# Properties

 Label 169.2.c.b Level $169$ Weight $2$ Character orbit 169.c 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(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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.64827.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1$$ x^6 - x^5 + 3*x^4 + 5*x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} + \beta_{4} - 1) q^{2} + ( - \beta_{5} - \beta_1 + 1) q^{3} + ( - \beta_{5} - 2 \beta_{4} + \cdots - \beta_1) q^{4}+ \cdots + (\beta_{5} + \beta_{4} + \beta_{3} + \cdots - \beta_1) q^{9}+O(q^{10})$$ q + (b5 + b4 - 1) * q^2 + (-b5 - b1 + 1) * q^3 + (-b5 - 2*b4 - 2*b3 + b2 - b1) * q^4 + (b2 + 1) * q^5 + (b4 + b3) * q^6 + (2*b4 + 2*b3 - b2 + b1) * q^7 + (b3 - 2*b2 + 2) * q^8 + (b5 + b4 + b3 + b2 - b1) * q^9 $$q + (\beta_{5} + \beta_{4} - 1) q^{2} + ( - \beta_{5} - \beta_1 + 1) q^{3} + ( - \beta_{5} - 2 \beta_{4} + \cdots - \beta_1) q^{4}+ \cdots + ( - 4 \beta_{3} - 2 \beta_{2} - 1) q^{99}+O(q^{100})$$ q + (b5 + b4 - 1) * q^2 + (-b5 - b1 + 1) * q^3 + (-b5 - 2*b4 - 2*b3 + b2 - b1) * q^4 + (b2 + 1) * q^5 + (b4 + b3) * q^6 + (2*b4 + 2*b3 - b2 + b1) * q^7 + (b3 - 2*b2 + 2) * q^8 + (b5 + b4 + b3 + b2 - b1) * q^9 + (2*b5 + b4 - 2) * q^10 + (3*b5 + b1 - 3) * q^11 + (-b3 - b2) * q^12 + (-2*b3 + 2*b2 - 3) * q^14 + (-b4 - b1) * q^15 + (-b4 - b1) * q^16 + (2*b5 + 3*b4 + 3*b3 - b2 + b1) * q^17 + (-2*b3 + b2 - 4) * q^18 + (-3*b5 - 3*b4 - 3*b3 + 2*b2 - 2*b1) * q^19 + (-2*b5 - 3*b4 - 3*b3 + 3*b2 - 3*b1) * q^20 + (b3 + 2*b2 - 1) * q^21 + (-2*b5 - 3*b4 - 3*b3) * q^22 + (-b5 + 2*b1 + 1) * q^23 + (-3*b5 + b4 - b1 + 3) * q^24 + (-b3 + 3*b2 - 3) * q^25 + (2*b3 - 3*b2 + 2) * q^27 + (-5*b5 - b4 + 5) * q^28 + (2*b5 + 2*b4 + 5*b1 - 2) * q^29 + (b5 + b4 + b3 - b2 + b1) * q^30 + (2*b3 - 5*b2 + 4) * q^31 + (5*b5 + 3*b4 + 3*b3 - 5*b2 + 5*b1) * q^32 + (4*b5 - b4 - b3 - 3*b2 + 3*b1) * q^33 + (-5*b3 + 3*b2 - 7) * q^34 + (b5 + 3*b4 + 3*b3 - 4*b2 + 4*b1) * q^35 + (-5*b5 - 4*b4 - 4*b1 + 5) * q^36 + (-5*b5 - b4 - 2*b1 + 5) * q^37 + (6*b3 - 3*b2 + 7) * q^38 + (3*b3 - 3*b2 + 1) * q^40 + (3*b5 - 2*b4 + 4*b1 - 3) * q^41 + (3*b5 + b1 - 3) * q^42 + (-4*b5 - b4 - b3 - 2*b2 + 2*b1) * q^43 + (5*b3 - b2 + 2) * q^44 + (3*b5 + 2*b2 - 2*b1) * q^45 + (3*b5 + b4 + b3) * q^46 + (-2*b3 + b2 + 5) * q^47 + (-b2 + b1) * q^48 + (-2*b5 - b4 - b1 + 2) * q^49 + (-2*b5 - 4*b4 - b1 + 2) * q^50 + (2*b3 + b2) * q^51 + (3*b3 + 4*b2) * q^53 + (3*b5 + 4*b4 + 2*b1 - 3) * q^54 + (2*b5 + b4 - b1 - 2) * q^55 + (b5 + 2*b4 + 2*b3 + 3*b2 - 3*b1) * q^56 + (-b3 - 2) * q^57 + (-b5 - 4*b4 - 4*b3 + 2*b2 - 2*b1) * q^58 + (-9*b5 - 4*b4 - 4*b3 + 4*b2 - 4*b1) * q^59 + (-b2 - 2) * q^60 + (-5*b5 - 5*b4 - 5*b3 + 6*b2 - 6*b1) * q^61 + (3*b5 + 6*b4 + 2*b1 - 3) * q^62 + (4*b5 + 3*b4 + 5*b1 - 4) * q^63 + (-6*b3 + b2 - 6) * q^64 + (-3*b3 - b2 + 1) * q^66 + (-2*b5 - b4 - 6*b1 + 2) * q^67 + (-10*b5 - 6*b4 - 3*b1 + 10) * q^68 + (b5 - 2*b4 - 2*b3 + b2 - b1) * q^69 + (-4*b3 + 3*b2 - 3) * q^70 + (-7*b5 + 3*b4 + 3*b3 - 3*b2 + 3*b1) * q^71 + (b5 + 5*b4 + 5*b3 - 2*b2 + 2*b1) * q^72 + (9*b3 - 6*b2 + 2) * q^73 + (5*b5 + 6*b4 + 6*b3 - b2 + b1) * q^74 + (5*b5 - 2*b4 + 2*b1 - 5) * q^75 + (10*b5 + 7*b4 + 2*b1 - 10) * q^76 + (-5*b3 + 1) * q^77 + (-9*b3 + b2 - 5) * q^79 + (-2*b4 - 3*b1) * q^80 + (4*b4 - 3*b1) * q^81 + (5*b5 - b4 - b3 - 2*b2 + 2*b1) * q^82 + (-7*b3 + 9*b2 - 3) * q^83 + (-4*b5 - b4 - b3 + 4*b2 - 4*b1) * q^84 + (4*b5 + 4*b4 + 4*b3 - 3*b2 + 3*b1) * q^85 + (5*b3 - b2 + 8) * q^86 + (5*b5 - 3*b4 - 3*b3) * q^87 + (7*b5 + b4 + 5*b1 - 7) * q^88 + (-b5 - 7*b4 - 7*b1 + 1) * q^89 + (-3*b3 - 5) * q^90 + (-4*b3 + 5*b2 - 3) * q^92 + (-7*b5 + 3*b4 - 2*b1 + 7) * q^93 + (2*b5 + 3*b4 - 2*b1 - 2) * q^94 + (-4*b5 - 5*b4 - 5*b3 + 4*b2 - 4*b1) * q^95 + (-2*b3 - 2*b2 + 7) * q^96 + (4*b5 + 6*b4 + 6*b3 + b2 - b1) * q^97 + (3*b5 + 3*b4 + 3*b3 - b2 + b1) * q^98 + (-4*b3 - 2*b2 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{2} + 2 q^{3} + 8 q^{5} - q^{6} - 3 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10})$$ 6 * q - 2 * q^2 + 2 * q^3 + 8 * q^5 - q^6 - 3 * q^7 + 6 * q^8 + 3 * q^9 $$6 q - 2 q^{2} + 2 q^{3} + 8 q^{5} - q^{6} - 3 q^{7} + 6 q^{8} + 3 q^{9} - 5 q^{10} - 8 q^{11} - 10 q^{14} - 2 q^{15} - 2 q^{16} + 2 q^{17} - 18 q^{18} - 4 q^{19} - 4 q^{21} - 3 q^{22} + 5 q^{23} + 9 q^{24} - 10 q^{25} + 2 q^{27} + 14 q^{28} + q^{29} + q^{30} + 10 q^{31} + 7 q^{32} + 10 q^{33} - 26 q^{34} - 4 q^{35} + 7 q^{36} + 12 q^{37} + 24 q^{38} - 6 q^{40} - 7 q^{41} - 8 q^{42} - 13 q^{43} + 11 q^{45} + 8 q^{46} + 36 q^{47} - q^{48} + 4 q^{49} + q^{50} - 2 q^{51} + 2 q^{53} - 3 q^{54} - 6 q^{55} + 4 q^{56} - 10 q^{57} + 3 q^{58} - 19 q^{59} - 14 q^{60} - 4 q^{61} - q^{62} - 4 q^{63} - 22 q^{64} + 10 q^{66} - q^{67} + 21 q^{68} + 6 q^{69} - 4 q^{70} - 27 q^{71} - 4 q^{72} - 18 q^{73} + 8 q^{74} - 15 q^{75} - 21 q^{76} + 16 q^{77} - 10 q^{79} - 5 q^{80} + q^{81} + 14 q^{82} + 14 q^{83} - 7 q^{84} + 5 q^{85} + 36 q^{86} + 18 q^{87} - 15 q^{88} - 11 q^{89} - 24 q^{90} + 22 q^{93} - 5 q^{94} - 3 q^{95} + 42 q^{96} + 7 q^{97} + 5 q^{98} - 2 q^{99}+O(q^{100})$$ 6 * q - 2 * q^2 + 2 * q^3 + 8 * q^5 - q^6 - 3 * q^7 + 6 * q^8 + 3 * q^9 - 5 * q^10 - 8 * q^11 - 10 * q^14 - 2 * q^15 - 2 * q^16 + 2 * q^17 - 18 * q^18 - 4 * q^19 - 4 * q^21 - 3 * q^22 + 5 * q^23 + 9 * q^24 - 10 * q^25 + 2 * q^27 + 14 * q^28 + q^29 + q^30 + 10 * q^31 + 7 * q^32 + 10 * q^33 - 26 * q^34 - 4 * q^35 + 7 * q^36 + 12 * q^37 + 24 * q^38 - 6 * q^40 - 7 * q^41 - 8 * q^42 - 13 * q^43 + 11 * q^45 + 8 * q^46 + 36 * q^47 - q^48 + 4 * q^49 + q^50 - 2 * q^51 + 2 * q^53 - 3 * q^54 - 6 * q^55 + 4 * q^56 - 10 * q^57 + 3 * q^58 - 19 * q^59 - 14 * q^60 - 4 * q^61 - q^62 - 4 * q^63 - 22 * q^64 + 10 * q^66 - q^67 + 21 * q^68 + 6 * q^69 - 4 * q^70 - 27 * q^71 - 4 * q^72 - 18 * q^73 + 8 * q^74 - 15 * q^75 - 21 * q^76 + 16 * q^77 - 10 * q^79 - 5 * q^80 + q^81 + 14 * q^82 + 14 * q^83 - 7 * q^84 + 5 * q^85 + 36 * q^86 + 18 * q^87 - 15 * q^88 - 11 * q^89 - 24 * q^90 + 22 * q^93 - 5 * q^94 - 3 * q^95 + 42 * q^96 + 7 * q^97 + 5 * q^98 - 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13$$ (-v^5 + 3*v^4 - 9*v^3 + 5*v^2 - 2*v + 6) / 13 $$\beta_{3}$$ $$=$$ $$( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13$$ (-3*v^5 + 9*v^4 - 14*v^3 + 15*v^2 - 6*v + 18) / 13 $$\beta_{4}$$ $$=$$ $$( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13$$ (-4*v^5 - v^4 - 10*v^3 - 6*v^2 - 34*v - 2) / 13 $$\beta_{5}$$ $$=$$ $$( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13$$ (-6*v^5 + 5*v^4 - 15*v^3 - 9*v^2 - 25*v + 10) / 13
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1$$ -b5 + b4 + b3 - b2 + b1 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3\beta_{2}$$ b3 - 3*b2 $$\nu^{4}$$ $$=$$ $$2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2$$ 2*b5 - 3*b4 - 4*b1 - 2 $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1$$ b5 - 4*b4 - 4*b3 + 9*b2 - 9*b1

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-\beta_{5}$$

## 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
 0.222521 − 0.385418i 0.900969 − 1.56052i −0.623490 + 1.07992i 0.222521 + 0.385418i 0.900969 + 1.56052i −0.623490 − 1.07992i
−1.12349 + 1.94594i 0.277479 0.480608i −1.52446 2.64044i 1.44504 0.623490 + 1.07992i 1.02446 + 1.77441i 2.35690 1.34601 + 2.33136i −1.62349 + 2.81197i
22.2 −0.277479 + 0.480608i −0.400969 + 0.694498i 0.846011 + 1.46533i 2.80194 −0.222521 0.385418i −1.34601 2.33136i −2.04892 1.17845 + 2.04113i −0.777479 + 1.34663i
22.3 0.400969 0.694498i 1.12349 1.94594i 0.678448 + 1.17511i −0.246980 −0.900969 1.56052i −1.17845 2.04113i 2.69202 −1.02446 1.77441i −0.0990311 + 0.171527i
146.1 −1.12349 1.94594i 0.277479 + 0.480608i −1.52446 + 2.64044i 1.44504 0.623490 1.07992i 1.02446 1.77441i 2.35690 1.34601 2.33136i −1.62349 2.81197i
146.2 −0.277479 0.480608i −0.400969 0.694498i 0.846011 1.46533i 2.80194 −0.222521 + 0.385418i −1.34601 + 2.33136i −2.04892 1.17845 2.04113i −0.777479 1.34663i
146.3 0.400969 + 0.694498i 1.12349 + 1.94594i 0.678448 1.17511i −0.246980 −0.900969 + 1.56052i −1.17845 + 2.04113i 2.69202 −1.02446 + 1.77441i −0.0990311 0.171527i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 22.3 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.c even 3 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 2T_{2}^{5} + 5T_{2}^{4} + 3T_{2}^{2} + T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 2 T^{5} + \cdots + 1$$
$3$ $$T^{6} - 2 T^{5} + \cdots + 1$$
$5$ $$(T^{3} - 4 T^{2} + 3 T + 1)^{2}$$
$7$ $$T^{6} + 3 T^{5} + \cdots + 169$$
$11$ $$T^{6} + 8 T^{5} + \cdots + 169$$
$13$ $$T^{6}$$
$17$ $$T^{6} - 2 T^{5} + \cdots + 169$$
$19$ $$T^{6} + 4 T^{5} + \cdots + 1$$
$23$ $$T^{6} - 5 T^{5} + \cdots + 169$$
$29$ $$T^{6} - T^{5} + \cdots + 6889$$
$31$ $$(T^{3} - 5 T^{2} + \cdots + 167)^{2}$$
$37$ $$T^{6} - 12 T^{5} + \cdots + 841$$
$41$ $$T^{6} + 7 T^{5} + \cdots + 2401$$
$43$ $$T^{6} + 13 T^{5} + \cdots + 169$$
$47$ $$(T^{3} - 18 T^{2} + \cdots - 167)^{2}$$
$53$ $$(T^{3} - T^{2} - 86 T + 337)^{2}$$
$59$ $$T^{6} + 19 T^{5} + \cdots + 1$$
$61$ $$T^{6} + 4 T^{5} + \cdots + 57121$$
$67$ $$T^{6} + T^{5} + \cdots + 1681$$
$71$ $$T^{6} + 27 T^{5} + \cdots + 299209$$
$73$ $$(T^{3} + 9 T^{2} + \cdots - 911)^{2}$$
$79$ $$(T^{3} + 5 T^{2} + \cdots + 127)^{2}$$
$83$ $$(T^{3} - 7 T^{2} + \cdots - 203)^{2}$$
$89$ $$T^{6} + 11 T^{5} + \cdots + 78961$$
$97$ $$T^{6} - 7 T^{5} + \cdots + 90601$$