Properties

Label 1080.2.q.a
Level $1080$
Weight $2$
Character orbit 1080.q
Analytic conductor $8.624$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1080.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.62384341830\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{5} +O(q^{10})\) \( q -\zeta_{6} q^{5} + ( -5 + 5 \zeta_{6} ) q^{11} -3 q^{17} + 5 q^{19} + 6 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + ( -10 + 10 \zeta_{6} ) q^{29} + 2 \zeta_{6} q^{31} + 4 q^{37} -3 \zeta_{6} q^{41} + ( -3 + 3 \zeta_{6} ) q^{43} + ( 4 - 4 \zeta_{6} ) q^{47} + 7 \zeta_{6} q^{49} + 6 q^{53} + 5 q^{55} -3 \zeta_{6} q^{59} + ( -2 + 2 \zeta_{6} ) q^{61} + 11 \zeta_{6} q^{67} + 14 q^{71} -15 q^{73} + ( -10 + 10 \zeta_{6} ) q^{79} + ( -12 + 12 \zeta_{6} ) q^{83} + 3 \zeta_{6} q^{85} -14 q^{89} -5 \zeta_{6} q^{95} + ( 13 - 13 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{5} + O(q^{10}) \) \( 2q - q^{5} - 5q^{11} - 6q^{17} + 10q^{19} + 6q^{23} - q^{25} - 10q^{29} + 2q^{31} + 8q^{37} - 3q^{41} - 3q^{43} + 4q^{47} + 7q^{49} + 12q^{53} + 10q^{55} - 3q^{59} - 2q^{61} + 11q^{67} + 28q^{71} - 30q^{73} - 10q^{79} - 12q^{83} + 3q^{85} - 28q^{89} - 5q^{95} + 13q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times\).

\(n\) \(217\) \(271\) \(541\) \(1001\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 −0.500000 + 0.866025i 0 0 0 0 0
721.1 0 0 0 −0.500000 0.866025i 0 0 0 0 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
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 1080.2.q.a 2
3.b odd 2 1 360.2.q.a 2
4.b odd 2 1 2160.2.q.c 2
9.c even 3 1 inner 1080.2.q.a 2
9.c even 3 1 3240.2.a.f 1
9.d odd 6 1 360.2.q.a 2
9.d odd 6 1 3240.2.a.b 1
12.b even 2 1 720.2.q.e 2
36.f odd 6 1 2160.2.q.c 2
36.f odd 6 1 6480.2.a.q 1
36.h even 6 1 720.2.q.e 2
36.h even 6 1 6480.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.q.a 2 3.b odd 2 1
360.2.q.a 2 9.d odd 6 1
720.2.q.e 2 12.b even 2 1
720.2.q.e 2 36.h even 6 1
1080.2.q.a 2 1.a even 1 1 trivial
1080.2.q.a 2 9.c even 3 1 inner
2160.2.q.c 2 4.b odd 2 1
2160.2.q.c 2 36.f odd 6 1
3240.2.a.b 1 9.d odd 6 1
3240.2.a.f 1 9.c even 3 1
6480.2.a.e 1 36.h even 6 1
6480.2.a.q 1 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} \) acting on \(S_{2}^{\mathrm{new}}(1080, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 25 + 5 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( 3 + T )^{2} \)
$19$ \( ( -5 + T )^{2} \)
$23$ \( 36 - 6 T + T^{2} \)
$29$ \( 100 + 10 T + T^{2} \)
$31$ \( 4 - 2 T + T^{2} \)
$37$ \( ( -4 + T )^{2} \)
$41$ \( 9 + 3 T + T^{2} \)
$43$ \( 9 + 3 T + T^{2} \)
$47$ \( 16 - 4 T + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( 9 + 3 T + T^{2} \)
$61$ \( 4 + 2 T + T^{2} \)
$67$ \( 121 - 11 T + T^{2} \)
$71$ \( ( -14 + T )^{2} \)
$73$ \( ( 15 + T )^{2} \)
$79$ \( 100 + 10 T + T^{2} \)
$83$ \( 144 + 12 T + T^{2} \)
$89$ \( ( 14 + T )^{2} \)
$97$ \( 169 - 13 T + T^{2} \)
show more
show less