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Properties

Label 39710.2.a.e
Level $39710$
Weight $2$
Character orbit 39710.a
Self dual yes
Analytic conductor $317.086$
Dimension $1$

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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] = [39710,2,Mod(1,39710)] 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("39710.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(39710, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 39710 = 2 \cdot 5 \cdot 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 39710.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,0,1,-1,0,-2,-1,-3,1,-1,0,-3,2,0,1,0,3,0,-1,0,1,0,0,1,3, 0,-2,1,0,-4,-1,0,0,2,-3,0,0,0,1,-12,0,1,-1,3,0,-1,0,-3,-1,0,-3,0,0,1,2, 0,-1,12,0,-11,4,6,1,3,0,14,0,0,-2,7,3,-12,0,0,0,2,0,-10,-1,9,12,3,0,0, -1,0,1,9,-3,6,0,0,1,0,0,-17,3,3,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(None)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(317.085946427\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: not computed
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)
 
\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} - 2 q^{7} - q^{8} - 3 q^{9} + q^{10} - q^{11} - 3 q^{13} + 2 q^{14} + q^{16} + 3 q^{18} - q^{20} + q^{22} + q^{25} + 3 q^{26} - 2 q^{28} + q^{29} - 4 q^{31} - q^{32}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(11\) \( +1 \)
\(19\) \( -1 \)

Inner twists

Inner twists of this newform have not been computed.

Twists

Twists of this newform have not been computed.