Properties

 Label 432.2.a.f Level $432$ Weight $2$ Character orbit 432.a Self dual yes Analytic conductor $3.450$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 432.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$3.44953736732$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 216) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{5} - 3q^{7} + O(q^{10})$$ $$q + q^{5} - 3q^{7} + 5q^{11} + 4q^{13} + 8q^{17} - 2q^{19} + 2q^{23} - 4q^{25} - 6q^{29} + 7q^{31} - 3q^{35} - 6q^{37} + 6q^{41} + 2q^{43} + 6q^{47} + 2q^{49} - 5q^{53} + 5q^{55} - 4q^{59} - 8q^{61} + 4q^{65} + 10q^{67} - 8q^{71} + q^{73} - 15q^{77} - 16q^{79} - 11q^{83} + 8q^{85} - 6q^{89} - 12q^{91} - 2q^{95} - q^{97} + O(q^{100})$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 1.00000 0 −3.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.2.a.f 1
3.b odd 2 1 432.2.a.c 1
4.b odd 2 1 216.2.a.c yes 1
8.b even 2 1 1728.2.a.i 1
8.d odd 2 1 1728.2.a.l 1
9.c even 3 2 1296.2.i.g 2
9.d odd 6 2 1296.2.i.l 2
12.b even 2 1 216.2.a.b 1
20.d odd 2 1 5400.2.a.e 1
20.e even 4 2 5400.2.f.b 2
24.f even 2 1 1728.2.a.s 1
24.h odd 2 1 1728.2.a.r 1
36.f odd 6 2 648.2.i.c 2
36.h even 6 2 648.2.i.e 2
60.h even 2 1 5400.2.a.h 1
60.l odd 4 2 5400.2.f.z 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.a.b 1 12.b even 2 1
216.2.a.c yes 1 4.b odd 2 1
432.2.a.c 1 3.b odd 2 1
432.2.a.f 1 1.a even 1 1 trivial
648.2.i.c 2 36.f odd 6 2
648.2.i.e 2 36.h even 6 2
1296.2.i.g 2 9.c even 3 2
1296.2.i.l 2 9.d odd 6 2
1728.2.a.i 1 8.b even 2 1
1728.2.a.l 1 8.d odd 2 1
1728.2.a.r 1 24.h odd 2 1
1728.2.a.s 1 24.f even 2 1
5400.2.a.e 1 20.d odd 2 1
5400.2.a.h 1 60.h even 2 1
5400.2.f.b 2 20.e even 4 2
5400.2.f.z 2 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(432))$$:

 $$T_{5} - 1$$ $$T_{7} + 3$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-1 + T$$
$7$ $$3 + T$$
$11$ $$-5 + T$$
$13$ $$-4 + T$$
$17$ $$-8 + T$$
$19$ $$2 + T$$
$23$ $$-2 + T$$
$29$ $$6 + T$$
$31$ $$-7 + T$$
$37$ $$6 + T$$
$41$ $$-6 + T$$
$43$ $$-2 + T$$
$47$ $$-6 + T$$
$53$ $$5 + T$$
$59$ $$4 + T$$
$61$ $$8 + T$$
$67$ $$-10 + T$$
$71$ $$8 + T$$
$73$ $$-1 + T$$
$79$ $$16 + T$$
$83$ $$11 + T$$
$89$ $$6 + T$$
$97$ $$1 + T$$