# Properties

 Label 2304.3.g.u Level $2304$ Weight $3$ Character orbit 2304.g Analytic conductor $62.779$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$62.7794529086$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 64) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{5} -8 \zeta_{12}^{3} q^{7} +O(q^{10})$$ $$q + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{5} -8 \zeta_{12}^{3} q^{7} + ( 2 - 4 \zeta_{12}^{2} ) q^{11} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{13} + 6 q^{17} + ( -14 + 28 \zeta_{12}^{2} ) q^{19} + 24 \zeta_{12}^{3} q^{23} + 23 q^{25} + ( 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{29} -32 \zeta_{12}^{3} q^{31} + ( -32 + 64 \zeta_{12}^{2} ) q^{35} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{37} + 66 q^{41} + ( -18 + 36 \zeta_{12}^{2} ) q^{43} -48 \zeta_{12}^{3} q^{47} -15 q^{49} + ( -104 \zeta_{12} + 52 \zeta_{12}^{3} ) q^{53} + 24 \zeta_{12}^{3} q^{55} + ( 18 - 36 \zeta_{12}^{2} ) q^{59} + ( 104 \zeta_{12} - 52 \zeta_{12}^{3} ) q^{61} -48 q^{65} + ( -46 + 92 \zeta_{12}^{2} ) q^{67} -120 \zeta_{12}^{3} q^{71} -58 q^{73} + ( -32 \zeta_{12} + 16 \zeta_{12}^{3} ) q^{77} + 16 \zeta_{12}^{3} q^{79} + ( -34 + 68 \zeta_{12}^{2} ) q^{83} + ( -48 \zeta_{12} + 24 \zeta_{12}^{3} ) q^{85} -102 q^{89} + ( 32 - 64 \zeta_{12}^{2} ) q^{91} -168 \zeta_{12}^{3} q^{95} + 26 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 24q^{17} + 92q^{25} + 264q^{41} - 60q^{49} - 192q^{65} - 232q^{73} - 408q^{89} + 104q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$1279$$ $$1793$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1279.1
 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i
0 0 0 −6.92820 0 8.00000i 0 0 0
1279.2 0 0 0 −6.92820 0 8.00000i 0 0 0
1279.3 0 0 0 6.92820 0 8.00000i 0 0 0
1279.4 0 0 0 6.92820 0 8.00000i 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
8.b even 2 1 inner
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.3.g.u 4
3.b odd 2 1 256.3.c.g 4
4.b odd 2 1 inner 2304.3.g.u 4
8.b even 2 1 inner 2304.3.g.u 4
8.d odd 2 1 inner 2304.3.g.u 4
12.b even 2 1 256.3.c.g 4
16.e even 4 2 576.3.b.d 4
16.f odd 4 2 576.3.b.d 4
24.f even 2 1 256.3.c.g 4
24.h odd 2 1 256.3.c.g 4
48.i odd 4 2 64.3.d.a 4
48.k even 4 2 64.3.d.a 4
240.t even 4 2 1600.3.g.c 4
240.z odd 4 1 1600.3.e.a 4
240.z odd 4 1 1600.3.e.f 4
240.bb even 4 1 1600.3.e.a 4
240.bb even 4 1 1600.3.e.f 4
240.bd odd 4 1 1600.3.e.a 4
240.bd odd 4 1 1600.3.e.f 4
240.bf even 4 1 1600.3.e.a 4
240.bf even 4 1 1600.3.e.f 4
240.bm odd 4 2 1600.3.g.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.3.d.a 4 48.i odd 4 2
64.3.d.a 4 48.k even 4 2
256.3.c.g 4 3.b odd 2 1
256.3.c.g 4 12.b even 2 1
256.3.c.g 4 24.f even 2 1
256.3.c.g 4 24.h odd 2 1
576.3.b.d 4 16.e even 4 2
576.3.b.d 4 16.f odd 4 2
1600.3.e.a 4 240.z odd 4 1
1600.3.e.a 4 240.bb even 4 1
1600.3.e.a 4 240.bd odd 4 1
1600.3.e.a 4 240.bf even 4 1
1600.3.e.f 4 240.z odd 4 1
1600.3.e.f 4 240.bb even 4 1
1600.3.e.f 4 240.bd odd 4 1
1600.3.e.f 4 240.bf even 4 1
1600.3.g.c 4 240.t even 4 2
1600.3.g.c 4 240.bm odd 4 2
2304.3.g.u 4 1.a even 1 1 trivial
2304.3.g.u 4 4.b odd 2 1 inner
2304.3.g.u 4 8.b even 2 1 inner
2304.3.g.u 4 8.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(2304, [\chi])$$:

 $$T_{5}^{2} - 48$$ $$T_{7}^{2} + 64$$ $$T_{11}^{2} + 12$$ $$T_{13}^{2} - 48$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -48 + T^{2} )^{2}$$
$7$ $$( 64 + T^{2} )^{2}$$
$11$ $$( 12 + T^{2} )^{2}$$
$13$ $$( -48 + T^{2} )^{2}$$
$17$ $$( -6 + T )^{4}$$
$19$ $$( 588 + T^{2} )^{2}$$
$23$ $$( 576 + T^{2} )^{2}$$
$29$ $$( -432 + T^{2} )^{2}$$
$31$ $$( 1024 + T^{2} )^{2}$$
$37$ $$( -48 + T^{2} )^{2}$$
$41$ $$( -66 + T )^{4}$$
$43$ $$( 972 + T^{2} )^{2}$$
$47$ $$( 2304 + T^{2} )^{2}$$
$53$ $$( -8112 + T^{2} )^{2}$$
$59$ $$( 972 + T^{2} )^{2}$$
$61$ $$( -8112 + T^{2} )^{2}$$
$67$ $$( 6348 + T^{2} )^{2}$$
$71$ $$( 14400 + T^{2} )^{2}$$
$73$ $$( 58 + T )^{4}$$
$79$ $$( 256 + T^{2} )^{2}$$
$83$ $$( 3468 + T^{2} )^{2}$$
$89$ $$( 102 + T )^{4}$$
$97$ $$( -26 + T )^{4}$$