# Properties

 Label 1089.3.b.h Level $1089$ Weight $3$ Character orbit 1089.b Analytic conductor $29.673$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# 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: $$8$$ Coefficient field: 8.0.3317760000.7 Defining polynomial: $$x^{8} + 12 x^{6} + 23 x^{4} + 12 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{7} q^{2} + \beta_{3} q^{4} -\beta_{1} q^{5} + ( \beta_{2} + \beta_{4} ) q^{7} + \beta_{6} q^{8} +O(q^{10})$$ $$q -\beta_{7} q^{2} + \beta_{3} q^{4} -\beta_{1} q^{5} + ( \beta_{2} + \beta_{4} ) q^{7} + \beta_{6} q^{8} + ( 2 \beta_{2} - \beta_{4} ) q^{10} + ( 5 \beta_{2} - 6 \beta_{4} ) q^{13} + \beta_{5} q^{14} + ( -1 + 4 \beta_{3} ) q^{16} + ( -10 \beta_{6} + 3 \beta_{7} ) q^{17} + ( -6 \beta_{2} - 2 \beta_{4} ) q^{19} + ( -\beta_{1} - 7 \beta_{5} ) q^{20} + ( -12 \beta_{1} + 13 \beta_{5} ) q^{23} + ( 17 - \beta_{3} ) q^{25} + ( 11 \beta_{1} - 28 \beta_{5} ) q^{26} + ( 5 \beta_{2} + 3 \beta_{4} ) q^{28} + 13 \beta_{7} q^{29} + ( 12 + 4 \beta_{3} ) q^{31} + ( 8 \beta_{6} + 17 \beta_{7} ) q^{32} + ( 22 - 3 \beta_{3} ) q^{34} + ( -5 \beta_{6} - \beta_{7} ) q^{35} + ( 2 + 9 \beta_{3} ) q^{37} + ( -4 \beta_{1} + 6 \beta_{5} ) q^{38} + ( 3 \beta_{2} + 2 \beta_{4} ) q^{40} + ( 26 \beta_{6} + 13 \beta_{7} ) q^{41} + ( 9 \beta_{2} - 19 \beta_{4} ) q^{43} + ( 37 \beta_{2} - 25 \beta_{4} ) q^{46} + ( -10 \beta_{1} - 6 \beta_{5} ) q^{47} + ( -41 + 2 \beta_{3} ) q^{49} + ( -\beta_{6} - 21 \beta_{7} ) q^{50} + ( -30 \beta_{2} + 15 \beta_{4} ) q^{52} + ( -3 \beta_{1} - 48 \beta_{5} ) q^{53} + ( 2 \beta_{1} + 3 \beta_{5} ) q^{56} + ( 52 - 13 \beta_{3} ) q^{58} + ( -6 \beta_{1} - 51 \beta_{5} ) q^{59} + ( 26 \beta_{2} - 2 \beta_{4} ) q^{61} + ( 4 \beta_{6} + 4 \beta_{7} ) q^{62} + ( 56 - \beta_{3} ) q^{64} + ( 8 \beta_{6} + 17 \beta_{7} ) q^{65} + ( 27 - 21 \beta_{3} ) q^{67} + ( -43 \beta_{6} - 22 \beta_{7} ) q^{68} + ( 1 + \beta_{3} ) q^{70} + ( 26 \beta_{1} + \beta_{5} ) q^{71} + ( 4 \beta_{2} - 52 \beta_{4} ) q^{73} + ( 9 \beta_{6} + 34 \beta_{7} ) q^{74} + ( -10 \beta_{2} - 18 \beta_{4} ) q^{76} + ( 44 \beta_{2} + 30 \beta_{4} ) q^{79} + ( -3 \beta_{1} - 28 \beta_{5} ) q^{80} + ( 26 - 13 \beta_{3} ) q^{82} + ( -20 \beta_{6} - 34 \beta_{7} ) q^{83} + ( -36 \beta_{2} - 17 \beta_{4} ) q^{85} + ( 28 \beta_{1} - 75 \beta_{5} ) q^{86} + ( 11 \beta_{1} + 19 \beta_{5} ) q^{89} + ( -15 - \beta_{3} ) q^{91} + ( 14 \beta_{1} - 97 \beta_{5} ) q^{92} + ( 14 \beta_{2} - 4 \beta_{4} ) q^{94} + ( 18 \beta_{6} - 2 \beta_{7} ) q^{95} + ( 44 - 7 \beta_{3} ) q^{97} + ( 2 \beta_{6} + 49 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + O(q^{10})$$ $$8 q - 8 q^{16} + 136 q^{25} + 96 q^{31} + 176 q^{34} + 16 q^{37} - 328 q^{49} + 416 q^{58} + 448 q^{64} + 216 q^{67} + 8 q^{70} + 208 q^{82} - 120 q^{91} + 352 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 12 x^{6} + 23 x^{4} + 12 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 13 \nu^{5} + 36 \nu^{3} + 48 \nu$$$$)/7$$ $$\beta_{2}$$ $$=$$ $$($$$$3 \nu^{6} + 32 \nu^{4} + 24 \nu^{2} - 10$$$$)/7$$ $$\beta_{3}$$ $$=$$ $$-\nu^{6} - 12 \nu^{4} - 22 \nu^{2} - 6$$ $$\beta_{4}$$ $$=$$ $$($$$$8 \nu^{6} + 90 \nu^{4} + 120 \nu^{2} + 27$$$$)/7$$ $$\beta_{5}$$ $$=$$ $$($$$$-6 \nu^{7} - 71 \nu^{5} - 125 \nu^{3} - 43 \nu$$$$)/7$$ $$\beta_{6}$$ $$=$$ $$\nu^{7} + 11 \nu^{5} + 12 \nu^{3}$$ $$\beta_{7}$$ $$=$$ $$\nu^{7} + 12 \nu^{5} + 23 \nu^{3} + 11 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} - \beta_{5} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{4} + \beta_{3} - 3 \beta_{2} - 6$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$11 \beta_{7} + \beta_{6} + 13 \beta_{5} - 6 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-21 \beta_{4} - 12 \beta_{3} + 28 \beta_{2} + 49$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-108 \beta_{7} - 13 \beta_{6} - 132 \beta_{5} + 55 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$104 \beta_{4} + 60 \beta_{3} - 135 \beta_{2} - 234$$ $$\nu^{7}$$ $$=$$ $$($$$$1056 \beta_{7} + 133 \beta_{6} + 1296 \beta_{5} - 533 \beta_{1}$$$$)/2$$

## 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
 0.319918i − 3.12580i 0.837556i − 1.19395i 1.19395i − 0.837556i 3.12580i − 0.319918i
2.80588i 0 −3.87298 2.03151i 0 0.504017 0.356394i 0 −5.70017
485.2 2.80588i 0 −3.87298 2.03151i 0 −0.504017 0.356394i 0 5.70017
485.3 0.356394i 0 3.87298 3.44572i 0 −3.96812 2.80588i 0 −1.22803
485.4 0.356394i 0 3.87298 3.44572i 0 3.96812 2.80588i 0 1.22803
485.5 0.356394i 0 3.87298 3.44572i 0 3.96812 2.80588i 0 1.22803
485.6 0.356394i 0 3.87298 3.44572i 0 −3.96812 2.80588i 0 −1.22803
485.7 2.80588i 0 −3.87298 2.03151i 0 −0.504017 0.356394i 0 5.70017
485.8 2.80588i 0 −3.87298 2.03151i 0 0.504017 0.356394i 0 −5.70017
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 485.8 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
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.b.h 8
3.b odd 2 1 inner 1089.3.b.h 8
11.b odd 2 1 inner 1089.3.b.h 8
33.d even 2 1 inner 1089.3.b.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.3.b.h 8 1.a even 1 1 trivial
1089.3.b.h 8 3.b odd 2 1 inner
1089.3.b.h 8 11.b odd 2 1 inner
1089.3.b.h 8 33.d even 2 1 inner

## Hecke kernels

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

 $$T_{2}^{4} + 8 T_{2}^{2} + 1$$ $$T_{7}^{4} - 16 T_{7}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 8 T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$( 49 + 16 T^{2} + T^{4} )^{2}$$
$7$ $$( 4 - 16 T^{2} + T^{4} )^{2}$$
$11$ $$T^{8}$$
$13$ $$( 11025 - 510 T^{2} + T^{4} )^{2}$$
$17$ $$( 121801 + 992 T^{2} + T^{4} )^{2}$$
$19$ $$( 7744 - 256 T^{2} + T^{4} )^{2}$$
$23$ $$( 964324 + 2356 T^{2} + T^{4} )^{2}$$
$29$ $$( 28561 + 1352 T^{2} + T^{4} )^{2}$$
$31$ $$( -96 - 24 T + T^{2} )^{4}$$
$37$ $$( -1211 - 4 T + T^{2} )^{4}$$
$41$ $$( 3455881 + 5408 T^{2} + T^{4} )^{2}$$
$43$ $$( 2439844 - 4096 T^{2} + T^{4} )^{2}$$
$47$ $$( 258064 + 1984 T^{2} + T^{4} )^{2}$$
$53$ $$( 23357889 + 9936 T^{2} + T^{4} )^{2}$$
$59$ $$( 30935844 + 12204 T^{2} + T^{4} )^{2}$$
$61$ $$( 4032064 - 4096 T^{2} + T^{4} )^{2}$$
$67$ $$( -5886 - 54 T + T^{2} )^{4}$$
$71$ $$( 21883684 + 10924 T^{2} + T^{4} )^{2}$$
$73$ $$( 181494784 - 27136 T^{2} + T^{4} )^{2}$$
$79$ $$( 1710864 - 20616 T^{2} + T^{4} )^{2}$$
$83$ $$( 15085456 + 9728 T^{2} + T^{4} )^{2}$$
$89$ $$( 85849 + 4216 T^{2} + T^{4} )^{2}$$
$97$ $$( 1201 - 88 T + T^{2} )^{4}$$