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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [45,8,Mod(1,45)] 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("45.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 45.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(14.0573261468\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
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)
 
\(f(q)\) \(=\) \( q + 13 q^{2} + 41 q^{4} + 125 q^{5} + 1380 q^{7} - 1131 q^{8} + 1625 q^{10} + 3304 q^{11} + 8506 q^{13} + 17940 q^{14} - 19951 q^{16} + 9994 q^{17} + 41236 q^{19} + 5125 q^{20} + 42952 q^{22} - 84120 q^{23}+ \cdots + 14051141 q^{98}+O(q^{100}) \) 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
13.0000 0 41.0000 125.000 0 1380.00 −1131.00 0 1625.00
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( -1 \)

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 45.8.a.e 1
3.b odd 2 1 15.8.a.b ✓ 1
5.b even 2 1 225.8.a.c 1
5.c odd 4 2 225.8.b.c 2
12.b even 2 1 240.8.a.h 1
15.d odd 2 1 75.8.a.b 1
15.e even 4 2 75.8.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.8.a.b ✓ 1 3.b odd 2 1
45.8.a.e 1 1.a even 1 1 trivial
75.8.a.b 1 15.d odd 2 1
75.8.b.b 2 15.e even 4 2
225.8.a.c 1 5.b even 2 1
225.8.b.c 2 5.c odd 4 2
240.8.a.h 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 13 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(45))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 13 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 125 \) Copy content Toggle raw display
$7$ \( T - 1380 \) Copy content Toggle raw display
$11$ \( T - 3304 \) Copy content Toggle raw display
$13$ \( T - 8506 \) Copy content Toggle raw display
$17$ \( T - 9994 \) Copy content Toggle raw display
$19$ \( T - 41236 \) Copy content Toggle raw display
$23$ \( T + 84120 \) Copy content Toggle raw display
$29$ \( T + 132802 \) Copy content Toggle raw display
$31$ \( T + 55800 \) Copy content Toggle raw display
$37$ \( T - 228170 \) Copy content Toggle raw display
$41$ \( T - 139670 \) Copy content Toggle raw display
$43$ \( T + 755492 \) Copy content Toggle raw display
$47$ \( T + 836984 \) Copy content Toggle raw display
$53$ \( T + 1641650 \) Copy content Toggle raw display
$59$ \( T - 989656 \) Copy content Toggle raw display
$61$ \( T + 1658162 \) Copy content Toggle raw display
$67$ \( T + 4523844 \) Copy content Toggle raw display
$71$ \( T - 389408 \) Copy content Toggle raw display
$73$ \( T - 5617330 \) Copy content Toggle raw display
$79$ \( T - 3901080 \) Copy content Toggle raw display
$83$ \( T - 9394116 \) Copy content Toggle raw display
$89$ \( T + 2803746 \) Copy content Toggle raw display
$97$ \( T - 5099426 \) Copy content Toggle raw display
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