# Properties

 Label 169.2.a.b Level $169$ Weight $2$ Character orbit 169.a Self dual yes Analytic conductor $1.349$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,2,Mod(1,169)]

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([0]))

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

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

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 169.a (trivial)

## 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: yes Analytic conductor: $$1.34947179416$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

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

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 1) q^{2} + (\beta_{2} - \beta_1) q^{3} + (\beta_{2} + \beta_1) q^{4} + ( - \beta_{2} + \beta_1 - 2) q^{5} + \beta_{2} q^{6} + (\beta_{2} + \beta_1 - 1) q^{7} + (\beta_{2} - 2 \beta_1) q^{8} + ( - 2 \beta_{2} + \beta_1 - 2) q^{9}+O(q^{10})$$ q + (-b2 - 1) * q^2 + (b2 - b1) * q^3 + (b2 + b1) * q^4 + (-b2 + b1 - 2) * q^5 + b2 * q^6 + (b2 + b1 - 1) * q^7 + (b2 - 2*b1) * q^8 + (-2*b2 + b1 - 2) * q^9 $$q + ( - \beta_{2} - 1) q^{2} + (\beta_{2} - \beta_1) q^{3} + (\beta_{2} + \beta_1) q^{4} + ( - \beta_{2} + \beta_1 - 2) q^{5} + \beta_{2} q^{6} + (\beta_{2} + \beta_1 - 1) q^{7} + (\beta_{2} - 2 \beta_1) q^{8} + ( - 2 \beta_{2} + \beta_1 - 2) q^{9} + (\beta_{2} + 2) q^{10} + (\beta_{2} - \beta_1 - 2) q^{11} + ( - 2 \beta_{2} + \beta_1 - 1) q^{12} + ( - 2 \beta_1 - 1) q^{14} + (\beta_1 - 1) q^{15} + ( - \beta_1 + 1) q^{16} + ( - 2 \beta_{2} - \beta_1 - 1) q^{17} + (\beta_{2} + \beta_1 + 3) q^{18} + ( - \beta_{2} - 2 \beta_1 - 1) q^{19} + ( - 3 \beta_1 + 1) q^{20} + ( - 3 \beta_{2} + 2 \beta_1 - 1) q^{21} + (3 \beta_{2} + 2) q^{22} + ( - 2 \beta_{2} + 2 \beta_1 - 3) q^{23} + ( - 2 \beta_{2} + \beta_1 + 2) q^{24} + (2 \beta_{2} - 3 \beta_1) q^{25} + ( - \beta_{2} + 3 \beta_1 - 1) q^{27} + (\beta_{2} + 5) q^{28} + ( - 3 \beta_{2} + 5 \beta_1 - 3) q^{29} - \beta_1 q^{30} + (3 \beta_{2} - 5 \beta_1 + 1) q^{31} + ( - 2 \beta_{2} + 5 \beta_1) q^{32} + ( - 4 \beta_{2} + 3 \beta_1 + 1) q^{33} + (2 \beta_{2} + 3 \beta_1 + 4) q^{34} + (\beta_{2} - 4 \beta_1 + 3) q^{35} + ( - 4 \beta_1 - 1) q^{36} + ( - \beta_{2} + 2 \beta_1 + 3) q^{37} + (3 \beta_{2} + 3 \beta_1 + 4) q^{38} + (3 \beta_1 - 2) q^{40} + (6 \beta_{2} - 4 \beta_1 + 1) q^{41} + ( - \beta_{2} + \beta_1 + 2) q^{42} + (3 \beta_{2} - 2 \beta_1 + 6) q^{43} + ( - 4 \beta_{2} - \beta_1 - 1) q^{44} + (2 \beta_{2} - 2 \beta_1 + 5) q^{45} + (\beta_{2} + 3) q^{46} + (\beta_{2} + \beta_1 - 6) q^{47} + (\beta_{2} - \beta_1 + 1) q^{48} + ( - \beta_1 - 1) q^{49} + (3 \beta_{2} + \beta_1 + 1) q^{50} + (3 \beta_{2} - \beta_1 + 1) q^{51} + (7 \beta_{2} - 4 \beta_1 + 4) q^{53} + ( - 2 \beta_{2} - 2 \beta_1 - 1) q^{54} + (2 \beta_{2} - \beta_1 + 3) q^{55} + ( - 5 \beta_{2} + 3 \beta_1 - 4) q^{56} + (\beta_{2} + 2) q^{57} + ( - 2 \beta_{2} - 2 \beta_1 + 1) q^{58} + ( - 4 \beta_1 - 5) q^{59} + (\beta_{2} - \beta_1 + 3) q^{60} + ( - \beta_{2} + 6 \beta_1 - 1) q^{61} + (4 \beta_{2} + 2 \beta_1 + 1) q^{62} + (2 \beta_{2} - 5 \beta_1 + 1) q^{63} + ( - 5 \beta_{2} - \beta_1 - 5) q^{64} + ( - 4 \beta_{2} + \beta_1) q^{66} + ( - 5 \beta_{2} + 6 \beta_1 - 4) q^{67} + ( - 3 \beta_{2} - 3 \beta_1 - 7) q^{68} + (\beta_{2} + \beta_1 - 2) q^{69} + (\beta_{2} + 3 \beta_1) q^{70} + (3 \beta_1 - 10) q^{71} + (3 \beta_{2} + 2 \beta_1 - 1) q^{72} + ( - 3 \beta_{2} - 6 \beta_1 + 4) q^{73} + ( - 5 \beta_{2} - \beta_1 - 4) q^{74} + ( - 4 \beta_{2} + 2 \beta_1 + 3) q^{75} + ( - 5 \beta_{2} - 2 \beta_1 - 8) q^{76} + ( - 5 \beta_{2} + 1) q^{77} + ( - 8 \beta_{2} - \beta_1 - 4) q^{79} + ( - \beta_{2} + 3 \beta_1 - 3) q^{80} + (7 \beta_{2} - 3 \beta_1 + 3) q^{81} + (3 \beta_{2} - 2 \beta_1 - 3) q^{82} + ( - 2 \beta_{2} + 9 \beta_1 - 6) q^{83} + (3 \beta_{2} - 4 \beta_1) q^{84} + (\beta_{2} + 3 \beta_1 + 1) q^{85} + ( - 4 \beta_{2} - \beta_1 - 7) q^{86} + (3 \beta_{2} - 5) q^{87} + ( - 4 \beta_{2} + 5 \beta_1 + 2) q^{88} + (7 \beta_1 - 6) q^{89} + ( - 3 \beta_{2} - 5) q^{90} + (\beta_{2} - 5 \beta_1 + 2) q^{92} + ( - 5 \beta_{2} + 2 \beta_1 + 5) q^{93} + (5 \beta_{2} - 2 \beta_1 + 4) q^{94} + (\beta_{2} + 4 \beta_1) q^{95} + (4 \beta_{2} - 2 \beta_1 - 5) q^{96} + (7 \beta_{2} - \beta_1 + 5) q^{97} + (2 \beta_{2} + \beta_1 + 2) q^{98} + (6 \beta_{2} - 2 \beta_1 + 3) q^{99}+O(q^{100})$$ q + (-b2 - 1) * q^2 + (b2 - b1) * q^3 + (b2 + b1) * q^4 + (-b2 + b1 - 2) * q^5 + b2 * q^6 + (b2 + b1 - 1) * q^7 + (b2 - 2*b1) * q^8 + (-2*b2 + b1 - 2) * q^9 + (b2 + 2) * q^10 + (b2 - b1 - 2) * q^11 + (-2*b2 + b1 - 1) * q^12 + (-2*b1 - 1) * q^14 + (b1 - 1) * q^15 + (-b1 + 1) * q^16 + (-2*b2 - b1 - 1) * q^17 + (b2 + b1 + 3) * q^18 + (-b2 - 2*b1 - 1) * q^19 + (-3*b1 + 1) * q^20 + (-3*b2 + 2*b1 - 1) * q^21 + (3*b2 + 2) * q^22 + (-2*b2 + 2*b1 - 3) * q^23 + (-2*b2 + b1 + 2) * q^24 + (2*b2 - 3*b1) * q^25 + (-b2 + 3*b1 - 1) * q^27 + (b2 + 5) * q^28 + (-3*b2 + 5*b1 - 3) * q^29 - b1 * q^30 + (3*b2 - 5*b1 + 1) * q^31 + (-2*b2 + 5*b1) * q^32 + (-4*b2 + 3*b1 + 1) * q^33 + (2*b2 + 3*b1 + 4) * q^34 + (b2 - 4*b1 + 3) * q^35 + (-4*b1 - 1) * q^36 + (-b2 + 2*b1 + 3) * q^37 + (3*b2 + 3*b1 + 4) * q^38 + (3*b1 - 2) * q^40 + (6*b2 - 4*b1 + 1) * q^41 + (-b2 + b1 + 2) * q^42 + (3*b2 - 2*b1 + 6) * q^43 + (-4*b2 - b1 - 1) * q^44 + (2*b2 - 2*b1 + 5) * q^45 + (b2 + 3) * q^46 + (b2 + b1 - 6) * q^47 + (b2 - b1 + 1) * q^48 + (-b1 - 1) * q^49 + (3*b2 + b1 + 1) * q^50 + (3*b2 - b1 + 1) * q^51 + (7*b2 - 4*b1 + 4) * q^53 + (-2*b2 - 2*b1 - 1) * q^54 + (2*b2 - b1 + 3) * q^55 + (-5*b2 + 3*b1 - 4) * q^56 + (b2 + 2) * q^57 + (-2*b2 - 2*b1 + 1) * q^58 + (-4*b1 - 5) * q^59 + (b2 - b1 + 3) * q^60 + (-b2 + 6*b1 - 1) * q^61 + (4*b2 + 2*b1 + 1) * q^62 + (2*b2 - 5*b1 + 1) * q^63 + (-5*b2 - b1 - 5) * q^64 + (-4*b2 + b1) * q^66 + (-5*b2 + 6*b1 - 4) * q^67 + (-3*b2 - 3*b1 - 7) * q^68 + (b2 + b1 - 2) * q^69 + (b2 + 3*b1) * q^70 + (3*b1 - 10) * q^71 + (3*b2 + 2*b1 - 1) * q^72 + (-3*b2 - 6*b1 + 4) * q^73 + (-5*b2 - b1 - 4) * q^74 + (-4*b2 + 2*b1 + 3) * q^75 + (-5*b2 - 2*b1 - 8) * q^76 + (-5*b2 + 1) * q^77 + (-8*b2 - b1 - 4) * q^79 + (-b2 + 3*b1 - 3) * q^80 + (7*b2 - 3*b1 + 3) * q^81 + (3*b2 - 2*b1 - 3) * q^82 + (-2*b2 + 9*b1 - 6) * q^83 + (3*b2 - 4*b1) * q^84 + (b2 + 3*b1 + 1) * q^85 + (-4*b2 - b1 - 7) * q^86 + (3*b2 - 5) * q^87 + (-4*b2 + 5*b1 + 2) * q^88 + (7*b1 - 6) * q^89 + (-3*b2 - 5) * q^90 + (b2 - 5*b1 + 2) * q^92 + (-5*b2 + 2*b1 + 5) * q^93 + (5*b2 - 2*b1 + 4) * q^94 + (b2 + 4*b1) * q^95 + (4*b2 - 2*b1 - 5) * q^96 + (7*b2 - b1 + 5) * q^97 + (2*b2 + b1 + 2) * q^98 + (6*b2 - 2*b1 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} - 2 q^{3} - 4 q^{5} - q^{6} - 3 q^{7} - 3 q^{8} - 3 q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 - 2 * q^3 - 4 * q^5 - q^6 - 3 * q^7 - 3 * q^8 - 3 * q^9 $$3 q - 2 q^{2} - 2 q^{3} - 4 q^{5} - q^{6} - 3 q^{7} - 3 q^{8} - 3 q^{9} + 5 q^{10} - 8 q^{11} - 5 q^{14} - 2 q^{15} + 2 q^{16} - 2 q^{17} + 9 q^{18} - 4 q^{19} + 2 q^{21} + 3 q^{22} - 5 q^{23} + 9 q^{24} - 5 q^{25} + q^{27} + 14 q^{28} - q^{29} - q^{30} - 5 q^{31} + 7 q^{32} + 10 q^{33} + 13 q^{34} + 4 q^{35} - 7 q^{36} + 12 q^{37} + 12 q^{38} - 3 q^{40} - 7 q^{41} + 8 q^{42} + 13 q^{43} + 11 q^{45} + 8 q^{46} - 18 q^{47} + q^{48} - 4 q^{49} + q^{50} - q^{51} + q^{53} - 3 q^{54} + 6 q^{55} - 4 q^{56} + 5 q^{57} + 3 q^{58} - 19 q^{59} + 7 q^{60} + 4 q^{61} + q^{62} - 4 q^{63} - 11 q^{64} + 5 q^{66} - q^{67} - 21 q^{68} - 6 q^{69} + 2 q^{70} - 27 q^{71} - 4 q^{72} + 9 q^{73} - 8 q^{74} + 15 q^{75} - 21 q^{76} + 8 q^{77} - 5 q^{79} - 5 q^{80} - q^{81} - 14 q^{82} - 7 q^{83} - 7 q^{84} + 5 q^{85} - 18 q^{86} - 18 q^{87} + 15 q^{88} - 11 q^{89} - 12 q^{90} + 22 q^{93} + 5 q^{94} + 3 q^{95} - 21 q^{96} + 7 q^{97} + 5 q^{98} + q^{99}+O(q^{100})$$ 3 * q - 2 * q^2 - 2 * q^3 - 4 * q^5 - q^6 - 3 * q^7 - 3 * q^8 - 3 * q^9 + 5 * q^10 - 8 * q^11 - 5 * q^14 - 2 * q^15 + 2 * q^16 - 2 * q^17 + 9 * q^18 - 4 * q^19 + 2 * q^21 + 3 * q^22 - 5 * q^23 + 9 * q^24 - 5 * q^25 + q^27 + 14 * q^28 - q^29 - q^30 - 5 * q^31 + 7 * q^32 + 10 * q^33 + 13 * q^34 + 4 * q^35 - 7 * q^36 + 12 * q^37 + 12 * q^38 - 3 * q^40 - 7 * q^41 + 8 * q^42 + 13 * q^43 + 11 * q^45 + 8 * q^46 - 18 * q^47 + q^48 - 4 * q^49 + q^50 - q^51 + q^53 - 3 * q^54 + 6 * q^55 - 4 * q^56 + 5 * q^57 + 3 * q^58 - 19 * q^59 + 7 * q^60 + 4 * q^61 + q^62 - 4 * q^63 - 11 * q^64 + 5 * q^66 - q^67 - 21 * q^68 - 6 * q^69 + 2 * q^70 - 27 * q^71 - 4 * q^72 + 9 * q^73 - 8 * q^74 + 15 * q^75 - 21 * q^76 + 8 * q^77 - 5 * q^79 - 5 * q^80 - q^81 - 14 * q^82 - 7 * q^83 - 7 * q^84 + 5 * q^85 - 18 * q^86 - 18 * q^87 + 15 * q^88 - 11 * q^89 - 12 * q^90 + 22 * q^93 + 5 * q^94 + 3 * q^95 - 21 * q^96 + 7 * q^97 + 5 * q^98 + q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{14} + \zeta_{14}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.80194 −1.24698 0.445042
−2.24698 −0.554958 3.04892 −1.44504 1.24698 2.04892 −2.35690 −2.69202 3.24698
1.2 −0.554958 0.801938 −1.69202 −2.80194 −0.445042 −2.69202 2.04892 −2.35690 1.55496
1.3 0.801938 −2.24698 −1.35690 0.246980 −1.80194 −2.35690 −2.69202 2.04892 0.198062
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$13$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.2.a.b 3 1.a even 1 1 trivial
169.2.a.c yes 3 13.b even 2 1
169.2.b.b 6 13.d odd 4 2
169.2.c.b 6 13.e even 6 2
169.2.c.c 6 13.c even 3 2
169.2.e.b 12 13.f odd 12 4
1521.2.a.o 3 39.d odd 2 1
1521.2.a.r 3 3.b odd 2 1
1521.2.b.l 6 39.f even 4 2
2704.2.a.z 3 4.b odd 2 1
2704.2.a.ba 3 52.b odd 2 1
2704.2.f.o 6 52.f even 4 2
4225.2.a.bb 3 65.d even 2 1
4225.2.a.bg 3 5.b even 2 1
8281.2.a.bf 3 7.b odd 2 1
8281.2.a.bj 3 91.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + 2T_{2}^{2} - T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(169))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 2T^{2} - T - 1$$
$3$ $$T^{3} + 2T^{2} - T - 1$$
$5$ $$T^{3} + 4 T^{2} + \cdots - 1$$
$7$ $$T^{3} + 3 T^{2} + \cdots - 13$$
$11$ $$T^{3} + 8 T^{2} + \cdots + 13$$
$13$ $$T^{3}$$
$17$ $$T^{3} + 2 T^{2} + \cdots + 13$$
$19$ $$T^{3} + 4 T^{2} + \cdots - 1$$
$23$ $$T^{3} + 5T^{2} - T - 13$$
$29$ $$T^{3} + T^{2} + \cdots + 83$$
$31$ $$T^{3} + 5 T^{2} + \cdots - 167$$
$37$ $$T^{3} - 12 T^{2} + \cdots - 29$$
$41$ $$T^{3} + 7 T^{2} + \cdots + 49$$
$43$ $$T^{3} - 13 T^{2} + \cdots + 13$$
$47$ $$T^{3} + 18 T^{2} + \cdots + 167$$
$53$ $$T^{3} - T^{2} + \cdots + 337$$
$59$ $$T^{3} + 19 T^{2} + \cdots + 1$$
$61$ $$T^{3} - 4 T^{2} + \cdots + 239$$
$67$ $$T^{3} + T^{2} + \cdots + 41$$
$71$ $$T^{3} + 27 T^{2} + \cdots + 547$$
$73$ $$T^{3} - 9 T^{2} + \cdots + 911$$
$79$ $$T^{3} + 5 T^{2} + \cdots + 127$$
$83$ $$T^{3} + 7 T^{2} + \cdots + 203$$
$89$ $$T^{3} + 11 T^{2} + \cdots - 281$$
$97$ $$T^{3} - 7 T^{2} + \cdots + 301$$