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Properties

Label 729.2.c.c
Level $729$
Weight $2$
Character orbit 729.c
Analytic conductor $5.821$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [729,2,Mod(244,729)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(729, 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("729.244");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.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: \(5.82109430735\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{36})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{3} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{10} - \beta_{8} - \beta_{7}) q^{2} - \beta_{11} q^{4} + (\beta_{10} + \beta_{5} - \beta_{2}) q^{5} + ( - \beta_{11} + \beta_{9} + \beta_{6} - \beta_{3} + 2 \beta_1) q^{7} + ( - \beta_{8} - \beta_{7} + \beta_{5}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{10} - \beta_{8} - \beta_{7}) q^{2} - \beta_{11} q^{4} + (\beta_{10} + \beta_{5} - \beta_{2}) q^{5} + ( - \beta_{11} + \beta_{9} + \beta_{6} - \beta_{3} + 2 \beta_1) q^{7} + ( - \beta_{8} - \beta_{7} + \beta_{5}) q^{8} + ( - 2 \beta_{9} - 3 \beta_{6} - 2) q^{10} + (\beta_{4} - 2 \beta_{2}) q^{11} + ( - \beta_{11} + 3 \beta_{6} - 3 \beta_{3} - 2 \beta_1 + 2) q^{13} + (3 \beta_{10} - \beta_{7} + \beta_{4}) q^{14} + ( - 2 \beta_{11} + 2 \beta_{9} + 2 \beta_{6} - \beta_{3} + 2 \beta_1) q^{16} + ( - 2 \beta_{8} + 2 \beta_{7} + \beta_{5}) q^{17} + ( - 2 \beta_{9} - \beta_{6} - 4) q^{19} + ( - 2 \beta_{10} + 2 \beta_{8} + 2 \beta_{7} + \beta_{4} + \beta_{2}) q^{20} + ( - 2 \beta_{11} - \beta_{6} + \beta_{3} - \beta_1 + 1) q^{22} + ( - 3 \beta_{10} + 5 \beta_{7} - 5 \beta_{4}) q^{23} + (3 \beta_{11} - 3 \beta_{9} - 3 \beta_{6} - 2 \beta_{3}) q^{25} + ( - \beta_{8} - 4 \beta_{7} - 2 \beta_{5}) q^{26} + ( - \beta_{9} - 2 \beta_{6} - 1) q^{28} + (2 \beta_{10} - 2 \beta_{8} - 2 \beta_{7} - \beta_{4} + \beta_{2}) q^{29} + ( - 2 \beta_{11} + 2 \beta_{6} - 2 \beta_{3} - 5 \beta_1 + 5) q^{31} + (2 \beta_{10} - \beta_{7} + 3 \beta_{5} + \beta_{4} - 3 \beta_{2}) q^{32} + ( - \beta_{11} + \beta_{9} + \beta_{6} - 5 \beta_{3}) q^{34} + (2 \beta_{8} + 3 \beta_{7} + 2 \beta_{5}) q^{35} + (\beta_{9} + 2 \beta_{6} - 1) q^{37} + ( - 6 \beta_{10} + 6 \beta_{8} + 6 \beta_{7} - \beta_{4} + \beta_{2}) q^{38} + ( - \beta_{11} - \beta_1 + 1) q^{40} + (3 \beta_{10} - 4 \beta_{7} + \beta_{5} + 4 \beta_{4} - \beta_{2}) q^{41} + ( - 2 \beta_{3} + 2 \beta_1) q^{43} + (\beta_{8} + 4 \beta_{7} - \beta_{5}) q^{44} + (3 \beta_{9} - 2 \beta_{6} + 1) q^{46} + (\beta_{10} - \beta_{8} - \beta_{7} + \beta_{2}) q^{47} + ( - 3 \beta_{11} + 4 \beta_{6} - 4 \beta_{3} - \beta_1 + 1) q^{49} + ( - 3 \beta_{10} - 2 \beta_{7} - 5 \beta_{5} + 2 \beta_{4} + 5 \beta_{2}) q^{50} + ( - 5 \beta_{11} + 5 \beta_{9} + 5 \beta_{6} - \beta_{3} + \beta_1) q^{52} + (2 \beta_{8} - 2 \beta_{7} - 2 \beta_{5}) q^{53} + ( - 2 \beta_{6} - 5) q^{55} + (4 \beta_{10} - 4 \beta_{8} - 4 \beta_{7} + 3 \beta_{4} + 2 \beta_{2}) q^{56} + ( - \beta_{11} + 5 \beta_1 - 5) q^{58} + ( - 5 \beta_{10} - 2 \beta_{5} + 2 \beta_{2}) q^{59} + (4 \beta_{11} - 4 \beta_{9} - 4 \beta_{6} + 6 \beta_{3} + 2 \beta_1) q^{61} + ( - 3 \beta_{8} - 5 \beta_{7}) q^{62} + ( - \beta_{9} - 5 \beta_{6} + 1) q^{64} + (6 \beta_{10} - 6 \beta_{8} - 6 \beta_{7} + \beta_{4} - 4 \beta_{2}) q^{65} + ( - 3 \beta_{11} - 5 \beta_{6} + 5 \beta_{3} + \beta_1 - 1) q^{67} + ( - 3 \beta_{10} + 3 \beta_{7} - 2 \beta_{5} - 3 \beta_{4} + 2 \beta_{2}) q^{68} + (4 \beta_{11} - 4 \beta_{9} - 4 \beta_{6} - 3 \beta_{3} - 5 \beta_1) q^{70} + (2 \beta_{8} - 2 \beta_{7}) q^{71} + ( - 6 \beta_{9} - 3 \beta_{6} + 2) q^{73} + ( - \beta_{4} - 2 \beta_{2}) q^{74} + (3 \beta_{11} + 2 \beta_{6} - 2 \beta_{3} - 3 \beta_1 + 3) q^{76} + ( - 2 \beta_{10} + 3 \beta_{7} + 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{2}) q^{77} + ( - 5 \beta_{11} + 5 \beta_{9} + 5 \beta_{6} - 3 \beta_{3} + 8 \beta_1) q^{79} + (4 \beta_{8} + 6 \beta_{7} + 3 \beta_{5}) q^{80} + ( - 4 \beta_{9} - \beta_{6} - 2) q^{82} + ( - 3 \beta_{10} + 3 \beta_{8} + 3 \beta_{7} - 8 \beta_{4} - 4 \beta_{2}) q^{83} + (5 \beta_{11} - \beta_{6} + \beta_{3} - 3 \beta_1 + 3) q^{85} + (2 \beta_{10} - 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2}) q^{86} + ( - 4 \beta_{11} + 4 \beta_{9} + 4 \beta_{6} - 7 \beta_1) q^{88} + (6 \beta_{8} + 3 \beta_{7} - 5 \beta_{5}) q^{89} + (\beta_{9} + 7 \beta_{6}) q^{91} + ( - 2 \beta_{10} + 2 \beta_{8} + 2 \beta_{7} - 5 \beta_{4} + 2 \beta_{2}) q^{92} + (\beta_{6} - \beta_{3} + 2 \beta_1 - 2) q^{94} + ( - 6 \beta_{10} - 2 \beta_{7} - 5 \beta_{5} + 2 \beta_{4} + 5 \beta_{2}) q^{95} + (3 \beta_{11} - 3 \beta_{9} - 3 \beta_{6} + 7 \beta_{3} + 2 \beta_1) q^{97} + (2 \beta_{8} - 2 \beta_{7} - \beta_{5}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{7} - 24 q^{10} + 12 q^{13} + 12 q^{16} - 48 q^{19} + 6 q^{22} - 12 q^{28} + 30 q^{31} - 12 q^{37} + 6 q^{40} + 12 q^{43} + 12 q^{46} + 6 q^{49} + 6 q^{52} - 60 q^{55} - 30 q^{58} + 12 q^{61} + 12 q^{64} - 6 q^{67} - 30 q^{70} + 24 q^{73} + 18 q^{76} + 48 q^{79} - 24 q^{82} + 18 q^{85} - 42 q^{88} - 12 q^{94} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{36}^{6} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{36}^{9} + \zeta_{36}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{36}^{10} + \zeta_{36}^{2} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{36}^{11} + \zeta_{36} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{36}^{9} + 2\zeta_{36}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{36}^{8} + \zeta_{36}^{4} + \zeta_{36}^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{36}^{7} + \zeta_{36}^{5} + \zeta_{36} \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -\zeta_{36}^{11} + \zeta_{36}^{7} \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( -\zeta_{36}^{10} + \zeta_{36}^{8} \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( -\zeta_{36}^{11} - \zeta_{36}^{7} + \zeta_{36} \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( -\zeta_{36}^{10} - \zeta_{36}^{8} + \zeta_{36}^{2} \) Copy content Toggle raw display

Display \(\zeta_{36}^j\) in terms of \(\beta_i\)

\(\zeta_{36}\)\(=\) \( ( \beta_{10} + \beta_{8} + 2\beta_{4} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{2}\)\(=\) \( ( \beta_{11} + \beta_{9} + 2\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{3}\)\(=\) \( ( \beta_{5} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{4}\)\(=\) \( ( -2\beta_{11} + \beta_{9} + 3\beta_{6} - \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{5}\)\(=\) \( ( -2\beta_{10} + \beta_{8} + 3\beta_{7} - \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{6}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{36}^{7}\)\(=\) \( ( -\beta_{10} + 2\beta_{8} + \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{8}\)\(=\) \( ( -\beta_{11} + 2\beta_{9} + \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{9}\)\(=\) \( ( -\beta_{5} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{10}\)\(=\) \( ( -\beta_{11} - \beta_{9} + \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{11}\)\(=\) \( ( -\beta_{10} - \beta_{8} + \beta_{4} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{36}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/729\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
244.1
0.984808 + 0.173648i
0.642788 − 0.766044i
0.342020 + 0.939693i
−0.342020 − 0.939693i
−0.642788 + 0.766044i
−0.984808 − 0.173648i
0.984808 − 0.173648i
0.642788 + 0.766044i
0.342020 − 0.939693i
−0.342020 + 0.939693i
−0.642788 − 0.766044i
−0.984808 + 0.173648i
−0.984808 − 1.70574i 0 −0.939693 + 1.62760i 1.85083 − 3.20574i 0 1.17365 + 2.03282i −0.237565 0 −7.29086
244.2 −0.642788 − 1.11334i 0 0.173648 − 0.300767i −0.223238 + 0.386659i 0 1.76604 + 3.05888i −3.01763 0 0.573978
244.3 −0.342020 − 0.592396i 0 0.766044 − 1.32683i −0.524005 + 0.907604i 0 0.0603074 + 0.104455i −2.41609 0 0.716881
244.4 0.342020 + 0.592396i 0 0.766044 − 1.32683i 0.524005 − 0.907604i 0 0.0603074 + 0.104455i 2.41609 0 0.716881
244.5 0.642788 + 1.11334i 0 0.173648 − 0.300767i 0.223238 − 0.386659i 0 1.76604 + 3.05888i 3.01763 0 0.573978
244.6 0.984808 + 1.70574i 0 −0.939693 + 1.62760i −1.85083 + 3.20574i 0 1.17365 + 2.03282i 0.237565 0 −7.29086
487.1 −0.984808 + 1.70574i 0 −0.939693 − 1.62760i 1.85083 + 3.20574i 0 1.17365 − 2.03282i −0.237565 0 −7.29086
487.2 −0.642788 + 1.11334i 0 0.173648 + 0.300767i −0.223238 − 0.386659i 0 1.76604 − 3.05888i −3.01763 0 0.573978
487.3 −0.342020 + 0.592396i 0 0.766044 + 1.32683i −0.524005 − 0.907604i 0 0.0603074 − 0.104455i −2.41609 0 0.716881
487.4 0.342020 − 0.592396i 0 0.766044 + 1.32683i 0.524005 + 0.907604i 0 0.0603074 − 0.104455i 2.41609 0 0.716881
487.5 0.642788 − 1.11334i 0 0.173648 + 0.300767i 0.223238 + 0.386659i 0 1.76604 − 3.05888i 3.01763 0 0.573978
487.6 0.984808 − 1.70574i 0 −0.939693 − 1.62760i −1.85083 − 3.20574i 0 1.17365 − 2.03282i 0.237565 0 −7.29086
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 487.6
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 729.2.c.c 12
3.b odd 2 1 inner 729.2.c.c 12
9.c even 3 1 729.2.a.c ✓ 6
9.c even 3 1 inner 729.2.c.c 12
9.d odd 6 1 729.2.a.c ✓ 6
9.d odd 6 1 inner 729.2.c.c 12
27.e even 9 2 729.2.e.m 12
27.e even 9 2 729.2.e.q 12
27.e even 9 2 729.2.e.r 12
27.f odd 18 2 729.2.e.m 12
27.f odd 18 2 729.2.e.q 12
27.f odd 18 2 729.2.e.r 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
729.2.a.c ✓ 6 9.c even 3 1
729.2.a.c ✓ 6 9.d odd 6 1
729.2.c.c 12 1.a even 1 1 trivial
729.2.c.c 12 3.b odd 2 1 inner
729.2.c.c 12 9.c even 3 1 inner
729.2.c.c 12 9.d odd 6 1 inner
729.2.e.m 12 27.e even 9 2
729.2.e.m 12 27.f odd 18 2
729.2.e.q 12 27.e even 9 2
729.2.e.q 12 27.f odd 18 2
729.2.e.r 12 27.e even 9 2
729.2.e.r 12 27.f odd 18 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 6T_{2}^{10} + 27T_{2}^{8} + 48T_{2}^{6} + 63T_{2}^{4} + 27T_{2}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(729, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 6 T^{10} + 27 T^{8} + 48 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 15 T^{10} + 207 T^{8} + 264 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( (T^{6} - 6 T^{5} + 27 T^{4} - 52 T^{3} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 42 T^{10} + 1359 T^{8} + \cdots + 1172889 \) Copy content Toggle raw display
$13$ \( (T^{6} - 6 T^{5} + 45 T^{4} - 88 T^{3} + \cdots + 5041)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 81 T^{4} + 1755 T^{2} + \cdots - 9747)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 12 T^{2} + 39 T + 19)^{4} \) Copy content Toggle raw display
$23$ \( T^{12} + 114 T^{10} + 9747 T^{8} + \cdots + 751689 \) Copy content Toggle raw display
$29$ \( T^{12} + 51 T^{10} + 1791 T^{8} + \cdots + 16867449 \) Copy content Toggle raw display
$31$ \( (T^{6} - 15 T^{5} + 162 T^{4} - 799 T^{3} + \cdots + 5329)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 3 T^{2} - 6 T - 17)^{4} \) Copy content Toggle raw display
$41$ \( T^{12} + 87 T^{10} + 5715 T^{8} + \cdots + 71014329 \) Copy content Toggle raw display
$43$ \( (T^{6} - 6 T^{5} + 36 T^{4} - 16 T^{3} + \cdots + 64)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 15 T^{10} + 171 T^{8} + 804 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$53$ \( (T^{6} - 108 T^{4} + 2592 T^{2} + \cdots - 15552)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 186 T^{10} + 33039 T^{8} + \cdots + 9 \) Copy content Toggle raw display
$61$ \( (T^{6} - 6 T^{5} + 108 T^{4} - 160 T^{3} + \cdots + 87616)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 3 T^{5} + 153 T^{4} - 934 T^{3} + \cdots + 63001)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 72 T^{4} + 1296 T^{2} + \cdots - 1728)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 6 T^{2} - 69 T - 89)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} - 24 T^{5} + 441 T^{4} + \cdots + 11449)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 438 T^{10} + \cdots + 2405238079689 \) Copy content Toggle raw display
$89$ \( (T^{6} - 387 T^{4} + 16146 T^{2} + \cdots - 7803)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 6 T^{5} + 135 T^{4} + \cdots + 418609)^{2} \) Copy content Toggle raw display
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