# Properties

 Label 2304.1.t.a Level $2304$ Weight $1$ Character orbit 2304.t Analytic conductor $1.150$ Analytic rank $0$ Dimension $4$ Projective image $A_{4}$ CM/RM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2304.t (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.14984578911$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 288) Projective image: $$A_{4}$$ Projective field: Galois closure of 4.0.5184.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - q^{3} + \zeta_{12}^{5} q^{5} -\zeta_{12} q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + \zeta_{12}^{5} q^{5} -\zeta_{12} q^{7} + q^{9} -\zeta_{12}^{4} q^{11} + \zeta_{12}^{5} q^{13} -\zeta_{12}^{5} q^{15} + \zeta_{12} q^{21} -\zeta_{12}^{5} q^{23} - q^{27} + \zeta_{12} q^{29} + \zeta_{12}^{5} q^{31} + \zeta_{12}^{4} q^{33} + q^{35} -\zeta_{12}^{5} q^{39} + \zeta_{12}^{2} q^{41} + \zeta_{12}^{4} q^{43} + \zeta_{12}^{5} q^{45} + \zeta_{12} q^{47} + \zeta_{12}^{3} q^{55} + \zeta_{12}^{2} q^{59} + \zeta_{12} q^{61} -\zeta_{12} q^{63} -\zeta_{12}^{4} q^{65} + \zeta_{12}^{2} q^{67} + \zeta_{12}^{5} q^{69} + 2 \zeta_{12}^{3} q^{71} + \zeta_{12}^{5} q^{77} -\zeta_{12} q^{79} + q^{81} -\zeta_{12}^{4} q^{83} -\zeta_{12} q^{87} + q^{91} -\zeta_{12}^{5} q^{93} + \zeta_{12}^{4} q^{97} -\zeta_{12}^{4} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} + 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{3} + 4 q^{9} + 2 q^{11} - 4 q^{27} - 2 q^{33} + 4 q^{35} + 2 q^{41} - 2 q^{43} + 2 q^{59} + 2 q^{65} + 2 q^{67} + 4 q^{81} + 2 q^{83} + 4 q^{91} - 2 q^{97} + 2 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$$.

 $$n$$ $$1279$$ $$1793$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
895.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 −1.00000 0 −0.866025 0.500000i 0 −0.866025 + 0.500000i 0 1.00000 0
895.2 0 −1.00000 0 0.866025 + 0.500000i 0 0.866025 0.500000i 0 1.00000 0
1663.1 0 −1.00000 0 −0.866025 + 0.500000i 0 −0.866025 0.500000i 0 1.00000 0
1663.2 0 −1.00000 0 0.866025 0.500000i 0 0.866025 + 0.500000i 0 1.00000 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
8.d odd 2 1 inner
9.c even 3 1 inner
72.p odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.1.t.a 4
4.b odd 2 1 2304.1.t.b 4
8.b even 2 1 2304.1.t.b 4
8.d odd 2 1 inner 2304.1.t.a 4
9.c even 3 1 inner 2304.1.t.a 4
16.e even 4 1 288.1.o.a 4
16.e even 4 1 576.1.o.a 4
16.f odd 4 1 288.1.o.a 4
16.f odd 4 1 576.1.o.a 4
36.f odd 6 1 2304.1.t.b 4
48.i odd 4 1 864.1.o.a 4
48.i odd 4 1 1728.1.o.a 4
48.k even 4 1 864.1.o.a 4
48.k even 4 1 1728.1.o.a 4
72.n even 6 1 2304.1.t.b 4
72.p odd 6 1 inner 2304.1.t.a 4
144.u even 12 1 864.1.o.a 4
144.u even 12 1 1728.1.o.a 4
144.u even 12 1 2592.1.g.b 2
144.v odd 12 1 288.1.o.a 4
144.v odd 12 1 576.1.o.a 4
144.v odd 12 1 2592.1.g.a 2
144.w odd 12 1 864.1.o.a 4
144.w odd 12 1 1728.1.o.a 4
144.w odd 12 1 2592.1.g.b 2
144.x even 12 1 288.1.o.a 4
144.x even 12 1 576.1.o.a 4
144.x even 12 1 2592.1.g.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.1.o.a 4 16.e even 4 1
288.1.o.a 4 16.f odd 4 1
288.1.o.a 4 144.v odd 12 1
288.1.o.a 4 144.x even 12 1
576.1.o.a 4 16.e even 4 1
576.1.o.a 4 16.f odd 4 1
576.1.o.a 4 144.v odd 12 1
576.1.o.a 4 144.x even 12 1
864.1.o.a 4 48.i odd 4 1
864.1.o.a 4 48.k even 4 1
864.1.o.a 4 144.u even 12 1
864.1.o.a 4 144.w odd 12 1
1728.1.o.a 4 48.i odd 4 1
1728.1.o.a 4 48.k even 4 1
1728.1.o.a 4 144.u even 12 1
1728.1.o.a 4 144.w odd 12 1
2304.1.t.a 4 1.a even 1 1 trivial
2304.1.t.a 4 8.d odd 2 1 inner
2304.1.t.a 4 9.c even 3 1 inner
2304.1.t.a 4 72.p odd 6 1 inner
2304.1.t.b 4 4.b odd 2 1
2304.1.t.b 4 8.b even 2 1
2304.1.t.b 4 36.f odd 6 1
2304.1.t.b 4 72.n even 6 1
2592.1.g.a 2 144.v odd 12 1
2592.1.g.a 2 144.x even 12 1
2592.1.g.b 2 144.u even 12 1
2592.1.g.b 2 144.w odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{2} - T_{11} + 1$$ acting on $$S_{1}^{\mathrm{new}}(2304, [\chi])$$.

## Hecke characteristic polynomials

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