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Properties

Label 819.2.ct.d
Level $819$
Weight $2$
Character orbit 819.ct
Analytic conductor $6.540$
Analytic rank $0$
Dimension $24$
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] = [819,2,Mod(127,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.ct (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: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{4} - 12 q^{10} - 4 q^{13} + 8 q^{16} + 36 q^{22} - 40 q^{25} + 24 q^{37} - 32 q^{40} - 20 q^{43} - 12 q^{46} + 12 q^{49} + 52 q^{52} - 12 q^{55} - 24 q^{58} - 8 q^{61} - 8 q^{64} + 60 q^{67} - 72 q^{76} - 88 q^{79} - 8 q^{82} + 12 q^{85} - 28 q^{88} - 8 q^{91} + 56 q^{94} + 84 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1 −2.16383 + 1.24929i 0 2.12144 − 3.67444i 2.08040i 0 −0.866025 − 0.500000i 5.60399i 0 −2.59902 − 4.50163i
127.2 −1.77210 + 1.02312i 0 1.09356 − 1.89410i − 2.45030i 0 0.866025 + 0.500000i 0.382896i 0 2.50696 + 4.34218i
127.3 −1.49298 + 0.861973i 0 0.485994 − 0.841767i 3.99211i 0 0.866025 + 0.500000i − 1.77224i 0 −3.44109 − 5.96014i
127.4 −0.999449 + 0.577032i 0 −0.334068 + 0.578623i 0.0693301i 0 −0.866025 − 0.500000i − 3.07920i 0 −0.0400057 − 0.0692919i
127.5 −0.794146 + 0.458500i 0 −0.579555 + 1.00382i 1.23417i 0 0.866025 + 0.500000i − 2.89691i 0 −0.565869 − 0.980115i
127.6 −0.564753 + 0.326060i 0 −0.787370 + 1.36376i − 3.49328i 0 −0.866025 − 0.500000i − 2.33116i 0 1.13902 + 1.97284i
127.7 0.564753 − 0.326060i 0 −0.787370 + 1.36376i 3.49328i 0 −0.866025 − 0.500000i 2.33116i 0 1.13902 + 1.97284i
127.8 0.794146 − 0.458500i 0 −0.579555 + 1.00382i − 1.23417i 0 0.866025 + 0.500000i 2.89691i 0 −0.565869 − 0.980115i
127.9 0.999449 − 0.577032i 0 −0.334068 + 0.578623i − 0.0693301i 0 −0.866025 − 0.500000i 3.07920i 0 −0.0400057 − 0.0692919i
127.10 1.49298 − 0.861973i 0 0.485994 − 0.841767i − 3.99211i 0 0.866025 + 0.500000i 1.77224i 0 −3.44109 − 5.96014i
127.11 1.77210 − 1.02312i 0 1.09356 − 1.89410i 2.45030i 0 0.866025 + 0.500000i − 0.382896i 0 2.50696 + 4.34218i
127.12 2.16383 − 1.24929i 0 2.12144 − 3.67444i − 2.08040i 0 −0.866025 − 0.500000i − 5.60399i 0 −2.59902 − 4.50163i
316.1 −2.16383 − 1.24929i 0 2.12144 + 3.67444i − 2.08040i 0 −0.866025 + 0.500000i − 5.60399i 0 −2.59902 + 4.50163i
316.2 −1.77210 − 1.02312i 0 1.09356 + 1.89410i 2.45030i 0 0.866025 − 0.500000i − 0.382896i 0 2.50696 − 4.34218i
316.3 −1.49298 − 0.861973i 0 0.485994 + 0.841767i − 3.99211i 0 0.866025 − 0.500000i 1.77224i 0 −3.44109 + 5.96014i
316.4 −0.999449 − 0.577032i 0 −0.334068 − 0.578623i − 0.0693301i 0 −0.866025 + 0.500000i 3.07920i 0 −0.0400057 + 0.0692919i
316.5 −0.794146 − 0.458500i 0 −0.579555 − 1.00382i − 1.23417i 0 0.866025 − 0.500000i 2.89691i 0 −0.565869 + 0.980115i
316.6 −0.564753 − 0.326060i 0 −0.787370 − 1.36376i 3.49328i 0 −0.866025 + 0.500000i 2.33116i 0 1.13902 − 1.97284i
316.7 0.564753 + 0.326060i 0 −0.787370 − 1.36376i − 3.49328i 0 −0.866025 + 0.500000i − 2.33116i 0 1.13902 − 1.97284i
316.8 0.794146 + 0.458500i 0 −0.579555 − 1.00382i 1.23417i 0 0.866025 − 0.500000i − 2.89691i 0 −0.565869 + 0.980115i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.e even 6 1 inner
39.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.ct.d ✓ 24
3.b odd 2 1 inner 819.2.ct.d ✓ 24
13.e even 6 1 inner 819.2.ct.d ✓ 24
39.h odd 6 1 inner 819.2.ct.d ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.ct.d ✓ 24 1.a even 1 1 trivial
819.2.ct.d ✓ 24 3.b odd 2 1 inner
819.2.ct.d ✓ 24 13.e even 6 1 inner
819.2.ct.d ✓ 24 39.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 16 T_{2}^{22} + 162 T_{2}^{20} - 996 T_{2}^{18} + 4447 T_{2}^{16} - 13662 T_{2}^{14} + \cdots + 1369 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\). Copy content Toggle raw display