# Properties

 Label 1089.3.b.c 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: $$x^{4} + 16x^{2} + 49$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} - 4) q^{4} + (2 \beta_{2} - \beta_1) q^{5} + (\beta_{3} - 5) q^{7} + (7 \beta_{2} - \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b3 - 4) * q^4 + (2*b2 - b1) * q^5 + (b3 - 5) * q^7 + (7*b2 - b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{3} - 4) q^{4} + (2 \beta_{2} - \beta_1) q^{5} + (\beta_{3} - 5) q^{7} + (7 \beta_{2} - \beta_1) q^{8} + ( - 3 \beta_{3} + 10) q^{10} + (2 \beta_{3} - 17) q^{13} + (7 \beta_{2} - 6 \beta_1) q^{14} + ( - 4 \beta_{3} - 1) q^{16} + (4 \beta_{2} + 3 \beta_1) q^{17} + ( - 6 \beta_{3} + 6) q^{19} + ( - 13 \beta_{2} + 9 \beta_1) q^{20} + (5 \beta_{2} - 6 \beta_1) q^{23} + (5 \beta_{3} + 5) q^{25} + (14 \beta_{2} - 19 \beta_1) q^{26} + ( - 9 \beta_{3} + 35) q^{28} + ( - 22 \beta_{2} + \beta_1) q^{29} + ( - 10 \beta_{3} - 14) q^{31} - \beta_1 q^{32} + ( - \beta_{3} - 20) q^{34} + ( - 15 \beta_{2} + 10 \beta_1) q^{35} + (5 \beta_{3} + 18) q^{37} + ( - 42 \beta_{2} + 12 \beta_1) q^{38} + (10 \beta_{3} - 45) q^{40} + (20 \beta_{2} + 23 \beta_1) q^{41} + ( - 17 \beta_{3} - 3) q^{43} + ( - 11 \beta_{3} + 53) q^{46} + (2 \beta_{2} - 6 \beta_1) q^{47} + ( - 10 \beta_{3} - 9) q^{49} + 35 \beta_{2} q^{50} + ( - 25 \beta_{3} + 98) q^{52} + (14 \beta_{2} + 7 \beta_1) q^{53} + ( - 35 \beta_{2} + 20 \beta_1) q^{56} + (23 \beta_{3} - 30) q^{58} - 49 \beta_{2} q^{59} + (8 \beta_{3} - 6) q^{61} + ( - 70 \beta_{2} - 4 \beta_1) q^{62} + ( - 17 \beta_{3} + 4) q^{64} + ( - 44 \beta_{2} + 27 \beta_1) q^{65} + ( - \beta_{3} - 13) q^{67} + (9 \beta_{2} - 7 \beta_1) q^{68} + (25 \beta_{3} - 95) q^{70} + ( - 43 \beta_{2} - 8 \beta_1) q^{71} + ( - 6 \beta_{3} - 32) q^{73} + (35 \beta_{2} + 13 \beta_1) q^{74} + (30 \beta_{3} - 114) q^{76} + ( - 8 \beta_{3} - 34) q^{79} + (18 \beta_{2} - 19 \beta_1) q^{80} + (3 \beta_{3} - 164) q^{82} + (42 \beta_{2} + 30 \beta_1) q^{83} + ( - 5 \beta_{3} + 10) q^{85} + ( - 119 \beta_{2} + 14 \beta_1) q^{86} + ( - 77 \beta_{2} - \beta_1) q^{89} + ( - 27 \beta_{3} + 115) q^{91} + ( - 57 \beta_{2} + 40 \beta_1) q^{92} + ( - 8 \beta_{3} + 50) q^{94} + (42 \beta_{2} - 36 \beta_1) q^{95} + ( - 15 \beta_{3} + 44) q^{97} + ( - 70 \beta_{2} + \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b3 - 4) * q^4 + (2*b2 - b1) * q^5 + (b3 - 5) * q^7 + (7*b2 - b1) * q^8 + (-3*b3 + 10) * q^10 + (2*b3 - 17) * q^13 + (7*b2 - 6*b1) * q^14 + (-4*b3 - 1) * q^16 + (4*b2 + 3*b1) * q^17 + (-6*b3 + 6) * q^19 + (-13*b2 + 9*b1) * q^20 + (5*b2 - 6*b1) * q^23 + (5*b3 + 5) * q^25 + (14*b2 - 19*b1) * q^26 + (-9*b3 + 35) * q^28 + (-22*b2 + b1) * q^29 + (-10*b3 - 14) * q^31 - b1 * q^32 + (-b3 - 20) * q^34 + (-15*b2 + 10*b1) * q^35 + (5*b3 + 18) * q^37 + (-42*b2 + 12*b1) * q^38 + (10*b3 - 45) * q^40 + (20*b2 + 23*b1) * q^41 + (-17*b3 - 3) * q^43 + (-11*b3 + 53) * q^46 + (2*b2 - 6*b1) * q^47 + (-10*b3 - 9) * q^49 + 35*b2 * q^50 + (-25*b3 + 98) * q^52 + (14*b2 + 7*b1) * q^53 + (-35*b2 + 20*b1) * q^56 + (23*b3 - 30) * q^58 - 49*b2 * q^59 + (8*b3 - 6) * q^61 + (-70*b2 - 4*b1) * q^62 + (-17*b3 + 4) * q^64 + (-44*b2 + 27*b1) * q^65 + (-b3 - 13) * q^67 + (9*b2 - 7*b1) * q^68 + (25*b3 - 95) * q^70 + (-43*b2 - 8*b1) * q^71 + (-6*b3 - 32) * q^73 + (35*b2 + 13*b1) * q^74 + (30*b3 - 114) * q^76 + (-8*b3 - 34) * q^79 + (18*b2 - 19*b1) * q^80 + (3*b3 - 164) * q^82 + (42*b2 + 30*b1) * q^83 + (-5*b3 + 10) * q^85 + (-119*b2 + 14*b1) * q^86 + (-77*b2 - b1) * q^89 + (-27*b3 + 115) * q^91 + (-57*b2 + 40*b1) * q^92 + (-8*b3 + 50) * q^94 + (42*b2 - 36*b1) * q^95 + (-15*b3 + 44) * q^97 + (-70*b2 + b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 16 q^{4} - 20 q^{7}+O(q^{10})$$ 4 * q - 16 * q^4 - 20 * q^7 $$4 q - 16 q^{4} - 20 q^{7} + 40 q^{10} - 68 q^{13} - 4 q^{16} + 24 q^{19} + 20 q^{25} + 140 q^{28} - 56 q^{31} - 80 q^{34} + 72 q^{37} - 180 q^{40} - 12 q^{43} + 212 q^{46} - 36 q^{49} + 392 q^{52} - 120 q^{58} - 24 q^{61} + 16 q^{64} - 52 q^{67} - 380 q^{70} - 128 q^{73} - 456 q^{76} - 136 q^{79} - 656 q^{82} + 40 q^{85} + 460 q^{91} + 200 q^{94} + 176 q^{97}+O(q^{100})$$ 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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 16x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 9\nu ) / 7$$ (v^3 + 9*v) / 7 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 8$$ v^2 + 8
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 8$$ b3 - 8 $$\nu^{3}$$ $$=$$ $$7\beta_{2} - 9\beta_1$$ 7*b2 - 9*b1

## 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.c 4
3.b odd 2 1 inner 1089.3.b.c 4
11.b odd 2 1 1089.3.b.d yes 4
33.d even 2 1 1089.3.b.d yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.3.b.c 4 1.a even 1 1 trivial
1089.3.b.c 4 3.b odd 2 1 inner
1089.3.b.d yes 4 11.b odd 2 1
1089.3.b.d yes 4 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 $$S_{3}^{\mathrm{new}}(1089, [\chi])$$:

 $$T_{2}^{4} + 16T_{2}^{2} + 49$$ T2^4 + 16*T2^2 + 49 $$T_{7}^{2} + 10T_{7} + 10$$ T7^2 + 10*T7 + 10

## Hecke characteristic polynomials

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