# Properties

 Label 7942.2.a.bd Level $7942$ Weight $2$ Character orbit 7942.a Self dual yes Analytic conductor $63.417$ Analytic rank $0$ Dimension $3$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1940.1 Defining polynomial: $$x^{3} - 8x - 2$$ x^3 - 8*x - 2 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - \beta_1 q^{3} + q^{4} + 2 q^{5} + \beta_1 q^{6} + (\beta_1 + 2) q^{7} - q^{8} + (\beta_{2} + 2) q^{9}+O(q^{10})$$ q - q^2 - b1 * q^3 + q^4 + 2 * q^5 + b1 * q^6 + (b1 + 2) * q^7 - q^8 + (b2 + 2) * q^9 $$q - q^{2} - \beta_1 q^{3} + q^{4} + 2 q^{5} + \beta_1 q^{6} + (\beta_1 + 2) q^{7} - q^{8} + (\beta_{2} + 2) q^{9} - 2 q^{10} + q^{11} - \beta_1 q^{12} + (\beta_{2} + 2) q^{13} + ( - \beta_1 - 2) q^{14} - 2 \beta_1 q^{15} + q^{16} + (\beta_{2} - \beta_1 + 3) q^{17} + ( - \beta_{2} - 2) q^{18} + 2 q^{20} + ( - \beta_{2} - 2 \beta_1 - 5) q^{21} - q^{22} + ( - 2 \beta_1 + 4) q^{23} + \beta_1 q^{24} - q^{25} + ( - \beta_{2} - 2) q^{26} + ( - 2 \beta_1 - 2) q^{27} + (\beta_1 + 2) q^{28} - q^{29} + 2 \beta_1 q^{30} + ( - \beta_{2} + 2 \beta_1 + 1) q^{31} - q^{32} - \beta_1 q^{33} + ( - \beta_{2} + \beta_1 - 3) q^{34} + (2 \beta_1 + 4) q^{35} + (\beta_{2} + 2) q^{36} + (\beta_{2} - 3 \beta_1 + 1) q^{37} + ( - 5 \beta_1 - 2) q^{39} - 2 q^{40} + (\beta_{2} + \beta_1 + 3) q^{41} + (\beta_{2} + 2 \beta_1 + 5) q^{42} + ( - \beta_{2} - \beta_1 + 2) q^{43} + q^{44} + (2 \beta_{2} + 4) q^{45} + (2 \beta_1 - 4) q^{46} + ( - 2 \beta_{2} + \beta_1 - 3) q^{47} - \beta_1 q^{48} + (\beta_{2} + 4 \beta_1 + 2) q^{49} + q^{50} + (\beta_{2} - 6 \beta_1 + 3) q^{51} + (\beta_{2} + 2) q^{52} + 4 \beta_1 q^{53} + (2 \beta_1 + 2) q^{54} + 2 q^{55} + ( - \beta_1 - 2) q^{56} + q^{58} - 2 \beta_1 q^{60} + ( - \beta_{2} - 2 \beta_1 + 4) q^{61} + (\beta_{2} - 2 \beta_1 - 1) q^{62} + (2 \beta_{2} + 5 \beta_1 + 6) q^{63} + q^{64} + (2 \beta_{2} + 4) q^{65} + \beta_1 q^{66} + 3 \beta_1 q^{67} + (\beta_{2} - \beta_1 + 3) q^{68} + (2 \beta_{2} - 4 \beta_1 + 10) q^{69} + ( - 2 \beta_1 - 4) q^{70} + (\beta_{2} + \beta_1 - 2) q^{71} + ( - \beta_{2} - 2) q^{72} + ( - \beta_{2} - \beta_1 + 3) q^{73} + ( - \beta_{2} + 3 \beta_1 - 1) q^{74} + \beta_1 q^{75} + (\beta_1 + 2) q^{77} + (5 \beta_1 + 2) q^{78} + ( - 3 \beta_1 + 2) q^{79} + 2 q^{80} + ( - \beta_{2} + 2 \beta_1 + 4) q^{81} + ( - \beta_{2} - \beta_1 - 3) q^{82} + (\beta_{2} + 3 \beta_1 + 2) q^{83} + ( - \beta_{2} - 2 \beta_1 - 5) q^{84} + (2 \beta_{2} - 2 \beta_1 + 6) q^{85} + (\beta_{2} + \beta_1 - 2) q^{86} + \beta_1 q^{87} - q^{88} + ( - 2 \beta_{2} + 2 \beta_1 + 1) q^{89} + ( - 2 \beta_{2} - 4) q^{90} + (2 \beta_{2} + 5 \beta_1 + 6) q^{91} + ( - 2 \beta_1 + 4) q^{92} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{93} + (2 \beta_{2} - \beta_1 + 3) q^{94} + \beta_1 q^{96} + (2 \beta_{2} - 2 \beta_1 - 3) q^{97} + ( - \beta_{2} - 4 \beta_1 - 2) q^{98} + (\beta_{2} + 2) q^{99}+O(q^{100})$$ q - q^2 - b1 * q^3 + q^4 + 2 * q^5 + b1 * q^6 + (b1 + 2) * q^7 - q^8 + (b2 + 2) * q^9 - 2 * q^10 + q^11 - b1 * q^12 + (b2 + 2) * q^13 + (-b1 - 2) * q^14 - 2*b1 * q^15 + q^16 + (b2 - b1 + 3) * q^17 + (-b2 - 2) * q^18 + 2 * q^20 + (-b2 - 2*b1 - 5) * q^21 - q^22 + (-2*b1 + 4) * q^23 + b1 * q^24 - q^25 + (-b2 - 2) * q^26 + (-2*b1 - 2) * q^27 + (b1 + 2) * q^28 - q^29 + 2*b1 * q^30 + (-b2 + 2*b1 + 1) * q^31 - q^32 - b1 * q^33 + (-b2 + b1 - 3) * q^34 + (2*b1 + 4) * q^35 + (b2 + 2) * q^36 + (b2 - 3*b1 + 1) * q^37 + (-5*b1 - 2) * q^39 - 2 * q^40 + (b2 + b1 + 3) * q^41 + (b2 + 2*b1 + 5) * q^42 + (-b2 - b1 + 2) * q^43 + q^44 + (2*b2 + 4) * q^45 + (2*b1 - 4) * q^46 + (-2*b2 + b1 - 3) * q^47 - b1 * q^48 + (b2 + 4*b1 + 2) * q^49 + q^50 + (b2 - 6*b1 + 3) * q^51 + (b2 + 2) * q^52 + 4*b1 * q^53 + (2*b1 + 2) * q^54 + 2 * q^55 + (-b1 - 2) * q^56 + q^58 - 2*b1 * q^60 + (-b2 - 2*b1 + 4) * q^61 + (b2 - 2*b1 - 1) * q^62 + (2*b2 + 5*b1 + 6) * q^63 + q^64 + (2*b2 + 4) * q^65 + b1 * q^66 + 3*b1 * q^67 + (b2 - b1 + 3) * q^68 + (2*b2 - 4*b1 + 10) * q^69 + (-2*b1 - 4) * q^70 + (b2 + b1 - 2) * q^71 + (-b2 - 2) * q^72 + (-b2 - b1 + 3) * q^73 + (-b2 + 3*b1 - 1) * q^74 + b1 * q^75 + (b1 + 2) * q^77 + (5*b1 + 2) * q^78 + (-3*b1 + 2) * q^79 + 2 * q^80 + (-b2 + 2*b1 + 4) * q^81 + (-b2 - b1 - 3) * q^82 + (b2 + 3*b1 + 2) * q^83 + (-b2 - 2*b1 - 5) * q^84 + (2*b2 - 2*b1 + 6) * q^85 + (b2 + b1 - 2) * q^86 + b1 * q^87 - q^88 + (-2*b2 + 2*b1 + 1) * q^89 + (-2*b2 - 4) * q^90 + (2*b2 + 5*b1 + 6) * q^91 + (-2*b1 + 4) * q^92 + (-2*b2 + 2*b1 - 8) * q^93 + (2*b2 - b1 + 3) * q^94 + b1 * q^96 + (2*b2 - 2*b1 - 3) * q^97 + (-b2 - 4*b1 - 2) * q^98 + (b2 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{7} - 3 q^{8} + 7 q^{9}+O(q^{10})$$ 3 * q - 3 * q^2 + 3 * q^4 + 6 * q^5 + 6 * q^7 - 3 * q^8 + 7 * q^9 $$3 q - 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{7} - 3 q^{8} + 7 q^{9} - 6 q^{10} + 3 q^{11} + 7 q^{13} - 6 q^{14} + 3 q^{16} + 10 q^{17} - 7 q^{18} + 6 q^{20} - 16 q^{21} - 3 q^{22} + 12 q^{23} - 3 q^{25} - 7 q^{26} - 6 q^{27} + 6 q^{28} - 3 q^{29} + 2 q^{31} - 3 q^{32} - 10 q^{34} + 12 q^{35} + 7 q^{36} + 4 q^{37} - 6 q^{39} - 6 q^{40} + 10 q^{41} + 16 q^{42} + 5 q^{43} + 3 q^{44} + 14 q^{45} - 12 q^{46} - 11 q^{47} + 7 q^{49} + 3 q^{50} + 10 q^{51} + 7 q^{52} + 6 q^{54} + 6 q^{55} - 6 q^{56} + 3 q^{58} + 11 q^{61} - 2 q^{62} + 20 q^{63} + 3 q^{64} + 14 q^{65} + 10 q^{68} + 32 q^{69} - 12 q^{70} - 5 q^{71} - 7 q^{72} + 8 q^{73} - 4 q^{74} + 6 q^{77} + 6 q^{78} + 6 q^{79} + 6 q^{80} + 11 q^{81} - 10 q^{82} + 7 q^{83} - 16 q^{84} + 20 q^{85} - 5 q^{86} - 3 q^{88} + q^{89} - 14 q^{90} + 20 q^{91} + 12 q^{92} - 26 q^{93} + 11 q^{94} - 7 q^{97} - 7 q^{98} + 7 q^{99}+O(q^{100})$$ 3 * q - 3 * q^2 + 3 * q^4 + 6 * q^5 + 6 * q^7 - 3 * q^8 + 7 * q^9 - 6 * q^10 + 3 * q^11 + 7 * q^13 - 6 * q^14 + 3 * q^16 + 10 * q^17 - 7 * q^18 + 6 * q^20 - 16 * q^21 - 3 * q^22 + 12 * q^23 - 3 * q^25 - 7 * q^26 - 6 * q^27 + 6 * q^28 - 3 * q^29 + 2 * q^31 - 3 * q^32 - 10 * q^34 + 12 * q^35 + 7 * q^36 + 4 * q^37 - 6 * q^39 - 6 * q^40 + 10 * q^41 + 16 * q^42 + 5 * q^43 + 3 * q^44 + 14 * q^45 - 12 * q^46 - 11 * q^47 + 7 * q^49 + 3 * q^50 + 10 * q^51 + 7 * q^52 + 6 * q^54 + 6 * q^55 - 6 * q^56 + 3 * q^58 + 11 * q^61 - 2 * q^62 + 20 * q^63 + 3 * q^64 + 14 * q^65 + 10 * q^68 + 32 * q^69 - 12 * q^70 - 5 * q^71 - 7 * q^72 + 8 * q^73 - 4 * q^74 + 6 * q^77 + 6 * q^78 + 6 * q^79 + 6 * q^80 + 11 * q^81 - 10 * q^82 + 7 * q^83 - 16 * q^84 + 20 * q^85 - 5 * q^86 - 3 * q^88 + q^89 - 14 * q^90 + 20 * q^91 + 12 * q^92 - 26 * q^93 + 11 * q^94 - 7 * q^97 - 7 * q^98 + 7 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 8x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 5$$ v^2 - 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 5$$ b2 + 5

## 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
 2.94600 −0.252000 −2.69399
−1.00000 −2.94600 1.00000 2.00000 2.94600 4.94600 −1.00000 5.67889 −2.00000
1.2 −1.00000 0.252000 1.00000 2.00000 −0.252000 1.74800 −1.00000 −2.93650 −2.00000
1.3 −1.00000 2.69399 1.00000 2.00000 −2.69399 −0.693995 −1.00000 4.25761 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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.bd 3
19.b odd 2 1 7942.2.a.bj 3
19.c even 3 2 418.2.e.i 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.e.i 6 19.c even 3 2
7942.2.a.bd 3 1.a even 1 1 trivial
7942.2.a.bj 3 19.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7942))$$:

 $$T_{3}^{3} - 8T_{3} + 2$$ T3^3 - 8*T3 + 2 $$T_{5} - 2$$ T5 - 2 $$T_{13}^{3} - 7T_{13}^{2} - 5T_{13} + 71$$ T13^3 - 7*T13^2 - 5*T13 + 71

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{3}$$
$3$ $$T^{3} - 8T + 2$$
$5$ $$(T - 2)^{3}$$
$7$ $$T^{3} - 6 T^{2} + 4 T + 6$$
$11$ $$(T - 1)^{3}$$
$13$ $$T^{3} - 7 T^{2} - 5 T + 71$$
$17$ $$T^{3} - 10 T^{2} + 10 T + 50$$
$19$ $$T^{3}$$
$23$ $$T^{3} - 12 T^{2} + 16 T + 80$$
$29$ $$(T + 1)^{3}$$
$31$ $$T^{3} - 2 T^{2} - 40 T + 116$$
$37$ $$T^{3} - 4 T^{2} - 70 T - 150$$
$41$ $$T^{3} - 10 T^{2} - 2 T + 54$$
$43$ $$T^{3} - 5 T^{2} - 27 T + 81$$
$47$ $$T^{3} + 11 T^{2} - 41 T - 501$$
$53$ $$T^{3} - 128T - 128$$
$59$ $$T^{3}$$
$61$ $$T^{3} - 11 T^{2} - 25 T + 375$$
$67$ $$T^{3} - 72T - 54$$
$71$ $$T^{3} + 5 T^{2} - 27 T - 81$$
$73$ $$T^{3} - 8 T^{2} - 14 T + 102$$
$79$ $$T^{3} - 6 T^{2} - 60 T + 190$$
$83$ $$T^{3} - 7 T^{2} - 95 T - 205$$
$89$ $$T^{3} - T^{2} - 93 T - 43$$
$97$ $$T^{3} + 7 T^{2} - 77 T - 131$$