# Properties

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

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## 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. 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 $$q - q^{3} - q^{5} + \beta q^{7} + q^{9} + \beta q^{11} + q^{13} + q^{15} + (\beta - 2) q^{17} + ( - 2 \beta + 4) q^{19} - \beta q^{21} + ( - \beta - 4) q^{23} + q^{25} - q^{27} + (2 \beta - 6) q^{29} + ( - 4 \beta + 4) q^{31} - \beta q^{33} - \beta q^{35} + ( - 3 \beta + 2) q^{37} - q^{39} + ( - 3 \beta - 2) q^{41} + (2 \beta + 4) q^{43} - q^{45} - 8 q^{47} + (\beta - 3) q^{49} + ( - \beta + 2) q^{51} + (\beta - 2) q^{53} - \beta q^{55} + (2 \beta - 4) q^{57} + 4 q^{59} + (3 \beta + 2) q^{61} + \beta q^{63} - q^{65} + ( - 4 \beta + 4) q^{67} + (\beta + 4) q^{69} + ( - 3 \beta + 4) q^{71} + ( - 4 \beta - 6) q^{73} - q^{75} + (\beta + 4) q^{77} + ( - \beta + 8) q^{79} + q^{81} + 4 q^{83} + ( - \beta + 2) q^{85} + ( - 2 \beta + 6) q^{87} + ( - \beta - 2) q^{89} + \beta q^{91} + (4 \beta - 4) q^{93} + (2 \beta - 4) q^{95} + (3 \beta - 2) q^{97} + \beta q^{99} +O(q^{100})$$ q - q^3 - q^5 + b * q^7 + q^9 + b * q^11 + q^13 + q^15 + (b - 2) * q^17 + (-2*b + 4) * q^19 - b * q^21 + (-b - 4) * q^23 + q^25 - q^27 + (2*b - 6) * q^29 + (-4*b + 4) * q^31 - b * q^33 - b * q^35 + (-3*b + 2) * q^37 - q^39 + (-3*b - 2) * q^41 + (2*b + 4) * q^43 - q^45 - 8 * q^47 + (b - 3) * q^49 + (-b + 2) * q^51 + (b - 2) * q^53 - b * q^55 + (2*b - 4) * q^57 + 4 * q^59 + (3*b + 2) * q^61 + b * q^63 - q^65 + (-4*b + 4) * q^67 + (b + 4) * q^69 + (-3*b + 4) * q^71 + (-4*b - 6) * q^73 - q^75 + (b + 4) * q^77 + (-b + 8) * q^79 + q^81 + 4 * q^83 + (-b + 2) * q^85 + (-2*b + 6) * q^87 + (-b - 2) * q^89 + b * q^91 + (4*b - 4) * q^93 + (2*b - 4) * q^95 + (3*b - 2) * q^97 + b * 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 $$2 q - 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9} + q^{11} + 2 q^{13} + 2 q^{15} - 3 q^{17} + 6 q^{19} - q^{21} - 9 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} + 4 q^{31} - q^{33} - q^{35} + q^{37} - 2 q^{39} - 7 q^{41} + 10 q^{43} - 2 q^{45} - 16 q^{47} - 5 q^{49} + 3 q^{51} - 3 q^{53} - q^{55} - 6 q^{57} + 8 q^{59} + 7 q^{61} + q^{63} - 2 q^{65} + 4 q^{67} + 9 q^{69} + 5 q^{71} - 16 q^{73} - 2 q^{75} + 9 q^{77} + 15 q^{79} + 2 q^{81} + 8 q^{83} + 3 q^{85} + 10 q^{87} - 5 q^{89} + q^{91} - 4 q^{93} - 6 q^{95} - q^{97} + q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^5 + q^7 + 2 * q^9 + q^11 + 2 * q^13 + 2 * q^15 - 3 * q^17 + 6 * q^19 - q^21 - 9 * q^23 + 2 * q^25 - 2 * q^27 - 10 * q^29 + 4 * q^31 - q^33 - q^35 + q^37 - 2 * q^39 - 7 * q^41 + 10 * q^43 - 2 * q^45 - 16 * q^47 - 5 * q^49 + 3 * q^51 - 3 * q^53 - q^55 - 6 * q^57 + 8 * q^59 + 7 * q^61 + q^63 - 2 * q^65 + 4 * q^67 + 9 * q^69 + 5 * q^71 - 16 * q^73 - 2 * q^75 + 9 * q^77 + 15 * q^79 + 2 * q^81 + 8 * q^83 + 3 * q^85 + 10 * q^87 - 5 * q^89 + q^91 - 4 * q^93 - 6 * q^95 - q^97 + 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
 −1.56155 2.56155
0 −1.00000 0 −1.00000 0 −1.56155 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 2.56155 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.bi 2
4.b odd 2 1 6240.2.a.bq yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6240.2.a.bi 2 1.a even 1 1 trivial
6240.2.a.bq yes 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6240))$$:

 $$T_{7}^{2} - T_{7} - 4$$ T7^2 - T7 - 4 $$T_{11}^{2} - T_{11} - 4$$ T11^2 - T11 - 4 $$T_{17}^{2} + 3T_{17} - 2$$ T17^2 + 3*T17 - 2 $$T_{19}^{2} - 6T_{19} - 8$$ T19^2 - 6*T19 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - T - 4$$
$11$ $$T^{2} - T - 4$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} + 3T - 2$$
$19$ $$T^{2} - 6T - 8$$
$23$ $$T^{2} + 9T + 16$$
$29$ $$T^{2} + 10T + 8$$
$31$ $$T^{2} - 4T - 64$$
$37$ $$T^{2} - T - 38$$
$41$ $$T^{2} + 7T - 26$$
$43$ $$T^{2} - 10T + 8$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} + 3T - 2$$
$59$ $$(T - 4)^{2}$$
$61$ $$T^{2} - 7T - 26$$
$67$ $$T^{2} - 4T - 64$$
$71$ $$T^{2} - 5T - 32$$
$73$ $$T^{2} + 16T - 4$$
$79$ $$T^{2} - 15T + 52$$
$83$ $$(T - 4)^{2}$$
$89$ $$T^{2} + 5T + 2$$
$97$ $$T^{2} + T - 38$$
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