Properties

Label 1089.3.b.b
Level $1089$
Weight $3$
Character orbit 1089.b
Analytic conductor $29.673$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 2 q^{4} + 4 \beta q^{5} + q^{7} + 6 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} + 2 q^{4} + 4 \beta q^{5} + q^{7} + 6 \beta q^{8} -8 q^{10} + 16 q^{13} + \beta q^{14} -4 q^{16} -8 \beta q^{17} + 33 q^{19} + 8 \beta q^{20} + 17 \beta q^{23} -7 q^{25} + 16 \beta q^{26} + 2 q^{28} + 9 \beta q^{29} + 31 q^{31} + 20 \beta q^{32} + 16 q^{34} + 4 \beta q^{35} -57 q^{37} + 33 \beta q^{38} -48 q^{40} -11 \beta q^{41} -34 q^{46} -43 \beta q^{47} -48 q^{49} -7 \beta q^{50} + 32 q^{52} + 63 \beta q^{53} + 6 \beta q^{56} -18 q^{58} -49 \beta q^{59} + 105 q^{61} + 31 \beta q^{62} -56 q^{64} + 64 \beta q^{65} -103 q^{67} -16 \beta q^{68} -8 q^{70} + 84 \beta q^{71} -47 q^{73} -57 \beta q^{74} + 66 q^{76} + 23 q^{79} -16 \beta q^{80} + 22 q^{82} -75 \beta q^{83} + 64 q^{85} -36 \beta q^{89} + 16 q^{91} + 34 \beta q^{92} + 86 q^{94} + 132 \beta q^{95} -25 q^{97} -48 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} + 2 q^{7} + O(q^{10}) \) \( 2 q + 4 q^{4} + 2 q^{7} - 16 q^{10} + 32 q^{13} - 8 q^{16} + 66 q^{19} - 14 q^{25} + 4 q^{28} + 62 q^{31} + 32 q^{34} - 114 q^{37} - 96 q^{40} - 68 q^{46} - 96 q^{49} + 64 q^{52} - 36 q^{58} + 210 q^{61} - 112 q^{64} - 206 q^{67} - 16 q^{70} - 94 q^{73} + 132 q^{76} + 46 q^{79} + 44 q^{82} + 128 q^{85} + 32 q^{91} + 172 q^{94} - 50 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(244\) \(848\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
485.1
1.41421i
1.41421i
1.41421i 0 2.00000 5.65685i 0 1.00000 8.48528i 0 −8.00000
485.2 1.41421i 0 2.00000 5.65685i 0 1.00000 8.48528i 0 −8.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.3.b.b yes 2
3.b odd 2 1 inner 1089.3.b.b yes 2
11.b odd 2 1 1089.3.b.a 2
33.d even 2 1 1089.3.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.3.b.a 2 11.b odd 2 1
1089.3.b.a 2 33.d even 2 1
1089.3.b.b yes 2 1.a even 1 1 trivial
1089.3.b.b yes 2 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1089, [\chi])\):

\( T_{2}^{2} + 2 \)
\( T_{7} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 32 + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -16 + T )^{2} \)
$17$ \( 128 + T^{2} \)
$19$ \( ( -33 + T )^{2} \)
$23$ \( 578 + T^{2} \)
$29$ \( 162 + T^{2} \)
$31$ \( ( -31 + T )^{2} \)
$37$ \( ( 57 + T )^{2} \)
$41$ \( 242 + T^{2} \)
$43$ \( T^{2} \)
$47$ \( 3698 + T^{2} \)
$53$ \( 7938 + T^{2} \)
$59$ \( 4802 + T^{2} \)
$61$ \( ( -105 + T )^{2} \)
$67$ \( ( 103 + T )^{2} \)
$71$ \( 14112 + T^{2} \)
$73$ \( ( 47 + T )^{2} \)
$79$ \( ( -23 + T )^{2} \)
$83$ \( 11250 + T^{2} \)
$89$ \( 2592 + T^{2} \)
$97$ \( ( 25 + T )^{2} \)
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