# Properties

 Label 7942.2.a.y Level $7942$ Weight $2$ Character orbit 7942.a Self dual yes Analytic conductor $63.417$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7942 = 2 \cdot 11 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7942.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$63.4171892853$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 418) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} - \beta q^{7} + q^{8} + 3 q^{9} +O(q^{10})$$ q + q^2 + b * q^3 + q^4 + b * q^6 - b * q^7 + q^8 + 3 * q^9 $$q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} - \beta q^{7} + q^{8} + 3 q^{9} - q^{11} + \beta q^{12} + ( - 2 \beta + 1) q^{13} - \beta q^{14} + q^{16} + ( - \beta - 2) q^{17} + 3 q^{18} - 6 q^{21} - q^{22} + (2 \beta - 4) q^{23} + \beta q^{24} - 5 q^{25} + ( - 2 \beta + 1) q^{26} - \beta q^{28} + (2 \beta - 3) q^{29} - 2 q^{31} + q^{32} - \beta q^{33} + ( - \beta - 2) q^{34} + 3 q^{36} + ( - \beta - 6) q^{37} + (\beta - 12) q^{39} + ( - \beta + 10) q^{41} - 6 q^{42} + (\beta + 7) q^{43} - q^{44} + (2 \beta - 4) q^{46} + (\beta - 9) q^{47} + \beta q^{48} - q^{49} - 5 q^{50} + ( - 2 \beta - 6) q^{51} + ( - 2 \beta + 1) q^{52} + (4 \beta - 2) q^{53} - \beta q^{56} + (2 \beta - 3) q^{58} + ( - 2 \beta + 2) q^{59} + (2 \beta + 3) q^{61} - 2 q^{62} - 3 \beta q^{63} + q^{64} - \beta q^{66} + (\beta - 12) q^{67} + ( - \beta - 2) q^{68} + ( - 4 \beta + 12) q^{69} + ( - \beta - 5) q^{71} + 3 q^{72} + ( - \beta - 12) q^{73} + ( - \beta - 6) q^{74} - 5 \beta q^{75} + \beta q^{77} + (\beta - 12) q^{78} + ( - \beta - 2) q^{79} - 9 q^{81} + ( - \beta + 10) q^{82} + ( - \beta - 3) q^{83} - 6 q^{84} + (\beta + 7) q^{86} + ( - 3 \beta + 12) q^{87} - q^{88} + 9 q^{89} + ( - \beta + 12) q^{91} + (2 \beta - 4) q^{92} - 2 \beta q^{93} + (\beta - 9) q^{94} + \beta q^{96} + ( - 2 \beta + 1) q^{97} - q^{98} - 3 q^{99} +O(q^{100})$$ q + q^2 + b * q^3 + q^4 + b * q^6 - b * q^7 + q^8 + 3 * q^9 - q^11 + b * q^12 + (-2*b + 1) * q^13 - b * q^14 + q^16 + (-b - 2) * q^17 + 3 * q^18 - 6 * q^21 - q^22 + (2*b - 4) * q^23 + b * q^24 - 5 * q^25 + (-2*b + 1) * q^26 - b * q^28 + (2*b - 3) * q^29 - 2 * q^31 + q^32 - b * q^33 + (-b - 2) * q^34 + 3 * q^36 + (-b - 6) * q^37 + (b - 12) * q^39 + (-b + 10) * q^41 - 6 * q^42 + (b + 7) * q^43 - q^44 + (2*b - 4) * q^46 + (b - 9) * q^47 + b * q^48 - q^49 - 5 * q^50 + (-2*b - 6) * q^51 + (-2*b + 1) * q^52 + (4*b - 2) * q^53 - b * q^56 + (2*b - 3) * q^58 + (-2*b + 2) * q^59 + (2*b + 3) * q^61 - 2 * q^62 - 3*b * q^63 + q^64 - b * q^66 + (b - 12) * q^67 + (-b - 2) * q^68 + (-4*b + 12) * q^69 + (-b - 5) * q^71 + 3 * q^72 + (-b - 12) * q^73 + (-b - 6) * q^74 - 5*b * q^75 + b * q^77 + (b - 12) * q^78 + (-b - 2) * q^79 - 9 * q^81 + (-b + 10) * q^82 + (-b - 3) * q^83 - 6 * q^84 + (b + 7) * q^86 + (-3*b + 12) * q^87 - q^88 + 9 * q^89 + (-b + 12) * q^91 + (2*b - 4) * q^92 - 2*b * q^93 + (b - 9) * q^94 + b * q^96 + (-2*b + 1) * q^97 - q^98 - 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 6 * q^9 $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 6 q^{9} - 2 q^{11} + 2 q^{13} + 2 q^{16} - 4 q^{17} + 6 q^{18} - 12 q^{21} - 2 q^{22} - 8 q^{23} - 10 q^{25} + 2 q^{26} - 6 q^{29} - 4 q^{31} + 2 q^{32} - 4 q^{34} + 6 q^{36} - 12 q^{37} - 24 q^{39} + 20 q^{41} - 12 q^{42} + 14 q^{43} - 2 q^{44} - 8 q^{46} - 18 q^{47} - 2 q^{49} - 10 q^{50} - 12 q^{51} + 2 q^{52} - 4 q^{53} - 6 q^{58} + 4 q^{59} + 6 q^{61} - 4 q^{62} + 2 q^{64} - 24 q^{67} - 4 q^{68} + 24 q^{69} - 10 q^{71} + 6 q^{72} - 24 q^{73} - 12 q^{74} - 24 q^{78} - 4 q^{79} - 18 q^{81} + 20 q^{82} - 6 q^{83} - 12 q^{84} + 14 q^{86} + 24 q^{87} - 2 q^{88} + 18 q^{89} + 24 q^{91} - 8 q^{92} - 18 q^{94} + 2 q^{97} - 2 q^{98} - 6 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 6 * q^9 - 2 * q^11 + 2 * q^13 + 2 * q^16 - 4 * q^17 + 6 * q^18 - 12 * q^21 - 2 * q^22 - 8 * q^23 - 10 * q^25 + 2 * q^26 - 6 * q^29 - 4 * q^31 + 2 * q^32 - 4 * q^34 + 6 * q^36 - 12 * q^37 - 24 * q^39 + 20 * q^41 - 12 * q^42 + 14 * q^43 - 2 * q^44 - 8 * q^46 - 18 * q^47 - 2 * q^49 - 10 * q^50 - 12 * q^51 + 2 * q^52 - 4 * q^53 - 6 * q^58 + 4 * q^59 + 6 * q^61 - 4 * q^62 + 2 * q^64 - 24 * q^67 - 4 * q^68 + 24 * q^69 - 10 * q^71 + 6 * q^72 - 24 * q^73 - 12 * q^74 - 24 * q^78 - 4 * q^79 - 18 * q^81 + 20 * q^82 - 6 * q^83 - 12 * q^84 + 14 * q^86 + 24 * q^87 - 2 * q^88 + 18 * q^89 + 24 * q^91 - 8 * q^92 - 18 * q^94 + 2 * q^97 - 2 * q^98 - 6 * q^99

## 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
 −2.44949 2.44949
1.00000 −2.44949 1.00000 0 −2.44949 2.44949 1.00000 3.00000 0
1.2 1.00000 2.44949 1.00000 0 2.44949 −2.44949 1.00000 3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$11$$ $$1$$
$$19$$ $$-1$$

## 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 7942.2.a.y 2
19.b odd 2 1 7942.2.a.v 2
19.d odd 6 2 418.2.e.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.e.g 4 19.d odd 6 2
7942.2.a.v 2 19.b odd 2 1
7942.2.a.y 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7942))$$:

 $$T_{3}^{2} - 6$$ T3^2 - 6 $$T_{5}$$ T5 $$T_{13}^{2} - 2T_{13} - 23$$ T13^2 - 2*T13 - 23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} - 6$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 6$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} - 2T - 23$$
$17$ $$T^{2} + 4T - 2$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 8T - 8$$
$29$ $$T^{2} + 6T - 15$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2} + 12T + 30$$
$41$ $$T^{2} - 20T + 94$$
$43$ $$T^{2} - 14T + 43$$
$47$ $$T^{2} + 18T + 75$$
$53$ $$T^{2} + 4T - 92$$
$59$ $$T^{2} - 4T - 20$$
$61$ $$T^{2} - 6T - 15$$
$67$ $$T^{2} + 24T + 138$$
$71$ $$T^{2} + 10T + 19$$
$73$ $$T^{2} + 24T + 138$$
$79$ $$T^{2} + 4T - 2$$
$83$ $$T^{2} + 6T + 3$$
$89$ $$(T - 9)^{2}$$
$97$ $$T^{2} - 2T - 23$$