# Properties

 Label 5184.2.a.cf Level $5184$ Weight $2$ Character orbit 5184.a Self dual yes Analytic conductor $41.394$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5184 = 2^{6} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5184.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$41.3944484078$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.13068.1 Defining polynomial: $$x^{4} - x^{3} - 6x^{2} - x + 1$$ x^4 - x^3 - 6*x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 288) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 1) q^{5} + \beta_{3} q^{7}+O(q^{10})$$ q + (b1 + 1) * q^5 + b3 * q^7 $$q + (\beta_1 + 1) q^{5} + \beta_{3} q^{7} + \beta_{2} q^{11} + (\beta_1 - 1) q^{13} - \beta_1 q^{17} + ( - \beta_{3} + \beta_{2}) q^{19} + \beta_{3} q^{23} + (\beta_1 + 4) q^{25} + (\beta_1 + 3) q^{29} + ( - \beta_{3} + 2 \beta_{2}) q^{31} + (3 \beta_{3} - 2 \beta_{2}) q^{35} - 4 q^{37} + q^{41} - \beta_{2} q^{43} + \beta_{3} q^{47} + (3 \beta_1 + 8) q^{49} + 4 q^{53} + ( - \beta_{3} - 2 \beta_{2}) q^{55} + ( - 2 \beta_{3} - \beta_{2}) q^{59} + (3 \beta_1 - 5) q^{61} + ( - \beta_1 + 7) q^{65} + (2 \beta_{3} + \beta_{2}) q^{67} + 2 \beta_{3} q^{71} + (\beta_1 + 8) q^{73} + ( - 3 \beta_1 + 3) q^{77} + (\beta_{3} - 2 \beta_{2}) q^{79} + (\beta_{3} + 2 \beta_{2}) q^{83} - 8 q^{85} + (2 \beta_1 + 8) q^{89} + (\beta_{3} - 2 \beta_{2}) q^{91} - 4 \beta_{3} q^{95} + 9 q^{97}+O(q^{100})$$ q + (b1 + 1) * q^5 + b3 * q^7 + b2 * q^11 + (b1 - 1) * q^13 - b1 * q^17 + (-b3 + b2) * q^19 + b3 * q^23 + (b1 + 4) * q^25 + (b1 + 3) * q^29 + (-b3 + 2*b2) * q^31 + (3*b3 - 2*b2) * q^35 - 4 * q^37 + q^41 - b2 * q^43 + b3 * q^47 + (3*b1 + 8) * q^49 + 4 * q^53 + (-b3 - 2*b2) * q^55 + (-2*b3 - b2) * q^59 + (3*b1 - 5) * q^61 + (-b1 + 7) * q^65 + (2*b3 + b2) * q^67 + 2*b3 * q^71 + (b1 + 8) * q^73 + (-3*b1 + 3) * q^77 + (b3 - 2*b2) * q^79 + (b3 + 2*b2) * q^83 - 8 * q^85 + (2*b1 + 8) * q^89 + (b3 - 2*b2) * q^91 - 4*b3 * q^95 + 9 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{5}+O(q^{10})$$ 4 * q + 2 * q^5 $$4 q + 2 q^{5} - 6 q^{13} + 2 q^{17} + 14 q^{25} + 10 q^{29} - 16 q^{37} + 4 q^{41} + 26 q^{49} + 16 q^{53} - 26 q^{61} + 30 q^{65} + 30 q^{73} + 18 q^{77} - 32 q^{85} + 28 q^{89} + 36 q^{97}+O(q^{100})$$ 4 * q + 2 * q^5 - 6 * q^13 + 2 * q^17 + 14 * q^25 + 10 * q^29 - 16 * q^37 + 4 * q^41 + 26 * q^49 + 16 * q^53 - 26 * q^61 + 30 * q^65 + 30 * q^73 + 18 * q^77 - 32 * q^85 + 28 * q^89 + 36 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 6x^{2} - x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} - \nu^{2} - 7\nu - 1$$ v^3 - v^2 - 7*v - 1 $$\beta_{2}$$ $$=$$ $$\nu^{3} - 6\nu - 4$$ v^3 - 6*v - 4 $$\beta_{3}$$ $$=$$ $$2\nu^{3} - 3\nu^{2} - 9\nu + 1$$ 2*v^3 - 3*v^2 - 9*v + 1
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 3\beta_1 ) / 6$$ (b3 + b2 - 3*b1) / 6 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 5\beta_{2} - 3\beta _1 + 18 ) / 6$$ (-b3 + 5*b2 - 3*b1 + 18) / 6 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2\beta_{2} - 3\beta _1 + 4$$ b3 + 2*b2 - 3*b1 + 4

## 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
 0.328543 3.04374 −1.82405 −0.548230
0 0 0 −2.37228 0 −2.20979 0 0 0
1.2 0 0 0 −2.37228 0 2.20979 0 0 0
1.3 0 0 0 3.37228 0 −4.70285 0 0 0
1.4 0 0 0 3.37228 0 4.70285 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5184.2.a.cf 4
3.b odd 2 1 5184.2.a.cc 4
4.b odd 2 1 inner 5184.2.a.cf 4
8.b even 2 1 2592.2.a.u 4
8.d odd 2 1 2592.2.a.u 4
9.c even 3 2 576.2.i.n 8
9.d odd 6 2 1728.2.i.n 8
12.b even 2 1 5184.2.a.cc 4
24.f even 2 1 2592.2.a.x 4
24.h odd 2 1 2592.2.a.x 4
36.f odd 6 2 576.2.i.n 8
36.h even 6 2 1728.2.i.n 8
72.j odd 6 2 864.2.i.f 8
72.l even 6 2 864.2.i.f 8
72.n even 6 2 288.2.i.f 8
72.p odd 6 2 288.2.i.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.f 8 72.n even 6 2
288.2.i.f 8 72.p odd 6 2
576.2.i.n 8 9.c even 3 2
576.2.i.n 8 36.f odd 6 2
864.2.i.f 8 72.j odd 6 2
864.2.i.f 8 72.l even 6 2
1728.2.i.n 8 9.d odd 6 2
1728.2.i.n 8 36.h even 6 2
2592.2.a.u 4 8.b even 2 1
2592.2.a.u 4 8.d odd 2 1
2592.2.a.x 4 24.f even 2 1
2592.2.a.x 4 24.h odd 2 1
5184.2.a.cc 4 3.b odd 2 1
5184.2.a.cc 4 12.b even 2 1
5184.2.a.cf 4 1.a even 1 1 trivial
5184.2.a.cf 4 4.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5184))$$:

 $$T_{5}^{2} - T_{5} - 8$$ T5^2 - T5 - 8 $$T_{7}^{4} - 27T_{7}^{2} + 108$$ T7^4 - 27*T7^2 + 108 $$T_{11}^{4} - 36T_{11}^{2} + 27$$ T11^4 - 36*T11^2 + 27

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - T - 8)^{2}$$
$7$ $$T^{4} - 27T^{2} + 108$$
$11$ $$T^{4} - 36T^{2} + 27$$
$13$ $$(T^{2} + 3 T - 6)^{2}$$
$17$ $$(T^{2} - T - 8)^{2}$$
$19$ $$T^{4} - 45T^{2} + 432$$
$23$ $$T^{4} - 27T^{2} + 108$$
$29$ $$(T^{2} - 5 T - 2)^{2}$$
$31$ $$T^{4} - 135T^{2} + 3888$$
$37$ $$(T + 4)^{4}$$
$41$ $$(T - 1)^{4}$$
$43$ $$T^{4} - 36T^{2} + 27$$
$47$ $$T^{4} - 27T^{2} + 108$$
$53$ $$(T - 4)^{4}$$
$59$ $$T^{4} - 180T^{2} + 7803$$
$61$ $$(T^{2} + 13 T - 32)^{2}$$
$67$ $$T^{4} - 180T^{2} + 7803$$
$71$ $$T^{4} - 108T^{2} + 1728$$
$73$ $$(T^{2} - 15 T + 48)^{2}$$
$79$ $$T^{4} - 135T^{2} + 3888$$
$83$ $$T^{4} - 207T^{2} + 1728$$
$89$ $$(T^{2} - 14 T + 16)^{2}$$
$97$ $$(T - 9)^{4}$$