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Properties

Label 320.8.a.h
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 = [1,0,48,0,-125,0,-1644,0,117] 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 5)
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)
 
\(f(q)\) \(=\) \( q + 48 q^{3} - 125 q^{5} - 1644 q^{7} + 117 q^{9} - 172 q^{11} - 3862 q^{13} - 6000 q^{15} - 12254 q^{17} + 25940 q^{19} - 78912 q^{21} + 12972 q^{23} + 15625 q^{25} - 99360 q^{27} + 81610 q^{29} - 156888 q^{31}+ \cdots - 20124 q^{99}+O(q^{100}) \) 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 48.0000 0 −125.000 0 −1644.00 0 117.000 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.h 1
4.b odd 2 1 320.8.a.a 1
8.b even 2 1 5.8.a.a ✓ 1
8.d odd 2 1 80.8.a.d 1
24.h odd 2 1 45.8.a.f 1
40.e odd 2 1 400.8.a.e 1
40.f even 2 1 25.8.a.a 1
40.i odd 4 2 25.8.b.a 2
40.k even 4 2 400.8.c.e 2
56.h odd 2 1 245.8.a.a 1
88.b odd 2 1 605.8.a.c 1
120.i odd 2 1 225.8.a.b 1
120.w even 4 2 225.8.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.8.a.a ✓ 1 8.b even 2 1
25.8.a.a 1 40.f even 2 1
25.8.b.a 2 40.i odd 4 2
45.8.a.f 1 24.h odd 2 1
80.8.a.d 1 8.d odd 2 1
225.8.a.b 1 120.i odd 2 1
225.8.b.b 2 120.w even 4 2
245.8.a.a 1 56.h odd 2 1
320.8.a.a 1 4.b odd 2 1
320.8.a.h 1 1.a even 1 1 trivial
400.8.a.e 1 40.e odd 2 1
400.8.c.e 2 40.k even 4 2
605.8.a.c 1 88.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 48 \) 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
$3$ \( T - 48 \) Copy content Toggle raw display
$5$ \( T + 125 \) Copy content Toggle raw display
$7$ \( T + 1644 \) Copy content Toggle raw display
$11$ \( T + 172 \) Copy content Toggle raw display
$13$ \( T + 3862 \) Copy content Toggle raw display
$17$ \( T + 12254 \) Copy content Toggle raw display
$19$ \( T - 25940 \) Copy content Toggle raw display
$23$ \( T - 12972 \) Copy content Toggle raw display
$29$ \( T - 81610 \) Copy content Toggle raw display
$31$ \( T + 156888 \) Copy content Toggle raw display
$37$ \( T + 110126 \) Copy content Toggle raw display
$41$ \( T - 467882 \) Copy content Toggle raw display
$43$ \( T - 499208 \) Copy content Toggle raw display
$47$ \( T + 396884 \) Copy content Toggle raw display
$53$ \( T - 1280498 \) Copy content Toggle raw display
$59$ \( T - 1337420 \) Copy content Toggle raw display
$61$ \( T - 923978 \) Copy content Toggle raw display
$67$ \( T - 797304 \) Copy content Toggle raw display
$71$ \( T - 5103392 \) Copy content Toggle raw display
$73$ \( T + 4267478 \) Copy content Toggle raw display
$79$ \( T + 960 \) Copy content Toggle raw display
$83$ \( T + 6140832 \) Copy content Toggle raw display
$89$ \( T - 2010570 \) Copy content Toggle raw display
$97$ \( T + 4881934 \) Copy content Toggle raw display
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