Properties

Label 1089.2.a.d
Level $1089$
Weight $2$
Character orbit 1089.a
Self dual yes
Analytic conductor $8.696$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1089.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.69570878012\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 99)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{4} + 4q^{5} + 2q^{7} + 3q^{8} + O(q^{10}) \) \( q - q^{2} - q^{4} + 4q^{5} + 2q^{7} + 3q^{8} - 4q^{10} + 2q^{13} - 2q^{14} - q^{16} + 2q^{17} + 6q^{19} - 4q^{20} - 4q^{23} + 11q^{25} - 2q^{26} - 2q^{28} - 6q^{29} + 4q^{31} - 5q^{32} - 2q^{34} + 8q^{35} - 6q^{37} - 6q^{38} + 12q^{40} - 10q^{41} - 6q^{43} + 4q^{46} + 8q^{47} - 3q^{49} - 11q^{50} - 2q^{52} + 6q^{56} + 6q^{58} - 4q^{59} + 6q^{61} - 4q^{62} + 7q^{64} + 8q^{65} + 8q^{67} - 2q^{68} - 8q^{70} + 2q^{73} + 6q^{74} - 6q^{76} + 10q^{79} - 4q^{80} + 10q^{82} + 12q^{83} + 8q^{85} + 6q^{86} + 4q^{91} + 4q^{92} - 8q^{94} + 24q^{95} + 2q^{97} + 3q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−1.00000 0 −1.00000 4.00000 0 2.00000 3.00000 0 −4.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.2.a.d 1
3.b odd 2 1 1089.2.a.h 1
11.b odd 2 1 99.2.a.c yes 1
33.d even 2 1 99.2.a.a 1
44.c even 2 1 1584.2.a.r 1
55.d odd 2 1 2475.2.a.c 1
55.e even 4 2 2475.2.c.g 2
77.b even 2 1 4851.2.a.o 1
88.b odd 2 1 6336.2.a.b 1
88.g even 2 1 6336.2.a.f 1
99.g even 6 2 891.2.e.j 2
99.h odd 6 2 891.2.e.c 2
132.d odd 2 1 1584.2.a.b 1
165.d even 2 1 2475.2.a.j 1
165.l odd 4 2 2475.2.c.b 2
231.h odd 2 1 4851.2.a.g 1
264.m even 2 1 6336.2.a.cl 1
264.p odd 2 1 6336.2.a.cm 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.a.a 1 33.d even 2 1
99.2.a.c yes 1 11.b odd 2 1
891.2.e.c 2 99.h odd 6 2
891.2.e.j 2 99.g even 6 2
1089.2.a.d 1 1.a even 1 1 trivial
1089.2.a.h 1 3.b odd 2 1
1584.2.a.b 1 132.d odd 2 1
1584.2.a.r 1 44.c even 2 1
2475.2.a.c 1 55.d odd 2 1
2475.2.a.j 1 165.d even 2 1
2475.2.c.b 2 165.l odd 4 2
2475.2.c.g 2 55.e even 4 2
4851.2.a.g 1 231.h odd 2 1
4851.2.a.o 1 77.b even 2 1
6336.2.a.b 1 88.b odd 2 1
6336.2.a.f 1 88.g even 2 1
6336.2.a.cl 1 264.m even 2 1
6336.2.a.cm 1 264.p odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1089))\):

\( T_{2} + 1 \)
\( T_{5} - 4 \)
\( T_{7} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( T \)
$5$ \( -4 + T \)
$7$ \( -2 + T \)
$11$ \( T \)
$13$ \( -2 + T \)
$17$ \( -2 + T \)
$19$ \( -6 + T \)
$23$ \( 4 + T \)
$29$ \( 6 + T \)
$31$ \( -4 + T \)
$37$ \( 6 + T \)
$41$ \( 10 + T \)
$43$ \( 6 + T \)
$47$ \( -8 + T \)
$53$ \( T \)
$59$ \( 4 + T \)
$61$ \( -6 + T \)
$67$ \( -8 + T \)
$71$ \( T \)
$73$ \( -2 + T \)
$79$ \( -10 + T \)
$83$ \( -12 + T \)
$89$ \( T \)
$97$ \( -2 + T \)
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