# Properties

 Label 1089.3.b.f 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: 8.0.1105425137664.10 Defining polynomial: $$x^{8} - 14 x^{6} + 71 x^{4} - 80 x^{2} + 121$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}\cdot 3^{3}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -4 - \beta_{2} ) q^{4} + ( -\beta_{6} - \beta_{7} ) q^{5} + ( \beta_{3} + \beta_{5} ) q^{7} + ( -5 \beta_{1} + \beta_{4} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -4 - \beta_{2} ) q^{4} + ( -\beta_{6} - \beta_{7} ) q^{5} + ( \beta_{3} + \beta_{5} ) q^{7} + ( -5 \beta_{1} + \beta_{4} ) q^{8} + ( 5 \beta_{3} + \beta_{5} ) q^{10} + \beta_{5} q^{13} + ( 3 \beta_{6} + 5 \beta_{7} ) q^{14} + ( 21 + 4 \beta_{2} ) q^{16} + ( -4 \beta_{1} - \beta_{4} ) q^{17} -5 \beta_{3} q^{19} + ( 11 \beta_{6} + 9 \beta_{7} ) q^{20} + 5 \beta_{7} q^{23} + ( -6 - 5 \beta_{2} ) q^{25} + 3 \beta_{7} q^{26} + ( -15 \beta_{3} - \beta_{5} ) q^{28} -10 \beta_{1} q^{29} + ( -25 - \beta_{2} ) q^{31} + 21 \beta_{1} q^{32} + ( 35 + \beta_{2} ) q^{34} + ( -16 \beta_{1} + \beta_{4} ) q^{35} + ( -15 - 5 \beta_{2} ) q^{37} + ( -15 \beta_{6} - 10 \beta_{7} ) q^{38} + ( -31 \beta_{3} - 5 \beta_{5} ) q^{40} + ( 10 \beta_{1} - 5 \beta_{4} ) q^{41} + ( 5 \beta_{3} - 6 \beta_{5} ) q^{43} + ( -10 \beta_{3} - 5 \beta_{5} ) q^{46} + ( -10 \beta_{6} + 5 \beta_{7} ) q^{47} + ( 24 - \beta_{2} ) q^{49} + ( -31 \beta_{1} + 5 \beta_{4} ) q^{50} + ( -6 \beta_{3} + \beta_{5} ) q^{52} + ( 5 \beta_{6} - 15 \beta_{7} ) q^{53} + ( -33 \beta_{6} - 13 \beta_{7} ) q^{56} + ( 80 + 10 \beta_{2} ) q^{58} -2 \beta_{6} q^{59} + ( -20 \beta_{3} + 5 \beta_{5} ) q^{61} + ( -30 \beta_{1} + \beta_{4} ) q^{62} + ( -84 - 5 \beta_{2} ) q^{64} -6 \beta_{1} q^{65} + ( 20 - 10 \beta_{2} ) q^{67} + ( 24 \beta_{1} - 5 \beta_{4} ) q^{68} + ( 125 + 19 \beta_{2} ) q^{70} + ( 8 \beta_{6} + 25 \beta_{7} ) q^{71} + ( 14 \beta_{3} + 4 \beta_{5} ) q^{73} + ( -40 \beta_{1} + 5 \beta_{4} ) q^{74} + ( 45 \beta_{3} + 10 \beta_{5} ) q^{76} + ( -15 \beta_{3} + 5 \beta_{5} ) q^{79} + ( -49 \beta_{6} - 41 \beta_{7} ) q^{80} + ( -65 - 25 \beta_{2} ) q^{82} + ( -16 \beta_{1} - 5 \beta_{4} ) q^{83} + ( -14 \beta_{3} - 4 \beta_{5} ) q^{85} + ( 15 \beta_{6} - 8 \beta_{7} ) q^{86} + ( -15 \beta_{6} + 10 \beta_{7} ) q^{89} + ( 60 - 6 \beta_{2} ) q^{91} + ( -30 \beta_{6} - 15 \beta_{7} ) q^{92} + ( 20 \beta_{3} - 5 \beta_{5} ) q^{94} + ( 50 \beta_{1} - 5 \beta_{4} ) q^{95} + ( -55 + 15 \beta_{2} ) q^{97} + ( 19 \beta_{1} + \beta_{4} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 32 q^{4} + O(q^{10})$$ $$8 q - 32 q^{4} + 168 q^{16} - 48 q^{25} - 200 q^{31} + 280 q^{34} - 120 q^{37} + 192 q^{49} + 640 q^{58} - 672 q^{64} + 160 q^{67} + 1000 q^{70} - 520 q^{82} + 480 q^{91} - 440 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 14 x^{6} + 71 x^{4} - 80 x^{2} + 121$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{6} + 19 \nu^{4} - 93 \nu^{2} + 70$$$$)/73$$ $$\beta_{2}$$ $$=$$ $$($$$$-4 \nu^{6} + 38 \nu^{4} - 40 \nu^{2} - 371$$$$)/73$$ $$\beta_{3}$$ $$=$$ $$($$$$13 \nu^{7} - 160 \nu^{5} + 714 \nu^{3} - 1623 \nu$$$$)/803$$ $$\beta_{4}$$ $$=$$ $$($$$$-14 \nu^{6} + 206 \nu^{4} - 1016 \nu^{2} + 782$$$$)/73$$ $$\beta_{5}$$ $$=$$ $$($$$$-25 \nu^{7} + 493 \nu^{5} - 3535 \nu^{3} + 8248 \nu$$$$)/803$$ $$\beta_{6}$$ $$=$$ $$($$$$39 \nu^{7} - 480 \nu^{5} + 2142 \nu^{3} - 51 \nu$$$$)/803$$ $$\beta_{7}$$ $$=$$ $$($$$$-43 \nu^{7} + 591 \nu^{5} - 2547 \nu^{3} + 118 \nu$$$$)/803$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} - 3 \beta_{3}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} - 2 \beta_{1} + 7$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$9 \beta_{7} + 13 \beta_{6} - 3 \beta_{5} - 15 \beta_{3}$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{4} + 5 \beta_{2} - 24 \beta_{1} + 27$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$105 \beta_{7} + 124 \beta_{6} - 9 \beta_{5} - 42 \beta_{3}$$$$)/6$$ $$\nu^{6}$$ $$=$$ $$($$$$19 \beta_{4} + \beta_{2} - 208 \beta_{1} + 1$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$798 \beta_{7} + 937 \beta_{6} + 54 \beta_{5} + 303 \beta_{3}$$$$)/6$$

## 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
 2.65356 + 0.707107i −2.65356 − 0.707107i −0.979091 − 0.707107i 0.979091 + 0.707107i 0.979091 − 0.707107i −0.979091 + 0.707107i −2.65356 + 0.707107i 2.65356 − 0.707107i
3.75270i 0 −10.0828 7.83670i 0 −8.18030 22.8268i 0 −29.4088
485.2 3.75270i 0 −10.0828 7.83670i 0 8.18030 22.8268i 0 29.4088
485.3 1.38464i 0 2.08276 0.765629i 0 −8.89285 8.42246i 0 −1.06012
485.4 1.38464i 0 2.08276 0.765629i 0 8.89285 8.42246i 0 1.06012
485.5 1.38464i 0 2.08276 0.765629i 0 8.89285 8.42246i 0 1.06012
485.6 1.38464i 0 2.08276 0.765629i 0 −8.89285 8.42246i 0 −1.06012
485.7 3.75270i 0 −10.0828 7.83670i 0 8.18030 22.8268i 0 29.4088
485.8 3.75270i 0 −10.0828 7.83670i 0 −8.18030 22.8268i 0 −29.4088
 $$n$$: 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.f 8
3.b odd 2 1 inner 1089.3.b.f 8
11.b odd 2 1 inner 1089.3.b.f 8
33.d even 2 1 inner 1089.3.b.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.3.b.f 8 1.a even 1 1 trivial
1089.3.b.f 8 3.b odd 2 1 inner
1089.3.b.f 8 11.b odd 2 1 inner
1089.3.b.f 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 $$S_{3}^{\mathrm{new}}(1089, [\chi])$$:

 $$T_{2}^{4} + 16 T_{2}^{2} + 27$$ $$T_{7}^{4} - 146 T_{7}^{2} + 5292$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 27 + 16 T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$( 36 + 62 T^{2} + T^{4} )^{2}$$
$7$ $$( 5292 - 146 T^{2} + T^{4} )^{2}$$
$11$ $$T^{8}$$
$13$ $$( 972 - 126 T^{2} + T^{4} )^{2}$$
$17$ $$( 52272 + 556 T^{2} + T^{4} )^{2}$$
$19$ $$( 67500 - 800 T^{2} + T^{4} )^{2}$$
$23$ $$( 202500 + 950 T^{2} + T^{4} )^{2}$$
$29$ $$( 270000 + 1600 T^{2} + T^{4} )^{2}$$
$31$ $$( 588 + 50 T + T^{2} )^{4}$$
$37$ $$( -700 + 30 T + T^{2} )^{4}$$
$41$ $$( 13230000 + 7300 T^{2} + T^{4} )^{2}$$
$43$ $$( 484812 - 5696 T^{2} + T^{4} )^{2}$$
$47$ $$( 2722500 + 5150 T^{2} + T^{4} )^{2}$$
$53$ $$( 9922500 + 10350 T^{2} + T^{4} )^{2}$$
$59$ $$( 72 + T^{2} )^{4}$$
$61$ $$( 73507500 - 17150 T^{2} + T^{4} )^{2}$$
$67$ $$( -3300 - 40 T + T^{2} )^{4}$$
$71$ $$( 127644804 + 23654 T^{2} + T^{4} )^{2}$$
$73$ $$( 1881792 - 7616 T^{2} + T^{4} )^{2}$$
$79$ $$( 29767500 - 11250 T^{2} + T^{4} )^{2}$$
$83$ $$( 15431472 + 11356 T^{2} + T^{4} )^{2}$$
$89$ $$( 9922500 + 13700 T^{2} + T^{4} )^{2}$$
$97$ $$( -5300 + 110 T + T^{2} )^{4}$$