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Properties

Label 760.2.bj.b
Level $760$
Weight $2$
Character orbit 760.bj
Analytic conductor $6.069$
Analytic rank $0$
Dimension $56$
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] = [760,2,Mod(49,760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(760, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("760.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.bj (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: \(6.06863055362\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 4 q^{5} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 4 q^{5} + 28 q^{9} + 24 q^{11} + 8 q^{15} - 4 q^{21} + 2 q^{25} - 4 q^{29} - 40 q^{31} + 8 q^{35} + 64 q^{39} - 4 q^{41} + 48 q^{45} - 72 q^{49} + 28 q^{51} + 6 q^{55} - 28 q^{59} - 24 q^{61} - 60 q^{65} - 72 q^{69} + 12 q^{71} + 4 q^{75} - 12 q^{81} - 10 q^{85} + 8 q^{89} - 24 q^{91} - 6 q^{95} + 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −2.66511 − 1.53870i 0 −1.05861 + 1.96961i 0 − 1.08824i 0 3.23519 + 5.60351i 0
49.2 0 −2.66390 − 1.53800i 0 2.09514 + 0.781263i 0 − 0.314722i 0 3.23091 + 5.59609i 0
49.3 0 −2.53438 − 1.46322i 0 −0.496798 − 2.18018i 0 3.65117i 0 2.78204 + 4.81864i 0
49.4 0 −2.29543 − 1.32527i 0 1.27509 − 1.83688i 0 − 4.82154i 0 2.01267 + 3.48605i 0
49.5 0 −2.01244 − 1.16188i 0 −2.04805 − 0.897495i 0 − 3.47788i 0 1.19994 + 2.07836i 0
49.6 0 −1.91924 − 1.10808i 0 1.45864 − 1.69481i 0 2.77177i 0 0.955666 + 1.65526i 0
49.7 0 −1.70195 − 0.982619i 0 −1.80558 − 1.31905i 0 3.04936i 0 0.431078 + 0.746650i 0
49.8 0 −1.48427 − 0.856942i 0 −2.20417 − 0.376367i 0 − 1.30006i 0 −0.0313006 − 0.0542142i 0
49.9 0 −1.17664 − 0.679334i 0 −0.389779 + 2.20183i 0 4.26649i 0 −0.577012 − 0.999414i 0
49.10 0 −0.827432 − 0.477718i 0 2.09280 + 0.787513i 0 − 3.43643i 0 −1.04357 − 1.80752i 0
49.11 0 −0.807150 − 0.466008i 0 2.22591 − 0.212903i 0 0.900311i 0 −1.06567 − 1.84580i 0
49.12 0 −0.564005 − 0.325628i 0 0.498921 − 2.17970i 0 − 2.88465i 0 −1.28793 − 2.23076i 0
49.13 0 −0.469891 − 0.271292i 0 −0.287581 + 2.21750i 0 2.45284i 0 −1.35280 − 2.34312i 0
49.14 0 −0.127242 − 0.0734631i 0 1.18414 + 1.89679i 0 − 1.48697i 0 −1.48921 − 2.57938i 0
49.15 0 0.127242 + 0.0734631i 0 −2.23474 − 0.0770965i 0 1.48697i 0 −1.48921 − 2.57938i 0
49.16 0 0.469891 + 0.271292i 0 −1.77662 + 1.35780i 0 − 2.45284i 0 −1.35280 − 2.34312i 0
49.17 0 0.564005 + 0.325628i 0 1.63821 − 1.52193i 0 2.88465i 0 −1.28793 − 2.23076i 0
49.18 0 0.807150 + 0.466008i 0 −0.928575 − 2.03415i 0 − 0.900311i 0 −1.06567 − 1.84580i 0
49.19 0 0.827432 + 0.477718i 0 −1.72841 − 1.41866i 0 3.43643i 0 −1.04357 − 1.80752i 0
49.20 0 1.17664 + 0.679334i 0 −1.71195 + 1.43848i 0 − 4.26649i 0 −0.577012 − 0.999414i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.2.bj.b ✓ 56
5.b even 2 1 inner 760.2.bj.b ✓ 56
19.c even 3 1 inner 760.2.bj.b ✓ 56
95.i even 6 1 inner 760.2.bj.b ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.bj.b ✓ 56 1.a even 1 1 trivial
760.2.bj.b ✓ 56 5.b even 2 1 inner
760.2.bj.b ✓ 56 19.c even 3 1 inner
760.2.bj.b ✓ 56 95.i even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{56} - 56 T_{3}^{54} + 1760 T_{3}^{52} - 37996 T_{3}^{50} + 622244 T_{3}^{48} - 8080122 T_{3}^{46} + \cdots + 40960000 \) acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\). Copy content Toggle raw display