# Properties

 Label 4608.2.a.s Level $4608$ Weight $2$ Character orbit 4608.a Self dual yes Analytic conductor $36.795$ Analytic rank $1$ Dimension $4$ CM discriminant -24 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4608 = 2^{9} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4608.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.7950652514$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{24})^+$$ Defining polynomial: $$x^{4} - 4x^{2} + 1$$ x^4 - 4*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 2) q^{5} + (\beta_{3} + \beta_1) q^{7}+O(q^{10})$$ q + (-b2 - 2) * q^5 + (b3 + b1) * q^7 $$q + ( - \beta_{2} - 2) q^{5} + (\beta_{3} + \beta_1) q^{7} - \beta_{3} q^{11} + (4 \beta_{2} + 5) q^{25} + (3 \beta_{2} - 2) q^{29} + ( - \beta_{3} - 5 \beta_1) q^{31} + ( - 3 \beta_{3} - 8 \beta_1) q^{35} + (4 \beta_{2} + 7) q^{49} + ( - \beta_{2} - 10) q^{53} + (2 \beta_{3} + 6 \beta_1) q^{55} + 8 \beta_1 q^{59} - 4 \beta_{2} q^{73} + ( - 2 \beta_{2} - 12) q^{77} + ( - 3 \beta_{3} + 5 \beta_1) q^{79} + 5 \beta_{3} q^{83} - 2 q^{97}+O(q^{100})$$ q + (-b2 - 2) * q^5 + (b3 + b1) * q^7 - b3 * q^11 + (4*b2 + 5) * q^25 + (3*b2 - 2) * q^29 + (-b3 - 5*b1) * q^31 + (-3*b3 - 8*b1) * q^35 + (4*b2 + 7) * q^49 + (-b2 - 10) * q^53 + (2*b3 + 6*b1) * q^55 + 8*b1 * q^59 - 4*b2 * q^73 + (-2*b2 - 12) * q^77 + (-3*b3 + 5*b1) * q^79 + 5*b3 * q^83 - 2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{5}+O(q^{10})$$ 4 * q - 8 * q^5 $$4 q - 8 q^{5} + 20 q^{25} - 8 q^{29} + 28 q^{49} - 40 q^{53} - 48 q^{77} - 8 q^{97}+O(q^{100})$$ 4 * q - 8 * q^5 + 20 * q^25 - 8 * q^29 + 28 * q^49 - 40 * q^53 - 48 * q^77 - 8 * q^97

Basis of coefficient ring in terms of $$\nu = \zeta_{24} + \zeta_{24}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu^{3} - 3\nu$$ v^3 - 3*v $$\beta_{2}$$ $$=$$ $$-\nu^{3} + 5\nu$$ -v^3 + 5*v $$\beta_{3}$$ $$=$$ $$2\nu^{2} - 4$$ 2*v^2 - 4
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 4 ) / 2$$ (b3 + 4) / 2 $$\nu^{3}$$ $$=$$ $$( 3\beta_{2} + 5\beta_1 ) / 2$$ (3*b2 + 5*b1) / 2

## 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
 0.517638 1.93185 −0.517638 −1.93185
0 0 0 −4.44949 0 −4.87832 0 0 0
1.2 0 0 0 −4.44949 0 4.87832 0 0 0
1.3 0 0 0 0.449490 0 −2.04989 0 0 0
1.4 0 0 0 0.449490 0 2.04989 0 0 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
$$2$$ $$1$$
$$3$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
4.b odd 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4608.2.a.s 4
3.b odd 2 1 4608.2.a.bc yes 4
4.b odd 2 1 inner 4608.2.a.s 4
8.b even 2 1 4608.2.a.bc yes 4
8.d odd 2 1 4608.2.a.bc yes 4
12.b even 2 1 4608.2.a.bc yes 4
16.e even 4 2 4608.2.d.q 8
16.f odd 4 2 4608.2.d.q 8
24.f even 2 1 inner 4608.2.a.s 4
24.h odd 2 1 CM 4608.2.a.s 4
48.i odd 4 2 4608.2.d.q 8
48.k even 4 2 4608.2.d.q 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4608.2.a.s 4 1.a even 1 1 trivial
4608.2.a.s 4 4.b odd 2 1 inner
4608.2.a.s 4 24.f even 2 1 inner
4608.2.a.s 4 24.h odd 2 1 CM
4608.2.a.bc yes 4 3.b odd 2 1
4608.2.a.bc yes 4 8.b even 2 1
4608.2.a.bc yes 4 8.d odd 2 1
4608.2.a.bc yes 4 12.b even 2 1
4608.2.d.q 8 16.e even 4 2
4608.2.d.q 8 16.f odd 4 2
4608.2.d.q 8 48.i odd 4 2
4608.2.d.q 8 48.k even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4608))$$:

 $$T_{5}^{2} + 4T_{5} - 2$$ T5^2 + 4*T5 - 2 $$T_{7}^{4} - 28T_{7}^{2} + 100$$ T7^4 - 28*T7^2 + 100 $$T_{11}^{2} - 12$$ T11^2 - 12 $$T_{17}$$ T17 $$T_{23}$$ T23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 4 T - 2)^{2}$$
$7$ $$T^{4} - 28T^{2} + 100$$
$11$ $$(T^{2} - 12)^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$(T^{2} + 4 T - 50)^{2}$$
$31$ $$T^{4} - 124T^{2} + 1444$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$(T^{2} + 20 T + 94)^{2}$$
$59$ $$(T^{2} - 128)^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$(T^{2} - 96)^{2}$$
$79$ $$T^{4} - 316T^{2} + 3364$$
$83$ $$(T^{2} - 300)^{2}$$
$89$ $$T^{4}$$
$97$ $$(T + 2)^{4}$$