# 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 $2$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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