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Properties

Label 110.4.f.a
Level $110$
Weight $4$
Character orbit 110.f
Analytic conductor $6.490$
Analytic rank $0$
Dimension $36$
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] = [110,4,Mod(43,110)] 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("110.43"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(110, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 110 = 2 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 110.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.49021010063\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 8 q^{3} - 32 q^{5} + 44 q^{11} + 32 q^{12} - 164 q^{15} - 576 q^{16} + 64 q^{20} - 312 q^{22} - 176 q^{23} + 880 q^{25} - 304 q^{26} + 364 q^{27} + 432 q^{31} + 832 q^{33} - 1808 q^{36} + 960 q^{37}+ \cdots - 4224 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.41421 − 1.41421i −6.61133 − 6.61133i 4.00000i −10.1738 − 4.63605i 18.6997i −17.6781 − 17.6781i 5.65685 − 5.65685i 60.4193i 7.83162 + 20.9443i
43.2 −1.41421 − 1.41421i −6.31079 − 6.31079i 4.00000i 9.13321 + 6.44860i 17.8496i 19.5854 + 19.5854i 5.65685 − 5.65685i 52.6522i −3.79660 − 22.0360i
43.3 −1.41421 − 1.41421i −2.46018 − 2.46018i 4.00000i 10.8948 − 2.51076i 6.95843i −4.79287 − 4.79287i 5.65685 − 5.65685i − 14.8951i −18.9583 − 11.8568i
43.4 −1.41421 − 1.41421i −1.88627 − 1.88627i 4.00000i −2.58046 − 10.8785i 5.33517i 3.12870 + 3.12870i 5.65685 − 5.65685i − 19.8840i −11.7352 + 19.0338i
43.5 −1.41421 − 1.41421i −1.04107 − 1.04107i 4.00000i −11.1552 + 0.749351i 2.94459i 18.3452 + 18.3452i 5.65685 − 5.65685i − 24.8323i 16.8356 + 14.7161i
43.6 −1.41421 − 1.41421i 0.813628 + 0.813628i 4.00000i −4.32381 + 10.3104i − 2.30129i −10.4760 − 10.4760i 5.65685 − 5.65685i − 25.6760i 20.6959 − 8.46633i
43.7 −1.41421 − 1.41421i 4.30697 + 4.30697i 4.00000i 1.51643 − 11.0770i − 12.1819i −17.0999 − 17.0999i 5.65685 − 5.65685i 10.1000i −17.8098 + 13.5207i
43.8 −1.41421 − 1.41421i 4.60438 + 4.60438i 4.00000i 9.68683 + 5.58258i − 13.0231i 0.880216 + 0.880216i 5.65685 − 5.65685i 15.4006i −5.80428 − 21.5942i
43.9 −1.41421 − 1.41421i 6.58466 + 6.58466i 4.00000i −10.9979 + 2.01136i − 18.6242i 9.52166 + 9.52166i 5.65685 − 5.65685i 59.7154i 18.3979 + 12.7089i
43.10 1.41421 + 1.41421i −6.61133 − 6.61133i 4.00000i −10.1738 − 4.63605i − 18.6997i 17.6781 + 17.6781i −5.65685 + 5.65685i 60.4193i −7.83162 − 20.9443i
43.11 1.41421 + 1.41421i −6.31079 − 6.31079i 4.00000i 9.13321 + 6.44860i − 17.8496i −19.5854 − 19.5854i −5.65685 + 5.65685i 52.6522i 3.79660 + 22.0360i
43.12 1.41421 + 1.41421i −2.46018 − 2.46018i 4.00000i 10.8948 − 2.51076i − 6.95843i 4.79287 + 4.79287i −5.65685 + 5.65685i − 14.8951i 18.9583 + 11.8568i
43.13 1.41421 + 1.41421i −1.88627 − 1.88627i 4.00000i −2.58046 − 10.8785i − 5.33517i −3.12870 − 3.12870i −5.65685 + 5.65685i − 19.8840i 11.7352 − 19.0338i
43.14 1.41421 + 1.41421i −1.04107 − 1.04107i 4.00000i −11.1552 + 0.749351i − 2.94459i −18.3452 − 18.3452i −5.65685 + 5.65685i − 24.8323i −16.8356 − 14.7161i
43.15 1.41421 + 1.41421i 0.813628 + 0.813628i 4.00000i −4.32381 + 10.3104i 2.30129i 10.4760 + 10.4760i −5.65685 + 5.65685i − 25.6760i −20.6959 + 8.46633i
43.16 1.41421 + 1.41421i 4.30697 + 4.30697i 4.00000i 1.51643 − 11.0770i 12.1819i 17.0999 + 17.0999i −5.65685 + 5.65685i 10.1000i 17.8098 − 13.5207i
43.17 1.41421 + 1.41421i 4.60438 + 4.60438i 4.00000i 9.68683 + 5.58258i 13.0231i −0.880216 − 0.880216i −5.65685 + 5.65685i 15.4006i 5.80428 + 21.5942i
43.18 1.41421 + 1.41421i 6.58466 + 6.58466i 4.00000i −10.9979 + 2.01136i 18.6242i −9.52166 − 9.52166i −5.65685 + 5.65685i 59.7154i −18.3979 − 12.7089i
87.1 −1.41421 + 1.41421i −6.61133 + 6.61133i − 4.00000i −10.1738 + 4.63605i − 18.6997i −17.6781 + 17.6781i 5.65685 + 5.65685i − 60.4193i 7.83162 − 20.9443i
87.2 −1.41421 + 1.41421i −6.31079 + 6.31079i − 4.00000i 9.13321 − 6.44860i − 17.8496i 19.5854 − 19.5854i 5.65685 + 5.65685i − 52.6522i −3.79660 + 22.0360i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.b odd 2 1 inner
55.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 110.4.f.a ✓ 36
5.c odd 4 1 inner 110.4.f.a ✓ 36
11.b odd 2 1 inner 110.4.f.a ✓ 36
55.e even 4 1 inner 110.4.f.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.4.f.a ✓ 36 1.a even 1 1 trivial
110.4.f.a ✓ 36 5.c odd 4 1 inner
110.4.f.a ✓ 36 11.b odd 2 1 inner
110.4.f.a ✓ 36 55.e even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(110, [\chi])\).