# Properties

 Label 432.5.e.b Level 432 Weight 5 Character orbit 432.e Self dual yes Analytic conductor 44.656 Analytic rank 0 Dimension 1 CM discriminant -3 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 432.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.6558240522$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 108) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 23q^{7} + O(q^{10})$$ $$q - 23q^{7} + 191q^{13} - 647q^{19} + 625q^{25} - 194q^{31} + 2591q^{37} + 3214q^{43} - 1872q^{49} - 5233q^{61} + 8809q^{67} + 9791q^{73} + 12361q^{79} - 4393q^{91} + 9743q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$$.

 $$n$$ $$271$$ $$325$$ $$353$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 0
0 0 0 0 0 −23.0000 0 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.5.e.b 1
3.b odd 2 1 CM 432.5.e.b 1
4.b odd 2 1 108.5.c.a 1
12.b even 2 1 108.5.c.a 1
36.f odd 6 2 324.5.g.a 2
36.h even 6 2 324.5.g.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.5.c.a 1 4.b odd 2 1
108.5.c.a 1 12.b even 2 1
324.5.g.a 2 36.f odd 6 2
324.5.g.a 2 36.h even 6 2
432.5.e.b 1 1.a even 1 1 trivial
432.5.e.b 1 3.b odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{5}^{\mathrm{new}}(432, [\chi])$$:

 $$T_{5}$$ $$T_{7} + 23$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 25 T )( 1 + 25 T )$$
$7$ $$1 + 23 T + 2401 T^{2}$$
$11$ $$( 1 - 121 T )( 1 + 121 T )$$
$13$ $$1 - 191 T + 28561 T^{2}$$
$17$ $$( 1 - 289 T )( 1 + 289 T )$$
$19$ $$1 + 647 T + 130321 T^{2}$$
$23$ $$( 1 - 529 T )( 1 + 529 T )$$
$29$ $$( 1 - 841 T )( 1 + 841 T )$$
$31$ $$1 + 194 T + 923521 T^{2}$$
$37$ $$1 - 2591 T + 1874161 T^{2}$$
$41$ $$( 1 - 1681 T )( 1 + 1681 T )$$
$43$ $$1 - 3214 T + 3418801 T^{2}$$
$47$ $$( 1 - 2209 T )( 1 + 2209 T )$$
$53$ $$( 1 - 2809 T )( 1 + 2809 T )$$
$59$ $$( 1 - 3481 T )( 1 + 3481 T )$$
$61$ $$1 + 5233 T + 13845841 T^{2}$$
$67$ $$1 - 8809 T + 20151121 T^{2}$$
$71$ $$( 1 - 5041 T )( 1 + 5041 T )$$
$73$ $$1 - 9791 T + 28398241 T^{2}$$
$79$ $$1 - 12361 T + 38950081 T^{2}$$
$83$ $$( 1 - 6889 T )( 1 + 6889 T )$$
$89$ $$( 1 - 7921 T )( 1 + 7921 T )$$
$97$ $$1 - 9743 T + 88529281 T^{2}$$