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Properties

Label 1710.2.t.e
Level $1710$
Weight $2$
Character orbit 1710.t
Analytic conductor $13.654$
Analytic rank $0$
Dimension $40$
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] = [1710,2,Mod(919,1710)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1710, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1710.919");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1710.t (of order \(6\), degree \(2\), 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: \(13.6544187456\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 20 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 20 q^{4} - 20 q^{16} + 28 q^{19} - 12 q^{25} + 16 q^{31} - 16 q^{34} - 8 q^{46} - 64 q^{49} - 48 q^{55} + 16 q^{61} - 40 q^{64} - 4 q^{70} + 8 q^{76} + 16 q^{79} + 48 q^{85} + 40 q^{91} + 112 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
919.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −2.20368 + 0.379207i 0 − 2.25125i 1.00000i 0 1.71884 − 1.43024i
919.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.43024 + 1.71884i 0 2.25125i 1.00000i 0 0.379207 − 2.20368i
919.3 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.00409545 − 2.23606i 0 − 2.96028i 1.00000i 0 1.12158 + 1.93444i
919.4 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 1.93444 + 1.12158i 0 2.96028i 1.00000i 0 −2.23606 − 0.00409545i
919.5 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.43439 − 1.71538i 0 − 0.826543i 1.00000i 0 2.09991 + 0.768368i
919.6 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.768368 + 2.09991i 0 0.826543i 1.00000i 0 −1.71538 − 1.43439i
919.7 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 1.54340 − 1.61799i 0 − 1.05892i 1.00000i 0 −0.527631 + 2.17293i
919.8 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 2.17293 − 0.527631i 0 1.05892i 1.00000i 0 −1.61799 + 1.54340i
919.9 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.72163 − 1.42688i 0 5.23107i 1.00000i 0 2.20442 + 0.374899i
919.10 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.374899 + 2.20442i 0 − 5.23107i 1.00000i 0 −1.42688 − 1.72163i
919.11 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.374899 − 2.20442i 0 − 5.23107i − 1.00000i 0 −1.42688 − 1.72163i
919.12 0.866025 − 0.500000i 0 0.500000 − 0.866025i 1.72163 + 1.42688i 0 5.23107i − 1.00000i 0 2.20442 + 0.374899i
919.13 0.866025 − 0.500000i 0 0.500000 − 0.866025i 1.43024 − 1.71884i 0 2.25125i − 1.00000i 0 0.379207 − 2.20368i
919.14 0.866025 − 0.500000i 0 0.500000 − 0.866025i 2.20368 − 0.379207i 0 − 2.25125i − 1.00000i 0 1.71884 − 1.43024i
919.15 0.866025 − 0.500000i 0 0.500000 − 0.866025i −1.93444 − 1.12158i 0 2.96028i − 1.00000i 0 −2.23606 − 0.00409545i
919.16 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.00409545 + 2.23606i 0 − 2.96028i − 1.00000i 0 1.12158 + 1.93444i
919.17 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.768368 − 2.09991i 0 0.826543i − 1.00000i 0 −1.71538 − 1.43439i
919.18 0.866025 − 0.500000i 0 0.500000 − 0.866025i 1.43439 + 1.71538i 0 − 0.826543i − 1.00000i 0 2.09991 + 0.768368i
919.19 0.866025 − 0.500000i 0 0.500000 − 0.866025i −2.17293 + 0.527631i 0 1.05892i − 1.00000i 0 −1.61799 + 1.54340i
919.20 0.866025 − 0.500000i 0 0.500000 − 0.866025i −1.54340 + 1.61799i 0 − 1.05892i − 1.00000i 0 −0.527631 + 2.17293i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 919.20
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
15.d odd 2 1 inner
19.c even 3 1 inner
57.h odd 6 1 inner
95.i even 6 1 inner
285.n odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1710.2.t.e ✓ 40
3.b odd 2 1 inner 1710.2.t.e ✓ 40
5.b even 2 1 inner 1710.2.t.e ✓ 40
15.d odd 2 1 inner 1710.2.t.e ✓ 40
19.c even 3 1 inner 1710.2.t.e ✓ 40
57.h odd 6 1 inner 1710.2.t.e ✓ 40
95.i even 6 1 inner 1710.2.t.e ✓ 40
285.n odd 6 1 inner 1710.2.t.e ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1710.2.t.e ✓ 40 1.a even 1 1 trivial
1710.2.t.e ✓ 40 3.b odd 2 1 inner
1710.2.t.e ✓ 40 5.b even 2 1 inner
1710.2.t.e ✓ 40 15.d odd 2 1 inner
1710.2.t.e ✓ 40 19.c even 3 1 inner
1710.2.t.e ✓ 40 57.h odd 6 1 inner
1710.2.t.e ✓ 40 95.i even 6 1 inner
1710.2.t.e ✓ 40 285.n odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} + 43T_{7}^{8} + 498T_{7}^{6} + 2010T_{7}^{4} + 2517T_{7}^{2} + 931 \) acting on \(S_{2}^{\mathrm{new}}(1710, [\chi])\). Copy content Toggle raw display