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Properties

Label 1458.2.c.h
Level $1458$
Weight $2$
Character orbit 1458.c
Analytic conductor $11.642$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1458,2,Mod(487,1458)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1458, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1458.487");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1458 = 2 \cdot 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1458.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: \(11.6421886147\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{7} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} - \beta_1 q^{4} + ( - \beta_{2} + \beta_1) q^{5} + ( - \beta_{10} + \beta_{9} + \cdots - 2 \beta_{3}) q^{7}+ \cdots - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} - \beta_1 q^{4} + ( - \beta_{2} + \beta_1) q^{5} + ( - \beta_{10} + \beta_{9} + \cdots - 2 \beta_{3}) q^{7}+ \cdots + ( - 2 \beta_{10} + 2 \beta_{8} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} - 6 q^{4} + 6 q^{5} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{2} - 6 q^{4} + 6 q^{5} - 12 q^{8} + 12 q^{10} + 6 q^{11} - 6 q^{13} - 6 q^{16} - 24 q^{17} + 12 q^{19} + 6 q^{20} - 6 q^{22} + 12 q^{23} - 6 q^{25} - 12 q^{26} + 6 q^{29} + 6 q^{31} + 6 q^{32} - 12 q^{34} - 24 q^{35} + 6 q^{38} - 6 q^{40} + 24 q^{41} + 6 q^{43} - 12 q^{44} + 24 q^{46} + 18 q^{47} - 6 q^{49} + 6 q^{50} - 6 q^{52} - 48 q^{53} - 36 q^{55} - 6 q^{58} + 12 q^{59} + 6 q^{61} + 12 q^{62} + 12 q^{64} + 12 q^{65} + 24 q^{67} + 12 q^{68} - 12 q^{70} + 12 q^{71} - 48 q^{73} - 6 q^{76} + 12 q^{77} + 12 q^{79} - 12 q^{80} + 48 q^{82} + 18 q^{83} - 6 q^{86} - 6 q^{88} - 24 q^{89} - 60 q^{91} + 12 q^{92} - 18 q^{94} - 6 q^{95} - 6 q^{97} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

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

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

\(\zeta_{36}\)\(=\) \( ( 3\beta_{11} + 2\beta_{10} + 2\beta_{8} + \beta_{7} - \beta_{6} + 3\beta_{5} + 3\beta_{4} + 4\beta_{3} + 2\beta_{2} ) / 9 \) Copy content Toggle raw display
\(\zeta_{36}^{2}\)\(=\) \( ( \beta_{10} - \beta_{8} - \beta_{7} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{3}\)\(=\) \( ( \beta_{10} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{3} + \beta_{2} ) / 9 \) Copy content Toggle raw display
\(\zeta_{36}^{4}\)\(=\) \( ( \beta_{8} + \beta_{7} + \beta_{6} + \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{5}\)\(=\) \( ( -\beta_{10} + 3\beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - 3\beta_{5} - 2\beta_{3} + 2\beta_{2} ) / 9 \) Copy content Toggle raw display
\(\zeta_{36}^{6}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{36}^{7}\)\(=\) \( ( 3\beta_{11} + \beta_{10} + 3\beta_{9} + \beta_{8} + 2\beta_{7} - 2\beta_{6} + 3\beta_{5} + 2\beta_{3} + 4\beta_{2} ) / 9 \) Copy content Toggle raw display
\(\zeta_{36}^{8}\)\(=\) \( ( -\beta_{8} + \beta_{6} - \beta_{3} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{9}\)\(=\) \( ( -\beta_{10} - \beta_{8} - 2\beta_{7} + 2\beta_{6} + \beta_{3} + 2\beta_{2} ) / 9 \) Copy content Toggle raw display
\(\zeta_{36}^{10}\)\(=\) \( ( -\beta_{10} + \beta_{8} + \beta_{6} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{11}\)\(=\) \( ( 3 \beta_{11} - 2 \beta_{10} + 3 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} - 3 \beta_{5} + \cdots + 4 \beta_{2} ) / 9 \) 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 \(\left(\mathbb{Z}/1458\mathbb{Z}\right)^\times\).

\(n\) \(731\)
\(\chi(n)\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
487.1
−0.642788 + 0.766044i
−0.342020 − 0.939693i
0.984808 + 0.173648i
0.642788 − 0.766044i
0.342020 + 0.939693i
−0.984808 − 0.173648i
−0.642788 − 0.766044i
−0.342020 + 0.939693i
0.984808 − 0.173648i
0.642788 + 0.766044i
0.342020 − 0.939693i
−0.984808 + 0.173648i
0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.30572 − 2.26157i 0 2.56729 − 4.44667i −1.00000 0 −2.61144
487.2 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.192377 − 0.333207i 0 −0.474416 + 0.821712i −1.00000 0 −0.384754
487.3 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.400019 + 0.692853i 0 −2.09287 + 3.62496i −1.00000 0 0.800038
487.4 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.426333 + 0.738430i 0 −0.687903 + 1.19148i −1.00000 0 0.852666
487.5 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 1.53967 + 2.66679i 0 0.127119 − 0.220177i −1.00000 0 3.07935
487.6 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 2.13207 + 3.69285i 0 0.560783 − 0.971305i −1.00000 0 4.26414
973.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.30572 + 2.26157i 0 2.56729 + 4.44667i −1.00000 0 −2.61144
973.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.192377 + 0.333207i 0 −0.474416 − 0.821712i −1.00000 0 −0.384754
973.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.400019 − 0.692853i 0 −2.09287 − 3.62496i −1.00000 0 0.800038
973.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.426333 − 0.738430i 0 −0.687903 − 1.19148i −1.00000 0 0.852666
973.5 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.53967 − 2.66679i 0 0.127119 + 0.220177i −1.00000 0 3.07935
973.6 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 2.13207 − 3.69285i 0 0.560783 + 0.971305i −1.00000 0 4.26414
\(n\): 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1458.2.c.h 12
3.b odd 2 1 1458.2.c.e 12
9.c even 3 1 1458.2.a.e ✓ 6
9.c even 3 1 inner 1458.2.c.h 12
9.d odd 6 1 1458.2.a.h yes 6
9.d odd 6 1 1458.2.c.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1458.2.a.e ✓ 6 9.c even 3 1
1458.2.a.h yes 6 9.d odd 6 1
1458.2.c.e 12 3.b odd 2 1
1458.2.c.e 12 9.d odd 6 1
1458.2.c.h 12 1.a even 1 1 trivial
1458.2.c.h 12 9.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 6 T_{5}^{11} + 36 T_{5}^{10} - 84 T_{5}^{9} + 297 T_{5}^{8} - 540 T_{5}^{7} + 1782 T_{5}^{6} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(1458, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 6 T^{11} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{12} + 24 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{12} - 6 T^{11} + \cdots + 5184 \) Copy content Toggle raw display
$13$ \( T^{12} + 6 T^{11} + \cdots + 32761 \) Copy content Toggle raw display
$17$ \( (T^{6} + 12 T^{5} + \cdots - 963)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 6 T^{5} + \cdots - 3176)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 12 T^{11} + \cdots + 59351616 \) Copy content Toggle raw display
$29$ \( T^{12} - 6 T^{11} + \cdots + 30437289 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 2870387776 \) Copy content Toggle raw display
$37$ \( (T^{6} - 96 T^{4} + \cdots + 8317)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 1913975001 \) Copy content Toggle raw display
$43$ \( T^{12} - 6 T^{11} + \cdots + 87616 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 574848576 \) Copy content Toggle raw display
$53$ \( (T^{6} + 24 T^{5} + \cdots - 43083)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 962985024 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 2591115409 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 1134881344 \) Copy content Toggle raw display
$71$ \( (T^{6} - 6 T^{5} + \cdots - 59688)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 24 T^{5} + \cdots + 252361)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} - 12 T^{11} + \cdots + 13957696 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 2237668416 \) Copy content Toggle raw display
$89$ \( (T^{6} + 12 T^{5} + \cdots + 71613)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 298206642889 \) Copy content Toggle raw display
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