Properties

Label 720.1.bu.a
Level $720$
Weight $1$
Character orbit 720.bu
Analytic conductor $0.359$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -20
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 720.bu (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.359326809096\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{6}\)
Projective field Galois closure of 6.0.10497600.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{12} q^{3} -\zeta_{12}^{4} q^{5} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{12} q^{3} -\zeta_{12}^{4} q^{5} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{2} q^{9} + \zeta_{12}^{5} q^{15} + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{21} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{23} -\zeta_{12}^{2} q^{25} -\zeta_{12}^{3} q^{27} + \zeta_{12}^{2} q^{29} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{35} + \zeta_{12}^{4} q^{41} + q^{45} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{47} + ( -1 + \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{49} -\zeta_{12}^{2} q^{61} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{63} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{67} + ( -1 + \zeta_{12}^{4} ) q^{69} + \zeta_{12}^{3} q^{75} + \zeta_{12}^{4} q^{81} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{83} -\zeta_{12}^{3} q^{87} - q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{5} + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{5} + 2q^{9} - 2q^{25} + 2q^{29} - 2q^{41} + 4q^{45} - 4q^{49} - 2q^{61} - 6q^{69} - 2q^{81} - 4q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(\zeta_{12}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 −0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0.866025 1.50000i 0 0.500000 0.866025i 0
79.2 0 0.866025 0.500000i 0 0.500000 + 0.866025i 0 −0.866025 + 1.50000i 0 0.500000 0.866025i 0
319.1 0 −0.866025 0.500000i 0 0.500000 0.866025i 0 0.866025 + 1.50000i 0 0.500000 + 0.866025i 0
319.2 0 0.866025 + 0.500000i 0 0.500000 0.866025i 0 −0.866025 1.50000i 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 \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner
45.j even 6 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 720.1.bu.a 4
3.b odd 2 1 2160.1.bu.a 4
4.b odd 2 1 inner 720.1.bu.a 4
5.b even 2 1 inner 720.1.bu.a 4
5.c odd 4 1 3600.1.cc.a 2
5.c odd 4 1 3600.1.cc.b 2
8.b even 2 1 2880.1.bu.c 4
8.d odd 2 1 2880.1.bu.c 4
9.c even 3 1 inner 720.1.bu.a 4
9.d odd 6 1 2160.1.bu.a 4
12.b even 2 1 2160.1.bu.a 4
15.d odd 2 1 2160.1.bu.a 4
20.d odd 2 1 CM 720.1.bu.a 4
20.e even 4 1 3600.1.cc.a 2
20.e even 4 1 3600.1.cc.b 2
36.f odd 6 1 inner 720.1.bu.a 4
36.h even 6 1 2160.1.bu.a 4
40.e odd 2 1 2880.1.bu.c 4
40.f even 2 1 2880.1.bu.c 4
45.h odd 6 1 2160.1.bu.a 4
45.j even 6 1 inner 720.1.bu.a 4
45.k odd 12 1 3600.1.cc.a 2
45.k odd 12 1 3600.1.cc.b 2
60.h even 2 1 2160.1.bu.a 4
72.n even 6 1 2880.1.bu.c 4
72.p odd 6 1 2880.1.bu.c 4
180.n even 6 1 2160.1.bu.a 4
180.p odd 6 1 inner 720.1.bu.a 4
180.x even 12 1 3600.1.cc.a 2
180.x even 12 1 3600.1.cc.b 2
360.z odd 6 1 2880.1.bu.c 4
360.bk even 6 1 2880.1.bu.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.1.bu.a 4 1.a even 1 1 trivial
720.1.bu.a 4 4.b odd 2 1 inner
720.1.bu.a 4 5.b even 2 1 inner
720.1.bu.a 4 9.c even 3 1 inner
720.1.bu.a 4 20.d odd 2 1 CM
720.1.bu.a 4 36.f odd 6 1 inner
720.1.bu.a 4 45.j even 6 1 inner
720.1.bu.a 4 180.p odd 6 1 inner
2160.1.bu.a 4 3.b odd 2 1
2160.1.bu.a 4 9.d odd 6 1
2160.1.bu.a 4 12.b even 2 1
2160.1.bu.a 4 15.d odd 2 1
2160.1.bu.a 4 36.h even 6 1
2160.1.bu.a 4 45.h odd 6 1
2160.1.bu.a 4 60.h even 2 1
2160.1.bu.a 4 180.n even 6 1
2880.1.bu.c 4 8.b even 2 1
2880.1.bu.c 4 8.d odd 2 1
2880.1.bu.c 4 40.e odd 2 1
2880.1.bu.c 4 40.f even 2 1
2880.1.bu.c 4 72.n even 6 1
2880.1.bu.c 4 72.p odd 6 1
2880.1.bu.c 4 360.z odd 6 1
2880.1.bu.c 4 360.bk even 6 1
3600.1.cc.a 2 5.c odd 4 1
3600.1.cc.a 2 20.e even 4 1
3600.1.cc.a 2 45.k odd 12 1
3600.1.cc.a 2 180.x even 12 1
3600.1.cc.b 2 5.c odd 4 1
3600.1.cc.b 2 20.e even 4 1
3600.1.cc.b 2 45.k odd 12 1
3600.1.cc.b 2 180.x even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(720, [\chi])\).

Hecke characteristic polynomials

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