# Properties

 Label 169.2.c.a Level $169$ Weight $2$ Character orbit 169.c Analytic conductor $1.349$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,2,Mod(22,169)]

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([4]))

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

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

chi := DirichletCharacter("169.22");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 169.c (of order $$3$$, 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} - 2 \beta_1 q^{3} + (\beta_1 - 1) q^{4} - \beta_{3} q^{5} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{6} - \beta_{3} q^{8} + (\beta_1 - 1) q^{9}+O(q^{10})$$ q - b2 * q^2 - 2*b1 * q^3 + (b1 - 1) * q^4 - b3 * q^5 + (-2*b3 + 2*b2) * q^6 - b3 * q^8 + (b1 - 1) * q^9 $$q - \beta_{2} q^{2} - 2 \beta_1 q^{3} + (\beta_1 - 1) q^{4} - \beta_{3} q^{5} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{6} - \beta_{3} q^{8} + (\beta_1 - 1) q^{9} + 3 \beta_1 q^{10} + 2 q^{12} + 2 \beta_{2} q^{15} + 5 \beta_1 q^{16} + (3 \beta_1 - 3) q^{17} + \beta_{3} q^{18} + (2 \beta_{3} - 2 \beta_{2}) q^{19} + (\beta_{3} - \beta_{2}) q^{20} - 6 \beta_1 q^{23} + 2 \beta_{2} q^{24} - 2 q^{25} - 4 q^{27} - 3 \beta_1 q^{29} + ( - 6 \beta_1 + 6) q^{30} + 2 \beta_{3} q^{31} + (3 \beta_{3} - 3 \beta_{2}) q^{32} + 3 \beta_{3} q^{34} - \beta_1 q^{36} - 5 \beta_{2} q^{37} - 6 q^{38} + 3 q^{40} - 3 \beta_{2} q^{41} + ( - 8 \beta_1 + 8) q^{43} + (\beta_{3} - \beta_{2}) q^{45} + ( - 6 \beta_{3} + 6 \beta_{2}) q^{46} + 2 \beta_{3} q^{47} + ( - 10 \beta_1 + 10) q^{48} + 7 \beta_1 q^{49} + 2 \beta_{2} q^{50} + 6 q^{51} - 3 q^{53} + 4 \beta_{2} q^{54} - 4 \beta_{3} q^{57} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{58} + (4 \beta_{3} - 4 \beta_{2}) q^{59} - 2 \beta_{3} q^{60} + (\beta_1 - 1) q^{61} - 6 \beta_1 q^{62} + q^{64} + 2 \beta_{2} q^{67} - 3 \beta_1 q^{68} + (12 \beta_1 - 12) q^{69} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{71} + (\beta_{3} - \beta_{2}) q^{72} - \beta_{3} q^{73} + (15 \beta_1 - 15) q^{74} + 4 \beta_1 q^{75} + 2 \beta_{2} q^{76} + 4 q^{79} - 5 \beta_{2} q^{80} + 11 \beta_1 q^{81} + (9 \beta_1 - 9) q^{82} + 8 \beta_{3} q^{83} + (3 \beta_{3} - 3 \beta_{2}) q^{85} - 8 \beta_{3} q^{86} + (6 \beta_1 - 6) q^{87} + 4 \beta_{2} q^{89} - 3 q^{90} + 6 q^{92} - 4 \beta_{2} q^{93} - 6 \beta_1 q^{94} + (6 \beta_1 - 6) q^{95} - 6 \beta_{3} q^{96} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{97} + (7 \beta_{3} - 7 \beta_{2}) q^{98}+O(q^{100})$$ q - b2 * q^2 - 2*b1 * q^3 + (b1 - 1) * q^4 - b3 * q^5 + (-2*b3 + 2*b2) * q^6 - b3 * q^8 + (b1 - 1) * q^9 + 3*b1 * q^10 + 2 * q^12 + 2*b2 * q^15 + 5*b1 * q^16 + (3*b1 - 3) * q^17 + b3 * q^18 + (2*b3 - 2*b2) * q^19 + (b3 - b2) * q^20 - 6*b1 * q^23 + 2*b2 * q^24 - 2 * q^25 - 4 * q^27 - 3*b1 * q^29 + (-6*b1 + 6) * q^30 + 2*b3 * q^31 + (3*b3 - 3*b2) * q^32 + 3*b3 * q^34 - b1 * q^36 - 5*b2 * q^37 - 6 * q^38 + 3 * q^40 - 3*b2 * q^41 + (-8*b1 + 8) * q^43 + (b3 - b2) * q^45 + (-6*b3 + 6*b2) * q^46 + 2*b3 * q^47 + (-10*b1 + 10) * q^48 + 7*b1 * q^49 + 2*b2 * q^50 + 6 * q^51 - 3 * q^53 + 4*b2 * q^54 - 4*b3 * q^57 + (-3*b3 + 3*b2) * q^58 + (4*b3 - 4*b2) * q^59 - 2*b3 * q^60 + (b1 - 1) * q^61 - 6*b1 * q^62 + q^64 + 2*b2 * q^67 - 3*b1 * q^68 + (12*b1 - 12) * q^69 + (-2*b3 + 2*b2) * q^71 + (b3 - b2) * q^72 - b3 * q^73 + (15*b1 - 15) * q^74 + 4*b1 * q^75 + 2*b2 * q^76 + 4 * q^79 - 5*b2 * q^80 + 11*b1 * q^81 + (9*b1 - 9) * q^82 + 8*b3 * q^83 + (3*b3 - 3*b2) * q^85 - 8*b3 * q^86 + (6*b1 - 6) * q^87 + 4*b2 * q^89 - 3 * q^90 + 6 * q^92 - 4*b2 * q^93 - 6*b1 * q^94 + (6*b1 - 6) * q^95 - 6*b3 * q^96 + (-4*b3 + 4*b2) * q^97 + (7*b3 - 7*b2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 2 q^{4} - 2 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 - 2 * q^4 - 2 * q^9 $$4 q - 4 q^{3} - 2 q^{4} - 2 q^{9} + 6 q^{10} + 8 q^{12} + 10 q^{16} - 6 q^{17} - 12 q^{23} - 8 q^{25} - 16 q^{27} - 6 q^{29} + 12 q^{30} - 2 q^{36} - 24 q^{38} + 12 q^{40} + 16 q^{43} + 20 q^{48} + 14 q^{49} + 24 q^{51} - 12 q^{53} - 2 q^{61} - 12 q^{62} + 4 q^{64} - 6 q^{68} - 24 q^{69} - 30 q^{74} + 8 q^{75} + 16 q^{79} + 22 q^{81} - 18 q^{82} - 12 q^{87} - 12 q^{90} + 24 q^{92} - 12 q^{94} - 12 q^{95}+O(q^{100})$$ 4 * q - 4 * q^3 - 2 * q^4 - 2 * q^9 + 6 * q^10 + 8 * q^12 + 10 * q^16 - 6 * q^17 - 12 * q^23 - 8 * q^25 - 16 * q^27 - 6 * q^29 + 12 * q^30 - 2 * q^36 - 24 * q^38 + 12 * q^40 + 16 * q^43 + 20 * q^48 + 14 * q^49 + 24 * q^51 - 12 * q^53 - 2 * q^61 - 12 * q^62 + 4 * q^64 - 6 * q^68 - 24 * q^69 - 30 * q^74 + 8 * q^75 + 16 * q^79 + 22 * q^81 - 18 * q^82 - 12 * q^87 - 12 * q^90 + 24 * q^92 - 12 * q^94 - 12 * q^95

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{3} + \zeta_{12}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{12}^{3}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} ) / 3$$ (-b3 + 2*b2) / 3

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-1 + \beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−0.866025 + 1.50000i −1.00000 + 1.73205i −0.500000 0.866025i −1.73205 −1.73205 3.00000i 0 −1.73205 −0.500000 0.866025i 1.50000 2.59808i
22.2 0.866025 1.50000i −1.00000 + 1.73205i −0.500000 0.866025i 1.73205 1.73205 + 3.00000i 0 1.73205 −0.500000 0.866025i 1.50000 2.59808i
146.1 −0.866025 1.50000i −1.00000 1.73205i −0.500000 + 0.866025i −1.73205 −1.73205 + 3.00000i 0 −1.73205 −0.500000 + 0.866025i 1.50000 + 2.59808i
146.2 0.866025 + 1.50000i −1.00000 1.73205i −0.500000 + 0.866025i 1.73205 1.73205 3.00000i 0 1.73205 −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.b even 2 1 inner
13.c even 3 1 inner
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.c.a 4
13.b even 2 1 inner 169.2.c.a 4
13.c even 3 1 169.2.a.a 2
13.c even 3 1 inner 169.2.c.a 4
13.d odd 4 1 13.2.e.a 2
13.d odd 4 1 169.2.e.a 2
13.e even 6 1 169.2.a.a 2
13.e even 6 1 inner 169.2.c.a 4
13.f odd 12 1 13.2.e.a 2
13.f odd 12 2 169.2.b.a 2
13.f odd 12 1 169.2.e.a 2
39.f even 4 1 117.2.q.c 2
39.h odd 6 1 1521.2.a.k 2
39.i odd 6 1 1521.2.a.k 2
39.k even 12 1 117.2.q.c 2
39.k even 12 2 1521.2.b.a 2
52.f even 4 1 208.2.w.b 2
52.i odd 6 1 2704.2.a.o 2
52.j odd 6 1 2704.2.a.o 2
52.l even 12 1 208.2.w.b 2
52.l even 12 2 2704.2.f.b 2
65.f even 4 1 325.2.m.a 4
65.g odd 4 1 325.2.n.a 2
65.k even 4 1 325.2.m.a 4
65.l even 6 1 4225.2.a.v 2
65.n even 6 1 4225.2.a.v 2
65.o even 12 1 325.2.m.a 4
65.s odd 12 1 325.2.n.a 2
65.t even 12 1 325.2.m.a 4
91.i even 4 1 637.2.q.a 2
91.n odd 6 1 8281.2.a.q 2
91.t odd 6 1 8281.2.a.q 2
91.w even 12 1 637.2.k.c 2
91.x odd 12 1 637.2.u.c 2
91.z odd 12 1 637.2.k.a 2
91.z odd 12 1 637.2.u.c 2
91.ba even 12 1 637.2.u.b 2
91.bb even 12 1 637.2.k.c 2
91.bb even 12 1 637.2.u.b 2
91.bc even 12 1 637.2.q.a 2
91.bd odd 12 1 637.2.k.a 2
104.j odd 4 1 832.2.w.d 2
104.m even 4 1 832.2.w.a 2
104.u even 12 1 832.2.w.a 2
104.x odd 12 1 832.2.w.d 2
156.l odd 4 1 1872.2.by.d 2
156.v odd 12 1 1872.2.by.d 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 3T_{2}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3T^{2} + 9$$
$3$ $$(T^{2} + 2 T + 4)^{2}$$
$5$ $$(T^{2} - 3)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 3 T + 9)^{2}$$
$19$ $$T^{4} + 12T^{2} + 144$$
$23$ $$(T^{2} + 6 T + 36)^{2}$$
$29$ $$(T^{2} + 3 T + 9)^{2}$$
$31$ $$(T^{2} - 12)^{2}$$
$37$ $$T^{4} + 75T^{2} + 5625$$
$41$ $$T^{4} + 27T^{2} + 729$$
$43$ $$(T^{2} - 8 T + 64)^{2}$$
$47$ $$(T^{2} - 12)^{2}$$
$53$ $$(T + 3)^{4}$$
$59$ $$T^{4} + 48T^{2} + 2304$$
$61$ $$(T^{2} + T + 1)^{2}$$
$67$ $$T^{4} + 12T^{2} + 144$$
$71$ $$T^{4} + 12T^{2} + 144$$
$73$ $$(T^{2} - 3)^{2}$$
$79$ $$(T - 4)^{4}$$
$83$ $$(T^{2} - 192)^{2}$$
$89$ $$T^{4} + 48T^{2} + 2304$$
$97$ $$T^{4} + 48T^{2} + 2304$$