# Properties

 Label 169.4.c.b 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.

 $$f(q)$$ $$=$$ $$q + ( -3 + 3 \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{4} -9 q^{5} + 3 \zeta_{6} q^{6} -15 \zeta_{6} q^{7} -21 q^{8} + 26 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -3 + 3 \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{4} -9 q^{5} + 3 \zeta_{6} q^{6} -15 \zeta_{6} q^{7} -21 q^{8} + 26 \zeta_{6} q^{9} + ( 27 - 27 \zeta_{6} ) q^{10} + ( 48 - 48 \zeta_{6} ) q^{11} - q^{12} + 45 q^{14} + ( -9 + 9 \zeta_{6} ) q^{15} + ( 71 - 71 \zeta_{6} ) q^{16} -45 \zeta_{6} q^{17} -78 q^{18} -6 \zeta_{6} q^{19} + 9 \zeta_{6} q^{20} -15 q^{21} + 144 \zeta_{6} q^{22} + ( 162 - 162 \zeta_{6} ) q^{23} + ( -21 + 21 \zeta_{6} ) q^{24} -44 q^{25} + 53 q^{27} + ( -15 + 15 \zeta_{6} ) q^{28} + ( 144 - 144 \zeta_{6} ) q^{29} -27 \zeta_{6} q^{30} + 264 q^{31} + 45 \zeta_{6} q^{32} -48 \zeta_{6} q^{33} + 135 q^{34} + 135 \zeta_{6} q^{35} + ( 26 - 26 \zeta_{6} ) q^{36} + ( -303 + 303 \zeta_{6} ) q^{37} + 18 q^{38} + 189 q^{40} + ( 192 - 192 \zeta_{6} ) q^{41} + ( 45 - 45 \zeta_{6} ) q^{42} -97 \zeta_{6} q^{43} -48 q^{44} -234 \zeta_{6} q^{45} + 486 \zeta_{6} q^{46} + 111 q^{47} -71 \zeta_{6} q^{48} + ( 118 - 118 \zeta_{6} ) q^{49} + ( 132 - 132 \zeta_{6} ) q^{50} -45 q^{51} -414 q^{53} + ( -159 + 159 \zeta_{6} ) q^{54} + ( -432 + 432 \zeta_{6} ) q^{55} + 315 \zeta_{6} q^{56} -6 q^{57} + 432 \zeta_{6} q^{58} -522 \zeta_{6} q^{59} + 9 q^{60} -376 \zeta_{6} q^{61} + ( -792 + 792 \zeta_{6} ) q^{62} + ( 390 - 390 \zeta_{6} ) q^{63} + 433 q^{64} + 144 q^{66} + ( 36 - 36 \zeta_{6} ) q^{67} + ( -45 + 45 \zeta_{6} ) q^{68} -162 \zeta_{6} q^{69} -405 q^{70} -357 \zeta_{6} q^{71} -546 \zeta_{6} q^{72} -1098 q^{73} -909 \zeta_{6} q^{74} + ( -44 + 44 \zeta_{6} ) q^{75} + ( -6 + 6 \zeta_{6} ) q^{76} -720 q^{77} -830 q^{79} + ( -639 + 639 \zeta_{6} ) q^{80} + ( -649 + 649 \zeta_{6} ) q^{81} + 576 \zeta_{6} q^{82} -438 q^{83} + 15 \zeta_{6} q^{84} + 405 \zeta_{6} q^{85} + 291 q^{86} -144 \zeta_{6} q^{87} + ( -1008 + 1008 \zeta_{6} ) q^{88} + ( 438 - 438 \zeta_{6} ) q^{89} + 702 q^{90} -162 q^{92} + ( 264 - 264 \zeta_{6} ) q^{93} + ( -333 + 333 \zeta_{6} ) q^{94} + 54 \zeta_{6} q^{95} + 45 q^{96} + 852 \zeta_{6} q^{97} + 354 \zeta_{6} q^{98} + 1248 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + q^{3} - q^{4} - 18 q^{5} + 3 q^{6} - 15 q^{7} - 42 q^{8} + 26 q^{9} + O(q^{10})$$ $$2 q - 3 q^{2} + q^{3} - q^{4} - 18 q^{5} + 3 q^{6} - 15 q^{7} - 42 q^{8} + 26 q^{9} + 27 q^{10} + 48 q^{11} - 2 q^{12} + 90 q^{14} - 9 q^{15} + 71 q^{16} - 45 q^{17} - 156 q^{18} - 6 q^{19} + 9 q^{20} - 30 q^{21} + 144 q^{22} + 162 q^{23} - 21 q^{24} - 88 q^{25} + 106 q^{27} - 15 q^{28} + 144 q^{29} - 27 q^{30} + 528 q^{31} + 45 q^{32} - 48 q^{33} + 270 q^{34} + 135 q^{35} + 26 q^{36} - 303 q^{37} + 36 q^{38} + 378 q^{40} + 192 q^{41} + 45 q^{42} - 97 q^{43} - 96 q^{44} - 234 q^{45} + 486 q^{46} + 222 q^{47} - 71 q^{48} + 118 q^{49} + 132 q^{50} - 90 q^{51} - 828 q^{53} - 159 q^{54} - 432 q^{55} + 315 q^{56} - 12 q^{57} + 432 q^{58} - 522 q^{59} + 18 q^{60} - 376 q^{61} - 792 q^{62} + 390 q^{63} + 866 q^{64} + 288 q^{66} + 36 q^{67} - 45 q^{68} - 162 q^{69} - 810 q^{70} - 357 q^{71} - 546 q^{72} - 2196 q^{73} - 909 q^{74} - 44 q^{75} - 6 q^{76} - 1440 q^{77} - 1660 q^{79} - 639 q^{80} - 649 q^{81} + 576 q^{82} - 876 q^{83} + 15 q^{84} + 405 q^{85} + 582 q^{86} - 144 q^{87} - 1008 q^{88} + 438 q^{89} + 1404 q^{90} - 324 q^{92} + 264 q^{93} - 333 q^{94} + 54 q^{95} + 90 q^{96} + 852 q^{97} + 354 q^{98} + 2496 q^{99} + O(q^{100})$$

## 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
−1.50000 + 2.59808i 0.500000 0.866025i −0.500000 0.866025i −9.00000 1.50000 + 2.59808i −7.50000 12.9904i −21.0000 13.0000 + 22.5167i 13.5000 23.3827i
146.1 −1.50000 2.59808i 0.500000 + 0.866025i −0.500000 + 0.866025i −9.00000 1.50000 2.59808i −7.50000 + 12.9904i −21.0000 13.0000 22.5167i 13.5000 + 23.3827i
 $$n$$: 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.b 2
13.b even 2 1 169.4.c.c 2
13.c even 3 1 169.4.a.c 1
13.c even 3 1 inner 169.4.c.b 2
13.d odd 4 2 169.4.e.d 4
13.e even 6 1 169.4.a.b 1
13.e even 6 1 169.4.c.c 2
13.f odd 12 2 13.4.b.a 2
13.f odd 12 2 169.4.e.d 4
39.h odd 6 1 1521.4.a.i 1
39.i odd 6 1 1521.4.a.d 1
39.k even 12 2 117.4.b.a 2
52.l even 12 2 208.4.f.b 2
65.o even 12 2 325.4.d.b 2
65.s odd 12 2 325.4.c.b 2
65.t even 12 2 325.4.d.a 2
104.u even 12 2 832.4.f.c 2
104.x odd 12 2 832.4.f.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.b.a 2 13.f odd 12 2
117.4.b.a 2 39.k even 12 2
169.4.a.b 1 13.e even 6 1
169.4.a.c 1 13.c even 3 1
169.4.c.b 2 1.a even 1 1 trivial
169.4.c.b 2 13.c even 3 1 inner
169.4.c.c 2 13.b even 2 1
169.4.c.c 2 13.e even 6 1
169.4.e.d 4 13.d odd 4 2
169.4.e.d 4 13.f odd 12 2
208.4.f.b 2 52.l even 12 2
325.4.c.b 2 65.s odd 12 2
325.4.d.a 2 65.t even 12 2
325.4.d.b 2 65.o even 12 2
832.4.f.c 2 104.u even 12 2
832.4.f.e 2 104.x odd 12 2
1521.4.a.d 1 39.i odd 6 1
1521.4.a.i 1 39.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3 T_{2} + 9$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 + 3 T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$( 9 + T )^{2}$$
$7$ $$225 + 15 T + T^{2}$$
$11$ $$2304 - 48 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$2025 + 45 T + T^{2}$$
$19$ $$36 + 6 T + T^{2}$$
$23$ $$26244 - 162 T + T^{2}$$
$29$ $$20736 - 144 T + T^{2}$$
$31$ $$( -264 + T )^{2}$$
$37$ $$91809 + 303 T + T^{2}$$
$41$ $$36864 - 192 T + T^{2}$$
$43$ $$9409 + 97 T + T^{2}$$
$47$ $$( -111 + T )^{2}$$
$53$ $$( 414 + T )^{2}$$
$59$ $$272484 + 522 T + T^{2}$$
$61$ $$141376 + 376 T + T^{2}$$
$67$ $$1296 - 36 T + T^{2}$$
$71$ $$127449 + 357 T + T^{2}$$
$73$ $$( 1098 + T )^{2}$$
$79$ $$( 830 + T )^{2}$$
$83$ $$( 438 + T )^{2}$$
$89$ $$191844 - 438 T + T^{2}$$
$97$ $$725904 - 852 T + T^{2}$$