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Properties

Label 1560.2.dx.a
Level $1560$
Weight $2$
Character orbit 1560.dx
Analytic conductor $12.457$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1560,2,Mod(289,1560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1560, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("1560.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1560.dx (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: \(12.4566627153\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{5} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{5} + 20 q^{9} - 8 q^{11} + 2 q^{19} + 4 q^{21} + 28 q^{25} + 2 q^{29} + 12 q^{31} + 2 q^{35} + 10 q^{39} + 12 q^{41} - 2 q^{45} - 6 q^{49} + 8 q^{51} - 32 q^{55} + 34 q^{59} + 28 q^{61} - 52 q^{65} - 10 q^{69} - 16 q^{71} + 8 q^{75} - 4 q^{79} - 20 q^{81} + 8 q^{85} + 46 q^{89} + 14 q^{91} + 26 q^{95} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1 0 −0.866025 + 0.500000i 0 2.23572 − 0.0396606i 0 3.13727 + 1.81131i 0 0.500000 − 0.866025i 0
289.2 0 −0.866025 + 0.500000i 0 0.0562471 + 2.23536i 0 1.37906 + 0.796199i 0 0.500000 − 0.866025i 0
289.3 0 −0.866025 + 0.500000i 0 −1.37698 + 1.76180i 0 −0.118695 − 0.0685284i 0 0.500000 − 0.866025i 0
289.4 0 −0.866025 + 0.500000i 0 −1.56342 − 1.59867i 0 0.480039 + 0.277151i 0 0.500000 − 0.866025i 0
289.5 0 −0.866025 + 0.500000i 0 1.74653 − 1.39629i 0 −1.71179 − 0.988304i 0 0.500000 − 0.866025i 0
289.6 0 −0.866025 + 0.500000i 0 −2.23049 − 0.157859i 0 2.32058 + 1.33979i 0 0.500000 − 0.866025i 0
289.7 0 −0.866025 + 0.500000i 0 1.55586 − 1.60602i 0 2.24978 + 1.29891i 0 0.500000 − 0.866025i 0
289.8 0 −0.866025 + 0.500000i 0 −1.85074 + 1.25490i 0 −2.84102 − 1.64026i 0 0.500000 − 0.866025i 0
289.9 0 −0.866025 + 0.500000i 0 1.82537 + 1.29152i 0 −2.55122 − 1.47295i 0 0.500000 − 0.866025i 0
289.10 0 −0.866025 + 0.500000i 0 −1.39811 − 1.74508i 0 −3.21003 − 1.85331i 0 0.500000 − 0.866025i 0
289.11 0 0.866025 − 0.500000i 0 −1.39811 + 1.74508i 0 3.21003 + 1.85331i 0 0.500000 − 0.866025i 0
289.12 0 0.866025 − 0.500000i 0 1.82537 − 1.29152i 0 2.55122 + 1.47295i 0 0.500000 − 0.866025i 0
289.13 0 0.866025 − 0.500000i 0 −1.85074 − 1.25490i 0 2.84102 + 1.64026i 0 0.500000 − 0.866025i 0
289.14 0 0.866025 − 0.500000i 0 1.55586 + 1.60602i 0 −2.24978 − 1.29891i 0 0.500000 − 0.866025i 0
289.15 0 0.866025 − 0.500000i 0 −2.23049 + 0.157859i 0 −2.32058 − 1.33979i 0 0.500000 − 0.866025i 0
289.16 0 0.866025 − 0.500000i 0 1.74653 + 1.39629i 0 1.71179 + 0.988304i 0 0.500000 − 0.866025i 0
289.17 0 0.866025 − 0.500000i 0 −1.56342 + 1.59867i 0 −0.480039 − 0.277151i 0 0.500000 − 0.866025i 0
289.18 0 0.866025 − 0.500000i 0 −1.37698 − 1.76180i 0 0.118695 + 0.0685284i 0 0.500000 − 0.866025i 0
289.19 0 0.866025 − 0.500000i 0 0.0562471 − 2.23536i 0 −1.37906 − 0.796199i 0 0.500000 − 0.866025i 0
289.20 0 0.866025 − 0.500000i 0 2.23572 + 0.0396606i 0 −3.13727 − 1.81131i 0 0.500000 − 0.866025i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1560.2.dx.a ✓ 40
5.b even 2 1 inner 1560.2.dx.a ✓ 40
13.c even 3 1 inner 1560.2.dx.a ✓ 40
65.n even 6 1 inner 1560.2.dx.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.dx.a ✓ 40 1.a even 1 1 trivial
1560.2.dx.a ✓ 40 5.b even 2 1 inner
1560.2.dx.a ✓ 40 13.c even 3 1 inner
1560.2.dx.a ✓ 40 65.n even 6 1 inner