# Properties

 Label 6240.2.a.bp Level $6240$ Weight $2$ Character orbit 6240.a Self dual yes Analytic conductor $49.827$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} - \beta q^{7} + q^{9} +O(q^{10})$$ q + q^3 - q^5 - b * q^7 + q^9 $$q + q^{3} - q^{5} - \beta q^{7} + q^{9} + (\beta - 4) q^{11} - q^{13} - q^{15} + ( - \beta - 2) q^{17} - 2 \beta q^{19} - \beta q^{21} + (\beta + 4) q^{23} + q^{25} + q^{27} + (2 \beta + 2) q^{29} + (\beta - 4) q^{33} + \beta q^{35} + ( - \beta - 6) q^{37} - q^{39} + ( - 3 \beta + 2) q^{41} + (2 \beta + 4) q^{43} - q^{45} + 4 q^{47} + (\beta + 1) q^{49} + ( - \beta - 2) q^{51} + ( - \beta + 2) q^{53} + ( - \beta + 4) q^{55} - 2 \beta q^{57} - 4 q^{59} + ( - \beta + 6) q^{61} - \beta q^{63} + q^{65} + ( - 4 \beta + 4) q^{67} + (\beta + 4) q^{69} + (\beta + 8) q^{71} + 10 q^{73} + q^{75} + (3 \beta - 8) q^{77} - \beta q^{79} + q^{81} + (4 \beta - 4) q^{83} + (\beta + 2) q^{85} + (2 \beta + 2) q^{87} + ( - \beta - 6) q^{89} + \beta q^{91} + 2 \beta q^{95} + (\beta + 2) q^{97} + (\beta - 4) q^{99} +O(q^{100})$$ q + q^3 - q^5 - b * q^7 + q^9 + (b - 4) * q^11 - q^13 - q^15 + (-b - 2) * q^17 - 2*b * q^19 - b * q^21 + (b + 4) * q^23 + q^25 + q^27 + (2*b + 2) * q^29 + (b - 4) * q^33 + b * q^35 + (-b - 6) * q^37 - q^39 + (-3*b + 2) * q^41 + (2*b + 4) * q^43 - q^45 + 4 * q^47 + (b + 1) * q^49 + (-b - 2) * q^51 + (-b + 2) * q^53 + (-b + 4) * q^55 - 2*b * q^57 - 4 * q^59 + (-b + 6) * q^61 - b * q^63 + q^65 + (-4*b + 4) * q^67 + (b + 4) * q^69 + (b + 8) * q^71 + 10 * q^73 + q^75 + (3*b - 8) * q^77 - b * q^79 + q^81 + (4*b - 4) * q^83 + (b + 2) * q^85 + (2*b + 2) * q^87 + (-b - 6) * q^89 + b * q^91 + 2*b * q^95 + (b + 2) * q^97 + (b - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9 $$2 q + 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9} - 7 q^{11} - 2 q^{13} - 2 q^{15} - 5 q^{17} - 2 q^{19} - q^{21} + 9 q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} - 7 q^{33} + q^{35} - 13 q^{37} - 2 q^{39} + q^{41} + 10 q^{43} - 2 q^{45} + 8 q^{47} + 3 q^{49} - 5 q^{51} + 3 q^{53} + 7 q^{55} - 2 q^{57} - 8 q^{59} + 11 q^{61} - q^{63} + 2 q^{65} + 4 q^{67} + 9 q^{69} + 17 q^{71} + 20 q^{73} + 2 q^{75} - 13 q^{77} - q^{79} + 2 q^{81} - 4 q^{83} + 5 q^{85} + 6 q^{87} - 13 q^{89} + q^{91} + 2 q^{95} + 5 q^{97} - 7 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9 - 7 * q^11 - 2 * q^13 - 2 * q^15 - 5 * q^17 - 2 * q^19 - q^21 + 9 * q^23 + 2 * q^25 + 2 * q^27 + 6 * q^29 - 7 * q^33 + q^35 - 13 * q^37 - 2 * q^39 + q^41 + 10 * q^43 - 2 * q^45 + 8 * q^47 + 3 * q^49 - 5 * q^51 + 3 * q^53 + 7 * q^55 - 2 * q^57 - 8 * q^59 + 11 * q^61 - q^63 + 2 * q^65 + 4 * q^67 + 9 * q^69 + 17 * q^71 + 20 * q^73 + 2 * q^75 - 13 * q^77 - q^79 + 2 * q^81 - 4 * q^83 + 5 * q^85 + 6 * q^87 - 13 * q^89 + q^91 + 2 * q^95 + 5 * q^97 - 7 * q^99

## 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
 3.37228 −2.37228
0 1.00000 0 −1.00000 0 −3.37228 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 2.37228 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.bp yes 2
4.b odd 2 1 6240.2.a.bj 2

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

## Hecke kernels

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

 $$T_{7}^{2} + T_{7} - 8$$ T7^2 + T7 - 8 $$T_{11}^{2} + 7T_{11} + 4$$ T11^2 + 7*T11 + 4 $$T_{17}^{2} + 5T_{17} - 2$$ T17^2 + 5*T17 - 2 $$T_{19}^{2} + 2T_{19} - 32$$ T19^2 + 2*T19 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + T - 8$$
$11$ $$T^{2} + 7T + 4$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} + 5T - 2$$
$19$ $$T^{2} + 2T - 32$$
$23$ $$T^{2} - 9T + 12$$
$29$ $$T^{2} - 6T - 24$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 13T + 34$$
$41$ $$T^{2} - T - 74$$
$43$ $$T^{2} - 10T - 8$$
$47$ $$(T - 4)^{2}$$
$53$ $$T^{2} - 3T - 6$$
$59$ $$(T + 4)^{2}$$
$61$ $$T^{2} - 11T + 22$$
$67$ $$T^{2} - 4T - 128$$
$71$ $$T^{2} - 17T + 64$$
$73$ $$(T - 10)^{2}$$
$79$ $$T^{2} + T - 8$$
$83$ $$T^{2} + 4T - 128$$
$89$ $$T^{2} + 13T + 34$$
$97$ $$T^{2} - 5T - 2$$