Properties

Label 243.2.c.d
Level $243$
Weight $2$
Character orbit 243.c
Analytic conductor $1.940$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [243,2,Mod(82,243)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(243, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("243.82");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 243 = 3^{5} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 243.c (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: \(1.94036476912\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_1 - 1) q^{4} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{5} + \beta_1 q^{7} - \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_1 - 1) q^{4} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{5} + \beta_1 q^{7} - \beta_{3} q^{8} + 6 q^{10} + 2 \beta_{2} q^{11} + (5 \beta_1 - 5) q^{13} + (\beta_{3} - \beta_{2}) q^{14} + 5 \beta_1 q^{16} - q^{19} - 2 \beta_{2} q^{20} + ( - 6 \beta_1 + 6) q^{22} + (4 \beta_{3} - 4 \beta_{2}) q^{23} - 7 \beta_1 q^{25} + 5 \beta_{3} q^{26} - q^{28} + 2 \beta_{2} q^{29} + (5 \beta_1 - 5) q^{31} + (3 \beta_{3} - 3 \beta_{2}) q^{32} - 2 \beta_{3} q^{35} - q^{37} + \beta_{2} q^{38} + ( - 6 \beta_1 + 6) q^{40} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{41} + \beta_1 q^{43} - 2 \beta_{3} q^{44} - 12 q^{46} + 2 \beta_{2} q^{47} + ( - 6 \beta_1 + 6) q^{49} + ( - 7 \beta_{3} + 7 \beta_{2}) q^{50} - 5 \beta_1 q^{52} - 6 \beta_{3} q^{53} - 12 q^{55} - \beta_{2} q^{56} + ( - 6 \beta_1 + 6) q^{58} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{59} - 2 \beta_1 q^{61} + 5 \beta_{3} q^{62} + q^{64} - 10 \beta_{2} q^{65} + (8 \beta_1 - 8) q^{67} + 6 \beta_1 q^{70} + 6 \beta_{3} q^{71} + 2 q^{73} + \beta_{2} q^{74} + ( - \beta_1 + 1) q^{76} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{77} + \beta_1 q^{79} - 10 \beta_{3} q^{80} + 6 q^{82} - 4 \beta_{2} q^{83} + (\beta_{3} - \beta_{2}) q^{86} - 6 \beta_1 q^{88} + 6 \beta_{3} q^{89} - 5 q^{91} + 4 \beta_{2} q^{92} + ( - 6 \beta_1 + 6) q^{94} + (2 \beta_{3} - 2 \beta_{2}) q^{95} - 17 \beta_1 q^{97} - 6 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{4} + 2 q^{7} + 24 q^{10} - 10 q^{13} + 10 q^{16} - 4 q^{19} + 12 q^{22} - 14 q^{25} - 4 q^{28} - 10 q^{31} - 4 q^{37} + 12 q^{40} + 2 q^{43} - 48 q^{46} + 12 q^{49} - 10 q^{52} - 48 q^{55} + 12 q^{58} - 4 q^{61} + 4 q^{64} - 16 q^{67} + 12 q^{70} + 8 q^{73} + 2 q^{76} + 2 q^{79} + 24 q^{82} - 12 q^{88} - 20 q^{91} + 12 q^{94} - 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/243\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 1.50000i 0 −0.500000 + 0.866025i −1.73205 + 3.00000i 0 0.500000 + 0.866025i −1.73205 0 6.00000
82.2 0.866025 + 1.50000i 0 −0.500000 + 0.866025i 1.73205 3.00000i 0 0.500000 + 0.866025i 1.73205 0 6.00000
163.1 −0.866025 + 1.50000i 0 −0.500000 0.866025i −1.73205 3.00000i 0 0.500000 0.866025i −1.73205 0 6.00000
163.2 0.866025 1.50000i 0 −0.500000 0.866025i 1.73205 + 3.00000i 0 0.500000 0.866025i 1.73205 0 6.00000
\(n\): 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 243.2.c.d 4
3.b odd 2 1 inner 243.2.c.d 4
9.c even 3 1 243.2.a.c 2
9.c even 3 1 inner 243.2.c.d 4
9.d odd 6 1 243.2.a.c 2
9.d odd 6 1 inner 243.2.c.d 4
27.e even 9 6 729.2.e.n 12
27.f odd 18 6 729.2.e.n 12
36.f odd 6 1 3888.2.a.ba 2
36.h even 6 1 3888.2.a.ba 2
45.h odd 6 1 6075.2.a.bm 2
45.j even 6 1 6075.2.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.2.a.c 2 9.c even 3 1
243.2.a.c 2 9.d odd 6 1
243.2.c.d 4 1.a even 1 1 trivial
243.2.c.d 4 3.b odd 2 1 inner
243.2.c.d 4 9.c even 3 1 inner
243.2.c.d 4 9.d odd 6 1 inner
729.2.e.n 12 27.e even 9 6
729.2.e.n 12 27.f odd 18 6
3888.2.a.ba 2 36.f odd 6 1
3888.2.a.ba 2 36.h even 6 1
6075.2.a.bm 2 45.h odd 6 1
6075.2.a.bm 2 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(243, [\chi])\):

\( T_{2}^{4} + 3T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$13$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T + 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$29$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$31$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$43$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$53$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$73$ \( (T - 2)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$89$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 17 T + 289)^{2} \) Copy content Toggle raw display
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