# Properties

 Label 1728.2.a.p Level $1728$ Weight $2$ Character orbit 1728.a Self dual yes Analytic conductor $13.798$ Analytic rank $0$ Dimension $1$ CM discriminant -3 Inner twists $2$

# 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 108) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 5q^{7} + O(q^{10})$$ $$q + 5q^{7} + 7q^{13} + q^{19} - 5q^{25} - 4q^{31} + q^{37} - 8q^{43} + 18q^{49} + 13q^{61} - 11q^{67} + 17q^{73} - 13q^{79} + 35q^{91} + 5q^{97} + O(q^{100})$$

## 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 0 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

## Atkin-Lehner signs

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

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.a.p 1
3.b odd 2 1 CM 1728.2.a.p 1
4.b odd 2 1 1728.2.a.m 1
8.b even 2 1 108.2.a.a 1
8.d odd 2 1 432.2.a.d 1
12.b even 2 1 1728.2.a.m 1
24.f even 2 1 432.2.a.d 1
24.h odd 2 1 108.2.a.a 1
40.f even 2 1 2700.2.a.b 1
40.i odd 4 2 2700.2.d.g 2
56.h odd 2 1 5292.2.a.j 1
72.j odd 6 2 324.2.e.b 2
72.l even 6 2 1296.2.i.j 2
72.n even 6 2 324.2.e.b 2
72.p odd 6 2 1296.2.i.j 2
120.i odd 2 1 2700.2.a.b 1
120.w even 4 2 2700.2.d.g 2
168.i even 2 1 5292.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.a.a 1 8.b even 2 1
108.2.a.a 1 24.h odd 2 1
324.2.e.b 2 72.j odd 6 2
324.2.e.b 2 72.n even 6 2
432.2.a.d 1 8.d odd 2 1
432.2.a.d 1 24.f even 2 1
1296.2.i.j 2 72.l even 6 2
1296.2.i.j 2 72.p odd 6 2
1728.2.a.m 1 4.b odd 2 1
1728.2.a.m 1 12.b even 2 1
1728.2.a.p 1 1.a even 1 1 trivial
1728.2.a.p 1 3.b odd 2 1 CM
2700.2.a.b 1 40.f even 2 1
2700.2.a.b 1 120.i odd 2 1
2700.2.d.g 2 40.i odd 4 2
2700.2.d.g 2 120.w even 4 2
5292.2.a.j 1 56.h odd 2 1
5292.2.a.j 1 168.i even 2 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{7} - 5$$ $$T_{11}$$

## Hecke characteristic polynomials

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