# Properties

 Label 4608.2.a.ba Level $4608$ Weight $2$ Character orbit 4608.a Self dual yes Analytic conductor $36.795$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# 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: 4.4.4352.1 Defining polynomial: $$x^{4} - 6x^{2} - 4x + 2$$ x^4 - 6*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 1536) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{5} + \beta_{3} q^{7}+O(q^{10})$$ q + b1 * q^5 + b3 * q^7 $$q + \beta_1 q^{5} + \beta_{3} q^{7} + (\beta_{2} + 2) q^{11} + (\beta_{3} - \beta_1) q^{13} + (\beta_{2} - 2) q^{17} + \beta_{2} q^{19} + 2 \beta_1 q^{23} + (2 \beta_{2} + 5) q^{25} + (2 \beta_{3} - \beta_1) q^{29} - \beta_{3} q^{31} + ( - 3 \beta_{2} + 2) q^{35} + (\beta_{3} + \beta_1) q^{37} + (\beta_{2} - 6) q^{41} + ( - 3 \beta_{2} + 4) q^{43} + 2 \beta_1 q^{47} + ( - 4 \beta_{2} + 7) q^{49} + ( - 2 \beta_{3} - \beta_1) q^{53} + ( - 2 \beta_{3} + 4 \beta_1) q^{55} + (2 \beta_{2} + 4) q^{59} + (\beta_{3} + \beta_1) q^{61} + ( - 5 \beta_{2} - 8) q^{65} + (2 \beta_{2} + 8) q^{67} + 2 \beta_{3} q^{71} + 4 q^{73} - 2 \beta_1 q^{77} + \beta_{3} q^{79} + (\beta_{2} + 6) q^{83} - 2 \beta_{3} q^{85} - 10 q^{89} + ( - \beta_{2} + 12) q^{91} + ( - 2 \beta_{3} + 2 \beta_1) q^{95} + (2 \beta_{2} + 6) q^{97}+O(q^{100})$$ q + b1 * q^5 + b3 * q^7 + (b2 + 2) * q^11 + (b3 - b1) * q^13 + (b2 - 2) * q^17 + b2 * q^19 + 2*b1 * q^23 + (2*b2 + 5) * q^25 + (2*b3 - b1) * q^29 - b3 * q^31 + (-3*b2 + 2) * q^35 + (b3 + b1) * q^37 + (b2 - 6) * q^41 + (-3*b2 + 4) * q^43 + 2*b1 * q^47 + (-4*b2 + 7) * q^49 + (-2*b3 - b1) * q^53 + (-2*b3 + 4*b1) * q^55 + (2*b2 + 4) * q^59 + (b3 + b1) * q^61 + (-5*b2 - 8) * q^65 + (2*b2 + 8) * q^67 + 2*b3 * q^71 + 4 * q^73 - 2*b1 * q^77 + b3 * q^79 + (b2 + 6) * q^83 - 2*b3 * q^85 - 10 * q^89 + (-b2 + 12) * q^91 + (-2*b3 + 2*b1) * q^95 + (2*b2 + 6) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 8 q^{11} - 8 q^{17} + 20 q^{25} + 8 q^{35} - 24 q^{41} + 16 q^{43} + 28 q^{49} + 16 q^{59} - 32 q^{65} + 32 q^{67} + 16 q^{73} + 24 q^{83} - 40 q^{89} + 48 q^{91} + 24 q^{97}+O(q^{100})$$ 4 * q + 8 * q^11 - 8 * q^17 + 20 * q^25 + 8 * q^35 - 24 * q^41 + 16 * q^43 + 28 * q^49 + 16 * q^59 - 32 * q^65 + 32 * q^67 + 16 * q^73 + 24 * q^83 - 40 * q^89 + 48 * q^91 + 24 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 6x^{2} - 4x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} - \nu^{2} - 6\nu$$ v^3 - v^2 - 6*v $$\beta_{2}$$ $$=$$ $$2\nu^{3} - 2\nu^{2} - 8\nu$$ 2*v^3 - 2*v^2 - 8*v $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 3\nu^{2} + 2\nu - 6$$ -v^3 + 3*v^2 + 2*v - 6
 $$\nu$$ $$=$$ $$( \beta_{2} - 2\beta_1 ) / 4$$ (b2 - 2*b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} - \beta _1 + 6 ) / 2$$ (b3 + b2 - b1 + 6) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{3} + 4\beta_{2} - 5\beta _1 + 6 ) / 2$$ (b3 + 4*b2 - 5*b1 + 6) / 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
 2.68554 0.334904 −1.74912 −1.27133
0 0 0 −3.95687 0 1.63899 0 0 0
1.2 0 0 0 −2.08402 0 −5.03127 0 0 0
1.3 0 0 0 2.08402 0 5.03127 0 0 0
1.4 0 0 0 3.95687 0 −1.63899 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
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.ba 4
3.b odd 2 1 1536.2.a.m 4
4.b odd 2 1 4608.2.a.t 4
8.b even 2 1 4608.2.a.t 4
8.d odd 2 1 inner 4608.2.a.ba 4
12.b even 2 1 1536.2.a.n yes 4
16.e even 4 2 4608.2.d.p 8
16.f odd 4 2 4608.2.d.p 8
24.f even 2 1 1536.2.a.m 4
24.h odd 2 1 1536.2.a.n yes 4
48.i odd 4 2 1536.2.d.g 8
48.k even 4 2 1536.2.d.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.a.m 4 3.b odd 2 1
1536.2.a.m 4 24.f even 2 1
1536.2.a.n yes 4 12.b even 2 1
1536.2.a.n yes 4 24.h odd 2 1
1536.2.d.g 8 48.i odd 4 2
1536.2.d.g 8 48.k even 4 2
4608.2.a.t 4 4.b odd 2 1
4608.2.a.t 4 8.b even 2 1
4608.2.a.ba 4 1.a even 1 1 trivial
4608.2.a.ba 4 8.d odd 2 1 inner
4608.2.d.p 8 16.e even 4 2
4608.2.d.p 8 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4608))$$:

 $$T_{5}^{4} - 20T_{5}^{2} + 68$$ T5^4 - 20*T5^2 + 68 $$T_{7}^{4} - 28T_{7}^{2} + 68$$ T7^4 - 28*T7^2 + 68 $$T_{11}^{2} - 4T_{11} - 4$$ T11^2 - 4*T11 - 4 $$T_{17}^{2} + 4T_{17} - 4$$ T17^2 + 4*T17 - 4 $$T_{23}^{4} - 80T_{23}^{2} + 1088$$ T23^4 - 80*T23^2 + 1088

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 20T^{2} + 68$$
$7$ $$T^{4} - 28T^{2} + 68$$
$11$ $$(T^{2} - 4 T - 4)^{2}$$
$13$ $$T^{4} - 40T^{2} + 272$$
$17$ $$(T^{2} + 4 T - 4)^{2}$$
$19$ $$(T^{2} - 8)^{2}$$
$23$ $$T^{4} - 80T^{2} + 1088$$
$29$ $$T^{4} - 116T^{2} + 3332$$
$31$ $$T^{4} - 28T^{2} + 68$$
$37$ $$T^{4} - 56T^{2} + 272$$
$41$ $$(T^{2} + 12 T + 28)^{2}$$
$43$ $$(T^{2} - 8 T - 56)^{2}$$
$47$ $$T^{4} - 80T^{2} + 1088$$
$53$ $$T^{4} - 148T^{2} + 68$$
$59$ $$(T^{2} - 8 T - 16)^{2}$$
$61$ $$T^{4} - 56T^{2} + 272$$
$67$ $$(T^{2} - 16 T + 32)^{2}$$
$71$ $$T^{4} - 112T^{2} + 1088$$
$73$ $$(T - 4)^{4}$$
$79$ $$T^{4} - 28T^{2} + 68$$
$83$ $$(T^{2} - 12 T + 28)^{2}$$
$89$ $$(T + 10)^{4}$$
$97$ $$(T^{2} - 12 T + 4)^{2}$$