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Properties

Label 1458.2.c.d
Level $1458$
Weight $2$
Character orbit 1458.c
Analytic conductor $11.642$
Analytic rank $0$
Dimension $6$
CM no
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 54)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_1 - 1) q^{4} + (\beta_{5} + \beta_1 - 1) q^{5} + (\beta_{2} + 2 \beta_1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_1 - 1) q^{4} + (\beta_{5} + \beta_1 - 1) q^{5} + (\beta_{2} + 2 \beta_1) q^{7} - q^{8} + (\beta_{4} + \beta_{3} - 1) q^{10} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - \beta_1) q^{11} + (2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 3) q^{13} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{14} - \beta_1 q^{16} + ( - \beta_{4} + 3 \beta_{3} + 2) q^{17} + (3 \beta_{4} + 4 \beta_{3} - 3) q^{19} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{20} + (3 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{22} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{23} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{25} + (2 \beta_{4} + 3) q^{26} + ( - \beta_{3} - 2) q^{28} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - \beta_1) q^{29} + ( - 3 \beta_{5} - 4 \beta_1 + 4) q^{31} + ( - \beta_1 + 1) q^{32} + (\beta_{5} - \beta_{4} - \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{34} + (3 \beta_{4} + 2 \beta_{3} - 1) q^{35} + (\beta_{4} - 2 \beta_{3} - 5) q^{37} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + \beta_{2} - 3 \beta_1) q^{38} + ( - \beta_{5} - \beta_1 + 1) q^{40} + (3 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{41} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{43} + (3 \beta_{4} + \beta_{3} + 1) q^{44} + ( - 2 \beta_{3} + 1) q^{46} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1) q^{47} + ( - \beta_{5} - 3 \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{49} + ( - 2 \beta_{5} + \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{50} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 3 \beta_1) q^{52} + ( - \beta_{4} + 2 \beta_{3} - 2) q^{53} + (4 \beta_{4} - \beta_{3} - 3) q^{55} + ( - \beta_{2} - 2 \beta_1) q^{56} + (2 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{58} + (4 \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{59} + ( - 5 \beta_{5} + 5 \beta_{4} + 5 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{61} + ( - 3 \beta_{4} - 3 \beta_{3} + 4) q^{62} + q^{64} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 4 \beta_{2} + \beta_1) q^{65} + ( - 3 \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{67} + (\beta_{5} - 4 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 2) q^{68} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - \beta_{2} - \beta_1) q^{70} + ( - \beta_{4} - 6 \beta_{3} - 4) q^{71} + (5 \beta_{4} - 2 \beta_{3} - 1) q^{73} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} - 5 \beta_1) q^{74} + ( - 3 \beta_{5} - \beta_{3} + \beta_{2} - 3 \beta_1 + 3) q^{76} + (7 \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{77} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 8 \beta_{2}) q^{79} + ( - \beta_{4} - \beta_{3} + 1) q^{80} + (3 \beta_{4} + 5 \beta_{3} - 1) q^{82} + ( - 4 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} - 2 \beta_{2}) q^{83} + (7 \beta_{5} - 5 \beta_{3} + 5 \beta_{2}) q^{85} + ( - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 4) q^{86} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + \beta_1) q^{88} + ( - \beta_{4} + 3 \beta_{3} + 5) q^{89} + (4 \beta_{4} + 5 \beta_{3} + 4) q^{91} + ( - 2 \beta_{2} + \beta_1) q^{92} + ( - \beta_{5} - \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{94} + ( - 5 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 10 \beta_1 + 10) q^{95} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 6 \beta_1) q^{97} + ( - \beta_{4} - 4 \beta_{3} + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} - 3 q^{4} - 3 q^{5} + 6 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{2} - 3 q^{4} - 3 q^{5} + 6 q^{7} - 6 q^{8} - 6 q^{10} - 3 q^{11} + 9 q^{13} - 6 q^{14} - 3 q^{16} + 12 q^{17} - 18 q^{19} - 3 q^{20} + 3 q^{22} + 3 q^{23} + 6 q^{25} + 18 q^{26} - 12 q^{28} - 3 q^{29} + 12 q^{31} + 3 q^{32} + 6 q^{34} - 6 q^{35} - 30 q^{37} - 9 q^{38} + 3 q^{40} - 3 q^{41} + 12 q^{43} + 6 q^{44} + 6 q^{46} + 9 q^{47} + 3 q^{49} - 6 q^{50} + 9 q^{52} - 12 q^{53} - 18 q^{55} - 6 q^{56} + 3 q^{58} + 3 q^{59} - 6 q^{61} + 24 q^{62} + 6 q^{64} + 3 q^{65} + 3 q^{67} - 6 q^{68} - 3 q^{70} - 24 q^{71} - 6 q^{73} - 15 q^{74} + 9 q^{76} + 3 q^{77} + 6 q^{80} - 6 q^{82} - 12 q^{86} + 3 q^{88} + 30 q^{89} + 24 q^{91} + 3 q^{92} - 9 q^{94} + 30 q^{95} + 18 q^{97} + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{18}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{18}^{5} + \zeta_{18} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{18}^{5} + \zeta_{18}^{4} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} \) Copy content Toggle raw display

Display \(\zeta_{18}^j\) in terms of \(\beta_i\)

\(\zeta_{18}\)\(=\) \( ( \beta_{5} + \beta_{4} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{2}\)\(=\) \( ( -2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{18}^{4}\)\(=\) \( ( -\beta_{5} + 2\beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{5}\)\(=\) \( ( -\beta_{5} - \beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{18}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1458\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
487.1
−0.766044 − 0.642788i
−0.173648 + 0.984808i
0.939693 − 0.342020i
−0.766044 + 0.642788i
−0.173648 − 0.984808i
0.939693 + 0.342020i
0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.26604 − 2.19285i 0 1.17365 − 2.03282i −1.00000 0 −2.53209
487.2 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.673648 − 1.16679i 0 0.0603074 − 0.104455i −1.00000 0 −1.34730
487.3 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.439693 + 0.761570i 0 1.76604 − 3.05888i −1.00000 0 0.879385
973.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.26604 + 2.19285i 0 1.17365 + 2.03282i −1.00000 0 −2.53209
973.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.673648 + 1.16679i 0 0.0603074 + 0.104455i −1.00000 0 −1.34730
973.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.439693 − 0.761570i 0 1.76604 + 3.05888i −1.00000 0 0.879385
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 487.3
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.d 6
3.b odd 2 1 1458.2.c.a 6
9.c even 3 1 1458.2.a.a 3
9.c even 3 1 inner 1458.2.c.d 6
9.d odd 6 1 1458.2.a.d 3
9.d odd 6 1 1458.2.c.a 6
27.e even 9 2 54.2.e.a ✓ 6
27.e even 9 2 486.2.e.b 6
27.e even 9 2 486.2.e.d 6
27.f odd 18 2 162.2.e.a 6
27.f odd 18 2 486.2.e.a 6
27.f odd 18 2 486.2.e.c 6
108.j odd 18 2 432.2.u.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.e.a ✓ 6 27.e even 9 2
162.2.e.a 6 27.f odd 18 2
432.2.u.a 6 108.j odd 18 2
486.2.e.a 6 27.f odd 18 2
486.2.e.b 6 27.e even 9 2
486.2.e.c 6 27.f odd 18 2
486.2.e.d 6 27.e even 9 2
1458.2.a.a 3 9.c even 3 1
1458.2.a.d 3 9.d odd 6 1
1458.2.c.a 6 3.b odd 2 1
1458.2.c.a 6 9.d odd 6 1
1458.2.c.d 6 1.a even 1 1 trivial
1458.2.c.d 6 9.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 3T_{5}^{5} + 9T_{5}^{4} + 6T_{5}^{3} + 9T_{5}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(1458, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{5} + 9 T^{4} + 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} - 6 T^{5} + 27 T^{4} - 52 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} + 3 T^{5} + 27 T^{4} + \cdots + 3249 \) Copy content Toggle raw display
$13$ \( T^{6} - 9 T^{5} + 66 T^{4} - 169 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$17$ \( (T^{3} - 6 T^{2} - 27 T + 159)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 9 T^{2} - 12 T - 179)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} - 3 T^{5} + 18 T^{4} + 21 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$29$ \( T^{6} + 3 T^{5} + 27 T^{4} - 48 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$31$ \( T^{6} - 12 T^{5} + 123 T^{4} + \cdots + 5041 \) Copy content Toggle raw display
$37$ \( (T^{3} + 15 T^{2} + 54 T - 17)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 3 T^{5} + 63 T^{4} + \cdots + 47961 \) Copy content Toggle raw display
$43$ \( T^{6} - 12 T^{5} + 132 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$47$ \( T^{6} - 9 T^{5} + 63 T^{4} - 144 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$53$ \( (T^{3} + 6 T^{2} - 9 T + 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 3 T^{5} + 45 T^{4} + \cdots + 3249 \) Copy content Toggle raw display
$61$ \( T^{6} + 6 T^{5} + 87 T^{4} + \cdots + 2809 \) Copy content Toggle raw display
$67$ \( T^{6} - 3 T^{5} + 33 T^{4} + 74 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( (T^{3} + 12 T^{2} - 45 T - 327)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 3 T^{2} - 114 T - 269)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 147 T^{4} - 1366 T^{3} + \cdots + 466489 \) Copy content Toggle raw display
$83$ \( T^{6} + 36 T^{4} - 144 T^{3} + \cdots + 5184 \) Copy content Toggle raw display
$89$ \( (T^{3} - 15 T^{2} + 36 T + 159)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} - 18 T^{5} + 237 T^{4} + \cdots + 16129 \) Copy content Toggle raw display
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