# Properties

 Label 1728.2.a.h 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

 $$f(q)$$ $$=$$ $$q - 2 q^{5} + 3 q^{7} + O(q^{10})$$ $$q - 2 q^{5} + 3 q^{7} - 6 q^{11} + 3 q^{13} - 2 q^{17} + 3 q^{19} + 6 q^{23} - q^{25} + 8 q^{29} - 6 q^{35} - 7 q^{37} + 8 q^{41} + 12 q^{43} + 6 q^{47} + 2 q^{49} - 4 q^{53} + 12 q^{55} - 6 q^{59} + q^{61} - 6 q^{65} + 3 q^{67} + 12 q^{71} - 15 q^{73} - 18 q^{77} + 9 q^{79} + 12 q^{83} + 4 q^{85} - 10 q^{89} + 9 q^{91} - 6 q^{95} + 9 q^{97} + O(q^{100})$$

## 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 −2.00000 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.h 1
3.b odd 2 1 1728.2.a.x 1
4.b odd 2 1 1728.2.a.e 1
8.b even 2 1 864.2.a.l yes 1
8.d odd 2 1 864.2.a.i yes 1
12.b even 2 1 1728.2.a.u 1
24.f even 2 1 864.2.a.a 1
24.h odd 2 1 864.2.a.d yes 1
72.j odd 6 2 2592.2.i.r 2
72.l even 6 2 2592.2.i.v 2
72.n even 6 2 2592.2.i.c 2
72.p odd 6 2 2592.2.i.g 2

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

## Hecke kernels

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

 $$T_{5} + 2$$ $$T_{7} - 3$$ $$T_{11} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$2 + T$$
$7$ $$-3 + T$$
$11$ $$6 + T$$
$13$ $$-3 + T$$
$17$ $$2 + T$$
$19$ $$-3 + T$$
$23$ $$-6 + T$$
$29$ $$-8 + T$$
$31$ $$T$$
$37$ $$7 + T$$
$41$ $$-8 + T$$
$43$ $$-12 + T$$
$47$ $$-6 + T$$
$53$ $$4 + T$$
$59$ $$6 + T$$
$61$ $$-1 + T$$
$67$ $$-3 + T$$
$71$ $$-12 + T$$
$73$ $$15 + T$$
$79$ $$-9 + T$$
$83$ $$-12 + T$$
$89$ $$10 + T$$
$97$ $$-9 + T$$