Properties

Label 108.6.a.d
Level $108$
Weight $6$
Character orbit 108.a
Self dual yes
Analytic conductor $17.321$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [108,6,Mod(1,108)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(108, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("108.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 108.a (trivial)

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: yes
Analytic conductor: \(17.3214525398\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 36\sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} + 29 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{5} + 29 q^{7} + \beta q^{11} + 329 q^{13} - 25 \beta q^{17} + 1799 q^{19} + 41 \beta q^{23} + 4651 q^{25} + 16 \beta q^{29} + 5228 q^{31} + 29 \beta q^{35} + 8783 q^{37} - 176 \beta q^{41} + 19976 q^{43} - 123 \beta q^{47} - 15966 q^{49} + 334 \beta q^{53} + 7776 q^{55} + 65 \beta q^{59} - 1069 q^{61} + 329 \beta q^{65} - 62077 q^{67} - 526 \beta q^{71} - 48079 q^{73} + 29 \beta q^{77} + 49979 q^{79} - 654 \beta q^{83} - 194400 q^{85} - 997 \beta q^{89} + 9541 q^{91} + 1799 \beta q^{95} + 12917 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 58 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 58 q^{7} + 658 q^{13} + 3598 q^{19} + 9302 q^{25} + 10456 q^{31} + 17566 q^{37} + 39952 q^{43} - 31932 q^{49} + 15552 q^{55} - 2138 q^{61} - 124154 q^{67} - 96158 q^{73} + 99958 q^{79} - 388800 q^{85} + 19082 q^{91} + 25834 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.44949
2.44949
0 0 0 −88.1816 0 29.0000 0 0 0
1.2 0 0 0 88.1816 0 29.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.6.a.d 2
3.b odd 2 1 inner 108.6.a.d 2
4.b odd 2 1 432.6.a.p 2
9.c even 3 2 324.6.e.e 4
9.d odd 6 2 324.6.e.e 4
12.b even 2 1 432.6.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.6.a.d 2 1.a even 1 1 trivial
108.6.a.d 2 3.b odd 2 1 inner
324.6.e.e 4 9.c even 3 2
324.6.e.e 4 9.d odd 6 2
432.6.a.p 2 4.b odd 2 1
432.6.a.p 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(108))\):

\( T_{5}^{2} - 7776 \) Copy content Toggle raw display
\( T_{11}^{2} - 7776 \) 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
$5$ \( T^{2} - 7776 \) Copy content Toggle raw display
$7$ \( (T - 29)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 7776 \) Copy content Toggle raw display
$13$ \( (T - 329)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 4860000 \) Copy content Toggle raw display
$19$ \( (T - 1799)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 13071456 \) Copy content Toggle raw display
$29$ \( T^{2} - 1990656 \) Copy content Toggle raw display
$31$ \( (T - 5228)^{2} \) Copy content Toggle raw display
$37$ \( (T - 8783)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 240869376 \) Copy content Toggle raw display
$43$ \( (T - 19976)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 117643104 \) Copy content Toggle raw display
$53$ \( T^{2} - 867459456 \) Copy content Toggle raw display
$59$ \( T^{2} - 32853600 \) Copy content Toggle raw display
$61$ \( (T + 1069)^{2} \) Copy content Toggle raw display
$67$ \( (T + 62077)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 2151432576 \) Copy content Toggle raw display
$73$ \( (T + 48079)^{2} \) Copy content Toggle raw display
$79$ \( (T - 49979)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 3325919616 \) Copy content Toggle raw display
$89$ \( T^{2} - 7729413984 \) Copy content Toggle raw display
$97$ \( (T - 12917)^{2} \) Copy content Toggle raw display
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