# Properties

 Label 1088.2.a.o Level $1088$ Weight $2$ Character orbit 1088.a Self dual yes Analytic conductor $8.688$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 136) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{3} - 2 q^{5} + ( - \beta - 1) q^{7} + (2 \beta + 3) q^{9}+O(q^{10})$$ q + (-b - 1) * q^3 - 2 * q^5 + (-b - 1) * q^7 + (2*b + 3) * q^9 $$q + ( - \beta - 1) q^{3} - 2 q^{5} + ( - \beta - 1) q^{7} + (2 \beta + 3) q^{9} + (\beta + 1) q^{11} + 2 \beta q^{13} + (2 \beta + 2) q^{15} + q^{17} + (2 \beta - 2) q^{19} + (2 \beta + 6) q^{21} + ( - \beta - 1) q^{23} - q^{25} + ( - 2 \beta - 10) q^{27} - 2 q^{29} + (\beta + 1) q^{31} + ( - 2 \beta - 6) q^{33} + (2 \beta + 2) q^{35} + ( - 4 \beta + 2) q^{37} + ( - 2 \beta - 10) q^{39} + 2 q^{41} + ( - 2 \beta - 6) q^{43} + ( - 4 \beta - 6) q^{45} + (4 \beta - 4) q^{47} + (2 \beta - 1) q^{49} + ( - \beta - 1) q^{51} + 2 q^{53} + ( - 2 \beta - 2) q^{55} - 8 q^{57} + ( - 2 \beta + 10) q^{59} + (4 \beta + 2) q^{61} + ( - 5 \beta - 13) q^{63} - 4 \beta q^{65} - 12 q^{67} + (2 \beta + 6) q^{69} + (\beta - 7) q^{71} + ( - 4 \beta + 6) q^{73} + (\beta + 1) q^{75} + ( - 2 \beta - 6) q^{77} + (3 \beta - 5) q^{79} + (6 \beta + 11) q^{81} + (2 \beta + 6) q^{83} - 2 q^{85} + (2 \beta + 2) q^{87} + ( - 2 \beta - 12) q^{89} + ( - 2 \beta - 10) q^{91} + ( - 2 \beta - 6) q^{93} + ( - 4 \beta + 4) q^{95} + 2 q^{97} + (5 \beta + 13) q^{99} +O(q^{100})$$ q + (-b - 1) * q^3 - 2 * q^5 + (-b - 1) * q^7 + (2*b + 3) * q^9 + (b + 1) * q^11 + 2*b * q^13 + (2*b + 2) * q^15 + q^17 + (2*b - 2) * q^19 + (2*b + 6) * q^21 + (-b - 1) * q^23 - q^25 + (-2*b - 10) * q^27 - 2 * q^29 + (b + 1) * q^31 + (-2*b - 6) * q^33 + (2*b + 2) * q^35 + (-4*b + 2) * q^37 + (-2*b - 10) * q^39 + 2 * q^41 + (-2*b - 6) * q^43 + (-4*b - 6) * q^45 + (4*b - 4) * q^47 + (2*b - 1) * q^49 + (-b - 1) * q^51 + 2 * q^53 + (-2*b - 2) * q^55 - 8 * q^57 + (-2*b + 10) * q^59 + (4*b + 2) * q^61 + (-5*b - 13) * q^63 - 4*b * q^65 - 12 * q^67 + (2*b + 6) * q^69 + (b - 7) * q^71 + (-4*b + 6) * q^73 + (b + 1) * q^75 + (-2*b - 6) * q^77 + (3*b - 5) * q^79 + (6*b + 11) * q^81 + (2*b + 6) * q^83 - 2 * q^85 + (2*b + 2) * q^87 + (-2*b - 12) * q^89 + (-2*b - 10) * q^91 + (-2*b - 6) * q^93 + (-4*b + 4) * q^95 + 2 * q^97 + (5*b + 13) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 4 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 4 * q^5 - 2 * q^7 + 6 * q^9 $$2 q - 2 q^{3} - 4 q^{5} - 2 q^{7} + 6 q^{9} + 2 q^{11} + 4 q^{15} + 2 q^{17} - 4 q^{19} + 12 q^{21} - 2 q^{23} - 2 q^{25} - 20 q^{27} - 4 q^{29} + 2 q^{31} - 12 q^{33} + 4 q^{35} + 4 q^{37} - 20 q^{39} + 4 q^{41} - 12 q^{43} - 12 q^{45} - 8 q^{47} - 2 q^{49} - 2 q^{51} + 4 q^{53} - 4 q^{55} - 16 q^{57} + 20 q^{59} + 4 q^{61} - 26 q^{63} - 24 q^{67} + 12 q^{69} - 14 q^{71} + 12 q^{73} + 2 q^{75} - 12 q^{77} - 10 q^{79} + 22 q^{81} + 12 q^{83} - 4 q^{85} + 4 q^{87} - 24 q^{89} - 20 q^{91} - 12 q^{93} + 8 q^{95} + 4 q^{97} + 26 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 4 * q^5 - 2 * q^7 + 6 * q^9 + 2 * q^11 + 4 * q^15 + 2 * q^17 - 4 * q^19 + 12 * q^21 - 2 * q^23 - 2 * q^25 - 20 * q^27 - 4 * q^29 + 2 * q^31 - 12 * q^33 + 4 * q^35 + 4 * q^37 - 20 * q^39 + 4 * q^41 - 12 * q^43 - 12 * q^45 - 8 * q^47 - 2 * q^49 - 2 * q^51 + 4 * q^53 - 4 * q^55 - 16 * q^57 + 20 * q^59 + 4 * q^61 - 26 * q^63 - 24 * q^67 + 12 * q^69 - 14 * q^71 + 12 * q^73 + 2 * q^75 - 12 * q^77 - 10 * q^79 + 22 * q^81 + 12 * q^83 - 4 * q^85 + 4 * q^87 - 24 * q^89 - 20 * q^91 - 12 * q^93 + 8 * q^95 + 4 * q^97 + 26 * 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
 1.61803 −0.618034
0 −3.23607 0 −2.00000 0 −3.23607 0 7.47214 0
1.2 0 1.23607 0 −2.00000 0 1.23607 0 −1.47214 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.o 2
3.b odd 2 1 9792.2.a.da 2
4.b odd 2 1 1088.2.a.s 2
8.b even 2 1 272.2.a.f 2
8.d odd 2 1 136.2.a.c 2
12.b even 2 1 9792.2.a.db 2
24.f even 2 1 1224.2.a.i 2
24.h odd 2 1 2448.2.a.u 2
40.e odd 2 1 3400.2.a.i 2
40.f even 2 1 6800.2.a.bd 2
40.k even 4 2 3400.2.e.f 4
56.e even 2 1 6664.2.a.i 2
136.e odd 2 1 2312.2.a.m 2
136.h even 2 1 4624.2.a.h 2
136.j odd 4 2 2312.2.b.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.a.c 2 8.d odd 2 1
272.2.a.f 2 8.b even 2 1
1088.2.a.o 2 1.a even 1 1 trivial
1088.2.a.s 2 4.b odd 2 1
1224.2.a.i 2 24.f even 2 1
2312.2.a.m 2 136.e odd 2 1
2312.2.b.g 4 136.j odd 4 2
2448.2.a.u 2 24.h odd 2 1
3400.2.a.i 2 40.e odd 2 1
3400.2.e.f 4 40.k even 4 2
4624.2.a.h 2 136.h even 2 1
6664.2.a.i 2 56.e even 2 1
6800.2.a.bd 2 40.f even 2 1
9792.2.a.da 2 3.b odd 2 1
9792.2.a.db 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1088))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T - 4$$
$5$ $$(T + 2)^{2}$$
$7$ $$T^{2} + 2T - 4$$
$11$ $$T^{2} - 2T - 4$$
$13$ $$T^{2} - 20$$
$17$ $$(T - 1)^{2}$$
$19$ $$T^{2} + 4T - 16$$
$23$ $$T^{2} + 2T - 4$$
$29$ $$(T + 2)^{2}$$
$31$ $$T^{2} - 2T - 4$$
$37$ $$T^{2} - 4T - 76$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} + 12T + 16$$
$47$ $$T^{2} + 8T - 64$$
$53$ $$(T - 2)^{2}$$
$59$ $$T^{2} - 20T + 80$$
$61$ $$T^{2} - 4T - 76$$
$67$ $$(T + 12)^{2}$$
$71$ $$T^{2} + 14T + 44$$
$73$ $$T^{2} - 12T - 44$$
$79$ $$T^{2} + 10T - 20$$
$83$ $$T^{2} - 12T + 16$$
$89$ $$T^{2} + 24T + 124$$
$97$ $$(T - 2)^{2}$$