Properties

 Label 169.2.b.a Level $169$ Weight $2$ Character orbit 169.b 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(168,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("169.168");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 169.b (of order $$2$$, degree $$1$$, 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: $$2$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 $$\beta = \sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + 2 q^{3} - q^{4} + \beta q^{5} - 2 \beta q^{6} - \beta q^{8} + q^{9} +O(q^{10})$$ q - b * q^2 + 2 * q^3 - q^4 + b * q^5 - 2*b * q^6 - b * q^8 + q^9 $$q - \beta q^{2} + 2 q^{3} - q^{4} + \beta q^{5} - 2 \beta q^{6} - \beta q^{8} + q^{9} + 3 q^{10} - 2 q^{12} + 2 \beta q^{15} - 5 q^{16} - 3 q^{17} - \beta q^{18} + 2 \beta q^{19} - \beta q^{20} - 6 q^{23} - 2 \beta q^{24} + 2 q^{25} - 4 q^{27} + 3 q^{29} + 6 q^{30} - 2 \beta q^{31} + 3 \beta q^{32} + 3 \beta q^{34} - q^{36} + 5 \beta q^{37} + 6 q^{38} + 3 q^{40} - 3 \beta q^{41} + 8 q^{43} + \beta q^{45} + 6 \beta q^{46} + 2 \beta q^{47} - 10 q^{48} + 7 q^{49} - 2 \beta q^{50} - 6 q^{51} - 3 q^{53} + 4 \beta q^{54} + 4 \beta q^{57} - 3 \beta q^{58} - 4 \beta q^{59} - 2 \beta q^{60} + q^{61} - 6 q^{62} - q^{64} + 2 \beta q^{67} + 3 q^{68} - 12 q^{69} - 2 \beta q^{71} - \beta q^{72} - \beta q^{73} + 15 q^{74} + 4 q^{75} - 2 \beta q^{76} + 4 q^{79} - 5 \beta q^{80} - 11 q^{81} - 9 q^{82} - 8 \beta q^{83} - 3 \beta q^{85} - 8 \beta q^{86} + 6 q^{87} - 4 \beta q^{89} + 3 q^{90} + 6 q^{92} - 4 \beta q^{93} + 6 q^{94} - 6 q^{95} + 6 \beta q^{96} - 4 \beta q^{97} - 7 \beta q^{98} +O(q^{100})$$ q - b * q^2 + 2 * q^3 - q^4 + b * q^5 - 2*b * q^6 - b * q^8 + q^9 + 3 * q^10 - 2 * q^12 + 2*b * q^15 - 5 * q^16 - 3 * q^17 - b * q^18 + 2*b * q^19 - b * q^20 - 6 * q^23 - 2*b * q^24 + 2 * q^25 - 4 * q^27 + 3 * q^29 + 6 * q^30 - 2*b * q^31 + 3*b * q^32 + 3*b * q^34 - q^36 + 5*b * q^37 + 6 * q^38 + 3 * q^40 - 3*b * q^41 + 8 * q^43 + b * q^45 + 6*b * q^46 + 2*b * q^47 - 10 * q^48 + 7 * q^49 - 2*b * q^50 - 6 * q^51 - 3 * q^53 + 4*b * q^54 + 4*b * q^57 - 3*b * q^58 - 4*b * q^59 - 2*b * q^60 + q^61 - 6 * q^62 - q^64 + 2*b * q^67 + 3 * q^68 - 12 * q^69 - 2*b * q^71 - b * q^72 - b * q^73 + 15 * q^74 + 4 * q^75 - 2*b * q^76 + 4 * q^79 - 5*b * q^80 - 11 * q^81 - 9 * q^82 - 8*b * q^83 - 3*b * q^85 - 8*b * q^86 + 6 * q^87 - 4*b * q^89 + 3 * q^90 + 6 * q^92 - 4*b * q^93 + 6 * q^94 - 6 * q^95 + 6*b * q^96 - 4*b * q^97 - 7*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q + 4 * q^3 - 2 * q^4 + 2 * q^9 $$2 q + 4 q^{3} - 2 q^{4} + 2 q^{9} + 6 q^{10} - 4 q^{12} - 10 q^{16} - 6 q^{17} - 12 q^{23} + 4 q^{25} - 8 q^{27} + 6 q^{29} + 12 q^{30} - 2 q^{36} + 12 q^{38} + 6 q^{40} + 16 q^{43} - 20 q^{48} + 14 q^{49} - 12 q^{51} - 6 q^{53} + 2 q^{61} - 12 q^{62} - 2 q^{64} + 6 q^{68} - 24 q^{69} + 30 q^{74} + 8 q^{75} + 8 q^{79} - 22 q^{81} - 18 q^{82} + 12 q^{87} + 6 q^{90} + 12 q^{92} + 12 q^{94} - 12 q^{95}+O(q^{100})$$ 2 * q + 4 * q^3 - 2 * q^4 + 2 * q^9 + 6 * q^10 - 4 * q^12 - 10 * q^16 - 6 * q^17 - 12 * q^23 + 4 * q^25 - 8 * q^27 + 6 * q^29 + 12 * q^30 - 2 * q^36 + 12 * q^38 + 6 * q^40 + 16 * q^43 - 20 * q^48 + 14 * q^49 - 12 * q^51 - 6 * q^53 + 2 * q^61 - 12 * q^62 - 2 * q^64 + 6 * q^68 - 24 * q^69 + 30 * q^74 + 8 * q^75 + 8 * q^79 - 22 * q^81 - 18 * q^82 + 12 * q^87 + 6 * q^90 + 12 * q^92 + 12 * q^94 - 12 * q^95

Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
168.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.73205i 2.00000 −1.00000 1.73205i 3.46410i 0 1.73205i 1.00000 3.00000
168.2 1.73205i 2.00000 −1.00000 1.73205i 3.46410i 0 1.73205i 1.00000 3.00000
 $$n$$: 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
13.b even 2 1 inner

Twists

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

Hecke kernels

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

Hecke characteristic polynomials

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