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

## 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6930))$$:

 $$T_{13} - 2$$ T13 - 2 $$T_{17}^{2} - 2T_{17} - 16$$ T17^2 - 2*T17 - 16 $$T_{19}^{2} + 6T_{19} - 8$$ T19^2 + 6*T19 - 8 $$T_{23}^{2} + 6T_{23} - 8$$ T23^2 + 6*T23 - 8 $$T_{29}^{2} - 68$$ T29^2 - 68 $$T_{31}^{2} + 6T_{31} - 8$$ T31^2 + 6*T31 - 8

## Hecke characteristic polynomials

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