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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9248,2,Mod(1,9248)] 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("9248.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9248, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9248 = 2^{5} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9248.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-2,0,-2,0,-2,0,1,0,2,0,2,0,4,0,0,0,-4,0,4,0,-2,0,-1,0,4, 0,-2,0,10,0,-4,0,4,0,-10,0,-4,0,-2,0,-4,0,-2,0,0,0,-3,0,0,0,6,0,-4,0,8, 0,-4,0,-2,0,-2,0,-4,0,16,0,4,0,10,0,6,0,2,0,-4,0,6,0,-11,0,4,0,0,0,4,0, 6,0,-4,0,-20,0,8,0,14,0,2,0,-6,0,16,0,-8,0,-18,0,6,0,20,0,6,0,4,0,2,0, 0,0,-7,0,4,0,12,0,-20,0,8,0,22,0,8,0,-8,0,22,0,-2,0,0,0,4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(145)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(73.8456517893\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 544)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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)
 
\(f(q)\) \(=\) \( q - 2 q^{3} - 2 q^{5} - 2 q^{7} + q^{9} + 2 q^{11} + 2 q^{13} + 4 q^{15} - 4 q^{19} + 4 q^{21} - 2 q^{23} - q^{25} + 4 q^{27} - 2 q^{29} + 10 q^{31} - 4 q^{33} + 4 q^{35} - 10 q^{37} - 4 q^{39} - 2 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 −2.00000 0 −2.00000 0 −2.00000 0 1.00000 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(17\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9248.2.a.b 1
4.b odd 2 1 9248.2.a.i 1
17.b even 2 1 544.2.a.e yes 1
51.c odd 2 1 4896.2.a.f 1
68.d odd 2 1 544.2.a.a ✓ 1
136.e odd 2 1 1088.2.a.k 1
136.h even 2 1 1088.2.a.b 1
204.h even 2 1 4896.2.a.c 1
408.b odd 2 1 9792.2.a.bw 1
408.h even 2 1 9792.2.a.bt 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
544.2.a.a ✓ 1 68.d odd 2 1
544.2.a.e yes 1 17.b even 2 1
1088.2.a.b 1 136.h even 2 1
1088.2.a.k 1 136.e odd 2 1
4896.2.a.c 1 204.h even 2 1
4896.2.a.f 1 51.c odd 2 1
9248.2.a.b 1 1.a even 1 1 trivial
9248.2.a.i 1 4.b odd 2 1
9792.2.a.bt 1 408.h even 2 1
9792.2.a.bw 1 408.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9248))\):

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{19} + 4 \) Copy content Toggle raw display
\( T_{43} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T - 2 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T + 2 \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T - 10 \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T + 2 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T + 4 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T - 16 \) Copy content Toggle raw display
$71$ \( T - 10 \) Copy content Toggle raw display
$73$ \( T - 6 \) Copy content Toggle raw display
$79$ \( T - 6 \) Copy content Toggle raw display
$83$ \( T - 4 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T - 14 \) Copy content Toggle raw display
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