Properties

Label 6422.2.a.bf
Level $6422$
Weight $2$
Character orbit 6422.a
Self dual yes
Analytic conductor $51.280$
Analytic rank $0$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 2 x^{6} - 16 x^{5} + 29 x^{4} + 60 x^{3} - 79 x^{2} - 47 x - 2\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_{1} q^{3} + q^{4} -\beta_{2} q^{5} + \beta_{1} q^{6} -\beta_{5} q^{7} + q^{8} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + q^{2} + \beta_{1} q^{3} + q^{4} -\beta_{2} q^{5} + \beta_{1} q^{6} -\beta_{5} q^{7} + q^{8} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} -\beta_{2} q^{10} + ( 1 + \beta_{6} ) q^{11} + \beta_{1} q^{12} -\beta_{5} q^{14} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{15} + q^{16} + ( 2 + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{17} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{18} + q^{19} -\beta_{2} q^{20} + ( -\beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{21} + ( 1 + \beta_{6} ) q^{22} + ( \beta_{1} - \beta_{3} - \beta_{6} ) q^{23} + \beta_{1} q^{24} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{25} + ( -1 + 3 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{27} -\beta_{5} q^{28} + ( -2 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{29} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{30} + ( 3 + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} ) q^{31} + q^{32} + ( 2 + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{33} + ( 2 + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{34} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{35} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{36} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{37} + q^{38} -\beta_{2} q^{40} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{5} ) q^{41} + ( -\beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{42} + ( 5 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} ) q^{43} + ( 1 + \beta_{6} ) q^{44} + ( -8 + 2 \beta_{1} - 4 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{45} + ( \beta_{1} - \beta_{3} - \beta_{6} ) q^{46} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{47} + \beta_{1} q^{48} + ( 3 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{6} ) q^{49} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{50} + ( -3 + 4 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{51} + ( 4 - \beta_{2} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{53} + ( -1 + 3 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{54} + ( 3 - 3 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{55} -\beta_{5} q^{56} + \beta_{1} q^{57} + ( -2 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{58} + ( -1 + 3 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} ) q^{59} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{60} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{61} + ( 3 + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} ) q^{62} + ( -2 - 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{63} + q^{64} + ( 2 + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{66} + ( 5 - 3 \beta_{1} - \beta_{3} ) q^{67} + ( 2 + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{68} + ( 7 + \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} ) q^{69} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{70} + ( 2 + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{71} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{72} + ( 3 + \beta_{2} - 3 \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{73} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{74} + ( -3 + 3 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{75} + q^{76} + ( -4 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{5} + 2 \beta_{6} ) q^{77} + ( -2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{79} -\beta_{2} q^{80} + ( 9 + 4 \beta_{2} - \beta_{3} + \beta_{4} + 4 \beta_{5} - 2 \beta_{6} ) q^{81} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{5} ) q^{82} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{83} + ( -\beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{84} + ( -6 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{85} + ( 5 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} ) q^{86} + ( -1 - 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{4} - 2 \beta_{5} ) q^{87} + ( 1 + \beta_{6} ) q^{88} + ( -5 + 2 \beta_{1} - 4 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} ) q^{89} + ( -8 + 2 \beta_{1} - 4 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{90} + ( \beta_{1} - \beta_{3} - \beta_{6} ) q^{92} + ( 1 + 3 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{93} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{94} -\beta_{2} q^{95} + \beta_{1} q^{96} + ( -5 - 2 \beta_{3} + 2 \beta_{5} - \beta_{6} ) q^{97} + ( 3 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{6} ) q^{98} + ( -4 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 7q^{2} + 2q^{3} + 7q^{4} + 2q^{5} + 2q^{6} + q^{7} + 7q^{8} + 15q^{9} + O(q^{10}) \) \( 7q + 7q^{2} + 2q^{3} + 7q^{4} + 2q^{5} + 2q^{6} + q^{7} + 7q^{8} + 15q^{9} + 2q^{10} + 5q^{11} + 2q^{12} + q^{14} + 13q^{15} + 7q^{16} + 16q^{17} + 15q^{18} + 7q^{19} + 2q^{20} - 3q^{21} + 5q^{22} + 3q^{23} + 2q^{24} + 7q^{25} + 5q^{27} + q^{28} - 7q^{29} + 13q^{30} + 11q^{31} + 7q^{32} + 6q^{33} + 16q^{34} + q^{35} + 15q^{36} - 3q^{37} + 7q^{38} + 2q^{40} + 15q^{41} - 3q^{42} + 23q^{43} + 5q^{44} - 38q^{45} + 3q^{46} - q^{47} + 2q^{48} + 14q^{49} + 7q^{50} - 16q^{51} + 21q^{53} + 5q^{54} + 8q^{55} + q^{56} + 2q^{57} - 7q^{58} + 8q^{59} + 13q^{60} - 6q^{61} + 11q^{62} - 22q^{63} + 7q^{64} + 6q^{66} + 28q^{67} + 16q^{68} + 48q^{69} + q^{70} + 4q^{71} + 15q^{72} + 12q^{73} - 3q^{74} + 5q^{75} + 7q^{76} - 26q^{77} - 4q^{79} + 2q^{80} + 51q^{81} + 15q^{82} + 13q^{83} - 3q^{84} - 39q^{85} + 23q^{86} - 16q^{87} + 5q^{88} - 15q^{89} - 38q^{90} + 3q^{92} + 6q^{93} - q^{94} + 2q^{95} + 2q^{96} - 37q^{97} + 14q^{98} - 15q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 2 x^{6} - 16 x^{5} + 29 x^{4} + 60 x^{3} - 79 x^{2} - 47 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - \nu^{5} - 15 \nu^{4} + 14 \nu^{3} + 50 \nu^{2} - 37 \nu - 30 \)\()/10\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} + \nu^{5} + 15 \nu^{4} - 14 \nu^{3} - 40 \nu^{2} + 27 \nu - 20 \)\()/10\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{6} + \nu^{5} + 17 \nu^{4} - 18 \nu^{3} - 66 \nu^{2} + 71 \nu + 16 \)\()/10\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} + 2 \nu^{5} + 16 \nu^{4} - 28 \nu^{3} - 57 \nu^{2} + 71 \nu + 24 \)\()/5\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} + 3 \nu^{5} + 15 \nu^{4} - 43 \nu^{3} - 48 \nu^{2} + 116 \nu + 27 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(-\beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 9 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-2 \beta_{6} + 4 \beta_{5} + \beta_{4} + 8 \beta_{3} + 13 \beta_{2} + 9 \beta_{1} + 45\)
\(\nu^{5}\)\(=\)\(-12 \beta_{6} + 29 \beta_{5} - 29 \beta_{4} - \beta_{3} + 4 \beta_{2} + 90 \beta_{1} - 18\)
\(\nu^{6}\)\(=\)\(-28 \beta_{6} + 61 \beta_{5} + 14 \beta_{4} + 69 \beta_{3} + 159 \beta_{2} + 86 \beta_{1} + 451\)

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
−3.29705
−1.95679
−0.431028
−0.0462763
1.77366
2.64643
3.31105
1.00000 −3.29705 1.00000 −3.53833 −3.29705 −0.150781 1.00000 7.87054 −3.53833
1.2 1.00000 −1.95679 1.00000 0.613661 −1.95679 0.465830 1.00000 0.829012 0.613661
1.3 1.00000 −0.431028 1.00000 0.638027 −0.431028 2.88689 1.00000 −2.81422 0.638027
1.4 1.00000 −0.0462763 1.00000 2.81822 −0.0462763 −4.11903 1.00000 −2.99786 2.81822
1.5 1.00000 1.77366 1.00000 −0.491653 1.77366 4.65991 1.00000 0.145874 −0.491653
1.6 1.00000 2.64643 1.00000 4.02894 2.64643 1.07639 1.00000 4.00358 4.02894
1.7 1.00000 3.31105 1.00000 −2.06886 3.31105 −3.81920 1.00000 7.96307 −2.06886
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.bf 7
13.b even 2 1 6422.2.a.be 7
13.d odd 4 2 494.2.d.c 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.d.c 14 13.d odd 4 2
6422.2.a.be 7 13.b even 2 1
6422.2.a.bf 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\):

\( T_{3}^{7} - 2 T_{3}^{6} - 16 T_{3}^{5} + 29 T_{3}^{4} + 60 T_{3}^{3} - 79 T_{3}^{2} - 47 T_{3} - 2 \)
\( T_{5}^{7} - 2 T_{5}^{6} - 19 T_{5}^{5} + 29 T_{5}^{4} + 77 T_{5}^{3} - 70 T_{5}^{2} - 16 T_{5} + 16 \)
\( T_{7}^{7} - T_{7}^{6} - 31 T_{7}^{5} + 31 T_{7}^{4} + 220 T_{7}^{3} - 300 T_{7}^{2} + 56 T_{7} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{7} \)
$3$ \( -2 - 47 T - 79 T^{2} + 60 T^{3} + 29 T^{4} - 16 T^{5} - 2 T^{6} + T^{7} \)
$5$ \( 16 - 16 T - 70 T^{2} + 77 T^{3} + 29 T^{4} - 19 T^{5} - 2 T^{6} + T^{7} \)
$7$ \( 16 + 56 T - 300 T^{2} + 220 T^{3} + 31 T^{4} - 31 T^{5} - T^{6} + T^{7} \)
$11$ \( 3328 - 1032 T - 1420 T^{2} + 318 T^{3} + 165 T^{4} - 34 T^{5} - 5 T^{6} + T^{7} \)
$13$ \( T^{7} \)
$17$ \( 22622 + 10479 T - 4089 T^{2} - 1434 T^{3} + 395 T^{4} + 42 T^{5} - 16 T^{6} + T^{7} \)
$19$ \( ( -1 + T )^{7} \)
$23$ \( 14944 - 6652 T - 3783 T^{2} + 1748 T^{3} + 164 T^{4} - 87 T^{5} - 3 T^{6} + T^{7} \)
$29$ \( -1352 + 3250 T + 6849 T^{2} + 480 T^{3} - 572 T^{4} - 73 T^{5} + 7 T^{6} + T^{7} \)
$31$ \( 173056 + 18592 T - 27016 T^{2} + 152 T^{3} + 1215 T^{4} - 88 T^{5} - 11 T^{6} + T^{7} \)
$37$ \( 36352 - 15488 T - 11296 T^{2} + 4936 T^{3} + 12 T^{4} - 134 T^{5} + 3 T^{6} + T^{7} \)
$41$ \( 4160 - 1088 T - 4748 T^{2} - 616 T^{3} + 603 T^{4} - 6 T^{5} - 15 T^{6} + T^{7} \)
$43$ \( 3328 - 19904 T + 27232 T^{2} - 11288 T^{3} + 1236 T^{4} + 94 T^{5} - 23 T^{6} + T^{7} \)
$47$ \( -107968 - 49664 T + 29856 T^{2} + 8328 T^{3} - 584 T^{4} - 194 T^{5} + T^{6} + T^{7} \)
$53$ \( 100256 + 54468 T - 38157 T^{2} - 5558 T^{3} + 2074 T^{4} - 9 T^{5} - 21 T^{6} + T^{7} \)
$59$ \( -2276864 - 1101312 T - 40960 T^{2} + 29136 T^{3} + 1422 T^{4} - 283 T^{5} - 8 T^{6} + T^{7} \)
$61$ \( -58240 - 39552 T + 10080 T^{2} + 5648 T^{3} - 344 T^{4} - 176 T^{5} + 6 T^{6} + T^{7} \)
$67$ \( -848704 + 317904 T + 16336 T^{2} - 16924 T^{3} + 1246 T^{4} + 185 T^{5} - 28 T^{6} + T^{7} \)
$71$ \( -78784 + 70724 T - 12748 T^{2} - 5099 T^{3} + 2095 T^{4} - 205 T^{5} - 4 T^{6} + T^{7} \)
$73$ \( -690304 - 709888 T - 140552 T^{2} + 27508 T^{3} + 3100 T^{4} - 319 T^{5} - 12 T^{6} + T^{7} \)
$79$ \( -2882560 - 621376 T + 96672 T^{2} + 22496 T^{3} - 1080 T^{4} - 264 T^{5} + 4 T^{6} + T^{7} \)
$83$ \( -6258400 + 3018568 T - 439620 T^{2} - 1534 T^{3} + 5237 T^{4} - 278 T^{5} - 13 T^{6} + T^{7} \)
$89$ \( -13514752 - 4080784 T + 640208 T^{2} + 74350 T^{3} - 5965 T^{4} - 476 T^{5} + 15 T^{6} + T^{7} \)
$97$ \( -84736 + 128984 T - 27996 T^{2} - 10370 T^{3} + 1041 T^{4} + 436 T^{5} + 37 T^{6} + T^{7} \)
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