# Properties

 Label 3744.1.gs.c 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: no (minimal twist has level 416) Projective image $$D_{12}$$ Projective field Galois closure of 12.0.469804094334435328.7

## $q$-expansion

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

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

## 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 −0.366025 0.366025i 0 0 0 0 0
1441.1 0 0 0 1.36603 + 1.36603i 0 0 0 0 0
2017.1 0 0 0 −0.366025 + 0.366025i 0 0 0 0 0
2593.1 0 0 0 1.36603 1.36603i 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.c 4
3.b odd 2 1 416.1.bl.a 4
4.b odd 2 1 CM 3744.1.gs.c 4
12.b even 2 1 416.1.bl.a 4
13.f odd 12 1 inner 3744.1.gs.c 4
24.f even 2 1 832.1.bl.a 4
24.h odd 2 1 832.1.bl.a 4
39.k even 12 1 416.1.bl.a 4
48.i odd 4 1 3328.1.bv.a 4
48.i odd 4 1 3328.1.bv.b 4
48.k even 4 1 3328.1.bv.a 4
48.k even 4 1 3328.1.bv.b 4
52.l even 12 1 inner 3744.1.gs.c 4
156.v odd 12 1 416.1.bl.a 4
312.bo even 12 1 832.1.bl.a 4
312.bq odd 12 1 832.1.bl.a 4
624.ce even 12 1 3328.1.bv.a 4
624.cg odd 12 1 3328.1.bv.a 4
624.cy odd 12 1 3328.1.bv.b 4
624.da even 12 1 3328.1.bv.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.1.bl.a 4 3.b odd 2 1
416.1.bl.a 4 12.b even 2 1
416.1.bl.a 4 39.k even 12 1
416.1.bl.a 4 156.v odd 12 1
832.1.bl.a 4 24.f even 2 1
832.1.bl.a 4 24.h odd 2 1
832.1.bl.a 4 312.bo even 12 1
832.1.bl.a 4 312.bq odd 12 1
3328.1.bv.a 4 48.i odd 4 1
3328.1.bv.a 4 48.k even 4 1
3328.1.bv.a 4 624.ce even 12 1
3328.1.bv.a 4 624.cg odd 12 1
3328.1.bv.b 4 48.i odd 4 1
3328.1.bv.b 4 48.k even 4 1
3328.1.bv.b 4 624.cy odd 12 1
3328.1.bv.b 4 624.da even 12 1
3744.1.gs.c 4 1.a even 1 1 trivial
3744.1.gs.c 4 4.b odd 2 1 CM
3744.1.gs.c 4 13.f odd 12 1 inner
3744.1.gs.c 4 52.l even 12 1 inner

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$1 + 2 T + 2 T^{2} - 2 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$1 - T^{2} + T^{4}$$
$17$ $$1 - T^{2} + T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$9 + 3 T^{2} + T^{4}$$
$31$ $$T^{4}$$
$37$ $$1 + 4 T + 5 T^{2} + 2 T^{3} + T^{4}$$
$41$ $$1 - 2 T + 5 T^{2} - 4 T^{3} + T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$( 1 + T )^{4}$$
$59$ $$T^{4}$$
$61$ $$( 1 + T + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$1 - 2 T + 2 T^{2} + 2 T^{3} + T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4}$$
$97$ $$4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4}$$