# Properties

 Label 1728.2.a.j Level $1728$ Weight $2$ Character orbit 1728.a Self dual yes Analytic conductor $13.798$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1728.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.7981494693$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 864) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{5} - 3q^{7} + O(q^{10})$$ $$q - q^{5} - 3q^{7} + 3q^{11} - 4q^{17} + 6q^{19} + 6q^{23} - 4q^{25} - 2q^{29} - 9q^{31} + 3q^{35} + 2q^{37} + 10q^{41} + 6q^{43} + 6q^{47} + 2q^{49} + 13q^{53} - 3q^{55} + 12q^{59} - 8q^{61} + 6q^{67} + 12q^{71} + 9q^{73} - 9q^{77} + 3q^{83} + 4q^{85} - 14q^{89} - 6q^{95} - 9q^{97} + O(q^{100})$$

## 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 1728.2.a.j 1
3.b odd 2 1 1728.2.a.q 1
4.b odd 2 1 1728.2.a.k 1
8.b even 2 1 864.2.a.g yes 1
8.d odd 2 1 864.2.a.h yes 1
12.b even 2 1 1728.2.a.t 1
24.f even 2 1 864.2.a.f yes 1
24.h odd 2 1 864.2.a.e 1
72.j odd 6 2 2592.2.i.p 2
72.l even 6 2 2592.2.i.m 2
72.n even 6 2 2592.2.i.l 2
72.p odd 6 2 2592.2.i.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.a.e 1 24.h odd 2 1
864.2.a.f yes 1 24.f even 2 1
864.2.a.g yes 1 8.b even 2 1
864.2.a.h yes 1 8.d odd 2 1
1728.2.a.j 1 1.a even 1 1 trivial
1728.2.a.k 1 4.b odd 2 1
1728.2.a.q 1 3.b odd 2 1
1728.2.a.t 1 12.b even 2 1
2592.2.i.i 2 72.p odd 6 2
2592.2.i.l 2 72.n even 6 2
2592.2.i.m 2 72.l even 6 2
2592.2.i.p 2 72.j odd 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1728))$$:

 $$T_{5} + 1$$ $$T_{7} + 3$$ $$T_{11} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$1 + T$$
$7$ $$3 + T$$
$11$ $$-3 + T$$
$13$ $$T$$
$17$ $$4 + T$$
$19$ $$-6 + T$$
$23$ $$-6 + T$$
$29$ $$2 + T$$
$31$ $$9 + T$$
$37$ $$-2 + T$$
$41$ $$-10 + T$$
$43$ $$-6 + T$$
$47$ $$-6 + T$$
$53$ $$-13 + T$$
$59$ $$-12 + T$$
$61$ $$8 + T$$
$67$ $$-6 + T$$
$71$ $$-12 + T$$
$73$ $$-9 + T$$
$79$ $$T$$
$83$ $$-3 + T$$
$89$ $$14 + T$$
$97$ $$9 + T$$