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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [137,4,Mod(136,137)] 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("137.136"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(137, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 137 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 137.b (of order \(2\), degree \(1\), 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: \(8.08326167079\)
Analytic rank: \(0\)
Dimension: \(34\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 34 q - 4 q^{2} + 112 q^{4} - 12 q^{7} - 84 q^{8} - 310 q^{9} - 4 q^{11} + 108 q^{14} + 176 q^{15} + 440 q^{16} - 340 q^{17} + 54 q^{18} - 12 q^{19} + 246 q^{22} - 1042 q^{25} - 210 q^{28} - 278 q^{30} - 1446 q^{32}+ \cdots - 3528 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1 −5.55017 − 6.15636i 22.8043 20.0731i 34.1688i 12.8574 −82.1665 −10.9008 − 111.409i
136.2 −5.55017 6.15636i 22.8043 − 20.0731i − 34.1688i 12.8574 −82.1665 −10.9008 111.409i
136.3 −4.83022 − 6.94533i 15.3310 − 10.1810i 33.5475i −36.5490 −35.4105 −21.2376 49.1766i
136.4 −4.83022 6.94533i 15.3310 10.1810i − 33.5475i −36.5490 −35.4105 −21.2376 − 49.1766i
136.5 −4.47866 − 3.09664i 12.0584 − 6.63880i 13.8688i 7.36915 −18.1760 17.4108 29.7329i
136.6 −4.47866 3.09664i 12.0584 6.63880i − 13.8688i 7.36915 −18.1760 17.4108 − 29.7329i
136.7 −2.90497 − 9.39232i 0.438858 6.56266i 27.2844i −4.11875 21.9649 −61.2157 − 19.0643i
136.8 −2.90497 9.39232i 0.438858 − 6.56266i − 27.2844i −4.11875 21.9649 −61.2157 19.0643i
136.9 −2.87687 − 6.69269i 0.276400 4.69070i 19.2540i 15.8564 22.2198 −17.7921 − 13.4946i
136.10 −2.87687 6.69269i 0.276400 − 4.69070i − 19.2540i 15.8564 22.2198 −17.7921 13.4946i
136.11 −2.46973 − 0.859292i −1.90043 16.0037i 2.12222i −24.0872 24.4514 26.2616 − 39.5250i
136.12 −2.46973 0.859292i −1.90043 − 16.0037i − 2.12222i −24.0872 24.4514 26.2616 39.5250i
136.13 −2.17714 − 3.75406i −3.26007 − 17.8233i 8.17312i 29.2462 24.5147 12.9070 38.8039i
136.14 −2.17714 3.75406i −3.26007 17.8233i − 8.17312i 29.2462 24.5147 12.9070 − 38.8039i
136.15 −0.566556 − 3.07957i −7.67901 4.63306i 1.74475i −13.1810 8.88304 17.5163 − 2.62489i
136.16 −0.566556 3.07957i −7.67901 − 4.63306i − 1.74475i −13.1810 8.88304 17.5163 2.62489i
136.17 −0.135063 − 8.95075i −7.98176 − 17.8985i 1.20892i −6.18769 2.15854 −53.1160 2.41743i
136.18 −0.135063 8.95075i −7.98176 17.8985i − 1.20892i −6.18769 2.15854 −53.1160 − 2.41743i
136.19 0.580738 − 4.32571i −7.66274 12.7900i − 2.51211i 24.3338 −9.09596 8.28824 7.42767i
136.20 0.580738 4.32571i −7.66274 − 12.7900i 2.51211i 24.3338 −9.09596 8.28824 − 7.42767i
See all 34 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 136.34
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
137.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 137.4.b.a ✓ 34
137.b even 2 1 inner 137.4.b.a ✓ 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
137.4.b.a ✓ 34 1.a even 1 1 trivial
137.4.b.a ✓ 34 137.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(137, [\chi])\).