# Properties

 Label 6864.2.a.bb Level $6864$ Weight $2$ Character orbit 6864.a Self dual yes Analytic conductor $54.809$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6864.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$54.8093159474$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1716) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( -2 + \beta ) q^{5} + 2 \beta q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + ( -2 + \beta ) q^{5} + 2 \beta q^{7} + q^{9} + q^{11} - q^{13} + ( 2 - \beta ) q^{15} + ( -2 + 3 \beta ) q^{17} -2 \beta q^{21} + ( 2 - 2 \beta ) q^{23} + ( 1 - 4 \beta ) q^{25} - q^{27} + ( -2 - \beta ) q^{29} + ( 4 - \beta ) q^{31} - q^{33} + ( 4 - 4 \beta ) q^{35} + ( -2 - 2 \beta ) q^{37} + q^{39} + ( -2 - 4 \beta ) q^{41} + ( -4 - 3 \beta ) q^{43} + ( -2 + \beta ) q^{45} + 8 q^{47} + q^{49} + ( 2 - 3 \beta ) q^{51} + ( -2 - 8 \beta ) q^{53} + ( -2 + \beta ) q^{55} + 2 \beta q^{59} + ( -6 - 2 \beta ) q^{61} + 2 \beta q^{63} + ( 2 - \beta ) q^{65} + 3 \beta q^{67} + ( -2 + 2 \beta ) q^{69} + ( 8 - 2 \beta ) q^{71} + 4 \beta q^{73} + ( -1 + 4 \beta ) q^{75} + 2 \beta q^{77} -5 \beta q^{79} + q^{81} + ( 12 - 4 \beta ) q^{83} + ( 10 - 8 \beta ) q^{85} + ( 2 + \beta ) q^{87} + ( -2 + 7 \beta ) q^{89} -2 \beta q^{91} + ( -4 + \beta ) q^{93} + ( -10 + 2 \beta ) q^{97} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 4q^{5} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 4q^{5} + 2q^{9} + 2q^{11} - 2q^{13} + 4q^{15} - 4q^{17} + 4q^{23} + 2q^{25} - 2q^{27} - 4q^{29} + 8q^{31} - 2q^{33} + 8q^{35} - 4q^{37} + 2q^{39} - 4q^{41} - 8q^{43} - 4q^{45} + 16q^{47} + 2q^{49} + 4q^{51} - 4q^{53} - 4q^{55} - 12q^{61} + 4q^{65} - 4q^{69} + 16q^{71} - 2q^{75} + 2q^{81} + 24q^{83} + 20q^{85} + 4q^{87} - 4q^{89} - 8q^{93} - 20q^{97} + 2q^{99} + O(q^{100})$$

## 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 −3.41421 0 −2.82843 0 1.00000 0
1.2 0 −1.00000 0 −0.585786 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$11$$ $$-1$$
$$13$$ $$1$$

## 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 6864.2.a.bb 2
4.b odd 2 1 1716.2.a.e 2
12.b even 2 1 5148.2.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1716.2.a.e 2 4.b odd 2 1
5148.2.a.j 2 12.b even 2 1
6864.2.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6864))$$:

 $$T_{5}^{2} + 4 T_{5} + 2$$ $$T_{7}^{2} - 8$$ $$T_{17}^{2} + 4 T_{17} - 14$$ $$T_{19}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$2 + 4 T + T^{2}$$
$7$ $$-8 + T^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$( 1 + T )^{2}$$
$17$ $$-14 + 4 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$-4 - 4 T + T^{2}$$
$29$ $$2 + 4 T + T^{2}$$
$31$ $$14 - 8 T + T^{2}$$
$37$ $$-4 + 4 T + T^{2}$$
$41$ $$-28 + 4 T + T^{2}$$
$43$ $$-2 + 8 T + T^{2}$$
$47$ $$( -8 + T )^{2}$$
$53$ $$-124 + 4 T + T^{2}$$
$59$ $$-8 + T^{2}$$
$61$ $$28 + 12 T + T^{2}$$
$67$ $$-18 + T^{2}$$
$71$ $$56 - 16 T + T^{2}$$
$73$ $$-32 + T^{2}$$
$79$ $$-50 + T^{2}$$
$83$ $$112 - 24 T + T^{2}$$
$89$ $$-94 + 4 T + T^{2}$$
$97$ $$92 + 20 T + T^{2}$$