# Properties

 Label 1728.3.e.c Level 1728 Weight 3 Character orbit 1728.e Self dual yes Analytic conductor 47.085 Analytic rank 0 Dimension 1 CM discriminant -3 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1728.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$47.0845896815$$ 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) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 11q^{7} + O(q^{10})$$ $$q + 11q^{7} - 23q^{13} + 37q^{19} + 25q^{25} - 46q^{31} + 73q^{37} + 22q^{43} + 72q^{49} - 47q^{61} + 13q^{67} + 143q^{73} + 11q^{79} - 253q^{91} - 169q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1025.1
 0
0 0 0 0 0 11.0000 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 1728.3.e.c 1
3.b odd 2 1 CM 1728.3.e.c 1
4.b odd 2 1 1728.3.e.b 1
8.b even 2 1 108.3.c.a 1
8.d odd 2 1 432.3.e.a 1
12.b even 2 1 1728.3.e.b 1
24.f even 2 1 432.3.e.a 1
24.h odd 2 1 108.3.c.a 1
40.f even 2 1 2700.3.g.b 1
40.i odd 4 2 2700.3.b.d 2
72.j odd 6 2 324.3.g.a 2
72.l even 6 2 1296.3.q.c 2
72.n even 6 2 324.3.g.a 2
72.p odd 6 2 1296.3.q.c 2
120.i odd 2 1 2700.3.g.b 1
120.w even 4 2 2700.3.b.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.c.a 1 8.b even 2 1
108.3.c.a 1 24.h odd 2 1
324.3.g.a 2 72.j odd 6 2
324.3.g.a 2 72.n even 6 2
432.3.e.a 1 8.d odd 2 1
432.3.e.a 1 24.f even 2 1
1296.3.q.c 2 72.l even 6 2
1296.3.q.c 2 72.p odd 6 2
1728.3.e.b 1 4.b odd 2 1
1728.3.e.b 1 12.b even 2 1
1728.3.e.c 1 1.a even 1 1 trivial
1728.3.e.c 1 3.b odd 2 1 CM
2700.3.b.d 2 40.i odd 4 2
2700.3.b.d 2 120.w even 4 2
2700.3.g.b 1 40.f even 2 1
2700.3.g.b 1 120.i odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1728, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 5 T )( 1 + 5 T )$$
$7$ $$1 - 11 T + 49 T^{2}$$
$11$ $$( 1 - 11 T )( 1 + 11 T )$$
$13$ $$1 + 23 T + 169 T^{2}$$
$17$ $$( 1 - 17 T )( 1 + 17 T )$$
$19$ $$1 - 37 T + 361 T^{2}$$
$23$ $$( 1 - 23 T )( 1 + 23 T )$$
$29$ $$( 1 - 29 T )( 1 + 29 T )$$
$31$ $$1 + 46 T + 961 T^{2}$$
$37$ $$1 - 73 T + 1369 T^{2}$$
$41$ $$( 1 - 41 T )( 1 + 41 T )$$
$43$ $$1 - 22 T + 1849 T^{2}$$
$47$ $$( 1 - 47 T )( 1 + 47 T )$$
$53$ $$( 1 - 53 T )( 1 + 53 T )$$
$59$ $$( 1 - 59 T )( 1 + 59 T )$$
$61$ $$1 + 47 T + 3721 T^{2}$$
$67$ $$1 - 13 T + 4489 T^{2}$$
$71$ $$( 1 - 71 T )( 1 + 71 T )$$
$73$ $$1 - 143 T + 5329 T^{2}$$
$79$ $$1 - 11 T + 6241 T^{2}$$
$83$ $$( 1 - 83 T )( 1 + 83 T )$$
$89$ $$( 1 - 89 T )( 1 + 89 T )$$
$97$ $$1 + 169 T + 9409 T^{2}$$