Properties

Label 1710.2.q.b
Level $1710$
Weight $2$
Character orbit 1710.q
Analytic conductor $13.654$
Analytic rank $0$
Dimension $64$
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(179,1710)]
 
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([3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1710.179");
 
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.q (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: \(64\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 32 q^{4} - 24 q^{10} - 32 q^{16} + 40 q^{19} - 8 q^{25} - 24 q^{34} - 24 q^{40} + 8 q^{55} + 104 q^{61} - 64 q^{64} - 48 q^{70} + 32 q^{76} + 48 q^{79} + 16 q^{85} + 96 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −1.65982 + 1.49833i 0 1.61258i 1.00000i 0 2.18661 0.467676i
179.2 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.467676 + 2.18661i 0 1.61258i 1.00000i 0 1.49833 1.65982i
179.3 −0.866025 0.500000i 0 0.500000 + 0.866025i −1.67574 1.48050i 0 4.01357i 1.00000i 0 0.710988 + 2.12002i
179.4 −0.866025 0.500000i 0 0.500000 + 0.866025i 2.12002 + 0.710988i 0 4.01357i 1.00000i 0 −1.48050 1.67574i
179.5 −0.866025 0.500000i 0 0.500000 + 0.866025i −2.17236 + 0.529970i 0 0.317852i 1.00000i 0 2.14630 + 0.627210i
179.6 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.627210 + 2.14630i 0 0.317852i 1.00000i 0 0.529970 2.17236i
179.7 −0.866025 0.500000i 0 0.500000 + 0.866025i −2.05514 0.881135i 0 0.180093i 1.00000i 0 1.33924 + 1.79066i
179.8 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.79066 + 1.33924i 0 0.180093i 1.00000i 0 −0.881135 2.05514i
179.9 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.861447 2.06347i 0 3.19371i 1.00000i 0 −1.77777 + 1.35629i
179.10 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.35629 1.77777i 0 3.19371i 1.00000i 0 −2.06347 + 0.861447i
179.11 −0.866025 0.500000i 0 0.500000 + 0.866025i −1.54820 1.61341i 0 1.47489i 1.00000i 0 0.534078 + 2.17135i
179.12 −0.866025 0.500000i 0 0.500000 + 0.866025i 2.17135 + 0.534078i 0 1.47489i 1.00000i 0 −1.61341 1.54820i
179.13 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.0434002 2.23565i 0 1.06029i 1.00000i 0 −1.08024 + 1.95783i
179.14 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.95783 1.08024i 0 1.06029i 1.00000i 0 −2.23565 0.0434002i
179.15 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.762301 2.10212i 0 4.86395i 1.00000i 0 −1.71123 + 1.43934i
179.16 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.43934 1.71123i 0 4.86395i 1.00000i 0 −2.10212 + 0.762301i
179.17 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.627210 2.14630i 0 0.317852i 1.00000i 0 0.529970 2.17236i
179.18 0.866025 + 0.500000i 0 0.500000 + 0.866025i 2.17236 0.529970i 0 0.317852i 1.00000i 0 2.14630 + 0.627210i
179.19 0.866025 + 0.500000i 0 0.500000 + 0.866025i −2.17135 0.534078i 0 1.47489i 1.00000i 0 −1.61341 1.54820i
179.20 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.54820 + 1.61341i 0 1.47489i 1.00000i 0 0.534078 + 2.17135i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 179.32
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.d odd 6 1 inner
57.f even 6 1 inner
95.h odd 6 1 inner
285.q even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1710.2.q.b 64
3.b odd 2 1 inner 1710.2.q.b 64
5.b even 2 1 inner 1710.2.q.b 64
15.d odd 2 1 inner 1710.2.q.b 64
19.d odd 6 1 inner 1710.2.q.b 64
57.f even 6 1 inner 1710.2.q.b 64
95.h odd 6 1 inner 1710.2.q.b 64
285.q even 6 1 inner 1710.2.q.b 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1710.2.q.b 64 1.a even 1 1 trivial
1710.2.q.b 64 3.b odd 2 1 inner
1710.2.q.b 64 5.b even 2 1 inner
1710.2.q.b 64 15.d odd 2 1 inner
1710.2.q.b 64 19.d odd 6 1 inner
1710.2.q.b 64 57.f even 6 1 inner
1710.2.q.b 64 95.h odd 6 1 inner
1710.2.q.b 64 285.q even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 56T_{7}^{14} + 1100T_{7}^{12} + 9232T_{7}^{10} + 33142T_{7}^{8} + 52152T_{7}^{6} + 31212T_{7}^{4} + 3456T_{7}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(1710, [\chi])\). Copy content Toggle raw display