# Properties

 Label 429.2.a.c Level $429$ Weight $2$ Character orbit 429.a Self dual yes Analytic conductor $3.426$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.41421 1.41421
−2.41421 1.00000 3.82843 −3.41421 −2.41421 0.828427 −4.41421 1.00000 8.24264
1.2 0.414214 1.00000 −1.82843 −0.585786 0.414214 −4.82843 −1.58579 1.00000 −0.242641
 $$n$$: 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.c 2
3.b odd 2 1 1287.2.a.g 2
4.b odd 2 1 6864.2.a.bc 2
11.b odd 2 1 4719.2.a.o 2
13.b even 2 1 5577.2.a.i 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + 2 T + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$2 + 4 T + T^{2}$$
$7$ $$-4 + 4 T + T^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$( 1 + T )^{2}$$
$17$ $$14 + 8 T + T^{2}$$
$19$ $$( 6 + T )^{2}$$
$23$ $$-4 - 4 T + T^{2}$$
$29$ $$-18 + T^{2}$$
$31$ $$-18 + T^{2}$$
$37$ $$-28 - 4 T + T^{2}$$
$41$ $$( 12 + T )^{2}$$
$43$ $$-46 + 4 T + T^{2}$$
$47$ $$-72 + T^{2}$$
$53$ $$-124 - 4 T + T^{2}$$
$59$ $$56 + 16 T + T^{2}$$
$61$ $$( 2 + T )^{2}$$
$67$ $$-34 - 8 T + T^{2}$$
$71$ $$-32 + T^{2}$$
$73$ $$-56 - 8 T + T^{2}$$
$79$ $$-46 + 4 T + T^{2}$$
$83$ $$-128 + T^{2}$$
$89$ $$194 + 28 T + T^{2}$$
$97$ $$( -10 + T )^{2}$$