# Properties

 Label 121.2.a.c Level $121$ Weight $2$ Character orbit 121.a Self dual yes Analytic conductor $0.966$ 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] = [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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + 2 q^{3} - q^{4} + q^{5} + 2 q^{6} - 2 q^{7} - 3 q^{8} + q^{9}+O(q^{10})$$ q + q^2 + 2 * q^3 - q^4 + q^5 + 2 * q^6 - 2 * q^7 - 3 * q^8 + q^9 $$q + q^{2} + 2 q^{3} - q^{4} + q^{5} + 2 q^{6} - 2 q^{7} - 3 q^{8} + q^{9} + q^{10} - 2 q^{12} + q^{13} - 2 q^{14} + 2 q^{15} - q^{16} - 5 q^{17} + q^{18} + 6 q^{19} - q^{20} - 4 q^{21} + 2 q^{23} - 6 q^{24} - 4 q^{25} + q^{26} - 4 q^{27} + 2 q^{28} + 9 q^{29} + 2 q^{30} - 2 q^{31} + 5 q^{32} - 5 q^{34} - 2 q^{35} - q^{36} - 3 q^{37} + 6 q^{38} + 2 q^{39} - 3 q^{40} - 5 q^{41} - 4 q^{42} + q^{45} + 2 q^{46} + 2 q^{47} - 2 q^{48} - 3 q^{49} - 4 q^{50} - 10 q^{51} - q^{52} + 9 q^{53} - 4 q^{54} + 6 q^{56} + 12 q^{57} + 9 q^{58} + 8 q^{59} - 2 q^{60} + 6 q^{61} - 2 q^{62} - 2 q^{63} + 7 q^{64} + q^{65} + 2 q^{67} + 5 q^{68} + 4 q^{69} - 2 q^{70} + 12 q^{71} - 3 q^{72} - 2 q^{73} - 3 q^{74} - 8 q^{75} - 6 q^{76} + 2 q^{78} - 10 q^{79} - q^{80} - 11 q^{81} - 5 q^{82} + 6 q^{83} + 4 q^{84} - 5 q^{85} + 18 q^{87} - 9 q^{89} + q^{90} - 2 q^{91} - 2 q^{92} - 4 q^{93} + 2 q^{94} + 6 q^{95} + 10 q^{96} - 13 q^{97} - 3 q^{98}+O(q^{100})$$ q + q^2 + 2 * q^3 - q^4 + q^5 + 2 * q^6 - 2 * q^7 - 3 * q^8 + q^9 + q^10 - 2 * q^12 + q^13 - 2 * q^14 + 2 * q^15 - q^16 - 5 * q^17 + q^18 + 6 * q^19 - q^20 - 4 * q^21 + 2 * q^23 - 6 * q^24 - 4 * q^25 + q^26 - 4 * q^27 + 2 * q^28 + 9 * q^29 + 2 * q^30 - 2 * q^31 + 5 * q^32 - 5 * q^34 - 2 * q^35 - q^36 - 3 * q^37 + 6 * q^38 + 2 * q^39 - 3 * q^40 - 5 * q^41 - 4 * q^42 + q^45 + 2 * q^46 + 2 * q^47 - 2 * q^48 - 3 * q^49 - 4 * q^50 - 10 * q^51 - q^52 + 9 * q^53 - 4 * q^54 + 6 * q^56 + 12 * q^57 + 9 * q^58 + 8 * q^59 - 2 * q^60 + 6 * q^61 - 2 * q^62 - 2 * q^63 + 7 * q^64 + q^65 + 2 * q^67 + 5 * q^68 + 4 * q^69 - 2 * q^70 + 12 * q^71 - 3 * q^72 - 2 * q^73 - 3 * q^74 - 8 * q^75 - 6 * q^76 + 2 * q^78 - 10 * q^79 - q^80 - 11 * q^81 - 5 * q^82 + 6 * q^83 + 4 * q^84 - 5 * q^85 + 18 * q^87 - 9 * q^89 + q^90 - 2 * q^91 - 2 * q^92 - 4 * q^93 + 2 * q^94 + 6 * q^95 + 10 * q^96 - 13 * 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 2.00000 −1.00000 1.00000 2.00000 −2.00000 −3.00000 1.00000 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
$$11$$ $$-1$$

## 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 121.2.a.c yes 1
3.b odd 2 1 1089.2.a.c 1
4.b odd 2 1 1936.2.a.b 1
5.b even 2 1 3025.2.a.b 1
7.b odd 2 1 5929.2.a.g 1
8.b even 2 1 7744.2.a.c 1
8.d odd 2 1 7744.2.a.bf 1
11.b odd 2 1 121.2.a.a 1
11.c even 5 4 121.2.c.b 4
11.d odd 10 4 121.2.c.d 4
33.d even 2 1 1089.2.a.i 1
44.c even 2 1 1936.2.a.a 1
55.d odd 2 1 3025.2.a.e 1
77.b even 2 1 5929.2.a.a 1
88.b odd 2 1 7744.2.a.f 1
88.g even 2 1 7744.2.a.be 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(121))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 2$$
$5$ $$T - 1$$
$7$ $$T + 2$$
$11$ $$T$$
$13$ $$T - 1$$
$17$ $$T + 5$$
$19$ $$T - 6$$
$23$ $$T - 2$$
$29$ $$T - 9$$
$31$ $$T + 2$$
$37$ $$T + 3$$
$41$ $$T + 5$$
$43$ $$T$$
$47$ $$T - 2$$
$53$ $$T - 9$$
$59$ $$T - 8$$
$61$ $$T - 6$$
$67$ $$T - 2$$
$71$ $$T - 12$$
$73$ $$T + 2$$
$79$ $$T + 10$$
$83$ $$T - 6$$
$89$ $$T + 9$$
$97$ $$T + 13$$