Properties

Label 1089.1.i.a
Level $1089$
Weight $1$
Character orbit 1089.i
Analytic conductor $0.543$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -11
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1089.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.543481798757\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
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 6.2.26198073.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6}^{2} q^{3} + \zeta_{6}^{2} q^{4} + ( 1 + \zeta_{6} ) q^{5} -\zeta_{6} q^{9} +O(q^{10})\) \( q -\zeta_{6}^{2} q^{3} + \zeta_{6}^{2} q^{4} + ( 1 + \zeta_{6} ) q^{5} -\zeta_{6} q^{9} + \zeta_{6} q^{12} + ( 1 - \zeta_{6}^{2} ) q^{15} -\zeta_{6} q^{16} + ( -1 + \zeta_{6}^{2} ) q^{20} + ( 1 + \zeta_{6} + \zeta_{6}^{2} ) q^{25} - q^{27} + \zeta_{6}^{2} q^{31} + q^{36} + q^{37} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{45} + ( -1 + \zeta_{6}^{2} ) q^{47} - q^{48} -\zeta_{6}^{2} q^{49} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{53} + ( -1 - \zeta_{6} ) q^{59} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{60} + q^{64} + \zeta_{6}^{2} q^{67} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{71} + ( 1 + \zeta_{6} - \zeta_{6}^{2} ) q^{75} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{80} + \zeta_{6}^{2} q^{81} + \zeta_{6} q^{93} + \zeta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - q^{4} + 3q^{5} - q^{9} + O(q^{10}) \) \( 2q + q^{3} - q^{4} + 3q^{5} - q^{9} + q^{12} + 3q^{15} - q^{16} - 3q^{20} + 2q^{25} - 2q^{27} - q^{31} + 2q^{36} + 2q^{37} - 3q^{47} - 2q^{48} + q^{49} - 3q^{59} + 2q^{64} - q^{67} + 4q^{75} - q^{81} + q^{93} + q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(1\) \(-\zeta_{6}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
122.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i −0.500000 + 0.866025i 1.50000 + 0.866025i 0 0 0 −0.500000 0.866025i 0
848.1 0 0.500000 + 0.866025i −0.500000 0.866025i 1.50000 0.866025i 0 0 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
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
9.d odd 6 1 inner
99.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.1.i.a 2
3.b odd 2 1 3267.1.i.a 2
9.c even 3 1 3267.1.i.a 2
9.d odd 6 1 inner 1089.1.i.a 2
11.b odd 2 1 CM 1089.1.i.a 2
11.c even 5 4 1089.1.r.a 8
11.d odd 10 4 1089.1.r.a 8
33.d even 2 1 3267.1.i.a 2
33.f even 10 4 3267.1.v.a 8
33.h odd 10 4 3267.1.v.a 8
99.g even 6 1 inner 1089.1.i.a 2
99.h odd 6 1 3267.1.i.a 2
99.m even 15 4 3267.1.v.a 8
99.n odd 30 4 1089.1.r.a 8
99.o odd 30 4 3267.1.v.a 8
99.p even 30 4 1089.1.r.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.1.i.a 2 1.a even 1 1 trivial
1089.1.i.a 2 9.d odd 6 1 inner
1089.1.i.a 2 11.b odd 2 1 CM
1089.1.i.a 2 99.g even 6 1 inner
1089.1.r.a 8 11.c even 5 4
1089.1.r.a 8 11.d odd 10 4
1089.1.r.a 8 99.n odd 30 4
1089.1.r.a 8 99.p even 30 4
3267.1.i.a 2 3.b odd 2 1
3267.1.i.a 2 9.c even 3 1
3267.1.i.a 2 33.d even 2 1
3267.1.i.a 2 99.h odd 6 1
3267.1.v.a 8 33.f even 10 4
3267.1.v.a 8 33.h odd 10 4
3267.1.v.a 8 99.m even 15 4
3267.1.v.a 8 99.o odd 30 4

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1089, [\chi])\).

Hecke characteristic polynomials

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