# Properties

 Label 6422.2.a.n Level $6422$ Weight $2$ Character orbit 6422.a Self dual yes Analytic conductor $51.280$ Analytic rank $0$ 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: $$0$$ 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.n 3
13.b even 2 1 6422.2.a.v yes 3

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

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

## Hecke characteristic polynomials

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