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 $1$

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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: \(0\)
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:

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

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