Properties

Label 1859.4.a.g
Level $1859$
Weight $4$
Character orbit 1859.a
Self dual yes
Analytic conductor $109.685$
Analytic rank $1$
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: \(1\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 93 x^{15} - 7 x^{14} + 3449 x^{13} + 406 x^{12} - 65242 x^{11} - 7942 x^{10} + 669163 x^{9} + 59532 x^{8} - 3663297 x^{7} - 79027 x^{6} + 9967603 x^{5} - 984554 x^{4} - 12177120 x^{3} + 3207432 x^{2} + 5215872 x - 2210688\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7} \)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} -\beta_{6} q^{3} + ( 3 + \beta_{2} ) q^{4} + ( -1 + \beta_{7} ) q^{5} + ( 1 + \beta_{1} + \beta_{5} ) q^{6} + ( -4 - \beta_{10} ) q^{7} + ( -1 - 3 \beta_{1} - \beta_{3} ) q^{8} + ( 7 - \beta_{2} + \beta_{6} - \beta_{7} + \beta_{14} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} -\beta_{6} q^{3} + ( 3 + \beta_{2} ) q^{4} + ( -1 + \beta_{7} ) q^{5} + ( 1 + \beta_{1} + \beta_{5} ) q^{6} + ( -4 - \beta_{10} ) q^{7} + ( -1 - 3 \beta_{1} - \beta_{3} ) q^{8} + ( 7 - \beta_{2} + \beta_{6} - \beta_{7} + \beta_{14} ) q^{9} + ( 1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{6} - \beta_{11} ) q^{10} + 11 q^{11} + ( -7 - 2 \beta_{1} - \beta_{6} + \beta_{8} - \beta_{11} + \beta_{13} ) q^{12} + ( 1 + 5 \beta_{1} + 2 \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} ) q^{14} + ( -10 - 6 \beta_{1} + \beta_{4} + 3 \beta_{6} + \beta_{8} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} ) q^{15} + ( 7 + 4 \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} + \beta_{12} + \beta_{14} ) q^{16} + ( -4 + 4 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} + \beta_{16} ) q^{17} + ( 5 - 2 \beta_{1} - \beta_{2} + \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{18} + ( -9 + 7 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{19} + ( -14 - 4 \beta_{1} - 5 \beta_{2} - \beta_{3} + 4 \beta_{5} + 3 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} - 2 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{20} + ( -13 - \beta_{1} - 4 \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} + 7 \beta_{6} + 2 \beta_{7} - \beta_{9} + 2 \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{16} ) q^{21} -11 \beta_{1} q^{22} + ( -13 + 7 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} + \beta_{16} ) q^{23} + ( 11 + 7 \beta_{1} - 4 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + 3 \beta_{10} + 2 \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{24} + ( 7 + 5 \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{14} - \beta_{15} + \beta_{16} ) q^{25} + ( -8 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 7 \beta_{6} + \beta_{7} - 3 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{13} - \beta_{15} + 2 \beta_{16} ) q^{27} + ( -24 - 11 \beta_{1} - 4 \beta_{3} - \beta_{4} - 3 \beta_{5} + 8 \beta_{6} - 4 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{13} + \beta_{14} - 2 \beta_{15} + 3 \beta_{16} ) q^{28} + ( -18 + 9 \beta_{1} - 12 \beta_{2} + \beta_{4} + \beta_{5} + 12 \beta_{6} - 10 \beta_{7} + \beta_{10} - \beta_{13} ) q^{29} + ( 46 + 5 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + 9 \beta_{6} - 2 \beta_{7} + \beta_{9} + 4 \beta_{10} - 2 \beta_{13} - \beta_{14} + 2 \beta_{15} - 2 \beta_{16} ) q^{30} + ( -30 - 7 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 11 \beta_{6} - 6 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + 2 \beta_{12} - 2 \beta_{14} ) q^{31} + ( -31 + 5 \beta_{1} - 2 \beta_{2} - 4 \beta_{4} + 8 \beta_{6} + \beta_{7} - 4 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - 2 \beta_{14} + 2 \beta_{16} ) q^{32} -11 \beta_{6} q^{33} + ( -46 + 5 \beta_{1} - 9 \beta_{2} + 6 \beta_{3} + 5 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} + 5 \beta_{7} + 5 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{34} + ( -2 + 8 \beta_{1} - 6 \beta_{2} - \beta_{4} - 7 \beta_{5} + 3 \beta_{6} - 12 \beta_{7} + \beta_{8} + 3 \beta_{10} + 4 \beta_{11} + \beta_{12} + \beta_{14} + \beta_{15} + \beta_{16} ) q^{35} + ( -43 + \beta_{1} + 3 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + \beta_{6} + 4 \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} + 3 \beta_{11} - \beta_{13} - 4 \beta_{14} - 2 \beta_{15} - \beta_{16} ) q^{36} + ( 9 + 11 \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{6} + 5 \beta_{7} - \beta_{8} - \beta_{9} + 3 \beta_{10} + 3 \beta_{11} + 3 \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} + 2 \beta_{16} ) q^{37} + ( -76 + 17 \beta_{1} - 15 \beta_{2} + 4 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} + 10 \beta_{7} - 3 \beta_{8} + \beta_{9} + 5 \beta_{10} - 2 \beta_{11} - 2 \beta_{13} - \beta_{14} + 2 \beta_{15} - 3 \beta_{16} ) q^{38} + ( 46 + 14 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} + \beta_{4} - 7 \beta_{5} - 13 \beta_{6} - 3 \beta_{7} + 5 \beta_{8} - 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} - 2 \beta_{16} ) q^{40} + ( -33 - 20 \beta_{1} + 5 \beta_{2} + \beta_{4} - 5 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} - \beta_{8} + 5 \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{13} - 2 \beta_{14} + 2 \beta_{15} - 2 \beta_{16} ) q^{41} + ( 9 + 29 \beta_{1} - 15 \beta_{2} + 4 \beta_{3} - \beta_{4} - 7 \beta_{5} + 12 \beta_{6} - 15 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} - \beta_{11} - 3 \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} ) q^{42} + ( 51 - 17 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 6 \beta_{5} + 18 \beta_{6} + \beta_{7} - 3 \beta_{8} + \beta_{9} + 8 \beta_{10} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - \beta_{14} - 2 \beta_{16} ) q^{43} + ( 33 + 11 \beta_{2} ) q^{44} + ( -85 + 22 \beta_{1} + 13 \beta_{2} + \beta_{3} + 4 \beta_{4} - \beta_{5} - 6 \beta_{6} + 18 \beta_{7} - \beta_{8} - 9 \beta_{10} - 3 \beta_{11} + 2 \beta_{12} + \beta_{13} - \beta_{14} - 3 \beta_{15} - 3 \beta_{16} ) q^{45} + ( -86 - 18 \beta_{1} - 18 \beta_{2} - 4 \beta_{3} + \beta_{4} + 6 \beta_{5} + 6 \beta_{6} - 15 \beta_{7} + 3 \beta_{8} - 5 \beta_{9} + 3 \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{14} - 6 \beta_{15} + 5 \beta_{16} ) q^{46} + ( -6 - 6 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 6 \beta_{4} - 8 \beta_{5} + 6 \beta_{6} + 7 \beta_{7} + \beta_{8} - 8 \beta_{9} - 6 \beta_{10} - \beta_{11} + 4 \beta_{13} + 4 \beta_{14} - 3 \beta_{15} + 3 \beta_{16} ) q^{47} + ( -11 + 11 \beta_{1} - 10 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 10 \beta_{5} - 12 \beta_{6} - 13 \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{48} + ( 117 - 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 8 \beta_{4} + 13 \beta_{5} + 4 \beta_{6} - 10 \beta_{7} - 8 \beta_{9} + 10 \beta_{10} - 4 \beta_{11} - 7 \beta_{12} + 4 \beta_{13} - 2 \beta_{14} + 6 \beta_{15} - 4 \beta_{16} ) q^{49} + ( -37 + 10 \beta_{1} - 16 \beta_{2} - \beta_{3} - \beta_{4} + 11 \beta_{5} + 21 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + 10 \beta_{10} - 3 \beta_{12} + \beta_{13} - \beta_{15} - 2 \beta_{16} ) q^{50} + ( -38 + 6 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} - 5 \beta_{4} - \beta_{5} + 13 \beta_{6} - 7 \beta_{7} - 2 \beta_{9} + 3 \beta_{10} + 4 \beta_{11} + 3 \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} + 2 \beta_{16} ) q^{51} + ( 9 + 8 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 6 \beta_{4} - 19 \beta_{6} - 9 \beta_{7} + 4 \beta_{8} + 6 \beta_{9} + 5 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} - 7 \beta_{13} - 7 \beta_{14} - \beta_{15} - 4 \beta_{16} ) q^{53} + ( -5 - \beta_{1} + 21 \beta_{2} - \beta_{3} - \beta_{4} - 17 \beta_{5} - \beta_{6} + 9 \beta_{7} - 3 \beta_{8} - 8 \beta_{10} + 7 \beta_{12} + 5 \beta_{13} + 4 \beta_{14} + 4 \beta_{15} + \beta_{16} ) q^{54} + ( -11 + 11 \beta_{7} ) q^{55} + ( 108 - 3 \beta_{1} + 29 \beta_{2} - 3 \beta_{4} - 12 \beta_{5} + 26 \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - 4 \beta_{10} + 3 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} + 4 \beta_{14} - 5 \beta_{15} + 2 \beta_{16} ) q^{56} + ( 66 + 7 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 11 \beta_{4} + 2 \beta_{5} + 39 \beta_{6} - 17 \beta_{7} - \beta_{8} - 4 \beta_{9} + 9 \beta_{10} - 2 \beta_{13} + 5 \beta_{15} - \beta_{16} ) q^{57} + ( -109 + 87 \beta_{1} - 18 \beta_{2} + 14 \beta_{3} + 3 \beta_{4} - 14 \beta_{5} + 8 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + 7 \beta_{11} - \beta_{13} - 2 \beta_{14} + \beta_{15} - \beta_{16} ) q^{58} + ( -11 - 23 \beta_{1} - 25 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} + 6 \beta_{5} - 15 \beta_{6} - 5 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} + 6 \beta_{10} - 4 \beta_{11} - \beta_{12} + 7 \beta_{13} - 4 \beta_{14} + 7 \beta_{15} - 3 \beta_{16} ) q^{59} + ( 5 - 58 \beta_{1} - 33 \beta_{2} - 10 \beta_{3} - 4 \beta_{4} + 7 \beta_{5} + 14 \beta_{6} - 5 \beta_{7} - 4 \beta_{8} + 7 \beta_{10} - 7 \beta_{11} - 3 \beta_{12} - 3 \beta_{13} + 2 \beta_{14} + 4 \beta_{15} + 4 \beta_{16} ) q^{60} + ( -31 + 18 \beta_{1} + 14 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 5 \beta_{5} - 30 \beta_{6} + 9 \beta_{7} - 5 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} + 3 \beta_{13} + \beta_{14} - \beta_{15} - 8 \beta_{16} ) q^{61} + ( 43 + 40 \beta_{1} + 24 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} - 16 \beta_{5} - 40 \beta_{6} + 4 \beta_{7} + 3 \beta_{8} + 6 \beta_{9} - 12 \beta_{10} + \beta_{11} + 3 \beta_{12} + 4 \beta_{14} - 3 \beta_{15} + 4 \beta_{16} ) q^{62} + ( -124 + 21 \beta_{1} + 10 \beta_{2} - \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 10 \beta_{6} + 29 \beta_{7} + 2 \beta_{8} - 6 \beta_{9} - 6 \beta_{10} - 7 \beta_{11} + 4 \beta_{12} + 10 \beta_{13} - 4 \beta_{14} + 6 \beta_{15} - 4 \beta_{16} ) q^{63} + ( -89 + 31 \beta_{1} - 8 \beta_{2} + 3 \beta_{3} - 6 \beta_{4} - 10 \beta_{5} + 10 \beta_{6} + 7 \beta_{7} + 3 \beta_{8} - 8 \beta_{9} + 3 \beta_{10} + 8 \beta_{11} + \beta_{13} - 4 \beta_{14} + \beta_{15} ) q^{64} + ( 11 + 11 \beta_{1} + 11 \beta_{5} ) q^{66} + ( -46 - 13 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} - 7 \beta_{4} - 11 \beta_{5} + 35 \beta_{6} - 4 \beta_{7} + 5 \beta_{8} - 9 \beta_{9} - \beta_{10} - 5 \beta_{11} + \beta_{12} + 10 \beta_{13} - 3 \beta_{14} + 14 \beta_{15} - 6 \beta_{16} ) q^{67} + ( -48 + 53 \beta_{1} - 38 \beta_{2} + 7 \beta_{3} + 15 \beta_{4} + 3 \beta_{5} + 12 \beta_{6} - 11 \beta_{7} - \beta_{8} + 10 \beta_{9} - 6 \beta_{10} - 5 \beta_{11} - 4 \beta_{12} - 4 \beta_{14} - \beta_{15} - 5 \beta_{16} ) q^{68} + ( 31 - 26 \beta_{1} + 23 \beta_{2} - \beta_{3} + 6 \beta_{4} - 10 \beta_{5} + 17 \beta_{6} - 8 \beta_{7} + 8 \beta_{8} - \beta_{9} - 11 \beta_{10} - 6 \beta_{11} + 5 \beta_{12} + 11 \beta_{13} + 3 \beta_{14} - 3 \beta_{15} + 7 \beta_{16} ) q^{69} + ( -107 + 31 \beta_{1} - 21 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + 77 \beta_{6} - 35 \beta_{7} - 5 \beta_{9} - 4 \beta_{10} + 11 \beta_{11} - 5 \beta_{12} - 3 \beta_{13} + 3 \beta_{14} - 12 \beta_{15} + 5 \beta_{16} ) q^{70} + ( -47 + 31 \beta_{1} - 15 \beta_{3} - 6 \beta_{4} + 7 \beta_{5} + 21 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} - 5 \beta_{9} - 8 \beta_{11} - 4 \beta_{12} + 6 \beta_{13} + 6 \beta_{14} - 3 \beta_{15} - 4 \beta_{16} ) q^{71} + ( -70 + 3 \beta_{1} - 16 \beta_{2} + 8 \beta_{4} - 8 \beta_{5} - 51 \beta_{6} + \beta_{7} + 4 \beta_{8} - 12 \beta_{10} - 10 \beta_{11} + 5 \beta_{12} + 9 \beta_{14} - \beta_{15} + 4 \beta_{16} ) q^{72} + ( -156 - 62 \beta_{1} - 11 \beta_{2} - 3 \beta_{3} + 7 \beta_{4} + 9 \beta_{5} + 27 \beta_{6} + 7 \beta_{7} - 4 \beta_{8} + 5 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 10 \beta_{12} - 8 \beta_{13} - 7 \beta_{14} - 2 \beta_{15} - 5 \beta_{16} ) q^{73} + ( -88 - 12 \beta_{1} + 19 \beta_{2} - 11 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} - 24 \beta_{6} - 16 \beta_{7} + 4 \beta_{8} - 9 \beta_{9} - 9 \beta_{10} - 12 \beta_{11} - 2 \beta_{12} - 11 \beta_{15} + 6 \beta_{16} ) q^{74} + ( 145 + 39 \beta_{1} + 12 \beta_{2} - 15 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} - 35 \beta_{6} - 20 \beta_{7} - 6 \beta_{8} + 7 \beta_{9} - 4 \beta_{11} + 10 \beta_{12} - 8 \beta_{13} + 9 \beta_{14} + 9 \beta_{15} + \beta_{16} ) q^{75} + ( -121 + 106 \beta_{1} - 25 \beta_{2} + 19 \beta_{3} + 19 \beta_{4} - 12 \beta_{5} - 26 \beta_{6} + 16 \beta_{7} + 9 \beta_{8} + 11 \beta_{9} - 2 \beta_{10} - 14 \beta_{11} - 6 \beta_{12} - 3 \beta_{13} - 3 \beta_{14} + 2 \beta_{15} - 6 \beta_{16} ) q^{76} + ( -44 - 11 \beta_{10} ) q^{77} + ( 149 + 79 \beta_{1} + 17 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} - 7 \beta_{5} - 35 \beta_{6} - 9 \beta_{7} + 8 \beta_{8} - 16 \beta_{9} + 3 \beta_{10} - 4 \beta_{11} - 6 \beta_{12} + \beta_{13} + \beta_{14} - 6 \beta_{15} ) q^{79} + ( -86 - 41 \beta_{1} - 14 \beta_{2} + 7 \beta_{3} + 2 \beta_{4} + 10 \beta_{5} + 73 \beta_{6} + \beta_{7} - 2 \beta_{8} - 7 \beta_{9} + 2 \beta_{10} + 6 \beta_{11} - 5 \beta_{12} + 2 \beta_{13} + 7 \beta_{14} - 10 \beta_{15} + 3 \beta_{16} ) q^{80} + ( 2 + 41 \beta_{1} - 46 \beta_{2} - 17 \beta_{3} - 3 \beta_{4} + 8 \beta_{5} + 69 \beta_{6} - 36 \beta_{7} + 7 \beta_{8} - 2 \beta_{9} + 9 \beta_{10} + \beta_{11} + 6 \beta_{12} - 4 \beta_{13} - 4 \beta_{14} + 7 \beta_{15} + \beta_{16} ) q^{81} + ( 173 - 30 \beta_{1} + 12 \beta_{2} - 8 \beta_{3} + 7 \beta_{4} - 4 \beta_{5} + 49 \beta_{6} + 6 \beta_{7} - 10 \beta_{8} - \beta_{9} + 5 \beta_{10} - 4 \beta_{11} - 3 \beta_{12} - 6 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} - 4 \beta_{16} ) q^{82} + ( 28 + 29 \beta_{1} + 10 \beta_{2} + 3 \beta_{3} + 9 \beta_{4} + 3 \beta_{5} + 10 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + 14 \beta_{9} - 4 \beta_{10} - 17 \beta_{11} + 4 \beta_{12} - 4 \beta_{13} + 13 \beta_{14} - 2 \beta_{15} - 5 \beta_{16} ) q^{83} + ( -228 + 107 \beta_{1} - 48 \beta_{2} - 2 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} + 50 \beta_{6} - 16 \beta_{7} - 9 \beta_{8} + 6 \beta_{9} + 14 \beta_{10} + 15 \beta_{11} + 9 \beta_{12} - 11 \beta_{13} + 8 \beta_{14} + 3 \beta_{15} + 4 \beta_{16} ) q^{84} + ( -115 - 31 \beta_{1} - 15 \beta_{2} + 5 \beta_{3} - 4 \beta_{4} - 9 \beta_{5} + 10 \beta_{6} - 7 \beta_{7} - 3 \beta_{8} - \beta_{9} - 3 \beta_{10} + 6 \beta_{11} + 6 \beta_{12} + 5 \beta_{13} - 4 \beta_{14} - 2 \beta_{15} - 4 \beta_{16} ) q^{85} + ( 181 - 130 \beta_{1} - 13 \beta_{2} - 8 \beta_{3} + 5 \beta_{4} - 7 \beta_{5} - 27 \beta_{6} + \beta_{7} + 14 \beta_{8} + 9 \beta_{9} - 2 \beta_{10} - 14 \beta_{11} - 11 \beta_{12} + \beta_{13} + 3 \beta_{14} - 2 \beta_{15} - 3 \beta_{16} ) q^{86} + ( -220 + 47 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} - 12 \beta_{4} - 10 \beta_{5} - 42 \beta_{6} + 14 \beta_{7} - 22 \beta_{8} + 6 \beta_{10} + 11 \beta_{11} + 9 \beta_{12} - 4 \beta_{13} - \beta_{14} + 7 \beta_{15} + 12 \beta_{16} ) q^{87} + ( -11 - 33 \beta_{1} - 11 \beta_{3} ) q^{88} + ( -87 + 34 \beta_{1} + 28 \beta_{2} - 13 \beta_{3} - 19 \beta_{4} - 6 \beta_{5} - 21 \beta_{6} + 4 \beta_{7} - 5 \beta_{8} + 7 \beta_{9} - 16 \beta_{10} - 4 \beta_{11} + 2 \beta_{12} + 10 \beta_{13} + 3 \beta_{14} + 12 \beta_{16} ) q^{89} + ( -290 + 27 \beta_{1} - 7 \beta_{2} - 12 \beta_{3} - 4 \beta_{4} - 5 \beta_{5} - 37 \beta_{6} + 24 \beta_{7} - 5 \beta_{8} + 3 \beta_{9} - 6 \beta_{10} - 3 \beta_{11} + 6 \beta_{12} + \beta_{13} - 6 \beta_{14} + 13 \beta_{15} + 5 \beta_{16} ) q^{90} + ( 326 + 203 \beta_{1} + 21 \beta_{2} + 17 \beta_{3} - 11 \beta_{4} - 12 \beta_{5} - 44 \beta_{6} - 10 \beta_{7} - 2 \beta_{8} - \beta_{9} - 9 \beta_{10} + 7 \beta_{11} + 9 \beta_{12} + 19 \beta_{13} + 19 \beta_{14} + 5 \beta_{15} + \beta_{16} ) q^{92} + ( -271 - 75 \beta_{1} + 6 \beta_{2} - 8 \beta_{4} + 11 \beta_{5} + 66 \beta_{6} - 24 \beta_{7} + \beta_{8} - \beta_{9} + 23 \beta_{10} + 11 \beta_{11} - 4 \beta_{12} + \beta_{13} - 7 \beta_{14} - 8 \beta_{15} + 12 \beta_{16} ) q^{93} + ( 103 + 58 \beta_{1} - 9 \beta_{3} - 28 \beta_{4} - 3 \beta_{5} + 60 \beta_{6} + 9 \beta_{7} - 8 \beta_{8} - 15 \beta_{9} + 3 \beta_{10} + 22 \beta_{11} + 6 \beta_{12} + \beta_{13} - 5 \beta_{14} + 3 \beta_{15} + 6 \beta_{16} ) q^{94} + ( 56 - 121 \beta_{1} - \beta_{2} - 2 \beta_{3} - 7 \beta_{4} + 19 \beta_{5} + 68 \beta_{6} - 2 \beta_{7} + 15 \beta_{8} - \beta_{9} + 10 \beta_{10} + 9 \beta_{11} - 9 \beta_{12} + 6 \beta_{13} - 13 \beta_{14} + 2 \beta_{15} ) q^{95} + ( -248 + 25 \beta_{1} - 35 \beta_{2} + 15 \beta_{3} + 5 \beta_{4} + 2 \beta_{5} + 99 \beta_{6} + 6 \beta_{7} - 5 \beta_{8} + 2 \beta_{9} - 11 \beta_{10} + 6 \beta_{11} + 7 \beta_{12} - 2 \beta_{13} - 9 \beta_{14} + 20 \beta_{15} - 10 \beta_{16} ) q^{96} + ( -107 + 92 \beta_{1} - 12 \beta_{2} - 25 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} - 10 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} + 13 \beta_{10} + \beta_{11} - 6 \beta_{12} + 3 \beta_{13} + 9 \beta_{14} + 11 \beta_{16} ) q^{97} + ( 105 - 109 \beta_{1} - 10 \beta_{2} + 5 \beta_{3} - 12 \beta_{4} + 17 \beta_{5} - 29 \beta_{6} + 23 \beta_{7} + 6 \beta_{8} + 15 \beta_{9} - 3 \beta_{10} - 5 \beta_{11} - 7 \beta_{12} - 14 \beta_{13} + 3 \beta_{14} + 4 \beta_{15} - 20 \beta_{16} ) q^{98} + ( 77 - 11 \beta_{2} + 11 \beta_{6} - 11 \beta_{7} + 11 \beta_{14} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q - 6 q^{3} + 50 q^{4} - 24 q^{5} + 16 q^{6} - 62 q^{7} - 21 q^{8} + 135 q^{9} + O(q^{10}) \) \( 17 q - 6 q^{3} + 50 q^{4} - 24 q^{5} + 16 q^{6} - 62 q^{7} - 21 q^{8} + 135 q^{9} + 2 q^{10} + 187 q^{11} - 127 q^{12} - 148 q^{15} + 126 q^{16} - 74 q^{17} + 90 q^{18} - 159 q^{19} - 222 q^{20} - 184 q^{21} - 215 q^{23} + 214 q^{24} + 95 q^{25} - 192 q^{27} - 358 q^{28} - 157 q^{29} + 829 q^{30} - 394 q^{31} - 553 q^{32} - 66 q^{33} - 702 q^{34} + 58 q^{35} - 700 q^{36} + 88 q^{37} - 1318 q^{38} + 733 q^{40} - 512 q^{41} + 337 q^{42} + 927 q^{43} + 550 q^{44} - 1482 q^{45} - 1361 q^{46} - 143 q^{47} - 178 q^{48} + 1835 q^{49} - 583 q^{50} - 568 q^{51} + 106 q^{53} - 67 q^{54} - 264 q^{55} + 2059 q^{56} + 1298 q^{57} - 1690 q^{58} - 266 q^{59} + 37 q^{60} - 624 q^{61} + 643 q^{62} - 2360 q^{63} - 1589 q^{64} + 176 q^{66} - 676 q^{67} - 413 q^{68} + 764 q^{69} - 1061 q^{70} - 763 q^{71} - 1366 q^{72} - 2374 q^{73} - 1649 q^{74} + 2420 q^{75} - 2101 q^{76} - 682 q^{77} + 2164 q^{79} - 1013 q^{80} + 537 q^{81} + 3152 q^{82} + 777 q^{83} - 3381 q^{84} - 1690 q^{85} + 2894 q^{86} - 4200 q^{87} - 231 q^{88} - 1687 q^{89} - 5399 q^{90} + 5542 q^{92} - 4310 q^{93} + 1777 q^{94} + 1124 q^{95} - 3465 q^{96} - 2047 q^{97} + 1553 q^{98} + 1485 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{17} - 93 x^{15} - 7 x^{14} + 3449 x^{13} + 406 x^{12} - 65242 x^{11} - 7942 x^{10} + 669163 x^{9} + 59532 x^{8} - 3663297 x^{7} - 79027 x^{6} + 9967603 x^{5} - 984554 x^{4} - 12177120 x^{3} + 3207432 x^{2} + 5215872 x - 2210688\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 11 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 19 \nu - 1 \)
\(\beta_{4}\)\(=\)\((\)\(41310869995 \nu^{16} + 278727206397 \nu^{15} - 4357449870192 \nu^{14} - 22798802169445 \nu^{13} + 178373934891776 \nu^{12} + 716365123007302 \nu^{11} - 3621438221595784 \nu^{10} - 10899842045845894 \nu^{9} + 38670820084482067 \nu^{8} + 83868393271353909 \nu^{7} - 212275314229813704 \nu^{6} - 316374785529478405 \nu^{5} + 551444746495558666 \nu^{4} + 517804542798383800 \nu^{3} - 589314874552179144 \nu^{2} - 245073579862158912 \nu + 193120824489777024\)\()/ 1272854848402176 \)
\(\beta_{5}\)\(=\)\((\)\(22792255353 \nu^{16} - 173935419925 \nu^{15} - 1445507747716 \nu^{14} + 13203532405837 \nu^{13} + 28731625134576 \nu^{12} - 382162547138290 \nu^{11} - 92877753246288 \nu^{10} + 5325334959642130 \nu^{9} - 3224598450547927 \nu^{8} - 37340323567706529 \nu^{7} + 36174674865580708 \nu^{6} + 129114534297550577 \nu^{5} - 135530553001133322 \nu^{4} - 203275618052083880 \nu^{3} + 197406569290912296 \nu^{2} + 110157361719943680 \nu - 92451555662157440\)\()/ 424284949467392 \)
\(\beta_{6}\)\(=\)\((\)\(-124885018663 \nu^{16} + 68376766059 \nu^{15} + 11092500475884 \nu^{14} - 3462328112507 \nu^{13} - 391117832151176 \nu^{12} + 35491557826550 \nu^{11} + 7001260746196576 \nu^{10} + 713203558482682 \nu^{9} - 67592428864662679 \nu^{8} - 17108450282689497 \nu^{7} + 345469943509992324 \nu^{6} + 118393312966623025 \nu^{5} - 857460683787723058 \nu^{4} - 283635614338668664 \nu^{3} + 910913004305338920 \nu^{2} + 191659502692433472 \nu - 319639332055585920\)\()/ 1272854848402176 \)
\(\beta_{7}\)\(=\)\((\)\(-294827222141 \nu^{16} + 219464620161 \nu^{15} + 25860702212988 \nu^{14} - 12725541872401 \nu^{13} - 898568125283296 \nu^{12} + 224527988550394 \nu^{11} + 15812490173164760 \nu^{10} - 535071081569314 \nu^{9} - 149677731066067037 \nu^{8} - 21195912703721763 \nu^{7} + 748323066818915820 \nu^{6} + 189194848643498195 \nu^{5} - 1813225525195868414 \nu^{4} - 463871483481226232 \nu^{3} + 1879900869637740024 \nu^{2} + 289420030139366400 \nu - 634975686686663808\)\()/ 2545709696804352 \)
\(\beta_{8}\)\(=\)\((\)\(177480472921 \nu^{16} - 398292066701 \nu^{15} - 14709832798652 \nu^{14} + 27780697951405 \nu^{13} + 478644757930352 \nu^{12} - 704218629973202 \nu^{11} - 7814228797171912 \nu^{10} + 7764238982009946 \nu^{9} + 67951199476640841 \nu^{8} - 32184328855692953 \nu^{7} - 308490898952635324 \nu^{6} - 1490025495625063 \nu^{5} + 654251360803069766 \nu^{4} + 175934653521669464 \nu^{3} - 544366907867948696 \nu^{2} - 121507588348011520 \nu + 139906562611909248\)\()/ 848569898934784 \)
\(\beta_{9}\)\(=\)\((\)\(-183146367977 \nu^{16} - 411803762979 \nu^{15} + 17660657765996 \nu^{14} + 35199004200787 \nu^{13} - 673331837979632 \nu^{12} - 1161942683655406 \nu^{11} + 12941087741494888 \nu^{10} + 18736788875497014 \nu^{9} - 132696889920181673 \nu^{8} - 155138782650201079 \nu^{7} + 708425883172414412 \nu^{6} + 641107897566669367 \nu^{5} - 1802610378563604534 \nu^{4} - 1149078031026157112 \nu^{3} + 1924628345505759128 \nu^{2} + 640196578222973568 \nu - 681865087899031680\)\()/ 848569898934784 \)
\(\beta_{10}\)\(=\)\((\)\(278509815379 \nu^{16} + 240283192845 \nu^{15} - 25773301577160 \nu^{14} - 23681279993365 \nu^{13} + 946899009302480 \nu^{12} + 891334111445566 \nu^{11} - 17639386439250904 \nu^{10} - 16298701719519886 \nu^{9} + 176776638852978331 \nu^{8} + 153218144056176621 \nu^{7} - 934804957212417696 \nu^{6} - 718119159583703341 \nu^{5} + 2408400419298443914 \nu^{4} + 1438407698117106136 \nu^{3} - 2718017488209778056 \nu^{2} - 894772647366121536 \nu + 1051903859121484416\)\()/ 1272854848402176 \)
\(\beta_{11}\)\(=\)\((\)\(719004694813 \nu^{16} - 1831736510361 \nu^{15} - 59159334330924 \nu^{14} + 132140276331041 \nu^{13} + 1908699169344344 \nu^{12} - 3564593685064562 \nu^{11} - 30881631864599440 \nu^{10} + 44756923149540218 \nu^{9} + 266725456943426965 \nu^{8} - 263282810437785069 \nu^{7} - 1215984236280607908 \nu^{6} + 651921926831937509 \nu^{5} + 2675697930801855886 \nu^{4} - 575703136285199240 \nu^{3} - 2406048010619033016 \nu^{2} + 138713065043428608 \nu + 601329229193857152\)\()/ 2545709696804352 \)
\(\beta_{12}\)\(=\)\((\)\(-483443617781 \nu^{16} - 896950422627 \nu^{15} + 47322682896696 \nu^{14} + 76549836795275 \nu^{13} - 1836186129603904 \nu^{12} - 2532242521755266 \nu^{11} + 36010561428074864 \nu^{10} + 41202147885040082 \nu^{9} - 377528556848834837 \nu^{8} - 348599223359990643 \nu^{7} + 2061848508132699672 \nu^{6} + 1501291433735641907 \nu^{5} - 5372195188778414894 \nu^{4} - 2856605594639303096 \nu^{3} + 5951886336261131832 \nu^{2} + 1768896994062648192 \nu - 2182304818142554752\)\()/ 1272854848402176 \)
\(\beta_{13}\)\(=\)\((\)\(-508877270167 \nu^{16} - 1725555260937 \nu^{15} + 52228351735548 \nu^{14} + 142874728959613 \nu^{13} - 2109429433149116 \nu^{12} - 4588948588615942 \nu^{11} + 42772820824260364 \nu^{10} + 72579932386081726 \nu^{9} - 460804354185067915 \nu^{8} - 596349391049937225 \nu^{7} + 2571226508294583624 \nu^{6} + 2481879935862400117 \nu^{5} - 6818167994267385274 \nu^{4} - 4529325870858228760 \nu^{3} + 7699312282746374088 \nu^{2} + 2611825420648195776 \nu - 2928198930087831936\)\()/ 1272854848402176 \)
\(\beta_{14}\)\(=\)\((\)\(675317839955 \nu^{16} + 243395485401 \nu^{15} - 61841470872860 \nu^{14} - 30140401950649 \nu^{13} + 2249668835964160 \nu^{12} + 1315078485106674 \nu^{11} - 41513401198814472 \nu^{10} - 26710083159987914 \nu^{9} + 411757727189966691 \nu^{8} + 272186564278781773 \nu^{7} - 2145114106598099740 \nu^{6} - 1354855667488657661 \nu^{5} + 5367091938556200850 \nu^{4} + 2788838231561584616 \nu^{3} - 5737153716470261896 \nu^{2} - 1715976290236193408 \nu + 2093833919768929152\)\()/ 848569898934784 \)
\(\beta_{15}\)\(=\)\((\)\(1065195885323 \nu^{16} + 1645100590137 \nu^{15} - 102145188288576 \nu^{14} - 145383659542541 \nu^{13} + 3886424400598324 \nu^{12} + 4982014321895306 \nu^{11} - 74822743750790228 \nu^{10} - 83967917983836698 \nu^{9} + 771059210673477359 \nu^{8} + 734380524240292341 \nu^{7} - 4145208641296852860 \nu^{6} - 3235272068009467961 \nu^{5} + 10630004849190802058 \nu^{4} + 6133948884109375400 \nu^{3} - 11602314997785132840 \nu^{2} - 3611265927827275008 \nu + 4275589606064369280\)\()/ 1272854848402176 \)
\(\beta_{16}\)\(=\)\((\)\(545813418179 \nu^{16} + 175570637925 \nu^{15} - 49939980916872 \nu^{14} - 23109325952993 \nu^{13} + 1816632666902092 \nu^{12} + 1043295102341930 \nu^{11} - 33564147748287224 \nu^{10} - 21700481283193418 \nu^{9} + 334010781638642795 \nu^{8} + 225318961399736769 \nu^{7} - 1752000049600615440 \nu^{6} - 1137776041803359045 \nu^{5} + 4443553755638853758 \nu^{4} + 2370999058449702464 \nu^{3} - 4877019690883807080 \nu^{2} - 1505709942259284384 \nu + 1836496519941294336\)\()/ 636427424201088 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 11\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 19 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{14} + \beta_{12} - \beta_{10} - \beta_{8} - \beta_{6} - \beta_{5} - \beta_{4} + 26 \beta_{2} + 4 \beta_{1} + 207\)
\(\nu^{5}\)\(=\)\(-2 \beta_{16} + 2 \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + 4 \beta_{9} - \beta_{7} - 8 \beta_{6} + 4 \beta_{4} + 32 \beta_{3} + 2 \beta_{2} + 411 \beta_{1} + 63\)
\(\nu^{6}\)\(=\)\(\beta_{15} + 36 \beta_{14} + \beta_{13} + 40 \beta_{12} + 8 \beta_{11} - 37 \beta_{10} - 8 \beta_{9} - 37 \beta_{8} + 7 \beta_{7} - 30 \beta_{6} - 50 \beta_{5} - 46 \beta_{4} + 3 \beta_{3} + 648 \beta_{2} + 191 \beta_{1} + 4479\)
\(\nu^{7}\)\(=\)\(-88 \beta_{16} + 16 \beta_{15} + 77 \beta_{14} - 28 \beta_{13} + 43 \beta_{12} - 44 \beta_{11} + 55 \beta_{10} + 176 \beta_{9} + \beta_{8} + 14 \beta_{7} - 407 \beta_{6} + \beta_{5} + 177 \beta_{4} + 892 \beta_{3} + 113 \beta_{2} + 9485 \beta_{1} + 2488\)
\(\nu^{8}\)\(=\)\(2 \beta_{16} + 70 \beta_{15} + 1024 \beta_{14} + 65 \beta_{13} + 1243 \beta_{12} + 411 \beta_{11} - 1055 \beta_{10} - 398 \beta_{9} - 1096 \beta_{8} + 315 \beta_{7} - 636 \beta_{6} - 1750 \beta_{5} - 1570 \beta_{4} + 109 \beta_{3} + 16265 \beta_{2} + 6678 \beta_{1} + 103475\)
\(\nu^{9}\)\(=\)\(-2892 \beta_{16} + 887 \beta_{15} + 2231 \beta_{14} - 603 \beta_{13} + 1393 \beta_{12} - 1450 \beta_{11} + 2076 \beta_{10} + 5790 \beta_{9} + 90 \beta_{8} + 1557 \beta_{7} - 14487 \beta_{6} + 107 \beta_{5} + 5847 \beta_{4} + 23948 \beta_{3} + 4106 \beta_{2} + 227800 \beta_{1} + 82167\)
\(\nu^{10}\)\(=\)\(154 \beta_{16} + 2932 \beta_{15} + 27330 \beta_{14} + 2713 \beta_{13} + 35465 \beta_{12} + 14693 \beta_{11} - 28161 \beta_{10} - 14078 \beta_{9} - 30310 \beta_{8} + 10457 \beta_{7} - 11940 \beta_{6} - 53662 \beta_{5} - 47634 \beta_{4} + 2872 \beta_{3} + 412481 \beta_{2} + 205178 \beta_{1} + 2487570\)
\(\nu^{11}\)\(=\)\(-85404 \beta_{16} + 33111 \beta_{15} + 59472 \beta_{14} - 11917 \beta_{13} + 40444 \beta_{12} - 43356 \beta_{11} + 66825 \beta_{10} + 171188 \beta_{9} + 4549 \beta_{8} + 65085 \beta_{7} - 449074 \beta_{6} + 4966 \beta_{5} + 173538 \beta_{4} + 633612 \beta_{3} + 126417 \beta_{2} + 5616564 \beta_{1} + 2465723\)
\(\nu^{12}\)\(=\)\(7496 \beta_{16} + 99188 \beta_{15} + 714934 \beta_{14} + 93336 \beta_{13} + 973360 \beta_{12} + 454612 \beta_{11} - 739022 \beta_{10} - 435808 \beta_{9} - 814256 \beta_{8} + 309022 \beta_{7} - 214708 \beta_{6} - 1544092 \beta_{5} - 1361140 \beta_{4} + 66965 \beta_{3} + 10549231 \beta_{2} + 5886766 \beta_{1} + 61381628\)
\(\nu^{13}\)\(=\)\(-2395480 \beta_{16} + 1052466 \beta_{15} + 1542287 \beta_{14} - 225820 \beta_{13} + 1112321 \beta_{12} - 1241998 \beta_{11} + 1978649 \beta_{10} + 4807826 \beta_{9} + 179927 \beta_{8} + 2143986 \beta_{7} - 13011165 \beta_{6} + 172653 \beta_{5} + 4892077 \beta_{4} + 16650981 \beta_{3} + 3588879 \beta_{2} + 140948257 \beta_{1} + 69948387\)
\(\nu^{14}\)\(=\)\(295642 \beta_{16} + 3011076 \beta_{15} + 18586881 \beta_{14} + 2905893 \beta_{13} + 26170874 \beta_{12} + 13077169 \beta_{11} - 19335396 \beta_{10} - 12623450 \beta_{9} - 21557907 \beta_{8} + 8607175 \beta_{7} - 3873925 \beta_{6} - 42909827 \beta_{5} - 37597319 \beta_{4} + 1451717 \beta_{3} + 271467880 \beta_{2} + 162251368 \beta_{1} + 1541335383\)
\(\nu^{15}\)\(=\)\(-65313734 \beta_{16} + 30847469 \beta_{15} + 39648325 \beta_{14} - 4120830 \beta_{13} + 29698754 \beta_{12} - 34787907 \beta_{11} + 55835459 \beta_{10} + 131277070 \beta_{9} + 6265300 \beta_{8} + 63581998 \beta_{7} - 363281423 \beta_{6} + 5194479 \beta_{5} + 134075399 \beta_{4} + 436056763 \beta_{3} + 97351878 \beta_{2} + 3579667870 \beta_{1} + 1916892151\)
\(\nu^{16}\)\(=\)\(10358698 \beta_{16} + 86056501 \beta_{15} + 482470073 \beta_{14} + 85537486 \beta_{13} + 695333740 \beta_{12} + 361031457 \beta_{11} - 506175581 \beta_{10} - 352335078 \beta_{9} - 566112138 \beta_{8} + 231707528 \beta_{7} - 73308841 \beta_{6} - 1168222151 \beta_{5} - 1017655863 \beta_{4} + 29349413 \beta_{3} + 7015991883 \beta_{2} + 4362296937 \beta_{1} + 39163943496\)

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
5.10562
5.08131
4.11212
3.17505
3.12700
1.71387
1.06843
0.664163
0.631226
−0.991815
−1.41378
−1.88904
−2.78419
−4.13824
−4.15571
−4.19083
−5.11519
−5.10562 4.57504 18.0674 −15.4387 −23.3584 −16.7622 −51.4001 −6.06900 78.8244
1.2 −5.08131 −7.39360 17.8197 8.39690 37.5692 −14.1925 −49.8972 27.6653 −42.6673
1.3 −4.11212 −4.01551 8.90957 −12.3424 16.5123 29.6370 −3.74025 −10.8757 50.7537
1.4 −3.17505 −0.600459 2.08094 12.4679 1.90649 10.6254 18.7933 −26.6394 −39.5863
1.5 −3.12700 −0.537604 1.77811 −0.516265 1.68109 −34.6419 19.4558 −26.7110 1.61436
1.6 −1.71387 7.58218 −5.06264 12.0244 −12.9949 1.14865 22.3877 30.4895 −20.6082
1.7 −1.06843 −9.85108 −6.85846 2.35929 10.5252 12.2506 15.8752 70.0437 −2.52073
1.8 −0.664163 −0.846704 −7.55889 −8.92433 0.562349 3.53056 10.3336 −26.2831 5.92721
1.9 −0.631226 9.24271 −7.60155 −19.0524 −5.83424 −25.4436 9.84810 58.4278 12.0263
1.10 0.991815 −5.36099 −7.01630 18.5970 −5.31711 −13.4049 −14.8934 1.74021 18.4447
1.11 1.41378 5.76511 −6.00123 −4.58681 8.15059 36.4239 −19.7947 6.23646 −6.48474
1.12 1.88904 1.56640 −4.43152 2.20557 2.95900 −1.80472 −23.4837 −24.5464 4.16640
1.13 2.78419 −8.94641 −0.248293 −17.0433 −24.9085 −30.9353 −22.9648 53.0382 −47.4517
1.14 4.13824 −5.47866 9.12507 −3.23279 −22.6720 3.07367 4.65582 3.01573 −13.3781
1.15 4.15571 7.17608 9.26990 −11.1721 29.8217 −7.78423 5.27734 24.4962 −46.4282
1.16 4.19083 4.70544 9.56305 14.4259 19.7197 −34.7012 6.55048 −4.85885 60.4566
1.17 5.11519 −3.58195 18.1651 −2.16776 −18.3223 20.9807 51.9966 −14.1696 −11.0885
\(n\): 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.g 17
13.b even 2 1 1859.4.a.h 17
13.c even 3 2 143.4.e.b 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.e.b 34 13.c even 3 2
1859.4.a.g 17 1.a even 1 1 trivial
1859.4.a.h 17 13.b even 2 1

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

$p$ $F_p(T)$
$2$ \( 2210688 + 5215872 T - 3207432 T^{2} - 12177120 T^{3} + 984554 T^{4} + 9967603 T^{5} + 79027 T^{6} - 3663297 T^{7} - 59532 T^{8} + 669163 T^{9} + 7942 T^{10} - 65242 T^{11} - 406 T^{12} + 3449 T^{13} + 7 T^{14} - 93 T^{15} + T^{17} \)
$3$ \( -7355831456 - 31850506672 T - 37852871232 T^{2} + 4584861152 T^{3} + 25812899408 T^{4} + 4228659588 T^{5} - 3818319924 T^{6} - 751894616 T^{7} + 251490944 T^{8} + 50687200 T^{9} - 8707814 T^{10} - 1714135 T^{11} + 163814 T^{12} + 30780 T^{13} - 1574 T^{14} - 279 T^{15} + 6 T^{16} + T^{17} \)
$5$ \( -179912814666900 - 378059620141055 T + 13516668334290 T^{2} + 150901810730739 T^{3} + 19001487093462 T^{4} - 16720771701703 T^{5} - 3417480391182 T^{6} + 363138981528 T^{7} + 105270937256 T^{8} - 1756318920 T^{9} - 1311914434 T^{10} - 18378005 T^{11} + 7855182 T^{12} + 224845 T^{13} - 22354 T^{14} - 822 T^{15} + 24 T^{16} + T^{17} \)
$7$ \( 1557722993557801824 - 1163894432377313680 T - 582028799774660480 T^{2} + 368391230499430224 T^{3} + 12314588696092032 T^{4} - 15916935244358556 T^{5} - 602920288964028 T^{6} + 209432984795024 T^{7} + 10634777029532 T^{8} - 1064216750056 T^{9} - 69168747206 T^{10} + 1840072191 T^{11} + 176279650 T^{12} - 91606 T^{13} - 180196 T^{14} - 1911 T^{15} + 62 T^{16} + T^{17} \)
$11$ \( ( -11 + T )^{17} \)
$13$ \( T^{17} \)
$17$ \( \)\(20\!\cdots\!00\)\( + \)\(47\!\cdots\!75\)\( T + \)\(46\!\cdots\!50\)\( T^{2} - \)\(80\!\cdots\!87\)\( T^{3} - \)\(92\!\cdots\!44\)\( T^{4} - \)\(38\!\cdots\!85\)\( T^{5} + 46376798482436784662 T^{6} + 52142104384310951042 T^{7} + 1331241404485947930 T^{8} - 2265205110593878 T^{9} - 459261582405494 T^{10} - 3039696033695 T^{11} + 59431687236 T^{12} + 588361907 T^{13} - 3417402 T^{14} - 40996 T^{15} + 74 T^{16} + T^{17} \)
$19$ \( \)\(20\!\cdots\!36\)\( - \)\(97\!\cdots\!20\)\( T + \)\(10\!\cdots\!64\)\( T^{2} - \)\(22\!\cdots\!20\)\( T^{3} - \)\(14\!\cdots\!32\)\( T^{4} + \)\(63\!\cdots\!12\)\( T^{5} + \)\(25\!\cdots\!56\)\( T^{6} - \)\(42\!\cdots\!40\)\( T^{7} + 2700913787991695192 T^{8} + 108856083972268336 T^{9} - 984537572264257 T^{10} - 14570970776029 T^{11} + 129531000067 T^{12} + 1157782765 T^{13} - 7535392 T^{14} - 52560 T^{15} + 159 T^{16} + T^{17} \)
$23$ \( \)\(38\!\cdots\!76\)\( + \)\(10\!\cdots\!00\)\( T + \)\(44\!\cdots\!68\)\( T^{2} - \)\(78\!\cdots\!60\)\( T^{3} - \)\(39\!\cdots\!20\)\( T^{4} - \)\(66\!\cdots\!92\)\( T^{5} + \)\(40\!\cdots\!44\)\( T^{6} + \)\(20\!\cdots\!48\)\( T^{7} + \)\(11\!\cdots\!08\)\( T^{8} - 1616078476167365324 T^{9} - 19635806386402185 T^{10} + 20977462050955 T^{11} + 1079387146394 T^{12} + 2305574788 T^{13} - 25278189 T^{14} - 92555 T^{15} + 215 T^{16} + T^{17} \)
$29$ \( \)\(17\!\cdots\!77\)\( - \)\(13\!\cdots\!95\)\( T - \)\(13\!\cdots\!82\)\( T^{2} + \)\(12\!\cdots\!30\)\( T^{3} + \)\(83\!\cdots\!65\)\( T^{4} - \)\(95\!\cdots\!95\)\( T^{5} - \)\(19\!\cdots\!07\)\( T^{6} - \)\(53\!\cdots\!19\)\( T^{7} + \)\(19\!\cdots\!00\)\( T^{8} + 8727746376191827676 T^{9} - 95710813941543632 T^{10} - 522083495270420 T^{11} + 2443813583721 T^{12} + 14720918677 T^{13} - 31145763 T^{14} - 196491 T^{15} + 157 T^{16} + T^{17} \)
$31$ \( \)\(11\!\cdots\!16\)\( + \)\(45\!\cdots\!52\)\( T + \)\(57\!\cdots\!48\)\( T^{2} + \)\(21\!\cdots\!44\)\( T^{3} - \)\(33\!\cdots\!76\)\( T^{4} - \)\(26\!\cdots\!40\)\( T^{5} - \)\(13\!\cdots\!56\)\( T^{6} + \)\(41\!\cdots\!64\)\( T^{7} + \)\(35\!\cdots\!64\)\( T^{8} - 20252297082754241464 T^{9} - 251596797144779320 T^{10} + 198485268402273 T^{11} + 7390788235266 T^{12} + 8234600895 T^{13} - 92583092 T^{14} - 188681 T^{15} + 394 T^{16} + T^{17} \)
$37$ \( \)\(60\!\cdots\!98\)\( + \)\(75\!\cdots\!49\)\( T - \)\(16\!\cdots\!12\)\( T^{2} - \)\(92\!\cdots\!22\)\( T^{3} + \)\(30\!\cdots\!96\)\( T^{4} + \)\(27\!\cdots\!85\)\( T^{5} + \)\(37\!\cdots\!60\)\( T^{6} - \)\(28\!\cdots\!25\)\( T^{7} - \)\(54\!\cdots\!10\)\( T^{8} + \)\(13\!\cdots\!40\)\( T^{9} + 227425381807597400 T^{10} - 3688217386790720 T^{11} - 4192572978312 T^{12} + 52873426449 T^{13} + 33948436 T^{14} - 376589 T^{15} - 88 T^{16} + T^{17} \)
$41$ \( \)\(56\!\cdots\!50\)\( - \)\(34\!\cdots\!45\)\( T - \)\(48\!\cdots\!00\)\( T^{2} + \)\(28\!\cdots\!61\)\( T^{3} + \)\(22\!\cdots\!62\)\( T^{4} - \)\(47\!\cdots\!77\)\( T^{5} - \)\(36\!\cdots\!90\)\( T^{6} + \)\(31\!\cdots\!50\)\( T^{7} + \)\(26\!\cdots\!98\)\( T^{8} - 95113381832271663214 T^{9} - 995836898840489252 T^{10} + 939290999878481 T^{11} + 19287273561822 T^{12} + 10890078943 T^{13} - 175191710 T^{14} - 258656 T^{15} + 512 T^{16} + T^{17} \)
$43$ \( -\)\(12\!\cdots\!00\)\( - \)\(18\!\cdots\!80\)\( T - \)\(25\!\cdots\!80\)\( T^{2} + \)\(31\!\cdots\!76\)\( T^{3} + \)\(37\!\cdots\!52\)\( T^{4} - \)\(13\!\cdots\!32\)\( T^{5} + \)\(32\!\cdots\!84\)\( T^{6} + \)\(76\!\cdots\!60\)\( T^{7} - \)\(33\!\cdots\!36\)\( T^{8} - \)\(15\!\cdots\!84\)\( T^{9} + 8545683034166989263 T^{10} + 11175196869160297 T^{11} - 94866309845971 T^{12} - 2417973069 T^{13} + 483704872 T^{14} - 318528 T^{15} - 927 T^{16} + T^{17} \)
$47$ \( -\)\(24\!\cdots\!72\)\( + \)\(37\!\cdots\!04\)\( T + \)\(12\!\cdots\!04\)\( T^{2} - \)\(39\!\cdots\!60\)\( T^{3} + \)\(21\!\cdots\!92\)\( T^{4} + \)\(16\!\cdots\!56\)\( T^{5} - \)\(25\!\cdots\!92\)\( T^{6} - \)\(40\!\cdots\!52\)\( T^{7} + \)\(66\!\cdots\!84\)\( T^{8} + \)\(60\!\cdots\!36\)\( T^{9} - 8132404700864719575 T^{10} - 55368674332107055 T^{11} + 50329210566093 T^{12} + 302602961061 T^{13} - 144960892 T^{14} - 873530 T^{15} + 143 T^{16} + T^{17} \)
$53$ \( -\)\(60\!\cdots\!50\)\( + \)\(10\!\cdots\!85\)\( T - \)\(38\!\cdots\!80\)\( T^{2} + \)\(58\!\cdots\!03\)\( T^{3} - \)\(31\!\cdots\!88\)\( T^{4} - \)\(10\!\cdots\!19\)\( T^{5} + \)\(16\!\cdots\!46\)\( T^{6} - \)\(24\!\cdots\!72\)\( T^{7} - \)\(29\!\cdots\!16\)\( T^{8} + \)\(84\!\cdots\!04\)\( T^{9} + 23912661771378324164 T^{10} - 91940392504326473 T^{11} - 97663135808460 T^{12} + 473574226541 T^{13} + 182914198 T^{14} - 1142478 T^{15} - 106 T^{16} + T^{17} \)
$59$ \( \)\(23\!\cdots\!00\)\( - \)\(18\!\cdots\!20\)\( T + \)\(62\!\cdots\!40\)\( T^{2} - \)\(48\!\cdots\!36\)\( T^{3} - \)\(22\!\cdots\!76\)\( T^{4} + \)\(30\!\cdots\!76\)\( T^{5} + \)\(13\!\cdots\!64\)\( T^{6} - \)\(66\!\cdots\!28\)\( T^{7} + \)\(29\!\cdots\!48\)\( T^{8} + \)\(69\!\cdots\!44\)\( T^{9} - 43789308359400400962 T^{10} - 384040751432503587 T^{11} + 212213024490218 T^{12} + 1154392582899 T^{13} - 418189950 T^{14} - 1744772 T^{15} + 266 T^{16} + T^{17} \)
$61$ \( -\)\(28\!\cdots\!14\)\( + \)\(67\!\cdots\!51\)\( T - \)\(78\!\cdots\!50\)\( T^{2} - \)\(43\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!32\)\( T^{4} + \)\(11\!\cdots\!09\)\( T^{5} - \)\(28\!\cdots\!42\)\( T^{6} - \)\(14\!\cdots\!21\)\( T^{7} + \)\(33\!\cdots\!90\)\( T^{8} + \)\(11\!\cdots\!58\)\( T^{9} - \)\(20\!\cdots\!14\)\( T^{10} - 496938469243789390 T^{11} + 642794994644688 T^{12} + 1282378725057 T^{13} - 1009738142 T^{14} - 1765853 T^{15} + 624 T^{16} + T^{17} \)
$67$ \( -\)\(56\!\cdots\!48\)\( - \)\(11\!\cdots\!52\)\( T - \)\(53\!\cdots\!64\)\( T^{2} + \)\(23\!\cdots\!44\)\( T^{3} + \)\(18\!\cdots\!52\)\( T^{4} - \)\(65\!\cdots\!84\)\( T^{5} - \)\(19\!\cdots\!28\)\( T^{6} - \)\(10\!\cdots\!24\)\( T^{7} + \)\(96\!\cdots\!88\)\( T^{8} + \)\(86\!\cdots\!72\)\( T^{9} - \)\(25\!\cdots\!68\)\( T^{10} - 2803551880889312383 T^{11} + 3583763078731896 T^{12} + 4450248390048 T^{13} - 2508598594 T^{14} - 3407337 T^{15} + 676 T^{16} + T^{17} \)
$71$ \( -\)\(60\!\cdots\!84\)\( + \)\(73\!\cdots\!88\)\( T + \)\(88\!\cdots\!68\)\( T^{2} - \)\(11\!\cdots\!64\)\( T^{3} - \)\(38\!\cdots\!88\)\( T^{4} + \)\(59\!\cdots\!08\)\( T^{5} + \)\(35\!\cdots\!20\)\( T^{6} - \)\(12\!\cdots\!96\)\( T^{7} + \)\(63\!\cdots\!44\)\( T^{8} + \)\(12\!\cdots\!92\)\( T^{9} - \)\(10\!\cdots\!33\)\( T^{10} - 631630877733482919 T^{11} + 561171004878584 T^{12} + 1625012291098 T^{13} - 1161966385 T^{14} - 2045341 T^{15} + 763 T^{16} + T^{17} \)
$73$ \( -\)\(66\!\cdots\!32\)\( - \)\(58\!\cdots\!84\)\( T + \)\(11\!\cdots\!96\)\( T^{2} + \)\(63\!\cdots\!56\)\( T^{3} - \)\(41\!\cdots\!88\)\( T^{4} - \)\(22\!\cdots\!16\)\( T^{5} + \)\(40\!\cdots\!68\)\( T^{6} + \)\(30\!\cdots\!09\)\( T^{7} + \)\(23\!\cdots\!58\)\( T^{8} - \)\(15\!\cdots\!75\)\( T^{9} - \)\(13\!\cdots\!60\)\( T^{10} + 3358352803917095334 T^{11} + 4687399682293756 T^{12} - 2304940840278 T^{13} - 5889714676 T^{14} - 869451 T^{15} + 2374 T^{16} + T^{17} \)
$79$ \( -\)\(82\!\cdots\!00\)\( - \)\(20\!\cdots\!00\)\( T + \)\(63\!\cdots\!00\)\( T^{2} + \)\(17\!\cdots\!04\)\( T^{3} - \)\(62\!\cdots\!04\)\( T^{4} - \)\(33\!\cdots\!32\)\( T^{5} + \)\(12\!\cdots\!08\)\( T^{6} + \)\(23\!\cdots\!12\)\( T^{7} - \)\(10\!\cdots\!68\)\( T^{8} - \)\(77\!\cdots\!04\)\( T^{9} + \)\(38\!\cdots\!88\)\( T^{10} + 958387594697175060 T^{11} - 7248974249265780 T^{12} + 554714847600 T^{13} + 6576993652 T^{14} - 2049084 T^{15} - 2164 T^{16} + T^{17} \)
$83$ \( -\)\(65\!\cdots\!56\)\( + \)\(14\!\cdots\!84\)\( T + \)\(21\!\cdots\!32\)\( T^{2} - \)\(13\!\cdots\!32\)\( T^{3} - \)\(13\!\cdots\!24\)\( T^{4} + \)\(19\!\cdots\!44\)\( T^{5} + \)\(17\!\cdots\!28\)\( T^{6} - \)\(12\!\cdots\!72\)\( T^{7} - \)\(98\!\cdots\!60\)\( T^{8} + \)\(42\!\cdots\!56\)\( T^{9} + \)\(27\!\cdots\!69\)\( T^{10} - 7804437480807587459 T^{11} - 3963207561752113 T^{12} + 8088211368987 T^{13} + 2819363456 T^{14} - 4425204 T^{15} - 777 T^{16} + T^{17} \)
$89$ \( \)\(19\!\cdots\!36\)\( + \)\(62\!\cdots\!12\)\( T - \)\(84\!\cdots\!44\)\( T^{2} - \)\(75\!\cdots\!84\)\( T^{3} + \)\(70\!\cdots\!12\)\( T^{4} - \)\(64\!\cdots\!64\)\( T^{5} - \)\(27\!\cdots\!64\)\( T^{6} - \)\(85\!\cdots\!48\)\( T^{7} + \)\(57\!\cdots\!44\)\( T^{8} + \)\(12\!\cdots\!08\)\( T^{9} - \)\(67\!\cdots\!55\)\( T^{10} - 24233171704436940463 T^{11} + 42898838177192722 T^{12} + 19879831080900 T^{13} - 13585840661 T^{14} - 7307789 T^{15} + 1687 T^{16} + T^{17} \)
$97$ \( -\)\(46\!\cdots\!84\)\( - \)\(28\!\cdots\!28\)\( T + \)\(34\!\cdots\!40\)\( T^{2} + \)\(15\!\cdots\!08\)\( T^{3} - \)\(75\!\cdots\!16\)\( T^{4} - \)\(31\!\cdots\!28\)\( T^{5} + \)\(57\!\cdots\!60\)\( T^{6} + \)\(31\!\cdots\!96\)\( T^{7} - \)\(17\!\cdots\!04\)\( T^{8} - \)\(12\!\cdots\!24\)\( T^{9} - \)\(10\!\cdots\!11\)\( T^{10} + 15902700333128185479 T^{11} + 25194968476768449 T^{12} + 1202166192207 T^{13} - 13416438197 T^{14} - 4721195 T^{15} + 2047 T^{16} + T^{17} \)
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