# Properties

 Label 1728.4.a.r Level 1728 Weight 4 Character orbit 1728.a Self dual yes Analytic conductor 101.955 Analytic rank 1 Dimension 1 CM discriminant -3 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1728.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$101.955300490$$ Analytic rank: $$1$$ 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

 $$f(q)$$ $$=$$ $$q + 37q^{7} + O(q^{10})$$ $$q + 37q^{7} + 19q^{13} - 163q^{19} - 125q^{25} - 308q^{31} - 323q^{37} - 520q^{43} + 1026q^{49} - 719q^{61} - 127q^{67} - 919q^{73} + 1387q^{79} + 703q^{91} - 523q^{97} + O(q^{100})$$

## 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 37.0000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 1728.4.a.r 1
3.b odd 2 1 CM 1728.4.a.r 1
4.b odd 2 1 1728.4.a.o 1
8.b even 2 1 432.4.a.h 1
8.d odd 2 1 108.4.a.b 1
12.b even 2 1 1728.4.a.o 1
24.f even 2 1 108.4.a.b 1
24.h odd 2 1 432.4.a.h 1
72.l even 6 2 324.4.e.e 2
72.p odd 6 2 324.4.e.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.a.b 1 8.d odd 2 1
108.4.a.b 1 24.f even 2 1
324.4.e.e 2 72.l even 6 2
324.4.e.e 2 72.p odd 6 2
432.4.a.h 1 8.b even 2 1
432.4.a.h 1 24.h odd 2 1
1728.4.a.o 1 4.b odd 2 1
1728.4.a.o 1 12.b even 2 1
1728.4.a.r 1 1.a even 1 1 trivial
1728.4.a.r 1 3.b odd 2 1 CM

## Atkin-Lehner signs

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1728))$$:

 $$T_{5}$$ $$T_{7} - 37$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 125 T^{2}$$
$7$ $$1 - 37 T + 343 T^{2}$$
$11$ $$1 + 1331 T^{2}$$
$13$ $$1 - 19 T + 2197 T^{2}$$
$17$ $$1 + 4913 T^{2}$$
$19$ $$1 + 163 T + 6859 T^{2}$$
$23$ $$1 + 12167 T^{2}$$
$29$ $$1 + 24389 T^{2}$$
$31$ $$1 + 308 T + 29791 T^{2}$$
$37$ $$1 + 323 T + 50653 T^{2}$$
$41$ $$1 + 68921 T^{2}$$
$43$ $$1 + 520 T + 79507 T^{2}$$
$47$ $$1 + 103823 T^{2}$$
$53$ $$1 + 148877 T^{2}$$
$59$ $$1 + 205379 T^{2}$$
$61$ $$1 + 719 T + 226981 T^{2}$$
$67$ $$1 + 127 T + 300763 T^{2}$$
$71$ $$1 + 357911 T^{2}$$
$73$ $$1 + 919 T + 389017 T^{2}$$
$79$ $$1 - 1387 T + 493039 T^{2}$$
$83$ $$1 + 571787 T^{2}$$
$89$ $$1 + 704969 T^{2}$$
$97$ $$1 + 523 T + 912673 T^{2}$$