# Properties

 Label 2304.2.a.j Level $2304$ Weight $2$ Character orbit 2304.a Self dual yes Analytic conductor $18.398$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

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## 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 384) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4q^{7} + O(q^{10})$$ $$q + 4q^{7} - 4q^{11} - 4q^{13} + 2q^{17} - 4q^{19} - 8q^{23} - 5q^{25} - 8q^{29} + 4q^{31} + 4q^{37} - 6q^{41} + 4q^{43} - 8q^{47} + 9q^{49} - 8q^{53} + 12q^{59} - 12q^{61} + 12q^{67} + 8q^{71} - 6q^{73} - 16q^{77} + 4q^{79} + 4q^{83} + 6q^{89} - 16q^{91} - 2q^{97} + O(q^{100})$$

## 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
 0
0 0 0 0 0 4.00000 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.a.j 1
3.b odd 2 1 768.2.a.c 1
4.b odd 2 1 2304.2.a.g 1
8.b even 2 1 2304.2.a.k 1
8.d odd 2 1 2304.2.a.f 1
12.b even 2 1 768.2.a.f 1
16.e even 4 2 1152.2.d.a 2
16.f odd 4 2 1152.2.d.f 2
24.f even 2 1 768.2.a.b 1
24.h odd 2 1 768.2.a.g 1
48.i odd 4 2 384.2.d.a 2
48.k even 4 2 384.2.d.b yes 2

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

## Hecke kernels

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

 $$T_{5}$$ $$T_{7} - 4$$ $$T_{11} + 4$$ $$T_{13} + 4$$ $$T_{19} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-4 + T$$
$11$ $$4 + T$$
$13$ $$4 + T$$
$17$ $$-2 + T$$
$19$ $$4 + T$$
$23$ $$8 + T$$
$29$ $$8 + T$$
$31$ $$-4 + T$$
$37$ $$-4 + T$$
$41$ $$6 + T$$
$43$ $$-4 + T$$
$47$ $$8 + T$$
$53$ $$8 + T$$
$59$ $$-12 + T$$
$61$ $$12 + T$$
$67$ $$-12 + T$$
$71$ $$-8 + T$$
$73$ $$6 + T$$
$79$ $$-4 + T$$
$83$ $$-4 + T$$
$89$ $$-6 + T$$
$97$ $$2 + T$$
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