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Properties

Label 64.2.b.a
Level $64$
Weight $2$
Character orbit 64.b
Analytic conductor $0.511$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $4$

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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] = [64,2,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: \(\Q(\sqrt{-1}) \)
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: $\mathrm{U}(1)[D_{2}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - q^{9} - 3 \beta q^{11} - 6 q^{17} + \beta q^{19} + 5 q^{25} + 2 \beta q^{27} + 12 q^{33} - 6 q^{41} + 5 \beta q^{43} - 7 q^{49} - 6 \beta q^{51} - 4 q^{57} - 3 \beta q^{59} - 7 \beta q^{67} + \cdots + 3 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9} - 12 q^{17} + 10 q^{25} + 24 q^{33} - 12 q^{41} - 14 q^{49} - 8 q^{57} + 4 q^{73} - 22 q^{81} + 36 q^{89} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
− 1.00000i
1.00000i
0 − 2.00000i 0 0 0 0 0 −1.00000 0
33.2 0 2.00000i 0 0 0 0 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 64.2.b.a ✓ 2
3.b odd 2 1 576.2.d.a 2
4.b odd 2 1 inner 64.2.b.a ✓ 2
5.b even 2 1 1600.2.d.a 2
5.c odd 4 1 1600.2.f.a 2
5.c odd 4 1 1600.2.f.b 2
7.b odd 2 1 3136.2.b.b 2
8.b even 2 1 inner 64.2.b.a ✓ 2
8.d odd 2 1 CM 64.2.b.a ✓ 2
12.b even 2 1 576.2.d.a 2
16.e even 4 1 256.2.a.a 1
16.e even 4 1 256.2.a.d 1
16.f odd 4 1 256.2.a.a 1
16.f odd 4 1 256.2.a.d 1
20.d odd 2 1 1600.2.d.a 2
20.e even 4 1 1600.2.f.a 2
20.e even 4 1 1600.2.f.b 2
24.f even 2 1 576.2.d.a 2
24.h odd 2 1 576.2.d.a 2
28.d even 2 1 3136.2.b.b 2
32.g even 8 4 1024.2.e.l 4
32.h odd 8 4 1024.2.e.l 4
40.e odd 2 1 1600.2.d.a 2
40.f even 2 1 1600.2.d.a 2
40.i odd 4 1 1600.2.f.a 2
40.i odd 4 1 1600.2.f.b 2
40.k even 4 1 1600.2.f.a 2
40.k even 4 1 1600.2.f.b 2
48.i odd 4 1 2304.2.a.h 1
48.i odd 4 1 2304.2.a.i 1
48.k even 4 1 2304.2.a.h 1
48.k even 4 1 2304.2.a.i 1
56.e even 2 1 3136.2.b.b 2
56.h odd 2 1 3136.2.b.b 2
80.k odd 4 1 6400.2.a.a 1
80.k odd 4 1 6400.2.a.x 1
80.q even 4 1 6400.2.a.a 1
80.q even 4 1 6400.2.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.2.b.a ✓ 2 1.a even 1 1 trivial
64.2.b.a ✓ 2 4.b odd 2 1 inner
64.2.b.a ✓ 2 8.b even 2 1 inner
64.2.b.a ✓ 2 8.d odd 2 1 CM
256.2.a.a 1 16.e even 4 1
256.2.a.a 1 16.f odd 4 1
256.2.a.d 1 16.e even 4 1
256.2.a.d 1 16.f odd 4 1
576.2.d.a 2 3.b odd 2 1
576.2.d.a 2 12.b even 2 1
576.2.d.a 2 24.f even 2 1
576.2.d.a 2 24.h odd 2 1
1024.2.e.l 4 32.g even 8 4
1024.2.e.l 4 32.h odd 8 4
1600.2.d.a 2 5.b even 2 1
1600.2.d.a 2 20.d odd 2 1
1600.2.d.a 2 40.e odd 2 1
1600.2.d.a 2 40.f even 2 1
1600.2.f.a 2 5.c odd 4 1
1600.2.f.a 2 20.e even 4 1
1600.2.f.a 2 40.i odd 4 1
1600.2.f.a 2 40.k even 4 1
1600.2.f.b 2 5.c odd 4 1
1600.2.f.b 2 20.e even 4 1
1600.2.f.b 2 40.i odd 4 1
1600.2.f.b 2 40.k even 4 1
2304.2.a.h 1 48.i odd 4 1
2304.2.a.h 1 48.k even 4 1
2304.2.a.i 1 48.i odd 4 1
2304.2.a.i 1 48.k even 4 1
3136.2.b.b 2 7.b odd 2 1
3136.2.b.b 2 28.d even 2 1
3136.2.b.b 2 56.e even 2 1
3136.2.b.b 2 56.h odd 2 1
6400.2.a.a 1 80.k odd 4 1
6400.2.a.a 1 80.q even 4 1
6400.2.a.x 1 80.k odd 4 1
6400.2.a.x 1 80.q even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(64, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 36 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 4 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 100 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 36 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 196 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 324 \) Copy content Toggle raw display
$89$ \( (T - 18)^{2} \) Copy content Toggle raw display
$97$ \( (T - 10)^{2} \) Copy content Toggle raw display
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