Properties

Label 64.22.a.c.1.1
Level $64$
Weight $22$
Character 64.1
Self dual yes
Analytic conductor $178.866$
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] = [64,22,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, 22); 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, 22, names="a")
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 64.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-59316] 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: \(178.865500344\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2)
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)
 
\(f(q)\) \(=\) \(q-59316.0 q^{3} -4.97535e6 q^{5} +1.42743e9 q^{7} -6.94197e9 q^{9} +1.06768e11 q^{11} +1.50151e11 q^{13} +2.95118e11 q^{15} -1.12040e13 q^{17} -1.10241e13 q^{19} -8.46692e13 q^{21} +1.29503e14 q^{23} -4.52083e14 q^{25} +1.03224e15 q^{27} -2.38237e15 q^{29} -8.78553e14 q^{31} -6.33304e15 q^{33} -7.10194e15 q^{35} -3.11300e16 q^{37} -8.90633e15 q^{39} -2.46129e16 q^{41} +1.33386e17 q^{43} +3.45387e16 q^{45} -1.92524e17 q^{47} +1.47900e18 q^{49} +6.64575e17 q^{51} +5.94166e17 q^{53} -5.31208e17 q^{55} +6.53903e17 q^{57} +2.95595e18 q^{59} -7.98415e18 q^{61} -9.90914e18 q^{63} -7.47052e17 q^{65} -4.83704e18 q^{67} -7.68159e18 q^{69} +8.84902e18 q^{71} +3.66844e19 q^{73} +2.68158e19 q^{75} +1.52403e20 q^{77} +3.38406e19 q^{79} +1.13873e19 q^{81} -2.04215e20 q^{83} +5.57437e19 q^{85} +1.41313e20 q^{87} -4.10241e19 q^{89} +2.14329e20 q^{91} +5.21122e19 q^{93} +5.48485e19 q^{95} -7.27592e20 q^{97} -7.41179e20 q^{99} +O(q^{100})\)

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
\(3\) −59316.0 −0.579961 −0.289980 − 0.957033i \(-0.593649\pi\)
−0.289980 + 0.957033i \(0.593649\pi\)
\(4\) 0 0
\(5\) −4.97535e6 −0.227845 −0.113922 − 0.993490i \(-0.536341\pi\)
−0.113922 + 0.993490i \(0.536341\pi\)
\(6\) 0 0
\(7\) 1.42743e9 1.90996 0.954980 − 0.296671i \(-0.0958765\pi\)
0.954980 + 0.296671i \(0.0958765\pi\)
\(8\) 0 0
\(9\) −6.94197e9 −0.663645
\(10\) 0 0
\(11\) 1.06768e11 1.24113 0.620565 − 0.784155i \(-0.286903\pi\)
0.620565 + 0.784155i \(0.286903\pi\)
\(12\) 0 0
\(13\) 1.50151e11 0.302080 0.151040 − 0.988528i \(-0.451738\pi\)
0.151040 + 0.988528i \(0.451738\pi\)
\(14\) 0 0
\(15\) 2.95118e11 0.132141
\(16\) 0 0
\(17\) −1.12040e13 −1.34790 −0.673952 − 0.738776i \(-0.735404\pi\)
−0.673952 + 0.738776i \(0.735404\pi\)
\(18\) 0 0
\(19\) −1.10241e13 −0.412504 −0.206252 − 0.978499i \(-0.566127\pi\)
−0.206252 + 0.978499i \(0.566127\pi\)
\(20\) 0 0
\(21\) −8.46692e13 −1.10770
\(22\) 0 0
\(23\) 1.29503e14 0.651834 0.325917 − 0.945398i \(-0.394327\pi\)
0.325917 + 0.945398i \(0.394327\pi\)
\(24\) 0 0
\(25\) −4.52083e14 −0.948087
\(26\) 0 0
\(27\) 1.03224e15 0.964849
\(28\) 0 0
\(29\) −2.38237e15 −1.05155 −0.525776 − 0.850623i \(-0.676225\pi\)
−0.525776 + 0.850623i \(0.676225\pi\)
\(30\) 0 0
\(31\) −8.78553e14 −0.192517 −0.0962587 − 0.995356i \(-0.530688\pi\)
−0.0962587 + 0.995356i \(0.530688\pi\)
\(32\) 0 0
\(33\) −6.33304e15 −0.719807
\(34\) 0 0
\(35\) −7.10194e15 −0.435174
\(36\) 0 0
\(37\) −3.11300e16 −1.06429 −0.532146 − 0.846652i \(-0.678614\pi\)
−0.532146 + 0.846652i \(0.678614\pi\)
\(38\) 0 0
\(39\) −8.90633e15 −0.175194
\(40\) 0 0
\(41\) −2.46129e16 −0.286373 −0.143187 − 0.989696i \(-0.545735\pi\)
−0.143187 + 0.989696i \(0.545735\pi\)
\(42\) 0 0
\(43\) 1.33386e17 0.941222 0.470611 − 0.882341i \(-0.344034\pi\)
0.470611 + 0.882341i \(0.344034\pi\)
\(44\) 0 0
\(45\) 3.45387e16 0.151208
\(46\) 0 0
\(47\) −1.92524e17 −0.533897 −0.266948 − 0.963711i \(-0.586015\pi\)
−0.266948 + 0.963711i \(0.586015\pi\)
\(48\) 0 0
\(49\) 1.47900e18 2.64794
\(50\) 0 0
\(51\) 6.64575e17 0.781731
\(52\) 0 0
\(53\) 5.94166e17 0.466672 0.233336 − 0.972396i \(-0.425036\pi\)
0.233336 + 0.972396i \(0.425036\pi\)
\(54\) 0 0
\(55\) −5.31208e17 −0.282785
\(56\) 0 0
\(57\) 6.53903e17 0.239236
\(58\) 0 0
\(59\) 2.95595e18 0.752925 0.376462 − 0.926432i \(-0.377140\pi\)
0.376462 + 0.926432i \(0.377140\pi\)
\(60\) 0 0
\(61\) −7.98415e18 −1.43306 −0.716532 − 0.697554i \(-0.754272\pi\)
−0.716532 + 0.697554i \(0.754272\pi\)
\(62\) 0 0
\(63\) −9.90914e18 −1.26754
\(64\) 0 0
\(65\) −7.47052e17 −0.0688272
\(66\) 0 0
\(67\) −4.83704e18 −0.324186 −0.162093 − 0.986775i \(-0.551824\pi\)
−0.162093 + 0.986775i \(0.551824\pi\)
\(68\) 0 0
\(69\) −7.68159e18 −0.378038
\(70\) 0 0
\(71\) 8.84902e18 0.322613 0.161307 − 0.986904i \(-0.448429\pi\)
0.161307 + 0.986904i \(0.448429\pi\)
\(72\) 0 0
\(73\) 3.66844e19 0.999060 0.499530 − 0.866297i \(-0.333506\pi\)
0.499530 + 0.866297i \(0.333506\pi\)
\(74\) 0 0
\(75\) 2.68158e19 0.549853
\(76\) 0 0
\(77\) 1.52403e20 2.37051
\(78\) 0 0
\(79\) 3.38406e19 0.402118 0.201059 − 0.979579i \(-0.435562\pi\)
0.201059 + 0.979579i \(0.435562\pi\)
\(80\) 0 0
\(81\) 1.13873e19 0.104071
\(82\) 0 0
\(83\) −2.04215e20 −1.44466 −0.722332 − 0.691547i \(-0.756930\pi\)
−0.722332 + 0.691547i \(0.756930\pi\)
\(84\) 0 0
\(85\) 5.57437e19 0.307112
\(86\) 0 0
\(87\) 1.41313e20 0.609858
\(88\) 0 0
\(89\) −4.10241e19 −0.139458 −0.0697290 − 0.997566i \(-0.522213\pi\)
−0.0697290 + 0.997566i \(0.522213\pi\)
\(90\) 0 0
\(91\) 2.14329e20 0.576960
\(92\) 0 0
\(93\) 5.21122e19 0.111653
\(94\) 0 0
\(95\) 5.48485e19 0.0939869
\(96\) 0 0
\(97\) −7.27592e20 −1.00181 −0.500905 − 0.865503i \(-0.666999\pi\)
−0.500905 + 0.865503i \(0.666999\pi\)
\(98\) 0 0
\(99\) −7.41179e20 −0.823671
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 64.22.a.c.1.1 1
4.3 odd 2 64.22.a.e.1.1 1
8.3 odd 2 16.22.a.b.1.1 1
8.5 even 2 2.22.a.b.1.1 ✓ 1
24.5 odd 2 18.22.a.b.1.1 1
40.13 odd 4 50.22.b.c.49.1 2
40.29 even 2 50.22.a.a.1.1 1
40.37 odd 4 50.22.b.c.49.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.22.a.b.1.1 ✓ 1 8.5 even 2
16.22.a.b.1.1 1 8.3 odd 2
18.22.a.b.1.1 1 24.5 odd 2
50.22.a.a.1.1 1 40.29 even 2
50.22.b.c.49.1 2 40.13 odd 4
50.22.b.c.49.2 2 40.37 odd 4
64.22.a.c.1.1 1 1.1 even 1 trivial
64.22.a.e.1.1 1 4.3 odd 2