# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,2,Mod(1,99)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("99.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$0.790518980011$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} - 4 q^{5} - 2 q^{7} + 3 q^{8}+O(q^{10})$$ q - q^2 - q^4 - 4 * q^5 - 2 * q^7 + 3 * q^8 $$q - q^{2} - q^{4} - 4 q^{5} - 2 q^{7} + 3 q^{8} + 4 q^{10} - q^{11} - 2 q^{13} + 2 q^{14} - q^{16} + 2 q^{17} - 6 q^{19} + 4 q^{20} + q^{22} + 4 q^{23} + 11 q^{25} + 2 q^{26} + 2 q^{28} - 6 q^{29} + 4 q^{31} - 5 q^{32} - 2 q^{34} + 8 q^{35} - 6 q^{37} + 6 q^{38} - 12 q^{40} - 10 q^{41} + 6 q^{43} + q^{44} - 4 q^{46} - 8 q^{47} - 3 q^{49} - 11 q^{50} + 2 q^{52} + 4 q^{55} - 6 q^{56} + 6 q^{58} + 4 q^{59} - 6 q^{61} - 4 q^{62} + 7 q^{64} + 8 q^{65} + 8 q^{67} - 2 q^{68} - 8 q^{70} - 2 q^{73} + 6 q^{74} + 6 q^{76} + 2 q^{77} - 10 q^{79} + 4 q^{80} + 10 q^{82} + 12 q^{83} - 8 q^{85} - 6 q^{86} - 3 q^{88} + 4 q^{91} - 4 q^{92} + 8 q^{94} + 24 q^{95} + 2 q^{97} + 3 q^{98}+O(q^{100})$$ q - q^2 - q^4 - 4 * q^5 - 2 * q^7 + 3 * q^8 + 4 * q^10 - q^11 - 2 * q^13 + 2 * q^14 - q^16 + 2 * q^17 - 6 * q^19 + 4 * q^20 + q^22 + 4 * q^23 + 11 * q^25 + 2 * q^26 + 2 * q^28 - 6 * q^29 + 4 * q^31 - 5 * q^32 - 2 * q^34 + 8 * q^35 - 6 * q^37 + 6 * q^38 - 12 * q^40 - 10 * q^41 + 6 * q^43 + q^44 - 4 * q^46 - 8 * q^47 - 3 * q^49 - 11 * q^50 + 2 * q^52 + 4 * q^55 - 6 * q^56 + 6 * q^58 + 4 * q^59 - 6 * q^61 - 4 * q^62 + 7 * q^64 + 8 * q^65 + 8 * q^67 - 2 * q^68 - 8 * q^70 - 2 * q^73 + 6 * q^74 + 6 * q^76 + 2 * q^77 - 10 * q^79 + 4 * q^80 + 10 * q^82 + 12 * q^83 - 8 * q^85 - 6 * q^86 - 3 * q^88 + 4 * q^91 - 4 * q^92 + 8 * q^94 + 24 * q^95 + 2 * q^97 + 3 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 −4.00000 0 −2.00000 3.00000 0 4.00000
 $$n$$: 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.a 1
3.b odd 2 1 99.2.a.c yes 1
4.b odd 2 1 1584.2.a.b 1
5.b even 2 1 2475.2.a.j 1
5.c odd 4 2 2475.2.c.b 2
7.b odd 2 1 4851.2.a.g 1
8.b even 2 1 6336.2.a.cl 1
8.d odd 2 1 6336.2.a.cm 1
9.c even 3 2 891.2.e.j 2
9.d odd 6 2 891.2.e.c 2
11.b odd 2 1 1089.2.a.h 1
12.b even 2 1 1584.2.a.r 1
15.d odd 2 1 2475.2.a.c 1
15.e even 4 2 2475.2.c.g 2
21.c even 2 1 4851.2.a.o 1
24.f even 2 1 6336.2.a.f 1
24.h odd 2 1 6336.2.a.b 1
33.d even 2 1 1089.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.a.a 1 1.a even 1 1 trivial
99.2.a.c yes 1 3.b odd 2 1
891.2.e.c 2 9.d odd 6 2
891.2.e.j 2 9.c even 3 2
1089.2.a.d 1 33.d even 2 1
1089.2.a.h 1 11.b odd 2 1
1584.2.a.b 1 4.b odd 2 1
1584.2.a.r 1 12.b even 2 1
2475.2.a.c 1 15.d odd 2 1
2475.2.a.j 1 5.b even 2 1
2475.2.c.b 2 5.c odd 4 2
2475.2.c.g 2 15.e even 4 2
4851.2.a.g 1 7.b odd 2 1
4851.2.a.o 1 21.c even 2 1
6336.2.a.b 1 24.h odd 2 1
6336.2.a.f 1 24.f even 2 1
6336.2.a.cl 1 8.b even 2 1
6336.2.a.cm 1 8.d 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 $$T_{5} + 4$$ T5 + 4

## Hecke characteristic polynomials

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