Properties

Label 6422.2.a.bd
Level $6422$
Weight $2$
Character orbit 6422.a
Self dual yes
Analytic conductor $51.280$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.316645497.1
Defining polynomial: \(x^{6} - 2 x^{5} - 12 x^{4} + 24 x^{3} + 13 x^{2} - 24 x + 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_{1} q^{3} + q^{4} + \beta_{2} q^{5} + \beta_{1} q^{6} -\beta_{3} q^{7} + q^{8} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})\) \( q + q^{2} + \beta_{1} q^{3} + q^{4} + \beta_{2} q^{5} + \beta_{1} q^{6} -\beta_{3} q^{7} + q^{8} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{9} + \beta_{2} q^{10} + \beta_{2} q^{11} + \beta_{1} q^{12} -\beta_{3} q^{14} + ( -2 + 2 \beta_{1} - \beta_{4} - \beta_{5} ) q^{15} + q^{16} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{17} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{18} - q^{19} + \beta_{2} q^{20} + ( 2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{21} + \beta_{2} q^{22} + ( 1 - \beta_{1} + \beta_{2} + \beta_{5} ) q^{23} + \beta_{1} q^{24} + ( 4 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{25} + ( -1 + 3 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{27} -\beta_{3} q^{28} + ( 3 + \beta_{5} ) q^{29} + ( -2 + 2 \beta_{1} - \beta_{4} - \beta_{5} ) q^{30} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{31} + q^{32} + ( -2 + 2 \beta_{1} - \beta_{4} - \beta_{5} ) q^{33} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{34} + ( \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{35} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{36} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{37} - q^{38} + \beta_{2} q^{40} + ( -2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{41} + ( 2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{42} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{43} + \beta_{2} q^{44} + ( 7 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{45} + ( 1 - \beta_{1} + \beta_{2} + \beta_{5} ) q^{46} + ( 3 + \beta_{2} ) q^{47} + \beta_{1} q^{48} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{49} + ( 4 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{50} + ( \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{51} + ( 3 - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{53} + ( -1 + 3 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{54} + ( 9 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{55} -\beta_{3} q^{56} -\beta_{1} q^{57} + ( 3 + \beta_{5} ) q^{58} + ( -1 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{59} + ( -2 + 2 \beta_{1} - \beta_{4} - \beta_{5} ) q^{60} + ( 1 + 3 \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{61} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{62} + ( 7 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{63} + q^{64} + ( -2 + 2 \beta_{1} - \beta_{4} - \beta_{5} ) q^{66} + ( -7 + 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{67} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{68} + ( -5 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{69} + ( \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{70} + ( 7 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{71} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{72} + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{73} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{74} + ( -2 + \beta_{1} - \beta_{2} - 3 \beta_{4} - 3 \beta_{5} ) q^{75} - q^{76} + ( \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{77} + ( 3 + 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{79} + \beta_{2} q^{80} + ( 9 - 3 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{81} + ( -2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{82} + ( -4 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} ) q^{83} + ( 2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{84} + ( 4 - 4 \beta_{1} - 6 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{85} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{86} + ( 1 + 3 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{87} + \beta_{2} q^{88} + ( 2 + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{89} + ( 7 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{90} + ( 1 - \beta_{1} + \beta_{2} + \beta_{5} ) q^{92} + ( -7 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{93} + ( 3 + \beta_{2} ) q^{94} -\beta_{2} q^{95} + \beta_{1} q^{96} + ( 2 \beta_{1} + 4 \beta_{3} + 4 \beta_{4} ) q^{97} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{98} + ( 7 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 6 q^{8} + 10 q^{9} + O(q^{10}) \) \( 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 6 q^{8} + 10 q^{9} + 2 q^{10} + 2 q^{11} + 2 q^{12} + q^{14} - 9 q^{15} + 6 q^{16} - 2 q^{17} + 10 q^{18} - 6 q^{19} + 2 q^{20} + q^{21} + 2 q^{22} + 8 q^{23} + 2 q^{24} + 16 q^{25} - 4 q^{27} + q^{28} + 20 q^{29} - 9 q^{30} + 3 q^{31} + 6 q^{32} - 9 q^{33} - 2 q^{34} + 7 q^{35} + 10 q^{36} - 9 q^{37} - 6 q^{38} + 2 q^{40} + 3 q^{41} + q^{42} + 13 q^{43} + 2 q^{44} + 38 q^{45} + 8 q^{46} + 20 q^{47} + 2 q^{48} - 7 q^{49} + 16 q^{50} + 2 q^{51} + 25 q^{53} - 4 q^{54} + 46 q^{55} + q^{56} - 2 q^{57} + 20 q^{58} - 9 q^{60} + 6 q^{61} + 3 q^{62} + 46 q^{63} + 6 q^{64} - 9 q^{66} - 32 q^{67} - 2 q^{68} - 29 q^{69} + 7 q^{70} + 39 q^{71} + 10 q^{72} + 7 q^{73} - 9 q^{74} - 15 q^{75} - 6 q^{76} + 7 q^{77} + 18 q^{79} + 2 q^{80} + 54 q^{81} + 3 q^{82} - 7 q^{83} + q^{84} - 2 q^{85} + 13 q^{86} + 12 q^{87} + 2 q^{88} + 9 q^{89} + 38 q^{90} + 8 q^{92} - 47 q^{93} + 20 q^{94} - 2 q^{95} + 2 q^{96} - 4 q^{97} - 7 q^{98} + 38 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 12 x^{4} + 24 x^{3} + 13 x^{2} - 24 x + 5\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 3 \nu^{4} - 12 \nu^{3} + 36 \nu^{2} + 10 \nu - 31 \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{5} - 3 \nu^{4} - 24 \nu^{3} + 33 \nu^{2} + 29 \nu - 14 \)\()/6\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{5} + 3 \nu^{4} + 27 \nu^{3} - 33 \nu^{2} - 56 \nu + 17 \)\()/6\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} - \nu^{4} - 13 \nu^{3} + 12 \nu^{2} + 25 \nu - 8 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(2 \beta_{4} + 2 \beta_{3} + 9 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(13 \beta_{5} + 13 \beta_{4} - 11 \beta_{3} + 9 \beta_{2} - 3 \beta_{1} + 36\)
\(\nu^{5}\)\(=\)\(3 \beta_{5} + 27 \beta_{4} + 27 \beta_{3} - 3 \beta_{2} + 89 \beta_{1} - 17\)

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
−3.28714
−1.13360
0.260281
0.724358
2.19765
3.23845
1.00000 −3.28714 1.00000 2.88198 −3.28714 3.02408 1.00000 7.80527 2.88198
1.2 1.00000 −1.13360 1.00000 2.43014 −1.13360 −3.63269 1.00000 −1.71494 2.43014
1.3 1.00000 0.260281 1.00000 −4.36375 0.260281 0.775135 1.00000 −2.93225 −4.36375
1.4 1.00000 0.724358 1.00000 −1.67578 0.724358 −2.46211 1.00000 −2.47531 −1.67578
1.5 1.00000 2.19765 1.00000 3.12703 2.19765 2.17944 1.00000 1.82965 3.12703
1.6 1.00000 3.23845 1.00000 −0.399620 3.23845 1.11615 1.00000 7.48758 −0.399620
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.bd 6
13.b even 2 1 6422.2.a.bc 6
13.c even 3 2 494.2.g.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.g.f 12 13.c even 3 2
6422.2.a.bc 6 13.b even 2 1
6422.2.a.bd 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\):

\( T_{3}^{6} - 2 T_{3}^{5} - 12 T_{3}^{4} + 24 T_{3}^{3} + 13 T_{3}^{2} - 24 T_{3} + 5 \)
\( T_{5}^{6} - 2 T_{5}^{5} - 21 T_{5}^{4} + 51 T_{5}^{3} + 64 T_{5}^{2} - 144 T_{5} - 64 \)
\( T_{7}^{6} - T_{7}^{5} - 17 T_{7}^{4} + 25 T_{7}^{3} + 57 T_{7}^{2} - 117 T_{7} + 51 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{6} \)
$3$ \( 5 - 24 T + 13 T^{2} + 24 T^{3} - 12 T^{4} - 2 T^{5} + T^{6} \)
$5$ \( -64 - 144 T + 64 T^{2} + 51 T^{3} - 21 T^{4} - 2 T^{5} + T^{6} \)
$7$ \( 51 - 117 T + 57 T^{2} + 25 T^{3} - 17 T^{4} - T^{5} + T^{6} \)
$11$ \( -64 - 144 T + 64 T^{2} + 51 T^{3} - 21 T^{4} - 2 T^{5} + T^{6} \)
$13$ \( T^{6} \)
$17$ \( 120 - 1368 T + 1614 T^{2} - 49 T^{3} - 79 T^{4} + 2 T^{5} + T^{6} \)
$19$ \( ( 1 + T )^{6} \)
$23$ \( -137 - 274 T + 7 T^{2} + 152 T^{3} - 16 T^{4} - 8 T^{5} + T^{6} \)
$29$ \( 37 - 274 T + 605 T^{2} - 446 T^{3} + 142 T^{4} - 20 T^{5} + T^{6} \)
$31$ \( -16376 - 6272 T + 2466 T^{2} + 331 T^{3} - 100 T^{4} - 3 T^{5} + T^{6} \)
$37$ \( -43096 + 13276 T + 3024 T^{2} - 741 T^{3} - 97 T^{4} + 9 T^{5} + T^{6} \)
$41$ \( -296 - 2288 T + 1914 T^{2} + 277 T^{3} - 100 T^{4} - 3 T^{5} + T^{6} \)
$43$ \( 200 + 92 T - 608 T^{2} + 237 T^{3} + 19 T^{4} - 13 T^{5} + T^{6} \)
$47$ \( -919 + 849 T + 226 T^{2} - 417 T^{3} + 144 T^{4} - 20 T^{5} + T^{6} \)
$53$ \( -99821 + 82519 T - 23641 T^{2} + 2333 T^{3} + 79 T^{4} - 25 T^{5} + T^{6} \)
$59$ \( 42513 - 60360 T + 12183 T^{2} + 578 T^{3} - 220 T^{4} + T^{6} \)
$61$ \( 184136 - 120784 T + 12294 T^{2} + 1743 T^{3} - 235 T^{4} - 6 T^{5} + T^{6} \)
$67$ \( 172488 - 50352 T - 25398 T^{2} - 1631 T^{3} + 243 T^{4} + 32 T^{5} + T^{6} \)
$71$ \( -116072 + 69488 T - 5118 T^{2} - 2497 T^{3} + 533 T^{4} - 39 T^{5} + T^{6} \)
$73$ \( 3033 + 6624 T + 3126 T^{2} + 139 T^{3} - 116 T^{4} - 7 T^{5} + T^{6} \)
$79$ \( -87032 + 78176 T - 25050 T^{2} + 3209 T^{3} - 61 T^{4} - 18 T^{5} + T^{6} \)
$83$ \( -92760 - 10068 T + 20052 T^{2} - 853 T^{3} - 266 T^{4} + 7 T^{5} + T^{6} \)
$89$ \( -648 - 108 T + 666 T^{2} - 25 T^{3} - 106 T^{4} - 9 T^{5} + T^{6} \)
$97$ \( 1150912 + 706944 T + 72944 T^{2} - 3632 T^{3} - 552 T^{4} + 4 T^{5} + T^{6} \)
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