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Properties

Label 81.6.c.f
Level $81$
Weight $6$
Character orbit 81.c
Analytic conductor $12.991$
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] = [81,6,Mod(28,81)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(81, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.28");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 81.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: \(12.9910894049\)
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: \( 3^{3} \)
Twist minimal: no (minimal twist has level 27)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (22 \beta_1 - 22) q^{4} + ( - 8 \beta_{3} + 8 \beta_{2}) q^{5} - 167 \beta_1 q^{7} - 10 \beta_{3} q^{8} + 432 q^{10} + 104 \beta_{2} q^{11} + ( - 235 \beta_1 + 235) q^{13} + ( - 167 \beta_{3} + 167 \beta_{2}) q^{14}+ \cdots + 11082 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 44 q^{4} - 334 q^{7} + 1728 q^{10} + 470 q^{13} + 2488 q^{16} + 5444 q^{19} + 11232 q^{22} - 662 q^{25} + 14696 q^{28} - 7000 q^{31} - 2592 q^{34} + 52460 q^{37} + 8640 q^{40} - 208 q^{43} - 70848 q^{46}+ \cdots + 99206 q^{97}+O(q^{100}) \) 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
\(\beta_{2}\)\(=\) \( ( 3\nu^{3} + 6\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{3} + 12\nu ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} + 4\beta_{2} ) / 9 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
1.22474 + 0.707107i
−1.22474 − 0.707107i
1.22474 − 0.707107i
−1.22474 + 0.707107i
−3.67423 − 6.36396i 0 −11.0000 + 19.0526i −29.3939 + 50.9117i 0 −83.5000 − 144.626i −73.4847 0 432.000
28.2 3.67423 + 6.36396i 0 −11.0000 + 19.0526i 29.3939 − 50.9117i 0 −83.5000 − 144.626i 73.4847 0 432.000
55.1 −3.67423 + 6.36396i 0 −11.0000 − 19.0526i −29.3939 − 50.9117i 0 −83.5000 + 144.626i −73.4847 0 432.000
55.2 3.67423 − 6.36396i 0 −11.0000 − 19.0526i 29.3939 + 50.9117i 0 −83.5000 + 144.626i 73.4847 0 432.000
\(n\): 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 81.6.c.f 4
3.b odd 2 1 inner 81.6.c.f 4
9.c even 3 1 27.6.a.c ✓ 2
9.c even 3 1 inner 81.6.c.f 4
9.d odd 6 1 27.6.a.c ✓ 2
9.d odd 6 1 inner 81.6.c.f 4
36.f odd 6 1 432.6.a.o 2
36.h even 6 1 432.6.a.o 2
45.h odd 6 1 675.6.a.j 2
45.j even 6 1 675.6.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.6.a.c ✓ 2 9.c even 3 1
27.6.a.c ✓ 2 9.d odd 6 1
81.6.c.f 4 1.a even 1 1 trivial
81.6.c.f 4 3.b odd 2 1 inner
81.6.c.f 4 9.c even 3 1 inner
81.6.c.f 4 9.d odd 6 1 inner
432.6.a.o 2 36.f odd 6 1
432.6.a.o 2 36.h even 6 1
675.6.a.j 2 45.h odd 6 1
675.6.a.j 2 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 54T_{2}^{2} + 2916 \) acting on \(S_{6}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 54T^{2} + 2916 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 3456 T^{2} + 11943936 \) Copy content Toggle raw display
$7$ \( (T^{2} + 167 T + 27889)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 341130756096 \) Copy content Toggle raw display
$13$ \( (T^{2} - 235 T + 55225)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 31104)^{2} \) Copy content Toggle raw display
$19$ \( (T - 1361)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 33750708535296 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 48922361856 \) Copy content Toggle raw display
$31$ \( (T^{2} + 3500 T + 12250000)^{2} \) Copy content Toggle raw display
$37$ \( (T - 13115)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{2} + 104 T + 10816)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{2} - 1119744)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 89\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{2} - 7393 T + 54656449)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 38861 T + 1510177321)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 6096384)^{2} \) Copy content Toggle raw display
$73$ \( (T - 5465)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 82903 T + 6872907409)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 31\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{2} - 8058455424)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 49603 T + 2460457609)^{2} \) Copy content Toggle raw display
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