# Properties

 Label 6240.2.a.bw Level $6240$ Weight $2$ Character orbit 6240.a Self dual yes Analytic conductor $49.827$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6240.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$49.8266508613$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{5} + ( - \beta_{2} - 2) q^{7} + q^{9}+O(q^{10})$$ q - q^3 - q^5 + (-b2 - 2) * q^7 + q^9 $$q - q^{3} - q^{5} + ( - \beta_{2} - 2) q^{7} + q^{9} - \beta_1 q^{11} + q^{13} + q^{15} + (\beta_{2} - 2 \beta_1) q^{17} + ( - \beta_{2} - \beta_1 - 2) q^{19} + (\beta_{2} + 2) q^{21} + ( - 3 \beta_{2} - 2) q^{23} + q^{25} - q^{27} + ( - 3 \beta_{2} + \beta_1) q^{29} - 4 q^{31} + \beta_1 q^{33} + (\beta_{2} + 2) q^{35} + ( - \beta_1 + 2) q^{37} - q^{39} + ( - \beta_1 - 2) q^{41} - 2 \beta_1 q^{43} - q^{45} + 4 \beta_{2} q^{47} + (4 \beta_{2} - \beta_1 + 1) q^{49} + ( - \beta_{2} + 2 \beta_1) q^{51} + (3 \beta_1 + 2) q^{53} + \beta_1 q^{55} + (\beta_{2} + \beta_1 + 2) q^{57} + ( - 2 \beta_{2} + 2 \beta_1) q^{59} + ( - 2 \beta_{2} - \beta_1 - 2) q^{61} + ( - \beta_{2} - 2) q^{63} - q^{65} + (3 \beta_{2} + 2) q^{69} + ( - 2 \beta_{2} + \beta_1 - 4) q^{71} + ( - 5 \beta_{2} + \beta_1) q^{73} - q^{75} + ( - 2 \beta_{2} + \beta_1) q^{77} + (2 \beta_{2} - \beta_1 - 8) q^{79} + q^{81} + (4 \beta_{2} + 8) q^{83} + ( - \beta_{2} + 2 \beta_1) q^{85} + (3 \beta_{2} - \beta_1) q^{87} + (2 \beta_{2} - \beta_1 - 2) q^{89} + ( - \beta_{2} - 2) q^{91} + 4 q^{93} + (\beta_{2} + \beta_1 + 2) q^{95} + (\beta_{2} + 8) q^{97} - \beta_1 q^{99}+O(q^{100})$$ q - q^3 - q^5 + (-b2 - 2) * q^7 + q^9 - b1 * q^11 + q^13 + q^15 + (b2 - 2*b1) * q^17 + (-b2 - b1 - 2) * q^19 + (b2 + 2) * q^21 + (-3*b2 - 2) * q^23 + q^25 - q^27 + (-3*b2 + b1) * q^29 - 4 * q^31 + b1 * q^33 + (b2 + 2) * q^35 + (-b1 + 2) * q^37 - q^39 + (-b1 - 2) * q^41 - 2*b1 * q^43 - q^45 + 4*b2 * q^47 + (4*b2 - b1 + 1) * q^49 + (-b2 + 2*b1) * q^51 + (3*b1 + 2) * q^53 + b1 * q^55 + (b2 + b1 + 2) * q^57 + (-2*b2 + 2*b1) * q^59 + (-2*b2 - b1 - 2) * q^61 + (-b2 - 2) * q^63 - q^65 + (3*b2 + 2) * q^69 + (-2*b2 + b1 - 4) * q^71 + (-5*b2 + b1) * q^73 - q^75 + (-2*b2 + b1) * q^77 + (2*b2 - b1 - 8) * q^79 + q^81 + (4*b2 + 8) * q^83 + (-b2 + 2*b1) * q^85 + (3*b2 - b1) * q^87 + (2*b2 - b1 - 2) * q^89 + (-b2 - 2) * q^91 + 4 * q^93 + (b2 + b1 + 2) * q^95 + (b2 + 8) * q^97 - b1 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9 $$3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9} + q^{11} + 3 q^{13} + 3 q^{15} + q^{17} - 4 q^{19} + 5 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} + 2 q^{29} - 12 q^{31} - q^{33} + 5 q^{35} + 7 q^{37} - 3 q^{39} - 5 q^{41} + 2 q^{43} - 3 q^{45} - 4 q^{47} - q^{51} + 3 q^{53} - q^{55} + 4 q^{57} - 3 q^{61} - 5 q^{63} - 3 q^{65} + 3 q^{69} - 11 q^{71} + 4 q^{73} - 3 q^{75} + q^{77} - 25 q^{79} + 3 q^{81} + 20 q^{83} - q^{85} - 2 q^{87} - 7 q^{89} - 5 q^{91} + 12 q^{93} + 4 q^{95} + 23 q^{97} + q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9 + q^11 + 3 * q^13 + 3 * q^15 + q^17 - 4 * q^19 + 5 * q^21 - 3 * q^23 + 3 * q^25 - 3 * q^27 + 2 * q^29 - 12 * q^31 - q^33 + 5 * q^35 + 7 * q^37 - 3 * q^39 - 5 * q^41 + 2 * q^43 - 3 * q^45 - 4 * q^47 - q^51 + 3 * q^53 - q^55 + 4 * q^57 - 3 * q^61 - 5 * q^63 - 3 * q^65 + 3 * q^69 - 11 * q^71 + 4 * q^73 - 3 * q^75 + q^77 - 25 * q^79 + 3 * q^81 + 20 * q^83 - q^85 - 2 * q^87 - 7 * q^89 - 5 * q^91 + 12 * q^93 + 4 * q^95 + 23 * q^97 + q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 3$$ v^2 + v - 3 $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta_1 ) / 2$$ (-b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + \beta _1 + 6 ) / 2$$ (b2 + 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
 −1.86081 2.11491 −0.254102
0 −1.00000 0 −1.00000 0 −4.32340 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 −1.35793 0 1.00000 0
1.3 0 −1.00000 0 −1.00000 0 0.681331 0 1.00000 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
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6240.2.a.bw 3
4.b odd 2 1 6240.2.a.cb yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6240.2.a.bw 3 1.a even 1 1 trivial
6240.2.a.cb yes 3 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6240))$$:

 $$T_{7}^{3} + 5T_{7}^{2} + 2T_{7} - 4$$ T7^3 + 5*T7^2 + 2*T7 - 4 $$T_{11}^{3} - T_{11}^{2} - 12T_{11} + 16$$ T11^3 - T11^2 - 12*T11 + 16 $$T_{17}^{3} - T_{17}^{2} - 50T_{17} + 148$$ T17^3 - T17^2 - 50*T17 + 148 $$T_{19}^{3} + 4T_{19}^{2} - 16T_{19} - 56$$ T19^3 + 4*T19^2 - 16*T19 - 56

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T + 1)^{3}$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} + 5 T^{2} + 2 T - 4$$
$11$ $$T^{3} - T^{2} - 12 T + 16$$
$13$ $$(T - 1)^{3}$$
$17$ $$T^{3} - T^{2} - 50 T + 148$$
$19$ $$T^{3} + 4 T^{2} - 16 T - 56$$
$23$ $$T^{3} + 3 T^{2} - 54 T - 4$$
$29$ $$T^{3} - 2 T^{2} - 60 T + 224$$
$31$ $$(T + 4)^{3}$$
$37$ $$T^{3} - 7 T^{2} + 4 T + 28$$
$41$ $$T^{3} + 5 T^{2} - 4 T - 4$$
$43$ $$T^{3} - 2 T^{2} - 48 T + 128$$
$47$ $$T^{3} + 4 T^{2} - 96 T - 256$$
$53$ $$T^{3} - 3 T^{2} - 108 T - 212$$
$59$ $$T^{3} - 64T - 64$$
$61$ $$T^{3} + 3 T^{2} - 40 T - 148$$
$67$ $$T^{3}$$
$71$ $$T^{3} + 11 T^{2} + 8 T - 16$$
$73$ $$T^{3} - 4 T^{2} - 152 T + 904$$
$79$ $$T^{3} + 25 T^{2} + 176 T + 256$$
$83$ $$T^{3} - 20 T^{2} + 32 T + 256$$
$89$ $$T^{3} + 7 T^{2} - 16 T - 116$$
$97$ $$T^{3} - 23 T^{2} + 170 T - 404$$