# Properties

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

# Learn more

## 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 216) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 4q^{5} - 3q^{7} + O(q^{10})$$ $$q - 4q^{5} - 3q^{7} - 4q^{11} - q^{13} - 4q^{17} + q^{19} + 4q^{23} + 11q^{25} - 4q^{31} + 12q^{35} + 9q^{37} + 8q^{43} - 12q^{47} + 2q^{49} + 8q^{53} + 16q^{55} - 4q^{59} + 5q^{61} + 4q^{65} - 11q^{67} + 8q^{71} + q^{73} + 12q^{77} - 5q^{79} - 8q^{83} + 16q^{85} + 12q^{89} + 3q^{91} - 4q^{95} + 5q^{97} + O(q^{100})$$

## 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 −4.00000 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.a 1
3.b odd 2 1 1728.2.a.ba 1
4.b odd 2 1 1728.2.a.b 1
8.b even 2 1 216.2.a.d yes 1
8.d odd 2 1 432.2.a.h 1
12.b even 2 1 1728.2.a.bb 1
24.f even 2 1 432.2.a.a 1
24.h odd 2 1 216.2.a.a 1
40.f even 2 1 5400.2.a.bp 1
40.i odd 4 2 5400.2.f.v 2
72.j odd 6 2 648.2.i.h 2
72.l even 6 2 1296.2.i.q 2
72.n even 6 2 648.2.i.a 2
72.p odd 6 2 1296.2.i.a 2
120.i odd 2 1 5400.2.a.bn 1
120.w even 4 2 5400.2.f.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.a.a 1 24.h odd 2 1
216.2.a.d yes 1 8.b even 2 1
432.2.a.a 1 24.f even 2 1
432.2.a.h 1 8.d odd 2 1
648.2.i.a 2 72.n even 6 2
648.2.i.h 2 72.j odd 6 2
1296.2.i.a 2 72.p odd 6 2
1296.2.i.q 2 72.l even 6 2
1728.2.a.a 1 1.a even 1 1 trivial
1728.2.a.b 1 4.b odd 2 1
1728.2.a.ba 1 3.b odd 2 1
1728.2.a.bb 1 12.b even 2 1
5400.2.a.bn 1 120.i odd 2 1
5400.2.a.bp 1 40.f even 2 1
5400.2.f.e 2 120.w even 4 2
5400.2.f.v 2 40.i odd 4 2

## Hecke kernels

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

 $$T_{5} + 4$$ $$T_{7} + 3$$ $$T_{11} + 4$$

## Hecke characteristic polynomials

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