# Properties

 Label 117.2.a.a Level $117$ Weight $2$ Character orbit 117.a Self dual yes Analytic conductor $0.934$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 117.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.934249703649$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} - 2 q^{5} - 4 q^{7} + 3 q^{8}+O(q^{10})$$ q - q^2 - q^4 - 2 * q^5 - 4 * q^7 + 3 * q^8 $$q - q^{2} - q^{4} - 2 q^{5} - 4 q^{7} + 3 q^{8} + 2 q^{10} - 4 q^{11} + q^{13} + 4 q^{14} - q^{16} - 2 q^{17} + 2 q^{20} + 4 q^{22} - q^{25} - q^{26} + 4 q^{28} + 10 q^{29} + 4 q^{31} - 5 q^{32} + 2 q^{34} + 8 q^{35} - 2 q^{37} - 6 q^{40} - 6 q^{41} - 12 q^{43} + 4 q^{44} + 9 q^{49} + q^{50} - q^{52} - 6 q^{53} + 8 q^{55} - 12 q^{56} - 10 q^{58} - 12 q^{59} - 2 q^{61} - 4 q^{62} + 7 q^{64} - 2 q^{65} - 8 q^{67} + 2 q^{68} - 8 q^{70} + 2 q^{73} + 2 q^{74} + 16 q^{77} + 8 q^{79} + 2 q^{80} + 6 q^{82} - 4 q^{83} + 4 q^{85} + 12 q^{86} - 12 q^{88} + 2 q^{89} - 4 q^{91} + 10 q^{97} - 9 q^{98}+O(q^{100})$$ q - q^2 - q^4 - 2 * q^5 - 4 * q^7 + 3 * q^8 + 2 * q^10 - 4 * q^11 + q^13 + 4 * q^14 - q^16 - 2 * q^17 + 2 * q^20 + 4 * q^22 - q^25 - q^26 + 4 * q^28 + 10 * q^29 + 4 * q^31 - 5 * q^32 + 2 * q^34 + 8 * q^35 - 2 * q^37 - 6 * q^40 - 6 * q^41 - 12 * q^43 + 4 * q^44 + 9 * q^49 + q^50 - q^52 - 6 * q^53 + 8 * q^55 - 12 * q^56 - 10 * q^58 - 12 * q^59 - 2 * q^61 - 4 * q^62 + 7 * q^64 - 2 * q^65 - 8 * q^67 + 2 * q^68 - 8 * q^70 + 2 * q^73 + 2 * q^74 + 16 * q^77 + 8 * q^79 + 2 * q^80 + 6 * q^82 - 4 * q^83 + 4 * q^85 + 12 * q^86 - 12 * q^88 + 2 * q^89 - 4 * q^91 + 10 * q^97 - 9 * q^98

## 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
−1.00000 0 −1.00000 −2.00000 0 −4.00000 3.00000 0 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$13$$ $$-1$$

## 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 117.2.a.a 1
3.b odd 2 1 39.2.a.a 1
4.b odd 2 1 1872.2.a.h 1
5.b even 2 1 2925.2.a.p 1
5.c odd 4 2 2925.2.c.e 2
7.b odd 2 1 5733.2.a.e 1
8.b even 2 1 7488.2.a.bl 1
8.d odd 2 1 7488.2.a.by 1
9.c even 3 2 1053.2.e.d 2
9.d odd 6 2 1053.2.e.b 2
12.b even 2 1 624.2.a.i 1
13.b even 2 1 1521.2.a.e 1
13.d odd 4 2 1521.2.b.b 2
15.d odd 2 1 975.2.a.f 1
15.e even 4 2 975.2.c.f 2
21.c even 2 1 1911.2.a.f 1
24.f even 2 1 2496.2.a.e 1
24.h odd 2 1 2496.2.a.q 1
33.d even 2 1 4719.2.a.c 1
39.d odd 2 1 507.2.a.a 1
39.f even 4 2 507.2.b.a 2
39.h odd 6 2 507.2.e.b 2
39.i odd 6 2 507.2.e.a 2
39.k even 12 4 507.2.j.e 4
156.h even 2 1 8112.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.a 1 3.b odd 2 1
117.2.a.a 1 1.a even 1 1 trivial
507.2.a.a 1 39.d odd 2 1
507.2.b.a 2 39.f even 4 2
507.2.e.a 2 39.i odd 6 2
507.2.e.b 2 39.h odd 6 2
507.2.j.e 4 39.k even 12 4
624.2.a.i 1 12.b even 2 1
975.2.a.f 1 15.d odd 2 1
975.2.c.f 2 15.e even 4 2
1053.2.e.b 2 9.d odd 6 2
1053.2.e.d 2 9.c even 3 2
1521.2.a.e 1 13.b even 2 1
1521.2.b.b 2 13.d odd 4 2
1872.2.a.h 1 4.b odd 2 1
1911.2.a.f 1 21.c even 2 1
2496.2.a.e 1 24.f even 2 1
2496.2.a.q 1 24.h odd 2 1
2925.2.a.p 1 5.b even 2 1
2925.2.c.e 2 5.c odd 4 2
4719.2.a.c 1 33.d even 2 1
5733.2.a.e 1 7.b odd 2 1
7488.2.a.bl 1 8.b even 2 1
7488.2.a.by 1 8.d odd 2 1
8112.2.a.s 1 156.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(117))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T + 2$$
$7$ $$T + 4$$
$11$ $$T + 4$$
$13$ $$T - 1$$
$17$ $$T + 2$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T - 10$$
$31$ $$T - 4$$
$37$ $$T + 2$$
$41$ $$T + 6$$
$43$ $$T + 12$$
$47$ $$T$$
$53$ $$T + 6$$
$59$ $$T + 12$$
$61$ $$T + 2$$
$67$ $$T + 8$$
$71$ $$T$$
$73$ $$T - 2$$
$79$ $$T - 8$$
$83$ $$T + 4$$
$89$ $$T - 2$$
$97$ $$T - 10$$