# Properties

 Label 2304.2.c.e Level $2304$ Weight $2$ Character orbit 2304.c Analytic conductor $18.398$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$18.3975326257$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1152) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + 3 \beta q^{5} +O(q^{10})$$ $$q + 3 \beta q^{5} + 6 q^{13} + 5 \beta q^{17} -13 q^{25} + 3 \beta q^{29} + 12 q^{37} -\beta q^{41} + 7 q^{49} -9 \beta q^{53} -12 q^{61} + 18 \beta q^{65} -16 q^{73} -30 q^{85} + 13 \beta q^{89} -8 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 12q^{13} - 26q^{25} + 24q^{37} + 14q^{49} - 24q^{61} - 32q^{73} - 60q^{85} - 16q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2303.1
 − 1.41421i 1.41421i
0 0 0 4.24264i 0 0 0 0 0
2303.2 0 0 0 4.24264i 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
3.b odd 2 1 inner
12.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.c.e 2
3.b odd 2 1 inner 2304.2.c.e 2
4.b odd 2 1 CM 2304.2.c.e 2
8.b even 2 1 2304.2.c.c 2
8.d odd 2 1 2304.2.c.c 2
12.b even 2 1 inner 2304.2.c.e 2
16.e even 4 2 1152.2.f.c 4
16.f odd 4 2 1152.2.f.c 4
24.f even 2 1 2304.2.c.c 2
24.h odd 2 1 2304.2.c.c 2
48.i odd 4 2 1152.2.f.c 4
48.k even 4 2 1152.2.f.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.f.c 4 16.e even 4 2
1152.2.f.c 4 16.f odd 4 2
1152.2.f.c 4 48.i odd 4 2
1152.2.f.c 4 48.k even 4 2
2304.2.c.c 2 8.b even 2 1
2304.2.c.c 2 8.d odd 2 1
2304.2.c.c 2 24.f even 2 1
2304.2.c.c 2 24.h odd 2 1
2304.2.c.e 2 1.a even 1 1 trivial
2304.2.c.e 2 3.b odd 2 1 inner
2304.2.c.e 2 4.b odd 2 1 CM
2304.2.c.e 2 12.b even 2 1 inner

## Hecke kernels

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

 $$T_{5}^{2} + 18$$ $$T_{7}$$ $$T_{11}$$ $$T_{13} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 8 T^{2} + 25 T^{4}$$
$7$ $$( 1 - 7 T^{2} )^{2}$$
$11$ $$( 1 + 11 T^{2} )^{2}$$
$13$ $$( 1 - 6 T + 13 T^{2} )^{2}$$
$17$ $$1 + 16 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 19 T^{2} )^{2}$$
$23$ $$( 1 + 23 T^{2} )^{2}$$
$29$ $$1 - 40 T^{2} + 841 T^{4}$$
$31$ $$( 1 - 31 T^{2} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )^{2}$$
$41$ $$1 - 80 T^{2} + 1681 T^{4}$$
$43$ $$( 1 - 43 T^{2} )^{2}$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$1 + 56 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 59 T^{2} )^{2}$$
$61$ $$( 1 + 12 T + 61 T^{2} )^{2}$$
$67$ $$( 1 - 67 T^{2} )^{2}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$( 1 + 16 T + 73 T^{2} )^{2}$$
$79$ $$( 1 - 79 T^{2} )^{2}$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$1 + 160 T^{2} + 7921 T^{4}$$
$97$ $$( 1 + 8 T + 97 T^{2} )^{2}$$