# Properties

 Label 169.4.a.i Level $169$ Weight $4$ Character orbit 169.a Self dual yes Analytic conductor $9.971$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.a (trivial)

## Newform invariants

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

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 2 q^{3} -5 q^{4} + \beta q^{5} + 2 \beta q^{6} -8 \beta q^{7} -13 \beta q^{8} -23 q^{9} +O(q^{10})$$ $$q + \beta q^{2} + 2 q^{3} -5 q^{4} + \beta q^{5} + 2 \beta q^{6} -8 \beta q^{7} -13 \beta q^{8} -23 q^{9} + 3 q^{10} -8 \beta q^{11} -10 q^{12} -24 q^{14} + 2 \beta q^{15} + q^{16} -117 q^{17} -23 \beta q^{18} + 66 \beta q^{19} -5 \beta q^{20} -16 \beta q^{21} -24 q^{22} + 78 q^{23} -26 \beta q^{24} -122 q^{25} -100 q^{27} + 40 \beta q^{28} -141 q^{29} + 6 q^{30} -90 \beta q^{31} + 105 \beta q^{32} -16 \beta q^{33} -117 \beta q^{34} -24 q^{35} + 115 q^{36} + 83 \beta q^{37} + 198 q^{38} -39 q^{40} + 157 \beta q^{41} -48 q^{42} -104 q^{43} + 40 \beta q^{44} -23 \beta q^{45} + 78 \beta q^{46} + 174 \beta q^{47} + 2 q^{48} -151 q^{49} -122 \beta q^{50} -234 q^{51} + 93 q^{53} -100 \beta q^{54} -24 q^{55} + 312 q^{56} + 132 \beta q^{57} -141 \beta q^{58} -164 \beta q^{59} -10 \beta q^{60} + 145 q^{61} -270 q^{62} + 184 \beta q^{63} + 307 q^{64} -48 q^{66} -454 \beta q^{67} + 585 q^{68} + 156 q^{69} -24 \beta q^{70} -610 \beta q^{71} + 299 \beta q^{72} + 265 \beta q^{73} + 249 q^{74} -244 q^{75} -330 \beta q^{76} + 192 q^{77} + 1276 q^{79} + \beta q^{80} + 421 q^{81} + 471 q^{82} + 456 \beta q^{83} + 80 \beta q^{84} -117 \beta q^{85} -104 \beta q^{86} -282 q^{87} + 312 q^{88} + 564 \beta q^{89} -69 q^{90} -390 q^{92} -180 \beta q^{93} + 522 q^{94} + 198 q^{95} + 210 \beta q^{96} -116 \beta q^{97} -151 \beta q^{98} + 184 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{3} - 10 q^{4} - 46 q^{9} + O(q^{10})$$ $$2 q + 4 q^{3} - 10 q^{4} - 46 q^{9} + 6 q^{10} - 20 q^{12} - 48 q^{14} + 2 q^{16} - 234 q^{17} - 48 q^{22} + 156 q^{23} - 244 q^{25} - 200 q^{27} - 282 q^{29} + 12 q^{30} - 48 q^{35} + 230 q^{36} + 396 q^{38} - 78 q^{40} - 96 q^{42} - 208 q^{43} + 4 q^{48} - 302 q^{49} - 468 q^{51} + 186 q^{53} - 48 q^{55} + 624 q^{56} + 290 q^{61} - 540 q^{62} + 614 q^{64} - 96 q^{66} + 1170 q^{68} + 312 q^{69} + 498 q^{74} - 488 q^{75} + 384 q^{77} + 2552 q^{79} + 842 q^{81} + 942 q^{82} - 564 q^{87} + 624 q^{88} - 138 q^{90} - 780 q^{92} + 1044 q^{94} + 396 q^{95} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
−1.73205 2.00000 −5.00000 −1.73205 −3.46410 13.8564 22.5167 −23.0000 3.00000
1.2 1.73205 2.00000 −5.00000 1.73205 3.46410 −13.8564 −22.5167 −23.0000 3.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$13$$ $$-1$$

## 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.a.i 2
3.b odd 2 1 1521.4.a.o 2
13.b even 2 1 inner 169.4.a.i 2
13.c even 3 2 169.4.c.h 4
13.d odd 4 2 169.4.b.d 2
13.e even 6 2 169.4.c.h 4
13.f odd 12 2 13.4.e.b 2
13.f odd 12 2 169.4.e.a 2
39.d odd 2 1 1521.4.a.o 2
39.k even 12 2 117.4.q.a 2
52.l even 12 2 208.4.w.b 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 3$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(169))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 + T^{2}$$
$3$ $$( -2 + T )^{2}$$
$5$ $$-3 + T^{2}$$
$7$ $$-192 + T^{2}$$
$11$ $$-192 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( 117 + T )^{2}$$
$19$ $$-13068 + T^{2}$$
$23$ $$( -78 + T )^{2}$$
$29$ $$( 141 + T )^{2}$$
$31$ $$-24300 + T^{2}$$
$37$ $$-20667 + T^{2}$$
$41$ $$-73947 + T^{2}$$
$43$ $$( 104 + T )^{2}$$
$47$ $$-90828 + T^{2}$$
$53$ $$( -93 + T )^{2}$$
$59$ $$-80688 + T^{2}$$
$61$ $$( -145 + T )^{2}$$
$67$ $$-618348 + T^{2}$$
$71$ $$-1116300 + T^{2}$$
$73$ $$-210675 + T^{2}$$
$79$ $$( -1276 + T )^{2}$$
$83$ $$-623808 + T^{2}$$
$89$ $$-954288 + T^{2}$$
$97$ $$-40368 + T^{2}$$