# Properties

 Label 4704.2.a.bi 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: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta q^{5} + q^{9} +O(q^{10})$$ $$q - q^{3} + \beta q^{5} + q^{9} + \beta q^{11} -\beta q^{15} -\beta q^{17} -4 q^{19} -3 \beta q^{23} + 3 q^{25} - q^{27} + 2 q^{29} -\beta q^{33} -6 q^{37} -3 \beta q^{41} -4 \beta q^{43} + \beta q^{45} -8 q^{47} + \beta q^{51} + 6 q^{53} + 8 q^{55} + 4 q^{57} -12 q^{59} -2 \beta q^{61} + 2 \beta q^{67} + 3 \beta q^{69} -\beta q^{71} + 2 \beta q^{73} -3 q^{75} -2 \beta q^{79} + q^{81} + 4 q^{83} -8 q^{85} -2 q^{87} + \beta q^{89} -4 \beta q^{95} + 6 \beta q^{97} + \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{9} - 8q^{19} + 6q^{25} - 2q^{27} + 4q^{29} - 12q^{37} - 16q^{47} + 12q^{53} + 16q^{55} + 8q^{57} - 24q^{59} - 6q^{75} + 2q^{81} + 8q^{83} - 16q^{85} - 4q^{87} + O(q^{100})$$

## 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 −2.82843 0 0 0 1.00000 0
1.2 0 −1.00000 0 2.82843 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.bi 2
4.b odd 2 1 4704.2.a.br yes 2
7.b odd 2 1 4704.2.a.br yes 2
8.b even 2 1 9408.2.a.eb 2
8.d odd 2 1 9408.2.a.dk 2
28.d even 2 1 inner 4704.2.a.bi 2
56.e even 2 1 9408.2.a.eb 2
56.h odd 2 1 9408.2.a.dk 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$-8 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-8 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$-8 + T^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$-72 + T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$( 6 + T )^{2}$$
$41$ $$-72 + T^{2}$$
$43$ $$-128 + T^{2}$$
$47$ $$( 8 + T )^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$( 12 + T )^{2}$$
$61$ $$-32 + T^{2}$$
$67$ $$-32 + T^{2}$$
$71$ $$-8 + T^{2}$$
$73$ $$-32 + T^{2}$$
$79$ $$-32 + T^{2}$$
$83$ $$( -4 + T )^{2}$$
$89$ $$-8 + T^{2}$$
$97$ $$-288 + T^{2}$$