# Properties

 Label 3744.1.ck.a Level $3744$ Weight $1$ Character orbit 3744.ck 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.ck (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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$A_{4}$$ Projective field: Galois closure of 4.0.876096.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{5} q^{3} -\zeta_{12}^{4} q^{5} -\zeta_{12}^{4} q^{9} +O(q^{10})$$ $$q + \zeta_{12}^{5} q^{3} -\zeta_{12}^{4} q^{5} -\zeta_{12}^{4} q^{9} + \zeta_{12}^{3} q^{11} + \zeta_{12}^{2} q^{13} + \zeta_{12}^{3} q^{15} -\zeta_{12}^{2} q^{17} -\zeta_{12}^{5} q^{19} + \zeta_{12}^{3} q^{27} + q^{29} -\zeta_{12} q^{31} -\zeta_{12}^{2} q^{33} + \zeta_{12}^{4} q^{37} -\zeta_{12} q^{39} + 2 \zeta_{12} q^{43} -\zeta_{12}^{2} q^{45} + \zeta_{12}^{5} q^{47} -\zeta_{12}^{2} q^{49} + \zeta_{12} q^{51} + \zeta_{12} q^{55} + \zeta_{12}^{4} q^{57} -\zeta_{12}^{3} q^{59} + q^{65} -\zeta_{12}^{5} q^{71} + 2 q^{73} -\zeta_{12}^{5} q^{79} -\zeta_{12}^{2} q^{81} -\zeta_{12}^{5} q^{83} - q^{85} + \zeta_{12}^{5} q^{87} + \zeta_{12}^{4} q^{89} + q^{93} -\zeta_{12}^{3} q^{95} + \zeta_{12} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{5} + 2q^{9} + O(q^{10})$$ $$4q + 2q^{5} + 2q^{9} + 2q^{13} - 2q^{17} + 4q^{29} - 2q^{33} - 2q^{37} - 2q^{45} - 2q^{49} - 2q^{57} + 4q^{65} + 8q^{73} - 2q^{81} - 4q^{85} - 2q^{89} + 4q^{93} + O(q^{100})$$

## Character values

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

 $$n$$ $$703$$ $$2017$$ $$2081$$ $$2341$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{2}$$ $$\zeta_{12}^{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
607.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 0 0.500000 0.866025i 0
607.2 0 0.866025 0.500000i 0 0.500000 0.866025i 0 0 0 0.500000 0.866025i 0
2239.1 0 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 0 0.500000 + 0.866025i 0
2239.2 0 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 0 0.500000 + 0.866025i 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
117.h even 3 1 inner
468.y 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.ck.a yes 4
4.b odd 2 1 inner 3744.1.ck.a yes 4
9.c even 3 1 3744.1.cc.a 4
13.c even 3 1 3744.1.cc.a 4
36.f odd 6 1 3744.1.cc.a 4
52.j odd 6 1 3744.1.cc.a 4
117.h even 3 1 inner 3744.1.ck.a yes 4
468.y odd 6 1 inner 3744.1.ck.a yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3744.1.cc.a 4 9.c even 3 1
3744.1.cc.a 4 13.c even 3 1
3744.1.cc.a 4 36.f odd 6 1
3744.1.cc.a 4 52.j odd 6 1
3744.1.ck.a yes 4 1.a even 1 1 trivial
3744.1.ck.a yes 4 4.b odd 2 1 inner
3744.1.ck.a yes 4 117.h even 3 1 inner
3744.1.ck.a yes 4 468.y odd 6 1 inner

## Hecke kernels

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

## Hecke characteristic polynomials

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