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Properties

Label 806.2.be.a
Level $806$
Weight $2$
Character orbit 806.be
Analytic conductor $6.436$
Analytic rank $0$
Dimension $144$
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] = [806,2,Mod(123,806)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(806, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([5, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("806.123");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.be (of order \(12\), degree \(4\), 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: \(6.43594240292\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(36\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 8 q^{7} + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 8 q^{7} + 64 q^{9} + 16 q^{14} + 72 q^{16} + 16 q^{18} + 8 q^{28} - 16 q^{31} - 56 q^{33} - 40 q^{35} - 8 q^{47} + 48 q^{49} - 36 q^{62} + 104 q^{63} - 64 q^{66} - 72 q^{67} + 96 q^{69} + 48 q^{70} + 56 q^{71} - 16 q^{72} - 96 q^{78} - 104 q^{81} + 64 q^{87} - 40 q^{93} - 8 q^{94} - 144 q^{97} + 16 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
123.1 −0.258819 − 0.965926i −2.91114 − 1.68075i −0.866025 + 0.500000i 0.233831 − 0.233831i −0.870019 + 3.24696i 1.01373 − 3.78329i 0.707107 + 0.707107i 4.14983 + 7.18771i −0.286384 − 0.165344i
123.2 −0.258819 − 0.965926i −2.52880 − 1.46000i −0.866025 + 0.500000i 2.22093 − 2.22093i −0.755753 + 2.82051i −0.326180 + 1.21732i 0.707107 + 0.707107i 2.76322 + 4.78604i −2.72007 − 1.57043i
123.3 −0.258819 − 0.965926i −2.32939 − 1.34487i −0.866025 + 0.500000i −0.536830 + 0.536830i −0.696158 + 2.59810i −0.595488 + 2.22239i 0.707107 + 0.707107i 2.11737 + 3.66739i 0.657480 + 0.379596i
123.4 −0.258819 − 0.965926i −2.27814 − 1.31528i −0.866025 + 0.500000i −2.71440 + 2.71440i −0.680841 + 2.54093i −0.466413 + 1.74068i 0.707107 + 0.707107i 1.95994 + 3.39472i 3.32445 + 1.91937i
123.5 −0.258819 − 0.965926i −1.52622 − 0.881166i −0.866025 + 0.500000i −0.425608 + 0.425608i −0.456125 + 1.70228i 0.238460 − 0.889944i 0.707107 + 0.707107i 0.0529061 + 0.0916360i 0.521262 + 0.300951i
123.6 −0.258819 − 0.965926i −0.959575 − 0.554011i −0.866025 + 0.500000i 2.29825 − 2.29825i −0.286777 + 1.07027i −1.05065 + 3.92108i 0.707107 + 0.707107i −0.886144 − 1.53485i −2.81477 − 1.62511i
123.7 −0.258819 − 0.965926i −0.607507 − 0.350744i −0.866025 + 0.500000i 1.54215 − 1.54215i −0.181559 + 0.677586i 0.910076 − 3.39645i 0.707107 + 0.707107i −1.25396 − 2.17192i −1.88874 − 1.09047i
123.8 −0.258819 − 0.965926i −0.168160 − 0.0970875i −0.866025 + 0.500000i −2.55164 + 2.55164i −0.0502562 + 0.187559i 0.330483 − 1.23338i 0.707107 + 0.707107i −1.48115 − 2.56542i 3.12511 + 1.80428i
123.9 −0.258819 − 0.965926i −0.121459 − 0.0701245i −0.866025 + 0.500000i −0.0666827 + 0.0666827i −0.0362991 + 0.135470i −0.678861 + 2.53354i 0.707107 + 0.707107i −1.49017 − 2.58104i 0.0816693 + 0.0471518i
123.10 −0.258819 − 0.965926i 0.121459 + 0.0701245i −0.866025 + 0.500000i −0.0666827 + 0.0666827i 0.0362991 − 0.135470i −0.678861 + 2.53354i 0.707107 + 0.707107i −1.49017 − 2.58104i 0.0816693 + 0.0471518i
123.11 −0.258819 − 0.965926i 0.168160 + 0.0970875i −0.866025 + 0.500000i −2.55164 + 2.55164i 0.0502562 − 0.187559i 0.330483 − 1.23338i 0.707107 + 0.707107i −1.48115 − 2.56542i 3.12511 + 1.80428i
123.12 −0.258819 − 0.965926i 0.607507 + 0.350744i −0.866025 + 0.500000i 1.54215 − 1.54215i 0.181559 − 0.677586i 0.910076 − 3.39645i 0.707107 + 0.707107i −1.25396 − 2.17192i −1.88874 − 1.09047i
123.13 −0.258819 − 0.965926i 0.959575 + 0.554011i −0.866025 + 0.500000i 2.29825 − 2.29825i 0.286777 − 1.07027i −1.05065 + 3.92108i 0.707107 + 0.707107i −0.886144 − 1.53485i −2.81477 − 1.62511i
123.14 −0.258819 − 0.965926i 1.52622 + 0.881166i −0.866025 + 0.500000i −0.425608 + 0.425608i 0.456125 − 1.70228i 0.238460 − 0.889944i 0.707107 + 0.707107i 0.0529061 + 0.0916360i 0.521262 + 0.300951i
123.15 −0.258819 − 0.965926i 2.27814 + 1.31528i −0.866025 + 0.500000i −2.71440 + 2.71440i 0.680841 − 2.54093i −0.466413 + 1.74068i 0.707107 + 0.707107i 1.95994 + 3.39472i 3.32445 + 1.91937i
123.16 −0.258819 − 0.965926i 2.32939 + 1.34487i −0.866025 + 0.500000i −0.536830 + 0.536830i 0.696158 − 2.59810i −0.595488 + 2.22239i 0.707107 + 0.707107i 2.11737 + 3.66739i 0.657480 + 0.379596i
123.17 −0.258819 − 0.965926i 2.52880 + 1.46000i −0.866025 + 0.500000i 2.22093 − 2.22093i 0.755753 − 2.82051i −0.326180 + 1.21732i 0.707107 + 0.707107i 2.76322 + 4.78604i −2.72007 − 1.57043i
123.18 −0.258819 − 0.965926i 2.91114 + 1.68075i −0.866025 + 0.500000i 0.233831 − 0.233831i 0.870019 − 3.24696i 1.01373 − 3.78329i 0.707107 + 0.707107i 4.14983 + 7.18771i −0.286384 − 0.165344i
123.19 0.258819 + 0.965926i −2.56678 − 1.48193i −0.866025 + 0.500000i −1.35878 + 1.35878i 0.767103 − 2.86287i 0.434637 − 1.62209i −0.707107 − 0.707107i 2.89223 + 5.00949i −1.66416 − 0.960801i
123.20 0.258819 + 0.965926i −2.31749 − 1.33800i −0.866025 + 0.500000i 0.301439 − 0.301439i 0.692602 − 2.58483i −0.225059 + 0.839932i −0.707107 − 0.707107i 2.08051 + 3.60355i 0.369186 + 0.213150i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 123.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner
31.b odd 2 1 inner
403.bg even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 806.2.be.a ✓ 144
13.f odd 12 1 inner 806.2.be.a ✓ 144
31.b odd 2 1 inner 806.2.be.a ✓ 144
403.bg even 12 1 inner 806.2.be.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
806.2.be.a ✓ 144 1.a even 1 1 trivial
806.2.be.a ✓ 144 13.f odd 12 1 inner
806.2.be.a ✓ 144 31.b odd 2 1 inner
806.2.be.a ✓ 144 403.bg even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(806, [\chi])\).