# Properties

 Label 1089.3.b.g Level $1089$ Weight $3$ Character orbit 1089.b Analytic conductor $29.673$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: 8.0.65306824704.6 Defining polynomial: $$x^{8} - 4 x^{7} - 2 x^{6} + 20 x^{5} + x^{4} - 40 x^{3} + 36 x^{2} - 12 x + 27$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}\cdot 3^{3}$$ Twist minimal: no (minimal twist has level 99) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + ( -2 - \beta_{3} ) q^{4} + ( -\beta_{1} + \beta_{2} ) q^{5} + ( -2 + \beta_{6} ) q^{7} + ( -\beta_{1} + 4 \beta_{2} + \beta_{5} - \beta_{7} ) q^{8} +O(q^{10})$$ $$q -\beta_{2} q^{2} + ( -2 - \beta_{3} ) q^{4} + ( -\beta_{1} + \beta_{2} ) q^{5} + ( -2 + \beta_{6} ) q^{7} + ( -\beta_{1} + 4 \beta_{2} + \beta_{5} - \beta_{7} ) q^{8} + ( 6 + 2 \beta_{3} + \beta_{6} ) q^{10} + ( 1 + \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{13} + ( \beta_{1} + 3 \beta_{2} + 2 \beta_{5} ) q^{14} + ( 13 + 4 \beta_{3} + 4 \beta_{4} ) q^{16} + ( -\beta_{1} - 5 \beta_{2} - \beta_{5} + \beta_{7} ) q^{17} + ( -5 - \beta_{3} + 3 \beta_{4} + \beta_{6} ) q^{19} + ( -\beta_{1} - 13 \beta_{2} + 2 \beta_{7} ) q^{20} + ( -\beta_{1} + \beta_{5} - 2 \beta_{7} ) q^{23} + ( -14 + \beta_{3} - 2 \beta_{4} ) q^{25} + ( -\beta_{1} - 8 \beta_{2} - 3 \beta_{5} ) q^{26} + ( 4 + 2 \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{28} + ( -2 \beta_{1} - 6 \beta_{2} + 6 \beta_{5} ) q^{29} + ( -7 + 3 \beta_{3} - 4 \beta_{4} + 4 \beta_{6} ) q^{31} + ( -25 \beta_{2} - 8 \beta_{5} + 4 \beta_{7} ) q^{32} + ( -27 - 7 \beta_{3} - 4 \beta_{4} + 2 \beta_{6} ) q^{34} + ( 3 \beta_{1} + 5 \beta_{2} + 3 \beta_{5} + 3 \beta_{7} ) q^{35} + ( 17 + \beta_{3} - 2 \beta_{4} + 6 \beta_{6} ) q^{37} + ( 9 \beta_{2} - 3 \beta_{5} + 2 \beta_{7} ) q^{38} + ( -54 - 10 \beta_{3} - 6 \beta_{4} + 5 \beta_{6} ) q^{40} + ( \beta_{1} + 13 \beta_{2} + 5 \beta_{5} + \beta_{7} ) q^{41} + ( 13 - 7 \beta_{3} - 5 \beta_{4} - \beta_{6} ) q^{43} + ( -3 + 7 \beta_{3} + 7 \beta_{4} ) q^{46} + ( -\beta_{1} - 2 \beta_{2} + 11 \beta_{5} + 2 \beta_{7} ) q^{47} + ( -12 - 3 \beta_{3} - 2 \beta_{4} - 8 \beta_{6} ) q^{49} + ( \beta_{1} + 10 \beta_{2} + 3 \beta_{5} - \beta_{7} ) q^{50} + ( -35 - 3 \beta_{3} - 7 \beta_{4} - 4 \beta_{6} ) q^{52} + ( 5 \beta_{1} - \beta_{2} + 8 \beta_{5} ) q^{53} + ( 7 \beta_{1} - 5 \beta_{2} + 4 \beta_{5} + 4 \beta_{7} ) q^{56} + ( -54 - 4 \beta_{3} + 6 \beta_{4} - 4 \beta_{6} ) q^{58} + ( 2 \beta_{1} - 12 \beta_{2} - 8 \beta_{5} - 2 \beta_{7} ) q^{59} + ( 1 - 3 \beta_{3} - 9 \beta_{4} + 2 \beta_{6} ) q^{61} + ( 7 \beta_{1} - 3 \beta_{2} + 13 \beta_{5} - \beta_{7} ) q^{62} + ( -74 - 21 \beta_{3} - 4 \beta_{4} + 8 \beta_{6} ) q^{64} + ( 2 \beta_{1} - 6 \beta_{2} - 14 \beta_{5} - 8 \beta_{7} ) q^{65} + ( 14 + 4 \beta_{3} + 2 \beta_{4} + 6 \beta_{6} ) q^{67} + ( -9 \beta_{1} + 55 \beta_{2} + 15 \beta_{5} - 7 \beta_{7} ) q^{68} + ( 21 - 7 \beta_{3} - 6 \beta_{4} - 6 \beta_{6} ) q^{70} + ( 11 \beta_{1} + 4 \beta_{2} + \beta_{5} - 2 \beta_{7} ) q^{71} + ( -56 + 6 \beta_{3} - 10 \beta_{4} + 6 \beta_{6} ) q^{73} + ( 7 \beta_{1} - 15 \beta_{2} + 15 \beta_{5} - \beta_{7} ) q^{74} + ( 43 - \beta_{3} + 3 \beta_{4} + 7 \beta_{6} ) q^{76} + ( -56 - 6 \beta_{3} + 4 \beta_{4} + \beta_{6} ) q^{79} + ( -9 \beta_{1} + 73 \beta_{2} + 32 \beta_{5} - 8 \beta_{7} ) q^{80} + ( 63 + 9 \beta_{3} + 2 \beta_{4} - 6 \beta_{6} ) q^{82} + ( 9 \beta_{1} - 25 \beta_{2} - 7 \beta_{5} - 3 \beta_{7} ) q^{83} + ( -6 + 18 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{85} + ( -8 \beta_{1} + 33 \beta_{2} + 15 \beta_{5} - 12 \beta_{7} ) q^{86} + ( -14 \beta_{1} - 27 \beta_{2} - 9 \beta_{5} + 4 \beta_{7} ) q^{89} + ( -68 + 6 \beta_{3} + 12 \beta_{6} ) q^{91} + ( 3 \beta_{1} - 46 \beta_{2} - 17 \beta_{5} + 6 \beta_{7} ) q^{92} + ( -45 - 7 \beta_{3} + 5 \beta_{4} - 10 \beta_{6} ) q^{94} + ( -3 \beta_{1} + 3 \beta_{2} + 27 \beta_{5} + 5 \beta_{7} ) q^{95} + ( -19 + \beta_{3} - 2 \beta_{4} - 16 \beta_{6} ) q^{97} + ( -11 \beta_{1} + 24 \beta_{2} - 9 \beta_{5} - 5 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 16 q^{4} - 16 q^{7} + O(q^{10})$$ $$8 q - 16 q^{4} - 16 q^{7} + 48 q^{10} + 8 q^{13} + 104 q^{16} - 40 q^{19} - 112 q^{25} + 32 q^{28} - 56 q^{31} - 216 q^{34} + 136 q^{37} - 432 q^{40} + 104 q^{43} - 24 q^{46} - 96 q^{49} - 280 q^{52} - 432 q^{58} + 8 q^{61} - 592 q^{64} + 112 q^{67} + 168 q^{70} - 448 q^{73} + 344 q^{76} - 448 q^{79} + 504 q^{82} - 48 q^{85} - 544 q^{91} - 360 q^{94} - 152 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} - 2 x^{6} + 20 x^{5} + x^{4} - 40 x^{3} + 36 x^{2} - 12 x + 27$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{6} + 3 \nu^{5} + 6 \nu^{4} - 17 \nu^{3} - 20 \nu^{2} + 29 \nu - 3$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$-7 \nu^{7} + 11 \nu^{6} + 60 \nu^{5} - 83 \nu^{4} - 170 \nu^{3} - 21 \nu^{2} + 21 \nu + 54$$$$)/324$$ $$\beta_{3}$$ $$=$$ $$($$$$-7 \nu^{7} + 11 \nu^{6} + 60 \nu^{5} - 83 \nu^{4} - 170 \nu^{3} + 303 \nu^{2} + 21 \nu - 756$$$$)/162$$ $$\beta_{4}$$ $$=$$ $$($$$$19 \nu^{7} - 53 \nu^{6} - 186 \nu^{5} + 503 \nu^{4} + 716 \nu^{3} - 1563 \nu^{2} - 219 \nu + 1080$$$$)/324$$ $$\beta_{5}$$ $$=$$ $$($$$$-35 \nu^{7} + 136 \nu^{6} + 57 \nu^{5} - 577 \nu^{4} - 121 \nu^{3} + 1191 \nu^{2} - 1596 \nu + 513$$$$)/324$$ $$\beta_{6}$$ $$=$$ $$($$$$47 \nu^{7} - 178 \nu^{6} - 183 \nu^{5} + 997 \nu^{4} + 667 \nu^{3} - 2127 \nu^{2} - 546 \nu - 27$$$$)/324$$ $$\beta_{7}$$ $$=$$ $$($$$$-8 \nu^{7} + 28 \nu^{6} + 30 \nu^{5} - 145 \nu^{4} - 40 \nu^{3} + 219 \nu^{2} - 300 \nu + 108$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 3$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 2 \beta_{2} + 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{7} - \beta_{6} - 7 \beta_{5} + 4 \beta_{4} + 7 \beta_{3} - 16 \beta_{2} + 21$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{7} - 2 \beta_{5} + 2 \beta_{4} + 5 \beta_{3} - 22 \beta_{2} + 2 \beta_{1} + 17$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$30 \beta_{7} + 2 \beta_{6} - 49 \beta_{5} + 7 \beta_{4} + 25 \beta_{3} - 184 \beta_{2} + 15 \beta_{1} + 93$$$$)/6$$ $$\nu^{6}$$ $$=$$ $$($$$$25 \beta_{7} - 2 \beta_{6} - 31 \beta_{5} + 6 \beta_{4} + 5 \beta_{3} - 195 \beta_{2} + 19 \beta_{1} - 1$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$231 \beta_{7} + 29 \beta_{6} - 328 \beta_{5} - 77 \beta_{4} - 116 \beta_{3} - 1588 \beta_{2} + 147 \beta_{1} - 312$$$$)/6$$

## 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
 2.75726 − 0.707107i −1.75726 + 0.707107i 1.13623 − 0.707107i −0.136233 + 0.707107i −0.136233 − 0.707107i 1.13623 + 0.707107i −1.75726 − 0.707107i 2.75726 + 0.707107i
3.89935i 0 −11.2049 6.10332i 0 −2.61086 28.0945i 0 23.7990
485.2 2.48514i 0 −2.17590 4.68911i 0 3.30128 4.53313i 0 11.6531
485.3 1.60688i 0 1.41795 6.21249i 0 −11.1468 8.70597i 0 −9.98270
485.4 0.192662i 0 3.96288 7.62670i 0 2.45638 1.53415i 0 −1.46938
485.5 0.192662i 0 3.96288 7.62670i 0 2.45638 1.53415i 0 −1.46938
485.6 1.60688i 0 1.41795 6.21249i 0 −11.1468 8.70597i 0 −9.98270
485.7 2.48514i 0 −2.17590 4.68911i 0 3.30128 4.53313i 0 11.6531
485.8 3.89935i 0 −11.2049 6.10332i 0 −2.61086 28.0945i 0 23.7990
 $$n$$: 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.3.b.g 8
3.b odd 2 1 inner 1089.3.b.g 8
11.b odd 2 1 99.3.b.a 8
33.d even 2 1 99.3.b.a 8
44.c even 2 1 1584.3.i.b 8
132.d odd 2 1 1584.3.i.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.3.b.a 8 11.b odd 2 1
99.3.b.a 8 33.d even 2 1
1089.3.b.g 8 1.a even 1 1 trivial
1089.3.b.g 8 3.b odd 2 1 inner
1584.3.i.b 8 44.c even 2 1
1584.3.i.b 8 132.d odd 2 1

## Hecke kernels

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

 $$T_{2}^{8} + 24 T_{2}^{6} + 150 T_{2}^{4} + 248 T_{2}^{2} + 9$$ $$T_{7}^{4} + 8 T_{7}^{3} - 42 T_{7}^{2} - 56 T_{7} + 236$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 + 248 T^{2} + 150 T^{4} + 24 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$1838736 + 212240 T^{2} + 8796 T^{4} + 156 T^{6} + T^{8}$$
$7$ $$( 236 - 56 T - 42 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$11$ $$T^{8}$$
$13$ $$( 2828 + 2416 T - 390 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$17$ $$472801536 + 20761472 T^{2} + 269232 T^{4} + 1080 T^{6} + T^{8}$$
$19$ $$( -37908 - 12744 T - 576 T^{2} + 20 T^{3} + T^{4} )^{2}$$
$23$ $$11052737424 + 247138064 T^{2} + 1093980 T^{4} + 1788 T^{6} + T^{8}$$
$29$ $$125180100864 + 1948831488 T^{2} + 6583392 T^{4} + 4848 T^{6} + T^{8}$$
$31$ $$( 244016 - 67312 T - 2412 T^{2} + 28 T^{3} + T^{4} )^{2}$$
$37$ $$( -914256 + 87312 T - 972 T^{2} - 68 T^{3} + T^{4} )^{2}$$
$41$ $$85142571264 + 2333096576 T^{2} + 6598512 T^{4} + 4824 T^{6} + T^{8}$$
$43$ $$( -3241332 + 265992 T - 3936 T^{2} - 52 T^{3} + T^{4} )^{2}$$
$47$ $$12929518743696 + 42615984080 T^{2} + 37225212 T^{4} + 11052 T^{6} + T^{8}$$
$53$ $$26983975824 + 2500597008 T^{2} + 16828380 T^{4} + 8124 T^{6} + T^{8}$$
$59$ $$1028100096 + 10901565440 T^{2} + 17185152 T^{4} + 7872 T^{6} + T^{8}$$
$61$ $$( 1712844 - 106176 T - 6198 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$67$ $$( -2846576 + 193888 T - 2520 T^{2} - 56 T^{3} + T^{4} )^{2}$$
$71$ $$13312019262096 + 110560776336 T^{2} + 88459164 T^{4} + 17916 T^{6} + T^{8}$$
$73$ $$( -37468144 - 904064 T + 7464 T^{2} + 224 T^{3} + T^{4} )^{2}$$
$79$ $$( -4606964 + 243368 T + 15318 T^{2} + 224 T^{3} + T^{4} )^{2}$$
$83$ $$177188471398656 + 616317268608 T^{2} + 239224752 T^{4} + 28584 T^{6} + T^{8}$$
$89$ $$13886211865011984 + 6188302325664 T^{2} + 870980184 T^{4} + 49704 T^{6} + T^{8}$$
$97$ $$( -9076976 - 1330448 T - 15060 T^{2} + 76 T^{3} + T^{4} )^{2}$$