# Properties

 Label 429.2.e.a Level $429$ Weight $2$ Character orbit 429.e Analytic conductor $3.426$ Analytic rank $0$ Dimension $8$ CM discriminant -39 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: 8.0.151613669376.21 Defining polynomial: $$x^{8} + 5 x^{4} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} -\beta_{2} q^{3} + ( -2 - \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{6} ) q^{5} + ( \beta_{1} - \beta_{3} + \beta_{6} ) q^{6} + ( \beta_{1} - \beta_{3} + \beta_{6} ) q^{8} + 3 q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{2} -\beta_{2} q^{3} + ( -2 - \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{6} ) q^{5} + ( \beta_{1} - \beta_{3} + \beta_{6} ) q^{6} + ( \beta_{1} - \beta_{3} + \beta_{6} ) q^{8} + 3 q^{9} + ( 1 - \beta_{2} + 2 \beta_{5} - 2 \beta_{7} ) q^{10} + ( -\beta_{1} + \beta_{3} + \beta_{4} ) q^{11} + ( 3 + 2 \beta_{2} ) q^{12} + \beta_{5} q^{13} + ( -\beta_{1} - 2 \beta_{4} + \beta_{6} ) q^{15} + ( -1 + 2 \beta_{2} ) q^{16} + 3 \beta_{3} q^{18} + ( -3 \beta_{1} - 2 \beta_{4} + 3 \beta_{6} ) q^{20} + ( -4 - 2 \beta_{2} + \beta_{7} ) q^{22} + 3 \beta_{3} q^{24} + ( 5 + 4 \beta_{2} ) q^{25} + ( -\beta_{1} - 3 \beta_{4} + \beta_{6} ) q^{26} -3 \beta_{2} q^{27} + ( -1 + \beta_{2} - 4 \beta_{5} + 2 \beta_{7} ) q^{30} -\beta_{3} q^{32} + ( -2 \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{33} + ( -6 - 3 \beta_{2} ) q^{36} + ( -1 + \beta_{2} - \beta_{5} + 2 \beta_{7} ) q^{39} + ( -1 + \beta_{2} - 4 \beta_{5} + 2 \beta_{7} ) q^{40} + ( -\beta_{1} - 4 \beta_{3} - \beta_{6} ) q^{41} + ( 2 - 2 \beta_{2} + 2 \beta_{5} - 4 \beta_{7} ) q^{43} + ( 2 \beta_{1} - 4 \beta_{3} + \beta_{6} ) q^{44} + ( 3 \beta_{1} - 3 \beta_{6} ) q^{45} + ( -3 \beta_{1} + 2 \beta_{4} + 3 \beta_{6} ) q^{47} + ( -6 + \beta_{2} ) q^{48} -7 q^{49} + ( -4 \beta_{1} + 9 \beta_{3} - 4 \beta_{6} ) q^{50} + ( -1 + \beta_{2} - 3 \beta_{5} + 2 \beta_{7} ) q^{52} + ( 3 \beta_{1} - 3 \beta_{3} + 3 \beta_{6} ) q^{54} + ( -3 + 3 \beta_{5} - 2 \beta_{7} ) q^{55} + ( 3 \beta_{1} + 2 \beta_{4} - 3 \beta_{6} ) q^{59} + ( 5 \beta_{1} + 4 \beta_{4} - 5 \beta_{6} ) q^{60} + 2 \beta_{5} q^{61} + ( 2 + 5 \beta_{2} ) q^{64} + ( -3 \beta_{1} + 4 \beta_{3} - 3 \beta_{6} ) q^{65} + ( 7 + 4 \beta_{2} + \beta_{5} + \beta_{7} ) q^{66} + ( \beta_{1} - 6 \beta_{4} - \beta_{6} ) q^{71} + ( 3 \beta_{1} - 3 \beta_{3} + 3 \beta_{6} ) q^{72} + ( -12 - 5 \beta_{2} ) q^{75} + ( 4 \beta_{1} - \beta_{4} - 4 \beta_{6} ) q^{78} -4 \beta_{5} q^{79} + ( \beta_{1} + 4 \beta_{4} - \beta_{6} ) q^{80} + 9 q^{81} + ( 17 + \beta_{2} ) q^{82} + ( -\beta_{1} - 6 \beta_{3} - \beta_{6} ) q^{83} + ( -8 \beta_{1} + 2 \beta_{4} + 8 \beta_{6} ) q^{86} + ( 7 + 4 \beta_{2} + \beta_{5} + \beta_{7} ) q^{88} + ( 3 \beta_{1} + 4 \beta_{4} - 3 \beta_{6} ) q^{89} + ( 3 - 3 \beta_{2} + 6 \beta_{5} - 6 \beta_{7} ) q^{90} + ( -3 + 3 \beta_{2} - 4 \beta_{5} + 6 \beta_{7} ) q^{94} + ( -\beta_{1} + \beta_{3} - \beta_{6} ) q^{96} -7 \beta_{3} q^{98} + ( -3 \beta_{1} + 3 \beta_{3} + 3 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 16q^{4} + 24q^{9} + O(q^{10})$$ $$8q - 16q^{4} + 24q^{9} + 24q^{12} - 8q^{16} - 28q^{22} + 40q^{25} - 48q^{36} - 48q^{48} - 56q^{49} - 32q^{55} + 16q^{64} + 60q^{66} - 96q^{75} + 72q^{81} + 136q^{82} + 60q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5 x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 3 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + \nu^{2}$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 5 \nu^{3} + 8 \nu$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - 5 \nu^{3} + 8 \nu$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{6} + 9 \nu^{2}$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} + 3 \nu^{3}$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$\nu^{4} + \nu^{2} + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{3}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} - \beta_{2}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{6} - \beta_{4} + \beta_{3}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{7} - \beta_{5} + \beta_{2} - 6$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-3 \beta_{4} - 3 \beta_{3} + 4 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-\beta_{5} + 9 \beta_{2}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-10 \beta_{6} - 3 \beta_{4} + 3 \beta_{3}$$$$)/2$$

## 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
 −0.752986 − 1.19709i 0.752986 − 1.19709i 1.19709 − 0.752986i −1.19709 − 0.752986i 1.19709 + 0.752986i −1.19709 + 0.752986i −0.752986 + 1.19709i 0.752986 + 1.19709i
2.39417i −1.73205 −3.73205 −4.11439 4.14682i 0 4.14682i 3.00000 9.85055i
428.2 2.39417i −1.73205 −3.73205 4.11439 4.14682i 0 4.14682i 3.00000 9.85055i
428.3 1.50597i 1.73205 −0.267949 −1.75265 2.60842i 0 2.60842i 3.00000 2.63945i
428.4 1.50597i 1.73205 −0.267949 1.75265 2.60842i 0 2.60842i 3.00000 2.63945i
428.5 1.50597i 1.73205 −0.267949 −1.75265 2.60842i 0 2.60842i 3.00000 2.63945i
428.6 1.50597i 1.73205 −0.267949 1.75265 2.60842i 0 2.60842i 3.00000 2.63945i
428.7 2.39417i −1.73205 −3.73205 −4.11439 4.14682i 0 4.14682i 3.00000 9.85055i
428.8 2.39417i −1.73205 −3.73205 4.11439 4.14682i 0 4.14682i 3.00000 9.85055i
 $$n$$: 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
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$
3.b odd 2 1 inner
11.b odd 2 1 inner
13.b even 2 1 inner
33.d even 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.a 8
3.b odd 2 1 inner 429.2.e.a 8
11.b odd 2 1 inner 429.2.e.a 8
13.b even 2 1 inner 429.2.e.a 8
33.d even 2 1 inner 429.2.e.a 8
39.d odd 2 1 CM 429.2.e.a 8
143.d odd 2 1 inner 429.2.e.a 8
429.e even 2 1 inner 429.2.e.a 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 8 T_{2}^{2} + 13$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 13 + 8 T^{2} + T^{4} )^{2}$$
$3$ $$( -3 + T^{2} )^{4}$$
$5$ $$( 52 - 20 T^{2} + T^{4} )^{2}$$
$7$ $$T^{8}$$
$11$ $$14641 - 190 T^{4} + T^{8}$$
$13$ $$( 13 + T^{2} )^{4}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$( 6292 + 164 T^{2} + T^{4} )^{2}$$
$43$ $$( 156 + T^{2} )^{4}$$
$47$ $$( 8788 - 188 T^{2} + T^{4} )^{2}$$
$53$ $$T^{8}$$
$59$ $$( 52 - 236 T^{2} + T^{4} )^{2}$$
$61$ $$( 52 + T^{2} )^{4}$$
$67$ $$T^{8}$$
$71$ $$( 6292 - 284 T^{2} + T^{4} )^{2}$$
$73$ $$T^{8}$$
$79$ $$( 208 + T^{2} )^{4}$$
$83$ $$( 27508 + 332 T^{2} + T^{4} )^{2}$$
$89$ $$( 6292 - 356 T^{2} + T^{4} )^{2}$$
$97$ $$T^{8}$$