Properties

Label 42.4.f.a
Level $42$
Weight $4$
Character orbit 42.f
Analytic conductor $2.478$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 42.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.47808022024\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 2 x^{15} - x^{14} - 2 x^{13} + 9 x^{12} - 24 x^{11} + 714 x^{10} - 1940 x^{9} - 2834 x^{8} - 17460 x^{7} + 57834 x^{6} - 17496 x^{5} + 59049 x^{4} - 118098 x^{3} - 531441 x^{2} - 9565938 x + 43046721\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + \beta_{1} q^{3} + ( 4 + 4 \beta_{5} ) q^{4} + ( -\beta_{2} - 2 \beta_{7} - \beta_{11} - \beta_{15} ) q^{5} + ( -\beta_{4} + \beta_{5} ) q^{6} + ( 5 - \beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{14} ) q^{7} + ( -4 \beta_{2} - 4 \beta_{7} ) q^{8} + ( -1 + 2 \beta_{2} - \beta_{4} - 3 \beta_{5} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} + \beta_{1} q^{3} + ( 4 + 4 \beta_{5} ) q^{4} + ( -\beta_{2} - 2 \beta_{7} - \beta_{11} - \beta_{15} ) q^{5} + ( -\beta_{4} + \beta_{5} ) q^{6} + ( 5 - \beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{14} ) q^{7} + ( -4 \beta_{2} - 4 \beta_{7} ) q^{8} + ( -1 + 2 \beta_{2} - \beta_{4} - 3 \beta_{5} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{9} + ( -1 + 2 \beta_{1} + 2 \beta_{5} + \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{12} + \beta_{14} ) q^{10} + ( 1 - 2 \beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 3 \beta_{7} - 3 \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} - \beta_{14} - 2 \beta_{15} ) q^{11} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{12} + ( -7 - 2 \beta_{3} + 2 \beta_{4} - 16 \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} ) q^{13} + ( -2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} - \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{14} + ( 1 + 2 \beta_{1} + 14 \beta_{2} - \beta_{3} + 2 \beta_{6} + 15 \beta_{7} - 4 \beta_{10} + 2 \beta_{11} + 2 \beta_{13} + \beta_{15} ) q^{15} + 16 \beta_{5} q^{16} + ( 1 - 9 \beta_{1} + 4 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} - \beta_{9} + 6 \beta_{10} + 4 \beta_{11} + \beta_{12} - 2 \beta_{13} + 3 \beta_{14} ) q^{17} + ( -6 + 2 \beta_{2} - 6 \beta_{5} + 2 \beta_{6} + 6 \beta_{7} - 3 \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + \beta_{12} - 2 \beta_{13} + 4 \beta_{15} ) q^{18} + ( -28 - 7 \beta_{1} + 7 \beta_{3} + 2 \beta_{4} - 15 \beta_{5} + 4 \beta_{6} - 4 \beta_{8} + 3 \beta_{9} - 7 \beta_{10} + 4 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} ) q^{19} + ( -4 \beta_{7} - 4 \beta_{15} ) q^{20} + ( -34 + 7 \beta_{1} + 24 \beta_{2} - \beta_{3} - \beta_{4} - 10 \beta_{5} - \beta_{6} + 12 \beta_{7} + 8 \beta_{8} + 3 \beta_{10} - 5 \beta_{12} + 2 \beta_{14} + 3 \beta_{15} ) q^{21} + ( 4 - 4 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + \beta_{6} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{12} + 3 \beta_{13} + 6 \beta_{14} ) q^{22} + ( 4 - \beta_{1} - 6 \beta_{2} - \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 3 \beta_{8} - 4 \beta_{9} + \beta_{10} + 4 \beta_{12} - 4 \beta_{13} + 4 \beta_{14} ) q^{23} + ( -4 - 4 \beta_{14} ) q^{24} + ( -25 + 10 \beta_{1} - 20 \beta_{3} - 8 \beta_{4} - 21 \beta_{5} - 3 \beta_{6} - 8 \beta_{8} - 4 \beta_{9} + 3 \beta_{10} + \beta_{12} + 3 \beta_{13} + 4 \beta_{14} ) q^{25} + ( 1 - 6 \beta_{1} + 5 \beta_{2} + 6 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} + 10 \beta_{7} - 3 \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} - 3 \beta_{14} - 2 \beta_{15} ) q^{26} + ( 15 + 6 \beta_{3} + 3 \beta_{4} + 27 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} + 7 \beta_{8} + 9 \beta_{9} - \beta_{10} - 9 \beta_{12} + 3 \beta_{13} + 3 \beta_{15} ) q^{27} + ( 16 + 4 \beta_{3} + 4 \beta_{4} + 20 \beta_{5} + 4 \beta_{8} + 4 \beta_{9} - 4 \beta_{14} ) q^{28} + ( -2 \beta_{1} - 26 \beta_{2} + \beta_{3} + 5 \beta_{4} - 5 \beta_{5} + 5 \beta_{6} - 27 \beta_{7} + \beta_{10} - 2 \beta_{11} + 5 \beta_{13} - 10 \beta_{14} - \beta_{15} ) q^{29} + ( -2 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 44 \beta_{5} - 5 \beta_{6} - 13 \beta_{8} - 3 \beta_{9} - 10 \beta_{10} + 5 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} ) q^{30} + ( 28 + 6 \beta_{1} - 37 \beta_{5} - 4 \beta_{6} - \beta_{8} + \beta_{9} + 2 \beta_{10} + 5 \beta_{12} - 4 \beta_{13} - 9 \beta_{14} ) q^{31} -16 \beta_{7} q^{32} + ( 112 + 3 \beta_{1} - 18 \beta_{2} - 3 \beta_{3} - 6 \beta_{4} + 59 \beta_{5} - 5 \beta_{6} - 36 \beta_{7} + 11 \beta_{8} - 7 \beta_{9} + 6 \beta_{10} + 3 \beta_{11} - 5 \beta_{12} + 7 \beta_{13} + 6 \beta_{14} + 3 \beta_{15} ) q^{33} + ( -14 + 4 \beta_{3} + 5 \beta_{4} - 33 \beta_{5} - \beta_{6} + 14 \beta_{8} + 6 \beta_{9} + 4 \beta_{10} - 6 \beta_{12} + 7 \beta_{13} ) q^{34} + ( -2 + 7 \beta_{1} - 56 \beta_{2} - 2 \beta_{3} - 7 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - 37 \beta_{7} + \beta_{8} + 2 \beta_{9} - 4 \beta_{10} - 8 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} + 8 \beta_{14} - 11 \beta_{15} ) q^{35} + ( 8 + 4 \beta_{2} + 4 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 4 \beta_{8} + 4 \beta_{9} - 8 \beta_{10} - 8 \beta_{11} + 4 \beta_{12} - 8 \beta_{14} - 4 \beta_{15} ) q^{36} + ( -7 + 7 \beta_{1} + 7 \beta_{3} - 7 \beta_{4} + 122 \beta_{5} - 7 \beta_{6} - 7 \beta_{8} - 14 \beta_{9} + 7 \beta_{10} + 7 \beta_{12} - 14 \beta_{13} - 7 \beta_{14} ) q^{37} + ( 6 \beta_{1} + 28 \beta_{2} + 7 \beta_{7} - 2 \beta_{8} - 4 \beta_{10} - 14 \beta_{11} ) q^{38} + ( 77 - 6 \beta_{1} - 6 \beta_{2} + 12 \beta_{3} + 4 \beta_{4} + 75 \beta_{5} - \beta_{6} + 30 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} + \beta_{10} - 6 \beta_{11} + 3 \beta_{12} + \beta_{13} - 2 \beta_{14} - 12 \beta_{15} ) q^{39} + ( -12 + 8 \beta_{1} - 8 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} + 4 \beta_{8} - 4 \beta_{9} + 8 \beta_{10} - 4 \beta_{12} + 4 \beta_{13} + 4 \beta_{14} ) q^{40} + ( -16 + 6 \beta_{3} - 24 \beta_{4} - 8 \beta_{5} + 8 \beta_{6} + 8 \beta_{7} + 12 \beta_{8} + 16 \beta_{9} - 2 \beta_{10} - 16 \beta_{12} + 8 \beta_{13} + 8 \beta_{15} ) q^{41} + ( -57 - 4 \beta_{1} + 26 \beta_{2} + 2 \beta_{3} + \beta_{4} - 87 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} + \beta_{9} + 6 \beta_{10} + 2 \beta_{11} + 5 \beta_{12} - 8 \beta_{13} - 3 \beta_{14} - 8 \beta_{15} ) q^{42} + ( 119 + 8 \beta_{1} - 4 \beta_{3} - 10 \beta_{4} + 10 \beta_{5} + 13 \beta_{6} - 3 \beta_{8} + 3 \beta_{9} + \beta_{10} + 3 \beta_{12} + 10 \beta_{13} + 20 \beta_{14} ) q^{43} + ( 4 + 8 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 12 \beta_{8} - 4 \beta_{9} - 8 \beta_{10} + 4 \beta_{11} + 4 \beta_{12} - 4 \beta_{13} + 4 \beta_{14} - 4 \beta_{15} ) q^{44} + ( -96 - 6 \beta_{1} - 132 \beta_{2} + 105 \beta_{5} - 3 \beta_{6} - 69 \beta_{7} + \beta_{8} + 5 \beta_{10} - 6 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} + 9 \beta_{14} ) q^{45} + ( 21 - 16 \beta_{1} + 32 \beta_{3} + 2 \beta_{4} + 20 \beta_{5} + 17 \beta_{8} + \beta_{9} - \beta_{12} - \beta_{14} ) q^{46} + ( 27 \beta_{1} + 56 \beta_{2} - 27 \beta_{3} + 112 \beta_{7} + 9 \beta_{8} - 9 \beta_{10} + 14 \beta_{11} + 14 \beta_{15} ) q^{47} -16 \beta_{3} q^{48} + ( 74 - 17 \beta_{1} - 12 \beta_{3} + 20 \beta_{4} + 47 \beta_{5} - 3 \beta_{6} - 19 \beta_{8} + 6 \beta_{9} - 21 \beta_{10} - 14 \beta_{13} - 23 \beta_{14} ) q^{49} + ( 20 \beta_{1} + 37 \beta_{2} - 10 \beta_{3} + 7 \beta_{4} - 7 \beta_{5} + 7 \beta_{6} + 43 \beta_{7} - 10 \beta_{10} + 12 \beta_{11} + 7 \beta_{13} - 14 \beta_{14} + 6 \beta_{15} ) q^{50} + ( 9 - 4 \beta_{1} + 17 \beta_{2} - 4 \beta_{3} + 9 \beta_{4} - 194 \beta_{5} - 11 \beta_{6} + 11 \beta_{8} - 14 \beta_{9} + 25 \beta_{10} + 11 \beta_{11} + 11 \beta_{12} - 14 \beta_{13} + 9 \beta_{14} - 11 \beta_{15} ) q^{51} + ( 36 - 8 \beta_{1} - 28 \beta_{5} + 4 \beta_{6} - 4 \beta_{10} - 4 \beta_{12} + 4 \beta_{13} + 8 \beta_{14} ) q^{52} + ( 6 + 11 \beta_{2} + 12 \beta_{4} - 63 \beta_{7} - 6 \beta_{8} - 6 \beta_{9} + 11 \beta_{11} + 6 \beta_{12} - 6 \beta_{14} + 22 \beta_{15} ) q^{53} + ( 16 - 12 \beta_{1} - 27 \beta_{2} + 12 \beta_{3} + 8 \beta_{5} + 4 \beta_{6} - 54 \beta_{7} - 22 \beta_{8} + 2 \beta_{9} + 12 \beta_{10} - 6 \beta_{11} + 4 \beta_{12} - 2 \beta_{13} - 6 \beta_{15} ) q^{54} + ( -52 - 12 \beta_{3} + 21 \beta_{4} - 125 \beta_{5} + 8 \beta_{6} + 3 \beta_{8} + 13 \beta_{9} - 5 \beta_{10} - 13 \beta_{12} + 5 \beta_{13} ) q^{55} + ( -4 - 16 \beta_{1} - 20 \beta_{2} + 8 \beta_{3} - 8 \beta_{4} - 24 \beta_{7} + 4 \beta_{8} + 4 \beta_{9} + 8 \beta_{10} - 4 \beta_{12} + 4 \beta_{14} + 8 \beta_{15} ) q^{56} + ( -156 - 28 \beta_{1} - 45 \beta_{2} + 14 \beta_{3} - 7 \beta_{4} + 7 \beta_{5} + 12 \beta_{6} - 42 \beta_{7} + 4 \beta_{8} - 4 \beta_{9} + 39 \beta_{10} + 6 \beta_{11} - 4 \beta_{12} + 16 \beta_{13} + 14 \beta_{14} + 3 \beta_{15} ) q^{57} + ( 2 - 18 \beta_{1} - 18 \beta_{3} + 2 \beta_{4} + 98 \beta_{5} + 2 \beta_{6} - 18 \beta_{8} - 18 \beta_{10} - 2 \beta_{12} + 2 \beta_{14} ) q^{58} + ( -1 + 42 \beta_{1} + 257 \beta_{2} - 2 \beta_{5} + 2 \beta_{6} + 131 \beta_{7} - 13 \beta_{8} + \beta_{9} - 28 \beta_{10} + 5 \beta_{11} - \beta_{12} + 2 \beta_{13} - 3 \beta_{14} ) q^{59} + ( 4 + 4 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} + 4 \beta_{5} + 60 \beta_{7} + 8 \beta_{8} - 8 \beta_{9} + 4 \beta_{11} + 8 \beta_{12} + 8 \beta_{15} ) q^{60} + ( -385 - 13 \beta_{1} + 13 \beta_{3} - 7 \beta_{4} - 189 \beta_{5} - 7 \beta_{6} + 16 \beta_{8} - 7 \beta_{9} + 5 \beta_{10} - 7 \beta_{12} + 7 \beta_{13} + 7 \beta_{14} ) q^{61} + ( -8 - 35 \beta_{2} - 24 \beta_{3} - 12 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 47 \beta_{7} + 24 \beta_{8} + 8 \beta_{9} + 8 \beta_{10} - 8 \beta_{12} + 4 \beta_{13} + 12 \beta_{15} ) q^{62} + ( -198 - 33 \beta_{1} + 103 \beta_{2} + 12 \beta_{3} + 10 \beta_{4} - 146 \beta_{5} - 10 \beta_{6} + 108 \beta_{7} - 35 \beta_{8} + 3 \beta_{9} - 29 \beta_{10} - 5 \beta_{11} + 5 \beta_{12} - 8 \beta_{13} + \beta_{14} - \beta_{15} ) q^{63} -64 q^{64} + ( -2 - 22 \beta_{1} - 16 \beta_{2} - 22 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 24 \beta_{8} + 2 \beta_{9} + 22 \beta_{10} - 10 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + 10 \beta_{15} ) q^{65} + ( -84 - 6 \beta_{1} - 102 \beta_{2} + 90 \beta_{5} - 12 \beta_{6} - 39 \beta_{7} - 13 \beta_{8} - 3 \beta_{9} - 14 \beta_{10} + 24 \beta_{11} + 9 \beta_{12} - 12 \beta_{13} + 6 \beta_{14} ) q^{66} + ( -85 - 11 \beta_{1} + 22 \beta_{3} - 85 \beta_{5} + 13 \beta_{6} - 15 \beta_{8} - 13 \beta_{10} + 13 \beta_{12} - 13 \beta_{13} ) q^{67} + ( -4 - 36 \beta_{1} + 16 \beta_{2} + 36 \beta_{3} - 12 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} + 32 \beta_{7} - 8 \beta_{8} + 4 \beta_{9} + 12 \beta_{10} + 16 \beta_{11} - 4 \beta_{12} - 4 \beta_{13} + 12 \beta_{14} + 16 \beta_{15} ) q^{68} + ( 1 + 105 \beta_{2} - 3 \beta_{4} + 5 \beta_{5} - 11 \beta_{6} - 132 \beta_{7} + 15 \beta_{8} - \beta_{9} + 8 \beta_{10} + \beta_{12} + 10 \beta_{13} - 27 \beta_{15} ) q^{69} + ( 103 + 50 \beta_{1} - 2 \beta_{3} - 8 \beta_{4} + 218 \beta_{5} - 6 \beta_{6} + 19 \beta_{8} - 15 \beta_{9} + 42 \beta_{10} - 7 \beta_{12} + 9 \beta_{14} ) q^{70} + ( -28 \beta_{1} + 102 \beta_{2} + 14 \beta_{3} + 10 \beta_{4} - 10 \beta_{5} + 10 \beta_{6} + 90 \beta_{7} + 14 \beta_{10} - 24 \beta_{11} + 10 \beta_{13} - 20 \beta_{14} - 12 \beta_{15} ) q^{71} + ( -8 \beta_{2} - 24 \beta_{5} + 4 \beta_{6} - 4 \beta_{8} - 4 \beta_{9} - 8 \beta_{11} - 4 \beta_{12} - 4 \beta_{13} + 8 \beta_{15} ) q^{72} + ( 136 - 41 \beta_{1} - 137 \beta_{5} - 3 \beta_{6} - 14 \beta_{8} - 5 \beta_{9} - 25 \beta_{10} - 2 \beta_{12} - 3 \beta_{13} - \beta_{14} ) q^{73} + ( 42 \beta_{1} - 14 \beta_{2} - 84 \beta_{3} - 115 \beta_{7} + 42 \beta_{8} - 14 \beta_{11} - 28 \beta_{15} ) q^{74} + ( 512 - 34 \beta_{1} - 48 \beta_{2} + 34 \beta_{3} + 30 \beta_{4} + 241 \beta_{5} + 20 \beta_{6} - 96 \beta_{7} - 13 \beta_{8} + 25 \beta_{9} - 52 \beta_{10} - 6 \beta_{11} + 20 \beta_{12} - 25 \beta_{13} - 30 \beta_{14} - 6 \beta_{15} ) q^{75} + ( -52 + 28 \beta_{3} + 8 \beta_{4} - 112 \beta_{5} + 12 \beta_{6} - 4 \beta_{8} - 4 \beta_{9} + 4 \beta_{12} - 16 \beta_{13} ) q^{76} + ( -6 + 50 \beta_{1} - 137 \beta_{2} + 23 \beta_{3} - 17 \beta_{4} + 5 \beta_{5} - 5 \beta_{6} - 150 \beta_{7} - 26 \beta_{8} + 6 \beta_{9} - 41 \beta_{10} + 31 \beta_{11} - 6 \beta_{12} - 5 \beta_{13} + 16 \beta_{14} + 29 \beta_{15} ) q^{77} + ( 184 + 4 \beta_{1} - 71 \beta_{2} - 2 \beta_{3} - 13 \beta_{4} + 13 \beta_{5} - 5 \beta_{6} - 69 \beta_{7} + 12 \beta_{8} - 12 \beta_{9} + 10 \beta_{10} + 4 \beta_{11} - 12 \beta_{12} + 7 \beta_{13} + 26 \beta_{14} + 2 \beta_{15} ) q^{78} + ( -11 + 26 \beta_{1} + 26 \beta_{3} - 11 \beta_{4} + 322 \beta_{5} - 11 \beta_{6} + 55 \beta_{8} + 29 \beta_{9} + 26 \beta_{10} + 11 \beta_{12} + 29 \beta_{13} - 11 \beta_{14} ) q^{79} + ( 16 \beta_{2} + 16 \beta_{7} + 16 \beta_{11} ) q^{80} + ( 12 + 21 \beta_{1} - 3 \beta_{2} - 42 \beta_{3} - 12 \beta_{4} + 18 \beta_{5} - 9 \beta_{6} + 306 \beta_{7} - 15 \beta_{8} - 3 \beta_{9} + 9 \beta_{10} - 3 \beta_{11} - 6 \beta_{12} + 9 \beta_{13} + 6 \beta_{14} - 6 \beta_{15} ) q^{81} + ( 38 + 80 \beta_{1} - 80 \beta_{3} + 2 \beta_{4} + 18 \beta_{5} + 10 \beta_{6} - 42 \beta_{8} + 6 \beta_{9} + 16 \beta_{10} + 10 \beta_{12} - 6 \beta_{13} - 2 \beta_{14} ) q^{82} + ( 18 - 210 \beta_{2} + 27 \beta_{3} + 27 \beta_{4} + 9 \beta_{5} - 9 \beta_{6} + 191 \beta_{7} - 36 \beta_{8} - 18 \beta_{9} - 9 \beta_{10} + 18 \beta_{12} - 9 \beta_{13} - 19 \beta_{15} ) q^{83} + ( -96 + 24 \beta_{1} + 48 \beta_{2} - 28 \beta_{3} - 8 \beta_{4} - 136 \beta_{5} + 84 \beta_{7} + 40 \beta_{8} + 4 \beta_{9} + 28 \beta_{10} - 12 \beta_{11} - 20 \beta_{12} + 20 \beta_{13} + 4 \beta_{14} ) q^{84} + ( 314 + 46 \beta_{1} - 23 \beta_{3} + 13 \beta_{4} - 13 \beta_{5} - 13 \beta_{6} + 23 \beta_{10} - 13 \beta_{13} - 26 \beta_{14} ) q^{85} + ( -7 + 46 \beta_{1} - 73 \beta_{2} + 46 \beta_{3} - 7 \beta_{4} - 7 \beta_{5} + 7 \beta_{6} - 39 \beta_{8} + 7 \beta_{9} - 46 \beta_{10} - 6 \beta_{11} - 7 \beta_{12} + 7 \beta_{13} - 7 \beta_{14} + 6 \beta_{15} ) q^{86} + ( 33 + 3 \beta_{1} - 276 \beta_{2} - 54 \beta_{5} + 3 \beta_{6} - 150 \beta_{7} + 4 \beta_{8} - 15 \beta_{9} + 5 \beta_{10} - 24 \beta_{11} - 18 \beta_{12} + 3 \beta_{13} - 21 \beta_{14} ) q^{87} + ( 4 - 8 \beta_{1} + 16 \beta_{3} - 24 \beta_{4} + 16 \beta_{5} - 8 \beta_{6} + 12 \beta_{8} - 12 \beta_{9} + 8 \beta_{10} + 4 \beta_{12} + 8 \beta_{13} + 12 \beta_{14} ) q^{88} + ( 9 + 33 \beta_{1} - 42 \beta_{2} - 33 \beta_{3} + 27 \beta_{4} - 9 \beta_{5} + 9 \beta_{6} - 84 \beta_{7} + 2 \beta_{8} - 9 \beta_{9} - 11 \beta_{10} - 84 \beta_{11} + 9 \beta_{12} + 9 \beta_{13} - 27 \beta_{14} - 84 \beta_{15} ) q^{89} + ( 256 + 105 \beta_{2} + 54 \beta_{3} + 9 \beta_{4} + 503 \beta_{5} + 10 \beta_{6} - 99 \beta_{7} + 20 \beta_{8} + 8 \beta_{9} + 6 \beta_{10} - 8 \beta_{12} - 2 \beta_{13} + 6 \beta_{15} ) q^{90} + ( 320 - 6 \beta_{1} - 16 \beta_{3} - 36 \beta_{4} + 127 \beta_{5} - 6 \beta_{6} + 19 \beta_{8} - 15 \beta_{9} + 14 \beta_{10} - 14 \beta_{12} + 35 \beta_{13} + 44 \beta_{14} ) q^{91} + ( -8 \beta_{1} - 24 \beta_{2} + 4 \beta_{3} - 16 \beta_{4} + 16 \beta_{5} - 16 \beta_{6} - 24 \beta_{7} + 4 \beta_{10} - 16 \beta_{13} + 32 \beta_{14} ) q^{92} + ( -6 + 41 \beta_{1} + 165 \beta_{2} + 41 \beta_{3} - 6 \beta_{4} - 264 \beta_{5} + 21 \beta_{6} + 33 \beta_{8} + 15 \beta_{9} + 18 \beta_{10} - 24 \beta_{11} - 21 \beta_{12} + 15 \beta_{13} - 6 \beta_{14} + 24 \beta_{15} ) q^{93} + ( 137 - 28 \beta_{1} - 178 \beta_{5} - 18 \beta_{6} - 23 \beta_{8} + 5 \beta_{9} - 28 \beta_{10} + 23 \beta_{12} - 18 \beta_{13} - 41 \beta_{14} ) q^{94} + ( -20 - 47 \beta_{1} + 94 \beta_{3} - 40 \beta_{4} - 534 \beta_{7} - 27 \beta_{8} + 20 \beta_{9} - 20 \beta_{12} + 20 \beta_{14} ) q^{95} + ( -16 + 16 \beta_{4} - 16 \beta_{5} - 16 \beta_{14} ) q^{96} + ( 118 + 7 \beta_{3} - 31 \beta_{4} + 267 \beta_{5} - 14 \beta_{6} - 5 \beta_{8} - 17 \beta_{9} + 6 \beta_{10} + 17 \beta_{12} - 3 \beta_{13} ) q^{97} + ( 15 - 58 \beta_{1} - 125 \beta_{2} - 22 \beta_{3} + 34 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 89 \beta_{7} + 19 \beta_{8} - 15 \beta_{9} + 46 \beta_{10} - 22 \beta_{11} + 15 \beta_{12} + 4 \beta_{13} - 23 \beta_{14} - 10 \beta_{15} ) q^{98} + ( -297 + 96 \beta_{1} - 114 \beta_{2} - 48 \beta_{3} + 24 \beta_{4} - 24 \beta_{5} - 33 \beta_{6} - 108 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} - 42 \beta_{10} + 12 \beta_{11} + 3 \beta_{12} - 36 \beta_{13} - 48 \beta_{14} + 6 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 32q^{4} + 80q^{7} + 18q^{9} + O(q^{10}) \) \( 16q + 32q^{4} + 80q^{7} + 18q^{9} - 36q^{10} - 128q^{16} - 48q^{18} - 342q^{19} - 450q^{21} + 24q^{22} - 48q^{24} - 194q^{25} + 88q^{28} + 360q^{30} + 804q^{31} + 1332q^{33} + 144q^{36} - 962q^{37} + 594q^{39} - 144q^{40} - 180q^{42} + 1732q^{43} - 2394q^{45} + 168q^{46} + 820q^{49} + 1638q^{51} + 744q^{52} + 180q^{54} - 2664q^{57} - 780q^{58} - 4620q^{61} - 2016q^{63} - 1024q^{64} - 2016q^{66} - 706q^{67} - 60q^{70} + 192q^{72} + 3294q^{73} + 6174q^{75} + 2832q^{78} - 2656q^{79} + 126q^{81} + 432q^{82} - 432q^{84} + 5232q^{85} + 1026q^{87} + 48q^{88} + 4098q^{91} + 2016q^{93} + 3888q^{94} - 192q^{96} - 4284q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 2 x^{15} - x^{14} - 2 x^{13} + 9 x^{12} - 24 x^{11} + 714 x^{10} - 1940 x^{9} - 2834 x^{8} - 17460 x^{7} + 57834 x^{6} - 17496 x^{5} + 59049 x^{4} - 118098 x^{3} - 531441 x^{2} - 9565938 x + 43046721\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-11783 \nu^{15} - 129119 \nu^{14} - 517471 \nu^{13} + 606928 \nu^{12} - 14603832 \nu^{11} + 41353032 \nu^{10} - 40265502 \nu^{9} + 53965450 \nu^{8} - 392955662 \nu^{7} + 1778268744 \nu^{6} - 3008422296 \nu^{5} + 39016272456 \nu^{4} + 110463994377 \nu^{3} + 163835052489 \nu^{2} - 800613209295 \nu + 756665695800\)\()/ 802467406944 \)
\(\beta_{2}\)\(=\)\((\)\(-99716 \nu^{15} - 525581 \nu^{14} - 4176229 \nu^{13} + 461611 \nu^{12} - 656424 \nu^{11} + 115779900 \nu^{10} - 5977428 \nu^{9} - 100103006 \nu^{8} - 2177986010 \nu^{7} + 3121868142 \nu^{6} + 5634915336 \nu^{5} - 6477453684 \nu^{4} + 27572576256 \nu^{3} + 28249454943 \nu^{2} - 1410719168997 \nu - 74485176237\)\()/ 4413570738192 \)
\(\beta_{3}\)\(=\)\((\)\(-212507 \nu^{15} - 5626388 \nu^{14} - 25621768 \nu^{13} - 99499205 \nu^{12} - 308400336 \nu^{11} + 890922000 \nu^{10} + 2273573622 \nu^{9} + 935214484 \nu^{8} - 11480207372 \nu^{7} - 20499423642 \nu^{6} + 20807095968 \nu^{5} + 1009683312480 \nu^{4} + 2258075337057 \nu^{3} + 4343313293208 \nu^{2} - 16875583713108 \nu - 21889683312741\)\()/ 8827141476384 \)
\(\beta_{4}\)\(=\)\((\)\(2357 \nu^{15} - 76480 \nu^{14} + 402562 \nu^{13} + 93215 \nu^{12} - 52902 \nu^{11} - 2581176 \nu^{10} - 10594434 \nu^{9} - 124678732 \nu^{8} - 129897586 \nu^{7} + 356616720 \nu^{6} + 1654725186 \nu^{5} - 6803786160 \nu^{4} - 1343266335 \nu^{3} - 16310987172 \nu^{2} - 72639481644 \nu + 475470165321\)\()/ 90072872208 \)
\(\beta_{5}\)\(=\)\((\)\(1025 \nu^{15} - 7522 \nu^{14} - 1448 \nu^{13} + 4529 \nu^{12} + 66042 \nu^{11} + 322620 \nu^{10} + 2446566 \nu^{9} + 2014304 \nu^{8} - 7719994 \nu^{7} - 24266412 \nu^{6} + 128621250 \nu^{5} + 43066404 \nu^{4} + 341572221 \nu^{3} + 683039466 \nu^{2} - 12045641706 \nu - 60620943429\)\()/ 30024290736 \)
\(\beta_{6}\)\(=\)\((\)\(38623 \nu^{15} + 1198056 \nu^{14} - 5469812 \nu^{13} + 5048117 \nu^{12} + 91977224 \nu^{11} + 273691056 \nu^{10} + 580611186 \nu^{9} - 137054516 \nu^{8} - 4409276604 \nu^{7} - 4021210726 \nu^{6} - 3106763496 \nu^{5} + 153718002720 \nu^{4} - 127900569213 \nu^{3} - 1310836230540 \nu^{2} - 2328952553784 \nu - 9719194424139\)\()/ 980793497376 \)
\(\beta_{7}\)\(=\)\((\)\(-286331 \nu^{15} - 1172447 \nu^{14} - 3463231 \nu^{13} - 17257904 \nu^{12} + 49540536 \nu^{11} + 76849140 \nu^{10} + 32041542 \nu^{9} - 621195842 \nu^{8} - 255995654 \nu^{7} + 277800696 \nu^{6} + 53069502888 \nu^{5} + 129384485892 \nu^{4} + 226114636401 \nu^{3} - 957931318899 \nu^{2} - 623668865463 \nu + 545449784760\)\()/ 4413570738192 \)
\(\beta_{8}\)\(=\)\((\)\(-80557 \nu^{15} - 475105 \nu^{14} + 29131 \nu^{13} + 26780 \nu^{12} + 12598524 \nu^{11} + 7246644 \nu^{10} - 32616894 \nu^{9} - 273397906 \nu^{8} + 153425198 \nu^{7} + 1266876720 \nu^{6} - 913564980 \nu^{5} + 3717856260 \nu^{4} + 1830354975 \nu^{3} - 162634704417 \nu^{2} - 114262472205 \nu + 476938536804\)\()/ 326931165792 \)
\(\beta_{9}\)\(=\)\((\)\(370944 \nu^{15} - 1248091 \nu^{14} - 538303 \nu^{13} - 14904299 \nu^{12} - 87963988 \nu^{11} - 357861492 \nu^{10} - 278588100 \nu^{9} + 734243010 \nu^{8} + 2333238362 \nu^{7} + 6714774578 \nu^{6} + 11983444956 \nu^{5} - 87923774052 \nu^{4} + 316363730508 \nu^{3} + 2331892439469 \nu^{2} + 6605115383241 \nu + 5989636082637\)\()/ 980793497376 \)
\(\beta_{10}\)\(=\)\((\)\(-193901 \nu^{15} - 416618 \nu^{14} - 1981174 \nu^{13} + 5790835 \nu^{12} + 7775244 \nu^{11} + 26275764 \nu^{10} - 130741998 \nu^{9} - 118606412 \nu^{8} - 524615396 \nu^{7} + 7736574438 \nu^{6} + 13819426524 \nu^{5} + 27002466180 \nu^{4} - 110194048593 \nu^{3} - 86204099826 \nu^{2} - 243730534302 \nu + 1369512296739\)\()/ 326931165792 \)
\(\beta_{11}\)\(=\)\((\)\(5816395 \nu^{15} + 4462603 \nu^{14} - 58985857 \nu^{13} - 70253912 \nu^{12} - 510053436 \nu^{11} - 1391063244 \nu^{10} + 9327548154 \nu^{9} + 12058070998 \nu^{8} - 21168407762 \nu^{7} - 253393254504 \nu^{6} - 66277109196 \nu^{5} + 614369803428 \nu^{4} + 4285317878271 \nu^{3} + 11486205988443 \nu^{2} + 24934743520479 \nu - 185904439092000\)\()/ 8827141476384 \)
\(\beta_{12}\)\(=\)\((\)\(-3030617 \nu^{15} - 6794630 \nu^{14} + 2799530 \nu^{13} - 11951165 \nu^{12} + 40715880 \nu^{11} - 63345192 \nu^{10} + 744805470 \nu^{9} - 3483960380 \nu^{8} + 6148152436 \nu^{7} + 77075174166 \nu^{6} + 38386369800 \nu^{5} - 73495997352 \nu^{4} + 1371421101591 \nu^{3} - 1364448589110 \nu^{2} - 8731794229398 \nu + 2314040260275\)\()/ 2942380492128 \)
\(\beta_{13}\)\(=\)\((\)\(-1111135 \nu^{15} - 1280964 \nu^{14} + 6759296 \nu^{13} + 6088015 \nu^{12} - 23922104 \nu^{11} + 62393280 \nu^{10} + 203938398 \nu^{9} - 806713708 \nu^{8} + 3160773852 \nu^{7} + 17612125126 \nu^{6} + 15062725080 \nu^{5} - 98632791504 \nu^{4} + 285502192749 \nu^{3} - 943343046936 \nu^{2} - 1746194429844 \nu - 890478819513\)\()/ 980793497376 \)
\(\beta_{14}\)\(=\)\((\)\(-28607 \nu^{15} + 29539 \nu^{14} + 231701 \nu^{13} + 96310 \nu^{12} - 379746 \nu^{11} - 1096566 \nu^{10} - 29136138 \nu^{9} - 10559702 \nu^{8} + 26686030 \nu^{7} + 707918058 \nu^{6} - 999264114 \nu^{5} - 2972265678 \nu^{4} - 2852007651 \nu^{3} - 5844020481 \nu^{2} - 3239132895 \nu + 576366896376\)\()/ 22518218052 \)
\(\beta_{15}\)\(=\)\((\)\(-4503491 \nu^{15} - 4051820 \nu^{14} + 29323772 \nu^{13} + 95485939 \nu^{12} + 204560964 \nu^{11} + 311170116 \nu^{10} - 3506282358 \nu^{9} - 6256386860 \nu^{8} + 14301802036 \nu^{7} + 116556367446 \nu^{6} + 51467300100 \nu^{5} - 477886674204 \nu^{4} - 1649137112883 \nu^{3} - 2790931616928 \nu^{2} - 8992178707464 \nu + 55258803118467\)\()/ 2942380492128 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{13} - \beta_{12} + \beta_{9} + \beta_{8} + \beta_{6} - 2 \beta_{5}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{15} - \beta_{13} + \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{6} + 2 \beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_{1} + 2\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{15} + 2 \beta_{12} + 4 \beta_{11} - 6 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + 27 \beta_{7} + 2 \beta_{6} - 16 \beta_{3} + 25 \beta_{2} + 32 \beta_{1} + 8\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{15} + \beta_{14} - \beta_{13} - 2 \beta_{12} + \beta_{11} + 7 \beta_{10} - \beta_{9} + 6 \beta_{8} + 2 \beta_{6} - 5 \beta_{5} + \beta_{4} + 10 \beta_{3} - 104 \beta_{2} + 10 \beta_{1} + 1\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(32 \beta_{15} - 13 \beta_{14} + 4 \beta_{13} - 12 \beta_{12} + 16 \beta_{11} + 4 \beta_{10} + 8 \beta_{9} - 116 \beta_{8} + 300 \beta_{7} - 4 \beta_{6} + 110 \beta_{5} + 26 \beta_{4} - 24 \beta_{3} + 16 \beta_{2} + 12 \beta_{1} + 123\)\()/6\)
\(\nu^{6}\)\(=\)\((\)\(-56 \beta_{15} - 16 \beta_{14} + 12 \beta_{13} + 66 \beta_{12} - 112 \beta_{11} + 28 \beta_{10} + 66 \beta_{9} - 66 \beta_{8} - 264 \beta_{7} + 78 \beta_{6} - 8 \beta_{5} + 8 \beta_{4} - 22 \beta_{3} - 208 \beta_{2} + 44 \beta_{1} - 715\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(368 \beta_{15} - 704 \beta_{14} - 9 \beta_{13} + 261 \beta_{12} - 368 \beta_{11} - 372 \beta_{10} - 9 \beta_{9} - 381 \beta_{8} - 261 \beta_{6} - 3646 \beta_{5} - 704 \beta_{4} + 148 \beta_{3} - 788 \beta_{2} + 148 \beta_{1} - 704\)\()/6\)
\(\nu^{8}\)\(=\)\((\)\(-642 \beta_{15} + 682 \beta_{14} + 259 \beta_{13} - 262 \beta_{12} - 321 \beta_{11} + 259 \beta_{10} + 3 \beta_{9} + 71 \beta_{8} + 906 \beta_{7} - 259 \beta_{6} + 10774 \beta_{5} - 1364 \beta_{4} - 1102 \beta_{3} - 321 \beta_{2} + 551 \beta_{1} + 10092\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(-370 \beta_{15} - 1520 \beta_{14} + 2312 \beta_{13} + 1554 \beta_{12} - 740 \beta_{11} + 814 \beta_{10} + 1554 \beta_{9} - 1554 \beta_{8} - 17763 \beta_{7} + 3866 \beta_{6} - 760 \beta_{5} + 760 \beta_{4} + 5812 \beta_{3} - 17393 \beta_{2} - 11624 \beta_{1} + 80552\)\()/6\)
\(\nu^{10}\)\(=\)\((\)\(4273 \beta_{15} - 2997 \beta_{14} + 7897 \beta_{13} - 8840 \beta_{12} - 4273 \beta_{11} - 6513 \beta_{10} + 7897 \beta_{9} + 1384 \beta_{8} + 8840 \beta_{6} - 6375 \beta_{5} - 2997 \beta_{4} + 5272 \beta_{3} + 42782 \beta_{2} + 5272 \beta_{1} - 2997\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(40896 \beta_{15} + 10137 \beta_{14} - 39840 \beta_{13} + 16424 \beta_{12} + 20448 \beta_{11} - 39840 \beta_{10} + 23416 \beta_{9} - 184 \beta_{8} + 113256 \beta_{7} + 39840 \beta_{6} + 218482 \beta_{5} - 20274 \beta_{4} - 33600 \beta_{3} + 20448 \beta_{2} + 16800 \beta_{1} + 208345\)\()/6\)
\(\nu^{12}\)\(=\)\((\)\(31536 \beta_{15} - 20440 \beta_{14} + 13868 \beta_{13} - 14848 \beta_{12} + 63072 \beta_{11} + 22456 \beta_{10} - 14848 \beta_{9} + 14848 \beta_{8} - 43824 \beta_{7} - 980 \beta_{6} - 10220 \beta_{5} + 10220 \beta_{4} - 127304 \beta_{3} - 75360 \beta_{2} + 254608 \beta_{1} + 622547\)\()/3\)
\(\nu^{13}\)\(=\)\((\)\(40992 \beta_{15} + 224312 \beta_{14} + 210401 \beta_{13} - 313729 \beta_{12} - 40992 \beta_{11} - 91808 \beta_{10} + 210401 \beta_{9} + 118593 \beta_{8} + 313729 \beta_{6} + 2983030 \beta_{5} + 224312 \beta_{4} + 222352 \beta_{3} - 3261720 \beta_{2} + 222352 \beta_{1} + 224312\)\()/6\)
\(\nu^{14}\)\(=\)\((\)\(432322 \beta_{15} - 100872 \beta_{14} - 339137 \beta_{13} + 382692 \beta_{12} + 216161 \beta_{11} - 339137 \beta_{10} - 43555 \beta_{9} - 1764237 \beta_{8} + 1316412 \beta_{7} + 339137 \beta_{6} - 4341902 \beta_{5} + 201744 \beta_{4} - 94142 \beta_{3} + 216161 \beta_{2} + 47071 \beta_{1} - 4241030\)\()/3\)
\(\nu^{15}\)\(=\)\((\)\(-2937310 \beta_{15} - 3314576 \beta_{14} - 576456 \beta_{13} + 1478274 \beta_{12} - 5874620 \beta_{11} - 833654 \beta_{10} + 1478274 \beta_{9} - 1478274 \beta_{8} - 7894293 \beta_{7} + 901818 \beta_{6} - 1657288 \beta_{5} + 1657288 \beta_{4} - 5727200 \beta_{3} - 4956983 \beta_{2} + 11454400 \beta_{1} - 16054616\)\()/6\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(31\)
\(\chi(n)\) \(-1\) \(-\beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
2.41164 1.78437i
2.30541 + 1.91966i
−1.62928 2.51902i
−2.58777 + 1.51770i
−2.81518 1.03671i
−0.0204843 + 2.99993i
0.339489 2.98073i
2.99617 + 0.151487i
2.41164 + 1.78437i
2.30541 1.91966i
−1.62928 + 2.51902i
−2.58777 1.51770i
−2.81518 + 1.03671i
−0.0204843 2.99993i
0.339489 + 2.98073i
2.99617 0.151487i
−1.73205 1.00000i −4.17709 3.09062i 2.00000 + 3.46410i −4.27911 + 7.41164i 4.14431 + 9.53020i −6.41772 + 17.3728i 8.00000i 7.89612 + 25.8196i 14.8233 8.55823i
5.2 −1.73205 1.00000i −3.99309 + 3.32495i 2.00000 + 3.46410i 9.90442 17.1550i 10.2412 1.76589i 18.4277 1.84901i 8.00000i 4.88947 26.5536i −34.3099 + 19.8088i
5.3 −1.73205 1.00000i 2.82199 4.36307i 2.00000 + 3.46410i 2.24534 3.88904i −9.25090 + 4.73506i −9.71288 15.7690i 8.00000i −11.0727 24.6251i −7.77808 + 4.49068i
5.4 −1.73205 1.00000i 4.48216 + 2.62874i 2.00000 + 3.46410i −5.27257 + 9.13236i −5.13458 9.03527i 17.7029 + 5.44135i 8.00000i 13.1794 + 23.5649i 18.2647 10.5451i
5.5 1.73205 + 1.00000i −4.87603 + 1.79564i 2.00000 + 3.46410i −9.90442 + 17.1550i −10.2412 1.76589i 18.4277 1.84901i 8.00000i 20.5514 17.5112i −34.3099 + 19.8088i
5.6 1.73205 + 1.00000i −0.0354799 5.19603i 2.00000 + 3.46410i 5.27257 9.13236i 5.13458 9.03527i 17.7029 + 5.44135i 8.00000i −26.9975 + 0.368709i 18.2647 10.5451i
5.7 1.73205 + 1.00000i 0.588012 + 5.16277i 2.00000 + 3.46410i 4.27911 7.41164i −4.14431 + 9.53020i −6.41772 + 17.3728i 8.00000i −26.3085 + 6.07155i 14.8233 8.55823i
5.8 1.73205 + 1.00000i 5.18952 0.262384i 2.00000 + 3.46410i −2.24534 + 3.88904i 9.25090 + 4.73506i −9.71288 15.7690i 8.00000i 26.8623 2.72329i −7.77808 + 4.49068i
17.1 −1.73205 + 1.00000i −4.17709 + 3.09062i 2.00000 3.46410i −4.27911 7.41164i 4.14431 9.53020i −6.41772 17.3728i 8.00000i 7.89612 25.8196i 14.8233 + 8.55823i
17.2 −1.73205 + 1.00000i −3.99309 3.32495i 2.00000 3.46410i 9.90442 + 17.1550i 10.2412 + 1.76589i 18.4277 + 1.84901i 8.00000i 4.88947 + 26.5536i −34.3099 19.8088i
17.3 −1.73205 + 1.00000i 2.82199 + 4.36307i 2.00000 3.46410i 2.24534 + 3.88904i −9.25090 4.73506i −9.71288 + 15.7690i 8.00000i −11.0727 + 24.6251i −7.77808 4.49068i
17.4 −1.73205 + 1.00000i 4.48216 2.62874i 2.00000 3.46410i −5.27257 9.13236i −5.13458 + 9.03527i 17.7029 5.44135i 8.00000i 13.1794 23.5649i 18.2647 + 10.5451i
17.5 1.73205 1.00000i −4.87603 1.79564i 2.00000 3.46410i −9.90442 17.1550i −10.2412 + 1.76589i 18.4277 + 1.84901i 8.00000i 20.5514 + 17.5112i −34.3099 19.8088i
17.6 1.73205 1.00000i −0.0354799 + 5.19603i 2.00000 3.46410i 5.27257 + 9.13236i 5.13458 + 9.03527i 17.7029 5.44135i 8.00000i −26.9975 0.368709i 18.2647 + 10.5451i
17.7 1.73205 1.00000i 0.588012 5.16277i 2.00000 3.46410i 4.27911 + 7.41164i −4.14431 9.53020i −6.41772 17.3728i 8.00000i −26.3085 6.07155i 14.8233 + 8.55823i
17.8 1.73205 1.00000i 5.18952 + 0.262384i 2.00000 3.46410i −2.24534 3.88904i 9.25090 4.73506i −9.71288 + 15.7690i 8.00000i 26.8623 + 2.72329i −7.77808 4.49068i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 42.4.f.a 16
3.b odd 2 1 inner 42.4.f.a 16
4.b odd 2 1 336.4.bc.e 16
7.b odd 2 1 294.4.f.a 16
7.c even 3 1 294.4.d.a 16
7.c even 3 1 294.4.f.a 16
7.d odd 6 1 inner 42.4.f.a 16
7.d odd 6 1 294.4.d.a 16
12.b even 2 1 336.4.bc.e 16
21.c even 2 1 294.4.f.a 16
21.g even 6 1 inner 42.4.f.a 16
21.g even 6 1 294.4.d.a 16
21.h odd 6 1 294.4.d.a 16
21.h odd 6 1 294.4.f.a 16
28.f even 6 1 336.4.bc.e 16
84.j odd 6 1 336.4.bc.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.f.a 16 1.a even 1 1 trivial
42.4.f.a 16 3.b odd 2 1 inner
42.4.f.a 16 7.d odd 6 1 inner
42.4.f.a 16 21.g even 6 1 inner
294.4.d.a 16 7.c even 3 1
294.4.d.a 16 7.d odd 6 1
294.4.d.a 16 21.g even 6 1
294.4.d.a 16 21.h odd 6 1
294.4.f.a 16 7.b odd 2 1
294.4.f.a 16 7.c even 3 1
294.4.f.a 16 21.c even 2 1
294.4.f.a 16 21.h odd 6 1
336.4.bc.e 16 4.b odd 2 1
336.4.bc.e 16 12.b even 2 1
336.4.bc.e 16 28.f even 6 1
336.4.bc.e 16 84.j odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(42, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16 - 4 T^{2} + T^{4} )^{4} \)
$3$ \( 282429536481 - 3486784401 T^{2} + 4782969 T^{4} - 89282088 T^{5} + 12951414 T^{6} + 244944 T^{7} - 442422 T^{8} + 9072 T^{9} + 17766 T^{10} - 4536 T^{11} + 9 T^{12} - 9 T^{14} + T^{16} \)
$5$ \( 4153645759078656 + 310619615073840 T^{2} + 17289861638841 T^{4} + 367182336789 T^{6} + 5550035922 T^{8} + 45374877 T^{10} + 264258 T^{12} + 597 T^{14} + T^{16} \)
$7$ \( ( 13841287201 - 1614144280 T + 70001155 T^{2} - 5858440 T^{3} + 498232 T^{4} - 17080 T^{5} + 595 T^{6} - 40 T^{7} + T^{8} )^{2} \)
$11$ \( \)\(24\!\cdots\!16\)\( - \)\(10\!\cdots\!76\)\( T^{2} + 4392246020177179521 T^{4} - 13464145104313743 T^{6} + 29836951734258 T^{8} - 30424674303 T^{10} + 22614066 T^{12} - 5391 T^{14} + T^{16} \)
$13$ \( ( 16498888704 + 570419712 T^{2} + 2979108 T^{4} + 4551 T^{6} + T^{8} )^{2} \)
$17$ \( \)\(87\!\cdots\!00\)\( + \)\(24\!\cdots\!00\)\( T^{2} + \)\(68\!\cdots\!00\)\( T^{4} + 10884654228190861980 T^{6} + 15419407094864361 T^{8} + 2831427288342 T^{10} + 387463563 T^{12} + 22782 T^{14} + T^{16} \)
$19$ \( ( 69773043356676 - 11621573426826 T + 548436272553 T^{2} + 16123787289 T^{3} + 46647738 T^{4} - 1981719 T^{5} - 1842 T^{6} + 171 T^{7} + T^{8} )^{2} \)
$23$ \( \)\(40\!\cdots\!96\)\( - \)\(40\!\cdots\!92\)\( T^{2} + \)\(28\!\cdots\!00\)\( T^{4} - \)\(96\!\cdots\!24\)\( T^{6} + 238601271153713481 T^{8} - 21745254238794 T^{10} + 1432520235 T^{12} - 44802 T^{14} + T^{16} \)
$29$ \( ( 20034685818372096 + 10537006777344 T^{2} + 1509134976 T^{4} + 72729 T^{6} + T^{8} )^{2} \)
$31$ \( ( 81628318996303401 - 2746763081118126 T + 34275384741024 T^{2} - 116636150232 T^{3} - 855373959 T^{4} + 4877064 T^{5} + 41736 T^{6} - 402 T^{7} + T^{8} )^{2} \)
$37$ \( ( 81666889425188053156 + 646161179732391370 T + 4356757010847979 T^{2} + 14673347636587 T^{3} + 50423553172 T^{4} + 102777379 T^{5} + 314992 T^{6} + 481 T^{7} + T^{8} )^{2} \)
$41$ \( ( 15703157914051608576 - 1898169992301312 T^{2} + 52068871440 T^{4} - 432408 T^{6} + T^{8} )^{2} \)
$43$ \( ( 966156928 + 37466096 T - 85728 T^{2} - 433 T^{3} + T^{4} )^{4} \)
$47$ \( \)\(20\!\cdots\!76\)\( + \)\(39\!\cdots\!00\)\( T^{2} + \)\(63\!\cdots\!56\)\( T^{4} + \)\(23\!\cdots\!56\)\( T^{6} + \)\(60\!\cdots\!37\)\( T^{8} + 8488943280174618 T^{10} + 85463986203 T^{12} + 339978 T^{14} + T^{16} \)
$53$ \( \)\(65\!\cdots\!00\)\( - \)\(23\!\cdots\!00\)\( T^{2} + \)\(52\!\cdots\!25\)\( T^{4} - \)\(72\!\cdots\!75\)\( T^{6} + \)\(71\!\cdots\!86\)\( T^{8} - 45756035835996519 T^{10} + 212669417970 T^{12} - 569799 T^{14} + T^{16} \)
$59$ \( \)\(11\!\cdots\!36\)\( + \)\(34\!\cdots\!48\)\( T^{2} + \)\(10\!\cdots\!93\)\( T^{4} + \)\(28\!\cdots\!93\)\( T^{6} + \)\(69\!\cdots\!22\)\( T^{8} + 271081786005979821 T^{10} + 775669902066 T^{12} + 1029045 T^{14} + T^{16} \)
$61$ \( ( 25760656722530386944 + 892159531070742144 T + 13303541774870928 T^{2} + 104045523589980 T^{3} + 480636786753 T^{4} + 1367323650 T^{5} + 2370615 T^{6} + 2310 T^{7} + T^{8} )^{2} \)
$67$ \( ( 3036229573514088004 - 13656359405763358 T + 297604295739955 T^{2} - 167895466285 T^{3} + 19396001164 T^{4} - 32172037 T^{5} + 260152 T^{6} + 353 T^{7} + T^{8} )^{2} \)
$71$ \( ( 182329765184802816 + 123867835918848 T^{2} + 20183760144 T^{4} + 678168 T^{6} + T^{8} )^{2} \)
$73$ \( ( \)\(13\!\cdots\!64\)\( - 1429202978440395444 T + 3817932715234431 T^{2} + 14737310733717 T^{3} - 42941956140 T^{4} - 194808807 T^{5} + 1022484 T^{6} - 1647 T^{7} + T^{8} )^{2} \)
$79$ \( ( \)\(11\!\cdots\!61\)\( + \)\(51\!\cdots\!48\)\( T + 1926071802117069670 T^{2} + 2192123600069456 T^{3} + 3059389735471 T^{4} + 1841095856 T^{5} + 2621710 T^{6} + 1328 T^{7} + T^{8} )^{2} \)
$83$ \( ( \)\(16\!\cdots\!56\)\( - 360446437983184128 T^{2} + 2014371869040 T^{4} - 2852067 T^{6} + T^{8} )^{2} \)
$89$ \( \)\(18\!\cdots\!36\)\( + \)\(93\!\cdots\!60\)\( T^{2} + \)\(29\!\cdots\!56\)\( T^{4} + \)\(59\!\cdots\!64\)\( T^{6} + \)\(88\!\cdots\!77\)\( T^{8} + 9111941852559810822 T^{10} + 6939745175403 T^{12} + 3316302 T^{14} + T^{16} \)
$97$ \( ( 10175730882067316736 + 2451916579670784 T^{2} + 84302904096 T^{4} + 832731 T^{6} + T^{8} )^{2} \)
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