Properties

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

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-11})\)
Defining polynomial: \( x^{4} - 2x^{3} + 11x^{2} - 10x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - 7 q^{4} - 2 \beta_1 q^{5} + 7 \beta_{2} q^{7} + 3 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - 7 q^{4} - 2 \beta_1 q^{5} + 7 \beta_{2} q^{7} + 3 \beta_{3} q^{8} + 22 \beta_{2} q^{10} + 9 \beta_{2} q^{13} + 7 \beta_1 q^{14} + 5 q^{16} - 3 \beta_{2} q^{19} + 14 \beta_1 q^{20} - 6 \beta_1 q^{23} + 63 q^{25} + 9 \beta_1 q^{26} - 49 \beta_{2} q^{28} - 4 \beta_{3} q^{29} - 44 q^{31} + 7 \beta_{3} q^{32} - 28 \beta_{3} q^{35} + 44 q^{37} - 3 \beta_1 q^{38} - 66 \beta_{2} q^{40} - 12 \beta_{3} q^{41} - 21 \beta_{2} q^{43} + 66 \beta_{2} q^{46} - 2 \beta_1 q^{47} - 49 q^{49} - 63 \beta_{3} q^{50} - 63 \beta_{2} q^{52} + 14 \beta_1 q^{53} - 21 \beta_1 q^{56} - 44 q^{58} - 4 \beta_1 q^{59} + 49 \beta_{2} q^{61} + 44 \beta_{3} q^{62} + 97 q^{64} - 36 \beta_{3} q^{65} - 88 q^{67} - 308 q^{70} + 10 \beta_1 q^{71} - 29 \beta_{2} q^{73} - 44 \beta_{3} q^{74} + 21 \beta_{2} q^{76} - 9 \beta_{2} q^{79} - 10 \beta_1 q^{80} - 132 q^{82} + 28 \beta_{3} q^{83} - 21 \beta_1 q^{86} - 24 \beta_1 q^{89} - 126 q^{91} + 42 \beta_1 q^{92} + 22 \beta_{2} q^{94} + 12 \beta_{3} q^{95} + 70 q^{97} + 49 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 28 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 28 q^{4} + 20 q^{16} + 252 q^{25} - 176 q^{31} + 176 q^{37} - 196 q^{49} - 176 q^{58} + 388 q^{64} - 352 q^{67} - 1232 q^{70} - 528 q^{82} - 504 q^{91} + 280 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} + 11x^{2} - 10x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu + 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{3} - 3\nu^{2} + 19\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{3} - 6\nu^{2} + 44\nu - 21 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - 2\beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 2\beta_{2} + 2\beta _1 - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -8\beta_{3} + 19\beta_{2} + 3\beta _1 - 14 ) / 2 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
604.1
0.500000 + 0.244099i
0.500000 + 3.07253i
0.500000 0.244099i
0.500000 3.07253i
3.31662i 0 −7.00000 −9.38083 0 9.89949i 9.94987i 0 31.1127i
604.2 3.31662i 0 −7.00000 9.38083 0 9.89949i 9.94987i 0 31.1127i
604.3 3.31662i 0 −7.00000 −9.38083 0 9.89949i 9.94987i 0 31.1127i
604.4 3.31662i 0 −7.00000 9.38083 0 9.89949i 9.94987i 0 31.1127i
\(n\): 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
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 11 \) acting on \(S_{3}^{\mathrm{new}}(1089, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 11)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 88)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 98)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 162)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 792)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 176)^{2} \) Copy content Toggle raw display
$31$ \( (T + 44)^{4} \) Copy content Toggle raw display
$37$ \( (T - 44)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 1584)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 882)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 88)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 4312)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 352)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 4802)^{2} \) Copy content Toggle raw display
$67$ \( (T + 88)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 2200)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 1682)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 162)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 8624)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 12672)^{2} \) Copy content Toggle raw display
$97$ \( (T - 70)^{4} \) Copy content Toggle raw display
show more
show less