# Properties

 Label 2304.2.c.g Level $2304$ Weight $2$ Character orbit 2304.c Analytic conductor $18.398$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1152) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} + 2 \beta q^{7} +O(q^{10})$$ $$q + \beta q^{5} + 2 \beta q^{7} + 4 q^{11} -2 q^{13} + \beta q^{17} + 4 \beta q^{19} -4 q^{23} + 3 q^{25} + 5 \beta q^{29} -6 \beta q^{31} -4 q^{35} -8 q^{37} + 3 \beta q^{41} -8 \beta q^{43} + 12 q^{47} - q^{49} + 9 \beta q^{53} + 4 \beta q^{55} -8 q^{61} -2 \beta q^{65} + 4 \beta q^{67} -4 q^{71} -8 q^{73} + 8 \beta q^{77} -2 \beta q^{79} -12 q^{83} -2 q^{85} -11 \beta q^{89} -4 \beta q^{91} -8 q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 8q^{11} - 4q^{13} - 8q^{23} + 6q^{25} - 8q^{35} - 16q^{37} + 24q^{47} - 2q^{49} - 16q^{61} - 8q^{71} - 16q^{73} - 24q^{83} - 4q^{85} - 16q^{95} + O(q^{100})$$

## 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 1.41421i 0 2.82843i 0 0 0
2303.2 0 0 0 1.41421i 0 2.82843i 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
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.g 2
3.b odd 2 1 2304.2.c.a 2
4.b odd 2 1 2304.2.c.a 2
8.b even 2 1 2304.2.c.b 2
8.d odd 2 1 2304.2.c.h 2
12.b even 2 1 inner 2304.2.c.g 2
16.e even 4 2 1152.2.f.d yes 4
16.f odd 4 2 1152.2.f.a 4
24.f even 2 1 2304.2.c.b 2
24.h odd 2 1 2304.2.c.h 2
48.i odd 4 2 1152.2.f.a 4
48.k even 4 2 1152.2.f.d yes 4

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

## Hecke kernels

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

 $$T_{5}^{2} + 2$$ $$T_{7}^{2} + 8$$ $$T_{11} - 4$$ $$T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 8 T^{2} + 25 T^{4}$$
$7$ $$1 - 6 T^{2} + 49 T^{4}$$
$11$ $$( 1 - 4 T + 11 T^{2} )^{2}$$
$13$ $$( 1 + 2 T + 13 T^{2} )^{2}$$
$17$ $$1 - 32 T^{2} + 289 T^{4}$$
$19$ $$1 - 6 T^{2} + 361 T^{4}$$
$23$ $$( 1 + 4 T + 23 T^{2} )^{2}$$
$29$ $$1 - 8 T^{2} + 841 T^{4}$$
$31$ $$1 + 10 T^{2} + 961 T^{4}$$
$37$ $$( 1 + 8 T + 37 T^{2} )^{2}$$
$41$ $$1 - 64 T^{2} + 1681 T^{4}$$
$43$ $$1 + 42 T^{2} + 1849 T^{4}$$
$47$ $$( 1 - 12 T + 47 T^{2} )^{2}$$
$53$ $$1 + 56 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 59 T^{2} )^{2}$$
$61$ $$( 1 + 8 T + 61 T^{2} )^{2}$$
$67$ $$1 - 102 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 4 T + 71 T^{2} )^{2}$$
$73$ $$( 1 + 8 T + 73 T^{2} )^{2}$$
$79$ $$1 - 150 T^{2} + 6241 T^{4}$$
$83$ $$( 1 + 12 T + 83 T^{2} )^{2}$$
$89$ $$1 + 64 T^{2} + 7921 T^{4}$$
$97$ $$( 1 + 97 T^{2} )^{2}$$