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Properties

Label 234.6.x.a
Level $234$
Weight $6$
Character orbit 234.x
Analytic conductor $37.530$
Analytic rank $0$
Dimension $40$
Inner twists $4$

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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] = [234,6,Mod(71,234)] 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("234.71"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(234, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 5])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 234.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(37.5298138362\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 1016 q^{7} - 288 q^{10} - 348 q^{13} + 5120 q^{16} - 9232 q^{19} - 1344 q^{22} - 16256 q^{28} + 11296 q^{31} + 11088 q^{34} + 9148 q^{37} + 7680 q^{40} - 84504 q^{43} + 20160 q^{46} - 3000 q^{49}+ \cdots + 225676 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1 −3.86370 + 1.03528i 0 13.8564 − 8.00000i −58.2528 − 58.2528i 0 −16.8114 + 62.7411i −45.2548 + 45.2548i 0 285.379 + 164.764i
71.2 −3.86370 + 1.03528i 0 13.8564 − 8.00000i 7.19815 + 7.19815i 0 −30.0642 + 112.201i −45.2548 + 45.2548i 0 −35.2636 − 20.3594i
71.3 −3.86370 + 1.03528i 0 13.8564 − 8.00000i 17.6983 + 17.6983i 0 −2.91811 + 10.8905i −45.2548 + 45.2548i 0 −86.7036 − 50.0583i
71.4 −3.86370 + 1.03528i 0 13.8564 − 8.00000i −38.0836 − 38.0836i 0 49.7363 − 185.618i −45.2548 + 45.2548i 0 186.571 + 107.717i
71.5 −3.86370 + 1.03528i 0 13.8564 − 8.00000i 68.1819 + 68.1819i 0 54.3113 − 202.693i −45.2548 + 45.2548i 0 −334.022 − 192.847i
71.6 3.86370 − 1.03528i 0 13.8564 − 8.00000i −68.1819 − 68.1819i 0 54.3113 − 202.693i 45.2548 − 45.2548i 0 −334.022 − 192.847i
71.7 3.86370 − 1.03528i 0 13.8564 − 8.00000i 38.0836 + 38.0836i 0 49.7363 − 185.618i 45.2548 − 45.2548i 0 186.571 + 107.717i
71.8 3.86370 − 1.03528i 0 13.8564 − 8.00000i −17.6983 − 17.6983i 0 −2.91811 + 10.8905i 45.2548 − 45.2548i 0 −86.7036 − 50.0583i
71.9 3.86370 − 1.03528i 0 13.8564 − 8.00000i −7.19815 − 7.19815i 0 −30.0642 + 112.201i 45.2548 − 45.2548i 0 −35.2636 − 20.3594i
71.10 3.86370 − 1.03528i 0 13.8564 − 8.00000i 58.2528 + 58.2528i 0 −16.8114 + 62.7411i 45.2548 − 45.2548i 0 285.379 + 164.764i
89.1 −3.86370 − 1.03528i 0 13.8564 + 8.00000i −58.2528 + 58.2528i 0 −16.8114 − 62.7411i −45.2548 − 45.2548i 0 285.379 − 164.764i
89.2 −3.86370 − 1.03528i 0 13.8564 + 8.00000i 7.19815 − 7.19815i 0 −30.0642 − 112.201i −45.2548 − 45.2548i 0 −35.2636 + 20.3594i
89.3 −3.86370 − 1.03528i 0 13.8564 + 8.00000i 17.6983 − 17.6983i 0 −2.91811 − 10.8905i −45.2548 − 45.2548i 0 −86.7036 + 50.0583i
89.4 −3.86370 − 1.03528i 0 13.8564 + 8.00000i −38.0836 + 38.0836i 0 49.7363 + 185.618i −45.2548 − 45.2548i 0 186.571 − 107.717i
89.5 −3.86370 − 1.03528i 0 13.8564 + 8.00000i 68.1819 − 68.1819i 0 54.3113 + 202.693i −45.2548 − 45.2548i 0 −334.022 + 192.847i
89.6 3.86370 + 1.03528i 0 13.8564 + 8.00000i −68.1819 + 68.1819i 0 54.3113 + 202.693i 45.2548 + 45.2548i 0 −334.022 + 192.847i
89.7 3.86370 + 1.03528i 0 13.8564 + 8.00000i 38.0836 − 38.0836i 0 49.7363 + 185.618i 45.2548 + 45.2548i 0 186.571 − 107.717i
89.8 3.86370 + 1.03528i 0 13.8564 + 8.00000i −17.6983 + 17.6983i 0 −2.91811 − 10.8905i 45.2548 + 45.2548i 0 −86.7036 + 50.0583i
89.9 3.86370 + 1.03528i 0 13.8564 + 8.00000i −7.19815 + 7.19815i 0 −30.0642 − 112.201i 45.2548 + 45.2548i 0 −35.2636 + 20.3594i
89.10 3.86370 + 1.03528i 0 13.8564 + 8.00000i 58.2528 − 58.2528i 0 −16.8114 − 62.7411i 45.2548 + 45.2548i 0 285.379 − 164.764i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.f odd 12 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 234.6.x.a ✓ 40
3.b odd 2 1 inner 234.6.x.a ✓ 40
13.f odd 12 1 inner 234.6.x.a ✓ 40
39.k even 12 1 inner 234.6.x.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
234.6.x.a ✓ 40 1.a even 1 1 trivial
234.6.x.a ✓ 40 3.b odd 2 1 inner
234.6.x.a ✓ 40 13.f odd 12 1 inner
234.6.x.a ✓ 40 39.k even 12 1 inner