# Properties

 Label 429.2.a.f 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.

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

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
 −1.48119 0.311108 2.17009
−1.48119 1.00000 0.193937 −1.67513 −1.48119 −4.15633 2.67513 1.00000 2.48119
1.2 0.311108 1.00000 −1.90321 2.21432 0.311108 1.52543 −1.21432 1.00000 0.688892
1.3 2.17009 1.00000 2.70928 −0.539189 2.17009 0.630898 1.53919 1.00000 −1.17009
 $$n$$: 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.f 3
3.b odd 2 1 1287.2.a.i 3
4.b odd 2 1 6864.2.a.bp 3
11.b odd 2 1 4719.2.a.t 3
13.b even 2 1 5577.2.a.k 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.a.f 3 1.a even 1 1 trivial
1287.2.a.i 3 3.b odd 2 1
4719.2.a.t 3 11.b odd 2 1
5577.2.a.k 3 13.b even 2 1
6864.2.a.bp 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(429))$$:

 $$T_{2}^{3} - T_{2}^{2} - 3 T_{2} + 1$$ $$T_{5}^{3} - 4 T_{5} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T - T^{2} + T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$-2 - 4 T + T^{3}$$
$7$ $$4 - 8 T + 2 T^{2} + T^{3}$$
$11$ $$( -1 + T )^{3}$$
$13$ $$( -1 + T )^{3}$$
$17$ $$26 - 2 T - 8 T^{2} + T^{3}$$
$19$ $$100 - 16 T - 6 T^{2} + T^{3}$$
$23$ $$-4 + 12 T - 8 T^{2} + T^{3}$$
$29$ $$-10 - 14 T - 2 T^{2} + T^{3}$$
$31$ $$2 + 8 T + 6 T^{2} + T^{3}$$
$37$ $$-16 + 32 T + 12 T^{2} + T^{3}$$
$41$ $$100 + 20 T - 12 T^{2} + T^{3}$$
$43$ $$74 - 74 T - 2 T^{2} + T^{3}$$
$47$ $$104 - 36 T - 2 T^{2} + T^{3}$$
$53$ $$296 - 148 T + 2 T^{2} + T^{3}$$
$59$ $$-80 - 64 T - 8 T^{2} + T^{3}$$
$61$ $$-32 - 16 T + 4 T^{2} + T^{3}$$
$67$ $$74 - 36 T - 2 T^{2} + T^{3}$$
$71$ $$184 - 52 T - 2 T^{2} + T^{3}$$
$73$ $$-412 - 84 T + 4 T^{2} + T^{3}$$
$79$ $$-10 + 42 T - 14 T^{2} + T^{3}$$
$83$ $$-16 - 8 T + 4 T^{2} + T^{3}$$
$89$ $$-1690 - 168 T + 10 T^{2} + T^{3}$$
$97$ $$-16 + 32 T + 12 T^{2} + T^{3}$$