Properties

Label 71.2.c.a
Level $71$
Weight $2$
Character orbit 71.c
Analytic conductor $0.567$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 71.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.566937854351\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 6 x^{19} + 28 x^{18} - 91 x^{17} + 268 x^{16} - 604 x^{15} + 1278 x^{14} - 1990 x^{13} + 3162 x^{12} - 4046 x^{11} + 6406 x^{10} - 8426 x^{9} + 12709 x^{8} - 13621 x^{7} + 17787 x^{6} - 13514 x^{5} + 9316 x^{4} - 1639 x^{3} - 228 x^{2} + 620 x + 961\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{14} q^{2} + \beta_{17} q^{3} + ( -\beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{10} ) q^{4} + ( -\beta_{3} - \beta_{11} + \beta_{14} ) q^{5} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{9} - \beta_{12} - \beta_{16} + \beta_{19} ) q^{6} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{14} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{7} + ( -1 + 2 \beta_{1} + 2 \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{9} + \beta_{10} + \beta_{13} - \beta_{18} ) q^{8} + ( 1 - 2 \beta_{1} - \beta_{4} - \beta_{5} + \beta_{8} - \beta_{9} + \beta_{11} - \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} - \beta_{19} ) q^{9} +O(q^{10})\) \( q -\beta_{14} q^{2} + \beta_{17} q^{3} + ( -\beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{10} ) q^{4} + ( -\beta_{3} - \beta_{11} + \beta_{14} ) q^{5} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{9} - \beta_{12} - \beta_{16} + \beta_{19} ) q^{6} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{14} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{7} + ( -1 + 2 \beta_{1} + 2 \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{9} + \beta_{10} + \beta_{13} - \beta_{18} ) q^{8} + ( 1 - 2 \beta_{1} - \beta_{4} - \beta_{5} + \beta_{8} - \beta_{9} + \beta_{11} - \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} - \beta_{19} ) q^{9} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{10} + \beta_{14} - \beta_{15} + \beta_{18} ) q^{10} + ( -1 + \beta_{5} + \beta_{6} + \beta_{14} - \beta_{17} - \beta_{18} ) q^{11} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{10} + \beta_{12} + \beta_{15} + \beta_{16} - \beta_{17} - \beta_{18} ) q^{12} + ( -1 + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{14} - \beta_{16} - \beta_{19} ) q^{13} + ( 1 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{11} - \beta_{15} + \beta_{16} - 2 \beta_{17} - \beta_{19} ) q^{14} + ( 1 - 5 \beta_{1} - \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + \beta_{12} - \beta_{13} - \beta_{15} + \beta_{16} + \beta_{18} - \beta_{19} ) q^{15} + ( 1 - 4 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} + \beta_{15} - \beta_{18} - \beta_{19} ) q^{16} + ( -1 + 4 \beta_{1} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} - 2 \beta_{16} + \beta_{19} ) q^{17} + ( 2 + \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{11} + \beta_{14} ) q^{18} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{9} - 2 \beta_{10} - \beta_{13} - \beta_{15} + \beta_{17} + 2 \beta_{18} ) q^{19} + ( -2 - 3 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{13} - 2 \beta_{14} - \beta_{16} ) q^{20} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{15} - \beta_{16} + \beta_{17} + \beta_{19} ) q^{21} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{15} + \beta_{16} + \beta_{17} - \beta_{19} ) q^{22} + ( \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{10} - \beta_{11} + 2 \beta_{16} - \beta_{17} - \beta_{19} ) q^{23} + ( -1 + 5 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{9} + \beta_{14} ) q^{24} + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{9} + \beta_{10} + \beta_{13} + \beta_{15} - \beta_{17} - 2 \beta_{18} ) q^{25} + ( -1 - 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{4} - \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - 3 \beta_{14} + \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{26} + ( 1 - \beta_{1} - \beta_{2} - 3 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{9} + \beta_{10} - \beta_{12} + 2 \beta_{13} - \beta_{15} + \beta_{16} - \beta_{17} - \beta_{18} ) q^{27} + ( 1 - \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{9} - \beta_{11} + \beta_{14} ) q^{28} + ( -1 + 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{14} + \beta_{15} - \beta_{18} ) q^{29} + ( -2 + 6 \beta_{1} + \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + 3 \beta_{9} + 3 \beta_{10} + 2 \beta_{11} - \beta_{12} - 3 \beta_{16} + 2 \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{30} + ( 1 - \beta_{3} - \beta_{4} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{17} - \beta_{19} ) q^{31} + ( 6 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + 4 \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{13} + 2 \beta_{14} + 2 \beta_{17} + 2 \beta_{19} ) q^{32} + ( -1 + 4 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} + \beta_{9} - 2 \beta_{11} + \beta_{15} - \beta_{16} + \beta_{19} ) q^{33} + ( 1 - 2 \beta_{1} - \beta_{2} - 4 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + 3 \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{34} + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} + \beta_{16} - \beta_{18} ) q^{35} + ( 2 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{14} - \beta_{15} - \beta_{16} + \beta_{18} - \beta_{19} ) q^{36} + ( -1 + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{11} + \beta_{13} - 2 \beta_{14} + 2 \beta_{17} + 2 \beta_{19} ) q^{37} + ( -2 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{9} + 3 \beta_{10} - 2 \beta_{16} + 2 \beta_{19} ) q^{38} + ( -5 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{39} + ( 1 + 5 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} - \beta_{11} + 3 \beta_{14} + \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{40} + ( -3 - \beta_{11} - \beta_{13} - \beta_{16} ) q^{41} + ( 1 + 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} - 2 \beta_{18} ) q^{42} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{16} + \beta_{17} ) q^{43} + ( 3 + \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{14} - \beta_{15} + 2 \beta_{18} ) q^{44} + ( -2 \beta_{1} + \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + \beta_{10} + 2 \beta_{11} - 3 \beta_{14} - \beta_{16} ) q^{45} + ( 1 - 5 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} - 4 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} - 3 \beta_{11} - \beta_{13} + 2 \beta_{14} - \beta_{15} + \beta_{17} - 2 \beta_{19} ) q^{46} + ( 5 + \beta_{2} - \beta_{3} - 5 \beta_{4} - 4 \beta_{5} + 4 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{14} + \beta_{16} - \beta_{19} ) q^{47} + ( 1 - 9 \beta_{1} - 3 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 5 \beta_{9} - 3 \beta_{10} - 4 \beta_{14} + 2 \beta_{16} - 2 \beta_{17} - 2 \beta_{19} ) q^{48} + ( 1 - 4 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - 4 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - \beta_{15} + \beta_{16} - 2 \beta_{17} - \beta_{19} ) q^{49} + ( 2 - 5 \beta_{1} - 4 \beta_{3} + 3 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - 3 \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{14} + \beta_{15} + 2 \beta_{16} - 2 \beta_{19} ) q^{50} + ( -3 + 5 \beta_{1} + 6 \beta_{4} + 5 \beta_{5} - \beta_{6} + 3 \beta_{9} - 3 \beta_{11} - \beta_{13} + 4 \beta_{14} - \beta_{16} + \beta_{17} + \beta_{19} ) q^{51} + ( 2 + 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} + 5 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} + \beta_{11} + \beta_{17} + 2 \beta_{19} ) q^{52} + ( 1 - 4 \beta_{1} - 5 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{11} - 3 \beta_{14} ) q^{53} + ( 3 - 9 \beta_{1} + \beta_{2} - 5 \beta_{3} - 4 \beta_{4} + \beta_{5} + 5 \beta_{6} - \beta_{7} - 5 \beta_{9} - 5 \beta_{10} + \beta_{12} - \beta_{13} + 2 \beta_{14} - 4 \beta_{19} ) q^{54} + ( -3 + 5 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + \beta_{14} - \beta_{16} + \beta_{19} ) q^{55} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{4} + 4 \beta_{5} - 4 \beta_{7} - 2 \beta_{8} + \beta_{12} - \beta_{13} + 3 \beta_{14} - 2 \beta_{16} - 2 \beta_{19} ) q^{56} + ( 2 - 5 \beta_{1} - 3 \beta_{4} - \beta_{5} + \beta_{7} - 2 \beta_{9} + \beta_{10} + 3 \beta_{11} + \beta_{12} - 2 \beta_{14} - \beta_{15} + 2 \beta_{16} - 2 \beta_{19} ) q^{57} + ( -3 + \beta_{1} + 3 \beta_{4} + \beta_{6} - \beta_{10} - \beta_{15} + \beta_{18} + \beta_{19} ) q^{58} + ( 3 - 3 \beta_{1} - 4 \beta_{2} - 3 \beta_{4} + 4 \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} - \beta_{18} ) q^{59} + ( -1 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{6} + 5 \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} - 3 \beta_{17} - 3 \beta_{18} + \beta_{19} ) q^{60} + ( -1 + 3 \beta_{1} + \beta_{2} - 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{8} + \beta_{12} - \beta_{13} + 2 \beta_{14} - 2 \beta_{16} ) q^{61} + ( -1 + 3 \beta_{1} + \beta_{2} + 2 \beta_{4} - 3 \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{14} - \beta_{15} - \beta_{16} + \beta_{17} + \beta_{19} ) q^{62} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{6} + \beta_{7} + 4 \beta_{8} - \beta_{9} + 2 \beta_{16} - 2 \beta_{17} ) q^{63} + ( -2 - 5 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + \beta_{8} - 4 \beta_{9} - 4 \beta_{10} + \beta_{11} - 2 \beta_{14} - \beta_{17} - \beta_{19} ) q^{64} + ( 1 - 3 \beta_{1} - 4 \beta_{2} + 5 \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{11} + 2 \beta_{12} - \beta_{14} + \beta_{16} + \beta_{17} - \beta_{19} ) q^{65} + ( -5 - 4 \beta_{1} - 2 \beta_{2} - \beta_{4} + 2 \beta_{6} - 5 \beta_{7} - 2 \beta_{9} + 2 \beta_{11} - 2 \beta_{14} + 2 \beta_{16} - 2 \beta_{17} ) q^{66} + ( -1 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} - 4 \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} - 2 \beta_{11} - \beta_{12} + 2 \beta_{14} + \beta_{15} - \beta_{16} - 2 \beta_{17} + \beta_{19} ) q^{67} + ( -3 - 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{5} + 3 \beta_{6} - 7 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} + 4 \beta_{14} + 3 \beta_{17} ) q^{68} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - 5 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{14} - 3 \beta_{17} + \beta_{18} + \beta_{19} ) q^{69} + ( \beta_{2} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{70} + ( 3 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{15} + 2 \beta_{18} + 2 \beta_{19} ) q^{71} + ( -1 + 4 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} - \beta_{12} + \beta_{13} + 2 \beta_{14} - 3 \beta_{16} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{72} + ( 1 - 6 \beta_{1} - 5 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} - \beta_{14} - 3 \beta_{17} - 4 \beta_{19} ) q^{73} + ( 1 - \beta_{1} - 5 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{6} + 7 \beta_{7} + \beta_{8} + 4 \beta_{11} + \beta_{14} - 3 \beta_{17} - \beta_{19} ) q^{74} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{5} + \beta_{6} - 2 \beta_{8} - 3 \beta_{10} - 3 \beta_{11} + \beta_{14} - \beta_{15} + 2 \beta_{17} ) q^{75} + ( \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{10} - \beta_{11} + \beta_{13} + \beta_{14} + 2 \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{76} + ( -1 + 3 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} + 2 \beta_{12} + \beta_{15} - \beta_{16} + 3 \beta_{17} + \beta_{19} ) q^{77} + ( -1 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 4 \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 3 \beta_{14} - \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{78} + ( -2 - 2 \beta_{1} - 3 \beta_{2} - 4 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{79} + ( 1 + \beta_{1} + 6 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{14} - 2 \beta_{15} + \beta_{16} - \beta_{17} - \beta_{19} ) q^{80} + ( -7 + 7 \beta_{1} + \beta_{2} + 2 \beta_{3} + 7 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + \beta_{14} + 2 \beta_{15} + 2 \beta_{16} - 2 \beta_{18} + \beta_{19} ) q^{81} + ( -1 + 6 \beta_{1} + 3 \beta_{3} + 3 \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} + 4 \beta_{10} - \beta_{11} + \beta_{13} + 3 \beta_{14} + \beta_{15} + \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{82} + ( -3 + 4 \beta_{1} + 4 \beta_{2} + \beta_{3} + 6 \beta_{4} - \beta_{6} - 4 \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} - \beta_{15} + \beta_{18} ) q^{83} + ( -3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{16} ) q^{84} + ( -3 + 7 \beta_{1} + \beta_{2} + 5 \beta_{3} + 6 \beta_{4} - \beta_{5} - 5 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} + 4 \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{14} + \beta_{15} - 3 \beta_{16} + 3 \beta_{17} + 3 \beta_{19} ) q^{85} + ( -1 - \beta_{1} + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + \beta_{15} + \beta_{16} - \beta_{18} ) q^{86} + ( 8 - 8 \beta_{1} + \beta_{2} - 3 \beta_{3} - 6 \beta_{4} - \beta_{5} + 4 \beta_{6} + \beta_{7} - 3 \beta_{9} - 4 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} - \beta_{14} - 2 \beta_{15} + 2 \beta_{18} - 2 \beta_{19} ) q^{87} + ( 3 - 4 \beta_{1} - 6 \beta_{2} - \beta_{3} - 3 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} + 6 \beta_{7} + 2 \beta_{8} - \beta_{9} + 2 \beta_{10} - 6 \beta_{14} + 2 \beta_{16} ) q^{88} + ( -1 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} - 2 \beta_{16} + 2 \beta_{17} + \beta_{18} ) q^{89} + ( 5 + 7 \beta_{1} + \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 6 \beta_{5} - 4 \beta_{6} + 8 \beta_{7} - \beta_{8} + 6 \beta_{9} + 6 \beta_{10} - 4 \beta_{11} + 2 \beta_{13} + 4 \beta_{14} + 2 \beta_{15} - 3 \beta_{18} + \beta_{19} ) q^{90} + ( -3 + 2 \beta_{1} + 4 \beta_{3} + 5 \beta_{4} + 4 \beta_{5} - \beta_{6} + 6 \beta_{9} + 4 \beta_{10} - \beta_{11} + 3 \beta_{13} - 3 \beta_{14} - \beta_{16} + \beta_{17} + \beta_{19} ) q^{91} + ( -2 + \beta_{1} + \beta_{3} + 10 \beta_{4} + 4 \beta_{5} - \beta_{6} - 8 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{14} + 3 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} + 2 \beta_{19} ) q^{92} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{14} - 2 \beta_{15} ) q^{93} + ( -4 - \beta_{3} + 4 \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{10} - 5 \beta_{11} - 2 \beta_{16} + \beta_{17} + \beta_{19} ) q^{94} + ( 6 - 5 \beta_{1} - \beta_{2} - 2 \beta_{3} - 5 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} + 3 \beta_{16} - \beta_{18} - 3 \beta_{19} ) q^{95} + ( 7 - 4 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 5 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} + 2 \beta_{14} + 2 \beta_{17} + 3 \beta_{18} - 2 \beta_{19} ) q^{96} + ( -6 + \beta_{1} - 4 \beta_{2} - \beta_{3} - \beta_{4} + 4 \beta_{5} + \beta_{6} + 4 \beta_{7} + 3 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} + 4 \beta_{12} - \beta_{13} + 3 \beta_{14} - \beta_{16} - \beta_{17} - 4 \beta_{18} - \beta_{19} ) q^{97} + ( 2 - 2 \beta_{1} - 3 \beta_{3} - 7 \beta_{4} - 4 \beta_{5} - 4 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 4 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} + 2 \beta_{18} ) q^{98} + ( 3 + \beta_{1} + 3 \beta_{2} - 4 \beta_{4} - \beta_{5} + \beta_{7} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} - 3 \beta_{11} - \beta_{12} - \beta_{13} + 3 \beta_{14} - \beta_{15} - 4 \beta_{16} + 4 \beta_{17} + 2 \beta_{18} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 6q^{2} - q^{3} - 10q^{4} - q^{5} - 8q^{6} - q^{7} - q^{8} - 10q^{9} + O(q^{10}) \) \( 20q - 6q^{2} - q^{3} - 10q^{4} - q^{5} - 8q^{6} - q^{7} - q^{8} - 10q^{9} + 15q^{10} - q^{11} + 4q^{12} + 3q^{13} + 5q^{14} - 5q^{15} - 16q^{16} - 2q^{17} + 36q^{18} + 3q^{19} - 72q^{20} + 25q^{21} + 10q^{22} - 22q^{23} + 19q^{24} + 14q^{25} - 42q^{26} + 2q^{27} + 4q^{28} - q^{29} + 20q^{30} + 6q^{31} + 52q^{32} + 10q^{33} - 34q^{34} + 3q^{35} + 20q^{36} - 6q^{37} + 25q^{38} - 86q^{39} + 65q^{40} - 60q^{41} + 27q^{42} + 23q^{43} + 37q^{44} - 44q^{45} - 19q^{46} + 29q^{47} - 96q^{48} + 22q^{49} + 36q^{50} + 34q^{51} + 43q^{52} + 2q^{53} + 4q^{54} + 10q^{55} + 31q^{56} - 18q^{57} - 33q^{58} + 31q^{59} - 38q^{60} - 2q^{61} + 5q^{62} + 23q^{63} - 65q^{64} + 54q^{65} - 94q^{66} - 38q^{67} - 3q^{68} + 11q^{69} - 34q^{70} + 45q^{71} - 10q^{72} - 21q^{73} + 21q^{74} + 13q^{75} - q^{76} - 12q^{77} - 6q^{78} - 59q^{79} - 16q^{80} - 35q^{81} + 53q^{82} - 15q^{83} - 33q^{84} + 13q^{85} - 19q^{86} + 49q^{87} - 64q^{88} + 16q^{89} + 86q^{90} - 18q^{91} + 86q^{92} - 62q^{94} + 15q^{95} + 107q^{96} - 58q^{97} - 30q^{98} + 30q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 6 x^{19} + 28 x^{18} - 91 x^{17} + 268 x^{16} - 604 x^{15} + 1278 x^{14} - 1990 x^{13} + 3162 x^{12} - 4046 x^{11} + 6406 x^{10} - 8426 x^{9} + 12709 x^{8} - 13621 x^{7} + 17787 x^{6} - 13514 x^{5} + 9316 x^{4} - 1639 x^{3} - 228 x^{2} + 620 x + 961\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(\)\(33\!\cdots\!52\)\( \nu^{19} - \)\(22\!\cdots\!67\)\( \nu^{18} + \)\(12\!\cdots\!39\)\( \nu^{17} - \)\(45\!\cdots\!58\)\( \nu^{16} + \)\(14\!\cdots\!74\)\( \nu^{15} - \)\(37\!\cdots\!58\)\( \nu^{14} + \)\(87\!\cdots\!02\)\( \nu^{13} - \)\(15\!\cdots\!14\)\( \nu^{12} + \)\(27\!\cdots\!82\)\( \nu^{11} - \)\(35\!\cdots\!18\)\( \nu^{10} + \)\(57\!\cdots\!62\)\( \nu^{9} - \)\(70\!\cdots\!24\)\( \nu^{8} + \)\(11\!\cdots\!62\)\( \nu^{7} - \)\(13\!\cdots\!17\)\( \nu^{6} + \)\(19\!\cdots\!98\)\( \nu^{5} - \)\(15\!\cdots\!84\)\( \nu^{4} + \)\(24\!\cdots\!60\)\( \nu^{3} - \)\(51\!\cdots\!62\)\( \nu^{2} + \)\(22\!\cdots\!35\)\( \nu + \)\(79\!\cdots\!35\)\(\)\()/ \)\(92\!\cdots\!38\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(59\!\cdots\!24\)\( \nu^{19} + \)\(83\!\cdots\!42\)\( \nu^{18} - \)\(42\!\cdots\!81\)\( \nu^{17} + \)\(16\!\cdots\!65\)\( \nu^{16} - \)\(47\!\cdots\!58\)\( \nu^{15} + \)\(12\!\cdots\!77\)\( \nu^{14} - \)\(24\!\cdots\!29\)\( \nu^{13} + \)\(44\!\cdots\!08\)\( \nu^{12} - \)\(56\!\cdots\!77\)\( \nu^{11} + \)\(75\!\cdots\!08\)\( \nu^{10} - \)\(97\!\cdots\!18\)\( \nu^{9} + \)\(15\!\cdots\!27\)\( \nu^{8} - \)\(21\!\cdots\!89\)\( \nu^{7} + \)\(24\!\cdots\!91\)\( \nu^{6} - \)\(23\!\cdots\!13\)\( \nu^{5} + \)\(22\!\cdots\!62\)\( \nu^{4} - \)\(11\!\cdots\!61\)\( \nu^{3} - \)\(39\!\cdots\!85\)\( \nu^{2} - \)\(11\!\cdots\!39\)\( \nu + \)\(20\!\cdots\!43\)\(\)\()/ \)\(14\!\cdots\!39\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(25\!\cdots\!85\)\( \nu^{19} - \)\(16\!\cdots\!22\)\( \nu^{18} + \)\(78\!\cdots\!57\)\( \nu^{17} - \)\(27\!\cdots\!44\)\( \nu^{16} + \)\(82\!\cdots\!78\)\( \nu^{15} - \)\(20\!\cdots\!34\)\( \nu^{14} + \)\(44\!\cdots\!28\)\( \nu^{13} - \)\(78\!\cdots\!12\)\( \nu^{12} + \)\(12\!\cdots\!04\)\( \nu^{11} - \)\(18\!\cdots\!52\)\( \nu^{10} + \)\(27\!\cdots\!68\)\( \nu^{9} - \)\(39\!\cdots\!32\)\( \nu^{8} + \)\(54\!\cdots\!09\)\( \nu^{7} - \)\(71\!\cdots\!07\)\( \nu^{6} + \)\(86\!\cdots\!22\)\( \nu^{5} - \)\(94\!\cdots\!28\)\( \nu^{4} + \)\(72\!\cdots\!64\)\( \nu^{3} - \)\(79\!\cdots\!75\)\( \nu^{2} + \)\(15\!\cdots\!42\)\( \nu - \)\(55\!\cdots\!85\)\(\)\()/ \)\(28\!\cdots\!78\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(26\!\cdots\!17\)\( \nu^{19} - \)\(16\!\cdots\!86\)\( \nu^{18} - \)\(95\!\cdots\!03\)\( \nu^{17} + \)\(14\!\cdots\!72\)\( \nu^{16} - \)\(52\!\cdots\!56\)\( \nu^{15} + \)\(20\!\cdots\!42\)\( \nu^{14} - \)\(45\!\cdots\!50\)\( \nu^{13} + \)\(11\!\cdots\!22\)\( \nu^{12} - \)\(16\!\cdots\!86\)\( \nu^{11} + \)\(28\!\cdots\!70\)\( \nu^{10} - \)\(31\!\cdots\!06\)\( \nu^{9} + \)\(55\!\cdots\!86\)\( \nu^{8} - \)\(68\!\cdots\!75\)\( \nu^{7} + \)\(11\!\cdots\!17\)\( \nu^{6} - \)\(11\!\cdots\!92\)\( \nu^{5} + \)\(17\!\cdots\!56\)\( \nu^{4} - \)\(11\!\cdots\!58\)\( \nu^{3} + \)\(80\!\cdots\!33\)\( \nu^{2} + \)\(45\!\cdots\!62\)\( \nu - \)\(46\!\cdots\!47\)\(\)\()/ \)\(28\!\cdots\!78\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(32\!\cdots\!14\)\( \nu^{19} + \)\(33\!\cdots\!87\)\( \nu^{18} - \)\(15\!\cdots\!73\)\( \nu^{17} + \)\(60\!\cdots\!98\)\( \nu^{16} - \)\(17\!\cdots\!18\)\( \nu^{15} + \)\(45\!\cdots\!28\)\( \nu^{14} - \)\(91\!\cdots\!00\)\( \nu^{13} + \)\(17\!\cdots\!58\)\( \nu^{12} - \)\(23\!\cdots\!16\)\( \nu^{11} + \)\(37\!\cdots\!58\)\( \nu^{10} - \)\(46\!\cdots\!84\)\( \nu^{9} + \)\(79\!\cdots\!94\)\( \nu^{8} - \)\(92\!\cdots\!98\)\( \nu^{7} + \)\(13\!\cdots\!25\)\( \nu^{6} - \)\(12\!\cdots\!74\)\( \nu^{5} + \)\(18\!\cdots\!74\)\( \nu^{4} - \)\(60\!\cdots\!62\)\( \nu^{3} + \)\(28\!\cdots\!24\)\( \nu^{2} + \)\(57\!\cdots\!09\)\( \nu - \)\(58\!\cdots\!91\)\(\)\()/ \)\(28\!\cdots\!78\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(52\!\cdots\!06\)\( \nu^{19} + \)\(42\!\cdots\!51\)\( \nu^{18} - \)\(20\!\cdots\!67\)\( \nu^{17} + \)\(72\!\cdots\!20\)\( \nu^{16} - \)\(21\!\cdots\!78\)\( \nu^{15} + \)\(53\!\cdots\!24\)\( \nu^{14} - \)\(11\!\cdots\!52\)\( \nu^{13} + \)\(19\!\cdots\!06\)\( \nu^{12} - \)\(28\!\cdots\!20\)\( \nu^{11} + \)\(41\!\cdots\!62\)\( \nu^{10} - \)\(56\!\cdots\!50\)\( \nu^{9} + \)\(87\!\cdots\!74\)\( \nu^{8} - \)\(11\!\cdots\!74\)\( \nu^{7} + \)\(15\!\cdots\!69\)\( \nu^{6} - \)\(15\!\cdots\!32\)\( \nu^{5} + \)\(18\!\cdots\!76\)\( \nu^{4} - \)\(91\!\cdots\!90\)\( \nu^{3} + \)\(36\!\cdots\!80\)\( \nu^{2} + \)\(47\!\cdots\!01\)\( \nu - \)\(11\!\cdots\!37\)\(\)\()/ \)\(28\!\cdots\!78\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(59\!\cdots\!02\)\( \nu^{19} - \)\(46\!\cdots\!85\)\( \nu^{18} + \)\(23\!\cdots\!45\)\( \nu^{17} - \)\(86\!\cdots\!00\)\( \nu^{16} + \)\(26\!\cdots\!02\)\( \nu^{15} - \)\(68\!\cdots\!20\)\( \nu^{14} + \)\(15\!\cdots\!88\)\( \nu^{13} - \)\(27\!\cdots\!62\)\( \nu^{12} + \)\(45\!\cdots\!08\)\( \nu^{11} - \)\(64\!\cdots\!66\)\( \nu^{10} + \)\(96\!\cdots\!46\)\( \nu^{9} - \)\(13\!\cdots\!86\)\( \nu^{8} + \)\(19\!\cdots\!68\)\( \nu^{7} - \)\(24\!\cdots\!15\)\( \nu^{6} + \)\(31\!\cdots\!08\)\( \nu^{5} - \)\(31\!\cdots\!92\)\( \nu^{4} + \)\(30\!\cdots\!46\)\( \nu^{3} - \)\(16\!\cdots\!72\)\( \nu^{2} + \)\(98\!\cdots\!97\)\( \nu - \)\(94\!\cdots\!39\)\(\)\()/ \)\(28\!\cdots\!78\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(22\!\cdots\!87\)\( \nu^{19} - \)\(16\!\cdots\!82\)\( \nu^{18} + \)\(80\!\cdots\!91\)\( \nu^{17} - \)\(27\!\cdots\!02\)\( \nu^{16} + \)\(82\!\cdots\!00\)\( \nu^{15} - \)\(19\!\cdots\!48\)\( \nu^{14} + \)\(41\!\cdots\!54\)\( \nu^{13} - \)\(70\!\cdots\!10\)\( \nu^{12} + \)\(10\!\cdots\!26\)\( \nu^{11} - \)\(14\!\cdots\!18\)\( \nu^{10} + \)\(20\!\cdots\!38\)\( \nu^{9} - \)\(29\!\cdots\!52\)\( \nu^{8} + \)\(42\!\cdots\!25\)\( \nu^{7} - \)\(51\!\cdots\!93\)\( \nu^{6} + \)\(57\!\cdots\!70\)\( \nu^{5} - \)\(54\!\cdots\!58\)\( \nu^{4} + \)\(31\!\cdots\!88\)\( \nu^{3} + \)\(11\!\cdots\!49\)\( \nu^{2} - \)\(12\!\cdots\!52\)\( \nu - \)\(12\!\cdots\!35\)\(\)\()/ \)\(92\!\cdots\!38\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(10\!\cdots\!85\)\( \nu^{19} - \)\(42\!\cdots\!64\)\( \nu^{18} + \)\(17\!\cdots\!79\)\( \nu^{17} - \)\(38\!\cdots\!82\)\( \nu^{16} + \)\(10\!\cdots\!44\)\( \nu^{15} - \)\(10\!\cdots\!48\)\( \nu^{14} + \)\(18\!\cdots\!44\)\( \nu^{13} + \)\(32\!\cdots\!20\)\( \nu^{12} - \)\(28\!\cdots\!34\)\( \nu^{11} + \)\(13\!\cdots\!92\)\( \nu^{10} - \)\(16\!\cdots\!28\)\( \nu^{9} + \)\(23\!\cdots\!16\)\( \nu^{8} - \)\(13\!\cdots\!85\)\( \nu^{7} + \)\(80\!\cdots\!69\)\( \nu^{6} - \)\(37\!\cdots\!70\)\( \nu^{5} + \)\(14\!\cdots\!26\)\( \nu^{4} - \)\(81\!\cdots\!62\)\( \nu^{3} + \)\(78\!\cdots\!81\)\( \nu^{2} + \)\(19\!\cdots\!44\)\( \nu + \)\(40\!\cdots\!67\)\(\)\()/ \)\(28\!\cdots\!78\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(34\!\cdots\!65\)\( \nu^{19} + \)\(17\!\cdots\!29\)\( \nu^{18} - \)\(79\!\cdots\!54\)\( \nu^{17} + \)\(23\!\cdots\!70\)\( \nu^{16} - \)\(68\!\cdots\!00\)\( \nu^{15} + \)\(13\!\cdots\!64\)\( \nu^{14} - \)\(29\!\cdots\!86\)\( \nu^{13} + \)\(38\!\cdots\!08\)\( \nu^{12} - \)\(66\!\cdots\!06\)\( \nu^{11} + \)\(73\!\cdots\!94\)\( \nu^{10} - \)\(13\!\cdots\!78\)\( \nu^{9} + \)\(15\!\cdots\!20\)\( \nu^{8} - \)\(25\!\cdots\!53\)\( \nu^{7} + \)\(21\!\cdots\!10\)\( \nu^{6} - \)\(37\!\cdots\!32\)\( \nu^{5} + \)\(13\!\cdots\!74\)\( \nu^{4} - \)\(91\!\cdots\!66\)\( \nu^{3} - \)\(11\!\cdots\!43\)\( \nu^{2} + \)\(25\!\cdots\!07\)\( \nu - \)\(16\!\cdots\!86\)\(\)\()/ \)\(92\!\cdots\!38\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(10\!\cdots\!50\)\( \nu^{19} - \)\(10\!\cdots\!63\)\( \nu^{18} + \)\(51\!\cdots\!43\)\( \nu^{17} - \)\(19\!\cdots\!10\)\( \nu^{16} + \)\(60\!\cdots\!38\)\( \nu^{15} - \)\(15\!\cdots\!24\)\( \nu^{14} + \)\(34\!\cdots\!24\)\( \nu^{13} - \)\(64\!\cdots\!20\)\( \nu^{12} + \)\(10\!\cdots\!10\)\( \nu^{11} - \)\(14\!\cdots\!62\)\( \nu^{10} + \)\(20\!\cdots\!08\)\( \nu^{9} - \)\(30\!\cdots\!04\)\( \nu^{8} + \)\(42\!\cdots\!54\)\( \nu^{7} - \)\(56\!\cdots\!67\)\( \nu^{6} + \)\(64\!\cdots\!72\)\( \nu^{5} - \)\(72\!\cdots\!80\)\( \nu^{4} + \)\(57\!\cdots\!70\)\( \nu^{3} - \)\(29\!\cdots\!02\)\( \nu^{2} + \)\(30\!\cdots\!27\)\( \nu + \)\(15\!\cdots\!37\)\(\)\()/ \)\(28\!\cdots\!78\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(45\!\cdots\!47\)\( \nu^{19} - \)\(37\!\cdots\!61\)\( \nu^{18} + \)\(18\!\cdots\!40\)\( \nu^{17} - \)\(66\!\cdots\!38\)\( \nu^{16} + \)\(19\!\cdots\!88\)\( \nu^{15} - \)\(49\!\cdots\!64\)\( \nu^{14} + \)\(10\!\cdots\!26\)\( \nu^{13} - \)\(18\!\cdots\!72\)\( \nu^{12} + \)\(27\!\cdots\!58\)\( \nu^{11} - \)\(38\!\cdots\!90\)\( \nu^{10} + \)\(52\!\cdots\!18\)\( \nu^{9} - \)\(78\!\cdots\!80\)\( \nu^{8} + \)\(10\!\cdots\!11\)\( \nu^{7} - \)\(13\!\cdots\!16\)\( \nu^{6} + \)\(14\!\cdots\!22\)\( \nu^{5} - \)\(14\!\cdots\!50\)\( \nu^{4} + \)\(85\!\cdots\!02\)\( \nu^{3} + \)\(27\!\cdots\!89\)\( \nu^{2} - \)\(34\!\cdots\!03\)\( \nu + \)\(30\!\cdots\!24\)\(\)\()/ \)\(92\!\cdots\!38\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(46\!\cdots\!36\)\( \nu^{19} + \)\(27\!\cdots\!09\)\( \nu^{18} - \)\(12\!\cdots\!49\)\( \nu^{17} + \)\(40\!\cdots\!52\)\( \nu^{16} - \)\(11\!\cdots\!10\)\( \nu^{15} + \)\(25\!\cdots\!96\)\( \nu^{14} - \)\(53\!\cdots\!92\)\( \nu^{13} + \)\(80\!\cdots\!40\)\( \nu^{12} - \)\(12\!\cdots\!92\)\( \nu^{11} + \)\(15\!\cdots\!68\)\( \nu^{10} - \)\(25\!\cdots\!88\)\( \nu^{9} + \)\(32\!\cdots\!88\)\( \nu^{8} - \)\(50\!\cdots\!54\)\( \nu^{7} + \)\(50\!\cdots\!41\)\( \nu^{6} - \)\(67\!\cdots\!74\)\( \nu^{5} + \)\(44\!\cdots\!30\)\( \nu^{4} - \)\(27\!\cdots\!16\)\( \nu^{3} - \)\(16\!\cdots\!98\)\( \nu^{2} + \)\(68\!\cdots\!77\)\( \nu + \)\(83\!\cdots\!27\)\(\)\()/ \)\(92\!\cdots\!38\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(27\!\cdots\!29\)\( \nu^{19} - \)\(15\!\cdots\!16\)\( \nu^{18} + \)\(74\!\cdots\!31\)\( \nu^{17} - \)\(24\!\cdots\!94\)\( \nu^{16} + \)\(72\!\cdots\!54\)\( \nu^{15} - \)\(16\!\cdots\!50\)\( \nu^{14} + \)\(35\!\cdots\!58\)\( \nu^{13} - \)\(56\!\cdots\!94\)\( \nu^{12} + \)\(95\!\cdots\!44\)\( \nu^{11} - \)\(12\!\cdots\!14\)\( \nu^{10} + \)\(19\!\cdots\!16\)\( \nu^{9} - \)\(25\!\cdots\!72\)\( \nu^{8} + \)\(39\!\cdots\!83\)\( \nu^{7} - \)\(42\!\cdots\!29\)\( \nu^{6} + \)\(58\!\cdots\!66\)\( \nu^{5} - \)\(44\!\cdots\!36\)\( \nu^{4} + \)\(39\!\cdots\!42\)\( \nu^{3} - \)\(12\!\cdots\!57\)\( \nu^{2} + \)\(66\!\cdots\!58\)\( \nu + \)\(19\!\cdots\!17\)\(\)\()/ \)\(28\!\cdots\!78\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(93\!\cdots\!90\)\( \nu^{19} + \)\(56\!\cdots\!03\)\( \nu^{18} - \)\(26\!\cdots\!61\)\( \nu^{17} + \)\(84\!\cdots\!96\)\( \nu^{16} - \)\(24\!\cdots\!40\)\( \nu^{15} + \)\(55\!\cdots\!92\)\( \nu^{14} - \)\(11\!\cdots\!20\)\( \nu^{13} + \)\(18\!\cdots\!60\)\( \nu^{12} - \)\(28\!\cdots\!04\)\( \nu^{11} + \)\(35\!\cdots\!76\)\( \nu^{10} - \)\(56\!\cdots\!44\)\( \nu^{9} + \)\(74\!\cdots\!32\)\( \nu^{8} - \)\(11\!\cdots\!24\)\( \nu^{7} + \)\(11\!\cdots\!17\)\( \nu^{6} - \)\(15\!\cdots\!54\)\( \nu^{5} + \)\(10\!\cdots\!06\)\( \nu^{4} - \)\(64\!\cdots\!72\)\( \nu^{3} - \)\(36\!\cdots\!16\)\( \nu^{2} + \)\(23\!\cdots\!09\)\( \nu - \)\(12\!\cdots\!43\)\(\)\()/ \)\(92\!\cdots\!38\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(29\!\cdots\!95\)\( \nu^{19} + \)\(17\!\cdots\!06\)\( \nu^{18} - \)\(81\!\cdots\!15\)\( \nu^{17} + \)\(26\!\cdots\!06\)\( \nu^{16} - \)\(79\!\cdots\!66\)\( \nu^{15} + \)\(17\!\cdots\!34\)\( \nu^{14} - \)\(38\!\cdots\!16\)\( \nu^{13} + \)\(60\!\cdots\!36\)\( \nu^{12} - \)\(99\!\cdots\!68\)\( \nu^{11} + \)\(12\!\cdots\!50\)\( \nu^{10} - \)\(20\!\cdots\!20\)\( \nu^{9} + \)\(26\!\cdots\!86\)\( \nu^{8} - \)\(40\!\cdots\!25\)\( \nu^{7} + \)\(43\!\cdots\!81\)\( \nu^{6} - \)\(58\!\cdots\!38\)\( \nu^{5} + \)\(45\!\cdots\!58\)\( \nu^{4} - \)\(36\!\cdots\!80\)\( \nu^{3} + \)\(97\!\cdots\!87\)\( \nu^{2} + \)\(71\!\cdots\!40\)\( \nu - \)\(40\!\cdots\!25\)\(\)\()/ \)\(28\!\cdots\!78\)\( \)
\(\beta_{18}\)\(=\)\((\)\(\)\(33\!\cdots\!15\)\( \nu^{19} - \)\(19\!\cdots\!85\)\( \nu^{18} + \)\(90\!\cdots\!24\)\( \nu^{17} - \)\(29\!\cdots\!86\)\( \nu^{16} + \)\(85\!\cdots\!98\)\( \nu^{15} - \)\(19\!\cdots\!06\)\( \nu^{14} + \)\(41\!\cdots\!82\)\( \nu^{13} - \)\(63\!\cdots\!70\)\( \nu^{12} + \)\(10\!\cdots\!16\)\( \nu^{11} - \)\(13\!\cdots\!02\)\( \nu^{10} + \)\(21\!\cdots\!14\)\( \nu^{9} - \)\(27\!\cdots\!46\)\( \nu^{8} + \)\(42\!\cdots\!35\)\( \nu^{7} - \)\(43\!\cdots\!02\)\( \nu^{6} + \)\(61\!\cdots\!06\)\( \nu^{5} - \)\(43\!\cdots\!00\)\( \nu^{4} + \)\(36\!\cdots\!08\)\( \nu^{3} - \)\(80\!\cdots\!47\)\( \nu^{2} - \)\(71\!\cdots\!23\)\( \nu + \)\(34\!\cdots\!80\)\(\)\()/ \)\(28\!\cdots\!78\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(40\!\cdots\!73\)\( \nu^{19} + \)\(23\!\cdots\!63\)\( \nu^{18} - \)\(10\!\cdots\!64\)\( \nu^{17} + \)\(33\!\cdots\!04\)\( \nu^{16} - \)\(95\!\cdots\!88\)\( \nu^{15} + \)\(20\!\cdots\!54\)\( \nu^{14} - \)\(41\!\cdots\!84\)\( \nu^{13} + \)\(60\!\cdots\!22\)\( \nu^{12} - \)\(92\!\cdots\!02\)\( \nu^{11} + \)\(11\!\cdots\!38\)\( \nu^{10} - \)\(18\!\cdots\!06\)\( \nu^{9} + \)\(24\!\cdots\!66\)\( \nu^{8} - \)\(35\!\cdots\!89\)\( \nu^{7} + \)\(35\!\cdots\!54\)\( \nu^{6} - \)\(45\!\cdots\!44\)\( \nu^{5} + \)\(29\!\cdots\!00\)\( \nu^{4} - \)\(65\!\cdots\!48\)\( \nu^{3} - \)\(12\!\cdots\!73\)\( \nu^{2} + \)\(57\!\cdots\!43\)\( \nu + \)\(32\!\cdots\!68\)\(\)\()/ \)\(28\!\cdots\!78\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{7} - 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} - 1\)
\(\nu^{3}\)\(=\)\(-\beta_{14} - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 5 \beta_{2}\)
\(\nu^{4}\)\(=\)\(-\beta_{17} - \beta_{15} - 7 \beta_{14} - \beta_{13} - \beta_{10} - \beta_{9} + 14 \beta_{7} - 6 \beta_{6} - 6 \beta_{5} - \beta_{4} - \beta_{3} - \beta_{1} + 6\)
\(\nu^{5}\)\(=\)\(-2 \beta_{19} - 8 \beta_{18} - 2 \beta_{17} - 2 \beta_{14} - \beta_{13} + 8 \beta_{12} - 19 \beta_{11} - 2 \beta_{10} - 12 \beta_{9} + 8 \beta_{7} + 2 \beta_{6} - 10 \beta_{4} - 10 \beta_{3} - 8 \beta_{2} - 14 \beta_{1} - 8\)
\(\nu^{6}\)\(=\)\(-10 \beta_{19} - 10 \beta_{18} - \beta_{16} + 10 \beta_{15} - 2 \beta_{14} - 10 \beta_{13} + 10 \beta_{12} - 44 \beta_{10} - 57 \beta_{9} - \beta_{8} - 66 \beta_{7} + 44 \beta_{6} + 66 \beta_{5} + 23 \beta_{4} - 57 \beta_{3} - 12 \beta_{2} - 114 \beta_{1} - 33\)
\(\nu^{7}\)\(=\)\(40 \beta_{18} + 20 \beta_{17} - 20 \beta_{16} + 14 \beta_{15} - 100 \beta_{14} - 54 \beta_{13} + 14 \beta_{12} + 100 \beta_{11} - 40 \beta_{10} - 100 \beta_{9} - 21 \beta_{8} + 3 \beta_{7} + 46 \beta_{6} - 54 \beta_{5} + 22 \beta_{4} - 124 \beta_{3} - 144 \beta_{2} - 230 \beta_{1} + 43\)
\(\nu^{8}\)\(=\)\(73 \beta_{19} + 88 \beta_{17} - 73 \beta_{16} + 80 \beta_{15} + 44 \beta_{14} + 79 \beta_{12} - 11 \beta_{11} + 352 \beta_{10} + 73 \beta_{9} - 88 \beta_{8} + 178 \beta_{7} - 29 \beta_{6} - 438 \beta_{5} - 105 \beta_{4} + 29 \beta_{3} - 401 \beta_{2} + 113 \beta_{1} - 73\)
\(\nu^{9}\)\(=\)\(153 \beta_{19} - 213 \beta_{18} + 174 \beta_{17} + 352 \beta_{15} + 1083 \beta_{14} + 352 \beta_{13} - 214 \beta_{11} + 818 \beta_{10} + 818 \beta_{9} - 153 \beta_{8} - 572 \beta_{7} + 59 \beta_{6} + 251 \beta_{5} + 505 \beta_{4} + 505 \beta_{3} - 99 \beta_{2} + 971 \beta_{1} - 404\)
\(\nu^{10}\)\(=\)\(160 \beta_{19} + 585 \beta_{18} + 160 \beta_{17} + 484 \beta_{16} + 1596 \beta_{14} + 605 \beta_{13} - 585 \beta_{12} + 645 \beta_{11} + 630 \beta_{10} + 2438 \beta_{9} - 585 \beta_{7} - 160 \beta_{6} + 965 \beta_{5} + 1710 \beta_{4} + 1215 \beta_{3} + 585 \beta_{2} + 2971 \beta_{1} + 1480\)
\(\nu^{11}\)\(=\)\(261 \beta_{19} + 2292 \beta_{18} + 1089 \beta_{16} - 2292 \beta_{15} + 1101 \beta_{14} + 1215 \beta_{13} - 1215 \beta_{12} + 1941 \beta_{10} + 5357 \beta_{9} + 1089 \beta_{8} + 2765 \beta_{7} - 1941 \beta_{6} - 2765 \beta_{5} - 2189 \beta_{4} + 5357 \beta_{3} + 2316 \beta_{2} + 11622 \beta_{1} + 3404\)
\(\nu^{12}\)\(=\)\(272 \beta_{18} - 1476 \beta_{17} + 1476 \beta_{16} - 4505 \beta_{15} + 4445 \beta_{14} + 4233 \beta_{13} - 4505 \beta_{12} - 4445 \beta_{11} - 272 \beta_{10} + 8229 \beta_{9} + 4596 \beta_{8} - 5162 \beta_{7} - 3996 \beta_{6} + 4233 \beta_{5} - 6435 \beta_{4} + 15945 \beta_{3} + 20256 \beta_{2} + 23980 \beta_{1} - 4890\)
\(\nu^{13}\)\(=\)\(-2589 \beta_{19} - 10214 \beta_{17} + 2589 \beta_{16} - 9977 \beta_{15} - 15811 \beta_{14} - 15069 \beta_{12} + 5399 \beta_{11} - 30332 \beta_{10} - 2589 \beta_{9} + 10214 \beta_{8} - 7229 \beta_{7} - 13222 \beta_{6} + 27166 \beta_{5} + 4640 \beta_{4} + 13222 \beta_{3} + 57196 \beta_{2} + 5234 \beta_{1} + 2589\)
\(\nu^{14}\)\(=\)\(-12566 \beta_{19} - 3081 \beta_{18} - 32671 \beta_{17} - 30332 \beta_{15} - 112847 \beta_{14} - 30332 \beta_{13} + 1395 \beta_{11} - 75961 \beta_{10} - 75961 \beta_{9} + 12566 \beta_{8} + 97890 \beta_{7} - 37355 \beta_{6} - 24169 \beta_{5} - 42898 \beta_{4} - 42898 \beta_{3} + 31668 \beta_{2} - 88527 \beta_{1} + 36735\)
\(\nu^{15}\)\(=\)\(-53518 \beta_{19} - 100358 \beta_{18} - 53518 \beta_{17} - 22793 \beta_{16} - 183741 \beta_{14} - 79042 \beta_{13} + 100358 \beta_{12} - 114398 \beta_{11} - 147204 \beta_{10} - 340820 \beta_{9} + 100358 \beta_{7} + 53518 \beta_{6} - 12548 \beta_{5} - 166424 \beta_{4} - 247562 \beta_{3} - 100358 \beta_{2} - 484821 \beta_{1} - 117737\)
\(\nu^{16}\)\(=\)\(-131083 \beta_{19} - 216409 \beta_{18} - 101835 \beta_{16} + 216409 \beta_{15} - 264879 \beta_{14} - 247562 \beta_{13} + 247562 \beta_{12} - 460841 \beta_{10} - 971186 \beta_{9} - 101835 \beta_{8} - 367285 \beta_{7} + 460841 \beta_{6} + 367285 \beta_{5} + 95559 \beta_{4} - 971186 \beta_{3} - 512441 \beta_{2} - 1894925 \beta_{1} - 343121\)
\(\nu^{17}\)\(=\)\(65070 \beta_{18} + 378645 \beta_{17} - 378645 \beta_{16} + 612180 \beta_{15} - 697043 \beta_{14} - 677250 \beta_{13} + 612180 \beta_{12} + 697043 \beta_{11} - 65070 \beta_{10} - 1519449 \beta_{9} - 565806 \beta_{8} + 62744 \beta_{7} + 842199 \beta_{6} - 677250 \beta_{5} + 499969 \beta_{4} - 2196660 \beta_{3} - 2734364 \beta_{2} - 3641633 \beta_{1} + 127814\)
\(\nu^{18}\)\(=\)\(868734 \beta_{19} + 1668075 \beta_{17} - 868734 \beta_{16} + 1833024 \beta_{15} + 2037321 \beta_{14} + 1541622 \beta_{12} + 102205 \beta_{11} + 4626848 \beta_{10} + 868734 \beta_{9} - 1668075 \beta_{8} + 407018 \beta_{7} + 1168587 \beta_{6} - 3918504 \beta_{5} + 461716 \beta_{4} - 1168587 \beta_{3} - 7202634 \beta_{2} - 402058 \beta_{1} - 868734\)
\(\nu^{19}\)\(=\)\(2701758 \beta_{19} + 42838 \beta_{18} + 4173987 \beta_{17} + 4626848 \beta_{15} + 15374951 \beta_{14} + 4626848 \beta_{13} - 202721 \beta_{11} + 13517648 \beta_{10} + 13517648 \beta_{9} - 2701758 \beta_{8} - 8370232 \beta_{7} + 2182179 \beta_{6} + 189012 \beta_{5} + 7328606 \beta_{4} + 7328606 \beta_{3} - 5986321 \beta_{2} + 16219406 \beta_{1} - 2890770\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/71\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(-\beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.699129 2.15170i
0.561200 1.72720i
0.300331 0.924322i
−0.420063 + 1.29282i
−0.758630 + 2.33482i
2.18637 + 1.58849i
1.27286 + 0.924789i
0.685174 + 0.497808i
−0.363618 0.264184i
−1.16276 0.844794i
2.18637 1.58849i
1.27286 0.924789i
0.685174 0.497808i
−0.363618 + 0.264184i
−1.16276 + 0.844794i
0.699129 + 2.15170i
0.561200 + 1.72720i
0.300331 + 0.924322i
−0.420063 1.29282i
−0.758630 2.33482i
−0.699129 2.15170i −0.874263 2.69071i −2.52299 + 1.83306i 3.04432 + 2.21183i −5.17837 + 3.76230i −0.0944386 0.290652i 2.04740 + 1.48752i −4.04851 + 2.94142i 2.63081 8.09681i
5.2 −0.561200 1.72720i 0.906993 + 2.79144i −1.05022 + 0.763032i 1.21045 + 0.879441i 4.31235 3.13311i −0.388444 1.19551i −1.03119 0.749203i −4.54244 + 3.30028i 0.839664 2.58422i
5.3 −0.300331 0.924322i −0.143416 0.441389i 0.853860 0.620366i −1.37660 1.00016i −0.364913 + 0.265125i 0.656494 + 2.02048i −2.40241 1.74545i 2.25280 1.63675i −0.511035 + 1.57280i
5.4 0.420063 + 1.29282i 0.672312 + 2.06916i 0.123097 0.0894354i −2.53185 1.83950i −2.39265 + 1.73836i −1.31457 4.04584i 2.36681 + 1.71959i −1.40238 + 1.01889i 1.31461 4.04595i
5.5 0.758630 + 2.33482i −0.252609 0.777449i −3.25785 + 2.36696i −0.0372909 0.0270934i 1.62357 1.17959i 0.331946 + 1.02163i −4.02570 2.92484i 1.88643 1.37058i 0.0349684 0.107622i
25.1 −2.18637 + 1.58849i 1.68338 1.22305i 1.63889 5.04397i −0.994748 3.06152i −1.73770 + 5.34809i 0.649950 0.472216i 2.75886 + 8.49090i 0.410878 1.26455i 7.03810 + 5.11348i
25.2 −1.27286 + 0.924789i −0.850399 + 0.617851i 0.146911 0.452144i 0.148297 + 0.456410i 0.511060 1.57288i −4.12180 + 2.99466i −0.741239 2.28130i −0.585612 + 1.80233i −0.610844 0.443804i
25.3 −0.685174 + 0.497808i 1.00262 0.728443i −0.396383 + 1.21994i 0.467113 + 1.43763i −0.324342 + 0.998220i 2.86367 2.08058i −0.859132 2.64414i −0.452442 + 1.39247i −1.03572 0.752493i
25.4 0.363618 0.264184i −2.65787 + 1.93106i −0.555609 + 1.70999i 0.225736 + 0.694745i −0.456296 + 1.40433i 1.83483 1.33308i 0.527502 + 1.62348i 2.40825 7.41182i 0.265622 + 0.192986i
25.5 1.16276 0.844794i 0.0132545 0.00962997i 0.0202979 0.0624704i −0.655415 2.01716i 0.00727647 0.0223947i −0.917632 + 0.666699i 0.859096 + 2.64402i −0.926968 + 2.85291i −2.46618 1.79178i
54.1 −2.18637 1.58849i 1.68338 + 1.22305i 1.63889 + 5.04397i −0.994748 + 3.06152i −1.73770 5.34809i 0.649950 + 0.472216i 2.75886 8.49090i 0.410878 + 1.26455i 7.03810 5.11348i
54.2 −1.27286 0.924789i −0.850399 0.617851i 0.146911 + 0.452144i 0.148297 0.456410i 0.511060 + 1.57288i −4.12180 2.99466i −0.741239 + 2.28130i −0.585612 1.80233i −0.610844 + 0.443804i
54.3 −0.685174 0.497808i 1.00262 + 0.728443i −0.396383 1.21994i 0.467113 1.43763i −0.324342 0.998220i 2.86367 + 2.08058i −0.859132 + 2.64414i −0.452442 1.39247i −1.03572 + 0.752493i
54.4 0.363618 + 0.264184i −2.65787 1.93106i −0.555609 1.70999i 0.225736 0.694745i −0.456296 1.40433i 1.83483 + 1.33308i 0.527502 1.62348i 2.40825 + 7.41182i 0.265622 0.192986i
54.5 1.16276 + 0.844794i 0.0132545 + 0.00962997i 0.0202979 + 0.0624704i −0.655415 + 2.01716i 0.00727647 + 0.0223947i −0.917632 0.666699i 0.859096 2.64402i −0.926968 2.85291i −2.46618 + 1.79178i
57.1 −0.699129 + 2.15170i −0.874263 + 2.69071i −2.52299 1.83306i 3.04432 2.21183i −5.17837 3.76230i −0.0944386 + 0.290652i 2.04740 1.48752i −4.04851 2.94142i 2.63081 + 8.09681i
57.2 −0.561200 + 1.72720i 0.906993 2.79144i −1.05022 0.763032i 1.21045 0.879441i 4.31235 + 3.13311i −0.388444 + 1.19551i −1.03119 + 0.749203i −4.54244 3.30028i 0.839664 + 2.58422i
57.3 −0.300331 + 0.924322i −0.143416 + 0.441389i 0.853860 + 0.620366i −1.37660 + 1.00016i −0.364913 0.265125i 0.656494 2.02048i −2.40241 + 1.74545i 2.25280 + 1.63675i −0.511035 1.57280i
57.4 0.420063 1.29282i 0.672312 2.06916i 0.123097 + 0.0894354i −2.53185 + 1.83950i −2.39265 1.73836i −1.31457 + 4.04584i 2.36681 1.71959i −1.40238 1.01889i 1.31461 + 4.04595i
57.5 0.758630 2.33482i −0.252609 + 0.777449i −3.25785 2.36696i −0.0372909 + 0.0270934i 1.62357 + 1.17959i 0.331946 1.02163i −4.02570 + 2.92484i 1.88643 + 1.37058i 0.0349684 + 0.107622i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 57.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 71.2.c.a 20
3.b odd 2 1 639.2.f.c 20
71.c even 5 1 inner 71.2.c.a 20
71.c even 5 1 5041.2.a.i 10
71.e odd 10 1 5041.2.a.j 10
213.h odd 10 1 639.2.f.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.2.c.a 20 1.a even 1 1 trivial
71.2.c.a 20 71.c even 5 1 inner
639.2.f.c 20 3.b odd 2 1
639.2.f.c 20 213.h odd 10 1
5041.2.a.i 10 71.c even 5 1
5041.2.a.j 10 71.e odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(71, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 961 - 620 T - 228 T^{2} + 1639 T^{3} + 9316 T^{4} + 13514 T^{5} + 17787 T^{6} + 13621 T^{7} + 12709 T^{8} + 8426 T^{9} + 6406 T^{10} + 4046 T^{11} + 3162 T^{12} + 1990 T^{13} + 1278 T^{14} + 604 T^{15} + 268 T^{16} + 91 T^{17} + 28 T^{18} + 6 T^{19} + T^{20} \)
$3$ \( 1 - 97 T + 3558 T^{2} + 5938 T^{3} + 23193 T^{4} + 10296 T^{5} + 17526 T^{6} - 7557 T^{7} + 9285 T^{8} + 4517 T^{9} + 12908 T^{10} - 5726 T^{11} + 5875 T^{12} - 1296 T^{13} + 851 T^{14} - 42 T^{15} + 117 T^{16} + 14 T^{17} + 13 T^{18} + T^{19} + T^{20} \)
$5$ \( 25 + 825 T + 10110 T^{2} - 18275 T^{3} + 83501 T^{4} - 66667 T^{5} + 118594 T^{6} - 10952 T^{7} + 35356 T^{8} + 1596 T^{9} + 18082 T^{10} - 596 T^{11} + 6422 T^{12} + 1422 T^{13} + 1943 T^{14} + 352 T^{15} + 106 T^{16} - 11 T^{17} + 6 T^{18} + T^{19} + T^{20} \)
$7$ \( 19321 + 6811 T + 180568 T^{2} - 292253 T^{3} + 396397 T^{4} - 227199 T^{5} + 330423 T^{6} - 130553 T^{7} + 305945 T^{8} - 139547 T^{9} + 182559 T^{10} - 121817 T^{11} + 74728 T^{12} - 25792 T^{13} + 7222 T^{14} - 640 T^{15} + 265 T^{16} - 46 T^{17} + 7 T^{18} + T^{19} + T^{20} \)
$11$ \( 72914521 + 43053638 T + 42354786 T^{2} + 34080400 T^{3} + 30219787 T^{4} + 8085958 T^{5} + 9987434 T^{6} + 4902038 T^{7} + 2951849 T^{8} + 1287869 T^{9} + 905418 T^{10} + 286964 T^{11} + 124319 T^{12} + 33048 T^{13} + 11439 T^{14} + 1048 T^{15} + 485 T^{16} - 27 T^{17} + 29 T^{18} + T^{19} + T^{20} \)
$13$ \( 27626096521 + 13235381930 T + 76974685019 T^{2} - 47492312900 T^{3} + 24953787844 T^{4} - 8516431104 T^{5} + 3465262193 T^{6} - 829373247 T^{7} + 268616379 T^{8} - 51436256 T^{9} + 16646812 T^{10} - 3334992 T^{11} + 1012193 T^{12} - 160552 T^{13} + 46402 T^{14} - 5392 T^{15} + 1075 T^{16} - 46 T^{17} + 24 T^{18} - 3 T^{19} + T^{20} \)
$17$ \( 13374691201 - 44716726691 T + 69737671642 T^{2} - 60167857639 T^{3} + 41627182393 T^{4} - 25326912807 T^{5} + 12756886119 T^{6} - 4426158199 T^{7} + 1212705175 T^{8} - 311208139 T^{9} + 84008012 T^{10} - 11490952 T^{11} + 3165230 T^{12} - 415548 T^{13} + 104849 T^{14} - 7754 T^{15} + 3937 T^{16} - 48 T^{17} + 97 T^{18} + 2 T^{19} + T^{20} \)
$19$ \( 375390625 + 346328125 T - 3859375 T^{2} - 75043750 T^{3} + 281353125 T^{4} + 378725750 T^{5} + 262080950 T^{6} + 130053365 T^{7} + 68600266 T^{8} + 23884102 T^{9} + 7349687 T^{10} + 1362076 T^{11} + 580280 T^{12} + 18418 T^{13} + 72072 T^{14} - 7027 T^{15} + 4630 T^{16} - 219 T^{17} + 107 T^{18} - 3 T^{19} + T^{20} \)
$23$ \( ( 2981269 + 1425795 T - 728569 T^{2} - 424543 T^{3} + 27274 T^{4} + 34340 T^{5} + 1185 T^{6} - 1061 T^{7} - 77 T^{8} + 11 T^{9} + T^{10} )^{2} \)
$29$ \( 84842521 + 143102096 T + 4581761930 T^{2} - 16536985650 T^{3} + 27084904473 T^{4} - 26227670048 T^{5} + 17330502292 T^{6} - 8433685182 T^{7} + 3205777059 T^{8} - 986535089 T^{9} + 253076066 T^{10} - 54468799 T^{11} + 9850773 T^{12} - 1415293 T^{13} + 167245 T^{14} - 17685 T^{15} + 3166 T^{16} - 441 T^{17} + 89 T^{18} + T^{19} + T^{20} \)
$31$ \( 195132961 + 393702296 T + 900117418 T^{2} + 1846744095 T^{3} + 3282017684 T^{4} + 2115197634 T^{5} + 2362173074 T^{6} + 1598237766 T^{7} + 743310626 T^{8} + 92626932 T^{9} + 84129096 T^{10} - 473630 T^{11} + 3589523 T^{12} - 409083 T^{13} + 86767 T^{14} + 3774 T^{15} + 3742 T^{16} - 225 T^{17} + 36 T^{18} - 6 T^{19} + T^{20} \)
$37$ \( ( 1974269 + 2537477 T - 119063 T^{2} - 953944 T^{3} - 175804 T^{4} + 44557 T^{5} + 10103 T^{6} - 678 T^{7} - 182 T^{8} + 3 T^{9} + T^{10} )^{2} \)
$41$ \( ( 52111 + 258248 T + 214521 T^{2} - 3834 T^{3} - 59971 T^{4} - 19042 T^{5} + 1071 T^{6} + 1625 T^{7} + 339 T^{8} + 30 T^{9} + T^{10} )^{2} \)
$43$ \( 285441025 - 1883623550 T + 4510928465 T^{2} + 3676158370 T^{3} + 7412946186 T^{4} + 4267660780 T^{5} + 3443033263 T^{6} + 1434283409 T^{7} + 687417437 T^{8} + 180674244 T^{9} + 38050651 T^{10} - 5943248 T^{11} + 1275660 T^{12} + 204807 T^{13} + 80453 T^{14} - 48723 T^{15} + 15672 T^{16} - 2686 T^{17} + 318 T^{18} - 23 T^{19} + T^{20} \)
$47$ \( 2910063025 + 15090304575 T + 189862227565 T^{2} + 38248636270 T^{3} - 49518433709 T^{4} - 23743247890 T^{5} + 92341474262 T^{6} - 960424523 T^{7} + 19752795357 T^{8} - 3861676687 T^{9} + 1553514969 T^{10} - 466275999 T^{11} + 108167340 T^{12} - 20942681 T^{13} + 4011222 T^{14} - 596639 T^{15} + 66452 T^{16} - 5842 T^{17} + 472 T^{18} - 29 T^{19} + T^{20} \)
$53$ \( 255064391521 - 707647515786 T + 620203694775 T^{2} + 873531365532 T^{3} + 1793167225454 T^{4} + 1131044320104 T^{5} + 473447434428 T^{6} + 34624678281 T^{7} + 28771443870 T^{8} - 2257136906 T^{9} + 5188500268 T^{10} + 504943333 T^{11} + 176188419 T^{12} + 21519359 T^{13} + 4005485 T^{14} + 165323 T^{15} + 26012 T^{16} - 1333 T^{17} + 163 T^{18} - 2 T^{19} + T^{20} \)
$59$ \( 18611898850801 - 73645883780421 T + 119368970431545 T^{2} - 34584136160397 T^{3} + 54583142417922 T^{4} - 17303962331569 T^{5} + 9826113043241 T^{6} - 3263182850627 T^{7} + 1053121033083 T^{8} - 242706007333 T^{9} + 37954822491 T^{10} - 3303233969 T^{11} + 329596294 T^{12} - 63889403 T^{13} + 11366262 T^{14} - 1237254 T^{15} + 125851 T^{16} - 10116 T^{17} + 685 T^{18} - 31 T^{19} + T^{20} \)
$61$ \( 422508025 - 1703598400 T + 4417457965 T^{2} - 6650931080 T^{3} + 6243156581 T^{4} - 2327236616 T^{5} + 832048784 T^{6} + 21812029 T^{7} + 91544504 T^{8} + 11417166 T^{9} + 33085023 T^{10} + 17841467 T^{11} + 7788831 T^{12} + 1963820 T^{13} + 382278 T^{14} + 43547 T^{15} + 3704 T^{16} - 5 T^{17} + 25 T^{18} + 2 T^{19} + T^{20} \)
$67$ \( 373215840252025 + 362257543570200 T + 1247350908058510 T^{2} - 87241128218310 T^{3} + 323745492292861 T^{4} - 202686950259114 T^{5} + 56442506395287 T^{6} + 1768275060025 T^{7} - 585214492860 T^{8} - 43915300366 T^{9} + 44344202886 T^{10} + 10074529685 T^{11} + 1914497864 T^{12} + 272632225 T^{13} + 32965384 T^{14} + 3107972 T^{15} + 251110 T^{16} + 16040 T^{17} + 908 T^{18} + 38 T^{19} + T^{20} \)
$71$ \( 3255243551009881201 - 2063182532330206395 T + 758760399213769175 T^{2} - 196909351429165150 T^{3} + 38973870881544645 T^{4} - 6041604515942229 T^{5} + 722568062219285 T^{6} - 61267831324250 T^{7} + 2299259205725 T^{8} + 282492962165 T^{9} - 60430064769 T^{10} + 3978774115 T^{11} + 456111725 T^{12} - 171181750 T^{13} + 28434485 T^{14} - 3348579 T^{15} + 304245 T^{16} - 21650 T^{17} + 1175 T^{18} - 45 T^{19} + T^{20} \)
$73$ \( 230480751714323881 + 150228668799574149 T + 27851375865181002 T^{2} - 15455614256635962 T^{3} + 8431419746380520 T^{4} - 1647693211086003 T^{5} + 310330662140422 T^{6} - 25867299604233 T^{7} + 4281079529566 T^{8} - 149132797573 T^{9} + 55376155895 T^{10} + 599441223 T^{11} + 652956667 T^{12} + 38686733 T^{13} + 10610538 T^{14} + 892612 T^{15} + 103827 T^{16} + 6670 T^{17} + 499 T^{18} + 21 T^{19} + T^{20} \)
$79$ \( 3714697296025 + 19983317752300 T + 60901189183715 T^{2} + 111281347034555 T^{3} + 136209460292006 T^{4} + 78375196068575 T^{5} + 36635251240449 T^{6} + 14117264963741 T^{7} + 4551253523382 T^{8} + 1211622913918 T^{9} + 265293347398 T^{10} + 47771732705 T^{11} + 7118565497 T^{12} + 884147966 T^{13} + 92337122 T^{14} + 8163806 T^{15} + 610351 T^{16} + 37652 T^{17} + 1802 T^{18} + 59 T^{19} + T^{20} \)
$83$ \( 11944136448841 + 29587074637087 T + 63628153787193 T^{2} + 63859130785852 T^{3} + 36705455463917 T^{4} + 8699481354842 T^{5} + 2018839239449 T^{6} - 403220932804 T^{7} + 316339258435 T^{8} + 122801860922 T^{9} + 23237940014 T^{10} + 2189829303 T^{11} + 227876361 T^{12} + 21978081 T^{13} + 3696386 T^{14} + 417158 T^{15} + 52107 T^{16} + 3752 T^{17} + 384 T^{18} + 15 T^{19} + T^{20} \)
$89$ \( 11245514665406641 + 2783915934485395 T + 5632249681937541 T^{2} - 955353976673566 T^{3} + 358993877606922 T^{4} - 31180608645311 T^{5} + 9270272029754 T^{6} - 978570926088 T^{7} + 368471430226 T^{8} - 28424850796 T^{9} + 9054714481 T^{10} - 867613301 T^{11} + 151924980 T^{12} - 11254537 T^{13} + 1800521 T^{14} - 156186 T^{15} + 26907 T^{16} - 1934 T^{17} + 207 T^{18} - 16 T^{19} + T^{20} \)
$97$ \( ( 3911261945 + 1208280730 T - 353312904 T^{2} - 82259513 T^{3} + 6068202 T^{4} + 1581282 T^{5} - 17407 T^{6} - 11719 T^{7} - 217 T^{8} + 29 T^{9} + T^{10} )^{2} \)
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