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Properties

Label 495.2.bj.b
Level $495$
Weight $2$
Character orbit 495.bj
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(28,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 15, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.28");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.bj (of order \(20\), degree \(8\), 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: \(12\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 20 q^{7} + 40 q^{16} + 8 q^{22} - 16 q^{25} - 60 q^{28} + 16 q^{31} - 40 q^{37} + 280 q^{46} + 40 q^{52} + 24 q^{55} - 68 q^{58} + 40 q^{61} + 16 q^{67} + 52 q^{70} - 60 q^{73} - 112 q^{82} + 80 q^{85} + 24 q^{88} - 56 q^{91} - 112 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1 −1.26218 + 2.47717i 0 −3.36769 − 4.63522i −1.14810 + 1.91882i 0 2.91528 + 0.461734i 10.2409 − 1.62200i 0 −3.30412 − 5.26593i
28.2 −0.978497 + 1.92041i 0 −1.55494 − 2.14019i −0.218645 − 2.22535i 0 2.13983 + 0.338916i 1.37397 − 0.217616i 0 4.48753 + 1.75761i
28.3 −0.717635 + 1.40844i 0 −0.293128 − 0.403456i −2.22726 + 0.198299i 0 −1.38435 − 0.219259i −2.34393 + 0.371242i 0 1.31907 − 3.27926i
28.4 −0.715913 + 1.40506i 0 −0.286089 − 0.393768i 1.24495 + 1.85744i 0 −2.43914 − 0.386321i −2.35696 + 0.373306i 0 −3.50110 + 0.419458i
28.5 −0.310497 + 0.609386i 0 0.900628 + 1.23961i −0.209914 − 2.22619i 0 −1.14995 − 0.182134i −2.38606 + 0.377915i 0 1.42179 + 0.563309i
28.6 −0.188252 + 0.369466i 0 1.07450 + 1.47893i 2.23606 + 0.00601689i 0 4.21724 + 0.667945i −1.56780 + 0.248316i 0 −0.423167 + 0.825016i
28.7 0.188252 − 0.369466i 0 1.07450 + 1.47893i −2.23606 − 0.00601689i 0 4.21724 + 0.667945i 1.56780 − 0.248316i 0 −0.423167 + 0.825016i
28.8 0.310497 − 0.609386i 0 0.900628 + 1.23961i 0.209914 + 2.22619i 0 −1.14995 − 0.182134i 2.38606 − 0.377915i 0 1.42179 + 0.563309i
28.9 0.715913 − 1.40506i 0 −0.286089 − 0.393768i −1.24495 − 1.85744i 0 −2.43914 − 0.386321i 2.35696 − 0.373306i 0 −3.50110 + 0.419458i
28.10 0.717635 − 1.40844i 0 −0.293128 − 0.403456i 2.22726 − 0.198299i 0 −1.38435 − 0.219259i 2.34393 − 0.371242i 0 1.31907 − 3.27926i
28.11 0.978497 − 1.92041i 0 −1.55494 − 2.14019i 0.218645 + 2.22535i 0 2.13983 + 0.338916i −1.37397 + 0.217616i 0 4.48753 + 1.75761i
28.12 1.26218 − 2.47717i 0 −3.36769 − 4.63522i 1.14810 − 1.91882i 0 2.91528 + 0.461734i −10.2409 + 1.62200i 0 −3.30412 − 5.26593i
73.1 −0.412283 − 2.60305i 0 −4.70380 + 1.52836i 2.07434 + 0.834939i 0 −1.01770 + 1.99734i 3.52471 + 6.91763i 0 1.31818 − 5.74384i
73.2 −0.366845 − 2.31617i 0 −3.32796 + 1.08132i −2.13649 + 0.659843i 0 2.11652 − 4.15391i 1.59611 + 3.13254i 0 2.31207 + 4.70642i
73.3 −0.298934 − 1.88740i 0 −1.57080 + 0.510383i 0.575351 − 2.16078i 0 0.270117 − 0.530134i −0.302223 − 0.593145i 0 −4.25024 − 0.439986i
73.4 −0.183826 − 1.16063i 0 0.588846 − 0.191328i −1.95019 − 1.09397i 0 −2.02438 + 3.97307i −1.39727 − 2.74230i 0 −0.911199 + 2.46454i
73.5 −0.113010 − 0.713516i 0 1.40578 − 0.456765i 0.472785 + 2.18551i 0 1.51831 − 2.97985i −1.14071 − 2.23877i 0 1.50597 − 0.584324i
73.6 −0.00368345 − 0.0232564i 0 1.90159 − 0.617863i −2.10616 + 0.751062i 0 −0.396402 + 0.777982i −0.0427533 − 0.0839080i 0 0.0252250 + 0.0462152i
73.7 0.00368345 + 0.0232564i 0 1.90159 − 0.617863i 2.10616 − 0.751062i 0 −0.396402 + 0.777982i 0.0427533 + 0.0839080i 0 0.0252250 + 0.0462152i
73.8 0.113010 + 0.713516i 0 1.40578 − 0.456765i −0.472785 − 2.18551i 0 1.51831 − 2.97985i 1.14071 + 2.23877i 0 1.50597 − 0.584324i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.12
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
11.d odd 10 1 inner
15.e even 4 1 inner
33.f even 10 1 inner
55.l even 20 1 inner
165.u odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.bj.b ✓ 96
3.b odd 2 1 inner 495.2.bj.b ✓ 96
5.c odd 4 1 inner 495.2.bj.b ✓ 96
11.d odd 10 1 inner 495.2.bj.b ✓ 96
15.e even 4 1 inner 495.2.bj.b ✓ 96
33.f even 10 1 inner 495.2.bj.b ✓ 96
55.l even 20 1 inner 495.2.bj.b ✓ 96
165.u odd 20 1 inner 495.2.bj.b ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.bj.b ✓ 96 1.a even 1 1 trivial
495.2.bj.b ✓ 96 3.b odd 2 1 inner
495.2.bj.b ✓ 96 5.c odd 4 1 inner
495.2.bj.b ✓ 96 11.d odd 10 1 inner
495.2.bj.b ✓ 96 15.e even 4 1 inner
495.2.bj.b ✓ 96 33.f even 10 1 inner
495.2.bj.b ✓ 96 55.l even 20 1 inner
495.2.bj.b ✓ 96 165.u odd 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{96} - 94 T_{2}^{92} + 6687 T_{2}^{88} - 443090 T_{2}^{84} + 24980953 T_{2}^{80} - 857829330 T_{2}^{76} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\). Copy content Toggle raw display