Properties

Label 6422.2.a.y
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: 3.3.1129.1
Defining polynomial: \(x^{3} - 7 x - 3\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 7 x - 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 5\)

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
−2.39766
−0.440808
2.83847
1.00000 −2.39766 1.00000 0 −2.39766 −4.54410 1.00000 2.74878 0
1.2 1.00000 −0.440808 1.00000 0 −0.440808 4.92407 1.00000 −2.80569 0
1.3 1.00000 2.83847 1.00000 0 2.83847 3.62003 1.00000 5.05691 0
\(n\): 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.y 3
13.b even 2 1 6422.2.a.o 3
13.e even 6 2 494.2.g.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.g.c 6 13.e even 6 2
6422.2.a.o 3 13.b even 2 1
6422.2.a.y 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\):

\( T_{3}^{3} - 7 T_{3} - 3 \)
\( T_{5} \)
\( T_{7}^{3} - 4 T_{7}^{2} - 21 T_{7} + 81 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( -3 - 7 T + T^{3} \)
$5$ \( T^{3} \)
$7$ \( 81 - 21 T - 4 T^{2} + T^{3} \)
$11$ \( T^{3} \)
$13$ \( T^{3} \)
$17$ \( 33 + 41 T + 12 T^{2} + T^{3} \)
$19$ \( ( 1 + T )^{3} \)
$23$ \( -261 - 57 T + 4 T^{2} + T^{3} \)
$29$ \( 99 - 51 T + 2 T^{2} + T^{3} \)
$31$ \( -24 - 56 T - 2 T^{2} + T^{3} \)
$37$ \( -9 + 47 T - 15 T^{2} + T^{3} \)
$41$ \( 408 - 128 T - 2 T^{2} + T^{3} \)
$43$ \( 72 + 20 T - 12 T^{2} + T^{3} \)
$47$ \( 303 - 109 T - 3 T^{2} + T^{3} \)
$53$ \( 111 + T - 10 T^{2} + T^{3} \)
$59$ \( -297 + 147 T - 22 T^{2} + T^{3} \)
$61$ \( 24 + 80 T + 18 T^{2} + T^{3} \)
$67$ \( -17 - 13 T + 2 T^{2} + T^{3} \)
$71$ \( -216 + 96 T + 22 T^{2} + T^{3} \)
$73$ \( -967 - 99 T + 12 T^{2} + T^{3} \)
$79$ \( 312 - 32 T - 10 T^{2} + T^{3} \)
$83$ \( -216 - 60 T + 4 T^{2} + T^{3} \)
$89$ \( -648 - 228 T + 2 T^{2} + T^{3} \)
$97$ \( -72 + 20 T + 12 T^{2} + T^{3} \)
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