# Properties

 Label 1089.2.a.o Level $1089$ Weight $2$ Character orbit 1089.a Self dual yes Analytic conductor $8.696$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1089.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.69570878012$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 363) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + q^{4} + 3 q^{5} + 2 \beta q^{7} - \beta q^{8} +O(q^{10})$$ q + b * q^2 + q^4 + 3 * q^5 + 2*b * q^7 - b * q^8 $$q + \beta q^{2} + q^{4} + 3 q^{5} + 2 \beta q^{7} - \beta q^{8} + 3 \beta q^{10} + \beta q^{13} + 6 q^{14} - 5 q^{16} + \beta q^{17} - 4 \beta q^{19} + 3 q^{20} + 6 q^{23} + 4 q^{25} + 3 q^{26} + 2 \beta q^{28} - \beta q^{29} + 4 q^{31} - 3 \beta q^{32} + 3 q^{34} + 6 \beta q^{35} - 11 q^{37} - 12 q^{38} - 3 \beta q^{40} - \beta q^{41} - 2 \beta q^{43} + 6 \beta q^{46} + 5 q^{49} + 4 \beta q^{50} + \beta q^{52} + 9 q^{53} - 6 q^{56} - 3 q^{58} + 6 q^{59} + 4 \beta q^{62} + q^{64} + 3 \beta q^{65} - 2 q^{67} + \beta q^{68} + 18 q^{70} + 6 q^{71} - 4 \beta q^{73} - 11 \beta q^{74} - 4 \beta q^{76} - 15 q^{80} - 3 q^{82} + 3 \beta q^{85} - 6 q^{86} - 9 q^{89} + 6 q^{91} + 6 q^{92} - 12 \beta q^{95} - 7 q^{97} + 5 \beta q^{98} +O(q^{100})$$ q + b * q^2 + q^4 + 3 * q^5 + 2*b * q^7 - b * q^8 + 3*b * q^10 + b * q^13 + 6 * q^14 - 5 * q^16 + b * q^17 - 4*b * q^19 + 3 * q^20 + 6 * q^23 + 4 * q^25 + 3 * q^26 + 2*b * q^28 - b * q^29 + 4 * q^31 - 3*b * q^32 + 3 * q^34 + 6*b * q^35 - 11 * q^37 - 12 * q^38 - 3*b * q^40 - b * q^41 - 2*b * q^43 + 6*b * q^46 + 5 * q^49 + 4*b * q^50 + b * q^52 + 9 * q^53 - 6 * q^56 - 3 * q^58 + 6 * q^59 + 4*b * q^62 + q^64 + 3*b * q^65 - 2 * q^67 + b * q^68 + 18 * q^70 + 6 * q^71 - 4*b * q^73 - 11*b * q^74 - 4*b * q^76 - 15 * q^80 - 3 * q^82 + 3*b * q^85 - 6 * q^86 - 9 * q^89 + 6 * q^91 + 6 * q^92 - 12*b * q^95 - 7 * q^97 + 5*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 6 q^{5}+O(q^{10})$$ 2 * q + 2 * q^4 + 6 * q^5 $$2 q + 2 q^{4} + 6 q^{5} + 12 q^{14} - 10 q^{16} + 6 q^{20} + 12 q^{23} + 8 q^{25} + 6 q^{26} + 8 q^{31} + 6 q^{34} - 22 q^{37} - 24 q^{38} + 10 q^{49} + 18 q^{53} - 12 q^{56} - 6 q^{58} + 12 q^{59} + 2 q^{64} - 4 q^{67} + 36 q^{70} + 12 q^{71} - 30 q^{80} - 6 q^{82} - 12 q^{86} - 18 q^{89} + 12 q^{91} + 12 q^{92} - 14 q^{97}+O(q^{100})$$ 2 * q + 2 * q^4 + 6 * q^5 + 12 * q^14 - 10 * q^16 + 6 * q^20 + 12 * q^23 + 8 * q^25 + 6 * q^26 + 8 * q^31 + 6 * q^34 - 22 * q^37 - 24 * q^38 + 10 * q^49 + 18 * q^53 - 12 * q^56 - 6 * q^58 + 12 * q^59 + 2 * q^64 - 4 * q^67 + 36 * q^70 + 12 * q^71 - 30 * q^80 - 6 * q^82 - 12 * q^86 - 18 * q^89 + 12 * q^91 + 12 * q^92 - 14 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
−1.73205 0 1.00000 3.00000 0 −3.46410 1.73205 0 −5.19615
1.2 1.73205 0 1.00000 3.00000 0 3.46410 −1.73205 0 5.19615
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$11$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.2.a.o 2
3.b odd 2 1 363.2.a.f 2
11.b odd 2 1 inner 1089.2.a.o 2
12.b even 2 1 5808.2.a.ca 2
15.d odd 2 1 9075.2.a.bo 2
33.d even 2 1 363.2.a.f 2
33.f even 10 4 363.2.e.m 8
33.h odd 10 4 363.2.e.m 8
132.d odd 2 1 5808.2.a.ca 2
165.d even 2 1 9075.2.a.bo 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.a.f 2 3.b odd 2 1
363.2.a.f 2 33.d even 2 1
363.2.e.m 8 33.f even 10 4
363.2.e.m 8 33.h odd 10 4
1089.2.a.o 2 1.a even 1 1 trivial
1089.2.a.o 2 11.b odd 2 1 inner
5808.2.a.ca 2 12.b even 2 1
5808.2.a.ca 2 132.d odd 2 1
9075.2.a.bo 2 15.d odd 2 1
9075.2.a.bo 2 165.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1089))$$:

 $$T_{2}^{2} - 3$$ T2^2 - 3 $$T_{5} - 3$$ T5 - 3 $$T_{7}^{2} - 12$$ T7^2 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3$$
$3$ $$T^{2}$$
$5$ $$(T - 3)^{2}$$
$7$ $$T^{2} - 12$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 3$$
$17$ $$T^{2} - 3$$
$19$ $$T^{2} - 48$$
$23$ $$(T - 6)^{2}$$
$29$ $$T^{2} - 3$$
$31$ $$(T - 4)^{2}$$
$37$ $$(T + 11)^{2}$$
$41$ $$T^{2} - 3$$
$43$ $$T^{2} - 12$$
$47$ $$T^{2}$$
$53$ $$(T - 9)^{2}$$
$59$ $$(T - 6)^{2}$$
$61$ $$T^{2}$$
$67$ $$(T + 2)^{2}$$
$71$ $$(T - 6)^{2}$$
$73$ $$T^{2} - 48$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$(T + 9)^{2}$$
$97$ $$(T + 7)^{2}$$
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