# Properties

 Label 6240.2.a.br 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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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 $$\beta = 2\sqrt{2}$$. 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 $$q + q^{3} - q^{5} + \beta q^{7} + q^{9} + \beta q^{11} + q^{13} - q^{15} + (2 \beta + 2) q^{17} + (\beta + 4) q^{19} + \beta q^{21} - 4 q^{23} + q^{25} + q^{27} + (2 \beta - 2) q^{29} + ( - \beta + 4) q^{31} + \beta q^{33} - \beta q^{35} - 2 q^{37} + q^{39} - 2 q^{41} + ( - 2 \beta + 4) q^{43} - q^{45} + ( - \beta + 4) q^{47} + q^{49} + (2 \beta + 2) q^{51} - 2 q^{53} - \beta q^{55} + (\beta + 4) q^{57} - 3 \beta q^{59} + 6 q^{61} + \beta q^{63} - q^{65} + \beta q^{67} - 4 q^{69} + ( - \beta - 8) q^{71} + ( - 2 \beta + 10) q^{73} + q^{75} + 8 q^{77} + ( - 2 \beta - 8) q^{79} + q^{81} + (\beta + 4) q^{83} + ( - 2 \beta - 2) q^{85} + (2 \beta - 2) q^{87} + ( - 4 \beta + 6) q^{89} + \beta q^{91} + ( - \beta + 4) q^{93} + ( - \beta - 4) q^{95} + (2 \beta + 2) q^{97} + \beta q^{99} +O(q^{100})$$ q + q^3 - q^5 + b * q^7 + q^9 + b * q^11 + q^13 - q^15 + (2*b + 2) * q^17 + (b + 4) * q^19 + b * q^21 - 4 * q^23 + q^25 + q^27 + (2*b - 2) * q^29 + (-b + 4) * q^31 + b * q^33 - b * q^35 - 2 * q^37 + q^39 - 2 * q^41 + (-2*b + 4) * q^43 - q^45 + (-b + 4) * q^47 + q^49 + (2*b + 2) * q^51 - 2 * q^53 - b * q^55 + (b + 4) * q^57 - 3*b * q^59 + 6 * q^61 + b * q^63 - q^65 + b * q^67 - 4 * q^69 + (-b - 8) * q^71 + (-2*b + 10) * q^73 + q^75 + 8 * q^77 + (-2*b - 8) * q^79 + q^81 + (b + 4) * q^83 + (-2*b - 2) * q^85 + (2*b - 2) * q^87 + (-4*b + 6) * q^89 + b * q^91 + (-b + 4) * q^93 + (-b - 4) * q^95 + (2*b + 2) * q^97 + b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9 $$2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} + 2 q^{13} - 2 q^{15} + 4 q^{17} + 8 q^{19} - 8 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} + 8 q^{31} - 4 q^{37} + 2 q^{39} - 4 q^{41} + 8 q^{43} - 2 q^{45} + 8 q^{47} + 2 q^{49} + 4 q^{51} - 4 q^{53} + 8 q^{57} + 12 q^{61} - 2 q^{65} - 8 q^{69} - 16 q^{71} + 20 q^{73} + 2 q^{75} + 16 q^{77} - 16 q^{79} + 2 q^{81} + 8 q^{83} - 4 q^{85} - 4 q^{87} + 12 q^{89} + 8 q^{93} - 8 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9 + 2 * q^13 - 2 * q^15 + 4 * q^17 + 8 * q^19 - 8 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^29 + 8 * q^31 - 4 * q^37 + 2 * q^39 - 4 * q^41 + 8 * q^43 - 2 * q^45 + 8 * q^47 + 2 * q^49 + 4 * q^51 - 4 * q^53 + 8 * q^57 + 12 * q^61 - 2 * q^65 - 8 * q^69 - 16 * q^71 + 20 * q^73 + 2 * q^75 + 16 * q^77 - 16 * q^79 + 2 * q^81 + 8 * q^83 - 4 * q^85 - 4 * q^87 + 12 * q^89 + 8 * q^93 - 8 * q^95 + 4 * q^97

## 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
 −1.41421 1.41421
0 1.00000 0 −1.00000 0 −2.82843 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 2.82843 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.br yes 2
4.b odd 2 1 6240.2.a.bh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6240.2.a.bh 2 4.b odd 2 1
6240.2.a.br 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6240))$$:

 $$T_{7}^{2} - 8$$ T7^2 - 8 $$T_{11}^{2} - 8$$ T11^2 - 8 $$T_{17}^{2} - 4T_{17} - 28$$ T17^2 - 4*T17 - 28 $$T_{19}^{2} - 8T_{19} + 8$$ T19^2 - 8*T19 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 8$$
$11$ $$T^{2} - 8$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} - 4T - 28$$
$19$ $$T^{2} - 8T + 8$$
$23$ $$(T + 4)^{2}$$
$29$ $$T^{2} + 4T - 28$$
$31$ $$T^{2} - 8T + 8$$
$37$ $$(T + 2)^{2}$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} - 8T - 16$$
$47$ $$T^{2} - 8T + 8$$
$53$ $$(T + 2)^{2}$$
$59$ $$T^{2} - 72$$
$61$ $$(T - 6)^{2}$$
$67$ $$T^{2} - 8$$
$71$ $$T^{2} + 16T + 56$$
$73$ $$T^{2} - 20T + 68$$
$79$ $$T^{2} + 16T + 32$$
$83$ $$T^{2} - 8T + 8$$
$89$ $$T^{2} - 12T - 92$$
$97$ $$T^{2} - 4T - 28$$