Properties

Label 6422.2.a.k
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: 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.

\(f(q)\) \(=\) \( q - q^{2} + ( -1 + \beta_{1} + \beta_{2} ) q^{3} + q^{4} + ( -2 + \beta_{1} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} ) q^{6} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{7} - q^{8} + ( 3 - \beta_{1} ) q^{9} +O(q^{10})\) \( q - q^{2} + ( -1 + \beta_{1} + \beta_{2} ) q^{3} + q^{4} + ( -2 + \beta_{1} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} ) q^{6} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{7} - q^{8} + ( 3 - \beta_{1} ) q^{9} + ( 2 - \beta_{1} ) q^{10} + ( 4 - 2 \beta_{1} + 3 \beta_{2} ) q^{11} + ( -1 + \beta_{1} + \beta_{2} ) q^{12} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{14} + ( 5 - 3 \beta_{1} ) q^{15} + q^{16} + ( -1 - 3 \beta_{1} + 3 \beta_{2} ) q^{17} + ( -3 + \beta_{1} ) q^{18} + q^{19} + ( -2 + \beta_{1} ) q^{20} + ( -4 + 6 \beta_{1} - 4 \beta_{2} ) q^{21} + ( -4 + 2 \beta_{1} - 3 \beta_{2} ) q^{22} + ( -4 \beta_{1} - \beta_{2} ) q^{23} + ( 1 - \beta_{1} - \beta_{2} ) q^{24} + ( 1 - 4 \beta_{1} + \beta_{2} ) q^{25} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{27} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{28} + ( -4 + 2 \beta_{1} - 3 \beta_{2} ) q^{29} + ( -5 + 3 \beta_{1} ) q^{30} + ( -4 + 3 \beta_{1} - 2 \beta_{2} ) q^{31} - q^{32} + ( -4 + 9 \beta_{1} - 3 \beta_{2} ) q^{33} + ( 1 + 3 \beta_{1} - 3 \beta_{2} ) q^{34} + ( -6 + 6 \beta_{1} - 4 \beta_{2} ) q^{35} + ( 3 - \beta_{1} ) q^{36} + ( -8 + 4 \beta_{1} - 8 \beta_{2} ) q^{37} - q^{38} + ( 2 - \beta_{1} ) q^{40} + ( 5 + 3 \beta_{1} - 3 \beta_{2} ) q^{41} + ( 4 - 6 \beta_{1} + 4 \beta_{2} ) q^{42} + ( -6 + 4 \beta_{1} + 4 \beta_{2} ) q^{43} + ( 4 - 2 \beta_{1} + 3 \beta_{2} ) q^{44} + ( -8 + 5 \beta_{1} - \beta_{2} ) q^{45} + ( 4 \beta_{1} + \beta_{2} ) q^{46} + ( -6 + 2 \beta_{1} - 4 \beta_{2} ) q^{47} + ( -1 + \beta_{1} + \beta_{2} ) q^{48} + ( 1 - 4 \beta_{1} ) q^{49} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{50} + ( -2 + 5 \beta_{1} - 10 \beta_{2} ) q^{51} + ( 2 - 3 \beta_{1} ) q^{53} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{54} + ( -9 + 8 \beta_{1} - 5 \beta_{2} ) q^{55} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{56} + ( -1 + \beta_{1} + \beta_{2} ) q^{57} + ( 4 - 2 \beta_{1} + 3 \beta_{2} ) q^{58} -4 \beta_{1} q^{59} + ( 5 - 3 \beta_{1} ) q^{60} + ( -8 \beta_{1} + 2 \beta_{2} ) q^{61} + ( 4 - 3 \beta_{1} + 2 \beta_{2} ) q^{62} + ( 8 - 8 \beta_{1} + 6 \beta_{2} ) q^{63} + q^{64} + ( 4 - 9 \beta_{1} + 3 \beta_{2} ) q^{66} + ( 2 \beta_{1} - 6 \beta_{2} ) q^{67} + ( -1 - 3 \beta_{1} + 3 \beta_{2} ) q^{68} + ( -14 + 3 \beta_{1} - 7 \beta_{2} ) q^{69} + ( 6 - 6 \beta_{1} + 4 \beta_{2} ) q^{70} + ( -4 + 3 \beta_{1} - 6 \beta_{2} ) q^{71} + ( -3 + \beta_{1} ) q^{72} + ( 6 - 6 \beta_{1} + 8 \beta_{2} ) q^{73} + ( 8 - 4 \beta_{1} + 8 \beta_{2} ) q^{74} + ( -11 + 6 \beta_{1} - 8 \beta_{2} ) q^{75} + q^{76} + ( 12 - 6 \beta_{1} + 2 \beta_{2} ) q^{77} + ( -14 + 6 \beta_{1} - 10 \beta_{2} ) q^{79} + ( -2 + \beta_{1} ) q^{80} + ( -7 - 3 \beta_{1} + \beta_{2} ) q^{81} + ( -5 - 3 \beta_{1} + 3 \beta_{2} ) q^{82} + ( 9 + 3 \beta_{1} - \beta_{2} ) q^{83} + ( -4 + 6 \beta_{1} - 4 \beta_{2} ) q^{84} + ( -1 + 5 \beta_{1} - 6 \beta_{2} ) q^{85} + ( 6 - 4 \beta_{1} - 4 \beta_{2} ) q^{86} + ( 4 - 9 \beta_{1} + 3 \beta_{2} ) q^{87} + ( -4 + 2 \beta_{1} - 3 \beta_{2} ) q^{88} + ( 8 - 4 \beta_{1} + 3 \beta_{2} ) q^{89} + ( 8 - 5 \beta_{1} + \beta_{2} ) q^{90} + ( -4 \beta_{1} - \beta_{2} ) q^{92} + ( 9 - 9 \beta_{1} + 4 \beta_{2} ) q^{93} + ( 6 - 2 \beta_{1} + 4 \beta_{2} ) q^{94} + ( -2 + \beta_{1} ) q^{95} + ( 1 - \beta_{1} - \beta_{2} ) q^{96} + ( -8 \beta_{1} + 5 \beta_{2} ) q^{97} + ( -1 + 4 \beta_{1} ) q^{98} + ( 13 - 10 \beta_{1} + 8 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} - 3q^{3} + 3q^{4} - 5q^{5} + 3q^{6} + 2q^{7} - 3q^{8} + 8q^{9} + O(q^{10}) \) \( 3q - 3q^{2} - 3q^{3} + 3q^{4} - 5q^{5} + 3q^{6} + 2q^{7} - 3q^{8} + 8q^{9} + 5q^{10} + 7q^{11} - 3q^{12} - 2q^{14} + 12q^{15} + 3q^{16} - 9q^{17} - 8q^{18} + 3q^{19} - 5q^{20} - 2q^{21} - 7q^{22} - 3q^{23} + 3q^{24} - 2q^{25} - 6q^{27} + 2q^{28} - 7q^{29} - 12q^{30} - 7q^{31} - 3q^{32} + 9q^{34} - 8q^{35} + 8q^{36} - 12q^{37} - 3q^{38} + 5q^{40} + 21q^{41} + 2q^{42} - 18q^{43} + 7q^{44} - 18q^{45} + 3q^{46} - 12q^{47} - 3q^{48} - q^{49} + 2q^{50} + 9q^{51} + 3q^{53} + 6q^{54} - 14q^{55} - 2q^{56} - 3q^{57} + 7q^{58} - 4q^{59} + 12q^{60} - 10q^{61} + 7q^{62} + 10q^{63} + 3q^{64} + 8q^{67} - 9q^{68} - 32q^{69} + 8q^{70} - 3q^{71} - 8q^{72} + 4q^{73} + 12q^{74} - 19q^{75} + 3q^{76} + 28q^{77} - 26q^{79} - 5q^{80} - 25q^{81} - 21q^{82} + 31q^{83} - 2q^{84} + 8q^{85} + 18q^{86} - 7q^{88} + 17q^{89} + 18q^{90} - 3q^{92} + 14q^{93} + 12q^{94} - 5q^{95} + 3q^{96} - 13q^{97} + q^{98} + 21q^{99} + O(q^{100}) \)

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
−1.24698
0.445042
1.80194
−1.00000 −2.69202 1.00000 −3.24698 2.69202 3.60388 −1.00000 4.24698 3.24698
1.2 −1.00000 −2.35690 1.00000 −1.55496 2.35690 −2.49396 −1.00000 2.55496 1.55496
1.3 −1.00000 2.04892 1.00000 −0.198062 −2.04892 0.890084 −1.00000 1.19806 0.198062
\(n\): 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.k 3
13.b even 2 1 6422.2.a.t 3
13.d odd 4 2 494.2.d.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.d.b 6 13.d odd 4 2
6422.2.a.k 3 1.a even 1 1 trivial
6422.2.a.t 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\):

\( T_{3}^{3} + 3 T_{3}^{2} - 4 T_{3} - 13 \)
\( T_{5}^{3} + 5 T_{5}^{2} + 6 T_{5} + 1 \)
\( T_{7}^{3} - 2 T_{7}^{2} - 8 T_{7} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{3} \)
$3$ \( -13 - 4 T + 3 T^{2} + T^{3} \)
$5$ \( 1 + 6 T + 5 T^{2} + T^{3} \)
$7$ \( 8 - 8 T - 2 T^{2} + T^{3} \)
$11$ \( 49 - 7 T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( -29 + 6 T + 9 T^{2} + T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( 1 - 46 T + 3 T^{2} + T^{3} \)
$29$ \( -49 + 7 T^{2} + T^{3} \)
$31$ \( -7 + 7 T^{2} + T^{3} \)
$37$ \( -832 - 64 T + 12 T^{2} + T^{3} \)
$41$ \( -203 + 126 T - 21 T^{2} + T^{3} \)
$43$ \( -904 - 4 T + 18 T^{2} + T^{3} \)
$47$ \( -104 + 20 T + 12 T^{2} + T^{3} \)
$53$ \( 13 - 18 T - 3 T^{2} + T^{3} \)
$59$ \( -64 - 32 T + 4 T^{2} + T^{3} \)
$61$ \( -776 - 88 T + 10 T^{2} + T^{3} \)
$67$ \( 8 - 44 T - 8 T^{2} + T^{3} \)
$71$ \( -251 - 60 T + 3 T^{2} + T^{3} \)
$73$ \( 568 - 116 T - 4 T^{2} + T^{3} \)
$79$ \( -1784 + 48 T + 26 T^{2} + T^{3} \)
$83$ \( -911 + 304 T - 31 T^{2} + T^{3} \)
$89$ \( -43 + 66 T - 17 T^{2} + T^{3} \)
$97$ \( -797 - 58 T + 13 T^{2} + T^{3} \)
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