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Properties

Label 64.3.c.a.63.1
Level $64$
Weight $3$
Character 64.63
Self dual yes
Analytic conductor $1.744$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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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] = [64,3,Mod(63,64)] 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("64.63"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(64, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 64.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1] 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: \(1.74387369191\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 63.1
Character \(\chi\) \(=\) 64.63

$q$-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)
 
\(f(q)\) \(=\) \(q+6.00000 q^{5} +9.00000 q^{9} -10.0000 q^{13} -30.0000 q^{17} +11.0000 q^{25} -42.0000 q^{29} +70.0000 q^{37} +18.0000 q^{41} +54.0000 q^{45} +49.0000 q^{49} -90.0000 q^{53} +22.0000 q^{61} -60.0000 q^{65} -110.000 q^{73} +81.0000 q^{81} -180.000 q^{85} -78.0000 q^{89} +130.000 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(63\)
\(\chi(n)\) \(1\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 6.00000 1.20000 0.600000 − 0.800000i \(-0.295167\pi\)
0.600000 + 0.800000i \(0.295167\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 9.00000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −10.0000 −0.769231 −0.384615 − 0.923077i \(-0.625666\pi\)
−0.384615 + 0.923077i \(0.625666\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −30.0000 −1.76471 −0.882353 − 0.470588i \(-0.844042\pi\)
−0.882353 + 0.470588i \(0.844042\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 11.0000 0.440000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −42.0000 −1.44828 −0.724138 − 0.689655i \(-0.757762\pi\)
−0.724138 + 0.689655i \(0.757762\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 70.0000 1.89189 0.945946 − 0.324324i \(-0.105137\pi\)
0.945946 + 0.324324i \(0.105137\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 18.0000 0.439024 0.219512 − 0.975610i \(-0.429553\pi\)
0.219512 + 0.975610i \(0.429553\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 54.0000 1.20000
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −90.0000 −1.69811 −0.849057 − 0.528302i \(-0.822829\pi\)
−0.849057 + 0.528302i \(0.822829\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 22.0000 0.360656 0.180328 − 0.983607i \(-0.442284\pi\)
0.180328 + 0.983607i \(0.442284\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −60.0000 −0.923077
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −110.000 −1.50685 −0.753425 − 0.657534i \(-0.771599\pi\)
−0.753425 + 0.657534i \(0.771599\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) −180.000 −2.11765
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −78.0000 −0.876404 −0.438202 − 0.898876i \(-0.644385\pi\)
−0.438202 + 0.898876i \(0.644385\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 130.000 1.34021 0.670103 − 0.742268i \(-0.266250\pi\)
0.670103 + 0.742268i \(0.266250\pi\)
\(98\) 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 64.3.c.a.63.1 1
3.2 odd 2 576.3.g.b.127.1 1
4.3 odd 2 CM 64.3.c.a.63.1 1
5.2 odd 4 1600.3.h.b.1599.2 2
5.3 odd 4 1600.3.h.b.1599.1 2
5.4 even 2 1600.3.b.b.1151.1 1
8.3 odd 2 16.3.c.a.15.1 ✓ 1
8.5 even 2 16.3.c.a.15.1 ✓ 1
12.11 even 2 576.3.g.b.127.1 1
16.3 odd 4 256.3.d.b.127.1 2
16.5 even 4 256.3.d.b.127.2 2
16.11 odd 4 256.3.d.b.127.2 2
16.13 even 4 256.3.d.b.127.1 2
20.3 even 4 1600.3.h.b.1599.1 2
20.7 even 4 1600.3.h.b.1599.2 2
20.19 odd 2 1600.3.b.b.1151.1 1
24.5 odd 2 144.3.g.a.127.1 1
24.11 even 2 144.3.g.a.127.1 1
40.3 even 4 400.3.h.a.399.2 2
40.13 odd 4 400.3.h.a.399.2 2
40.19 odd 2 400.3.b.a.351.1 1
40.27 even 4 400.3.h.a.399.1 2
40.29 even 2 400.3.b.a.351.1 1
40.37 odd 4 400.3.h.a.399.1 2
48.5 odd 4 2304.3.b.f.127.1 2
48.11 even 4 2304.3.b.f.127.1 2
48.29 odd 4 2304.3.b.f.127.2 2
48.35 even 4 2304.3.b.f.127.2 2
56.3 even 6 784.3.r.d.79.1 2
56.5 odd 6 784.3.r.d.655.1 2
56.11 odd 6 784.3.r.e.79.1 2
56.13 odd 2 784.3.d.b.687.1 1
56.19 even 6 784.3.r.d.655.1 2
56.27 even 2 784.3.d.b.687.1 1
56.37 even 6 784.3.r.e.655.1 2
56.45 odd 6 784.3.r.d.79.1 2
56.51 odd 6 784.3.r.e.655.1 2
56.53 even 6 784.3.r.e.79.1 2
72.5 odd 6 1296.3.o.b.703.1 2
72.11 even 6 1296.3.o.b.271.1 2
72.13 even 6 1296.3.o.o.703.1 2
72.29 odd 6 1296.3.o.b.271.1 2
72.43 odd 6 1296.3.o.o.271.1 2
72.59 even 6 1296.3.o.b.703.1 2
72.61 even 6 1296.3.o.o.271.1 2
72.67 odd 6 1296.3.o.o.703.1 2
120.29 odd 2 3600.3.e.c.3151.1 1
120.53 even 4 3600.3.j.a.1999.2 2
120.59 even 2 3600.3.e.c.3151.1 1
120.77 even 4 3600.3.j.a.1999.1 2
120.83 odd 4 3600.3.j.a.1999.2 2
120.107 odd 4 3600.3.j.a.1999.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.3.c.a.15.1 ✓ 1 8.3 odd 2
16.3.c.a.15.1 ✓ 1 8.5 even 2
64.3.c.a.63.1 1 1.1 even 1 trivial
64.3.c.a.63.1 1 4.3 odd 2 CM
144.3.g.a.127.1 1 24.5 odd 2
144.3.g.a.127.1 1 24.11 even 2
256.3.d.b.127.1 2 16.3 odd 4
256.3.d.b.127.1 2 16.13 even 4
256.3.d.b.127.2 2 16.5 even 4
256.3.d.b.127.2 2 16.11 odd 4
400.3.b.a.351.1 1 40.19 odd 2
400.3.b.a.351.1 1 40.29 even 2
400.3.h.a.399.1 2 40.27 even 4
400.3.h.a.399.1 2 40.37 odd 4
400.3.h.a.399.2 2 40.3 even 4
400.3.h.a.399.2 2 40.13 odd 4
576.3.g.b.127.1 1 3.2 odd 2
576.3.g.b.127.1 1 12.11 even 2
784.3.d.b.687.1 1 56.13 odd 2
784.3.d.b.687.1 1 56.27 even 2
784.3.r.d.79.1 2 56.3 even 6
784.3.r.d.79.1 2 56.45 odd 6
784.3.r.d.655.1 2 56.5 odd 6
784.3.r.d.655.1 2 56.19 even 6
784.3.r.e.79.1 2 56.11 odd 6
784.3.r.e.79.1 2 56.53 even 6
784.3.r.e.655.1 2 56.37 even 6
784.3.r.e.655.1 2 56.51 odd 6
1296.3.o.b.271.1 2 72.11 even 6
1296.3.o.b.271.1 2 72.29 odd 6
1296.3.o.b.703.1 2 72.5 odd 6
1296.3.o.b.703.1 2 72.59 even 6
1296.3.o.o.271.1 2 72.43 odd 6
1296.3.o.o.271.1 2 72.61 even 6
1296.3.o.o.703.1 2 72.13 even 6
1296.3.o.o.703.1 2 72.67 odd 6
1600.3.b.b.1151.1 1 5.4 even 2
1600.3.b.b.1151.1 1 20.19 odd 2
1600.3.h.b.1599.1 2 5.3 odd 4
1600.3.h.b.1599.1 2 20.3 even 4
1600.3.h.b.1599.2 2 5.2 odd 4
1600.3.h.b.1599.2 2 20.7 even 4
2304.3.b.f.127.1 2 48.5 odd 4
2304.3.b.f.127.1 2 48.11 even 4
2304.3.b.f.127.2 2 48.29 odd 4
2304.3.b.f.127.2 2 48.35 even 4
3600.3.e.c.3151.1 1 120.29 odd 2
3600.3.e.c.3151.1 1 120.59 even 2
3600.3.j.a.1999.1 2 120.77 even 4
3600.3.j.a.1999.1 2 120.107 odd 4
3600.3.j.a.1999.2 2 120.53 even 4
3600.3.j.a.1999.2 2 120.83 odd 4