# Properties

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

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## 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: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + q^{2} -\beta_{1} q^{3} + q^{4} + ( -3 + \beta_{1} - \beta_{2} ) q^{5} -\beta_{1} q^{6} + ( -2 - 2 \beta_{2} ) q^{7} + q^{8} + ( -1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + q^{2} -\beta_{1} q^{3} + q^{4} + ( -3 + \beta_{1} - \beta_{2} ) q^{5} -\beta_{1} q^{6} + ( -2 - 2 \beta_{2} ) q^{7} + q^{8} + ( -1 + \beta_{2} ) q^{9} + ( -3 + \beta_{1} - \beta_{2} ) q^{10} -2 \beta_{2} q^{11} -\beta_{1} q^{12} + ( -2 - 2 \beta_{2} ) q^{14} + ( -1 + 3 \beta_{1} ) q^{15} + q^{16} + ( 4 + 3 \beta_{2} ) q^{17} + ( -1 + \beta_{2} ) q^{18} + q^{19} + ( -3 + \beta_{1} - \beta_{2} ) q^{20} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{21} -2 \beta_{2} q^{22} + ( 4 - 4 \beta_{1} + 6 \beta_{2} ) q^{23} -\beta_{1} q^{24} + ( 5 - 5 \beta_{1} + 4 \beta_{2} ) q^{25} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{27} + ( -2 - 2 \beta_{2} ) q^{28} + ( 4 - 4 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -1 + 3 \beta_{1} ) q^{30} + ( 2 - 3 \beta_{1} + 2 \beta_{2} ) q^{31} + q^{32} + ( 2 + 2 \beta_{2} ) q^{33} + ( 4 + 3 \beta_{2} ) q^{34} + ( 6 + 4 \beta_{2} ) q^{35} + ( -1 + \beta_{2} ) q^{36} + ( -4 + 6 \beta_{1} ) q^{37} + q^{38} + ( -3 + \beta_{1} - \beta_{2} ) q^{40} + ( -8 + 4 \beta_{1} - 2 \beta_{2} ) q^{41} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{42} + 4 \beta_{1} q^{43} -2 \beta_{2} q^{44} + ( 3 - 2 \beta_{1} ) q^{45} + ( 4 - 4 \beta_{1} + 6 \beta_{2} ) q^{46} + ( -2 + 4 \beta_{1} ) q^{47} -\beta_{1} q^{48} + ( 1 + 4 \beta_{1} + 4 \beta_{2} ) q^{49} + ( 5 - 5 \beta_{1} + 4 \beta_{2} ) q^{50} + ( -3 - 4 \beta_{1} - 3 \beta_{2} ) q^{51} + ( -10 + 2 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{54} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{55} + ( -2 - 2 \beta_{2} ) q^{56} -\beta_{1} q^{57} + ( 4 - 4 \beta_{1} + 2 \beta_{2} ) q^{58} + ( -1 - 5 \beta_{1} - \beta_{2} ) q^{59} + ( -1 + 3 \beta_{1} ) q^{60} + ( -1 - \beta_{1} + 5 \beta_{2} ) q^{61} + ( 2 - 3 \beta_{1} + 2 \beta_{2} ) q^{62} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{63} + q^{64} + ( 2 + 2 \beta_{2} ) q^{66} + ( 4 - 3 \beta_{1} ) q^{67} + ( 4 + 3 \beta_{2} ) q^{68} + ( 2 - 4 \beta_{1} - 2 \beta_{2} ) q^{69} + ( 6 + 4 \beta_{2} ) q^{70} + ( 4 + 2 \beta_{1} - 3 \beta_{2} ) q^{71} + ( -1 + \beta_{2} ) q^{72} + ( -4 - 3 \beta_{2} ) q^{73} + ( -4 + 6 \beta_{1} ) q^{74} + ( 6 - 5 \beta_{1} + \beta_{2} ) q^{75} + q^{76} + ( 4 + 4 \beta_{1} ) q^{77} + ( -4 + 6 \beta_{1} - 7 \beta_{2} ) q^{79} + ( -3 + \beta_{1} - \beta_{2} ) q^{80} + ( -4 + \beta_{1} - 6 \beta_{2} ) q^{81} + ( -8 + 4 \beta_{1} - 2 \beta_{2} ) q^{82} + ( -4 + 2 \beta_{1} ) q^{83} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{84} + ( -12 + \beta_{1} - 7 \beta_{2} ) q^{85} + 4 \beta_{1} q^{86} + ( 6 - 4 \beta_{1} + 2 \beta_{2} ) q^{87} -2 \beta_{2} q^{88} + ( -14 + 4 \beta_{1} - 8 \beta_{2} ) q^{89} + ( 3 - 2 \beta_{1} ) q^{90} + ( 4 - 4 \beta_{1} + 6 \beta_{2} ) q^{92} + ( 4 - 2 \beta_{1} + \beta_{2} ) q^{93} + ( -2 + 4 \beta_{1} ) q^{94} + ( -3 + \beta_{1} - \beta_{2} ) q^{95} -\beta_{1} q^{96} + ( 4 - 6 \beta_{1} + 8 \beta_{2} ) q^{97} + ( 1 + 4 \beta_{1} + 4 \beta_{2} ) q^{98} + ( -2 - 2 \beta_{1} + 4 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} - q^{3} + 3 q^{4} - 7 q^{5} - q^{6} - 4 q^{7} + 3 q^{8} - 4 q^{9} + O(q^{10})$$ $$3 q + 3 q^{2} - q^{3} + 3 q^{4} - 7 q^{5} - q^{6} - 4 q^{7} + 3 q^{8} - 4 q^{9} - 7 q^{10} + 2 q^{11} - q^{12} - 4 q^{14} + 3 q^{16} + 9 q^{17} - 4 q^{18} + 3 q^{19} - 7 q^{20} + 6 q^{21} + 2 q^{22} + 2 q^{23} - q^{24} + 6 q^{25} + 2 q^{27} - 4 q^{28} + 6 q^{29} + q^{31} + 3 q^{32} + 4 q^{33} + 9 q^{34} + 14 q^{35} - 4 q^{36} - 6 q^{37} + 3 q^{38} - 7 q^{40} - 18 q^{41} + 6 q^{42} + 4 q^{43} + 2 q^{44} + 7 q^{45} + 2 q^{46} - 2 q^{47} - q^{48} + 3 q^{49} + 6 q^{50} - 10 q^{51} - 26 q^{53} + 2 q^{54} - 4 q^{56} - q^{57} + 6 q^{58} - 7 q^{59} - 9 q^{61} + q^{62} - 4 q^{63} + 3 q^{64} + 4 q^{66} + 9 q^{67} + 9 q^{68} + 4 q^{69} + 14 q^{70} + 17 q^{71} - 4 q^{72} - 9 q^{73} - 6 q^{74} + 12 q^{75} + 3 q^{76} + 16 q^{77} + q^{79} - 7 q^{80} - 5 q^{81} - 18 q^{82} - 10 q^{83} + 6 q^{84} - 28 q^{85} + 4 q^{86} + 12 q^{87} + 2 q^{88} - 30 q^{89} + 7 q^{90} + 2 q^{92} + 9 q^{93} - 2 q^{94} - 7 q^{95} - q^{96} - 2 q^{97} + 3 q^{98} - 12 q^{99} + O(q^{100})$$

## 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
 1.80194 0.445042 −1.24698
1.00000 −1.80194 1.00000 −2.44504 −1.80194 −4.49396 1.00000 0.246980 −2.44504
1.2 1.00000 −0.445042 1.00000 −0.753020 −0.445042 1.60388 1.00000 −2.80194 −0.753020
1.3 1.00000 1.24698 1.00000 −3.80194 1.24698 −1.10992 1.00000 −1.44504 −3.80194
 $$n$$: 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.v yes 3
13.b even 2 1 6422.2.a.n 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6422.2.a.n 3 13.b even 2 1
6422.2.a.v yes 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6422))$$:

 $$T_{3}^{3} + T_{3}^{2} - 2 T_{3} - 1$$ $$T_{5}^{3} + 7 T_{5}^{2} + 14 T_{5} + 7$$ $$T_{7}^{3} + 4 T_{7}^{2} - 4 T_{7} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$-1 - 2 T + T^{2} + T^{3}$$
$5$ $$7 + 14 T + 7 T^{2} + T^{3}$$
$7$ $$-8 - 4 T + 4 T^{2} + T^{3}$$
$11$ $$8 - 8 T - 2 T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$29 + 6 T - 9 T^{2} + T^{3}$$
$19$ $$( -1 + T )^{3}$$
$23$ $$232 - 64 T - 2 T^{2} + T^{3}$$
$29$ $$-8 - 16 T - 6 T^{2} + T^{3}$$
$31$ $$-13 - 16 T - T^{2} + T^{3}$$
$37$ $$-104 - 72 T + 6 T^{2} + T^{3}$$
$41$ $$104 + 80 T + 18 T^{2} + T^{3}$$
$43$ $$64 - 32 T - 4 T^{2} + T^{3}$$
$47$ $$-8 - 36 T + 2 T^{2} + T^{3}$$
$53$ $$568 + 216 T + 26 T^{2} + T^{3}$$
$59$ $$-91 - 56 T + 7 T^{2} + T^{3}$$
$61$ $$-71 - 22 T + 9 T^{2} + T^{3}$$
$67$ $$29 + 6 T - 9 T^{2} + T^{3}$$
$71$ $$-113 + 80 T - 17 T^{2} + T^{3}$$
$73$ $$-29 + 6 T + 9 T^{2} + T^{3}$$
$79$ $$-181 - 100 T - T^{2} + T^{3}$$
$83$ $$8 + 24 T + 10 T^{2} + T^{3}$$
$89$ $$-568 + 188 T + 30 T^{2} + T^{3}$$
$97$ $$328 - 120 T + 2 T^{2} + T^{3}$$
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