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Properties

Label 135.4.q.a
Level $135$
Weight $4$
Character orbit 135.q
Analytic conductor $7.965$
Analytic rank $0$
Dimension $624$
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] = [135,4,Mod(2,135)] 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("135.2"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(135, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([2, 9])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 135.q (of order \(36\), degree \(12\), 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: \(7.96525785077\)
Analytic rank: \(0\)
Dimension: \(624\)
Relative dimension: \(52\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 624 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 12 q^{7} - 18 q^{8} - 6 q^{10} - 12 q^{12} - 12 q^{13} - 12 q^{15} - 24 q^{16} - 18 q^{17} + 702 q^{18} + 756 q^{20} - 24 q^{21} - 12 q^{22} - 324 q^{23} + 420 q^{25}+ \cdots - 5832 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −3.16779 − 4.52407i −1.14475 + 5.06848i −7.69615 + 21.1450i 10.9280 + 2.36206i 26.5565 − 10.8769i −9.94919 − 4.63938i 77.3637 − 20.7295i −24.3791 − 11.6043i −23.9314 − 56.9214i
2.2 −2.96307 − 4.23171i −2.99584 − 4.24558i −6.39139 + 17.5602i −9.21366 + 6.33312i −9.08914 + 25.2575i −30.9655 − 14.4394i 53.3283 − 14.2893i −9.04984 + 25.4382i 54.1007 + 20.2240i
2.3 −2.96137 − 4.22927i −4.49084 − 2.61387i −6.38087 + 17.5313i 1.14050 − 11.1220i 2.24426 + 26.7336i 27.4008 + 12.7772i 53.1441 − 14.2399i 13.3353 + 23.4770i −50.4154 + 28.1129i
2.4 −2.93471 − 4.19120i 5.07973 + 1.09377i −6.21748 + 17.0824i −1.88316 − 11.0206i −10.3233 − 24.5001i −5.76190 − 2.68682i 50.3049 − 13.4792i 24.6073 + 11.1121i −40.6631 + 40.2350i
2.5 −2.90715 − 4.15184i 2.56106 − 4.52117i −6.05010 + 16.6225i 8.37390 + 7.40795i −26.2166 + 2.51061i 22.1632 + 10.3349i 47.4366 − 12.7106i −13.8820 − 23.1580i 6.41243 − 56.3032i
2.6 −2.65055 − 3.78538i −4.48346 + 2.62651i −4.56750 + 12.5491i −8.38072 + 7.40024i 21.8260 + 10.0099i 17.1288 + 7.98731i 23.9004 − 6.40410i 13.2029 − 23.5517i 50.2262 + 12.1095i
2.7 −2.52998 − 3.61318i 5.15143 + 0.680276i −3.91814 + 10.7650i 1.78125 + 11.0375i −10.5750 − 20.3341i −14.6995 − 6.85449i 14.7240 − 3.94529i 26.0744 + 7.00879i 35.3741 − 34.3607i
2.8 −2.52862 − 3.61125i 3.43038 − 3.90288i −3.91101 + 10.7454i −11.1746 − 0.356708i −22.7684 − 2.51904i 6.75344 + 3.14918i 14.6274 − 3.91941i −3.46498 − 26.7767i 26.9683 + 41.2564i
2.9 −2.42105 − 3.45762i 0.155936 + 5.19381i −3.35749 + 9.22463i −9.86867 − 5.25445i 17.5807 − 13.1137i −3.93222 − 1.83363i 7.40674 − 1.98463i −26.9514 + 1.61981i 5.72466 + 46.8435i
2.10 −2.22231 − 3.17379i −4.88164 + 1.78032i −2.39812 + 6.58877i 2.90764 − 10.7956i 16.4989 + 11.5369i −22.7294 − 10.5989i −3.69897 + 0.991136i 20.6609 − 17.3818i −40.7248 + 14.7630i
2.11 −2.13703 − 3.05199i −2.02505 − 4.78531i −2.01159 + 5.52681i 11.1789 − 0.181310i −10.2771 + 16.4067i −9.89486 − 4.61405i −7.62411 + 2.04287i −18.7984 + 19.3810i −24.4429 − 33.7303i
2.12 −2.02675 − 2.89450i −5.19001 + 0.252519i −1.53426 + 4.21535i 8.20133 + 7.59857i 11.2498 + 14.5107i −2.20959 − 1.03035i −11.9942 + 3.21383i 26.8725 − 2.62115i 5.37201 − 39.1392i
2.13 −2.00986 − 2.87038i 3.51242 + 3.82922i −1.46337 + 4.02057i 9.68106 − 5.59258i 3.93182 − 17.7782i 26.9868 + 12.5841i −12.5958 + 3.37503i −2.32580 + 26.8996i −35.5104 − 16.5480i
2.14 −1.82395 − 2.60488i 1.58111 + 4.94976i −0.722414 + 1.98482i −0.423302 + 11.1723i 10.0096 − 13.1467i −0.283638 − 0.132262i −18.0851 + 4.84588i −22.0002 + 15.6522i 29.8746 − 19.2752i
2.15 −1.64725 − 2.35252i 0.144169 − 5.19415i −0.0847548 + 0.232862i −5.80231 − 9.55684i −12.4568 + 8.21693i 2.69088 + 1.25478i −21.5049 + 5.76222i −26.9584 − 1.49767i −12.9248 + 29.3926i
2.16 −1.61530 − 2.30689i 4.58656 − 2.44202i 0.0236138 − 0.0648784i 9.09040 − 6.50881i −13.0422 − 6.63610i −10.8081 − 5.03990i −21.9497 + 5.88140i 15.0731 − 22.4009i −29.6989 − 10.4569i
2.17 −1.30893 − 1.86935i −3.30544 − 4.00925i 0.955006 − 2.62386i −3.32738 + 10.6737i −3.16807 + 11.4268i 19.6356 + 9.15625i −23.7893 + 6.37431i −5.14809 + 26.5047i 24.3082 − 7.75114i
2.18 −1.06457 − 1.52036i 5.19185 − 0.211419i 1.55797 − 4.28050i −9.89272 + 5.20905i −5.84850 − 7.66840i 32.2257 + 15.0271i −22.5086 + 6.03117i 26.9106 − 2.19531i 18.4511 + 9.49509i
2.19 −0.989369 − 1.41297i 4.43349 + 2.71001i 1.71854 − 4.72165i −10.9624 − 2.19697i −0.557206 − 8.94557i −19.4135 − 9.05267i −21.7009 + 5.81474i 12.3117 + 24.0296i 7.74158 + 17.6631i
2.20 −0.879779 − 1.25645i −3.10345 + 4.16757i 1.93149 − 5.30674i −4.25616 − 10.3385i 7.96671 + 0.232803i 20.7658 + 9.68325i −20.2196 + 5.41783i −7.73723 − 25.8676i −9.24540 + 14.4433i
See next 80 embeddings (of 624 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 2.52
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
27.f odd 18 1 inner
135.q even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.4.q.a ✓ 624
5.c odd 4 1 inner 135.4.q.a ✓ 624
27.f odd 18 1 inner 135.4.q.a ✓ 624
135.q even 36 1 inner 135.4.q.a ✓ 624
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.q.a ✓ 624 1.a even 1 1 trivial
135.4.q.a ✓ 624 5.c odd 4 1 inner
135.4.q.a ✓ 624 27.f odd 18 1 inner
135.4.q.a ✓ 624 135.q even 36 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(135, [\chi])\).