# Properties

 Label 4704.2.a.bh Level $4704$ Weight $2$ Character orbit 4704.a Self dual yes Analytic conductor $37.562$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$4704 = 2^{5} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4704.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$37.5616291108$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 672) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} + 4q^{5} + q^{9} + O(q^{10})$$ $$q + q^{3} + 4q^{5} + q^{9} + 6q^{11} - 5q^{13} + 4q^{15} - 2q^{17} - q^{19} - 6q^{23} + 11q^{25} + q^{27} + 3q^{31} + 6q^{33} + 3q^{37} - 5q^{39} + 6q^{41} + 5q^{43} + 4q^{45} + 4q^{47} - 2q^{51} - 6q^{53} + 24q^{55} - q^{57} + 6q^{59} + 2q^{61} - 20q^{65} + 7q^{67} - 6q^{69} + 16q^{71} + 3q^{73} + 11q^{75} + 11q^{79} + q^{81} - 12q^{83} - 8q^{85} - 4q^{89} + 3q^{93} - 4q^{95} + 6q^{97} + 6q^{99} + O(q^{100})$$

## 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 1.00000 0 4.00000 0 0 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

## 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 4704.2.a.bh 1
4.b odd 2 1 4704.2.a.p 1
7.b odd 2 1 4704.2.a.a 1
7.d odd 6 2 672.2.q.j yes 2
8.b even 2 1 9408.2.a.a 1
8.d odd 2 1 9408.2.a.bs 1
21.g even 6 2 2016.2.s.a 2
28.d even 2 1 4704.2.a.r 1
28.f even 6 2 672.2.q.e 2
56.e even 2 1 9408.2.a.bp 1
56.h odd 2 1 9408.2.a.dd 1
56.j odd 6 2 1344.2.q.a 2
56.m even 6 2 1344.2.q.l 2
84.j odd 6 2 2016.2.s.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.e 2 28.f even 6 2
672.2.q.j yes 2 7.d odd 6 2
1344.2.q.a 2 56.j odd 6 2
1344.2.q.l 2 56.m even 6 2
2016.2.s.a 2 21.g even 6 2
2016.2.s.b 2 84.j odd 6 2
4704.2.a.a 1 7.b odd 2 1
4704.2.a.p 1 4.b odd 2 1
4704.2.a.r 1 28.d even 2 1
4704.2.a.bh 1 1.a even 1 1 trivial
9408.2.a.a 1 8.b even 2 1
9408.2.a.bp 1 56.e even 2 1
9408.2.a.bs 1 8.d odd 2 1
9408.2.a.dd 1 56.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4704))$$:

 $$T_{5} - 4$$ $$T_{11} - 6$$ $$T_{13} + 5$$ $$T_{19} + 1$$ $$T_{31} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$-4 + T$$
$7$ $$T$$
$11$ $$-6 + T$$
$13$ $$5 + T$$
$17$ $$2 + T$$
$19$ $$1 + T$$
$23$ $$6 + T$$
$29$ $$T$$
$31$ $$-3 + T$$
$37$ $$-3 + T$$
$41$ $$-6 + T$$
$43$ $$-5 + T$$
$47$ $$-4 + T$$
$53$ $$6 + T$$
$59$ $$-6 + T$$
$61$ $$-2 + T$$
$67$ $$-7 + T$$
$71$ $$-16 + T$$
$73$ $$-3 + T$$
$79$ $$-11 + T$$
$83$ $$12 + T$$
$89$ $$4 + T$$
$97$ $$-6 + T$$
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