# Properties

 Label 2304.3.e.a Level $2304$ Weight $3$ Character orbit 2304.e Analytic conductor $62.779$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$62.7794529086$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ 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.

 $$f(q)$$ $$=$$ $$q + 7 \beta q^{5} +O(q^{10})$$ $$q + 7 \beta q^{5} -10 q^{13} + 23 \beta q^{17} -73 q^{25} -41 \beta q^{29} -24 q^{37} + 49 \beta q^{41} -49 q^{49} -17 \beta q^{53} + 120 q^{61} -70 \beta q^{65} -96 q^{73} -322 q^{85} + 119 \beta q^{89} + 144 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 20q^{13} - 146q^{25} - 48q^{37} - 98q^{49} + 240q^{61} - 192q^{73} - 644q^{85} + 288q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1025.1
 − 1.41421i 1.41421i
0 0 0 9.89949i 0 0 0 0 0
1025.2 0 0 0 9.89949i 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 2304.3.e.a 2
3.b odd 2 1 inner 2304.3.e.a 2
4.b odd 2 1 CM 2304.3.e.a 2
8.b even 2 1 2304.3.e.c 2
8.d odd 2 1 2304.3.e.c 2
12.b even 2 1 inner 2304.3.e.a 2
16.e even 4 2 1152.3.h.c 4
16.f odd 4 2 1152.3.h.c 4
24.f even 2 1 2304.3.e.c 2
24.h odd 2 1 2304.3.e.c 2
48.i odd 4 2 1152.3.h.c 4
48.k even 4 2 1152.3.h.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.h.c 4 16.e even 4 2
1152.3.h.c 4 16.f odd 4 2
1152.3.h.c 4 48.i odd 4 2
1152.3.h.c 4 48.k even 4 2
2304.3.e.a 2 1.a even 1 1 trivial
2304.3.e.a 2 3.b odd 2 1 inner
2304.3.e.a 2 4.b odd 2 1 CM
2304.3.e.a 2 12.b even 2 1 inner
2304.3.e.c 2 8.b even 2 1
2304.3.e.c 2 8.d odd 2 1
2304.3.e.c 2 24.f even 2 1
2304.3.e.c 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 $$S_{3}^{\mathrm{new}}(2304, [\chi])$$:

 $$T_{5}^{2} + 98$$ $$T_{7}$$ $$T_{13} + 10$$ $$T_{19}$$ $$T_{31}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$98 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$( 10 + T )^{2}$$
$17$ $$1058 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$3362 + T^{2}$$
$31$ $$T^{2}$$
$37$ $$( 24 + T )^{2}$$
$41$ $$4802 + T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$578 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -120 + T )^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 96 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$28322 + T^{2}$$
$97$ $$( -144 + T )^{2}$$