# Properties

 Label 429.2.i.b Level $429$ Weight $2$ Character orbit 429.i Analytic conductor $3.426$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + 2 q^{5} + 3 \zeta_{6} q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + 2 q^{5} + 3 \zeta_{6} q^{7} -\zeta_{6} q^{9} + ( -1 + \zeta_{6} ) q^{11} + 2 q^{12} + ( -4 + 3 \zeta_{6} ) q^{13} + ( 2 - 2 \zeta_{6} ) q^{15} + ( -4 + 4 \zeta_{6} ) q^{16} -2 \zeta_{6} q^{17} -4 \zeta_{6} q^{19} + 4 \zeta_{6} q^{20} + 3 q^{21} + ( 4 - 4 \zeta_{6} ) q^{23} - q^{25} - q^{27} + ( -6 + 6 \zeta_{6} ) q^{28} + 5 q^{31} + \zeta_{6} q^{33} + 6 \zeta_{6} q^{35} + ( 2 - 2 \zeta_{6} ) q^{36} + ( 6 - 6 \zeta_{6} ) q^{37} + ( -1 + 4 \zeta_{6} ) q^{39} + ( 8 - 8 \zeta_{6} ) q^{41} -5 \zeta_{6} q^{43} -2 q^{44} -2 \zeta_{6} q^{45} + 4 \zeta_{6} q^{48} + ( -2 + 2 \zeta_{6} ) q^{49} -2 q^{51} + ( -6 - 2 \zeta_{6} ) q^{52} + 14 q^{53} + ( -2 + 2 \zeta_{6} ) q^{55} -4 q^{57} + 4 q^{60} + 7 \zeta_{6} q^{61} + ( 3 - 3 \zeta_{6} ) q^{63} -8 q^{64} + ( -8 + 6 \zeta_{6} ) q^{65} + ( -3 + 3 \zeta_{6} ) q^{67} + ( 4 - 4 \zeta_{6} ) q^{68} -4 \zeta_{6} q^{69} -10 \zeta_{6} q^{71} -9 q^{73} + ( -1 + \zeta_{6} ) q^{75} + ( 8 - 8 \zeta_{6} ) q^{76} -3 q^{77} -11 q^{79} + ( -8 + 8 \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} -6 q^{83} + 6 \zeta_{6} q^{84} -4 \zeta_{6} q^{85} + ( 8 - 8 \zeta_{6} ) q^{89} + ( -9 - 3 \zeta_{6} ) q^{91} + 8 q^{92} + ( 5 - 5 \zeta_{6} ) q^{93} -8 \zeta_{6} q^{95} + 7 \zeta_{6} q^{97} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} + 2q^{4} + 4q^{5} + 3q^{7} - q^{9} + O(q^{10})$$ $$2q + q^{3} + 2q^{4} + 4q^{5} + 3q^{7} - q^{9} - q^{11} + 4q^{12} - 5q^{13} + 2q^{15} - 4q^{16} - 2q^{17} - 4q^{19} + 4q^{20} + 6q^{21} + 4q^{23} - 2q^{25} - 2q^{27} - 6q^{28} + 10q^{31} + q^{33} + 6q^{35} + 2q^{36} + 6q^{37} + 2q^{39} + 8q^{41} - 5q^{43} - 4q^{44} - 2q^{45} + 4q^{48} - 2q^{49} - 4q^{51} - 14q^{52} + 28q^{53} - 2q^{55} - 8q^{57} + 8q^{60} + 7q^{61} + 3q^{63} - 16q^{64} - 10q^{65} - 3q^{67} + 4q^{68} - 4q^{69} - 10q^{71} - 18q^{73} - q^{75} + 8q^{76} - 6q^{77} - 22q^{79} - 8q^{80} - q^{81} - 12q^{83} + 6q^{84} - 4q^{85} + 8q^{89} - 21q^{91} + 16q^{92} + 5q^{93} - 8q^{95} + 7q^{97} + 2q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/429\mathbb{Z}\right)^\times$$.

 $$n$$ $$67$$ $$79$$ $$287$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
100.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i 1.00000 + 1.73205i 2.00000 0 1.50000 + 2.59808i 0 −0.500000 0.866025i 0
133.1 0 0.500000 + 0.866025i 1.00000 1.73205i 2.00000 0 1.50000 2.59808i 0 −0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.i.b 2
13.c even 3 1 inner 429.2.i.b 2
13.c even 3 1 5577.2.a.d 1
13.e even 6 1 5577.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.i.b 2 1.a even 1 1 trivial
429.2.i.b 2 13.c even 3 1 inner
5577.2.a.c 1 13.e even 6 1
5577.2.a.d 1 13.c even 3 1

## Hecke kernels

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

 $$T_{2}$$ $$T_{7}^{2} - 3 T_{7} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$( -2 + T )^{2}$$
$7$ $$9 - 3 T + T^{2}$$
$11$ $$1 + T + T^{2}$$
$13$ $$13 + 5 T + T^{2}$$
$17$ $$4 + 2 T + T^{2}$$
$19$ $$16 + 4 T + T^{2}$$
$23$ $$16 - 4 T + T^{2}$$
$29$ $$T^{2}$$
$31$ $$( -5 + T )^{2}$$
$37$ $$36 - 6 T + T^{2}$$
$41$ $$64 - 8 T + T^{2}$$
$43$ $$25 + 5 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$( -14 + T )^{2}$$
$59$ $$T^{2}$$
$61$ $$49 - 7 T + T^{2}$$
$67$ $$9 + 3 T + T^{2}$$
$71$ $$100 + 10 T + T^{2}$$
$73$ $$( 9 + T )^{2}$$
$79$ $$( 11 + T )^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$64 - 8 T + T^{2}$$
$97$ $$49 - 7 T + T^{2}$$