# Properties

 Label 6930.2.a.bv Level $6930$ Weight $2$ Character orbit 6930.a Self dual yes Analytic conductor $55.336$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 770) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} - q^{10} - q^{11} + ( 2 + 2 \beta ) q^{13} - q^{14} + q^{16} -2 \beta q^{17} + ( 5 - \beta ) q^{19} + q^{20} + q^{22} + ( 3 - 3 \beta ) q^{23} + q^{25} + ( -2 - 2 \beta ) q^{26} + q^{28} + ( 3 - \beta ) q^{29} + 2 q^{31} - q^{32} + 2 \beta q^{34} + q^{35} + ( -1 - \beta ) q^{37} + ( -5 + \beta ) q^{38} - q^{40} + ( -3 - 3 \beta ) q^{41} + 2 q^{43} - q^{44} + ( -3 + 3 \beta ) q^{46} + 4 \beta q^{47} + q^{49} - q^{50} + ( 2 + 2 \beta ) q^{52} + ( 9 + \beta ) q^{53} - q^{55} - q^{56} + ( -3 + \beta ) q^{58} + 4 \beta q^{59} + ( 2 + 4 \beta ) q^{61} -2 q^{62} + q^{64} + ( 2 + 2 \beta ) q^{65} -4 q^{67} -2 \beta q^{68} - q^{70} + ( -6 + 2 \beta ) q^{71} + ( -4 + 6 \beta ) q^{73} + ( 1 + \beta ) q^{74} + ( 5 - \beta ) q^{76} - q^{77} + ( -7 + 3 \beta ) q^{79} + q^{80} + ( 3 + 3 \beta ) q^{82} + ( 6 - 6 \beta ) q^{83} -2 \beta q^{85} -2 q^{86} + q^{88} + 2 \beta q^{89} + ( 2 + 2 \beta ) q^{91} + ( 3 - 3 \beta ) q^{92} -4 \beta q^{94} + ( 5 - \beta ) q^{95} + ( -1 - 9 \beta ) q^{97} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} - 2 q^{8} + O(q^{10})$$ $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} - 2 q^{8} - 2 q^{10} - 2 q^{11} + 4 q^{13} - 2 q^{14} + 2 q^{16} + 10 q^{19} + 2 q^{20} + 2 q^{22} + 6 q^{23} + 2 q^{25} - 4 q^{26} + 2 q^{28} + 6 q^{29} + 4 q^{31} - 2 q^{32} + 2 q^{35} - 2 q^{37} - 10 q^{38} - 2 q^{40} - 6 q^{41} + 4 q^{43} - 2 q^{44} - 6 q^{46} + 2 q^{49} - 2 q^{50} + 4 q^{52} + 18 q^{53} - 2 q^{55} - 2 q^{56} - 6 q^{58} + 4 q^{61} - 4 q^{62} + 2 q^{64} + 4 q^{65} - 8 q^{67} - 2 q^{70} - 12 q^{71} - 8 q^{73} + 2 q^{74} + 10 q^{76} - 2 q^{77} - 14 q^{79} + 2 q^{80} + 6 q^{82} + 12 q^{83} - 4 q^{86} + 2 q^{88} + 4 q^{91} + 6 q^{92} + 10 q^{95} - 2 q^{97} - 2 q^{98} + O(q^{100})$$

## 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
 −1.73205 1.73205
−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
 $$n$$: 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.bv 2
3.b odd 2 1 770.2.a.j 2
12.b even 2 1 6160.2.a.t 2
15.d odd 2 1 3850.2.a.bd 2
15.e even 4 2 3850.2.c.x 4
21.c even 2 1 5390.2.a.bs 2
33.d even 2 1 8470.2.a.br 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.j 2 3.b odd 2 1
3850.2.a.bd 2 15.d odd 2 1
3850.2.c.x 4 15.e even 4 2
5390.2.a.bs 2 21.c even 2 1
6160.2.a.t 2 12.b even 2 1
6930.2.a.bv 2 1.a even 1 1 trivial
8470.2.a.br 2 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6930))$$:

 $$T_{13}^{2} - 4 T_{13} - 8$$ $$T_{17}^{2} - 12$$ $$T_{19}^{2} - 10 T_{19} + 22$$ $$T_{23}^{2} - 6 T_{23} - 18$$ $$T_{29}^{2} - 6 T_{29} + 6$$ $$T_{31} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$-8 - 4 T + T^{2}$$
$17$ $$-12 + T^{2}$$
$19$ $$22 - 10 T + T^{2}$$
$23$ $$-18 - 6 T + T^{2}$$
$29$ $$6 - 6 T + T^{2}$$
$31$ $$( -2 + T )^{2}$$
$37$ $$-2 + 2 T + T^{2}$$
$41$ $$-18 + 6 T + T^{2}$$
$43$ $$( -2 + T )^{2}$$
$47$ $$-48 + T^{2}$$
$53$ $$78 - 18 T + T^{2}$$
$59$ $$-48 + T^{2}$$
$61$ $$-44 - 4 T + T^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$24 + 12 T + T^{2}$$
$73$ $$-92 + 8 T + T^{2}$$
$79$ $$22 + 14 T + T^{2}$$
$83$ $$-72 - 12 T + T^{2}$$
$89$ $$-12 + T^{2}$$
$97$ $$-242 + 2 T + T^{2}$$