# Properties

 Label 429.2.a.g Level $429$ Weight $2$ Character orbit 429.a Self dual yes Analytic conductor $3.426$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{2} ) q^{2} + q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{1} ) q^{5} + ( 1 + \beta_{2} ) q^{6} + ( -1 + \beta_{1} - \beta_{2} ) q^{7} + ( 4 - 3 \beta_{1} ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{2} ) q^{2} + q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{1} ) q^{5} + ( 1 + \beta_{2} ) q^{6} + ( -1 + \beta_{1} - \beta_{2} ) q^{7} + ( 4 - 3 \beta_{1} ) q^{8} + q^{9} + ( 2 \beta_{1} + \beta_{2} ) q^{10} - q^{11} + ( 2 - \beta_{1} + \beta_{2} ) q^{12} - q^{13} + ( -5 + 3 \beta_{1} - \beta_{2} ) q^{14} + ( 1 + \beta_{1} ) q^{15} + ( 3 - 4 \beta_{1} + 2 \beta_{2} ) q^{16} -\beta_{2} q^{17} + ( 1 + \beta_{2} ) q^{18} + ( 1 - \beta_{1} + \beta_{2} ) q^{19} + ( -1 + \beta_{1} ) q^{20} + ( -1 + \beta_{1} - \beta_{2} ) q^{21} + ( -1 - \beta_{2} ) q^{22} + ( 1 + \beta_{1} - 3 \beta_{2} ) q^{23} + ( 4 - 3 \beta_{1} ) q^{24} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{25} + ( -1 - \beta_{2} ) q^{26} + q^{27} + ( -9 + 5 \beta_{1} - 3 \beta_{2} ) q^{28} + ( 2 - 4 \beta_{1} - 3 \beta_{2} ) q^{29} + ( 2 \beta_{1} + \beta_{2} ) q^{30} + ( 5 - \beta_{1} ) q^{31} + ( 5 - 4 \beta_{1} + 3 \beta_{2} ) q^{32} - q^{33} + ( -3 + \beta_{1} ) q^{34} + 2 q^{35} + ( 2 - \beta_{1} + \beta_{2} ) q^{36} -4 \beta_{2} q^{37} + ( 5 - 3 \beta_{1} + \beta_{2} ) q^{38} - q^{39} + ( -2 - 2 \beta_{1} - 3 \beta_{2} ) q^{40} + ( -1 - \beta_{1} + \beta_{2} ) q^{41} + ( -5 + 3 \beta_{1} - \beta_{2} ) q^{42} + ( -4 + 6 \beta_{1} + 3 \beta_{2} ) q^{43} + ( -2 + \beta_{1} - \beta_{2} ) q^{44} + ( 1 + \beta_{1} ) q^{45} + ( -9 + 5 \beta_{1} + \beta_{2} ) q^{46} + ( 6 - 4 \beta_{1} - 2 \beta_{2} ) q^{47} + ( 3 - 4 \beta_{1} + 2 \beta_{2} ) q^{48} + ( 1 - 4 \beta_{1} + 2 \beta_{2} ) q^{49} + ( -2 + 5 \beta_{1} - 2 \beta_{2} ) q^{50} -\beta_{2} q^{51} + ( -2 + \beta_{1} - \beta_{2} ) q^{52} + ( -2 + 4 \beta_{1} - 2 \beta_{2} ) q^{53} + ( 1 + \beta_{2} ) q^{54} + ( -1 - \beta_{1} ) q^{55} + ( -13 + 7 \beta_{1} - 7 \beta_{2} ) q^{56} + ( 1 - \beta_{1} + \beta_{2} ) q^{57} + ( -3 - 5 \beta_{1} + 2 \beta_{2} ) q^{58} + ( 10 - 2 \beta_{1} ) q^{59} + ( -1 + \beta_{1} ) q^{60} + ( -2 - 2 \beta_{1} + 4 \beta_{2} ) q^{61} + ( 6 - 2 \beta_{1} + 5 \beta_{2} ) q^{62} + ( -1 + \beta_{1} - \beta_{2} ) q^{63} + ( 12 - 3 \beta_{1} + \beta_{2} ) q^{64} + ( -1 - \beta_{1} ) q^{65} + ( -1 - \beta_{2} ) q^{66} + ( 3 - 7 \beta_{1} - 2 \beta_{2} ) q^{67} + ( -4 + 2 \beta_{1} - \beta_{2} ) q^{68} + ( 1 + \beta_{1} - 3 \beta_{2} ) q^{69} + ( 2 + 2 \beta_{2} ) q^{70} + ( 2 + 4 \beta_{1} + 8 \beta_{2} ) q^{71} + ( 4 - 3 \beta_{1} ) q^{72} + ( -7 - 3 \beta_{1} - 5 \beta_{2} ) q^{73} + ( -12 + 4 \beta_{1} ) q^{74} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{75} + ( 9 - 5 \beta_{1} + 3 \beta_{2} ) q^{76} + ( 1 - \beta_{1} + \beta_{2} ) q^{77} + ( -1 - \beta_{2} ) q^{78} + ( -6 - 4 \beta_{1} - \beta_{2} ) q^{79} + ( -7 - 3 \beta_{1} - 2 \beta_{2} ) q^{80} + q^{81} + ( 3 - 3 \beta_{1} - \beta_{2} ) q^{82} + ( 6 - 2 \beta_{1} ) q^{83} + ( -9 + 5 \beta_{1} - 3 \beta_{2} ) q^{84} + ( 1 - \beta_{1} - \beta_{2} ) q^{85} + ( -1 + 9 \beta_{1} - 4 \beta_{2} ) q^{86} + ( 2 - 4 \beta_{1} - 3 \beta_{2} ) q^{87} + ( -4 + 3 \beta_{1} ) q^{88} + ( 1 - \beta_{1} ) q^{89} + ( 2 \beta_{1} + \beta_{2} ) q^{90} + ( 1 - \beta_{1} + \beta_{2} ) q^{91} + ( -13 + 7 \beta_{1} - 3 \beta_{2} ) q^{92} + ( 5 - \beta_{1} ) q^{93} + ( 4 - 6 \beta_{1} + 6 \beta_{2} ) q^{94} -2 q^{95} + ( 5 - 4 \beta_{1} + 3 \beta_{2} ) q^{96} + ( -4 + 4 \beta_{2} ) q^{97} + ( 11 - 10 \beta_{1} + \beta_{2} ) q^{98} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} + 3q^{3} + 5q^{4} + 4q^{5} + 3q^{6} - 2q^{7} + 9q^{8} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{2} + 3q^{3} + 5q^{4} + 4q^{5} + 3q^{6} - 2q^{7} + 9q^{8} + 3q^{9} + 2q^{10} - 3q^{11} + 5q^{12} - 3q^{13} - 12q^{14} + 4q^{15} + 5q^{16} + 3q^{18} + 2q^{19} - 2q^{20} - 2q^{21} - 3q^{22} + 4q^{23} + 9q^{24} - 3q^{25} - 3q^{26} + 3q^{27} - 22q^{28} + 2q^{29} + 2q^{30} + 14q^{31} + 11q^{32} - 3q^{33} - 8q^{34} + 6q^{35} + 5q^{36} + 12q^{38} - 3q^{39} - 8q^{40} - 4q^{41} - 12q^{42} - 6q^{43} - 5q^{44} + 4q^{45} - 22q^{46} + 14q^{47} + 5q^{48} - q^{49} - q^{50} - 5q^{52} - 2q^{53} + 3q^{54} - 4q^{55} - 32q^{56} + 2q^{57} - 14q^{58} + 28q^{59} - 2q^{60} - 8q^{61} + 16q^{62} - 2q^{63} + 33q^{64} - 4q^{65} - 3q^{66} + 2q^{67} - 10q^{68} + 4q^{69} + 6q^{70} + 10q^{71} + 9q^{72} - 24q^{73} - 32q^{74} - 3q^{75} + 22q^{76} + 2q^{77} - 3q^{78} - 22q^{79} - 24q^{80} + 3q^{81} + 6q^{82} + 16q^{83} - 22q^{84} + 2q^{85} + 6q^{86} + 2q^{87} - 9q^{88} + 2q^{89} + 2q^{90} + 2q^{91} - 32q^{92} + 14q^{93} + 6q^{94} - 6q^{95} + 11q^{96} - 12q^{97} + 23q^{98} - 3q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$

## 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
 0.311108 2.17009 −1.48119
−1.21432 1.00000 −0.525428 1.31111 −1.21432 1.52543 3.06668 1.00000 −1.59210
1.2 1.53919 1.00000 0.369102 3.17009 1.53919 0.630898 −2.51026 1.00000 4.87936
1.3 2.67513 1.00000 5.15633 −0.481194 2.67513 −4.15633 8.44358 1.00000 −1.28726
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$11$$ $$1$$
$$13$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.a.g 3
3.b odd 2 1 1287.2.a.h 3
4.b odd 2 1 6864.2.a.bs 3
11.b odd 2 1 4719.2.a.q 3
13.b even 2 1 5577.2.a.j 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.a.g 3 1.a even 1 1 trivial
1287.2.a.h 3 3.b odd 2 1
4719.2.a.q 3 11.b odd 2 1
5577.2.a.j 3 13.b even 2 1
6864.2.a.bs 3 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(429))$$:

 $$T_{2}^{3} - 3 T_{2}^{2} - T_{2} + 5$$ $$T_{5}^{3} - 4 T_{5}^{2} + 2 T_{5} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$5 - T - 3 T^{2} + T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$2 + 2 T - 4 T^{2} + T^{3}$$
$7$ $$4 - 8 T + 2 T^{2} + T^{3}$$
$11$ $$( 1 + T )^{3}$$
$13$ $$( 1 + T )^{3}$$
$17$ $$-2 - 4 T + T^{3}$$
$19$ $$-4 - 8 T - 2 T^{2} + T^{3}$$
$23$ $$68 - 40 T - 4 T^{2} + T^{3}$$
$29$ $$178 - 64 T - 2 T^{2} + T^{3}$$
$31$ $$-86 + 62 T - 14 T^{2} + T^{3}$$
$37$ $$-128 - 64 T + T^{3}$$
$41$ $$-20 - 4 T + 4 T^{2} + T^{3}$$
$43$ $$-734 - 108 T + 6 T^{2} + T^{3}$$
$47$ $$296 + 12 T - 14 T^{2} + T^{3}$$
$53$ $$232 - 84 T + 2 T^{2} + T^{3}$$
$59$ $$-688 + 248 T - 28 T^{2} + T^{3}$$
$61$ $$-368 - 72 T + 8 T^{2} + T^{3}$$
$67$ $$698 - 150 T - 2 T^{2} + T^{3}$$
$71$ $$2056 - 212 T - 10 T^{2} + T^{3}$$
$73$ $$-556 + 92 T + 24 T^{2} + T^{3}$$
$79$ $$134 + 112 T + 22 T^{2} + T^{3}$$
$83$ $$-80 + 72 T - 16 T^{2} + T^{3}$$
$89$ $$2 - 2 T - 2 T^{2} + T^{3}$$
$97$ $$-64 - 16 T + 12 T^{2} + T^{3}$$