↑

Properties

Label 64.22.a.d.1.1
Level $64$
Weight $22$
Character 64.1
Self dual yes
Analytic conductor $178.866$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

Related objects

Downloads

Learn more

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 = [1,0,0] 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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

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)
 
\(f(q)\) \(=\) \(q-1.33986e7 q^{5} -1.04604e10 q^{9} -9.70515e11 q^{13} -1.50959e13 q^{17} -2.97314e14 q^{25} -6.17267e14 q^{29} -1.87640e16 q^{37} -5.56716e15 q^{41} +1.40154e17 q^{45} -5.58546e17 q^{49} -2.30506e18 q^{53} +1.00221e19 q^{61} +1.30036e19 q^{65} -6.97145e19 q^{73} +1.09419e20 q^{81} +2.02265e20 q^{85} -4.36441e20 q^{89} -1.16329e21 q^{97} +O(q^{100})\)

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
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) −1.33986e7 −0.613586 −0.306793 − 0.951776i \(-0.599256\pi\)
−0.306793 + 0.951776i \(0.599256\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −1.04604e10 −1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −9.70515e11 −1.95253 −0.976264 − 0.216586i \(-0.930508\pi\)
−0.976264 + 0.216586i \(0.930508\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.50959e13 −1.81612 −0.908062 − 0.418836i \(-0.862438\pi\)
−0.908062 + 0.418836i \(0.862438\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −2.97314e14 −0.623512
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.17267e14 −0.272455 −0.136227 − 0.990678i \(-0.543498\pi\)
−0.136227 + 0.990678i \(0.543498\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.87640e16 −0.641516 −0.320758 − 0.947161i \(-0.603938\pi\)
−0.320758 + 0.947161i \(0.603938\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.56716e15 −0.0647744 −0.0323872 − 0.999475i \(-0.510311\pi\)
−0.0323872 + 0.999475i \(0.510311\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 1.40154e17 0.613586
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −5.58546e17 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.30506e18 −1.81045 −0.905223 − 0.424937i \(-0.860296\pi\)
−0.905223 + 0.424937i \(0.860296\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 1.00221e19 1.79886 0.899428 − 0.437069i \(-0.143983\pi\)
0.899428 + 0.437069i \(0.143983\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.30036e19 1.19804
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −6.97145e19 −1.89860 −0.949299 − 0.314374i \(-0.898205\pi\)
−0.949299 + 0.314374i \(0.898205\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.09419e20 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 2.02265e20 1.11435
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4.36441e20 −1.48365 −0.741823 − 0.670596i \(-0.766039\pi\)
−0.741823 + 0.670596i \(0.766039\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.16329e21 −1.60172 −0.800860 − 0.598852i \(-0.795624\pi\)
−0.800860 + 0.598852i \(0.795624\pi\)
\(98\) 0 0
\(99\) 0 0
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.d.1.1 1
4.3 odd 2 CM 64.22.a.d.1.1 1
8.3 odd 2 32.22.a.a.1.1 ✓ 1
8.5 even 2 32.22.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.22.a.a.1.1 ✓ 1 8.3 odd 2
32.22.a.a.1.1 ✓ 1 8.5 even 2
64.22.a.d.1.1 1 1.1 even 1 trivial
64.22.a.d.1.1 1 4.3 odd 2 CM