# Properties

 Label 2304.2.a.q Level $2304$ Weight $2$ Character orbit 2304.a Self dual yes Analytic conductor $18.398$ Analytic rank $1$ Dimension $2$ CM discriminant -24 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2304.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$18.3975326257$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 72) Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{5} -2 q^{7} +O(q^{10})$$ $$q + 2 \beta q^{5} -2 q^{7} -4 \beta q^{11} + 3 q^{25} -2 \beta q^{29} -10 q^{31} -4 \beta q^{35} -3 q^{49} -10 \beta q^{53} -16 q^{55} + 8 \beta q^{59} -14 q^{73} + 8 \beta q^{77} -10 q^{79} + 4 \beta q^{83} + 2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{7} + O(q^{10})$$ $$2q - 4q^{7} + 6q^{25} - 20q^{31} - 6q^{49} - 32q^{55} - 28q^{73} - 20q^{79} + 4q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
0 0 0 −2.82843 0 −2.00000 0 0 0
1.2 0 0 0 2.82843 0 −2.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
3.b odd 2 1 inner
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.a.q 2
3.b odd 2 1 inner 2304.2.a.q 2
4.b odd 2 1 2304.2.a.y 2
8.b even 2 1 inner 2304.2.a.q 2
8.d odd 2 1 2304.2.a.y 2
12.b even 2 1 2304.2.a.y 2
16.e even 4 2 72.2.d.a 2
16.f odd 4 2 288.2.d.a 2
24.f even 2 1 2304.2.a.y 2
24.h odd 2 1 CM 2304.2.a.q 2
48.i odd 4 2 72.2.d.a 2
48.k even 4 2 288.2.d.a 2
80.i odd 4 2 1800.2.d.n 4
80.j even 4 2 7200.2.d.p 4
80.k odd 4 2 7200.2.k.h 2
80.q even 4 2 1800.2.k.e 2
80.s even 4 2 7200.2.d.p 4
80.t odd 4 2 1800.2.d.n 4
144.u even 12 4 2592.2.r.i 4
144.v odd 12 4 2592.2.r.i 4
144.w odd 12 4 648.2.n.h 4
144.x even 12 4 648.2.n.h 4
240.t even 4 2 7200.2.k.h 2
240.z odd 4 2 7200.2.d.p 4
240.bb even 4 2 1800.2.d.n 4
240.bd odd 4 2 7200.2.d.p 4
240.bf even 4 2 1800.2.d.n 4
240.bm odd 4 2 1800.2.k.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.d.a 2 16.e even 4 2
72.2.d.a 2 48.i odd 4 2
288.2.d.a 2 16.f odd 4 2
288.2.d.a 2 48.k even 4 2
648.2.n.h 4 144.w odd 12 4
648.2.n.h 4 144.x even 12 4
1800.2.d.n 4 80.i odd 4 2
1800.2.d.n 4 80.t odd 4 2
1800.2.d.n 4 240.bb even 4 2
1800.2.d.n 4 240.bf even 4 2
1800.2.k.e 2 80.q even 4 2
1800.2.k.e 2 240.bm odd 4 2
2304.2.a.q 2 1.a even 1 1 trivial
2304.2.a.q 2 3.b odd 2 1 inner
2304.2.a.q 2 8.b even 2 1 inner
2304.2.a.q 2 24.h odd 2 1 CM
2304.2.a.y 2 4.b odd 2 1
2304.2.a.y 2 8.d odd 2 1
2304.2.a.y 2 12.b even 2 1
2304.2.a.y 2 24.f even 2 1
2592.2.r.i 4 144.u even 12 4
2592.2.r.i 4 144.v odd 12 4
7200.2.d.p 4 80.j even 4 2
7200.2.d.p 4 80.s even 4 2
7200.2.d.p 4 240.z odd 4 2
7200.2.d.p 4 240.bd odd 4 2
7200.2.k.h 2 80.k odd 4 2
7200.2.k.h 2 240.t even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2304))$$:

 $$T_{5}^{2} - 8$$ $$T_{7} + 2$$ $$T_{11}^{2} - 32$$ $$T_{13}$$ $$T_{19}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 2 T^{2} + 25 T^{4}$$
$7$ $$( 1 + 2 T + 7 T^{2} )^{2}$$
$11$ $$1 - 10 T^{2} + 121 T^{4}$$
$13$ $$( 1 + 13 T^{2} )^{2}$$
$17$ $$( 1 + 17 T^{2} )^{2}$$
$19$ $$( 1 + 19 T^{2} )^{2}$$
$23$ $$( 1 + 23 T^{2} )^{2}$$
$29$ $$1 + 50 T^{2} + 841 T^{4}$$
$31$ $$( 1 + 10 T + 31 T^{2} )^{2}$$
$37$ $$( 1 + 37 T^{2} )^{2}$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 + 43 T^{2} )^{2}$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$1 - 94 T^{2} + 2809 T^{4}$$
$59$ $$1 - 10 T^{2} + 3481 T^{4}$$
$61$ $$( 1 + 61 T^{2} )^{2}$$
$67$ $$( 1 + 67 T^{2} )^{2}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$( 1 + 14 T + 73 T^{2} )^{2}$$
$79$ $$( 1 + 10 T + 79 T^{2} )^{2}$$
$83$ $$1 + 134 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 89 T^{2} )^{2}$$
$97$ $$( 1 - 2 T + 97 T^{2} )^{2}$$