# Properties

 Label 6840.2.a.bb Level $6840$ Weight $2$ Character orbit 6840.a Self dual yes Analytic conductor $54.618$ Analytic rank $0$ 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: $$0$$ 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: $$1$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{5} + \beta q^{7}+O(q^{10})$$ q + q^5 + b * q^7 $$q + q^{5} + \beta q^{7} + ( - \beta + 2) q^{11} + ( - \beta + 2) q^{13} + 2 \beta q^{17} + q^{19} + (2 \beta + 2) q^{23} + q^{25} + 5 \beta q^{29} + ( - 2 \beta - 2) q^{31} + \beta q^{35} + (3 \beta + 2) q^{37} - 7 \beta q^{41} + (5 \beta + 4) q^{43} + ( - 2 \beta + 6) q^{47} - 5 q^{49} - 6 \beta q^{53} + ( - \beta + 2) q^{55} + ( - 2 \beta + 4) q^{59} + ( - 4 \beta + 4) q^{61} + ( - \beta + 2) q^{65} - 8 \beta q^{67} + (2 \beta + 12) q^{71} + ( - 4 \beta - 6) q^{73} + (2 \beta - 2) q^{77} + ( - 4 \beta - 8) q^{79} + (4 \beta + 2) q^{83} + 2 \beta q^{85} + (3 \beta + 8) q^{89} + (2 \beta - 2) q^{91} + q^{95} + (7 \beta + 2) q^{97} +O(q^{100})$$ q + q^5 + b * q^7 + (-b + 2) * q^11 + (-b + 2) * q^13 + 2*b * q^17 + q^19 + (2*b + 2) * q^23 + q^25 + 5*b * q^29 + (-2*b - 2) * q^31 + b * q^35 + (3*b + 2) * q^37 - 7*b * q^41 + (5*b + 4) * q^43 + (-2*b + 6) * q^47 - 5 * q^49 - 6*b * q^53 + (-b + 2) * q^55 + (-2*b + 4) * q^59 + (-4*b + 4) * q^61 + (-b + 2) * q^65 - 8*b * q^67 + (2*b + 12) * q^71 + (-4*b - 6) * q^73 + (2*b - 2) * q^77 + (-4*b - 8) * q^79 + (4*b + 2) * q^83 + 2*b * q^85 + (3*b + 8) * q^89 + (2*b - 2) * q^91 + q^95 + (7*b + 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5}+O(q^{10})$$ 2 * q + 2 * q^5 $$2 q + 2 q^{5} + 4 q^{11} + 4 q^{13} + 2 q^{19} + 4 q^{23} + 2 q^{25} - 4 q^{31} + 4 q^{37} + 8 q^{43} + 12 q^{47} - 10 q^{49} + 4 q^{55} + 8 q^{59} + 8 q^{61} + 4 q^{65} + 24 q^{71} - 12 q^{73} - 4 q^{77} - 16 q^{79} + 4 q^{83} + 16 q^{89} - 4 q^{91} + 2 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 + 4 * q^11 + 4 * q^13 + 2 * q^19 + 4 * q^23 + 2 * q^25 - 4 * q^31 + 4 * q^37 + 8 * q^43 + 12 * q^47 - 10 * q^49 + 4 * q^55 + 8 * q^59 + 8 * q^61 + 4 * q^65 + 24 * q^71 - 12 * q^73 - 4 * q^77 - 16 * q^79 + 4 * q^83 + 16 * q^89 - 4 * q^91 + 2 * 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 0 0 1.00000 0 −1.41421 0 0 0
1.2 0 0 0 1.00000 0 1.41421 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.bb 2
3.b odd 2 1 2280.2.a.k 2
12.b even 2 1 4560.2.a.bn 2

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

 $$T_{7}^{2} - 2$$ T7^2 - 2 $$T_{11}^{2} - 4T_{11} + 2$$ T11^2 - 4*T11 + 2 $$T_{13}^{2} - 4T_{13} + 2$$ T13^2 - 4*T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 2$$
$11$ $$T^{2} - 4T + 2$$
$13$ $$T^{2} - 4T + 2$$
$17$ $$T^{2} - 8$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} - 4T - 4$$
$29$ $$T^{2} - 50$$
$31$ $$T^{2} + 4T - 4$$
$37$ $$T^{2} - 4T - 14$$
$41$ $$T^{2} - 98$$
$43$ $$T^{2} - 8T - 34$$
$47$ $$T^{2} - 12T + 28$$
$53$ $$T^{2} - 72$$
$59$ $$T^{2} - 8T + 8$$
$61$ $$T^{2} - 8T - 16$$
$67$ $$T^{2} - 128$$
$71$ $$T^{2} - 24T + 136$$
$73$ $$T^{2} + 12T + 4$$
$79$ $$T^{2} + 16T + 32$$
$83$ $$T^{2} - 4T - 28$$
$89$ $$T^{2} - 16T + 46$$
$97$ $$T^{2} - 4T - 94$$