Properties

Label 117.6.g.a
Level $117$
Weight $6$
Character orbit 117.g
Analytic conductor $18.765$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,6,Mod(55,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.55");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 117.g (of order \(3\), 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: \(18.7649069181\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 32 \zeta_{6} q^{4} + 211 \zeta_{6} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 32 \zeta_{6} q^{4} + 211 \zeta_{6} q^{7} + ( - 543 \zeta_{6} - 116) q^{13} + (1024 \zeta_{6} - 1024) q^{16} + 1432 \zeta_{6} q^{19} - 3125 q^{25} + (6752 \zeta_{6} - 6752) q^{28} + 2723 q^{31} + (16550 \zeta_{6} - 16550) q^{37} + 19123 \zeta_{6} q^{43} + (27714 \zeta_{6} - 27714) q^{49} + ( - 21088 \zeta_{6} + 17376) q^{52} - 56927 \zeta_{6} q^{61} - 32768 q^{64} + ( - 37939 \zeta_{6} + 37939) q^{67} + 79577 q^{73} + (45824 \zeta_{6} - 45824) q^{76} + 90857 q^{79} + ( - 139049 \zeta_{6} + 114573) q^{91} - 177725 \zeta_{6} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 32 q^{4} + 211 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 32 q^{4} + 211 q^{7} - 775 q^{13} - 1024 q^{16} + 1432 q^{19} - 6250 q^{25} - 6752 q^{28} + 5446 q^{31} - 16550 q^{37} + 19123 q^{43} - 27714 q^{49} + 13664 q^{52} - 56927 q^{61} - 65536 q^{64} + 37939 q^{67} + 159154 q^{73} - 45824 q^{76} + 181714 q^{79} + 90097 q^{91} - 177725 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 16.0000 27.7128i 0 0 105.500 182.731i 0 0 0
100.1 0 0 16.0000 + 27.7128i 0 0 105.500 + 182.731i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
13.c even 3 1 inner
39.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.6.g.a 2
3.b odd 2 1 CM 117.6.g.a 2
13.c even 3 1 inner 117.6.g.a 2
39.i odd 6 1 inner 117.6.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.6.g.a 2 1.a even 1 1 trivial
117.6.g.a 2 3.b odd 2 1 CM
117.6.g.a 2 13.c even 3 1 inner
117.6.g.a 2 39.i odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{6}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 211T + 44521 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 775T + 371293 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 1432 T + 2050624 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 2723)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16550 T + 273902500 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 19123 T + 365689129 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 56927 T + 3240683329 \) Copy content Toggle raw display
$67$ \( T^{2} - 37939 T + 1439367721 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 79577)^{2} \) Copy content Toggle raw display
$79$ \( (T - 90857)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 177725 T + 31586175625 \) Copy content Toggle raw display
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