Properties

 Label 117.2.t.a Level $117$ Weight $2$ Character orbit 117.t 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(25,117)]

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, 3]))

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

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

chi := DirichletCharacter("117.25");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 117.t (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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} - 1) q^{3} + (2 \zeta_{6} - 2) q^{4} + ( - 2 \zeta_{6} - 2) q^{5} + (2 \zeta_{6} - 4) q^{7} + 3 \zeta_{6} q^{9}+O(q^{10})$$ q + (-z - 1) * q^3 + (2*z - 2) * q^4 + (-2*z - 2) * q^5 + (2*z - 4) * q^7 + 3*z * q^9 $$q + ( - \zeta_{6} - 1) q^{3} + (2 \zeta_{6} - 2) q^{4} + ( - 2 \zeta_{6} - 2) q^{5} + (2 \zeta_{6} - 4) q^{7} + 3 \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 4) q^{11} + ( - 2 \zeta_{6} + 4) q^{12} + ( - 3 \zeta_{6} - 1) q^{13} + 6 \zeta_{6} q^{15} - 4 \zeta_{6} q^{16} - 3 q^{17} + (4 \zeta_{6} - 2) q^{19} + ( - 4 \zeta_{6} + 8) q^{20} + 6 q^{21} + (3 \zeta_{6} - 3) q^{23} + 7 \zeta_{6} q^{25} + ( - 6 \zeta_{6} + 3) q^{27} + ( - 8 \zeta_{6} + 4) q^{28} - 6 \zeta_{6} q^{29} + (2 \zeta_{6} + 2) q^{31} - 6 q^{33} + 12 q^{35} - 6 q^{36} + (8 \zeta_{6} - 4) q^{37} + (7 \zeta_{6} - 2) q^{39} + ( - 4 \zeta_{6} - 4) q^{41} + \zeta_{6} q^{43} + (8 \zeta_{6} - 4) q^{44} + ( - 12 \zeta_{6} + 6) q^{45} + (4 \zeta_{6} - 8) q^{47} + (8 \zeta_{6} - 4) q^{48} + ( - 5 \zeta_{6} + 5) q^{49} + (3 \zeta_{6} + 3) q^{51} + ( - 2 \zeta_{6} + 8) q^{52} - 9 q^{53} - 12 q^{55} + ( - 6 \zeta_{6} + 6) q^{57} + (2 \zeta_{6} + 2) q^{59} - 12 q^{60} - 7 \zeta_{6} q^{61} + ( - 6 \zeta_{6} - 6) q^{63} + 8 q^{64} + (14 \zeta_{6} - 4) q^{65} + ( - 6 \zeta_{6} + 6) q^{68} + ( - 3 \zeta_{6} + 6) q^{69} + ( - 12 \zeta_{6} + 6) q^{73} + ( - 14 \zeta_{6} + 7) q^{75} + ( - 4 \zeta_{6} - 4) q^{76} + (12 \zeta_{6} - 12) q^{77} - \zeta_{6} q^{79} + (16 \zeta_{6} - 8) q^{80} + (9 \zeta_{6} - 9) q^{81} + ( - 4 \zeta_{6} + 8) q^{83} + (12 \zeta_{6} - 12) q^{84} + (6 \zeta_{6} + 6) q^{85} + (12 \zeta_{6} - 6) q^{87} + (4 \zeta_{6} - 2) q^{89} + (4 \zeta_{6} + 10) q^{91} - 6 \zeta_{6} q^{92} - 6 \zeta_{6} q^{93} + ( - 12 \zeta_{6} + 12) q^{95} + ( - 4 \zeta_{6} + 8) q^{97} + (6 \zeta_{6} + 6) q^{99} +O(q^{100})$$ q + (-z - 1) * q^3 + (2*z - 2) * q^4 + (-2*z - 2) * q^5 + (2*z - 4) * q^7 + 3*z * q^9 + (-2*z + 4) * q^11 + (-2*z + 4) * q^12 + (-3*z - 1) * q^13 + 6*z * q^15 - 4*z * q^16 - 3 * q^17 + (4*z - 2) * q^19 + (-4*z + 8) * q^20 + 6 * q^21 + (3*z - 3) * q^23 + 7*z * q^25 + (-6*z + 3) * q^27 + (-8*z + 4) * q^28 - 6*z * q^29 + (2*z + 2) * q^31 - 6 * q^33 + 12 * q^35 - 6 * q^36 + (8*z - 4) * q^37 + (7*z - 2) * q^39 + (-4*z - 4) * q^41 + z * q^43 + (8*z - 4) * q^44 + (-12*z + 6) * q^45 + (4*z - 8) * q^47 + (8*z - 4) * q^48 + (-5*z + 5) * q^49 + (3*z + 3) * q^51 + (-2*z + 8) * q^52 - 9 * q^53 - 12 * q^55 + (-6*z + 6) * q^57 + (2*z + 2) * q^59 - 12 * q^60 - 7*z * q^61 + (-6*z - 6) * q^63 + 8 * q^64 + (14*z - 4) * q^65 + (-6*z + 6) * q^68 + (-3*z + 6) * q^69 + (-12*z + 6) * q^73 + (-14*z + 7) * q^75 + (-4*z - 4) * q^76 + (12*z - 12) * q^77 - z * q^79 + (16*z - 8) * q^80 + (9*z - 9) * q^81 + (-4*z + 8) * q^83 + (12*z - 12) * q^84 + (6*z + 6) * q^85 + (12*z - 6) * q^87 + (4*z - 2) * q^89 + (4*z + 10) * q^91 - 6*z * q^92 - 6*z * q^93 + (-12*z + 12) * q^95 + (-4*z + 8) * q^97 + (6*z + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} - 2 q^{4} - 6 q^{5} - 6 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 - 2 * q^4 - 6 * q^5 - 6 * q^7 + 3 * q^9 $$2 q - 3 q^{3} - 2 q^{4} - 6 q^{5} - 6 q^{7} + 3 q^{9} + 6 q^{11} + 6 q^{12} - 5 q^{13} + 6 q^{15} - 4 q^{16} - 6 q^{17} + 12 q^{20} + 12 q^{21} - 3 q^{23} + 7 q^{25} - 6 q^{29} + 6 q^{31} - 12 q^{33} + 24 q^{35} - 12 q^{36} + 3 q^{39} - 12 q^{41} + q^{43} - 12 q^{47} + 5 q^{49} + 9 q^{51} + 14 q^{52} - 18 q^{53} - 24 q^{55} + 6 q^{57} + 6 q^{59} - 24 q^{60} - 7 q^{61} - 18 q^{63} + 16 q^{64} + 6 q^{65} + 6 q^{68} + 9 q^{69} - 12 q^{76} - 12 q^{77} - q^{79} - 9 q^{81} + 12 q^{83} - 12 q^{84} + 18 q^{85} + 24 q^{91} - 6 q^{92} - 6 q^{93} + 12 q^{95} + 12 q^{97} + 18 q^{99}+O(q^{100})$$ 2 * q - 3 * q^3 - 2 * q^4 - 6 * q^5 - 6 * q^7 + 3 * q^9 + 6 * q^11 + 6 * q^12 - 5 * q^13 + 6 * q^15 - 4 * q^16 - 6 * q^17 + 12 * q^20 + 12 * q^21 - 3 * q^23 + 7 * q^25 - 6 * q^29 + 6 * q^31 - 12 * q^33 + 24 * q^35 - 12 * q^36 + 3 * q^39 - 12 * q^41 + q^43 - 12 * q^47 + 5 * q^49 + 9 * q^51 + 14 * q^52 - 18 * q^53 - 24 * q^55 + 6 * q^57 + 6 * q^59 - 24 * q^60 - 7 * q^61 - 18 * q^63 + 16 * q^64 + 6 * q^65 + 6 * q^68 + 9 * q^69 - 12 * q^76 - 12 * q^77 - q^79 - 9 * q^81 + 12 * q^83 - 12 * q^84 + 18 * q^85 + 24 * q^91 - 6 * q^92 - 6 * q^93 + 12 * q^95 + 12 * q^97 + 18 * q^99

Character values

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

 $$n$$ $$28$$ $$92$$ $$\chi(n)$$ $$-1$$ $$-1 + \zeta_{6}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −1.50000 + 0.866025i −1.00000 1.73205i −3.00000 + 1.73205i 0 −3.00000 1.73205i 0 1.50000 2.59808i 0
103.1 0 −1.50000 0.866025i −1.00000 + 1.73205i −3.00000 1.73205i 0 −3.00000 + 1.73205i 0 1.50000 + 2.59808i 0
 $$n$$: 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.t 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.t.a 2
3.b odd 2 1 351.2.t.b 2
9.c even 3 1 117.2.t.b yes 2
9.c even 3 1 1053.2.b.e 2
9.d odd 6 1 351.2.t.a 2
9.d odd 6 1 1053.2.b.f 2
13.b even 2 1 117.2.t.b yes 2
39.d odd 2 1 351.2.t.a 2
117.n odd 6 1 351.2.t.b 2
117.n odd 6 1 1053.2.b.f 2
117.t even 6 1 inner 117.2.t.a 2
117.t even 6 1 1053.2.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.t.a 2 1.a even 1 1 trivial
117.2.t.a 2 117.t even 6 1 inner
117.2.t.b yes 2 9.c even 3 1
117.2.t.b yes 2 13.b even 2 1
351.2.t.a 2 9.d odd 6 1
351.2.t.a 2 39.d odd 2 1
351.2.t.b 2 3.b odd 2 1
351.2.t.b 2 117.n odd 6 1
1053.2.b.e 2 9.c even 3 1
1053.2.b.e 2 117.t even 6 1
1053.2.b.f 2 9.d odd 6 1
1053.2.b.f 2 117.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(117, [\chi])$$:

 $$T_{2}$$ T2 $$T_{5}^{2} + 6T_{5} + 12$$ T5^2 + 6*T5 + 12

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3T + 3$$
$5$ $$T^{2} + 6T + 12$$
$7$ $$T^{2} + 6T + 12$$
$11$ $$T^{2} - 6T + 12$$
$13$ $$T^{2} + 5T + 13$$
$17$ $$(T + 3)^{2}$$
$19$ $$T^{2} + 12$$
$23$ $$T^{2} + 3T + 9$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$T^{2} - 6T + 12$$
$37$ $$T^{2} + 48$$
$41$ $$T^{2} + 12T + 48$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2} + 12T + 48$$
$53$ $$(T + 9)^{2}$$
$59$ $$T^{2} - 6T + 12$$
$61$ $$T^{2} + 7T + 49$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 108$$
$79$ $$T^{2} + T + 1$$
$83$ $$T^{2} - 12T + 48$$
$89$ $$T^{2} + 12$$
$97$ $$T^{2} - 12T + 48$$