# Properties

 Label 4608.2.a.z Level $4608$ Weight $2$ Character orbit 4608.a Self dual yes Analytic conductor $36.795$ Analytic rank $0$ Dimension $4$ CM no 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{24})^+$$ Defining polynomial: $$x^{4} - 4x^{2} + 1$$ x^4 - 4*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5} + \beta_1 q^{7}+O(q^{10})$$ q + b2 * q^5 + b1 * q^7 $$q + \beta_{2} q^{5} + \beta_1 q^{7} - \beta_{3} q^{11} + 2 \beta_{2} q^{13} + 4 q^{17} - 2 \beta_{3} q^{19} + 4 \beta_1 q^{23} + q^{25} + \beta_{2} q^{29} - \beta_1 q^{31} + \beta_{3} q^{35} + 2 \beta_{2} q^{37} + 4 q^{41} + 2 \beta_{3} q^{43} - 4 \beta_1 q^{47} - 5 q^{49} - 3 \beta_{2} q^{53} - 6 \beta_1 q^{55} + 4 \beta_{3} q^{59} + 2 \beta_{2} q^{61} + 12 q^{65} + 8 \beta_1 q^{71} - 4 q^{73} - 2 \beta_{2} q^{77} + 5 \beta_1 q^{79} - 3 \beta_{3} q^{83} + 4 \beta_{2} q^{85} + 16 q^{89} + 2 \beta_{3} q^{91} - 12 \beta_1 q^{95} + 6 q^{97}+O(q^{100})$$ q + b2 * q^5 + b1 * q^7 - b3 * q^11 + 2*b2 * q^13 + 4 * q^17 - 2*b3 * q^19 + 4*b1 * q^23 + q^25 + b2 * q^29 - b1 * q^31 + b3 * q^35 + 2*b2 * q^37 + 4 * q^41 + 2*b3 * q^43 - 4*b1 * q^47 - 5 * q^49 - 3*b2 * q^53 - 6*b1 * q^55 + 4*b3 * q^59 + 2*b2 * q^61 + 12 * q^65 + 8*b1 * q^71 - 4 * q^73 - 2*b2 * q^77 + 5*b1 * q^79 - 3*b3 * q^83 + 4*b2 * q^85 + 16 * q^89 + 2*b3 * q^91 - 12*b1 * q^95 + 6 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 16 q^{17} + 4 q^{25} + 16 q^{41} - 20 q^{49} + 48 q^{65} - 16 q^{73} + 64 q^{89} + 24 q^{97}+O(q^{100})$$ 4 * q + 16 * q^17 + 4 * q^25 + 16 * q^41 - 20 * q^49 + 48 * q^65 - 16 * q^73 + 64 * q^89 + 24 * 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
 −1.93185 −0.517638 0.517638 1.93185
0 0 0 −2.44949 0 −1.41421 0 0 0
1.2 0 0 0 −2.44949 0 1.41421 0 0 0
1.3 0 0 0 2.44949 0 −1.41421 0 0 0
1.4 0 0 0 2.44949 0 1.41421 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 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.z yes 4
3.b odd 2 1 4608.2.a.v 4
4.b odd 2 1 inner 4608.2.a.z yes 4
8.b even 2 1 inner 4608.2.a.z yes 4
8.d odd 2 1 inner 4608.2.a.z yes 4
12.b even 2 1 4608.2.a.v 4
16.e even 4 2 4608.2.d.m 4
16.f odd 4 2 4608.2.d.m 4
24.f even 2 1 4608.2.a.v 4
24.h odd 2 1 4608.2.a.v 4
48.i odd 4 2 4608.2.d.e 4
48.k even 4 2 4608.2.d.e 4

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

## Hecke kernels

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

 $$T_{5}^{2} - 6$$ T5^2 - 6 $$T_{7}^{2} - 2$$ T7^2 - 2 $$T_{11}^{2} - 12$$ T11^2 - 12 $$T_{17} - 4$$ T17 - 4 $$T_{23}^{2} - 32$$ T23^2 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 6)^{2}$$
$7$ $$(T^{2} - 2)^{2}$$
$11$ $$(T^{2} - 12)^{2}$$
$13$ $$(T^{2} - 24)^{2}$$
$17$ $$(T - 4)^{4}$$
$19$ $$(T^{2} - 48)^{2}$$
$23$ $$(T^{2} - 32)^{2}$$
$29$ $$(T^{2} - 6)^{2}$$
$31$ $$(T^{2} - 2)^{2}$$
$37$ $$(T^{2} - 24)^{2}$$
$41$ $$(T - 4)^{4}$$
$43$ $$(T^{2} - 48)^{2}$$
$47$ $$(T^{2} - 32)^{2}$$
$53$ $$(T^{2} - 54)^{2}$$
$59$ $$(T^{2} - 192)^{2}$$
$61$ $$(T^{2} - 24)^{2}$$
$67$ $$T^{4}$$
$71$ $$(T^{2} - 128)^{2}$$
$73$ $$(T + 4)^{4}$$
$79$ $$(T^{2} - 50)^{2}$$
$83$ $$(T^{2} - 108)^{2}$$
$89$ $$(T - 16)^{4}$$
$97$ $$(T - 6)^{4}$$