# Properties

 Label 121.2.a.d 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: no (minimal twist has level 11) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{9}+O(q^{10})$$ q + 2 * q^2 - q^3 + 2 * q^4 + q^5 - 2 * q^6 + 2 * q^7 - 2 * q^9 $$q + 2 q^{2} - q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{9} + 2 q^{10} - 2 q^{12} - 4 q^{13} + 4 q^{14} - q^{15} - 4 q^{16} + 2 q^{17} - 4 q^{18} + 2 q^{20} - 2 q^{21} - q^{23} - 4 q^{25} - 8 q^{26} + 5 q^{27} + 4 q^{28} - 2 q^{30} + 7 q^{31} - 8 q^{32} + 4 q^{34} + 2 q^{35} - 4 q^{36} + 3 q^{37} + 4 q^{39} + 8 q^{41} - 4 q^{42} + 6 q^{43} - 2 q^{45} - 2 q^{46} + 8 q^{47} + 4 q^{48} - 3 q^{49} - 8 q^{50} - 2 q^{51} - 8 q^{52} - 6 q^{53} + 10 q^{54} + 5 q^{59} - 2 q^{60} - 12 q^{61} + 14 q^{62} - 4 q^{63} - 8 q^{64} - 4 q^{65} - 7 q^{67} + 4 q^{68} + q^{69} + 4 q^{70} - 3 q^{71} - 4 q^{73} + 6 q^{74} + 4 q^{75} + 8 q^{78} + 10 q^{79} - 4 q^{80} + q^{81} + 16 q^{82} + 6 q^{83} - 4 q^{84} + 2 q^{85} + 12 q^{86} + 15 q^{89} - 4 q^{90} - 8 q^{91} - 2 q^{92} - 7 q^{93} + 16 q^{94} + 8 q^{96} - 7 q^{97} - 6 q^{98}+O(q^{100})$$ q + 2 * q^2 - q^3 + 2 * q^4 + q^5 - 2 * q^6 + 2 * q^7 - 2 * q^9 + 2 * q^10 - 2 * q^12 - 4 * q^13 + 4 * q^14 - q^15 - 4 * q^16 + 2 * q^17 - 4 * q^18 + 2 * q^20 - 2 * q^21 - q^23 - 4 * q^25 - 8 * q^26 + 5 * q^27 + 4 * q^28 - 2 * q^30 + 7 * q^31 - 8 * q^32 + 4 * q^34 + 2 * q^35 - 4 * q^36 + 3 * q^37 + 4 * q^39 + 8 * q^41 - 4 * q^42 + 6 * q^43 - 2 * q^45 - 2 * q^46 + 8 * q^47 + 4 * q^48 - 3 * q^49 - 8 * q^50 - 2 * q^51 - 8 * q^52 - 6 * q^53 + 10 * q^54 + 5 * q^59 - 2 * q^60 - 12 * q^61 + 14 * q^62 - 4 * q^63 - 8 * q^64 - 4 * q^65 - 7 * q^67 + 4 * q^68 + q^69 + 4 * q^70 - 3 * q^71 - 4 * q^73 + 6 * q^74 + 4 * q^75 + 8 * q^78 + 10 * q^79 - 4 * q^80 + q^81 + 16 * q^82 + 6 * q^83 - 4 * q^84 + 2 * q^85 + 12 * q^86 + 15 * q^89 - 4 * q^90 - 8 * q^91 - 2 * q^92 - 7 * q^93 + 16 * q^94 + 8 * q^96 - 7 * 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
2.00000 −1.00000 2.00000 1.00000 −2.00000 2.00000 0 −2.00000 2.00000
 $$n$$: 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.d 1
3.b odd 2 1 1089.2.a.b 1
4.b odd 2 1 1936.2.a.i 1
5.b even 2 1 3025.2.a.a 1
7.b odd 2 1 5929.2.a.h 1
8.b even 2 1 7744.2.a.x 1
8.d odd 2 1 7744.2.a.k 1
11.b odd 2 1 11.2.a.a 1
11.c even 5 4 121.2.c.a 4
11.d odd 10 4 121.2.c.e 4
33.d even 2 1 99.2.a.d 1
44.c even 2 1 176.2.a.b 1
55.d odd 2 1 275.2.a.b 1
55.e even 4 2 275.2.b.a 2
77.b even 2 1 539.2.a.a 1
77.h odd 6 2 539.2.e.h 2
77.i even 6 2 539.2.e.g 2
88.b odd 2 1 704.2.a.h 1
88.g even 2 1 704.2.a.c 1
99.g even 6 2 891.2.e.b 2
99.h odd 6 2 891.2.e.k 2
132.d odd 2 1 1584.2.a.g 1
143.d odd 2 1 1859.2.a.b 1
165.d even 2 1 2475.2.a.a 1
165.l odd 4 2 2475.2.c.a 2
176.i even 4 2 2816.2.c.f 2
176.l odd 4 2 2816.2.c.j 2
187.b odd 2 1 3179.2.a.a 1
209.d even 2 1 3971.2.a.b 1
220.g even 2 1 4400.2.a.i 1
220.i odd 4 2 4400.2.b.h 2
231.h odd 2 1 4851.2.a.t 1
253.b even 2 1 5819.2.a.a 1
264.m even 2 1 6336.2.a.br 1
264.p odd 2 1 6336.2.a.bu 1
308.g odd 2 1 8624.2.a.j 1
319.d odd 2 1 9251.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 11.b odd 2 1
99.2.a.d 1 33.d even 2 1
121.2.a.d 1 1.a even 1 1 trivial
121.2.c.a 4 11.c even 5 4
121.2.c.e 4 11.d odd 10 4
176.2.a.b 1 44.c even 2 1
275.2.a.b 1 55.d odd 2 1
275.2.b.a 2 55.e even 4 2
539.2.a.a 1 77.b even 2 1
539.2.e.g 2 77.i even 6 2
539.2.e.h 2 77.h odd 6 2
704.2.a.c 1 88.g even 2 1
704.2.a.h 1 88.b odd 2 1
891.2.e.b 2 99.g even 6 2
891.2.e.k 2 99.h odd 6 2
1089.2.a.b 1 3.b odd 2 1
1584.2.a.g 1 132.d odd 2 1
1859.2.a.b 1 143.d odd 2 1
1936.2.a.i 1 4.b odd 2 1
2475.2.a.a 1 165.d even 2 1
2475.2.c.a 2 165.l odd 4 2
2816.2.c.f 2 176.i even 4 2
2816.2.c.j 2 176.l odd 4 2
3025.2.a.a 1 5.b even 2 1
3179.2.a.a 1 187.b odd 2 1
3971.2.a.b 1 209.d even 2 1
4400.2.a.i 1 220.g even 2 1
4400.2.b.h 2 220.i odd 4 2
4851.2.a.t 1 231.h odd 2 1
5819.2.a.a 1 253.b even 2 1
5929.2.a.h 1 7.b odd 2 1
6336.2.a.br 1 264.m even 2 1
6336.2.a.bu 1 264.p odd 2 1
7744.2.a.k 1 8.d odd 2 1
7744.2.a.x 1 8.b even 2 1
8624.2.a.j 1 308.g odd 2 1
9251.2.a.d 1 319.d odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T + 1$$
$5$ $$T - 1$$
$7$ $$T - 2$$
$11$ $$T$$
$13$ $$T + 4$$
$17$ $$T - 2$$
$19$ $$T$$
$23$ $$T + 1$$
$29$ $$T$$
$31$ $$T - 7$$
$37$ $$T - 3$$
$41$ $$T - 8$$
$43$ $$T - 6$$
$47$ $$T - 8$$
$53$ $$T + 6$$
$59$ $$T - 5$$
$61$ $$T + 12$$
$67$ $$T + 7$$
$71$ $$T + 3$$
$73$ $$T + 4$$
$79$ $$T - 10$$
$83$ $$T - 6$$
$89$ $$T - 15$$
$97$ $$T + 7$$