Properties

Label 540.1.p.b
Level $540$
Weight $1$
Character orbit 540.p
Analytic conductor $0.269$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -20
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 540.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.269495106822\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 180)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.1620.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} -\zeta_{6}^{2} q^{5} + \zeta_{6} q^{7} - q^{8} +O(q^{10})\) \( q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} -\zeta_{6}^{2} q^{5} + \zeta_{6} q^{7} - q^{8} + q^{10} + \zeta_{6}^{2} q^{14} -\zeta_{6} q^{16} + \zeta_{6} q^{20} + \zeta_{6}^{2} q^{23} -\zeta_{6} q^{25} - q^{28} -\zeta_{6} q^{29} -\zeta_{6}^{2} q^{32} + q^{35} + \zeta_{6}^{2} q^{40} + \zeta_{6}^{2} q^{41} -2 \zeta_{6} q^{43} - q^{46} -\zeta_{6} q^{47} -\zeta_{6}^{2} q^{50} -\zeta_{6} q^{56} -\zeta_{6}^{2} q^{58} + \zeta_{6} q^{61} + q^{64} -\zeta_{6}^{2} q^{67} + \zeta_{6} q^{70} - q^{80} - q^{82} -\zeta_{6} q^{83} -2 \zeta_{6}^{2} q^{86} + q^{89} -\zeta_{6} q^{92} -\zeta_{6}^{2} q^{94} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} + q^{5} + q^{7} - 2q^{8} + O(q^{10}) \) \( 2q + q^{2} - q^{4} + q^{5} + q^{7} - 2q^{8} + 2q^{10} - q^{14} - q^{16} + q^{20} - q^{23} - q^{25} - 2q^{28} - q^{29} + q^{32} + 2q^{35} - q^{40} - q^{41} - 2q^{43} - 2q^{46} - q^{47} + q^{50} - q^{56} + q^{58} + q^{61} + 2q^{64} + q^{67} + q^{70} - 2q^{80} - 2q^{82} - q^{83} + 2q^{86} + 2q^{89} - q^{92} + q^{94} + O(q^{100}) \)

Character values

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

\(n\) \(217\) \(271\) \(461\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{6}^{2}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i 0.500000 + 0.866025i 0 0.500000 0.866025i −1.00000 0 1.00000
199.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 0.866025i 0 0.500000 + 0.866025i −1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
9.c even 3 1 inner
180.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 540.1.p.b 2
3.b odd 2 1 180.1.p.a 2
4.b odd 2 1 540.1.p.a 2
5.b even 2 1 540.1.p.a 2
5.c odd 4 2 2700.1.t.a 4
9.c even 3 1 inner 540.1.p.b 2
9.c even 3 1 1620.1.f.a 1
9.d odd 6 1 180.1.p.a 2
9.d odd 6 1 1620.1.f.d 1
12.b even 2 1 180.1.p.b yes 2
15.d odd 2 1 180.1.p.b yes 2
15.e even 4 2 900.1.t.a 4
20.d odd 2 1 CM 540.1.p.b 2
20.e even 4 2 2700.1.t.a 4
24.f even 2 1 2880.1.bu.a 2
24.h odd 2 1 2880.1.bu.b 2
36.f odd 6 1 540.1.p.a 2
36.f odd 6 1 1620.1.f.c 1
36.h even 6 1 180.1.p.b yes 2
36.h even 6 1 1620.1.f.b 1
45.h odd 6 1 180.1.p.b yes 2
45.h odd 6 1 1620.1.f.b 1
45.j even 6 1 540.1.p.a 2
45.j even 6 1 1620.1.f.c 1
45.k odd 12 2 2700.1.t.a 4
45.l even 12 2 900.1.t.a 4
60.h even 2 1 180.1.p.a 2
60.l odd 4 2 900.1.t.a 4
72.j odd 6 1 2880.1.bu.b 2
72.l even 6 1 2880.1.bu.a 2
120.i odd 2 1 2880.1.bu.a 2
120.m even 2 1 2880.1.bu.b 2
180.n even 6 1 180.1.p.a 2
180.n even 6 1 1620.1.f.d 1
180.p odd 6 1 inner 540.1.p.b 2
180.p odd 6 1 1620.1.f.a 1
180.v odd 12 2 900.1.t.a 4
180.x even 12 2 2700.1.t.a 4
360.bd even 6 1 2880.1.bu.b 2
360.bh odd 6 1 2880.1.bu.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.1.p.a 2 3.b odd 2 1
180.1.p.a 2 9.d odd 6 1
180.1.p.a 2 60.h even 2 1
180.1.p.a 2 180.n even 6 1
180.1.p.b yes 2 12.b even 2 1
180.1.p.b yes 2 15.d odd 2 1
180.1.p.b yes 2 36.h even 6 1
180.1.p.b yes 2 45.h odd 6 1
540.1.p.a 2 4.b odd 2 1
540.1.p.a 2 5.b even 2 1
540.1.p.a 2 36.f odd 6 1
540.1.p.a 2 45.j even 6 1
540.1.p.b 2 1.a even 1 1 trivial
540.1.p.b 2 9.c even 3 1 inner
540.1.p.b 2 20.d odd 2 1 CM
540.1.p.b 2 180.p odd 6 1 inner
900.1.t.a 4 15.e even 4 2
900.1.t.a 4 45.l even 12 2
900.1.t.a 4 60.l odd 4 2
900.1.t.a 4 180.v odd 12 2
1620.1.f.a 1 9.c even 3 1
1620.1.f.a 1 180.p odd 6 1
1620.1.f.b 1 36.h even 6 1
1620.1.f.b 1 45.h odd 6 1
1620.1.f.c 1 36.f odd 6 1
1620.1.f.c 1 45.j even 6 1
1620.1.f.d 1 9.d odd 6 1
1620.1.f.d 1 180.n even 6 1
2700.1.t.a 4 5.c odd 4 2
2700.1.t.a 4 20.e even 4 2
2700.1.t.a 4 45.k odd 12 2
2700.1.t.a 4 180.x even 12 2
2880.1.bu.a 2 24.f even 2 1
2880.1.bu.a 2 72.l even 6 1
2880.1.bu.a 2 120.i odd 2 1
2880.1.bu.a 2 360.bh odd 6 1
2880.1.bu.b 2 24.h odd 2 1
2880.1.bu.b 2 72.j odd 6 1
2880.1.bu.b 2 120.m even 2 1
2880.1.bu.b 2 360.bd even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - T_{7} + 1 \) acting on \(S_{1}^{\mathrm{new}}(540, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 1 - T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( 1 + T + T^{2} \)
$29$ \( 1 + T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( 1 + T + T^{2} \)
$43$ \( 4 + 2 T + T^{2} \)
$47$ \( 1 + T + T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 1 - T + T^{2} \)
$67$ \( 1 - T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( T^{2} \)
$79$ \( T^{2} \)
$83$ \( 1 + T + T^{2} \)
$89$ \( ( -1 + T )^{2} \)
$97$ \( T^{2} \)
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