# Properties

 Label 1089.3.b.a Level $1089$ Weight $3$ Character orbit 1089.b Analytic conductor $29.673$ Analytic rank $0$ Dimension $2$ 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: $$2$$ 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: 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.a 2
3.b odd 2 1 inner 1089.3.b.a 2
11.b odd 2 1 1089.3.b.b yes 2
33.d even 2 1 1089.3.b.b yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.3.b.a 2 1.a even 1 1 trivial
1089.3.b.a 2 3.b odd 2 1 inner
1089.3.b.b yes 2 11.b odd 2 1
1089.3.b.b yes 2 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 $$S_{3}^{\mathrm{new}}(1089, [\chi])$$:

 $$T_{2}^{2} + 2$$ $$T_{7} + 1$$

## Hecke characteristic polynomials

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