Properties

Label 2880.1.bu.a
Level 2880
Weight 1
Character orbit 2880.bu
Analytic conductor 1.437
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM discriminant -20
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 2880.bu (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.43730723638\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{3} -\zeta_{6}^{2} q^{5} -\zeta_{6} q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + \zeta_{6}^{2} q^{3} -\zeta_{6}^{2} q^{5} -\zeta_{6} q^{7} -\zeta_{6} q^{9} + \zeta_{6} q^{15} + q^{21} + \zeta_{6}^{2} q^{23} -\zeta_{6} q^{25} + q^{27} -\zeta_{6} q^{29} - q^{35} -\zeta_{6}^{2} q^{41} -2 \zeta_{6} q^{43} - q^{45} -\zeta_{6} q^{47} -\zeta_{6} q^{61} + \zeta_{6}^{2} q^{63} -\zeta_{6}^{2} q^{67} -\zeta_{6} q^{69} + q^{75} + \zeta_{6}^{2} q^{81} + \zeta_{6} q^{83} + q^{87} - q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + q^{5} - q^{7} - q^{9} + O(q^{10}) \) \( 2q - q^{3} + q^{5} - q^{7} - q^{9} + q^{15} + 2q^{21} - q^{23} - q^{25} + 2q^{27} - q^{29} - 2q^{35} + q^{41} - 2q^{43} - 2q^{45} - q^{47} - q^{61} - q^{63} + q^{67} - q^{69} + 2q^{75} - q^{81} + q^{83} + 2q^{87} - 2q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(-1\) \(\zeta_{6}^{2}\) \(1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −0.500000 + 0.866025i 0 0.500000 0.866025i 0 −0.500000 0.866025i 0 −0.500000 0.866025i 0
2239.1 0 −0.500000 0.866025i 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0
\(n\): 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 2880.1.bu.a 2
4.b odd 2 1 2880.1.bu.b 2
5.b even 2 1 2880.1.bu.b 2
8.b even 2 1 180.1.p.b yes 2
8.d odd 2 1 180.1.p.a 2
9.c even 3 1 inner 2880.1.bu.a 2
20.d odd 2 1 CM 2880.1.bu.a 2
24.f even 2 1 540.1.p.b 2
24.h odd 2 1 540.1.p.a 2
36.f odd 6 1 2880.1.bu.b 2
40.e odd 2 1 180.1.p.b yes 2
40.f even 2 1 180.1.p.a 2
40.i odd 4 2 900.1.t.a 4
40.k even 4 2 900.1.t.a 4
45.j even 6 1 2880.1.bu.b 2
72.j odd 6 1 540.1.p.a 2
72.j odd 6 1 1620.1.f.c 1
72.l even 6 1 540.1.p.b 2
72.l even 6 1 1620.1.f.a 1
72.n even 6 1 180.1.p.b yes 2
72.n even 6 1 1620.1.f.b 1
72.p odd 6 1 180.1.p.a 2
72.p odd 6 1 1620.1.f.d 1
120.i odd 2 1 540.1.p.b 2
120.m even 2 1 540.1.p.a 2
120.q odd 4 2 2700.1.t.a 4
120.w even 4 2 2700.1.t.a 4
180.p odd 6 1 inner 2880.1.bu.a 2
360.z odd 6 1 180.1.p.b yes 2
360.z odd 6 1 1620.1.f.b 1
360.bd even 6 1 540.1.p.a 2
360.bd even 6 1 1620.1.f.c 1
360.bh odd 6 1 540.1.p.b 2
360.bh odd 6 1 1620.1.f.a 1
360.bk even 6 1 180.1.p.a 2
360.bk even 6 1 1620.1.f.d 1
360.bo even 12 2 900.1.t.a 4
360.br even 12 2 2700.1.t.a 4
360.bt odd 12 2 2700.1.t.a 4
360.bu odd 12 2 900.1.t.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.1.p.a 2 8.d odd 2 1
180.1.p.a 2 40.f even 2 1
180.1.p.a 2 72.p odd 6 1
180.1.p.a 2 360.bk even 6 1
180.1.p.b yes 2 8.b even 2 1
180.1.p.b yes 2 40.e odd 2 1
180.1.p.b yes 2 72.n even 6 1
180.1.p.b yes 2 360.z odd 6 1
540.1.p.a 2 24.h odd 2 1
540.1.p.a 2 72.j odd 6 1
540.1.p.a 2 120.m even 2 1
540.1.p.a 2 360.bd even 6 1
540.1.p.b 2 24.f even 2 1
540.1.p.b 2 72.l even 6 1
540.1.p.b 2 120.i odd 2 1
540.1.p.b 2 360.bh odd 6 1
900.1.t.a 4 40.i odd 4 2
900.1.t.a 4 40.k even 4 2
900.1.t.a 4 360.bo even 12 2
900.1.t.a 4 360.bu odd 12 2
1620.1.f.a 1 72.l even 6 1
1620.1.f.a 1 360.bh odd 6 1
1620.1.f.b 1 72.n even 6 1
1620.1.f.b 1 360.z odd 6 1
1620.1.f.c 1 72.j odd 6 1
1620.1.f.c 1 360.bd even 6 1
1620.1.f.d 1 72.p odd 6 1
1620.1.f.d 1 360.bk even 6 1
2700.1.t.a 4 120.q odd 4 2
2700.1.t.a 4 120.w even 4 2
2700.1.t.a 4 360.br even 12 2
2700.1.t.a 4 360.bt odd 12 2
2880.1.bu.a 2 1.a even 1 1 trivial
2880.1.bu.a 2 9.c even 3 1 inner
2880.1.bu.a 2 20.d odd 2 1 CM
2880.1.bu.a 2 180.p odd 6 1 inner
2880.1.bu.b 2 4.b odd 2 1
2880.1.bu.b 2 5.b even 2 1
2880.1.bu.b 2 36.f odd 6 1
2880.1.bu.b 2 45.j 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 \(S_{1}^{\mathrm{new}}(2880, [\chi])\).

Hecke Characteristic Polynomials

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