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Properties

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

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1024,2,Mod(257,1024)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.e (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-4,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content 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: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 512)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1 - 1) q^{3} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{9} + ( - \beta_{3} - 3 \beta_{2} + 3) q^{11} + (2 \beta_{3} - 2 \beta_1) q^{17} + ( - \beta_{2} - 3 \beta_1 - 1) q^{19}+ \cdots + (\beta_{2} - 9 \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 12 q^{11} - 4 q^{19} + 32 q^{27} - 8 q^{33} + 20 q^{43} + 28 q^{49} - 32 q^{51} + 12 q^{59} - 28 q^{67} + 20 q^{75} - 92 q^{81} + 36 q^{83} + 4 q^{99}+O(q^{100}) \) 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

Display \(\zeta_{8}^j\) in terms of \(\beta_i\)

\(\zeta_{8}\)\(=\) \( ( \beta_1 ) / 2 \) 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

Display \(\beta_i\) in terms of \(\zeta_{8}^j\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 −2.41421 − 2.41421i 0 0 0 0 0 8.65685i 0
257.2 0 0.414214 + 0.414214i 0 0 0 0 0 − 2.65685i 0
769.1 0 −2.41421 + 2.41421i 0 0 0 0 0 − 8.65685i 0
769.2 0 0.414214 − 0.414214i 0 0 0 0 0 2.65685i 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
16.e even 4 1 inner
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1024.2.e.g 4
4.b odd 2 1 1024.2.e.o 4
8.b even 2 1 1024.2.e.o 4
8.d odd 2 1 CM 1024.2.e.g 4
16.e even 4 1 inner 1024.2.e.g 4
16.e even 4 1 1024.2.e.o 4
16.f odd 4 1 inner 1024.2.e.g 4
16.f odd 4 1 1024.2.e.o 4
32.g even 8 1 512.2.a.a ✓ 2
32.g even 8 1 512.2.a.f yes 2
32.g even 8 2 512.2.b.c 4
32.h odd 8 1 512.2.a.a ✓ 2
32.h odd 8 1 512.2.a.f yes 2
32.h odd 8 2 512.2.b.c 4
96.o even 8 1 4608.2.a.i 2
96.o even 8 1 4608.2.a.k 2
96.o even 8 2 4608.2.d.k 4
96.p odd 8 1 4608.2.a.i 2
96.p odd 8 1 4608.2.a.k 2
96.p odd 8 2 4608.2.d.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.a.a ✓ 2 32.g even 8 1
512.2.a.a ✓ 2 32.h odd 8 1
512.2.a.f yes 2 32.g even 8 1
512.2.a.f yes 2 32.h odd 8 1
512.2.b.c 4 32.g even 8 2
512.2.b.c 4 32.h odd 8 2
1024.2.e.g 4 1.a even 1 1 trivial
1024.2.e.g 4 8.d odd 2 1 CM
1024.2.e.g 4 16.e even 4 1 inner
1024.2.e.g 4 16.f odd 4 1 inner
1024.2.e.o 4 4.b odd 2 1
1024.2.e.o 4 8.b even 2 1
1024.2.e.o 4 16.e even 4 1
1024.2.e.o 4 16.f odd 4 1
4608.2.a.i 2 96.o even 8 1
4608.2.a.i 2 96.p odd 8 1
4608.2.a.k 2 96.o even 8 1
4608.2.a.k 2 96.p odd 8 1
4608.2.d.k 4 96.o even 8 2
4608.2.d.k 4 96.p odd 8 2

Hecke kernels

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

\( T_{3}^{4} + 4T_{3}^{3} + 8T_{3}^{2} - 8T_{3} + 4 \) 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
$3$ \( T^{4} + 4 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 12 T^{3} + \cdots + 196 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} + \cdots + 1156 \) 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
$43$ \( T^{4} - 20 T^{3} + \cdots + 196 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 12 T^{3} + \cdots + 6724 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 28 T^{3} + \cdots + 3844 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 288)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} - 36 T^{3} + \cdots + 24964 \) Copy content Toggle raw display
$89$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 288)^{2} \) Copy content Toggle raw display
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