Properties

Label 256.2.b.b
Level $256$
Weight $2$
Character orbit 256.b
Analytic conductor $2.044$
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] = [256,2,Mod(129,256)]
 
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([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 256.b (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: \(2.04417029174\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 32)
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.

\(f(q)\) \(=\) \( q + \beta q^{5} + 3 q^{9} + 3 \beta q^{13} + 2 q^{17} + q^{25} - 5 \beta q^{29} + \beta q^{37} - 10 q^{41} + 3 \beta q^{45} - 7 q^{49} - 7 \beta q^{53} - 5 \beta q^{61} - 12 q^{65} + 6 q^{73} + 9 q^{81} + \cdots + 18 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{9} + 4 q^{17} + 2 q^{25} - 20 q^{41} - 14 q^{49} - 24 q^{65} + 12 q^{73} + 18 q^{81} - 20 q^{89} + 36 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
1.00000i
1.00000i
0 0 0 2.00000i 0 0 0 3.00000 0
129.2 0 0 0 2.00000i 0 0 0 3.00000 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
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 256.2.b.b 2
3.b odd 2 1 2304.2.d.j 2
4.b odd 2 1 CM 256.2.b.b 2
8.b even 2 1 inner 256.2.b.b 2
8.d odd 2 1 inner 256.2.b.b 2
12.b even 2 1 2304.2.d.j 2
16.e even 4 1 32.2.a.a 1
16.e even 4 1 64.2.a.a 1
16.f odd 4 1 32.2.a.a 1
16.f odd 4 1 64.2.a.a 1
24.f even 2 1 2304.2.d.j 2
24.h odd 2 1 2304.2.d.j 2
32.g even 8 4 1024.2.e.j 4
32.h odd 8 4 1024.2.e.j 4
48.i odd 4 1 288.2.a.d 1
48.i odd 4 1 576.2.a.c 1
48.k even 4 1 288.2.a.d 1
48.k even 4 1 576.2.a.c 1
80.i odd 4 1 800.2.c.e 2
80.i odd 4 1 1600.2.c.l 2
80.j even 4 1 800.2.c.e 2
80.j even 4 1 1600.2.c.l 2
80.k odd 4 1 800.2.a.d 1
80.k odd 4 1 1600.2.a.n 1
80.q even 4 1 800.2.a.d 1
80.q even 4 1 1600.2.a.n 1
80.s even 4 1 800.2.c.e 2
80.s even 4 1 1600.2.c.l 2
80.t odd 4 1 800.2.c.e 2
80.t odd 4 1 1600.2.c.l 2
112.j even 4 1 1568.2.a.e 1
112.j even 4 1 3136.2.a.m 1
112.l odd 4 1 1568.2.a.e 1
112.l odd 4 1 3136.2.a.m 1
112.u odd 12 2 1568.2.i.g 2
112.v even 12 2 1568.2.i.f 2
112.w even 12 2 1568.2.i.g 2
112.x odd 12 2 1568.2.i.f 2
144.u even 12 2 2592.2.i.e 2
144.v odd 12 2 2592.2.i.t 2
144.w odd 12 2 2592.2.i.e 2
144.x even 12 2 2592.2.i.t 2
176.i even 4 1 3872.2.a.f 1
176.i even 4 1 7744.2.a.v 1
176.l odd 4 1 3872.2.a.f 1
176.l odd 4 1 7744.2.a.v 1
208.o odd 4 1 5408.2.a.g 1
208.p even 4 1 5408.2.a.g 1
240.t even 4 1 7200.2.a.v 1
240.z odd 4 1 7200.2.f.m 2
240.bb even 4 1 7200.2.f.m 2
240.bd odd 4 1 7200.2.f.m 2
240.bf even 4 1 7200.2.f.m 2
240.bm odd 4 1 7200.2.a.v 1
272.k odd 4 1 9248.2.a.f 1
272.r even 4 1 9248.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a 1 16.e even 4 1
32.2.a.a 1 16.f odd 4 1
64.2.a.a 1 16.e even 4 1
64.2.a.a 1 16.f odd 4 1
256.2.b.b 2 1.a even 1 1 trivial
256.2.b.b 2 4.b odd 2 1 CM
256.2.b.b 2 8.b even 2 1 inner
256.2.b.b 2 8.d odd 2 1 inner
288.2.a.d 1 48.i odd 4 1
288.2.a.d 1 48.k even 4 1
576.2.a.c 1 48.i odd 4 1
576.2.a.c 1 48.k even 4 1
800.2.a.d 1 80.k odd 4 1
800.2.a.d 1 80.q even 4 1
800.2.c.e 2 80.i odd 4 1
800.2.c.e 2 80.j even 4 1
800.2.c.e 2 80.s even 4 1
800.2.c.e 2 80.t odd 4 1
1024.2.e.j 4 32.g even 8 4
1024.2.e.j 4 32.h odd 8 4
1568.2.a.e 1 112.j even 4 1
1568.2.a.e 1 112.l odd 4 1
1568.2.i.f 2 112.v even 12 2
1568.2.i.f 2 112.x odd 12 2
1568.2.i.g 2 112.u odd 12 2
1568.2.i.g 2 112.w even 12 2
1600.2.a.n 1 80.k odd 4 1
1600.2.a.n 1 80.q even 4 1
1600.2.c.l 2 80.i odd 4 1
1600.2.c.l 2 80.j even 4 1
1600.2.c.l 2 80.s even 4 1
1600.2.c.l 2 80.t odd 4 1
2304.2.d.j 2 3.b odd 2 1
2304.2.d.j 2 12.b even 2 1
2304.2.d.j 2 24.f even 2 1
2304.2.d.j 2 24.h odd 2 1
2592.2.i.e 2 144.u even 12 2
2592.2.i.e 2 144.w odd 12 2
2592.2.i.t 2 144.v odd 12 2
2592.2.i.t 2 144.x even 12 2
3136.2.a.m 1 112.j even 4 1
3136.2.a.m 1 112.l odd 4 1
3872.2.a.f 1 176.i even 4 1
3872.2.a.f 1 176.l odd 4 1
5408.2.a.g 1 208.o odd 4 1
5408.2.a.g 1 208.p even 4 1
7200.2.a.v 1 240.t even 4 1
7200.2.a.v 1 240.bm odd 4 1
7200.2.f.m 2 240.z odd 4 1
7200.2.f.m 2 240.bb even 4 1
7200.2.f.m 2 240.bd odd 4 1
7200.2.f.m 2 240.bf even 4 1
7744.2.a.v 1 176.i even 4 1
7744.2.a.v 1 176.l odd 4 1
9248.2.a.f 1 272.k odd 4 1
9248.2.a.f 1 272.r even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(256, [\chi])\):

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