Properties

Label 256.11.c.c
Level $256$
Weight $11$
Character orbit 256.c
Analytic conductor $162.651$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [256,11,Mod(255,256)]
 
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, 0]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.255");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 256.c (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: \(162.651456684\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 8)
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.

\(f(q)\) \(=\) \( q + 241 \beta q^{3} - 173275 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 241 \beta q^{3} - 173275 q^{9} - 48713 \beta q^{11} + 823682 q^{17} - 1676863 \beta q^{19} - 9765625 q^{25} - 27528466 \beta q^{27} + 46959332 q^{33} + 37778926 q^{41} + 107242807 \beta q^{43} + 282475249 q^{49} + 198507362 \beta q^{51} + 1616495932 q^{57} + 460521799 \beta q^{59} - 906854191 \beta q^{67} + 1605781582 q^{73} - 2353515625 \beta q^{75} + 16305725749 q^{81} - 48025759 \beta q^{83} + 11116019374 q^{89} - 9872978014 q^{97} + 8440745075 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 346550 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 346550 q^{9} + 1647364 q^{17} - 19531250 q^{25} + 93918664 q^{33} + 75557852 q^{41} + 564950498 q^{49} + 3232991864 q^{57} + 3211563164 q^{73} + 32611451498 q^{81} + 22232038748 q^{89} - 19745956028 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
255.1
1.00000i
1.00000i
0 482.000i 0 0 0 0 0 −173275. 0
255.2 0 482.000i 0 0 0 0 0 −173275. 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 256.11.c.c 2
4.b odd 2 1 inner 256.11.c.c 2
8.b even 2 1 inner 256.11.c.c 2
8.d odd 2 1 CM 256.11.c.c 2
16.e even 4 1 8.11.d.a 1
16.e even 4 1 32.11.d.a 1
16.f odd 4 1 8.11.d.a 1
16.f odd 4 1 32.11.d.a 1
48.i odd 4 1 72.11.b.a 1
48.i odd 4 1 288.11.b.a 1
48.k even 4 1 72.11.b.a 1
48.k even 4 1 288.11.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.11.d.a 1 16.e even 4 1
8.11.d.a 1 16.f odd 4 1
32.11.d.a 1 16.e even 4 1
32.11.d.a 1 16.f odd 4 1
72.11.b.a 1 48.i odd 4 1
72.11.b.a 1 48.k even 4 1
256.11.c.c 2 1.a even 1 1 trivial
256.11.c.c 2 4.b odd 2 1 inner
256.11.c.c 2 8.b even 2 1 inner
256.11.c.c 2 8.d odd 2 1 CM
288.11.b.a 1 48.i odd 4 1
288.11.b.a 1 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{11}^{\mathrm{new}}(256, [\chi])\):

\( T_{3}^{2} + 232324 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 232324 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 9491825476 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 823682)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 11247478083076 \) 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
$41$ \( (T - 37778926)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 46\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 84\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 32\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 1605781582)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 92\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T - 11116019374)^{2} \) Copy content Toggle raw display
$97$ \( (T + 9872978014)^{2} \) Copy content Toggle raw display
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