LMFDB - Embedded newform 39.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [39,2,Mod(1,39)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(39, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("39.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 39 = 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	39.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:	$0.311416567883$
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	$\chi$	$=$	39.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} -4.00000 q^{14} -2.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -2.00000 q^{20} +4.00000 q^{21} +4.00000 q^{22} +3.00000 q^{24} -1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} -10.0000 q^{29} -2.00000 q^{30} +4.00000 q^{31} +5.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} -8.00000 q^{35} -1.00000 q^{36} -2.00000 q^{37} -1.00000 q^{39} -6.00000 q^{40} +6.00000 q^{41} +4.00000 q^{42} -12.0000 q^{43} -4.00000 q^{44} +2.00000 q^{45} +1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} -1.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +8.00000 q^{55} +12.0000 q^{56} -10.0000 q^{58} +12.0000 q^{59} +2.00000 q^{60} -2.00000 q^{61} +4.00000 q^{62} -4.00000 q^{63} +7.00000 q^{64} +2.00000 q^{65} -4.00000 q^{66} -8.00000 q^{67} -2.00000 q^{68} -8.00000 q^{70} -3.00000 q^{72} +2.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -16.0000 q^{77} -1.00000 q^{78} +8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +4.00000 q^{83} -4.00000 q^{84} +4.00000 q^{85} -12.0000 q^{86} +10.0000 q^{87} -12.0000 q^{88} -2.00000 q^{89} +2.00000 q^{90} -4.00000 q^{91} -4.00000 q^{93} -5.00000 q^{96} +10.0000 q^{97} +9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	−1.00000	−0.408248
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	1.00000	0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−4.00000	−1.06904
$15$	−2.00000	−0.516398
$16$	−1.00000	−0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	1.00000	0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−2.00000	−0.447214
$21$	4.00000	0.872872
$22$	4.00000	0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	3.00000	0.612372
$25$	−1.00000	−0.200000
$26$	1.00000	0.196116
$27$	−1.00000	−0.192450
$28$	4.00000	0.755929
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	−2.00000	−0.365148
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	5.00000	0.883883
$33$	−4.00000	−0.696311
$34$	2.00000	0.342997
$35$	−8.00000	−1.35225
$36$	−1.00000	−0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	−6.00000	−0.948683
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	4.00000	0.617213
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	−4.00000	−0.603023
$45$	2.00000	0.298142
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	−1.00000	−0.141421
$51$	−2.00000	−0.280056
$52$	−1.00000	−0.138675
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−1.00000	−0.136083
$55$	8.00000	1.07872
$56$	12.0000	1.60357
$57$	0	0
$58$	−10.0000	−1.31306
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	2.00000	0.258199
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	4.00000	0.508001
$63$	−4.00000	−0.503953
$64$	7.00000	0.875000
$65$	2.00000	0.248069
$66$	−4.00000	−0.492366
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	−8.00000	−0.956183
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−3.00000	−0.353553
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−2.00000	−0.232495
$75$	1.00000	0.115470
$76$	0	0
$77$	−16.0000	−1.82337
$78$	−1.00000	−0.113228
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	−4.00000	−0.436436
$85$	4.00000	0.433861
$86$	−12.0000	−1.29399
$87$	10.0000	1.07211
$88$	−12.0000	−1.27920
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	2.00000	0.210819
$91$	−4.00000	−0.419314
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	0	0
$96$	−5.00000	−0.510310
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	9.00000	0.909137
$99$	4.00000	0.402015
$100$	1.00000	0.100000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	−2.00000	−0.198030
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−3.00000	−0.294174
$105$	8.00000	0.780720
$106$	6.00000	0.582772
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	1.00000	0.0962250
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	8.00000	0.762770
$111$	2.00000	0.189832
$112$	4.00000	0.377964
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	10.0000	0.928477
$117$	1.00000	0.0924500
$118$	12.0000	1.10469
$119$	−8.00000	−0.733359
$120$	6.00000	0.547723
$121$	5.00000	0.454545
$122$	−2.00000	−0.181071
$123$	−6.00000	−0.541002
$124$	−4.00000	−0.359211
$125$	−12.0000	−1.07331
$126$	−4.00000	−0.356348
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	−3.00000	−0.265165
$129$	12.0000	1.05654
$130$	2.00000	0.175412
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	−8.00000	−0.691095
$135$	−2.00000	−0.172133
$136$	−6.00000	−0.514496
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	8.00000	0.676123
$141$	0	0
$142$	0	0
$143$	4.00000	0.334497
$144$	−1.00000	−0.0833333
$145$	−20.0000	−1.66091
$146$	2.00000	0.165521
$147$	−9.00000	−0.742307
$148$	2.00000	0.164399
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	1.00000	0.0816497
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	−16.0000	−1.28932
$155$	8.00000	0.642575
$156$	1.00000	0.0800641
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	8.00000	0.636446
$159$	−6.00000	−0.475831
$160$	10.0000	0.790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	−6.00000	−0.468521
$165$	−8.00000	−0.622799
$166$	4.00000	0.310460
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	−12.0000	−0.925820
$169$	1.00000	0.0769231
$170$	4.00000	0.306786
$171$	0	0
$172$	12.0000	0.914991
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	10.0000	0.758098
$175$	4.00000	0.302372
$176$	−4.00000	−0.301511
$177$	−12.0000	−0.901975
$178$	−2.00000	−0.149906
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	−2.00000	−0.149071
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	−4.00000	−0.296500
$183$	2.00000	0.147844
$184$	0	0
$185$	−4.00000	−0.294086
$186$	−4.00000	−0.293294
$187$	8.00000	0.585018
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	−7.00000	−0.505181
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	10.0000	0.717958
$195$	−2.00000	−0.143223
$196$	−9.00000	−0.642857
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	4.00000	0.284268
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	3.00000	0.212132
$201$	8.00000	0.564276
$202$	−18.0000	−1.26648
$203$	40.0000	2.80745
$204$	2.00000	0.140028
$205$	12.0000	0.838116
$206$	0	0
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	0	0
$210$	8.00000	0.552052
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	12.0000	0.820303
$215$	−24.0000	−1.63679
$216$	3.00000	0.204124
$217$	−16.0000	−1.08615
$218$	−2.00000	−0.135457
$219$	−2.00000	−0.135147
$220$	−8.00000	−0.539360
$221$	2.00000	0.134535
$222$	2.00000	0.134231
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	−20.0000	−1.33631
$225$	−1.00000	−0.0666667
$226$	−6.00000	−0.399114
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	30.0000	1.96960
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	−12.0000	−0.781133
$237$	−8.00000	−0.519656
$238$	−8.00000	−0.518563
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	2.00000	0.129099
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	5.00000	0.321412
$243$	−1.00000	−0.0641500
$244$	2.00000	0.128037
$245$	18.0000	1.14998
$246$	−6.00000	−0.382546
$247$	0	0
$248$	−12.0000	−0.762001
$249$	−4.00000	−0.253490
$250$	−12.0000	−0.758947
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	4.00000	0.251976
$253$	0	0
$254$	−16.0000	−1.00393
$255$	−4.00000	−0.250490
$256$	−17.0000	−1.06250
$257$	26.0000	1.62184	0.810918	−	0.585160i	$-0.198968\pi$
$257$	26.0000	1.62184	0.810918	+	0.585160i	$0.198968\pi$
$258$	12.0000	0.747087
$259$	8.00000	0.497096
$260$	−2.00000	−0.124035
$261$	−10.0000	−0.618984
$262$	4.00000	0.247121
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	12.0000	0.738549
$265$	12.0000	0.737154
$266$	0	0
$267$	2.00000	0.122398
$268$	8.00000	0.488678
$269$	22.0000	1.34136	0.670682	−	0.741745i	$-0.266002\pi$
$269$	22.0000	1.34136	0.670682	+	0.741745i	$0.266002\pi$
$270$	−2.00000	−0.121716
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	−2.00000	−0.121268
$273$	4.00000	0.242091
$274$	6.00000	0.362473
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	12.0000	0.719712
$279$	4.00000	0.239474
$280$	24.0000	1.43427
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	0	0
$285$	0	0
$286$	4.00000	0.236525
$287$	−24.0000	−1.41668
$288$	5.00000	0.294628
$289$	−13.0000	−0.764706
$290$	−20.0000	−1.17444
$291$	−10.0000	−0.586210
$292$	−2.00000	−0.117041
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	−9.00000	−0.524891
$295$	24.0000	1.39733
$296$	6.00000	0.348743
$297$	−4.00000	−0.232104
$298$	−6.00000	−0.347571
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	48.0000	2.76667
$302$	4.00000	0.230174
$303$	18.0000	1.03407
$304$	0	0
$305$	−4.00000	−0.229039
$306$	2.00000	0.114332
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	16.0000	0.911685
$309$	0	0
$310$	8.00000	0.454369
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	3.00000	0.169842
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−18.0000	−1.01580
$315$	−8.00000	−0.450749
$316$	−8.00000	−0.450035
$317$	26.0000	1.46031	0.730153	−	0.683284i	$-0.239449\pi$
$317$	26.0000	1.46031	0.730153	+	0.683284i	$0.239449\pi$
$318$	−6.00000	−0.336463
$319$	−40.0000	−2.23957
$320$	14.0000	0.782624
$321$	−12.0000	−0.669775
$322$	0	0
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	−1.00000	−0.0554700
$326$	8.00000	0.443079
$327$	2.00000	0.110600
$328$	−18.0000	−0.993884
$329$	0	0
$330$	−8.00000	−0.440386
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	−4.00000	−0.219529
$333$	−2.00000	−0.109599
$334$	−8.00000	−0.437741
$335$	−16.0000	−0.874173
$336$	−4.00000	−0.218218
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	1.00000	0.0543928
$339$	6.00000	0.325875
$340$	−4.00000	−0.216930
$341$	16.0000	0.866449
$342$	0	0
$343$	−8.00000	−0.431959
$344$	36.0000	1.94099
$345$	0	0
$346$	6.00000	0.322562
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−10.0000	−0.536056
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	4.00000	0.213809
$351$	−1.00000	−0.0533761
$352$	20.0000	1.06600
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	2.00000	0.106000
$357$	8.00000	0.423405
$358$	4.00000	0.211407
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	−6.00000	−0.316228
$361$	−19.0000	−1.00000
$362$	−10.0000	−0.525588
$363$	−5.00000	−0.262432
$364$	4.00000	0.209657
$365$	4.00000	0.209370
$366$	2.00000	0.104542
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	−4.00000	−0.207950
$371$	−24.0000	−1.24602
$372$	4.00000	0.207390
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	8.00000	0.413670
$375$	12.0000	0.619677
$376$	0	0
$377$	−10.0000	−0.515026
$378$	4.00000	0.205738
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	8.00000	0.409316
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	3.00000	0.153093
$385$	−32.0000	−1.63087
$386$	18.0000	0.916176
$387$	−12.0000	−0.609994
$388$	−10.0000	−0.507673
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	−2.00000	−0.101274
$391$	0	0
$392$	−27.0000	−1.36371
$393$	−4.00000	−0.201773
$394$	18.0000	0.906827
$395$	16.0000	0.805047
$396$	−4.00000	−0.201008
$397$	38.0000	1.90717	0.953583	−	0.301131i	$-0.0973643\pi$
$397$	38.0000	1.90717	0.953583	+	0.301131i	$0.0973643\pi$
$398$	8.00000	0.401004
$399$	0	0
$400$	1.00000	0.0500000
$401$	22.0000	1.09863	0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000	1.09863	0.549314	+	0.835616i	$0.314889\pi$
$402$	8.00000	0.399004
$403$	4.00000	0.199254
$404$	18.0000	0.895533
$405$	2.00000	0.0993808
$406$	40.0000	1.98517
$407$	−8.00000	−0.396545
$408$	6.00000	0.297044
$409$	34.0000	1.68119	0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000	1.68119	0.840596	+	0.541663i	$0.182205\pi$
$410$	12.0000	0.592638
$411$	−6.00000	−0.295958
$412$	0	0
$413$	−48.0000	−2.36193
$414$	0	0
$415$	8.00000	0.392705
$416$	5.00000	0.245145
$417$	−12.0000	−0.587643
$418$	0	0
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	−8.00000	−0.390360
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−20.0000	−0.973585
$423$	0	0
$424$	−18.0000	−0.874157
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	8.00000	0.387147
$428$	−12.0000	−0.580042
$429$	−4.00000	−0.193122
$430$	−24.0000	−1.15738
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	1.00000	0.0481125
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	−16.0000	−0.768025
$435$	20.0000	0.958927
$436$	2.00000	0.0957826
$437$	0	0
$438$	−2.00000	−0.0955637
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	−24.0000	−1.14416
$441$	9.00000	0.428571
$442$	2.00000	0.0951303
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	−2.00000	−0.0949158
$445$	−4.00000	−0.189618
$446$	4.00000	0.189405
$447$	6.00000	0.283790
$448$	−28.0000	−1.32288
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	−1.00000	−0.0471405
$451$	24.0000	1.13012
$452$	6.00000	0.282216
$453$	−4.00000	−0.187936
$454$	−20.0000	−0.938647
$455$	−8.00000	−0.375046
$456$	0	0
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	−10.0000	−0.467269
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−38.0000	−1.76984	−0.884918	−	0.465746i	$-0.845786\pi$
$461$	−38.0000	−1.76984	−0.884918	+	0.465746i	$0.845786\pi$
$462$	16.0000	0.744387
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	10.0000	0.464238
$465$	−8.00000	−0.370991
$466$	−14.0000	−0.648537
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	−1.00000	−0.0462250
$469$	32.0000	1.47762
$470$	0	0
$471$	18.0000	0.829396
$472$	−36.0000	−1.65703
$473$	−48.0000	−2.20704
$474$	−8.00000	−0.367452
$475$	0	0
$476$	8.00000	0.366679
$477$	6.00000	0.274721
$478$	−24.0000	−1.09773
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	−10.0000	−0.456435
$481$	−2.00000	−0.0911922
$482$	10.0000	0.455488
$483$	0	0
$484$	−5.00000	−0.227273
$485$	20.0000	0.908153
$486$	−1.00000	−0.0453609
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	6.00000	0.271607
$489$	−8.00000	−0.361773
$490$	18.0000	0.813157
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	6.00000	0.270501
$493$	−20.0000	−0.900755
$494$	0	0
$495$	8.00000	0.359573
$496$	−4.00000	−0.179605
$497$	0	0
$498$	−4.00000	−0.179244
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	12.0000	0.536656
$501$	8.00000	0.357414
$502$	−12.0000	−0.535586
$503$	−8.00000	−0.356702	−0.178351	−	0.983967i	$-0.557076\pi$
$503$	−8.00000	−0.356702	−0.178351	+	0.983967i	$0.557076\pi$
$504$	12.0000	0.534522
$505$	−36.0000	−1.60198
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	16.0000	0.709885
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	−4.00000	−0.177123
$511$	−8.00000	−0.353899
$512$	−11.0000	−0.486136
$513$	0	0
$514$	26.0000	1.14681
$515$	0	0
$516$	−12.0000	−0.528271
$517$	0	0
$518$	8.00000	0.351500
$519$	−6.00000	−0.263371
$520$	−6.00000	−0.263117
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	−10.0000	−0.437688
$523$	44.0000	1.92399	0.961993	−	0.273075i	$-0.0880406\pi$
$523$	44.0000	1.92399	0.961993	+	0.273075i	$0.0880406\pi$
$524$	−4.00000	−0.174741
$525$	−4.00000	−0.174574
$526$	24.0000	1.04645
$527$	8.00000	0.348485
$528$	4.00000	0.174078
$529$	−23.0000	−1.00000
$530$	12.0000	0.521247
$531$	12.0000	0.520756
$532$	0	0
$533$	6.00000	0.259889
$534$	2.00000	0.0865485
$535$	24.0000	1.03761
$536$	24.0000	1.03664
$537$	−4.00000	−0.172613
$538$	22.0000	0.948487
$539$	36.0000	1.55063
$540$	2.00000	0.0860663
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	−12.0000	−0.515444
$543$	10.0000	0.429141
$544$	10.0000	0.428746
$545$	−4.00000	−0.171341
$546$	4.00000	0.171184
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	−6.00000	−0.256307
$549$	−2.00000	−0.0853579
$550$	−4.00000	−0.170561
$551$	0	0
$552$	0	0
$553$	−32.0000	−1.36078
$554$	−10.0000	−0.424859
$555$	4.00000	0.169791
$556$	−12.0000	−0.508913
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	4.00000	0.169334
$559$	−12.0000	−0.507546
$560$	8.00000	0.338062
$561$	−8.00000	−0.337760
$562$	−10.0000	−0.421825
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	−12.0000	−0.504844
$566$	12.0000	0.504398
$567$	−4.00000	−0.167984
$568$	0	0
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	−4.00000	−0.167248
$573$	−8.00000	−0.334205
$574$	−24.0000	−1.00174
$575$	0	0
$576$	7.00000	0.291667
$577$	−46.0000	−1.91501	−0.957503	−	0.288425i	$-0.906868\pi$
$577$	−46.0000	−1.91501	−0.957503	+	0.288425i	$0.906868\pi$
$578$	−13.0000	−0.540729
$579$	−18.0000	−0.748054
$580$	20.0000	0.830455
$581$	−16.0000	−0.663792
$582$	−10.0000	−0.414513
$583$	24.0000	0.993978
$584$	−6.00000	−0.248282
$585$	2.00000	0.0826898
$586$	−6.00000	−0.247858
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	9.00000	0.371154
$589$	0	0
$590$	24.0000	0.988064
$591$	−18.0000	−0.740421
$592$	2.00000	0.0821995
$593$	−26.0000	−1.06769	−0.533846	−	0.845582i	$-0.679254\pi$
$593$	−26.0000	−1.06769	−0.533846	+	0.845582i	$0.679254\pi$
$594$	−4.00000	−0.164122
$595$	−16.0000	−0.655936
$596$	6.00000	0.245770
$597$	−8.00000	−0.327418
$598$	0	0
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	−3.00000	−0.122474
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	48.0000	1.95633
$603$	−8.00000	−0.325785
$604$	−4.00000	−0.162758
$605$	10.0000	0.406558
$606$	18.0000	0.731200
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	0	0
$609$	−40.0000	−1.62088
$610$	−4.00000	−0.161955
$611$	0	0
$612$	−2.00000	−0.0808452
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	−16.0000	−0.645707
$615$	−12.0000	−0.483887
$616$	48.0000	1.93398
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	24.0000	0.964641	0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000	0.964641	0.482321	+	0.875995i	$0.339794\pi$
$620$	−8.00000	−0.321288
$621$	0	0
$622$	0	0
$623$	8.00000	0.320513
$624$	1.00000	0.0400320
$625$	−19.0000	−0.760000
$626$	−6.00000	−0.239808
$627$	0	0
$628$	18.0000	0.718278
$629$	−4.00000	−0.159490
$630$	−8.00000	−0.318728
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	−24.0000	−0.954669
$633$	20.0000	0.794929
$634$	26.0000	1.03259
$635$	−32.0000	−1.26988
$636$	6.00000	0.237915
$637$	9.00000	0.356593
$638$	−40.0000	−1.58362
$639$	0	0
$640$	−6.00000	−0.237171
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	−12.0000	−0.473602
$643$	−40.0000	−1.57745	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	−40.0000	−1.57745	−0.788723	+	0.614749i	$0.789257\pi$
$644$	0	0
$645$	24.0000	0.944999
$646$	0	0
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	−3.00000	−0.117851
$649$	48.0000	1.88416
$650$	−1.00000	−0.0392232
$651$	16.0000	0.627089
$652$	−8.00000	−0.313304
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	2.00000	0.0782062
$655$	8.00000	0.312586
$656$	−6.00000	−0.234261
$657$	2.00000	0.0780274
$658$	0	0
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	8.00000	0.311400
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	−16.0000	−0.621858
$663$	−2.00000	−0.0776736
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	8.00000	0.309529
$669$	−4.00000	−0.154649
$670$	−16.0000	−0.618134
$671$	−8.00000	−0.308837
$672$	20.0000	0.771517
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	18.0000	0.693334
$675$	1.00000	0.0384900
$676$	−1.00000	−0.0384615
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	6.00000	0.230429
$679$	−40.0000	−1.53506
$680$	−12.0000	−0.460179
$681$	20.0000	0.766402
$682$	16.0000	0.612672
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	−8.00000	−0.305441
$687$	10.0000	0.381524
$688$	12.0000	0.457496
$689$	6.00000	0.228582
$690$	0	0
$691$	−24.0000	−0.913003	−0.456502	−	0.889723i	$-0.650898\pi$
$691$	−24.0000	−0.913003	−0.456502	+	0.889723i	$0.650898\pi$
$692$	−6.00000	−0.228086
$693$	−16.0000	−0.607790
$694$	−12.0000	−0.455514
$695$	24.0000	0.910372
$696$	−30.0000	−1.13715
$697$	12.0000	0.454532
$698$	−26.0000	−0.984115
$699$	14.0000	0.529529
$700$	−4.00000	−0.151186
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	−1.00000	−0.0377426
$703$	0	0
$704$	28.0000	1.05529
$705$	0	0
$706$	−2.00000	−0.0752710
$707$	72.0000	2.70784
$708$	12.0000	0.450988
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	6.00000	0.224860
$713$	0	0
$714$	8.00000	0.299392
$715$	8.00000	0.299183
$716$	−4.00000	−0.149487
$717$	24.0000	0.896296
$718$	24.0000	0.895672
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	−19.0000	−0.707107
$723$	−10.0000	−0.371904
$724$	10.0000	0.371647
$725$	10.0000	0.371391
$726$	−5.00000	−0.185567
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	12.0000	0.444750
$729$	1.00000	0.0370370
$730$	4.00000	0.148047
$731$	−24.0000	−0.887672
$732$	−2.00000	−0.0739221
$733$	30.0000	1.10808	0.554038	−	0.832492i	$-0.313086\pi$
$733$	30.0000	1.10808	0.554038	+	0.832492i	$0.313086\pi$
$734$	16.0000	0.590571
$735$	−18.0000	−0.663940
$736$	0	0
$737$	−32.0000	−1.17874
$738$	6.00000	0.220863
$739$	32.0000	1.17714	0.588570	−	0.808447i	$-0.299691\pi$
$739$	32.0000	1.17714	0.588570	+	0.808447i	$0.299691\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	−24.0000	−0.881068
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	12.0000	0.439941
$745$	−12.0000	−0.439646
$746$	−26.0000	−0.951928
$747$	4.00000	0.146352
$748$	−8.00000	−0.292509
$749$	−48.0000	−1.75388
$750$	12.0000	0.438178
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	12.0000	0.437304
$754$	−10.0000	−0.364179
$755$	8.00000	0.291150
$756$	−4.00000	−0.145479
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	−24.0000	−0.871719
$759$	0	0
$760$	0	0
$761$	−10.0000	−0.362500	−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000	−0.362500	−0.181250	+	0.983437i	$0.558014\pi$
$762$	16.0000	0.579619
$763$	8.00000	0.289619
$764$	−8.00000	−0.289430
$765$	4.00000	0.144620
$766$	−16.0000	−0.578103
$767$	12.0000	0.433295
$768$	17.0000	0.613435
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	−32.0000	−1.15320
$771$	−26.0000	−0.936367
$772$	−18.0000	−0.647834
$773$	10.0000	0.359675	0.179838	−	0.983696i	$-0.442443\pi$
$773$	10.0000	0.359675	0.179838	+	0.983696i	$0.442443\pi$
$774$	−12.0000	−0.431331
$775$	−4.00000	−0.143684
$776$	−30.0000	−1.07694
$777$	−8.00000	−0.286998
$778$	22.0000	0.788738
$779$	0	0
$780$	2.00000	0.0716115
$781$	0	0
$782$	0	0
$783$	10.0000	0.357371
$784$	−9.00000	−0.321429
$785$	−36.0000	−1.28490
$786$	−4.00000	−0.142675
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	−18.0000	−0.641223
$789$	−24.0000	−0.854423
$790$	16.0000	0.569254
$791$	24.0000	0.853342
$792$	−12.0000	−0.426401
$793$	−2.00000	−0.0710221
$794$	38.0000	1.34857
$795$	−12.0000	−0.425596
$796$	−8.00000	−0.283552
$797$	46.0000	1.62940	0.814702	−	0.579880i	$-0.196901\pi$
$797$	46.0000	1.62940	0.814702	+	0.579880i	$0.196901\pi$
$798$	0	0
$799$	0	0
$800$	−5.00000	−0.176777
$801$	−2.00000	−0.0706665
$802$	22.0000	0.776847
$803$	8.00000	0.282314
$804$	−8.00000	−0.282138
$805$	0	0
$806$	4.00000	0.140894
$807$	−22.0000	−0.774437
$808$	54.0000	1.89971
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	2.00000	0.0702728
$811$	8.00000	0.280918	0.140459	−	0.990086i	$-0.455142\pi$
$811$	8.00000	0.280918	0.140459	+	0.990086i	$0.455142\pi$
$812$	−40.0000	−1.40372
$813$	12.0000	0.420858
$814$	−8.00000	−0.280400
$815$	16.0000	0.560456
$816$	2.00000	0.0700140
$817$	0	0
$818$	34.0000	1.18878
$819$	−4.00000	−0.139771
$820$	−12.0000	−0.419058
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	−6.00000	−0.209274
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	−48.0000	−1.67013
$827$	4.00000	0.139094	0.0695468	−	0.997579i	$-0.477845\pi$
$827$	4.00000	0.139094	0.0695468	+	0.997579i	$0.477845\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	8.00000	0.277684
$831$	10.0000	0.346896
$832$	7.00000	0.242681
$833$	18.0000	0.623663
$834$	−12.0000	−0.415526
$835$	−16.0000	−0.553703
$836$	0	0
$837$	−4.00000	−0.138260
$838$	4.00000	0.138178
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	−24.0000	−0.828079
$841$	71.0000	2.44828
$842$	−10.0000	−0.344623
$843$	10.0000	0.344418
$844$	20.0000	0.688428
$845$	2.00000	0.0688021
$846$	0	0
$847$	−20.0000	−0.687208
$848$	−6.00000	−0.206041
$849$	−12.0000	−0.411839
$850$	−2.00000	−0.0685994
$851$	0	0
$852$	0	0
$853$	30.0000	1.02718	0.513590	−	0.858036i	$-0.328315\pi$
$853$	30.0000	1.02718	0.513590	+	0.858036i	$0.328315\pi$
$854$	8.00000	0.273754
$855$	0	0
$856$	−36.0000	−1.23045
$857$	−46.0000	−1.57133	−0.785665	−	0.618652i	$-0.787679\pi$
$857$	−46.0000	−1.57133	−0.785665	+	0.618652i	$0.787679\pi$
$858$	−4.00000	−0.136558
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	24.0000	0.818393
$861$	24.0000	0.817918
$862$	0	0
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	−5.00000	−0.170103
$865$	12.0000	0.408012
$866$	34.0000	1.15537
$867$	13.0000	0.441503
$868$	16.0000	0.543075
$869$	32.0000	1.08553
$870$	20.0000	0.678064
$871$	−8.00000	−0.271070
$872$	6.00000	0.203186
$873$	10.0000	0.338449
$874$	0	0
$875$	48.0000	1.62270
$876$	2.00000	0.0675737
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	32.0000	1.07995
$879$	6.00000	0.202375
$880$	−8.00000	−0.269680
$881$	58.0000	1.95407	0.977035	−	0.213080i	$-0.0683494\pi$
$881$	58.0000	1.95407	0.977035	+	0.213080i	$0.0683494\pi$
$882$	9.00000	0.303046
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	−2.00000	−0.0672673
$885$	−24.0000	−0.806751
$886$	−4.00000	−0.134383
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	−6.00000	−0.201347
$889$	64.0000	2.14649
$890$	−4.00000	−0.134080
$891$	4.00000	0.134005
$892$	−4.00000	−0.133930
$893$	0	0
$894$	6.00000	0.200670
$895$	8.00000	0.267411
$896$	12.0000	0.400892
$897$	0	0
$898$	22.0000	0.734150
$899$	−40.0000	−1.33407
$900$	1.00000	0.0333333
$901$	12.0000	0.399778
$902$	24.0000	0.799113
$903$	−48.0000	−1.59734
$904$	18.0000	0.598671
$905$	−20.0000	−0.664822
$906$	−4.00000	−0.132891
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	20.0000	0.663723
$909$	−18.0000	−0.597022
$910$	−8.00000	−0.265197
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	2.00000	0.0661541
$915$	4.00000	0.132236
$916$	10.0000	0.330409
$917$	−16.0000	−0.528367
$918$	−2.00000	−0.0660098
$919$	−48.0000	−1.58337	−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−48.0000	−1.58337	−0.791687	+	0.610927i	$0.790797\pi$
$920$	0	0
$921$	16.0000	0.527218
$922$	−38.0000	−1.25146
$923$	0	0
$924$	−16.0000	−0.526361
$925$	2.00000	0.0657596
$926$	4.00000	0.131448
$927$	0	0
$928$	−50.0000	−1.64133
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	−8.00000	−0.262330
$931$	0	0
$932$	14.0000	0.458585
$933$	0	0
$934$	12.0000	0.392652
$935$	16.0000	0.523256
$936$	−3.00000	−0.0980581
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	32.0000	1.04484
$939$	6.00000	0.195803
$940$	0	0
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	18.0000	0.586472
$943$	0	0
$944$	−12.0000	−0.390567
$945$	8.00000	0.260240
$946$	−48.0000	−1.56061
$947$	−60.0000	−1.94974	−0.974869	−	0.222779i	$-0.928487\pi$
$947$	−60.0000	−1.94974	−0.974869	+	0.222779i	$0.928487\pi$
$948$	8.00000	0.259828
$949$	2.00000	0.0649227
$950$	0	0
$951$	−26.0000	−0.843108
$952$	24.0000	0.777844
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	6.00000	0.194257
$955$	16.0000	0.517748
$956$	24.0000	0.776215
$957$	40.0000	1.29302
$958$	−24.0000	−0.775405
$959$	−24.0000	−0.775000
$960$	−14.0000	−0.451848
$961$	−15.0000	−0.483871
$962$	−2.00000	−0.0644826
$963$	12.0000	0.386695
$964$	−10.0000	−0.322078
$965$	36.0000	1.15888
$966$	0	0
$967$	−52.0000	−1.67221	−0.836104	−	0.548572i	$-0.815172\pi$
$967$	−52.0000	−1.67221	−0.836104	+	0.548572i	$0.815172\pi$
$968$	−15.0000	−0.482118
$969$	0	0
$970$	20.0000	0.642161
$971$	−60.0000	−1.92549	−0.962746	−	0.270408i	$-0.912841\pi$
$971$	−60.0000	−1.92549	−0.962746	+	0.270408i	$0.912841\pi$
$972$	1.00000	0.0320750
$973$	−48.0000	−1.53881
$974$	12.0000	0.384505
$975$	1.00000	0.0320256
$976$	2.00000	0.0640184
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	−8.00000	−0.255812
$979$	−8.00000	−0.255681
$980$	−18.0000	−0.574989
$981$	−2.00000	−0.0638551
$982$	−12.0000	−0.382935
$983$	16.0000	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$984$	18.0000	0.573819
$985$	36.0000	1.14706
$986$	−20.0000	−0.636930
$987$	0	0
$988$	0	0
$989$	0	0
$990$	8.00000	0.254257
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	20.0000	0.635001
$993$	16.0000	0.507745
$994$	0	0
$995$	16.0000	0.507234
$996$	4.00000	0.126745
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	−24.0000	−0.759707
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.2.a.a.1.1	✓	1
3.2	odd	2		117.2.a.a.1.1		1
4.3	odd	2		624.2.a.i.1.1		1
5.2	odd	4		975.2.c.f.274.2		2
5.3	odd	4		975.2.c.f.274.1		2
5.4	even	2		975.2.a.f.1.1		1
7.6	odd	2		1911.2.a.f.1.1		1
8.3	odd	2		2496.2.a.e.1.1		1
8.5	even	2		2496.2.a.q.1.1		1
9.2	odd	6		1053.2.e.d.352.1		2
9.4	even	3		1053.2.e.b.703.1		2
9.5	odd	6		1053.2.e.d.703.1		2
9.7	even	3		1053.2.e.b.352.1		2
11.10	odd	2		4719.2.a.c.1.1		1
12.11	even	2		1872.2.a.h.1.1		1
13.2	odd	12		507.2.j.e.316.2		4
13.3	even	3		507.2.e.a.22.1		2
13.4	even	6		507.2.e.b.484.1		2
13.5	odd	4		507.2.b.a.337.1		2
13.6	odd	12		507.2.j.e.361.1		4
13.7	odd	12		507.2.j.e.361.2		4
13.8	odd	4		507.2.b.a.337.2		2
13.9	even	3		507.2.e.a.484.1		2
13.10	even	6		507.2.e.b.22.1		2
13.11	odd	12		507.2.j.e.316.1		4
13.12	even	2		507.2.a.a.1.1		1
15.2	even	4		2925.2.c.e.2224.1		2
15.8	even	4		2925.2.c.e.2224.2		2
15.14	odd	2		2925.2.a.p.1.1		1
21.20	even	2		5733.2.a.e.1.1		1
24.5	odd	2		7488.2.a.bl.1.1		1
24.11	even	2		7488.2.a.by.1.1		1
39.5	even	4		1521.2.b.b.1351.2		2
39.8	even	4		1521.2.b.b.1351.1		2
39.38	odd	2		1521.2.a.e.1.1		1
52.51	odd	2		8112.2.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.a.a.1.1	✓	1	1.1	even	1	trivial
117.2.a.a.1.1		1	3.2	odd	2
507.2.a.a.1.1		1	13.12	even	2
507.2.b.a.337.1		2	13.5	odd	4
507.2.b.a.337.2		2	13.8	odd	4
507.2.e.a.22.1		2	13.3	even	3
507.2.e.a.484.1		2	13.9	even	3
507.2.e.b.22.1		2	13.10	even	6
507.2.e.b.484.1		2	13.4	even	6
507.2.j.e.316.1		4	13.11	odd	12
507.2.j.e.316.2		4	13.2	odd	12
507.2.j.e.361.1		4	13.6	odd	12
507.2.j.e.361.2		4	13.7	odd	12
624.2.a.i.1.1		1	4.3	odd	2
975.2.a.f.1.1		1	5.4	even	2
975.2.c.f.274.1		2	5.3	odd	4
975.2.c.f.274.2		2	5.2	odd	4
1053.2.e.b.352.1		2	9.7	even	3
1053.2.e.b.703.1		2	9.4	even	3
1053.2.e.d.352.1		2	9.2	odd	6
1053.2.e.d.703.1		2	9.5	odd	6
1521.2.a.e.1.1		1	39.38	odd	2
1521.2.b.b.1351.1		2	39.8	even	4
1521.2.b.b.1351.2		2	39.5	even	4
1872.2.a.h.1.1		1	12.11	even	2
1911.2.a.f.1.1		1	7.6	odd	2
2496.2.a.e.1.1		1	8.3	odd	2
2496.2.a.q.1.1		1	8.5	even	2
2925.2.a.p.1.1		1	15.14	odd	2
2925.2.c.e.2224.1		2	15.2	even	4
2925.2.c.e.2224.2		2	15.8	even	4
4719.2.a.c.1.1		1	11.10	odd	2
5733.2.a.e.1.1		1	21.20	even	2
7488.2.a.bl.1.1		1	24.5	odd	2
7488.2.a.by.1.1		1	24.11	even	2
8112.2.a.s.1.1		1	52.51	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	39.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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