Properties

Label 1089.3.b.c
Level $1089$
Weight $3$
Character orbit 1089.b
Analytic conductor $29.673$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \( x^{4} + 16x^{2} + 49 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - 4) q^{4} + (2 \beta_{2} - \beta_1) q^{5} + (\beta_{3} - 5) q^{7} + (7 \beta_{2} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} - 4) q^{4} + (2 \beta_{2} - \beta_1) q^{5} + (\beta_{3} - 5) q^{7} + (7 \beta_{2} - \beta_1) q^{8} + ( - 3 \beta_{3} + 10) q^{10} + (2 \beta_{3} - 17) q^{13} + (7 \beta_{2} - 6 \beta_1) q^{14} + ( - 4 \beta_{3} - 1) q^{16} + (4 \beta_{2} + 3 \beta_1) q^{17} + ( - 6 \beta_{3} + 6) q^{19} + ( - 13 \beta_{2} + 9 \beta_1) q^{20} + (5 \beta_{2} - 6 \beta_1) q^{23} + (5 \beta_{3} + 5) q^{25} + (14 \beta_{2} - 19 \beta_1) q^{26} + ( - 9 \beta_{3} + 35) q^{28} + ( - 22 \beta_{2} + \beta_1) q^{29} + ( - 10 \beta_{3} - 14) q^{31} - \beta_1 q^{32} + ( - \beta_{3} - 20) q^{34} + ( - 15 \beta_{2} + 10 \beta_1) q^{35} + (5 \beta_{3} + 18) q^{37} + ( - 42 \beta_{2} + 12 \beta_1) q^{38} + (10 \beta_{3} - 45) q^{40} + (20 \beta_{2} + 23 \beta_1) q^{41} + ( - 17 \beta_{3} - 3) q^{43} + ( - 11 \beta_{3} + 53) q^{46} + (2 \beta_{2} - 6 \beta_1) q^{47} + ( - 10 \beta_{3} - 9) q^{49} + 35 \beta_{2} q^{50} + ( - 25 \beta_{3} + 98) q^{52} + (14 \beta_{2} + 7 \beta_1) q^{53} + ( - 35 \beta_{2} + 20 \beta_1) q^{56} + (23 \beta_{3} - 30) q^{58} - 49 \beta_{2} q^{59} + (8 \beta_{3} - 6) q^{61} + ( - 70 \beta_{2} - 4 \beta_1) q^{62} + ( - 17 \beta_{3} + 4) q^{64} + ( - 44 \beta_{2} + 27 \beta_1) q^{65} + ( - \beta_{3} - 13) q^{67} + (9 \beta_{2} - 7 \beta_1) q^{68} + (25 \beta_{3} - 95) q^{70} + ( - 43 \beta_{2} - 8 \beta_1) q^{71} + ( - 6 \beta_{3} - 32) q^{73} + (35 \beta_{2} + 13 \beta_1) q^{74} + (30 \beta_{3} - 114) q^{76} + ( - 8 \beta_{3} - 34) q^{79} + (18 \beta_{2} - 19 \beta_1) q^{80} + (3 \beta_{3} - 164) q^{82} + (42 \beta_{2} + 30 \beta_1) q^{83} + ( - 5 \beta_{3} + 10) q^{85} + ( - 119 \beta_{2} + 14 \beta_1) q^{86} + ( - 77 \beta_{2} - \beta_1) q^{89} + ( - 27 \beta_{3} + 115) q^{91} + ( - 57 \beta_{2} + 40 \beta_1) q^{92} + ( - 8 \beta_{3} + 50) q^{94} + (42 \beta_{2} - 36 \beta_1) q^{95} + ( - 15 \beta_{3} + 44) q^{97} + ( - 70 \beta_{2} + \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{4} - 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{4} - 20 q^{7} + 40 q^{10} - 68 q^{13} - 4 q^{16} + 24 q^{19} + 20 q^{25} + 140 q^{28} - 56 q^{31} - 80 q^{34} + 72 q^{37} - 180 q^{40} - 12 q^{43} + 212 q^{46} - 36 q^{49} + 392 q^{52} - 120 q^{58} - 24 q^{61} + 16 q^{64} - 52 q^{67} - 380 q^{70} - 128 q^{73} - 456 q^{76} - 136 q^{79} - 656 q^{82} + 40 q^{85} + 460 q^{91} + 200 q^{94} + 176 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 9\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{2} - 9\beta_1 \) Copy content Toggle raw display

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.c 4
3.b odd 2 1 inner 1089.3.b.c 4
11.b odd 2 1 1089.3.b.d yes 4
33.d even 2 1 1089.3.b.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.3.b.c 4 1.a even 1 1 trivial
1089.3.b.c 4 3.b odd 2 1 inner
1089.3.b.d yes 4 11.b odd 2 1
1089.3.b.d yes 4 33.d even 2 1

Hecke kernels

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

\( T_{2}^{4} + 16T_{2}^{2} + 49 \) Copy content Toggle raw display
\( T_{7}^{2} + 10T_{7} + 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

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