# Properties

 Label 6422.2.a.m Level $6422$ Weight $2$ Character orbit 6422.a Self dual yes Analytic conductor $51.280$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6422 = 2 \cdot 13^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6422.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$51.2799281781$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.169.1 Defining polynomial: $$x^{3} - x^{2} - 4x - 1$$ x^3 - x^2 - 4*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 494) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + (\beta_1 - 1) q^{3} + q^{4} + (\beta_{2} - \beta_1) q^{5} + ( - \beta_1 + 1) q^{6} + \beta_1 q^{7} - q^{8} + (\beta_{2} - \beta_1 + 1) q^{9}+O(q^{10})$$ q - q^2 + (b1 - 1) * q^3 + q^4 + (b2 - b1) * q^5 + (-b1 + 1) * q^6 + b1 * q^7 - q^8 + (b2 - b1 + 1) * q^9 $$q - q^{2} + (\beta_1 - 1) q^{3} + q^{4} + (\beta_{2} - \beta_1) q^{5} + ( - \beta_1 + 1) q^{6} + \beta_1 q^{7} - q^{8} + (\beta_{2} - \beta_1 + 1) q^{9} + ( - \beta_{2} + \beta_1) q^{10} + ( - \beta_{2} + \beta_1 + 2) q^{11} + (\beta_1 - 1) q^{12} - \beta_1 q^{14} + ( - 2 \beta_{2} + \beta_1 - 2) q^{15} + q^{16} - 2 \beta_1 q^{17} + ( - \beta_{2} + \beta_1 - 1) q^{18} - q^{19} + (\beta_{2} - \beta_1) q^{20} + (\beta_{2} + 3) q^{21} + (\beta_{2} - \beta_1 - 2) q^{22} + (2 \beta_{2} + \beta_1 + 1) q^{23} + ( - \beta_1 + 1) q^{24} + ( - \beta_{2} - 2) q^{25} + ( - 2 \beta_{2} - \beta_1) q^{27} + \beta_1 q^{28} + ( - 3 \beta_{2} + \beta_1 - 6) q^{29} + (2 \beta_{2} - \beta_1 + 2) q^{30} + (3 \beta_{2} - 4 \beta_1 + 4) q^{31} - q^{32} + (2 \beta_{2} + \beta_1) q^{33} + 2 \beta_1 q^{34} + ( - \beta_{2} - 2) q^{35} + (\beta_{2} - \beta_1 + 1) q^{36} + ( - 4 \beta_{2} + 2 \beta_1 - 2) q^{37} + q^{38} + ( - \beta_{2} + \beta_1) q^{40} + (2 \beta_{2} - \beta_1 - 4) q^{41} + ( - \beta_{2} - 3) q^{42} + ( - 2 \beta_1 - 1) q^{43} + ( - \beta_{2} + \beta_1 + 2) q^{44} + ( - \beta_1 + 3) q^{45} + ( - 2 \beta_{2} - \beta_1 - 1) q^{46} + (\beta_{2} + 3 \beta_1 + 3) q^{47} + (\beta_1 - 1) q^{48} + (\beta_{2} + \beta_1 - 4) q^{49} + (\beta_{2} + 2) q^{50} + ( - 2 \beta_{2} - 6) q^{51} + (3 \beta_{2} - 4 \beta_1 + 4) q^{53} + (2 \beta_{2} + \beta_1) q^{54} + (3 \beta_{2} - 2 \beta_1 - 3) q^{55} - \beta_1 q^{56} + ( - \beta_1 + 1) q^{57} + (3 \beta_{2} - \beta_1 + 6) q^{58} + ( - 5 \beta_{2} - \beta_1) q^{59} + ( - 2 \beta_{2} + \beta_1 - 2) q^{60} + ( - \beta_{2} - 2 \beta_1 + 1) q^{61} + ( - 3 \beta_{2} + 4 \beta_1 - 4) q^{62} + ( - \beta_{2} + \beta_1 - 2) q^{63} + q^{64} + ( - 2 \beta_{2} - \beta_1) q^{66} + (4 \beta_{2} - 2 \beta_1) q^{67} - 2 \beta_1 q^{68} + ( - \beta_{2} + 3 \beta_1 + 4) q^{69} + (\beta_{2} + 2) q^{70} + 3 q^{71} + ( - \beta_{2} + \beta_1 - 1) q^{72} + ( - 8 \beta_{2} + 4 \beta_1 - 9) q^{73} + (4 \beta_{2} - 2 \beta_1 + 2) q^{74} + (\beta_{2} - 3 \beta_1 + 1) q^{75} - q^{76} + (\beta_{2} + 2 \beta_1 + 2) q^{77} + ( - 5 \beta_1 + 1) q^{79} + (\beta_{2} - \beta_1) q^{80} + ( - 2 \beta_{2} + \beta_1 - 8) q^{81} + ( - 2 \beta_{2} + \beta_1 + 4) q^{82} + ( - 9 \beta_{2} + 4 \beta_1 - 6) q^{83} + (\beta_{2} + 3) q^{84} + (2 \beta_{2} + 4) q^{85} + (2 \beta_1 + 1) q^{86} + (4 \beta_{2} - 9 \beta_1 + 6) q^{87} + (\beta_{2} - \beta_1 - 2) q^{88} + (3 \beta_{2} + 10) q^{89} + (\beta_1 - 3) q^{90} + (2 \beta_{2} + \beta_1 + 1) q^{92} + ( - 7 \beta_{2} + 7 \beta_1 - 13) q^{93} + ( - \beta_{2} - 3 \beta_1 - 3) q^{94} + ( - \beta_{2} + \beta_1) q^{95} + ( - \beta_1 + 1) q^{96} + (2 \beta_1 + 6) q^{97} + ( - \beta_{2} - \beta_1 + 4) q^{98} + (2 \beta_{2} - \beta_1 - 1) q^{99}+O(q^{100})$$ q - q^2 + (b1 - 1) * q^3 + q^4 + (b2 - b1) * q^5 + (-b1 + 1) * q^6 + b1 * q^7 - q^8 + (b2 - b1 + 1) * q^9 + (-b2 + b1) * q^10 + (-b2 + b1 + 2) * q^11 + (b1 - 1) * q^12 - b1 * q^14 + (-2*b2 + b1 - 2) * q^15 + q^16 - 2*b1 * q^17 + (-b2 + b1 - 1) * q^18 - q^19 + (b2 - b1) * q^20 + (b2 + 3) * q^21 + (b2 - b1 - 2) * q^22 + (2*b2 + b1 + 1) * q^23 + (-b1 + 1) * q^24 + (-b2 - 2) * q^25 + (-2*b2 - b1) * q^27 + b1 * q^28 + (-3*b2 + b1 - 6) * q^29 + (2*b2 - b1 + 2) * q^30 + (3*b2 - 4*b1 + 4) * q^31 - q^32 + (2*b2 + b1) * q^33 + 2*b1 * q^34 + (-b2 - 2) * q^35 + (b2 - b1 + 1) * q^36 + (-4*b2 + 2*b1 - 2) * q^37 + q^38 + (-b2 + b1) * q^40 + (2*b2 - b1 - 4) * q^41 + (-b2 - 3) * q^42 + (-2*b1 - 1) * q^43 + (-b2 + b1 + 2) * q^44 + (-b1 + 3) * q^45 + (-2*b2 - b1 - 1) * q^46 + (b2 + 3*b1 + 3) * q^47 + (b1 - 1) * q^48 + (b2 + b1 - 4) * q^49 + (b2 + 2) * q^50 + (-2*b2 - 6) * q^51 + (3*b2 - 4*b1 + 4) * q^53 + (2*b2 + b1) * q^54 + (3*b2 - 2*b1 - 3) * q^55 - b1 * q^56 + (-b1 + 1) * q^57 + (3*b2 - b1 + 6) * q^58 + (-5*b2 - b1) * q^59 + (-2*b2 + b1 - 2) * q^60 + (-b2 - 2*b1 + 1) * q^61 + (-3*b2 + 4*b1 - 4) * q^62 + (-b2 + b1 - 2) * q^63 + q^64 + (-2*b2 - b1) * q^66 + (4*b2 - 2*b1) * q^67 - 2*b1 * q^68 + (-b2 + 3*b1 + 4) * q^69 + (b2 + 2) * q^70 + 3 * q^71 + (-b2 + b1 - 1) * q^72 + (-8*b2 + 4*b1 - 9) * q^73 + (4*b2 - 2*b1 + 2) * q^74 + (b2 - 3*b1 + 1) * q^75 - q^76 + (b2 + 2*b1 + 2) * q^77 + (-5*b1 + 1) * q^79 + (b2 - b1) * q^80 + (-2*b2 + b1 - 8) * q^81 + (-2*b2 + b1 + 4) * q^82 + (-9*b2 + 4*b1 - 6) * q^83 + (b2 + 3) * q^84 + (2*b2 + 4) * q^85 + (2*b1 + 1) * q^86 + (4*b2 - 9*b1 + 6) * q^87 + (b2 - b1 - 2) * q^88 + (3*b2 + 10) * q^89 + (b1 - 3) * q^90 + (2*b2 + b1 + 1) * q^92 + (-7*b2 + 7*b1 - 13) * q^93 + (-b2 - 3*b1 - 3) * q^94 + (-b2 + b1) * q^95 + (-b1 + 1) * q^96 + (2*b1 + 6) * q^97 + (-b2 - b1 + 4) * q^98 + (2*b2 - b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 2 q^{5} + 2 q^{6} + q^{7} - 3 q^{8} + q^{9}+O(q^{10})$$ 3 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 - 2 * q^5 + 2 * q^6 + q^7 - 3 * q^8 + q^9 $$3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 2 q^{5} + 2 q^{6} + q^{7} - 3 q^{8} + q^{9} + 2 q^{10} + 8 q^{11} - 2 q^{12} - q^{14} - 3 q^{15} + 3 q^{16} - 2 q^{17} - q^{18} - 3 q^{19} - 2 q^{20} + 8 q^{21} - 8 q^{22} + 2 q^{23} + 2 q^{24} - 5 q^{25} + q^{27} + q^{28} - 14 q^{29} + 3 q^{30} + 5 q^{31} - 3 q^{32} - q^{33} + 2 q^{34} - 5 q^{35} + q^{36} + 3 q^{38} + 2 q^{40} - 15 q^{41} - 8 q^{42} - 5 q^{43} + 8 q^{44} + 8 q^{45} - 2 q^{46} + 11 q^{47} - 2 q^{48} - 12 q^{49} + 5 q^{50} - 16 q^{51} + 5 q^{53} - q^{54} - 14 q^{55} - q^{56} + 2 q^{57} + 14 q^{58} + 4 q^{59} - 3 q^{60} + 2 q^{61} - 5 q^{62} - 4 q^{63} + 3 q^{64} + q^{66} - 6 q^{67} - 2 q^{68} + 16 q^{69} + 5 q^{70} + 9 q^{71} - q^{72} - 15 q^{73} - q^{75} - 3 q^{76} + 7 q^{77} - 2 q^{79} - 2 q^{80} - 21 q^{81} + 15 q^{82} - 5 q^{83} + 8 q^{84} + 10 q^{85} + 5 q^{86} + 5 q^{87} - 8 q^{88} + 27 q^{89} - 8 q^{90} + 2 q^{92} - 25 q^{93} - 11 q^{94} + 2 q^{95} + 2 q^{96} + 20 q^{97} + 12 q^{98} - 6 q^{99}+O(q^{100})$$ 3 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 - 2 * q^5 + 2 * q^6 + q^7 - 3 * q^8 + q^9 + 2 * q^10 + 8 * q^11 - 2 * q^12 - q^14 - 3 * q^15 + 3 * q^16 - 2 * q^17 - q^18 - 3 * q^19 - 2 * q^20 + 8 * q^21 - 8 * q^22 + 2 * q^23 + 2 * q^24 - 5 * q^25 + q^27 + q^28 - 14 * q^29 + 3 * q^30 + 5 * q^31 - 3 * q^32 - q^33 + 2 * q^34 - 5 * q^35 + q^36 + 3 * q^38 + 2 * q^40 - 15 * q^41 - 8 * q^42 - 5 * q^43 + 8 * q^44 + 8 * q^45 - 2 * q^46 + 11 * q^47 - 2 * q^48 - 12 * q^49 + 5 * q^50 - 16 * q^51 + 5 * q^53 - q^54 - 14 * q^55 - q^56 + 2 * q^57 + 14 * q^58 + 4 * q^59 - 3 * q^60 + 2 * q^61 - 5 * q^62 - 4 * q^63 + 3 * q^64 + q^66 - 6 * q^67 - 2 * q^68 + 16 * q^69 + 5 * q^70 + 9 * q^71 - q^72 - 15 * q^73 - q^75 - 3 * q^76 + 7 * q^77 - 2 * q^79 - 2 * q^80 - 21 * q^81 + 15 * q^82 - 5 * q^83 + 8 * q^84 + 10 * q^85 + 5 * q^86 + 5 * q^87 - 8 * q^88 + 27 * q^89 - 8 * q^90 + 2 * q^92 - 25 * q^93 - 11 * q^94 + 2 * q^95 + 2 * q^96 + 20 * q^97 + 12 * q^98 - 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3

## 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
 −1.37720 −0.273891 2.65109
−1.00000 −2.37720 1.00000 1.65109 2.37720 −1.37720 −1.00000 2.65109 −1.65109
1.2 −1.00000 −1.27389 1.00000 −2.37720 1.27389 −0.273891 −1.00000 −1.37720 2.37720
1.3 −1.00000 1.65109 1.00000 −1.27389 −1.65109 2.65109 −1.00000 −0.273891 1.27389
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$13$$ $$1$$
$$19$$ $$1$$

## 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 6422.2.a.m 3
13.b even 2 1 6422.2.a.u 3
13.e even 6 2 494.2.g.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.g.b 6 13.e even 6 2
6422.2.a.m 3 1.a even 1 1 trivial
6422.2.a.u 3 13.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6422))$$:

 $$T_{3}^{3} + 2T_{3}^{2} - 3T_{3} - 5$$ T3^3 + 2*T3^2 - 3*T3 - 5 $$T_{5}^{3} + 2T_{5}^{2} - 3T_{5} - 5$$ T5^3 + 2*T5^2 - 3*T5 - 5 $$T_{7}^{3} - T_{7}^{2} - 4T_{7} - 1$$ T7^3 - T7^2 - 4*T7 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{3}$$
$3$ $$T^{3} + 2 T^{2} - 3 T - 5$$
$5$ $$T^{3} + 2 T^{2} - 3 T - 5$$
$7$ $$T^{3} - T^{2} - 4T - 1$$
$11$ $$T^{3} - 8 T^{2} + 17 T - 5$$
$13$ $$T^{3}$$
$17$ $$T^{3} + 2 T^{2} - 16 T + 8$$
$19$ $$(T + 1)^{3}$$
$23$ $$T^{3} - 2 T^{2} - 29 T + 5$$
$29$ $$T^{3} + 14 T^{2} + 35 T - 103$$
$31$ $$T^{3} - 5 T^{2} - 48 T - 73$$
$37$ $$T^{3} - 52T - 104$$
$41$ $$T^{3} + 15 T^{2} + 62 T + 73$$
$43$ $$T^{3} + 5 T^{2} - 9 T - 5$$
$47$ $$T^{3} - 11 T^{2} - 16 T - 5$$
$53$ $$T^{3} - 5 T^{2} - 48 T - 73$$
$59$ $$T^{3} - 4 T^{2} - 129 T + 1$$
$61$ $$T^{3} - 2 T^{2} - 29 T + 83$$
$67$ $$T^{3} + 6 T^{2} - 40 T + 8$$
$71$ $$(T - 3)^{3}$$
$73$ $$T^{3} + 15 T^{2} - 133 T - 1747$$
$79$ $$T^{3} + 2 T^{2} - 107 T + 229$$
$83$ $$T^{3} + 5 T^{2} - 256 T - 1825$$
$89$ $$T^{3} - 27 T^{2} + 204 T - 313$$
$97$ $$T^{3} - 20 T^{2} + 116 T - 200$$