# Properties

 Label 1088.2.a.t Level $1088$ Weight $2$ Character orbit 1088.a Self dual yes Analytic conductor $8.688$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 68) 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.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{3} + 2 \beta q^{5} + (\beta + 1) q^{7} + (2 \beta + 1) q^{9}+O(q^{10})$$ q + (b + 1) * q^3 + 2*b * q^5 + (b + 1) * q^7 + (2*b + 1) * q^9 $$q + (\beta + 1) q^{3} + 2 \beta q^{5} + (\beta + 1) q^{7} + (2 \beta + 1) q^{9} + (\beta - 3) q^{11} + ( - 2 \beta - 2) q^{13} + (2 \beta + 6) q^{15} - q^{17} + ( - 2 \beta + 2) q^{19} + (2 \beta + 4) q^{21} + ( - \beta + 3) q^{23} + 7 q^{25} + 4 q^{27} - 2 \beta q^{29} + ( - 3 \beta + 1) q^{31} - 2 \beta q^{33} + (2 \beta + 6) q^{35} + (2 \beta - 8) q^{37} + ( - 4 \beta - 8) q^{39} - 6 q^{41} + ( - 6 \beta + 2) q^{43} + (2 \beta + 12) q^{45} + 4 \beta q^{47} + (2 \beta - 3) q^{49} + ( - \beta - 1) q^{51} + ( - 4 \beta - 6) q^{53} + ( - 6 \beta + 6) q^{55} - 4 q^{57} + ( - 2 \beta + 6) q^{59} + ( - 2 \beta + 4) q^{61} + (3 \beta + 7) q^{63} + ( - 4 \beta - 12) q^{65} + (4 \beta + 8) q^{67} + 2 \beta q^{69} + (3 \beta + 3) q^{71} + 2 q^{73} + (7 \beta + 7) q^{75} - 2 \beta q^{77} + (3 \beta + 7) q^{79} + ( - 2 \beta + 1) q^{81} + (2 \beta - 6) q^{83} - 2 \beta q^{85} + ( - 2 \beta - 6) q^{87} + ( - 2 \beta + 6) q^{89} + ( - 4 \beta - 8) q^{91} + ( - 2 \beta - 8) q^{93} + (4 \beta - 12) q^{95} + ( - 4 \beta + 2) q^{97} + ( - 5 \beta + 3) q^{99} +O(q^{100})$$ q + (b + 1) * q^3 + 2*b * q^5 + (b + 1) * q^7 + (2*b + 1) * q^9 + (b - 3) * q^11 + (-2*b - 2) * q^13 + (2*b + 6) * q^15 - q^17 + (-2*b + 2) * q^19 + (2*b + 4) * q^21 + (-b + 3) * q^23 + 7 * q^25 + 4 * q^27 - 2*b * q^29 + (-3*b + 1) * q^31 - 2*b * q^33 + (2*b + 6) * q^35 + (2*b - 8) * q^37 + (-4*b - 8) * q^39 - 6 * q^41 + (-6*b + 2) * q^43 + (2*b + 12) * q^45 + 4*b * q^47 + (2*b - 3) * q^49 + (-b - 1) * q^51 + (-4*b - 6) * q^53 + (-6*b + 6) * q^55 - 4 * q^57 + (-2*b + 6) * q^59 + (-2*b + 4) * q^61 + (3*b + 7) * q^63 + (-4*b - 12) * q^65 + (4*b + 8) * q^67 + 2*b * q^69 + (3*b + 3) * q^71 + 2 * q^73 + (7*b + 7) * q^75 - 2*b * q^77 + (3*b + 7) * q^79 + (-2*b + 1) * q^81 + (2*b - 6) * q^83 - 2*b * q^85 + (-2*b - 6) * q^87 + (-2*b + 6) * q^89 + (-4*b - 8) * q^91 + (-2*b - 8) * q^93 + (4*b - 12) * q^95 + (-4*b + 2) * q^97 + (-5*b + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{7} + 2 q^{9} - 6 q^{11} - 4 q^{13} + 12 q^{15} - 2 q^{17} + 4 q^{19} + 8 q^{21} + 6 q^{23} + 14 q^{25} + 8 q^{27} + 2 q^{31} + 12 q^{35} - 16 q^{37} - 16 q^{39} - 12 q^{41} + 4 q^{43} + 24 q^{45} - 6 q^{49} - 2 q^{51} - 12 q^{53} + 12 q^{55} - 8 q^{57} + 12 q^{59} + 8 q^{61} + 14 q^{63} - 24 q^{65} + 16 q^{67} + 6 q^{71} + 4 q^{73} + 14 q^{75} + 14 q^{79} + 2 q^{81} - 12 q^{83} - 12 q^{87} + 12 q^{89} - 16 q^{91} - 16 q^{93} - 24 q^{95} + 4 q^{97} + 6 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9 - 6 * q^11 - 4 * q^13 + 12 * q^15 - 2 * q^17 + 4 * q^19 + 8 * q^21 + 6 * q^23 + 14 * q^25 + 8 * q^27 + 2 * q^31 + 12 * q^35 - 16 * q^37 - 16 * q^39 - 12 * q^41 + 4 * q^43 + 24 * q^45 - 6 * q^49 - 2 * q^51 - 12 * q^53 + 12 * q^55 - 8 * q^57 + 12 * q^59 + 8 * q^61 + 14 * q^63 - 24 * q^65 + 16 * q^67 + 6 * q^71 + 4 * q^73 + 14 * q^75 + 14 * q^79 + 2 * q^81 - 12 * q^83 - 12 * q^87 + 12 * q^89 - 16 * q^91 - 16 * q^93 - 24 * q^95 + 4 * q^97 + 6 * q^99

## 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
0 −0.732051 0 −3.46410 0 −0.732051 0 −2.46410 0
1.2 0 2.73205 0 3.46410 0 2.73205 0 4.46410 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.t 2
3.b odd 2 1 9792.2.a.cs 2
4.b odd 2 1 1088.2.a.p 2
8.b even 2 1 272.2.a.e 2
8.d odd 2 1 68.2.a.a 2
12.b even 2 1 9792.2.a.cr 2
24.f even 2 1 612.2.a.e 2
24.h odd 2 1 2448.2.a.y 2
40.e odd 2 1 1700.2.a.d 2
40.f even 2 1 6800.2.a.bh 2
40.k even 4 2 1700.2.e.c 4
56.e even 2 1 3332.2.a.h 2
88.g even 2 1 8228.2.a.k 2
136.e odd 2 1 1156.2.a.a 2
136.h even 2 1 4624.2.a.x 2
136.j odd 4 2 1156.2.b.c 4
136.p odd 8 4 1156.2.e.d 8
136.s even 16 8 1156.2.h.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.2.a.a 2 8.d odd 2 1
272.2.a.e 2 8.b even 2 1
612.2.a.e 2 24.f even 2 1
1088.2.a.p 2 4.b odd 2 1
1088.2.a.t 2 1.a even 1 1 trivial
1156.2.a.a 2 136.e odd 2 1
1156.2.b.c 4 136.j odd 4 2
1156.2.e.d 8 136.p odd 8 4
1156.2.h.f 16 136.s even 16 8
1700.2.a.d 2 40.e odd 2 1
1700.2.e.c 4 40.k even 4 2
2448.2.a.y 2 24.h odd 2 1
3332.2.a.h 2 56.e even 2 1
4624.2.a.x 2 136.h even 2 1
6800.2.a.bh 2 40.f even 2 1
8228.2.a.k 2 88.g even 2 1
9792.2.a.cr 2 12.b even 2 1
9792.2.a.cs 2 3.b odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T - 2$$
$5$ $$T^{2} - 12$$
$7$ $$T^{2} - 2T - 2$$
$11$ $$T^{2} + 6T + 6$$
$13$ $$T^{2} + 4T - 8$$
$17$ $$(T + 1)^{2}$$
$19$ $$T^{2} - 4T - 8$$
$23$ $$T^{2} - 6T + 6$$
$29$ $$T^{2} - 12$$
$31$ $$T^{2} - 2T - 26$$
$37$ $$T^{2} + 16T + 52$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} - 4T - 104$$
$47$ $$T^{2} - 48$$
$53$ $$T^{2} + 12T - 12$$
$59$ $$T^{2} - 12T + 24$$
$61$ $$T^{2} - 8T + 4$$
$67$ $$T^{2} - 16T + 16$$
$71$ $$T^{2} - 6T - 18$$
$73$ $$(T - 2)^{2}$$
$79$ $$T^{2} - 14T + 22$$
$83$ $$T^{2} + 12T + 24$$
$89$ $$T^{2} - 12T + 24$$
$97$ $$T^{2} - 4T - 44$$