# Properties

 Label 990.2.a.f Level $990$ Weight $2$ Character orbit 990.a Self dual yes Analytic conductor $7.905$ Analytic rank $0$ 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] = [990,2,Mod(1,990)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(990, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("990.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$990 = 2 \cdot 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 990.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: $$7.90518980011$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 110) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + q^{5} - q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + q^5 - q^7 - q^8 $$q - q^{2} + q^{4} + q^{5} - q^{7} - q^{8} - q^{10} + q^{11} + 2 q^{13} + q^{14} + q^{16} + 3 q^{17} - q^{19} + q^{20} - q^{22} - 6 q^{23} + q^{25} - 2 q^{26} - q^{28} + 9 q^{29} + 5 q^{31} - q^{32} - 3 q^{34} - q^{35} + 5 q^{37} + q^{38} - q^{40} + 6 q^{41} + 8 q^{43} + q^{44} + 6 q^{46} - 6 q^{47} - 6 q^{49} - q^{50} + 2 q^{52} - 9 q^{53} + q^{55} + q^{56} - 9 q^{58} - 6 q^{59} + 5 q^{61} - 5 q^{62} + q^{64} + 2 q^{65} + 8 q^{67} + 3 q^{68} + q^{70} + 9 q^{71} - 10 q^{73} - 5 q^{74} - q^{76} - q^{77} + 14 q^{79} + q^{80} - 6 q^{82} + 6 q^{83} + 3 q^{85} - 8 q^{86} - q^{88} + 15 q^{89} - 2 q^{91} - 6 q^{92} + 6 q^{94} - q^{95} + 8 q^{97} + 6 q^{98}+O(q^{100})$$ q - q^2 + q^4 + q^5 - q^7 - q^8 - q^10 + q^11 + 2 * q^13 + q^14 + q^16 + 3 * q^17 - q^19 + q^20 - q^22 - 6 * q^23 + q^25 - 2 * q^26 - q^28 + 9 * q^29 + 5 * q^31 - q^32 - 3 * q^34 - q^35 + 5 * q^37 + q^38 - q^40 + 6 * q^41 + 8 * q^43 + q^44 + 6 * q^46 - 6 * q^47 - 6 * q^49 - q^50 + 2 * q^52 - 9 * q^53 + q^55 + q^56 - 9 * q^58 - 6 * q^59 + 5 * q^61 - 5 * q^62 + q^64 + 2 * q^65 + 8 * q^67 + 3 * q^68 + q^70 + 9 * q^71 - 10 * q^73 - 5 * q^74 - q^76 - q^77 + 14 * q^79 + q^80 - 6 * q^82 + 6 * q^83 + 3 * q^85 - 8 * q^86 - q^88 + 15 * q^89 - 2 * q^91 - 6 * q^92 + 6 * q^94 - q^95 + 8 * q^97 + 6 * 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 1.00000 0 −1.00000 −1.00000 0 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$11$$ $$-1$$

## 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 990.2.a.f 1
3.b odd 2 1 110.2.a.c 1
4.b odd 2 1 7920.2.a.bc 1
5.b even 2 1 4950.2.a.bm 1
5.c odd 4 2 4950.2.c.s 2
12.b even 2 1 880.2.a.d 1
15.d odd 2 1 550.2.a.d 1
15.e even 4 2 550.2.b.c 2
21.c even 2 1 5390.2.a.x 1
24.f even 2 1 3520.2.a.ba 1
24.h odd 2 1 3520.2.a.k 1
33.d even 2 1 1210.2.a.e 1
60.h even 2 1 4400.2.a.t 1
60.l odd 4 2 4400.2.b.j 2
132.d odd 2 1 9680.2.a.g 1
165.d even 2 1 6050.2.a.bc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.c 1 3.b odd 2 1
550.2.a.d 1 15.d odd 2 1
550.2.b.c 2 15.e even 4 2
880.2.a.d 1 12.b even 2 1
990.2.a.f 1 1.a even 1 1 trivial
1210.2.a.e 1 33.d even 2 1
3520.2.a.k 1 24.h odd 2 1
3520.2.a.ba 1 24.f even 2 1
4400.2.a.t 1 60.h even 2 1
4400.2.b.j 2 60.l odd 4 2
4950.2.a.bm 1 5.b even 2 1
4950.2.c.s 2 5.c odd 4 2
5390.2.a.x 1 21.c even 2 1
6050.2.a.bc 1 165.d even 2 1
7920.2.a.bc 1 4.b odd 2 1
9680.2.a.g 1 132.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(990))$$:

 $$T_{7} + 1$$ T7 + 1 $$T_{13} - 2$$ T13 - 2 $$T_{17} - 3$$ T17 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T + 1$$
$11$ $$T - 1$$
$13$ $$T - 2$$
$17$ $$T - 3$$
$19$ $$T + 1$$
$23$ $$T + 6$$
$29$ $$T - 9$$
$31$ $$T - 5$$
$37$ $$T - 5$$
$41$ $$T - 6$$
$43$ $$T - 8$$
$47$ $$T + 6$$
$53$ $$T + 9$$
$59$ $$T + 6$$
$61$ $$T - 5$$
$67$ $$T - 8$$
$71$ $$T - 9$$
$73$ $$T + 10$$
$79$ $$T - 14$$
$83$ $$T - 6$$
$89$ $$T - 15$$
$97$ $$T - 8$$