Properties

Label 1859.4.a.i
Level $1859$
Weight $4$
Character orbit 1859.a
Self dual yes
Analytic conductor $109.685$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1859.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.684550701\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 4 x^{16} - 99 x^{15} + 375 x^{14} + 3949 x^{13} - 13998 x^{12} - 81750 x^{11} + 267574 x^{10} + 941923 x^{9} - 2799440 x^{8} - 6021311 x^{7} + 15765187 x^{6} + 20197463 x^{5} - 42560950 x^{4} - 32766128 x^{3} + 37775816 x^{2} + 27375104 x + 2596992\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 5 \)
Twist minimal: no (minimal twist has level 143)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{8} q^{3} + ( 5 + \beta_{2} ) q^{4} + ( 1 + \beta_{6} ) q^{5} + ( 1 - \beta_{1} + \beta_{4} - \beta_{6} - \beta_{8} ) q^{6} -\beta_{7} q^{7} + ( 3 + 4 \beta_{1} + \beta_{2} + \beta_{3} ) q^{8} + ( 7 + 2 \beta_{1} + \beta_{8} - \beta_{15} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{8} q^{3} + ( 5 + \beta_{2} ) q^{4} + ( 1 + \beta_{6} ) q^{5} + ( 1 - \beta_{1} + \beta_{4} - \beta_{6} - \beta_{8} ) q^{6} -\beta_{7} q^{7} + ( 3 + 4 \beta_{1} + \beta_{2} + \beta_{3} ) q^{8} + ( 7 + 2 \beta_{1} + \beta_{8} - \beta_{15} ) q^{9} + ( -1 + \beta_{1} - \beta_{6} + 2 \beta_{8} + \beta_{13} - \beta_{15} ) q^{10} -11 q^{11} + ( -4 + \beta_{1} + \beta_{4} - \beta_{7} - 4 \beta_{8} + \beta_{11} + \beta_{15} ) q^{12} + ( -4 - 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{10} + \beta_{11} ) q^{14} + ( -1 - 6 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + \beta_{7} - 3 \beta_{8} - \beta_{14} ) q^{15} + ( 22 + 5 \beta_{1} + 4 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{16} + ( 8 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{17} + ( 29 + 7 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 6 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{12} + \beta_{13} - 2 \beta_{15} + \beta_{16} ) q^{18} + ( 22 + 2 \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{19} + ( 10 - \beta_{1} + 4 \beta_{2} - 3 \beta_{4} + 7 \beta_{6} - \beta_{9} + \beta_{11} + \beta_{14} - \beta_{15} ) q^{20} + ( 13 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{21} -11 \beta_{1} q^{22} + ( -15 + 9 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} + 2 \beta_{6} + 3 \beta_{8} + \beta_{10} - \beta_{11} - 2 \beta_{13} - 2 \beta_{15} - \beta_{16} ) q^{23} + ( 7 + \beta_{1} + 3 \beta_{2} + \beta_{3} + 2 \beta_{4} - 6 \beta_{6} - 5 \beta_{7} - 6 \beta_{8} - \beta_{9} + 2 \beta_{11} + 2 \beta_{13} + \beta_{15} - 2 \beta_{16} ) q^{24} + ( 18 - 6 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{7} + 5 \beta_{8} + \beta_{9} - 3 \beta_{10} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{25} + ( -24 + 5 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} - 10 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} ) q^{27} + ( -17 - 10 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{5} - 5 \beta_{6} - 2 \beta_{7} - 6 \beta_{8} + \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} - 2 \beta_{16} ) q^{28} + ( 3 - \beta_{1} + 5 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - 7 \beta_{8} - \beta_{10} + \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + \beta_{14} + 2 \beta_{15} - \beta_{16} ) q^{29} + ( -67 + 3 \beta_{1} - 8 \beta_{2} + \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 6 \beta_{7} - 23 \beta_{8} - \beta_{9} + 4 \beta_{10} - \beta_{12} - 2 \beta_{13} + 2 \beta_{15} ) q^{30} + ( 41 - 4 \beta_{1} + \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} ) q^{31} + ( 45 + 21 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 6 \beta_{8} - 4 \beta_{9} + \beta_{11} + \beta_{12} ) q^{32} + 11 \beta_{8} q^{33} + ( 43 + 6 \beta_{1} + 10 \beta_{2} + 5 \beta_{4} - 7 \beta_{6} - \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} - 3 \beta_{12} + 4 \beta_{13} - 3 \beta_{15} + \beta_{16} ) q^{34} + ( -8 - 4 \beta_{2} - \beta_{3} + 7 \beta_{4} - 3 \beta_{6} + 7 \beta_{8} - 2 \beta_{11} + 3 \beta_{13} - \beta_{14} + 2 \beta_{15} + 3 \beta_{16} ) q^{35} + ( 19 + 25 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} - \beta_{5} + 4 \beta_{6} + 19 \beta_{8} - \beta_{9} - 4 \beta_{10} + \beta_{11} + \beta_{12} + 4 \beta_{13} + 2 \beta_{14} - 4 \beta_{15} ) q^{36} + ( 60 + 11 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + 3 \beta_{16} ) q^{37} + ( 18 + 27 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - \beta_{6} - 3 \beta_{7} + 7 \beta_{8} + 3 \beta_{10} - 6 \beta_{11} - 4 \beta_{12} - 3 \beta_{13} - \beta_{14} - \beta_{15} + 3 \beta_{16} ) q^{38} + ( -9 + 26 \beta_{1} - 2 \beta_{2} + \beta_{3} + 4 \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 20 \beta_{8} - 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + \beta_{14} - 3 \beta_{15} ) q^{40} + ( 13 - 15 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} + 2 \beta_{6} + 5 \beta_{7} - 9 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} + 3 \beta_{12} + 2 \beta_{14} + 2 \beta_{16} ) q^{41} + ( 29 + 20 \beta_{1} + 14 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + 13 \beta_{6} + 2 \beta_{7} - 16 \beta_{8} - 2 \beta_{13} + \beta_{14} + 3 \beta_{15} + \beta_{16} ) q^{42} + ( -38 + 6 \beta_{1} - 10 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{9} - 3 \beta_{10} - 4 \beta_{11} - \beta_{12} - 2 \beta_{13} - 3 \beta_{14} - \beta_{15} + 3 \beta_{16} ) q^{43} + ( -55 - 11 \beta_{2} ) q^{44} + ( 63 + 65 \beta_{1} + 11 \beta_{2} + \beta_{3} - 10 \beta_{4} + \beta_{5} + 12 \beta_{6} + 2 \beta_{7} + 7 \beta_{8} - \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - 2 \beta_{13} + 4 \beta_{14} - 6 \beta_{15} + \beta_{16} ) q^{45} + ( 99 - 26 \beta_{1} + 11 \beta_{2} - 10 \beta_{3} - 7 \beta_{4} - \beta_{5} + 7 \beta_{6} + 8 \beta_{7} + 19 \beta_{8} + 4 \beta_{9} + 2 \beta_{10} - 4 \beta_{11} - \beta_{12} + \beta_{13} - 6 \beta_{14} - \beta_{15} + 4 \beta_{16} ) q^{46} + ( 43 - \beta_{1} + 18 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 5 \beta_{5} - \beta_{6} + 3 \beta_{7} - 11 \beta_{8} - 4 \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} - 3 \beta_{13} + 2 \beta_{14} - 5 \beta_{15} ) q^{47} + ( 58 + 4 \beta_{1} + 13 \beta_{2} + 8 \beta_{3} + 14 \beta_{4} + \beta_{5} - 6 \beta_{6} - 6 \beta_{7} - 6 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} + 2 \beta_{13} + 3 \beta_{14} + \beta_{15} - 7 \beta_{16} ) q^{48} + ( 12 + 12 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} - \beta_{7} + 12 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} - 3 \beta_{11} - \beta_{12} - 6 \beta_{14} + 3 \beta_{15} + 3 \beta_{16} ) q^{49} + ( -50 + 43 \beta_{1} + 3 \beta_{2} - \beta_{3} - 10 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 8 \beta_{7} + 27 \beta_{8} + 4 \beta_{9} - \beta_{10} - 5 \beta_{11} - 6 \beta_{12} - \beta_{13} + 4 \beta_{14} - 3 \beta_{15} ) q^{50} + ( 106 - 14 \beta_{1} + 10 \beta_{2} + 5 \beta_{3} + 7 \beta_{4} - \beta_{5} - 15 \beta_{6} - 5 \beta_{7} - 17 \beta_{8} - \beta_{9} + 5 \beta_{11} + 5 \beta_{12} + 2 \beta_{13} + \beta_{14} + 3 \beta_{15} - 5 \beta_{16} ) q^{51} + ( -54 + 35 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + 9 \beta_{4} - 2 \beta_{5} - 5 \beta_{6} - \beta_{7} + 28 \beta_{8} - 3 \beta_{9} + \beta_{10} - 3 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} - \beta_{15} - 2 \beta_{16} ) q^{53} + ( 38 - 51 \beta_{1} - 16 \beta_{2} - 6 \beta_{3} + 8 \beta_{4} + 6 \beta_{5} - 11 \beta_{6} + 5 \beta_{7} - 64 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} + 5 \beta_{11} + 6 \beta_{12} - 4 \beta_{13} - 5 \beta_{14} + 6 \beta_{15} + 3 \beta_{16} ) q^{54} + ( -11 - 11 \beta_{6} ) q^{55} + ( -96 - 30 \beta_{1} - 8 \beta_{2} + 15 \beta_{3} + 9 \beta_{4} - \beta_{5} + 2 \beta_{6} - 7 \beta_{7} - 35 \beta_{8} - 3 \beta_{9} - 10 \beta_{10} + 10 \beta_{11} + 7 \beta_{12} + 3 \beta_{13} + 9 \beta_{14} + 3 \beta_{15} - 6 \beta_{16} ) q^{56} + ( -3 - 18 \beta_{1} + 4 \beta_{2} + 5 \beta_{3} - 5 \beta_{4} + 4 \beta_{5} + 16 \beta_{6} - 7 \beta_{7} - 47 \beta_{8} + \beta_{11} + 5 \beta_{12} - 7 \beta_{13} + 5 \beta_{14} + 2 \beta_{15} - 4 \beta_{16} ) q^{57} + ( 36 + 29 \beta_{1} + 8 \beta_{2} - 6 \beta_{3} + 4 \beta_{4} - \beta_{5} - 22 \beta_{6} - 2 \beta_{7} - 11 \beta_{8} + 5 \beta_{9} + 5 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} + 6 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{58} + ( 136 - 9 \beta_{1} + 23 \beta_{2} + 8 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} + 10 \beta_{6} - 6 \beta_{7} - 8 \beta_{8} - \beta_{10} + 3 \beta_{11} + 4 \beta_{12} - 5 \beta_{13} + 2 \beta_{14} + 3 \beta_{15} - 6 \beta_{16} ) q^{59} + ( 5 - 102 \beta_{1} + 22 \beta_{4} - 29 \beta_{6} + 4 \beta_{7} - 45 \beta_{8} + 7 \beta_{11} + 7 \beta_{12} - 4 \beta_{14} + 12 \beta_{15} ) q^{60} + ( 28 + 7 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} + 8 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} - 5 \beta_{7} - 13 \beta_{8} + 12 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} - \beta_{14} + 5 \beta_{15} ) q^{61} + ( -56 + 39 \beta_{1} - 6 \beta_{2} - 3 \beta_{4} + 4 \beta_{5} + 14 \beta_{6} - 6 \beta_{7} + 3 \beta_{8} - \beta_{9} - \beta_{10} + 3 \beta_{11} - 3 \beta_{12} - \beta_{13} + 4 \beta_{14} - \beta_{15} + \beta_{16} ) q^{62} + ( 52 + 43 \beta_{1} - 12 \beta_{4} + 5 \beta_{5} + 2 \beta_{6} - \beta_{7} - 19 \beta_{8} - 6 \beta_{9} + 5 \beta_{10} + \beta_{11} - 6 \beta_{13} + \beta_{14} - 5 \beta_{15} ) q^{63} + ( 210 + 43 \beta_{1} + 20 \beta_{2} - 5 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 16 \beta_{6} - 16 \beta_{7} - 37 \beta_{8} - 5 \beta_{9} + 11 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + 5 \beta_{13} + 2 \beta_{14} - \beta_{15} - \beta_{16} ) q^{64} + ( -11 + 11 \beta_{1} - 11 \beta_{4} + 11 \beta_{6} + 11 \beta_{8} ) q^{66} + ( -12 - 15 \beta_{1} - 14 \beta_{2} + 4 \beta_{3} - 5 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - 5 \beta_{9} + 4 \beta_{10} - 12 \beta_{11} - 8 \beta_{12} - 5 \beta_{13} + 2 \beta_{14} - 6 \beta_{15} ) q^{67} + ( 121 + 80 \beta_{1} + 36 \beta_{2} + 10 \beta_{3} + 4 \beta_{4} + 20 \beta_{6} - 6 \beta_{7} - 35 \beta_{8} - 6 \beta_{9} - 4 \beta_{10} + 9 \beta_{11} - \beta_{12} + 2 \beta_{13} + 5 \beta_{14} - 3 \beta_{15} - 6 \beta_{16} ) q^{68} + ( -91 - 55 \beta_{1} - 23 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} - 20 \beta_{6} + 5 \beta_{7} - 20 \beta_{8} + 9 \beta_{9} - \beta_{11} + \beta_{12} + 4 \beta_{13} - 7 \beta_{14} - 5 \beta_{15} - 3 \beta_{16} ) q^{69} + ( 45 - 12 \beta_{1} + 10 \beta_{2} + 3 \beta_{3} - 14 \beta_{4} - \beta_{5} + 32 \beta_{6} + 7 \beta_{7} - 53 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} + 3 \beta_{11} + 3 \beta_{12} - 4 \beta_{13} + 3 \beta_{15} ) q^{70} + ( 32 + 21 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} + \beta_{4} - 5 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} - 30 \beta_{8} + 10 \beta_{9} - 6 \beta_{10} + 13 \beta_{11} + 11 \beta_{12} + \beta_{13} + 5 \beta_{14} + 12 \beta_{15} - 4 \beta_{16} ) q^{71} + ( 183 - 39 \beta_{1} + 46 \beta_{2} + 6 \beta_{3} - 4 \beta_{4} - 8 \beta_{5} + 14 \beta_{6} - 9 \beta_{7} + 32 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} - 3 \beta_{12} + 10 \beta_{13} + 10 \beta_{14} - 3 \beta_{15} - 10 \beta_{16} ) q^{72} + ( 52 - 83 \beta_{1} + 29 \beta_{2} + 7 \beta_{3} - 12 \beta_{4} - 3 \beta_{5} - 6 \beta_{6} + 6 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 6 \beta_{11} + 2 \beta_{12} - 10 \beta_{13} + 2 \beta_{14} - \beta_{15} - 8 \beta_{16} ) q^{73} + ( 135 + 90 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} - 10 \beta_{4} + 5 \beta_{5} + 30 \beta_{6} + 6 \beta_{7} + 13 \beta_{8} - 2 \beta_{9} + 10 \beta_{10} - 8 \beta_{11} - 3 \beta_{12} - 11 \beta_{13} - \beta_{14} - 2 \beta_{15} + 5 \beta_{16} ) q^{74} + ( -149 - 33 \beta_{1} - 24 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} + 12 \beta_{6} - 5 \beta_{7} - 26 \beta_{8} + \beta_{9} + 6 \beta_{10} + 6 \beta_{11} - 13 \beta_{13} - \beta_{14} + 20 \beta_{15} - 3 \beta_{16} ) q^{75} + ( 121 + 29 \beta_{1} + 14 \beta_{2} + 11 \beta_{3} - 7 \beta_{4} + 2 \beta_{5} + 17 \beta_{6} + 19 \beta_{7} + 12 \beta_{8} + \beta_{9} - 8 \beta_{10} - 3 \beta_{11} + 11 \beta_{12} - 11 \beta_{13} - 4 \beta_{14} + 12 \beta_{15} + 6 \beta_{16} ) q^{76} + 11 \beta_{7} q^{77} + ( -64 + 29 \beta_{1} + 7 \beta_{2} - \beta_{3} + 11 \beta_{4} - \beta_{5} - 19 \beta_{6} + 2 \beta_{7} + 40 \beta_{8} + 2 \beta_{9} - 5 \beta_{10} + 3 \beta_{11} - 4 \beta_{13} - \beta_{14} - 5 \beta_{15} - 13 \beta_{16} ) q^{79} + ( 295 - 15 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} - 15 \beta_{6} - 6 \beta_{7} - 11 \beta_{8} + 4 \beta_{9} - 9 \beta_{10} - 3 \beta_{12} + 7 \beta_{13} - 3 \beta_{14} + 2 \beta_{15} - \beta_{16} ) q^{80} + ( 115 + 90 \beta_{1} + 16 \beta_{2} - 20 \beta_{3} - 7 \beta_{4} + 30 \beta_{6} + 5 \beta_{7} - \beta_{8} + 10 \beta_{9} - 6 \beta_{10} - 7 \beta_{11} - \beta_{12} + 4 \beta_{13} - 3 \beta_{14} - 11 \beta_{15} + 18 \beta_{16} ) q^{81} + ( -83 - 17 \beta_{1} - 17 \beta_{2} - 20 \beta_{3} - 11 \beta_{4} - 2 \beta_{5} + 11 \beta_{6} + 4 \beta_{7} - 51 \beta_{8} + 3 \beta_{9} + 6 \beta_{10} - 5 \beta_{11} - 6 \beta_{12} + 10 \beta_{13} + 2 \beta_{14} - 8 \beta_{15} + 7 \beta_{16} ) q^{82} + ( 13 - 9 \beta_{1} + 7 \beta_{2} - 16 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} + 2 \beta_{7} - 53 \beta_{8} - 9 \beta_{9} + 5 \beta_{10} - 14 \beta_{11} - 13 \beta_{12} - 9 \beta_{14} + 3 \beta_{15} + 9 \beta_{16} ) q^{83} + ( 220 + 104 \beta_{1} + 21 \beta_{2} + 3 \beta_{3} + 8 \beta_{4} + 3 \beta_{5} - 23 \beta_{6} - \beta_{7} + 8 \beta_{8} - 9 \beta_{9} + 2 \beta_{10} - 10 \beta_{11} - 3 \beta_{12} + 3 \beta_{13} - 5 \beta_{14} - 7 \beta_{15} - 2 \beta_{16} ) q^{84} + ( 275 - 40 \beta_{1} + 21 \beta_{2} + 10 \beta_{4} - \beta_{5} + 18 \beta_{6} + 7 \beta_{7} + 38 \beta_{8} + \beta_{9} - 11 \beta_{10} - \beta_{12} + 4 \beta_{13} - 5 \beta_{14} + 5 \beta_{15} + 10 \beta_{16} ) q^{85} + ( 107 - 107 \beta_{1} + 10 \beta_{2} - 20 \beta_{3} - 17 \beta_{4} - 7 \beta_{5} + 16 \beta_{6} - 5 \beta_{7} - 20 \beta_{8} + 10 \beta_{9} + 6 \beta_{10} - 9 \beta_{11} - 7 \beta_{12} - 13 \beta_{13} - 4 \beta_{14} - 4 \beta_{15} + 4 \beta_{16} ) q^{86} + ( 161 + 33 \beta_{1} - 9 \beta_{2} - 13 \beta_{3} - 8 \beta_{4} - 6 \beta_{5} + 16 \beta_{6} - \beta_{7} + 29 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} - 7 \beta_{12} + \beta_{13} - 8 \beta_{14} - 6 \beta_{15} + 4 \beta_{16} ) q^{87} + ( -33 - 44 \beta_{1} - 11 \beta_{2} - 11 \beta_{3} ) q^{88} + ( 244 + \beta_{1} + 16 \beta_{2} - 9 \beta_{3} + 3 \beta_{4} - 14 \beta_{5} - 9 \beta_{6} + 15 \beta_{7} + 4 \beta_{8} + 7 \beta_{9} + 2 \beta_{10} - 5 \beta_{11} - \beta_{12} + 2 \beta_{13} - 2 \beta_{14} - 12 \beta_{15} - 5 \beta_{16} ) q^{89} + ( 843 + 95 \beta_{1} + 69 \beta_{2} + 5 \beta_{3} - 12 \beta_{4} - 14 \beta_{5} + 16 \beta_{6} + 20 \beta_{7} + 106 \beta_{8} + 11 \beta_{9} - 20 \beta_{10} - 5 \beta_{11} + 4 \beta_{12} + 7 \beta_{13} + \beta_{14} - 24 \beta_{15} + 3 \beta_{16} ) q^{90} + ( -316 + 191 \beta_{1} - 52 \beta_{2} + 13 \beta_{3} - 27 \beta_{4} + 5 \beta_{5} + 23 \beta_{6} + 24 \beta_{7} + 82 \beta_{8} + \beta_{9} + 5 \beta_{10} - 12 \beta_{11} + 7 \beta_{12} - 12 \beta_{13} - 2 \beta_{14} + 8 \beta_{15} + 17 \beta_{16} ) q^{92} + ( 75 + 12 \beta_{1} - 11 \beta_{2} + 7 \beta_{3} - 6 \beta_{4} - \beta_{5} + 19 \beta_{6} - 5 \beta_{7} - 74 \beta_{8} + 5 \beta_{9} + 9 \beta_{10} - \beta_{12} - 11 \beta_{13} + \beta_{14} - \beta_{15} - 4 \beta_{16} ) q^{93} + ( 115 + 144 \beta_{1} + 21 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} - 11 \beta_{5} + 8 \beta_{6} + 8 \beta_{7} - 24 \beta_{8} + 8 \beta_{9} + 8 \beta_{10} - 6 \beta_{11} - \beta_{12} + 6 \beta_{13} - 11 \beta_{14} - 5 \beta_{15} + 4 \beta_{16} ) q^{94} + ( -188 - 45 \beta_{1} + 6 \beta_{2} - 10 \beta_{3} - 25 \beta_{4} + 4 \beta_{5} + 65 \beta_{6} + 24 \beta_{7} - 59 \beta_{8} - 9 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} - 6 \beta_{12} - \beta_{13} + 2 \beta_{14} - 8 \beta_{15} + 4 \beta_{16} ) q^{95} + ( 181 + 105 \beta_{1} + 38 \beta_{2} + 2 \beta_{3} + 14 \beta_{4} - 7 \beta_{5} + 32 \beta_{6} - 6 \beta_{7} - 79 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + 13 \beta_{11} - 7 \beta_{12} + 17 \beta_{13} + 7 \beta_{14} + 4 \beta_{15} ) q^{96} + ( 139 + 112 \beta_{1} - 4 \beta_{2} - 7 \beta_{3} + 4 \beta_{4} + \beta_{5} + 7 \beta_{6} - 8 \beta_{7} - 16 \beta_{8} - 7 \beta_{9} - 6 \beta_{10} - 7 \beta_{11} - 5 \beta_{12} - 8 \beta_{14} + 15 \beta_{15} + 6 \beta_{16} ) q^{97} + ( 25 - 30 \beta_{1} + 12 \beta_{2} - 2 \beta_{3} - 12 \beta_{4} + 11 \beta_{5} - 9 \beta_{6} + 15 \beta_{7} + 33 \beta_{8} - 11 \beta_{9} + 15 \beta_{10} - 18 \beta_{11} - 6 \beta_{12} - 12 \beta_{13} - 11 \beta_{14} + 2 \beta_{15} + 9 \beta_{16} ) q^{98} + ( -77 - 22 \beta_{1} - 11 \beta_{8} + 11 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 4 q^{2} - 6 q^{3} + 78 q^{4} + 16 q^{5} + 14 q^{6} - 6 q^{7} + 63 q^{8} + 135 q^{9} + O(q^{10}) \) \( 17 q + 4 q^{2} - 6 q^{3} + 78 q^{4} + 16 q^{5} + 14 q^{6} - 6 q^{7} + 63 q^{8} + 135 q^{9} + 2 q^{10} - 187 q^{11} - 95 q^{12} - 60 q^{14} - 28 q^{15} + 350 q^{16} + 118 q^{17} + 478 q^{18} + 403 q^{19} + 98 q^{20} + 220 q^{21} - 44 q^{22} - 215 q^{23} + 26 q^{24} + 319 q^{25} - 384 q^{27} - 396 q^{28} - 7 q^{29} - 1269 q^{30} + 682 q^{31} + 813 q^{32} + 66 q^{33} + 738 q^{34} + 10 q^{35} + 560 q^{36} + 1084 q^{37} + 410 q^{38} + 95 q^{40} + 240 q^{41} + 393 q^{42} - 435 q^{43} - 858 q^{44} + 1242 q^{45} + 1671 q^{46} + 549 q^{47} + 894 q^{48} + 403 q^{49} - 651 q^{50} + 1552 q^{51} - 566 q^{53} + 311 q^{54} - 176 q^{55} - 1925 q^{56} - 534 q^{57} + 618 q^{58} + 2010 q^{59} - 411 q^{60} + 460 q^{61} - 823 q^{62} + 820 q^{63} + 3171 q^{64} - 154 q^{66} - 232 q^{67} + 1795 q^{68} - 1608 q^{69} + 207 q^{70} + 489 q^{71} + 2556 q^{72} + 290 q^{73} + 2653 q^{74} - 2852 q^{75} + 2421 q^{76} + 66 q^{77} - 732 q^{79} + 4915 q^{80} + 2393 q^{81} - 1772 q^{82} - 117 q^{83} + 4161 q^{84} + 4858 q^{85} + 1034 q^{86} + 3032 q^{87} - 693 q^{88} + 4113 q^{89} + 15145 q^{90} - 3554 q^{92} + 802 q^{93} + 2325 q^{94} - 3924 q^{95} + 2601 q^{96} + 2793 q^{97} + 533 q^{98} - 1485 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{17} - 4 x^{16} - 99 x^{15} + 375 x^{14} + 3949 x^{13} - 13998 x^{12} - 81750 x^{11} + 267574 x^{10} + 941923 x^{9} - 2799440 x^{8} - 6021311 x^{7} + 15765187 x^{6} + 20197463 x^{5} - 42560950 x^{4} - 32766128 x^{3} + 37775816 x^{2} + 27375104 x + 2596992\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 13 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 20 \nu + 10 \)
\(\beta_{4}\)\(=\)\((\)\(320755191500311577 \nu^{16} + 12220861324930981551 \nu^{15} - 86276209144188559950 \nu^{14} - 1089691170983042995419 \nu^{13} + 5833727324126602849724 \nu^{12} + 37643143106635550853998 \nu^{11} - 171164136459150704236180 \nu^{10} - 637097390027371710941150 \nu^{9} + 2482441345101946425892713 \nu^{8} + 5505225111267290509992731 \nu^{7} - 17774686946610383545589102 \nu^{6} - 23335646798509388610951591 \nu^{5} + 56492005825843061507150314 \nu^{4} + 45976808470153155586111784 \nu^{3} - 55604933131807308939165272 \nu^{2} - 46459625359141949016244480 \nu - 4005776073270108633041280\)\()/ \)\(19\!\cdots\!24\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-386045941214303951 \nu^{16} - 16780707919544935689 \nu^{15} + 107019848753033523354 \nu^{14} + 1519209013952630432517 \nu^{13} - 7264914351002868800972 \nu^{12} - 53562836030236774388930 \nu^{11} + 212515466194045461357988 \nu^{10} + 931492962122600546754578 \nu^{9} - 3050211553396792419142647 \nu^{8} - 8331921762884182908956669 \nu^{7} + 21260617620147652922491442 \nu^{6} + 36559198506852545757974505 \nu^{5} - 62426016709271278113819526 \nu^{4} - 71259894123286134590648504 \nu^{3} + 42403000983433033556775464 \nu^{2} + 63056700801129872480984704 \nu + 16500463079882723556101760\)\()/ \)\(19\!\cdots\!24\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-834355344101763163 \nu^{16} + 5377304972460401163 \nu^{15} + 67527700045633927674 \nu^{14} - 464620731506214219951 \nu^{13} - 1999857507681436152316 \nu^{12} + 15435666144060616993790 \nu^{11} + 25515832962439663658804 \nu^{10} - 249259690591585967807054 \nu^{9} - 104307193522747761536547 \nu^{8} + 2040905566609312580237903 \nu^{7} - 447842633376516758769758 \nu^{6} - 8138969644016812017528963 \nu^{5} + 4110265225736896062855778 \nu^{4} + 14346725921463554772267176 \nu^{3} - 6716099813670673792069688 \nu^{2} - 10436961863781423151253632 \nu + 946935233399516690044032\)\()/ \)\(19\!\cdots\!24\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-246001677822851943 \nu^{16} - 677699904112737401 \nu^{15} + 29116573867575102514 \nu^{14} + 68695718062154067565 \nu^{13} - 1373270106845628091124 \nu^{12} - 2776257706572940996546 \nu^{11} + 33021141054423613533156 \nu^{10} + 57101739172667830425634 \nu^{9} - 430194131871573684828351 \nu^{8} - 633588781178425639716957 \nu^{7} + 2978640746162419862606938 \nu^{6} + 3736749823076970811763313 \nu^{5} - 9922801033207085479851006 \nu^{4} - 10968382728342737928642472 \nu^{3} + 11478063945715241514885672 \nu^{2} + 13048986773425192113396800 \nu + 1857642607657073539939200\)\()/ \)\(33\!\cdots\!04\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-794945999869239535 \nu^{16} + 1807282731806681235 \nu^{15} + 77085158542626105006 \nu^{14} - 144117636813427476807 \nu^{13} - 2970824170558640012788 \nu^{12} + 4240037519706955497122 \nu^{11} + 58123134527729820553268 \nu^{10} - 56243761730486894837330 \nu^{9} - 611102833047427733796087 \nu^{8} + 320926527514169096349683 \nu^{7} + 3375383848603830705202654 \nu^{6} - 493666337619772718455719 \nu^{5} - 8951220179730454768044110 \nu^{4} - 1308433526648827103033128 \nu^{3} + 8923827583809858356515048 \nu^{2} + 3338510421881759195977280 \nu - 312466578555150408503424\)\()/ \)\(99\!\cdots\!12\)\( \)
\(\beta_{9}\)\(=\)\((\)\(155486252094825627 \nu^{16} - 520614068795656775 \nu^{15} - 14599544582538301484 \nu^{14} + 44632339955601634477 \nu^{13} + 539898869913703807450 \nu^{12} - 1477492620570823726966 \nu^{11} - 10006999854303408125730 \nu^{10} + 24095649275694889714462 \nu^{9} + 97828074288180730269825 \nu^{8} - 206016269428743313291251 \nu^{7} - 490373581971114473381906 \nu^{6} + 910563429895306171658973 \nu^{5} + 1178777557344139423194594 \nu^{4} - 1739525836522434296317624 \nu^{3} - 1278541649684379721427448 \nu^{2} + 263461682627912690185664 \nu + 445812494432237802387840\)\()/ \)\(16\!\cdots\!52\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-401684023324405389 \nu^{16} + 1792552784613818351 \nu^{15} + 35161134110441480840 \nu^{14} - 149319496393491484627 \nu^{13} - 1187252250085449887296 \nu^{12} + 4683435812869384390948 \nu^{11} + 19577823932393552550384 \nu^{10} - 68569101746942057247100 \nu^{9} - 165631634438522277703593 \nu^{8} + 463858933659058379880837 \nu^{7} + 719868763535121102545312 \nu^{6} - 1142012868968443514824737 \nu^{5} - 1789792709700367081277082 \nu^{4} - 305814498176742101214632 \nu^{3} + 3137680051681207126664184 \nu^{2} + 1596507155218615036240384 \nu - 461415185777467950047232\)\()/ \)\(16\!\cdots\!52\)\( \)
\(\beta_{11}\)\(=\)\((\)\(4829276221031694793 \nu^{16} - 19284674488241722281 \nu^{15} - 457468265004854947110 \nu^{14} + 1708714101000259789197 \nu^{13} + 17237980831271333470684 \nu^{12} - 58904936706635741320058 \nu^{11} - 331454329038764362490708 \nu^{10} + 1005145130013904108707722 \nu^{9} + 3463513350492767939837433 \nu^{8} - 8923246732173942392520677 \nu^{7} - 19276961548576070622697078 \nu^{6} + 39356639855540743272837129 \nu^{5} + 50661168686216023509934634 \nu^{4} - 70151831340015541353681176 \nu^{3} - 40595470245958879516187224 \nu^{2} + 14463609179121479652373504 \nu - 7153964600185080668547456\)\()/ \)\(19\!\cdots\!24\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-1453038091328142239 \nu^{16} + 7942108156575237441 \nu^{15} + 128039367790348748568 \nu^{14} - 704677004901730046361 \nu^{13} - 4366138031426659256228 \nu^{12} + 24312102165820772638288 \nu^{11} + 72958850156528943501796 \nu^{10} - 414728277741166000467232 \nu^{9} - 623546378582022859241151 \nu^{8} + 3679657207272629908937815 \nu^{7} + 2601771794010579321439772 \nu^{6} - 16401285927440756711216487 \nu^{5} - 4706082324872649705752926 \nu^{4} + 32174626963205953829221624 \nu^{3} + 4238097793521018709476680 \nu^{2} - 19010609048725512549715520 \nu - 5138725070565031353389952\)\()/ \)\(49\!\cdots\!56\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-3555216097341445111 \nu^{16} - 3385940474786076861 \nu^{15} + 404863097090957807754 \nu^{14} + 338203908811421232909 \nu^{13} - 18409681132571854969384 \nu^{12} - 13187166866038469263726 \nu^{11} + 426437228613283366401560 \nu^{10} + 252696123365868605562142 \nu^{9} - 5320684556007493253013747 \nu^{8} - 2472464321682009327529213 \nu^{7} + 34756651463343465655682494 \nu^{6} + 12095714859611388303880485 \nu^{5} - 105845741991384190307302358 \nu^{4} - 32152647990081680920035688 \nu^{3} + 105658747314575401830992392 \nu^{2} + 55469087455397896100328512 \nu + 6517735619048495503435392\)\()/ \)\(99\!\cdots\!12\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-119629311245531367 \nu^{16} + 445526327537030795 \nu^{15} + 11255478868614860042 \nu^{14} - 37916402873948071123 \nu^{13} - 422086926091684992160 \nu^{12} + 1229771504057779472410 \nu^{11} + 8132893409488820723376 \nu^{10} - 19038545289400276194442 \nu^{9} - 86750295822367095797883 \nu^{8} + 142732466580791125364691 \nu^{7} + 515710907077155482290694 \nu^{6} - 444293491713271477435443 \nu^{5} - 1611689526537873451363206 \nu^{4} + 152978406136835152639192 \nu^{3} + 2090299834438029090161736 \nu^{2} + 959678662302754067843392 \nu - 57296721716171638867584\)\()/ \)\(30\!\cdots\!64\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-5747872223055716855 \nu^{16} + 5076712012817397759 \nu^{15} + 601138302887419570842 \nu^{14} - 365226872150539899915 \nu^{13} - 25229580738510485037860 \nu^{12} + 8921433310284870691846 \nu^{11} + 542929528061973571293940 \nu^{10} - 75957202429120851331942 \nu^{9} - 6343325546399511195442239 \nu^{8} - 115186227595703219180573 \nu^{7} + 39223941287140969807033922 \nu^{6} + 4696803797837985984577905 \nu^{5} - 115221199883171356814809630 \nu^{4} - 24915531096114567203929256 \nu^{3} + 116323706342218721755151144 \nu^{2} + 55470839455481591137640896 \nu + 4236507095656427146743168\)\()/ \)\(99\!\cdots\!12\)\( \)
\(\beta_{16}\)\(=\)\((\)\(6986541270776725381 \nu^{16} - 8426738008742616801 \nu^{15} - 707170283228976671250 \nu^{14} + 587779790581119353253 \nu^{13} + 28610574430359347732356 \nu^{12} - 13003027896057641526566 \nu^{11} - 590746390376011896797012 \nu^{10} + 58623640258067664718886 \nu^{9} + 6582150944399490450099237 \nu^{8} + 1378626587778159489856783 \nu^{7} - 38499356501626419899083930 \nu^{6} - 17473398284762458322961771 \nu^{5} + 105841107352771081370713226 \nu^{4} + 70481065908800217157672024 \nu^{3} - 96065753360577250702798648 \nu^{2} - 99324566713284533917102016 \nu - 21080850440440604229280896\)\()/ \)\(99\!\cdots\!12\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 13\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 20 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(-\beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} - 2 \beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 28 \beta_{2} + 5 \beta_{1} + 270\)
\(\nu^{5}\)\(=\)\(\beta_{12} + \beta_{11} - 4 \beta_{9} - 6 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 36 \beta_{3} + 37 \beta_{2} + 469 \beta_{1} + 141\)
\(\nu^{6}\)\(=\)\(-\beta_{16} - \beta_{15} - 38 \beta_{14} + 45 \beta_{13} - 42 \beta_{12} - 38 \beta_{11} + 51 \beta_{10} - 5 \beta_{9} - 117 \beta_{8} - 16 \beta_{7} + 24 \beta_{6} - 36 \beta_{5} + 42 \beta_{4} - 45 \beta_{3} + 756 \beta_{2} + 243 \beta_{1} + 6530\)
\(\nu^{7}\)\(=\)\(-4 \beta_{16} + 24 \beta_{15} - 13 \beta_{14} + 5 \beta_{13} + 63 \beta_{12} + 75 \beta_{11} - 7 \beta_{10} - 178 \beta_{9} - 392 \beta_{8} + 108 \beta_{7} - 89 \beta_{6} + 87 \beta_{5} + 195 \beta_{4} + 1093 \beta_{3} + 1137 \beta_{2} + 11906 \beta_{1} + 5057\)
\(\nu^{8}\)\(=\)\(-76 \beta_{16} - 28 \beta_{15} - 1170 \beta_{14} + 1574 \beta_{13} - 1413 \beta_{12} - 1173 \beta_{11} + 1886 \beta_{10} - 268 \beta_{9} - 4822 \beta_{8} - 946 \beta_{7} + 264 \beta_{6} - 1044 \beta_{5} + 1533 \beta_{4} - 1525 \beta_{3} + 20693 \beta_{2} + 8745 \beta_{1} + 169491\)
\(\nu^{9}\)\(=\)\(-345 \beta_{16} + 1579 \beta_{15} - 723 \beta_{14} + 250 \beta_{13} + 2655 \beta_{12} + 3491 \beta_{11} - 488 \beta_{10} - 6081 \beta_{9} - 17799 \beta_{8} + 3992 \beta_{7} - 3147 \beta_{6} + 2875 \beta_{5} + 8373 \beta_{4} + 32005 \beta_{3} + 33767 \beta_{2} + 315678 \beta_{1} + 163801\)
\(\nu^{10}\)\(=\)\(-3718 \beta_{16} - 190 \beta_{15} - 34032 \beta_{14} + 50710 \beta_{13} - 44323 \beta_{12} - 33991 \beta_{11} + 61894 \beta_{10} - 10388 \beta_{9} - 172282 \beta_{8} - 39246 \beta_{7} - 6864 \beta_{6} - 28990 \beta_{5} + 53453 \beta_{4} - 47030 \beta_{3} + 575212 \beta_{2} + 283904 \beta_{1} + 4568062\)
\(\nu^{11}\)\(=\)\(-18009 \beta_{16} + 70427 \beta_{15} - 27372 \beta_{14} + 9395 \beta_{13} + 94656 \beta_{12} + 134756 \beta_{11} - 22823 \beta_{10} - 191401 \beta_{9} - 687577 \beta_{8} + 125952 \beta_{7} - 103450 \beta_{6} + 87018 \beta_{5} + 306908 \beta_{4} + 929278 \beta_{3} + 996042 \beta_{2} + 8595757 \beta_{1} + 5080318\)
\(\nu^{12}\)\(=\)\(-153116 \beta_{16} + 18360 \beta_{15} - 970618 \beta_{14} + 1577410 \beta_{13} - 1347192 \beta_{12} - 956692 \beta_{11} + 1917774 \beta_{10} - 357466 \beta_{9} - 5735262 \beta_{8} - 1417528 \beta_{7} - 604632 \beta_{6} - 804426 \beta_{5} + 1803290 \beta_{4} - 1391367 \beta_{3} + 16185896 \beta_{2} + 8808157 \beta_{1} + 125942716\)
\(\nu^{13}\)\(=\)\(-764378 \beta_{16} + 2660158 \beta_{15} - 879847 \beta_{14} + 342061 \beta_{13} + 3096891 \beta_{12} + 4739891 \beta_{11} - 902335 \beta_{10} - 5832894 \beta_{9} - 24310996 \beta_{8} + 3640240 \beta_{7} - 3295729 \beta_{6} + 2542917 \beta_{5} + 10421449 \beta_{4} + 26941012 \beta_{3} + 29360982 \beta_{2} + 238229705 \beta_{1} + 154334378\)
\(\nu^{14}\)\(=\)\(-5757958 \beta_{16} + 1212598 \beta_{15} - 27452667 \beta_{14} + 48227045 \beta_{13} - 40253180 \beta_{12} - 26435548 \beta_{11} + 57576305 \beta_{10} - 11639418 \beta_{9} - 183617470 \beta_{8} - 47773774 \beta_{7} - 27999751 \beta_{6} - 22548257 \beta_{5} + 59231568 \beta_{4} - 40274070 \beta_{3} + 459636526 \beta_{2} + 267153666 \beta_{1} + 3524491166\)
\(\nu^{15}\)\(=\)\(-29143851 \beta_{16} + 91765041 \beta_{15} - 25797947 \beta_{14} + 12824784 \beta_{13} + 96456020 \beta_{12} + 158182544 \beta_{11} - 32630158 \beta_{10} - 175195819 \beta_{9} - 814759379 \beta_{8} + 99149538 \beta_{7} - 103414949 \beta_{6} + 73102897 \beta_{5} + 339364280 \beta_{4} + 781231128 \beta_{3} + 866184407 \beta_{2} + 6684307207 \beta_{1} + 4638375275\)
\(\nu^{16}\)\(=\)\(-204737235 \beta_{16} + 52327109 \beta_{15} - 772903021 \beta_{14} + 1460641142 \beta_{13} - 1190547912 \beta_{12} - 720048736 \beta_{11} + 1696387564 \beta_{10} - 368360961 \beta_{9} - 5743338205 \beta_{8} - 1548115966 \beta_{7} - 1071111125 \beta_{6} - 639110345 \beta_{5} + 1904974870 \beta_{4} - 1150457764 \beta_{3} + 13141462830 \beta_{2} + 8005816364 \beta_{1} + 99659560772\)

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
−5.36576
−4.61453
−4.18927
−3.21160
−3.17483
−2.47248
−1.07101
−0.549217
−0.114461
1.75097
2.02345
2.74484
2.79502
3.98991
4.70666
5.30056
5.45173
−5.36576 −0.836231 20.7914 1.92861 4.48702 17.6723 −68.6355 −26.3007 −10.3485
1.2 −4.61453 3.68758 13.2939 19.7910 −17.0165 −18.7869 −24.4287 −13.4017 −91.3259
1.3 −4.18927 −8.20774 9.54998 −18.8743 34.3844 −14.8122 −6.49330 40.3670 79.0697
1.4 −3.21160 −6.73941 2.31437 9.24473 21.6443 8.56428 18.2600 18.4196 −29.6904
1.5 −3.17483 2.79441 2.07954 −8.68483 −8.87177 −6.73544 18.7965 −19.1913 27.5728
1.6 −2.47248 6.49330 −1.88686 −10.7576 −16.0545 24.9760 24.4450 15.1629 26.5979
1.7 −1.07101 −5.99193 −6.85294 0.973634 6.41740 −28.3191 15.9076 8.90321 −1.04277
1.8 −0.549217 −0.561969 −7.69836 16.1600 0.308643 31.1335 8.62181 −26.6842 −8.87537
1.9 −0.114461 5.80602 −7.98690 11.2238 −0.664561 −15.9656 1.82987 6.70989 −1.28469
1.10 1.75097 3.47555 −4.93410 −8.23074 6.08559 −17.9650 −22.6472 −14.9205 −14.4118
1.11 2.02345 −7.47211 −3.90564 −6.22433 −15.1195 28.5679 −24.0905 28.8324 −12.5946
1.12 2.74484 −4.19208 −0.465832 5.57379 −11.5066 −6.88148 −23.2374 −9.42645 15.2992
1.13 2.79502 10.1456 −0.187860 5.45712 28.3572 7.73594 −22.8852 75.9332 15.2528
1.14 3.98991 1.67407 7.91939 −20.8270 6.67941 9.96761 −0.321630 −24.1975 −83.0979
1.15 4.70666 −10.1964 14.1527 19.7628 −47.9909 −2.86520 28.9586 76.9660 93.0167
1.16 5.30056 −2.59422 20.0959 −4.49343 −13.7508 −33.2644 64.1151 −20.2700 −23.8177
1.17 5.45173 6.71552 21.7213 3.97680 36.6112 10.9778 74.8051 18.0982 21.6804
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(1\)
\(13\) \(1\)

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 1859.4.a.i 17
13.b even 2 1 1859.4.a.f 17
13.c even 3 2 143.4.e.a 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.e.a 34 13.c even 3 2
1859.4.a.f 17 13.b even 2 1
1859.4.a.i 17 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{17} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1859))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2596992 + 27375104 T + 37775816 T^{2} - 32766128 T^{3} - 42560950 T^{4} + 20197463 T^{5} + 15765187 T^{6} - 6021311 T^{7} - 2799440 T^{8} + 941923 T^{9} + 267574 T^{10} - 81750 T^{11} - 13998 T^{12} + 3949 T^{13} + 375 T^{14} - 99 T^{15} - 4 T^{16} + T^{17} \)
$3$ \( 19875034496 + 43337722688 T - 8606865840 T^{2} - 40329425288 T^{3} + 6272375440 T^{4} + 9931166228 T^{5} - 1537314792 T^{6} - 1041883992 T^{7} + 144789952 T^{8} + 56978696 T^{9} - 6386698 T^{10} - 1758711 T^{11} + 141610 T^{12} + 30700 T^{13} - 1510 T^{14} - 279 T^{15} + 6 T^{16} + T^{17} \)
$5$ \( 1259401303114488 - 2148764531729503 T + 826552886571026 T^{2} + 156936892794827 T^{3} - 110824802801286 T^{4} - 198397009263 T^{5} + 5099695627382 T^{6} - 212359703704 T^{7} - 111298639196 T^{8} + 6163659096 T^{9} + 1242580022 T^{10} - 74450349 T^{11} - 6962198 T^{12} + 427957 T^{13} + 17618 T^{14} - 1094 T^{15} - 16 T^{16} + T^{17} \)
$7$ \( 28417206769411190400 + 9832880282647008960 T - 1670586427614725744 T^{2} - 554028354144370424 T^{3} + 42201629654650208 T^{4} + 11927496854608580 T^{5} - 557779189251656 T^{6} - 125509327596396 T^{7} + 3743474156584 T^{8} + 712586573128 T^{9} - 12630596922 T^{10} - 2221741277 T^{11} + 21512454 T^{12} + 3720802 T^{13} - 18024 T^{14} - 3099 T^{15} + 6 T^{16} + T^{17} \)
$11$ \( ( 11 + T )^{17} \)
$13$ \( T^{17} \)
$17$ \( \)\(14\!\cdots\!36\)\( - \)\(26\!\cdots\!53\)\( T - \)\(18\!\cdots\!66\)\( T^{2} - \)\(19\!\cdots\!67\)\( T^{3} - \)\(14\!\cdots\!12\)\( T^{4} + \)\(65\!\cdots\!63\)\( T^{5} + \)\(17\!\cdots\!74\)\( T^{6} - 7657844435477853270 T^{7} - 256502970146744734 T^{8} + 4390876272620766 T^{9} + 138554519703938 T^{10} - 1590095202603 T^{11} - 31722490296 T^{12} + 310105039 T^{13} + 3203154 T^{14} - 28776 T^{15} - 118 T^{16} + T^{17} \)
$19$ \( \)\(13\!\cdots\!00\)\( - \)\(40\!\cdots\!00\)\( T - \)\(21\!\cdots\!48\)\( T^{2} + \)\(65\!\cdots\!24\)\( T^{3} + \)\(11\!\cdots\!80\)\( T^{4} - \)\(42\!\cdots\!16\)\( T^{5} - \)\(27\!\cdots\!16\)\( T^{6} + \)\(14\!\cdots\!76\)\( T^{7} + 851256490864299804 T^{8} - 272135569943596580 T^{9} + 834697185100289 T^{10} + 28724679111991 T^{11} - 170015837387 T^{12} - 1460881127 T^{13} + 13574484 T^{14} + 14100 T^{15} - 403 T^{16} + T^{17} \)
$23$ \( \)\(12\!\cdots\!64\)\( - \)\(21\!\cdots\!76\)\( T - \)\(78\!\cdots\!88\)\( T^{2} + \)\(35\!\cdots\!04\)\( T^{3} + \)\(11\!\cdots\!04\)\( T^{4} - \)\(18\!\cdots\!92\)\( T^{5} - \)\(64\!\cdots\!56\)\( T^{6} + \)\(24\!\cdots\!88\)\( T^{7} + \)\(14\!\cdots\!32\)\( T^{8} + 75192626526360976 T^{9} - 15428546602643201 T^{10} - 39507502601589 T^{11} + 830001283330 T^{12} + 3025512496 T^{13} - 21593941 T^{14} - 92231 T^{15} + 215 T^{16} + T^{17} \)
$29$ \( -\)\(40\!\cdots\!57\)\( + \)\(62\!\cdots\!33\)\( T - \)\(11\!\cdots\!22\)\( T^{2} - \)\(14\!\cdots\!66\)\( T^{3} + \)\(27\!\cdots\!43\)\( T^{4} + \)\(15\!\cdots\!57\)\( T^{5} - \)\(14\!\cdots\!01\)\( T^{6} - \)\(67\!\cdots\!59\)\( T^{7} + 25603001240068774596 T^{8} + 1391374654866798768 T^{9} - 2125869936490072 T^{10} - 136591862759412 T^{11} + 98778457427 T^{12} + 6438361937 T^{13} - 2084133 T^{14} - 136775 T^{15} + 7 T^{16} + T^{17} \)
$31$ \( \)\(59\!\cdots\!24\)\( - \)\(84\!\cdots\!12\)\( T + \)\(44\!\cdots\!92\)\( T^{2} - \)\(10\!\cdots\!84\)\( T^{3} + \)\(60\!\cdots\!00\)\( T^{4} + \)\(21\!\cdots\!56\)\( T^{5} - \)\(40\!\cdots\!88\)\( T^{6} + \)\(32\!\cdots\!04\)\( T^{7} + \)\(49\!\cdots\!20\)\( T^{8} - 3388581485581925296 T^{9} - 20570030794864556 T^{10} + 295106910220281 T^{11} - 166601222250 T^{12} - 9891820713 T^{13} + 34479328 T^{14} + 90975 T^{15} - 682 T^{16} + T^{17} \)
$37$ \( \)\(64\!\cdots\!74\)\( + \)\(68\!\cdots\!85\)\( T - \)\(50\!\cdots\!52\)\( T^{2} - \)\(80\!\cdots\!06\)\( T^{3} + \)\(79\!\cdots\!60\)\( T^{4} + \)\(24\!\cdots\!61\)\( T^{5} - \)\(34\!\cdots\!04\)\( T^{6} - \)\(15\!\cdots\!09\)\( T^{7} + \)\(45\!\cdots\!50\)\( T^{8} - \)\(17\!\cdots\!76\)\( T^{9} - 948439871126508356 T^{10} + 7699111149434776 T^{11} - 4988522718820 T^{12} - 85141122831 T^{13} + 211300676 T^{14} + 164691 T^{15} - 1084 T^{16} + T^{17} \)
$41$ \( -\)\(87\!\cdots\!50\)\( - \)\(42\!\cdots\!85\)\( T - \)\(56\!\cdots\!88\)\( T^{2} + \)\(20\!\cdots\!49\)\( T^{3} + \)\(98\!\cdots\!18\)\( T^{4} + \)\(42\!\cdots\!19\)\( T^{5} - \)\(46\!\cdots\!66\)\( T^{6} - \)\(37\!\cdots\!86\)\( T^{7} + \)\(69\!\cdots\!50\)\( T^{8} + \)\(11\!\cdots\!50\)\( T^{9} + 164381272178736844 T^{10} - 17281918699483619 T^{11} - 13189448352874 T^{12} + 142553216867 T^{13} + 106707722 T^{14} - 607064 T^{15} - 240 T^{16} + T^{17} \)
$43$ \( -\)\(89\!\cdots\!56\)\( - \)\(71\!\cdots\!72\)\( T - \)\(12\!\cdots\!76\)\( T^{2} + \)\(15\!\cdots\!52\)\( T^{3} + \)\(38\!\cdots\!00\)\( T^{4} - \)\(25\!\cdots\!72\)\( T^{5} - \)\(25\!\cdots\!24\)\( T^{6} - \)\(30\!\cdots\!36\)\( T^{7} + \)\(70\!\cdots\!76\)\( T^{8} + \)\(13\!\cdots\!44\)\( T^{9} - 10151157141205254799 T^{10} - 21396021100094927 T^{11} + 77510693024099 T^{12} + 172582881491 T^{13} - 296235676 T^{14} - 675152 T^{15} + 435 T^{16} + T^{17} \)
$47$ \( -\)\(74\!\cdots\!68\)\( - \)\(13\!\cdots\!76\)\( T + \)\(85\!\cdots\!96\)\( T^{2} + \)\(82\!\cdots\!36\)\( T^{3} - \)\(26\!\cdots\!04\)\( T^{4} - \)\(16\!\cdots\!64\)\( T^{5} + \)\(32\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!92\)\( T^{7} - \)\(18\!\cdots\!00\)\( T^{8} - \)\(43\!\cdots\!92\)\( T^{9} + 5658791380093564173 T^{10} + 2593041028316081 T^{11} - 81012937271051 T^{12} + 74741138249 T^{13} + 391573028 T^{14} - 584574 T^{15} - 549 T^{16} + T^{17} \)
$53$ \( -\)\(44\!\cdots\!54\)\( + \)\(94\!\cdots\!61\)\( T + \)\(31\!\cdots\!88\)\( T^{2} - \)\(15\!\cdots\!45\)\( T^{3} - \)\(46\!\cdots\!68\)\( T^{4} + \)\(31\!\cdots\!41\)\( T^{5} - \)\(31\!\cdots\!98\)\( T^{6} - \)\(17\!\cdots\!44\)\( T^{7} + \)\(24\!\cdots\!08\)\( T^{8} + \)\(43\!\cdots\!48\)\( T^{9} - 6579666543709460964 T^{10} - 50449726944467825 T^{11} + 73200100541652 T^{12} + 293900225893 T^{13} - 348124650 T^{14} - 837646 T^{15} + 566 T^{16} + T^{17} \)
$59$ \( \)\(68\!\cdots\!32\)\( + \)\(63\!\cdots\!12\)\( T + \)\(12\!\cdots\!68\)\( T^{2} - \)\(21\!\cdots\!52\)\( T^{3} - \)\(49\!\cdots\!32\)\( T^{4} + \)\(47\!\cdots\!96\)\( T^{5} + \)\(56\!\cdots\!56\)\( T^{6} - \)\(65\!\cdots\!60\)\( T^{7} - \)\(53\!\cdots\!24\)\( T^{8} + \)\(30\!\cdots\!48\)\( T^{9} - \)\(15\!\cdots\!66\)\( T^{10} + 256790924139158629 T^{11} + 191558310899458 T^{12} - 1283484867501 T^{13} + 1344174814 T^{14} + 645756 T^{15} - 2010 T^{16} + T^{17} \)
$61$ \( \)\(13\!\cdots\!50\)\( + \)\(89\!\cdots\!35\)\( T - \)\(86\!\cdots\!18\)\( T^{2} - \)\(23\!\cdots\!60\)\( T^{3} + \)\(64\!\cdots\!44\)\( T^{4} + \)\(13\!\cdots\!21\)\( T^{5} + \)\(39\!\cdots\!78\)\( T^{6} - \)\(22\!\cdots\!77\)\( T^{7} - \)\(17\!\cdots\!62\)\( T^{8} + \)\(17\!\cdots\!38\)\( T^{9} + \)\(13\!\cdots\!86\)\( T^{10} - 714493764984025534 T^{11} - 472555564015860 T^{12} + 1657351152637 T^{13} + 756483726 T^{14} - 2014701 T^{15} - 460 T^{16} + T^{17} \)
$67$ \( \)\(28\!\cdots\!28\)\( - \)\(28\!\cdots\!04\)\( T - \)\(34\!\cdots\!32\)\( T^{2} + \)\(86\!\cdots\!04\)\( T^{3} - \)\(64\!\cdots\!12\)\( T^{4} - \)\(30\!\cdots\!76\)\( T^{5} + \)\(24\!\cdots\!88\)\( T^{6} - \)\(69\!\cdots\!08\)\( T^{7} - \)\(23\!\cdots\!60\)\( T^{8} + \)\(11\!\cdots\!08\)\( T^{9} + 70493544439722649320 T^{10} - 674902555999652183 T^{11} - 9386765254604 T^{12} + 1803796939880 T^{13} - 227602606 T^{14} - 2205377 T^{15} + 232 T^{16} + T^{17} \)
$71$ \( -\)\(30\!\cdots\!48\)\( - \)\(20\!\cdots\!16\)\( T - \)\(28\!\cdots\!80\)\( T^{2} - \)\(31\!\cdots\!08\)\( T^{3} + \)\(13\!\cdots\!64\)\( T^{4} + \)\(48\!\cdots\!60\)\( T^{5} - \)\(16\!\cdots\!88\)\( T^{6} - \)\(83\!\cdots\!44\)\( T^{7} + \)\(62\!\cdots\!24\)\( T^{8} + \)\(56\!\cdots\!24\)\( T^{9} + 6778212049625350619 T^{10} - 1863150642788736343 T^{11} - 483739050896712 T^{12} + 3229954128274 T^{13} + 904299643 T^{14} - 2836765 T^{15} - 489 T^{16} + T^{17} \)
$73$ \( \)\(18\!\cdots\!00\)\( + \)\(99\!\cdots\!60\)\( T + \)\(20\!\cdots\!48\)\( T^{2} - \)\(36\!\cdots\!60\)\( T^{3} - \)\(60\!\cdots\!08\)\( T^{4} + \)\(52\!\cdots\!28\)\( T^{5} + \)\(63\!\cdots\!24\)\( T^{6} - \)\(38\!\cdots\!91\)\( T^{7} - \)\(33\!\cdots\!70\)\( T^{8} + \)\(15\!\cdots\!97\)\( T^{9} + \)\(92\!\cdots\!88\)\( T^{10} - 3774304635645063710 T^{11} - 1399738154673036 T^{12} + 5089560585122 T^{13} + 1042817956 T^{14} - 3572763 T^{15} - 290 T^{16} + T^{17} \)
$79$ \( -\)\(30\!\cdots\!88\)\( - \)\(33\!\cdots\!72\)\( T + \)\(12\!\cdots\!44\)\( T^{2} + \)\(22\!\cdots\!40\)\( T^{3} + \)\(88\!\cdots\!20\)\( T^{4} - \)\(37\!\cdots\!56\)\( T^{5} - \)\(27\!\cdots\!28\)\( T^{6} - \)\(84\!\cdots\!04\)\( T^{7} + \)\(16\!\cdots\!12\)\( T^{8} + \)\(14\!\cdots\!36\)\( T^{9} - \)\(42\!\cdots\!96\)\( T^{10} - 4676047213406911380 T^{11} + 5268249062266760 T^{12} + 6406718902860 T^{13} - 3157368248 T^{14} - 4096092 T^{15} + 732 T^{16} + T^{17} \)
$83$ \( \)\(81\!\cdots\!36\)\( - \)\(13\!\cdots\!32\)\( T + \)\(28\!\cdots\!48\)\( T^{2} + \)\(43\!\cdots\!44\)\( T^{3} - \)\(22\!\cdots\!76\)\( T^{4} + \)\(21\!\cdots\!92\)\( T^{5} + \)\(17\!\cdots\!08\)\( T^{6} - \)\(22\!\cdots\!24\)\( T^{7} - \)\(48\!\cdots\!32\)\( T^{8} + \)\(92\!\cdots\!60\)\( T^{9} + \)\(57\!\cdots\!47\)\( T^{10} - 15845005319889122779 T^{11} - 2933302683131607 T^{12} + 13667786282947 T^{13} + 326401428 T^{14} - 5861224 T^{15} + 117 T^{16} + T^{17} \)
$89$ \( \)\(12\!\cdots\!04\)\( - \)\(36\!\cdots\!92\)\( T + \)\(29\!\cdots\!64\)\( T^{2} - \)\(24\!\cdots\!24\)\( T^{3} - \)\(62\!\cdots\!28\)\( T^{4} + \)\(24\!\cdots\!68\)\( T^{5} - \)\(25\!\cdots\!80\)\( T^{6} - \)\(17\!\cdots\!48\)\( T^{7} + \)\(31\!\cdots\!36\)\( T^{8} + \)\(17\!\cdots\!56\)\( T^{9} - \)\(11\!\cdots\!23\)\( T^{10} + 9500261909205056905 T^{11} + 8155053645402866 T^{12} - 18278249686148 T^{13} + 8313791371 T^{14} + 3433179 T^{15} - 4113 T^{16} + T^{17} \)
$97$ \( -\)\(49\!\cdots\!76\)\( - \)\(29\!\cdots\!16\)\( T + \)\(18\!\cdots\!12\)\( T^{2} + \)\(92\!\cdots\!40\)\( T^{3} - \)\(26\!\cdots\!84\)\( T^{4} - \)\(10\!\cdots\!28\)\( T^{5} + \)\(20\!\cdots\!32\)\( T^{6} + \)\(53\!\cdots\!40\)\( T^{7} - \)\(94\!\cdots\!00\)\( T^{8} - \)\(12\!\cdots\!96\)\( T^{9} + \)\(23\!\cdots\!89\)\( T^{10} + 9970550312003947287 T^{11} - 27412659470832111 T^{12} + 84422695919 T^{13} + 14499016731 T^{14} - 3313883 T^{15} - 2793 T^{16} + T^{17} \)
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