# Properties

 Label 1088.2.a.v Level $1088$ Weight $2$ Character orbit 1088.a Self dual yes Analytic conductor $8.688$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# 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: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{3} + (\beta_{2} + \beta_1) q^{5} + (\beta_{2} - 1) q^{7} + (\beta_{2} - \beta_1 + 1) q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^3 + (b2 + b1) * q^5 + (b2 - 1) * q^7 + (b2 - b1 + 1) * q^9 $$q + ( - \beta_1 + 1) q^{3} + (\beta_{2} + \beta_1) q^{5} + (\beta_{2} - 1) q^{7} + (\beta_{2} - \beta_1 + 1) q^{9} + (\beta_1 + 3) q^{11} + ( - \beta_{2} + \beta_1 + 2) q^{13} + ( - 2 \beta_1 - 2) q^{15} + q^{17} + ( - 2 \beta_1 + 2) q^{19} + (\beta_{2} - \beta_1) q^{21} + ( - 3 \beta_{2} - 1) q^{23} + (4 \beta_1 + 3) q^{25} + (2 \beta_{2} + 2) q^{27} + ( - \beta_{2} - \beta_1 + 4) q^{29} + (\beta_{2} - 2 \beta_1 - 3) q^{31} + ( - \beta_{2} - 3 \beta_1) q^{33} + ( - 2 \beta_{2} + 6) q^{35} + ( - \beta_{2} - \beta_1 + 4) q^{37} + ( - 2 \beta_{2} - 2) q^{39} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{41} + ( - 2 \beta_{2} - 2) q^{43} + ( - \beta_{2} - \beta_1 + 4) q^{45} + (2 \beta_{2} + 2 \beta_1) q^{47} + ( - 3 \beta_{2} - \beta_1 + 1) q^{49} + ( - \beta_1 + 1) q^{51} + (2 \beta_{2} - 2 \beta_1 + 2) q^{53} + (4 \beta_{2} + 6 \beta_1 + 2) q^{55} + (2 \beta_{2} - 2 \beta_1 + 8) q^{57} + (2 \beta_{2} + 4 \beta_1 - 2) q^{59} + ( - \beta_{2} - 5 \beta_1 + 8) q^{61} + ( - \beta_{2} - 2 \beta_1 + 7) q^{63} + (4 \beta_{2} + 4 \beta_1 - 4) q^{65} + (4 \beta_1 - 4) q^{67} + ( - 3 \beta_{2} + 7 \beta_1 - 4) q^{69} + (\beta_{2} + 2 \beta_1 - 11) q^{71} + ( - 4 \beta_1 + 6) q^{73} + ( - 4 \beta_{2} - 3 \beta_1 - 9) q^{75} + (3 \beta_{2} + \beta_1 - 4) q^{77} + ( - 3 \beta_{2} + 4 \beta_1 + 3) q^{79} + ( - \beta_{2} - 3 \beta_1 + 1) q^{81} + (2 \beta_{2} + 2) q^{83} + (\beta_{2} + \beta_1) q^{85} + ( - 2 \beta_1 + 6) q^{87} + (3 \beta_{2} + \beta_1 + 2) q^{89} + (4 \beta_{2} + 2 \beta_1 - 10) q^{91} + (3 \beta_{2} + \beta_1 + 4) q^{93} + ( - 4 \beta_1 - 4) q^{95} + ( - 4 \beta_1 - 2) q^{97} + (2 \beta_{2} - \beta_1 - 1) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^3 + (b2 + b1) * q^5 + (b2 - 1) * q^7 + (b2 - b1 + 1) * q^9 + (b1 + 3) * q^11 + (-b2 + b1 + 2) * q^13 + (-2*b1 - 2) * q^15 + q^17 + (-2*b1 + 2) * q^19 + (b2 - b1) * q^21 + (-3*b2 - 1) * q^23 + (4*b1 + 3) * q^25 + (2*b2 + 2) * q^27 + (-b2 - b1 + 4) * q^29 + (b2 - 2*b1 - 3) * q^31 + (-b2 - 3*b1) * q^33 + (-2*b2 + 6) * q^35 + (-b2 - b1 + 4) * q^37 + (-2*b2 - 2) * q^39 + (-2*b2 + 2*b1 + 2) * q^41 + (-2*b2 - 2) * q^43 + (-b2 - b1 + 4) * q^45 + (2*b2 + 2*b1) * q^47 + (-3*b2 - b1 + 1) * q^49 + (-b1 + 1) * q^51 + (2*b2 - 2*b1 + 2) * q^53 + (4*b2 + 6*b1 + 2) * q^55 + (2*b2 - 2*b1 + 8) * q^57 + (2*b2 + 4*b1 - 2) * q^59 + (-b2 - 5*b1 + 8) * q^61 + (-b2 - 2*b1 + 7) * q^63 + (4*b2 + 4*b1 - 4) * q^65 + (4*b1 - 4) * q^67 + (-3*b2 + 7*b1 - 4) * q^69 + (b2 + 2*b1 - 11) * q^71 + (-4*b1 + 6) * q^73 + (-4*b2 - 3*b1 - 9) * q^75 + (3*b2 + b1 - 4) * q^77 + (-3*b2 + 4*b1 + 3) * q^79 + (-b2 - 3*b1 + 1) * q^81 + (2*b2 + 2) * q^83 + (b2 + b1) * q^85 + (-2*b1 + 6) * q^87 + (3*b2 + b1 + 2) * q^89 + (4*b2 + 2*b1 - 10) * q^91 + (3*b2 + b1 + 4) * q^93 + (-4*b1 - 4) * q^95 + (-4*b1 - 2) * q^97 + (2*b2 - b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 3 * q^9 $$3 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9} + 10 q^{11} + 6 q^{13} - 8 q^{15} + 3 q^{17} + 4 q^{19} - 6 q^{23} + 13 q^{25} + 8 q^{27} + 10 q^{29} - 10 q^{31} - 4 q^{33} + 16 q^{35} + 10 q^{37} - 8 q^{39} + 6 q^{41} - 8 q^{43} + 10 q^{45} + 4 q^{47} - q^{49} + 2 q^{51} + 6 q^{53} + 16 q^{55} + 24 q^{57} + 18 q^{61} + 18 q^{63} - 4 q^{65} - 8 q^{67} - 8 q^{69} - 30 q^{71} + 14 q^{73} - 34 q^{75} - 8 q^{77} + 10 q^{79} - q^{81} + 8 q^{83} + 2 q^{85} + 16 q^{87} + 10 q^{89} - 24 q^{91} + 16 q^{93} - 16 q^{95} - 10 q^{97} - 2 q^{99}+O(q^{100})$$ 3 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 3 * q^9 + 10 * q^11 + 6 * q^13 - 8 * q^15 + 3 * q^17 + 4 * q^19 - 6 * q^23 + 13 * q^25 + 8 * q^27 + 10 * q^29 - 10 * q^31 - 4 * q^33 + 16 * q^35 + 10 * q^37 - 8 * q^39 + 6 * q^41 - 8 * q^43 + 10 * q^45 + 4 * q^47 - q^49 + 2 * q^51 + 6 * q^53 + 16 * q^55 + 24 * q^57 + 18 * q^61 + 18 * q^63 - 4 * q^65 - 8 * q^67 - 8 * q^69 - 30 * q^71 + 14 * q^73 - 34 * q^75 - 8 * q^77 + 10 * q^79 - q^81 + 8 * q^83 + 2 * q^85 + 16 * q^87 + 10 * q^89 - 24 * q^91 + 16 * q^93 - 16 * q^95 - 10 * q^97 - 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2\nu + 2$$ -v^2 + 2*v + 2
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 + 2$$ b1 + 2

## 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
 2.17009 −1.48119 0.311108
0 −1.70928 0 4.34017 0 0.630898 0 −0.0783777 0
1.2 0 0.806063 0 −2.96239 0 −4.15633 0 −2.35026 0
1.3 0 2.90321 0 0.622216 0 1.52543 0 5.42864 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1088.2.a.v 3
3.b odd 2 1 9792.2.a.dc 3
4.b odd 2 1 1088.2.a.u 3
8.b even 2 1 544.2.a.i 3
8.d odd 2 1 544.2.a.j yes 3
12.b even 2 1 9792.2.a.dd 3
24.f even 2 1 4896.2.a.bf 3
24.h odd 2 1 4896.2.a.be 3
136.e odd 2 1 9248.2.a.t 3
136.h even 2 1 9248.2.a.u 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
544.2.a.i 3 8.b even 2 1
544.2.a.j yes 3 8.d odd 2 1
1088.2.a.u 3 4.b odd 2 1
1088.2.a.v 3 1.a even 1 1 trivial
4896.2.a.be 3 24.h odd 2 1
4896.2.a.bf 3 24.f even 2 1
9248.2.a.t 3 136.e odd 2 1
9248.2.a.u 3 136.h even 2 1
9792.2.a.dc 3 3.b odd 2 1
9792.2.a.dd 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1088))$$:

 $$T_{3}^{3} - 2T_{3}^{2} - 4T_{3} + 4$$ T3^3 - 2*T3^2 - 4*T3 + 4 $$T_{5}^{3} - 2T_{5}^{2} - 12T_{5} + 8$$ T5^3 - 2*T5^2 - 12*T5 + 8 $$T_{7}^{3} + 2T_{7}^{2} - 8T_{7} + 4$$ T7^3 + 2*T7^2 - 8*T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 2 T^{2} - 4 T + 4$$
$5$ $$T^{3} - 2 T^{2} - 12 T + 8$$
$7$ $$T^{3} + 2 T^{2} - 8 T + 4$$
$11$ $$T^{3} - 10 T^{2} + 28 T - 20$$
$13$ $$T^{3} - 6 T^{2} - 4 T + 40$$
$17$ $$(T - 1)^{3}$$
$19$ $$T^{3} - 4 T^{2} - 16 T + 32$$
$23$ $$T^{3} + 6 T^{2} - 72 T - 428$$
$29$ $$T^{3} - 10 T^{2} + 20 T + 8$$
$31$ $$T^{3} + 10T^{2} - 148$$
$37$ $$T^{3} - 10 T^{2} + 20 T + 8$$
$41$ $$T^{3} - 6 T^{2} - 52 T + 248$$
$43$ $$T^{3} + 8 T^{2} - 16 T - 160$$
$47$ $$T^{3} - 4 T^{2} - 48 T + 64$$
$53$ $$T^{3} - 6 T^{2} - 52 T - 8$$
$59$ $$T^{3} - 112T - 416$$
$61$ $$T^{3} - 18 T^{2} - 28 T + 1096$$
$67$ $$T^{3} + 8 T^{2} - 64 T - 256$$
$71$ $$T^{3} + 30 T^{2} + 272 T + 668$$
$73$ $$T^{3} - 14 T^{2} - 20 T + 344$$
$79$ $$T^{3} - 10 T^{2} - 152 T + 1444$$
$83$ $$T^{3} - 8 T^{2} - 16 T + 160$$
$89$ $$T^{3} - 10 T^{2} - 52 T + 536$$
$97$ $$T^{3} + 10 T^{2} - 52 T - 200$$