# Properties

 Label 429.2.e.b Level $429$ Weight $2$ Character orbit 429.e Analytic conductor $3.426$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.3317760000.2 Defining polynomial: $$x^{8} - 25 x^{4} + 625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 25 x^{4} + 625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{6}$$$$/125$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{4} - 25$$$$)/25$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{5} - 5 \nu^{3} + 25 \nu$$$$)/25$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} + 50 \nu^{2}$$$$)/125$$ $$\beta_{5}$$ $$=$$ $$($$$$-2 \nu^{7} + 5 \nu^{5} + 25 \nu^{3} + 125 \nu$$$$)/125$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{5} + 5 \nu^{3} + 25 \nu$$$$)/25$$ $$\beta_{7}$$ $$=$$ $$($$$$2 \nu^{7} + 5 \nu^{5} - 25 \nu^{3} + 125 \nu$$$$)/125$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + \beta_{5} + \beta_{3}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$5 \beta_{4} + 5 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{6} - 5 \beta_{3}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$25 \beta_{2} + 25$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$25 \beta_{7} - 25 \beta_{6} + 25 \beta_{5} - 25 \beta_{3}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$125 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$($$$$125 \beta_{7} + 125 \beta_{6} - 125 \beta_{5} - 125 \beta_{3}$$$$)/4$$

## Character values

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

 $$n$$ $$67$$ $$79$$ $$287$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
428.1
 −2.15988 + 0.578737i 2.15988 − 0.578737i 0.578737 − 2.15988i −0.578737 + 2.15988i 0.578737 + 2.15988i −0.578737 − 2.15988i −2.15988 − 0.578737i 2.15988 + 0.578737i
1.00000i 1.73205i 1.00000 −3.16228 −1.73205 −3.46410 3.00000i −3.00000 3.16228i
428.2 1.00000i 1.73205i 1.00000 3.16228 −1.73205 −3.46410 3.00000i −3.00000 3.16228i
428.3 1.00000i 1.73205i 1.00000 −3.16228 1.73205 3.46410 3.00000i −3.00000 3.16228i
428.4 1.00000i 1.73205i 1.00000 3.16228 1.73205 3.46410 3.00000i −3.00000 3.16228i
428.5 1.00000i 1.73205i 1.00000 −3.16228 1.73205 3.46410 3.00000i −3.00000 3.16228i
428.6 1.00000i 1.73205i 1.00000 3.16228 1.73205 3.46410 3.00000i −3.00000 3.16228i
428.7 1.00000i 1.73205i 1.00000 −3.16228 −1.73205 −3.46410 3.00000i −3.00000 3.16228i
428.8 1.00000i 1.73205i 1.00000 3.16228 −1.73205 −3.46410 3.00000i −3.00000 3.16228i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 428.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
13.b even 2 1 inner
33.d even 2 1 inner
39.d odd 2 1 inner
143.d odd 2 1 inner
429.e even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.e.b 8
3.b odd 2 1 inner 429.2.e.b 8
11.b odd 2 1 inner 429.2.e.b 8
13.b even 2 1 inner 429.2.e.b 8
33.d even 2 1 inner 429.2.e.b 8
39.d odd 2 1 inner 429.2.e.b 8
143.d odd 2 1 inner 429.2.e.b 8
429.e even 2 1 inner 429.2.e.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.e.b 8 1.a even 1 1 trivial
429.2.e.b 8 3.b odd 2 1 inner
429.2.e.b 8 11.b odd 2 1 inner
429.2.e.b 8 13.b even 2 1 inner
429.2.e.b 8 33.d even 2 1 inner
429.2.e.b 8 39.d odd 2 1 inner
429.2.e.b 8 143.d odd 2 1 inner
429.2.e.b 8 429.e even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{4}$$
$3$ $$( 3 + T^{2} )^{4}$$
$5$ $$( -10 + T^{2} )^{4}$$
$7$ $$( -12 + T^{2} )^{4}$$
$11$ $$( 121 - 18 T^{2} + T^{4} )^{2}$$
$13$ $$( 169 + 14 T^{2} + T^{4} )^{2}$$
$17$ $$( -30 + T^{2} )^{4}$$
$19$ $$( -12 + T^{2} )^{4}$$
$23$ $$( 48 + T^{2} )^{4}$$
$29$ $$( -30 + T^{2} )^{4}$$
$31$ $$( 30 + T^{2} )^{4}$$
$37$ $$T^{8}$$
$41$ $$( 4 + T^{2} )^{4}$$
$43$ $$( 90 + T^{2} )^{4}$$
$47$ $$( -40 + T^{2} )^{4}$$
$53$ $$( 48 + T^{2} )^{4}$$
$59$ $$( -40 + T^{2} )^{4}$$
$61$ $$( 160 + T^{2} )^{4}$$
$67$ $$( 30 + T^{2} )^{4}$$
$71$ $$( -40 + T^{2} )^{4}$$
$73$ $$( -48 + T^{2} )^{4}$$
$79$ $$( 10 + T^{2} )^{4}$$
$83$ $$( 4 + T^{2} )^{4}$$
$89$ $$( -10 + T^{2} )^{4}$$
$97$ $$( 120 + T^{2} )^{4}$$