# Properties

 Label 2304.3.h.a Level $2304$ Weight $3$ Character orbit 2304.h 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.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$62.7794529086$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 72) 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 $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{5} -12 q^{7} +O(q^{10})$$ $$q + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{5} -12 q^{7} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{11} -8 \zeta_{8}^{2} q^{13} + ( -7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{17} -16 \zeta_{8}^{2} q^{19} + ( -28 \zeta_{8} - 28 \zeta_{8}^{3} ) q^{23} + 25 q^{25} + ( -21 \zeta_{8} + 21 \zeta_{8}^{3} ) q^{29} -4 q^{31} + ( 60 \zeta_{8} - 60 \zeta_{8}^{3} ) q^{35} -30 \zeta_{8}^{2} q^{37} + ( -15 \zeta_{8} - 15 \zeta_{8}^{3} ) q^{41} + 8 \zeta_{8}^{2} q^{43} + ( 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{47} + 95 q^{49} + ( -35 \zeta_{8} + 35 \zeta_{8}^{3} ) q^{53} -40 q^{55} + ( -56 \zeta_{8} + 56 \zeta_{8}^{3} ) q^{59} -14 \zeta_{8}^{2} q^{61} + ( 40 \zeta_{8} + 40 \zeta_{8}^{3} ) q^{65} -88 \zeta_{8}^{2} q^{67} + ( -20 \zeta_{8} - 20 \zeta_{8}^{3} ) q^{71} + 80 q^{73} + ( -48 \zeta_{8} + 48 \zeta_{8}^{3} ) q^{77} + 100 q^{79} + ( -92 \zeta_{8} + 92 \zeta_{8}^{3} ) q^{83} + 70 \zeta_{8}^{2} q^{85} + ( 105 \zeta_{8} + 105 \zeta_{8}^{3} ) q^{89} + 96 \zeta_{8}^{2} q^{91} + ( 80 \zeta_{8} + 80 \zeta_{8}^{3} ) q^{95} -112 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 48q^{7} + O(q^{10})$$ $$4q - 48q^{7} + 100q^{25} - 16q^{31} + 380q^{49} - 160q^{55} + 320q^{73} + 400q^{79} - 448q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2177.1
 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i
0 0 0 −7.07107 0 −12.0000 0 0 0
2177.2 0 0 0 −7.07107 0 −12.0000 0 0 0
2177.3 0 0 0 7.07107 0 −12.0000 0 0 0
2177.4 0 0 0 7.07107 0 −12.0000 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
3.b odd 2 1 inner
8.b even 2 1 inner
24.h 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.h.a 4
3.b odd 2 1 inner 2304.3.h.a 4
4.b odd 2 1 2304.3.h.h 4
8.b even 2 1 inner 2304.3.h.a 4
8.d odd 2 1 2304.3.h.h 4
12.b even 2 1 2304.3.h.h 4
16.e even 4 1 72.3.e.a 2
16.e even 4 1 576.3.e.h 2
16.f odd 4 1 144.3.e.a 2
16.f odd 4 1 576.3.e.a 2
24.f even 2 1 2304.3.h.h 4
24.h odd 2 1 inner 2304.3.h.a 4
48.i odd 4 1 72.3.e.a 2
48.i odd 4 1 576.3.e.h 2
48.k even 4 1 144.3.e.a 2
48.k even 4 1 576.3.e.a 2
80.i odd 4 1 1800.3.c.a 4
80.j even 4 1 3600.3.c.c 4
80.k odd 4 1 3600.3.l.l 2
80.q even 4 1 1800.3.l.a 2
80.s even 4 1 3600.3.c.c 4
80.t odd 4 1 1800.3.c.a 4
112.l odd 4 1 3528.3.d.a 2
144.u even 12 2 1296.3.q.k 4
144.v odd 12 2 1296.3.q.k 4
144.w odd 12 2 648.3.m.a 4
144.x even 12 2 648.3.m.a 4
240.t even 4 1 3600.3.l.l 2
240.z odd 4 1 3600.3.c.c 4
240.bb even 4 1 1800.3.c.a 4
240.bd odd 4 1 3600.3.c.c 4
240.bf even 4 1 1800.3.c.a 4
240.bm odd 4 1 1800.3.l.a 2
336.y even 4 1 3528.3.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.e.a 2 16.e even 4 1
72.3.e.a 2 48.i odd 4 1
144.3.e.a 2 16.f odd 4 1
144.3.e.a 2 48.k even 4 1
576.3.e.a 2 16.f odd 4 1
576.3.e.a 2 48.k even 4 1
576.3.e.h 2 16.e even 4 1
576.3.e.h 2 48.i odd 4 1
648.3.m.a 4 144.w odd 12 2
648.3.m.a 4 144.x even 12 2
1296.3.q.k 4 144.u even 12 2
1296.3.q.k 4 144.v odd 12 2
1800.3.c.a 4 80.i odd 4 1
1800.3.c.a 4 80.t odd 4 1
1800.3.c.a 4 240.bb even 4 1
1800.3.c.a 4 240.bf even 4 1
1800.3.l.a 2 80.q even 4 1
1800.3.l.a 2 240.bm odd 4 1
2304.3.h.a 4 1.a even 1 1 trivial
2304.3.h.a 4 3.b odd 2 1 inner
2304.3.h.a 4 8.b even 2 1 inner
2304.3.h.a 4 24.h odd 2 1 inner
2304.3.h.h 4 4.b odd 2 1
2304.3.h.h 4 8.d odd 2 1
2304.3.h.h 4 12.b even 2 1
2304.3.h.h 4 24.f even 2 1
3528.3.d.a 2 112.l odd 4 1
3528.3.d.a 2 336.y even 4 1
3600.3.c.c 4 80.j even 4 1
3600.3.c.c 4 80.s even 4 1
3600.3.c.c 4 240.z odd 4 1
3600.3.c.c 4 240.bd odd 4 1
3600.3.l.l 2 80.k odd 4 1
3600.3.l.l 2 240.t even 4 1

## Hecke kernels

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

 $$T_{5}^{2} - 50$$ $$T_{7} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -50 + T^{2} )^{2}$$
$7$ $$( 12 + T )^{4}$$
$11$ $$( -32 + T^{2} )^{2}$$
$13$ $$( 64 + T^{2} )^{2}$$
$17$ $$( 98 + T^{2} )^{2}$$
$19$ $$( 256 + T^{2} )^{2}$$
$23$ $$( 1568 + T^{2} )^{2}$$
$29$ $$( -882 + T^{2} )^{2}$$
$31$ $$( 4 + T )^{4}$$
$37$ $$( 900 + T^{2} )^{2}$$
$41$ $$( 450 + T^{2} )^{2}$$
$43$ $$( 64 + T^{2} )^{2}$$
$47$ $$( 288 + T^{2} )^{2}$$
$53$ $$( -2450 + T^{2} )^{2}$$
$59$ $$( -6272 + T^{2} )^{2}$$
$61$ $$( 196 + T^{2} )^{2}$$
$67$ $$( 7744 + T^{2} )^{2}$$
$71$ $$( 800 + T^{2} )^{2}$$
$73$ $$( -80 + T )^{4}$$
$79$ $$( -100 + T )^{4}$$
$83$ $$( -16928 + T^{2} )^{2}$$
$89$ $$( 22050 + T^{2} )^{2}$$
$97$ $$( 112 + T )^{4}$$