# Properties

 Label 2304.3.b.a Level $2304$ Weight $3$ Character orbit 2304.b Analytic conductor $62.779$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$62.7794529086$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 288) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 i q^{5} -4 i q^{7} +O(q^{10})$$ $$q + 4 i q^{5} -4 i q^{7} -16 q^{11} -2 i q^{13} -24 q^{17} + 24 q^{19} + 32 i q^{23} + 9 q^{25} -44 i q^{29} + 52 i q^{31} + 16 q^{35} -18 i q^{37} + 8 q^{41} -56 q^{43} -32 i q^{47} + 33 q^{49} -36 i q^{53} -64 i q^{55} + 32 q^{59} -62 i q^{61} + 8 q^{65} + 80 q^{67} -128 i q^{71} + 66 q^{73} + 64 i q^{77} + 20 i q^{79} -16 q^{83} -96 i q^{85} -144 q^{89} -8 q^{91} + 96 i q^{95} + 94 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 32q^{11} - 48q^{17} + 48q^{19} + 18q^{25} + 32q^{35} + 16q^{41} - 112q^{43} + 66q^{49} + 64q^{59} + 16q^{65} + 160q^{67} + 132q^{73} - 32q^{83} - 288q^{89} - 16q^{91} + 188q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 − 1.00000i 1.00000i
0 0 0 4.00000i 0 4.00000i 0 0 0
127.2 0 0 0 4.00000i 0 4.00000i 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
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.b.a 2
3.b odd 2 1 2304.3.b.i 2
4.b odd 2 1 2304.3.b.h 2
8.b even 2 1 2304.3.b.h 2
8.d odd 2 1 inner 2304.3.b.a 2
12.b even 2 1 2304.3.b.b 2
16.e even 4 1 288.3.g.a 2
16.e even 4 1 576.3.g.h 2
16.f odd 4 1 288.3.g.a 2
16.f odd 4 1 576.3.g.h 2
24.f even 2 1 2304.3.b.i 2
24.h odd 2 1 2304.3.b.b 2
48.i odd 4 1 288.3.g.c yes 2
48.i odd 4 1 576.3.g.d 2
48.k even 4 1 288.3.g.c yes 2
48.k even 4 1 576.3.g.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.3.g.a 2 16.e even 4 1
288.3.g.a 2 16.f odd 4 1
288.3.g.c yes 2 48.i odd 4 1
288.3.g.c yes 2 48.k even 4 1
576.3.g.d 2 48.i odd 4 1
576.3.g.d 2 48.k even 4 1
576.3.g.h 2 16.e even 4 1
576.3.g.h 2 16.f odd 4 1
2304.3.b.a 2 1.a even 1 1 trivial
2304.3.b.a 2 8.d odd 2 1 inner
2304.3.b.b 2 12.b even 2 1
2304.3.b.b 2 24.h odd 2 1
2304.3.b.h 2 4.b odd 2 1
2304.3.b.h 2 8.b even 2 1
2304.3.b.i 2 3.b odd 2 1
2304.3.b.i 2 24.f even 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 16$$ $$T_{7}^{2} + 16$$ $$T_{11} + 16$$ $$T_{17} + 24$$ $$T_{19} - 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$16 + T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$( 16 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$( 24 + T )^{2}$$
$19$ $$( -24 + T )^{2}$$
$23$ $$1024 + T^{2}$$
$29$ $$1936 + T^{2}$$
$31$ $$2704 + T^{2}$$
$37$ $$324 + T^{2}$$
$41$ $$( -8 + T )^{2}$$
$43$ $$( 56 + T )^{2}$$
$47$ $$1024 + T^{2}$$
$53$ $$1296 + T^{2}$$
$59$ $$( -32 + T )^{2}$$
$61$ $$3844 + T^{2}$$
$67$ $$( -80 + T )^{2}$$
$71$ $$16384 + T^{2}$$
$73$ $$( -66 + T )^{2}$$
$79$ $$400 + T^{2}$$
$83$ $$( 16 + T )^{2}$$
$89$ $$( 144 + T )^{2}$$
$97$ $$( -94 + T )^{2}$$