# Properties

 Label 121.2.a.b Level $121$ Weight $2$ Character orbit 121.a Self dual yes Analytic conductor $0.966$ Analytic rank $1$ Dimension $1$ CM discriminant -11 Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [121,2,Mod(1,121)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(121, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("121.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$121 = 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 121.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: $$0.966189864457$$ 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: $N(\mathrm{U}(1))$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{3} - 2 q^{4} - 3 q^{5} - 2 q^{9}+O(q^{10})$$ q - q^3 - 2 * q^4 - 3 * q^5 - 2 * q^9 $$q - q^{3} - 2 q^{4} - 3 q^{5} - 2 q^{9} + 2 q^{12} + 3 q^{15} + 4 q^{16} + 6 q^{20} - 9 q^{23} + 4 q^{25} + 5 q^{27} - 5 q^{31} + 4 q^{36} + 7 q^{37} + 6 q^{45} - 12 q^{47} - 4 q^{48} - 7 q^{49} + 6 q^{53} - 15 q^{59} - 6 q^{60} - 8 q^{64} + 13 q^{67} + 9 q^{69} - 3 q^{71} - 4 q^{75} - 12 q^{80} + q^{81} - 9 q^{89} + 18 q^{92} + 5 q^{93} + 17 q^{97}+O(q^{100})$$ q - q^3 - 2 * q^4 - 3 * q^5 - 2 * q^9 + 2 * q^12 + 3 * q^15 + 4 * q^16 + 6 * q^20 - 9 * q^23 + 4 * q^25 + 5 * q^27 - 5 * q^31 + 4 * q^36 + 7 * q^37 + 6 * q^45 - 12 * q^47 - 4 * q^48 - 7 * q^49 + 6 * q^53 - 15 * q^59 - 6 * q^60 - 8 * q^64 + 13 * q^67 + 9 * q^69 - 3 * q^71 - 4 * q^75 - 12 * q^80 + q^81 - 9 * q^89 + 18 * q^92 + 5 * q^93 + 17 * q^97

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

## Atkin-Lehner signs

$$p$$ Sign
$$11$$ $$+1$$

## 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 121.2.a.b 1
3.b odd 2 1 1089.2.a.g 1
4.b odd 2 1 1936.2.a.h 1
5.b even 2 1 3025.2.a.d 1
7.b odd 2 1 5929.2.a.e 1
8.b even 2 1 7744.2.a.bb 1
8.d odd 2 1 7744.2.a.n 1
11.b odd 2 1 CM 121.2.a.b 1
11.c even 5 4 121.2.c.c 4
11.d odd 10 4 121.2.c.c 4
33.d even 2 1 1089.2.a.g 1
44.c even 2 1 1936.2.a.h 1
55.d odd 2 1 3025.2.a.d 1
77.b even 2 1 5929.2.a.e 1
88.b odd 2 1 7744.2.a.bb 1
88.g even 2 1 7744.2.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.2.a.b 1 1.a even 1 1 trivial
121.2.a.b 1 11.b odd 2 1 CM
121.2.c.c 4 11.c even 5 4
121.2.c.c 4 11.d odd 10 4
1089.2.a.g 1 3.b odd 2 1
1089.2.a.g 1 33.d even 2 1
1936.2.a.h 1 4.b odd 2 1
1936.2.a.h 1 44.c even 2 1
3025.2.a.d 1 5.b even 2 1
3025.2.a.d 1 55.d odd 2 1
5929.2.a.e 1 7.b odd 2 1
5929.2.a.e 1 77.b even 2 1
7744.2.a.n 1 8.d odd 2 1
7744.2.a.n 1 88.g even 2 1
7744.2.a.bb 1 8.b even 2 1
7744.2.a.bb 1 88.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(121))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T + 3$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T + 9$$
$29$ $$T$$
$31$ $$T + 5$$
$37$ $$T - 7$$
$41$ $$T$$
$43$ $$T$$
$47$ $$T + 12$$
$53$ $$T - 6$$
$59$ $$T + 15$$
$61$ $$T$$
$67$ $$T - 13$$
$71$ $$T + 3$$
$73$ $$T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T + 9$$
$97$ $$T - 17$$
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