# Properties

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

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## 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 34) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{3} + 4 q^{7} + q^{9}+O(q^{10})$$ q - 2 * q^3 + 4 * q^7 + q^9 $$q - 2 q^{3} + 4 q^{7} + q^{9} + 6 q^{11} - 2 q^{13} - q^{17} - 4 q^{19} - 8 q^{21} - 5 q^{25} + 4 q^{27} + 4 q^{31} - 12 q^{33} + 4 q^{37} + 4 q^{39} + 6 q^{41} + 8 q^{43} + 9 q^{49} + 2 q^{51} + 6 q^{53} + 8 q^{57} + 4 q^{61} + 4 q^{63} + 8 q^{67} + 2 q^{73} + 10 q^{75} + 24 q^{77} - 8 q^{79} - 11 q^{81} - 6 q^{89} - 8 q^{91} - 8 q^{93} + 14 q^{97} + 6 q^{99}+O(q^{100})$$ q - 2 * q^3 + 4 * q^7 + q^9 + 6 * q^11 - 2 * q^13 - q^17 - 4 * q^19 - 8 * q^21 - 5 * q^25 + 4 * q^27 + 4 * q^31 - 12 * q^33 + 4 * q^37 + 4 * q^39 + 6 * q^41 + 8 * q^43 + 9 * q^49 + 2 * q^51 + 6 * q^53 + 8 * q^57 + 4 * q^61 + 4 * q^63 + 8 * q^67 + 2 * q^73 + 10 * q^75 + 24 * q^77 - 8 * q^79 - 11 * q^81 - 6 * q^89 - 8 * q^91 - 8 * q^93 + 14 * 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
 0
0 −2.00000 0 0 0 4.00000 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

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.d 1
3.b odd 2 1 9792.2.a.bj 1
4.b odd 2 1 1088.2.a.l 1
8.b even 2 1 272.2.a.d 1
8.d odd 2 1 34.2.a.a 1
12.b even 2 1 9792.2.a.y 1
24.f even 2 1 306.2.a.a 1
24.h odd 2 1 2448.2.a.k 1
40.e odd 2 1 850.2.a.e 1
40.f even 2 1 6800.2.a.b 1
40.k even 4 2 850.2.c.b 2
56.e even 2 1 1666.2.a.m 1
88.g even 2 1 4114.2.a.a 1
104.h odd 2 1 5746.2.a.b 1
120.m even 2 1 7650.2.a.ci 1
136.e odd 2 1 578.2.a.a 1
136.h even 2 1 4624.2.a.a 1
136.j odd 4 2 578.2.b.a 2
136.p odd 8 4 578.2.c.e 4
136.s even 16 8 578.2.d.e 8
408.h even 2 1 5202.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.2.a.a 1 8.d odd 2 1
272.2.a.d 1 8.b even 2 1
306.2.a.a 1 24.f even 2 1
578.2.a.a 1 136.e odd 2 1
578.2.b.a 2 136.j odd 4 2
578.2.c.e 4 136.p odd 8 4
578.2.d.e 8 136.s even 16 8
850.2.a.e 1 40.e odd 2 1
850.2.c.b 2 40.k even 4 2
1088.2.a.d 1 1.a even 1 1 trivial
1088.2.a.l 1 4.b odd 2 1
1666.2.a.m 1 56.e even 2 1
2448.2.a.k 1 24.h odd 2 1
4114.2.a.a 1 88.g even 2 1
4624.2.a.a 1 136.h even 2 1
5202.2.a.d 1 408.h even 2 1
5746.2.a.b 1 104.h odd 2 1
6800.2.a.b 1 40.f even 2 1
7650.2.a.ci 1 120.m even 2 1
9792.2.a.y 1 12.b even 2 1
9792.2.a.bj 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1088))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 2$$
$5$ $$T$$
$7$ $$T - 4$$
$11$ $$T - 6$$
$13$ $$T + 2$$
$17$ $$T + 1$$
$19$ $$T + 4$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T - 4$$
$37$ $$T - 4$$
$41$ $$T - 6$$
$43$ $$T - 8$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T$$
$61$ $$T - 4$$
$67$ $$T - 8$$
$71$ $$T$$
$73$ $$T - 2$$
$79$ $$T + 8$$
$83$ $$T$$
$89$ $$T + 6$$
$97$ $$T - 14$$
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