# Properties

 Label 121.4.a.a Level $121$ Weight $4$ Character orbit 121.a Self dual yes Analytic conductor $7.139$ 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] = [121,4,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, 4, names="a")

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

chi := DirichletCharacter("121.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$121 = 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$7.13923111069$$ Analytic rank: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + 8 q^{3} - 8 q^{4} + 18 q^{5} + 37 q^{9}+O(q^{10})$$ q + 8 * q^3 - 8 * q^4 + 18 * q^5 + 37 * q^9 $$q + 8 q^{3} - 8 q^{4} + 18 q^{5} + 37 q^{9} - 64 q^{12} + 144 q^{15} + 64 q^{16} - 144 q^{20} - 108 q^{23} + 199 q^{25} + 80 q^{27} + 340 q^{31} - 296 q^{36} - 434 q^{37} + 666 q^{45} - 36 q^{47} + 512 q^{48} - 343 q^{49} - 738 q^{53} - 720 q^{59} - 1152 q^{60} - 512 q^{64} - 416 q^{67} - 864 q^{69} + 612 q^{71} + 1592 q^{75} + 1152 q^{80} - 359 q^{81} + 1674 q^{89} + 864 q^{92} + 2720 q^{93} - 34 q^{97}+O(q^{100})$$ q + 8 * q^3 - 8 * q^4 + 18 * q^5 + 37 * q^9 - 64 * q^12 + 144 * q^15 + 64 * q^16 - 144 * q^20 - 108 * q^23 + 199 * q^25 + 80 * q^27 + 340 * q^31 - 296 * q^36 - 434 * q^37 + 666 * q^45 - 36 * q^47 + 512 * q^48 - 343 * q^49 - 738 * q^53 - 720 * q^59 - 1152 * q^60 - 512 * q^64 - 416 * q^67 - 864 * q^69 + 612 * q^71 + 1592 * q^75 + 1152 * q^80 - 359 * q^81 + 1674 * q^89 + 864 * q^92 + 2720 * q^93 - 34 * 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 8.00000 −8.00000 18.0000 0 0 0 37.0000 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 121.4.a.a 1
3.b odd 2 1 1089.4.a.f 1
4.b odd 2 1 1936.4.a.a 1
11.b odd 2 1 CM 121.4.a.a 1
11.c even 5 4 121.4.c.a 4
11.d odd 10 4 121.4.c.a 4
33.d even 2 1 1089.4.a.f 1
44.c even 2 1 1936.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.4.a.a 1 1.a even 1 1 trivial
121.4.a.a 1 11.b odd 2 1 CM
121.4.c.a 4 11.c even 5 4
121.4.c.a 4 11.d odd 10 4
1089.4.a.f 1 3.b odd 2 1
1089.4.a.f 1 33.d even 2 1
1936.4.a.a 1 4.b odd 2 1
1936.4.a.a 1 44.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(121))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 8$$
$5$ $$T - 18$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T + 108$$
$29$ $$T$$
$31$ $$T - 340$$
$37$ $$T + 434$$
$41$ $$T$$
$43$ $$T$$
$47$ $$T + 36$$
$53$ $$T + 738$$
$59$ $$T + 720$$
$61$ $$T$$
$67$ $$T + 416$$
$71$ $$T - 612$$
$73$ $$T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T - 1674$$
$97$ $$T + 34$$