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Properties

Label 495.4.f.a
Level $495$
Weight $4$
Character orbit 495.f
Analytic conductor $29.206$
Analytic rank $0$
Dimension $48$
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] = [495,4,Mod(296,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.296");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 495.f (of order \(2\), degree \(1\), 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: \(29.2059454528\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 192 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 192 q^{4} + 528 q^{16} - 528 q^{22} - 1200 q^{25} + 48 q^{31} + 1488 q^{34} + 528 q^{37} - 3360 q^{49} - 120 q^{55} + 4752 q^{58} - 864 q^{64} + 1632 q^{67} + 2160 q^{70} - 11088 q^{82} - 624 q^{88} - 1680 q^{91} - 3312 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
296.1 −5.38256 0 20.9720 5.00000i 0 − 5.04612i −69.8223 0 − 26.9128i
296.2 −5.38256 0 20.9720 − 5.00000i 0 5.04612i −69.8223 0 26.9128i
296.3 −5.10614 0 18.0726 − 5.00000i 0 15.1134i −51.4322 0 25.5307i
296.4 −5.10614 0 18.0726 5.00000i 0 − 15.1134i −51.4322 0 − 25.5307i
296.5 −4.60456 0 13.2020 − 5.00000i 0 − 30.1055i −23.9529 0 23.0228i
296.6 −4.60456 0 13.2020 5.00000i 0 30.1055i −23.9529 0 − 23.0228i
296.7 −4.01299 0 8.10411 − 5.00000i 0 24.6825i −0.417795 0 20.0650i
296.8 −4.01299 0 8.10411 5.00000i 0 − 24.6825i −0.417795 0 − 20.0650i
296.9 −3.86234 0 6.91763 − 5.00000i 0 − 26.9979i 4.18047 0 19.3117i
296.10 −3.86234 0 6.91763 5.00000i 0 26.9979i 4.18047 0 − 19.3117i
296.11 −3.70803 0 5.74950 − 5.00000i 0 4.00229i 8.34492 0 18.5402i
296.12 −3.70803 0 5.74950 5.00000i 0 − 4.00229i 8.34492 0 − 18.5402i
296.13 −3.39046 0 3.49521 5.00000i 0 25.9001i 15.2733 0 − 16.9523i
296.14 −3.39046 0 3.49521 − 5.00000i 0 − 25.9001i 15.2733 0 16.9523i
296.15 −2.42118 0 −2.13787 5.00000i 0 4.23163i 24.5456 0 − 12.1059i
296.16 −2.42118 0 −2.13787 − 5.00000i 0 − 4.23163i 24.5456 0 12.1059i
296.17 −1.85993 0 −4.54065 − 5.00000i 0 10.6877i 23.3248 0 9.29966i
296.18 −1.85993 0 −4.54065 5.00000i 0 − 10.6877i 23.3248 0 − 9.29966i
296.19 −1.01557 0 −6.96862 − 5.00000i 0 19.1759i 15.2016 0 5.07783i
296.20 −1.01557 0 −6.96862 5.00000i 0 − 19.1759i 15.2016 0 − 5.07783i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 296.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.4.f.a ✓ 48
3.b odd 2 1 inner 495.4.f.a ✓ 48
11.b odd 2 1 inner 495.4.f.a ✓ 48
33.d even 2 1 inner 495.4.f.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.4.f.a ✓ 48 1.a even 1 1 trivial
495.4.f.a ✓ 48 3.b odd 2 1 inner
495.4.f.a ✓ 48 11.b odd 2 1 inner
495.4.f.a ✓ 48 33.d even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(495, [\chi])\).