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Properties

Label 495.2.z.a
Level $495$
Weight $2$
Character orbit 495.z
Analytic conductor $3.953$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [495,2,Mod(134,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.134");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.z (of order \(10\), degree \(4\), 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: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(24\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 24 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 24 q^{4} + 24 q^{16} - 12 q^{25} + 40 q^{31} + 16 q^{34} - 60 q^{40} - 40 q^{46} - 112 q^{49} - 32 q^{55} - 40 q^{61} - 96 q^{64} - 28 q^{70} - 40 q^{79} + 100 q^{85} - 32 q^{91} - 40 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
134.1 −2.57582 − 0.836935i 0 4.31635 + 3.13602i −1.05723 + 1.97035i 0 −0.990823 − 0.719875i −5.30962 − 7.30807i 0 4.37229 − 4.19042i
134.2 −2.57582 − 0.836935i 0 4.31635 + 3.13602i 2.01346 + 0.972618i 0 0.990823 + 0.719875i −5.30962 − 7.30807i 0 −4.37229 − 4.19042i
134.3 −2.19661 − 0.713721i 0 2.69765 + 1.95996i −1.58387 − 1.57840i 0 0.351455 + 0.255347i −1.81166 − 2.49353i 0 2.35261 + 4.59756i
134.4 −2.19661 − 0.713721i 0 2.69765 + 1.95996i 0.353622 − 2.20793i 0 −0.351455 − 0.255347i −1.81166 − 2.49353i 0 −2.35261 + 4.59756i
134.5 −1.47952 − 0.480724i 0 0.339840 + 0.246909i 0.501072 + 2.17920i 0 −2.92188 − 2.12287i 1.44468 + 1.98843i 0 0.306251 − 3.46505i
134.6 −1.47952 − 0.480724i 0 0.339840 + 0.246909i 0.875527 + 2.05754i 0 2.92188 + 2.12287i 1.44468 + 1.98843i 0 −0.306251 − 3.46505i
134.7 −1.24412 − 0.404240i 0 −0.233601 − 0.169721i −1.92953 + 1.13001i 0 1.31808 + 0.957639i 1.75984 + 2.42221i 0 2.85737 − 0.625874i
134.8 −1.24412 − 0.404240i 0 −0.233601 − 0.169721i 2.22522 − 0.219955i 0 −1.31808 − 0.957639i 1.75984 + 2.42221i 0 −2.85737 − 0.625874i
134.9 −1.03885 − 0.337543i 0 −0.652758 − 0.474257i −2.23171 − 0.139598i 0 −3.21685 − 2.33718i 1.80213 + 2.48041i 0 2.27129 + 0.898318i
134.10 −1.03885 − 0.337543i 0 −0.652758 − 0.474257i 1.72343 − 1.42470i 0 3.21685 + 2.33718i 1.80213 + 2.48041i 0 −2.27129 + 0.898318i
134.11 −0.0721074 − 0.0234291i 0 −1.61338 − 1.17219i −1.44765 − 1.70420i 0 0.638096 + 0.463604i 0.178003 + 0.245000i 0 0.0644586 + 0.156803i
134.12 −0.0721074 − 0.0234291i 0 −1.61338 − 1.17219i 0.169472 − 2.22964i 0 −0.638096 − 0.463604i 0.178003 + 0.245000i 0 −0.0644586 + 0.156803i
134.13 0.0721074 + 0.0234291i 0 −1.61338 − 1.17219i −0.169472 + 2.22964i 0 −0.638096 − 0.463604i −0.178003 − 0.245000i 0 −0.0644586 + 0.156803i
134.14 0.0721074 + 0.0234291i 0 −1.61338 − 1.17219i 1.44765 + 1.70420i 0 0.638096 + 0.463604i −0.178003 − 0.245000i 0 0.0644586 + 0.156803i
134.15 1.03885 + 0.337543i 0 −0.652758 − 0.474257i −1.72343 + 1.42470i 0 3.21685 + 2.33718i −1.80213 − 2.48041i 0 −2.27129 + 0.898318i
134.16 1.03885 + 0.337543i 0 −0.652758 − 0.474257i 2.23171 + 0.139598i 0 −3.21685 − 2.33718i −1.80213 − 2.48041i 0 2.27129 + 0.898318i
134.17 1.24412 + 0.404240i 0 −0.233601 − 0.169721i −2.22522 + 0.219955i 0 −1.31808 − 0.957639i −1.75984 − 2.42221i 0 −2.85737 − 0.625874i
134.18 1.24412 + 0.404240i 0 −0.233601 − 0.169721i 1.92953 − 1.13001i 0 1.31808 + 0.957639i −1.75984 − 2.42221i 0 2.85737 − 0.625874i
134.19 1.47952 + 0.480724i 0 0.339840 + 0.246909i −0.875527 − 2.05754i 0 2.92188 + 2.12287i −1.44468 − 1.98843i 0 −0.306251 − 3.46505i
134.20 1.47952 + 0.480724i 0 0.339840 + 0.246909i −0.501072 − 2.17920i 0 −2.92188 − 2.12287i −1.44468 − 1.98843i 0 0.306251 − 3.46505i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 134.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
11.d odd 10 1 inner
15.d odd 2 1 inner
33.f even 10 1 inner
55.h odd 10 1 inner
165.r even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.z.a ✓ 96
3.b odd 2 1 inner 495.2.z.a ✓ 96
5.b even 2 1 inner 495.2.z.a ✓ 96
11.d odd 10 1 inner 495.2.z.a ✓ 96
15.d odd 2 1 inner 495.2.z.a ✓ 96
33.f even 10 1 inner 495.2.z.a ✓ 96
55.h odd 10 1 inner 495.2.z.a ✓ 96
165.r even 10 1 inner 495.2.z.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.z.a ✓ 96 1.a even 1 1 trivial
495.2.z.a ✓ 96 3.b odd 2 1 inner
495.2.z.a ✓ 96 5.b even 2 1 inner
495.2.z.a ✓ 96 11.d odd 10 1 inner
495.2.z.a ✓ 96 15.d odd 2 1 inner
495.2.z.a ✓ 96 33.f even 10 1 inner
495.2.z.a ✓ 96 55.h odd 10 1 inner
495.2.z.a ✓ 96 165.r even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(495, [\chi])\).