# Properties

 Label 4704.2.a.bw Level $4704$ Weight $2$ Character orbit 4704.a Self dual yes Analytic conductor $37.562$ Analytic rank $1$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.2624.1 Defining polynomial: $$x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( -1 + \beta_{3} ) q^{5} + q^{9} +O(q^{10})$$ $$q - q^{3} + ( -1 + \beta_{3} ) q^{5} + q^{9} + ( 1 - \beta_{2} ) q^{11} + ( -2 - \beta_{2} - \beta_{3} ) q^{13} + ( 1 - \beta_{3} ) q^{15} + ( -1 - \beta_{1} + \beta_{2} ) q^{17} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{19} + ( 3 + \beta_{2} ) q^{23} + ( 3 + 2 \beta_{1} - 2 \beta_{3} ) q^{25} - q^{27} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{29} + ( 2 + 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{31} + ( -1 + \beta_{2} ) q^{33} -4 \beta_{1} q^{37} + ( 2 + \beta_{2} + \beta_{3} ) q^{39} + ( -5 + 3 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{41} + ( -2 + 2 \beta_{1} + 2 \beta_{3} ) q^{43} + ( -1 + \beta_{3} ) q^{45} + ( 4 - \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{47} + ( 1 + \beta_{1} - \beta_{2} ) q^{51} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{53} + ( 2 - 5 \beta_{1} + \beta_{2} + \beta_{3} ) q^{55} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{57} + ( 3 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{59} + ( -4 + \beta_{2} - \beta_{3} ) q^{61} + ( -2 - 7 \beta_{1} + \beta_{2} - \beta_{3} ) q^{65} + ( -2 - 4 \beta_{1} + 2 \beta_{2} ) q^{67} + ( -3 - \beta_{2} ) q^{69} + ( 3 - 4 \beta_{1} + \beta_{2} ) q^{71} + ( -2 - \beta_{1} + 2 \beta_{3} ) q^{73} + ( -3 - 2 \beta_{1} + 2 \beta_{3} ) q^{75} + ( 4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{79} + q^{81} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{83} + ( -2 + 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{85} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{87} + ( -7 - 5 \beta_{1} - \beta_{2} ) q^{89} + ( -2 - 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{93} + ( 6 + 8 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{95} + ( -10 + 3 \beta_{1} + 2 \beta_{3} ) q^{97} + ( 1 - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 4 q^{5} + 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{3} - 4 q^{5} + 4 q^{9} + 4 q^{11} - 8 q^{13} + 4 q^{15} - 4 q^{17} - 8 q^{19} + 12 q^{23} + 12 q^{25} - 4 q^{27} + 8 q^{31} - 4 q^{33} + 8 q^{39} - 20 q^{41} - 8 q^{43} - 4 q^{45} + 16 q^{47} + 4 q^{51} + 8 q^{55} + 8 q^{57} - 16 q^{61} - 8 q^{65} - 8 q^{67} - 12 q^{69} + 12 q^{71} - 8 q^{73} - 12 q^{75} + 16 q^{79} + 4 q^{81} - 8 q^{85} - 28 q^{89} - 8 q^{93} + 24 q^{95} - 40 q^{97} + 4 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 2 \nu + 1$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} - 4 \nu - 3$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 2 \nu^{2} + 4 \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3} + \beta_{2} + 4 \beta_{1} + 5$$

## 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
 −0.360409 −1.22833 0.814115 2.77462
0 −1.00000 0 −4.13503 0 0 0 1.00000 0
1.2 0 −1.00000 0 −3.04244 0 0 0 1.00000 0
1.3 0 −1.00000 0 1.04244 0 0 0 1.00000 0
1.4 0 −1.00000 0 2.13503 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.bw 4
4.b odd 2 1 4704.2.a.by yes 4
7.b odd 2 1 4704.2.a.bz yes 4
8.b even 2 1 9408.2.a.en 4
8.d odd 2 1 9408.2.a.el 4
28.d even 2 1 4704.2.a.bx yes 4
56.e even 2 1 9408.2.a.em 4
56.h odd 2 1 9408.2.a.ek 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4704.2.a.bw 4 1.a even 1 1 trivial
4704.2.a.bx yes 4 28.d even 2 1
4704.2.a.by yes 4 4.b odd 2 1
4704.2.a.bz yes 4 7.b odd 2 1
9408.2.a.ek 4 56.h odd 2 1
9408.2.a.el 4 8.d odd 2 1
9408.2.a.em 4 56.e even 2 1
9408.2.a.en 4 8.b even 2 1

## Hecke kernels

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

 $$T_{5}^{4} + 4 T_{5}^{3} - 8 T_{5}^{2} - 24 T_{5} + 28$$ $$T_{11}^{4} - 4 T_{11}^{3} - 20 T_{11}^{2} + 48 T_{11} + 16$$ $$T_{13}^{4} + 8 T_{13}^{3} - 4 T_{13}^{2} - 80 T_{13} + 68$$ $$T_{19}^{4} + 8 T_{19}^{3} - 8 T_{19}^{2} - 64 T_{19} + 64$$ $$T_{31}^{4} - 8 T_{31}^{3} - 40 T_{31}^{2} + 128 T_{31} - 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$28 - 24 T - 8 T^{2} + 4 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$16 + 48 T - 20 T^{2} - 4 T^{3} + T^{4}$$
$13$ $$68 - 80 T - 4 T^{2} + 8 T^{3} + T^{4}$$
$17$ $$-100 - 120 T - 24 T^{2} + 4 T^{3} + T^{4}$$
$19$ $$64 - 64 T - 8 T^{2} + 8 T^{3} + T^{4}$$
$23$ $$-112 + 48 T + 28 T^{2} - 12 T^{3} + T^{4}$$
$29$ $$64 + 128 T - 56 T^{2} + T^{4}$$
$31$ $$-64 + 128 T - 40 T^{2} - 8 T^{3} + T^{4}$$
$37$ $$( -32 + T^{2} )^{2}$$
$41$ $$-4804 - 920 T + 56 T^{2} + 20 T^{3} + T^{4}$$
$43$ $$-256 - 384 T - 48 T^{2} + 8 T^{3} + T^{4}$$
$47$ $$-3008 + 1024 T - 24 T^{2} - 16 T^{3} + T^{4}$$
$53$ $$272 - 128 T - 72 T^{2} + T^{4}$$
$59$ $$-1600 - 960 T - 152 T^{2} + T^{4}$$
$61$ $$-412 - 160 T + 44 T^{2} + 16 T^{3} + T^{4}$$
$67$ $$-4352 - 1664 T - 144 T^{2} + 8 T^{3} + T^{4}$$
$71$ $$272 + 176 T - 36 T^{2} - 12 T^{3} + T^{4}$$
$73$ $$452 - 144 T - 36 T^{2} + 8 T^{3} + T^{4}$$
$79$ $$-1024 + 1024 T - 32 T^{2} - 16 T^{3} + T^{4}$$
$83$ $$1792 - 256 T - 128 T^{2} + T^{4}$$
$89$ $$-4772 - 712 T + 168 T^{2} + 28 T^{3} + T^{4}$$
$97$ $$-1148 + 1968 T + 508 T^{2} + 40 T^{3} + T^{4}$$