Properties

Label 324.6.e.e
Level $324$
Weight $6$
Character orbit 324.e
Analytic conductor $51.964$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [324,6,Mod(109,324)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(324, 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("324.109");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 324.e (of order \(3\), degree \(2\), not 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: \(51.9643576194\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} + (29 \beta_1 - 29) q^{7} + ( - \beta_{3} + \beta_{2}) q^{11} - 329 \beta_1 q^{13} - 25 \beta_{3} q^{17} + 1799 q^{19} - 41 \beta_{2} q^{23} + (4651 \beta_1 - 4651) q^{25} + ( - 16 \beta_{3} + 16 \beta_{2}) q^{29}+ \cdots + (12917 \beta_1 - 12917) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 58 q^{7} - 658 q^{13} + 7196 q^{19} - 9302 q^{25} - 10456 q^{31} + 35132 q^{37} - 39952 q^{43} + 31932 q^{49} + 31104 q^{55} + 2138 q^{61} + 124154 q^{67} - 192316 q^{73} - 99958 q^{79} + 388800 q^{85}+ \cdots - 25834 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 18\nu^{3} + 36\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -18\nu^{3} + 72\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 108 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 54 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
1.22474 0.707107i
−1.22474 + 0.707107i
1.22474 + 0.707107i
−1.22474 0.707107i
0 0 0 −44.0908 + 76.3675i 0 −14.5000 25.1147i 0 0 0
109.2 0 0 0 44.0908 76.3675i 0 −14.5000 25.1147i 0 0 0
217.1 0 0 0 −44.0908 76.3675i 0 −14.5000 + 25.1147i 0 0 0
217.2 0 0 0 44.0908 + 76.3675i 0 −14.5000 + 25.1147i 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 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.6.e.e 4
3.b odd 2 1 inner 324.6.e.e 4
9.c even 3 1 108.6.a.d 2
9.c even 3 1 inner 324.6.e.e 4
9.d odd 6 1 108.6.a.d 2
9.d odd 6 1 inner 324.6.e.e 4
36.f odd 6 1 432.6.a.p 2
36.h even 6 1 432.6.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.6.a.d 2 9.c even 3 1
108.6.a.d 2 9.d odd 6 1
324.6.e.e 4 1.a even 1 1 trivial
324.6.e.e 4 3.b odd 2 1 inner
324.6.e.e 4 9.c even 3 1 inner
324.6.e.e 4 9.d odd 6 1 inner
432.6.a.p 2 36.f odd 6 1
432.6.a.p 2 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{4} + 7776T_{5}^{2} + 60466176 \) Copy content Toggle raw display
\( T_{7}^{2} + 29T_{7} + 841 \) Copy content Toggle raw display
\( T_{11}^{4} + 7776T_{11}^{2} + 60466176 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 7776 T^{2} + 60466176 \) Copy content Toggle raw display
$7$ \( (T^{2} + 29 T + 841)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 7776 T^{2} + 60466176 \) Copy content Toggle raw display
$13$ \( (T^{2} + 329 T + 108241)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 4860000)^{2} \) Copy content Toggle raw display
$19$ \( (T - 1799)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 170862961959936 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 3962711310336 \) Copy content Toggle raw display
$31$ \( (T^{2} + 5228 T + 27331984)^{2} \) Copy content Toggle raw display
$37$ \( (T - 8783)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 58\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{2} + 19976 T + 399040576)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{2} - 867459456)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{2} - 1069 T + 1142761)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 62077 T + 3853553929)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 2151432576)^{2} \) Copy content Toggle raw display
$73$ \( (T + 48079)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 49979 T + 2497900441)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{2} - 7729413984)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 12917 T + 166848889)^{2} \) Copy content Toggle raw display
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