Properties

Label 324.3.f.k
Level $324$
Weight $3$
Character orbit 324.f
Analytic conductor $8.828$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [324,3,Mod(55,324)]
 
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([3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.55");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 324.f (of order \(6\), 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: \(8.82836056527\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (2 \beta_1 - 2) q^{2} - 4 \beta_1 q^{4} + ( - \beta_{2} - 4 \beta_1) q^{5} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_1 - 2) q^{2} - 4 \beta_1 q^{4} + ( - \beta_{2} - 4 \beta_1) q^{5} + 8 q^{8} + (2 \beta_{3} + 8) q^{10} + (4 \beta_{2} - 5 \beta_1) q^{13} + (16 \beta_1 - 16) q^{16} + ( - 5 \beta_{3} - 8) q^{17} + ( - 4 \beta_{3} + 4 \beta_{2} + 16 \beta_1 - 16) q^{20} + ( - 8 \beta_{3} + 8 \beta_{2} + 18 \beta_1 - 18) q^{25} + ( - 8 \beta_{3} + 10) q^{26} + ( - 7 \beta_{3} + 7 \beta_{2} - 20 \beta_1 + 20) q^{29} - 32 \beta_1 q^{32} + (10 \beta_{3} - 10 \beta_{2} - 16 \beta_1 + 16) q^{34} + ( - 4 \beta_{3} + 35) q^{37} + ( - 8 \beta_{2} - 32 \beta_1) q^{40} + 80 \beta_1 q^{41} - 49 \beta_1 q^{49} + ( - 16 \beta_{2} - 36 \beta_1) q^{50} + (16 \beta_{3} - 16 \beta_{2} + 20 \beta_1 - 20) q^{52} - 56 q^{53} + ( - 14 \beta_{2} + 40 \beta_1) q^{58} + ( - 20 \beta_{3} + 20 \beta_{2} + 11 \beta_1 - 11) q^{61} + 64 q^{64} + (11 \beta_{3} - 11 \beta_{2} - 88 \beta_1 + 88) q^{65} + (20 \beta_{2} + 32 \beta_1) q^{68} + ( - 16 \beta_{3} - 55) q^{73} + (8 \beta_{3} - 8 \beta_{2} + 70 \beta_1 - 70) q^{74} + (16 \beta_{3} + 64) q^{80} - 160 q^{82} + (28 \beta_{2} + 167 \beta_1) q^{85} + (13 \beta_{3} - 80) q^{89} + ( - 130 \beta_1 + 130) q^{97} + 98 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 8 q^{4} - 8 q^{5} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 8 q^{4} - 8 q^{5} + 32 q^{8} + 32 q^{10} - 10 q^{13} - 32 q^{16} - 32 q^{17} - 32 q^{20} - 36 q^{25} + 40 q^{26} + 40 q^{29} - 64 q^{32} + 32 q^{34} + 140 q^{37} - 64 q^{40} + 160 q^{41} - 98 q^{49} - 72 q^{50} - 40 q^{52} - 224 q^{53} + 80 q^{58} - 22 q^{61} + 256 q^{64} + 176 q^{65} + 64 q^{68} - 220 q^{73} - 140 q^{74} + 256 q^{80} - 640 q^{82} + 334 q^{85} - 320 q^{89} + 260 q^{97} + 392 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\zeta_{12}^{3} + 3\zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\zeta_{12}^{3} + 6\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 9 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 9 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−1.00000 + 1.73205i 0 −2.00000 3.46410i −4.59808 7.96410i 0 0 8.00000 0 18.3923
55.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i 0.598076 + 1.03590i 0 0 8.00000 0 −2.39230
271.1 −1.00000 1.73205i 0 −2.00000 + 3.46410i −4.59808 + 7.96410i 0 0 8.00000 0 18.3923
271.2 −1.00000 1.73205i 0 −2.00000 + 3.46410i 0.598076 1.03590i 0 0 8.00000 0 −2.39230
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.3.f.k 4
3.b odd 2 1 324.3.f.n 4
4.b odd 2 1 CM 324.3.f.k 4
9.c even 3 1 324.3.d.d yes 2
9.c even 3 1 inner 324.3.f.k 4
9.d odd 6 1 324.3.d.a 2
9.d odd 6 1 324.3.f.n 4
12.b even 2 1 324.3.f.n 4
36.f odd 6 1 324.3.d.d yes 2
36.f odd 6 1 inner 324.3.f.k 4
36.h even 6 1 324.3.d.a 2
36.h even 6 1 324.3.f.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.3.d.a 2 9.d odd 6 1
324.3.d.a 2 36.h even 6 1
324.3.d.d yes 2 9.c even 3 1
324.3.d.d yes 2 36.f odd 6 1
324.3.f.k 4 1.a even 1 1 trivial
324.3.f.k 4 4.b odd 2 1 CM
324.3.f.k 4 9.c even 3 1 inner
324.3.f.k 4 36.f odd 6 1 inner
324.3.f.n 4 3.b odd 2 1
324.3.f.n 4 9.d odd 6 1
324.3.f.n 4 12.b even 2 1
324.3.f.n 4 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 \(S_{3}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{4} + 8T_{5}^{3} + 75T_{5}^{2} - 88T_{5} + 121 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 8 T^{3} + 75 T^{2} - 88 T + 121 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 10 T^{3} + 507 T^{2} + \cdots + 165649 \) Copy content Toggle raw display
$17$ \( (T^{2} + 16 T - 611)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 40 T^{3} + 2523 T^{2} + \cdots + 851929 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 70 T + 793)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 80 T + 6400)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T + 56)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 22 T^{3} + \cdots + 114041041 \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 110 T - 3887)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 160 T + 1837)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 130 T + 16900)^{2} \) Copy content Toggle raw display
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