Properties

Label 6422.2.a.bb
Level $6422$
Weight $2$
Character orbit 6422.a
Self dual yes
Analytic conductor $51.280$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.10273.1
Defining polynomial: \(x^{4} - 2 x^{3} - 5 x^{2} + x + 2\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 5 x^{2} + x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 4 \nu \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 3 \nu^{2} - 2 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + 2 \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(-2 \beta_{3} + 3 \beta_{2} + 8 \beta_{1} + 6\)

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
3.35017
0.673533
−0.641043
−1.38266
1.00000 −3.35017 1.00000 3.12030 −3.35017 1.75319 1.00000 8.22366 3.12030
1.2 1.00000 −0.673533 1.00000 0.0759953 −0.673533 −3.29588 1.00000 −2.54635 0.0759953
1.3 1.00000 0.641043 1.00000 −3.42689 0.641043 1.47887 1.00000 −2.58906 −3.42689
1.4 1.00000 1.38266 1.00000 1.23060 1.38266 −0.936179 1.00000 −1.08824 1.23060
\(n\): 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.bb 4
13.b even 2 1 6422.2.a.z 4
13.e even 6 2 494.2.g.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.g.e 8 13.e even 6 2
6422.2.a.z 4 13.b even 2 1
6422.2.a.bb 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\):

\( T_{3}^{4} + 2 T_{3}^{3} - 5 T_{3}^{2} - T_{3} + 2 \)
\( T_{5}^{4} - T_{5}^{3} - 11 T_{5}^{2} + 14 T_{5} - 1 \)
\( T_{7}^{4} + T_{7}^{3} - 8 T_{7}^{2} + T_{7} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( 2 - T - 5 T^{2} + 2 T^{3} + T^{4} \)
$5$ \( -1 + 14 T - 11 T^{2} - T^{3} + T^{4} \)
$7$ \( 8 + T - 8 T^{2} + T^{3} + T^{4} \)
$11$ \( 6 - 17 T - 9 T^{2} + 2 T^{3} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( -72 + 104 T - 34 T^{2} - T^{3} + T^{4} \)
$19$ \( ( -1 + T )^{4} \)
$23$ \( 12 + 35 T + 31 T^{2} + 10 T^{3} + T^{4} \)
$29$ \( 1129 - 24 T - 69 T^{2} + T^{3} + T^{4} \)
$31$ \( 8 + 45 T + 38 T^{2} + 11 T^{3} + T^{4} \)
$37$ \( -392 + 196 T - 4 T^{2} - 9 T^{3} + T^{4} \)
$41$ \( -47 - 111 T - 25 T^{2} + 4 T^{3} + T^{4} \)
$43$ \( 128 - 167 T + 31 T^{2} + 15 T^{3} + T^{4} \)
$47$ \( -11166 - 1675 T + 94 T^{2} + 25 T^{3} + T^{4} \)
$53$ \( -1 - 19 T + 5 T^{2} + 8 T^{3} + T^{4} \)
$59$ \( -2866 + 133 T + 223 T^{2} + 28 T^{3} + T^{4} \)
$61$ \( 73 - 170 T + 33 T^{2} + 15 T^{3} + T^{4} \)
$67$ \( -1168 - 952 T - 164 T^{2} + T^{4} \)
$71$ \( 822 - 119 T - 135 T^{2} - T^{3} + T^{4} \)
$73$ \( -2117 - 1094 T - 124 T^{2} + 6 T^{3} + T^{4} \)
$79$ \( -5294 - 2819 T - 213 T^{2} + 12 T^{3} + T^{4} \)
$83$ \( -538 + 17 T + 78 T^{2} - 17 T^{3} + T^{4} \)
$89$ \( -1250 + 875 T - 100 T^{2} - 15 T^{3} + T^{4} \)
$97$ \( 8752 - 1408 T - 300 T^{2} + 2 T^{3} + T^{4} \)
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