Properties

Label 1089.3.b.d
Level $1089$
Weight $3$
Character orbit 1089.b
Analytic conductor $29.673$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{15})\)
Defining polynomial: \(x^{4} + 16 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -4 + \beta_{3} ) q^{4} + ( \beta_{1} - 2 \beta_{2} ) q^{5} + ( 5 - \beta_{3} ) q^{7} + ( -\beta_{1} + 7 \beta_{2} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -4 + \beta_{3} ) q^{4} + ( \beta_{1} - 2 \beta_{2} ) q^{5} + ( 5 - \beta_{3} ) q^{7} + ( -\beta_{1} + 7 \beta_{2} ) q^{8} + ( -10 + 3 \beta_{3} ) q^{10} + ( 17 - 2 \beta_{3} ) q^{13} + ( 6 \beta_{1} - 7 \beta_{2} ) q^{14} + ( -1 - 4 \beta_{3} ) q^{16} + ( 3 \beta_{1} + 4 \beta_{2} ) q^{17} + ( -6 + 6 \beta_{3} ) q^{19} + ( -9 \beta_{1} + 13 \beta_{2} ) q^{20} + ( 6 \beta_{1} - 5 \beta_{2} ) q^{23} + ( 5 + 5 \beta_{3} ) q^{25} + ( 19 \beta_{1} - 14 \beta_{2} ) q^{26} + ( -35 + 9 \beta_{3} ) q^{28} + ( \beta_{1} - 22 \beta_{2} ) q^{29} + ( -14 - 10 \beta_{3} ) q^{31} -\beta_{1} q^{32} + ( -20 - \beta_{3} ) q^{34} + ( 10 \beta_{1} - 15 \beta_{2} ) q^{35} + ( 18 + 5 \beta_{3} ) q^{37} + ( -12 \beta_{1} + 42 \beta_{2} ) q^{38} + ( 45 - 10 \beta_{3} ) q^{40} + ( 23 \beta_{1} + 20 \beta_{2} ) q^{41} + ( 3 + 17 \beta_{3} ) q^{43} + ( -53 + 11 \beta_{3} ) q^{46} + ( 6 \beta_{1} - 2 \beta_{2} ) q^{47} + ( -9 - 10 \beta_{3} ) q^{49} + 35 \beta_{2} q^{50} + ( -98 + 25 \beta_{3} ) q^{52} + ( -7 \beta_{1} - 14 \beta_{2} ) q^{53} + ( -20 \beta_{1} + 35 \beta_{2} ) q^{56} + ( -30 + 23 \beta_{3} ) q^{58} + 49 \beta_{2} q^{59} + ( 6 - 8 \beta_{3} ) q^{61} + ( -4 \beta_{1} - 70 \beta_{2} ) q^{62} + ( 4 - 17 \beta_{3} ) q^{64} + ( 27 \beta_{1} - 44 \beta_{2} ) q^{65} + ( -13 - \beta_{3} ) q^{67} + ( -7 \beta_{1} + 9 \beta_{2} ) q^{68} + ( -95 + 25 \beta_{3} ) q^{70} + ( 8 \beta_{1} + 43 \beta_{2} ) q^{71} + ( 32 + 6 \beta_{3} ) q^{73} + ( 13 \beta_{1} + 35 \beta_{2} ) q^{74} + ( 114 - 30 \beta_{3} ) q^{76} + ( 34 + 8 \beta_{3} ) q^{79} + ( 19 \beta_{1} - 18 \beta_{2} ) q^{80} + ( -164 + 3 \beta_{3} ) q^{82} + ( 30 \beta_{1} + 42 \beta_{2} ) q^{83} + ( -10 + 5 \beta_{3} ) q^{85} + ( -14 \beta_{1} + 119 \beta_{2} ) q^{86} + ( \beta_{1} + 77 \beta_{2} ) q^{89} + ( 115 - 27 \beta_{3} ) q^{91} + ( -40 \beta_{1} + 57 \beta_{2} ) q^{92} + ( -50 + 8 \beta_{3} ) q^{94} + ( -36 \beta_{1} + 42 \beta_{2} ) q^{95} + ( 44 - 15 \beta_{3} ) q^{97} + ( \beta_{1} - 70 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{4} + 20 q^{7} + O(q^{10}) \) \( 4 q - 16 q^{4} + 20 q^{7} - 40 q^{10} + 68 q^{13} - 4 q^{16} - 24 q^{19} + 20 q^{25} - 140 q^{28} - 56 q^{31} - 80 q^{34} + 72 q^{37} + 180 q^{40} + 12 q^{43} - 212 q^{46} - 36 q^{49} - 392 q^{52} - 120 q^{58} + 24 q^{61} + 16 q^{64} - 52 q^{67} - 380 q^{70} + 128 q^{73} + 456 q^{76} + 136 q^{79} - 656 q^{82} - 40 q^{85} + 460 q^{91} - 200 q^{94} + 176 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 16 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 9 \nu \)\()/7\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 8 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 8\)
\(\nu^{3}\)\(=\)\(7 \beta_{2} - 9 \beta_{1}\)

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
3.44572i
2.03151i
2.03151i
3.44572i
3.44572i 0 −7.87298 6.27415i 0 8.87298 13.3452i 0 −21.6190
485.2 2.03151i 0 −0.127017 0.796921i 0 1.12702 7.86799i 0 1.61895
485.3 2.03151i 0 −0.127017 0.796921i 0 1.12702 7.86799i 0 1.61895
485.4 3.44572i 0 −7.87298 6.27415i 0 8.87298 13.3452i 0 −21.6190
\(n\): 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.d yes 4
3.b odd 2 1 inner 1089.3.b.d yes 4
11.b odd 2 1 1089.3.b.c 4
33.d even 2 1 1089.3.b.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.3.b.c 4 11.b odd 2 1
1089.3.b.c 4 33.d even 2 1
1089.3.b.d yes 4 1.a even 1 1 trivial
1089.3.b.d yes 4 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 \(S_{3}^{\mathrm{new}}(1089, [\chi])\):

\( T_{2}^{4} + 16 T_{2}^{2} + 49 \)
\( T_{7}^{2} - 10 T_{7} + 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 49 + 16 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 25 + 40 T^{2} + T^{4} \)
$7$ \( ( 10 - 10 T + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( ( 229 - 34 T + T^{2} )^{2} \)
$17$ \( 3025 + 160 T^{2} + T^{4} \)
$19$ \( ( -504 + 12 T + T^{2} )^{2} \)
$23$ \( 20164 + 796 T^{2} + T^{4} \)
$29$ \( 1010025 + 2040 T^{2} + T^{4} \)
$31$ \( ( -1304 + 28 T + T^{2} )^{2} \)
$37$ \( ( -51 - 36 T + T^{2} )^{2} \)
$41$ \( 14615329 + 8224 T^{2} + T^{4} \)
$43$ \( ( -4326 - 6 T + T^{2} )^{2} \)
$47$ \( 48400 + 640 T^{2} + T^{4} \)
$53$ \( 21609 + 1176 T^{2} + T^{4} \)
$59$ \( ( 4802 + T^{2} )^{2} \)
$61$ \( ( -924 - 12 T + T^{2} )^{2} \)
$67$ \( ( 154 + 26 T + T^{2} )^{2} \)
$71$ \( 6563844 + 7044 T^{2} + T^{4} \)
$73$ \( ( 484 - 64 T + T^{2} )^{2} \)
$79$ \( ( 196 - 68 T + T^{2} )^{2} \)
$83$ \( 28005264 + 16416 T^{2} + T^{4} \)
$89$ \( 136819809 + 23424 T^{2} + T^{4} \)
$97$ \( ( -1439 - 88 T + T^{2} )^{2} \)
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