Properties

Label 137.4.e.a
Level $137$
Weight $4$
Character orbit 137.e
Analytic conductor $8.083$
Analytic rank $0$
Dimension $544$
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(16,137)] 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("137.16"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(137, base_ring=CyclotomicField(34)) chi = DirichletCharacter(H, H._module([10])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 137 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 137.e (of order \(17\), degree \(16\), 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: \(544\)
Relative dimension: \(34\) over \(\Q(\zeta_{17})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{17}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 544 q - 17 q^{2} - 17 q^{3} - 51 q^{4} - 7 q^{5} + 33 q^{6} - 59 q^{7} + 19 q^{8} - 319 q^{9} - 32 q^{10} + 39 q^{11} - 213 q^{12} + 69 q^{13} + 15 q^{14} - 117 q^{15} - 1627 q^{16} + 377 q^{17} + 917 q^{18}+ \cdots + 2609 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −5.44354 + 1.01757i −2.08930 − 2.76668i 21.1369 − 8.18849i −11.4651 − 2.14320i 14.1885 + 12.9345i −14.1547 − 28.4264i −69.0606 + 42.7605i 4.09956 − 14.4084i 64.5918
16.2 −5.08975 + 0.951439i 0.350398 + 0.464001i 17.5405 − 6.79524i 15.0161 + 2.80700i −2.22491 − 2.02827i 14.1377 + 28.3923i −47.5929 + 29.4683i 7.29638 − 25.6441i −79.0991
16.3 −5.03636 + 0.941459i 4.90209 + 6.49142i 17.0188 − 6.59312i −0.297592 − 0.0556296i −30.8001 − 28.0780i −2.30867 − 4.63644i −44.6564 + 27.6501i −10.7191 + 37.6737i 1.55115
16.4 −4.64285 + 0.867899i −5.00619 − 6.62927i 13.3430 − 5.16912i 15.9338 + 2.97855i 28.9965 + 26.4338i −4.10321 − 8.24036i −25.3370 + 15.6880i −11.4964 + 40.4055i −76.5634
16.5 −4.58072 + 0.856284i −0.238522 − 0.315854i 12.7900 − 4.95486i −7.69603 − 1.43864i 1.36306 + 1.24260i 7.92384 + 15.9132i −22.6479 + 14.0230i 7.34603 − 25.8186i 36.4852
16.6 −4.19842 + 0.784820i −6.15159 − 8.14602i 9.55100 − 3.70008i −16.2441 − 3.03655i 32.2201 + 29.3725i 13.9972 + 28.1103i −8.14401 + 5.04256i −21.1267 + 74.2526i 70.5828
16.7 −3.73693 + 0.698553i 2.41298 + 3.19530i 6.01688 − 2.33095i 14.4564 + 2.70237i −11.2492 − 10.2550i −11.2438 − 22.5806i 5.00153 − 3.09682i 3.00143 − 10.5489i −55.9102
16.8 −3.66664 + 0.685414i −1.39947 − 1.85319i 5.51467 − 2.13639i −4.18441 − 0.782202i 6.40154 + 5.83577i −0.356994 − 0.716940i 6.61552 − 4.09616i 5.91308 − 20.7823i 15.8789
16.9 −3.49254 + 0.652868i 3.22380 + 4.26900i 4.31179 − 1.67040i −15.7150 − 2.93765i −14.0463 − 12.8049i −2.60271 − 5.22695i 10.1983 − 6.31449i −0.442592 + 1.55555i 56.8032
16.10 −2.98531 + 0.558052i 5.47433 + 7.24918i 1.14089 − 0.441984i 6.99172 + 1.30698i −20.3880 − 18.5861i 10.5771 + 21.2417i 17.4978 − 10.8342i −15.1935 + 53.3995i −21.6018
16.11 −2.85644 + 0.533961i −3.19734 − 4.23397i 0.414364 − 0.160526i 5.84318 + 1.09228i 11.3938 + 10.3868i −3.07513 − 6.17569i 18.6674 − 11.5584i −0.314563 + 1.10558i −17.2739
16.12 −1.78477 + 0.333632i −4.20790 − 5.57217i −4.38567 + 1.69902i −14.0262 − 2.62195i 9.36920 + 8.54116i −12.9047 − 25.9162i 19.6104 − 12.1423i −5.95367 + 20.9250i 25.9083
16.13 −1.51419 + 0.283052i 1.58768 + 2.10242i −5.24711 + 2.03274i 13.6921 + 2.55951i −2.99915 − 2.73408i 6.28265 + 12.6173i 17.8473 − 11.0506i 5.48943 − 19.2933i −21.4570
16.14 −1.44560 + 0.270230i 2.38252 + 3.15496i −5.44303 + 2.10864i −11.4014 − 2.13129i −4.29674 − 3.91700i 12.8644 + 25.8352i 17.3016 − 10.7127i 3.11150 − 10.9358i 17.0579
16.15 −1.03223 + 0.192956i −4.28839 − 5.67875i −6.43152 + 2.49158i 11.6679 + 2.18110i 5.52234 + 5.03428i 9.18770 + 18.4514i 13.3006 − 8.23536i −6.46897 + 22.7361i −12.4647
16.16 −0.704735 + 0.131738i 1.64134 + 2.17349i −6.98048 + 2.70425i 4.52717 + 0.846275i −1.44304 − 1.31551i −12.7786 − 25.6629i 9.43959 − 5.84475i 5.35885 − 18.8344i −3.30194
16.17 −0.488805 + 0.0913735i 5.41372 + 7.16892i −7.22920 + 2.80061i −7.46078 − 1.39466i −3.30130 − 3.00953i −8.95977 − 17.9936i 6.66008 − 4.12375i −14.6962 + 51.6517i 3.77430
16.18 −0.410351 + 0.0767078i −2.48305 − 3.28809i −7.29727 + 2.82698i −16.0067 − 2.99218i 1.27114 + 1.15880i 4.27155 + 8.57843i 5.61703 − 3.47792i 2.74291 − 9.64034i 6.79790
16.19 0.505711 − 0.0945337i 0.668551 + 0.885304i −7.21297 + 2.79432i −3.89334 − 0.727791i 0.421785 + 0.384507i −0.467244 − 0.938353i −6.88281 + 4.26166i 7.05210 − 24.7856i −2.03771
16.20 1.30259 − 0.243497i −5.97457 − 7.91161i −5.82232 + 2.25558i 2.89331 + 0.540854i −9.70888 − 8.85081i −7.27677 − 14.6137i −16.0482 + 9.93665i −19.5092 + 68.5677i 3.90050
See next 80 embeddings (of 544 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 16.34
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
137.e even 17 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 137.4.e.a ✓ 544
137.e even 17 1 inner 137.4.e.a ✓ 544
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
137.4.e.a ✓ 544 1.a even 1 1 trivial
137.4.e.a ✓ 544 137.e even 17 1 inner

Hecke kernels

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