LMFDB - Embedded newform 390.4.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,4,Mod(1,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("390.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 390 = 2 \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	390.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$23.0107449022$
Analytic rank:	$1$
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	$\chi$	$=$	390.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} -6.00000 q^{6} +8.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} -6.00000 q^{6} +8.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -10.0000 q^{10} -40.0000 q^{11} -12.0000 q^{12} -13.0000 q^{13} +16.0000 q^{14} +15.0000 q^{15} +16.0000 q^{16} +10.0000 q^{17} +18.0000 q^{18} -20.0000 q^{20} -24.0000 q^{21} -80.0000 q^{22} -180.000 q^{23} -24.0000 q^{24} +25.0000 q^{25} -26.0000 q^{26} -27.0000 q^{27} +32.0000 q^{28} +22.0000 q^{29} +30.0000 q^{30} -144.000 q^{31} +32.0000 q^{32} +120.000 q^{33} +20.0000 q^{34} -40.0000 q^{35} +36.0000 q^{36} +34.0000 q^{37} +39.0000 q^{39} -40.0000 q^{40} -502.000 q^{41} -48.0000 q^{42} -76.0000 q^{43} -160.000 q^{44} -45.0000 q^{45} -360.000 q^{46} -168.000 q^{47} -48.0000 q^{48} -279.000 q^{49} +50.0000 q^{50} -30.0000 q^{51} -52.0000 q^{52} -422.000 q^{53} -54.0000 q^{54} +200.000 q^{55} +64.0000 q^{56} +44.0000 q^{58} +104.000 q^{59} +60.0000 q^{60} -82.0000 q^{61} -288.000 q^{62} +72.0000 q^{63} +64.0000 q^{64} +65.0000 q^{65} +240.000 q^{66} -540.000 q^{67} +40.0000 q^{68} +540.000 q^{69} -80.0000 q^{70} +512.000 q^{71} +72.0000 q^{72} +622.000 q^{73} +68.0000 q^{74} -75.0000 q^{75} -320.000 q^{77} +78.0000 q^{78} +104.000 q^{79} -80.0000 q^{80} +81.0000 q^{81} -1004.00 q^{82} +348.000 q^{83} -96.0000 q^{84} -50.0000 q^{85} -152.000 q^{86} -66.0000 q^{87} -320.000 q^{88} -286.000 q^{89} -90.0000 q^{90} -104.000 q^{91} -720.000 q^{92} +432.000 q^{93} -336.000 q^{94} -96.0000 q^{96} +494.000 q^{97} -558.000 q^{98} -360.000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−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
$10$	−10.0000	−0.316228
$11$	−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
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	16.0000	0.305441
$15$	15.0000	0.258199
$16$	16.0000	0.250000
$17$	10.0000	0.142668	0.0713340	−	0.997452i	$-0.477274\pi$
$17$	10.0000	0.142668	0.0713340	+	0.997452i	$0.477274\pi$
$18$	18.0000	0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−20.0000	−0.223607
$21$	−24.0000	−0.249392
$22$	−80.0000	−0.775275
$23$	−180.000	−1.63185	−0.815926	−	0.578156i	$-0.803772\pi$
$23$	−180.000	−1.63185	−0.815926	+	0.578156i	$0.803772\pi$
$24$	−24.0000	−0.204124
$25$	25.0000	0.200000
$26$	−26.0000	−0.196116
$27$	−27.0000	−0.192450
$28$	32.0000	0.215980
$29$	22.0000	0.140872	0.0704362	−	0.997516i	$-0.477561\pi$
$29$	22.0000	0.140872	0.0704362	+	0.997516i	$0.477561\pi$
$30$	30.0000	0.182574
$31$	−144.000	−0.834296	−0.417148	−	0.908839i	$-0.636970\pi$
$31$	−144.000	−0.834296	−0.417148	+	0.908839i	$0.636970\pi$
$32$	32.0000	0.176777
$33$	120.000	0.633010
$34$	20.0000	0.100882
$35$	−40.0000	−0.193178
$36$	36.0000	0.166667
$37$	34.0000	0.151069	0.0755347	−	0.997143i	$-0.475934\pi$
$37$	34.0000	0.151069	0.0755347	+	0.997143i	$0.475934\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	−40.0000	−0.158114
$41$	−502.000	−1.91218	−0.956088	−	0.293079i	$-0.905320\pi$
$41$	−502.000	−1.91218	−0.956088	+	0.293079i	$0.905320\pi$
$42$	−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
$45$	−45.0000	−0.149071
$46$	−360.000	−1.15389
$47$	−168.000	−0.521390	−0.260695	−	0.965421i	$-0.583952\pi$
$47$	−168.000	−0.521390	−0.260695	+	0.965421i	$0.583952\pi$
$48$	−48.0000	−0.144338
$49$	−279.000	−0.813411
$50$	50.0000	0.141421
$51$	−30.0000	−0.0823694
$52$	−52.0000	−0.138675
$53$	−422.000	−1.09370	−0.546851	−	0.837230i	$-0.684173\pi$
$53$	−422.000	−1.09370	−0.546851	+	0.837230i	$0.684173\pi$
$54$	−54.0000	−0.136083
$55$	200.000	0.490327
$56$	64.0000	0.152721
$57$	0	0
$58$	44.0000	0.0996118
$59$	104.000	0.229486	0.114743	−	0.993395i	$-0.463396\pi$
$59$	104.000	0.229486	0.114743	+	0.993395i	$0.463396\pi$
$60$	60.0000	0.129099
$61$	−82.0000	−0.172115	−0.0860576	−	0.996290i	$-0.527427\pi$
$61$	−82.0000	−0.172115	−0.0860576	+	0.996290i	$0.527427\pi$
$62$	−288.000	−0.589936
$63$	72.0000	0.143986
$64$	64.0000	0.125000
$65$	65.0000	0.124035
$66$	240.000	0.447605
$67$	−540.000	−0.984649	−0.492325	−	0.870412i	$-0.663853\pi$
$67$	−540.000	−0.984649	−0.492325	+	0.870412i	$0.663853\pi$
$68$	40.0000	0.0713340
$69$	540.000	0.942150
$70$	−80.0000	−0.136598
$71$	512.000	0.855820	0.427910	−	0.903821i	$-0.359250\pi$
$71$	512.000	0.855820	0.427910	+	0.903821i	$0.359250\pi$
$72$	72.0000	0.117851
$73$	622.000	0.997255	0.498627	−	0.866816i	$-0.333838\pi$
$73$	622.000	0.997255	0.498627	+	0.866816i	$0.333838\pi$
$74$	68.0000	0.106822
$75$	−75.0000	−0.115470
$76$	0	0
$77$	−320.000	−0.473602
$78$	78.0000	0.113228
$79$	104.000	0.148113	0.0740564	−	0.997254i	$-0.476406\pi$
$79$	104.000	0.148113	0.0740564	+	0.997254i	$0.476406\pi$
$80$	−80.0000	−0.111803
$81$	81.0000	0.111111
$82$	−1004.00	−1.35211
$83$	348.000	0.460216	0.230108	−	0.973165i	$-0.426092\pi$
$83$	348.000	0.460216	0.230108	+	0.973165i	$0.426092\pi$
$84$	−96.0000	−0.124696
$85$	−50.0000	−0.0638031
$86$	−152.000	−0.190588
$87$	−66.0000	−0.0813327
$88$	−320.000	−0.387638
$89$	−286.000	−0.340629	−0.170314	−	0.985390i	$-0.554478\pi$
$89$	−286.000	−0.340629	−0.170314	+	0.985390i	$0.554478\pi$
$90$	−90.0000	−0.105409
$91$	−104.000	−0.119804
$92$	−720.000	−0.815926
$93$	432.000	0.481681
$94$	−336.000	−0.368678
$95$	0	0
$96$	−96.0000	−0.102062
$97$	494.000	0.517094	0.258547	−	0.965999i	$-0.416756\pi$
$97$	494.000	0.517094	0.258547	+	0.965999i	$0.416756\pi$
$98$	−558.000	−0.575168
$99$	−360.000	−0.365468
$100$	100.000	0.100000
$101$	1078.00	1.06203	0.531015	−	0.847362i	$-0.321811\pi$
$101$	1078.00	1.06203	0.531015	+	0.847362i	$0.321811\pi$
$102$	−60.0000	−0.0582440
$103$	−12.0000	−0.0114796	−0.00573978	−	0.999984i	$-0.501827\pi$
$103$	−12.0000	−0.0114796	−0.00573978	+	0.999984i	$0.501827\pi$
$104$	−104.000	−0.0980581
$105$	120.000	0.111531
$106$	−844.000	−0.773363
$107$	−828.000	−0.748091	−0.374046	−	0.927410i	$-0.622030\pi$
$107$	−828.000	−0.748091	−0.374046	+	0.927410i	$0.622030\pi$
$108$	−108.000	−0.0962250
$109$	1674.00	1.47101	0.735505	−	0.677519i	$-0.236945\pi$
$109$	1674.00	1.47101	0.735505	+	0.677519i	$0.236945\pi$
$110$	400.000	0.346714
$111$	−102.000	−0.0872199
$112$	128.000	0.107990
$113$	−134.000	−0.111555	−0.0557773	−	0.998443i	$-0.517764\pi$
$113$	−134.000	−0.111555	−0.0557773	+	0.998443i	$0.517764\pi$
$114$	0	0
$115$	900.000	0.729786
$116$	88.0000	0.0704362
$117$	−117.000	−0.0924500
$118$	208.000	0.162271
$119$	80.0000	0.0616268
$120$	120.000	0.0912871
$121$	269.000	0.202104
$122$	−164.000	−0.121704
$123$	1506.00	1.10400
$124$	−576.000	−0.417148
$125$	−125.000	−0.0894427
$126$	144.000	0.101814
$127$	−836.000	−0.584118	−0.292059	−	0.956400i	$-0.594340\pi$
$127$	−836.000	−0.584118	−0.292059	+	0.956400i	$0.594340\pi$
$128$	128.000	0.0883883
$129$	228.000	0.155615
$130$	130.000	0.0877058
$131$	1868.00	1.24586	0.622931	−	0.782277i	$-0.285942\pi$
$131$	1868.00	1.24586	0.622931	+	0.782277i	$0.285942\pi$
$132$	480.000	0.316505
$133$	0	0
$134$	−1080.00	−0.696252
$135$	135.000	0.0860663
$136$	80.0000	0.0504408
$137$	−1026.00	−0.639833	−0.319916	−	0.947446i	$-0.603655\pi$
$137$	−1026.00	−0.639833	−0.319916	+	0.947446i	$0.603655\pi$
$138$	1080.00	0.666201
$139$	532.000	0.324631	0.162315	−	0.986739i	$-0.448104\pi$
$139$	532.000	0.324631	0.162315	+	0.986739i	$0.448104\pi$
$140$	−160.000	−0.0965891
$141$	504.000	0.301025
$142$	1024.00	0.605156
$143$	520.000	0.304088
$144$	144.000	0.0833333
$145$	−110.000	−0.0630000
$146$	1244.00	0.705166
$147$	837.000	0.469623
$148$	136.000	0.0755347
$149$	3082.00	1.69455	0.847273	−	0.531158i	$-0.178243\pi$
$149$	3082.00	1.69455	0.847273	+	0.531158i	$0.178243\pi$
$150$	−150.000	−0.0816497
$151$	−576.000	−0.310425	−0.155213	−	0.987881i	$-0.549606\pi$
$151$	−576.000	−0.310425	−0.155213	+	0.987881i	$0.549606\pi$
$152$	0	0
$153$	90.0000	0.0475560
$154$	−640.000	−0.334887
$155$	720.000	0.373108
$156$	156.000	0.0800641
$157$	−230.000	−0.116917	−0.0584586	−	0.998290i	$-0.518619\pi$
$157$	−230.000	−0.116917	−0.0584586	+	0.998290i	$0.518619\pi$
$158$	208.000	0.104732
$159$	1266.00	0.631449
$160$	−160.000	−0.0790569
$161$	−1440.00	−0.704894
$162$	162.000	0.0785674
$163$	1388.00	0.666973	0.333486	−	0.942755i	$-0.391775\pi$
$163$	1388.00	0.666973	0.333486	+	0.942755i	$0.391775\pi$
$164$	−2008.00	−0.956088
$165$	−600.000	−0.283091
$166$	696.000	0.325422
$167$	−3488.00	−1.61622	−0.808112	−	0.589028i	$-0.799511\pi$
$167$	−3488.00	−1.61622	−0.808112	+	0.589028i	$0.799511\pi$
$168$	−192.000	−0.0881733
$169$	169.000	0.0769231
$170$	−100.000	−0.0451156
$171$	0	0
$172$	−304.000	−0.134766
$173$	−958.000	−0.421014	−0.210507	−	0.977592i	$-0.567511\pi$
$173$	−958.000	−0.421014	−0.210507	+	0.977592i	$0.567511\pi$
$174$	−132.000	−0.0575109
$175$	200.000	0.0863919
$176$	−640.000	−0.274101
$177$	−312.000	−0.132494
$178$	−572.000	−0.240861
$179$	−1412.00	−0.589597	−0.294798	−	0.955559i	$-0.595253\pi$
$179$	−1412.00	−0.589597	−0.294798	+	0.955559i	$0.595253\pi$
$180$	−180.000	−0.0745356
$181$	4414.00	1.81265	0.906326	−	0.422579i	$-0.138875\pi$
$181$	4414.00	1.81265	0.906326	+	0.422579i	$0.138875\pi$
$182$	−208.000	−0.0847142
$183$	246.000	0.0993707
$184$	−1440.00	−0.576947
$185$	−170.000	−0.0675603
$186$	864.000	0.340600
$187$	−400.000	−0.156422
$188$	−672.000	−0.260695
$189$	−216.000	−0.0831306
$190$	0	0
$191$	−4632.00	−1.75476	−0.877382	−	0.479793i	$-0.840712\pi$
$191$	−4632.00	−1.75476	−0.877382	+	0.479793i	$0.840712\pi$
$192$	−192.000	−0.0721688
$193$	2750.00	1.02564	0.512822	−	0.858495i	$-0.328600\pi$
$193$	2750.00	1.02564	0.512822	+	0.858495i	$0.328600\pi$
$194$	988.000	0.365641
$195$	−195.000	−0.0716115
$196$	−1116.00	−0.406706
$197$	−318.000	−0.115008	−0.0575040	−	0.998345i	$-0.518314\pi$
$197$	−318.000	−0.115008	−0.0575040	+	0.998345i	$0.518314\pi$
$198$	−720.000	−0.258425
$199$	−408.000	−0.145338	−0.0726692	−	0.997356i	$-0.523152\pi$
$199$	−408.000	−0.145338	−0.0726692	+	0.997356i	$0.523152\pi$
$200$	200.000	0.0707107
$201$	1620.00	0.568488
$202$	2156.00	0.750968
$203$	176.000	0.0608511
$204$	−120.000	−0.0411847
$205$	2510.00	0.855151
$206$	−24.0000	−0.00811728
$207$	−1620.00	−0.543951
$208$	−208.000	−0.0693375
$209$	0	0
$210$	240.000	0.0788646
$211$	4756.00	1.55174	0.775869	−	0.630895i	$-0.217312\pi$
$211$	4756.00	1.55174	0.775869	+	0.630895i	$0.217312\pi$
$212$	−1688.00	−0.546851
$213$	−1536.00	−0.494108
$214$	−1656.00	−0.528981
$215$	380.000	0.120539
$216$	−216.000	−0.0680414
$217$	−1152.00	−0.360382
$218$	3348.00	1.04016
$219$	−1866.00	−0.575765
$220$	800.000	0.245164
$221$	−130.000	−0.0395690
$222$	−204.000	−0.0616738
$223$	808.000	0.242635	0.121318	−	0.992614i	$-0.461288\pi$
$223$	808.000	0.242635	0.121318	+	0.992614i	$0.461288\pi$
$224$	256.000	0.0763604
$225$	225.000	0.0666667
$226$	−268.000	−0.0788810
$227$	−212.000	−0.0619865	−0.0309932	−	0.999520i	$-0.509867\pi$
$227$	−212.000	−0.0619865	−0.0309932	+	0.999520i	$0.509867\pi$
$228$	0	0
$229$	−1718.00	−0.495758	−0.247879	−	0.968791i	$-0.579734\pi$
$229$	−1718.00	−0.495758	−0.247879	+	0.968791i	$0.579734\pi$
$230$	1800.00	0.516037
$231$	960.000	0.273434
$232$	176.000	0.0498059
$233$	18.0000	0.00506103	0.00253051	−	0.999997i	$-0.499195\pi$
$233$	18.0000	0.00506103	0.00253051	+	0.999997i	$0.499195\pi$
$234$	−234.000	−0.0653720
$235$	840.000	0.233173
$236$	416.000	0.114743
$237$	−312.000	−0.0855130
$238$	160.000	0.0435767
$239$	−7216.00	−1.95299	−0.976494	−	0.215544i	$-0.930848\pi$
$239$	−7216.00	−1.95299	−0.976494	+	0.215544i	$0.930848\pi$
$240$	240.000	0.0645497
$241$	5378.00	1.43746	0.718729	−	0.695290i	$-0.244724\pi$
$241$	5378.00	1.43746	0.718729	+	0.695290i	$0.244724\pi$
$242$	538.000	0.142909
$243$	−243.000	−0.0641500
$244$	−328.000	−0.0860576
$245$	1395.00	0.363768
$246$	3012.00	0.780643
$247$	0	0
$248$	−1152.00	−0.294968
$249$	−1044.00	−0.265706
$250$	−250.000	−0.0632456
$251$	−4596.00	−1.15576	−0.577882	−	0.816120i	$-0.696121\pi$
$251$	−4596.00	−1.15576	−0.577882	+	0.816120i	$0.696121\pi$
$252$	288.000	0.0719932
$253$	7200.00	1.78917
$254$	−1672.00	−0.413034
$255$	150.000	0.0368367
$256$	256.000	0.0625000
$257$	234.000	0.0567958	0.0283979	−	0.999597i	$-0.490959\pi$
$257$	234.000	0.0567958	0.0283979	+	0.999597i	$0.490959\pi$
$258$	456.000	0.110036
$259$	272.000	0.0652558
$260$	260.000	0.0620174
$261$	198.000	0.0469574
$262$	3736.00	0.880957
$263$	−3932.00	−0.921892	−0.460946	−	0.887428i	$-0.652490\pi$
$263$	−3932.00	−0.921892	−0.460946	+	0.887428i	$0.652490\pi$
$264$	960.000	0.223803
$265$	2110.00	0.489118
$266$	0	0
$267$	858.000	0.196662
$268$	−2160.00	−0.492325
$269$	5646.00	1.27971	0.639856	−	0.768495i	$-0.278994\pi$
$269$	5646.00	1.27971	0.639856	+	0.768495i	$0.278994\pi$
$270$	270.000	0.0608581
$271$	−488.000	−0.109387	−0.0546935	−	0.998503i	$-0.517418\pi$
$271$	−488.000	−0.109387	−0.0546935	+	0.998503i	$0.517418\pi$
$272$	160.000	0.0356670
$273$	312.000	0.0691689
$274$	−2052.00	−0.452430
$275$	−1000.00	−0.219281
$276$	2160.00	0.471075
$277$	−2518.00	−0.546180	−0.273090	−	0.961988i	$-0.588046\pi$
$277$	−2518.00	−0.546180	−0.273090	+	0.961988i	$0.588046\pi$
$278$	1064.00	0.229548
$279$	−1296.00	−0.278099
$280$	−320.000	−0.0682988
$281$	8370.00	1.77691	0.888456	−	0.458962i	$-0.151778\pi$
$281$	8370.00	1.77691	0.888456	+	0.458962i	$0.151778\pi$
$282$	1008.00	0.212856
$283$	2684.00	0.563771	0.281886	−	0.959448i	$-0.409040\pi$
$283$	2684.00	0.563771	0.281886	+	0.959448i	$0.409040\pi$
$284$	2048.00	0.427910
$285$	0	0
$286$	1040.00	0.215023
$287$	−4016.00	−0.825983
$288$	288.000	0.0589256
$289$	−4813.00	−0.979646
$290$	−220.000	−0.0445477
$291$	−1482.00	−0.298544
$292$	2488.00	0.498627
$293$	−6142.00	−1.22464	−0.612320	−	0.790610i	$-0.709764\pi$
$293$	−6142.00	−1.22464	−0.612320	+	0.790610i	$0.709764\pi$
$294$	1674.00	0.332074
$295$	−520.000	−0.102629
$296$	272.000	0.0534111
$297$	1080.00	0.211003
$298$	6164.00	1.19822
$299$	2340.00	0.452594
$300$	−300.000	−0.0577350
$301$	−608.000	−0.116427
$302$	−1152.00	−0.219504
$303$	−3234.00	−0.613163
$304$	0	0
$305$	410.000	0.0769722
$306$	180.000	0.0336272
$307$	−4836.00	−0.899039	−0.449520	−	0.893270i	$-0.648405\pi$
$307$	−4836.00	−0.899039	−0.449520	+	0.893270i	$0.648405\pi$
$308$	−1280.00	−0.236801
$309$	36.0000	0.00662773
$310$	1440.00	0.263827
$311$	672.000	0.122526	0.0612631	−	0.998122i	$-0.480487\pi$
$311$	672.000	0.122526	0.0612631	+	0.998122i	$0.480487\pi$
$312$	312.000	0.0566139
$313$	−1302.00	−0.235123	−0.117561	−	0.993066i	$-0.537508\pi$
$313$	−1302.00	−0.235123	−0.117561	+	0.993066i	$0.537508\pi$
$314$	−460.000	−0.0826729
$315$	−360.000	−0.0643927
$316$	416.000	0.0740564
$317$	−5430.00	−0.962079	−0.481040	−	0.876699i	$-0.659741\pi$
$317$	−5430.00	−0.962079	−0.481040	+	0.876699i	$0.659741\pi$
$318$	2532.00	0.446502
$319$	−880.000	−0.154453
$320$	−320.000	−0.0559017
$321$	2484.00	0.431911
$322$	−2880.00	−0.498435
$323$	0	0
$324$	324.000	0.0555556
$325$	−325.000	−0.0554700
$326$	2776.00	0.471621
$327$	−5022.00	−0.849288
$328$	−4016.00	−0.676056
$329$	−1344.00	−0.225219
$330$	−1200.00	−0.200175
$331$	2200.00	0.365326	0.182663	−	0.983176i	$-0.441528\pi$
$331$	2200.00	0.365326	0.182663	+	0.983176i	$0.441528\pi$
$332$	1392.00	0.230108
$333$	306.000	0.0503564
$334$	−6976.00	−1.14284
$335$	2700.00	0.440349
$336$	−384.000	−0.0623480
$337$	−1006.00	−0.162612	−0.0813061	−	0.996689i	$-0.525909\pi$
$337$	−1006.00	−0.162612	−0.0813061	+	0.996689i	$0.525909\pi$
$338$	338.000	0.0543928
$339$	402.000	0.0644060
$340$	−200.000	−0.0319015
$341$	5760.00	0.914726
$342$	0	0
$343$	−4976.00	−0.783320
$344$	−608.000	−0.0952941
$345$	−2700.00	−0.421342
$346$	−1916.00	−0.297702
$347$	−7212.00	−1.11574	−0.557868	−	0.829930i	$-0.688380\pi$
$347$	−7212.00	−1.11574	−0.557868	+	0.829930i	$0.688380\pi$
$348$	−264.000	−0.0406663
$349$	7418.00	1.13775	0.568877	−	0.822422i	$-0.307378\pi$
$349$	7418.00	1.13775	0.568877	+	0.822422i	$0.307378\pi$
$350$	400.000	0.0610883
$351$	351.000	0.0533761
$352$	−1280.00	−0.193819
$353$	−11746.0	−1.77104	−0.885519	−	0.464603i	$-0.846197\pi$
$353$	−11746.0	−1.77104	−0.885519	+	0.464603i	$0.846197\pi$
$354$	−624.000	−0.0936871
$355$	−2560.00	−0.382734
$356$	−1144.00	−0.170314
$357$	−240.000	−0.0355802
$358$	−2824.00	−0.416908
$359$	−552.000	−0.0811517	−0.0405758	−	0.999176i	$-0.512919\pi$
$359$	−552.000	−0.0811517	−0.0405758	+	0.999176i	$0.512919\pi$
$360$	−360.000	−0.0527046
$361$	−6859.00	−1.00000
$362$	8828.00	1.28174
$363$	−807.000	−0.116685
$364$	−416.000	−0.0599020
$365$	−3110.00	−0.445986
$366$	492.000	0.0702657
$367$	3948.00	0.561537	0.280768	−	0.959776i	$-0.409411\pi$
$367$	3948.00	0.561537	0.280768	+	0.959776i	$0.409411\pi$
$368$	−2880.00	−0.407963
$369$	−4518.00	−0.637392
$370$	−340.000	−0.0477723
$371$	−3376.00	−0.472434
$372$	1728.00	0.240840
$373$	−3326.00	−0.461699	−0.230850	−	0.972989i	$-0.574151\pi$
$373$	−3326.00	−0.461699	−0.230850	+	0.972989i	$0.574151\pi$
$374$	−800.000	−0.110607
$375$	375.000	0.0516398
$376$	−1344.00	−0.184339
$377$	−286.000	−0.0390710
$378$	−432.000	−0.0587822
$379$	−13320.0	−1.80528	−0.902642	−	0.430393i	$-0.858375\pi$
$379$	−13320.0	−1.80528	−0.902642	+	0.430393i	$0.858375\pi$
$380$	0	0
$381$	2508.00	0.337241
$382$	−9264.00	−1.24080
$383$	−9600.00	−1.28078	−0.640388	−	0.768052i	$-0.721226\pi$
$383$	−9600.00	−1.28078	−0.640388	+	0.768052i	$0.721226\pi$
$384$	−384.000	−0.0510310
$385$	1600.00	0.211801
$386$	5500.00	0.725240
$387$	−684.000	−0.0898441
$388$	1976.00	0.258547
$389$	−2.00000	−0.000260679 0	−0.000130339	−	1.00000i	$-0.500041\pi$
$389$	−2.00000	−0.000260679 0	−0.000130339		1.00000i	$0.500041\pi$
$390$	−390.000	−0.0506370
$391$	−1800.00	−0.232813
$392$	−2232.00	−0.287584
$393$	−5604.00	−0.719299
$394$	−636.000	−0.0813229
$395$	−520.000	−0.0662381
$396$	−1440.00	−0.182734
$397$	4682.00	0.591896	0.295948	−	0.955204i	$-0.404364\pi$
$397$	4682.00	0.591896	0.295948	+	0.955204i	$0.404364\pi$
$398$	−816.000	−0.102770
$399$	0	0
$400$	400.000	0.0500000
$401$	−5190.00	−0.646325	−0.323162	−	0.946344i	$-0.604746\pi$
$401$	−5190.00	−0.646325	−0.323162	+	0.946344i	$0.604746\pi$
$402$	3240.00	0.401981
$403$	1872.00	0.231392
$404$	4312.00	0.531015
$405$	−405.000	−0.0496904
$406$	352.000	0.0430282
$407$	−1360.00	−0.165633
$408$	−240.000	−0.0291220
$409$	4354.00	0.526385	0.263192	−	0.964743i	$-0.415225\pi$
$409$	4354.00	0.526385	0.263192	+	0.964743i	$0.415225\pi$
$410$	5020.00	0.604683
$411$	3078.00	0.369408
$412$	−48.0000	−0.00573978
$413$	832.000	0.0991284
$414$	−3240.00	−0.384631
$415$	−1740.00	−0.205815
$416$	−416.000	−0.0490290
$417$	−1596.00	−0.187426
$418$	0	0
$419$	−628.000	−0.0732215	−0.0366107	−	0.999330i	$-0.511656\pi$
$419$	−628.000	−0.0732215	−0.0366107	+	0.999330i	$0.511656\pi$
$420$	480.000	0.0557657
$421$	−12782.0	−1.47971	−0.739853	−	0.672768i	$-0.765105\pi$
$421$	−12782.0	−1.47971	−0.739853	+	0.672768i	$0.765105\pi$
$422$	9512.00	1.09724
$423$	−1512.00	−0.173797
$424$	−3376.00	−0.386682
$425$	250.000	0.0285336
$426$	−3072.00	−0.349387
$427$	−656.000	−0.0743468
$428$	−3312.00	−0.374046
$429$	−1560.00	−0.175565
$430$	760.000	0.0852336
$431$	10152.0	1.13458	0.567291	−	0.823518i	$-0.307992\pi$
$431$	10152.0	1.13458	0.567291	+	0.823518i	$0.307992\pi$
$432$	−432.000	−0.0481125
$433$	1762.00	0.195557	0.0977787	−	0.995208i	$-0.468826\pi$
$433$	1762.00	0.195557	0.0977787	+	0.995208i	$0.468826\pi$
$434$	−2304.00	−0.254828
$435$	330.000	0.0363731
$436$	6696.00	0.735505
$437$	0	0
$438$	−3732.00	−0.407128
$439$	3616.00	0.393126	0.196563	−	0.980491i	$-0.437022\pi$
$439$	3616.00	0.393126	0.196563	+	0.980491i	$0.437022\pi$
$440$	1600.00	0.173357
$441$	−2511.00	−0.271137
$442$	−260.000	−0.0279795
$443$	−8476.00	−0.909045	−0.454522	−	0.890735i	$-0.650190\pi$
$443$	−8476.00	−0.909045	−0.454522	+	0.890735i	$0.650190\pi$
$444$	−408.000	−0.0436100
$445$	1430.00	0.152334
$446$	1616.00	0.171569
$447$	−9246.00	−0.978346
$448$	512.000	0.0539949
$449$	13914.0	1.46245	0.731227	−	0.682134i	$-0.238948\pi$
$449$	13914.0	1.46245	0.731227	+	0.682134i	$0.238948\pi$
$450$	450.000	0.0471405
$451$	20080.0	2.09652
$452$	−536.000	−0.0557773
$453$	1728.00	0.179224
$454$	−424.000	−0.0438311
$455$	520.000	0.0535780
$456$	0	0
$457$	−17386.0	−1.77961	−0.889806	−	0.456339i	$-0.849161\pi$
$457$	−17386.0	−1.77961	−0.889806	+	0.456339i	$0.849161\pi$
$458$	−3436.00	−0.350554
$459$	−270.000	−0.0274565
$460$	3600.00	0.364893
$461$	17514.0	1.76943	0.884716	−	0.466130i	$-0.154352\pi$
$461$	17514.0	1.76943	0.884716	+	0.466130i	$0.154352\pi$
$462$	1920.00	0.193347
$463$	11632.0	1.16757	0.583785	−	0.811908i	$-0.301571\pi$
$463$	11632.0	1.16757	0.583785	+	0.811908i	$0.301571\pi$
$464$	352.000	0.0352181
$465$	−2160.00	−0.215414
$466$	36.0000	0.00357869
$467$	−244.000	−0.0241777	−0.0120888	−	0.999927i	$-0.503848\pi$
$467$	−244.000	−0.0241777	−0.0120888	+	0.999927i	$0.503848\pi$
$468$	−468.000	−0.0462250
$469$	−4320.00	−0.425328
$470$	1680.00	0.164878
$471$	690.000	0.0675022
$472$	832.000	0.0811354
$473$	3040.00	0.295517
$474$	−624.000	−0.0604668
$475$	0	0
$476$	320.000	0.0308134
$477$	−3798.00	−0.364567
$478$	−14432.0	−1.38097
$479$	14416.0	1.37512	0.687561	−	0.726126i	$-0.258681\pi$
$479$	14416.0	1.37512	0.687561	+	0.726126i	$0.258681\pi$
$480$	480.000	0.0456435
$481$	−442.000	−0.0418991
$482$	10756.0	1.01644
$483$	4320.00	0.406971
$484$	1076.00	0.101052
$485$	−2470.00	−0.231251
$486$	−486.000	−0.0453609
$487$	−8992.00	−0.836687	−0.418343	−	0.908289i	$-0.637389\pi$
$487$	−8992.00	−0.836687	−0.418343	+	0.908289i	$0.637389\pi$
$488$	−656.000	−0.0608519
$489$	−4164.00	−0.385077
$490$	2790.00	0.257223
$491$	420.000	0.0386035	0.0193018	−	0.999814i	$-0.493856\pi$
$491$	420.000	0.0386035	0.0193018	+	0.999814i	$0.493856\pi$
$492$	6024.00	0.551998
$493$	220.000	0.0200980
$494$	0	0
$495$	1800.00	0.163442
$496$	−2304.00	−0.208574
$497$	4096.00	0.369679
$498$	−2088.00	−0.187883
$499$	−17176.0	−1.54089	−0.770444	−	0.637507i	$-0.779966\pi$
$499$	−17176.0	−1.54089	−0.770444	+	0.637507i	$0.779966\pi$
$500$	−500.000	−0.0447214
$501$	10464.0	0.933128
$502$	−9192.00	−0.817249
$503$	7964.00	0.705959	0.352979	−	0.935631i	$-0.385169\pi$
$503$	7964.00	0.705959	0.352979	+	0.935631i	$0.385169\pi$
$504$	576.000	0.0509069
$505$	−5390.00	−0.474954
$506$	14400.0	1.26513
$507$	−507.000	−0.0444116
$508$	−3344.00	−0.292059
$509$	9954.00	0.866804	0.433402	−	0.901201i	$-0.357313\pi$
$509$	9954.00	0.866804	0.433402	+	0.901201i	$0.357313\pi$
$510$	300.000	0.0260475
$511$	4976.00	0.430774
$512$	512.000	0.0441942
$513$	0	0
$514$	468.000	0.0401607
$515$	60.0000	0.00513382
$516$	912.000	0.0778073
$517$	6720.00	0.571654
$518$	544.000	0.0461428
$519$	2874.00	0.243072
$520$	520.000	0.0438529
$521$	14714.0	1.23730	0.618649	−	0.785668i	$-0.287680\pi$
$521$	14714.0	1.23730	0.618649	+	0.785668i	$0.287680\pi$
$522$	396.000	0.0332039
$523$	17076.0	1.42769	0.713844	−	0.700304i	$-0.246952\pi$
$523$	17076.0	1.42769	0.713844	+	0.700304i	$0.246952\pi$
$524$	7472.00	0.622931
$525$	−600.000	−0.0498784
$526$	−7864.00	−0.651876
$527$	−1440.00	−0.119027
$528$	1920.00	0.158252
$529$	20233.0	1.66294
$530$	4220.00	0.345859
$531$	936.000	0.0764952
$532$	0	0
$533$	6526.00	0.530342
$534$	1716.00	0.139061
$535$	4140.00	0.334557
$536$	−4320.00	−0.348126
$537$	4236.00	0.340404
$538$	11292.0	0.904893
$539$	11160.0	0.891828
$540$	540.000	0.0430331
$541$	−12542.0	−0.996715	−0.498357	−	0.866972i	$-0.666063\pi$
$541$	−12542.0	−0.996715	−0.498357	+	0.866972i	$0.666063\pi$
$542$	−976.000	−0.0773483
$543$	−13242.0	−1.04654
$544$	320.000	0.0252204
$545$	−8370.00	−0.657856
$546$	624.000	0.0489098
$547$	−18044.0	−1.41043	−0.705215	−	0.708993i	$-0.749150\pi$
$547$	−18044.0	−1.41043	−0.705215	+	0.708993i	$0.749150\pi$
$548$	−4104.00	−0.319916
$549$	−738.000	−0.0573717
$550$	−2000.00	−0.155055
$551$	0	0
$552$	4320.00	0.333100
$553$	832.000	0.0639787
$554$	−5036.00	−0.386208
$555$	510.000	0.0390059
$556$	2128.00	0.162315
$557$	13354.0	1.01585	0.507924	−	0.861402i	$-0.330413\pi$
$557$	13354.0	1.01585	0.507924	+	0.861402i	$0.330413\pi$
$558$	−2592.00	−0.196645
$559$	988.000	0.0747548
$560$	−640.000	−0.0482945
$561$	1200.00	0.0903102
$562$	16740.0	1.25647
$563$	−17172.0	−1.28546	−0.642730	−	0.766093i	$-0.722198\pi$
$563$	−17172.0	−1.28546	−0.642730	+	0.766093i	$0.722198\pi$
$564$	2016.00	0.150512
$565$	670.000	0.0498887
$566$	5368.00	0.398646
$567$	648.000	0.0479955
$568$	4096.00	0.302578
$569$	16074.0	1.18428	0.592142	−	0.805834i	$-0.298283\pi$
$569$	16074.0	1.18428	0.592142	+	0.805834i	$0.298283\pi$
$570$	0	0
$571$	−4052.00	−0.296972	−0.148486	−	0.988915i	$-0.547440\pi$
$571$	−4052.00	−0.296972	−0.148486	+	0.988915i	$0.547440\pi$
$572$	2080.00	0.152044
$573$	13896.0	1.01311
$574$	−8032.00	−0.584058
$575$	−4500.00	−0.326370
$576$	576.000	0.0416667
$577$	−12474.0	−0.899999	−0.449999	−	0.893029i	$-0.648576\pi$
$577$	−12474.0	−0.899999	−0.449999	+	0.893029i	$0.648576\pi$
$578$	−9626.00	−0.692714
$579$	−8250.00	−0.592156
$580$	−440.000	−0.0315000
$581$	2784.00	0.198795
$582$	−2964.00	−0.211103
$583$	16880.0	1.19914
$584$	4976.00	0.352583
$585$	585.000	0.0413449
$586$	−12284.0	−0.865951
$587$	2940.00	0.206724	0.103362	−	0.994644i	$-0.467040\pi$
$587$	2940.00	0.206724	0.103362	+	0.994644i	$0.467040\pi$
$588$	3348.00	0.234812
$589$	0	0
$590$	−1040.00	−0.0725697
$591$	954.000	0.0663999
$592$	544.000	0.0377673
$593$	10286.0	0.712303	0.356151	−	0.934428i	$-0.384089\pi$
$593$	10286.0	0.712303	0.356151	+	0.934428i	$0.384089\pi$
$594$	2160.00	0.149202
$595$	−400.000	−0.0275603
$596$	12328.0	0.847273
$597$	1224.00	0.0839112
$598$	4680.00	0.320032
$599$	11784.0	0.803808	0.401904	−	0.915682i	$-0.368348\pi$
$599$	11784.0	0.803808	0.401904	+	0.915682i	$0.368348\pi$
$600$	−600.000	−0.0408248
$601$	8410.00	0.570800	0.285400	−	0.958408i	$-0.407873\pi$
$601$	8410.00	0.570800	0.285400	+	0.958408i	$0.407873\pi$
$602$	−1216.00	−0.0823263
$603$	−4860.00	−0.328216
$604$	−2304.00	−0.155213
$605$	−1345.00	−0.0903835
$606$	−6468.00	−0.433572
$607$	3284.00	0.219594	0.109797	−	0.993954i	$-0.464980\pi$
$607$	3284.00	0.219594	0.109797	+	0.993954i	$0.464980\pi$
$608$	0	0
$609$	−528.000	−0.0351324
$610$	820.000	0.0544276
$611$	2184.00	0.144608
$612$	360.000	0.0237780
$613$	14546.0	0.958413	0.479207	−	0.877702i	$-0.340924\pi$
$613$	14546.0	0.958413	0.479207	+	0.877702i	$0.340924\pi$
$614$	−9672.00	−0.635717
$615$	−7530.00	−0.493722
$616$	−2560.00	−0.167444
$617$	22374.0	1.45988	0.729938	−	0.683514i	$-0.239549\pi$
$617$	22374.0	1.45988	0.729938	+	0.683514i	$0.239549\pi$
$618$	72.0000	0.00468651
$619$	11288.0	0.732961	0.366481	−	0.930426i	$-0.380563\pi$
$619$	11288.0	0.732961	0.366481	+	0.930426i	$0.380563\pi$
$620$	2880.00	0.186554
$621$	4860.00	0.314050
$622$	1344.00	0.0866391
$623$	−2288.00	−0.147138
$624$	624.000	0.0400320
$625$	625.000	0.0400000
$626$	−2604.00	−0.166257
$627$	0	0
$628$	−920.000	−0.0584586
$629$	340.000	0.0215528
$630$	−720.000	−0.0455325
$631$	−25536.0	−1.61105	−0.805525	−	0.592562i	$-0.798116\pi$
$631$	−25536.0	−1.61105	−0.805525	+	0.592562i	$0.798116\pi$
$632$	832.000	0.0523658
$633$	−14268.0	−0.895896
$634$	−10860.0	−0.680293
$635$	4180.00	0.261226
$636$	5064.00	0.315724
$637$	3627.00	0.225600
$638$	−1760.00	−0.109215
$639$	4608.00	0.285273
$640$	−640.000	−0.0395285
$641$	−29262.0	−1.80309	−0.901544	−	0.432687i	$-0.857565\pi$
$641$	−29262.0	−1.80309	−0.901544	+	0.432687i	$0.857565\pi$
$642$	4968.00	0.305407
$643$	9956.00	0.610616	0.305308	−	0.952254i	$-0.401241\pi$
$643$	9956.00	0.610616	0.305308	+	0.952254i	$0.401241\pi$
$644$	−5760.00	−0.352447
$645$	−1140.00	−0.0695930
$646$	0	0
$647$	−14676.0	−0.891767	−0.445883	−	0.895091i	$-0.647110\pi$
$647$	−14676.0	−0.891767	−0.445883	+	0.895091i	$0.647110\pi$
$648$	648.000	0.0392837
$649$	−4160.00	−0.251609
$650$	−650.000	−0.0392232
$651$	3456.00	0.208067
$652$	5552.00	0.333486
$653$	3114.00	0.186616	0.0933080	−	0.995637i	$-0.470256\pi$
$653$	3114.00	0.186616	0.0933080	+	0.995637i	$0.470256\pi$
$654$	−10044.0	−0.600537
$655$	−9340.00	−0.557166
$656$	−8032.00	−0.478044
$657$	5598.00	0.332418
$658$	−2688.00	−0.159254
$659$	−19604.0	−1.15882	−0.579411	−	0.815036i	$-0.696717\pi$
$659$	−19604.0	−1.15882	−0.579411	+	0.815036i	$0.696717\pi$
$660$	−2400.00	−0.141545
$661$	−12150.0	−0.714947	−0.357474	−	0.933923i	$-0.616362\pi$
$661$	−12150.0	−0.714947	−0.357474	+	0.933923i	$0.616362\pi$
$662$	4400.00	0.258325
$663$	390.000	0.0228452
$664$	2784.00	0.162711
$665$	0	0
$666$	612.000	0.0356074
$667$	−3960.00	−0.229883
$668$	−13952.0	−0.808112
$669$	−2424.00	−0.140086
$670$	5400.00	0.311373
$671$	3280.00	0.188708
$672$	−768.000	−0.0440867
$673$	8138.00	0.466117	0.233059	−	0.972463i	$-0.425127\pi$
$673$	8138.00	0.466117	0.233059	+	0.972463i	$0.425127\pi$
$674$	−2012.00	−0.114984
$675$	−675.000	−0.0384900
$676$	676.000	0.0384615
$677$	−9294.00	−0.527618	−0.263809	−	0.964575i	$-0.584979\pi$
$677$	−9294.00	−0.527618	−0.263809	+	0.964575i	$0.584979\pi$
$678$	804.000	0.0455419
$679$	3952.00	0.223364
$680$	−400.000	−0.0225578
$681$	636.000	0.0357879
$682$	11520.0	0.646809
$683$	30932.0	1.73291	0.866457	−	0.499252i	$-0.166392\pi$
$683$	30932.0	1.73291	0.866457	+	0.499252i	$0.166392\pi$
$684$	0	0
$685$	5130.00	0.286142
$686$	−9952.00	−0.553891
$687$	5154.00	0.286226
$688$	−1216.00	−0.0673831
$689$	5486.00	0.303338
$690$	−5400.00	−0.297934
$691$	15712.0	0.864997	0.432498	−	0.901635i	$-0.357632\pi$
$691$	15712.0	0.864997	0.432498	+	0.901635i	$0.357632\pi$
$692$	−3832.00	−0.210507
$693$	−2880.00	−0.157867
$694$	−14424.0	−0.788945
$695$	−2660.00	−0.145179
$696$	−528.000	−0.0287554
$697$	−5020.00	−0.272806
$698$	14836.0	0.804514
$699$	−54.0000	−0.00292199
$700$	800.000	0.0431959
$701$	−16610.0	−0.894937	−0.447469	−	0.894300i	$-0.647674\pi$
$701$	−16610.0	−0.894937	−0.447469	+	0.894300i	$0.647674\pi$
$702$	702.000	0.0377426
$703$	0	0
$704$	−2560.00	−0.137051
$705$	−2520.00	−0.134622
$706$	−23492.0	−1.25231
$707$	8624.00	0.458754
$708$	−1248.00	−0.0662468
$709$	14666.0	0.776859	0.388430	−	0.921478i	$-0.373018\pi$
$709$	14666.0	0.776859	0.388430	+	0.921478i	$0.373018\pi$
$710$	−5120.00	−0.270634
$711$	936.000	0.0493709
$712$	−2288.00	−0.120430
$713$	25920.0	1.36145
$714$	−480.000	−0.0251590
$715$	−2600.00	−0.135992
$716$	−5648.00	−0.294798
$717$	21648.0	1.12756
$718$	−1104.00	−0.0573829
$719$	11688.0	0.606243	0.303122	−	0.952952i	$-0.401971\pi$
$719$	11688.0	0.606243	0.303122	+	0.952952i	$0.401971\pi$
$720$	−720.000	−0.0372678
$721$	−96.0000	−0.00495871
$722$	−13718.0	−0.707107
$723$	−16134.0	−0.829917
$724$	17656.0	0.906326
$725$	550.000	0.0281745
$726$	−1614.00	−0.0825085
$727$	−28316.0	−1.44454	−0.722271	−	0.691610i	$-0.756902\pi$
$727$	−28316.0	−1.44454	−0.722271	+	0.691610i	$0.756902\pi$
$728$	−832.000	−0.0423571
$729$	729.000	0.0370370
$730$	−6220.00	−0.315360
$731$	−760.000	−0.0384536
$732$	984.000	0.0496854
$733$	4490.00	0.226251	0.113125	−	0.993581i	$-0.463914\pi$
$733$	4490.00	0.226251	0.113125	+	0.993581i	$0.463914\pi$
$734$	7896.00	0.397066
$735$	−4185.00	−0.210022
$736$	−5760.00	−0.288473
$737$	21600.0	1.07957
$738$	−9036.00	−0.450704
$739$	17592.0	0.875686	0.437843	−	0.899051i	$-0.355743\pi$
$739$	17592.0	0.875686	0.437843	+	0.899051i	$0.355743\pi$
$740$	−680.000	−0.0337801
$741$	0	0
$742$	−6752.00	−0.334062
$743$	−1848.00	−0.0912470	−0.0456235	−	0.998959i	$-0.514527\pi$
$743$	−1848.00	−0.0912470	−0.0456235	+	0.998959i	$0.514527\pi$
$744$	3456.00	0.170300
$745$	−15410.0	−0.757824
$746$	−6652.00	−0.326471
$747$	3132.00	0.153405
$748$	−1600.00	−0.0782110
$749$	−6624.00	−0.323145
$750$	750.000	0.0365148
$751$	−14080.0	−0.684136	−0.342068	−	0.939675i	$-0.611127\pi$
$751$	−14080.0	−0.684136	−0.342068	+	0.939675i	$0.611127\pi$
$752$	−2688.00	−0.130347
$753$	13788.0	0.667281
$754$	−572.000	−0.0276273
$755$	2880.00	0.138826
$756$	−864.000	−0.0415653
$757$	7146.00	0.343099	0.171549	−	0.985176i	$-0.445123\pi$
$757$	7146.00	0.343099	0.171549	+	0.985176i	$0.445123\pi$
$758$	−26640.0	−1.27653
$759$	−21600.0	−1.03298
$760$	0	0
$761$	−8398.00	−0.400036	−0.200018	−	0.979792i	$-0.564100\pi$
$761$	−8398.00	−0.400036	−0.200018	+	0.979792i	$0.564100\pi$
$762$	5016.00	0.238465
$763$	13392.0	0.635417
$764$	−18528.0	−0.877382
$765$	−450.000	−0.0212677
$766$	−19200.0	−0.905645
$767$	−1352.00	−0.0636478
$768$	−768.000	−0.0360844
$769$	5698.00	0.267198	0.133599	−	0.991035i	$-0.457347\pi$
$769$	5698.00	0.267198	0.133599	+	0.991035i	$0.457347\pi$
$770$	3200.00	0.149766
$771$	−702.000	−0.0327911
$772$	11000.0	0.512822
$773$	5602.00	0.260660	0.130330	−	0.991471i	$-0.458396\pi$
$773$	5602.00	0.260660	0.130330	+	0.991471i	$0.458396\pi$
$774$	−1368.00	−0.0635294
$775$	−3600.00	−0.166859
$776$	3952.00	0.182820
$777$	−816.000	−0.0376755
$778$	−4.00000	−0.000184328 0
$779$	0	0
$780$	−780.000	−0.0358057
$781$	−20480.0	−0.938325
$782$	−3600.00	−0.164624
$783$	−594.000	−0.0271109
$784$	−4464.00	−0.203353
$785$	1150.00	0.0522870
$786$	−11208.0	−0.508621
$787$	11500.0	0.520878	0.260439	−	0.965490i	$-0.416133\pi$
$787$	11500.0	0.520878	0.260439	+	0.965490i	$0.416133\pi$
$788$	−1272.00	−0.0575040
$789$	11796.0	0.532254
$790$	−1040.00	−0.0468374
$791$	−1072.00	−0.0481870
$792$	−2880.00	−0.129213
$793$	1066.00	0.0477362
$794$	9364.00	0.418534
$795$	−6330.00	−0.282392
$796$	−1632.00	−0.0726692
$797$	−686.000	−0.0304885	−0.0152443	−	0.999884i	$-0.504853\pi$
$797$	−686.000	−0.0304885	−0.0152443	+	0.999884i	$0.504853\pi$
$798$	0	0
$799$	−1680.00	−0.0743856
$800$	800.000	0.0353553
$801$	−2574.00	−0.113543
$802$	−10380.0	−0.457021
$803$	−24880.0	−1.09339
$804$	6480.00	0.284244
$805$	7200.00	0.315238
$806$	3744.00	0.163619
$807$	−16938.0	−0.738842
$808$	8624.00	0.375484
$809$	−30806.0	−1.33879	−0.669395	−	0.742907i	$-0.733447\pi$
$809$	−30806.0	−1.33879	−0.669395	+	0.742907i	$0.733447\pi$
$810$	−810.000	−0.0351364
$811$	−28064.0	−1.21512	−0.607559	−	0.794275i	$-0.707851\pi$
$811$	−28064.0	−1.21512	−0.607559	+	0.794275i	$0.707851\pi$
$812$	704.000	0.0304256
$813$	1464.00	0.0631546
$814$	−2720.00	−0.117120
$815$	−6940.00	−0.298279
$816$	−480.000	−0.0205924
$817$	0	0
$818$	8708.00	0.372210
$819$	−936.000	−0.0399347
$820$	10040.0	0.427576
$821$	19954.0	0.848233	0.424117	−	0.905608i	$-0.360585\pi$
$821$	19954.0	0.848233	0.424117	+	0.905608i	$0.360585\pi$
$822$	6156.00	0.261211
$823$	7676.00	0.325114	0.162557	−	0.986699i	$-0.448026\pi$
$823$	7676.00	0.325114	0.162557	+	0.986699i	$0.448026\pi$
$824$	−96.0000	−0.00405864
$825$	3000.00	0.126602
$826$	1664.00	0.0700944
$827$	19788.0	0.832039	0.416019	−	0.909356i	$-0.363425\pi$
$827$	19788.0	0.832039	0.416019	+	0.909356i	$0.363425\pi$
$828$	−6480.00	−0.271975
$829$	−2578.00	−0.108007	−0.0540034	−	0.998541i	$-0.517198\pi$
$829$	−2578.00	−0.108007	−0.0540034	+	0.998541i	$0.517198\pi$
$830$	−3480.00	−0.145533
$831$	7554.00	0.315337
$832$	−832.000	−0.0346688
$833$	−2790.00	−0.116048
$834$	−3192.00	−0.132530
$835$	17440.0	0.722798
$836$	0	0
$837$	3888.00	0.160560
$838$	−1256.00	−0.0517754
$839$	−6464.00	−0.265986	−0.132993	−	0.991117i	$-0.542459\pi$
$839$	−6464.00	−0.265986	−0.132993	+	0.991117i	$0.542459\pi$
$840$	960.000	0.0394323
$841$	−23905.0	−0.980155
$842$	−25564.0	−1.04631
$843$	−25110.0	−1.02590
$844$	19024.0	0.775869
$845$	−845.000	−0.0344010
$846$	−3024.00	−0.122893
$847$	2152.00	0.0873006
$848$	−6752.00	−0.273425
$849$	−8052.00	−0.325493
$850$	500.000	0.0201763
$851$	−6120.00	−0.246523
$852$	−6144.00	−0.247054
$853$	5026.00	0.201743	0.100872	−	0.994899i	$-0.467837\pi$
$853$	5026.00	0.201743	0.100872	+	0.994899i	$0.467837\pi$
$854$	−1312.00	−0.0525711
$855$	0	0
$856$	−6624.00	−0.264490
$857$	−32662.0	−1.30188	−0.650941	−	0.759128i	$-0.725625\pi$
$857$	−32662.0	−1.30188	−0.650941	+	0.759128i	$0.725625\pi$
$858$	−3120.00	−0.124143
$859$	−25236.0	−1.00238	−0.501188	−	0.865338i	$-0.667104\pi$
$859$	−25236.0	−1.00238	−0.501188	+	0.865338i	$0.667104\pi$
$860$	1520.00	0.0602693
$861$	12048.0	0.476881
$862$	20304.0	0.802270
$863$	−44664.0	−1.76174	−0.880869	−	0.473360i	$-0.843041\pi$
$863$	−44664.0	−1.76174	−0.880869	+	0.473360i	$0.843041\pi$
$864$	−864.000	−0.0340207
$865$	4790.00	0.188283
$866$	3524.00	0.138280
$867$	14439.0	0.565599
$868$	−4608.00	−0.180191
$869$	−4160.00	−0.162392
$870$	660.000	0.0257197
$871$	7020.00	0.273093
$872$	13392.0	0.520081
$873$	4446.00	0.172365
$874$	0	0
$875$	−1000.00	−0.0386356
$876$	−7464.00	−0.287883
$877$	15370.0	0.591799	0.295900	−	0.955219i	$-0.404381\pi$
$877$	15370.0	0.591799	0.295900	+	0.955219i	$0.404381\pi$
$878$	7232.00	0.277982
$879$	18426.0	0.707046
$880$	3200.00	0.122582
$881$	−24558.0	−0.939137	−0.469569	−	0.882896i	$-0.655591\pi$
$881$	−24558.0	−0.939137	−0.469569	+	0.882896i	$0.655591\pi$
$882$	−5022.00	−0.191723
$883$	1916.00	0.0730221	0.0365111	−	0.999333i	$-0.488376\pi$
$883$	1916.00	0.0730221	0.0365111	+	0.999333i	$0.488376\pi$
$884$	−520.000	−0.0197845
$885$	1560.00	0.0592529
$886$	−16952.0	−0.642792
$887$	−3412.00	−0.129159	−0.0645793	−	0.997913i	$-0.520571\pi$
$887$	−3412.00	−0.129159	−0.0645793	+	0.997913i	$0.520571\pi$
$888$	−816.000	−0.0308369
$889$	−6688.00	−0.252315
$890$	2860.00	0.107716
$891$	−3240.00	−0.121823
$892$	3232.00	0.121318
$893$	0	0
$894$	−18492.0	−0.691795
$895$	7060.00	0.263676
$896$	1024.00	0.0381802
$897$	−7020.00	−0.261305
$898$	27828.0	1.03411
$899$	−3168.00	−0.117529
$900$	900.000	0.0333333
$901$	−4220.00	−0.156036
$902$	40160.0	1.48246
$903$	1824.00	0.0672192
$904$	−1072.00	−0.0394405
$905$	−22070.0	−0.810643
$906$	3456.00	0.126731
$907$	−20780.0	−0.760737	−0.380369	−	0.924835i	$-0.624203\pi$
$907$	−20780.0	−0.760737	−0.380369	+	0.924835i	$0.624203\pi$
$908$	−848.000	−0.0309932
$909$	9702.00	0.354010
$910$	1040.00	0.0378853
$911$	−36040.0	−1.31071	−0.655356	−	0.755320i	$-0.727481\pi$
$911$	−36040.0	−1.31071	−0.655356	+	0.755320i	$0.727481\pi$
$912$	0	0
$913$	−13920.0	−0.504584
$914$	−34772.0	−1.25838
$915$	−1230.00	−0.0444399
$916$	−6872.00	−0.247879
$917$	14944.0	0.538162
$918$	−540.000	−0.0194147
$919$	13960.0	0.501086	0.250543	−	0.968105i	$-0.419391\pi$
$919$	13960.0	0.501086	0.250543	+	0.968105i	$0.419391\pi$
$920$	7200.00	0.258018
$921$	14508.0	0.519061
$922$	35028.0	1.25118
$923$	−6656.00	−0.237362
$924$	3840.00	0.136717
$925$	850.000	0.0302139
$926$	23264.0	0.825597
$927$	−108.000	−0.00382652
$928$	704.000	0.0249029
$929$	23562.0	0.832125	0.416063	−	0.909336i	$-0.363410\pi$
$929$	23562.0	0.832125	0.416063	+	0.909336i	$0.363410\pi$
$930$	−4320.00	−0.152321
$931$	0	0
$932$	72.0000	0.00253051
$933$	−2016.00	−0.0707405
$934$	−488.000	−0.0170962
$935$	2000.00	0.0699540
$936$	−936.000	−0.0326860
$937$	−41478.0	−1.44613	−0.723067	−	0.690778i	$-0.757268\pi$
$937$	−41478.0	−1.44613	−0.723067	+	0.690778i	$0.757268\pi$
$938$	−8640.00	−0.300753
$939$	3906.00	0.135748
$940$	3360.00	0.116586
$941$	−42062.0	−1.45715	−0.728577	−	0.684964i	$-0.759818\pi$
$941$	−42062.0	−1.45715	−0.728577	+	0.684964i	$0.759818\pi$
$942$	1380.00	0.0477312
$943$	90360.0	3.12039
$944$	1664.00	0.0573714
$945$	1080.00	0.0371771
$946$	6080.00	0.208962
$947$	38948.0	1.33647	0.668237	−	0.743949i	$-0.267049\pi$
$947$	38948.0	1.33647	0.668237	+	0.743949i	$0.267049\pi$
$948$	−1248.00	−0.0427565
$949$	−8086.00	−0.276589
$950$	0	0
$951$	16290.0	0.555457
$952$	640.000	0.0217884
$953$	−28902.0	−0.982400	−0.491200	−	0.871047i	$-0.663442\pi$
$953$	−28902.0	−0.982400	−0.491200	+	0.871047i	$0.663442\pi$
$954$	−7596.00	−0.257788
$955$	23160.0	0.784754
$956$	−28864.0	−0.976494
$957$	2640.00	0.0891735
$958$	28832.0	0.972359
$959$	−8208.00	−0.276382
$960$	960.000	0.0322749
$961$	−9055.00	−0.303951
$962$	−884.000	−0.0296271
$963$	−7452.00	−0.249364
$964$	21512.0	0.718729
$965$	−13750.0	−0.458682
$966$	8640.00	0.287772
$967$	−30824.0	−1.02506	−0.512530	−	0.858669i	$-0.671292\pi$
$967$	−30824.0	−1.02506	−0.512530	+	0.858669i	$0.671292\pi$
$968$	2152.00	0.0714544
$969$	0	0
$970$	−4940.00	−0.163519
$971$	−23684.0	−0.782756	−0.391378	−	0.920230i	$-0.628001\pi$
$971$	−23684.0	−0.782756	−0.391378	+	0.920230i	$0.628001\pi$
$972$	−972.000	−0.0320750
$973$	4256.00	0.140227
$974$	−17984.0	−0.591627
$975$	975.000	0.0320256
$976$	−1312.00	−0.0430288
$977$	31446.0	1.02973	0.514865	−	0.857271i	$-0.327842\pi$
$977$	31446.0	1.02973	0.514865	+	0.857271i	$0.327842\pi$
$978$	−8328.00	−0.272290
$979$	11440.0	0.373467
$980$	5580.00	0.181884
$981$	15066.0	0.490337
$982$	840.000	0.0272968
$983$	54952.0	1.78301	0.891504	−	0.453013i	$-0.149651\pi$
$983$	54952.0	1.78301	0.891504	+	0.453013i	$0.149651\pi$
$984$	12048.0	0.390321
$985$	1590.00	0.0514331
$986$	440.000	0.0142114
$987$	4032.00	0.130030
$988$	0	0
$989$	13680.0	0.439837
$990$	3600.00	0.115571
$991$	−39504.0	−1.26628	−0.633141	−	0.774036i	$-0.718235\pi$
$991$	−39504.0	−1.26628	−0.633141	+	0.774036i	$0.718235\pi$
$992$	−4608.00	−0.147484
$993$	−6600.00	−0.210921
$994$	8192.00	0.261403
$995$	2040.00	0.0649973
$996$	−4176.00	−0.132853
$997$	42658.0	1.35506	0.677529	−	0.735496i	$-0.263051\pi$
$997$	42658.0	1.35506	0.677529	+	0.735496i	$0.263051\pi$
$998$	−34352.0	−1.08957
$999$	−918.000	−0.0290733

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	390.4.a.h.1.1	✓	1
3.2	odd	2		1170.4.a.h.1.1		1
5.4	even	2		1950.4.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.4.a.h.1.1	✓	1	1.1	even	1	trivial
1170.4.a.h.1.1		1	3.2	odd	2
1950.4.a.d.1.1		1	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	390.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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