Properties

Label 28.4.f.a
Level $28$
Weight $4$
Character orbit 28.f
Analytic conductor $1.652$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 28.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.65205348016\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 2 x^{18} - 24 x^{17} + 28 x^{16} + 56 x^{15} - 192 x^{14} + 352 x^{13} - 448 x^{12} + 5376 x^{11} - 41472 x^{10} + 43008 x^{9} - 28672 x^{8} + 180224 x^{7} - 786432 x^{6} + 1835008 x^{5} + 7340032 x^{4} - 50331648 x^{3} - 33554432 x^{2} + 1073741824\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{24} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{5} ) q^{2} -\beta_{10} q^{3} -\beta_{14} q^{4} + ( \beta_{4} - \beta_{7} ) q^{5} + ( -1 + 2 \beta_{3} - \beta_{4} - \beta_{16} + \beta_{19} ) q^{6} + ( 1 - \beta_{3} + \beta_{9} - \beta_{11} + \beta_{13} + \beta_{14} - \beta_{17} ) q^{7} + ( 4 + \beta_{10} - \beta_{11} - \beta_{15} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{8} + ( -1 - \beta_{2} - 5 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{11} - \beta_{12} - \beta_{14} + \beta_{16} + 2 \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{5} ) q^{2} -\beta_{10} q^{3} -\beta_{14} q^{4} + ( \beta_{4} - \beta_{7} ) q^{5} + ( -1 + 2 \beta_{3} - \beta_{4} - \beta_{16} + \beta_{19} ) q^{6} + ( 1 - \beta_{3} + \beta_{9} - \beta_{11} + \beta_{13} + \beta_{14} - \beta_{17} ) q^{7} + ( 4 + \beta_{10} - \beta_{11} - \beta_{15} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{8} + ( -1 - \beta_{2} - 5 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{11} - \beta_{12} - \beta_{14} + \beta_{16} + 2 \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{9} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{5} - \beta_{6} + 2 \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} + \beta_{16} + \beta_{17} - \beta_{19} ) q^{10} + ( 1 + 2 \beta_{1} - \beta_{3} + \beta_{6} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{13} + 3 \beta_{14} + \beta_{16} - \beta_{19} ) q^{11} + ( -13 + 6 \beta_{3} - 2 \beta_{4} + 3 \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} - \beta_{18} ) q^{12} + ( -3 + 4 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{11} + 2 \beta_{14} - \beta_{16} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{13} + ( -4 - 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{9} - \beta_{10} + 5 \beta_{11} - 2 \beta_{12} + \beta_{15} + \beta_{16} + 3 \beta_{17} - \beta_{19} ) q^{14} + ( -1 + \beta_{2} - 8 \beta_{5} + \beta_{10} + \beta_{12} - \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} - 2 \beta_{18} ) q^{15} + ( 2 - 4 \beta_{1} + 2 \beta_{2} - 12 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} - 4 \beta_{17} ) q^{16} + ( 1 + 4 \beta_{1} - 5 \beta_{2} - 8 \beta_{5} + \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{11} + 3 \beta_{12} + 3 \beta_{14} - \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{17} + ( 9 + 4 \beta_{1} - \beta_{2} - 8 \beta_{3} + 4 \beta_{4} - 3 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} + 4 \beta_{10} - 12 \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{16} + \beta_{17} + \beta_{19} ) q^{18} + ( -2 + 6 \beta_{1} + \beta_{2} + 3 \beta_{3} + 6 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} - 6 \beta_{14} - \beta_{15} - 3 \beta_{16} + 2 \beta_{18} ) q^{19} + ( -4 - \beta_{2} + 8 \beta_{3} + 6 \beta_{4} + 4 \beta_{8} - 4 \beta_{9} + 4 \beta_{11} - 2 \beta_{12} - \beta_{14} + 2 \beta_{16} - 2 \beta_{18} - 2 \beta_{19} ) q^{20} + ( 13 - 14 \beta_{1} + 7 \beta_{2} - \beta_{3} - \beta_{4} + 14 \beta_{5} - \beta_{6} - 4 \beta_{7} + 3 \beta_{8} - \beta_{11} - 3 \beta_{12} - 5 \beta_{14} + \beta_{16} - 2 \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{21} + ( -7 + 3 \beta_{4} - 2 \beta_{6} - 6 \beta_{7} - 11 \beta_{10} + 5 \beta_{11} + 3 \beta_{15} + \beta_{16} + \beta_{17} + 4 \beta_{18} - 2 \beta_{19} ) q^{22} + ( -1 - 4 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} + 4 \beta_{5} - \beta_{6} - 2 \beta_{9} + 3 \beta_{10} + 2 \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} - 2 \beta_{15} - 3 \beta_{16} + 2 \beta_{17} + 6 \beta_{19} ) q^{23} + ( 13 + 4 \beta_{1} + \beta_{2} + 10 \beta_{3} - 8 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + 4 \beta_{9} - 14 \beta_{10} - 4 \beta_{11} + 5 \beta_{12} + \beta_{13} + \beta_{14} - 2 \beta_{15} - \beta_{16} - 3 \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{24} + ( -8 - 4 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} - 8 \beta_{4} + 4 \beta_{6} + 4 \beta_{7} + 8 \beta_{9} - 4 \beta_{11} + 2 \beta_{12} - 6 \beta_{14} - 4 \beta_{16} - 2 \beta_{17} + 2 \beta_{19} ) q^{25} + ( 27 - 2 \beta_{1} - \beta_{2} - 16 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} - 5 \beta_{10} + 7 \beta_{11} + 3 \beta_{12} + \beta_{13} + 7 \beta_{14} + \beta_{15} - \beta_{16} - 4 \beta_{18} ) q^{26} + ( -3 - 36 \beta_{1} - 5 \beta_{2} + 6 \beta_{3} + 18 \beta_{5} + \beta_{6} - 8 \beta_{8} - 6 \beta_{9} - \beta_{10} + 7 \beta_{11} - 3 \beta_{12} - 2 \beta_{13} - 5 \beta_{14} + \beta_{15} + 2 \beta_{16} + 4 \beta_{18} - 3 \beta_{19} ) q^{27} + ( 41 - 3 \beta_{2} - 34 \beta_{3} + 6 \beta_{4} + \beta_{6} - 4 \beta_{7} - 5 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 14 \beta_{11} - 5 \beta_{12} + \beta_{13} - 2 \beta_{15} + \beta_{16} + \beta_{17} + 4 \beta_{18} + \beta_{19} ) q^{28} + ( -19 - 4 \beta_{2} - 3 \beta_{4} + 30 \beta_{5} + \beta_{6} + 6 \beta_{7} + \beta_{11} + 2 \beta_{12} + 4 \beta_{14} - \beta_{16} + \beta_{17} - 9 \beta_{18} - \beta_{19} ) q^{29} + ( 1 + 7 \beta_{2} + 61 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} - 5 \beta_{10} - 3 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} - 2 \beta_{17} ) q^{30} + ( -6 - 4 \beta_{1} + 17 \beta_{2} - 3 \beta_{3} + 8 \beta_{5} - \beta_{6} - 2 \beta_{8} + 3 \beta_{9} + 7 \beta_{10} - 3 \beta_{11} + 9 \beta_{12} - \beta_{13} - 10 \beta_{14} + 2 \beta_{15} + \beta_{16} + 3 \beta_{17} + 4 \beta_{18} ) q^{31} + ( -2 + 16 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{6} - 2 \beta_{8} + 4 \beta_{9} - 16 \beta_{10} + 24 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{16} + 2 \beta_{17} - 2 \beta_{19} ) q^{32} + ( 6 - 20 \beta_{1} - 4 \beta_{2} - \beta_{3} + 4 \beta_{4} - 20 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} + 6 \beta_{8} - 4 \beta_{9} + 4 \beta_{14} + 4 \beta_{16} + 6 \beta_{18} ) q^{33} + ( 24 - 6 \beta_{1} - 2 \beta_{2} - 48 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} + 16 \beta_{8} - 12 \beta_{9} - \beta_{10} + 3 \beta_{11} - 6 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + \beta_{15} + 6 \beta_{16} + 5 \beta_{17} - 8 \beta_{18} - 7 \beta_{19} ) q^{34} + ( 3 + 42 \beta_{1} - 12 \beta_{2} + 5 \beta_{3} - 28 \beta_{5} + \beta_{6} + 10 \beta_{8} - 5 \beta_{9} + \beta_{10} + 8 \beta_{11} - 8 \beta_{12} - 7 \beta_{13} + 9 \beta_{14} + \beta_{16} + 2 \beta_{17} - 8 \beta_{18} - 5 \beta_{19} ) q^{35} + ( 4 + 2 \beta_{2} - 6 \beta_{4} + 12 \beta_{7} + 28 \beta_{10} - 12 \beta_{11} + 2 \beta_{12} - 2 \beta_{14} + 4 \beta_{15} - 2 \beta_{16} + 4 \beta_{17} + 10 \beta_{18} - 6 \beta_{19} ) q^{36} + ( 3 + 14 \beta_{1} + 7 \beta_{2} + 21 \beta_{3} - 14 \beta_{5} + 3 \beta_{6} - 7 \beta_{8} + 4 \beta_{9} - \beta_{11} + \beta_{12} + 3 \beta_{14} - 3 \beta_{16} - 6 \beta_{17} + 7 \beta_{18} + 6 \beta_{19} ) q^{37} + ( -52 - 9 \beta_{2} - 57 \beta_{3} + 5 \beta_{6} + 3 \beta_{7} + 4 \beta_{8} + 6 \beta_{9} + 30 \beta_{10} - 6 \beta_{11} + 7 \beta_{12} + 5 \beta_{13} + 7 \beta_{14} - 10 \beta_{15} - 5 \beta_{16} - 5 \beta_{17} - 8 \beta_{18} + 6 \beta_{19} ) q^{38} + ( 9 + 20 \beta_{1} + 4 \beta_{2} - 13 \beta_{3} - \beta_{6} + 4 \beta_{8} + 13 \beta_{9} + 10 \beta_{10} - 25 \beta_{11} + 4 \beta_{12} + 9 \beta_{13} + 13 \beta_{14} - \beta_{16} - 4 \beta_{17} + 5 \beta_{19} ) q^{39} + ( -99 - 8 \beta_{1} + \beta_{2} + 50 \beta_{3} + 4 \beta_{4} - 8 \beta_{5} - 11 \beta_{6} - 4 \beta_{7} - 5 \beta_{8} - 2 \beta_{9} + 11 \beta_{10} - 15 \beta_{11} + \beta_{12} + 5 \beta_{13} - 3 \beta_{14} + 5 \beta_{15} + 7 \beta_{16} - 5 \beta_{18} ) q^{40} + ( 15 + 64 \beta_{1} + 2 \beta_{2} - 30 \beta_{3} + 3 \beta_{4} - 32 \beta_{5} - 5 \beta_{6} - 14 \beta_{8} - 12 \beta_{9} + 7 \beta_{11} - 6 \beta_{12} + 2 \beta_{14} + 5 \beta_{16} + 5 \beta_{17} + 7 \beta_{18} - 5 \beta_{19} ) q^{41} + ( -105 + 9 \beta_{2} + 112 \beta_{3} - 4 \beta_{4} + 7 \beta_{5} + 11 \beta_{6} - 2 \beta_{7} - 20 \beta_{8} + 10 \beta_{9} + \beta_{10} + 17 \beta_{11} + \beta_{12} + \beta_{13} - 3 \beta_{14} - 5 \beta_{15} - 7 \beta_{16} - 14 \beta_{17} + 16 \beta_{18} + 8 \beta_{19} ) q^{42} + ( 4 - 4 \beta_{2} - 64 \beta_{5} + 2 \beta_{6} - 12 \beta_{10} + 4 \beta_{11} - 4 \beta_{12} + 4 \beta_{14} - 8 \beta_{15} - 6 \beta_{16} - 6 \beta_{17} - 16 \beta_{18} + 2 \beta_{19} ) q^{43} + ( -7 + 16 \beta_{1} - 4 \beta_{2} - 114 \beta_{3} - 16 \beta_{5} - 7 \beta_{6} + 3 \beta_{8} - 10 \beta_{9} + 5 \beta_{10} + 19 \beta_{11} - 3 \beta_{12} - 11 \beta_{13} - 7 \beta_{14} + 11 \beta_{15} - \beta_{16} + 14 \beta_{17} - 3 \beta_{18} + 2 \beta_{19} ) q^{44} + ( -20 + 18 \beta_{1} - 11 \beta_{2} - 11 \beta_{3} - 36 \beta_{5} - 5 \beta_{7} - \beta_{8} - 4 \beta_{9} + 4 \beta_{11} + \beta_{12} + \beta_{14} + 9 \beta_{17} + 2 \beta_{18} - 9 \beta_{19} ) q^{45} + ( 17 + 2 \beta_{2} - 19 \beta_{3} - 10 \beta_{4} + 10 \beta_{6} + 5 \beta_{7} - 8 \beta_{8} + 2 \beta_{9} + 16 \beta_{10} - 30 \beta_{11} + 2 \beta_{12} - 6 \beta_{13} + 4 \beta_{16} - 2 \beta_{17} - 5 \beta_{19} ) q^{46} + ( 3 + 28 \beta_{1} + 4 \beta_{2} - 11 \beta_{3} + 28 \beta_{5} + 5 \beta_{6} + 6 \beta_{8} + 11 \beta_{9} - 18 \beta_{10} + 19 \beta_{11} + 8 \beta_{12} + 7 \beta_{13} + 11 \beta_{14} + 7 \beta_{15} + 12 \beta_{16} + 6 \beta_{18} ) q^{47} + ( -56 - 24 \beta_{1} + 4 \beta_{2} + 112 \beta_{3} - 18 \beta_{4} + 12 \beta_{5} + 20 \beta_{8} + 4 \beta_{9} + 12 \beta_{11} + 2 \beta_{12} + 4 \beta_{14} - 14 \beta_{16} - 10 \beta_{18} + 14 \beta_{19} ) q^{48} + ( -56 - 56 \beta_{1} + 28 \beta_{3} + 7 \beta_{4} + 56 \beta_{5} + 7 \beta_{6} + 14 \beta_{7} + 14 \beta_{8} + 7 \beta_{11} - 7 \beta_{16} + 7 \beta_{17} - 21 \beta_{18} - 7 \beta_{19} ) q^{49} + ( -48 + 4 \beta_{2} - 4 \beta_{4} - 2 \beta_{5} + 8 \beta_{6} + 8 \beta_{7} - 16 \beta_{10} + 8 \beta_{11} + 4 \beta_{12} - 4 \beta_{14} - 16 \beta_{15} - 4 \beta_{16} - 8 \beta_{17} + 24 \beta_{18} + 12 \beta_{19} ) q^{50} + ( 2 - 14 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} + 14 \beta_{5} + 2 \beta_{6} - 6 \beta_{8} - 3 \beta_{9} - 24 \beta_{10} - 6 \beta_{11} - 5 \beta_{12} - 13 \beta_{13} + 2 \beta_{14} + 13 \beta_{15} + 15 \beta_{16} - 4 \beta_{17} + 6 \beta_{18} - 30 \beta_{19} ) q^{51} + ( 80 - 16 \beta_{1} - 4 \beta_{2} + 92 \beta_{3} + 32 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} + 6 \beta_{8} - 6 \beta_{9} - 24 \beta_{10} + 6 \beta_{11} - 4 \beta_{13} - 4 \beta_{14} + 8 \beta_{15} + 4 \beta_{16} + 12 \beta_{17} - 12 \beta_{18} - 2 \beta_{19} ) q^{52} + ( 66 - 22 \beta_{1} - 9 \beta_{2} - 57 \beta_{3} + 40 \beta_{4} - 18 \beta_{6} - 20 \beta_{7} + 15 \beta_{8} - 16 \beta_{9} - 2 \beta_{11} - 9 \beta_{12} - 13 \beta_{14} + 18 \beta_{16} + 9 \beta_{17} - 9 \beta_{19} ) q^{53} + ( 295 + 9 \beta_{2} - 137 \beta_{3} + 7 \beta_{4} - 11 \beta_{6} - 7 \beta_{7} - 4 \beta_{8} - 14 \beta_{9} - 7 \beta_{10} + 5 \beta_{11} - 7 \beta_{12} - 5 \beta_{13} - 39 \beta_{14} - 5 \beta_{15} + 6 \beta_{16} - 4 \beta_{18} ) q^{54} + ( 8 - 96 \beta_{1} + 8 \beta_{2} - 16 \beta_{3} + 48 \beta_{5} - 3 \beta_{6} - 24 \beta_{8} + 16 \beta_{9} + 8 \beta_{10} - 22 \beta_{11} + 8 \beta_{12} + 16 \beta_{13} + 8 \beta_{14} - 8 \beta_{15} - 11 \beta_{16} - 5 \beta_{17} + 12 \beta_{18} + 19 \beta_{19} ) q^{55} + ( 166 + 10 \beta_{2} - 148 \beta_{3} - 18 \beta_{4} + 28 \beta_{5} - 6 \beta_{6} + 12 \beta_{7} - 22 \beta_{8} + 4 \beta_{9} + 5 \beta_{10} - 21 \beta_{11} + 4 \beta_{12} - 6 \beta_{13} - 2 \beta_{14} + 11 \beta_{15} - 8 \beta_{16} + \beta_{17} + 19 \beta_{18} - 7 \beta_{19} ) q^{56} + ( 83 + 8 \beta_{2} + 16 \beta_{4} + 100 \beta_{5} - 4 \beta_{6} - 32 \beta_{7} - 4 \beta_{11} - 4 \beta_{12} - 8 \beta_{14} + 4 \beta_{16} - 4 \beta_{17} - 20 \beta_{18} + 4 \beta_{19} ) q^{57} + ( -3 + 8 \beta_{1} - 39 \beta_{2} + 240 \beta_{3} + 2 \beta_{4} - 8 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + 12 \beta_{8} - 6 \beta_{9} - 9 \beta_{10} - 9 \beta_{11} - 3 \beta_{12} + 3 \beta_{13} - 3 \beta_{14} - 3 \beta_{15} + \beta_{16} + 6 \beta_{17} - 12 \beta_{18} - 2 \beta_{19} ) q^{58} + ( 16 - 36 \beta_{1} - 22 \beta_{2} - 2 \beta_{3} + 72 \beta_{5} + 6 \beta_{6} - 8 \beta_{8} + 2 \beta_{9} - 17 \beta_{10} - 2 \beta_{11} - 14 \beta_{12} + 6 \beta_{13} + 20 \beta_{14} - 12 \beta_{15} - 6 \beta_{16} - 18 \beta_{17} + 16 \beta_{18} ) q^{59} + ( -73 - 64 \beta_{1} + 7 \beta_{2} + 66 \beta_{3} - 8 \beta_{4} + 3 \beta_{6} + 4 \beta_{7} - 11 \beta_{8} + 6 \beta_{9} + 8 \beta_{11} + 7 \beta_{12} + 11 \beta_{13} + 4 \beta_{14} - 11 \beta_{16} - 7 \beta_{17} + 11 \beta_{19} ) q^{60} + ( 5 - 54 \beta_{1} + 3 \beta_{2} - 9 \beta_{3} - 32 \beta_{4} - 54 \beta_{5} + 13 \beta_{6} + 32 \beta_{7} + 9 \beta_{8} + 8 \beta_{9} + 5 \beta_{11} + 5 \beta_{12} + 7 \beta_{14} - 13 \beta_{16} + 9 \beta_{18} ) q^{61} + ( 57 + 8 \beta_{2} - 114 \beta_{3} + 9 \beta_{4} + 20 \beta_{6} + 16 \beta_{8} + 32 \beta_{9} + 7 \beta_{10} - 17 \beta_{11} + 16 \beta_{12} + 14 \beta_{13} + 8 \beta_{14} - 7 \beta_{15} - 11 \beta_{16} - 27 \beta_{17} - 8 \beta_{18} + 18 \beta_{19} ) q^{62} + ( -10 + 84 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} - 84 \beta_{5} - 6 \beta_{6} + 24 \beta_{8} + 3 \beta_{9} - 13 \beta_{10} - 21 \beta_{11} + 13 \beta_{12} + 15 \beta_{13} - 18 \beta_{14} + \beta_{15} - 5 \beta_{16} + 8 \beta_{17} - 22 \beta_{18} + 22 \beta_{19} ) q^{63} + ( 144 - 8 \beta_{2} + 20 \beta_{4} - 8 \beta_{5} - 40 \beta_{7} - 4 \beta_{12} + 8 \beta_{14} + 12 \beta_{16} + 28 \beta_{18} + 12 \beta_{19} ) q^{64} + ( 3 + 8 \beta_{1} + 3 \beta_{2} - 7 \beta_{3} - 15 \beta_{4} - 8 \beta_{5} + 3 \beta_{6} - 15 \beta_{7} - 5 \beta_{8} + 6 \beta_{9} - 3 \beta_{11} + 3 \beta_{12} + 3 \beta_{14} - 3 \beta_{16} - 6 \beta_{17} + 5 \beta_{18} + 6 \beta_{19} ) q^{65} + ( -148 - \beta_{1} + 36 \beta_{2} - 144 \beta_{3} + 2 \beta_{5} - 4 \beta_{6} - 8 \beta_{7} + 8 \beta_{8} - 8 \beta_{9} + 8 \beta_{10} + 8 \beta_{11} - 12 \beta_{12} - 4 \beta_{13} - 20 \beta_{14} + 8 \beta_{15} + 4 \beta_{16} + 4 \beta_{17} - 16 \beta_{18} - 12 \beta_{19} ) q^{66} + ( -26 + 40 \beta_{1} - 20 \beta_{2} + 46 \beta_{3} - 10 \beta_{6} + 14 \beta_{8} - 46 \beta_{9} - 39 \beta_{10} + 84 \beta_{11} - 20 \beta_{12} - 30 \beta_{13} - 42 \beta_{14} - 10 \beta_{16} + 20 \beta_{17} - 10 \beta_{19} ) q^{67} + ( -396 + 32 \beta_{1} - 7 \beta_{2} + 204 \beta_{3} + 10 \beta_{4} + 32 \beta_{5} - 4 \beta_{6} - 10 \beta_{7} - 10 \beta_{8} + 2 \beta_{9} - 12 \beta_{10} + 2 \beta_{11} - 14 \beta_{12} - 4 \beta_{13} + 2 \beta_{14} - 4 \beta_{15} - 14 \beta_{16} - 10 \beta_{18} ) q^{68} + ( -58 + 136 \beta_{1} - 8 \beta_{2} + 116 \beta_{3} + \beta_{4} - 68 \beta_{5} + 8 \beta_{6} - 36 \beta_{8} + 24 \beta_{9} - 16 \beta_{11} + 12 \beta_{12} - 8 \beta_{14} - 8 \beta_{16} - 8 \beta_{17} + 18 \beta_{18} + 8 \beta_{19} ) q^{69} + ( -307 - 12 \beta_{2} + 229 \beta_{3} + 20 \beta_{4} - 14 \beta_{6} - 25 \beta_{7} - 8 \beta_{8} - 20 \beta_{9} - 13 \beta_{10} + 15 \beta_{11} - 4 \beta_{12} - 8 \beta_{13} + 24 \beta_{14} + 5 \beta_{15} + 14 \beta_{16} + 15 \beta_{17} + 12 \beta_{18} - 22 \beta_{19} ) q^{70} + ( -14 + 22 \beta_{2} - 112 \beta_{5} - 8 \beta_{6} + 54 \beta_{10} - 24 \beta_{11} + 14 \beta_{12} - 22 \beta_{14} + 22 \beta_{15} + 14 \beta_{16} + 14 \beta_{17} - 26 \beta_{18} - 8 \beta_{19} ) q^{71} + ( -4 - 24 \beta_{1} - 16 \beta_{2} - 480 \beta_{3} + 28 \beta_{4} + 24 \beta_{5} - 4 \beta_{6} + 28 \beta_{7} + 12 \beta_{8} - 4 \beta_{9} - 4 \beta_{10} - 16 \beta_{11} + 12 \beta_{13} - 4 \beta_{14} - 12 \beta_{15} + 8 \beta_{16} + 8 \beta_{17} - 12 \beta_{18} - 16 \beta_{19} ) q^{72} + ( 39 + 16 \beta_{1} + 40 \beta_{2} + 15 \beta_{3} - 32 \beta_{5} + 14 \beta_{7} - 14 \beta_{8} + 8 \beta_{9} - 8 \beta_{11} - 8 \beta_{12} - 8 \beta_{14} - 24 \beta_{17} + 28 \beta_{18} + 24 \beta_{19} ) q^{73} + ( 86 - 25 \beta_{1} + 10 \beta_{2} - 96 \beta_{3} - 12 \beta_{4} + 10 \beta_{6} + 6 \beta_{7} - 12 \beta_{8} + 2 \beta_{9} + 18 \beta_{11} + 10 \beta_{12} + 10 \beta_{13} + 20 \beta_{14} + 2 \beta_{16} - 10 \beta_{17} + 4 \beta_{19} ) q^{74} + ( -2 + 44 \beta_{1} - 14 \beta_{2} + 12 \beta_{3} + 44 \beta_{5} + 10 \beta_{6} + 10 \beta_{8} - 12 \beta_{9} + 66 \beta_{10} - 60 \beta_{11} - 10 \beta_{12} - 16 \beta_{13} + 6 \beta_{14} - 16 \beta_{15} - 6 \beta_{16} + 10 \beta_{18} ) q^{75} + ( -70 + 96 \beta_{1} + \beta_{2} + 140 \beta_{3} + 8 \beta_{4} - 48 \beta_{5} - 2 \beta_{6} + 30 \beta_{8} + 12 \beta_{9} + \beta_{10} - 29 \beta_{11} + 6 \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} + 34 \beta_{16} + \beta_{17} - 15 \beta_{18} - 33 \beta_{19} ) q^{76} + ( 59 - 84 \beta_{1} - 42 \beta_{2} - 130 \beta_{3} - 25 \beta_{4} + 42 \beta_{5} - 11 \beta_{6} + 5 \beta_{7} + 26 \beta_{8} - 11 \beta_{11} + 16 \beta_{12} + 22 \beta_{14} + 11 \beta_{16} - \beta_{17} - 11 \beta_{18} + \beta_{19} ) q^{77} + ( -212 - 24 \beta_{2} - 20 \beta_{4} + 2 \beta_{6} + 40 \beta_{7} - 9 \beta_{10} + 15 \beta_{11} - 8 \beta_{12} + 24 \beta_{14} + 17 \beta_{15} - 2 \beta_{16} + 19 \beta_{17} + 8 \beta_{18} - 19 \beta_{19} ) q^{78} + ( 5 - 40 \beta_{1} + 33 \beta_{2} - 30 \beta_{3} + 40 \beta_{5} + 5 \beta_{6} - 12 \beta_{8} + 30 \beta_{9} + 71 \beta_{10} + 24 \beta_{11} + 25 \beta_{12} + 22 \beta_{13} + 5 \beta_{14} - 22 \beta_{15} - 17 \beta_{16} - 10 \beta_{17} + 12 \beta_{18} + 34 \beta_{19} ) q^{79} + ( 230 + 52 \beta_{1} - 2 \beta_{2} + 236 \beta_{3} - 104 \beta_{5} - 2 \beta_{6} + 22 \beta_{7} + 8 \beta_{8} - 18 \beta_{9} + 28 \beta_{10} + 18 \beta_{11} - 30 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + 4 \beta_{15} + 2 \beta_{16} + 6 \beta_{17} - 16 \beta_{18} - 16 \beta_{19} ) q^{80} + ( -129 - 48 \beta_{1} + 7 \beta_{2} + 122 \beta_{3} - 62 \beta_{4} + 14 \beta_{6} + 31 \beta_{7} - \beta_{8} - 10 \beta_{9} + 24 \beta_{11} + 7 \beta_{12} + 55 \beta_{14} - 14 \beta_{16} - 7 \beta_{17} + 7 \beta_{19} ) q^{81} + ( 551 + 18 \beta_{1} - 33 \beta_{2} - 280 \beta_{3} + 10 \beta_{4} + 18 \beta_{5} - \beta_{6} - 10 \beta_{7} - 16 \beta_{8} + 14 \beta_{9} + 7 \beta_{10} - 17 \beta_{11} - 5 \beta_{12} + 5 \beta_{13} + 75 \beta_{14} + 5 \beta_{15} - 9 \beta_{16} - 16 \beta_{18} ) q^{82} + ( -14 - 64 \beta_{1} - 6 \beta_{2} + 28 \beta_{3} + 32 \beta_{5} - 4 \beta_{6} - 20 \beta_{8} - 28 \beta_{9} - 22 \beta_{10} + 50 \beta_{11} - 14 \beta_{12} - 44 \beta_{13} - 6 \beta_{14} + 22 \beta_{15} + 18 \beta_{16} + 26 \beta_{17} + 10 \beta_{18} - 40 \beta_{19} ) q^{83} + ( 508 + 9 \beta_{2} - 388 \beta_{3} - 112 \beta_{5} + 8 \beta_{6} - 14 \beta_{7} - 18 \beta_{8} + 10 \beta_{9} - 20 \beta_{10} + 42 \beta_{11} + 20 \beta_{12} + 8 \beta_{13} + 10 \beta_{14} - 12 \beta_{15} + 24 \beta_{16} - 20 \beta_{17} + 20 \beta_{18} + 14 \beta_{19} ) q^{84} + ( -169 + 12 \beta_{2} - 26 \beta_{4} + 78 \beta_{5} + \beta_{6} + 52 \beta_{7} + \beta_{11} - 6 \beta_{12} - 12 \beta_{14} - \beta_{16} + \beta_{17} - 19 \beta_{18} - \beta_{19} ) q^{85} + ( 76 \beta_{2} + 542 \beta_{3} - 14 \beta_{4} - 14 \beta_{7} + 12 \beta_{9} + 24 \beta_{10} + 4 \beta_{11} + 12 \beta_{12} + 8 \beta_{13} - 8 \beta_{15} - 6 \beta_{16} + 12 \beta_{19} ) q^{86} + ( -7 - 24 \beta_{1} - 48 \beta_{2} + 23 \beta_{3} + 48 \beta_{5} - 7 \beta_{6} - 2 \beta_{8} - 23 \beta_{9} + 18 \beta_{10} + 23 \beta_{11} - 16 \beta_{12} - 7 \beta_{13} + 9 \beta_{14} + 14 \beta_{15} + 7 \beta_{16} + 30 \beta_{17} + 4 \beta_{18} + 9 \beta_{19} ) q^{87} + ( -143 + 124 \beta_{1} + 5 \beta_{2} + 138 \beta_{3} + 28 \beta_{4} + 21 \beta_{6} - 14 \beta_{7} - 11 \beta_{8} - 24 \beta_{9} + 32 \beta_{10} - 30 \beta_{11} + 5 \beta_{12} - 11 \beta_{13} + \beta_{14} + 15 \beta_{16} - 5 \beta_{17} - 13 \beta_{19} ) q^{88} + ( -216 + 4 \beta_{1} + 22 \beta_{2} + 103 \beta_{3} + 24 \beta_{4} + 4 \beta_{5} + 10 \beta_{6} - 24 \beta_{7} - 2 \beta_{8} + 16 \beta_{9} - 6 \beta_{11} - 6 \beta_{12} - 34 \beta_{14} - 10 \beta_{16} - 2 \beta_{18} ) q^{89} + ( 160 + 40 \beta_{1} + 2 \beta_{2} - 320 \beta_{3} + 18 \beta_{4} - 20 \beta_{5} - 20 \beta_{6} + 24 \beta_{8} - 4 \beta_{9} - 15 \beta_{10} - 11 \beta_{11} - 2 \beta_{12} - 30 \beta_{13} + 2 \beta_{14} + 15 \beta_{15} + 2 \beta_{16} + 35 \beta_{17} - 12 \beta_{18} - 17 \beta_{19} ) q^{90} + ( -9 + 43 \beta_{2} - 8 \beta_{3} - 42 \beta_{5} + 11 \beta_{6} - 2 \beta_{8} + 8 \beta_{9} + 67 \beta_{10} + 18 \beta_{11} + 17 \beta_{12} - 27 \beta_{14} - 7 \beta_{15} + 4 \beta_{16} - 20 \beta_{17} - 4 \beta_{18} - 13 \beta_{19} ) q^{91} + ( 172 - 3 \beta_{2} - 6 \beta_{4} + 32 \beta_{5} - 4 \beta_{6} + 12 \beta_{7} - 83 \beta_{10} + 31 \beta_{11} - 2 \beta_{12} + 3 \beta_{14} - 13 \beta_{15} - 22 \beta_{16} - 17 \beta_{17} + 15 \beta_{18} - 9 \beta_{19} ) q^{92} + ( -11 - 6 \beta_{1} - 31 \beta_{2} - 21 \beta_{3} + 16 \beta_{4} + 6 \beta_{5} - 11 \beta_{6} + 16 \beta_{7} + 15 \beta_{8} - 12 \beta_{9} + \beta_{11} - \beta_{12} - 11 \beta_{14} + 11 \beta_{16} + 22 \beta_{17} - 15 \beta_{18} - 22 \beta_{19} ) q^{93} + ( -208 - 77 \beta_{2} - 197 \beta_{3} - 15 \beta_{6} - 7 \beta_{7} - 8 \beta_{8} - 20 \beta_{9} - 98 \beta_{10} + 20 \beta_{11} - 29 \beta_{12} - 15 \beta_{13} + 33 \beta_{14} + 30 \beta_{15} + 15 \beta_{16} + 11 \beta_{17} + 16 \beta_{18} + 4 \beta_{19} ) q^{94} + ( -5 - 28 \beta_{1} + 21 \beta_{2} - 16 \beta_{3} + 6 \beta_{6} - 2 \beta_{8} + 16 \beta_{9} + 71 \beta_{10} - 116 \beta_{11} + 21 \beta_{12} + 36 \beta_{13} - 25 \beta_{14} + 6 \beta_{16} - 21 \beta_{17} + 15 \beta_{19} ) q^{95} + ( -814 - 40 \beta_{1} + 14 \beta_{2} + 396 \beta_{3} - 12 \beta_{4} - 40 \beta_{5} + 50 \beta_{6} + 12 \beta_{7} - 10 \beta_{8} + 8 \beta_{9} - 2 \beta_{10} + 30 \beta_{11} + 14 \beta_{12} - 14 \beta_{13} - 6 \beta_{14} - 14 \beta_{15} - 6 \beta_{16} - 10 \beta_{18} ) q^{96} + ( 175 + 16 \beta_{1} - 30 \beta_{2} - 350 \beta_{3} - 5 \beta_{4} - 8 \beta_{5} - 5 \beta_{6} + 6 \beta_{8} + 20 \beta_{9} - 25 \beta_{11} + 10 \beta_{12} - 30 \beta_{14} + 5 \beta_{16} + 5 \beta_{17} - 3 \beta_{18} - 5 \beta_{19} ) q^{97} + ( -469 - 7 \beta_{1} + 7 \beta_{2} + 392 \beta_{3} + 14 \beta_{4} - 49 \beta_{5} - 21 \beta_{6} + 14 \beta_{7} + 14 \beta_{9} + 21 \beta_{10} - 91 \beta_{11} + 35 \beta_{12} + 21 \beta_{13} - 77 \beta_{14} + 7 \beta_{15} + 7 \beta_{16} + 14 \beta_{17} + 14 \beta_{19} ) q^{98} + ( 31 - 45 \beta_{2} + 22 \beta_{5} - \beta_{6} - 105 \beta_{10} + 44 \beta_{11} - 31 \beta_{12} + 45 \beta_{14} - 15 \beta_{15} - 16 \beta_{16} - 16 \beta_{17} + 2 \beta_{18} - \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{4} - 6q^{5} + 72q^{8} - 56q^{9} + O(q^{10}) \) \( 20q + 4q^{4} - 6q^{5} + 72q^{8} - 56q^{9} - 12q^{10} - 168q^{12} - 56q^{14} - 104q^{16} - 6q^{17} + 68q^{18} + 238q^{21} - 184q^{22} + 348q^{24} - 36q^{25} + 396q^{26} + 448q^{28} - 352q^{29} + 644q^{30} - 40q^{32} + 30q^{33} + 208q^{36} + 258q^{37} - 1620q^{38} - 1548q^{40} - 980q^{42} - 1248q^{44} - 504q^{45} + 232q^{46} - 644q^{49} - 864q^{50} + 2592q^{52} + 570q^{53} + 4572q^{54} + 1904q^{56} + 1452q^{57} + 2244q^{58} - 736q^{60} + 294q^{61} + 2560q^{64} - 124q^{65} - 4272q^{66} - 6084q^{68} - 4144q^{70} - 4672q^{72} + 966q^{73} + 832q^{74} - 378q^{77} - 4056q^{78} + 7032q^{80} - 1262q^{81} + 7692q^{82} + 6188q^{84} - 2980q^{85} + 5696q^{86} - 1396q^{88} - 3186q^{89} + 3312q^{92} - 306q^{93} - 6780q^{94} - 11784q^{96} - 4900q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 2 x^{18} - 24 x^{17} + 28 x^{16} + 56 x^{15} - 192 x^{14} + 352 x^{13} - 448 x^{12} + 5376 x^{11} - 41472 x^{10} + 43008 x^{9} - 28672 x^{8} + 180224 x^{7} - 786432 x^{6} + 1835008 x^{5} + 7340032 x^{4} - 50331648 x^{3} - 33554432 x^{2} + 1073741824\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\((\)\(793 \nu^{19} + 7760 \nu^{18} + 70462 \nu^{17} - 82360 \nu^{16} + 36188 \nu^{15} - 29384 \nu^{14} + 1591040 \nu^{13} - 2565664 \nu^{12} - 4427712 \nu^{11} + 24367872 \nu^{10} - 49737216 \nu^{9} - 1005568 \nu^{8} - 274354176 \nu^{7} + 2761342976 \nu^{6} - 9211805696 \nu^{5} + 2364801024 \nu^{4} + 4386193408 \nu^{3} + 76466356224 \nu^{2} - 466305941504 \nu - 940060966912\)\()/ 2280224980992 \)
\(\beta_{4}\)\(=\)\((\)\(647 \nu^{19} - 13656 \nu^{18} - 111550 \nu^{17} + 323784 \nu^{16} - 943772 \nu^{15} + 2117352 \nu^{14} - 21505984 \nu^{13} + 22798880 \nu^{12} - 67722048 \nu^{11} + 20075264 \nu^{10} - 7834112 \nu^{9} - 1015257088 \nu^{8} + 4428140544 \nu^{7} - 17098948608 \nu^{6} + 57840173056 \nu^{5} + 16812605440 \nu^{4} + 291370958848 \nu^{3} - 524946505728 \nu^{2} + 1932701728768 \nu + 3620321886208\)\()/ 1140112490496 \)
\(\beta_{5}\)\(=\)\((\)\(485 \nu^{19} + 4503 \nu^{18} - 3958 \nu^{17} + 874 \nu^{16} - 4612 \nu^{15} + 108956 \nu^{14} - 177800 \nu^{13} - 254528 \nu^{12} + 1256544 \nu^{11} - 1053120 \nu^{10} - 2194432 \nu^{9} - 15726080 \nu^{8} + 163651584 \nu^{7} - 536760320 \nu^{6} + 56852480 \nu^{5} - 89653248 \nu^{4} + 7273709568 \nu^{3} - 27481079808 \nu^{2} - 58753810432 \nu - 53217329152\)\()/ 142514061312 \)
\(\beta_{6}\)\(=\)\((\)\(3865 \nu^{19} - 96776 \nu^{18} + 881390 \nu^{17} + 176568 \nu^{16} - 2032324 \nu^{15} - 4568680 \nu^{14} - 7501056 \nu^{13} + 129312608 \nu^{12} - 436760256 \nu^{11} + 697100032 \nu^{10} - 1448702464 \nu^{9} + 2301237248 \nu^{8} - 31126745088 \nu^{7} + 57269895168 \nu^{6} + 160131186688 \nu^{5} - 472266833920 \nu^{4} - 255910215680 \nu^{3} - 1494531178496 \nu^{2} + 12339373932544 \nu - 46140162572288\)\()/ 2280224980992 \)
\(\beta_{7}\)\(=\)\((\)\(15263 \nu^{19} - 111128 \nu^{18} + 99154 \nu^{17} + 513800 \nu^{16} + 294276 \nu^{15} + 3535656 \nu^{14} - 26252480 \nu^{13} + 56933664 \nu^{12} - 301108544 \nu^{11} + 339575552 \nu^{10} - 898681344 \nu^{9} + 877848576 \nu^{8} - 2951573504 \nu^{7} - 37703860224 \nu^{6} + 90402652160 \nu^{5} - 222738776064 \nu^{4} + 845205733376 \nu^{3} - 1554060935168 \nu^{2} + 7149476380672 \nu - 3317191147520\)\()/ 2280224980992 \)
\(\beta_{8}\)\(=\)\((\)\(1751 \nu^{19} + 1586 \nu^{18} + 12018 \nu^{17} + 98900 \nu^{16} - 115692 \nu^{15} + 170432 \nu^{14} - 394960 \nu^{13} + 3798432 \nu^{12} - 5915776 \nu^{11} + 557952 \nu^{10} - 23881728 \nu^{9} - 24167424 \nu^{8} - 52215808 \nu^{7} - 233136128 \nu^{6} + 4145643520 \nu^{5} - 15210512384 \nu^{4} + 17581998080 \nu^{3} - 79358328832 \nu^{2} + 94178902016 \nu - 932611883008\)\()/ 142514061312 \)
\(\beta_{9}\)\(=\)\((\)\(-15231 \nu^{19} - 44416 \nu^{18} + 27030 \nu^{17} + 237384 \nu^{16} + 2711148 \nu^{15} - 3039944 \nu^{14} - 2002976 \nu^{13} - 13699808 \nu^{12} + 25418560 \nu^{11} + 22260224 \nu^{10} - 341555200 \nu^{9} + 2602584064 \nu^{8} - 1976213504 \nu^{7} + 5934596096 \nu^{6} - 71584874496 \nu^{5} + 95326830592 \nu^{4} - 211721125888 \nu^{3} - 54679044096 \nu^{2} + 1616937746432 \nu + 2908095512576\)\()/ 1140112490496 \)
\(\beta_{10}\)\(=\)\((\)\(39171 \nu^{19} + 88016 \nu^{18} + 123178 \nu^{17} - 159272 \nu^{16} + 596340 \nu^{15} + 927592 \nu^{14} - 1870720 \nu^{13} + 39336864 \nu^{12} + 15506112 \nu^{11} + 177052928 \nu^{10} - 527573504 \nu^{9} - 984066048 \nu^{8} - 2413375488 \nu^{7} - 6866649088 \nu^{6} - 48440344576 \nu^{5} - 4242276352 \nu^{4} + 193131970560 \nu^{3} - 1463598186496 \nu^{2} - 4227120234496 \nu - 10618098679808\)\()/ 2280224980992 \)
\(\beta_{11}\)\(=\)\((\)\(-31255 \nu^{19} + 175232 \nu^{18} - 16402 \nu^{17} + 13480 \nu^{16} - 1137668 \nu^{15} + 1560568 \nu^{14} + 9625152 \nu^{13} - 77024160 \nu^{12} + 150946880 \nu^{11} - 101624576 \nu^{10} + 827952640 \nu^{9} - 6066911232 \nu^{8} + 8012533760 \nu^{7} + 1314111488 \nu^{6} - 84968734720 \nu^{5} - 25482231808 \nu^{4} + 70970769408 \nu^{3} + 1809372413952 \nu^{2} - 7700943470592 \nu - 84422950912\)\()/ 2280224980992 \)
\(\beta_{12}\)\(=\)\((\)\(-11957 \nu^{19} - 10004 \nu^{18} - 13414 \nu^{17} + 144736 \nu^{16} - 1989964 \nu^{15} + 1508664 \nu^{14} - 769888 \nu^{13} + 409504 \nu^{12} - 49522880 \nu^{11} + 15729664 \nu^{10} + 558631424 \nu^{9} - 278478848 \nu^{8} + 1439739904 \nu^{7} - 3816161280 \nu^{6} + 14886633472 \nu^{5} - 92190015488 \nu^{4} + 170458611712 \nu^{3} + 308222623744 \nu^{2} + 1501023961088 \nu - 1149675503616\)\()/ 570056245248 \)
\(\beta_{13}\)\(=\)\((\)\(54759 \nu^{19} - 23200 \nu^{18} - 57294 \nu^{17} - 913128 \nu^{16} - 334268 \nu^{15} - 5765624 \nu^{14} - 1845312 \nu^{13} + 47062944 \nu^{12} - 33817664 \nu^{11} + 245775104 \nu^{10} - 1908473344 \nu^{9} + 1030551552 \nu^{8} + 8323133440 \nu^{7} + 17012768768 \nu^{6} + 62018813952 \nu^{5} - 164064657408 \nu^{4} - 836733239296 \nu^{3} - 2192211705856 \nu^{2} - 58451820544 \nu + 4940420349952\)\()/ 2280224980992 \)
\(\beta_{14}\)\(=\)\((\)\(4503 \nu^{19} - 2988 \nu^{18} + 12514 \nu^{17} - 18192 \nu^{16} + 81796 \nu^{15} - 84680 \nu^{14} - 425248 \nu^{13} + 1473824 \nu^{12} - 3660480 \nu^{11} + 17919488 \nu^{10} - 36584960 \nu^{9} + 177557504 \nu^{8} - 624168960 \nu^{7} + 438272000 \nu^{6} - 979632128 \nu^{5} + 3713794048 \nu^{4} - 3070230528 \nu^{3} - 42479910912 \nu^{2} - 53217329152 \nu - 520764784640\)\()/ 142514061312 \)
\(\beta_{15}\)\(=\)\((\)\(24923 \nu^{19} + 200112 \nu^{18} + 109250 \nu^{17} + 300344 \nu^{16} - 1935100 \nu^{15} + 13575080 \nu^{14} - 11050144 \nu^{13} + 34796256 \nu^{12} + 21024192 \nu^{11} + 430531584 \nu^{10} - 143023104 \nu^{9} - 5979150336 \nu^{8} + 8655618048 \nu^{7} - 20404060160 \nu^{6} + 61665411072 \nu^{5} - 259367895040 \nu^{4} + 313351208960 \nu^{3} - 2131287343104 \nu^{2} - 6828863782912 \nu - 15698508120064\)\()/ 1140112490496 \)
\(\beta_{16}\)\(=\)\((\)\(-36627 \nu^{19} - 138788 \nu^{18} - 194058 \nu^{17} + 142160 \nu^{16} + 1741996 \nu^{15} - 3189272 \nu^{14} - 3599072 \nu^{13} - 12142240 \nu^{12} - 68280896 \nu^{11} - 386566656 \nu^{10} + 867991040 \nu^{9} + 2098810880 \nu^{8} + 921997312 \nu^{7} + 8445198336 \nu^{6} + 20815937536 \nu^{5} - 79807381504 \nu^{4} - 120546394112 \nu^{3} + 1242529005568 \nu^{2} + 6717932830720 \nu + 13000597569536\)\()/ 1140112490496 \)
\(\beta_{17}\)\(=\)\((\)\(-22105 \nu^{19} - 251020 \nu^{18} + 371394 \nu^{17} + 18352 \nu^{16} + 1301572 \nu^{15} - 10654920 \nu^{14} + 14708192 \nu^{13} + 27009312 \nu^{12} - 332834496 \nu^{11} + 287667712 \nu^{10} - 198053376 \nu^{9} + 6669791232 \nu^{8} - 20072632320 \nu^{7} + 55113973760 \nu^{6} - 3085697024 \nu^{5} - 52089847808 \nu^{4} - 700046114816 \nu^{3} + 1709346652160 \nu^{2} + 12029196763136 \nu - 11577352781824\)\()/ 1140112490496 \)
\(\beta_{18}\)\(=\)\((\)\(23993 \nu^{19} + 6344 \nu^{18} + 14094 \nu^{17} - 12136 \nu^{16} + 12924 \nu^{15} + 1633112 \nu^{14} - 4841728 \nu^{13} + 21173856 \nu^{12} - 31274176 \nu^{11} + 93564672 \nu^{10} - 800094720 \nu^{9} + 633993216 \nu^{8} - 695971840 \nu^{7} + 2129281024 \nu^{6} + 3221880832 \nu^{5} - 29667098624 \nu^{4} + 195027795968 \nu^{3} - 1172517683200 \nu^{2} - 193340637184 \nu - 3730447532032\)\()/ 570056245248 \)
\(\beta_{19}\)\(=\)\((\)\(-80381 \nu^{19} - 19988 \nu^{18} + 382666 \nu^{17} + 429344 \nu^{16} - 1526380 \nu^{15} - 29576 \nu^{14} + 19089440 \nu^{13} - 38722656 \nu^{12} - 181541568 \nu^{11} + 391731200 \nu^{10} + 1936876032 \nu^{9} - 3118768128 \nu^{8} - 4812115968 \nu^{7} + 34541731840 \nu^{6} + 33871757312 \nu^{5} - 262743523328 \nu^{4} + 302281392128 \nu^{3} + 3559579975680 \nu^{2} + 3850102636544 \nu - 19108577935360\)\()/ 1140112490496 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{19} + \beta_{18} - \beta_{17} - \beta_{15} - \beta_{11} + \beta_{10} + 4\)
\(\nu^{4}\)\(=\)\(2 \beta_{17} - 2 \beta_{16} - 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - 2 \beta_{9} - 2 \beta_{7} - 2 \beta_{6} + 4 \beta_{4} + 12 \beta_{3} - 2 \beta_{2} + 4 \beta_{1} - 10\)
\(\nu^{5}\)\(=\)\(4 \beta_{19} - 2 \beta_{18} - 4 \beta_{17} - 2 \beta_{16} - 2 \beta_{15} + 2 \beta_{14} + 2 \beta_{13} - 6 \beta_{12} - 10 \beta_{11} - 14 \beta_{10} - 4 \beta_{9} + 2 \beta_{8} + 2 \beta_{6} + 16 \beta_{5} - 4 \beta_{3} - 2 \beta_{2} - 16 \beta_{1} + 2\)
\(\nu^{6}\)\(=\)\(12 \beta_{19} + 28 \beta_{18} + 12 \beta_{16} + 8 \beta_{14} - 4 \beta_{12} - 40 \beta_{7} - 8 \beta_{5} + 20 \beta_{4} - 8 \beta_{2} + 144\)
\(\nu^{7}\)\(=\)\(28 \beta_{19} - 4 \beta_{17} + 12 \beta_{16} - 36 \beta_{14} + 68 \beta_{13} + 4 \beta_{12} - 72 \beta_{11} + 32 \beta_{10} + 16 \beta_{9} + 44 \beta_{8} - 24 \beta_{7} - 60 \beta_{6} + 48 \beta_{4} + 696 \beta_{3} + 4 \beta_{2} + 80 \beta_{1} - 700\)
\(\nu^{8}\)\(=\)\(32 \beta_{19} - 80 \beta_{17} - 16 \beta_{16} - 40 \beta_{15} + 40 \beta_{14} + 40 \beta_{13} + 144 \beta_{12} - 336 \beta_{11} - 152 \beta_{10} + 184 \beta_{9} - 88 \beta_{7} + 40 \beta_{6} + 784 \beta_{5} - 88 \beta_{4} + 656 \beta_{3} + 168 \beta_{2} - 784 \beta_{1} + 40\)
\(\nu^{9}\)\(=\)\(232 \beta_{19} - 360 \beta_{18} - 72 \beta_{17} + 288 \beta_{16} + 56 \beta_{15} + 688 \beta_{14} + 192 \beta_{12} - 520 \beta_{11} + 840 \beta_{10} - 448 \beta_{7} - 128 \beta_{6} + 576 \beta_{5} + 224 \beta_{4} - 688 \beta_{2} + 1344\)
\(\nu^{10}\)\(=\)\(1360 \beta_{19} - 448 \beta_{17} - 1504 \beta_{16} + 1184 \beta_{14} + 1216 \beta_{13} + 448 \beta_{12} - 1360 \beta_{11} + 256 \beta_{10} + 1744 \beta_{9} - 976 \beta_{8} - 16 \beta_{7} - 320 \beta_{6} + 32 \beta_{4} - 15680 \beta_{3} + 448 \beta_{2} + 864 \beta_{1} + 15232\)
\(\nu^{11}\)\(=\)\(2016 \beta_{19} + 80 \beta_{18} - 6944 \beta_{17} - 1008 \beta_{16} - 1680 \beta_{15} + 3472 \beta_{14} + 1680 \beta_{13} + 1968 \beta_{12} - 5680 \beta_{11} - 2032 \beta_{10} + 5440 \beta_{9} - 80 \beta_{8} - 2976 \beta_{7} + 3472 \beta_{6} - 13888 \beta_{5} - 2976 \beta_{4} - 18976 \beta_{3} + 4080 \beta_{2} + 13888 \beta_{1} + 3472\)
\(\nu^{12}\)\(=\)\(1344 \beta_{19} - 768 \beta_{18} + 352 \beta_{17} + 2720 \beta_{16} + 1376 \beta_{15} - 20224 \beta_{14} + 1696 \beta_{12} - 21408 \beta_{11} + 42144 \beta_{10} - 11456 \beta_{7} - 1024 \beta_{6} - 22464 \beta_{5} + 5728 \beta_{4} + 20224 \beta_{2} + 70016\)
\(\nu^{13}\)\(=\)\(7264 \beta_{19} + 17184 \beta_{17} - 16864 \beta_{16} - 45664 \beta_{14} - 2336 \beta_{13} - 17184 \beta_{12} - 30976 \beta_{11} + 16128 \beta_{10} - 35648 \beta_{9} + 18528 \beta_{8} + 19072 \beta_{7} - 32032 \beta_{6} - 38144 \beta_{4} - 43712 \beta_{3} - 17184 \beta_{2} + 54528 \beta_{1} + 60896\)
\(\nu^{14}\)\(=\)\(-62848 \beta_{19} + 49984 \beta_{18} + 66048 \beta_{17} + 31424 \beta_{16} + 120064 \beta_{15} - 33024 \beta_{14} - 120064 \beta_{13} + 1088 \beta_{12} - 22592 \beta_{11} - 141568 \beta_{10} - 31936 \beta_{9} - 49984 \beta_{8} - 30656 \beta_{7} - 33024 \beta_{6} - 136576 \beta_{5} - 30656 \beta_{4} + 383744 \beta_{3} + 66176 \beta_{2} + 136576 \beta_{1} - 33024\)
\(\nu^{15}\)\(=\)\(90048 \beta_{19} - 32448 \beta_{18} - 5696 \beta_{17} + 148864 \beta_{16} + 58816 \beta_{15} + 11904 \beta_{14} - 227200 \beta_{12} - 121920 \beta_{11} + 173632 \beta_{10} + 82688 \beta_{7} - 64512 \beta_{6} + 259840 \beta_{5} - 41344 \beta_{4} - 11904 \beta_{2} - 216064\)
\(\nu^{16}\)\(=\)\(86784 \beta_{19} + 209536 \beta_{17} - 297088 \beta_{16} + 119168 \beta_{14} - 123520 \beta_{13} - 209536 \beta_{12} + 316032 \beta_{11} - 502784 \beta_{10} + 270464 \beta_{9} + 1149952 \beta_{8} - 63872 \beta_{7} - 295552 \beta_{6} + 127744 \beta_{4} - 3588352 \beta_{3} - 209536 \beta_{2} - 319232 \beta_{1} + 3797888\)
\(\nu^{17}\)\(=\)\(-220416 \beta_{19} - 1540480 \beta_{18} - 1132288 \beta_{17} + 110208 \beta_{16} - 101248 \beta_{15} + 566144 \beta_{14} + 101248 \beta_{13} + 16256 \beta_{12} - 1457024 \beta_{11} - 1339520 \beta_{10} + 582400 \beta_{9} + 1540480 \beta_{8} + 329216 \beta_{7} + 566144 \beta_{6} - 3126272 \beta_{5} + 329216 \beta_{4} + 35370240 \beta_{3} - 694656 \beta_{2} + 3126272 \beta_{1} + 566144\)
\(\nu^{18}\)\(=\)\(509184 \beta_{19} - 1090304 \beta_{18} + 1152000 \beta_{17} - 349952 \beta_{16} - 859136 \beta_{15} - 2764288 \beta_{14} - 944896 \beta_{12} - 236544 \beta_{11} + 3636224 \beta_{10} - 2440704 \beta_{7} + 2011136 \beta_{6} + 36940288 \beta_{5} + 1220352 \beta_{4} + 2764288 \beta_{2} + 57238528\)
\(\nu^{19}\)\(=\)\(-155392 \beta_{19} + 2562304 \beta_{17} + 3511552 \beta_{16} + 28631296 \beta_{14} + 3200768 \beta_{13} - 2562304 \beta_{12} + 9671168 \beta_{11} - 1763328 \beta_{10} - 11269120 \beta_{9} + 11527424 \beta_{8} + 370176 \beta_{7} - 8325376 \beta_{6} - 740352 \beta_{4} - 2891264 \beta_{3} - 2562304 \beta_{2} + 52632576 \beta_{1} + 5453568\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\)
\(\chi(n)\) \(-1\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
1.31147 + 2.50600i
2.59951 + 1.11469i
−1.03939 + 2.63053i
2.82698 + 0.0903966i
−1.75840 + 2.21540i
2.19431 1.78465i
−2.82600 + 0.117237i
0.448398 2.79266i
−2.26510 1.69390i
−1.49178 2.40304i
1.31147 2.50600i
2.59951 1.11469i
−1.03939 2.63053i
2.82698 0.0903966i
−1.75840 2.21540i
2.19431 + 1.78465i
−2.82600 0.117237i
0.448398 + 2.79266i
−2.26510 + 1.69390i
−1.49178 + 2.40304i
−2.82600 0.117237i 0.0469307 0.0812864i 7.97251 + 0.662623i 12.4861 7.20883i −0.142156 + 0.224213i 15.0686 10.7674i −22.4526 2.80725i 13.4956 + 23.3751i −36.1307 + 18.9083i
3.2 −2.26510 + 1.69390i −1.67134 + 2.89484i 2.26140 7.67373i −15.9583 + 9.21354i −1.11782 9.38820i −15.4841 10.1609i 7.87623 + 21.2124i 7.91327 + 13.7062i 20.5404 47.9014i
3.3 −1.75840 2.21540i −3.44104 + 5.96006i −1.81602 + 7.79115i −4.17670 + 2.41142i 19.2547 2.85690i −5.03893 + 17.8216i 20.4539 9.67677i −10.1816 17.6350i 12.6866 + 5.01283i
3.4 −1.49178 + 2.40304i 4.65345 8.06002i −3.54920 7.16960i 5.11538 2.95337i 12.4266 + 23.2062i 1.92900 + 18.4195i 22.5235 + 2.16656i −29.8092 51.6311i −0.533949 + 16.6982i
3.5 −1.03939 2.63053i 3.44104 5.96006i −5.83932 + 5.46830i −4.17670 + 2.41142i −19.2547 2.85690i 5.03893 17.8216i 20.4539 + 9.67677i −10.1816 17.6350i 10.6845 + 8.48051i
3.6 0.448398 + 2.79266i −2.11164 + 3.65747i −7.59788 + 2.50445i 1.03358 0.596737i −11.1609 4.25709i 16.7651 + 7.86959i −10.4009 20.0953i 4.58193 + 7.93614i 2.12994 + 2.61886i
3.7 1.31147 2.50600i −0.0469307 + 0.0812864i −4.56010 6.57309i 12.4861 7.20883i 0.142156 + 0.224213i −15.0686 + 10.7674i −22.4526 + 2.80725i 13.4956 + 23.3751i −1.69029 40.7443i
3.8 2.19431 + 1.78465i 2.11164 3.65747i 1.63002 + 7.83218i 1.03358 0.596737i 11.1609 4.25709i −16.7651 7.86959i −10.4009 + 20.0953i 4.58193 + 7.93614i 3.33297 + 0.535152i
3.9 2.59951 1.11469i 1.67134 2.89484i 5.51494 5.79529i −15.9583 + 9.21354i 1.11782 9.38820i 15.4841 + 10.1609i 7.87623 21.2124i 7.91327 + 13.7062i −31.2137 + 41.7393i
3.10 2.82698 0.0903966i −4.65345 + 8.06002i 7.98366 0.511099i 5.11538 2.95337i −12.4266 + 23.2062i −1.92900 18.4195i 22.5235 2.16656i −29.8092 51.6311i 14.1941 8.81153i
19.1 −2.82600 + 0.117237i 0.0469307 + 0.0812864i 7.97251 0.662623i 12.4861 + 7.20883i −0.142156 0.224213i 15.0686 + 10.7674i −22.4526 + 2.80725i 13.4956 23.3751i −36.1307 18.9083i
19.2 −2.26510 1.69390i −1.67134 2.89484i 2.26140 + 7.67373i −15.9583 9.21354i −1.11782 + 9.38820i −15.4841 + 10.1609i 7.87623 21.2124i 7.91327 13.7062i 20.5404 + 47.9014i
19.3 −1.75840 + 2.21540i −3.44104 5.96006i −1.81602 7.79115i −4.17670 2.41142i 19.2547 + 2.85690i −5.03893 17.8216i 20.4539 + 9.67677i −10.1816 + 17.6350i 12.6866 5.01283i
19.4 −1.49178 2.40304i 4.65345 + 8.06002i −3.54920 + 7.16960i 5.11538 + 2.95337i 12.4266 23.2062i 1.92900 18.4195i 22.5235 2.16656i −29.8092 + 51.6311i −0.533949 16.6982i
19.5 −1.03939 + 2.63053i 3.44104 + 5.96006i −5.83932 5.46830i −4.17670 2.41142i −19.2547 + 2.85690i 5.03893 + 17.8216i 20.4539 9.67677i −10.1816 + 17.6350i 10.6845 8.48051i
19.6 0.448398 2.79266i −2.11164 3.65747i −7.59788 2.50445i 1.03358 + 0.596737i −11.1609 + 4.25709i 16.7651 7.86959i −10.4009 + 20.0953i 4.58193 7.93614i 2.12994 2.61886i
19.7 1.31147 + 2.50600i −0.0469307 0.0812864i −4.56010 + 6.57309i 12.4861 + 7.20883i 0.142156 0.224213i −15.0686 10.7674i −22.4526 2.80725i 13.4956 23.3751i −1.69029 + 40.7443i
19.8 2.19431 1.78465i 2.11164 + 3.65747i 1.63002 7.83218i 1.03358 + 0.596737i 11.1609 + 4.25709i −16.7651 + 7.86959i −10.4009 20.0953i 4.58193 7.93614i 3.33297 0.535152i
19.9 2.59951 + 1.11469i 1.67134 + 2.89484i 5.51494 + 5.79529i −15.9583 9.21354i 1.11782 + 9.38820i 15.4841 10.1609i 7.87623 + 21.2124i 7.91327 13.7062i −31.2137 41.7393i
19.10 2.82698 + 0.0903966i −4.65345 8.06002i 7.98366 + 0.511099i 5.11538 + 2.95337i −12.4266 23.2062i −1.92900 + 18.4195i 22.5235 + 2.16656i −29.8092 + 51.6311i 14.1941 + 8.81153i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 28.4.f.a 20
4.b odd 2 1 inner 28.4.f.a 20
7.b odd 2 1 196.4.f.d 20
7.c even 3 1 196.4.d.b 20
7.c even 3 1 196.4.f.d 20
7.d odd 6 1 inner 28.4.f.a 20
7.d odd 6 1 196.4.d.b 20
8.b even 2 1 448.4.p.h 20
8.d odd 2 1 448.4.p.h 20
28.d even 2 1 196.4.f.d 20
28.f even 6 1 inner 28.4.f.a 20
28.f even 6 1 196.4.d.b 20
28.g odd 6 1 196.4.d.b 20
28.g odd 6 1 196.4.f.d 20
56.j odd 6 1 448.4.p.h 20
56.m even 6 1 448.4.p.h 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.f.a 20 1.a even 1 1 trivial
28.4.f.a 20 4.b odd 2 1 inner
28.4.f.a 20 7.d odd 6 1 inner
28.4.f.a 20 28.f even 6 1 inner
196.4.d.b 20 7.c even 3 1
196.4.d.b 20 7.d odd 6 1
196.4.d.b 20 28.f even 6 1
196.4.d.b 20 28.g odd 6 1
196.4.f.d 20 7.b odd 2 1
196.4.f.d 20 7.c even 3 1
196.4.f.d 20 28.d even 2 1
196.4.f.d 20 28.g odd 6 1
448.4.p.h 20 8.b even 2 1
448.4.p.h 20 8.d odd 2 1
448.4.p.h 20 56.j odd 6 1
448.4.p.h 20 56.m even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(28, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1073741824 - 33554432 T^{2} - 50331648 T^{3} + 7340032 T^{4} + 1835008 T^{5} - 786432 T^{6} + 180224 T^{7} - 28672 T^{8} + 43008 T^{9} - 41472 T^{10} + 5376 T^{11} - 448 T^{12} + 352 T^{13} - 192 T^{14} + 56 T^{15} + 28 T^{16} - 24 T^{17} - 2 T^{18} + T^{20} \)
$3$ \( 51883209 + 5898399843 T^{2} + 669515995603 T^{4} + 119263400326 T^{6} + 14545748317 T^{8} + 927039337 T^{10} + 42494101 T^{12} + 1043398 T^{14} + 18379 T^{16} + 163 T^{18} + T^{20} \)
$5$ \( ( 81588675 - 115162845 T + 49396951 T^{2} + 6757398 T^{3} - 1926469 T^{4} - 261591 T^{5} + 84761 T^{6} - 906 T^{7} - 299 T^{8} + 3 T^{9} + T^{10} )^{2} \)
$7$ \( \)\(22\!\cdots\!49\)\( + \)\(61\!\cdots\!22\)\( T^{2} + \)\(14\!\cdots\!57\)\( T^{4} + 1487961738200295288 T^{6} + 4321067385141346 T^{8} + 5950089233676 T^{10} + 36728466754 T^{12} + 107501688 T^{14} + 86093 T^{16} + 322 T^{18} + T^{20} \)
$11$ \( \)\(20\!\cdots\!25\)\( - \)\(21\!\cdots\!25\)\( T^{2} + \)\(22\!\cdots\!79\)\( T^{4} - \)\(61\!\cdots\!46\)\( T^{6} + \)\(10\!\cdots\!77\)\( T^{8} - 116554315700537683 T^{10} + 94391335342469 T^{12} - 53599840914 T^{14} + 22516499 T^{16} - 5985 T^{18} + T^{20} \)
$13$ \( ( 10072189747200 + 940129122304 T^{2} + 5983444672 T^{4} + 10544432 T^{6} + 5924 T^{8} + T^{10} )^{2} \)
$17$ \( ( 60677626846401075 + 1667036543705715 T - 91515356083889 T^{2} - 2933689388994 T^{3} + 145581511771 T^{4} + 7626792369 T^{5} + 110389001 T^{6} - 32178 T^{7} - 10723 T^{8} + 3 T^{9} + T^{10} )^{2} \)
$19$ \( \)\(12\!\cdots\!25\)\( + \)\(10\!\cdots\!75\)\( T^{2} + \)\(51\!\cdots\!39\)\( T^{4} + \)\(17\!\cdots\!54\)\( T^{6} + \)\(41\!\cdots\!05\)\( T^{8} + \)\(74\!\cdots\!57\)\( T^{10} + 101688106164103285 T^{12} + 10090710253910 T^{14} + 739321531 T^{16} + 34667 T^{18} + T^{20} \)
$23$ \( \)\(17\!\cdots\!61\)\( - \)\(14\!\cdots\!61\)\( T^{2} + \)\(86\!\cdots\!55\)\( T^{4} - \)\(22\!\cdots\!18\)\( T^{6} + \)\(42\!\cdots\!81\)\( T^{8} - \)\(19\!\cdots\!03\)\( T^{10} + 64011688751415749 T^{12} - 10322924418402 T^{14} + 1195146467 T^{16} - 38721 T^{18} + T^{20} \)
$29$ \( ( 5066614016 - 2408128 T - 4923568 T^{2} - 40796 T^{3} + 88 T^{4} + T^{5} )^{4} \)
$31$ \( \)\(43\!\cdots\!25\)\( + \)\(42\!\cdots\!75\)\( T^{2} + \)\(34\!\cdots\!99\)\( T^{4} + \)\(67\!\cdots\!78\)\( T^{6} + \)\(10\!\cdots\!21\)\( T^{8} + \)\(23\!\cdots\!69\)\( T^{10} + 3712965269092946901 T^{12} + 271342758337398 T^{14} + 14481693531 T^{16} + 129411 T^{18} + T^{20} \)
$37$ \( ( \)\(48\!\cdots\!25\)\( - 2736352858160219745 T + 109671722097362959 T^{2} - 810307003720758 T^{3} + 19350195624517 T^{4} - 120978951207 T^{5} + 1364598141 T^{6} - 4625430 T^{7} + 47239 T^{8} - 129 T^{9} + T^{10} )^{2} \)
$41$ \( ( \)\(14\!\cdots\!72\)\( + 31103998837585985536 T^{2} + 2106458435887808 T^{4} + 45452939696 T^{6} + 371044 T^{8} + T^{10} )^{2} \)
$43$ \( ( \)\(55\!\cdots\!56\)\( + \)\(10\!\cdots\!80\)\( T^{2} + 4759034131340160 T^{4} + 76172353504 T^{6} + 481188 T^{8} + T^{10} )^{2} \)
$47$ \( \)\(90\!\cdots\!25\)\( + \)\(88\!\cdots\!75\)\( T^{2} + \)\(84\!\cdots\!59\)\( T^{4} + \)\(18\!\cdots\!18\)\( T^{6} + \)\(35\!\cdots\!61\)\( T^{8} + \)\(13\!\cdots\!49\)\( T^{10} + \)\(39\!\cdots\!21\)\( T^{12} + 35108367873452838 T^{14} + 229434171691 T^{16} + 548571 T^{18} + T^{20} \)
$53$ \( ( \)\(23\!\cdots\!25\)\( - \)\(17\!\cdots\!65\)\( T + \)\(19\!\cdots\!51\)\( T^{2} - 466225986823511566 T^{3} + 4698483953203645 T^{4} - 6987261591931 T^{5} + 92220981965 T^{6} + 7091442 T^{7} + 385439 T^{8} - 285 T^{9} + T^{10} )^{2} \)
$59$ \( \)\(53\!\cdots\!25\)\( + \)\(86\!\cdots\!75\)\( T^{2} + \)\(11\!\cdots\!59\)\( T^{4} + \)\(28\!\cdots\!58\)\( T^{6} + \)\(45\!\cdots\!61\)\( T^{8} + \)\(43\!\cdots\!49\)\( T^{10} + \)\(30\!\cdots\!21\)\( T^{12} + 134460135107508598 T^{14} + 432699299611 T^{16} + 814891 T^{18} + T^{20} \)
$61$ \( ( \)\(25\!\cdots\!47\)\( - \)\(18\!\cdots\!39\)\( T + \)\(22\!\cdots\!63\)\( T^{2} + 15895203677629438566 T^{3} - 24061957047836909 T^{4} - 123028492312221 T^{5} + 318077925929 T^{6} + 92530326 T^{7} - 622255 T^{8} - 147 T^{9} + T^{10} )^{2} \)
$67$ \( \)\(16\!\cdots\!21\)\( - \)\(56\!\cdots\!69\)\( T^{2} + \)\(14\!\cdots\!27\)\( T^{4} - \)\(12\!\cdots\!62\)\( T^{6} + \)\(72\!\cdots\!37\)\( T^{8} - \)\(27\!\cdots\!39\)\( T^{10} + \)\(77\!\cdots\!29\)\( T^{12} - 1494540359131738850 T^{14} + 2113861632803 T^{16} - 1823609 T^{18} + T^{20} \)
$71$ \( ( \)\(16\!\cdots\!00\)\( + \)\(38\!\cdots\!08\)\( T^{2} + 21869277178736512 T^{4} + 350254218720 T^{6} + 1630916 T^{8} + T^{10} )^{2} \)
$73$ \( ( \)\(39\!\cdots\!75\)\( - \)\(31\!\cdots\!55\)\( T + \)\(63\!\cdots\!71\)\( T^{2} + \)\(13\!\cdots\!34\)\( T^{3} - 253040332806530029 T^{4} - 393556648905309 T^{5} + 811349980841 T^{6} + 522521958 T^{7} - 1004063 T^{8} - 483 T^{9} + T^{10} )^{2} \)
$79$ \( \)\(14\!\cdots\!25\)\( - \)\(33\!\cdots\!25\)\( T^{2} + \)\(49\!\cdots\!19\)\( T^{4} - \)\(41\!\cdots\!78\)\( T^{6} + \)\(25\!\cdots\!49\)\( T^{8} - \)\(86\!\cdots\!51\)\( T^{10} + \)\(21\!\cdots\!05\)\( T^{12} - 3268164132038273570 T^{14} + 3655335450275 T^{16} - 2364593 T^{18} + T^{20} \)
$83$ \( ( -\)\(41\!\cdots\!08\)\( + \)\(70\!\cdots\!40\)\( T^{2} - 457320228679630848 T^{4} + 1383233136896 T^{6} - 1929696 T^{8} + T^{10} )^{2} \)
$89$ \( ( \)\(99\!\cdots\!27\)\( + \)\(48\!\cdots\!49\)\( T + \)\(92\!\cdots\!07\)\( T^{2} + 70991208413939653182 T^{3} + 253640187212613659 T^{4} + 319508558427303 T^{5} - 187812567367 T^{6} - 620890866 T^{7} + 456121 T^{8} + 1593 T^{9} + T^{10} )^{2} \)
$97$ \( ( \)\(12\!\cdots\!00\)\( + \)\(34\!\cdots\!04\)\( T^{2} + 349801161086178496 T^{4} + 1444821395120 T^{6} + 2197892 T^{8} + T^{10} )^{2} \)
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