# Properties

 Label 1728.1.e.b Level 1728 Weight 1 Character orbit 1728.e Self dual yes Analytic conductor 0.862 Analytic rank 0 Dimension 1 Projective image $$D_{3}$$ CM discriminant -3 Inner twists 2

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## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$0.862384341830$$ 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) Projective image $$D_{3}$$ Projective field Galois closure of 3.1.108.1 Artin image $D_6$ Artin field Galois closure of 6.0.1492992.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{7} + O(q^{10})$$ $$q + q^{7} + q^{13} - q^{19} + q^{25} - 2q^{31} + q^{37} + 2q^{43} + q^{61} - q^{67} - q^{73} + q^{79} + q^{91} - q^{97} + O(q^{100})$$

## 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 1.00000 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 1728.1.e.b 1
3.b odd 2 1 CM 1728.1.e.b 1
4.b odd 2 1 1728.1.e.a 1
8.b even 2 1 432.1.e.a 1
8.d odd 2 1 108.1.c.a 1
12.b even 2 1 1728.1.e.a 1
24.f even 2 1 108.1.c.a 1
24.h odd 2 1 432.1.e.a 1
40.e odd 2 1 2700.1.g.b 1
40.k even 4 2 2700.1.b.b 2
72.j odd 6 2 1296.1.q.a 2
72.l even 6 2 324.1.g.a 2
72.n even 6 2 1296.1.q.a 2
72.p odd 6 2 324.1.g.a 2
120.m even 2 1 2700.1.g.b 1
120.q odd 4 2 2700.1.b.b 2
216.r odd 18 6 2916.1.k.c 6
216.v even 18 6 2916.1.k.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.1.c.a 1 8.d odd 2 1
108.1.c.a 1 24.f even 2 1
324.1.g.a 2 72.l even 6 2
324.1.g.a 2 72.p odd 6 2
432.1.e.a 1 8.b even 2 1
432.1.e.a 1 24.h odd 2 1
1296.1.q.a 2 72.j odd 6 2
1296.1.q.a 2 72.n even 6 2
1728.1.e.a 1 4.b odd 2 1
1728.1.e.a 1 12.b even 2 1
1728.1.e.b 1 1.a even 1 1 trivial
1728.1.e.b 1 3.b odd 2 1 CM
2700.1.b.b 2 40.k even 4 2
2700.1.b.b 2 120.q odd 4 2
2700.1.g.b 1 40.e odd 2 1
2700.1.g.b 1 120.m even 2 1
2916.1.k.c 6 216.r odd 18 6
2916.1.k.c 6 216.v even 18 6

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} - 1$$ acting on $$S_{1}^{\mathrm{new}}(1728, [\chi])$$.

## Hecke characteristic polynomials

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