# Properties

 Label 169.2.e.a Level $169$ Weight $2$ Character orbit 169.e Analytic conductor $1.349$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,2,Mod(23,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5]))

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

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

chi := DirichletCharacter("169.23");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 169.e (of order $$6$$, degree $$2$$, 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: $$1.34947179416$$ Analytic rank: $$0$$ 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: no (minimal twist has level 13) 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.50000 0.866025i −1.00000 1.73205i 0.500000 0.866025i 1.73205i −3.00000 1.73205i 0 1.73205i −0.500000 + 0.866025i −1.50000 2.59808i
147.1 1.50000 + 0.866025i −1.00000 + 1.73205i 0.500000 + 0.866025i 1.73205i −3.00000 + 1.73205i 0 1.73205i −0.500000 0.866025i −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
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.2.e.a 2
13.b even 2 1 13.2.e.a 2
13.c even 3 1 13.2.e.a 2
13.c even 3 1 169.2.b.a 2
13.d odd 4 2 169.2.c.a 4
13.e even 6 1 169.2.b.a 2
13.e even 6 1 inner 169.2.e.a 2
13.f odd 12 2 169.2.a.a 2
13.f odd 12 2 169.2.c.a 4
39.d odd 2 1 117.2.q.c 2
39.h odd 6 1 1521.2.b.a 2
39.i odd 6 1 117.2.q.c 2
39.i odd 6 1 1521.2.b.a 2
39.k even 12 2 1521.2.a.k 2
52.b odd 2 1 208.2.w.b 2
52.i odd 6 1 2704.2.f.b 2
52.j odd 6 1 208.2.w.b 2
52.j odd 6 1 2704.2.f.b 2
52.l even 12 2 2704.2.a.o 2
65.d even 2 1 325.2.n.a 2
65.h odd 4 2 325.2.m.a 4
65.n even 6 1 325.2.n.a 2
65.q odd 12 2 325.2.m.a 4
65.s odd 12 2 4225.2.a.v 2
91.b odd 2 1 637.2.q.a 2
91.g even 3 1 637.2.k.a 2
91.h even 3 1 637.2.u.c 2
91.m odd 6 1 637.2.k.c 2
91.n odd 6 1 637.2.q.a 2
91.r even 6 1 637.2.k.a 2
91.r even 6 1 637.2.u.c 2
91.s odd 6 1 637.2.k.c 2
91.s odd 6 1 637.2.u.b 2
91.v odd 6 1 637.2.u.b 2
91.bc even 12 2 8281.2.a.q 2
104.e even 2 1 832.2.w.d 2
104.h odd 2 1 832.2.w.a 2
104.n odd 6 1 832.2.w.a 2
104.r even 6 1 832.2.w.d 2
156.h even 2 1 1872.2.by.d 2
156.p even 6 1 1872.2.by.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.2.e.a 2 13.b even 2 1
13.2.e.a 2 13.c even 3 1
117.2.q.c 2 39.d odd 2 1
117.2.q.c 2 39.i odd 6 1
169.2.a.a 2 13.f odd 12 2
169.2.b.a 2 13.c even 3 1
169.2.b.a 2 13.e even 6 1
169.2.c.a 4 13.d odd 4 2
169.2.c.a 4 13.f odd 12 2
169.2.e.a 2 1.a even 1 1 trivial
169.2.e.a 2 13.e even 6 1 inner
208.2.w.b 2 52.b odd 2 1
208.2.w.b 2 52.j odd 6 1
325.2.m.a 4 65.h odd 4 2
325.2.m.a 4 65.q odd 12 2
325.2.n.a 2 65.d even 2 1
325.2.n.a 2 65.n even 6 1
637.2.k.a 2 91.g even 3 1
637.2.k.a 2 91.r even 6 1
637.2.k.c 2 91.m odd 6 1
637.2.k.c 2 91.s odd 6 1
637.2.q.a 2 91.b odd 2 1
637.2.q.a 2 91.n odd 6 1
637.2.u.b 2 91.s odd 6 1
637.2.u.b 2 91.v odd 6 1
637.2.u.c 2 91.h even 3 1
637.2.u.c 2 91.r even 6 1
832.2.w.a 2 104.h odd 2 1
832.2.w.a 2 104.n odd 6 1
832.2.w.d 2 104.e even 2 1
832.2.w.d 2 104.r even 6 1
1521.2.a.k 2 39.k even 12 2
1521.2.b.a 2 39.h odd 6 1
1521.2.b.a 2 39.i odd 6 1
1872.2.by.d 2 156.h even 2 1
1872.2.by.d 2 156.p even 6 1
2704.2.a.o 2 52.l even 12 2
2704.2.f.b 2 52.i odd 6 1
2704.2.f.b 2 52.j odd 6 1
4225.2.a.v 2 65.s odd 12 2
8281.2.a.q 2 91.bc even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 3T_{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3T + 3$$
$3$ $$T^{2} + 2T + 4$$
$5$ $$T^{2} + 3$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2} - 6T + 12$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$T^{2} + 12$$
$37$ $$T^{2} + 15T + 75$$
$41$ $$T^{2} - 9T + 27$$
$43$ $$T^{2} + 8T + 64$$
$47$ $$T^{2} + 12$$
$53$ $$(T + 3)^{2}$$
$59$ $$T^{2} + 12T + 48$$
$61$ $$T^{2} + T + 1$$
$67$ $$T^{2} + 6T + 12$$
$71$ $$T^{2} + 6T + 12$$
$73$ $$T^{2} + 3$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} + 192$$
$89$ $$T^{2} - 12T + 48$$
$97$ $$T^{2} + 12T + 48$$