# Properties

 Label 1088.2.a.q Level $1088$ Weight $2$ Character orbit 1088.a Self dual yes Analytic conductor $8.688$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1088 = 2^{6} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1088.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.68772373992$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 544) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 2 q^{5} + 3 \beta q^{7} - q^{9}+O(q^{10})$$ q + b * q^3 + 2 * q^5 + 3*b * q^7 - q^9 $$q + \beta q^{3} + 2 q^{5} + 3 \beta q^{7} - q^{9} + \beta q^{11} + 4 q^{13} + 2 \beta q^{15} - q^{17} - 2 \beta q^{19} + 6 q^{21} - 3 \beta q^{23} - q^{25} - 4 \beta q^{27} + 6 q^{29} - 5 \beta q^{31} + 2 q^{33} + 6 \beta q^{35} + 2 q^{37} + 4 \beta q^{39} - 6 q^{41} + 6 \beta q^{43} - 2 q^{45} - 8 \beta q^{47} + 11 q^{49} - \beta q^{51} + 6 q^{53} + 2 \beta q^{55} - 4 q^{57} - 6 \beta q^{59} - 6 q^{61} - 3 \beta q^{63} + 8 q^{65} - 4 \beta q^{67} - 6 q^{69} + 5 \beta q^{71} + 10 q^{73} - \beta q^{75} + 6 q^{77} + 9 \beta q^{79} - 5 q^{81} - 10 \beta q^{83} - 2 q^{85} + 6 \beta q^{87} - 8 q^{89} + 12 \beta q^{91} - 10 q^{93} - 4 \beta q^{95} + 14 q^{97} - \beta q^{99} +O(q^{100})$$ q + b * q^3 + 2 * q^5 + 3*b * q^7 - q^9 + b * q^11 + 4 * q^13 + 2*b * q^15 - q^17 - 2*b * q^19 + 6 * q^21 - 3*b * q^23 - q^25 - 4*b * q^27 + 6 * q^29 - 5*b * q^31 + 2 * q^33 + 6*b * q^35 + 2 * q^37 + 4*b * q^39 - 6 * q^41 + 6*b * q^43 - 2 * q^45 - 8*b * q^47 + 11 * q^49 - b * q^51 + 6 * q^53 + 2*b * q^55 - 4 * q^57 - 6*b * q^59 - 6 * q^61 - 3*b * q^63 + 8 * q^65 - 4*b * q^67 - 6 * q^69 + 5*b * q^71 + 10 * q^73 - b * q^75 + 6 * q^77 + 9*b * q^79 - 5 * q^81 - 10*b * q^83 - 2 * q^85 + 6*b * q^87 - 8 * q^89 + 12*b * q^91 - 10 * q^93 - 4*b * q^95 + 14 * q^97 - b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5} - 2 q^{9}+O(q^{10})$$ 2 * q + 4 * q^5 - 2 * q^9 $$2 q + 4 q^{5} - 2 q^{9} + 8 q^{13} - 2 q^{17} + 12 q^{21} - 2 q^{25} + 12 q^{29} + 4 q^{33} + 4 q^{37} - 12 q^{41} - 4 q^{45} + 22 q^{49} + 12 q^{53} - 8 q^{57} - 12 q^{61} + 16 q^{65} - 12 q^{69} + 20 q^{73} + 12 q^{77} - 10 q^{81} - 4 q^{85} - 16 q^{89} - 20 q^{93} + 28 q^{97}+O(q^{100})$$ 2 * q + 4 * q^5 - 2 * q^9 + 8 * q^13 - 2 * q^17 + 12 * q^21 - 2 * q^25 + 12 * q^29 + 4 * q^33 + 4 * q^37 - 12 * q^41 - 4 * q^45 + 22 * q^49 + 12 * q^53 - 8 * q^57 - 12 * q^61 + 16 * q^65 - 12 * q^69 + 20 * q^73 + 12 * q^77 - 10 * q^81 - 4 * q^85 - 16 * q^89 - 20 * q^93 + 28 * 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
 −1.41421 1.41421
0 −1.41421 0 2.00000 0 −4.24264 0 −1.00000 0
1.2 0 1.41421 0 2.00000 0 4.24264 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$17$$ $$1$$

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1088.2.a.q 2
3.b odd 2 1 9792.2.a.cg 2
4.b odd 2 1 inner 1088.2.a.q 2
8.b even 2 1 544.2.a.g 2
8.d odd 2 1 544.2.a.g 2
12.b even 2 1 9792.2.a.cg 2
24.f even 2 1 4896.2.a.z 2
24.h odd 2 1 4896.2.a.z 2
136.e odd 2 1 9248.2.a.p 2
136.h even 2 1 9248.2.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
544.2.a.g 2 8.b even 2 1
544.2.a.g 2 8.d odd 2 1
1088.2.a.q 2 1.a even 1 1 trivial
1088.2.a.q 2 4.b odd 2 1 inner
4896.2.a.z 2 24.f even 2 1
4896.2.a.z 2 24.h odd 2 1
9248.2.a.p 2 136.e odd 2 1
9248.2.a.p 2 136.h even 2 1
9792.2.a.cg 2 3.b odd 2 1
9792.2.a.cg 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1088))$$:

 $$T_{3}^{2} - 2$$ T3^2 - 2 $$T_{5} - 2$$ T5 - 2 $$T_{7}^{2} - 18$$ T7^2 - 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2$$
$5$ $$(T - 2)^{2}$$
$7$ $$T^{2} - 18$$
$11$ $$T^{2} - 2$$
$13$ $$(T - 4)^{2}$$
$17$ $$(T + 1)^{2}$$
$19$ $$T^{2} - 8$$
$23$ $$T^{2} - 18$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} - 50$$
$37$ $$(T - 2)^{2}$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} - 72$$
$47$ $$T^{2} - 128$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} - 72$$
$61$ $$(T + 6)^{2}$$
$67$ $$T^{2} - 32$$
$71$ $$T^{2} - 50$$
$73$ $$(T - 10)^{2}$$
$79$ $$T^{2} - 162$$
$83$ $$T^{2} - 200$$
$89$ $$(T + 8)^{2}$$
$97$ $$(T - 14)^{2}$$