# Properties

 Label 169.2.b.a Level $169$ Weight $2$ Character orbit 169.b Analytic conductor $1.349$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
1.73205i 2.00000 −1.00000 1.73205i 3.46410i 0 1.73205i 1.00000 3.00000
168.2 1.73205i 2.00000 −1.00000 1.73205i 3.46410i 0 1.73205i 1.00000 3.00000
 $$n$$: 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

## Twists

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

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

## Hecke kernels

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

## Hecke characteristic polynomials

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