# Properties

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

# 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: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 672) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + \beta q^{5} + q^{9} +O(q^{10})$$ $$q + q^{3} + \beta q^{5} + q^{9} + ( -2 - \beta ) q^{11} -2 q^{13} + \beta q^{15} + ( -4 - \beta ) q^{17} -2 \beta q^{19} + ( -2 - \beta ) q^{23} + 7 q^{25} + q^{27} + ( 2 + 2 \beta ) q^{29} + ( -4 + 2 \beta ) q^{31} + ( -2 - \beta ) q^{33} -2 q^{37} -2 q^{39} + ( -8 + \beta ) q^{41} + 8 q^{43} + \beta q^{45} + ( 4 - 2 \beta ) q^{47} + ( -4 - \beta ) q^{51} -2 q^{53} + ( -12 - 2 \beta ) q^{55} -2 \beta q^{57} + ( -8 - 2 \beta ) q^{59} + ( 2 - 2 \beta ) q^{61} -2 \beta q^{65} + ( -4 - 2 \beta ) q^{67} + ( -2 - \beta ) q^{69} + ( -6 + \beta ) q^{71} + ( -6 + 2 \beta ) q^{73} + 7 q^{75} + ( 4 - 2 \beta ) q^{79} + q^{81} -4 q^{83} + ( -12 - 4 \beta ) q^{85} + ( 2 + 2 \beta ) q^{87} + \beta q^{89} + ( -4 + 2 \beta ) q^{93} -24 q^{95} + ( 2 - 2 \beta ) q^{97} + ( -2 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 2q^{9} - 4q^{11} - 4q^{13} - 8q^{17} - 4q^{23} + 14q^{25} + 2q^{27} + 4q^{29} - 8q^{31} - 4q^{33} - 4q^{37} - 4q^{39} - 16q^{41} + 16q^{43} + 8q^{47} - 8q^{51} - 4q^{53} - 24q^{55} - 16q^{59} + 4q^{61} - 8q^{67} - 4q^{69} - 12q^{71} - 12q^{73} + 14q^{75} + 8q^{79} + 2q^{81} - 8q^{83} - 24q^{85} + 4q^{87} - 8q^{93} - 48q^{95} + 4q^{97} - 4q^{99} + O(q^{100})$$

## 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
 −1.73205 1.73205
0 1.00000 0 −3.46410 0 0 0 1.00000 0
1.2 0 1.00000 0 3.46410 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.bn 2
4.b odd 2 1 4704.2.a.bm 2
7.b odd 2 1 672.2.a.i 2
8.b even 2 1 9408.2.a.do 2
8.d odd 2 1 9408.2.a.dx 2
21.c even 2 1 2016.2.a.t 2
28.d even 2 1 672.2.a.j yes 2
56.e even 2 1 1344.2.a.u 2
56.h odd 2 1 1344.2.a.v 2
84.h odd 2 1 2016.2.a.s 2
112.j even 4 2 5376.2.c.bn 4
112.l odd 4 2 5376.2.c.bh 4
168.e odd 2 1 4032.2.a.br 2
168.i even 2 1 4032.2.a.bs 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.a.i 2 7.b odd 2 1
672.2.a.j yes 2 28.d even 2 1
1344.2.a.u 2 56.e even 2 1
1344.2.a.v 2 56.h odd 2 1
2016.2.a.s 2 84.h odd 2 1
2016.2.a.t 2 21.c even 2 1
4032.2.a.br 2 168.e odd 2 1
4032.2.a.bs 2 168.i even 2 1
4704.2.a.bm 2 4.b odd 2 1
4704.2.a.bn 2 1.a even 1 1 trivial
5376.2.c.bh 4 112.l odd 4 2
5376.2.c.bn 4 112.j even 4 2
9408.2.a.do 2 8.b even 2 1
9408.2.a.dx 2 8.d odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} - 12$$ $$T_{11}^{2} + 4 T_{11} - 8$$ $$T_{13} + 2$$ $$T_{19}^{2} - 48$$ $$T_{31}^{2} + 8 T_{31} - 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$-12 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-8 + 4 T + T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$4 + 8 T + T^{2}$$
$19$ $$-48 + T^{2}$$
$23$ $$-8 + 4 T + T^{2}$$
$29$ $$-44 - 4 T + T^{2}$$
$31$ $$-32 + 8 T + T^{2}$$
$37$ $$( 2 + T )^{2}$$
$41$ $$52 + 16 T + T^{2}$$
$43$ $$( -8 + T )^{2}$$
$47$ $$-32 - 8 T + T^{2}$$
$53$ $$( 2 + T )^{2}$$
$59$ $$16 + 16 T + T^{2}$$
$61$ $$-44 - 4 T + T^{2}$$
$67$ $$-32 + 8 T + T^{2}$$
$71$ $$24 + 12 T + T^{2}$$
$73$ $$-12 + 12 T + T^{2}$$
$79$ $$-32 - 8 T + T^{2}$$
$83$ $$( 4 + T )^{2}$$
$89$ $$-12 + T^{2}$$
$97$ $$-44 - 4 T + T^{2}$$