Properties

 Label 4704.2.a.bp 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: $$\Z[a_1, \ldots, a_{11}]$$ 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 $$\beta = 4\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

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 0 0 0 0 1.00000 0
1.2 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

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.bp yes 2
4.b odd 2 1 4704.2.a.bk 2
7.b odd 2 1 4704.2.a.bk 2
8.b even 2 1 9408.2.a.dm 2
8.d odd 2 1 9408.2.a.dz 2
28.d even 2 1 inner 4704.2.a.bp yes 2
56.e even 2 1 9408.2.a.dm 2
56.h odd 2 1 9408.2.a.dz 2

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - T )^{2}$$
$5$ $$( 1 + 5 T^{2} )^{2}$$
$7$ 1
$11$ $$1 - 10 T^{2} + 121 T^{4}$$
$13$ $$1 - 6 T^{2} + 169 T^{4}$$
$17$ $$1 + 2 T^{2} + 289 T^{4}$$
$19$ $$( 1 + 4 T + 19 T^{2} )^{2}$$
$23$ $$1 + 14 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 6 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 8 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 2 T + 37 T^{2} )^{2}$$
$41$ $$1 + 50 T^{2} + 1681 T^{4}$$
$43$ $$( 1 + 43 T^{2} )^{2}$$
$47$ $$( 1 + 8 T + 47 T^{2} )^{2}$$
$53$ $$( 1 + 2 T + 53 T^{2} )^{2}$$
$59$ $$( 1 + 4 T + 59 T^{2} )^{2}$$
$61$ $$1 + 90 T^{2} + 3721 T^{4}$$
$67$ $$1 + 6 T^{2} + 4489 T^{4}$$
$71$ $$1 + 110 T^{2} + 5041 T^{4}$$
$73$ $$1 + 18 T^{2} + 5329 T^{4}$$
$79$ $$1 + 30 T^{2} + 6241 T^{4}$$
$83$ $$( 1 - 12 T + 83 T^{2} )^{2}$$
$89$ $$1 + 146 T^{2} + 7921 T^{4}$$
$97$ $$( 1 + 97 T^{2} )^{2}$$