Properties

Label 128.5.d.a
Level $128$
Weight $5$
Character orbit 128.d
Analytic conductor $13.231$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [128,5,Mod(63,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.63");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 128.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.2313552747\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 48i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{5} - 81 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{5} - 81 q^{9} + 5 \beta q^{13} - 322 q^{17} - 1679 q^{25} - 35 \beta q^{29} + 35 \beta q^{37} - 3038 q^{41} + 81 \beta q^{45} + 2401 q^{49} - 105 \beta q^{53} - 55 \beta q^{61} + 11520 q^{65} - 1442 q^{73} + 6561 q^{81} + 322 \beta q^{85} + 9758 q^{89} + 1918 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 162 q^{9} - 644 q^{17} - 3358 q^{25} - 6076 q^{41} + 4802 q^{49} + 23040 q^{65} - 2884 q^{73} + 13122 q^{81} + 19516 q^{89} + 3836 q^{97}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
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 48.0000i 0 0 0 −81.0000 0
63.2 0 0 0 48.0000i 0 0 0 −81.0000 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 128.5.d.a 2
3.b odd 2 1 1152.5.b.c 2
4.b odd 2 1 CM 128.5.d.a 2
8.b even 2 1 inner 128.5.d.a 2
8.d odd 2 1 inner 128.5.d.a 2
12.b even 2 1 1152.5.b.c 2
16.e even 4 1 256.5.c.a 1
16.e even 4 1 256.5.c.b 1
16.f odd 4 1 256.5.c.a 1
16.f odd 4 1 256.5.c.b 1
24.f even 2 1 1152.5.b.c 2
24.h odd 2 1 1152.5.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.5.d.a 2 1.a even 1 1 trivial
128.5.d.a 2 4.b odd 2 1 CM
128.5.d.a 2 8.b even 2 1 inner
128.5.d.a 2 8.d odd 2 1 inner
256.5.c.a 1 16.e even 4 1
256.5.c.a 1 16.f odd 4 1
256.5.c.b 1 16.e even 4 1
256.5.c.b 1 16.f odd 4 1
1152.5.b.c 2 3.b odd 2 1
1152.5.b.c 2 12.b even 2 1
1152.5.b.c 2 24.f even 2 1
1152.5.b.c 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{5}^{\mathrm{new}}(128, [\chi])\). 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
$5$ \( T^{2} + 2304 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 57600 \) Copy content Toggle raw display
$17$ \( (T + 322)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 2822400 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2822400 \) Copy content Toggle raw display
$41$ \( (T + 3038)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 25401600 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 6969600 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 1442)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 9758)^{2} \) Copy content Toggle raw display
$97$ \( (T - 1918)^{2} \) Copy content Toggle raw display
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