# Properties

 Label 429.1.v.b Level $429$ Weight $1$ Character orbit 429.v Analytic conductor $0.214$ Analytic rank $0$ Dimension $4$ Projective image $D_{5}$ CM discriminant -39 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 429.v (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.214098890420$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{5}$$ Projective field Galois closure of 5.1.22268961.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

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

## Character values

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

 $$n$$ $$67$$ $$79$$ $$287$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{10}^{3}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
38.1
 −0.309017 + 0.951057i 0.809017 + 0.587785i −0.309017 − 0.951057i 0.809017 − 0.587785i
0.190983 + 0.587785i −0.809017 0.587785i 0.500000 0.363271i −0.500000 + 1.53884i 0.190983 0.587785i 0 0.809017 + 0.587785i 0.309017 + 0.951057i −1.00000
311.1 1.30902 0.951057i 0.309017 + 0.951057i 0.500000 1.53884i −0.500000 0.363271i 1.30902 + 0.951057i 0 −0.309017 0.951057i −0.809017 + 0.587785i −1.00000
350.1 0.190983 0.587785i −0.809017 + 0.587785i 0.500000 + 0.363271i −0.500000 1.53884i 0.190983 + 0.587785i 0 0.809017 0.587785i 0.309017 0.951057i −1.00000
389.1 1.30902 + 0.951057i 0.309017 0.951057i 0.500000 + 1.53884i −0.500000 + 0.363271i 1.30902 0.951057i 0 −0.309017 + 0.951057i −0.809017 0.587785i −1.00000
 $$n$$: 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
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$
11.c even 5 1 inner
429.v odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.1.v.b yes 4
3.b odd 2 1 429.1.v.a 4
11.c even 5 1 inner 429.1.v.b yes 4
13.b even 2 1 429.1.v.a 4
33.h odd 10 1 429.1.v.a 4
39.d odd 2 1 CM 429.1.v.b yes 4
143.n even 10 1 429.1.v.a 4
429.v odd 10 1 inner 429.1.v.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.1.v.a 4 3.b odd 2 1
429.1.v.a 4 13.b even 2 1
429.1.v.a 4 33.h odd 10 1
429.1.v.a 4 143.n even 10 1
429.1.v.b yes 4 1.a even 1 1 trivial
429.1.v.b yes 4 11.c even 5 1 inner
429.1.v.b yes 4 39.d odd 2 1 CM
429.1.v.b yes 4 429.v odd 10 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 3 T_{2}^{3} + 4 T_{2}^{2} - 2 T_{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(429, [\chi])$$.

## Hecke characteristic polynomials

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