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Properties

Label 135.4.m.a
Level $135$
Weight $4$
Character orbit 135.m
Analytic conductor $7.965$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [135,4,Mod(8,135)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(135, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([2, 9]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("135.8");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 135.m (of order \(12\), degree \(4\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.96525785077\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 6 q^{2} + 6 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 6 q^{2} + 6 q^{5} - 2 q^{7} - 8 q^{10} + 36 q^{11} - 2 q^{13} + 316 q^{16} - 378 q^{20} - 34 q^{22} - 306 q^{23} - 146 q^{25} - 232 q^{28} - 4 q^{31} + 1770 q^{32} + 136 q^{37} - 114 q^{38} + 126 q^{40} - 1992 q^{41} - 2 q^{43} - 952 q^{46} - 3462 q^{47} - 666 q^{50} - 242 q^{52} + 284 q^{55} + 7128 q^{56} + 534 q^{58} + 32 q^{61} + 2094 q^{65} + 610 q^{67} + 2694 q^{68} + 498 q^{70} - 8 q^{73} + 1368 q^{76} + 6486 q^{77} - 3784 q^{82} - 2814 q^{83} - 1658 q^{85} - 12480 q^{86} - 1338 q^{88} + 992 q^{91} - 13152 q^{92} - 4284 q^{95} + 358 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −4.88245 + 1.30825i 0 15.1986 − 8.77490i 0.942427 − 11.1405i 0 −7.62587 − 28.4601i −34.1329 + 34.1329i 0 9.97324 + 55.6261i
8.2 −4.29910 + 1.15194i 0 10.2271 − 5.90461i −8.15241 + 7.65103i 0 1.91190 + 7.13532i −11.9883 + 11.9883i 0 26.2345 − 42.2836i
8.3 −4.14355 + 1.11026i 0 9.00816 − 5.20086i 9.46549 − 5.95016i 0 3.90791 + 14.5845i −7.28515 + 7.28515i 0 −32.6146 + 35.1640i
8.4 −3.56967 + 0.956489i 0 4.89944 − 2.82870i 4.76715 + 10.1131i 0 1.11867 + 4.17492i 6.12165 − 6.12165i 0 −26.6902 − 31.5406i
8.5 −2.15214 + 0.576664i 0 −2.62904 + 1.51788i −10.1671 − 4.65088i 0 −3.70051 − 13.8105i 17.3866 − 17.3866i 0 24.5629 + 4.14635i
8.6 −1.43923 + 0.385641i 0 −5.00553 + 2.88994i −11.1260 − 1.10058i 0 6.62758 + 24.7344i 14.5184 − 14.5184i 0 16.4374 − 2.70668i
8.7 −1.25351 + 0.335877i 0 −5.46973 + 3.15795i 11.1151 − 1.20586i 0 −0.803840 − 2.99997i 13.1367 − 13.1367i 0 −13.5279 + 5.24487i
8.8 −0.145161 + 0.0388957i 0 −6.90864 + 3.98871i 0.303237 + 11.1762i 0 −4.62436 − 17.2584i 1.69784 − 1.69784i 0 −0.478726 − 1.61056i
8.9 0.471331 − 0.126293i 0 −6.72200 + 3.88095i 6.42805 − 9.14768i 0 −1.53228 − 5.71856i −5.43846 + 5.43846i 0 1.87445 − 5.12340i
8.10 1.60134 − 0.429078i 0 −4.54802 + 2.62580i −2.86176 − 10.8079i 0 8.15082 + 30.4193i −15.5344 + 15.5344i 0 −9.22008 − 16.0792i
8.11 2.57169 − 0.689083i 0 −0.789441 + 0.455784i 4.19938 + 10.3617i 0 −0.400135 − 1.49332i −16.7770 + 16.7770i 0 17.9396 + 23.7534i
8.12 2.79258 − 0.748271i 0 0.310415 − 0.179218i −7.05208 + 8.67573i 0 5.03374 + 18.7862i −15.6218 + 15.6218i 0 −13.2017 + 29.5046i
8.13 2.87860 − 0.771317i 0 0.763180 − 0.440622i −10.4896 − 3.86889i 0 −8.39055 − 31.3140i −15.0012 + 15.0012i 0 −33.1795 − 3.04617i
8.14 3.92041 − 1.05047i 0 7.33795 − 4.23656i 9.56119 − 5.79513i 0 −6.13411 − 22.8928i 1.35786 − 1.35786i 0 31.3962 − 32.7631i
8.15 4.73973 − 1.27001i 0 13.9239 − 8.03899i 8.68621 + 7.03916i 0 5.69629 + 21.2589i 28.0284 − 28.0284i 0 50.1101 + 22.3322i
8.16 5.27515 − 1.41347i 0 18.9011 − 10.9125i −7.58342 − 8.21534i 0 1.13077 + 4.22008i 53.3880 − 53.3880i 0 −51.6158 − 32.6182i
17.1 −4.88245 − 1.30825i 0 15.1986 + 8.77490i 0.942427 + 11.1405i 0 −7.62587 + 28.4601i −34.1329 − 34.1329i 0 9.97324 − 55.6261i
17.2 −4.29910 − 1.15194i 0 10.2271 + 5.90461i −8.15241 − 7.65103i 0 1.91190 − 7.13532i −11.9883 − 11.9883i 0 26.2345 + 42.2836i
17.3 −4.14355 − 1.11026i 0 9.00816 + 5.20086i 9.46549 + 5.95016i 0 3.90791 − 14.5845i −7.28515 − 7.28515i 0 −32.6146 − 35.1640i
17.4 −3.56967 − 0.956489i 0 4.89944 + 2.82870i 4.76715 − 10.1131i 0 1.11867 − 4.17492i 6.12165 + 6.12165i 0 −26.6902 + 31.5406i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
9.d odd 6 1 inner
45.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.4.m.a 64
3.b odd 2 1 45.4.l.a ✓ 64
5.c odd 4 1 inner 135.4.m.a 64
9.c even 3 1 45.4.l.a ✓ 64
9.d odd 6 1 inner 135.4.m.a 64
15.d odd 2 1 225.4.p.b 64
15.e even 4 1 45.4.l.a ✓ 64
15.e even 4 1 225.4.p.b 64
45.j even 6 1 225.4.p.b 64
45.k odd 12 1 45.4.l.a ✓ 64
45.k odd 12 1 225.4.p.b 64
45.l even 12 1 inner 135.4.m.a 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.l.a ✓ 64 3.b odd 2 1
45.4.l.a ✓ 64 9.c even 3 1
45.4.l.a ✓ 64 15.e even 4 1
45.4.l.a ✓ 64 45.k odd 12 1
135.4.m.a 64 1.a even 1 1 trivial
135.4.m.a 64 5.c odd 4 1 inner
135.4.m.a 64 9.d odd 6 1 inner
135.4.m.a 64 45.l even 12 1 inner
225.4.p.b 64 15.d odd 2 1
225.4.p.b 64 15.e even 4 1
225.4.p.b 64 45.j even 6 1
225.4.p.b 64 45.k odd 12 1

Hecke kernels

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