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 $1$

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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: 3.3.361.1
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:

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

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