Properties

Label 256.9.d.a
Level $256$
Weight $9$
Character orbit 256.d
Analytic conductor $104.289$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [256,9,Mod(127,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.127");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 256.d (of order \(2\), degree \(1\), not 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: \(104.288924176\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 527 \beta q^{5} - 6561 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 527 \beta q^{5} - 6561 q^{9} + 239 \beta q^{13} - 63358 q^{17} - 720291 q^{25} + 703919 \beta q^{29} + 462961 \beta q^{37} - 3577922 q^{41} + 3457647 \beta q^{45} + 5764801 q^{49} - 4810319 \beta q^{53} - 10361041 \beta q^{61} + 503812 q^{65} + 54717118 q^{73} + 43046721 q^{81} + 33389666 \beta q^{85} + 30265918 q^{89} - 173379838 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 13122 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 13122 q^{9} - 126716 q^{17} - 1440582 q^{25} - 7155844 q^{41} + 11529602 q^{49} + 1007624 q^{65} + 109434236 q^{73} + 86093442 q^{81} + 60531836 q^{89} - 346759676 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
1.00000i
1.00000i
0 0 0 1054.00i 0 0 0 −6561.00 0
127.2 0 0 0 1054.00i 0 0 0 −6561.00 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 256.9.d.a 2
4.b odd 2 1 CM 256.9.d.a 2
8.b even 2 1 inner 256.9.d.a 2
8.d odd 2 1 inner 256.9.d.a 2
16.e even 4 1 4.9.b.a 1
16.e even 4 1 64.9.c.a 1
16.f odd 4 1 4.9.b.a 1
16.f odd 4 1 64.9.c.a 1
48.i odd 4 1 36.9.d.a 1
48.k even 4 1 36.9.d.a 1
80.i odd 4 1 100.9.d.a 2
80.j even 4 1 100.9.d.a 2
80.k odd 4 1 100.9.b.a 1
80.q even 4 1 100.9.b.a 1
80.s even 4 1 100.9.d.a 2
80.t odd 4 1 100.9.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.9.b.a 1 16.e even 4 1
4.9.b.a 1 16.f odd 4 1
36.9.d.a 1 48.i odd 4 1
36.9.d.a 1 48.k even 4 1
64.9.c.a 1 16.e even 4 1
64.9.c.a 1 16.f odd 4 1
100.9.b.a 1 80.k odd 4 1
100.9.b.a 1 80.q even 4 1
100.9.d.a 2 80.i odd 4 1
100.9.d.a 2 80.j even 4 1
100.9.d.a 2 80.s even 4 1
100.9.d.a 2 80.t odd 4 1
256.9.d.a 2 1.a even 1 1 trivial
256.9.d.a 2 4.b odd 2 1 CM
256.9.d.a 2 8.b even 2 1 inner
256.9.d.a 2 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{9}^{\mathrm{new}}(256, [\chi])\). 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
$5$ \( T^{2} + 1110916 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 228484 \) Copy content Toggle raw display
$17$ \( (T + 63358)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 1982007834244 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 857331550084 \) Copy content Toggle raw display
$41$ \( (T + 3577922)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 92556675527044 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 429404682414724 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 54717118)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 30265918)^{2} \) Copy content Toggle raw display
$97$ \( (T + 173379838)^{2} \) Copy content Toggle raw display
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