Properties

 Label 6930.2.a.bt 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 2310) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

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.bt 2
3.b odd 2 1 2310.2.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2310.2.a.bb 2 3.b odd 2 1
6930.2.a.bt 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6930))$$:

 $$T_{13}^{2} - 12$$ $$T_{17}^{2} - 12$$ $$T_{19}$$ $$T_{23}^{2} + 4 T_{23} - 8$$ $$T_{29} + 2$$ $$T_{31}^{2} - 48$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$-12 + T^{2}$$
$17$ $$-12 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$-8 + 4 T + T^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$-48 + T^{2}$$
$37$ $$4 - 8 T + T^{2}$$
$41$ $$-44 - 4 T + T^{2}$$
$43$ $$-32 - 8 T + T^{2}$$
$47$ $$-48 + T^{2}$$
$53$ $$-12 + 12 T + T^{2}$$
$59$ $$-32 + 8 T + T^{2}$$
$61$ $$-44 - 4 T + T^{2}$$
$67$ $$24 - 12 T + T^{2}$$
$71$ $$( -4 + T )^{2}$$
$73$ $$-12 - 12 T + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$-48 + T^{2}$$
$89$ $$-12 + T^{2}$$
$97$ $$-188 - 4 T + T^{2}$$