# Properties

 Label 6422.2.a.y 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: 3.3.1129.1 Defining polynomial: $$x^{3} - 7 x - 3$$ 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 5$$

## 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
 −2.39766 −0.440808 2.83847
1.00000 −2.39766 1.00000 0 −2.39766 −4.54410 1.00000 2.74878 0
1.2 1.00000 −0.440808 1.00000 0 −0.440808 4.92407 1.00000 −2.80569 0
1.3 1.00000 2.83847 1.00000 0 2.83847 3.62003 1.00000 5.05691 0
 $$n$$: 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.y 3
13.b even 2 1 6422.2.a.o 3
13.e even 6 2 494.2.g.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.g.c 6 13.e even 6 2
6422.2.a.o 3 13.b even 2 1
6422.2.a.y 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6422))$$:

 $$T_{3}^{3} - 7 T_{3} - 3$$ $$T_{5}$$ $$T_{7}^{3} - 4 T_{7}^{2} - 21 T_{7} + 81$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$-3 - 7 T + T^{3}$$
$5$ $$T^{3}$$
$7$ $$81 - 21 T - 4 T^{2} + T^{3}$$
$11$ $$T^{3}$$
$13$ $$T^{3}$$
$17$ $$33 + 41 T + 12 T^{2} + T^{3}$$
$19$ $$( 1 + T )^{3}$$
$23$ $$-261 - 57 T + 4 T^{2} + T^{3}$$
$29$ $$99 - 51 T + 2 T^{2} + T^{3}$$
$31$ $$-24 - 56 T - 2 T^{2} + T^{3}$$
$37$ $$-9 + 47 T - 15 T^{2} + T^{3}$$
$41$ $$408 - 128 T - 2 T^{2} + T^{3}$$
$43$ $$72 + 20 T - 12 T^{2} + T^{3}$$
$47$ $$303 - 109 T - 3 T^{2} + T^{3}$$
$53$ $$111 + T - 10 T^{2} + T^{3}$$
$59$ $$-297 + 147 T - 22 T^{2} + T^{3}$$
$61$ $$24 + 80 T + 18 T^{2} + T^{3}$$
$67$ $$-17 - 13 T + 2 T^{2} + T^{3}$$
$71$ $$-216 + 96 T + 22 T^{2} + T^{3}$$
$73$ $$-967 - 99 T + 12 T^{2} + T^{3}$$
$79$ $$312 - 32 T - 10 T^{2} + T^{3}$$
$83$ $$-216 - 60 T + 4 T^{2} + T^{3}$$
$89$ $$-648 - 228 T + 2 T^{2} + T^{3}$$
$97$ $$-72 + 20 T + 12 T^{2} + T^{3}$$