# Properties

 Label 4704.2.a.i 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

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{9} + O(q^{10})$$ $$q - q^{3} + q^{9} + 2q^{11} - 5q^{13} + 2q^{17} - 3q^{19} + 2q^{23} - 5q^{25} - q^{27} + 8q^{29} + q^{31} - 2q^{33} - 5q^{37} + 5q^{39} - 2q^{41} + 7q^{43} + 8q^{47} - 2q^{51} - 2q^{53} + 3q^{57} - 10q^{59} + 2q^{61} - 11q^{67} - 2q^{69} + 12q^{71} + 3q^{73} + 5q^{75} + 17q^{79} + q^{81} + 16q^{83} - 8q^{87} - 12q^{89} - q^{93} + 14q^{97} + 2q^{99} + O(q^{100})$$

## 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 0 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.i 1
4.b odd 2 1 4704.2.a.x 1
7.b odd 2 1 4704.2.a.bb 1
7.d odd 6 2 672.2.q.c 2
8.b even 2 1 9408.2.a.cj 1
8.d odd 2 1 9408.2.a.y 1
21.g even 6 2 2016.2.s.g 2
28.d even 2 1 4704.2.a.h 1
28.f even 6 2 672.2.q.i yes 2
56.e even 2 1 9408.2.a.cm 1
56.h odd 2 1 9408.2.a.s 1
56.j odd 6 2 1344.2.q.p 2
56.m even 6 2 1344.2.q.f 2
84.j odd 6 2 2016.2.s.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.c 2 7.d odd 6 2
672.2.q.i yes 2 28.f even 6 2
1344.2.q.f 2 56.m even 6 2
1344.2.q.p 2 56.j odd 6 2
2016.2.s.f 2 84.j odd 6 2
2016.2.s.g 2 21.g even 6 2
4704.2.a.h 1 28.d even 2 1
4704.2.a.i 1 1.a even 1 1 trivial
4704.2.a.x 1 4.b odd 2 1
4704.2.a.bb 1 7.b odd 2 1
9408.2.a.s 1 56.h odd 2 1
9408.2.a.y 1 8.d odd 2 1
9408.2.a.cj 1 8.b even 2 1
9408.2.a.cm 1 56.e even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4704))$$:

 $$T_{5}$$ $$T_{11} - 2$$ $$T_{13} + 5$$ $$T_{19} + 3$$ $$T_{31} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$-2 + T$$
$13$ $$5 + T$$
$17$ $$-2 + T$$
$19$ $$3 + T$$
$23$ $$-2 + T$$
$29$ $$-8 + T$$
$31$ $$-1 + T$$
$37$ $$5 + T$$
$41$ $$2 + T$$
$43$ $$-7 + T$$
$47$ $$-8 + T$$
$53$ $$2 + T$$
$59$ $$10 + T$$
$61$ $$-2 + T$$
$67$ $$11 + T$$
$71$ $$-12 + T$$
$73$ $$-3 + T$$
$79$ $$-17 + T$$
$83$ $$-16 + T$$
$89$ $$12 + T$$
$97$ $$-14 + T$$
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