# Properties

 Label 432.5.e.d Level 432 Weight 5 Character orbit 432.e Analytic conductor 44.656 Analytic rank 0 Dimension 2 CM no 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: no Analytic conductor: $$44.6558240522$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 108) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 9 i q^{5} -5 q^{7} +O(q^{10})$$ $$q + 9 i q^{5} -5 q^{7} -117 i q^{11} -34 q^{13} + 450 i q^{17} + 64 q^{19} -612 i q^{23} + 544 q^{25} + 1062 i q^{29} + 697 q^{31} -45 i q^{35} -748 q^{37} + 684 i q^{41} -2618 q^{43} + 2646 i q^{47} -2376 q^{49} -1071 i q^{53} + 1053 q^{55} + 5814 i q^{59} + 6404 q^{61} -306 i q^{65} + 5218 q^{67} + 6570 i q^{71} -4519 q^{73} + 585 i q^{77} -7502 q^{79} + 5481 i q^{83} -4050 q^{85} + 8874 i q^{89} + 170 q^{91} + 576 i q^{95} + 10571 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 10q^{7} + O(q^{10})$$ $$2q - 10q^{7} - 68q^{13} + 128q^{19} + 1088q^{25} + 1394q^{31} - 1496q^{37} - 5236q^{43} - 4752q^{49} + 2106q^{55} + 12808q^{61} + 10436q^{67} - 9038q^{73} - 15004q^{79} - 8100q^{85} + 340q^{91} + 21142q^{97} + O(q^{100})$$

## 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
 − 1.00000i 1.00000i
0 0 0 9.00000i 0 −5.00000 0 0 0
161.2 0 0 0 9.00000i 0 −5.00000 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 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{5}^{2} + 81$$ $$T_{7} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 1169 T^{2} + 390625 T^{4}$$
$7$ $$( 1 + 5 T + 2401 T^{2} )^{2}$$
$11$ $$1 - 15593 T^{2} + 214358881 T^{4}$$
$13$ $$( 1 + 34 T + 28561 T^{2} )^{2}$$
$17$ $$1 + 35458 T^{2} + 6975757441 T^{4}$$
$19$ $$( 1 - 64 T + 130321 T^{2} )^{2}$$
$23$ $$1 - 185138 T^{2} + 78310985281 T^{4}$$
$29$ $$1 - 286718 T^{2} + 500246412961 T^{4}$$
$31$ $$( 1 - 697 T + 923521 T^{2} )^{2}$$
$37$ $$( 1 + 748 T + 1874161 T^{2} )^{2}$$
$41$ $$1 - 5183666 T^{2} + 7984925229121 T^{4}$$
$43$ $$( 1 + 2618 T + 3418801 T^{2} )^{2}$$
$47$ $$1 - 2758046 T^{2} + 23811286661761 T^{4}$$
$53$ $$1 - 14633921 T^{2} + 62259690411361 T^{4}$$
$59$ $$1 + 9567874 T^{2} + 146830437604321 T^{4}$$
$61$ $$( 1 - 6404 T + 13845841 T^{2} )^{2}$$
$67$ $$( 1 - 5218 T + 20151121 T^{2} )^{2}$$
$71$ $$1 - 7658462 T^{2} + 645753531245761 T^{4}$$
$73$ $$( 1 + 4519 T + 28398241 T^{2} )^{2}$$
$79$ $$( 1 + 7502 T + 38950081 T^{2} )^{2}$$
$83$ $$1 - 64875281 T^{2} + 2252292232139041 T^{4}$$
$89$ $$1 - 46736606 T^{2} + 3936588805702081 T^{4}$$
$97$ $$( 1 - 10571 T + 88529281 T^{2} )^{2}$$