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Properties

Label 128.7.d.b
Level $128$
Weight $7$
Character orbit 128.d
Analytic conductor $29.447$
Analytic rank $0$
Dimension $2$
CM discriminant -4
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] = [128,7,Mod(63,128)] 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("128.63"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(128, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 128.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,-1458] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(29.4469227033\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
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 \(\beta = 8i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 11 \beta q^{5} - 729 q^{9} - 207 \beta q^{13} + 990 q^{17} + 7881 q^{25} - 4615 \beta q^{29} - 10593 \beta q^{37} - 84942 q^{41} - 8019 \beta q^{45} + 117649 q^{49} - 37037 \beta q^{53} - 48555 \beta q^{61} + \cdots + 1472510 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1458 q^{9} + 1980 q^{17} + 15762 q^{25} - 169884 q^{41} + 235298 q^{49} + 291456 q^{65} - 855140 q^{73} + 1062882 q^{81} - 2757924 q^{89} + 2945020 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
− 1.00000i
1.00000i
0 0 0 − 88.0000i 0 0 0 −729.000 0
63.2 0 0 0 88.0000i 0 0 0 −729.000 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.7.d.b ✓ 2
4.b odd 2 1 CM 128.7.d.b ✓ 2
8.b even 2 1 inner 128.7.d.b ✓ 2
8.d odd 2 1 inner 128.7.d.b ✓ 2
16.e even 4 1 256.7.c.a 1
16.e even 4 1 256.7.c.b 1
16.f odd 4 1 256.7.c.a 1
16.f odd 4 1 256.7.c.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.7.d.b ✓ 2 1.a even 1 1 trivial
128.7.d.b ✓ 2 4.b odd 2 1 CM
128.7.d.b ✓ 2 8.b even 2 1 inner
128.7.d.b ✓ 2 8.d odd 2 1 inner
256.7.c.a 1 16.e even 4 1
256.7.c.a 1 16.f odd 4 1
256.7.c.b 1 16.e even 4 1
256.7.c.b 1 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{7}^{\mathrm{new}}(128, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 7744 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2742336 \) Copy content Toggle raw display
$17$ \( (T - 990)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 1363086400 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 7181545536 \) Copy content Toggle raw display
$41$ \( (T + 84942)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 87791319616 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 150885633600 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 427570)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T + 1378962)^{2} \) Copy content Toggle raw display
$97$ \( (T - 1472510)^{2} \) Copy content Toggle raw display
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