# Properties

 Label 121.2.c.d Level $121$ Weight $2$ Character orbit 121.c Analytic conductor $0.966$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [121,2,Mod(3,121)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(121, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([8]))

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

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

chi := DirichletCharacter("121.3");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$121 = 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 121.c (of order $$5$$, degree $$4$$, not 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: no Analytic conductor: $$0.966189864457$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/121\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\zeta_{10}^{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 0.809017 + 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 − 0.587785i
0.809017 0.587785i 0.618034 + 1.90211i −0.309017 + 0.951057i −0.809017 0.587785i 1.61803 + 1.17557i 0.618034 1.90211i 0.927051 + 2.85317i −0.809017 + 0.587785i −1.00000
9.1 −0.309017 + 0.951057i −1.61803 + 1.17557i 0.809017 + 0.587785i 0.309017 + 0.951057i −0.618034 1.90211i −1.61803 1.17557i −2.42705 + 1.76336i 0.309017 0.951057i −1.00000
27.1 −0.309017 0.951057i −1.61803 1.17557i 0.809017 0.587785i 0.309017 0.951057i −0.618034 + 1.90211i −1.61803 + 1.17557i −2.42705 1.76336i 0.309017 + 0.951057i −1.00000
81.1 0.809017 + 0.587785i 0.618034 1.90211i −0.309017 0.951057i −0.809017 + 0.587785i 1.61803 1.17557i 0.618034 + 1.90211i 0.927051 2.85317i −0.809017 0.587785i −1.00000
 $$n$$: 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.c even 5 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.2.c.d 4
11.b odd 2 1 121.2.c.b 4
11.c even 5 1 121.2.a.a 1
11.c even 5 3 inner 121.2.c.d 4
11.d odd 10 1 121.2.a.c yes 1
11.d odd 10 3 121.2.c.b 4
33.f even 10 1 1089.2.a.c 1
33.h odd 10 1 1089.2.a.i 1
44.g even 10 1 1936.2.a.b 1
44.h odd 10 1 1936.2.a.a 1
55.h odd 10 1 3025.2.a.b 1
55.j even 10 1 3025.2.a.e 1
77.j odd 10 1 5929.2.a.a 1
77.l even 10 1 5929.2.a.g 1
88.k even 10 1 7744.2.a.bf 1
88.l odd 10 1 7744.2.a.be 1
88.o even 10 1 7744.2.a.f 1
88.p odd 10 1 7744.2.a.c 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - T_{2}^{3} + T_{2}^{2} - T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(121, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$3$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$5$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$7$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$11$ $$T^{4}$$
$13$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$17$ $$T^{4} + 5 T^{3} + 25 T^{2} + 125 T + 625$$
$19$ $$T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$23$ $$(T - 2)^{4}$$
$29$ $$T^{4} - 9 T^{3} + 81 T^{2} + \cdots + 6561$$
$31$ $$T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16$$
$37$ $$T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81$$
$41$ $$T^{4} + 5 T^{3} + 25 T^{2} + 125 T + 625$$
$43$ $$T^{4}$$
$47$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$53$ $$T^{4} + 9 T^{3} + 81 T^{2} + \cdots + 6561$$
$59$ $$T^{4} + 8 T^{3} + 64 T^{2} + \cdots + 4096$$
$61$ $$T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$67$ $$(T - 2)^{4}$$
$71$ $$T^{4} + 12 T^{3} + 144 T^{2} + \cdots + 20736$$
$73$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$79$ $$T^{4} + 10 T^{3} + 100 T^{2} + \cdots + 10000$$
$83$ $$T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$89$ $$(T + 9)^{4}$$
$97$ $$T^{4} - 13 T^{3} + 169 T^{2} + \cdots + 28561$$