Properties

Label 6422.2.a.u
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: 3.3.169.1
Defining polynomial: \(x^{3} - x^{2} - 4 x - 1\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)

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:

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

\( T_{3}^{3} + 2 T_{3}^{2} - 3 T_{3} - 5 \)
\( T_{5}^{3} - 2 T_{5}^{2} - 3 T_{5} + 5 \)
\( T_{7}^{3} + T_{7}^{2} - 4 T_{7} + 1 \)

Hecke characteristic polynomials

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