# Properties

 Label 169.2.b.b Level $169$ Weight $2$ Character orbit 169.b Analytic conductor $1.349$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.153664.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 5x^{4} + 6x^{2} + 1$$ x^6 + 5*x^4 + 6*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} + \beta_{3}) q^{2} + (\beta_{4} - 1) q^{3} + ( - \beta_{4} - 2 \beta_{2} + 1) q^{4} + (\beta_{5} + \beta_1) q^{5} - \beta_{3} q^{6} + (2 \beta_{3} - \beta_1) q^{7} + ( - 2 \beta_{5} - \beta_{3} + 2 \beta_1) q^{8} + ( - 2 \beta_{4} - \beta_{2}) q^{9}+O(q^{10})$$ q + (b5 + b3) * q^2 + (b4 - 1) * q^3 + (-b4 - 2*b2 + 1) * q^4 + (b5 + b1) * q^5 - b3 * q^6 + (2*b3 - b1) * q^7 + (-2*b5 - b3 + 2*b1) * q^8 + (-2*b4 - b2) * q^9 $$q + (\beta_{5} + \beta_{3}) q^{2} + (\beta_{4} - 1) q^{3} + ( - \beta_{4} - 2 \beta_{2} + 1) q^{4} + (\beta_{5} + \beta_1) q^{5} - \beta_{3} q^{6} + (2 \beta_{3} - \beta_1) q^{7} + ( - 2 \beta_{5} - \beta_{3} + 2 \beta_1) q^{8} + ( - 2 \beta_{4} - \beta_{2}) q^{9} + ( - \beta_{4} - \beta_{2} - 1) q^{10} + ( - 3 \beta_{5} + \beta_1) q^{11} + (2 \beta_{4} + \beta_{2} - 1) q^{12} + ( - 2 \beta_{2} - 1) q^{14} + ( - \beta_{3} + \beta_1) q^{15} + ( - \beta_{2} + 1) q^{16} + (2 \beta_{4} + 3 \beta_{2} - 1) q^{17} + ( - 4 \beta_{5} - 2 \beta_{3} + \beta_1) q^{18} + (3 \beta_{5} + 3 \beta_{3} - 2 \beta_1) q^{19} + ( - 2 \beta_{5} - 3 \beta_{3} + 3 \beta_1) q^{20} + (\beta_{5} - \beta_{3} - 2 \beta_1) q^{21} + (3 \beta_{4} + 3 \beta_{2} - 1) q^{22} + (2 \beta_{4} + 1) q^{23} + (3 \beta_{5} - \beta_{3} - \beta_1) q^{24} + ( - 2 \beta_{4} + \beta_{2} + 2) q^{25} + ( - \beta_{4} + 2 \beta_{2}) q^{27} + ( - 5 \beta_{5} - \beta_{3}) q^{28} + ( - 3 \beta_{4} + 2 \beta_{2}) q^{29} + \beta_{2} q^{30} + (4 \beta_{5} + 2 \beta_{3} - 5 \beta_1) q^{31} + ( - 5 \beta_{5} - 3 \beta_{3} + 5 \beta_1) q^{32} + (4 \beta_{5} - \beta_{3} - 3 \beta_1) q^{33} + (7 \beta_{5} + 5 \beta_{3} - 3 \beta_1) q^{34} + (\beta_{4} - 3 \beta_{2} + 2) q^{35} + (4 \beta_{2} + 1) q^{36} + (5 \beta_{5} + \beta_{3} - 2 \beta_1) q^{37} + ( - 3 \beta_{4} - 6 \beta_{2} - 1) q^{38} + (3 \beta_{2} - 2) q^{40} + (3 \beta_{5} - 2 \beta_{3} - 4 \beta_1) q^{41} + ( - \beta_{4} + 3) q^{42} + ( - 3 \beta_{4} - \beta_{2} - 3) q^{43} + (2 \beta_{5} + 5 \beta_{3} - \beta_1) q^{44} + ( - 3 \beta_{5} - 2 \beta_1) q^{45} + (3 \beta_{5} + \beta_{3}) q^{46} + ( - 5 \beta_{5} + 2 \beta_{3} - \beta_1) q^{47} + \beta_{4} q^{48} + (\beta_{2} + 1) q^{49} + (2 \beta_{5} + 4 \beta_{3} - \beta_1) q^{50} + ( - 3 \beta_{4} - 2 \beta_{2} + 2) q^{51} + (7 \beta_{4} + 3 \beta_{2} - 3) q^{53} + (3 \beta_{5} + 4 \beta_{3} - 2 \beta_1) q^{54} + (2 \beta_{4} + \beta_{2} + 1) q^{55} + (5 \beta_{4} + 2 \beta_{2} - 1) q^{56} + ( - 2 \beta_{5} - \beta_{3}) q^{57} + (\beta_{5} + 4 \beta_{3} - 2 \beta_1) q^{58} + ( - 9 \beta_{5} - 4 \beta_{3} + 4 \beta_1) q^{59} + (2 \beta_{5} + \beta_1) q^{60} + ( - \beta_{4} + 5 \beta_{2}) q^{61} + ( - 4 \beta_{4} - 6 \beta_{2} + 3) q^{62} + ( - 4 \beta_{5} - 3 \beta_{3} + 5 \beta_1) q^{63} + (5 \beta_{4} + 6 \beta_{2}) q^{64} + ( - 4 \beta_{4} - 3 \beta_{2} + 4) q^{66} + ( - 2 \beta_{5} - \beta_{3} + 6 \beta_1) q^{67} + ( - 3 \beta_{4} - 6 \beta_{2} - 4) q^{68} + ( - \beta_{4} - 2 \beta_{2} + 3) q^{69} + ( - 3 \beta_{5} - 4 \beta_{3} + 3 \beta_1) q^{70} + (7 \beta_{5} - 3 \beta_{3} + 3 \beta_1) q^{71} + (\beta_{5} + 5 \beta_{3} - 2 \beta_1) q^{72} + ( - 2 \beta_{5} - 9 \beta_{3} + 6 \beta_1) q^{73} + ( - 5 \beta_{4} - 6 \beta_{2} + 1) q^{74} + (4 \beta_{4} + 2 \beta_{2} - 7) q^{75} + ( - 10 \beta_{5} - 7 \beta_{3} + 2 \beta_1) q^{76} + (5 \beta_{4} + 5 \beta_{2} - 6) q^{77} + ( - 8 \beta_{4} - 9 \beta_{2} + 4) q^{79} + ( - 2 \beta_{3} + 3 \beta_1) q^{80} + (7 \beta_{4} + 4 \beta_{2} - 4) q^{81} + ( - 3 \beta_{4} - \beta_{2} + 6) q^{82} + ( - 3 \beta_{5} - 7 \beta_{3} + 9 \beta_1) q^{83} + (4 \beta_{5} + \beta_{3} - 4 \beta_1) q^{84} + (4 \beta_{5} + 4 \beta_{3} - 3 \beta_1) q^{85} + ( - 8 \beta_{5} - 5 \beta_{3} + \beta_1) q^{86} + (3 \beta_{4} + 3 \beta_{2} - 8) q^{87} + (4 \beta_{4} - \beta_{2} - 6) q^{88} + (\beta_{5} + 7 \beta_{3} - 7 \beta_1) q^{89} + (3 \beta_{4} + 3 \beta_{2} + 2) q^{90} + (\beta_{4} - 4 \beta_{2} + 1) q^{92} + ( - 7 \beta_{5} + 3 \beta_{3} + 2 \beta_1) q^{93} + (5 \beta_{4} + 3 \beta_{2} - 1) q^{94} + ( - \beta_{4} - 5 \beta_{2} + 1) q^{95} + (7 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{96} + ( - 4 \beta_{5} - 6 \beta_{3} - \beta_1) q^{97} + (3 \beta_{5} + 3 \beta_{3} - \beta_1) q^{98} + (\beta_{5} + 4 \beta_{3} + 2 \beta_1) q^{99}+O(q^{100})$$ q + (b5 + b3) * q^2 + (b4 - 1) * q^3 + (-b4 - 2*b2 + 1) * q^4 + (b5 + b1) * q^5 - b3 * q^6 + (2*b3 - b1) * q^7 + (-2*b5 - b3 + 2*b1) * q^8 + (-2*b4 - b2) * q^9 + (-b4 - b2 - 1) * q^10 + (-3*b5 + b1) * q^11 + (2*b4 + b2 - 1) * q^12 + (-2*b2 - 1) * q^14 + (-b3 + b1) * q^15 + (-b2 + 1) * q^16 + (2*b4 + 3*b2 - 1) * q^17 + (-4*b5 - 2*b3 + b1) * q^18 + (3*b5 + 3*b3 - 2*b1) * q^19 + (-2*b5 - 3*b3 + 3*b1) * q^20 + (b5 - b3 - 2*b1) * q^21 + (3*b4 + 3*b2 - 1) * q^22 + (2*b4 + 1) * q^23 + (3*b5 - b3 - b1) * q^24 + (-2*b4 + b2 + 2) * q^25 + (-b4 + 2*b2) * q^27 + (-5*b5 - b3) * q^28 + (-3*b4 + 2*b2) * q^29 + b2 * q^30 + (4*b5 + 2*b3 - 5*b1) * q^31 + (-5*b5 - 3*b3 + 5*b1) * q^32 + (4*b5 - b3 - 3*b1) * q^33 + (7*b5 + 5*b3 - 3*b1) * q^34 + (b4 - 3*b2 + 2) * q^35 + (4*b2 + 1) * q^36 + (5*b5 + b3 - 2*b1) * q^37 + (-3*b4 - 6*b2 - 1) * q^38 + (3*b2 - 2) * q^40 + (3*b5 - 2*b3 - 4*b1) * q^41 + (-b4 + 3) * q^42 + (-3*b4 - b2 - 3) * q^43 + (2*b5 + 5*b3 - b1) * q^44 + (-3*b5 - 2*b1) * q^45 + (3*b5 + b3) * q^46 + (-5*b5 + 2*b3 - b1) * q^47 + b4 * q^48 + (b2 + 1) * q^49 + (2*b5 + 4*b3 - b1) * q^50 + (-3*b4 - 2*b2 + 2) * q^51 + (7*b4 + 3*b2 - 3) * q^53 + (3*b5 + 4*b3 - 2*b1) * q^54 + (2*b4 + b2 + 1) * q^55 + (5*b4 + 2*b2 - 1) * q^56 + (-2*b5 - b3) * q^57 + (b5 + 4*b3 - 2*b1) * q^58 + (-9*b5 - 4*b3 + 4*b1) * q^59 + (2*b5 + b1) * q^60 + (-b4 + 5*b2) * q^61 + (-4*b4 - 6*b2 + 3) * q^62 + (-4*b5 - 3*b3 + 5*b1) * q^63 + (5*b4 + 6*b2) * q^64 + (-4*b4 - 3*b2 + 4) * q^66 + (-2*b5 - b3 + 6*b1) * q^67 + (-3*b4 - 6*b2 - 4) * q^68 + (-b4 - 2*b2 + 3) * q^69 + (-3*b5 - 4*b3 + 3*b1) * q^70 + (7*b5 - 3*b3 + 3*b1) * q^71 + (b5 + 5*b3 - 2*b1) * q^72 + (-2*b5 - 9*b3 + 6*b1) * q^73 + (-5*b4 - 6*b2 + 1) * q^74 + (4*b4 + 2*b2 - 7) * q^75 + (-10*b5 - 7*b3 + 2*b1) * q^76 + (5*b4 + 5*b2 - 6) * q^77 + (-8*b4 - 9*b2 + 4) * q^79 + (-2*b3 + 3*b1) * q^80 + (7*b4 + 4*b2 - 4) * q^81 + (-3*b4 - b2 + 6) * q^82 + (-3*b5 - 7*b3 + 9*b1) * q^83 + (4*b5 + b3 - 4*b1) * q^84 + (4*b5 + 4*b3 - 3*b1) * q^85 + (-8*b5 - 5*b3 + b1) * q^86 + (3*b4 + 3*b2 - 8) * q^87 + (4*b4 - b2 - 6) * q^88 + (b5 + 7*b3 - 7*b1) * q^89 + (3*b4 + 3*b2 + 2) * q^90 + (b4 - 4*b2 + 1) * q^92 + (-7*b5 + 3*b3 + 2*b1) * q^93 + (5*b4 + 3*b2 - 1) * q^94 + (-b4 - 5*b2 + 1) * q^95 + (7*b5 - 2*b3 - 2*b1) * q^96 + (-4*b5 - 6*b3 - b1) * q^97 + (3*b5 + 3*b3 - b1) * q^98 + (b5 + 4*b3 + 2*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 4 q^{3} - 6 q^{9}+O(q^{10})$$ 6 * q - 4 * q^3 - 6 * q^9 $$6 q - 4 q^{3} - 6 q^{9} - 10 q^{10} - 10 q^{14} + 4 q^{16} + 4 q^{17} + 6 q^{22} + 10 q^{23} + 10 q^{25} + 2 q^{27} - 2 q^{29} + 2 q^{30} + 8 q^{35} + 14 q^{36} - 24 q^{38} - 6 q^{40} + 16 q^{42} - 26 q^{43} + 2 q^{48} + 8 q^{49} + 2 q^{51} + 2 q^{53} + 12 q^{55} + 8 q^{56} + 8 q^{61} - 2 q^{62} + 22 q^{64} + 10 q^{66} - 42 q^{68} + 12 q^{69} - 16 q^{74} - 30 q^{75} - 16 q^{77} - 10 q^{79} - 2 q^{81} + 28 q^{82} - 36 q^{87} - 30 q^{88} + 24 q^{90} + 10 q^{94} - 6 q^{95}+O(q^{100})$$ 6 * q - 4 * q^3 - 6 * q^9 - 10 * q^10 - 10 * q^14 + 4 * q^16 + 4 * q^17 + 6 * q^22 + 10 * q^23 + 10 * q^25 + 2 * q^27 - 2 * q^29 + 2 * q^30 + 8 * q^35 + 14 * q^36 - 24 * q^38 - 6 * q^40 + 16 * q^42 - 26 * q^43 + 2 * q^48 + 8 * q^49 + 2 * q^51 + 2 * q^53 + 12 * q^55 + 8 * q^56 + 8 * q^61 - 2 * q^62 + 22 * q^64 + 10 * q^66 - 42 * q^68 + 12 * q^69 - 16 * q^74 - 30 * q^75 - 16 * q^77 - 10 * q^79 - 2 * q^81 + 28 * q^82 - 36 * q^87 - 30 * q^88 + 24 * q^90 + 10 * q^94 - 6 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 5x^{4} + 6x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ v^2 + 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 3\nu$$ v^3 + 3*v $$\beta_{4}$$ $$=$$ $$\nu^{4} + 3\nu^{2} + 1$$ v^4 + 3*v^2 + 1 $$\beta_{5}$$ $$=$$ $$\nu^{5} + 4\nu^{3} + 3\nu$$ v^5 + 4*v^3 + 3*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ b2 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3\beta_1$$ b3 - 3*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} - 3\beta_{2} + 5$$ b4 - 3*b2 + 5 $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4\beta_{3} + 9\beta_1$$ b5 - 4*b3 + 9*b1

## 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
 − 0.445042i − 1.24698i − 1.80194i 1.80194i 1.24698i 0.445042i
2.24698i −0.554958 −3.04892 1.44504i 1.24698i 2.04892i 2.35690i −2.69202 −3.24698
168.2 0.801938i −2.24698 1.35690 0.246980i 1.80194i 2.35690i 2.69202i 2.04892 −0.198062
168.3 0.554958i 0.801938 1.69202 2.80194i 0.445042i 2.69202i 2.04892i −2.35690 −1.55496
168.4 0.554958i 0.801938 1.69202 2.80194i 0.445042i 2.69202i 2.04892i −2.35690 −1.55496
168.5 0.801938i −2.24698 1.35690 0.246980i 1.80194i 2.35690i 2.69202i 2.04892 −0.198062
168.6 2.24698i −0.554958 −3.04892 1.44504i 1.24698i 2.04892i 2.35690i −2.69202 −3.24698
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 168.6 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.b 6
3.b odd 2 1 1521.2.b.l 6
4.b odd 2 1 2704.2.f.o 6
13.b even 2 1 inner 169.2.b.b 6
13.c even 3 2 169.2.e.b 12
13.d odd 4 1 169.2.a.b 3
13.d odd 4 1 169.2.a.c yes 3
13.e even 6 2 169.2.e.b 12
13.f odd 12 2 169.2.c.b 6
13.f odd 12 2 169.2.c.c 6
39.d odd 2 1 1521.2.b.l 6
39.f even 4 1 1521.2.a.o 3
39.f even 4 1 1521.2.a.r 3
52.b odd 2 1 2704.2.f.o 6
52.f even 4 1 2704.2.a.z 3
52.f even 4 1 2704.2.a.ba 3
65.g odd 4 1 4225.2.a.bb 3
65.g odd 4 1 4225.2.a.bg 3
91.i even 4 1 8281.2.a.bf 3
91.i even 4 1 8281.2.a.bj 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.2.a.b 3 13.d odd 4 1
169.2.a.c yes 3 13.d odd 4 1
169.2.b.b 6 1.a even 1 1 trivial
169.2.b.b 6 13.b even 2 1 inner
169.2.c.b 6 13.f odd 12 2
169.2.c.c 6 13.f odd 12 2
169.2.e.b 12 13.c even 3 2
169.2.e.b 12 13.e even 6 2
1521.2.a.o 3 39.f even 4 1
1521.2.a.r 3 39.f even 4 1
1521.2.b.l 6 3.b odd 2 1
1521.2.b.l 6 39.d odd 2 1
2704.2.a.z 3 52.f even 4 1
2704.2.a.ba 3 52.f even 4 1
2704.2.f.o 6 4.b odd 2 1
2704.2.f.o 6 52.b odd 2 1
4225.2.a.bb 3 65.g odd 4 1
4225.2.a.bg 3 65.g odd 4 1
8281.2.a.bf 3 91.i even 4 1
8281.2.a.bj 3 91.i even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 6T_{2}^{4} + 5T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 6 T^{4} + \cdots + 1$$
$3$ $$(T^{3} + 2 T^{2} - T - 1)^{2}$$
$5$ $$T^{6} + 10 T^{4} + \cdots + 1$$
$7$ $$T^{6} + 17 T^{4} + \cdots + 169$$
$11$ $$T^{6} + 26 T^{4} + \cdots + 169$$
$13$ $$T^{6}$$
$17$ $$(T^{3} - 2 T^{2} - 15 T - 13)^{2}$$
$19$ $$T^{6} + 38 T^{4} + \cdots + 1$$
$23$ $$(T^{3} - 5 T^{2} - T + 13)^{2}$$
$29$ $$(T^{3} + T^{2} - 44 T + 83)^{2}$$
$31$ $$T^{6} + 97 T^{4} + \cdots + 27889$$
$37$ $$T^{6} + 62 T^{4} + \cdots + 841$$
$41$ $$T^{6} + 147 T^{4} + \cdots + 2401$$
$43$ $$(T^{3} + 13 T^{2} + \cdots - 13)^{2}$$
$47$ $$T^{6} + 122 T^{4} + \cdots + 27889$$
$53$ $$(T^{3} - T^{2} - 86 T + 337)^{2}$$
$59$ $$T^{6} + 195 T^{4} + \cdots + 1$$
$61$ $$(T^{3} - 4 T^{2} + \cdots + 239)^{2}$$
$67$ $$T^{6} + 145 T^{4} + \cdots + 1681$$
$71$ $$T^{6} + 285 T^{4} + \cdots + 299209$$
$73$ $$T^{6} + 321 T^{4} + \cdots + 829921$$
$79$ $$(T^{3} + 5 T^{2} + \cdots + 127)^{2}$$
$83$ $$T^{6} + 329 T^{4} + \cdots + 41209$$
$89$ $$T^{6} + 269 T^{4} + \cdots + 78961$$
$97$ $$T^{6} + 217 T^{4} + \cdots + 90601$$