201$	2100.00	0.736928
$202$	1036.00	0.360855
$203$	−144.000	−0.0497873
$204$	1560.00	0.535401
$205$	−3620.00	−1.23333
$206$	224.000	0.0757613
$207$	0	0
$208$	0	0
$209$	−800.000	−0.264771
$210$	−480.000	−0.157729
$211$	−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
$215$	−760.000	−0.241077
$216$	216.000	0.0680414
$217$	1472.00	0.460488
$218$	−1868.00	−0.580353
$219$	−3174.00	−0.979356
$220$	1600.00	0.490327
$221$	0	0
$222$	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
$225$	−225.000	−0.0666667
$226$	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	$-0.307644\pi$
$229$	3938.00	1.13638	0.568189	+	0.822898i	$0.307644\pi$
$230$	0	0
$231$	−960.000	−0.273434
$232$	−144.000	−0.0407503
$233$	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
$236$	−1856.00	−0.511929
$237$	−2928.00	−0.802506
$238$	2080.00	0.566497
$239$	−7164.00	−1.93891	−0.969457	−	0.245260i	$-0.921127\pi$
$239$	−7164.00	−1.93891	−0.969457	+	0.245260i	$0.921127\pi$
$240$	−480.000	−0.129099
$241$	6182.00	1.65236	0.826178	−	0.563410i	$-0.190511\pi$
$241$	6182.00	1.65236	0.826178	+	0.563410i	$0.190511\pi$
$242$	538.000	0.142909
$243$	243.000	0.0641500
$244$	1432.00	0.375715
$245$	2790.00	0.727537
$246$	2172.00	0.562934
$247$	0	0
$248$	1472.00	0.376904
$249$	3024.00	0.769631
$250$	3000.00	0.758947
$251$	−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
$255$	−3900.00	−0.957755
$256$	256.000	0.0625000
$257$	6906.00	1.67620	0.838102	−	0.545514i	$-0.183665\pi$
$257$	6906.00	1.67620	0.838102	+	0.545514i	$0.183665\pi$
$258$	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	$-0.796650\pi$
$263$	−6848.00	−1.60557	−0.802787	+	0.596266i	$0.796650\pi$
$264$	−960.000	−0.223803
$265$	−3820.00	−0.885512
$266$	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	$0.739686\pi$
$270$	−540.000	−0.121716
$271$	−4832.00	−1.08311	−0.541556	−	0.840665i	$-0.682164\pi$
$271$	−4832.00	−1.08311	−0.541556	+	0.840665i	$0.682164\pi$
$272$	2080.00	0.463671
$273$	0	0
$274$	−4652.00	−1.02568
$275$	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	$0.645984\pi$
$278$	3864.00	0.833623
$279$	1656.00	0.355348
$280$	−640.000	−0.136598
$281$	−3350.00	−0.711189	−0.355595	−	0.934640i	$-0.615722\pi$
$281$	−3350.00	−0.711189	−0.355595	+	0.934640i	$0.615722\pi$
$282$	2712.00	0.572685
$283$	7796.00	1.63754	0.818770	−	0.574121i	$-0.194656\pi$
$283$	7796.00	1.63754	0.818770	+	0.574121i	$0.194656\pi$
$284$	2992.00	0.625150
$285$	−600.000	−0.124705
$286$	0	0
$287$	2896.00	0.595629
$288$	288.000	0.0589256
$289$	11987.0	2.43985
$290$	360.000	0.0728963
$291$	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
$295$	4640.00	0.915767
$296$	592.000	0.116248
$297$	−1080.00	−0.211003
$298$	−1764.00	−0.342905
$299$	0	0
$300$	−300.000	−0.0577350
$301$	608.000	0.116427
$302$	3552.00	0.676803
$303$	1554.00	0.294637
$304$	320.000	0.0603726
$305$	−3580.00	−0.672099
$306$	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
$310$	−3680.00	−0.674226
$311$	2352.00	0.428841	0.214421	−	0.976741i	$-0.431214\pi$
$311$	2352.00	0.428841	0.214421	+	0.976741i	$0.431214\pi$
$312$	0	0
$313$	8442.00	1.52450	0.762252	−	0.647280i	$-0.224093\pi$
$313$	8442.00	1.52450	0.762252	+	0.647280i	$0.224093\pi$
$314$	−4820.00	−0.866269
$315$	−720.000	−0.128785
$316$	−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
$320$	−640.000	−0.111803
$321$	−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
$330$	2400.00	0.400350
$331$	−140.000	−0.0232480	−0.0116240	−	0.999932i	$-0.503700\pi$
$331$	−140.000	−0.0232480	−0.0116240	+	0.999932i	$0.503700\pi$
$332$	4032.00	0.666520
$333$	666.000	0.109599
$334$	−3336.00	−0.546520
$335$	−7000.00	−1.14164
$336$	384.000	0.0623480
$337$	−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
$340$	−5200.00	−0.829440
$341$	−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	$-0.574242\pi$
$347$	−2988.00	−0.462260	−0.231130	+	0.972923i	$0.574242\pi$
$348$	−216.000	−0.0332725
$349$	162.000	0.0248472	0.0124236	−	0.999923i	$-0.496045\pi$
$349$	162.000	0.0248472	0.0124236	+	0.999923i	$0.496045\pi$
$350$	−400.000	−0.0610883
$351$	0	0
$352$	−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
$355$	−7480.00	−1.11830
$356$	1544.00	0.229865
$357$	3120.00	0.462543
$358$	2136.00	0.315338
$359$	−3588.00	−0.527486	−0.263743	−	0.964593i	$-0.584957\pi$
$359$	−3588.00	−0.527486	−0.263743	+	0.964593i	$0.584957\pi$
$360$	−720.000	−0.105409
$361$	−6459.00	−0.941682
$362$	−9572.00	−1.38976
$363$	807.000	0.116685
$364$	0	0
$365$	10580.0	1.51721
$366$	2148.00	0.306770
$367$	11272.0	1.60325	0.801626	−	0.597826i	$-0.203968\pi$
$367$	11272.0	1.60325	0.801626	+	0.597826i	$0.203968\pi$
$368$	0	0
$369$	3258.00	0.459633
$370$	−1480.00	−0.207950
$371$	3056.00	0.427654
$372$	2208.00	0.307741
$373$	−10914.0	−1.51503	−0.757514	−	0.652819i	$-0.773586\pi$
$373$	−10914.0	−1.51503	−0.757514	+	0.652819i	$0.773586\pi$
$374$	−10400.0	−1.43789
$375$	4500.00	0.619677
$376$	3616.00	0.495960
$377$	0	0
$378$	432.000	0.0587822
$379$	−8100.00	−1.09781	−0.548904	−	0.835886i	$-0.684955\pi$
$379$	−8100.00	−1.09781	−0.548904	+	0.835886i	$0.684955\pi$
$380$	−800.000	−0.107998
$381$	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
$385$	3200.00	0.423603
$386$	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
$395$	9760.00	1.24324
$396$	−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
$400$	−400.000	−0.0500000
$401$	−1830.00	−0.227895	−0.113947	−	0.993487i	$-0.536350\pi$
$401$	−1830.00	−0.227895	−0.113947	+	0.993487i	$0.536350\pi$
$402$	4200.00	0.521087
$403$	0	0
$404$	2072.00	0.255163
$405$	−810.000	−0.0993808
$406$	−288.000	−0.0352049
$407$	−2960.00	−0.360496
$408$	3120.00	0.378586
$409$	−12434.0	−1.50323	−0.751616	−	0.659601i	$-0.770725\pi$
$409$	−12434.0	−1.50323	−0.751616	+	0.659601i	$0.770725\pi$
$410$	−7240.00	−0.872093
$411$	−6978.00	−0.837468
$412$	448.000	0.0535713
$413$	−3712.00	−0.442265
$414$	0	0
$415$	−10080.0	−1.19231
$416$	0	0
$417$	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	$0.810024\pi$
$420$	−960.000	−0.111531
$421$	−8638.00	−0.999977	−0.499989	−	0.866032i	$-0.666662\pi$
$421$	−8638.00	−0.999977	−0.499989	+	0.866032i	$0.666662\pi$
$422$	−4008.00	−0.462337
$423$	4068.00	0.467596
$424$	3056.00	0.350029
$425$	−3250.00	−0.370937
$426$	4488.00	0.510433
$427$	2864.00	0.324587
$428$	−1488.00	−0.168050
$429$	0	0
$430$	−1520.00	−0.170467
$431$	−4292.00	−0.479671	−0.239836	−	0.970813i	$-0.577094\pi$
$431$	−4292.00	−0.479671	−0.239836	+	0.970813i	$0.577094\pi$
$432$	432.000	0.0481125
$433$	−5982.00	−0.663918	−0.331959	−	0.943294i	$-0.607710\pi$
$433$	−5982.00	−0.663918	−0.331959	+	0.943294i	$0.607710\pi$
$434$	2944.00	0.325614
$435$	540.000	0.0595196
$436$	−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	$0.495570\pi$
$440$	3200.00	0.346714
$441$	−2511.00	−0.271137
$442$	0	0
$443$	12556.0	1.34662	0.673311	−	0.739359i	$-0.264872\pi$
$443$	12556.0	1.34662	0.673311	+	0.739359i	$0.264872\pi$
$444$	888.000	0.0949158
$445$	−3860.00	−0.411194
$446$	11216.0	1.19079
$447$	−2646.00	−0.279981
$448$	512.000	0.0539949
$449$	−5574.00	−0.585865	−0.292932	−	0.956133i	$-0.594631\pi$
$449$	−5574.00	−0.585865	−0.292932	+	0.956133i	$0.594631\pi$
$450$	−450.000	−0.0471405
$451$	−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	$-0.624623\pi$
$461$	−7554.00	−0.763178	−0.381589	+	0.924332i	$0.624623\pi$
$462$	−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
$465$	−5520.00	−0.550503
$466$	5124.00	0.509366
$467$	7924.00	0.785180	0.392590	−	0.919714i	$-0.371579\pi$
$467$	7924.00	0.785180	0.392590	+	0.919714i	$0.371579\pi$
$468$	0	0
$469$	5600.00	0.551352
$470$	−9040.00	−0.887200
$471$	−7230.00	−0.707305
$472$	−3712.00	−0.361989
$473$	−3040.00	−0.295517
$474$	−5856.00	−0.567458
$475$	−500.000	−0.0482980
$476$	4160.00	0.400574
$477$	3438.00	0.330011
$478$	−14328.0	−1.37102
$479$	11084.0	1.05729	0.528644	−	0.848844i	$-0.322701\pi$
$479$	11084.0	1.05729	0.528644	+	0.848844i	$0.322701\pi$
$480$	−960.000	−0.0912871
$481$	0	0
$482$	12364.0	1.16839
$483$	0	0
$484$	1076.00	0.101052
$485$	−6140.00	−0.574852
$486$	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
$490$	5580.00	0.514446
$491$	−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
$494$	0	0
$495$	3600.00	0.326885
$496$	2944.00	0.266511
$497$	5984.00	0.540079
$498$	6048.00	0.544212
$499$	−1764.00	−0.158251	−0.0791257	−	0.996865i	$-0.525213\pi$
$499$	−1764.00	−0.158251	−0.0791257	+	0.996865i	$0.525213\pi$
$500$	6000.00	0.536656
$501$	−5004.00	−0.446232
$502$	−2792.00	−0.248233
$503$	16976.0	1.50482	0.752408	−	0.658697i	$-0.228892\pi$
$503$	16976.0	1.50482	0.752408	+	0.658697i	$0.228892\pi$
$504$	576.000	0.0509069
$505$	−5180.00	−0.456449
$506$	0	0
$507$	0	0
$508$	5184.00	0.452761
$509$	−9474.00	−0.825005	−0.412503	−	0.910956i	$-0.635345\pi$
$509$	−9474.00	−0.825005	−0.412503	+	0.910956i	$0.635345\pi$
$510$	−7800.00	−0.677235
$511$	−8464.00	−0.732731
$512$	512.000	0.0441942
$513$	540.000	0.0464748
$514$	13812.0	1.18526
$515$	−1120.00	−0.0958313
$516$	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	$-0.177836\pi$
$523$	20284.0	1.69590	0.847952	+	0.530074i	$0.177836\pi$
$524$	−3568.00	−0.297460
$525$	−600.000	−0.0498784
$526$	−13696.0	−1.13531
$527$	23920.0	1.97718
$528$	−1920.00	−0.158252
$529$	−12167.0	−1.00000
$530$	−7640.00	−0.626152
$531$	−4176.00	−0.341286
$532$	640.000	0.0521570
$533$	0	0
$534$	2316.00	0.187684
$535$	3720.00	0.300616
$536$	5600.00	0.451275
$537$	3204.00	0.257473
$538$	−12068.0	−0.967079
$539$	11160.0	0.891828
$540$	−1080.00	−0.0860663
$541$	14362.0	1.14135	0.570675	−	0.821176i	$-0.306682\pi$
$541$	14362.0	1.14135	0.570675	+	0.821176i	$0.306682\pi$
$542$	−9664.00	−0.765875
$543$	−14358.0	−1.13473
$544$	4160.00	0.327865
$545$	9340.00	0.734095
$546$	0	0
$547$	−20956.0	−1.63805	−0.819025	−	0.573757i	$-0.805485\pi$
$547$	−20956.0	−1.63805	−0.819025	+	0.573757i	$0.805485\pi$
$548$	−9304.00	−0.725269
$549$	3222.00	0.250477
$550$	2000.00	0.155055
$551$	−360.000	−0.0278340
$552$	0	0
$553$	−7808.00	−0.600416
$554$	−8164.00	−0.626092
$555$	−2220.00	−0.169791
$556$	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
$559$	0	0
$560$	−1280.00	−0.0965891
$561$	−15600.0	−1.17403
$562$	−6700.00	−0.502887
$563$	−16228.0	−1.21479	−0.607397	−	0.794399i	$-0.707786\pi$
$563$	−16228.0	−1.21479	−0.607397	+	0.794399i	$0.707786\pi$
$564$	5424.00	0.404950
$565$	−19140.0	−1.42518
$566$	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	$0.470478\pi$
$570$	−1200.00	−0.0881798
$571$	−11612.0	−0.851046	−0.425523	−	0.904948i	$-0.639910\pi$
$571$	−11612.0	−0.851046	−0.425523	+	0.904948i	$0.639910\pi$
$572$	0	0
$573$	−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
$580$	720.000	0.0515455
$581$	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
$590$	9280.00	0.647545
$591$	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
$595$	−10400.0	−0.716569
$596$	−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	$0.383839\pi$
$600$	−600.000	−0.0408248
$601$	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
$605$	−2690.00	−0.180767
$606$	3108.00	0.208340
$607$	−6664.00	−0.445607	−0.222803	−	0.974863i	$-0.571521\pi$
$607$	−6664.00	−0.445607	−0.222803	+	0.974863i	$0.571521\pi$
$608$	640.000	0.0426898
$609$	−432.000	−0.0287447
$610$	−7160.00	−0.475246
$611$	0	0
$612$	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
$615$	−10860.0	−0.712061
$616$	−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	$-0.897830\pi$
$619$	−29228.0	−1.89786	−0.948928	+	0.315494i	$0.897830\pi$
$620$	−7360.00	−0.476750
$621$	0	0
$622$	4704.00	0.303237
$623$	3088.00	0.198584
$624$	0	0
$625$	−11875.0	−0.760000
$626$	16884.0	1.07799
$627$	−2400.00	−0.152866
$628$	−9640.00	−0.612544
$629$	9620.00	0.609816
$630$	−1440.00	−0.0910650
$631$	13536.0	0.853977	0.426989	−	0.904257i	$-0.359574\pi$
$631$	13536.0	0.853977	0.426989	+	0.904257i	$0.359574\pi$
$632$	−7808.00	−0.491433
$633$	−6012.00	−0.377497
$634$	11100.0	0.695327
$635$	−12960.0	−0.809924
$636$	4584.00	0.285798
$637$	0	0
$638$	1440.00	0.0893576
$639$	6732.00	0.416767
$640$	−1280.00	−0.0790569
$641$	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
$645$	−2280.00	−0.139186
$646$	5200.00	0.316705
$647$	25176.0	1.52978	0.764892	−	0.644158i	$-0.222792\pi$
$647$	25176.0	1.52978	0.764892	+	0.644158i	$0.222792\pi$
$648$	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	$-0.659524\pi$
$653$	−16034.0	−0.960887	−0.480443	+	0.877026i	$0.659524\pi$
$654$	−5604.00	−0.335067
$655$	8920.00	0.532112
$656$	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	$0.230335\pi$
$660$	4800.00	0.283091
$661$	−18310.0	−1.07742	−0.538711	−	0.842490i	$-0.681089\pi$
$661$	−18310.0	−1.07742	−0.538711	+	0.842490i	$0.681089\pi$
$662$	−280.000	−0.0164388
$663$	0	0
$664$	8064.00	0.471301
$665$	−1600.00	−0.0933013
$666$	1332.00	0.0774984
$667$	0	0
$668$	−6672.00	−0.386448
$669$	16824.0	0.972277
$670$	−14000.0	−0.807264
$671$	−14320.0	−0.823871
$672$	768.000	0.0440867
$673$	24802.0	1.42057	0.710287	−	0.703912i	$-0.248565\pi$
$673$	24802.0	1.42057	0.710287	+	0.703912i	$0.248565\pi$
$674$	−12348.0	−0.705678
$675$	−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	$0.722937\pi$
$678$	11484.0	0.650502
$679$	4912.00	0.277622
$680$	−10400.0	−0.586503
$681$	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
$685$	23260.0	1.29740
$686$	−9952.00	−0.553891
$687$	11814.0	0.656088
$688$	1216.00	0.0673831
$689$	0	0
$690$	0	0
$691$	1148.00	0.0632011	0.0316006	−	0.999501i	$-0.489940\pi$
$691$	1148.00	0.0632011	0.0316006	+	0.999501i	$0.489940\pi$
$692$	14392.0	0.790609
$693$	−2880.00	−0.157867
$694$	−5976.00	−0.326867
$695$	−19320.0	−1.05446
$696$	−432.000	−0.0235272
$697$	47060.0	2.55742
$698$	324.000	0.0175696
$699$	7686.00	0.415896
$700$	−800.000	−0.0431959
$701$	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
$704$	−2560.00	−0.137051
$705$	−13560.0	−0.724396
$706$	21508.0	1.14655
$707$	4144.00	0.220440
$708$	−5568.00	−0.295563
$709$	6354.00	0.336572	0.168286	−	0.985738i	$-0.446177\pi$
$709$	6354.00	0.336572	0.168286	+	0.985738i	$0.446177\pi$
$710$	−14960.0	−0.790759
$711$	−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	$0.422564\pi$
$720$	−1440.00	−0.0745356
$721$	896.000	0.0462813
$722$	−12918.0	−0.665870
$723$	18546.0	0.953988
$724$	−19144.0	−0.982709
$725$	450.000	0.0230518
$726$	1614.00	0.0825085
$727$	−21544.0	−1.09907	−0.549534	−	0.835471i	$-0.685195\pi$
$727$	−21544.0	−1.09907	−0.549534	+	0.835471i	$0.685195\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	21160.0	1.07283
$731$	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
$735$	8370.00	0.420044
$736$	0	0
$737$	−28000.0	−1.39945
$738$	6516.00	0.325010
$739$	−532.000	−0.0264816	−0.0132408	−	0.999912i	$-0.504215\pi$
$739$	−532.000	−0.0264816	−0.0132408	+	0.999912i	$0.504215\pi$
$740$	−2960.00	−0.147043
$741$	0	0
$742$	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
$745$	8820.00	0.433745
$746$	−21828.0	−1.07129
$747$	9072.00	0.444347
$748$	−20800.0	−1.01674
$749$	−2976.00	−0.145181
$750$	9000.00	0.438178
$751$	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
$754$	0	0
$755$	−17760.0	−0.856096
$756$	864.000	0.0415653
$757$	−786.000	−0.0377380	−0.0188690	−	0.999822i	$-0.506007\pi$
$757$	−786.000	−0.0377380	−0.0188690	+	0.999822i	$0.506007\pi$
$758$	−16200.0	−0.776267
$759$	0	0
$760$	−1600.00	−0.0763659
$761$	1498.00	0.0713567	0.0356784	−	0.999363i	$-0.488641\pi$
$761$	1498.00	0.0713567	0.0356784	+	0.999363i	$0.488641\pi$
$762$	7776.00	0.369678
$763$	−7472.00	−0.354528
$764$	−5248.00	−0.248516
$765$	−11700.0	−0.552960
$766$	−12360.0	−0.583009
$767$	0	0
$768$	768.000	0.0360844
$769$	−14738.0	−0.691113	−0.345556	−	0.938398i	$-0.612310\pi$
$769$	−14738.0	−0.691113	−0.345556	+	0.938398i	$0.612310\pi$
$770$	6400.00	0.299532
$771$	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
$775$	−4600.00	−0.213209
$776$	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
$785$	24100.0	1.09575
$786$	−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
$790$	19520.0	0.879102
$791$	15312.0	0.688283
$792$	−2880.00	−0.129213
$793$	0	0
$794$	−12156.0	−0.543325
$795$	−11460.0	−0.511251
$796$	−13472.0	−0.599877
$797$	−21274.0	−0.945500	−0.472750	−	0.881197i	$-0.656739\pi$
$797$	−21274.0	−0.945500	−0.472750	+	0.881197i	$0.656739\pi$
$798$	960.000	0.0425860
$799$	58760.0	2.60173
$800$	−800.000	−0.0353553
$801$	3474.00	0.153243
$802$	−3660.00	−0.161146
$803$	42320.0	1.85983
$804$	8400.00	0.368464
$805$	0	0
$806$	0	0
$807$	−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	$0.704432\pi$
$810$	−1620.00	−0.0702728
$811$	11244.0	0.486844	0.243422	−	0.969921i	$-0.421730\pi$
$811$	11244.0	0.486844	0.243422	+	0.969921i	$0.421730\pi$
$812$	−576.000	−0.0248936
$813$	−14496.0	−0.625334
$814$	−5920.00	−0.254909
$815$	32120.0	1.38051
$816$	6240.00	0.267701
$817$	1520.00	0.0650894
$818$	−24868.0	−1.06295
$819$	0	0
$820$	−14480.0	−0.616663
$821$	−13554.0	−0.576173	−0.288086	−	0.957604i	$-0.593019\pi$
$821$	−13554.0	−0.576173	−0.288086	+	0.957604i	$0.593019\pi$
$822$	−13956.0	−0.592179
$823$	14384.0	0.609228	0.304614	−	0.952476i	$-0.401473\pi$
$823$	14384.0	0.609228	0.304614	+	0.952476i	$0.401473\pi$
$824$	896.000	0.0378806
$825$	3000.00	0.126602
$826$	−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	$0.643934\pi$
$830$	−20160.0	−0.843089
$831$	−12246.0	−0.511202
$832$	0	0
$833$	−36270.0	−1.50862
$834$	11592.0	0.481293
$835$	16680.0	0.691300
$836$	−3200.00	−0.132386
$837$	4968.00	0.205160
$838$	−28376.0	−1.16973
$839$	−23116.0	−0.951195	−0.475598	−	0.879663i	$-0.657768\pi$
$839$	−23116.0	−0.951195	−0.475598	+	0.879663i	$0.657768\pi$
$840$	−1920.00	−0.0788646
$841$	−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
$850$	−6500.00	−0.262292
$851$	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
$855$	−1800.00	−0.0719985
$856$	−2976.00	−0.118829
$857$	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	$-0.649550\pi$
$859$	−22796.0	−0.905459	−0.452730	+	0.891648i	$0.649550\pi$
$860$	−3040.00	−0.120539
$861$	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
$865$	−35980.0	−1.41429
$866$	−11964.0	−0.469461
$867$	35961.0	1.40865
$868$	5888.00	0.230244
$869$	39040.0	1.52398
$870$	1080.00	0.0420867
$871$	0	0
$872$	−7472.00	−0.290176
$873$	5526.00	0.214235
$874$	0	0
$875$	12000.0	0.463627
$876$	−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
$880$	6400.00	0.245164
$881$	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	$-0.141119\pi$
$883$	47404.0	1.80665	0.903325	+	0.428957i	$0.141119\pi$
$884$	0	0
$885$	13920.0	0.528718
$886$	25112.0	0.952206
$887$	33672.0	1.27463	0.637314	−	0.770604i	$-0.280045\pi$
$887$	33672.0	1.27463	0.637314	+	0.770604i	$0.280045\pi$
$888$	1776.00	0.0671156
$889$	10368.0	0.391149
$890$	−7720.00	−0.290758
$891$	−3240.00	−0.121823
$892$	22432.0	0.842017
$893$	9040.00	0.338759
$894$	−5292.00	−0.197976
$895$	−10680.0	−0.398875
$896$	1024.00	0.0381802
$897$	0	0
$898$	−11148.0	−0.414269
$899$	−3312.00	−0.122871
$900$	−900.000	−0.0333333
$901$	49660.0	1.83620
$902$	−28960.0	−1.06903
$903$	1824.00	0.0672192
$904$	15312.0	0.563351
$905$	47860.0	1.75792
$906$	10656.0	0.390753
$907$	−14540.0	−0.532296	−0.266148	−	0.963932i	$-0.585751\pi$
$907$	−14540.0	−0.532296	−0.266148	+	0.963932i	$0.585751\pi$
$908$	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
$915$	−10740.0	−0.388037
$916$	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
$923$	0	0
$924$	−3840.00	−0.136717
$925$	−1850.00	−0.0657596
$926$	13504.0	0.479232
$927$	1008.00	0.0357142
$928$	−576.000	−0.0203751
$929$	−7502.00	−0.264944	−0.132472	−	0.991187i	$-0.542291\pi$
$929$	−7502.00	−0.264944	−0.132472	+	0.991187i	$0.542291\pi$
$930$	−11040.0	−0.389264
$931$	−5580.00	−0.196431
$932$	10248.0	0.360176
$933$	7056.00	0.247592
$934$	15848.0	0.555206
$935$	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	$0.374365\pi$
$938$	11200.0	0.389865
$939$	25326.0	0.880173
$940$	−18080.0	−0.627345
$941$	−23338.0	−0.808498	−0.404249	−	0.914649i	$-0.632467\pi$
$941$	−23338.0	−0.808498	−0.404249	+	0.914649i	$0.632467\pi$
$942$	−14460.0	−0.500140
$943$	0	0
$944$	−7424.00	−0.255965
$945$	−2160.00	−0.0743543
$946$	−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
$949$	0	0
$950$	−1000.00	−0.0341519
$951$	16650.0	0.567732
$952$	8320.00	0.283249
$953$	9522.00	0.323660	0.161830	−	0.986819i	$-0.448260\pi$
$953$	9522.00	0.323660	0.161830	+	0.986819i	$0.448260\pi$
$954$	6876.00	0.233353
$955$	13120.0	0.444558
$956$	−28656.0	−0.969457
$957$	2160.00	0.0729602
$958$	22168.0	0.747615
$959$	−18608.0	−0.626573
$960$	−1920.00	−0.0645497
$961$	4065.00	0.136451
$962$	0	0
$963$	−3348.00	−0.112033
$964$	24728.0	0.826178
$965$	−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	$0.459582\pi$
$968$	2152.00	0.0714544
$969$	7800.00	0.258588
$970$	−12280.0	−0.406481
$971$	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
$980$	11160.0	0.363768
$981$	−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
$985$	−3420.00	−0.110630
$986$	−4680.00	−0.151158
$987$	10848.0	0.349844
$988$	0	0
$989$	0	0
$990$	7200.00	0.231142
$991$	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
$995$	33680.0	1.07309
$996$	12096.0	0.384816
$997$	48862.0	1.55213	0.776066	−	0.630652i	$-0.217212\pi$
$997$	48862.0	1.55213	0.776066	+	0.630652i	$0.217212\pi$
$998$	−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	1014.4.a.j.1.1		1
13.5	odd	4		1014.4.b.h.337.1		2
13.8	odd	4		1014.4.b.h.337.2		2
13.12	even	2		78.4.a.c.1.1	✓	1
39.38	odd	2		234.4.a.h.1.1		1
52.51	odd	2		624.4.a.d.1.1		1
65.64	even	2		1950.4.a.l.1.1		1
104.51	odd	2		2496.4.a.j.1.1		1
104.77	even	2		2496.4.a.a.1.1		1
156.155	even	2		1872.4.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.4.a.c.1.1	✓	1	13.12	even	2
234.4.a.h.1.1		1	39.38	odd	2
624.4.a.d.1.1		1	52.51	odd	2
1014.4.a.j.1.1		1	1.1	even	1	trivial
1014.4.b.h.337.1		2	13.5	odd	4
1014.4.b.h.337.2		2	13.8	odd	4
1872.4.a.d.1.1		1	156.155	even	2
1950.4.a.l.1.1		1	65.64	even	2
2496.4.a.a.1.1		1	104.77	even	2
2496.4.a.j.1.1		1	104.51	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:	\( N \)	\(=\)	\( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1014.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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