Properties

Label 69.2.e.c
Level $69$
Weight $2$
Character orbit 69.e
Analytic conductor $0.551$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 69.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.550967773947\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{11})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 7 x^{19} + 24 x^{18} - 70 x^{17} + 209 x^{16} - 527 x^{15} + 1115 x^{14} - 2187 x^{13} + 4165 x^{12} - 7040 x^{11} + 10649 x^{10} - 13519 x^{9} + 15111 x^{8} - 12101 x^{7} + 8244 x^{6} - 2837 x^{5} + 1859 x^{4} - 1730 x^{3} + 1957 x^{2} - 1656 x + 529\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} + 2 \beta_{17} - \beta_{18} - \beta_{19} ) q^{2} + \beta_{6} q^{3} + ( \beta_{2} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} - 2 \beta_{13} - \beta_{14} - \beta_{16} - 2 \beta_{19} ) q^{4} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{15} + \beta_{17} - \beta_{19} ) q^{5} + ( 1 + \beta_{2} + \beta_{4} - \beta_{13} - \beta_{16} ) q^{6} + ( -\beta_{1} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} + \beta_{17} + \beta_{18} ) q^{7} + ( -\beta_{1} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} - \beta_{9} + \beta_{10} - 3 \beta_{11} + \beta_{12} + 2 \beta_{13} - 2 \beta_{15} - 3 \beta_{17} + 3 \beta_{19} ) q^{8} + ( -1 - \beta_{2} - \beta_{6} - \beta_{7} - \beta_{11} + \beta_{13} - \beta_{15} + \beta_{16} - \beta_{17} + \beta_{19} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} + 2 \beta_{17} - \beta_{18} - \beta_{19} ) q^{2} + \beta_{6} q^{3} + ( \beta_{2} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} - 2 \beta_{13} - \beta_{14} - \beta_{16} - 2 \beta_{19} ) q^{4} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{15} + \beta_{17} - \beta_{19} ) q^{5} + ( 1 + \beta_{2} + \beta_{4} - \beta_{13} - \beta_{16} ) q^{6} + ( -\beta_{1} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} + \beta_{17} + \beta_{18} ) q^{7} + ( -\beta_{1} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} - \beta_{9} + \beta_{10} - 3 \beta_{11} + \beta_{12} + 2 \beta_{13} - 2 \beta_{15} - 3 \beta_{17} + 3 \beta_{19} ) q^{8} + ( -1 - \beta_{2} - \beta_{6} - \beta_{7} - \beta_{11} + \beta_{13} - \beta_{15} + \beta_{16} - \beta_{17} + \beta_{19} ) q^{9} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} - 3 \beta_{13} - \beta_{14} - \beta_{15} - 3 \beta_{16} - \beta_{17} - 2 \beta_{19} ) q^{10} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{6} - 2 \beta_{7} - 2 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{11} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{8} + \beta_{9} + 2 \beta_{13} + \beta_{14} + \beta_{15} + 2 \beta_{16} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{12} + ( -\beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + 3 \beta_{13} + 2 \beta_{16} - \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{13} + ( -1 + \beta_{6} + 3 \beta_{7} - 2 \beta_{8} - \beta_{11} - 3 \beta_{13} - \beta_{15} + \beta_{16} - \beta_{19} ) q^{14} + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{11} - \beta_{17} + \beta_{19} ) q^{15} + ( -2 + \beta_{1} - 4 \beta_{2} - \beta_{4} - 2 \beta_{6} + \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} + 4 \beta_{13} + \beta_{15} + 4 \beta_{16} - \beta_{17} + 4 \beta_{19} ) q^{16} + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} + \beta_{19} ) q^{17} + \beta_{18} q^{18} + ( -1 - \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{13} + \beta_{14} - \beta_{18} - 3 \beta_{19} ) q^{19} + ( -2 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - 6 \beta_{6} - 8 \beta_{7} + \beta_{10} - 8 \beta_{11} + 2 \beta_{12} + 7 \beta_{13} - 3 \beta_{15} + 4 \beta_{16} - 5 \beta_{17} + \beta_{18} + 7 \beta_{19} ) q^{20} + ( \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} - 2 \beta_{16} + \beta_{17} - \beta_{18} ) q^{21} + ( 2 + \beta_{4} + \beta_{7} - \beta_{8} - \beta_{10} - 2 \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{22} + ( 2 + 3 \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{15} - 3 \beta_{16} + 2 \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{23} + ( -1 + \beta_{1} - \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{18} - 3 \beta_{19} ) q^{24} + ( 2 \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} + \beta_{17} - \beta_{19} ) q^{25} + ( 2 \beta_{1} + \beta_{3} - \beta_{4} + 3 \beta_{5} + 5 \beta_{6} + 4 \beta_{7} + 6 \beta_{11} - 2 \beta_{12} - 6 \beta_{13} - 4 \beta_{16} + 4 \beta_{17} - 4 \beta_{19} ) q^{26} -\beta_{19} q^{27} + ( 1 - 2 \beta_{1} + 5 \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{11} - \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{28} + ( -2 - \beta_{1} - 4 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{15} + 4 \beta_{16} - 3 \beta_{17} + \beta_{19} ) q^{29} + ( -1 + 2 \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + 4 \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} + 3 \beta_{15} + \beta_{16} + 4 \beta_{17} - \beta_{18} - \beta_{19} ) q^{30} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} + \beta_{12} + \beta_{14} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{31} + ( 1 + 4 \beta_{2} + 4 \beta_{6} + 3 \beta_{7} - \beta_{8} - 2 \beta_{9} + 5 \beta_{11} - 5 \beta_{13} - \beta_{14} + \beta_{15} - 7 \beta_{16} + 2 \beta_{17} + 2 \beta_{18} - 6 \beta_{19} ) q^{32} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{6} - \beta_{7} - \beta_{9} - \beta_{14} - \beta_{15} - \beta_{17} + \beta_{19} ) q^{33} + ( 3 + \beta_{1} + 5 \beta_{2} + 2 \beta_{3} + 6 \beta_{6} + 6 \beta_{7} + \beta_{8} + \beta_{9} + 4 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} + 3 \beta_{15} - 5 \beta_{16} + \beta_{17} - \beta_{18} - 3 \beta_{19} ) q^{34} + ( -1 + \beta_{1} + 3 \beta_{2} + \beta_{3} + 3 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - \beta_{12} - 4 \beta_{13} + \beta_{15} - 2 \beta_{16} + \beta_{17} - \beta_{18} - 4 \beta_{19} ) q^{35} + ( 1 + \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - 3 \beta_{13} - 2 \beta_{16} - 2 \beta_{19} ) q^{36} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{5} + \beta_{6} - \beta_{7} + 4 \beta_{8} + 3 \beta_{9} + \beta_{11} - 3 \beta_{12} + 4 \beta_{13} + 3 \beta_{14} - 3 \beta_{15} + 2 \beta_{16} + \beta_{17} - 3 \beta_{18} + 2 \beta_{19} ) q^{37} + ( 5 - 3 \beta_{1} + 5 \beta_{2} - \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{12} + 2 \beta_{13} - \beta_{14} - 2 \beta_{15} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{38} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - 2 \beta_{8} - 2 \beta_{9} - \beta_{11} - \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} - 2 \beta_{17} ) q^{39} + ( 3 \beta_{1} - 3 \beta_{2} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + 6 \beta_{6} + 7 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} + 7 \beta_{11} - 2 \beta_{12} + \beta_{14} + 5 \beta_{15} + 3 \beta_{16} + 7 \beta_{17} - \beta_{19} ) q^{40} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + 5 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 5 \beta_{11} + \beta_{17} - \beta_{18} ) q^{41} + ( 2 - 2 \beta_{2} - 2 \beta_{6} - \beta_{7} - 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + \beta_{16} + 2 \beta_{17} + \beta_{19} ) q^{42} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{11} - \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} - 3 \beta_{16} + 2 \beta_{17} + \beta_{18} - \beta_{19} ) q^{43} + ( 1 + 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 6 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + 3 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} + 2 \beta_{14} + 4 \beta_{15} + 3 \beta_{17} - 2 \beta_{18} - 3 \beta_{19} ) q^{44} + ( 1 + \beta_{7} - \beta_{8} - \beta_{10} - \beta_{13} - \beta_{14} + \beta_{18} ) q^{45} + ( -5 - 8 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - 3 \beta_{6} - 4 \beta_{7} + \beta_{8} + 3 \beta_{9} - 2 \beta_{10} + \beta_{12} + 9 \beta_{13} + 6 \beta_{16} - 2 \beta_{17} - 2 \beta_{18} + 5 \beta_{19} ) q^{46} + ( 1 - 2 \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{15} - \beta_{16} - 3 \beta_{17} + \beta_{18} ) q^{47} + ( 2 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - 3 \beta_{11} + \beta_{12} - 3 \beta_{13} - \beta_{14} - 2 \beta_{15} - \beta_{16} - 3 \beta_{17} + \beta_{18} - \beta_{19} ) q^{48} + ( \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 5 \beta_{7} + \beta_{8} + 3 \beta_{10} - 2 \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} - 3 \beta_{17} + 2 \beta_{18} + 2 \beta_{19} ) q^{49} + ( -1 - 5 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - 6 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - 5 \beta_{11} + \beta_{12} + 3 \beta_{13} + \beta_{14} + 6 \beta_{16} - \beta_{17} + 3 \beta_{19} ) q^{50} + ( -1 - \beta_{3} - \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - 3 \beta_{11} + \beta_{12} - \beta_{15} + \beta_{16} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{51} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} - 5 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 8 \beta_{11} + 2 \beta_{12} - 3 \beta_{13} - \beta_{15} - 2 \beta_{16} - 5 \beta_{17} ) q^{52} + ( -2 + \beta_{1} - 3 \beta_{2} + \beta_{4} - 2 \beta_{6} + \beta_{8} + \beta_{10} - 2 \beta_{11} - 2 \beta_{15} + 2 \beta_{17} + 3 \beta_{19} ) q^{53} + ( -\beta_{1} - \beta_{6} - \beta_{7} - \beta_{11} + \beta_{13} - \beta_{15} - \beta_{17} + \beta_{19} ) q^{54} + ( 1 + \beta_{1} + \beta_{3} - \beta_{5} + 3 \beta_{6} + 3 \beta_{7} + 2 \beta_{9} + \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{15} + 2 \beta_{16} + 2 \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{55} + ( -2 + \beta_{1} - 7 \beta_{2} - 4 \beta_{4} + \beta_{5} - 5 \beta_{6} - \beta_{7} + 4 \beta_{8} + 4 \beta_{9} - 3 \beta_{10} + \beta_{11} + 10 \beta_{13} + \beta_{14} + 2 \beta_{15} + 4 \beta_{16} + 3 \beta_{17} - \beta_{18} + 3 \beta_{19} ) q^{56} + ( 1 - 2 \beta_{3} + \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{11} + 2 \beta_{12} - \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{57} + ( -1 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - 3 \beta_{11} - 2 \beta_{12} + 3 \beta_{13} - \beta_{14} - 4 \beta_{15} - 4 \beta_{16} - 2 \beta_{17} + 2 \beta_{18} + 4 \beta_{19} ) q^{58} + ( 2 - \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{13} - \beta_{14} + \beta_{15} + \beta_{18} + \beta_{19} ) q^{59} + ( -2 - \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} - 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} - \beta_{16} - 3 \beta_{17} + \beta_{18} - 3 \beta_{19} ) q^{60} + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + 7 \beta_{7} - \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} + 2 \beta_{15} + 4 \beta_{17} + \beta_{18} - 4 \beta_{19} ) q^{61} + ( -10 + 2 \beta_{1} - 6 \beta_{2} - 3 \beta_{4} + 2 \beta_{5} - 6 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} + 8 \beta_{13} + 2 \beta_{14} - \beta_{15} + 8 \beta_{16} - 4 \beta_{17} - 2 \beta_{18} + 4 \beta_{19} ) q^{62} + ( -\beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{11} - \beta_{14} + \beta_{19} ) q^{63} + ( -2 - 4 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 6 \beta_{6} - 10 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} - 10 \beta_{11} + 4 \beta_{12} - 5 \beta_{15} + 2 \beta_{16} - 6 \beta_{17} + \beta_{18} + 5 \beta_{19} ) q^{64} + ( 2 + 5 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 7 \beta_{6} + 6 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 5 \beta_{11} - \beta_{12} - 5 \beta_{13} + 2 \beta_{14} - 6 \beta_{16} + 2 \beta_{17} - 4 \beta_{19} ) q^{65} + ( -\beta_{1} - 3 \beta_{2} - \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{10} - 3 \beta_{11} + \beta_{12} + 2 \beta_{13} - 3 \beta_{15} - \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{66} + ( -2 - 2 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} - \beta_{10} - 4 \beta_{11} + 3 \beta_{12} - 3 \beta_{14} + \beta_{15} - \beta_{16} - 4 \beta_{17} + 3 \beta_{18} + 5 \beta_{19} ) q^{67} + ( -1 - 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 7 \beta_{6} - 7 \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - \beta_{15} + \beta_{16} - \beta_{17} - \beta_{18} + 3 \beta_{19} ) q^{68} + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{9} - \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} + \beta_{14} + 2 \beta_{15} - \beta_{16} + \beta_{18} - \beta_{19} ) q^{69} + ( -2 - 5 \beta_{1} - 4 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} - 14 \beta_{6} - 12 \beta_{7} - \beta_{8} - 4 \beta_{9} - 2 \beta_{10} - 7 \beta_{11} + 3 \beta_{12} + 10 \beta_{13} - 2 \beta_{14} - 7 \beta_{15} + 7 \beta_{16} - 11 \beta_{17} + \beta_{18} + 8 \beta_{19} ) q^{70} + ( -2 - 3 \beta_{1} + 3 \beta_{2} - \beta_{3} + 4 \beta_{4} - 5 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} - 5 \beta_{8} - 4 \beta_{9} - 6 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} - 3 \beta_{14} - 5 \beta_{15} - \beta_{16} - 6 \beta_{17} + \beta_{18} + 5 \beta_{19} ) q^{71} + ( -\beta_{1} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{11} + \beta_{12} + 3 \beta_{13} + \beta_{14} + 2 \beta_{16} - \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{72} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 4 \beta_{6} - \beta_{7} - \beta_{8} - 3 \beta_{9} - 2 \beta_{10} - \beta_{12} - 3 \beta_{14} + \beta_{18} - \beta_{19} ) q^{73} + ( -2 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 4 \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} + 7 \beta_{15} + 2 \beta_{16} + 4 \beta_{17} + 3 \beta_{18} - 7 \beta_{19} ) q^{74} + ( -1 + \beta_{2} + \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{16} + 2 \beta_{17} ) q^{75} + ( -2 + \beta_{1} - 3 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} + 3 \beta_{9} + \beta_{10} + 4 \beta_{11} - \beta_{12} - \beta_{13} + 3 \beta_{14} + 3 \beta_{15} - \beta_{16} + 5 \beta_{17} - \beta_{18} ) q^{76} + ( -3 + \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{7} + \beta_{9} + \beta_{11} - \beta_{12} + 3 \beta_{13} - \beta_{14} - 3 \beta_{15} - 4 \beta_{17} + \beta_{18} + 4 \beta_{19} ) q^{77} + ( -1 + \beta_{2} - 2 \beta_{6} - 4 \beta_{7} + 3 \beta_{8} + \beta_{9} + 5 \beta_{13} + 2 \beta_{14} - \beta_{15} + \beta_{16} + 2 \beta_{17} - \beta_{18} + 3 \beta_{19} ) q^{78} + ( -1 + 4 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + 4 \beta_{9} + 3 \beta_{11} - \beta_{13} + 4 \beta_{14} + \beta_{15} + 4 \beta_{16} + 4 \beta_{17} - 3 \beta_{18} ) q^{79} + ( 6 - 2 \beta_{1} + 9 \beta_{2} + \beta_{3} + 5 \beta_{4} + 6 \beta_{6} + 7 \beta_{7} - 5 \beta_{8} - 5 \beta_{9} + 4 \beta_{11} - \beta_{12} - 14 \beta_{13} - 5 \beta_{14} - 3 \beta_{15} - 13 \beta_{16} - 2 \beta_{17} + 2 \beta_{18} - 9 \beta_{19} ) q^{80} + \beta_{15} q^{81} + ( 3 + 2 \beta_{2} + \beta_{3} + 2 \beta_{6} + 3 \beta_{7} - 4 \beta_{9} + 4 \beta_{10} - 8 \beta_{13} - 8 \beta_{16} - 8 \beta_{19} ) q^{82} + ( -2 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - \beta_{5} - 2 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} - \beta_{9} - 5 \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} - 2 \beta_{15} + 4 \beta_{16} + 3 \beta_{17} + \beta_{18} + \beta_{19} ) q^{83} + ( 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 8 \beta_{7} + 3 \beta_{9} - 2 \beta_{10} + 4 \beta_{11} - 3 \beta_{12} - 3 \beta_{13} + 3 \beta_{15} + 3 \beta_{17} - 3 \beta_{19} ) q^{84} + ( 3 - 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - 2 \beta_{11} + \beta_{12} - 3 \beta_{14} - \beta_{15} - 5 \beta_{17} + \beta_{18} - \beta_{19} ) q^{85} + ( 6 - 2 \beta_{1} + 6 \beta_{2} + \beta_{3} + 5 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} - 7 \beta_{11} + 3 \beta_{12} - 6 \beta_{13} + \beta_{14} + 3 \beta_{15} - 6 \beta_{16} - 3 \beta_{17} - \beta_{19} ) q^{86} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{11} + \beta_{15} + 2 \beta_{16} + \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{87} + ( -\beta_{4} - \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} - 3 \beta_{12} + \beta_{13} - \beta_{14} + \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{88} + ( -2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{10} - 2 \beta_{11} + 5 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} + \beta_{16} + 2 \beta_{17} - 3 \beta_{18} + 3 \beta_{19} ) q^{89} + ( 1 + 4 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} - 4 \beta_{13} + \beta_{14} + \beta_{15} - 4 \beta_{16} + \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{90} + ( -1 + 4 \beta_{1} + \beta_{3} - 4 \beta_{4} + 4 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} - \beta_{8} + \beta_{9} + 6 \beta_{11} - 4 \beta_{12} - 3 \beta_{13} + 5 \beta_{15} - 5 \beta_{16} + 2 \beta_{17} + \beta_{18} - 7 \beta_{19} ) q^{91} + ( 3 + 3 \beta_{2} + \beta_{3} + 2 \beta_{5} + 11 \beta_{6} + 7 \beta_{7} - \beta_{8} - \beta_{9} + 5 \beta_{11} - \beta_{12} - 5 \beta_{13} - \beta_{14} - 7 \beta_{16} + 3 \beta_{17} + 3 \beta_{18} - 7 \beta_{19} ) q^{92} + ( 1 + 2 \beta_{1} + 3 \beta_{3} - \beta_{4} + 2 \beta_{5} + 8 \beta_{6} + 5 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} - \beta_{12} - \beta_{13} + 3 \beta_{14} + \beta_{15} - \beta_{16} + 3 \beta_{17} - 2 \beta_{18} - 4 \beta_{19} ) q^{93} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{7} + 3 \beta_{8} + 3 \beta_{9} - \beta_{10} + 10 \beta_{11} + \beta_{13} + 2 \beta_{16} + 10 \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{94} + ( -2 \beta_{1} - 8 \beta_{2} - \beta_{5} - 4 \beta_{6} - 5 \beta_{7} - \beta_{8} - 2 \beta_{10} - 6 \beta_{11} + 2 \beta_{12} + 7 \beta_{13} - \beta_{14} - 8 \beta_{15} + 8 \beta_{16} - 7 \beta_{17} - \beta_{18} + 7 \beta_{19} ) q^{95} + ( -2 + \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} - \beta_{12} + 6 \beta_{13} + 2 \beta_{14} + 3 \beta_{16} - 2 \beta_{18} + 5 \beta_{19} ) q^{96} + ( 1 + 4 \beta_{1} + \beta_{2} + 3 \beta_{3} - 4 \beta_{4} + 6 \beta_{5} + 10 \beta_{6} + 8 \beta_{7} + 3 \beta_{8} + 6 \beta_{9} - 2 \beta_{10} + 8 \beta_{11} - 2 \beta_{12} + 4 \beta_{15} - \beta_{16} + 10 \beta_{17} - 4 \beta_{18} - 4 \beta_{19} ) q^{97} + ( 5 + 3 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + \beta_{8} + 3 \beta_{9} + \beta_{10} + 4 \beta_{11} - \beta_{12} - 5 \beta_{13} - \beta_{14} + 4 \beta_{15} + 3 \beta_{16} + \beta_{19} ) q^{98} + ( \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{15} + 2 \beta_{17} - \beta_{18} - \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 4q^{2} - 2q^{3} - 6q^{4} - 6q^{5} + 7q^{6} - 6q^{7} + 10q^{8} - 2q^{9} + O(q^{10}) \) \( 20q - 4q^{2} - 2q^{3} - 6q^{4} - 6q^{5} + 7q^{6} - 6q^{7} + 10q^{8} - 2q^{9} - 18q^{10} - 16q^{11} - 6q^{12} + 14q^{13} - 22q^{14} + 5q^{15} - 8q^{16} + 11q^{17} - 4q^{18} - 11q^{19} + 57q^{20} - 6q^{21} + 26q^{22} - 12q^{24} - 4q^{25} - 14q^{26} - 2q^{27} - 14q^{28} + 12q^{29} - 18q^{30} + 41q^{31} - 46q^{32} - 5q^{33} - 3q^{34} - 26q^{35} + 5q^{36} - 18q^{37} + 70q^{38} + 14q^{39} - 13q^{40} + 44q^{42} - 10q^{43} - 3q^{44} + 16q^{45} - 24q^{46} + 18q^{47} + 25q^{48} - 10q^{49} + 33q^{50} + 61q^{52} - 20q^{53} + 7q^{54} - 17q^{55} + 6q^{56} + 11q^{57} - 37q^{58} + 40q^{59} - 31q^{60} - 12q^{61} - 89q^{62} - 6q^{63} - 2q^{64} - 51q^{65} + 15q^{66} - 47q^{67} - 12q^{68} - 22q^{69} + 32q^{70} - 47q^{71} + 10q^{72} + 39q^{73} - 50q^{74} - 26q^{75} - 39q^{76} + 22q^{77} - 14q^{78} - 2q^{79} + 12q^{80} - 2q^{81} + 26q^{82} - 52q^{83} - 14q^{84} + 35q^{85} + 34q^{86} - 32q^{87} + 30q^{88} + 36q^{89} - 18q^{90} + 8q^{91} - 19q^{92} - 14q^{93} + 21q^{94} + 89q^{95} - 2q^{96} + 85q^{98} - 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 7 x^{19} + 24 x^{18} - 70 x^{17} + 209 x^{16} - 527 x^{15} + 1115 x^{14} - 2187 x^{13} + 4165 x^{12} - 7040 x^{11} + 10649 x^{10} - 13519 x^{9} + 15111 x^{8} - 12101 x^{7} + 8244 x^{6} - 2837 x^{5} + 1859 x^{4} - 1730 x^{3} + 1957 x^{2} - 1656 x + 529\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(\)\(44\!\cdots\!58\)\( \nu^{19} - \)\(18\!\cdots\!31\)\( \nu^{18} + \)\(34\!\cdots\!88\)\( \nu^{17} - \)\(10\!\cdots\!28\)\( \nu^{16} + \)\(35\!\cdots\!55\)\( \nu^{15} - \)\(60\!\cdots\!53\)\( \nu^{14} + \)\(97\!\cdots\!11\)\( \nu^{13} - \)\(20\!\cdots\!41\)\( \nu^{12} + \)\(39\!\cdots\!90\)\( \nu^{11} - \)\(36\!\cdots\!91\)\( \nu^{10} + \)\(50\!\cdots\!41\)\( \nu^{9} - \)\(91\!\cdots\!33\)\( \nu^{8} + \)\(49\!\cdots\!54\)\( \nu^{7} + \)\(13\!\cdots\!01\)\( \nu^{6} + \)\(96\!\cdots\!32\)\( \nu^{5} + \)\(23\!\cdots\!80\)\( \nu^{4} + \)\(84\!\cdots\!76\)\( \nu^{3} + \)\(49\!\cdots\!44\)\( \nu^{2} + \)\(17\!\cdots\!37\)\( \nu + \)\(45\!\cdots\!20\)\(\)\()/ \)\(24\!\cdots\!19\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(75\!\cdots\!64\)\( \nu^{19} + \)\(46\!\cdots\!53\)\( \nu^{18} - \)\(15\!\cdots\!83\)\( \nu^{17} + \)\(48\!\cdots\!59\)\( \nu^{16} - \)\(14\!\cdots\!05\)\( \nu^{15} + \)\(34\!\cdots\!55\)\( \nu^{14} - \)\(76\!\cdots\!95\)\( \nu^{13} + \)\(15\!\cdots\!45\)\( \nu^{12} - \)\(28\!\cdots\!99\)\( \nu^{11} + \)\(48\!\cdots\!46\)\( \nu^{10} - \)\(77\!\cdots\!02\)\( \nu^{9} + \)\(97\!\cdots\!28\)\( \nu^{8} - \)\(11\!\cdots\!47\)\( \nu^{7} + \)\(90\!\cdots\!24\)\( \nu^{6} - \)\(82\!\cdots\!25\)\( \nu^{5} - \)\(36\!\cdots\!59\)\( \nu^{4} - \)\(29\!\cdots\!10\)\( \nu^{3} - \)\(53\!\cdots\!83\)\( \nu^{2} - \)\(35\!\cdots\!33\)\( \nu + \)\(26\!\cdots\!36\)\(\)\()/ \)\(24\!\cdots\!19\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(12\!\cdots\!75\)\( \nu^{19} + \)\(71\!\cdots\!04\)\( \nu^{18} - \)\(20\!\cdots\!32\)\( \nu^{17} + \)\(56\!\cdots\!67\)\( \nu^{16} - \)\(17\!\cdots\!13\)\( \nu^{15} + \)\(39\!\cdots\!59\)\( \nu^{14} - \)\(75\!\cdots\!05\)\( \nu^{13} + \)\(14\!\cdots\!80\)\( \nu^{12} - \)\(27\!\cdots\!29\)\( \nu^{11} + \)\(41\!\cdots\!01\)\( \nu^{10} - \)\(58\!\cdots\!69\)\( \nu^{9} + \)\(61\!\cdots\!84\)\( \nu^{8} - \)\(67\!\cdots\!59\)\( \nu^{7} + \)\(26\!\cdots\!20\)\( \nu^{6} - \)\(35\!\cdots\!26\)\( \nu^{5} - \)\(23\!\cdots\!54\)\( \nu^{4} - \)\(56\!\cdots\!84\)\( \nu^{3} + \)\(68\!\cdots\!69\)\( \nu^{2} - \)\(11\!\cdots\!68\)\( \nu + \)\(23\!\cdots\!82\)\(\)\()/ \)\(24\!\cdots\!19\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(14\!\cdots\!14\)\( \nu^{19} + \)\(11\!\cdots\!04\)\( \nu^{18} - \)\(40\!\cdots\!72\)\( \nu^{17} + \)\(11\!\cdots\!57\)\( \nu^{16} - \)\(34\!\cdots\!26\)\( \nu^{15} + \)\(90\!\cdots\!50\)\( \nu^{14} - \)\(19\!\cdots\!71\)\( \nu^{13} + \)\(38\!\cdots\!09\)\( \nu^{12} - \)\(72\!\cdots\!71\)\( \nu^{11} + \)\(12\!\cdots\!97\)\( \nu^{10} - \)\(19\!\cdots\!47\)\( \nu^{9} + \)\(24\!\cdots\!44\)\( \nu^{8} - \)\(28\!\cdots\!40\)\( \nu^{7} + \)\(24\!\cdots\!83\)\( \nu^{6} - \)\(15\!\cdots\!99\)\( \nu^{5} + \)\(68\!\cdots\!00\)\( \nu^{4} - \)\(29\!\cdots\!73\)\( \nu^{3} + \)\(29\!\cdots\!49\)\( \nu^{2} - \)\(42\!\cdots\!99\)\( \nu + \)\(30\!\cdots\!65\)\(\)\()/ \)\(24\!\cdots\!19\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(16\!\cdots\!92\)\( \nu^{19} - \)\(11\!\cdots\!81\)\( \nu^{18} + \)\(40\!\cdots\!33\)\( \nu^{17} - \)\(12\!\cdots\!31\)\( \nu^{16} + \)\(36\!\cdots\!24\)\( \nu^{15} - \)\(93\!\cdots\!53\)\( \nu^{14} + \)\(20\!\cdots\!22\)\( \nu^{13} - \)\(41\!\cdots\!62\)\( \nu^{12} + \)\(78\!\cdots\!48\)\( \nu^{11} - \)\(13\!\cdots\!62\)\( \nu^{10} + \)\(21\!\cdots\!42\)\( \nu^{9} - \)\(27\!\cdots\!55\)\( \nu^{8} + \)\(32\!\cdots\!44\)\( \nu^{7} - \)\(28\!\cdots\!58\)\( \nu^{6} + \)\(21\!\cdots\!40\)\( \nu^{5} - \)\(77\!\cdots\!90\)\( \nu^{4} + \)\(42\!\cdots\!48\)\( \nu^{3} - \)\(32\!\cdots\!95\)\( \nu^{2} + \)\(42\!\cdots\!34\)\( \nu - \)\(41\!\cdots\!10\)\(\)\()/ \)\(24\!\cdots\!19\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(18\!\cdots\!74\)\( \nu^{19} - \)\(12\!\cdots\!98\)\( \nu^{18} + \)\(38\!\cdots\!56\)\( \nu^{17} - \)\(10\!\cdots\!58\)\( \nu^{16} + \)\(31\!\cdots\!82\)\( \nu^{15} - \)\(77\!\cdots\!62\)\( \nu^{14} + \)\(15\!\cdots\!12\)\( \nu^{13} - \)\(28\!\cdots\!13\)\( \nu^{12} + \)\(55\!\cdots\!74\)\( \nu^{11} - \)\(88\!\cdots\!66\)\( \nu^{10} + \)\(12\!\cdots\!00\)\( \nu^{9} - \)\(14\!\cdots\!08\)\( \nu^{8} + \)\(14\!\cdots\!38\)\( \nu^{7} - \)\(88\!\cdots\!58\)\( \nu^{6} + \)\(41\!\cdots\!44\)\( \nu^{5} - \)\(21\!\cdots\!04\)\( \nu^{4} + \)\(62\!\cdots\!88\)\( \nu^{3} - \)\(24\!\cdots\!84\)\( \nu^{2} - \)\(16\!\cdots\!85\)\( \nu - \)\(80\!\cdots\!94\)\(\)\()/ \)\(24\!\cdots\!19\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(25\!\cdots\!75\)\( \nu^{19} - \)\(18\!\cdots\!65\)\( \nu^{18} + \)\(63\!\cdots\!12\)\( \nu^{17} - \)\(18\!\cdots\!00\)\( \nu^{16} + \)\(53\!\cdots\!94\)\( \nu^{15} - \)\(13\!\cdots\!32\)\( \nu^{14} + \)\(28\!\cdots\!75\)\( \nu^{13} - \)\(55\!\cdots\!34\)\( \nu^{12} + \)\(10\!\cdots\!89\)\( \nu^{11} - \)\(17\!\cdots\!62\)\( \nu^{10} + \)\(26\!\cdots\!84\)\( \nu^{9} - \)\(32\!\cdots\!40\)\( \nu^{8} + \)\(34\!\cdots\!77\)\( \nu^{7} - \)\(25\!\cdots\!71\)\( \nu^{6} + \)\(13\!\cdots\!04\)\( \nu^{5} + \)\(30\!\cdots\!17\)\( \nu^{4} + \)\(14\!\cdots\!21\)\( \nu^{3} - \)\(36\!\cdots\!28\)\( \nu^{2} + \)\(49\!\cdots\!79\)\( \nu - \)\(32\!\cdots\!90\)\(\)\()/ \)\(24\!\cdots\!19\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(28\!\cdots\!38\)\( \nu^{19} - \)\(18\!\cdots\!74\)\( \nu^{18} + \)\(61\!\cdots\!08\)\( \nu^{17} - \)\(17\!\cdots\!22\)\( \nu^{16} + \)\(53\!\cdots\!23\)\( \nu^{15} - \)\(13\!\cdots\!20\)\( \nu^{14} + \)\(27\!\cdots\!69\)\( \nu^{13} - \)\(54\!\cdots\!90\)\( \nu^{12} + \)\(10\!\cdots\!76\)\( \nu^{11} - \)\(17\!\cdots\!80\)\( \nu^{10} + \)\(25\!\cdots\!84\)\( \nu^{9} - \)\(31\!\cdots\!93\)\( \nu^{8} + \)\(35\!\cdots\!87\)\( \nu^{7} - \)\(25\!\cdots\!32\)\( \nu^{6} + \)\(19\!\cdots\!07\)\( \nu^{5} - \)\(32\!\cdots\!63\)\( \nu^{4} + \)\(50\!\cdots\!82\)\( \nu^{3} - \)\(30\!\cdots\!30\)\( \nu^{2} + \)\(53\!\cdots\!74\)\( \nu - \)\(25\!\cdots\!46\)\(\)\()/ \)\(24\!\cdots\!19\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(32\!\cdots\!70\)\( \nu^{19} + \)\(20\!\cdots\!42\)\( \nu^{18} - \)\(62\!\cdots\!94\)\( \nu^{17} + \)\(17\!\cdots\!83\)\( \nu^{16} - \)\(53\!\cdots\!60\)\( \nu^{15} + \)\(12\!\cdots\!71\)\( \nu^{14} - \)\(25\!\cdots\!59\)\( \nu^{13} + \)\(49\!\cdots\!00\)\( \nu^{12} - \)\(94\!\cdots\!00\)\( \nu^{11} + \)\(15\!\cdots\!63\)\( \nu^{10} - \)\(21\!\cdots\!80\)\( \nu^{9} + \)\(25\!\cdots\!18\)\( \nu^{8} - \)\(27\!\cdots\!42\)\( \nu^{7} + \)\(16\!\cdots\!93\)\( \nu^{6} - \)\(11\!\cdots\!72\)\( \nu^{5} + \)\(54\!\cdots\!31\)\( \nu^{4} - \)\(47\!\cdots\!76\)\( \nu^{3} + \)\(81\!\cdots\!69\)\( \nu^{2} - \)\(22\!\cdots\!55\)\( \nu + \)\(21\!\cdots\!72\)\(\)\()/ \)\(24\!\cdots\!19\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(35\!\cdots\!92\)\( \nu^{19} + \)\(20\!\cdots\!04\)\( \nu^{18} - \)\(55\!\cdots\!01\)\( \nu^{17} + \)\(15\!\cdots\!91\)\( \nu^{16} - \)\(47\!\cdots\!49\)\( \nu^{15} + \)\(10\!\cdots\!52\)\( \nu^{14} - \)\(20\!\cdots\!47\)\( \nu^{13} + \)\(39\!\cdots\!87\)\( \nu^{12} - \)\(74\!\cdots\!99\)\( \nu^{11} + \)\(11\!\cdots\!09\)\( \nu^{10} - \)\(15\!\cdots\!85\)\( \nu^{9} + \)\(15\!\cdots\!17\)\( \nu^{8} - \)\(16\!\cdots\!29\)\( \nu^{7} + \)\(54\!\cdots\!71\)\( \nu^{6} - \)\(79\!\cdots\!40\)\( \nu^{5} - \)\(33\!\cdots\!44\)\( \nu^{4} - \)\(92\!\cdots\!52\)\( \nu^{3} + \)\(19\!\cdots\!18\)\( \nu^{2} - \)\(25\!\cdots\!14\)\( \nu + \)\(38\!\cdots\!17\)\(\)\()/ \)\(24\!\cdots\!19\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(40\!\cdots\!23\)\( \nu^{19} - \)\(25\!\cdots\!16\)\( \nu^{18} + \)\(77\!\cdots\!73\)\( \nu^{17} - \)\(21\!\cdots\!65\)\( \nu^{16} + \)\(66\!\cdots\!94\)\( \nu^{15} - \)\(15\!\cdots\!00\)\( \nu^{14} + \)\(31\!\cdots\!92\)\( \nu^{13} - \)\(60\!\cdots\!37\)\( \nu^{12} + \)\(11\!\cdots\!65\)\( \nu^{11} - \)\(18\!\cdots\!32\)\( \nu^{10} + \)\(26\!\cdots\!28\)\( \nu^{9} - \)\(30\!\cdots\!24\)\( \nu^{8} + \)\(31\!\cdots\!07\)\( \nu^{7} - \)\(18\!\cdots\!21\)\( \nu^{6} + \)\(13\!\cdots\!85\)\( \nu^{5} - \)\(10\!\cdots\!64\)\( \nu^{4} + \)\(72\!\cdots\!39\)\( \nu^{3} - \)\(57\!\cdots\!13\)\( \nu^{2} + \)\(38\!\cdots\!20\)\( \nu - \)\(23\!\cdots\!35\)\(\)\()/ \)\(24\!\cdots\!19\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(41\!\cdots\!68\)\( \nu^{19} + \)\(25\!\cdots\!06\)\( \nu^{18} - \)\(78\!\cdots\!90\)\( \nu^{17} + \)\(22\!\cdots\!66\)\( \nu^{16} - \)\(68\!\cdots\!29\)\( \nu^{15} + \)\(16\!\cdots\!76\)\( \nu^{14} - \)\(33\!\cdots\!49\)\( \nu^{13} + \)\(64\!\cdots\!57\)\( \nu^{12} - \)\(12\!\cdots\!20\)\( \nu^{11} + \)\(19\!\cdots\!20\)\( \nu^{10} - \)\(28\!\cdots\!69\)\( \nu^{9} + \)\(33\!\cdots\!12\)\( \nu^{8} - \)\(36\!\cdots\!30\)\( \nu^{7} + \)\(22\!\cdots\!26\)\( \nu^{6} - \)\(17\!\cdots\!99\)\( \nu^{5} + \)\(23\!\cdots\!44\)\( \nu^{4} - \)\(71\!\cdots\!81\)\( \nu^{3} + \)\(24\!\cdots\!64\)\( \nu^{2} - \)\(72\!\cdots\!07\)\( \nu + \)\(46\!\cdots\!53\)\(\)\()/ \)\(24\!\cdots\!19\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(43\!\cdots\!69\)\( \nu^{19} + \)\(31\!\cdots\!59\)\( \nu^{18} - \)\(11\!\cdots\!05\)\( \nu^{17} + \)\(33\!\cdots\!32\)\( \nu^{16} - \)\(98\!\cdots\!23\)\( \nu^{15} + \)\(25\!\cdots\!00\)\( \nu^{14} - \)\(54\!\cdots\!39\)\( \nu^{13} + \)\(10\!\cdots\!25\)\( \nu^{12} - \)\(19\!\cdots\!46\)\( \nu^{11} + \)\(34\!\cdots\!44\)\( \nu^{10} - \)\(52\!\cdots\!97\)\( \nu^{9} + \)\(65\!\cdots\!75\)\( \nu^{8} - \)\(72\!\cdots\!38\)\( \nu^{7} + \)\(58\!\cdots\!90\)\( \nu^{6} - \)\(38\!\cdots\!23\)\( \nu^{5} + \)\(12\!\cdots\!45\)\( \nu^{4} - \)\(11\!\cdots\!14\)\( \nu^{3} + \)\(12\!\cdots\!95\)\( \nu^{2} - \)\(11\!\cdots\!79\)\( \nu + \)\(61\!\cdots\!51\)\(\)\()/ \)\(24\!\cdots\!19\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(44\!\cdots\!15\)\( \nu^{19} + \)\(27\!\cdots\!82\)\( \nu^{18} - \)\(81\!\cdots\!44\)\( \nu^{17} + \)\(23\!\cdots\!77\)\( \nu^{16} - \)\(71\!\cdots\!70\)\( \nu^{15} + \)\(16\!\cdots\!11\)\( \nu^{14} - \)\(33\!\cdots\!25\)\( \nu^{13} + \)\(66\!\cdots\!13\)\( \nu^{12} - \)\(12\!\cdots\!38\)\( \nu^{11} + \)\(19\!\cdots\!35\)\( \nu^{10} - \)\(29\!\cdots\!03\)\( \nu^{9} + \)\(34\!\cdots\!57\)\( \nu^{8} - \)\(37\!\cdots\!41\)\( \nu^{7} + \)\(22\!\cdots\!08\)\( \nu^{6} - \)\(18\!\cdots\!39\)\( \nu^{5} - \)\(37\!\cdots\!30\)\( \nu^{4} - \)\(72\!\cdots\!21\)\( \nu^{3} + \)\(44\!\cdots\!11\)\( \nu^{2} - \)\(29\!\cdots\!42\)\( \nu + \)\(35\!\cdots\!20\)\(\)\()/ \)\(24\!\cdots\!19\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(47\!\cdots\!74\)\( \nu^{19} + \)\(30\!\cdots\!80\)\( \nu^{18} - \)\(94\!\cdots\!02\)\( \nu^{17} + \)\(26\!\cdots\!72\)\( \nu^{16} - \)\(80\!\cdots\!44\)\( \nu^{15} + \)\(19\!\cdots\!75\)\( \nu^{14} - \)\(39\!\cdots\!90\)\( \nu^{13} + \)\(75\!\cdots\!69\)\( \nu^{12} - \)\(14\!\cdots\!20\)\( \nu^{11} + \)\(23\!\cdots\!84\)\( \nu^{10} - \)\(33\!\cdots\!46\)\( \nu^{9} + \)\(38\!\cdots\!22\)\( \nu^{8} - \)\(39\!\cdots\!21\)\( \nu^{7} + \)\(22\!\cdots\!87\)\( \nu^{6} - \)\(13\!\cdots\!24\)\( \nu^{5} - \)\(56\!\cdots\!69\)\( \nu^{4} - \)\(55\!\cdots\!03\)\( \nu^{3} + \)\(31\!\cdots\!38\)\( \nu^{2} - \)\(61\!\cdots\!88\)\( \nu + \)\(24\!\cdots\!70\)\(\)\()/ \)\(24\!\cdots\!19\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(50\!\cdots\!84\)\( \nu^{19} + \)\(34\!\cdots\!24\)\( \nu^{18} - \)\(11\!\cdots\!63\)\( \nu^{17} + \)\(33\!\cdots\!97\)\( \nu^{16} - \)\(10\!\cdots\!97\)\( \nu^{15} + \)\(25\!\cdots\!63\)\( \nu^{14} - \)\(52\!\cdots\!05\)\( \nu^{13} + \)\(10\!\cdots\!13\)\( \nu^{12} - \)\(19\!\cdots\!15\)\( \nu^{11} + \)\(32\!\cdots\!61\)\( \nu^{10} - \)\(48\!\cdots\!70\)\( \nu^{9} + \)\(60\!\cdots\!94\)\( \nu^{8} - \)\(66\!\cdots\!96\)\( \nu^{7} + \)\(49\!\cdots\!37\)\( \nu^{6} - \)\(32\!\cdots\!72\)\( \nu^{5} + \)\(60\!\cdots\!83\)\( \nu^{4} - \)\(94\!\cdots\!15\)\( \nu^{3} + \)\(58\!\cdots\!10\)\( \nu^{2} - \)\(10\!\cdots\!71\)\( \nu + \)\(48\!\cdots\!71\)\(\)\()/ \)\(24\!\cdots\!19\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(57\!\cdots\!30\)\( \nu^{19} + \)\(37\!\cdots\!69\)\( \nu^{18} - \)\(11\!\cdots\!97\)\( \nu^{17} + \)\(33\!\cdots\!83\)\( \nu^{16} - \)\(10\!\cdots\!76\)\( \nu^{15} + \)\(24\!\cdots\!98\)\( \nu^{14} - \)\(49\!\cdots\!43\)\( \nu^{13} + \)\(94\!\cdots\!61\)\( \nu^{12} - \)\(17\!\cdots\!86\)\( \nu^{11} + \)\(29\!\cdots\!68\)\( \nu^{10} - \)\(41\!\cdots\!11\)\( \nu^{9} + \)\(47\!\cdots\!24\)\( \nu^{8} - \)\(49\!\cdots\!59\)\( \nu^{7} + \)\(28\!\cdots\!04\)\( \nu^{6} - \)\(17\!\cdots\!62\)\( \nu^{5} - \)\(13\!\cdots\!77\)\( \nu^{4} - \)\(14\!\cdots\!67\)\( \nu^{3} + \)\(73\!\cdots\!52\)\( \nu^{2} - \)\(12\!\cdots\!86\)\( \nu + \)\(12\!\cdots\!30\)\(\)\()/ \)\(24\!\cdots\!19\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(60\!\cdots\!10\)\( \nu^{19} + \)\(39\!\cdots\!95\)\( \nu^{18} - \)\(12\!\cdots\!75\)\( \nu^{17} + \)\(36\!\cdots\!88\)\( \nu^{16} - \)\(10\!\cdots\!90\)\( \nu^{15} + \)\(26\!\cdots\!76\)\( \nu^{14} - \)\(53\!\cdots\!18\)\( \nu^{13} + \)\(10\!\cdots\!95\)\( \nu^{12} - \)\(19\!\cdots\!16\)\( \nu^{11} + \)\(32\!\cdots\!11\)\( \nu^{10} - \)\(46\!\cdots\!28\)\( \nu^{9} + \)\(55\!\cdots\!06\)\( \nu^{8} - \)\(59\!\cdots\!70\)\( \nu^{7} + \)\(38\!\cdots\!33\)\( \nu^{6} - \)\(24\!\cdots\!69\)\( \nu^{5} + \)\(36\!\cdots\!66\)\( \nu^{4} - \)\(11\!\cdots\!07\)\( \nu^{3} + \)\(90\!\cdots\!79\)\( \nu^{2} - \)\(82\!\cdots\!42\)\( \nu + \)\(50\!\cdots\!81\)\(\)\()/ \)\(24\!\cdots\!19\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{19} + \beta_{17} + \beta_{15} + \beta_{11} + \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(2 \beta_{17} + \beta_{15} - \beta_{13} - \beta_{12} + 2 \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} - 5 \beta_{4} + 2 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(4 \beta_{19} - \beta_{18} + 2 \beta_{17} - \beta_{16} + \beta_{14} + 5 \beta_{13} - \beta_{12} + 2 \beta_{11} - 7 \beta_{10} + 8 \beta_{9} + 2 \beta_{8} + \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 8 \beta_{4} + \beta_{3} - 5 \beta_{2} + 8 \beta_{1} - 5\)
\(\nu^{5}\)\(=\)\(-7 \beta_{19} + 6 \beta_{17} + 8 \beta_{16} + 7 \beta_{15} + 8 \beta_{14} + 8 \beta_{13} + 7 \beta_{11} - 9 \beta_{10} + 29 \beta_{9} + 10 \beta_{8} + 2 \beta_{6} + 8 \beta_{5} - 17 \beta_{4} - \beta_{2} + 10 \beta_{1} - 6\)
\(\nu^{6}\)\(=\)\(15 \beta_{19} - 3 \beta_{18} + 13 \beta_{17} + 29 \beta_{16} + 10 \beta_{15} + 14 \beta_{14} + 7 \beta_{13} + 3 \beta_{11} - 36 \beta_{10} + 50 \beta_{9} + 48 \beta_{8} + 2 \beta_{7} - 9 \beta_{6} + 3 \beta_{5} - 48 \beta_{4} - 9 \beta_{3} - 26 \beta_{2} + 14 \beta_{1} - 1\)
\(\nu^{7}\)\(=\)\(21 \beta_{19} - 56 \beta_{18} + 56 \beta_{17} + 61 \beta_{16} - 7 \beta_{15} + 81 \beta_{14} + 102 \beta_{13} + 4 \beta_{12} + 14 \beta_{11} - 100 \beta_{10} + 125 \beta_{9} + 125 \beta_{8} - 23 \beta_{7} + 76 \beta_{6} - 81 \beta_{4} - 4 \beta_{3} - 79 \beta_{2} + 56 \beta_{1} - 42\)
\(\nu^{8}\)\(=\)\(-116 \beta_{19} - 92 \beta_{18} + 19 \beta_{17} + 225 \beta_{16} + 291 \beta_{14} + 262 \beta_{13} + 106 \beta_{12} + 2 \beta_{11} - 28 \beta_{10} + 291 \beta_{9} + 319 \beta_{8} - 133 \beta_{7} + 121 \beta_{6} - 43 \beta_{5} - 92 \beta_{4} - 43 \beta_{3} - 90 \beta_{2} - 73\)
\(\nu^{9}\)\(=\)\(145 \beta_{19} - 227 \beta_{18} + 13 \beta_{17} + 455 \beta_{16} + 34 \beta_{15} + 509 \beta_{14} + 419 \beta_{13} + 442 \beta_{12} - 228 \beta_{11} + 227 \beta_{9} + 795 \beta_{8} - 192 \beta_{7} + 82 \beta_{6} - 442 \beta_{5} - 380 \beta_{3} - 214 \beta_{2} - 380 \beta_{1} + 34\)
\(\nu^{10}\)\(=\)\(297 \beta_{19} - 1211 \beta_{18} + 537 \beta_{17} + 674 \beta_{16} - 483 \beta_{15} + 1485 \beta_{14} + 1457 \beta_{13} + 1131 \beta_{12} - 246 \beta_{11} + 274 \beta_{10} + 1485 \beta_{8} - 571 \beta_{7} + 1259 \beta_{6} - 1545 \beta_{5} + 710 \beta_{4} - 710 \beta_{3} - 483 \beta_{2} - 1131 \beta_{1}\)
\(\nu^{11}\)\(=\)\(-122 \beta_{19} - 1870 \beta_{18} - 1240 \beta_{17} + 1500 \beta_{16} - 1500 \beta_{15} + 3426 \beta_{14} + 3110 \beta_{13} + 4398 \beta_{12} - 1434 \beta_{11} + 3426 \beta_{10} - 1833 \beta_{9} + 1870 \beta_{8} - 2807 \beta_{7} + 619 \beta_{6} - 5046 \beta_{5} + 4398 \beta_{4} - 1833 \beta_{3} - 5046 \beta_{1} - 118\)
\(\nu^{12}\)\(=\)\(4262 \beta_{19} - 2330 \beta_{18} - 4262 \beta_{17} - 1786 \beta_{15} + 2330 \beta_{14} + 1786 \beta_{13} + 12936 \beta_{12} - 6790 \beta_{11} + 9960 \beta_{10} - 12936 \beta_{9} - 5511 \beta_{7} - 4460 \beta_{6} - 16482 \beta_{5} + 16482 \beta_{4} - 7630 \beta_{3} + 3102 \beta_{2} - 19165 \beta_{1} + 3102\)
\(\nu^{13}\)\(=\)\(10883 \beta_{19} - 8909 \beta_{18} - 5526 \beta_{17} - 9390 \beta_{16} - 10883 \beta_{15} + 26744 \beta_{12} - 13802 \beta_{11} + 26744 \beta_{10} - 44123 \beta_{9} - 12738 \beta_{8} - 13802 \beta_{7} - 5526 \beta_{6} - 44123 \beta_{5} + 49873 \beta_{4} - 12738 \beta_{3} + 9330 \beta_{2} - 49873 \beta_{1} + 9390\)
\(\nu^{14}\)\(=\)\(18468 \beta_{19} - 49417 \beta_{17} - 35000 \beta_{16} - 30770 \beta_{15} - 18468 \beta_{14} - 15115 \beta_{13} + 67948 \beta_{12} - 45605 \beta_{11} + 88470 \beta_{10} - 134683 \beta_{9} - 67948 \beta_{8} - 49417 \beta_{7} - 49238 \beta_{6} - 106938 \beta_{5} + 152122 \beta_{4} - 18468 \beta_{3} + 35000 \beta_{2} - 134683 \beta_{1} + 15115\)
\(\nu^{15}\)\(=\)\(67441 \beta_{19} + 53292 \beta_{18} - 140549 \beta_{17} - 138979 \beta_{16} - 35723 \beta_{15} - 141583 \beta_{14} - 134867 \beta_{13} + 141583 \beta_{12} - 140549 \beta_{11} + 222452 \beta_{10} - 410885 \beta_{9} - 245496 \beta_{8} - 89015 \beta_{7} - 194875 \beta_{6} - 245496 \beta_{5} + 410885 \beta_{4} - 53292 \beta_{3} + 134867 \beta_{2} - 364035 \beta_{1} + 74142\)
\(\nu^{16}\)\(=\)\(145408 \beta_{19} + 140670 \beta_{18} - 272160 \beta_{17} - 452326 \beta_{16} - 134161 \beta_{15} - 468982 \beta_{14} - 452326 \beta_{13} + 140670 \beta_{12} - 274831 \beta_{11} + 492028 \beta_{10} - 1075437 \beta_{9} - 752433 \beta_{8} - 140670 \beta_{7} - 406593 \beta_{6} - 468982 \beta_{5} + 961010 \beta_{4} + 323574 \beta_{2} - 752433 \beta_{1} + 203059\)
\(\nu^{17}\)\(=\)\(160116 \beta_{19} + 618272 \beta_{18} - 948695 \beta_{17} - 1216107 \beta_{16} - 330423 \beta_{15} - 1376223 \beta_{14} - 1270704 \beta_{13} - 618272 \beta_{11} + 1177528 \beta_{10} - 2553751 \beta_{9} - 2190475 \beta_{8} - 313886 \beta_{7} - 1181574 \beta_{6} - 618272 \beta_{5} + 2190475 \beta_{4} + 330983 \beta_{3} + 812921 \beta_{2} - 1376223 \beta_{1} + 304386\)
\(\nu^{18}\)\(=\)\(168627 \beta_{19} + 2217072 \beta_{18} - 2217072 \beta_{17} - 3248010 \beta_{16} - 95985 \beta_{15} - 4262288 \beta_{14} - 4093661 \beta_{13} - 1045800 \beta_{12} - 1263141 \beta_{11} + 2291948 \beta_{10} - 5794581 \beta_{9} - 5794581 \beta_{8} + 105691 \beta_{7} - 3231350 \beta_{6} + 4262288 \beta_{4} + 1045800 \beta_{3} + 2322763 \beta_{2} - 2217072 \beta_{1} + 953931\)
\(\nu^{19}\)\(=\)\(-241892 \beta_{19} + 5471549 \beta_{18} - 3300861 \beta_{17} - 8086529 \beta_{16} - 11100538 \beta_{14} - 10816524 \beta_{13} - 6475244 \beta_{12} - 894624 \beta_{11} + 2731858 \beta_{10} - 11100538 \beta_{9} - 13832396 \beta_{8} + 2614980 \beta_{7} - 5755563 \beta_{6} + 4354157 \beta_{5} + 5471549 \beta_{4} + 4354157 \beta_{3} + 4576925 \beta_{2} + 2170688\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(-\beta_{16}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.545935 + 0.160301i
2.55199 + 0.749331i
−0.591994 + 0.380451i
1.76796 1.13620i
−0.591994 0.380451i
1.76796 + 1.13620i
−0.201819 + 1.40368i
0.124087 0.863041i
−0.932269 + 2.04138i
0.480233 1.05156i
−0.932269 2.04138i
0.480233 + 1.05156i
0.545935 0.160301i
2.55199 0.749331i
1.05639 1.21914i
−1.30051 + 1.50087i
−0.201819 1.40368i
0.124087 + 0.863041i
1.05639 + 1.21914i
−1.30051 1.50087i
−1.10489 2.41937i −0.959493 + 0.281733i −3.32285 + 3.83478i −2.44470 1.57111i 1.74175 + 2.01009i −0.326826 2.27313i 7.84516 + 2.30355i 0.841254 0.540641i −1.09998 + 7.65055i
4.2 −0.236364 0.517565i −0.959493 + 0.281733i 1.09772 1.26683i 2.27341 + 1.46103i 0.372604 + 0.430008i −0.481879 3.35154i −2.00700 0.589308i 0.841254 0.540641i 0.218827 1.52197i
13.1 −1.37624 1.58827i 0.841254 + 0.540641i −0.343924 + 2.39205i 1.43308 3.13801i −0.299086 2.08019i −1.51768 0.445631i 0.736612 0.473392i 0.415415 + 0.909632i −6.95628 + 2.04255i
13.2 0.460828 + 0.531824i 0.841254 + 0.540641i 0.214155 1.48948i −0.700489 + 1.53386i 0.100148 + 0.696541i −3.11747 0.915371i 2.07482 1.33341i 0.415415 + 0.909632i −1.13855 + 0.334308i
16.1 −1.37624 + 1.58827i 0.841254 0.540641i −0.343924 2.39205i 1.43308 + 3.13801i −0.299086 + 2.08019i −1.51768 + 0.445631i 0.736612 + 0.473392i 0.415415 0.909632i −6.95628 2.04255i
16.2 0.460828 0.531824i 0.841254 0.540641i 0.214155 + 1.48948i −0.700489 1.53386i 0.100148 0.696541i −3.11747 + 0.915371i 2.07482 + 1.33341i 0.415415 0.909632i −1.13855 0.334308i
25.1 −0.733503 0.471394i −0.142315 0.989821i −0.515016 1.12773i −1.20424 0.353596i −0.362207 + 0.793123i 1.95301 2.25389i −0.402011 + 2.79605i −0.959493 + 0.281733i 0.716629 + 0.827034i
25.2 1.19300 + 0.766692i −0.142315 0.989821i 0.00459227 + 0.0100557i 0.815022 + 0.239312i 0.589107 1.28996i −3.31578 + 3.82661i 0.401407 2.79185i −0.959493 + 0.281733i 0.788839 + 0.910368i
31.1 −0.164520 + 1.14426i 0.415415 + 0.909632i 0.636711 + 0.186955i −1.13298 1.30753i −1.10920 + 0.325692i −0.589836 0.379064i −1.27914 + 2.80093i −0.654861 + 0.755750i 1.68256 1.08131i
31.2 0.319381 2.22134i 0.415415 + 0.909632i −2.91338 0.855446i −1.82263 2.10342i 2.15328 0.632261i 2.85304 + 1.83354i −0.966182 + 2.11564i −0.654861 + 0.755750i −5.25454 + 3.37689i
49.1 −0.164520 1.14426i 0.415415 0.909632i 0.636711 0.186955i −1.13298 + 1.30753i −1.10920 0.325692i −0.589836 + 0.379064i −1.27914 2.80093i −0.654861 0.755750i 1.68256 + 1.08131i
49.2 0.319381 + 2.22134i 0.415415 0.909632i −2.91338 + 0.855446i −1.82263 + 2.10342i 2.15328 + 0.632261i 2.85304 1.83354i −0.966182 2.11564i −0.654861 0.755750i −5.25454 3.37689i
52.1 −1.10489 + 2.41937i −0.959493 0.281733i −3.32285 3.83478i −2.44470 + 1.57111i 1.74175 2.01009i −0.326826 + 2.27313i 7.84516 2.30355i 0.841254 + 0.540641i −1.09998 7.65055i
52.2 −0.236364 + 0.517565i −0.959493 0.281733i 1.09772 + 1.26683i 2.27341 1.46103i 0.372604 0.430008i −0.481879 + 3.35154i −2.00700 + 0.589308i 0.841254 + 0.540641i 0.218827 + 1.52197i
55.1 −1.90549 + 0.559503i −0.654861 0.755750i 1.63535 1.05098i −0.291446 2.02705i 1.67068 + 1.07368i 1.34249 2.93963i 0.0729013 0.0841325i −0.142315 + 0.989821i 1.68949 + 3.69946i
55.2 1.54781 0.454477i −0.654861 0.755750i 0.506650 0.325604i 0.0749695 + 0.521424i −1.35707 0.872135i 0.200934 0.439985i −1.47656 + 1.70404i −0.142315 + 0.989821i 0.353014 + 0.772992i
58.1 −0.733503 + 0.471394i −0.142315 + 0.989821i −0.515016 + 1.12773i −1.20424 + 0.353596i −0.362207 0.793123i 1.95301 + 2.25389i −0.402011 2.79605i −0.959493 0.281733i 0.716629 0.827034i
58.2 1.19300 0.766692i −0.142315 + 0.989821i 0.00459227 0.0100557i 0.815022 0.239312i 0.589107 + 1.28996i −3.31578 3.82661i 0.401407 + 2.79185i −0.959493 0.281733i 0.788839 0.910368i
64.1 −1.90549 0.559503i −0.654861 + 0.755750i 1.63535 + 1.05098i −0.291446 + 2.02705i 1.67068 1.07368i 1.34249 + 2.93963i 0.0729013 + 0.0841325i −0.142315 0.989821i 1.68949 3.69946i
64.2 1.54781 + 0.454477i −0.654861 + 0.755750i 0.506650 + 0.325604i 0.0749695 0.521424i −1.35707 + 0.872135i 0.200934 + 0.439985i −1.47656 1.70404i −0.142315 0.989821i 0.353014 0.772992i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.2.e.c 20
3.b odd 2 1 207.2.i.d 20
23.c even 11 1 inner 69.2.e.c 20
23.c even 11 1 1587.2.a.u 10
23.d odd 22 1 1587.2.a.t 10
69.g even 22 1 4761.2.a.bu 10
69.h odd 22 1 207.2.i.d 20
69.h odd 22 1 4761.2.a.bt 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.2.e.c 20 1.a even 1 1 trivial
69.2.e.c 20 23.c even 11 1 inner
207.2.i.d 20 3.b odd 2 1
207.2.i.d 20 69.h odd 22 1
1587.2.a.t 10 23.d odd 22 1
1587.2.a.u 10 23.c even 11 1
4761.2.a.bt 10 69.h odd 22 1
4761.2.a.bu 10 69.g even 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{20} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(69, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 529 + 621 T + 1451 T^{2} + 217 T^{3} + 2090 T^{4} + 3378 T^{5} + 3360 T^{6} - 518 T^{7} - 3 T^{8} - 1463 T^{9} + 584 T^{10} + 891 T^{11} + 238 T^{12} - 130 T^{13} - 117 T^{14} - 32 T^{15} + 11 T^{16} + 18 T^{17} + 13 T^{18} + 4 T^{19} + T^{20} \)
$3$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
$5$ \( 64009 + 19481 T + 145805 T^{2} + 115434 T^{3} - 189002 T^{4} - 321354 T^{5} - 108988 T^{6} + 191290 T^{7} + 315601 T^{8} + 239228 T^{9} + 121342 T^{10} + 41252 T^{11} + 11873 T^{12} + 3528 T^{13} + 1897 T^{14} + 868 T^{15} + 317 T^{16} + 84 T^{17} + 25 T^{18} + 6 T^{19} + T^{20} \)
$7$ \( 5031049 + 8855364 T + 17109883 T^{2} + 48683984 T^{3} + 86987141 T^{4} + 66450774 T^{5} + 30044087 T^{6} + 13805173 T^{7} + 5386991 T^{8} + 1645479 T^{9} + 874237 T^{10} + 217569 T^{11} + 61312 T^{12} + 23923 T^{13} + 5216 T^{14} + 1601 T^{15} + 475 T^{16} + 78 T^{17} + 30 T^{18} + 6 T^{19} + T^{20} \)
$11$ \( 212521 - 650932 T + 10742767 T^{2} + 49827015 T^{3} + 98337129 T^{4} + 115131941 T^{5} + 88596105 T^{6} + 46615032 T^{7} + 17526879 T^{8} + 5825193 T^{9} + 2788787 T^{10} + 1763355 T^{11} + 962288 T^{12} + 406647 T^{13} + 134524 T^{14} + 35190 T^{15} + 7359 T^{16} + 1240 T^{17} + 164 T^{18} + 16 T^{19} + T^{20} \)
$13$ \( 27867841 + 247305313 T + 687886189 T^{2} + 515954831 T^{3} + 273907480 T^{4} - 30965637 T^{5} + 114335707 T^{6} - 17272254 T^{7} + 8646260 T^{8} - 5805569 T^{9} + 3099922 T^{10} - 1207316 T^{11} + 602621 T^{12} - 277492 T^{13} + 101450 T^{14} - 28511 T^{15} + 6281 T^{16} - 1059 T^{17} + 137 T^{18} - 14 T^{19} + T^{20} \)
$17$ \( 21743569 - 43137413 T + 48476041 T^{2} - 7983613 T^{3} - 29379918 T^{4} - 18459463 T^{5} + 47446006 T^{6} - 12216325 T^{7} + 6678083 T^{8} - 4986542 T^{9} + 1649825 T^{10} - 703538 T^{11} + 302443 T^{12} - 82005 T^{13} + 27779 T^{14} - 7491 T^{15} + 2071 T^{16} - 473 T^{17} + 93 T^{18} - 11 T^{19} + T^{20} \)
$19$ \( 417211354561 + 1167689138605 T + 1682292045723 T^{2} + 1433687867084 T^{3} + 811387501395 T^{4} + 329174049314 T^{5} + 110527289603 T^{6} + 34913262439 T^{7} + 9456358850 T^{8} + 2081466519 T^{9} + 460920780 T^{10} + 85076937 T^{11} + 13167971 T^{12} + 2017598 T^{13} + 261853 T^{14} + 27082 T^{15} + 3701 T^{16} + 506 T^{17} + 70 T^{18} + 11 T^{19} + T^{20} \)
$23$ \( 41426511213649 + 5246836013827 T^{2} - 299624639336 T^{3} + 413760309755 T^{4} - 44887056082 T^{5} + 26707185517 T^{6} - 3174345966 T^{7} + 1552323521 T^{8} - 157655432 T^{9} + 74911893 T^{10} - 6854584 T^{11} + 2934449 T^{12} - 260898 T^{13} + 95437 T^{14} - 6974 T^{15} + 2795 T^{16} - 88 T^{17} + 67 T^{18} + T^{20} \)
$29$ \( 3983644752649 + 973224212270 T + 94251604200 T^{2} - 792189719668 T^{3} + 330488683877 T^{4} + 10384382832 T^{5} - 3619091775 T^{6} - 16373509081 T^{7} + 756055141 T^{8} + 1921283221 T^{9} + 544882757 T^{10} - 3333165 T^{11} + 10019154 T^{12} + 2168676 T^{13} - 226104 T^{14} + 35694 T^{15} + 6809 T^{16} - 1113 T^{17} + 216 T^{18} - 12 T^{19} + T^{20} \)
$31$ \( 811602438117769 - 1846497616518784 T + 1872887881219736 T^{2} - 1027486171939647 T^{3} + 398774487936204 T^{4} - 124579680707311 T^{5} + 32891050772434 T^{6} - 7604737856759 T^{7} + 1591921061317 T^{8} - 306024162026 T^{9} + 54208195317 T^{10} - 8840598096 T^{11} + 1311579404 T^{12} - 173686914 T^{13} + 20316350 T^{14} - 2098762 T^{15} + 190758 T^{16} - 14843 T^{17} + 926 T^{18} - 41 T^{19} + T^{20} \)
$37$ \( 185232969769 - 1756398156938 T + 6635693247222 T^{2} - 10304387023177 T^{3} + 3386245214811 T^{4} + 2892061876742 T^{5} + 2846110565799 T^{6} + 577450156873 T^{7} + 278618953746 T^{8} + 41382636537 T^{9} + 6033509472 T^{10} + 542204146 T^{11} + 74590506 T^{12} + 15460427 T^{13} + 2837236 T^{14} + 400439 T^{15} + 43139 T^{16} + 3657 T^{17} + 292 T^{18} + 18 T^{19} + T^{20} \)
$41$ \( 664350515929 + 2151543270437 T + 4435405724115 T^{2} - 6206735687405 T^{3} + 3150551989121 T^{4} - 234814713430 T^{5} + 215580808888 T^{6} + 92520558658 T^{7} + 9197553644 T^{8} + 1155635140 T^{9} + 519465937 T^{10} + 32715188 T^{11} - 9777127 T^{12} - 1290751 T^{13} + 110878 T^{14} - 4367 T^{15} + 302 T^{16} + 517 T^{17} + 92 T^{18} + T^{20} \)
$43$ \( 131462437009849 + 120467936170428 T + 18987329537704 T^{2} - 5740979127169 T^{3} + 2442809848182 T^{4} + 1729517040119 T^{5} + 324877483029 T^{6} + 68880571485 T^{7} + 37909003957 T^{8} + 6012378658 T^{9} + 2865540668 T^{10} + 473110703 T^{11} + 89091641 T^{12} + 14115991 T^{13} + 1983873 T^{14} + 227303 T^{15} + 24036 T^{16} + 2452 T^{17} + 177 T^{18} + 10 T^{19} + T^{20} \)
$47$ \( ( 87731039 - 27995385 T - 11778349 T^{2} + 4905780 T^{3} + 149039 T^{4} - 213863 T^{5} + 12852 T^{6} + 2715 T^{7} - 247 T^{8} - 9 T^{9} + T^{10} )^{2} \)
$53$ \( 828516832441 + 3909806748890 T + 7076214821723 T^{2} + 6594580754285 T^{3} + 4338128713385 T^{4} + 1727068326175 T^{5} + 515773867396 T^{6} + 95130368281 T^{7} + 15318461849 T^{8} + 1429790440 T^{9} + 622787253 T^{10} + 205009333 T^{11} + 48359617 T^{12} + 7779456 T^{13} + 676012 T^{14} + 23028 T^{15} + 1688 T^{16} + 887 T^{17} + 223 T^{18} + 20 T^{19} + T^{20} \)
$59$ \( 57652331881 + 110161048764 T + 100084329616 T^{2} + 39625142487 T^{3} - 19391688439 T^{4} - 9931944327 T^{5} + 10066551886 T^{6} - 6008254903 T^{7} + 2964084876 T^{8} - 1527150108 T^{9} + 876983064 T^{10} - 425624815 T^{11} + 155331218 T^{12} - 42110293 T^{13} + 8589677 T^{14} - 1325866 T^{15} + 155225 T^{16} - 13657 T^{17} + 890 T^{18} - 40 T^{19} + T^{20} \)
$61$ \( 559921052347801 - 1154348449468164 T + 806981863275915 T^{2} + 93863827675685 T^{3} + 29344336060767 T^{4} + 7585345272390 T^{5} + 529202840696 T^{6} + 303497378875 T^{7} + 84966227190 T^{8} + 9039355534 T^{9} + 3011670982 T^{10} + 417125302 T^{11} + 116294128 T^{12} + 3462423 T^{13} + 2094924 T^{14} + 138814 T^{15} + 15773 T^{16} + 2164 T^{17} + 109 T^{18} + 12 T^{19} + T^{20} \)
$67$ \( 2647129 + 36552182 T + 543359398 T^{2} + 2137736189 T^{3} + 9600272213 T^{4} + 1489358731 T^{5} + 13698171880 T^{6} + 2426760132 T^{7} + 2964727742 T^{8} + 1419485936 T^{9} + 89799821 T^{10} + 68751518 T^{11} + 59010099 T^{12} + 9921221 T^{13} - 410514 T^{14} - 141969 T^{15} + 34648 T^{16} + 9805 T^{17} + 964 T^{18} + 47 T^{19} + T^{20} \)
$71$ \( 1821721454093641 - 1830722241380419 T + 1003139847880820 T^{2} - 635636305555544 T^{3} - 39960729406213 T^{4} + 93890000504563 T^{5} + 97337704577511 T^{6} + 38923383044875 T^{7} + 8229807343325 T^{8} + 1341037196455 T^{9} + 261345406642 T^{10} + 52566087761 T^{11} + 8362735895 T^{12} + 1012547564 T^{13} + 97178716 T^{14} + 7539597 T^{15} + 489049 T^{16} + 27162 T^{17} + 1303 T^{18} + 47 T^{19} + T^{20} \)
$73$ \( 11485321 + 2251624488 T + 154361335484 T^{2} - 371749127445 T^{3} + 718578720842 T^{4} - 777433596759 T^{5} + 912350474910 T^{6} - 569941735282 T^{7} + 306014278123 T^{8} - 90102650635 T^{9} + 21939058383 T^{10} - 4000778448 T^{11} + 487575760 T^{12} - 24817062 T^{13} - 3522575 T^{14} + 777188 T^{15} - 44024 T^{16} - 2979 T^{17} + 621 T^{18} - 39 T^{19} + T^{20} \)
$79$ \( 39874302504589081 + 26488514204354931 T + 3459956022868490 T^{2} + 2061148260614538 T^{3} + 2138876125691477 T^{4} + 622929500570041 T^{5} + 151916833707022 T^{6} + 28462467861757 T^{7} + 4667502220967 T^{8} + 619902055003 T^{9} + 76119563884 T^{10} + 7995477853 T^{11} + 751866713 T^{12} + 60789653 T^{13} + 4775460 T^{14} + 268912 T^{15} + 17045 T^{16} + 1692 T^{17} + 51 T^{18} + 2 T^{19} + T^{20} \)
$83$ \( 798327809318302561 + 126906732472212559 T + 33681219145998552 T^{2} - 16361144210713322 T^{3} + 8994641235973971 T^{4} + 1991863161501339 T^{5} + 1046744052678420 T^{6} + 199525685414329 T^{7} + 46388868610077 T^{8} + 7949630245051 T^{9} + 1064505795266 T^{10} + 144298142417 T^{11} + 15684694589 T^{12} + 1218619319 T^{13} + 87684090 T^{14} + 6651486 T^{15} + 425641 T^{16} + 23016 T^{17} + 1245 T^{18} + 52 T^{19} + T^{20} \)
$89$ \( 586753729 - 494875890 T + 4463167167 T^{2} - 8242699772 T^{3} + 6362643694 T^{4} - 10827229018 T^{5} + 14322007318 T^{6} - 5606741159 T^{7} + 728466628 T^{8} - 73857124 T^{9} + 277723469 T^{10} - 98074647 T^{11} + 7431698 T^{12} + 3070235 T^{13} + 1182503 T^{14} - 376179 T^{15} + 71445 T^{16} - 8447 T^{17} + 668 T^{18} - 36 T^{19} + T^{20} \)
$97$ \( 48681166341721 + 30168220977004 T + 62978375011036 T^{2} + 23690798477368 T^{3} + 25944095157652 T^{4} + 6113719304440 T^{5} + 6037836503997 T^{6} + 667484775056 T^{7} + 543036804344 T^{8} - 104195023306 T^{9} + 46007430195 T^{10} + 4849489238 T^{11} - 790087143 T^{12} - 90565552 T^{13} + 9911969 T^{14} + 595837 T^{15} - 47959 T^{16} - 1419 T^{17} + 12 T^{18} + T^{20} \)
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