# Properties

 Label 3744.1.gs.a Level $3744$ Weight $1$ Character orbit 3744.gs Analytic conductor $1.868$ Analytic rank $0$ Dimension $4$ Projective image $D_{12}$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3744 = 2^{5} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3744.gs (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.86849940730$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{12}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{12} + \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{12} - \zeta_{12}^{2} ) q^{5} +O(q^{10})$$ $$q + ( \zeta_{12} - \zeta_{12}^{2} ) q^{5} -\zeta_{12} q^{13} + ( 1 - \zeta_{12}^{4} ) q^{17} + ( \zeta_{12}^{2} - \zeta_{12}^{3} + \zeta_{12}^{4} ) q^{25} -\zeta_{12}^{4} q^{29} + ( -\zeta_{12}^{2} + \zeta_{12}^{3} ) q^{37} + ( -\zeta_{12}^{2} - \zeta_{12}^{3} ) q^{41} + \zeta_{12} q^{49} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{53} -\zeta_{12}^{2} q^{61} + ( -\zeta_{12}^{2} + \zeta_{12}^{3} ) q^{65} + ( -\zeta_{12}^{4} + \zeta_{12}^{5} ) q^{73} + ( -1 + \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{5} ) q^{85} + ( \zeta_{12} - \zeta_{12}^{4} ) q^{89} + ( \zeta_{12}^{2} - \zeta_{12}^{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{5} + O(q^{10})$$ $$4 q - 2 q^{5} + 6 q^{17} + 2 q^{29} - 2 q^{37} - 2 q^{41} - 2 q^{61} - 2 q^{65} + 2 q^{73} - 6 q^{85} + 2 q^{89} + 2 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times$$.

 $$n$$ $$703$$ $$2017$$ $$2081$$ $$2341$$ $$\chi(n)$$ $$1$$ $$\zeta_{12}^{5}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
865.1
 −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i
0 0 0 −1.36603 1.36603i 0 0 0 0 0
1441.1 0 0 0 0.366025 + 0.366025i 0 0 0 0 0
2017.1 0 0 0 −1.36603 + 1.36603i 0 0 0 0 0
2593.1 0 0 0 0.366025 0.366025i 0 0 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
13.f odd 12 1 inner
52.l even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.1.gs.a 4
3.b odd 2 1 3744.1.gs.b yes 4
4.b odd 2 1 CM 3744.1.gs.a 4
12.b even 2 1 3744.1.gs.b yes 4
13.f odd 12 1 inner 3744.1.gs.a 4
39.k even 12 1 3744.1.gs.b yes 4
52.l even 12 1 inner 3744.1.gs.a 4
156.v odd 12 1 3744.1.gs.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3744.1.gs.a 4 1.a even 1 1 trivial
3744.1.gs.a 4 4.b odd 2 1 CM
3744.1.gs.a 4 13.f odd 12 1 inner
3744.1.gs.a 4 52.l even 12 1 inner
3744.1.gs.b yes 4 3.b odd 2 1
3744.1.gs.b yes 4 12.b even 2 1
3744.1.gs.b yes 4 39.k even 12 1
3744.1.gs.b yes 4 156.v odd 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3744, [\chi])$$:

 $$T_{5}^{4} + 2 T_{5}^{3} + 2 T_{5}^{2} - 2 T_{5} + 1$$ $$T_{17}^{2} - 3 T_{17} + 3$$

## Hecke characteristic polynomials

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