# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1089.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$29.6731007888$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - 2x^{3} + 11x^{2} - 10x + 3$$ x^4 - 2*x^3 + 11*x^2 - 10*x + 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} - 7 q^{4} - 2 \beta_1 q^{5} + 7 \beta_{2} q^{7} + 3 \beta_{3} q^{8}+O(q^{10})$$ q - b3 * q^2 - 7 * q^4 - 2*b1 * q^5 + 7*b2 * q^7 + 3*b3 * q^8 $$q - \beta_{3} q^{2} - 7 q^{4} - 2 \beta_1 q^{5} + 7 \beta_{2} q^{7} + 3 \beta_{3} q^{8} + 22 \beta_{2} q^{10} + 9 \beta_{2} q^{13} + 7 \beta_1 q^{14} + 5 q^{16} - 3 \beta_{2} q^{19} + 14 \beta_1 q^{20} - 6 \beta_1 q^{23} + 63 q^{25} + 9 \beta_1 q^{26} - 49 \beta_{2} q^{28} - 4 \beta_{3} q^{29} - 44 q^{31} + 7 \beta_{3} q^{32} - 28 \beta_{3} q^{35} + 44 q^{37} - 3 \beta_1 q^{38} - 66 \beta_{2} q^{40} - 12 \beta_{3} q^{41} - 21 \beta_{2} q^{43} + 66 \beta_{2} q^{46} - 2 \beta_1 q^{47} - 49 q^{49} - 63 \beta_{3} q^{50} - 63 \beta_{2} q^{52} + 14 \beta_1 q^{53} - 21 \beta_1 q^{56} - 44 q^{58} - 4 \beta_1 q^{59} + 49 \beta_{2} q^{61} + 44 \beta_{3} q^{62} + 97 q^{64} - 36 \beta_{3} q^{65} - 88 q^{67} - 308 q^{70} + 10 \beta_1 q^{71} - 29 \beta_{2} q^{73} - 44 \beta_{3} q^{74} + 21 \beta_{2} q^{76} - 9 \beta_{2} q^{79} - 10 \beta_1 q^{80} - 132 q^{82} + 28 \beta_{3} q^{83} - 21 \beta_1 q^{86} - 24 \beta_1 q^{89} - 126 q^{91} + 42 \beta_1 q^{92} + 22 \beta_{2} q^{94} + 12 \beta_{3} q^{95} + 70 q^{97} + 49 \beta_{3} q^{98}+O(q^{100})$$ q - b3 * q^2 - 7 * q^4 - 2*b1 * q^5 + 7*b2 * q^7 + 3*b3 * q^8 + 22*b2 * q^10 + 9*b2 * q^13 + 7*b1 * q^14 + 5 * q^16 - 3*b2 * q^19 + 14*b1 * q^20 - 6*b1 * q^23 + 63 * q^25 + 9*b1 * q^26 - 49*b2 * q^28 - 4*b3 * q^29 - 44 * q^31 + 7*b3 * q^32 - 28*b3 * q^35 + 44 * q^37 - 3*b1 * q^38 - 66*b2 * q^40 - 12*b3 * q^41 - 21*b2 * q^43 + 66*b2 * q^46 - 2*b1 * q^47 - 49 * q^49 - 63*b3 * q^50 - 63*b2 * q^52 + 14*b1 * q^53 - 21*b1 * q^56 - 44 * q^58 - 4*b1 * q^59 + 49*b2 * q^61 + 44*b3 * q^62 + 97 * q^64 - 36*b3 * q^65 - 88 * q^67 - 308 * q^70 + 10*b1 * q^71 - 29*b2 * q^73 - 44*b3 * q^74 + 21*b2 * q^76 - 9*b2 * q^79 - 10*b1 * q^80 - 132 * q^82 + 28*b3 * q^83 - 21*b1 * q^86 - 24*b1 * q^89 - 126 * q^91 + 42*b1 * q^92 + 22*b2 * q^94 + 12*b3 * q^95 + 70 * q^97 + 49*b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 28 q^{4}+O(q^{10})$$ 4 * q - 28 * q^4 $$4 q - 28 q^{4} + 20 q^{16} + 252 q^{25} - 176 q^{31} + 176 q^{37} - 196 q^{49} - 176 q^{58} + 388 q^{64} - 352 q^{67} - 1232 q^{70} - 528 q^{82} - 504 q^{91} + 280 q^{97}+O(q^{100})$$ 4 * q - 28 * q^4 + 20 * q^16 + 252 * q^25 - 176 * q^31 + 176 * q^37 - 196 * q^49 - 176 * q^58 + 388 * q^64 - 352 * q^67 - 1232 * q^70 - 528 * q^82 - 504 * q^91 + 280 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} + 11x^{2} - 10x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 5$$ v^2 - v + 5 $$\beta_{2}$$ $$=$$ $$( 2\nu^{3} - 3\nu^{2} + 19\nu - 9 ) / 3$$ (2*v^3 - 3*v^2 + 19*v - 9) / 3 $$\beta_{3}$$ $$=$$ $$( 4\nu^{3} - 6\nu^{2} + 44\nu - 21 ) / 3$$ (4*v^3 - 6*v^2 + 44*v - 21) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} - 2\beta_{2} + 1 ) / 2$$ (b3 - 2*b2 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 2\beta_{2} + 2\beta _1 - 9 ) / 2$$ (b3 - 2*b2 + 2*b1 - 9) / 2 $$\nu^{3}$$ $$=$$ $$( -8\beta_{3} + 19\beta_{2} + 3\beta _1 - 14 ) / 2$$ (-8*b3 + 19*b2 + 3*b1 - 14) / 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
604.1
 0.5 + 0.244099i 0.5 + 3.07253i 0.5 − 0.244099i 0.5 − 3.07253i
3.31662i 0 −7.00000 −9.38083 0 9.89949i 9.94987i 0 31.1127i
604.2 3.31662i 0 −7.00000 9.38083 0 9.89949i 9.94987i 0 31.1127i
604.3 3.31662i 0 −7.00000 −9.38083 0 9.89949i 9.94987i 0 31.1127i
604.4 3.31662i 0 −7.00000 9.38083 0 9.89949i 9.94987i 0 31.1127i
 $$n$$: 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
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.c.b 4
3.b odd 2 1 inner 1089.3.c.b 4
11.b odd 2 1 inner 1089.3.c.b 4
33.d even 2 1 inner 1089.3.c.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.3.c.b 4 1.a even 1 1 trivial
1089.3.c.b 4 3.b odd 2 1 inner
1089.3.c.b 4 11.b odd 2 1 inner
1089.3.c.b 4 33.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 11$$ acting on $$S_{3}^{\mathrm{new}}(1089, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 11)^{2}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 88)^{2}$$
$7$ $$(T^{2} + 98)^{2}$$
$11$ $$T^{4}$$
$13$ $$(T^{2} + 162)^{2}$$
$17$ $$T^{4}$$
$19$ $$(T^{2} + 18)^{2}$$
$23$ $$(T^{2} - 792)^{2}$$
$29$ $$(T^{2} + 176)^{2}$$
$31$ $$(T + 44)^{4}$$
$37$ $$(T - 44)^{4}$$
$41$ $$(T^{2} + 1584)^{2}$$
$43$ $$(T^{2} + 882)^{2}$$
$47$ $$(T^{2} - 88)^{2}$$
$53$ $$(T^{2} - 4312)^{2}$$
$59$ $$(T^{2} - 352)^{2}$$
$61$ $$(T^{2} + 4802)^{2}$$
$67$ $$(T + 88)^{4}$$
$71$ $$(T^{2} - 2200)^{2}$$
$73$ $$(T^{2} + 1682)^{2}$$
$79$ $$(T^{2} + 162)^{2}$$
$83$ $$(T^{2} + 8624)^{2}$$
$89$ $$(T^{2} - 12672)^{2}$$
$97$ $$(T - 70)^{4}$$