# Properties

 Label 99.2.a.b Level $99$ Weight $2$ Character orbit 99.a Self dual yes Analytic conductor $0.791$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 99.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.790518980011$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) 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 $$q - q^{2} - q^{4} + 2 q^{5} + 4 q^{7} + 3 q^{8} - 2 q^{10} - q^{11} - 2 q^{13} - 4 q^{14} - q^{16} + 2 q^{17} - 2 q^{20} + q^{22} - 8 q^{23} - q^{25} + 2 q^{26} - 4 q^{28} + 6 q^{29} - 8 q^{31} - 5 q^{32} - 2 q^{34} + 8 q^{35} + 6 q^{37} + 6 q^{40} + 2 q^{41} + q^{44} + 8 q^{46} - 8 q^{47} + 9 q^{49} + q^{50} + 2 q^{52} - 6 q^{53} - 2 q^{55} + 12 q^{56} - 6 q^{58} + 4 q^{59} + 6 q^{61} + 8 q^{62} + 7 q^{64} - 4 q^{65} - 4 q^{67} - 2 q^{68} - 8 q^{70} - 14 q^{73} - 6 q^{74} - 4 q^{77} - 4 q^{79} - 2 q^{80} - 2 q^{82} - 12 q^{83} + 4 q^{85} - 3 q^{88} + 6 q^{89} - 8 q^{91} + 8 q^{92} + 8 q^{94} + 2 q^{97} - 9 q^{98}+O(q^{100})$$ q - q^2 - q^4 + 2 * q^5 + 4 * q^7 + 3 * q^8 - 2 * q^10 - q^11 - 2 * q^13 - 4 * q^14 - q^16 + 2 * q^17 - 2 * q^20 + q^22 - 8 * q^23 - q^25 + 2 * q^26 - 4 * q^28 + 6 * q^29 - 8 * q^31 - 5 * q^32 - 2 * q^34 + 8 * q^35 + 6 * q^37 + 6 * q^40 + 2 * q^41 + q^44 + 8 * q^46 - 8 * q^47 + 9 * q^49 + q^50 + 2 * q^52 - 6 * q^53 - 2 * q^55 + 12 * q^56 - 6 * q^58 + 4 * q^59 + 6 * q^61 + 8 * q^62 + 7 * q^64 - 4 * q^65 - 4 * q^67 - 2 * q^68 - 8 * q^70 - 14 * q^73 - 6 * q^74 - 4 * q^77 - 4 * q^79 - 2 * q^80 - 2 * q^82 - 12 * q^83 + 4 * q^85 - 3 * q^88 + 6 * q^89 - 8 * q^91 + 8 * q^92 + 8 * q^94 + 2 * 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
$$3$$ $$-1$$
$$11$$ $$1$$

## 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 99.2.a.b 1
3.b odd 2 1 33.2.a.a 1
4.b odd 2 1 1584.2.a.o 1
5.b even 2 1 2475.2.a.g 1
5.c odd 4 2 2475.2.c.d 2
7.b odd 2 1 4851.2.a.b 1
8.b even 2 1 6336.2.a.x 1
8.d odd 2 1 6336.2.a.n 1
9.c even 3 2 891.2.e.g 2
9.d odd 6 2 891.2.e.e 2
11.b odd 2 1 1089.2.a.j 1
12.b even 2 1 528.2.a.g 1
15.d odd 2 1 825.2.a.a 1
15.e even 4 2 825.2.c.a 2
21.c even 2 1 1617.2.a.j 1
24.f even 2 1 2112.2.a.j 1
24.h odd 2 1 2112.2.a.bb 1
33.d even 2 1 363.2.a.b 1
33.f even 10 4 363.2.e.g 4
33.h odd 10 4 363.2.e.e 4
39.d odd 2 1 5577.2.a.a 1
51.c odd 2 1 9537.2.a.m 1
132.d odd 2 1 5808.2.a.t 1
165.d even 2 1 9075.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 3.b odd 2 1
99.2.a.b 1 1.a even 1 1 trivial
363.2.a.b 1 33.d even 2 1
363.2.e.e 4 33.h odd 10 4
363.2.e.g 4 33.f even 10 4
528.2.a.g 1 12.b even 2 1
825.2.a.a 1 15.d odd 2 1
825.2.c.a 2 15.e even 4 2
891.2.e.e 2 9.d odd 6 2
891.2.e.g 2 9.c even 3 2
1089.2.a.j 1 11.b odd 2 1
1584.2.a.o 1 4.b odd 2 1
1617.2.a.j 1 21.c even 2 1
2112.2.a.j 1 24.f even 2 1
2112.2.a.bb 1 24.h odd 2 1
2475.2.a.g 1 5.b even 2 1
2475.2.c.d 2 5.c odd 4 2
4851.2.a.b 1 7.b odd 2 1
5577.2.a.a 1 39.d odd 2 1
5808.2.a.t 1 132.d odd 2 1
6336.2.a.n 1 8.d odd 2 1
6336.2.a.x 1 8.b even 2 1
9075.2.a.q 1 165.d even 2 1
9537.2.a.m 1 51.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(99))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{5} - 2$$ T5 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T - 2$$
$7$ $$T - 4$$
$11$ $$T + 1$$
$13$ $$T + 2$$
$17$ $$T - 2$$
$19$ $$T$$
$23$ $$T + 8$$
$29$ $$T - 6$$
$31$ $$T + 8$$
$37$ $$T - 6$$
$41$ $$T - 2$$
$43$ $$T$$
$47$ $$T + 8$$
$53$ $$T + 6$$
$59$ $$T - 4$$
$61$ $$T - 6$$
$67$ $$T + 4$$
$71$ $$T$$
$73$ $$T + 14$$
$79$ $$T + 4$$
$83$ $$T + 12$$
$89$ $$T - 6$$
$97$ $$T - 2$$