# Properties

 Label 6422.2.a.w 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: 3.3.361.1 Defining polynomial: $$x^{3} - x^{2} - 6 x + 7$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6 x + 7$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$

## 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
 2.28514 1.22188 −2.50702
1.00000 −2.28514 1.00000 −1.22188 −2.28514 5.01404 1.00000 2.22188 −1.22188
1.2 1.00000 −1.22188 1.00000 2.50702 −1.22188 −4.57028 1.00000 −1.50702 2.50702
1.3 1.00000 2.50702 1.00000 −2.28514 2.50702 −2.44375 1.00000 3.28514 −2.28514
 $$n$$: 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.w 3
13.b even 2 1 494.2.a.f 3
39.d odd 2 1 4446.2.a.bk 3
52.b odd 2 1 3952.2.a.o 3
247.d odd 2 1 9386.2.a.bc 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.a.f 3 13.b even 2 1
3952.2.a.o 3 52.b odd 2 1
4446.2.a.bk 3 39.d odd 2 1
6422.2.a.w 3 1.a even 1 1 trivial
9386.2.a.bc 3 247.d odd 2 1

## Hecke kernels

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

 $$T_{3}^{3} + T_{3}^{2} - 6 T_{3} - 7$$ $$T_{5}^{3} + T_{5}^{2} - 6 T_{5} - 7$$ $$T_{7}^{3} + 2 T_{7}^{2} - 24 T_{7} - 56$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$-7 - 6 T + T^{2} + T^{3}$$
$5$ $$-7 - 6 T + T^{2} + T^{3}$$
$7$ $$-56 - 24 T + 2 T^{2} + T^{3}$$
$11$ $$7 - 6 T - T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$-1 + 2 T + 5 T^{2} + T^{3}$$
$19$ $$( -1 + T )^{3}$$
$23$ $$-121 - 44 T + T^{2} + T^{3}$$
$29$ $$83 - 28 T - 7 T^{2} + T^{3}$$
$31$ $$-133 + 76 T + 19 T^{2} + T^{3}$$
$37$ $$88 + 108 T + 20 T^{2} + T^{3}$$
$41$ $$-11 + 8 T + 9 T^{2} + T^{3}$$
$43$ $$-88 - 4 T + 8 T^{2} + T^{3}$$
$47$ $$8 - 20 T + 4 T^{2} + T^{3}$$
$53$ $$-11 + 56 T - 15 T^{2} + T^{3}$$
$59$ $$-56 - 24 T + 2 T^{2} + T^{3}$$
$61$ $$8 - 20 T + 4 T^{2} + T^{3}$$
$67$ $$-704 - 16 T + 16 T^{2} + T^{3}$$
$71$ $$463 - 120 T - T^{2} + T^{3}$$
$73$ $$-1304 - 156 T + 8 T^{2} + T^{3}$$
$79$ $$-64 - 80 T - 8 T^{2} + T^{3}$$
$83$ $$683 - 168 T - 3 T^{2} + T^{3}$$
$89$ $$-7 - 6 T + T^{2} + T^{3}$$
$97$ $$83 + 186 T + 27 T^{2} + T^{3}$$