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Properties

Label 495.6.f.a
Level $495$
Weight $6$
Character orbit 495.f
Analytic conductor $79.390$
Analytic rank $0$
Dimension $80$
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] = [495,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.296");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(79.3899908074\)
Analytic rank: \(0\)
Dimension: \(80\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 1280 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 1280 q^{4} + 23824 q^{16} + 14384 q^{22} - 50000 q^{25} - 4224 q^{31} - 77872 q^{34} + 14912 q^{37} - 109424 q^{49} + 31600 q^{55} + 66544 q^{58} + 237216 q^{64} - 48928 q^{67} - 13200 q^{70} + 920720 q^{82} + 872720 q^{88} - 120064 q^{91} - 64800 q^{97}+O(q^{100}) \) 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 −11.2175 0 93.8328 − 25.0000i 0 110.671i −693.611 0 280.438i
296.2 −11.2175 0 93.8328 25.0000i 0 − 110.671i −693.611 0 − 280.438i
296.3 −10.2344 0 72.7422 25.0000i 0 46.7646i −416.971 0 − 255.859i
296.4 −10.2344 0 72.7422 − 25.0000i 0 − 46.7646i −416.971 0 255.859i
296.5 −10.0723 0 69.4505 25.0000i 0 224.840i −377.211 0 − 251.807i
296.6 −10.0723 0 69.4505 − 25.0000i 0 − 224.840i −377.211 0 251.807i
296.7 −9.86356 0 65.2898 − 25.0000i 0 − 61.8625i −328.356 0 246.589i
296.8 −9.86356 0 65.2898 25.0000i 0 61.8625i −328.356 0 − 246.589i
296.9 −9.23184 0 53.2270 − 25.0000i 0 − 55.7598i −195.964 0 230.796i
296.10 −9.23184 0 53.2270 25.0000i 0 55.7598i −195.964 0 − 230.796i
296.11 −8.70131 0 43.7128 25.0000i 0 14.7846i −101.916 0 − 217.533i
296.12 −8.70131 0 43.7128 − 25.0000i 0 − 14.7846i −101.916 0 217.533i
296.13 −8.69824 0 43.6594 25.0000i 0 − 178.786i −101.416 0 − 217.456i
296.14 −8.69824 0 43.6594 − 25.0000i 0 178.786i −101.416 0 217.456i
296.15 −8.33486 0 37.4699 25.0000i 0 − 174.233i −45.5912 0 − 208.372i
296.16 −8.33486 0 37.4699 − 25.0000i 0 174.233i −45.5912 0 208.372i
296.17 −7.55940 0 25.1446 25.0000i 0 123.011i 51.8228 0 − 188.985i
296.18 −7.55940 0 25.1446 − 25.0000i 0 − 123.011i 51.8228 0 188.985i
296.19 −6.24647 0 7.01836 25.0000i 0 − 25.7362i 156.047 0 − 156.162i
296.20 −6.24647 0 7.01836 − 25.0000i 0 25.7362i 156.047 0 156.162i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 296.80
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 495.6.f.a ✓ 80
3.b odd 2 1 inner 495.6.f.a ✓ 80
11.b odd 2 1 inner 495.6.f.a ✓ 80
33.d even 2 1 inner 495.6.f.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.6.f.a ✓ 80 1.a even 1 1 trivial
495.6.f.a ✓ 80 3.b odd 2 1 inner
495.6.f.a ✓ 80 11.b odd 2 1 inner
495.6.f.a ✓ 80 33.d even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(495, [\chi])\).