# Properties

 Label 540.3.b.a Level $540$ Weight $3$ Character orbit 540.b Analytic conductor $14.714$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$540 = 2^{2} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 540.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.7139342755$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-11}, \sqrt{-19})$$ Defining polynomial: $$x^{4} + 15x^{2} + 4$$ x^4 + 15*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - 1) q^{5} - \beta_{2} q^{7}+O(q^{10})$$ q + (-b3 - 1) * q^5 - b2 * q^7 $$q + ( - \beta_{3} - 1) q^{5} - \beta_{2} q^{7} + (2 \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{11} + ( - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{13} + (\beta_{3} + \beta_1 - 1) q^{17} + (3 \beta_{3} + 3 \beta_1 + 3) q^{19} + ( - \beta_{3} - \beta_1 + 7) q^{23} + (3 \beta_{3} - 5 \beta_{2} + 3) q^{25} + ( - 3 \beta_{3} + 9 \beta_{2} + 3 \beta_1) q^{29} + (6 \beta_{3} + 6 \beta_1 + 7) q^{31} + (5 \beta_1 - 10) q^{35} + ( - 6 \beta_{3} + 2 \beta_{2} + 6 \beta_1) q^{37} + ( - \beta_{3} + 3 \beta_{2} + \beta_1) q^{41} + ( - 9 \beta_{3} + \beta_{2} + 9 \beta_1) q^{43} + ( - 4 \beta_{3} - 4 \beta_1 - 14) q^{47} + (3 \beta_{3} + 3 \beta_1 + 21) q^{49} + (6 \beta_{3} + 6 \beta_1 + 51) q^{53} + ( - 6 \beta_{3} + 10 \beta_{2} + 15 \beta_1 + 14) q^{55} + (11 \beta_{3} - 9 \beta_{2} - 11 \beta_1) q^{59} + (3 \beta_{3} + 3 \beta_1 + 11) q^{61} + (9 \beta_{3} - 15 \beta_{2} - 15 \beta_1 - 36) q^{65} + ( - 6 \beta_{3} - 8 \beta_{2} + 6 \beta_1) q^{67} + ( - 15 \beta_{3} + 3 \beta_{2} + 15 \beta_1) q^{71} + (12 \beta_{3} - 17 \beta_{2} - 12 \beta_1) q^{73} + ( - \beta_{3} - \beta_1 - 44) q^{77} + (9 \beta_{3} + 9 \beta_1 + 3) q^{79} + ( - 10 \beta_{3} - 10 \beta_1 + 67) q^{83} + (5 \beta_{2} - 25) q^{85} + ( - 11 \beta_{3} - 9 \beta_{2} + 11 \beta_1) q^{89} + (6 \beta_{3} + 6 \beta_1 + 24) q^{91} + ( - 6 \beta_{3} + 15 \beta_{2} - 81) q^{95} + ( - 3 \beta_{3} + 28 \beta_{2} + 3 \beta_1) q^{97}+O(q^{100})$$ q + (-b3 - 1) * q^5 - b2 * q^7 + (2*b3 - 3*b2 - 2*b1) * q^11 + (-3*b3 + 3*b2 + 3*b1) * q^13 + (b3 + b1 - 1) * q^17 + (3*b3 + 3*b1 + 3) * q^19 + (-b3 - b1 + 7) * q^23 + (3*b3 - 5*b2 + 3) * q^25 + (-3*b3 + 9*b2 + 3*b1) * q^29 + (6*b3 + 6*b1 + 7) * q^31 + (5*b1 - 10) * q^35 + (-6*b3 + 2*b2 + 6*b1) * q^37 + (-b3 + 3*b2 + b1) * q^41 + (-9*b3 + b2 + 9*b1) * q^43 + (-4*b3 - 4*b1 - 14) * q^47 + (3*b3 + 3*b1 + 21) * q^49 + (6*b3 + 6*b1 + 51) * q^53 + (-6*b3 + 10*b2 + 15*b1 + 14) * q^55 + (11*b3 - 9*b2 - 11*b1) * q^59 + (3*b3 + 3*b1 + 11) * q^61 + (9*b3 - 15*b2 - 15*b1 - 36) * q^65 + (-6*b3 - 8*b2 + 6*b1) * q^67 + (-15*b3 + 3*b2 + 15*b1) * q^71 + (12*b3 - 17*b2 - 12*b1) * q^73 + (-b3 - b1 - 44) * q^77 + (9*b3 + 9*b1 + 3) * q^79 + (-10*b3 - 10*b1 + 67) * q^83 + (5*b2 - 25) * q^85 + (-11*b3 - 9*b2 + 11*b1) * q^89 + (6*b3 + 6*b1 + 24) * q^91 + (-6*b3 + 15*b2 - 81) * q^95 + (-3*b3 + 28*b2 + 3*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{5}+O(q^{10})$$ 4 * q - 3 * q^5 $$4 q - 3 q^{5} - 6 q^{17} + 6 q^{19} + 30 q^{23} + 9 q^{25} + 16 q^{31} - 45 q^{35} - 48 q^{47} + 78 q^{49} + 192 q^{53} + 47 q^{55} + 38 q^{61} - 138 q^{65} - 174 q^{77} - 6 q^{79} + 288 q^{83} - 100 q^{85} + 84 q^{91} - 318 q^{95}+O(q^{100})$$ 4 * q - 3 * q^5 - 6 * q^17 + 6 * q^19 + 30 * q^23 + 9 * q^25 + 16 * q^31 - 45 * q^35 - 48 * q^47 + 78 * q^49 + 192 * q^53 + 47 * q^55 + 38 * q^61 - 138 * q^65 - 174 * q^77 - 6 * q^79 + 288 * q^83 - 100 * q^85 + 84 * q^91 - 318 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 15x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} + 19\nu + 14 ) / 4$$ (v^3 + 2*v^2 + 19*v + 14) / 4 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 11\nu ) / 2$$ (v^3 + 11*v) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 2\nu^{2} - 19\nu + 14 ) / 4$$ (-v^3 + 2*v^2 - 19*v + 14) / 4
 $$\nu$$ $$=$$ $$( -\beta_{3} - \beta_{2} + \beta_1 ) / 4$$ (-b3 - b2 + b1) / 4 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta _1 - 7$$ b3 + b1 - 7 $$\nu^{3}$$ $$=$$ $$( 11\beta_{3} + 19\beta_{2} - 11\beta_1 ) / 4$$ (11*b3 + 19*b2 - 11*b1) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/540\mathbb{Z}\right)^\times$$.

 $$n$$ $$217$$ $$271$$ $$461$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
269.1
 − 0.521137i 0.521137i − 3.83776i 3.83776i
0 0 0 −4.36421 2.44002i 0 2.79549i 0 0 0
269.2 0 0 0 −4.36421 + 2.44002i 0 2.79549i 0 0 0
269.3 0 0 0 2.86421 4.09833i 0 7.15439i 0 0 0
269.4 0 0 0 2.86421 + 4.09833i 0 7.15439i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 540.3.b.a 4
3.b odd 2 1 540.3.b.b yes 4
4.b odd 2 1 2160.3.c.h 4
5.b even 2 1 540.3.b.b yes 4
5.c odd 4 2 2700.3.g.s 8
9.c even 3 2 1620.3.t.d 8
9.d odd 6 2 1620.3.t.a 8
12.b even 2 1 2160.3.c.l 4
15.d odd 2 1 inner 540.3.b.a 4
15.e even 4 2 2700.3.g.s 8
20.d odd 2 1 2160.3.c.l 4
45.h odd 6 2 1620.3.t.d 8
45.j even 6 2 1620.3.t.a 8
60.h even 2 1 2160.3.c.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.3.b.a 4 1.a even 1 1 trivial
540.3.b.a 4 15.d odd 2 1 inner
540.3.b.b yes 4 3.b odd 2 1
540.3.b.b yes 4 5.b even 2 1
1620.3.t.a 8 9.d odd 6 2
1620.3.t.a 8 45.j even 6 2
1620.3.t.d 8 9.c even 3 2
1620.3.t.d 8 45.h odd 6 2
2160.3.c.h 4 4.b odd 2 1
2160.3.c.h 4 60.h even 2 1
2160.3.c.l 4 12.b even 2 1
2160.3.c.l 4 20.d odd 2 1
2700.3.g.s 8 5.c odd 4 2
2700.3.g.s 8 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(540, [\chi])$$:

 $$T_{7}^{4} + 59T_{7}^{2} + 400$$ T7^4 + 59*T7^2 + 400 $$T_{17}^{2} + 3T_{17} - 50$$ T17^2 + 3*T17 - 50

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 3 T^{3} + 75 T + 625$$
$7$ $$T^{4} + 59T^{2} + 400$$
$11$ $$T^{4} + 355T^{2} + 8464$$
$13$ $$T^{4} + 540T^{2} + 5184$$
$17$ $$(T^{2} + 3 T - 50)^{2}$$
$19$ $$(T^{2} - 3 T - 468)^{2}$$
$23$ $$(T^{2} - 15 T + 4)^{2}$$
$29$ $$(T^{2} + 1584)^{2}$$
$31$ $$(T^{2} - 8 T - 1865)^{2}$$
$37$ $$(T^{2} + 1216)^{2}$$
$41$ $$(T^{2} + 176)^{2}$$
$43$ $$T^{4} + 6620 T^{2} + \cdots + 9684544$$
$47$ $$(T^{2} + 24 T - 692)^{2}$$
$53$ $$(T^{2} - 96 T + 423)^{2}$$
$59$ $$T^{4} + 6880 T^{2} + \cdots + 4129024$$
$61$ $$(T^{2} - 19 T - 380)^{2}$$
$67$ $$T^{4} + 11372 T^{2} + \cdots + 541696$$
$71$ $$T^{4} + 16956 T^{2} + \cdots + 68558400$$
$73$ $$T^{4} + 11795 T^{2} + \cdots + 6091024$$
$79$ $$(T^{2} + 3 T - 4230)^{2}$$
$83$ $$(T^{2} - 144 T - 41)^{2}$$
$89$ $$T^{4} + 24700 T^{2} + \cdots + 19430464$$
$97$ $$T^{4} + 39515 T^{2} + \cdots + 266603584$$