Properties

Label 137.5.d.a
Level $137$
Weight $5$
Character orbit 137.d
Analytic conductor $14.162$
Analytic rank $0$
Dimension $180$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [137,5,Mod(10,137)] 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("137.10"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(137, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 137 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 137.d (of order \(8\), degree \(4\), 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: \(14.1616849425\)
Analytic rank: \(0\)
Dimension: \(180\)
Relative dimension: \(45\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 180 q - 16 q^{2} - 64 q^{5} - 68 q^{6} - 4 q^{7} + 72 q^{8} - 200 q^{9} + 388 q^{10} + 56 q^{11} + 320 q^{12} + 356 q^{13} - 9520 q^{16} + 272 q^{17} + 812 q^{19} + 3324 q^{20} + 1016 q^{21} + 1928 q^{23}+ \cdots - 9068 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −5.58834 − 5.58834i −1.45898 + 3.52228i 46.4592i −22.1588 + 9.17849i 27.8370 − 11.5304i −20.6874 − 20.6874i 170.216 − 170.216i 46.9978 + 46.9978i 175.124 + 72.5386i
10.2 −5.16993 − 5.16993i 6.49471 − 15.6796i 37.4563i −21.1621 + 8.76563i −114.640 + 47.4853i −43.8992 − 43.8992i 110.928 − 110.928i −146.394 − 146.394i 154.724 + 64.0888i
10.3 −5.08437 − 5.08437i 0.923031 − 2.22839i 35.7016i 20.9645 − 8.68377i −16.0230 + 6.63695i −9.61738 − 9.61738i 100.170 − 100.170i 53.1619 + 53.1619i −150.743 − 62.4397i
10.4 −4.97208 − 4.97208i −4.75549 + 11.4808i 33.4431i 20.3645 − 8.43526i 80.7279 − 33.4386i 58.1347 + 58.1347i 86.7284 − 86.7284i −51.9177 − 51.9177i −143.195 − 59.3132i
10.5 −4.85784 − 4.85784i 4.35516 − 10.5143i 31.1972i 34.6127 − 14.3371i −72.2334 + 29.9200i 22.0561 + 22.0561i 73.8258 − 73.8258i −34.3070 − 34.3070i −237.790 − 98.4960i
10.6 −4.57797 − 4.57797i −6.23911 + 15.0626i 25.9155i −6.65935 + 2.75839i 97.5183 − 40.3934i −54.5484 − 54.5484i 45.3930 − 45.3930i −130.678 − 130.678i 43.1141 + 17.8585i
10.7 −4.33213 − 4.33213i 0.205908 − 0.497106i 21.5348i −7.08866 + 2.93622i −3.04555 + 1.26151i −3.93314 − 3.93314i 23.9773 − 23.9773i 57.0709 + 57.0709i 43.4291 + 17.9889i
10.8 −4.30796 − 4.30796i 3.22032 − 7.77454i 21.1170i −32.9153 + 13.6339i −47.3654 + 19.6194i 58.9245 + 58.9245i 22.0439 − 22.0439i 7.20270 + 7.20270i 200.532 + 83.0631i
10.9 −4.05814 − 4.05814i −3.62249 + 8.74546i 16.9371i −40.5584 + 16.7999i 50.1909 − 20.7898i 30.9564 + 30.9564i 3.80276 − 3.80276i −6.08496 − 6.08496i 232.768 + 96.4158i
10.10 −3.81358 − 3.81358i −2.81004 + 6.78404i 13.0868i 37.4500 − 15.5123i 36.5878 − 15.1552i −45.6917 − 45.6917i −11.1096 + 11.1096i 19.1488 + 19.1488i −201.976 − 83.6613i
10.11 −3.76253 − 3.76253i 4.29771 − 10.3756i 12.3133i −7.01674 + 2.90643i −55.2088 + 22.8682i 5.59073 + 5.59073i −13.8714 + 13.8714i −31.9070 − 31.9070i 37.3362 + 15.4652i
10.12 −2.86511 − 2.86511i 2.12679 − 5.13454i 0.417748i −29.0005 + 12.0124i −20.8045 + 8.61752i −61.6280 − 61.6280i −44.6449 + 44.6449i 35.4354 + 35.4354i 117.507 + 48.6728i
10.13 −2.80492 − 2.80492i −2.46248 + 5.94494i − 0.264873i 7.92797 − 3.28387i 23.5821 − 9.76803i 5.69452 + 5.69452i −45.6216 + 45.6216i 27.9971 + 27.9971i −31.4483 − 13.0263i
10.14 −2.65345 − 2.65345i −5.06829 + 12.2359i − 1.91843i −0.905630 + 0.375124i 45.9159 − 19.0190i 21.8990 + 21.8990i −47.5456 + 47.5456i −66.7551 − 66.7551i 3.39841 + 1.40767i
10.15 −2.30255 − 2.30255i 0.230755 − 0.557093i − 5.39655i 24.2289 − 10.0359i −1.81406 + 0.751407i 45.2355 + 45.2355i −49.2666 + 49.2666i 57.0185 + 57.0185i −78.8964 − 32.6800i
10.16 −2.29081 − 2.29081i 6.06532 − 14.6430i − 5.50435i 15.2121 − 6.30107i −47.4389 + 19.6498i 33.6295 + 33.6295i −49.2624 + 49.2624i −120.353 − 120.353i −49.2827 − 20.4136i
10.17 −2.25253 − 2.25253i 3.96951 − 9.58324i − 5.85225i 25.2969 − 10.4783i −30.5279 + 12.6451i −47.0995 − 47.0995i −49.2228 + 49.2228i −18.8058 − 18.8058i −80.5845 − 33.3792i
10.18 −2.05670 − 2.05670i −2.98797 + 7.21360i − 7.54001i −34.4924 + 14.2872i 20.9815 − 8.69084i −22.0575 − 22.0575i −48.4146 + 48.4146i 14.1676 + 14.1676i 100.325 + 41.5560i
10.19 −1.02248 − 1.02248i 0.901553 − 2.17654i − 13.9091i −7.49576 + 3.10484i −3.14729 + 1.30365i 37.4357 + 37.4357i −30.5814 + 30.5814i 53.3511 + 53.3511i 10.8389 + 4.48962i
10.20 −0.870053 − 0.870053i −5.61995 + 13.5677i − 14.4860i 41.4638 − 17.1748i 16.6943 − 6.91501i 5.57316 + 5.57316i −26.5244 + 26.5244i −95.2244 − 95.2244i −51.0187 − 21.1326i
See next 80 embeddings (of 180 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 10.45
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
137.d odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 137.5.d.a ✓ 180
137.d odd 8 1 inner 137.5.d.a ✓ 180
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
137.5.d.a ✓ 180 1.a even 1 1 trivial
137.5.d.a ✓ 180 137.d odd 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(137, [\chi])\).