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Properties

Label 8013.2.a.c.1.19
Level $8013$
Weight $2$
Character 8013.1
Self dual yes
Analytic conductor $63.984$
Analytic rank $1$
Dimension $116$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8013,2,Mod(1,8013)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8013.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8013, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8013 = 3 \cdot 2671 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8013.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [116] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.9841271397\)
Analytic rank: \(1\)
Dimension: \(116\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.19
Character \(\chi\) \(=\) 8013.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.18030 q^{2} -1.00000 q^{3} +2.75372 q^{4} +1.60452 q^{5} +2.18030 q^{6} +4.21617 q^{7} -1.64333 q^{8} +1.00000 q^{9} -3.49835 q^{10} +2.92472 q^{11} -2.75372 q^{12} +0.434613 q^{13} -9.19252 q^{14} -1.60452 q^{15} -1.92448 q^{16} -7.66422 q^{17} -2.18030 q^{18} -0.461164 q^{19} +4.41840 q^{20} -4.21617 q^{21} -6.37677 q^{22} -6.81500 q^{23} +1.64333 q^{24} -2.42550 q^{25} -0.947587 q^{26} -1.00000 q^{27} +11.6101 q^{28} +4.62126 q^{29} +3.49835 q^{30} +5.68311 q^{31} +7.48260 q^{32} -2.92472 q^{33} +16.7103 q^{34} +6.76494 q^{35} +2.75372 q^{36} -1.51311 q^{37} +1.00548 q^{38} -0.434613 q^{39} -2.63676 q^{40} -3.16081 q^{41} +9.19252 q^{42} +3.60474 q^{43} +8.05384 q^{44} +1.60452 q^{45} +14.8588 q^{46} -4.57809 q^{47} +1.92448 q^{48} +10.7761 q^{49} +5.28833 q^{50} +7.66422 q^{51} +1.19680 q^{52} -2.49252 q^{53} +2.18030 q^{54} +4.69278 q^{55} -6.92855 q^{56} +0.461164 q^{57} -10.0758 q^{58} +3.63804 q^{59} -4.41840 q^{60} +0.336312 q^{61} -12.3909 q^{62} +4.21617 q^{63} -12.4654 q^{64} +0.697347 q^{65} +6.37677 q^{66} -6.98241 q^{67} -21.1051 q^{68} +6.81500 q^{69} -14.7496 q^{70} -10.4205 q^{71} -1.64333 q^{72} -9.92098 q^{73} +3.29903 q^{74} +2.42550 q^{75} -1.26991 q^{76} +12.3311 q^{77} +0.947587 q^{78} -7.48776 q^{79} -3.08787 q^{80} +1.00000 q^{81} +6.89151 q^{82} -6.86095 q^{83} -11.6101 q^{84} -12.2974 q^{85} -7.85942 q^{86} -4.62126 q^{87} -4.80627 q^{88} +0.0993936 q^{89} -3.49835 q^{90} +1.83240 q^{91} -18.7666 q^{92} -5.68311 q^{93} +9.98162 q^{94} -0.739948 q^{95} -7.48260 q^{96} -15.3372 q^{97} -23.4951 q^{98} +2.92472 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 116 q - 16 q^{2} - 116 q^{3} + 116 q^{4} - 20 q^{5} + 16 q^{6} - 33 q^{7} - 45 q^{8} + 116 q^{9} + 3 q^{10} - 57 q^{11} - 116 q^{12} + 6 q^{13} - 9 q^{14} + 20 q^{15} + 112 q^{16} - 30 q^{17} - 16 q^{18}+ \cdots - 57 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.18030 −1.54171 −0.770853 − 0.637013i \(-0.780170\pi\)
−0.770853 + 0.637013i \(0.780170\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.75372 1.37686
\(5\) 1.60452 0.717565 0.358783 − 0.933421i \(-0.383192\pi\)
0.358783 + 0.933421i \(0.383192\pi\)
\(6\) 2.18030 0.890105
\(7\) 4.21617 1.59356 0.796781 − 0.604269i \(-0.206535\pi\)
0.796781 + 0.604269i \(0.206535\pi\)
\(8\) −1.64333 −0.581005
\(9\) 1.00000 0.333333
\(10\) −3.49835 −1.10627
\(11\) 2.92472 0.881835 0.440918 − 0.897548i \(-0.354653\pi\)
0.440918 + 0.897548i \(0.354653\pi\)
\(12\) −2.75372 −0.794929
\(13\) 0.434613 0.120540 0.0602699 − 0.998182i \(-0.480804\pi\)
0.0602699 + 0.998182i \(0.480804\pi\)
\(14\) −9.19252 −2.45680
\(15\) −1.60452 −0.414286
\(16\) −1.92448 −0.481120
\(17\) −7.66422 −1.85885 −0.929423 − 0.369017i \(-0.879694\pi\)
−0.929423 + 0.369017i \(0.879694\pi\)
\(18\) −2.18030 −0.513902
\(19\) −0.461164 −0.105798 −0.0528991 − 0.998600i \(-0.516846\pi\)
−0.0528991 + 0.998600i \(0.516846\pi\)
\(20\) 4.41840 0.987985
\(21\) −4.21617 −0.920043
\(22\) −6.37677 −1.35953
\(23\) −6.81500 −1.42103 −0.710513 − 0.703684i \(-0.751537\pi\)
−0.710513 + 0.703684i \(0.751537\pi\)
\(24\) 1.64333 0.335443
\(25\) −2.42550 −0.485100
\(26\) −0.947587 −0.185837
\(27\) −1.00000 −0.192450
\(28\) 11.6101 2.19411
\(29\) 4.62126 0.858147 0.429074 − 0.903270i \(-0.358840\pi\)
0.429074 + 0.903270i \(0.358840\pi\)
\(30\) 3.49835 0.638708
\(31\) 5.68311 1.02072 0.510359 − 0.859962i \(-0.329513\pi\)
0.510359 + 0.859962i \(0.329513\pi\)
\(32\) 7.48260 1.32275
\(33\) −2.92472 −0.509128
\(34\) 16.7103 2.86579
\(35\) 6.76494 1.14348
\(36\) 2.75372 0.458953
\(37\) −1.51311 −0.248753 −0.124377 − 0.992235i \(-0.539693\pi\)
−0.124377 + 0.992235i \(0.539693\pi\)
\(38\) 1.00548 0.163110
\(39\) −0.434613 −0.0695937
\(40\) −2.63676 −0.416909
\(41\) −3.16081 −0.493635 −0.246818 − 0.969062i \(-0.579385\pi\)
−0.246818 + 0.969062i \(0.579385\pi\)
\(42\) 9.19252 1.41844
\(43\) 3.60474 0.549717 0.274859 − 0.961485i \(-0.411369\pi\)
0.274859 + 0.961485i \(0.411369\pi\)
\(44\) 8.05384 1.21416
\(45\) 1.60452 0.239188
\(46\) 14.8588 2.19080
\(47\) −4.57809 −0.667783 −0.333892 − 0.942611i \(-0.608362\pi\)
−0.333892 + 0.942611i \(0.608362\pi\)
\(48\) 1.92448 0.277775
\(49\) 10.7761 1.53944
\(50\) 5.28833 0.747882
\(51\) 7.66422 1.07320
\(52\) 1.19680 0.165966
\(53\) −2.49252 −0.342374 −0.171187 − 0.985239i \(-0.554760\pi\)
−0.171187 + 0.985239i \(0.554760\pi\)
\(54\) 2.18030 0.296702
\(55\) 4.69278 0.632774
\(56\) −6.92855 −0.925866
\(57\) 0.461164 0.0610826
\(58\) −10.0758 −1.32301
\(59\) 3.63804 0.473632 0.236816 − 0.971555i \(-0.423896\pi\)
0.236816 + 0.971555i \(0.423896\pi\)
\(60\) −4.41840 −0.570414
\(61\) 0.336312 0.0430604 0.0215302 − 0.999768i \(-0.493146\pi\)
0.0215302 + 0.999768i \(0.493146\pi\)
\(62\) −12.3909 −1.57365
\(63\) 4.21617 0.531187
\(64\) −12.4654 −1.55817
\(65\) 0.697347 0.0864952
\(66\) 6.37677 0.784926
\(67\) −6.98241 −0.853038 −0.426519 − 0.904479i \(-0.640260\pi\)
−0.426519 + 0.904479i \(0.640260\pi\)
\(68\) −21.1051 −2.55937
\(69\) 6.81500 0.820430
\(70\) −14.7496 −1.76292
\(71\) −10.4205 −1.23668 −0.618342 − 0.785910i \(-0.712195\pi\)
−0.618342 + 0.785910i \(0.712195\pi\)
\(72\) −1.64333 −0.193668
\(73\) −9.92098 −1.16116 −0.580581 − 0.814202i \(-0.697175\pi\)
−0.580581 + 0.814202i \(0.697175\pi\)
\(74\) 3.29903 0.383505
\(75\) 2.42550 0.280073
\(76\) −1.26991 −0.145669
\(77\) 12.3311 1.40526
\(78\) 0.947587 0.107293
\(79\) −7.48776 −0.842439 −0.421219 − 0.906959i \(-0.638398\pi\)
−0.421219 + 0.906959i \(0.638398\pi\)
\(80\) −3.08787 −0.345235
\(81\) 1.00000 0.111111
\(82\) 6.89151 0.761040
\(83\) −6.86095 −0.753087 −0.376543 − 0.926399i \(-0.622887\pi\)
−0.376543 + 0.926399i \(0.622887\pi\)
\(84\) −11.6101 −1.26677
\(85\) −12.2974 −1.33384
\(86\) −7.85942 −0.847503
\(87\) −4.62126 −0.495452
\(88\) −4.80627 −0.512350
\(89\) 0.0993936 0.0105357 0.00526785 − 0.999986i \(-0.498323\pi\)
0.00526785 + 0.999986i \(0.498323\pi\)
\(90\) −3.49835 −0.368758
\(91\) 1.83240 0.192088
\(92\) −18.7666 −1.95655
\(93\) −5.68311 −0.589312
\(94\) 9.98162 1.02953
\(95\) −0.739948 −0.0759171
\(96\) −7.48260 −0.763690
\(97\) −15.3372 −1.55726 −0.778629 − 0.627484i \(-0.784085\pi\)
−0.778629 + 0.627484i \(0.784085\pi\)
\(98\) −23.4951 −2.37336
\(99\) 2.92472 0.293945
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8013.2.a.c.1.19 ✓ 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8013.2.a.c.1.19 ✓ 116 1.1 even 1 trivial