# Properties

 Label 6930.2.a.cj Level $6930$ Weight $2$ Character orbit 6930.a Self dual yes Analytic conductor $55.336$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6930.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$55.3363286007$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ Defining polynomial: $$x^{3} - 3 x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} - q^{10} - q^{11} + \beta_{1} q^{13} + q^{14} + q^{16} + ( -2 + \beta_{1} ) q^{17} + ( -2 - \beta_{1} - \beta_{2} ) q^{19} - q^{20} - q^{22} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{23} + q^{25} + \beta_{1} q^{26} + q^{28} + ( -2 - 2 \beta_{1} ) q^{29} + ( \beta_{1} + \beta_{2} ) q^{31} + q^{32} + ( -2 + \beta_{1} ) q^{34} - q^{35} + ( -2 - \beta_{1} ) q^{37} + ( -2 - \beta_{1} - \beta_{2} ) q^{38} - q^{40} + ( -2 + \beta_{1} + \beta_{2} ) q^{41} + ( -2 - \beta_{1} - \beta_{2} ) q^{43} - q^{44} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{46} -6 q^{47} + q^{49} + q^{50} + \beta_{1} q^{52} + ( -2 + \beta_{1} - \beta_{2} ) q^{53} + q^{55} + q^{56} + ( -2 - 2 \beta_{1} ) q^{58} + ( -4 - \beta_{1} - \beta_{2} ) q^{59} -4 q^{61} + ( \beta_{1} + \beta_{2} ) q^{62} + q^{64} -\beta_{1} q^{65} + ( 4 - \beta_{1} + 2 \beta_{2} ) q^{67} + ( -2 + \beta_{1} ) q^{68} - q^{70} + ( -2 + 4 \beta_{1} ) q^{71} + ( -2 - \beta_{1} - \beta_{2} ) q^{73} + ( -2 - \beta_{1} ) q^{74} + ( -2 - \beta_{1} - \beta_{2} ) q^{76} - q^{77} + 4 \beta_{1} q^{79} - q^{80} + ( -2 + \beta_{1} + \beta_{2} ) q^{82} + ( 3 \beta_{1} - \beta_{2} ) q^{83} + ( 2 - \beta_{1} ) q^{85} + ( -2 - \beta_{1} - \beta_{2} ) q^{86} - q^{88} + ( -4 - 4 \beta_{1} + \beta_{2} ) q^{89} + \beta_{1} q^{91} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{92} -6 q^{94} + ( 2 + \beta_{1} + \beta_{2} ) q^{95} + ( 2 + 3 \beta_{1} + \beta_{2} ) q^{97} + q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8} + O(q^{10})$$ $$3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8} - 3 q^{10} - 3 q^{11} + 3 q^{14} + 3 q^{16} - 6 q^{17} - 6 q^{19} - 3 q^{20} - 3 q^{22} - 6 q^{23} + 3 q^{25} + 3 q^{28} - 6 q^{29} + 3 q^{32} - 6 q^{34} - 3 q^{35} - 6 q^{37} - 6 q^{38} - 3 q^{40} - 6 q^{41} - 6 q^{43} - 3 q^{44} - 6 q^{46} - 18 q^{47} + 3 q^{49} + 3 q^{50} - 6 q^{53} + 3 q^{55} + 3 q^{56} - 6 q^{58} - 12 q^{59} - 12 q^{61} + 3 q^{64} + 12 q^{67} - 6 q^{68} - 3 q^{70} - 6 q^{71} - 6 q^{73} - 6 q^{74} - 6 q^{76} - 3 q^{77} - 3 q^{80} - 6 q^{82} + 6 q^{85} - 6 q^{86} - 3 q^{88} - 12 q^{89} - 6 q^{92} - 18 q^{94} + 6 q^{95} + 6 q^{97} + 3 q^{98} + O(q^{100})$$

## 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
 −1.53209 −0.347296 1.87939
1.00000 0 1.00000 −1.00000 0 1.00000 1.00000 0 −1.00000
1.2 1.00000 0 1.00000 −1.00000 0 1.00000 1.00000 0 −1.00000
1.3 1.00000 0 1.00000 −1.00000 0 1.00000 1.00000 0 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$7$$ $$-1$$
$$11$$ $$1$$

## 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 6930.2.a.cj yes 3
3.b odd 2 1 6930.2.a.ci 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6930.2.a.ci 3 3.b odd 2 1
6930.2.a.cj yes 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6930))$$:

 $$T_{13}^{3} - 12 T_{13} - 8$$ $$T_{17}^{3} + 6 T_{17}^{2} - 24$$ $$T_{19}^{3} + 6 T_{19}^{2} - 36 T_{19} - 152$$ $$T_{23}^{3} + 6 T_{23}^{2} - 72 T_{23} - 456$$ $$T_{29}^{3} + 6 T_{29}^{2} - 36 T_{29} - 24$$ $$T_{31}^{3} - 48 T_{31} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$T^{3}$$
$5$ $$( 1 + T )^{3}$$
$7$ $$( -1 + T )^{3}$$
$11$ $$( 1 + T )^{3}$$
$13$ $$-8 - 12 T + T^{3}$$
$17$ $$-24 + 6 T^{2} + T^{3}$$
$19$ $$-152 - 36 T + 6 T^{2} + T^{3}$$
$23$ $$-456 - 72 T + 6 T^{2} + T^{3}$$
$29$ $$-24 - 36 T + 6 T^{2} + T^{3}$$
$31$ $$64 - 48 T + T^{3}$$
$37$ $$-8 + 6 T^{2} + T^{3}$$
$41$ $$-24 - 36 T + 6 T^{2} + T^{3}$$
$43$ $$-152 - 36 T + 6 T^{2} + T^{3}$$
$47$ $$( 6 + T )^{3}$$
$53$ $$-24 - 36 T + 6 T^{2} + T^{3}$$
$59$ $$-192 + 12 T^{2} + T^{3}$$
$61$ $$( 4 + T )^{3}$$
$67$ $$712 - 108 T - 12 T^{2} + T^{3}$$
$71$ $$-888 - 180 T + 6 T^{2} + T^{3}$$
$73$ $$-152 - 36 T + 6 T^{2} + T^{3}$$
$79$ $$-512 - 192 T + T^{3}$$
$83$ $$576 - 144 T + T^{3}$$
$89$ $$-1704 - 180 T + 12 T^{2} + T^{3}$$
$97$ $$-296 - 132 T - 6 T^{2} + T^{3}$$