Properties

Label 67.2.e.c
Level $67$
Weight $2$
Character orbit 67.e
Analytic conductor $0.535$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 67.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.534997693543\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{11})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 7 x^{19} + 39 x^{18} - 148 x^{17} + 492 x^{16} - 1282 x^{15} + 2921 x^{14} - 4316 x^{13} + 2696 x^{12} + 9361 x^{11} - 20998 x^{10} + 10813 x^{9} + 44155 x^{8} - 46933 x^{7} - 36976 x^{6} - 49131 x^{5} + 105828 x^{4} + 108234 x^{3} + 143215 x^{2} + 41004 x + 4489\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 7 x^{19} + 39 x^{18} - 148 x^{17} + 492 x^{16} - 1282 x^{15} + 2921 x^{14} - 4316 x^{13} + 2696 x^{12} + 9361 x^{11} - 20998 x^{10} + 10813 x^{9} + 44155 x^{8} - 46933 x^{7} - 36976 x^{6} - 49131 x^{5} + 105828 x^{4} + 108234 x^{3} + 143215 x^{2} + 41004 x + 4489\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(\)\(46\!\cdots\!60\)\( \nu^{19} - \)\(36\!\cdots\!45\)\( \nu^{18} + \)\(25\!\cdots\!61\)\( \nu^{17} - \)\(11\!\cdots\!87\)\( \nu^{16} + \)\(46\!\cdots\!10\)\( \nu^{15} - \)\(14\!\cdots\!31\)\( \nu^{14} + \)\(41\!\cdots\!61\)\( \nu^{13} - \)\(92\!\cdots\!81\)\( \nu^{12} + \)\(17\!\cdots\!96\)\( \nu^{11} - \)\(19\!\cdots\!61\)\( \nu^{10} + \)\(52\!\cdots\!59\)\( \nu^{9} + \)\(43\!\cdots\!43\)\( \nu^{8} - \)\(68\!\cdots\!72\)\( \nu^{7} + \)\(22\!\cdots\!09\)\( \nu^{6} + \)\(15\!\cdots\!75\)\( \nu^{5} - \)\(20\!\cdots\!98\)\( \nu^{4} + \)\(91\!\cdots\!60\)\( \nu^{3} - \)\(20\!\cdots\!79\)\( \nu^{2} + \)\(34\!\cdots\!90\)\( \nu + \)\(13\!\cdots\!10\)\(\)\()/ \)\(41\!\cdots\!51\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(12\!\cdots\!39\)\( \nu^{19} - \)\(97\!\cdots\!11\)\( \nu^{18} + \)\(57\!\cdots\!38\)\( \nu^{17} - \)\(23\!\cdots\!00\)\( \nu^{16} + \)\(82\!\cdots\!41\)\( \nu^{15} - \)\(22\!\cdots\!97\)\( \nu^{14} + \)\(56\!\cdots\!34\)\( \nu^{13} - \)\(98\!\cdots\!51\)\( \nu^{12} + \)\(11\!\cdots\!30\)\( \nu^{11} + \)\(60\!\cdots\!81\)\( \nu^{10} - \)\(39\!\cdots\!54\)\( \nu^{9} + \)\(66\!\cdots\!99\)\( \nu^{8} + \)\(17\!\cdots\!25\)\( \nu^{7} - \)\(11\!\cdots\!93\)\( \nu^{6} + \)\(22\!\cdots\!23\)\( \nu^{5} - \)\(21\!\cdots\!01\)\( \nu^{4} + \)\(99\!\cdots\!61\)\( \nu^{3} + \)\(26\!\cdots\!57\)\( \nu^{2} + \)\(30\!\cdots\!30\)\( \nu + \)\(12\!\cdots\!12\)\(\)\()/ \)\(41\!\cdots\!51\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(15\!\cdots\!64\)\( \nu^{19} - \)\(15\!\cdots\!86\)\( \nu^{18} + \)\(92\!\cdots\!85\)\( \nu^{17} - \)\(40\!\cdots\!98\)\( \nu^{16} + \)\(14\!\cdots\!08\)\( \nu^{15} - \)\(43\!\cdots\!01\)\( \nu^{14} + \)\(11\!\cdots\!67\)\( \nu^{13} - \)\(21\!\cdots\!42\)\( \nu^{12} + \)\(29\!\cdots\!51\)\( \nu^{11} - \)\(74\!\cdots\!09\)\( \nu^{10} - \)\(61\!\cdots\!53\)\( \nu^{9} + \)\(12\!\cdots\!85\)\( \nu^{8} - \)\(20\!\cdots\!30\)\( \nu^{7} - \)\(22\!\cdots\!94\)\( \nu^{6} + \)\(23\!\cdots\!20\)\( \nu^{5} - \)\(23\!\cdots\!67\)\( \nu^{4} + \)\(26\!\cdots\!35\)\( \nu^{3} - \)\(46\!\cdots\!90\)\( \nu^{2} - \)\(17\!\cdots\!29\)\( \nu - \)\(10\!\cdots\!89\)\(\)\()/ \)\(41\!\cdots\!51\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(35\!\cdots\!18\)\( \nu^{19} - \)\(28\!\cdots\!24\)\( \nu^{18} + \)\(16\!\cdots\!06\)\( \nu^{17} - \)\(67\!\cdots\!71\)\( \nu^{16} + \)\(22\!\cdots\!45\)\( \nu^{15} - \)\(62\!\cdots\!35\)\( \nu^{14} + \)\(14\!\cdots\!24\)\( \nu^{13} - \)\(24\!\cdots\!64\)\( \nu^{12} + \)\(21\!\cdots\!98\)\( \nu^{11} + \)\(33\!\cdots\!82\)\( \nu^{10} - \)\(12\!\cdots\!11\)\( \nu^{9} + \)\(12\!\cdots\!47\)\( \nu^{8} + \)\(15\!\cdots\!14\)\( \nu^{7} - \)\(44\!\cdots\!51\)\( \nu^{6} + \)\(18\!\cdots\!88\)\( \nu^{5} + \)\(87\!\cdots\!17\)\( \nu^{4} + \)\(24\!\cdots\!34\)\( \nu^{3} + \)\(17\!\cdots\!69\)\( \nu^{2} + \)\(20\!\cdots\!69\)\( \nu - \)\(45\!\cdots\!74\)\(\)\()/ \)\(41\!\cdots\!51\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(53\!\cdots\!99\)\( \nu^{19} - \)\(37\!\cdots\!27\)\( \nu^{18} + \)\(21\!\cdots\!84\)\( \nu^{17} - \)\(80\!\cdots\!28\)\( \nu^{16} + \)\(26\!\cdots\!95\)\( \nu^{15} - \)\(69\!\cdots\!37\)\( \nu^{14} + \)\(15\!\cdots\!46\)\( \nu^{13} - \)\(22\!\cdots\!10\)\( \nu^{12} + \)\(11\!\cdots\!72\)\( \nu^{11} + \)\(56\!\cdots\!73\)\( \nu^{10} - \)\(12\!\cdots\!75\)\( \nu^{9} + \)\(56\!\cdots\!39\)\( \nu^{8} + \)\(28\!\cdots\!93\)\( \nu^{7} - \)\(33\!\cdots\!00\)\( \nu^{6} - \)\(25\!\cdots\!46\)\( \nu^{5} + \)\(17\!\cdots\!40\)\( \nu^{4} + \)\(39\!\cdots\!45\)\( \nu^{3} + \)\(33\!\cdots\!32\)\( \nu^{2} + \)\(67\!\cdots\!37\)\( \nu + \)\(15\!\cdots\!77\)\(\)\()/ \)\(41\!\cdots\!51\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(67\!\cdots\!43\)\( \nu^{19} - \)\(49\!\cdots\!50\)\( \nu^{18} + \)\(27\!\cdots\!77\)\( \nu^{17} - \)\(10\!\cdots\!66\)\( \nu^{16} + \)\(36\!\cdots\!78\)\( \nu^{15} - \)\(98\!\cdots\!50\)\( \nu^{14} + \)\(23\!\cdots\!28\)\( \nu^{13} - \)\(37\!\cdots\!46\)\( \nu^{12} + \)\(32\!\cdots\!46\)\( \nu^{11} + \)\(45\!\cdots\!81\)\( \nu^{10} - \)\(14\!\cdots\!61\)\( \nu^{9} + \)\(11\!\cdots\!99\)\( \nu^{8} + \)\(23\!\cdots\!47\)\( \nu^{7} - \)\(29\!\cdots\!11\)\( \nu^{6} - \)\(19\!\cdots\!36\)\( \nu^{5} - \)\(39\!\cdots\!61\)\( \nu^{4} + \)\(10\!\cdots\!15\)\( \nu^{3} + \)\(47\!\cdots\!40\)\( \nu^{2} + \)\(82\!\cdots\!85\)\( \nu - \)\(14\!\cdots\!22\)\(\)\()/ \)\(28\!\cdots\!17\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(67\!\cdots\!22\)\( \nu^{19} - \)\(45\!\cdots\!48\)\( \nu^{18} + \)\(24\!\cdots\!50\)\( \nu^{17} - \)\(89\!\cdots\!54\)\( \nu^{16} + \)\(28\!\cdots\!67\)\( \nu^{15} - \)\(71\!\cdots\!89\)\( \nu^{14} + \)\(15\!\cdots\!17\)\( \nu^{13} - \)\(19\!\cdots\!44\)\( \nu^{12} + \)\(17\!\cdots\!24\)\( \nu^{11} + \)\(77\!\cdots\!08\)\( \nu^{10} - \)\(12\!\cdots\!62\)\( \nu^{9} - \)\(10\!\cdots\!51\)\( \nu^{8} + \)\(38\!\cdots\!59\)\( \nu^{7} - \)\(21\!\cdots\!88\)\( \nu^{6} - \)\(54\!\cdots\!89\)\( \nu^{5} - \)\(20\!\cdots\!86\)\( \nu^{4} + \)\(77\!\cdots\!55\)\( \nu^{3} + \)\(89\!\cdots\!26\)\( \nu^{2} + \)\(98\!\cdots\!53\)\( \nu + \)\(41\!\cdots\!11\)\(\)\()/ \)\(28\!\cdots\!17\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(15\!\cdots\!67\)\( \nu^{19} + \)\(10\!\cdots\!81\)\( \nu^{18} - \)\(59\!\cdots\!51\)\( \nu^{17} + \)\(22\!\cdots\!21\)\( \nu^{16} - \)\(74\!\cdots\!98\)\( \nu^{15} + \)\(19\!\cdots\!58\)\( \nu^{14} - \)\(42\!\cdots\!40\)\( \nu^{13} + \)\(59\!\cdots\!83\)\( \nu^{12} - \)\(27\!\cdots\!18\)\( \nu^{11} - \)\(16\!\cdots\!04\)\( \nu^{10} + \)\(33\!\cdots\!69\)\( \nu^{9} - \)\(12\!\cdots\!20\)\( \nu^{8} - \)\(77\!\cdots\!80\)\( \nu^{7} + \)\(74\!\cdots\!21\)\( \nu^{6} + \)\(72\!\cdots\!90\)\( \nu^{5} + \)\(61\!\cdots\!37\)\( \nu^{4} - \)\(16\!\cdots\!87\)\( \nu^{3} - \)\(18\!\cdots\!23\)\( \nu^{2} - \)\(19\!\cdots\!75\)\( \nu - \)\(52\!\cdots\!25\)\(\)\()/ \)\(28\!\cdots\!17\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(16\!\cdots\!11\)\( \nu^{19} - \)\(12\!\cdots\!87\)\( \nu^{18} + \)\(69\!\cdots\!58\)\( \nu^{17} - \)\(27\!\cdots\!71\)\( \nu^{16} + \)\(91\!\cdots\!03\)\( \nu^{15} - \)\(24\!\cdots\!78\)\( \nu^{14} + \)\(56\!\cdots\!61\)\( \nu^{13} - \)\(89\!\cdots\!58\)\( \nu^{12} + \)\(72\!\cdots\!34\)\( \nu^{11} + \)\(13\!\cdots\!43\)\( \nu^{10} - \)\(40\!\cdots\!36\)\( \nu^{9} + \)\(30\!\cdots\!88\)\( \nu^{8} + \)\(65\!\cdots\!24\)\( \nu^{7} - \)\(10\!\cdots\!68\)\( \nu^{6} - \)\(32\!\cdots\!15\)\( \nu^{5} - \)\(65\!\cdots\!86\)\( \nu^{4} + \)\(20\!\cdots\!64\)\( \nu^{3} + \)\(12\!\cdots\!00\)\( \nu^{2} + \)\(16\!\cdots\!70\)\( \nu + \)\(20\!\cdots\!59\)\(\)\()/ \)\(28\!\cdots\!17\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(18\!\cdots\!36\)\( \nu^{19} + \)\(13\!\cdots\!65\)\( \nu^{18} - \)\(73\!\cdots\!41\)\( \nu^{17} + \)\(28\!\cdots\!74\)\( \nu^{16} - \)\(93\!\cdots\!12\)\( \nu^{15} + \)\(24\!\cdots\!99\)\( \nu^{14} - \)\(56\!\cdots\!55\)\( \nu^{13} + \)\(84\!\cdots\!54\)\( \nu^{12} - \)\(56\!\cdots\!73\)\( \nu^{11} - \)\(16\!\cdots\!86\)\( \nu^{10} + \)\(39\!\cdots\!55\)\( \nu^{9} - \)\(22\!\cdots\!86\)\( \nu^{8} - \)\(77\!\cdots\!47\)\( \nu^{7} + \)\(87\!\cdots\!63\)\( \nu^{6} + \)\(61\!\cdots\!05\)\( \nu^{5} + \)\(10\!\cdots\!57\)\( \nu^{4} - \)\(21\!\cdots\!75\)\( \nu^{3} - \)\(20\!\cdots\!37\)\( \nu^{2} - \)\(24\!\cdots\!21\)\( \nu - \)\(56\!\cdots\!34\)\(\)\()/ \)\(28\!\cdots\!17\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(19\!\cdots\!03\)\( \nu^{19} + \)\(13\!\cdots\!78\)\( \nu^{18} - \)\(76\!\cdots\!14\)\( \nu^{17} + \)\(28\!\cdots\!99\)\( \nu^{16} - \)\(94\!\cdots\!42\)\( \nu^{15} + \)\(24\!\cdots\!71\)\( \nu^{14} - \)\(54\!\cdots\!71\)\( \nu^{13} + \)\(77\!\cdots\!71\)\( \nu^{12} - \)\(35\!\cdots\!62\)\( \nu^{11} - \)\(21\!\cdots\!17\)\( \nu^{10} + \)\(43\!\cdots\!02\)\( \nu^{9} - \)\(17\!\cdots\!99\)\( \nu^{8} - \)\(10\!\cdots\!82\)\( \nu^{7} + \)\(10\!\cdots\!07\)\( \nu^{6} + \)\(83\!\cdots\!45\)\( \nu^{5} + \)\(70\!\cdots\!51\)\( \nu^{4} - \)\(21\!\cdots\!23\)\( \nu^{3} - \)\(23\!\cdots\!60\)\( \nu^{2} - \)\(22\!\cdots\!81\)\( \nu - \)\(95\!\cdots\!77\)\(\)\()/ \)\(28\!\cdots\!17\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(21\!\cdots\!24\)\( \nu^{19} - \)\(15\!\cdots\!08\)\( \nu^{18} + \)\(86\!\cdots\!62\)\( \nu^{17} - \)\(33\!\cdots\!37\)\( \nu^{16} + \)\(11\!\cdots\!64\)\( \nu^{15} - \)\(29\!\cdots\!66\)\( \nu^{14} + \)\(67\!\cdots\!16\)\( \nu^{13} - \)\(10\!\cdots\!23\)\( \nu^{12} + \)\(73\!\cdots\!02\)\( \nu^{11} + \)\(19\!\cdots\!95\)\( \nu^{10} - \)\(50\!\cdots\!30\)\( \nu^{9} + \)\(31\!\cdots\!99\)\( \nu^{8} + \)\(95\!\cdots\!03\)\( \nu^{7} - \)\(13\!\cdots\!26\)\( \nu^{6} - \)\(62\!\cdots\!74\)\( \nu^{5} - \)\(65\!\cdots\!35\)\( \nu^{4} + \)\(24\!\cdots\!15\)\( \nu^{3} + \)\(16\!\cdots\!08\)\( \nu^{2} + \)\(23\!\cdots\!84\)\( \nu + \)\(17\!\cdots\!99\)\(\)\()/ \)\(28\!\cdots\!17\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(21\!\cdots\!22\)\( \nu^{19} - \)\(15\!\cdots\!26\)\( \nu^{18} + \)\(89\!\cdots\!18\)\( \nu^{17} - \)\(34\!\cdots\!75\)\( \nu^{16} + \)\(11\!\cdots\!24\)\( \nu^{15} - \)\(30\!\cdots\!25\)\( \nu^{14} + \)\(71\!\cdots\!01\)\( \nu^{13} - \)\(11\!\cdots\!63\)\( \nu^{12} + \)\(84\!\cdots\!01\)\( \nu^{11} + \)\(18\!\cdots\!66\)\( \nu^{10} - \)\(50\!\cdots\!89\)\( \nu^{9} + \)\(35\!\cdots\!41\)\( \nu^{8} + \)\(89\!\cdots\!46\)\( \nu^{7} - \)\(12\!\cdots\!45\)\( \nu^{6} - \)\(55\!\cdots\!85\)\( \nu^{5} - \)\(89\!\cdots\!14\)\( \nu^{4} + \)\(25\!\cdots\!74\)\( \nu^{3} + \)\(15\!\cdots\!77\)\( \nu^{2} + \)\(27\!\cdots\!52\)\( \nu + \)\(30\!\cdots\!98\)\(\)\()/ \)\(28\!\cdots\!17\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(23\!\cdots\!31\)\( \nu^{19} + \)\(17\!\cdots\!50\)\( \nu^{18} - \)\(95\!\cdots\!18\)\( \nu^{17} + \)\(36\!\cdots\!16\)\( \nu^{16} - \)\(12\!\cdots\!28\)\( \nu^{15} + \)\(32\!\cdots\!07\)\( \nu^{14} - \)\(74\!\cdots\!30\)\( \nu^{13} + \)\(11\!\cdots\!78\)\( \nu^{12} - \)\(79\!\cdots\!46\)\( \nu^{11} - \)\(21\!\cdots\!67\)\( \nu^{10} + \)\(53\!\cdots\!29\)\( \nu^{9} - \)\(34\!\cdots\!28\)\( \nu^{8} - \)\(10\!\cdots\!92\)\( \nu^{7} + \)\(13\!\cdots\!54\)\( \nu^{6} + \)\(65\!\cdots\!56\)\( \nu^{5} + \)\(10\!\cdots\!79\)\( \nu^{4} - \)\(25\!\cdots\!88\)\( \nu^{3} - \)\(23\!\cdots\!39\)\( \nu^{2} - \)\(31\!\cdots\!21\)\( \nu - \)\(52\!\cdots\!45\)\(\)\()/ \)\(28\!\cdots\!17\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(24\!\cdots\!58\)\( \nu^{19} - \)\(17\!\cdots\!07\)\( \nu^{18} + \)\(10\!\cdots\!10\)\( \nu^{17} - \)\(39\!\cdots\!39\)\( \nu^{16} + \)\(13\!\cdots\!05\)\( \nu^{15} - \)\(35\!\cdots\!94\)\( \nu^{14} + \)\(81\!\cdots\!73\)\( \nu^{13} - \)\(12\!\cdots\!78\)\( \nu^{12} + \)\(10\!\cdots\!05\)\( \nu^{11} + \)\(20\!\cdots\!92\)\( \nu^{10} - \)\(58\!\cdots\!58\)\( \nu^{9} + \)\(45\!\cdots\!37\)\( \nu^{8} + \)\(94\!\cdots\!69\)\( \nu^{7} - \)\(14\!\cdots\!94\)\( \nu^{6} - \)\(39\!\cdots\!67\)\( \nu^{5} - \)\(10\!\cdots\!34\)\( \nu^{4} + \)\(27\!\cdots\!33\)\( \nu^{3} + \)\(18\!\cdots\!08\)\( \nu^{2} + \)\(30\!\cdots\!30\)\( \nu + \)\(42\!\cdots\!38\)\(\)\()/ \)\(28\!\cdots\!17\)\( \)
\(\beta_{17}\)\(=\)\((\)\(\)\(65\!\cdots\!88\)\( \nu^{19} - \)\(46\!\cdots\!31\)\( \nu^{18} + \)\(26\!\cdots\!60\)\( \nu^{17} - \)\(10\!\cdots\!52\)\( \nu^{16} + \)\(33\!\cdots\!19\)\( \nu^{15} - \)\(88\!\cdots\!14\)\( \nu^{14} + \)\(20\!\cdots\!28\)\( \nu^{13} - \)\(31\!\cdots\!68\)\( \nu^{12} + \)\(22\!\cdots\!81\)\( \nu^{11} + \)\(57\!\cdots\!54\)\( \nu^{10} - \)\(14\!\cdots\!65\)\( \nu^{9} + \)\(95\!\cdots\!83\)\( \nu^{8} + \)\(26\!\cdots\!68\)\( \nu^{7} - \)\(35\!\cdots\!06\)\( \nu^{6} - \)\(17\!\cdots\!31\)\( \nu^{5} - \)\(30\!\cdots\!78\)\( \nu^{4} + \)\(72\!\cdots\!01\)\( \nu^{3} + \)\(62\!\cdots\!71\)\( \nu^{2} + \)\(80\!\cdots\!97\)\( \nu + \)\(90\!\cdots\!00\)\(\)\()/ \)\(28\!\cdots\!17\)\( \)
\(\beta_{18}\)\(=\)\((\)\(\)\(68\!\cdots\!04\)\( \nu^{19} - \)\(48\!\cdots\!95\)\( \nu^{18} + \)\(26\!\cdots\!60\)\( \nu^{17} - \)\(10\!\cdots\!07\)\( \nu^{16} + \)\(34\!\cdots\!27\)\( \nu^{15} - \)\(89\!\cdots\!16\)\( \nu^{14} + \)\(20\!\cdots\!30\)\( \nu^{13} - \)\(30\!\cdots\!17\)\( \nu^{12} + \)\(19\!\cdots\!42\)\( \nu^{11} + \)\(63\!\cdots\!56\)\( \nu^{10} - \)\(14\!\cdots\!00\)\( \nu^{9} + \)\(74\!\cdots\!72\)\( \nu^{8} + \)\(30\!\cdots\!33\)\( \nu^{7} - \)\(33\!\cdots\!74\)\( \nu^{6} - \)\(26\!\cdots\!92\)\( \nu^{5} - \)\(29\!\cdots\!31\)\( \nu^{4} + \)\(79\!\cdots\!31\)\( \nu^{3} + \)\(68\!\cdots\!47\)\( \nu^{2} + \)\(87\!\cdots\!84\)\( \nu + \)\(12\!\cdots\!28\)\(\)\()/ \)\(28\!\cdots\!17\)\( \)
\(\beta_{19}\)\(=\)\((\)\(\)\(69\!\cdots\!63\)\( \nu^{19} - \)\(49\!\cdots\!86\)\( \nu^{18} + \)\(27\!\cdots\!86\)\( \nu^{17} - \)\(10\!\cdots\!74\)\( \nu^{16} + \)\(35\!\cdots\!27\)\( \nu^{15} - \)\(93\!\cdots\!30\)\( \nu^{14} + \)\(21\!\cdots\!45\)\( \nu^{13} - \)\(32\!\cdots\!08\)\( \nu^{12} + \)\(23\!\cdots\!08\)\( \nu^{11} + \)\(61\!\cdots\!36\)\( \nu^{10} - \)\(15\!\cdots\!97\)\( \nu^{9} + \)\(94\!\cdots\!28\)\( \nu^{8} + \)\(29\!\cdots\!95\)\( \nu^{7} - \)\(36\!\cdots\!51\)\( \nu^{6} - \)\(22\!\cdots\!62\)\( \nu^{5} - \)\(30\!\cdots\!01\)\( \nu^{4} + \)\(83\!\cdots\!39\)\( \nu^{3} + \)\(61\!\cdots\!18\)\( \nu^{2} + \)\(84\!\cdots\!86\)\( \nu + \)\(14\!\cdots\!60\)\(\)\()/ \)\(28\!\cdots\!17\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{19} + \beta_{16} - 4 \beta_{14} + \beta_{13} - \beta_{10} + \beta_{9} + \beta_{6} + \beta_{4} + \beta_{1} - 1\)
\(\nu^{3}\)\(=\)\(6 \beta_{19} + \beta_{17} - \beta_{15} - 7 \beta_{14} + 8 \beta_{11} - 9 \beta_{10} - 8 \beta_{8} + \beta_{6} - \beta_{3} - \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(7 \beta_{19} + \beta_{18} + 9 \beta_{17} - 7 \beta_{16} + 5 \beta_{15} + 10 \beta_{14} - 16 \beta_{13} + 29 \beta_{11} - \beta_{10} - 3 \beta_{8} - 18 \beta_{7} - 2 \beta_{6} - 9 \beta_{5} - \beta_{4} + 7 \beta_{3} + 2\)
\(\nu^{5}\)\(=\)\(2 \beta_{18} - 2 \beta_{17} + 8 \beta_{16} + 24 \beta_{15} - 8 \beta_{14} + 24 \beta_{13} + 15 \beta_{12} + 7 \beta_{11} - 14 \beta_{9} + 20 \beta_{8} + 7 \beta_{7} + 8 \beta_{6} - 8 \beta_{5} - 3 \beta_{4} + 58 \beta_{3} + 3 \beta_{2} + 14\)
\(\nu^{6}\)\(=\)\(-45 \beta_{19} - 77 \beta_{18} - 19 \beta_{17} - 104 \beta_{16} - 96 \beta_{15} + 51 \beta_{14} + 98 \beta_{13} - 45 \beta_{12} - 168 \beta_{11} + 12 \beta_{10} - 26 \beta_{9} + 24 \beta_{8} + 225 \beta_{7} + 77 \beta_{6} + 32 \beta_{5} - 2 \beta_{4} + 19 \beta_{2} - 2 \beta_{1} - 12\)
\(\nu^{7}\)\(=\)\(-325 \beta_{19} - 221 \beta_{18} + 122 \beta_{17} - 430 \beta_{16} - 355 \beta_{15} + 817 \beta_{14} - 474 \beta_{13} - 502 \beta_{12} - 576 \beta_{11} + 695 \beta_{10} + 654 \beta_{9} + 546 \beta_{8} - 209 \beta_{7} + 134 \beta_{6} - 221 \beta_{5} + 177 \beta_{4} - 502 \beta_{3} - 134 \beta_{2} + 122 \beta_{1} - 857\)
\(\nu^{8}\)\(=\)\(-309 \beta_{19} + 680 \beta_{18} - 309 \beta_{17} + 1732 \beta_{16} + 585 \beta_{15} - 585 \beta_{14} - 136 \beta_{11} + 136 \beta_{10} + 1141 \beta_{9} + 490 \beta_{8} - 1141 \beta_{7} - 526 \beta_{6} - 386 \beta_{5} - 280 \beta_{4} - 280 \beta_{3} - 386 \beta_{2} - 526 \beta_{1} - 490\)
\(\nu^{9}\)\(=\)\(1200 \beta_{19} + 2398 \beta_{18} - 2534 \beta_{17} + 4853 \beta_{16} + 1827 \beta_{15} - 5725 \beta_{14} + 4737 \beta_{13} + 4451 \beta_{12} + 3027 \beta_{11} - 4498 \beta_{10} - 6906 \beta_{9} - 5725 \beta_{8} + 3434 \beta_{7} - 3251 \beta_{6} + 3134 \beta_{5} - 4451 \beta_{4} + 1917 \beta_{3} + 2398 \beta_{2} - 3134 \beta_{1} + 9052\)
\(\nu^{10}\)\(=\)\(-2760 \beta_{19} - 4199 \beta_{18} - 1672 \beta_{17} - 14575 \beta_{16} - 7226 \beta_{15} + 14584 \beta_{14} - 7625 \beta_{13} + 1672 \beta_{12} - 7067 \beta_{11} + 19007 \beta_{10} - 10802 \beta_{9} + 5395 \beta_{8} - 1355 \beta_{7} - 1962 \beta_{6} + 4432 \beta_{5} - 290 \beta_{3} + 6170 \beta_{2} + 4199 \beta_{1} - 4199\)
\(\nu^{11}\)\(=\)\(-7068 \beta_{19} - 12830 \beta_{18} - 24282 \beta_{16} - 24282 \beta_{15} + 19898 \beta_{14} - 25959 \beta_{13} - 19898 \beta_{12} - 45857 \beta_{11} + 43444 \beta_{10} + 43444 \beta_{9} + 47434 \beta_{8} - 27536 \beta_{7} + 19898 \beta_{6} - 11716 \beta_{5} + 40178 \beta_{4} - 8182 \beta_{3} - 19007 \beta_{2} + 19007 \beta_{1} - 79273\)
\(\nu^{12}\)\(=\)\(55107 \beta_{19} + 28866 \beta_{18} + 9300 \beta_{17} + 88243 \beta_{16} + 11392 \beta_{15} - 152170 \beta_{14} + 80072 \beta_{13} + 2363 \beta_{12} + 93273 \beta_{11} - 242876 \beta_{10} + 24411 \beta_{9} - 143350 \beta_{8} + 80072 \beta_{7} + 2363 \beta_{5} + 9300 \beta_{4} - 26503 \beta_{3} - 52744 \beta_{2} - 47700 \beta_{1} + 108726\)
\(\nu^{13}\)\(=\)\(62564 \beta_{19} + 87418 \beta_{18} + 166296 \beta_{17} + 73420 \beta_{16} + 272576 \beta_{15} + 47963 \beta_{14} - 13998 \beta_{13} + 87418 \beta_{12} + 570642 \beta_{11} - 335140 \beta_{10} - 238657 \beta_{9} - 301221 \beta_{8} - 201435 \beta_{6} - 201435 \beta_{4} + 10838 \beta_{3} + 76580 \beta_{2} - 24854 \beta_{1} + 595496\)
\(\nu^{14}\)\(=\)\(-197672 \beta_{19} + 60285 \beta_{17} + 706584 \beta_{15} + 290566 \beta_{14} - 146365 \beta_{13} + 274855 \beta_{12} - 230281 \beta_{11} + 1106079 \beta_{10} - 128490 \beta_{9} + 1166364 \beta_{8} - 708144 \beta_{7} + 60285 \beta_{6} - 15528 \beta_{5} + 15528 \beta_{4} + 772077 \beta_{3} + 274855 \beta_{2} + 497222 \beta_{1} - 373004\)
\(\nu^{15}\)\(=\)\(433106 \beta_{19} - 900567 \beta_{18} - 1106079 \beta_{17} - 784612 \beta_{16} - 1733161 \beta_{15} - 2117640 \beta_{14} + 1637241 \beta_{13} - 108690 \beta_{12} - 3350956 \beta_{11} - 63701 \beta_{10} + 108690 \beta_{9} + 193976 \beta_{8} + 2585101 \beta_{7} + 2295495 \beta_{6} + 1106079 \beta_{5} + 900567 \beta_{4} + 433106 \beta_{3} + 108690 \beta_{1} - 1999381\)
\(\nu^{16}\)\(=\)\(-2854959 \beta_{18} + 2854959 \beta_{17} - 8164703 \beta_{16} - 7847972 \beta_{15} + 8164703 \beta_{14} - 7847972 \beta_{13} - 7313083 \beta_{12} + 279847 \beta_{11} + 6288794 \beta_{9} - 4225170 \beta_{8} + 279847 \beta_{7} + 2994821 \beta_{6} - 2994821 \beta_{5} + 2791258 \beta_{4} - 9337631 \beta_{3} - 2791258 \beta_{2} - 6288794\)
\(\nu^{17}\)\(=\)\(-3628744 \beta_{19} + 11812158 \beta_{18} + 7961140 \beta_{17} + 20664705 \beta_{16} + 19773298 \beta_{15} + 6787974 \beta_{14} - 13567272 \beta_{13} - 3628744 \beta_{12} + 24997101 \beta_{11} - 285470 \beta_{10} + 18600132 \beta_{9} + 5383858 \beta_{8} - 33007342 \beta_{7} - 11812158 \beta_{6} - 19906519 \beta_{5} - 1118912 \beta_{4} - 7961140 \beta_{2} - 1118912 \beta_{1} + 285470\)
\(\nu^{18}\)\(=\)\(20352800 \beta_{19} + 42193117 \beta_{18} - 41404666 \beta_{17} + 117996676 \beta_{16} + 75682468 \beta_{15} - 144428287 \beta_{14} + 138958874 \beta_{13} + 89723120 \beta_{12} + 45884479 \beta_{11} - 103023621 \beta_{10} - 114466348 \beta_{9} - 62545917 \beta_{8} + 75803559 \beta_{7} - 41690136 \beta_{6} + 42193117 \beta_{5} - 69370320 \beta_{4} + 89723120 \beta_{3} + 41690136 \beta_{2} - 41404666 \beta_{1} + 165405588\)
\(\nu^{19}\)\(=\)\(-42150258 \beta_{19} - 152375454 \beta_{18} - 42150258 \beta_{17} - 336873110 \beta_{16} - 213804591 \beta_{15} + 213804591 \beta_{14} - 249911180 \beta_{11} + 249911180 \beta_{10} - 215731531 \beta_{9} + 62459923 \beta_{8} + 215731531 \beta_{7} + 48197363 \beta_{6} + 145173879 \beta_{5} - 30422061 \beta_{4} - 30422061 \beta_{3} + 145173879 \beta_{2} + 48197363 \beta_{1} - 62459923\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/67\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{10}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−1.27213 2.78557i
1.07319 + 2.34996i
−0.625078 + 0.721379i
1.66916 1.92631i
−1.27213 + 2.78557i
1.07319 2.34996i
−0.165981 + 0.106669i
2.28033 1.46548i
−0.625078 0.721379i
1.66916 + 1.92631i
1.58638 0.465803i
−1.35948 + 0.399179i
−0.127556 + 0.887172i
0.441163 3.06836i
1.58638 + 0.465803i
−1.35948 0.399179i
−0.127556 0.887172i
0.441163 + 3.06836i
−0.165981 0.106669i
2.28033 + 1.46548i
−0.239446 0.153882i −2.47877 0.727832i −0.797176 1.74557i 1.21919 1.40702i 0.481530 + 0.555715i −1.14774 0.737606i −0.158746 + 1.10411i 3.09080 + 1.98634i −0.508444 + 0.149293i
9.2 −0.239446 0.153882i 2.93826 + 0.862752i −0.797176 1.74557i −1.85253 + 2.13793i −0.570792 0.658729i −3.42222 2.19933i −0.158746 + 1.10411i 5.36529 + 3.44806i 0.772569 0.226847i
14.1 −0.797176 1.74557i −2.14425 1.37802i −1.10181 + 1.27155i 0.201991 + 1.40488i −0.696097 + 4.84146i −0.918948 2.01222i −0.584585 0.171650i 1.45260 + 3.18076i 2.29130 1.47253i
14.2 −0.797176 1.74557i 0.802994 + 0.516053i −1.10181 + 1.27155i −0.451016 3.13689i 0.260680 1.81307i 1.40141 + 3.06866i −0.584585 0.171650i −0.867756 1.90012i −5.11612 + 3.28793i
15.1 −0.239446 + 0.153882i −2.47877 + 0.727832i −0.797176 + 1.74557i 1.21919 + 1.40702i 0.481530 0.555715i −1.14774 + 0.737606i −0.158746 1.10411i 3.09080 1.98634i −0.508444 0.149293i
15.2 −0.239446 + 0.153882i 2.93826 0.862752i −0.797176 + 1.74557i −1.85253 2.13793i −0.570792 + 0.658729i −3.42222 + 2.19933i −0.158746 1.10411i 5.36529 3.44806i 0.772569 + 0.226847i
22.1 1.25667 0.368991i −0.385764 2.68305i −0.239446 + 0.153882i −1.35049 + 2.95717i −1.47480 3.22936i 4.00914 1.17719i −1.95949 + 2.26138i −4.17146 + 1.22485i −0.605953 + 4.21450i
22.2 1.25667 0.368991i 0.0280790 + 0.195293i −0.239446 + 0.153882i 0.681980 1.49333i 0.107348 + 0.235058i −3.00345 + 0.881891i −1.95949 + 2.26138i 2.84113 0.834230i 0.305998 2.12826i
24.1 −0.797176 + 1.74557i −2.14425 + 1.37802i −1.10181 1.27155i 0.201991 1.40488i −0.696097 4.84146i −0.918948 + 2.01222i −0.584585 + 0.171650i 1.45260 3.18076i 2.29130 + 1.47253i
24.2 −0.797176 + 1.74557i 0.802994 0.516053i −1.10181 1.27155i −0.451016 + 3.13689i 0.260680 + 1.81307i 1.40141 3.06866i −0.584585 + 0.171650i −0.867756 + 1.90012i −5.11612 3.28793i
25.1 −0.118239 0.822373i −0.927853 + 1.07080i 1.25667 0.368991i 1.58839 1.02080i 0.990306 + 0.636431i −0.0441131 0.306813i −1.14231 2.50132i 0.141244 + 0.982373i −1.02729 1.18555i
25.2 −0.118239 0.822373i 1.08271 1.24952i 1.25667 0.368991i −3.36803 + 2.16450i −1.15559 0.742653i 0.0592135 + 0.411839i −1.14231 2.50132i 0.0379173 + 0.263721i 2.17826 + 2.51385i
40.1 −1.10181 1.27155i −1.28775 + 2.81978i −0.118239 + 0.822373i 2.46094 + 0.722597i 5.00435 1.46941i 2.41674 + 2.78906i −1.65486 + 1.06351i −4.32827 4.99509i −1.79266 3.92538i
40.2 −1.10181 1.27155i 0.372334 0.815299i −0.118239 + 0.822373i 1.36958 + 0.402144i −1.44694 + 0.424859i −3.35003 3.86614i −1.65486 + 1.06351i 1.43850 + 1.66012i −0.997662 2.18457i
59.1 −0.118239 + 0.822373i −0.927853 1.07080i 1.25667 + 0.368991i 1.58839 + 1.02080i 0.990306 0.636431i −0.0441131 + 0.306813i −1.14231 + 2.50132i 0.141244 0.982373i −1.02729 + 1.18555i
59.2 −0.118239 + 0.822373i 1.08271 + 1.24952i 1.25667 + 0.368991i −3.36803 2.16450i −1.15559 + 0.742653i 0.0592135 0.411839i −1.14231 + 2.50132i 0.0379173 0.263721i 2.17826 2.51385i
62.1 −1.10181 + 1.27155i −1.28775 2.81978i −0.118239 0.822373i 2.46094 0.722597i 5.00435 + 1.46941i 2.41674 2.78906i −1.65486 1.06351i −4.32827 + 4.99509i −1.79266 + 3.92538i
62.2 −1.10181 + 1.27155i 0.372334 + 0.815299i −0.118239 0.822373i 1.36958 0.402144i −1.44694 0.424859i −3.35003 + 3.86614i −1.65486 1.06351i 1.43850 1.66012i −0.997662 + 2.18457i
64.1 1.25667 + 0.368991i −0.385764 + 2.68305i −0.239446 0.153882i −1.35049 2.95717i −1.47480 + 3.22936i 4.00914 + 1.17719i −1.95949 2.26138i −4.17146 1.22485i −0.605953 4.21450i
64.2 1.25667 + 0.368991i 0.0280790 0.195293i −0.239446 0.153882i 0.681980 + 1.49333i 0.107348 0.235058i −3.00345 0.881891i −1.95949 2.26138i 2.84113 + 0.834230i 0.305998 + 2.12826i
\(n\): 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
67.e even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 67.2.e.c 20
3.b odd 2 1 603.2.u.c 20
67.e even 11 1 inner 67.2.e.c 20
67.e even 11 1 4489.2.a.l 10
67.f odd 22 1 4489.2.a.m 10
201.k odd 22 1 603.2.u.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
67.2.e.c 20 1.a even 1 1 trivial
67.2.e.c 20 67.e even 11 1 inner
603.2.u.c 20 3.b odd 2 1
603.2.u.c 20 201.k odd 22 1
4489.2.a.l 10 67.e even 11 1
4489.2.a.m 10 67.f odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{10} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(67, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 6 T + 14 T^{2} + 7 T^{3} + 9 T^{4} - 12 T^{5} - 6 T^{6} - 3 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} )^{2} \)
$3$ \( 4489 - 12797 T + 130686 T^{2} - 172013 T^{3} + 161774 T^{4} - 17000 T^{5} + 11468 T^{6} - 23074 T^{7} + 69290 T^{8} + 28534 T^{9} + 1167 T^{10} + 4004 T^{11} + 5721 T^{12} + 2625 T^{13} + 534 T^{14} - 193 T^{15} - 146 T^{16} - 27 T^{17} + 6 T^{18} + 4 T^{19} + T^{20} \)
$5$ \( 12243001 - 38751425 T + 64417909 T^{2} - 73103531 T^{3} + 61743495 T^{4} - 40426860 T^{5} + 21256728 T^{6} - 9393478 T^{7} + 3770223 T^{8} - 1457269 T^{9} + 530058 T^{10} - 166705 T^{11} + 46917 T^{12} - 13273 T^{13} + 3587 T^{14} - 672 T^{15} + 154 T^{16} - 57 T^{17} + 7 T^{18} - T^{19} + T^{20} \)
$7$ \( 1739761 + 3592956 T + 30421885 T^{2} + 48115435 T^{3} + 139995442 T^{4} + 204122487 T^{5} + 159617730 T^{6} + 66827206 T^{7} + 17843461 T^{8} + 3029675 T^{9} + 953470 T^{10} + 389598 T^{11} + 11093 T^{12} - 22574 T^{13} + 1238 T^{14} + 1290 T^{15} - 178 T^{16} - 81 T^{17} + 18 T^{18} + 8 T^{19} + T^{20} \)
$11$ \( 4190209 + 19071899 T + 165359346 T^{2} - 172911585 T^{3} + 89582871 T^{4} + 6491969 T^{5} - 9051105 T^{6} + 3362755 T^{7} - 2017901 T^{8} + 1173843 T^{9} + 321520 T^{10} - 545171 T^{11} + 264295 T^{12} - 81136 T^{13} + 20717 T^{14} - 3476 T^{15} + 267 T^{16} - 22 T^{17} + 5 T^{18} + T^{20} \)
$13$ \( 4489 + 117317 T + 43983286 T^{2} - 93628046 T^{3} + 100286186 T^{4} - 91675024 T^{5} + 109056297 T^{6} - 131334264 T^{7} + 119253591 T^{8} - 75044530 T^{9} + 32454225 T^{10} - 10109858 T^{11} + 2614856 T^{12} - 633607 T^{13} + 134817 T^{14} - 23299 T^{15} + 4037 T^{16} - 552 T^{17} + 73 T^{18} - 12 T^{19} + T^{20} \)
$17$ \( 876811321 + 3586365876 T + 4780842296 T^{2} - 1968690111 T^{3} + 1717318889 T^{4} + 379204145 T^{5} + 21924787 T^{6} + 218224049 T^{7} + 88695890 T^{8} + 34988327 T^{9} + 21229891 T^{10} + 7173254 T^{11} + 1716524 T^{12} + 373507 T^{13} + 80898 T^{14} + 16688 T^{15} + 3014 T^{16} + 480 T^{17} + 57 T^{18} + 4 T^{19} + T^{20} \)
$19$ \( 23319241 + 112134209 T + 936759857 T^{2} - 379730186 T^{3} + 1328554953 T^{4} - 471067014 T^{5} + 422169748 T^{6} - 204772403 T^{7} + 111519870 T^{8} - 12746349 T^{9} - 197174 T^{10} - 1119033 T^{11} + 1363998 T^{12} - 417708 T^{13} + 72802 T^{14} - 11760 T^{15} + 465 T^{16} + 431 T^{17} - 61 T^{18} - 4 T^{19} + T^{20} \)
$23$ \( 1098458449 + 6410651632 T + 12456296600 T^{2} - 6205229569 T^{3} + 12119529279 T^{4} - 6215090607 T^{5} + 3139859668 T^{6} - 1229454809 T^{7} + 307830402 T^{8} - 54274693 T^{9} + 5136605 T^{10} - 1520547 T^{11} + 1123972 T^{12} - 15638 T^{13} - 48611 T^{14} + 6722 T^{15} + 1248 T^{16} - 140 T^{17} + 15 T^{18} - 2 T^{19} + T^{20} \)
$29$ \( ( 737 - 1067 T - 7469 T^{2} + 19063 T^{3} - 11726 T^{4} - 989 T^{5} + 2069 T^{6} - 70 T^{7} - 91 T^{8} + 3 T^{9} + T^{10} )^{2} \)
$31$ \( 867288026089 + 1044971234791 T + 1126793392847 T^{2} + 902797631428 T^{3} + 549693049632 T^{4} + 212623989391 T^{5} + 64235476779 T^{6} + 11393629962 T^{7} + 2979052847 T^{8} + 590016570 T^{9} + 88040459 T^{10} + 8277346 T^{11} + 7389768 T^{12} - 670032 T^{13} - 87889 T^{14} + 22693 T^{15} + 408 T^{16} - 242 T^{17} - 21 T^{18} + T^{20} \)
$37$ \( ( -279553 - 1318714 T + 119670 T^{2} + 318196 T^{3} - 12341 T^{4} - 25994 T^{5} + 1046 T^{6} + 912 T^{7} - 54 T^{8} - 12 T^{9} + T^{10} )^{2} \)
$41$ \( 132185354901721 - 64739521540100 T + 3816898736255 T^{2} + 6535665495304 T^{3} + 3639325109178 T^{4} - 6318749198498 T^{5} + 3562751449026 T^{6} - 1160878548654 T^{7} + 243380687143 T^{8} - 34659723318 T^{9} + 3742248555 T^{10} - 299849966 T^{11} + 6112187 T^{12} + 2794118 T^{13} - 215947 T^{14} + 22373 T^{15} + 1593 T^{16} + 161 T^{17} + 135 T^{18} + 6 T^{19} + T^{20} \)
$43$ \( 638971773407881 + 562079588722256 T + 535869419330391 T^{2} + 271061066245684 T^{3} + 91405475037655 T^{4} + 20915806382009 T^{5} + 2024782408214 T^{6} - 306805613400 T^{7} - 87046272452 T^{8} - 1202467497 T^{9} + 1682399665 T^{10} - 187012254 T^{11} + 26195326 T^{12} + 10376300 T^{13} + 1007180 T^{14} - 74063 T^{15} - 6056 T^{16} + 957 T^{17} + 290 T^{18} + 22 T^{19} + T^{20} \)
$47$ \( 2571253941169 + 442119000847 T + 1378653652444 T^{2} - 187982809186 T^{3} + 184601905664 T^{4} - 92137025894 T^{5} + 25633130362 T^{6} - 3067634022 T^{7} + 1076913790 T^{8} - 294483937 T^{9} - 12470941 T^{10} + 17082769 T^{11} + 132118 T^{12} - 666709 T^{13} + 21279 T^{14} + 11637 T^{15} + 1925 T^{16} - 976 T^{17} + 153 T^{18} - 16 T^{19} + T^{20} \)
$53$ \( 51457746649 - 11639314330 T + 22522336967 T^{2} - 359472235201 T^{3} + 702180170648 T^{4} - 811515176396 T^{5} + 701017970164 T^{6} - 364455852011 T^{7} + 148926003404 T^{8} - 25130098818 T^{9} + 2509548273 T^{10} + 398530209 T^{11} - 36351672 T^{12} - 13657614 T^{13} + 1234782 T^{14} + 59190 T^{15} - 1837 T^{16} - 292 T^{17} + 102 T^{18} + T^{19} + T^{20} \)
$59$ \( 279020843142889 + 254336880354894 T + 222915563949810 T^{2} + 123003168930119 T^{3} + 50241828715669 T^{4} + 5038406363183 T^{5} - 4341127135889 T^{6} - 2940534696304 T^{7} - 270988231359 T^{8} + 144383952042 T^{9} + 73167848865 T^{10} + 9243716108 T^{11} + 1072907694 T^{12} + 123899816 T^{13} + 12331215 T^{14} + 1274014 T^{15} + 117290 T^{16} + 6980 T^{17} + 380 T^{18} + 26 T^{19} + T^{20} \)
$61$ \( 76356191669209 - 43596327832292 T + 44514199075176 T^{2} - 21219949944232 T^{3} + 12279348643955 T^{4} - 3613007470176 T^{5} + 2807020092509 T^{6} + 17789585620 T^{7} + 238773705831 T^{8} + 1969879208 T^{9} + 7370921680 T^{10} + 4307694600 T^{11} + 458617952 T^{12} - 11928866 T^{13} - 4550540 T^{14} - 341534 T^{15} - 5008 T^{16} + 1436 T^{17} + 314 T^{18} + 26 T^{19} + T^{20} \)
$67$ \( 1822837804551761449 + 598543756718488834 T + 134408401271248171 T^{2} + 13466901187027706 T^{3} - 225784121893824 T^{4} - 442571010074600 T^{5} - 83380260615508 T^{6} - 8342463639158 T^{7} + 63788932406 T^{8} + 148729633492 T^{9} + 26753077319 T^{10} + 2219845276 T^{11} + 14210054 T^{12} - 27737666 T^{13} - 4137748 T^{14} - 327800 T^{15} - 2496 T^{16} + 2222 T^{17} + 331 T^{18} + 22 T^{19} + T^{20} \)
$71$ \( 1590305111329 + 22878612098067 T + 192739190609768 T^{2} - 13295246183109 T^{3} + 53829422659650 T^{4} - 42370947376153 T^{5} + 15184624283906 T^{6} - 3628051769705 T^{7} + 747758070169 T^{8} - 108959197149 T^{9} - 2767877254 T^{10} + 2082803564 T^{11} + 1181820445 T^{12} - 107866345 T^{13} + 7596488 T^{14} - 264870 T^{15} - 13415 T^{16} - 1103 T^{17} + 279 T^{18} - 20 T^{19} + T^{20} \)
$73$ \( 2420034779434801 + 5282234106002194 T + 5531875691713050 T^{2} + 3525180717523982 T^{3} + 1498295474107953 T^{4} + 430564267933635 T^{5} + 79879396404503 T^{6} + 6991792011039 T^{7} - 742934709831 T^{8} - 359329983507 T^{9} - 54892966952 T^{10} - 1649597136 T^{11} + 1157592709 T^{12} + 321490070 T^{13} + 51412398 T^{14} + 5889114 T^{15} + 514385 T^{16} + 34034 T^{17} + 1666 T^{18} + 55 T^{19} + T^{20} \)
$79$ \( 4559598209891209 - 3614541611092700 T + 2080752308937561 T^{2} - 629924613705810 T^{3} + 204575012393048 T^{4} - 28504320835495 T^{5} + 2642128297961 T^{6} + 1554323775340 T^{7} + 123378245711 T^{8} - 29782996265 T^{9} - 15073141863 T^{10} - 1006326145 T^{11} + 304035525 T^{12} + 63741113 T^{13} + 10488545 T^{14} + 1366751 T^{15} + 124509 T^{16} + 8472 T^{17} + 601 T^{18} + 34 T^{19} + T^{20} \)
$83$ \( 25849243729 + 169042063239 T + 4816007661847 T^{2} + 12574534145700 T^{3} + 1827640733019 T^{4} - 3974864631546 T^{5} + 19876050389670 T^{6} - 12290053064265 T^{7} + 4215057380334 T^{8} - 947740364725 T^{9} + 150542199380 T^{10} - 17055751911 T^{11} + 1418728712 T^{12} - 102580442 T^{13} + 11782932 T^{14} - 1964462 T^{15} + 267057 T^{16} - 24901 T^{17} + 1517 T^{18} - 56 T^{19} + T^{20} \)
$89$ \( 4126921988840047849 - 4260900774121910166 T + 2109455022029745626 T^{2} - 658092132985919927 T^{3} + 144595954488604672 T^{4} - 23704821826542566 T^{5} + 2988347030093149 T^{6} - 284983628846087 T^{7} + 18698410439578 T^{8} - 420317619014 T^{9} - 66681515131 T^{10} + 10523665408 T^{11} - 309984990 T^{12} + 833386 T^{13} + 8096622 T^{14} - 209897 T^{15} + 35765 T^{16} + 2615 T^{17} + 15 T^{18} + 3 T^{19} + T^{20} \)
$97$ \( ( -294045257 + 90831807 T + 54143463 T^{2} - 8875999 T^{3} - 2630430 T^{4} + 245563 T^{5} + 50873 T^{6} - 2470 T^{7} - 405 T^{8} + 7 T^{9} + T^{10} )^{2} \)
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