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Properties

Label 495.2.ba.b
Level $495$
Weight $2$
Character orbit 495.ba
Analytic conductor $3.953$
Analytic rank $0$
Dimension $48$
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(64,495)]
 
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([0, 5, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.ba (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: \(48\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{4} - 12 q^{10} - 28 q^{16} + 22 q^{25} - 20 q^{31} + 40 q^{34} + 52 q^{40} - 52 q^{46} + 44 q^{49} + 60 q^{55} + 16 q^{61} - 64 q^{64} - 74 q^{70} + 152 q^{76} + 28 q^{79} - 38 q^{85} + 40 q^{91} - 64 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 −2.32725 − 0.756170i 0 3.22628 + 2.34403i −1.31177 − 1.81087i 0 −0.207582 + 0.285712i −2.85924 − 3.93540i 0 1.68350 + 5.20628i
64.2 −2.32725 − 0.756170i 0 3.22628 + 2.34403i 0.00315652 + 2.23607i 0 0.207582 − 0.285712i −2.85924 − 3.93540i 0 1.68350 − 5.20628i
64.3 −1.45312 − 0.472148i 0 0.270606 + 0.196606i 1.95618 − 1.08321i 0 −2.73622 + 3.76608i 1.49576 + 2.05874i 0 −3.35401 + 0.650431i
64.4 −1.45312 − 0.472148i 0 0.270606 + 0.196606i 2.21928 − 0.273479i 0 2.73622 − 3.76608i 1.49576 + 2.05874i 0 −3.35401 − 0.650431i
64.5 −0.782884 − 0.254374i 0 −1.06983 − 0.777279i −2.12044 − 0.709728i 0 −1.01280 + 1.39399i 1.60753 + 2.21258i 0 1.47953 + 1.09502i
64.6 −0.782884 − 0.254374i 0 −1.06983 − 0.777279i −1.29831 + 1.82055i 0 1.01280 − 1.39399i 1.60753 + 2.21258i 0 1.47953 − 1.09502i
64.7 0.782884 + 0.254374i 0 −1.06983 − 0.777279i 1.29831 − 1.82055i 0 1.01280 − 1.39399i −1.60753 − 2.21258i 0 1.47953 − 1.09502i
64.8 0.782884 + 0.254374i 0 −1.06983 − 0.777279i 2.12044 + 0.709728i 0 −1.01280 + 1.39399i −1.60753 − 2.21258i 0 1.47953 + 1.09502i
64.9 1.45312 + 0.472148i 0 0.270606 + 0.196606i −2.21928 + 0.273479i 0 2.73622 − 3.76608i −1.49576 − 2.05874i 0 −3.35401 − 0.650431i
64.10 1.45312 + 0.472148i 0 0.270606 + 0.196606i −1.95618 + 1.08321i 0 −2.73622 + 3.76608i −1.49576 − 2.05874i 0 −3.35401 + 0.650431i
64.11 2.32725 + 0.756170i 0 3.22628 + 2.34403i −0.00315652 − 2.23607i 0 0.207582 − 0.285712i 2.85924 + 3.93540i 0 1.68350 − 5.20628i
64.12 2.32725 + 0.756170i 0 3.22628 + 2.34403i 1.31177 + 1.81087i 0 −0.207582 + 0.285712i 2.85924 + 3.93540i 0 1.68350 + 5.20628i
289.1 −1.46574 − 2.01742i 0 −1.30356 + 4.01194i −0.696888 − 2.12470i 0 3.38046 + 1.09838i 5.26123 − 1.70948i 0 −3.26496 + 4.52018i
289.2 −1.46574 − 2.01742i 0 −1.30356 + 4.01194i 2.23606 + 0.00621162i 0 −3.38046 − 1.09838i 5.26123 − 1.70948i 0 −3.26496 − 4.52018i
289.3 −0.974862 − 1.34178i 0 −0.231989 + 0.713990i −2.20493 − 0.371880i 0 −1.36188 − 0.442501i −1.97054 + 0.640268i 0 1.65052 + 3.32106i
289.4 −0.974862 − 1.34178i 0 −0.231989 + 0.713990i 1.03504 + 1.98209i 0 1.36188 + 0.442501i −1.97054 + 0.640268i 0 1.65052 − 3.32106i
289.5 −0.103256 − 0.142120i 0 0.608498 − 1.87276i −2.15951 + 0.580089i 0 3.62795 + 1.17879i −0.663132 + 0.215465i 0 0.305425 + 0.247012i
289.6 −0.103256 − 0.142120i 0 0.608498 − 1.87276i 0.115629 + 2.23308i 0 −3.62795 − 1.17879i −0.663132 + 0.215465i 0 0.305425 − 0.247012i
289.7 0.103256 + 0.142120i 0 0.608498 − 1.87276i −0.115629 − 2.23308i 0 −3.62795 − 1.17879i 0.663132 − 0.215465i 0 0.305425 − 0.247012i
289.8 0.103256 + 0.142120i 0 0.608498 − 1.87276i 2.15951 − 0.580089i 0 3.62795 + 1.17879i 0.663132 − 0.215465i 0 0.305425 + 0.247012i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.12
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.c even 5 1 inner
15.d odd 2 1 inner
33.h odd 10 1 inner
55.j even 10 1 inner
165.o odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.ba.b ✓ 48
3.b odd 2 1 inner 495.2.ba.b ✓ 48
5.b even 2 1 inner 495.2.ba.b ✓ 48
11.c even 5 1 inner 495.2.ba.b ✓ 48
15.d odd 2 1 inner 495.2.ba.b ✓ 48
33.h odd 10 1 inner 495.2.ba.b ✓ 48
55.j even 10 1 inner 495.2.ba.b ✓ 48
165.o odd 10 1 inner 495.2.ba.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.ba.b ✓ 48 1.a even 1 1 trivial
495.2.ba.b ✓ 48 3.b odd 2 1 inner
495.2.ba.b ✓ 48 5.b even 2 1 inner
495.2.ba.b ✓ 48 11.c even 5 1 inner
495.2.ba.b ✓ 48 15.d odd 2 1 inner
495.2.ba.b ✓ 48 33.h odd 10 1 inner
495.2.ba.b ✓ 48 55.j even 10 1 inner
495.2.ba.b ✓ 48 165.o odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 9 T_{2}^{22} + 65 T_{2}^{20} - 436 T_{2}^{18} + 2704 T_{2}^{16} - 8021 T_{2}^{14} + \cdots + 25 \) acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\). Copy content Toggle raw display