# Properties

 Label 99.9.c.a Level $99$ Weight $9$ Character orbit 99.c Self dual yes Analytic conductor $40.330$ Analytic rank $0$ Dimension $1$ CM discriminant -11 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,9,Mod(10,99)]

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, 1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("99.10");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 99.c (of order $$2$$, degree $$1$$, minimal)

## 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: $$40.3304823961$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 256 q^{4} - 1151 q^{5}+O(q^{10})$$ q + 256 * q^4 - 1151 * q^5 $$q + 256 q^{4} - 1151 q^{5} - 14641 q^{11} + 65536 q^{16} - 294656 q^{20} + 531793 q^{23} + 934176 q^{25} - 1541233 q^{31} + 716447 q^{37} - 3748096 q^{44} + 6080638 q^{47} + 5764801 q^{49} + 15265438 q^{53} + 16851791 q^{55} + 4101553 q^{59} + 16777216 q^{64} + 19806767 q^{67} - 7043087 q^{71} - 75431936 q^{80} + 84100993 q^{89} + 136139008 q^{92} - 81155713 q^{97}+O(q^{100})$$ q + 256 * q^4 - 1151 * q^5 - 14641 * q^11 + 65536 * q^16 - 294656 * q^20 + 531793 * q^23 + 934176 * q^25 - 1541233 * q^31 + 716447 * q^37 - 3748096 * q^44 + 6080638 * q^47 + 5764801 * q^49 + 15265438 * q^53 + 16851791 * q^55 + 4101553 * q^59 + 16777216 * q^64 + 19806767 * q^67 - 7043087 * q^71 - 75431936 * q^80 + 84100993 * q^89 + 136139008 * q^92 - 81155713 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/99\mathbb{Z}\right)^\times$$.

 $$n$$ $$46$$ $$56$$ $$\chi(n)$$ $$1$$ $$0$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 0
0 0 256.000 −1151.00 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.9.c.a 1
3.b odd 2 1 11.9.b.a 1
11.b odd 2 1 CM 99.9.c.a 1
12.b even 2 1 176.9.h.a 1
33.d even 2 1 11.9.b.a 1
132.d odd 2 1 176.9.h.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.9.b.a 1 3.b odd 2 1
11.9.b.a 1 33.d even 2 1
99.9.c.a 1 1.a even 1 1 trivial
99.9.c.a 1 11.b odd 2 1 CM
176.9.h.a 1 12.b even 2 1
176.9.h.a 1 132.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{9}^{\mathrm{new}}(99, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 1151$$
$7$ $$T$$
$11$ $$T + 14641$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T - 531793$$
$29$ $$T$$
$31$ $$T + 1541233$$
$37$ $$T - 716447$$
$41$ $$T$$
$43$ $$T$$
$47$ $$T - 6080638$$
$53$ $$T - 15265438$$
$59$ $$T - 4101553$$
$61$ $$T$$
$67$ $$T - 19806767$$
$71$ $$T + 7043087$$
$73$ $$T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T - 84100993$$
$97$ $$T + 81155713$$