# Properties

 Label 3744.1.dy.a Level $3744$ Weight $1$ Character orbit 3744.dy Analytic conductor $1.868$ Analytic rank $0$ Dimension $4$ Projective image $A_{4}$ CM/RM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.86849940730$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 416) Projective image $$A_{4}$$ Projective field Galois closure of 4.0.10816.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{7} +O(q^{10})$$ $$q + \zeta_{12} q^{7} -\zeta_{12}^{5} q^{11} + q^{13} -\zeta_{12}^{4} q^{17} + \zeta_{12} q^{19} + \zeta_{12}^{5} q^{23} - q^{25} -\zeta_{12}^{2} q^{29} -\zeta_{12}^{2} q^{37} + \zeta_{12}^{2} q^{41} -\zeta_{12} q^{43} + 2 \zeta_{12}^{3} q^{47} -\zeta_{12} q^{59} -\zeta_{12}^{4} q^{61} -\zeta_{12}^{5} q^{67} + \zeta_{12} q^{71} + q^{77} + 2 \zeta_{12}^{3} q^{79} -\zeta_{12}^{2} q^{89} + \zeta_{12} q^{91} -\zeta_{12}^{4} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 4q^{13} + 2q^{17} - 4q^{25} - 2q^{29} - 2q^{37} + 2q^{41} + 2q^{61} + 4q^{77} - 2q^{89} + 2q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$703$$ $$2017$$ $$2081$$ $$2341$$ $$\chi(n)$$ $$-1$$ $$\zeta_{12}^{4}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
991.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 0 0 0 0 −0.866025 0.500000i 0 0 0
991.2 0 0 0 0 0 0.866025 + 0.500000i 0 0 0
1855.1 0 0 0 0 0 −0.866025 + 0.500000i 0 0 0
1855.2 0 0 0 0 0 0.866025 0.500000i 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 inner
13.c even 3 1 inner
52.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.1.dy.a 4
3.b odd 2 1 416.1.bb.a 4
4.b odd 2 1 inner 3744.1.dy.a 4
12.b even 2 1 416.1.bb.a 4
13.c even 3 1 inner 3744.1.dy.a 4
24.f even 2 1 832.1.bb.b 4
24.h odd 2 1 832.1.bb.b 4
39.i odd 6 1 416.1.bb.a 4
48.i odd 4 1 3328.1.v.a 4
48.i odd 4 1 3328.1.v.c 4
48.k even 4 1 3328.1.v.a 4
48.k even 4 1 3328.1.v.c 4
52.j odd 6 1 inner 3744.1.dy.a 4
156.p even 6 1 416.1.bb.a 4
312.bh odd 6 1 832.1.bb.b 4
312.bn even 6 1 832.1.bb.b 4
624.cl even 12 1 3328.1.v.a 4
624.cl even 12 1 3328.1.v.c 4
624.cw odd 12 1 3328.1.v.a 4
624.cw odd 12 1 3328.1.v.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.1.bb.a 4 3.b odd 2 1
416.1.bb.a 4 12.b even 2 1
416.1.bb.a 4 39.i odd 6 1
416.1.bb.a 4 156.p even 6 1
832.1.bb.b 4 24.f even 2 1
832.1.bb.b 4 24.h odd 2 1
832.1.bb.b 4 312.bh odd 6 1
832.1.bb.b 4 312.bn even 6 1
3328.1.v.a 4 48.i odd 4 1
3328.1.v.a 4 48.k even 4 1
3328.1.v.a 4 624.cl even 12 1
3328.1.v.a 4 624.cw odd 12 1
3328.1.v.c 4 48.i odd 4 1
3328.1.v.c 4 48.k even 4 1
3328.1.v.c 4 624.cl even 12 1
3328.1.v.c 4 624.cw odd 12 1
3744.1.dy.a 4 1.a even 1 1 trivial
3744.1.dy.a 4 4.b odd 2 1 inner
3744.1.dy.a 4 13.c even 3 1 inner
3744.1.dy.a 4 52.j odd 6 1 inner

## Hecke kernels

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

## Hecke characteristic polynomials

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