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Properties

Label 243.4.c.b
Level $243$
Weight $4$
Character orbit 243.c
Analytic conductor $14.337$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [243,4,Mod(82,243)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("243.82"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(243, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 243 = 3^{5} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 243.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,8,0,0,-17] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.3374641314\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content 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

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + 8 \zeta_{6} q^{4} + (17 \zeta_{6} - 17) q^{7} + 19 \zeta_{6} q^{13} + (64 \zeta_{6} - 64) q^{16} - 163 q^{19} + ( - 125 \zeta_{6} + 125) q^{25} - 136 q^{28} + 289 \zeta_{6} q^{31} - 433 q^{37} + (449 \zeta_{6} - 449) q^{43}+ \cdots + ( - 523 \zeta_{6} + 523) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{4} - 17 q^{7} + 19 q^{13} - 64 q^{16} - 326 q^{19} + 125 q^{25} - 272 q^{28} + 289 q^{31} - 866 q^{37} - 449 q^{43} + 54 q^{49} - 152 q^{52} - 182 q^{61} - 1024 q^{64} + 880 q^{67} + 2380 q^{73}+ \cdots + 523 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-\zeta_{6}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
0.500000 − 0.866025i
0.500000 + 0.866025i
0 0 4.00000 − 6.92820i 0 0 −8.50000 − 14.7224i 0 0 0
163.1 0 0 4.00000 + 6.92820i 0 0 −8.50000 + 14.7224i 0 0 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
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 243.4.c.b 2
3.b odd 2 1 CM 243.4.c.b 2
9.c even 3 1 243.4.a.c ✓ 1
9.c even 3 1 inner 243.4.c.b 2
9.d odd 6 1 243.4.a.c ✓ 1
9.d odd 6 1 inner 243.4.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.4.a.c ✓ 1 9.c even 3 1
243.4.a.c ✓ 1 9.d odd 6 1
243.4.c.b 2 1.a even 1 1 trivial
243.4.c.b 2 3.b odd 2 1 CM
243.4.c.b 2 9.c even 3 1 inner
243.4.c.b 2 9.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(243, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{7}^{2} + 17T_{7} + 289 \) 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
$7$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 19T + 361 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 163)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 289T + 83521 \) Copy content Toggle raw display
$37$ \( (T + 433)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 449T + 201601 \) 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
$61$ \( T^{2} + 182T + 33124 \) Copy content Toggle raw display
$67$ \( T^{2} - 880T + 774400 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 1190)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 1387 T + 1923769 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 523T + 273529 \) Copy content Toggle raw display
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