# Properties

 Label 1089.3.c.a Level $1089$ Weight $3$ Character orbit 1089.c Analytic conductor $29.673$ 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: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1089.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$29.6731007888$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 121) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 2 q^{4} + 7 q^{5} - 5 \beta q^{7} + 6 \beta q^{8} +O(q^{10})$$ q + b * q^2 + 2 * q^4 + 7 * q^5 - 5*b * q^7 + 6*b * q^8 $$q + \beta q^{2} + 2 q^{4} + 7 q^{5} - 5 \beta q^{7} + 6 \beta q^{8} + 7 \beta q^{10} - 12 \beta q^{13} + 10 q^{14} - 4 q^{16} - 3 \beta q^{17} + 12 \beta q^{19} + 14 q^{20} + 9 q^{23} + 24 q^{25} + 24 q^{26} - 10 \beta q^{28} + 16 \beta q^{29} + 49 q^{31} + 20 \beta q^{32} + 6 q^{34} - 35 \beta q^{35} + 17 q^{37} - 24 q^{38} + 42 \beta q^{40} - 12 \beta q^{41} - 33 \beta q^{43} + 9 \beta q^{46} - 32 q^{47} - q^{49} + 24 \beta q^{50} - 24 \beta q^{52} - 16 q^{53} + 60 q^{56} - 32 q^{58} + 71 q^{59} - 8 \beta q^{61} + 49 \beta q^{62} - 56 q^{64} - 84 \beta q^{65} - 31 q^{67} - 6 \beta q^{68} + 70 q^{70} + 73 q^{71} + 28 \beta q^{73} + 17 \beta q^{74} + 24 \beta q^{76} - 111 \beta q^{79} - 28 q^{80} + 24 q^{82} - 25 \beta q^{83} - 21 \beta q^{85} + 66 q^{86} + 9 q^{89} - 120 q^{91} + 18 q^{92} - 32 \beta q^{94} + 84 \beta q^{95} - 17 q^{97} - \beta q^{98} +O(q^{100})$$ q + b * q^2 + 2 * q^4 + 7 * q^5 - 5*b * q^7 + 6*b * q^8 + 7*b * q^10 - 12*b * q^13 + 10 * q^14 - 4 * q^16 - 3*b * q^17 + 12*b * q^19 + 14 * q^20 + 9 * q^23 + 24 * q^25 + 24 * q^26 - 10*b * q^28 + 16*b * q^29 + 49 * q^31 + 20*b * q^32 + 6 * q^34 - 35*b * q^35 + 17 * q^37 - 24 * q^38 + 42*b * q^40 - 12*b * q^41 - 33*b * q^43 + 9*b * q^46 - 32 * q^47 - q^49 + 24*b * q^50 - 24*b * q^52 - 16 * q^53 + 60 * q^56 - 32 * q^58 + 71 * q^59 - 8*b * q^61 + 49*b * q^62 - 56 * q^64 - 84*b * q^65 - 31 * q^67 - 6*b * q^68 + 70 * q^70 + 73 * q^71 + 28*b * q^73 + 17*b * q^74 + 24*b * q^76 - 111*b * q^79 - 28 * q^80 + 24 * q^82 - 25*b * q^83 - 21*b * q^85 + 66 * q^86 + 9 * q^89 - 120 * q^91 + 18 * q^92 - 32*b * q^94 + 84*b * q^95 - 17 * q^97 - b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4} + 14 q^{5}+O(q^{10})$$ 2 * q + 4 * q^4 + 14 * q^5 $$2 q + 4 q^{4} + 14 q^{5} + 20 q^{14} - 8 q^{16} + 28 q^{20} + 18 q^{23} + 48 q^{25} + 48 q^{26} + 98 q^{31} + 12 q^{34} + 34 q^{37} - 48 q^{38} - 64 q^{47} - 2 q^{49} - 32 q^{53} + 120 q^{56} - 64 q^{58} + 142 q^{59} - 112 q^{64} - 62 q^{67} + 140 q^{70} + 146 q^{71} - 56 q^{80} + 48 q^{82} + 132 q^{86} + 18 q^{89} - 240 q^{91} + 36 q^{92} - 34 q^{97}+O(q^{100})$$ 2 * q + 4 * q^4 + 14 * q^5 + 20 * q^14 - 8 * q^16 + 28 * q^20 + 18 * q^23 + 48 * q^25 + 48 * q^26 + 98 * q^31 + 12 * q^34 + 34 * q^37 - 48 * q^38 - 64 * q^47 - 2 * q^49 - 32 * q^53 + 120 * q^56 - 64 * q^58 + 142 * q^59 - 112 * q^64 - 62 * q^67 + 140 * q^70 + 146 * q^71 - 56 * q^80 + 48 * q^82 + 132 * q^86 + 18 * q^89 - 240 * q^91 + 36 * q^92 - 34 * q^97

## 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
 − 1.41421i 1.41421i
1.41421i 0 2.00000 7.00000 0 7.07107i 8.48528i 0 9.89949i
604.2 1.41421i 0 2.00000 7.00000 0 7.07107i 8.48528i 0 9.89949i
 $$n$$: 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
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 1089.3.c.a 2
3.b odd 2 1 121.3.b.a 2
11.b odd 2 1 inner 1089.3.c.a 2
33.d even 2 1 121.3.b.a 2
33.f even 10 4 121.3.d.e 8
33.h odd 10 4 121.3.d.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.3.b.a 2 3.b odd 2 1
121.3.b.a 2 33.d even 2 1
121.3.d.e 8 33.f even 10 4
121.3.d.e 8 33.h odd 10 4
1089.3.c.a 2 1.a even 1 1 trivial
1089.3.c.a 2 11.b odd 2 1 inner

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2$$
$3$ $$T^{2}$$
$5$ $$(T - 7)^{2}$$
$7$ $$T^{2} + 50$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 288$$
$17$ $$T^{2} + 18$$
$19$ $$T^{2} + 288$$
$23$ $$(T - 9)^{2}$$
$29$ $$T^{2} + 512$$
$31$ $$(T - 49)^{2}$$
$37$ $$(T - 17)^{2}$$
$41$ $$T^{2} + 288$$
$43$ $$T^{2} + 2178$$
$47$ $$(T + 32)^{2}$$
$53$ $$(T + 16)^{2}$$
$59$ $$(T - 71)^{2}$$
$61$ $$T^{2} + 128$$
$67$ $$(T + 31)^{2}$$
$71$ $$(T - 73)^{2}$$
$73$ $$T^{2} + 1568$$
$79$ $$T^{2} + 24642$$
$83$ $$T^{2} + 1250$$
$89$ $$(T - 9)^{2}$$
$97$ $$(T + 17)^{2}$$
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