# Properties

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

# Related objects

## 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: $$1$$ 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: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

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

## 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
 −1.41421 1.41421
0 1.00000 0 −1.41421 0 0 0 1.00000 0
1.2 0 1.00000 0 1.41421 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4704.2.a.bo yes 2
4.b odd 2 1 4704.2.a.bl 2
7.b odd 2 1 4704.2.a.bl 2
8.b even 2 1 9408.2.a.dn 2
8.d odd 2 1 9408.2.a.dy 2
28.d even 2 1 inner 4704.2.a.bo yes 2
56.e even 2 1 9408.2.a.dn 2
56.h odd 2 1 9408.2.a.dy 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4704.2.a.bl 2 4.b odd 2 1
4704.2.a.bl 2 7.b odd 2 1
4704.2.a.bo yes 2 1.a even 1 1 trivial
4704.2.a.bo yes 2 28.d even 2 1 inner
9408.2.a.dn 2 8.b even 2 1
9408.2.a.dn 2 56.e even 2 1
9408.2.a.dy 2 8.d odd 2 1
9408.2.a.dy 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4704))$$:

 $$T_{5}^{2} - 2$$ $$T_{11}^{2} - 8$$ $$T_{13}^{2} - 18$$ $$T_{19} + 8$$ $$T_{31} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$-2 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-8 + T^{2}$$
$13$ $$-18 + T^{2}$$
$17$ $$-18 + T^{2}$$
$19$ $$( 8 + T )^{2}$$
$23$ $$-8 + T^{2}$$
$29$ $$T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$( 4 + T )^{2}$$
$41$ $$-50 + T^{2}$$
$43$ $$-32 + T^{2}$$
$47$ $$( -4 + T )^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$( 8 + T )^{2}$$
$61$ $$-18 + T^{2}$$
$67$ $$T^{2}$$
$71$ $$-200 + T^{2}$$
$73$ $$-2 + T^{2}$$
$79$ $$-128 + T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$-2 + T^{2}$$
$97$ $$-162 + T^{2}$$