LMFDB - Embedded newform 78.4.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [78,4,Mod(1,78)] 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("78.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(78, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 78 = 2 \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	78.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:	$4.60214898045$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	78.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +10.0000 q^{5} -6.00000 q^{6} -8.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -20.0000 q^{10} +40.0000 q^{11} +12.0000 q^{12} +13.0000 q^{13} +16.0000 q^{14} +30.0000 q^{15} +16.0000 q^{16} +130.000 q^{17} -18.0000 q^{18} -20.0000 q^{19} +40.0000 q^{20} -24.0000 q^{21} -80.0000 q^{22} -24.0000 q^{24} -25.0000 q^{25} -26.0000 q^{26} +27.0000 q^{27} -32.0000 q^{28} -18.0000 q^{29} -60.0000 q^{30} -184.000 q^{31} -32.0000 q^{32} +120.000 q^{33} -260.000 q^{34} -80.0000 q^{35} +36.0000 q^{36} -74.0000 q^{37} +40.0000 q^{38} +39.0000 q^{39} -80.0000 q^{40} -362.000 q^{41} +48.0000 q^{42} +76.0000 q^{43} +160.000 q^{44} +90.0000 q^{45} -452.000 q^{47} +48.0000 q^{48} -279.000 q^{49} +50.0000 q^{50} +390.000 q^{51} +52.0000 q^{52} +382.000 q^{53} -54.0000 q^{54} +400.000 q^{55} +64.0000 q^{56} -60.0000 q^{57} +36.0000 q^{58} +464.000 q^{59} +120.000 q^{60} +358.000 q^{61} +368.000 q^{62} -72.0000 q^{63} +64.0000 q^{64} +130.000 q^{65} -240.000 q^{66} -700.000 q^{67} +520.000 q^{68} +160.000 q^{70} -748.000 q^{71} -72.0000 q^{72} +1058.00 q^{73} +148.000 q^{74} -75.0000 q^{75} -80.0000 q^{76} -320.000 q^{77} -78.0000 q^{78} -976.000 q^{79} +160.000 q^{80} +81.0000 q^{81} +724.000 q^{82} -1008.00 q^{83} -96.0000 q^{84} +1300.00 q^{85} -152.000 q^{86} -54.0000 q^{87} -320.000 q^{88} -386.000 q^{89} -180.000 q^{90} -104.000 q^{91} -552.000 q^{93} +904.000 q^{94} -200.000 q^{95} -96.0000 q^{96} -614.000 q^{97} +558.000 q^{98} +360.000 q^{99} +O(q^{100})\)

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	$-0.352416\pi$
$5$	10.0000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	−6.00000	−0.408248
$7$	−8.00000	−0.431959	−0.215980	−	0.976398i	$-0.569295\pi$
$7$	−8.00000	−0.431959	−0.215980	+	0.976398i	$0.569295\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−20.0000	−0.632456
$11$	40.0000	1.09640	0.548202	−	0.836346i	$-0.315312\pi$
$11$	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
$15$	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	$-0.538528\pi$
$19$	−20.0000	−0.241490	−0.120745	+	0.992684i	$0.538528\pi$
$20$	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
$26$	−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	$0.518354\pi$
$30$	−60.0000	−0.365148
$31$	−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
$35$	−80.0000	−0.386356
$36$	36.0000	0.166667
$37$	−74.0000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−74.0000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	40.0000	0.170759
$39$	39.0000	0.160128
$40$	−80.0000	−0.316228
$41$	−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	$-0.456972\pi$
$43$	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	$-0.747438\pi$
$47$	−452.000	−1.40279	−0.701393	+	0.712774i	$0.747438\pi$
$48$	48.0000	0.144338
$49$	−279.000	−0.813411
$50$	50.0000	0.141421
$51$	390.000	1.07080
$52$	52.0000	0.138675
$53$	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	$-0.328931\pi$
$59$	464.000	1.02386	0.511929	+	0.859028i	$0.328931\pi$
$60$	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
$65$	130.000	0.248069
$66$	−240.000	−0.447605
$67$	−700.000	−1.27640	−0.638199	−	0.769872i	$-0.720320\pi$
$67$	−700.000	−1.27640	−0.638199	+	0.769872i	$0.720320\pi$
$68$	520.000	0.927342
$69$	0	0
$70$	160.000	0.273195
$71$	−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	$-0.177718\pi$
$73$	1058.00	1.69629	0.848147	+	0.529760i	$0.177718\pi$
$74$	148.000	0.232495
$75$	−75.0000	−0.115470
$76$	−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	$0.744592\pi$
$80$	160.000	0.223607
$81$	81.0000	0.111111
$82$	724.000	0.975030
$83$	−1008.00	−1.33304	−0.666520	−	0.745487i	$-0.732217\pi$
$83$	−1008.00	−1.33304	−0.666520	+	0.745487i	$0.732217\pi$
$84$	−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	$-0.573828\pi$
$89$	−386.000	−0.459729	−0.229865	+	0.973223i	$0.573828\pi$
$90$	−180.000	−0.210819
$91$	−104.000	−0.119804
$92$	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	$-0.604137\pi$
$97$	−614.000	−0.642704	−0.321352	+	0.946960i	$0.604137\pi$
$98$	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	$0.482940\pi$
$104$	−104.000	−0.0980581
$105$	−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	$-0.365399\pi$
$109$	934.000	0.820743	0.410371	+	0.911918i	$0.365399\pi$
$110$	−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
$117$	117.000	0.0924500
$118$	−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
$130$	−260.000	−0.175412
$131$	−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	$-0.241716\pi$
$137$	2326.00	1.45054	0.725269	+	0.688466i	$0.241716\pi$
$138$	0	0
$139$	1932.00	1.17892	0.589461	−	0.807797i	$-0.299340\pi$
$139$	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
$143$	520.000	0.304088
$144$	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	$-0.422042\pi$
$149$	882.000	0.484941	0.242471	+	0.970159i	$0.422042\pi$
$150$	150.000	0.0816497
$151$	−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
$155$	−1840.00	−0.953499
$156$	156.000	0.0800641
$157$	−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	$-0.219393\pi$
$163$	3212.00	1.54346	0.771728	+	0.635953i	$0.219393\pi$
$164$	−1448.00	−0.689450
$165$	1200.00	0.566181
$166$	2016.00	0.942602
$167$	1668.00	0.772896	0.386448	−	0.922311i	$-0.373702\pi$
$167$	1668.00	0.772896	0.386448	+	0.922311i	$0.373702\pi$
$168$	192.000	0.0881733
$169$	169.000	0.0769231
$170$	−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	$0.940720\pi$
$182$	208.000	0.0847142
$183$	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	$-0.520790\pi$
$193$	−350.000	−0.130537	−0.0652683	+	0.997868i	$0.520790\pi$
$194$	1228.00	0.454460
$195$	390.000	0.143223
$196$	−1116.00	−0.406706
$197$	−342.000	−0.123688	−0.0618439	−	0.998086i	$-0.519698\pi$
$197$	−342.000	−0.123688	−0.0618439	+	0.998086i	$0.519698\pi$
$198$	−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
$208$	208.000	0.0693375
$209$	−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
$221$	1690.00	0.514397
$222$	444.000	0.134231
$223$	−5608.00	−1.68403	−0.842017	−	0.539451i	$-0.818632\pi$
$223$	−5608.00	−1.68403	−0.842017	+	0.539451i	$0.818632\pi$
$224$	256.000	0.0763604
$225$	−225.000	−0.0666667
$226$	−3828.00	−1.12670
$227$	−1928.00	−0.563726	−0.281863	−	0.959455i	$-0.590952\pi$
$227$	−1928.00	−0.563726	−0.281863	+	0.959455i	$0.590952\pi$
$228$	−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	$-0.382717\pi$
$233$	2562.00	0.720353	0.360176	+	0.932884i	$0.382717\pi$
$234$	−234.000	−0.0653720
$235$	−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	$-0.0788733\pi$
$239$	7164.00	1.93891	0.969457	+	0.245260i	$0.0788733\pi$
$240$	480.000	0.129099
$241$	−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
$245$	−2790.00	−0.727537
$246$	2172.00	0.562934
$247$	−260.000	−0.0669773
$248$	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
$260$	520.000	0.124035
$261$	−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	$-0.317836\pi$
$271$	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
$275$	−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	$-0.384278\pi$
$281$	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	$-0.194656\pi$
$283$	7796.00	1.63754	0.818770	+	0.574121i	$0.194656\pi$
$284$	−2992.00	−0.625150
$285$	−600.000	−0.124705
$286$	−1040.00	−0.215023
$287$	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	$-0.372129\pi$
$293$	3922.00	0.781999	0.390999	+	0.920391i	$0.372129\pi$
$294$	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	$-0.313243\pi$
$307$	5956.00	1.10725	0.553627	+	0.832765i	$0.313243\pi$
$308$	−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	$0.431214\pi$
$312$	−312.000	−0.0566139
$313$	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	$-0.663614\pi$
$317$	−5550.00	−0.983341	−0.491670	+	0.870781i	$0.663614\pi$
$318$	−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
$325$	−325.000	−0.0554700
$326$	−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	$-0.496300\pi$
$331$	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
$335$	−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	$0.666296\pi$
$338$	−338.000	−0.0543928
$339$	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	$-0.503955\pi$
$349$	−162.000	−0.0248472	−0.0124236	+	0.999923i	$0.503955\pi$
$350$	−400.000	−0.0610883
$351$	351.000	0.0533761
$352$	−1280.00	−0.193819
$353$	−10754.0	−1.62147	−0.810733	−	0.585416i	$-0.800931\pi$
$353$	−10754.0	−1.62147	−0.810733	+	0.585416i	$0.800931\pi$
$354$	−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	$-0.415043\pi$
$359$	3588.00	0.527486	0.263743	+	0.964593i	$0.415043\pi$
$360$	−720.000	−0.105409
$361$	−6459.00	−0.941682
$362$	9572.00	1.38976
$363$	807.000	0.116685
$364$	−416.000	−0.0599020
$365$	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
$377$	−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	$0.315045\pi$
$380$	−800.000	−0.107998
$381$	3888.00	0.522804
$382$	2624.00	0.351454
$383$	6180.00	0.824499	0.412250	−	0.911071i	$-0.364743\pi$
$383$	6180.00	0.824499	0.412250	+	0.911071i	$0.364743\pi$
$384$	−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	$0.663079\pi$
$390$	−780.000	−0.101274
$391$	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	$-0.374481\pi$
$397$	6078.00	0.768378	0.384189	+	0.923254i	$0.374481\pi$
$398$	6736.00	0.848355
$399$	480.000	0.0602257
$400$	−400.000	−0.0500000
$401$	1830.00	0.227895	0.113947	−	0.993487i	$-0.463650\pi$
$401$	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
$405$	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	$-0.229275\pi$
$409$	12434.0	1.50323	0.751616	+	0.659601i	$0.229275\pi$
$410$	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
$416$	−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	$0.810024\pi$
$420$	−960.000	−0.111531
$421$	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
$425$	−3250.00	−0.370937
$426$	4488.00	0.510433
$427$	−2864.00	−0.324587
$428$	−1488.00	−0.168050
$429$	1560.00	0.175565
$430$	−1520.00	−0.170467
$431$	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	$-0.607710\pi$
$433$	−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
$442$	−3380.00	−0.363733
$443$	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	$-0.405369\pi$
$449$	5574.00	0.585865	0.292932	+	0.956133i	$0.405369\pi$
$450$	450.000	0.0471405
$451$	−14480.0	−1.51183
$452$	7656.00	0.796699
$453$	−5328.00	−0.552608
$454$	3856.00	0.398615
$455$	−1040.00	−0.107156
$456$	480.000	0.0492940
$457$	1266.00	0.129586	0.0647932	−	0.997899i	$-0.479361\pi$
$457$	1266.00	0.129586	0.0647932	+	0.997899i	$0.479361\pi$
$458$	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	$-0.610044\pi$
$463$	−6752.00	−0.677737	−0.338868	+	0.940834i	$0.610044\pi$
$464$	−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	$0.371579\pi$
$468$	468.000	0.0462250
$469$	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	$-0.677299\pi$
$479$	−11084.0	−1.05729	−0.528644	+	0.848844i	$0.677299\pi$
$480$	−960.000	−0.0912871
$481$	−962.000	−0.0911922
$482$	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	$-0.433892\pi$
$487$	4432.00	0.412388	0.206194	+	0.978511i	$0.433892\pi$
$488$	−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
$494$	520.000	0.0473601
$495$	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	$-0.474787\pi$
$499$	1764.00	0.158251	0.0791257	+	0.996865i	$0.474787\pi$
$500$	−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
$507$	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	$0.364655\pi$
$510$	−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
$520$	−1040.00	−0.0877058
$521$	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
$533$	−4706.00	−0.382438
$534$	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	$-0.693318\pi$
$541$	−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
$545$	9340.00	0.734095
$546$	624.000	0.0489098
$547$	−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	$-0.550259\pi$
$557$	−4134.00	−0.314476	−0.157238	+	0.987561i	$0.550259\pi$
$558$	3312.00	0.251269
$559$	988.000	0.0747548
$560$	−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	$0.639910\pi$
$572$	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	$-0.426382\pi$
$577$	6354.00	0.458441	0.229221	+	0.973375i	$0.426382\pi$
$578$	−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
$585$	1170.00	0.0826898
$586$	−7844.00	−0.552957
$587$	−13240.0	−0.930960	−0.465480	−	0.885059i	$-0.654118\pi$
$587$	−13240.0	−0.930960	−0.465480	+	0.885059i	$0.654118\pi$
$588$	−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	$-0.512634\pi$
$593$	−1146.00	−0.0793602	−0.0396801	+	0.999212i	$0.512634\pi$
$594$	−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
$611$	−5876.00	−0.389063
$612$	4680.00	0.309114
$613$	2134.00	0.140606	0.0703030	−	0.997526i	$-0.477603\pi$
$613$	2134.00	0.140606	0.0703030	+	0.997526i	$0.477603\pi$
$614$	−11912.0	−0.782947
$615$	−10860.0	−0.712061
$616$	2560.00	0.167444
$617$	−714.000	−0.0465876	−0.0232938	−	0.999729i	$-0.507415\pi$
$617$	−714.000	−0.0465876	−0.0232938	+	0.999729i	$0.507415\pi$
$618$	−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	$0.102170\pi$
$620$	−7360.00	−0.476750
$621$	0	0
$622$	−4704.00	−0.303237
$623$	3088.00	0.198584
$624$	624.000	0.0400320
$625$	−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	$-0.640426\pi$
$631$	−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
$635$	12960.0	0.809924
$636$	4584.00	0.285798
$637$	−3627.00	−0.225600
$638$	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	$-0.347370\pi$
$643$	15044.0	0.922671	0.461335	+	0.887226i	$0.347370\pi$
$644$	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
$650$	650.000	0.0392232
$651$	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	$-0.318911\pi$
$661$	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
$665$	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
$676$	676.000	0.0384615
$677$	−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	$-0.635991\pi$
$683$	−14792.0	−0.828697	−0.414349	+	0.910118i	$0.635991\pi$
$684$	−720.000	−0.0402484
$685$	23260.0	1.29740
$686$	−9952.00	−0.553891
$687$	−11814.0	−0.656088
$688$	1216.00	0.0673831
$689$	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
$695$	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	$0.368804\pi$
$702$	−702.000	−0.0377426
$703$	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	$-0.553823\pi$
$709$	−6354.00	−0.336572	−0.168286	+	0.985738i	$0.553823\pi$
$710$	14960.0	0.790759
$711$	−8784.00	−0.463327
$712$	3088.00	0.162539
$713$	0	0
$714$	6240.00	0.327067
$715$	5200.00	0.271985
$716$	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	$0.685195\pi$
$728$	832.000	0.0423571
$729$	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	$-0.331991\pi$
$733$	19990.0	1.00730	0.503648	+	0.863909i	$0.331991\pi$
$734$	−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	$-0.495785\pi$
$739$	532.000	0.0264816	0.0132408	+	0.999912i	$0.495785\pi$
$740$	−2960.00	−0.147043
$741$	−780.000	−0.0386694
$742$	6112.00	0.302397
$743$	−25452.0	−1.25672	−0.628360	−	0.777922i	$-0.716274\pi$
$743$	−25452.0	−1.25672	−0.628360	+	0.777922i	$0.716274\pi$
$744$	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
$754$	468.000	0.0226042
$755$	−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	$-0.511359\pi$
$761$	−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
$765$	11700.0	0.552960
$766$	−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	$0.387690\pi$
$770$	6400.00	0.299532
$771$	20718.0	0.967757
$772$	−1400.00	−0.0652683
$773$	−3822.00	−0.177837	−0.0889184	−	0.996039i	$-0.528341\pi$
$773$	−3822.00	−0.177837	−0.0889184	+	0.996039i	$0.528341\pi$
$774$	−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
$780$	1560.00	0.0716115
$781$	−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	$-0.586858\pi$
$787$	−11900.0	−0.538995	−0.269498	+	0.963001i	$0.586858\pi$
$788$	−1368.00	−0.0618439
$789$	−20544.0	−0.926978
$790$	19520.0	0.879102
$791$	−15312.0	−0.688283
$792$	−2880.00	−0.129213
$793$	4654.00	0.208409
$794$	−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
$806$	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	$0.704432\pi$
$810$	−1620.00	−0.0702728
$811$	−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
$815$	32120.0	1.38051
$816$	6240.00	0.267701
$817$	−1520.00	−0.0650894
$818$	−24868.0	−1.06295
$819$	−936.000	−0.0399347
$820$	−14480.0	−0.616663
$821$	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	$-0.401473\pi$
$823$	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	$-0.516658\pi$
$827$	−2488.00	−0.104615	−0.0523073	+	0.998631i	$0.516658\pi$
$828$	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
$832$	832.000	0.0346688
$833$	−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	$-0.342232\pi$
$839$	23116.0	0.951195	0.475598	+	0.879663i	$0.342232\pi$
$840$	1920.00	0.0788646
$841$	−24065.0	−0.986715
$842$	−17276.0	−0.707091
$843$	10050.0	0.410605
$844$	−8016.00	−0.326922
$845$	1690.00	0.0688021
$846$	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	$-0.494033\pi$
$853$	934.000	0.0374907	0.0187453	+	0.999824i	$0.494033\pi$
$854$	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	$0.418928\pi$
$858$	−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	$0.649550\pi$
$860$	3040.00	0.120539
$861$	8688.00	0.343886
$862$	−8584.00	−0.339179
$863$	−76.0000	−0.00299776	−0.00149888	−	0.999999i	$-0.500477\pi$
$863$	−76.0000	−0.00299776	−0.00149888	+	0.999999i	$0.500477\pi$
$864$	−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
$871$	−9100.00	−0.354009
$872$	−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	$-0.847964\pi$
$877$	−46130.0	−1.77617	−0.888084	+	0.459681i	$0.847964\pi$
$878$	−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	$0.141119\pi$
$884$	6760.00	0.257198
$885$	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
$910$	2080.00	0.0757707
$911$	−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
$923$	−9724.00	−0.346771
$924$	−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	$-0.457709\pi$
$929$	7502.00	0.264944	0.132472	+	0.991187i	$0.457709\pi$
$930$	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
$936$	−936.000	−0.0326860
$937$	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	$-0.367533\pi$
$941$	23338.0	0.808498	0.404249	+	0.914649i	$0.367533\pi$
$942$	14460.0	0.500140
$943$	0	0
$944$	7424.00	0.255965
$945$	−2160.00	−0.0743543
$946$	−6080.00	−0.208962
$947$	−30488.0	−1.04617	−0.523087	−	0.852279i	$-0.675220\pi$
$947$	−30488.0	−1.04617	−0.523087	+	0.852279i	$0.675220\pi$
$948$	−11712.0	−0.401253
$949$	13754.0	0.470468
$950$	−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
$962$	1924.00	0.0644826
$963$	−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	$-0.540418\pi$
$967$	−7616.00	−0.253272	−0.126636	+	0.991949i	$0.540418\pi$
$968$	−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
$975$	−975.000	−0.0320256
$976$	5728.00	0.187857
$977$	−48666.0	−1.59362	−0.796808	−	0.604232i	$-0.793480\pi$
$977$	−48666.0	−1.59362	−0.796808	+	0.604232i	$0.793480\pi$
$978$	−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	$-0.408972\pi$
$983$	17388.0	0.564182	0.282091	+	0.959388i	$0.408972\pi$
$984$	8688.00	0.281467
$985$	−3420.00	−0.110630
$986$	4680.00	0.151158
$987$	10848.0	0.349844
$988$	−1040.00	−0.0334887
$989$	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	78.4.a.c.1.1	✓	1
3.2	odd	2		234.4.a.h.1.1		1
4.3	odd	2		624.4.a.d.1.1		1
5.4	even	2		1950.4.a.l.1.1		1
8.3	odd	2		2496.4.a.j.1.1		1
8.5	even	2		2496.4.a.a.1.1		1
12.11	even	2		1872.4.a.d.1.1		1
13.5	odd	4		1014.4.b.h.337.2		2
13.8	odd	4		1014.4.b.h.337.1		2
13.12	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	1.1	even	1	trivial
234.4.a.h.1.1		1	3.2	odd	2
624.4.a.d.1.1		1	4.3	odd	2
1014.4.a.j.1.1		1	13.12	even	2
1014.4.b.h.337.1		2	13.8	odd	4
1014.4.b.h.337.2		2	13.5	odd	4
1872.4.a.d.1.1		1	12.11	even	2
1950.4.a.l.1.1		1	5.4	even	2
2496.4.a.a.1.1		1	8.5	even	2
2496.4.a.j.1.1		1	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	78.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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