# Properties

 Label 169.4.a.h 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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{2} -7 q^{3} + 4 q^{4} + 8 \beta q^{5} -14 \beta q^{6} -13 \beta q^{7} -8 \beta q^{8} + 22 q^{9} +O(q^{10})$$ $$q + 2 \beta q^{2} -7 q^{3} + 4 q^{4} + 8 \beta q^{5} -14 \beta q^{6} -13 \beta q^{7} -8 \beta q^{8} + 22 q^{9} + 48 q^{10} -13 \beta q^{11} -28 q^{12} -78 q^{14} -56 \beta q^{15} -80 q^{16} -27 q^{17} + 44 \beta q^{18} -51 \beta q^{19} + 32 \beta q^{20} + 91 \beta q^{21} -78 q^{22} -57 q^{23} + 56 \beta q^{24} + 67 q^{25} + 35 q^{27} -52 \beta q^{28} -69 q^{29} -336 q^{30} -42 \beta q^{31} -96 \beta q^{32} + 91 \beta q^{33} -54 \beta q^{34} -312 q^{35} + 88 q^{36} -23 \beta q^{37} -306 q^{38} -192 q^{40} + 227 \beta q^{41} + 546 q^{42} + 85 q^{43} -52 \beta q^{44} + 176 \beta q^{45} -114 \beta q^{46} + 198 \beta q^{47} + 560 q^{48} + 164 q^{49} + 134 \beta q^{50} + 189 q^{51} + 426 q^{53} + 70 \beta q^{54} -312 q^{55} + 312 q^{56} + 357 \beta q^{57} -138 \beta q^{58} + 11 \beta q^{59} -224 \beta q^{60} -17 q^{61} -252 q^{62} -286 \beta q^{63} + 64 q^{64} + 546 q^{66} -95 \beta q^{67} -108 q^{68} + 399 q^{69} -624 \beta q^{70} + 337 \beta q^{71} -176 \beta q^{72} -580 \beta q^{73} -138 q^{74} -469 q^{75} -204 \beta q^{76} + 507 q^{77} -1244 q^{79} -640 \beta q^{80} -839 q^{81} + 1362 q^{82} -246 \beta q^{83} + 364 \beta q^{84} -216 \beta q^{85} + 170 \beta q^{86} + 483 q^{87} + 312 q^{88} + 177 \beta q^{89} + 1056 q^{90} -228 q^{92} + 294 \beta q^{93} + 1188 q^{94} -1224 q^{95} + 672 \beta q^{96} + 713 \beta q^{97} + 328 \beta q^{98} -286 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 14 q^{3} + 8 q^{4} + 44 q^{9} + O(q^{10})$$ $$2 q - 14 q^{3} + 8 q^{4} + 44 q^{9} + 96 q^{10} - 56 q^{12} - 156 q^{14} - 160 q^{16} - 54 q^{17} - 156 q^{22} - 114 q^{23} + 134 q^{25} + 70 q^{27} - 138 q^{29} - 672 q^{30} - 624 q^{35} + 176 q^{36} - 612 q^{38} - 384 q^{40} + 1092 q^{42} + 170 q^{43} + 1120 q^{48} + 328 q^{49} + 378 q^{51} + 852 q^{53} - 624 q^{55} + 624 q^{56} - 34 q^{61} - 504 q^{62} + 128 q^{64} + 1092 q^{66} - 216 q^{68} + 798 q^{69} - 276 q^{74} - 938 q^{75} + 1014 q^{77} - 2488 q^{79} - 1678 q^{81} + 2724 q^{82} + 966 q^{87} + 624 q^{88} + 2112 q^{90} - 456 q^{92} + 2376 q^{94} - 2448 q^{95} + O(q^{100})$$

## 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
−3.46410 −7.00000 4.00000 −13.8564 24.2487 22.5167 13.8564 22.0000 48.0000
1.2 3.46410 −7.00000 4.00000 13.8564 −24.2487 −22.5167 −13.8564 22.0000 48.0000
 $$n$$: 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.h 2
3.b odd 2 1 1521.4.a.q 2
13.b even 2 1 inner 169.4.a.h 2
13.c even 3 2 169.4.c.i 4
13.d odd 4 2 169.4.b.b 2
13.e even 6 2 169.4.c.i 4
13.f odd 12 2 13.4.e.a 2
13.f odd 12 2 169.4.e.b 2
39.d odd 2 1 1521.4.a.q 2
39.k even 12 2 117.4.q.c 2
52.l even 12 2 208.4.w.a 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 12$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(169))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-12 + T^{2}$$
$3$ $$( 7 + T )^{2}$$
$5$ $$-192 + T^{2}$$
$7$ $$-507 + T^{2}$$
$11$ $$-507 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( 27 + T )^{2}$$
$19$ $$-7803 + T^{2}$$
$23$ $$( 57 + T )^{2}$$
$29$ $$( 69 + T )^{2}$$
$31$ $$-5292 + T^{2}$$
$37$ $$-1587 + T^{2}$$
$41$ $$-154587 + T^{2}$$
$43$ $$( -85 + T )^{2}$$
$47$ $$-117612 + T^{2}$$
$53$ $$( -426 + T )^{2}$$
$59$ $$-363 + T^{2}$$
$61$ $$( 17 + T )^{2}$$
$67$ $$-27075 + T^{2}$$
$71$ $$-340707 + T^{2}$$
$73$ $$-1009200 + T^{2}$$
$79$ $$( 1244 + T )^{2}$$
$83$ $$-181548 + T^{2}$$
$89$ $$-93987 + T^{2}$$
$97$ $$-1525107 + T^{2}$$