# Properties

 Label 169.4.c.a Level $169$ Weight $4$ Character orbit 169.c 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.c (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.97132279097$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -5 + 5 \zeta_{6} ) q^{2} + ( 7 - 7 \zeta_{6} ) q^{3} -17 \zeta_{6} q^{4} + 7 q^{5} + 35 \zeta_{6} q^{6} -13 \zeta_{6} q^{7} + 45 q^{8} -22 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -5 + 5 \zeta_{6} ) q^{2} + ( 7 - 7 \zeta_{6} ) q^{3} -17 \zeta_{6} q^{4} + 7 q^{5} + 35 \zeta_{6} q^{6} -13 \zeta_{6} q^{7} + 45 q^{8} -22 \zeta_{6} q^{9} + ( -35 + 35 \zeta_{6} ) q^{10} + ( -26 + 26 \zeta_{6} ) q^{11} -119 q^{12} + 65 q^{14} + ( 49 - 49 \zeta_{6} ) q^{15} + ( -89 + 89 \zeta_{6} ) q^{16} -77 \zeta_{6} q^{17} + 110 q^{18} -126 \zeta_{6} q^{19} -119 \zeta_{6} q^{20} -91 q^{21} -130 \zeta_{6} q^{22} + ( 96 - 96 \zeta_{6} ) q^{23} + ( 315 - 315 \zeta_{6} ) q^{24} -76 q^{25} + 35 q^{27} + ( -221 + 221 \zeta_{6} ) q^{28} + ( 82 - 82 \zeta_{6} ) q^{29} + 245 \zeta_{6} q^{30} -196 q^{31} -85 \zeta_{6} q^{32} + 182 \zeta_{6} q^{33} + 385 q^{34} -91 \zeta_{6} q^{35} + ( -374 + 374 \zeta_{6} ) q^{36} + ( -131 + 131 \zeta_{6} ) q^{37} + 630 q^{38} + 315 q^{40} + ( 336 - 336 \zeta_{6} ) q^{41} + ( 455 - 455 \zeta_{6} ) q^{42} + 201 \zeta_{6} q^{43} + 442 q^{44} -154 \zeta_{6} q^{45} + 480 \zeta_{6} q^{46} + 105 q^{47} + 623 \zeta_{6} q^{48} + ( 174 - 174 \zeta_{6} ) q^{49} + ( 380 - 380 \zeta_{6} ) q^{50} -539 q^{51} -432 q^{53} + ( -175 + 175 \zeta_{6} ) q^{54} + ( -182 + 182 \zeta_{6} ) q^{55} -585 \zeta_{6} q^{56} -882 q^{57} + 410 \zeta_{6} q^{58} -294 \zeta_{6} q^{59} -833 q^{60} + 56 \zeta_{6} q^{61} + ( 980 - 980 \zeta_{6} ) q^{62} + ( -286 + 286 \zeta_{6} ) q^{63} -287 q^{64} -910 q^{66} + ( 478 - 478 \zeta_{6} ) q^{67} + ( -1309 + 1309 \zeta_{6} ) q^{68} -672 \zeta_{6} q^{69} + 455 q^{70} + 9 \zeta_{6} q^{71} -990 \zeta_{6} q^{72} -98 q^{73} -655 \zeta_{6} q^{74} + ( -532 + 532 \zeta_{6} ) q^{75} + ( -2142 + 2142 \zeta_{6} ) q^{76} + 338 q^{77} + 1304 q^{79} + ( -623 + 623 \zeta_{6} ) q^{80} + ( 839 - 839 \zeta_{6} ) q^{81} + 1680 \zeta_{6} q^{82} + 308 q^{83} + 1547 \zeta_{6} q^{84} -539 \zeta_{6} q^{85} -1005 q^{86} -574 \zeta_{6} q^{87} + ( -1170 + 1170 \zeta_{6} ) q^{88} + ( -1190 + 1190 \zeta_{6} ) q^{89} + 770 q^{90} -1632 q^{92} + ( -1372 + 1372 \zeta_{6} ) q^{93} + ( -525 + 525 \zeta_{6} ) q^{94} -882 \zeta_{6} q^{95} -595 q^{96} + 70 \zeta_{6} q^{97} + 870 \zeta_{6} q^{98} + 572 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 5 q^{2} + 7 q^{3} - 17 q^{4} + 14 q^{5} + 35 q^{6} - 13 q^{7} + 90 q^{8} - 22 q^{9} + O(q^{10})$$ $$2 q - 5 q^{2} + 7 q^{3} - 17 q^{4} + 14 q^{5} + 35 q^{6} - 13 q^{7} + 90 q^{8} - 22 q^{9} - 35 q^{10} - 26 q^{11} - 238 q^{12} + 130 q^{14} + 49 q^{15} - 89 q^{16} - 77 q^{17} + 220 q^{18} - 126 q^{19} - 119 q^{20} - 182 q^{21} - 130 q^{22} + 96 q^{23} + 315 q^{24} - 152 q^{25} + 70 q^{27} - 221 q^{28} + 82 q^{29} + 245 q^{30} - 392 q^{31} - 85 q^{32} + 182 q^{33} + 770 q^{34} - 91 q^{35} - 374 q^{36} - 131 q^{37} + 1260 q^{38} + 630 q^{40} + 336 q^{41} + 455 q^{42} + 201 q^{43} + 884 q^{44} - 154 q^{45} + 480 q^{46} + 210 q^{47} + 623 q^{48} + 174 q^{49} + 380 q^{50} - 1078 q^{51} - 864 q^{53} - 175 q^{54} - 182 q^{55} - 585 q^{56} - 1764 q^{57} + 410 q^{58} - 294 q^{59} - 1666 q^{60} + 56 q^{61} + 980 q^{62} - 286 q^{63} - 574 q^{64} - 1820 q^{66} + 478 q^{67} - 1309 q^{68} - 672 q^{69} + 910 q^{70} + 9 q^{71} - 990 q^{72} - 196 q^{73} - 655 q^{74} - 532 q^{75} - 2142 q^{76} + 676 q^{77} + 2608 q^{79} - 623 q^{80} + 839 q^{81} + 1680 q^{82} + 616 q^{83} + 1547 q^{84} - 539 q^{85} - 2010 q^{86} - 574 q^{87} - 1170 q^{88} - 1190 q^{89} + 1540 q^{90} - 3264 q^{92} - 1372 q^{93} - 525 q^{94} - 882 q^{95} - 1190 q^{96} + 70 q^{97} + 870 q^{98} + 1144 q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1
 0.5 + 0.866025i 0.5 − 0.866025i
−2.50000 + 4.33013i 3.50000 6.06218i −8.50000 14.7224i 7.00000 17.5000 + 30.3109i −6.50000 11.2583i 45.0000 −11.0000 19.0526i −17.5000 + 30.3109i
146.1 −2.50000 4.33013i 3.50000 + 6.06218i −8.50000 + 14.7224i 7.00000 17.5000 30.3109i −6.50000 + 11.2583i 45.0000 −11.0000 + 19.0526i −17.5000 30.3109i
 $$n$$: 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.c even 3 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 5 T_{2} + 25$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$25 + 5 T + T^{2}$$
$3$ $$49 - 7 T + T^{2}$$
$5$ $$( -7 + T )^{2}$$
$7$ $$169 + 13 T + T^{2}$$
$11$ $$676 + 26 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$5929 + 77 T + T^{2}$$
$19$ $$15876 + 126 T + T^{2}$$
$23$ $$9216 - 96 T + T^{2}$$
$29$ $$6724 - 82 T + T^{2}$$
$31$ $$( 196 + T )^{2}$$
$37$ $$17161 + 131 T + T^{2}$$
$41$ $$112896 - 336 T + T^{2}$$
$43$ $$40401 - 201 T + T^{2}$$
$47$ $$( -105 + T )^{2}$$
$53$ $$( 432 + T )^{2}$$
$59$ $$86436 + 294 T + T^{2}$$
$61$ $$3136 - 56 T + T^{2}$$
$67$ $$228484 - 478 T + T^{2}$$
$71$ $$81 - 9 T + T^{2}$$
$73$ $$( 98 + T )^{2}$$
$79$ $$( -1304 + T )^{2}$$
$83$ $$( -308 + T )^{2}$$
$89$ $$1416100 + 1190 T + T^{2}$$
$97$ $$4900 - 70 T + T^{2}$$