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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [64,2,Mod(33,64)] 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("64.33"); S:= CuspForms(chi, 2); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 64.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.511042572936\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 33.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 64.33
Dual form 64.2.b.a.33.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{3} -1.00000 q^{9} +6.00000i q^{11} -6.00000 q^{17} -2.00000i q^{19} +5.00000 q^{25} -4.00000i q^{27} +12.0000 q^{33} -6.00000 q^{41} -10.0000i q^{43} -7.00000 q^{49} +12.0000i q^{51} -4.00000 q^{57} +6.00000i q^{59} +14.0000i q^{67} +2.00000 q^{73} -10.0000i q^{75} -11.0000 q^{81} -18.0000i q^{83} +18.0000 q^{89} +10.0000 q^{97} -6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9} - 12 q^{17} + 10 q^{25} + 24 q^{33} - 12 q^{41} - 14 q^{49} - 8 q^{57} + 4 q^{73} - 22 q^{81} + 36 q^{89} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) − 2.00000i − 1.15470i −0.816497 − 0.577350i \(-0.804087\pi\)
0.816497 − 0.577350i \(-0.195913\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 6.00000i 1.80907i 0.426401 + 0.904534i \(0.359781\pi\)
−0.426401 + 0.904534i \(0.640219\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) − 2.00000i − 0.458831i −0.973329 − 0.229416i \(-0.926318\pi\)
0.973329 − 0.229416i \(-0.0736815\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) − 4.00000i − 0.769800i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 12.0000 2.08893
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) − 10.0000i − 1.52499i −0.646997 − 0.762493i \(-0.723975\pi\)
0.646997 − 0.762493i \(-0.276025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 12.0000i 1.68034i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 6.00000i 0.781133i 0.920575 + 0.390567i \(0.127721\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 14.0000i 1.71037i 0.518321 + 0.855186i \(0.326557\pi\)
−0.518321 + 0.855186i \(0.673443\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) − 10.0000i − 1.15470i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) − 18.0000i − 1.97576i −0.155230 − 0.987878i \(-0.549612\pi\)
0.155230 − 0.987878i \(-0.450388\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 18.0000 1.90800 0.953998 − 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) − 6.00000i − 0.603023i
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 64.2.b.a.33.1 ✓ 2
3.2 odd 2 576.2.d.a.289.1 2
4.3 odd 2 inner 64.2.b.a.33.2 yes 2
5.2 odd 4 1600.2.f.a.1249.2 2
5.3 odd 4 1600.2.f.b.1249.2 2
5.4 even 2 1600.2.d.a.801.2 2
7.6 odd 2 3136.2.b.b.1569.2 2
8.3 odd 2 CM 64.2.b.a.33.1 ✓ 2
8.5 even 2 inner 64.2.b.a.33.2 yes 2
12.11 even 2 576.2.d.a.289.2 2
16.3 odd 4 256.2.a.a.1.1 1
16.5 even 4 256.2.a.a.1.1 1
16.11 odd 4 256.2.a.d.1.1 1
16.13 even 4 256.2.a.d.1.1 1
20.3 even 4 1600.2.f.a.1249.1 2
20.7 even 4 1600.2.f.b.1249.1 2
20.19 odd 2 1600.2.d.a.801.1 2
24.5 odd 2 576.2.d.a.289.2 2
24.11 even 2 576.2.d.a.289.1 2
28.27 even 2 3136.2.b.b.1569.1 2
32.3 odd 8 1024.2.e.l.769.1 4
32.5 even 8 1024.2.e.l.257.2 4
32.11 odd 8 1024.2.e.l.257.2 4
32.13 even 8 1024.2.e.l.769.1 4
32.19 odd 8 1024.2.e.l.769.2 4
32.21 even 8 1024.2.e.l.257.1 4
32.27 odd 8 1024.2.e.l.257.1 4
32.29 even 8 1024.2.e.l.769.2 4
40.3 even 4 1600.2.f.b.1249.2 2
40.13 odd 4 1600.2.f.a.1249.1 2
40.19 odd 2 1600.2.d.a.801.2 2
40.27 even 4 1600.2.f.a.1249.2 2
40.29 even 2 1600.2.d.a.801.1 2
40.37 odd 4 1600.2.f.b.1249.1 2
48.5 odd 4 2304.2.a.i.1.1 1
48.11 even 4 2304.2.a.h.1.1 1
48.29 odd 4 2304.2.a.h.1.1 1
48.35 even 4 2304.2.a.i.1.1 1
56.13 odd 2 3136.2.b.b.1569.1 2
56.27 even 2 3136.2.b.b.1569.2 2
80.19 odd 4 6400.2.a.x.1.1 1
80.29 even 4 6400.2.a.a.1.1 1
80.59 odd 4 6400.2.a.a.1.1 1
80.69 even 4 6400.2.a.x.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
64.2.b.a.33.1 ✓ 2 1.1 even 1 trivial
64.2.b.a.33.1 ✓ 2 8.3 odd 2 CM
64.2.b.a.33.2 yes 2 4.3 odd 2 inner
64.2.b.a.33.2 yes 2 8.5 even 2 inner
256.2.a.a.1.1 1 16.3 odd 4
256.2.a.a.1.1 1 16.5 even 4
256.2.a.d.1.1 1 16.11 odd 4
256.2.a.d.1.1 1 16.13 even 4
576.2.d.a.289.1 2 3.2 odd 2
576.2.d.a.289.1 2 24.11 even 2
576.2.d.a.289.2 2 12.11 even 2
576.2.d.a.289.2 2 24.5 odd 2
1024.2.e.l.257.1 4 32.21 even 8
1024.2.e.l.257.1 4 32.27 odd 8
1024.2.e.l.257.2 4 32.5 even 8
1024.2.e.l.257.2 4 32.11 odd 8
1024.2.e.l.769.1 4 32.3 odd 8
1024.2.e.l.769.1 4 32.13 even 8
1024.2.e.l.769.2 4 32.19 odd 8
1024.2.e.l.769.2 4 32.29 even 8
1600.2.d.a.801.1 2 20.19 odd 2
1600.2.d.a.801.1 2 40.29 even 2
1600.2.d.a.801.2 2 5.4 even 2
1600.2.d.a.801.2 2 40.19 odd 2
1600.2.f.a.1249.1 2 20.3 even 4
1600.2.f.a.1249.1 2 40.13 odd 4
1600.2.f.a.1249.2 2 5.2 odd 4
1600.2.f.a.1249.2 2 40.27 even 4
1600.2.f.b.1249.1 2 20.7 even 4
1600.2.f.b.1249.1 2 40.37 odd 4
1600.2.f.b.1249.2 2 5.3 odd 4
1600.2.f.b.1249.2 2 40.3 even 4
2304.2.a.h.1.1 1 48.11 even 4
2304.2.a.h.1.1 1 48.29 odd 4
2304.2.a.i.1.1 1 48.5 odd 4
2304.2.a.i.1.1 1 48.35 even 4
3136.2.b.b.1569.1 2 28.27 even 2
3136.2.b.b.1569.1 2 56.13 odd 2
3136.2.b.b.1569.2 2 7.6 odd 2
3136.2.b.b.1569.2 2 56.27 even 2
6400.2.a.a.1.1 1 80.29 even 4
6400.2.a.a.1.1 1 80.59 odd 4
6400.2.a.x.1.1 1 80.19 odd 4
6400.2.a.x.1.1 1 80.69 even 4