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Properties

Label 30345.2.a.bm
Level $30345$
Weight $2$
Character orbit 30345.a
Self dual yes
Analytic conductor $242.306$
Dimension $2$

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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] = [30345,2,Mod(1,30345)] 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("30345.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(30345, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 30345 = 3 \cdot 5 \cdot 7 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 30345.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,2,2,2,-2,-2,-6,2,-2,4,2,4,2,2,6,0,-2,8,2,-2,0,8,-6,2,4, 2,-2,12,-2,-20,6,4,0,-2,2,-4,-12,4,-6,0,2,-12,-4,2,-20,-16,6,2,-2,0,-12, -16,-2,4,6,8,-12,0,2,0,20,-2,-14,4,0,-12,0,8,2,-4,-6,-12,-8,2,16,-4,4, -8,6,2,-8,-16,-2,0,20,12,-8,20,-2,-4,32,-20,16,8,6,-12,-2,4,2] 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: \(242.306044934\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Twist minimal: not computed
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 6 q^{8} + 2 q^{9} - 2 q^{10} + 4 q^{11} + 2 q^{12} + 4 q^{13} + 2 q^{14} + 2 q^{15} + 6 q^{16} - 2 q^{18} + 8 q^{19} + 2 q^{20} - 2 q^{21}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( -1 \)
\(7\) \( +1 \)
\(17\) \( +1 \)

Inner twists

Inner twists of this newform have not been computed.

Twists

Twists of this newform have not been computed.