# Properties

 Label 169.2.e.a Level $169$ Weight $2$ Character orbit 169.e 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.e (of order $$6$$, degree $$2$$, 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: $$1$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

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

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.50000 0.866025i −1.00000 1.73205i 0.500000 0.866025i 1.73205i −3.00000 1.73205i 0 1.73205i −0.500000 + 0.866025i −1.50000 2.59808i
147.1 1.50000 + 0.866025i −1.00000 + 1.73205i 0.500000 + 0.866025i 1.73205i −3.00000 + 1.73205i 0 1.73205i −0.500000 0.866025i −1.50000 + 2.59808i
 $$n$$: 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.e even 6 1 inner

## Twists

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 - 3 T + T^{2}$$
$3$ $$4 + 2 T + T^{2}$$
$5$ $$3 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$9 - 3 T + T^{2}$$
$19$ $$12 - 6 T + T^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$9 + 3 T + T^{2}$$
$31$ $$12 + T^{2}$$
$37$ $$75 + 15 T + T^{2}$$
$41$ $$27 - 9 T + T^{2}$$
$43$ $$64 + 8 T + T^{2}$$
$47$ $$12 + T^{2}$$
$53$ $$( 3 + T )^{2}$$
$59$ $$48 + 12 T + T^{2}$$
$61$ $$1 + T + T^{2}$$
$67$ $$12 + 6 T + T^{2}$$
$71$ $$12 + 6 T + T^{2}$$
$73$ $$3 + T^{2}$$
$79$ $$( -4 + T )^{2}$$
$83$ $$192 + T^{2}$$
$89$ $$48 - 12 T + T^{2}$$
$97$ $$48 + 12 T + T^{2}$$