# Properties

 Label 3744.1.dx.a Level $3744$ Weight $1$ Character orbit 3744.dx Analytic conductor $1.868$ Analytic rank $0$ Dimension $8$ Projective image $D_{12}$ CM discriminant -4 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3744 = 2^{5} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3744.dx (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.86849940730$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{24} + \zeta_{24}^{11} ) q^{5} +O(q^{10})$$ $$q + ( -\zeta_{24} + \zeta_{24}^{11} ) q^{5} -\zeta_{24}^{2} q^{13} + ( \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{17} + ( 1 + \zeta_{24}^{2} - \zeta_{24}^{10} ) q^{25} + ( -\zeta_{24}^{7} + \zeta_{24}^{9} ) q^{29} + ( 1 + \zeta_{24}^{4} ) q^{37} + ( \zeta_{24}^{7} + \zeta_{24}^{9} ) q^{41} + \zeta_{24}^{8} q^{49} + ( \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{53} + \zeta_{24}^{4} q^{61} + ( \zeta_{24} + \zeta_{24}^{3} ) q^{65} + \zeta_{24}^{6} q^{73} + ( -\zeta_{24}^{2} + \zeta_{24}^{6} ) q^{85} + ( -\zeta_{24}^{5} - \zeta_{24}^{11} ) q^{89} + 2 \zeta_{24}^{10} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 8q^{25} + 12q^{37} - 4q^{49} + 4q^{61} + O(q^{100})$$

## Character values

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

 $$n$$ $$703$$ $$2017$$ $$2081$$ $$2341$$ $$\chi(n)$$ $$1$$ $$\zeta_{24}^{4}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1889.1
 0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i −0.965926 − 0.258819i 0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i −0.965926 + 0.258819i
0 0 0 −1.93185 0 0 0 0 0
1889.2 0 0 0 −0.517638 0 0 0 0 0
1889.3 0 0 0 0.517638 0 0 0 0 0
1889.4 0 0 0 1.93185 0 0 0 0 0
2753.1 0 0 0 −1.93185 0 0 0 0 0
2753.2 0 0 0 −0.517638 0 0 0 0 0
2753.3 0 0 0 0.517638 0 0 0 0 0
2753.4 0 0 0 1.93185 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2753.4 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 $$\Q(\sqrt{-1})$$
3.b odd 2 1 inner
12.b even 2 1 inner
13.e even 6 1 inner
39.h odd 6 1 inner
52.i odd 6 1 inner
156.r even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.1.dx.a 8
3.b odd 2 1 inner 3744.1.dx.a 8
4.b odd 2 1 CM 3744.1.dx.a 8
12.b even 2 1 inner 3744.1.dx.a 8
13.e even 6 1 inner 3744.1.dx.a 8
39.h odd 6 1 inner 3744.1.dx.a 8
52.i odd 6 1 inner 3744.1.dx.a 8
156.r even 6 1 inner 3744.1.dx.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3744.1.dx.a 8 1.a even 1 1 trivial
3744.1.dx.a 8 3.b odd 2 1 inner
3744.1.dx.a 8 4.b odd 2 1 CM
3744.1.dx.a 8 12.b even 2 1 inner
3744.1.dx.a 8 13.e even 6 1 inner
3744.1.dx.a 8 39.h odd 6 1 inner
3744.1.dx.a 8 52.i odd 6 1 inner
3744.1.dx.a 8 156.r even 6 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3744, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 1 - 4 T^{2} + T^{4} )^{2}$$
$7$ $$T^{8}$$
$11$ $$T^{8}$$
$13$ $$( 1 - T^{2} + T^{4} )^{2}$$
$17$ $$1 - 4 T^{2} + 15 T^{4} - 4 T^{6} + T^{8}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$1 - 4 T^{2} + 15 T^{4} - 4 T^{6} + T^{8}$$
$31$ $$T^{8}$$
$37$ $$( 3 - 3 T + T^{2} )^{4}$$
$41$ $$1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8}$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$( 1 + 4 T^{2} + T^{4} )^{2}$$
$59$ $$T^{8}$$
$61$ $$( 1 - T + T^{2} )^{4}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$( 1 + T^{2} )^{4}$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$( 4 + 2 T^{2} + T^{4} )^{2}$$
$97$ $$( 16 - 4 T^{2} + T^{4} )^{2}$$