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: \( x^{3} - x^{2} - 4x - 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + q^{2} + (\beta_1 - 1) q^{3} + q^{4} + ( - \beta_{2} + \beta_1) q^{5} + (\beta_1 - 1) q^{6} - \beta_1 q^{7} + q^{8} + (\beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\beta_1 - 1) q^{3} + q^{4} + ( - \beta_{2} + \beta_1) q^{5} + (\beta_1 - 1) q^{6} - \beta_1 q^{7} + q^{8} + (\beta_{2} - \beta_1 + 1) q^{9} + ( - \beta_{2} + \beta_1) q^{10} + (\beta_{2} - \beta_1 - 2) q^{11} + (\beta_1 - 1) q^{12} - \beta_1 q^{14} + (2 \beta_{2} - \beta_1 + 2) q^{15} + q^{16} - 2 \beta_1 q^{17} + (\beta_{2} - \beta_1 + 1) q^{18} + q^{19} + ( - \beta_{2} + \beta_1) q^{20} + ( - \beta_{2} - 3) q^{21} + (\beta_{2} - \beta_1 - 2) q^{22} + (2 \beta_{2} + \beta_1 + 1) q^{23} + (\beta_1 - 1) q^{24} + ( - \beta_{2} - 2) q^{25} + ( - 2 \beta_{2} - \beta_1) q^{27} - \beta_1 q^{28} + ( - 3 \beta_{2} + \beta_1 - 6) q^{29} + (2 \beta_{2} - \beta_1 + 2) q^{30} + ( - 3 \beta_{2} + 4 \beta_1 - 4) q^{31} + q^{32} + ( - 2 \beta_{2} - \beta_1) q^{33} - 2 \beta_1 q^{34} + ( - \beta_{2} - 2) q^{35} + (\beta_{2} - \beta_1 + 1) q^{36} + (4 \beta_{2} - 2 \beta_1 + 2) q^{37} + q^{38} + ( - \beta_{2} + \beta_1) q^{40} + ( - 2 \beta_{2} + \beta_1 + 4) q^{41} + ( - \beta_{2} - 3) q^{42} + ( - 2 \beta_1 - 1) q^{43} + (\beta_{2} - \beta_1 - 2) q^{44} + (\beta_1 - 3) q^{45} + (2 \beta_{2} + \beta_1 + 1) q^{46} + ( - \beta_{2} - 3 \beta_1 - 3) q^{47} + (\beta_1 - 1) q^{48} + (\beta_{2} + \beta_1 - 4) q^{49} + ( - \beta_{2} - 2) q^{50} + ( - 2 \beta_{2} - 6) q^{51} + (3 \beta_{2} - 4 \beta_1 + 4) q^{53} + ( - 2 \beta_{2} - \beta_1) q^{54} + (3 \beta_{2} - 2 \beta_1 - 3) q^{55} - \beta_1 q^{56} + (\beta_1 - 1) q^{57} + ( - 3 \beta_{2} + \beta_1 - 6) q^{58} + (5 \beta_{2} + \beta_1) q^{59} + (2 \beta_{2} - \beta_1 + 2) q^{60} + ( - \beta_{2} - 2 \beta_1 + 1) q^{61} + ( - 3 \beta_{2} + 4 \beta_1 - 4) q^{62} + (\beta_{2} - \beta_1 + 2) q^{63} + q^{64} + ( - 2 \beta_{2} - \beta_1) q^{66} + ( - 4 \beta_{2} + 2 \beta_1) q^{67} - 2 \beta_1 q^{68} + ( - \beta_{2} + 3 \beta_1 + 4) q^{69} + ( - \beta_{2} - 2) q^{70} - 3 q^{71} + (\beta_{2} - \beta_1 + 1) q^{72} + (8 \beta_{2} - 4 \beta_1 + 9) q^{73} + (4 \beta_{2} - 2 \beta_1 + 2) q^{74} + (\beta_{2} - 3 \beta_1 + 1) q^{75} + q^{76} + (\beta_{2} + 2 \beta_1 + 2) q^{77} + ( - 5 \beta_1 + 1) q^{79} + ( - \beta_{2} + \beta_1) q^{80} + ( - 2 \beta_{2} + \beta_1 - 8) q^{81} + ( - 2 \beta_{2} + \beta_1 + 4) q^{82} + (9 \beta_{2} - 4 \beta_1 + 6) q^{83} + ( - \beta_{2} - 3) q^{84} + ( - 2 \beta_{2} - 4) q^{85} + ( - 2 \beta_1 - 1) q^{86} + (4 \beta_{2} - 9 \beta_1 + 6) q^{87} + (\beta_{2} - \beta_1 - 2) q^{88} + ( - 3 \beta_{2} - 10) q^{89} + (\beta_1 - 3) q^{90} + (2 \beta_{2} + \beta_1 + 1) q^{92} + (7 \beta_{2} - 7 \beta_1 + 13) q^{93} + ( - \beta_{2} - 3 \beta_1 - 3) q^{94} + ( - \beta_{2} + \beta_1) q^{95} + (\beta_1 - 1) q^{96} + ( - 2 \beta_1 - 6) q^{97} + (\beta_{2} + \beta_1 - 4) q^{98} + ( - 2 \beta_{2} + \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{5} - 2 q^{6} - q^{7} + 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{5} - 2 q^{6} - q^{7} + 3 q^{8} + q^{9} + 2 q^{10} - 8 q^{11} - 2 q^{12} - q^{14} + 3 q^{15} + 3 q^{16} - 2 q^{17} + q^{18} + 3 q^{19} + 2 q^{20} - 8 q^{21} - 8 q^{22} + 2 q^{23} - 2 q^{24} - 5 q^{25} + q^{27} - q^{28} - 14 q^{29} + 3 q^{30} - 5 q^{31} + 3 q^{32} + q^{33} - 2 q^{34} - 5 q^{35} + q^{36} + 3 q^{38} + 2 q^{40} + 15 q^{41} - 8 q^{42} - 5 q^{43} - 8 q^{44} - 8 q^{45} + 2 q^{46} - 11 q^{47} - 2 q^{48} - 12 q^{49} - 5 q^{50} - 16 q^{51} + 5 q^{53} + q^{54} - 14 q^{55} - q^{56} - 2 q^{57} - 14 q^{58} - 4 q^{59} + 3 q^{60} + 2 q^{61} - 5 q^{62} + 4 q^{63} + 3 q^{64} + q^{66} + 6 q^{67} - 2 q^{68} + 16 q^{69} - 5 q^{70} - 9 q^{71} + q^{72} + 15 q^{73} - q^{75} + 3 q^{76} + 7 q^{77} - 2 q^{79} + 2 q^{80} - 21 q^{81} + 15 q^{82} + 5 q^{83} - 8 q^{84} - 10 q^{85} - 5 q^{86} + 5 q^{87} - 8 q^{88} - 27 q^{89} - 8 q^{90} + 2 q^{92} + 25 q^{93} - 11 q^{94} + 2 q^{95} - 2 q^{96} - 20 q^{97} - 12 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\):

\( T_{3}^{3} + 2T_{3}^{2} - 3T_{3} - 5 \) Copy content Toggle raw display
\( T_{5}^{3} - 2T_{5}^{2} - 3T_{5} + 5 \) Copy content Toggle raw display
\( T_{7}^{3} + T_{7}^{2} - 4T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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