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Properties

Label 45.8.a.g
Level $45$
Weight $8$
Character orbit 45.a
Self dual yes
Analytic conductor $14.057$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

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 = [1,22] 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)
 
\(f(q)\) \(=\) \( q + 22 q^{2} + 356 q^{4} + 125 q^{5} - 420 q^{7} + 5016 q^{8} + 2750 q^{10} + 2944 q^{11} - 11006 q^{13} - 9240 q^{14} + 64784 q^{16} + 16546 q^{17} - 25364 q^{19} + 44500 q^{20} + 64768 q^{22} + 5880 q^{23}+ \cdots - 14237146 q^{98}+O(q^{100}) \) 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
22.0000 0 356.000 125.000 0 −420.000 5016.00 0 2750.00
\(n\): e.g. 2-40 or 990-1000
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.g 1
3.b odd 2 1 15.8.a.a ✓ 1
5.b even 2 1 225.8.a.a 1
5.c odd 4 2 225.8.b.a 2
12.b even 2 1 240.8.a.c 1
15.d odd 2 1 75.8.a.c 1
15.e even 4 2 75.8.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.8.a.a ✓ 1 3.b odd 2 1
45.8.a.g 1 1.a even 1 1 trivial
75.8.a.c 1 15.d odd 2 1
75.8.b.a 2 15.e even 4 2
225.8.a.a 1 5.b even 2 1
225.8.b.a 2 5.c odd 4 2
240.8.a.c 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 22 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(45))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 22 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 125 \) Copy content Toggle raw display
$7$ \( T + 420 \) Copy content Toggle raw display
$11$ \( T - 2944 \) Copy content Toggle raw display
$13$ \( T + 11006 \) Copy content Toggle raw display
$17$ \( T - 16546 \) Copy content Toggle raw display
$19$ \( T + 25364 \) Copy content Toggle raw display
$23$ \( T - 5880 \) Copy content Toggle raw display
$29$ \( T + 163042 \) Copy content Toggle raw display
$31$ \( T + 201600 \) Copy content Toggle raw display
$37$ \( T - 120530 \) Copy content Toggle raw display
$41$ \( T - 115910 \) Copy content Toggle raw display
$43$ \( T + 19148 \) Copy content Toggle raw display
$47$ \( T + 841016 \) Copy content Toggle raw display
$53$ \( T + 501890 \) Copy content Toggle raw display
$59$ \( T - 1586176 \) Copy content Toggle raw display
$61$ \( T + 372962 \) Copy content Toggle raw display
$67$ \( T - 4561044 \) Copy content Toggle raw display
$71$ \( T + 1512832 \) Copy content Toggle raw display
$73$ \( T + 1522910 \) Copy content Toggle raw display
$79$ \( T - 4231920 \) Copy content Toggle raw display
$83$ \( T - 1854204 \) Copy content Toggle raw display
$89$ \( T - 6888174 \) Copy content Toggle raw display
$97$ \( T - 3700034 \) Copy content Toggle raw display
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