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Properties

Label 64.16.a.c.1.1
Level $64$
Weight $16$
Character 64.1
Self dual yes
Analytic conductor $91.324$
Analytic rank $1$
Dimension $1$
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] = [64,16,Mod(1,64)] 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("64.1"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(64, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 64.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-3348] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(91.3238432639\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 64.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3348.00 q^{3} -52110.0 q^{5} -2.82246e6 q^{7} -3.13980e6 q^{9} +2.05869e7 q^{11} +1.90073e8 q^{13} +1.74464e8 q^{15} +1.64653e9 q^{17} +1.56326e9 q^{19} +9.44958e9 q^{21} -9.45112e9 q^{23} -2.78021e10 q^{25} +5.85522e10 q^{27} +3.69026e10 q^{29} -7.15885e10 q^{31} -6.89248e10 q^{33} +1.47078e11 q^{35} +1.03365e12 q^{37} -6.36366e11 q^{39} +1.64197e12 q^{41} -4.92403e11 q^{43} +1.63615e11 q^{45} +3.41068e12 q^{47} +3.21870e12 q^{49} -5.51258e12 q^{51} -6.79715e12 q^{53} -1.07278e12 q^{55} -5.23379e12 q^{57} +9.85886e12 q^{59} -4.93184e12 q^{61} +8.86196e12 q^{63} -9.90472e12 q^{65} -2.88378e13 q^{67} +3.16423e13 q^{69} -1.25050e14 q^{71} -8.21715e13 q^{73} +9.30815e13 q^{75} -5.81055e13 q^{77} +2.54131e13 q^{79} -1.50980e14 q^{81} -2.81737e14 q^{83} -8.58006e13 q^{85} -1.23550e14 q^{87} +7.15619e14 q^{89} -5.36474e14 q^{91} +2.39678e14 q^{93} -8.14613e13 q^{95} +6.12786e14 q^{97} -6.46387e13 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3348.00 −0.883845 −0.441922 − 0.897053i \(-0.645703\pi\)
−0.441922 + 0.897053i \(0.645703\pi\)
\(4\) 0 0
\(5\) −52110.0 −0.298295 −0.149148 − 0.988815i \(-0.547653\pi\)
−0.149148 + 0.988815i \(0.547653\pi\)
\(6\) 0 0
\(7\) −2.82246e6 −1.29536 −0.647682 − 0.761911i \(-0.724261\pi\)
−0.647682 + 0.761911i \(0.724261\pi\)
\(8\) 0 0
\(9\) −3.13980e6 −0.218818
\(10\) 0 0
\(11\) 2.05869e7 0.318526 0.159263 − 0.987236i \(-0.449088\pi\)
0.159263 + 0.987236i \(0.449088\pi\)
\(12\) 0 0
\(13\) 1.90073e8 0.840129 0.420065 − 0.907494i \(-0.362007\pi\)
0.420065 + 0.907494i \(0.362007\pi\)
\(14\) 0 0
\(15\) 1.74464e8 0.263647
\(16\) 0 0
\(17\) 1.64653e9 0.973200 0.486600 − 0.873625i \(-0.338237\pi\)
0.486600 + 0.873625i \(0.338237\pi\)
\(18\) 0 0
\(19\) 1.56326e9 0.401216 0.200608 − 0.979672i \(-0.435708\pi\)
0.200608 + 0.979672i \(0.435708\pi\)
\(20\) 0 0
\(21\) 9.44958e9 1.14490
\(22\) 0 0
\(23\) −9.45112e9 −0.578794 −0.289397 − 0.957209i \(-0.593455\pi\)
−0.289397 + 0.957209i \(0.593455\pi\)
\(24\) 0 0
\(25\) −2.78021e10 −0.911020
\(26\) 0 0
\(27\) 5.85522e10 1.07725
\(28\) 0 0
\(29\) 3.69026e10 0.397257 0.198629 − 0.980075i \(-0.436351\pi\)
0.198629 + 0.980075i \(0.436351\pi\)
\(30\) 0 0
\(31\) −7.15885e10 −0.467337 −0.233669 − 0.972316i \(-0.575073\pi\)
−0.233669 + 0.972316i \(0.575073\pi\)
\(32\) 0 0
\(33\) −6.89248e10 −0.281528
\(34\) 0 0
\(35\) 1.47078e11 0.386401
\(36\) 0 0
\(37\) 1.03365e12 1.79003 0.895017 − 0.446031i \(-0.147163\pi\)
0.895017 + 0.446031i \(0.147163\pi\)
\(38\) 0 0
\(39\) −6.36366e11 −0.742544
\(40\) 0 0
\(41\) 1.64197e12 1.31670 0.658351 − 0.752711i \(-0.271254\pi\)
0.658351 + 0.752711i \(0.271254\pi\)
\(42\) 0 0
\(43\) −4.92403e11 −0.276253 −0.138127 − 0.990415i \(-0.544108\pi\)
−0.138127 + 0.990415i \(0.544108\pi\)
\(44\) 0 0
\(45\) 1.63615e11 0.0652724
\(46\) 0 0
\(47\) 3.41068e12 0.981991 0.490996 − 0.871162i \(-0.336633\pi\)
0.490996 + 0.871162i \(0.336633\pi\)
\(48\) 0 0
\(49\) 3.21870e12 0.677968
\(50\) 0 0
\(51\) −5.51258e12 −0.860158
\(52\) 0 0
\(53\) −6.79715e12 −0.794800 −0.397400 − 0.917645i \(-0.630087\pi\)
−0.397400 + 0.917645i \(0.630087\pi\)
\(54\) 0 0
\(55\) −1.07278e12 −0.0950147
\(56\) 0 0
\(57\) −5.23379e12 −0.354613
\(58\) 0 0
\(59\) 9.85886e12 0.515747 0.257873 − 0.966179i \(-0.416978\pi\)
0.257873 + 0.966179i \(0.416978\pi\)
\(60\) 0 0
\(61\) −4.93184e12 −0.200926 −0.100463 − 0.994941i \(-0.532032\pi\)
−0.100463 + 0.994941i \(0.532032\pi\)
\(62\) 0 0
\(63\) 8.86196e12 0.283449
\(64\) 0 0
\(65\) −9.90472e12 −0.250606
\(66\) 0 0
\(67\) −2.88378e13 −0.581302 −0.290651 − 0.956829i \(-0.593872\pi\)
−0.290651 + 0.956829i \(0.593872\pi\)
\(68\) 0 0
\(69\) 3.16423e13 0.511564
\(70\) 0 0
\(71\) −1.25050e14 −1.63172 −0.815862 − 0.578247i \(-0.803737\pi\)
−0.815862 + 0.578247i \(0.803737\pi\)
\(72\) 0 0
\(73\) −8.21715e13 −0.870562 −0.435281 − 0.900295i \(-0.643351\pi\)
−0.435281 + 0.900295i \(0.643351\pi\)
\(74\) 0 0
\(75\) 9.30815e13 0.805200
\(76\) 0 0
\(77\) −5.81055e13 −0.412607
\(78\) 0 0
\(79\) 2.54131e13 0.148886 0.0744430 − 0.997225i \(-0.476282\pi\)
0.0744430 + 0.997225i \(0.476282\pi\)
\(80\) 0 0
\(81\) −1.50980e14 −0.733300
\(82\) 0 0
\(83\) −2.81737e14 −1.13961 −0.569807 − 0.821779i \(-0.692982\pi\)
−0.569807 + 0.821779i \(0.692982\pi\)
\(84\) 0 0
\(85\) −8.58006e13 −0.290301
\(86\) 0 0
\(87\) −1.23550e14 −0.351114
\(88\) 0 0
\(89\) 7.15619e14 1.71497 0.857485 − 0.514509i \(-0.172026\pi\)
0.857485 + 0.514509i \(0.172026\pi\)
\(90\) 0 0
\(91\) −5.36474e14 −1.08827
\(92\) 0 0
\(93\) 2.39678e14 0.413054
\(94\) 0 0
\(95\) −8.14613e13 −0.119681
\(96\) 0 0
\(97\) 6.12786e14 0.770054 0.385027 − 0.922905i \(-0.374192\pi\)
0.385027 + 0.922905i \(0.374192\pi\)
\(98\) 0 0
\(99\) −6.46387e13 −0.0696993
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 64.16.a.c.1.1 1
4.3 odd 2 64.16.a.i.1.1 1
8.3 odd 2 1.16.a.a.1.1 ✓ 1
8.5 even 2 16.16.a.d.1.1 1
24.5 odd 2 144.16.a.f.1.1 1
24.11 even 2 9.16.a.a.1.1 1
40.3 even 4 25.16.b.a.24.1 2
40.19 odd 2 25.16.a.a.1.1 1
40.27 even 4 25.16.b.a.24.2 2
56.3 even 6 49.16.c.b.30.1 2
56.11 odd 6 49.16.c.c.30.1 2
56.19 even 6 49.16.c.b.18.1 2
56.27 even 2 49.16.a.a.1.1 1
56.51 odd 6 49.16.c.c.18.1 2
88.43 even 2 121.16.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.16.a.a.1.1 ✓ 1 8.3 odd 2
9.16.a.a.1.1 1 24.11 even 2
16.16.a.d.1.1 1 8.5 even 2
25.16.a.a.1.1 1 40.19 odd 2
25.16.b.a.24.1 2 40.3 even 4
25.16.b.a.24.2 2 40.27 even 4
49.16.a.a.1.1 1 56.27 even 2
49.16.c.b.18.1 2 56.19 even 6
49.16.c.b.30.1 2 56.3 even 6
49.16.c.c.18.1 2 56.51 odd 6
49.16.c.c.30.1 2 56.11 odd 6
64.16.a.c.1.1 1 1.1 even 1 trivial
64.16.a.i.1.1 1 4.3 odd 2
121.16.a.a.1.1 1 88.43 even 2
144.16.a.f.1.1 1 24.5 odd 2