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Properties

Label 135.4.f.a
Level $135$
Weight $4$
Character orbit 135.f
Analytic conductor $7.965$
Analytic rank $0$
Dimension $24$
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(53,135)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(135, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("135.53");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 135.f (of order \(4\), degree \(2\), 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: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{7} + 24 q^{10} + 72 q^{13} + 72 q^{16} + 24 q^{22} - 72 q^{25} - 720 q^{28} + 168 q^{31} + 1008 q^{37} - 1488 q^{40} + 384 q^{43} - 1176 q^{46} - 2064 q^{52} + 2808 q^{55} + 3504 q^{58} + 2496 q^{61} + 960 q^{67} + 888 q^{70} - 4656 q^{73} - 2256 q^{76} + 576 q^{82} - 6312 q^{85} - 1776 q^{88} - 1584 q^{91} - 4896 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −3.34457 − 3.34457i 0 14.3723i 5.57929 − 9.68873i 0 −10.5838 + 10.5838i 21.3126 − 21.3126i 0 −51.0650 + 13.7443i
53.2 −3.29870 − 3.29870i 0 13.7628i −10.9596 + 2.21063i 0 −10.4627 + 10.4627i 19.0097 − 19.0097i 0 43.4446 + 28.8602i
53.3 −2.54897 − 2.54897i 0 4.99449i −1.47107 + 11.0831i 0 18.5356 − 18.5356i −7.66096 + 7.66096i 0 32.0003 − 24.5009i
53.4 −1.55448 − 1.55448i 0 − 3.16717i 9.41268 + 6.03336i 0 −15.9667 + 15.9667i −17.3592 + 17.3592i 0 −5.25309 − 24.0106i
53.5 −1.40622 − 1.40622i 0 − 4.04511i −0.0894959 − 11.1800i 0 18.5304 − 18.5304i −16.9380 + 16.9380i 0 −15.5956 + 15.8473i
53.6 −0.203341 − 0.203341i 0 − 7.91730i −11.1350 + 1.00608i 0 −6.05280 + 6.05280i −3.23664 + 3.23664i 0 2.46877 + 2.05962i
53.7 0.203341 + 0.203341i 0 − 7.91730i 11.1350 − 1.00608i 0 −6.05280 + 6.05280i 3.23664 − 3.23664i 0 2.46877 + 2.05962i
53.8 1.40622 + 1.40622i 0 − 4.04511i 0.0894959 + 11.1800i 0 18.5304 − 18.5304i 16.9380 − 16.9380i 0 −15.5956 + 15.8473i
53.9 1.55448 + 1.55448i 0 − 3.16717i −9.41268 − 6.03336i 0 −15.9667 + 15.9667i 17.3592 − 17.3592i 0 −5.25309 − 24.0106i
53.10 2.54897 + 2.54897i 0 4.99449i 1.47107 − 11.0831i 0 18.5356 − 18.5356i 7.66096 − 7.66096i 0 32.0003 − 24.5009i
53.11 3.29870 + 3.29870i 0 13.7628i 10.9596 − 2.21063i 0 −10.4627 + 10.4627i −19.0097 + 19.0097i 0 43.4446 + 28.8602i
53.12 3.34457 + 3.34457i 0 14.3723i −5.57929 + 9.68873i 0 −10.5838 + 10.5838i −21.3126 + 21.3126i 0 −51.0650 + 13.7443i
107.1 −3.34457 + 3.34457i 0 − 14.3723i 5.57929 + 9.68873i 0 −10.5838 − 10.5838i 21.3126 + 21.3126i 0 −51.0650 − 13.7443i
107.2 −3.29870 + 3.29870i 0 − 13.7628i −10.9596 − 2.21063i 0 −10.4627 − 10.4627i 19.0097 + 19.0097i 0 43.4446 − 28.8602i
107.3 −2.54897 + 2.54897i 0 − 4.99449i −1.47107 − 11.0831i 0 18.5356 + 18.5356i −7.66096 − 7.66096i 0 32.0003 + 24.5009i
107.4 −1.55448 + 1.55448i 0 3.16717i 9.41268 − 6.03336i 0 −15.9667 − 15.9667i −17.3592 − 17.3592i 0 −5.25309 + 24.0106i
107.5 −1.40622 + 1.40622i 0 4.04511i −0.0894959 + 11.1800i 0 18.5304 + 18.5304i −16.9380 − 16.9380i 0 −15.5956 − 15.8473i
107.6 −0.203341 + 0.203341i 0 7.91730i −11.1350 − 1.00608i 0 −6.05280 − 6.05280i −3.23664 − 3.23664i 0 2.46877 − 2.05962i
107.7 0.203341 − 0.203341i 0 7.91730i 11.1350 + 1.00608i 0 −6.05280 − 6.05280i 3.23664 + 3.23664i 0 2.46877 − 2.05962i
107.8 1.40622 − 1.40622i 0 4.04511i 0.0894959 − 11.1800i 0 18.5304 + 18.5304i 16.9380 + 16.9380i 0 −15.5956 − 15.8473i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.4.f.a ✓ 24
3.b odd 2 1 inner 135.4.f.a ✓ 24
5.c odd 4 1 inner 135.4.f.a ✓ 24
15.e even 4 1 inner 135.4.f.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.f.a ✓ 24 1.a even 1 1 trivial
135.4.f.a ✓ 24 3.b odd 2 1 inner
135.4.f.a ✓ 24 5.c odd 4 1 inner
135.4.f.a ✓ 24 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 1182T_{2}^{20} + 446493T_{2}^{16} + 56108324T_{2}^{12} + 1708083300T_{2}^{8} + 14634840000T_{2}^{4} + 100000000 \) acting on \(S_{4}^{\mathrm{new}}(135, [\chi])\). Copy content Toggle raw display