Properties

 Label 6930.2.a.bs Level $6930$ Weight $2$ Character orbit 6930.a Self dual yes Analytic conductor $55.336$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{17}$$. We also show the integral $$q$$-expansion of the trace form.

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

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
 2.56155 −1.56155
−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.bs 2
3.b odd 2 1 6930.2.a.bx yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6930.2.a.bs 2 1.a even 1 1 trivial
6930.2.a.bx yes 2 3.b odd 2 1

Hecke kernels

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

 $$T_{13} - 2$$ $$T_{17}^{2} - 2 T_{17} - 16$$ $$T_{19}^{2} + 6 T_{19} - 8$$ $$T_{23}^{2} + 6 T_{23} - 8$$ $$T_{29}^{2} - 68$$ $$T_{31}^{2} + 6 T_{31} - 8$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$-16 - 2 T + T^{2}$$
$19$ $$-8 + 6 T + T^{2}$$
$23$ $$-8 + 6 T + T^{2}$$
$29$ $$-68 + T^{2}$$
$31$ $$-8 + 6 T + T^{2}$$
$37$ $$-8 - 6 T + T^{2}$$
$41$ $$8 + 10 T + T^{2}$$
$43$ $$-16 + 2 T + T^{2}$$
$47$ $$-152 + 2 T + T^{2}$$
$53$ $$( 10 + T )^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$-16 + 2 T + T^{2}$$
$67$ $$8 + 10 T + T^{2}$$
$71$ $$-8 + 6 T + T^{2}$$
$73$ $$-4 - 16 T + T^{2}$$
$79$ $$-16 - 2 T + T^{2}$$
$83$ $$32 + 14 T + T^{2}$$
$89$ $$-68 + T^{2}$$
$97$ $$32 + 14 T + T^{2}$$