# Properties

 Label 2304.1.b.a Level 2304 Weight 1 Character orbit 2304.b Analytic conductor 1.150 Analytic rank 0 Dimension 2 Projective image $$D_{2}$$ CM/RM discs -3, -4, 12 Inner twists 8

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.14984578911$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 144) Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\zeta_{12})$$ Artin image $D_4:C_2$ Artin field Galois closure of 8.0.3057647616.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q +O(q^{10})$$ $$q -2 i q^{13} + q^{25} -2 i q^{37} + q^{49} + 2 i q^{61} + 2 q^{73} -2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 2q^{25} + 2q^{49} + 4q^{73} - 4q^{97} + O(q^{100})$$

## 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 0 0 0 0 0 0
127.2 0 0 0 0 0 0 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 CM by $$\Q(\sqrt{-3})$$
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
12.b even 2 1 RM by $$\Q(\sqrt{3})$$
8.b even 2 1 inner
8.d odd 2 1 inner
24.f 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 2304.1.b.a 2
3.b odd 2 1 CM 2304.1.b.a 2
4.b odd 2 1 CM 2304.1.b.a 2
8.b even 2 1 inner 2304.1.b.a 2
8.d odd 2 1 inner 2304.1.b.a 2
12.b even 2 1 RM 2304.1.b.a 2
16.e even 4 1 144.1.g.a 1
16.e even 4 1 576.1.g.a 1
16.f odd 4 1 144.1.g.a 1
16.f odd 4 1 576.1.g.a 1
24.f even 2 1 inner 2304.1.b.a 2
24.h odd 2 1 inner 2304.1.b.a 2
48.i odd 4 1 144.1.g.a 1
48.i odd 4 1 576.1.g.a 1
48.k even 4 1 144.1.g.a 1
48.k even 4 1 576.1.g.a 1
80.i odd 4 1 3600.1.j.a 2
80.j even 4 1 3600.1.j.a 2
80.k odd 4 1 3600.1.e.b 1
80.q even 4 1 3600.1.e.b 1
80.s even 4 1 3600.1.j.a 2
80.t odd 4 1 3600.1.j.a 2
144.u even 12 2 1296.1.o.b 2
144.v odd 12 2 1296.1.o.b 2
144.w odd 12 2 1296.1.o.b 2
144.x even 12 2 1296.1.o.b 2
240.t even 4 1 3600.1.e.b 1
240.z odd 4 1 3600.1.j.a 2
240.bb even 4 1 3600.1.j.a 2
240.bd odd 4 1 3600.1.j.a 2
240.bf even 4 1 3600.1.j.a 2
240.bm odd 4 1 3600.1.e.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.1.g.a 1 16.e even 4 1
144.1.g.a 1 16.f odd 4 1
144.1.g.a 1 48.i odd 4 1
144.1.g.a 1 48.k even 4 1
576.1.g.a 1 16.e even 4 1
576.1.g.a 1 16.f odd 4 1
576.1.g.a 1 48.i odd 4 1
576.1.g.a 1 48.k even 4 1
1296.1.o.b 2 144.u even 12 2
1296.1.o.b 2 144.v odd 12 2
1296.1.o.b 2 144.w odd 12 2
1296.1.o.b 2 144.x even 12 2
2304.1.b.a 2 1.a even 1 1 trivial
2304.1.b.a 2 3.b odd 2 1 CM
2304.1.b.a 2 4.b odd 2 1 CM
2304.1.b.a 2 8.b even 2 1 inner
2304.1.b.a 2 8.d odd 2 1 inner
2304.1.b.a 2 12.b even 2 1 RM
2304.1.b.a 2 24.f even 2 1 inner
2304.1.b.a 2 24.h odd 2 1 inner
3600.1.e.b 1 80.k odd 4 1
3600.1.e.b 1 80.q even 4 1
3600.1.e.b 1 240.t even 4 1
3600.1.e.b 1 240.bm odd 4 1
3600.1.j.a 2 80.i odd 4 1
3600.1.j.a 2 80.j even 4 1
3600.1.j.a 2 80.s even 4 1
3600.1.j.a 2 80.t odd 4 1
3600.1.j.a 2 240.z odd 4 1
3600.1.j.a 2 240.bb even 4 1
3600.1.j.a 2 240.bd odd 4 1
3600.1.j.a 2 240.bf even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(2304, [\chi])$$.

## Hecke characteristic polynomials

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