Properties

Label 137.4.c.a
Level $137$
Weight $4$
Character orbit 137.c
Analytic conductor $8.083$
Analytic rank $0$
Dimension $66$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [137,4,Mod(37,137)] 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("137.37"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(137, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 137 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 137.c (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: \(8.08326167079\)
Analytic rank: \(0\)
Dimension: \(66\)
Relative dimension: \(33\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 66 q - 272 q^{4} - 4 q^{5} + 6 q^{6} - 174 q^{10} - 104 q^{12} - 56 q^{13} + 140 q^{14} + 276 q^{15} + 1616 q^{16} + 136 q^{18} - 18 q^{20} - 12 q^{21} + 1124 q^{22} + 170 q^{23} + 132 q^{24} + 68 q^{26}+ \cdots + 11068 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 − 5.41173i −0.127460 + 0.127460i −21.2868 12.9936 + 12.9936i 0.689776 + 0.689776i 2.13081i 71.9045i 26.9675i 70.3177 − 70.3177i
37.2 − 5.37457i −4.87243 + 4.87243i −20.8860 −11.1754 − 11.1754i 26.1872 + 26.1872i − 16.9880i 69.2570i − 20.4811i −60.0629 + 60.0629i
37.3 − 5.11342i 7.27241 − 7.27241i −18.1471 −3.16138 − 3.16138i −37.1869 − 37.1869i 21.3809i 51.8865i − 78.7759i −16.1655 + 16.1655i
37.4 − 4.72173i −0.383754 + 0.383754i −14.2948 −8.95469 − 8.95469i 1.81198 + 1.81198i 30.1134i 29.7223i 26.7055i −42.2816 + 42.2816i
37.5 − 4.52282i 3.03224 − 3.03224i −12.4559 −1.60654 − 1.60654i −13.7143 − 13.7143i − 24.8720i 20.1532i 8.61100i −7.26611 + 7.26611i
37.6 − 4.16498i −4.74102 + 4.74102i −9.34708 5.85868 + 5.85868i 19.7463 + 19.7463i − 4.01344i 5.61056i − 17.9545i 24.4013 − 24.4013i
37.7 − 3.30918i −0.495203 + 0.495203i −2.95070 4.52750 + 4.52750i 1.63872 + 1.63872i 26.4125i − 16.7091i 26.5095i 14.9823 − 14.9823i
37.8 − 3.18907i 2.81798 − 2.81798i −2.17016 −13.3712 − 13.3712i −8.98672 − 8.98672i − 6.93118i − 18.5918i 11.1180i −42.6418 + 42.6418i
37.9 − 3.13354i 4.78908 − 4.78908i −1.81908 10.0003 + 10.0003i −15.0068 − 15.0068i 2.57943i − 19.3682i − 18.8705i 31.3362 − 31.3362i
37.10 − 3.03423i −4.63612 + 4.63612i −1.20656 −2.73835 − 2.73835i 14.0671 + 14.0671i − 12.0978i − 20.6129i − 15.9873i −8.30877 + 8.30877i
37.11 − 1.81460i −6.57149 + 6.57149i 4.70724 −10.8258 − 10.8258i 11.9246 + 11.9246i 31.1189i − 23.0585i − 59.3688i −19.6445 + 19.6445i
37.12 − 1.50913i −0.901085 + 0.901085i 5.72252 −5.90463 − 5.90463i 1.35986 + 1.35986i − 9.94089i − 20.7091i 25.3761i −8.91087 + 8.91087i
37.13 − 1.19333i 5.67681 − 5.67681i 6.57597 −8.25881 − 8.25881i −6.77431 − 6.77431i 10.5067i − 17.3939i − 37.4524i −9.85547 + 9.85547i
37.14 − 1.06484i −0.527295 + 0.527295i 6.86612 4.75373 + 4.75373i 0.561483 + 0.561483i − 29.2079i − 15.8300i 26.4439i 5.06195 − 5.06195i
37.15 − 0.785234i 0.837673 − 0.837673i 7.38341 9.19025 + 9.19025i −0.657769 − 0.657769i 16.0566i − 12.0796i 25.5966i 7.21650 − 7.21650i
37.16 − 0.763466i −5.76139 + 5.76139i 7.41712 13.4611 + 13.4611i 4.39863 + 4.39863i 4.18111i − 11.7704i − 39.3873i 10.2771 − 10.2771i
37.17 − 0.281875i 6.11344 − 6.11344i 7.92055 4.41128 + 4.41128i −1.72323 − 1.72323i − 30.6988i − 4.48761i − 47.7483i 1.24343 − 1.24343i
37.18 0.313646i −1.80329 + 1.80329i 7.90163 −6.04842 − 6.04842i −0.565595 − 0.565595i 17.4049i 4.98748i 20.4963i 1.89706 − 1.89706i
37.19 1.25013i −6.12174 + 6.12174i 6.43717 −5.62020 − 5.62020i −7.65299 − 7.65299i − 29.5241i 18.0484i − 47.9515i 7.02599 − 7.02599i
37.20 1.29879i 4.09959 − 4.09959i 6.31315 1.69081 + 1.69081i 5.32449 + 5.32449i 23.8036i 18.5898i − 6.61319i −2.19601 + 2.19601i
See all 66 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 37.33
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
137.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 137.4.c.a ✓ 66
137.c even 4 1 inner 137.4.c.a ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
137.4.c.a ✓ 66 1.a even 1 1 trivial
137.4.c.a ✓ 66 137.c even 4 1 inner

Hecke kernels

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