Properties

Label 540.1.bf.a
Level $540$
Weight $1$
Character orbit 540.bf
Analytic conductor $0.269$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -20
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [540,1,Mod(79,540)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(540, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 10, 9]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("540.79");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 540.bf (of order \(18\), degree \(6\), 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: \(0.269495106822\)
Analytic rank: \(0\)
Dimension: \(6\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.5020969537440000.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{18} q^{2} - \zeta_{18}^{5} q^{3} + \zeta_{18}^{2} q^{4} - \zeta_{18}^{5} q^{5} + \zeta_{18}^{6} q^{6} + ( - \zeta_{18}^{3} + \zeta_{18}^{2}) q^{7} - \zeta_{18}^{3} q^{8} - \zeta_{18} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{18} q^{2} - \zeta_{18}^{5} q^{3} + \zeta_{18}^{2} q^{4} - \zeta_{18}^{5} q^{5} + \zeta_{18}^{6} q^{6} + ( - \zeta_{18}^{3} + \zeta_{18}^{2}) q^{7} - \zeta_{18}^{3} q^{8} - \zeta_{18} q^{9} + \zeta_{18}^{6} q^{10} - \zeta_{18}^{7} q^{12} + (\zeta_{18}^{4} - \zeta_{18}^{3}) q^{14} - \zeta_{18} q^{15} + \zeta_{18}^{4} q^{16} + \zeta_{18}^{2} q^{18} - \zeta_{18}^{7} q^{20} + (\zeta_{18}^{8} - \zeta_{18}^{7}) q^{21} + (\zeta_{18}^{4} + 1) q^{23} + \zeta_{18}^{8} q^{24} - \zeta_{18} q^{25} + \zeta_{18}^{6} q^{27} + ( - \zeta_{18}^{5} + \zeta_{18}^{4}) q^{28} + (\zeta_{18}^{8} - \zeta_{18}^{3}) q^{29} + \zeta_{18}^{2} q^{30} - \zeta_{18}^{5} q^{32} + (\zeta_{18}^{8} - \zeta_{18}^{7}) q^{35} - \zeta_{18}^{3} q^{36} + \zeta_{18}^{8} q^{40} + ( - \zeta_{18}^{7} + 1) q^{41} + (\zeta_{18}^{8} + 1) q^{42} - \zeta_{18}^{4} q^{43} + \zeta_{18}^{6} q^{45} + ( - \zeta_{18}^{5} - \zeta_{18}) q^{46} + (\zeta_{18}^{8} + \zeta_{18}^{6}) q^{47} + q^{48} + (\zeta_{18}^{6} - \zeta_{18}^{5} + \zeta_{18}^{4}) q^{49} + \zeta_{18}^{2} q^{50} - \zeta_{18}^{7} q^{54} + (\zeta_{18}^{6} - \zeta_{18}^{5}) q^{56} + (\zeta_{18}^{4} + 1) q^{58} - \zeta_{18}^{3} q^{60} + ( - \zeta_{18}^{3} + \zeta_{18}^{2}) q^{61} + (\zeta_{18}^{4} - \zeta_{18}^{3}) q^{63} + \zeta_{18}^{6} q^{64} + ( - \zeta_{18}^{7} + 1) q^{67} + ( - \zeta_{18}^{5} + 1) q^{69} + (\zeta_{18}^{8} + 1) q^{70} + \zeta_{18}^{4} q^{72} + \zeta_{18}^{6} q^{75} + q^{80} + \zeta_{18}^{2} q^{81} + (\zeta_{18}^{8} - \zeta_{18}) q^{82} + (\zeta_{18}^{2} + 1) q^{83} + ( - \zeta_{18} + 1) q^{84} + \zeta_{18}^{5} q^{86} + (\zeta_{18}^{8} + \zeta_{18}^{4}) q^{87} + (\zeta_{18}^{4} + \zeta_{18}^{2}) q^{89} - \zeta_{18}^{7} q^{90} + (\zeta_{18}^{6} + \zeta_{18}^{2}) q^{92} + ( - \zeta_{18}^{7} + 1) q^{94} - \zeta_{18} q^{96} + ( - \zeta_{18}^{7} + \zeta_{18}^{6} - \zeta_{18}^{5}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{6} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{6} - 3 q^{7} - 3 q^{8} - 3 q^{10} - 3 q^{14} + 6 q^{23} - 3 q^{27} - 3 q^{29} - 3 q^{36} + 6 q^{41} + 6 q^{42} - 3 q^{45} - 3 q^{47} + 6 q^{48} - 3 q^{49} - 3 q^{56} + 6 q^{58} - 3 q^{60} - 3 q^{61} - 3 q^{63} - 3 q^{64} + 6 q^{67} + 6 q^{69} + 6 q^{70} - 3 q^{75} + 6 q^{80} + 6 q^{83} + 6 q^{84} - 3 q^{92} + 6 q^{94} - 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/540\mathbb{Z}\right)^\times\).

\(n\) \(217\) \(271\) \(461\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{18}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−0.173648 + 0.984808i
0.939693 + 0.342020i
−0.766044 0.642788i
−0.766044 + 0.642788i
0.939693 0.342020i
−0.173648 0.984808i
0.173648 0.984808i 0.766044 0.642788i −0.939693 0.342020i 0.766044 0.642788i −0.500000 0.866025i −1.43969 + 0.524005i −0.500000 + 0.866025i 0.173648 0.984808i −0.500000 0.866025i
139.1 −0.939693 0.342020i 0.173648 0.984808i 0.766044 + 0.642788i 0.173648 0.984808i −0.500000 + 0.866025i 0.266044 0.223238i −0.500000 0.866025i −0.939693 0.342020i −0.500000 + 0.866025i
259.1 0.766044 + 0.642788i −0.939693 0.342020i 0.173648 + 0.984808i −0.939693 0.342020i −0.500000 0.866025i −0.326352 + 1.85083i −0.500000 + 0.866025i 0.766044 + 0.642788i −0.500000 0.866025i
319.1 0.766044 0.642788i −0.939693 + 0.342020i 0.173648 0.984808i −0.939693 + 0.342020i −0.500000 + 0.866025i −0.326352 1.85083i −0.500000 0.866025i 0.766044 0.642788i −0.500000 + 0.866025i
439.1 −0.939693 + 0.342020i 0.173648 + 0.984808i 0.766044 0.642788i 0.173648 + 0.984808i −0.500000 0.866025i 0.266044 + 0.223238i −0.500000 + 0.866025i −0.939693 + 0.342020i −0.500000 0.866025i
499.1 0.173648 + 0.984808i 0.766044 + 0.642788i −0.939693 + 0.342020i 0.766044 + 0.642788i −0.500000 + 0.866025i −1.43969 0.524005i −0.500000 0.866025i 0.173648 + 0.984808i −0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
27.e even 9 1 inner
540.bf odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 540.1.bf.a 6
3.b odd 2 1 1620.1.bf.a 6
4.b odd 2 1 540.1.bf.b yes 6
5.b even 2 1 540.1.bf.b yes 6
5.c odd 4 2 2700.1.bj.a 12
12.b even 2 1 1620.1.bf.b 6
15.d odd 2 1 1620.1.bf.b 6
20.d odd 2 1 CM 540.1.bf.a 6
20.e even 4 2 2700.1.bj.a 12
27.e even 9 1 inner 540.1.bf.a 6
27.f odd 18 1 1620.1.bf.a 6
60.h even 2 1 1620.1.bf.a 6
108.j odd 18 1 540.1.bf.b yes 6
108.l even 18 1 1620.1.bf.b 6
135.n odd 18 1 1620.1.bf.b 6
135.p even 18 1 540.1.bf.b yes 6
135.r odd 36 2 2700.1.bj.a 12
540.bb even 18 1 1620.1.bf.a 6
540.bf odd 18 1 inner 540.1.bf.a 6
540.bh even 36 2 2700.1.bj.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.1.bf.a 6 1.a even 1 1 trivial
540.1.bf.a 6 20.d odd 2 1 CM
540.1.bf.a 6 27.e even 9 1 inner
540.1.bf.a 6 540.bf odd 18 1 inner
540.1.bf.b yes 6 4.b odd 2 1
540.1.bf.b yes 6 5.b even 2 1
540.1.bf.b yes 6 108.j odd 18 1
540.1.bf.b yes 6 135.p even 18 1
1620.1.bf.a 6 3.b odd 2 1
1620.1.bf.a 6 27.f odd 18 1
1620.1.bf.a 6 60.h even 2 1
1620.1.bf.a 6 540.bb even 18 1
1620.1.bf.b 6 12.b even 2 1
1620.1.bf.b 6 15.d odd 2 1
1620.1.bf.b 6 108.l even 18 1
1620.1.bf.b 6 135.n odd 18 1
2700.1.bj.a 12 5.c odd 4 2
2700.1.bj.a 12 20.e even 4 2
2700.1.bj.a 12 135.r odd 36 2
2700.1.bj.a 12 540.bh even 36 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 3T_{7}^{5} + 6T_{7}^{4} + 8T_{7}^{3} + 3T_{7}^{2} - 3T_{7} + 1 \) acting on \(S_{1}^{\mathrm{new}}(540, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$3$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$5$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$7$ \( T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 6 T^{5} + 15 T^{4} - 19 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{6} \) Copy content Toggle raw display
$37$ \( T^{6} \) Copy content Toggle raw display
$41$ \( T^{6} - 6 T^{5} + 15 T^{4} - 19 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$47$ \( T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{6} - 6 T^{5} + 15 T^{4} - 19 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( T^{6} - 6 T^{5} + 15 T^{4} - 19 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{6} \) Copy content Toggle raw display
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