# Properties

 Label 169.4.b.a Level $169$ Weight $4$ Character orbit 169.b Analytic conductor $9.971$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.97132279097$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 5 i q^{2} -7 q^{3} -17 q^{4} + 7 i q^{5} -35 i q^{6} -13 i q^{7} -45 i q^{8} + 22 q^{9} +O(q^{10})$$ $$q + 5 i q^{2} -7 q^{3} -17 q^{4} + 7 i q^{5} -35 i q^{6} -13 i q^{7} -45 i q^{8} + 22 q^{9} -35 q^{10} -26 i q^{11} + 119 q^{12} + 65 q^{14} -49 i q^{15} + 89 q^{16} -77 q^{17} + 110 i q^{18} + 126 i q^{19} -119 i q^{20} + 91 i q^{21} + 130 q^{22} + 96 q^{23} + 315 i q^{24} + 76 q^{25} + 35 q^{27} + 221 i q^{28} -82 q^{29} + 245 q^{30} -196 i q^{31} + 85 i q^{32} + 182 i q^{33} -385 i q^{34} + 91 q^{35} -374 q^{36} -131 i q^{37} -630 q^{38} + 315 q^{40} -336 i q^{41} -455 q^{42} + 201 q^{43} + 442 i q^{44} + 154 i q^{45} + 480 i q^{46} -105 i q^{47} -623 q^{48} + 174 q^{49} + 380 i q^{50} + 539 q^{51} -432 q^{53} + 175 i q^{54} + 182 q^{55} -585 q^{56} -882 i q^{57} -410 i q^{58} -294 i q^{59} + 833 i q^{60} -56 q^{61} + 980 q^{62} -286 i q^{63} + 287 q^{64} -910 q^{66} -478 i q^{67} + 1309 q^{68} -672 q^{69} + 455 i q^{70} -9 i q^{71} -990 i q^{72} + 98 i q^{73} + 655 q^{74} -532 q^{75} -2142 i q^{76} -338 q^{77} + 1304 q^{79} + 623 i q^{80} -839 q^{81} + 1680 q^{82} + 308 i q^{83} -1547 i q^{84} -539 i q^{85} + 1005 i q^{86} + 574 q^{87} -1170 q^{88} -1190 i q^{89} -770 q^{90} -1632 q^{92} + 1372 i q^{93} + 525 q^{94} -882 q^{95} -595 i q^{96} -70 i q^{97} + 870 i q^{98} -572 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 14 q^{3} - 34 q^{4} + 44 q^{9} + O(q^{10})$$ $$2 q - 14 q^{3} - 34 q^{4} + 44 q^{9} - 70 q^{10} + 238 q^{12} + 130 q^{14} + 178 q^{16} - 154 q^{17} + 260 q^{22} + 192 q^{23} + 152 q^{25} + 70 q^{27} - 164 q^{29} + 490 q^{30} + 182 q^{35} - 748 q^{36} - 1260 q^{38} + 630 q^{40} - 910 q^{42} + 402 q^{43} - 1246 q^{48} + 348 q^{49} + 1078 q^{51} - 864 q^{53} + 364 q^{55} - 1170 q^{56} - 112 q^{61} + 1960 q^{62} + 574 q^{64} - 1820 q^{66} + 2618 q^{68} - 1344 q^{69} + 1310 q^{74} - 1064 q^{75} - 676 q^{77} + 2608 q^{79} - 1678 q^{81} + 3360 q^{82} + 1148 q^{87} - 2340 q^{88} - 1540 q^{90} - 3264 q^{92} + 1050 q^{94} - 1764 q^{95} + O(q^{100})$$

## 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
 − 1.00000i 1.00000i
5.00000i −7.00000 −17.0000 7.00000i 35.0000i 13.0000i 45.0000i 22.0000 −35.0000
168.2 5.00000i −7.00000 −17.0000 7.00000i 35.0000i 13.0000i 45.0000i 22.0000 −35.0000
 $$n$$: 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 169.4.b.a 2
13.b even 2 1 inner 169.4.b.a 2
13.c even 3 2 169.4.e.e 4
13.d odd 4 1 13.4.a.a 1
13.d odd 4 1 169.4.a.e 1
13.e even 6 2 169.4.e.e 4
13.f odd 12 2 169.4.c.a 2
13.f odd 12 2 169.4.c.e 2
39.f even 4 1 117.4.a.b 1
39.f even 4 1 1521.4.a.a 1
52.f even 4 1 208.4.a.g 1
65.f even 4 1 325.4.b.b 2
65.g odd 4 1 325.4.a.d 1
65.k even 4 1 325.4.b.b 2
91.i even 4 1 637.4.a.a 1
104.j odd 4 1 832.4.a.r 1
104.m even 4 1 832.4.a.a 1
143.g even 4 1 1573.4.a.a 1
156.l odd 4 1 1872.4.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.a 1 13.d odd 4 1
117.4.a.b 1 39.f even 4 1
169.4.a.e 1 13.d odd 4 1
169.4.b.a 2 1.a even 1 1 trivial
169.4.b.a 2 13.b even 2 1 inner
169.4.c.a 2 13.f odd 12 2
169.4.c.e 2 13.f odd 12 2
169.4.e.e 4 13.c even 3 2
169.4.e.e 4 13.e even 6 2
208.4.a.g 1 52.f even 4 1
325.4.a.d 1 65.g odd 4 1
325.4.b.b 2 65.f even 4 1
325.4.b.b 2 65.k even 4 1
637.4.a.a 1 91.i even 4 1
832.4.a.a 1 104.m even 4 1
832.4.a.r 1 104.j odd 4 1
1521.4.a.a 1 39.f even 4 1
1573.4.a.a 1 143.g even 4 1
1872.4.a.k 1 156.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 25$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$25 + T^{2}$$
$3$ $$( 7 + T )^{2}$$
$5$ $$49 + T^{2}$$
$7$ $$169 + T^{2}$$
$11$ $$676 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( 77 + T )^{2}$$
$19$ $$15876 + T^{2}$$
$23$ $$( -96 + T )^{2}$$
$29$ $$( 82 + T )^{2}$$
$31$ $$38416 + T^{2}$$
$37$ $$17161 + T^{2}$$
$41$ $$112896 + T^{2}$$
$43$ $$( -201 + T )^{2}$$
$47$ $$11025 + T^{2}$$
$53$ $$( 432 + T )^{2}$$
$59$ $$86436 + T^{2}$$
$61$ $$( 56 + T )^{2}$$
$67$ $$228484 + T^{2}$$
$71$ $$81 + T^{2}$$
$73$ $$9604 + T^{2}$$
$79$ $$( -1304 + T )^{2}$$
$83$ $$94864 + T^{2}$$
$89$ $$1416100 + T^{2}$$
$97$ $$4900 + T^{2}$$