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Properties

Label 1710.2.bq.a
Level $1710$
Weight $2$
Character orbit 1710.bq
Analytic conductor $13.654$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1710,2,Mod(521,1710)]
 
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, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1710.521");
 
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.bq (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: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 16 q^{2} - 16 q^{4} + 8 q^{7} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 16 q^{2} - 16 q^{4} + 8 q^{7} + 32 q^{8} - 12 q^{13} - 4 q^{14} - 16 q^{16} - 24 q^{17} - 8 q^{19} + 16 q^{25} - 4 q^{28} - 16 q^{32} + 24 q^{34} + 4 q^{38} + 12 q^{43} - 24 q^{47} + 72 q^{49} - 32 q^{50} + 12 q^{52} - 4 q^{53} + 8 q^{56} - 16 q^{59} - 12 q^{61} + 12 q^{62} + 32 q^{64} - 16 q^{65} + 36 q^{67} - 24 q^{71} - 4 q^{73} - 36 q^{74} + 4 q^{76} - 12 q^{79} + 12 q^{86} + 16 q^{89} - 48 q^{91} + 12 q^{95} + 24 q^{97} - 36 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
521.1 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.866025 − 0.500000i 0 2.95827 1.00000 0 0.866025 − 0.500000i
521.2 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.866025 − 0.500000i 0 −0.804842 1.00000 0 0.866025 − 0.500000i
521.3 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.866025 − 0.500000i 0 2.89868 1.00000 0 0.866025 − 0.500000i
521.4 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.866025 − 0.500000i 0 2.62414 1.00000 0 0.866025 − 0.500000i
521.5 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.866025 − 0.500000i 0 −2.25923 1.00000 0 0.866025 − 0.500000i
521.6 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.866025 − 0.500000i 0 −2.84082 1.00000 0 0.866025 − 0.500000i
521.7 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.866025 − 0.500000i 0 4.52668 1.00000 0 0.866025 − 0.500000i
521.8 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.866025 − 0.500000i 0 −5.10288 1.00000 0 0.866025 − 0.500000i
521.9 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.866025 + 0.500000i 0 4.73349 1.00000 0 −0.866025 + 0.500000i
521.10 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.866025 + 0.500000i 0 2.66725 1.00000 0 −0.866025 + 0.500000i
521.11 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.866025 + 0.500000i 0 −1.50239 1.00000 0 −0.866025 + 0.500000i
521.12 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.866025 + 0.500000i 0 0.903384 1.00000 0 −0.866025 + 0.500000i
521.13 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.866025 + 0.500000i 0 −0.534903 1.00000 0 −0.866025 + 0.500000i
521.14 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.866025 + 0.500000i 0 −2.17012 1.00000 0 −0.866025 + 0.500000i
521.15 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.866025 + 0.500000i 0 2.40277 1.00000 0 −0.866025 + 0.500000i
521.16 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.866025 + 0.500000i 0 −4.49949 1.00000 0 −0.866025 + 0.500000i
791.1 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −0.866025 + 0.500000i 0 2.95827 1.00000 0 0.866025 + 0.500000i
791.2 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −0.866025 + 0.500000i 0 −0.804842 1.00000 0 0.866025 + 0.500000i
791.3 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −0.866025 + 0.500000i 0 2.89868 1.00000 0 0.866025 + 0.500000i
791.4 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −0.866025 + 0.500000i 0 2.62414 1.00000 0 0.866025 + 0.500000i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 521.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.f 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.bq.a ✓ 32
3.b odd 2 1 1710.2.bq.b yes 32
19.d odd 6 1 1710.2.bq.b yes 32
57.f even 6 1 inner 1710.2.bq.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1710.2.bq.a ✓ 32 1.a even 1 1 trivial
1710.2.bq.a ✓ 32 57.f even 6 1 inner
1710.2.bq.b yes 32 3.b odd 2 1
1710.2.bq.b yes 32 19.d odd 6 1