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Properties

Label 320.8.a.g
Level $320$
Weight $8$
Character orbit 320.a
Self dual yes
Analytic conductor $99.963$
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] = [320,8,Mod(1,320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("320.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 320.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,36,0,-125,0,776,0,-891] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(99.9632081549\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
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)
 
\(f(q)\) \(=\) \( q + 36 q^{3} - 125 q^{5} + 776 q^{7} - 891 q^{9} + 124 q^{11} + 13082 q^{13} - 4500 q^{15} - 15950 q^{17} + 20516 q^{19} + 27936 q^{21} - 29224 q^{23} + 15625 q^{25} - 110808 q^{27} + 221482 q^{29} - 109760 q^{31}+ \cdots - 110484 q^{99}+O(q^{100}) \) 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 36.0000 0 −125.000 0 776.000 0 −891.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)

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 320.8.a.g 1
4.b odd 2 1 320.8.a.b 1
8.b even 2 1 40.8.a.a ✓ 1
8.d odd 2 1 80.8.a.c 1
24.h odd 2 1 360.8.a.b 1
40.e odd 2 1 400.8.a.g 1
40.f even 2 1 200.8.a.g 1
40.i odd 4 2 200.8.c.d 2
40.k even 4 2 400.8.c.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.8.a.a ✓ 1 8.b even 2 1
80.8.a.c 1 8.d odd 2 1
200.8.a.g 1 40.f even 2 1
200.8.c.d 2 40.i odd 4 2
320.8.a.b 1 4.b odd 2 1
320.8.a.g 1 1.a even 1 1 trivial
360.8.a.b 1 24.h odd 2 1
400.8.a.g 1 40.e odd 2 1
400.8.c.h 2 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 36 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(320))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 36 \) Copy content Toggle raw display
$5$ \( T + 125 \) Copy content Toggle raw display
$7$ \( T - 776 \) Copy content Toggle raw display
$11$ \( T - 124 \) Copy content Toggle raw display
$13$ \( T - 13082 \) Copy content Toggle raw display
$17$ \( T + 15950 \) Copy content Toggle raw display
$19$ \( T - 20516 \) Copy content Toggle raw display
$23$ \( T + 29224 \) Copy content Toggle raw display
$29$ \( T - 221482 \) Copy content Toggle raw display
$31$ \( T + 109760 \) Copy content Toggle raw display
$37$ \( T + 73422 \) Copy content Toggle raw display
$41$ \( T - 12762 \) Copy content Toggle raw display
$43$ \( T + 290548 \) Copy content Toggle raw display
$47$ \( T - 1269152 \) Copy content Toggle raw display
$53$ \( T - 395778 \) Copy content Toggle raw display
$59$ \( T + 421492 \) Copy content Toggle raw display
$61$ \( T - 2122250 \) Copy content Toggle raw display
$67$ \( T - 3132868 \) Copy content Toggle raw display
$71$ \( T + 5376552 \) Copy content Toggle raw display
$73$ \( T - 4985466 \) Copy content Toggle raw display
$79$ \( T - 3867504 \) Copy content Toggle raw display
$83$ \( T - 6190196 \) Copy content Toggle raw display
$89$ \( T - 1124394 \) Copy content Toggle raw display
$97$ \( T - 9968098 \) Copy content Toggle raw display
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