# Properties

 Label 6840.2.a.v Level $6840$ Weight $2$ Character orbit 6840.a Self dual yes Analytic conductor $54.618$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6840.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$54.6176749826$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2280) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5} + (\beta - 3) q^{7}+O(q^{10})$$ q - q^5 + (b - 3) * q^7 $$q - q^{5} + (\beta - 3) q^{7} + (3 \beta + 1) q^{11} + ( - \beta - 3) q^{13} - 4 \beta q^{17} + q^{19} + 6 q^{23} + q^{25} + ( - 3 \beta + 5) q^{29} - 2 q^{31} + ( - \beta + 3) q^{35} + ( - \beta + 1) q^{37} + (5 \beta - 3) q^{41} + (\beta - 3) q^{43} + 10 q^{47} + ( - 6 \beta + 5) q^{49} + ( - 3 \beta - 1) q^{55} + ( - 2 \beta + 6) q^{59} + ( - 2 \beta + 2) q^{61} + (\beta + 3) q^{65} + (4 \beta - 8) q^{67} + (6 \beta - 2) q^{71} + ( - 4 \beta - 6) q^{73} + ( - 8 \beta + 6) q^{77} - 8 q^{79} + ( - 2 \beta - 12) q^{83} + 4 \beta q^{85} + ( - 5 \beta - 1) q^{89} + 6 q^{91} - q^{95} + ( - 5 \beta - 7) q^{97} +O(q^{100})$$ q - q^5 + (b - 3) * q^7 + (3*b + 1) * q^11 + (-b - 3) * q^13 - 4*b * q^17 + q^19 + 6 * q^23 + q^25 + (-3*b + 5) * q^29 - 2 * q^31 + (-b + 3) * q^35 + (-b + 1) * q^37 + (5*b - 3) * q^41 + (b - 3) * q^43 + 10 * q^47 + (-6*b + 5) * q^49 + (-3*b - 1) * q^55 + (-2*b + 6) * q^59 + (-2*b + 2) * q^61 + (b + 3) * q^65 + (4*b - 8) * q^67 + (6*b - 2) * q^71 + (-4*b - 6) * q^73 + (-8*b + 6) * q^77 - 8 * q^79 + (-2*b - 12) * q^83 + 4*b * q^85 + (-5*b - 1) * q^89 + 6 * q^91 - q^95 + (-5*b - 7) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 6 q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 - 6 * q^7 $$2 q - 2 q^{5} - 6 q^{7} + 2 q^{11} - 6 q^{13} + 2 q^{19} + 12 q^{23} + 2 q^{25} + 10 q^{29} - 4 q^{31} + 6 q^{35} + 2 q^{37} - 6 q^{41} - 6 q^{43} + 20 q^{47} + 10 q^{49} - 2 q^{55} + 12 q^{59} + 4 q^{61} + 6 q^{65} - 16 q^{67} - 4 q^{71} - 12 q^{73} + 12 q^{77} - 16 q^{79} - 24 q^{83} - 2 q^{89} + 12 q^{91} - 2 q^{95} - 14 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 - 6 * q^7 + 2 * q^11 - 6 * q^13 + 2 * q^19 + 12 * q^23 + 2 * q^25 + 10 * q^29 - 4 * q^31 + 6 * q^35 + 2 * q^37 - 6 * q^41 - 6 * q^43 + 20 * q^47 + 10 * q^49 - 2 * q^55 + 12 * q^59 + 4 * q^61 + 6 * q^65 - 16 * q^67 - 4 * q^71 - 12 * q^73 + 12 * q^77 - 16 * q^79 - 24 * q^83 - 2 * q^89 + 12 * q^91 - 2 * q^95 - 14 * 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
 −1.73205 1.73205
0 0 0 −1.00000 0 −4.73205 0 0 0
1.2 0 0 0 −1.00000 0 −1.26795 0 0 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$19$$ $$-1$$

## 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 6840.2.a.v 2
3.b odd 2 1 2280.2.a.q 2
12.b even 2 1 4560.2.a.bi 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.a.q 2 3.b odd 2 1
4560.2.a.bi 2 12.b even 2 1
6840.2.a.v 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6840))$$:

 $$T_{7}^{2} + 6T_{7} + 6$$ T7^2 + 6*T7 + 6 $$T_{11}^{2} - 2T_{11} - 26$$ T11^2 - 2*T11 - 26 $$T_{13}^{2} + 6T_{13} + 6$$ T13^2 + 6*T13 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + 6T + 6$$
$11$ $$T^{2} - 2T - 26$$
$13$ $$T^{2} + 6T + 6$$
$17$ $$T^{2} - 48$$
$19$ $$(T - 1)^{2}$$
$23$ $$(T - 6)^{2}$$
$29$ $$T^{2} - 10T - 2$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2} - 2T - 2$$
$41$ $$T^{2} + 6T - 66$$
$43$ $$T^{2} + 6T + 6$$
$47$ $$(T - 10)^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2} - 12T + 24$$
$61$ $$T^{2} - 4T - 8$$
$67$ $$T^{2} + 16T + 16$$
$71$ $$T^{2} + 4T - 104$$
$73$ $$T^{2} + 12T - 12$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 24T + 132$$
$89$ $$T^{2} + 2T - 74$$
$97$ $$T^{2} + 14T - 26$$