Properties

Label 1024.2.e.l
Level $1024$
Weight $2$
Character orbit 1024.e
Analytic conductor $8.177$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1024,2,Mod(257,1024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1024, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1024.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.e (of order \(4\), degree \(2\), 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: \(8.17668116698\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 64)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{2} q^{9} - 3 \beta_{3} q^{11} + 6 q^{17} - \beta_1 q^{19} + 5 \beta_{2} q^{25} - 2 \beta_{3} q^{27} + 12 q^{33} + 6 \beta_{2} q^{41} + 5 \beta_{3} q^{43} + 7 q^{49} + 6 \beta_1 q^{51} - 4 \beta_{2} q^{57} + 3 \beta_{3} q^{59} - 7 \beta_1 q^{67} - 2 \beta_{2} q^{73} + 5 \beta_{3} q^{75} + 11 q^{81} - 9 \beta_1 q^{83} + 18 \beta_{2} q^{89} + 10 q^{97} + 3 \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 24 q^{17} + 48 q^{33} + 28 q^{49} + 44 q^{81} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{8}^{3} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(-\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
0 −1.41421 1.41421i 0 0 0 0 0 1.00000i 0
257.2 0 1.41421 + 1.41421i 0 0 0 0 0 1.00000i 0
769.1 0 −1.41421 + 1.41421i 0 0 0 0 0 1.00000i 0
769.2 0 1.41421 1.41421i 0 0 0 0 0 1.00000i 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
16.e even 4 2 inner
16.f odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1024.2.e.l 4
4.b odd 2 1 inner 1024.2.e.l 4
8.b even 2 1 inner 1024.2.e.l 4
8.d odd 2 1 CM 1024.2.e.l 4
16.e even 4 2 inner 1024.2.e.l 4
16.f odd 4 2 inner 1024.2.e.l 4
32.g even 8 2 64.2.b.a 2
32.g even 8 1 256.2.a.a 1
32.g even 8 1 256.2.a.d 1
32.h odd 8 2 64.2.b.a 2
32.h odd 8 1 256.2.a.a 1
32.h odd 8 1 256.2.a.d 1
96.o even 8 2 576.2.d.a 2
96.o even 8 1 2304.2.a.h 1
96.o even 8 1 2304.2.a.i 1
96.p odd 8 2 576.2.d.a 2
96.p odd 8 1 2304.2.a.h 1
96.p odd 8 1 2304.2.a.i 1
160.u even 8 1 1600.2.f.a 2
160.u even 8 1 1600.2.f.b 2
160.v odd 8 1 1600.2.f.a 2
160.v odd 8 1 1600.2.f.b 2
160.y odd 8 2 1600.2.d.a 2
160.y odd 8 1 6400.2.a.a 1
160.y odd 8 1 6400.2.a.x 1
160.z even 8 2 1600.2.d.a 2
160.z even 8 1 6400.2.a.a 1
160.z even 8 1 6400.2.a.x 1
160.ba even 8 1 1600.2.f.a 2
160.ba even 8 1 1600.2.f.b 2
160.bb odd 8 1 1600.2.f.a 2
160.bb odd 8 1 1600.2.f.b 2
224.v odd 8 2 3136.2.b.b 2
224.x even 8 2 3136.2.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.2.b.a 2 32.g even 8 2
64.2.b.a 2 32.h odd 8 2
256.2.a.a 1 32.g even 8 1
256.2.a.a 1 32.h odd 8 1
256.2.a.d 1 32.g even 8 1
256.2.a.d 1 32.h odd 8 1
576.2.d.a 2 96.o even 8 2
576.2.d.a 2 96.p odd 8 2
1024.2.e.l 4 1.a even 1 1 trivial
1024.2.e.l 4 4.b odd 2 1 inner
1024.2.e.l 4 8.b even 2 1 inner
1024.2.e.l 4 8.d odd 2 1 CM
1024.2.e.l 4 16.e even 4 2 inner
1024.2.e.l 4 16.f odd 4 2 inner
1600.2.d.a 2 160.y odd 8 2
1600.2.d.a 2 160.z even 8 2
1600.2.f.a 2 160.u even 8 1
1600.2.f.a 2 160.v odd 8 1
1600.2.f.a 2 160.ba even 8 1
1600.2.f.a 2 160.bb odd 8 1
1600.2.f.b 2 160.u even 8 1
1600.2.f.b 2 160.v odd 8 1
1600.2.f.b 2 160.ba even 8 1
1600.2.f.b 2 160.bb odd 8 1
2304.2.a.h 1 96.o even 8 1
2304.2.a.h 1 96.p odd 8 1
2304.2.a.i 1 96.o even 8 1
2304.2.a.i 1 96.p odd 8 1
3136.2.b.b 2 224.v odd 8 2
3136.2.b.b 2 224.x even 8 2
6400.2.a.a 1 160.y odd 8 1
6400.2.a.a 1 160.z even 8 1
6400.2.a.x 1 160.y odd 8 1
6400.2.a.x 1 160.z even 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1024, [\chi])\):

\( T_{3}^{4} + 16 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{47} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 1296 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T - 6)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 16 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 10000 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 1296 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 38416 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 104976 \) Copy content Toggle raw display
$89$ \( (T^{2} + 324)^{2} \) Copy content Toggle raw display
$97$ \( (T - 10)^{4} \) Copy content Toggle raw display
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