# Properties

 Label 169.2.b.b Level $169$ Weight $2$ Character orbit 169.b Analytic conductor $1.349$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 169.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.34947179416$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.153664.1 Defining polynomial: $$x^{6} + 5 x^{4} + 6 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 3 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} + 3 \nu^{2} + 1$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} + 4 \nu^{3} + 3 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} - 3 \beta_{2} + 5$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4 \beta_{3} + 9 \beta_{1}$$

## 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.

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 $$T_{2}^{6} + 6 T_{2}^{4} + 5 T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 5 T^{2} + 6 T^{4} + T^{6}$$
$3$ $$( -1 - T + 2 T^{2} + T^{3} )^{2}$$
$5$ $$1 + 17 T^{2} + 10 T^{4} + T^{6}$$
$7$ $$169 + 94 T^{2} + 17 T^{4} + T^{6}$$
$11$ $$169 + 153 T^{2} + 26 T^{4} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$( -13 - 15 T - 2 T^{2} + T^{3} )^{2}$$
$19$ $$1 + 129 T^{2} + 38 T^{4} + T^{6}$$
$23$ $$( 13 - T - 5 T^{2} + T^{3} )^{2}$$
$29$ $$( 83 - 44 T + T^{2} + T^{3} )^{2}$$
$31$ $$27889 + 2966 T^{2} + 97 T^{4} + T^{6}$$
$37$ $$841 + 985 T^{2} + 62 T^{4} + T^{6}$$
$41$ $$2401 + 1715 T^{2} + 147 T^{4} + T^{6}$$
$43$ $$( -13 + 40 T + 13 T^{2} + T^{3} )^{2}$$
$47$ $$27889 + 4189 T^{2} + 122 T^{4} + T^{6}$$
$53$ $$( 337 - 86 T - T^{2} + T^{3} )^{2}$$
$59$ $$1 + 6851 T^{2} + 195 T^{4} + T^{6}$$
$61$ $$( 239 - 67 T - 4 T^{2} + T^{3} )^{2}$$
$67$ $$1681 + 5102 T^{2} + 145 T^{4} + T^{6}$$
$71$ $$299209 + 19746 T^{2} + 285 T^{4} + T^{6}$$
$73$ $$829921 + 30798 T^{2} + 321 T^{4} + T^{6}$$
$79$ $$( 127 - 162 T + 5 T^{2} + T^{3} )^{2}$$
$83$ $$41209 + 16758 T^{2} + 329 T^{4} + T^{6}$$
$89$ $$78961 + 11658 T^{2} + 269 T^{4} + T^{6}$$
$97$ $$90601 + 11270 T^{2} + 217 T^{4} + T^{6}$$