# Properties

 Label 117.2.r.a Level $117$ Weight $2$ Character orbit 117.r Analytic conductor $0.934$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,2,Mod(43,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 1]))

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

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

chi := DirichletCharacter("117.43");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 117.r (of order $$6$$, degree $$2$$, 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: $$0.934249703649$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 2) q^{2} + (\zeta_{6} - 2) q^{3} + ( - \zeta_{6} + 1) q^{4} + (\zeta_{6} - 2) q^{5} + ( - 3 \zeta_{6} + 3) q^{6} + ( - 2 \zeta_{6} + 1) q^{7} + ( - 2 \zeta_{6} + 1) q^{8} + ( - 3 \zeta_{6} + 3) q^{9}+O(q^{10})$$ q + (z - 2) * q^2 + (z - 2) * q^3 + (-z + 1) * q^4 + (z - 2) * q^5 + (-3*z + 3) * q^6 + (-2*z + 1) * q^7 + (-2*z + 1) * q^8 + (-3*z + 3) * q^9 $$q + (\zeta_{6} - 2) q^{2} + (\zeta_{6} - 2) q^{3} + ( - \zeta_{6} + 1) q^{4} + (\zeta_{6} - 2) q^{5} + ( - 3 \zeta_{6} + 3) q^{6} + ( - 2 \zeta_{6} + 1) q^{7} + ( - 2 \zeta_{6} + 1) q^{8} + ( - 3 \zeta_{6} + 3) q^{9} + ( - 3 \zeta_{6} + 3) q^{10} + (2 \zeta_{6} - 4) q^{11} + (2 \zeta_{6} - 1) q^{12} + ( - 3 \zeta_{6} - 1) q^{13} + 3 \zeta_{6} q^{14} + ( - 3 \zeta_{6} + 3) q^{15} + 5 \zeta_{6} q^{16} - 3 \zeta_{6} q^{17} + (6 \zeta_{6} - 3) q^{18} + (\zeta_{6} - 2) q^{19} + (2 \zeta_{6} - 1) q^{20} + 3 \zeta_{6} q^{21} + ( - 6 \zeta_{6} + 6) q^{22} - 3 q^{23} + 3 \zeta_{6} q^{24} + (2 \zeta_{6} - 2) q^{25} + (2 \zeta_{6} + 5) q^{26} + (6 \zeta_{6} - 3) q^{27} + ( - \zeta_{6} - 1) q^{28} + 6 \zeta_{6} q^{29} + (6 \zeta_{6} - 3) q^{30} + (5 \zeta_{6} - 10) q^{31} + ( - 3 \zeta_{6} - 3) q^{32} + ( - 6 \zeta_{6} + 6) q^{33} + (3 \zeta_{6} + 3) q^{34} + 3 \zeta_{6} q^{35} - 3 \zeta_{6} q^{36} + ( - 3 \zeta_{6} - 3) q^{37} + ( - 3 \zeta_{6} + 3) q^{38} + (2 \zeta_{6} + 5) q^{39} + 3 \zeta_{6} q^{40} + ( - 14 \zeta_{6} + 7) q^{41} + ( - 3 \zeta_{6} - 3) q^{42} - q^{43} + (4 \zeta_{6} - 2) q^{44} + (6 \zeta_{6} - 3) q^{45} + ( - 3 \zeta_{6} + 6) q^{46} + ( - 3 \zeta_{6} - 3) q^{47} + ( - 5 \zeta_{6} - 5) q^{48} + 4 q^{49} + ( - 4 \zeta_{6} + 2) q^{50} + (3 \zeta_{6} + 3) q^{51} + (\zeta_{6} - 4) q^{52} + 6 q^{53} - 9 \zeta_{6} q^{54} + ( - 6 \zeta_{6} + 6) q^{55} - 3 q^{56} + ( - 3 \zeta_{6} + 3) q^{57} + ( - 6 \zeta_{6} - 6) q^{58} + (2 \zeta_{6} + 2) q^{59} - 3 \zeta_{6} q^{60} - 5 q^{61} + ( - 15 \zeta_{6} + 15) q^{62} + ( - 3 \zeta_{6} - 3) q^{63} - q^{64} + (2 \zeta_{6} + 5) q^{65} + (12 \zeta_{6} - 6) q^{66} + (14 \zeta_{6} - 7) q^{67} - 3 q^{68} + ( - 3 \zeta_{6} + 6) q^{69} + ( - 3 \zeta_{6} - 3) q^{70} + (5 \zeta_{6} - 10) q^{71} + ( - 3 \zeta_{6} - 3) q^{72} + (8 \zeta_{6} - 4) q^{73} + 9 q^{74} + ( - 4 \zeta_{6} + 2) q^{75} + (2 \zeta_{6} - 1) q^{76} + 6 \zeta_{6} q^{77} + (3 \zeta_{6} - 12) q^{78} + ( - 11 \zeta_{6} + 11) q^{79} + ( - 5 \zeta_{6} - 5) q^{80} - 9 \zeta_{6} q^{81} + 21 \zeta_{6} q^{82} + ( - 3 \zeta_{6} - 3) q^{83} + 3 q^{84} + (3 \zeta_{6} + 3) q^{85} + ( - \zeta_{6} + 2) q^{86} + ( - 6 \zeta_{6} - 6) q^{87} + 6 \zeta_{6} q^{88} + (9 \zeta_{6} + 9) q^{89} - 9 \zeta_{6} q^{90} + (5 \zeta_{6} - 7) q^{91} + (3 \zeta_{6} - 3) q^{92} + ( - 15 \zeta_{6} + 15) q^{93} + 9 q^{94} + ( - 3 \zeta_{6} + 3) q^{95} + 9 q^{96} + ( - 18 \zeta_{6} + 9) q^{97} + (4 \zeta_{6} - 8) q^{98} + (12 \zeta_{6} - 6) q^{99} +O(q^{100})$$ q + (z - 2) * q^2 + (z - 2) * q^3 + (-z + 1) * q^4 + (z - 2) * q^5 + (-3*z + 3) * q^6 + (-2*z + 1) * q^7 + (-2*z + 1) * q^8 + (-3*z + 3) * q^9 + (-3*z + 3) * q^10 + (2*z - 4) * q^11 + (2*z - 1) * q^12 + (-3*z - 1) * q^13 + 3*z * q^14 + (-3*z + 3) * q^15 + 5*z * q^16 - 3*z * q^17 + (6*z - 3) * q^18 + (z - 2) * q^19 + (2*z - 1) * q^20 + 3*z * q^21 + (-6*z + 6) * q^22 - 3 * q^23 + 3*z * q^24 + (2*z - 2) * q^25 + (2*z + 5) * q^26 + (6*z - 3) * q^27 + (-z - 1) * q^28 + 6*z * q^29 + (6*z - 3) * q^30 + (5*z - 10) * q^31 + (-3*z - 3) * q^32 + (-6*z + 6) * q^33 + (3*z + 3) * q^34 + 3*z * q^35 - 3*z * q^36 + (-3*z - 3) * q^37 + (-3*z + 3) * q^38 + (2*z + 5) * q^39 + 3*z * q^40 + (-14*z + 7) * q^41 + (-3*z - 3) * q^42 - q^43 + (4*z - 2) * q^44 + (6*z - 3) * q^45 + (-3*z + 6) * q^46 + (-3*z - 3) * q^47 + (-5*z - 5) * q^48 + 4 * q^49 + (-4*z + 2) * q^50 + (3*z + 3) * q^51 + (z - 4) * q^52 + 6 * q^53 - 9*z * q^54 + (-6*z + 6) * q^55 - 3 * q^56 + (-3*z + 3) * q^57 + (-6*z - 6) * q^58 + (2*z + 2) * q^59 - 3*z * q^60 - 5 * q^61 + (-15*z + 15) * q^62 + (-3*z - 3) * q^63 - q^64 + (2*z + 5) * q^65 + (12*z - 6) * q^66 + (14*z - 7) * q^67 - 3 * q^68 + (-3*z + 6) * q^69 + (-3*z - 3) * q^70 + (5*z - 10) * q^71 + (-3*z - 3) * q^72 + (8*z - 4) * q^73 + 9 * q^74 + (-4*z + 2) * q^75 + (2*z - 1) * q^76 + 6*z * q^77 + (3*z - 12) * q^78 + (-11*z + 11) * q^79 + (-5*z - 5) * q^80 - 9*z * q^81 + 21*z * q^82 + (-3*z - 3) * q^83 + 3 * q^84 + (3*z + 3) * q^85 + (-z + 2) * q^86 + (-6*z - 6) * q^87 + 6*z * q^88 + (9*z + 9) * q^89 - 9*z * q^90 + (5*z - 7) * q^91 + (3*z - 3) * q^92 + (-15*z + 15) * q^93 + 9 * q^94 + (-3*z + 3) * q^95 + 9 * q^96 + (-18*z + 9) * q^97 + (4*z - 8) * q^98 + (12*z - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} - 3 q^{3} + q^{4} - 3 q^{5} + 3 q^{6} + 3 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 - 3 * q^3 + q^4 - 3 * q^5 + 3 * q^6 + 3 * q^9 $$2 q - 3 q^{2} - 3 q^{3} + q^{4} - 3 q^{5} + 3 q^{6} + 3 q^{9} + 3 q^{10} - 6 q^{11} - 5 q^{13} + 3 q^{14} + 3 q^{15} + 5 q^{16} - 3 q^{17} - 3 q^{19} + 3 q^{21} + 6 q^{22} - 6 q^{23} + 3 q^{24} - 2 q^{25} + 12 q^{26} - 3 q^{28} + 6 q^{29} - 15 q^{31} - 9 q^{32} + 6 q^{33} + 9 q^{34} + 3 q^{35} - 3 q^{36} - 9 q^{37} + 3 q^{38} + 12 q^{39} + 3 q^{40} - 9 q^{42} - 2 q^{43} + 9 q^{46} - 9 q^{47} - 15 q^{48} + 8 q^{49} + 9 q^{51} - 7 q^{52} + 12 q^{53} - 9 q^{54} + 6 q^{55} - 6 q^{56} + 3 q^{57} - 18 q^{58} + 6 q^{59} - 3 q^{60} - 10 q^{61} + 15 q^{62} - 9 q^{63} - 2 q^{64} + 12 q^{65} - 6 q^{68} + 9 q^{69} - 9 q^{70} - 15 q^{71} - 9 q^{72} + 18 q^{74} + 6 q^{77} - 21 q^{78} + 11 q^{79} - 15 q^{80} - 9 q^{81} + 21 q^{82} - 9 q^{83} + 6 q^{84} + 9 q^{85} + 3 q^{86} - 18 q^{87} + 6 q^{88} + 27 q^{89} - 9 q^{90} - 9 q^{91} - 3 q^{92} + 15 q^{93} + 18 q^{94} + 3 q^{95} + 18 q^{96} - 12 q^{98}+O(q^{100})$$ 2 * q - 3 * q^2 - 3 * q^3 + q^4 - 3 * q^5 + 3 * q^6 + 3 * q^9 + 3 * q^10 - 6 * q^11 - 5 * q^13 + 3 * q^14 + 3 * q^15 + 5 * q^16 - 3 * q^17 - 3 * q^19 + 3 * q^21 + 6 * q^22 - 6 * q^23 + 3 * q^24 - 2 * q^25 + 12 * q^26 - 3 * q^28 + 6 * q^29 - 15 * q^31 - 9 * q^32 + 6 * q^33 + 9 * q^34 + 3 * q^35 - 3 * q^36 - 9 * q^37 + 3 * q^38 + 12 * q^39 + 3 * q^40 - 9 * q^42 - 2 * q^43 + 9 * q^46 - 9 * q^47 - 15 * q^48 + 8 * q^49 + 9 * q^51 - 7 * q^52 + 12 * q^53 - 9 * q^54 + 6 * q^55 - 6 * q^56 + 3 * q^57 - 18 * q^58 + 6 * q^59 - 3 * q^60 - 10 * q^61 + 15 * q^62 - 9 * q^63 - 2 * q^64 + 12 * q^65 - 6 * q^68 + 9 * q^69 - 9 * q^70 - 15 * q^71 - 9 * q^72 + 18 * q^74 + 6 * q^77 - 21 * q^78 + 11 * q^79 - 15 * q^80 - 9 * q^81 + 21 * q^82 - 9 * q^83 + 6 * q^84 + 9 * q^85 + 3 * q^86 - 18 * q^87 + 6 * q^88 + 27 * q^89 - 9 * q^90 - 9 * q^91 - 3 * q^92 + 15 * q^93 + 18 * q^94 + 3 * q^95 + 18 * q^96 - 12 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$$.

 $$n$$ $$28$$ $$92$$ $$\chi(n)$$ $$\zeta_{6}$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.50000 + 0.866025i −1.50000 + 0.866025i 0.500000 0.866025i −1.50000 + 0.866025i 1.50000 2.59808i 1.73205i 1.73205i 1.50000 2.59808i 1.50000 2.59808i
49.1 −1.50000 0.866025i −1.50000 0.866025i 0.500000 + 0.866025i −1.50000 0.866025i 1.50000 + 2.59808i 1.73205i 1.73205i 1.50000 + 2.59808i 1.50000 + 2.59808i
 $$n$$: 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
117.r even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.2.r.a yes 2
3.b odd 2 1 351.2.r.a 2
9.c even 3 1 117.2.l.a 2
9.d odd 6 1 351.2.l.a 2
13.e even 6 1 117.2.l.a 2
39.h odd 6 1 351.2.l.a 2
117.m odd 6 1 351.2.r.a 2
117.r even 6 1 inner 117.2.r.a yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.l.a 2 9.c even 3 1
117.2.l.a 2 13.e even 6 1
117.2.r.a yes 2 1.a even 1 1 trivial
117.2.r.a yes 2 117.r even 6 1 inner
351.2.l.a 2 9.d odd 6 1
351.2.l.a 2 39.h odd 6 1
351.2.r.a 2 3.b odd 2 1
351.2.r.a 2 117.m odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3T_{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(117, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T + 3$$
$3$ $$T^{2} + 3T + 3$$
$5$ $$T^{2} + 3T + 3$$
$7$ $$T^{2} + 3$$
$11$ $$T^{2} + 6T + 12$$
$13$ $$T^{2} + 5T + 13$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} + 3T + 3$$
$23$ $$(T + 3)^{2}$$
$29$ $$T^{2} - 6T + 36$$
$31$ $$T^{2} + 15T + 75$$
$37$ $$T^{2} + 9T + 27$$
$41$ $$T^{2} + 147$$
$43$ $$(T + 1)^{2}$$
$47$ $$T^{2} + 9T + 27$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} - 6T + 12$$
$61$ $$(T + 5)^{2}$$
$67$ $$T^{2} + 147$$
$71$ $$T^{2} + 15T + 75$$
$73$ $$T^{2} + 48$$
$79$ $$T^{2} - 11T + 121$$
$83$ $$T^{2} + 9T + 27$$
$89$ $$T^{2} - 27T + 243$$
$97$ $$T^{2} + 243$$