Properties

Label 77.2.f.b
Level $77$
Weight $2$
Character orbit 77.f
Analytic conductor $0.615$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 77.f (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.614848095564\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 3 x^{15} + 14 x^{14} - 32 x^{13} + 86 x^{12} - 145 x^{11} + 245 x^{10} - 245 x^{9} + 640 x^{8} - 1175 x^{7} + 2135 x^{6} - 2300 x^{5} + 1850 x^{4} - 925 x^{3} + 700 x^{2} - 200 x + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 3 x^{15} + 14 x^{14} - 32 x^{13} + 86 x^{12} - 145 x^{11} + 245 x^{10} - 245 x^{9} + 640 x^{8} - 1175 x^{7} + 2135 x^{6} - 2300 x^{5} + 1850 x^{4} - 925 x^{3} + 700 x^{2} - 200 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-33722118880975 \nu^{15} - 215936836813390 \nu^{14} + 154302737507927 \nu^{13} - 3234633146529285 \nu^{12} + 4494612950061958 \nu^{11} - 20835947263746904 \nu^{10} + 25956864481307657 \nu^{9} - 65098048812157320 \nu^{8} + 30869277708222543 \nu^{7} - 160665000409949210 \nu^{6} + 100524107363786610 \nu^{5} - 560336279122558890 \nu^{4} + 546963632352958070 \nu^{3} - 434474896057227745 \nu^{2} + 120133073008741205 \nu + 88528161377575350\)\()/ 454580475630153760 \)
\(\beta_{2}\)\(=\)\((\)\(-686277383720070 \nu^{15} - 1360333162376645 \nu^{14} + 444742620384542 \nu^{13} - 24553779304461089 \nu^{12} + 45808084826141647 \nu^{11} - 179218800902324756 \nu^{10} + 292393929070812298 \nu^{9} - 592228758909440833 \nu^{8} + 274072637813433960 \nu^{7} - 1226946693965489055 \nu^{6} + 2313198962618101340 \nu^{5} - 5164313069616781590 \nu^{4} + 5505326380247646890 \nu^{3} - 4265354995237286060 \nu^{2} + 1164454349991851775 \nu - 1569079812936964435\)\()/ 454580475630153760 \)
\(\beta_{3}\)\(=\)\((\)\(-874936745197272 \nu^{15} - 514706296938285 \nu^{14} - 5527222230184928 \nu^{13} - 8597109141630243 \nu^{12} - 9687442744709873 \nu^{11} - 69609203132183412 \nu^{10} + 41469078152400808 \nu^{9} - 248365491806816213 \nu^{8} - 292096370375405502 \nu^{7} - 583922916320059255 \nu^{6} + 395306407846606840 \nu^{5} - 2070346335621533450 \nu^{4} + 1021707328353270230 \nu^{3} - 1173049633586345730 \nu^{2} + 368306587037923705 \nu - 928639412535219835\)\()/ 454580475630153760 \)
\(\beta_{4}\)\(=\)\((\)\(-1565624412275039 \nu^{15} + 4832412997019746 \nu^{14} - 21672367495495401 \nu^{13} + 49577176484430299 \nu^{12} - 129795369286809034 \nu^{11} + 214474805906670840 \nu^{10} - 347621794857624183 \nu^{9} + 312652728515985000 \nu^{8} - 887685879699924177 \nu^{7} + 1770758763681867190 \nu^{6} - 3159142418183242750 \nu^{5} + 2957115228664641670 \nu^{4} - 1979053855141622330 \nu^{3} + 366777001484809055 \nu^{2} - 739274040704445675 \nu + 95214593589332550\)\()/ 454580475630153760 \)
\(\beta_{5}\)\(=\)\((\)\(1676337845170790 \nu^{15} - 5194753922189191 \nu^{14} + 23996414540642882 \nu^{13} - 55793399026072099 \nu^{12} + 149577475159066613 \nu^{11} - 254981761232364844 \nu^{10} + 433123005758956374 \nu^{9} - 436889234085257675 \nu^{8} + 1101433166283370416 \nu^{7} - 2030901870569891045 \nu^{6} + 3769863137668677300 \nu^{5} - 4085868252644216690 \nu^{4} + 3585858111334770990 \nu^{3} - 1476295694818421740 \nu^{2} + 1479871527393288525 \nu - 189874590198959025\)\()/ 454580475630153760 \)
\(\beta_{6}\)\(=\)\((\)\(-1676337845170790 \nu^{15} + 5194753922189191 \nu^{14} - 23996414540642882 \nu^{13} + 55793399026072099 \nu^{12} - 149577475159066613 \nu^{11} + 254981761232364844 \nu^{10} - 433123005758956374 \nu^{9} + 436889234085257675 \nu^{8} - 1101433166283370416 \nu^{7} + 2030901870569891045 \nu^{6} - 3769863137668677300 \nu^{5} + 4085868252644216690 \nu^{4} - 3585858111334770990 \nu^{3} + 1476295694818421740 \nu^{2} - 1025291051763134765 \nu + 189874590198959025\)\()/ 454580475630153760 \)
\(\beta_{7}\)\(=\)\((\)\(-3541126455103014 \nu^{15} + 10589657246428067 \nu^{14} - 49791707208255586 \nu^{13} + 113470349300804375 \nu^{12} - 307771508285388489 \nu^{11} + 517957948939998988 \nu^{10} - 888411928763985334 \nu^{9} + 893532845981546087 \nu^{8} - 2331418980078086280 \nu^{7} + 4191692862454263993 \nu^{6} - 7720969982054884100 \nu^{5} + 8245114954100718810 \nu^{4} - 7111420221063134790 \nu^{3} + 3822505603323246020 \nu^{2} - 2913263414629337545 \nu + 828358364029344005\)\()/ 454580475630153760 \)
\(\beta_{8}\)\(=\)\((\)\(-3808583743573302 \nu^{15} + 9860126818444867 \nu^{14} - 48487759413006482 \nu^{13} + 100202312298850263 \nu^{12} - 277961025462873673 \nu^{11} + 422449273531319756 \nu^{10} - 718628211268788150 \nu^{9} + 585481222317834807 \nu^{8} - 2124840867370928280 \nu^{7} + 3587400018998705673 \nu^{6} - 6360567528847132580 \nu^{5} + 5600600192035351850 \nu^{4} - 4088764696945967030 \nu^{3} + 1543886107663682020 \nu^{2} - 2299231619016502345 \nu + 22442708010214725\)\()/ 454580475630153760 \)
\(\beta_{9}\)\(=\)\((\)\(4147870816217860 \nu^{15} - 9098179878471885 \nu^{14} + 48642002600512068 \nu^{13} - 88958591156351927 \nu^{12} + 262233539631762351 \nu^{11} - 349871456206210068 \nu^{10} + 623455379026927044 \nu^{9} - 381247122554708893 \nu^{8} + 2150910817199786310 \nu^{7} - 3112048315103081495 \nu^{6} + 5535211606668989440 \nu^{5} - 3712550967753264370 \nu^{4} + 2767110524781279550 \nu^{3} - 1335895153189756830 \nu^{2} + 2975718288695695725 \nu - 29045982804284875\)\()/ 454580475630153760 \)
\(\beta_{10}\)\(=\)\((\)\(7327526319488073 \nu^{15} - 21838143406687493 \nu^{14} + 102438964384476967 \nu^{13} - 232311097915978782 \nu^{12} + 627185674080861763 \nu^{11} - 1047203823403574964 \nu^{10} + 1775572940212630785 \nu^{9} - 1735699601854553351 \nu^{8} + 4630478387715251365 \nu^{7} - 8426965416523262079 \nu^{6} + 15544790585628961210 \nu^{5} - 16343113922700919960 \nu^{4} + 12987506946195918920 \nu^{3} - 5718121221686276935 \nu^{2} + 4454224626365266160 \nu - 845040850217512955\)\()/ 454580475630153760 \)
\(\beta_{11}\)\(=\)\((\)\(-5481281688546819 \nu^{15} + 16493950936678582 \nu^{14} - 76461534295292341 \nu^{13} + 174584775915853027 \nu^{12} - 466393479528723526 \nu^{11} + 782325573976058616 \nu^{10} - 1307780222870847083 \nu^{9} + 1276995934654789444 \nu^{8} - 3395227103875767205 \nu^{7} + 6340299488823316290 \nu^{6} - 11480502622702818310 \nu^{5} + 12158198218393584390 \nu^{4} - 9161804501671007210 \nu^{3} + 3888216659798809835 \nu^{2} - 2965788298433688075 \nu + 864066394282347650\)\()/ 227290237815076880 \)
\(\beta_{12}\)\(=\)\((\)\(-11668286039317543 \nu^{15} + 32209452556198329 \nu^{14} - 155422622027191961 \nu^{13} + 335017774059801892 \nu^{12} - 918061503999498391 \nu^{11} + 1456783386223745044 \nu^{10} - 2469198487594261567 \nu^{9} + 2179375478325730279 \nu^{8} - 6793193349504836855 \nu^{7} + 11856118418058363915 \nu^{6} - 21771993760224740470 \nu^{5} + 21003963276182506660 \nu^{4} - 15299497235328621900 \nu^{3} + 5270338292583504165 \nu^{2} - 5226852994856502950 \nu + 59418569180970775\)\()/ 454580475630153760 \)
\(\beta_{13}\)\(=\)\((\)\(-16296167766296901 \nu^{15} + 46657645630438849 \nu^{14} - 220693813391828059 \nu^{13} + 490255203026234886 \nu^{12} - 1324081568271118439 \nu^{11} + 2171662124721910708 \nu^{10} - 3646287371621077981 \nu^{9} + 3462410165728884427 \nu^{8} - 9848220683260194529 \nu^{7} + 17797250626139187779 \nu^{6} - 31784433905471092610 \nu^{5} + 32902654497697219800 \nu^{4} - 24876887445936951080 \nu^{3} + 11522918100224582795 \nu^{2} - 8819030598088243120 \nu + 1667167131638041055\)\()/ 454580475630153760 \)
\(\beta_{14}\)\(=\)\((\)\(16576781827677814 \nu^{15} - 45453688208119129 \nu^{14} + 219284175557671186 \nu^{13} - 470575312000685073 \nu^{12} + 1289205279920317419 \nu^{11} - 2038073676823909444 \nu^{10} + 3445236405584454006 \nu^{9} - 3032211154151816061 \nu^{8} + 9585093068819660548 \nu^{7} - 16816185325336642299 \nu^{6} + 30431971572347844100 \nu^{5} - 29120858031490839590 \nu^{4} + 20842243045981534410 \nu^{3} - 8417551544734639360 \nu^{2} + 7973236770827366535 \nu - 1025213227739747135\)\()/ 454580475630153760 \)
\(\beta_{15}\)\(=\)\((\)\(-31742657223153564 \nu^{15} + 90930749048576029 \nu^{14} - 431347940936808556 \nu^{13} + 955931937354509231 \nu^{12} - 2590293343054165503 \nu^{11} + 4236554416589668948 \nu^{10} - 7143632052250437628 \nu^{9} + 6742540241983255677 \nu^{8} - 19234203982822940782 \nu^{7} + 34610408690995495111 \nu^{6} - 62595008084751532400 \nu^{5} + 64110835668533354210 \nu^{4} - 48322167102593217550 \nu^{3} + 21980961042064417270 \nu^{2} - 18197077383887239845 \nu + 3492178872565443195\)\()/ 454580475630153760 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{6} + \beta_{5}\)
\(\nu^{2}\)\(=\)\(\beta_{11} + 2 \beta_{10} - 2 \beta_{8} + 3 \beta_{7} + \beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(-\beta_{15} + \beta_{13} + \beta_{12} - \beta_{10} + \beta_{8} - 5 \beta_{6} - \beta_{5} - \beta_{2} + \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{15} - \beta_{14} - \beta_{11} - 8 \beta_{10} - 6 \beta_{9} - 15 \beta_{7} + \beta_{5} + \beta_{4} - 6 \beta_{2} - \beta_{1} + 1\)
\(\nu^{5}\)\(=\)\(-7 \beta_{14} - 7 \beta_{13} - 7 \beta_{12} + 7 \beta_{11} - 8 \beta_{8} + 8 \beta_{7} + 10 \beta_{6} - 10 \beta_{4} - \beta_{3} + 8 \beta_{2} - 18 \beta_{1} - 3\)
\(\nu^{6}\)\(=\)\(7 \beta_{15} + \beta_{14} - 7 \beta_{13} + 11 \beta_{12} + 54 \beta_{10} + 35 \beta_{9} + 40 \beta_{8} + 47 \beta_{7} - 10 \beta_{6} - 10 \beta_{5} - 2 \beta_{4} + 6 \beta_{3} + 2 \beta_{1}\)
\(\nu^{7}\)\(=\)\(42 \beta_{15} + 54 \beta_{14} + 13 \beta_{12} - 45 \beta_{11} + 12 \beta_{10} - 13 \beta_{9} + 30 \beta_{8} - 86 \beta_{7} + 76 \beta_{5} + 165 \beta_{4} + 42 \beta_{3} - 58 \beta_{2} + 28\)
\(\nu^{8}\)\(=\)\(-9 \beta_{15} + 38 \beta_{13} - 210 \beta_{12} + 89 \beta_{11} - 234 \beta_{10} - 89 \beta_{9} - 308 \beta_{8} + 80 \beta_{6} + 28 \beta_{5} + 210 \beta_{2} - 28 \beta_{1} - 98\)
\(\nu^{9}\)\(=\)\(-346 \beta_{15} - 346 \beta_{14} + 248 \beta_{13} + 117 \beta_{11} + 38 \beta_{10} + 290 \beta_{9} + 614 \beta_{7} - 524 \beta_{6} - 1000 \beta_{5} - 1000 \beta_{4} - 248 \beta_{3} + 290 \beta_{2} + 476 \beta_{1} - 94\)
\(\nu^{10}\)\(=\)\(-56 \beta_{14} - 56 \beta_{13} + 1290 \beta_{12} - 1290 \beta_{11} + 1992 \beta_{8} - 1992 \beta_{7} - 272 \beta_{6} + 272 \beta_{4} - 131 \beta_{3} - 1931 \beta_{2} + 328 \beta_{1} + 1406\)
\(\nu^{11}\)\(=\)\(2174 \beta_{15} + 1477 \beta_{14} - 2174 \beta_{13} - 913 \beta_{12} - 447 \beta_{10} - 1890 \beta_{9} - 1625 \beta_{8} - 2621 \beta_{7} + 6160 \beta_{6} + 6160 \beta_{5} + 3461 \beta_{4} + 697 \beta_{3} - 3461 \beta_{1}\)
\(\nu^{12}\)\(=\)\(284 \beta_{15} + 848 \beta_{14} - 4374 \beta_{12} + 8050 \beta_{11} + 8579 \beta_{10} + 4374 \beta_{9} - 8295 \beta_{8} + 21093 \beta_{7} - 2273 \beta_{5} - 4375 \beta_{4} + 284 \beta_{3} + 12424 \beta_{2} - 8669\)
\(\nu^{13}\)\(=\)\(-8898 \beta_{15} + 13556 \beta_{13} + 12425 \beta_{12} - 6647 \beta_{11} + 2237 \beta_{10} + 6647 \beta_{9} + 17819 \beta_{8} - 38325 \beta_{6} - 22394 \beta_{5} - 12425 \beta_{2} + 22394 \beta_{1} + 5394\)
\(\nu^{14}\)\(=\)\(-3382 \beta_{15} - 3382 \beta_{14} + 1131 \beta_{13} - 29041 \beta_{11} - 85364 \beta_{10} - 50750 \beta_{9} - 133987 \beta_{7} + 17557 \beta_{6} + 31350 \beta_{5} + 31350 \beta_{4} - 1131 \beta_{3} - 50750 \beta_{2} - 13793 \beta_{1} + 31232\)
\(\nu^{15}\)\(=\)\(-54132 \beta_{14} - 54132 \beta_{13} - 82100 \beta_{12} + 82100 \beta_{11} - 120596 \beta_{8} + 120596 \beta_{7} + 143446 \beta_{6} - 143446 \beta_{4} - 30172 \beta_{3} + 128698 \beta_{2} - 96554 \beta_{1} - 67213\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\)
\(\chi(n)\) \(1\) \(-1 + \beta_{7} - \beta_{8} + \beta_{10}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
1.60551 + 1.16647i
0.901622 + 0.655067i
0.183009 + 0.132964i
−1.38112 1.00344i
1.60551 1.16647i
0.901622 0.655067i
0.183009 0.132964i
−1.38112 + 1.00344i
0.751051 2.31150i
0.435488 1.34029i
−0.206962 + 0.636964i
−0.788594 + 2.42704i
0.751051 + 2.31150i
0.435488 + 1.34029i
−0.206962 0.636964i
−0.788594 2.42704i
−1.60551 1.16647i 0.861043 2.65002i 0.598967 + 1.84343i 0.0217822 0.0158257i −4.47357 + 3.25024i 0.309017 + 0.951057i −0.0378378 + 0.116453i −3.85415 2.80020i −0.0534317
15.2 −0.901622 0.655067i −0.883423 + 2.71890i −0.234224 0.720867i −2.79603 + 2.03143i 2.57757 1.87272i 0.309017 + 0.951057i −0.949813 + 2.92322i −4.18492 3.04052i 3.85168
15.3 −0.183009 0.132964i −0.0677147 + 0.208405i −0.602221 1.85345i 2.01892 1.46683i 0.0401026 0.0291363i 0.309017 + 0.951057i −0.276036 + 0.849550i 2.38820 + 1.73513i −0.564516
15.4 1.38112 + 1.00344i 0.708129 2.17940i 0.282562 + 0.869638i −3.28976 + 2.39015i 3.16491 2.29944i 0.309017 + 0.951057i 0.572703 1.76260i −1.82128 1.32323i −6.94194
36.1 −1.60551 + 1.16647i 0.861043 + 2.65002i 0.598967 1.84343i 0.0217822 + 0.0158257i −4.47357 3.25024i 0.309017 0.951057i −0.0378378 0.116453i −3.85415 + 2.80020i −0.0534317
36.2 −0.901622 + 0.655067i −0.883423 2.71890i −0.234224 + 0.720867i −2.79603 2.03143i 2.57757 + 1.87272i 0.309017 0.951057i −0.949813 2.92322i −4.18492 + 3.04052i 3.85168
36.3 −0.183009 + 0.132964i −0.0677147 0.208405i −0.602221 + 1.85345i 2.01892 + 1.46683i 0.0401026 + 0.0291363i 0.309017 0.951057i −0.276036 0.849550i 2.38820 1.73513i −0.564516
36.4 1.38112 1.00344i 0.708129 + 2.17940i 0.282562 0.869638i −3.28976 2.39015i 3.16491 + 2.29944i 0.309017 0.951057i 0.572703 + 1.76260i −1.82128 + 1.32323i −6.94194
64.1 −0.751051 + 2.31150i −1.16030 + 0.843005i −3.16091 2.29654i 0.388938 + 1.19703i −1.07716 3.31516i −0.809017 0.587785i 3.74989 2.72445i −0.291419 + 0.896896i −3.05904
64.2 −0.435488 + 1.34029i 1.75021 1.27160i 0.0112975 + 0.00820814i −0.565930 1.74175i 0.942126 + 2.89957i −0.809017 0.587785i −2.29616 + 1.66826i 0.519216 1.59798i 2.58091
64.3 0.206962 0.636964i −2.54013 + 1.84551i 1.25514 + 0.911915i 0.662464 + 2.03885i 0.649815 + 1.99992i −0.809017 0.587785i 1.92429 1.39808i 2.11929 6.52251i 1.43578
64.4 0.788594 2.42704i 0.332181 0.241344i −3.65062 2.65233i 1.05961 + 3.26115i −0.323795 0.996539i −0.809017 0.587785i −5.18703 + 3.76860i −0.874954 + 2.69283i 8.75055
71.1 −0.751051 2.31150i −1.16030 0.843005i −3.16091 + 2.29654i 0.388938 1.19703i −1.07716 + 3.31516i −0.809017 + 0.587785i 3.74989 + 2.72445i −0.291419 0.896896i −3.05904
71.2 −0.435488 1.34029i 1.75021 + 1.27160i 0.0112975 0.00820814i −0.565930 + 1.74175i 0.942126 2.89957i −0.809017 + 0.587785i −2.29616 1.66826i 0.519216 + 1.59798i 2.58091
71.3 0.206962 + 0.636964i −2.54013 1.84551i 1.25514 0.911915i 0.662464 2.03885i 0.649815 1.99992i −0.809017 + 0.587785i 1.92429 + 1.39808i 2.11929 + 6.52251i 1.43578
71.4 0.788594 + 2.42704i 0.332181 + 0.241344i −3.65062 + 2.65233i 1.05961 3.26115i −0.323795 + 0.996539i −0.809017 + 0.587785i −5.18703 3.76860i −0.874954 2.69283i 8.75055
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.2.f.b 16
3.b odd 2 1 693.2.m.i 16
7.b odd 2 1 539.2.f.e 16
7.c even 3 2 539.2.q.g 32
7.d odd 6 2 539.2.q.f 32
11.b odd 2 1 847.2.f.x 16
11.c even 5 1 inner 77.2.f.b 16
11.c even 5 1 847.2.a.p 8
11.c even 5 2 847.2.f.w 16
11.d odd 10 1 847.2.a.o 8
11.d odd 10 2 847.2.f.v 16
11.d odd 10 1 847.2.f.x 16
33.f even 10 1 7623.2.a.cw 8
33.h odd 10 1 693.2.m.i 16
33.h odd 10 1 7623.2.a.ct 8
77.j odd 10 1 539.2.f.e 16
77.j odd 10 1 5929.2.a.bt 8
77.l even 10 1 5929.2.a.bs 8
77.m even 15 2 539.2.q.g 32
77.p odd 30 2 539.2.q.f 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.b 16 1.a even 1 1 trivial
77.2.f.b 16 11.c even 5 1 inner
539.2.f.e 16 7.b odd 2 1
539.2.f.e 16 77.j odd 10 1
539.2.q.f 32 7.d odd 6 2
539.2.q.f 32 77.p odd 30 2
539.2.q.g 32 7.c even 3 2
539.2.q.g 32 77.m even 15 2
693.2.m.i 16 3.b odd 2 1
693.2.m.i 16 33.h odd 10 1
847.2.a.o 8 11.d odd 10 1
847.2.a.p 8 11.c even 5 1
847.2.f.v 16 11.d odd 10 2
847.2.f.w 16 11.c even 5 2
847.2.f.x 16 11.b odd 2 1
847.2.f.x 16 11.d odd 10 1
5929.2.a.bs 8 77.l even 10 1
5929.2.a.bt 8 77.j odd 10 1
7623.2.a.ct 8 33.h odd 10 1
7623.2.a.cw 8 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{16} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(77, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 25 + 200 T + 700 T^{2} + 925 T^{3} + 1850 T^{4} + 2300 T^{5} + 2135 T^{6} + 1175 T^{7} + 640 T^{8} + 245 T^{9} + 245 T^{10} + 145 T^{11} + 86 T^{12} + 32 T^{13} + 14 T^{14} + 3 T^{15} + T^{16} \)
$3$ \( 256 - 128 T + 3904 T^{2} - 14304 T^{3} + 22464 T^{4} + 14920 T^{5} + 8452 T^{6} + 872 T^{7} + 4253 T^{8} - 178 T^{9} + 747 T^{10} + 100 T^{11} + 124 T^{12} + 26 T^{13} + 14 T^{14} + 2 T^{15} + T^{16} \)
$5$ \( 256 - 15488 T + 359616 T^{2} - 157984 T^{3} + 349536 T^{4} - 47144 T^{5} + 109516 T^{6} - 3764 T^{7} + 15101 T^{8} + 2706 T^{9} + 1349 T^{10} + 357 T^{11} + 314 T^{12} + 59 T^{13} + 19 T^{14} + 5 T^{15} + T^{16} \)
$7$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
$11$ \( 214358881 + 58461513 T + 33659659 T^{2} - 12239876 T^{3} - 4187326 T^{4} - 2988095 T^{5} + 127292 T^{6} + 84348 T^{7} + 114073 T^{8} + 7668 T^{9} + 1052 T^{10} - 2245 T^{11} - 286 T^{12} - 76 T^{13} + 19 T^{14} + 3 T^{15} + T^{16} \)
$13$ \( 188897536 + 11105152 T + 54886464 T^{2} - 1175584 T^{3} + 10501024 T^{4} + 2287000 T^{5} + 2166132 T^{6} + 687552 T^{7} + 357453 T^{8} + 87462 T^{9} + 25007 T^{10} + 2465 T^{11} + 484 T^{12} + 81 T^{13} + 39 T^{14} + 7 T^{15} + T^{16} \)
$17$ \( 2611456 - 155136 T - 457536 T^{2} - 8512 T^{3} + 804176 T^{4} + 328912 T^{5} + 91164 T^{6} - 50192 T^{7} + 44721 T^{8} + 54357 T^{9} + 34011 T^{10} + 8601 T^{11} + 2694 T^{12} + 333 T^{13} + 81 T^{14} + 5 T^{15} + T^{16} \)
$19$ \( 62726400 - 102643200 T + 135696960 T^{2} - 105813600 T^{3} + 65590816 T^{4} - 27634936 T^{5} + 9110804 T^{6} - 2315850 T^{7} + 805255 T^{8} - 357261 T^{9} + 156481 T^{10} - 42779 T^{11} + 8310 T^{12} - 1315 T^{13} + 189 T^{14} - 19 T^{15} + T^{16} \)
$23$ \( ( 859 + 1310 T - 1702 T^{2} - 3298 T^{3} - 765 T^{4} + 342 T^{5} + 40 T^{6} - 16 T^{7} + T^{8} )^{2} \)
$29$ \( 245025 - 393525 T + 3459735 T^{2} - 14183400 T^{3} + 28254151 T^{4} - 31103248 T^{5} + 19614271 T^{6} - 6296330 T^{7} + 656200 T^{8} - 5008 T^{9} + 92399 T^{10} + 397 T^{11} + 4935 T^{12} - 75 T^{13} + 116 T^{14} - 3 T^{15} + T^{16} \)
$31$ \( 2629638400 + 7127920000 T + 9806984000 T^{2} + 7429634400 T^{3} + 4473688800 T^{4} + 1280520600 T^{5} + 472113060 T^{6} + 121888290 T^{7} + 20851115 T^{8} + 1170135 T^{9} + 287250 T^{10} + 40055 T^{11} + 9191 T^{12} + 943 T^{13} + 124 T^{14} + 7 T^{15} + T^{16} \)
$37$ \( 212521 - 804445 T + 4750971 T^{2} - 6462403 T^{3} + 12286174 T^{4} - 1090481 T^{5} - 1701204 T^{6} + 173633 T^{7} + 1195421 T^{8} + 176392 T^{9} + 49989 T^{10} - 4752 T^{11} + 421 T^{12} + 207 T^{13} + 84 T^{14} - 4 T^{15} + T^{16} \)
$41$ \( 13424896 - 92244864 T + 272362304 T^{2} - 327853088 T^{3} + 200671616 T^{4} - 62218152 T^{5} + 44819044 T^{6} - 10134798 T^{7} + 1356301 T^{8} + 192138 T^{9} + 235971 T^{10} + 39244 T^{11} + 10194 T^{12} + 1012 T^{13} + 196 T^{14} + 10 T^{15} + T^{16} \)
$43$ \( ( -971 - 8740 T - 10617 T^{2} + 812 T^{3} + 2780 T^{4} - 268 T^{5} - 105 T^{6} + 4 T^{7} + T^{8} )^{2} \)
$47$ \( 5345713926400 + 2875857587200 T + 1043738689600 T^{2} + 242392886400 T^{3} + 49474230000 T^{4} + 9254568000 T^{5} + 2203267060 T^{6} + 445925000 T^{7} + 89155845 T^{8} + 15909825 T^{9} + 2670120 T^{10} + 354745 T^{11} + 43531 T^{12} + 4317 T^{13} + 374 T^{14} + 23 T^{15} + T^{16} \)
$53$ \( 310840815961 - 469830816169 T + 923740577175 T^{2} - 470995458115 T^{3} + 138301768640 T^{4} - 21601047907 T^{5} + 1881024338 T^{6} + 14963425 T^{7} + 41741545 T^{8} - 13315320 T^{9} + 2123903 T^{10} - 38642 T^{11} + 11525 T^{12} + 885 T^{13} + 190 T^{14} - 4 T^{15} + T^{16} \)
$59$ \( 187142400 - 764985600 T + 1324491840 T^{2} - 486404160 T^{3} + 1686670096 T^{4} - 522422992 T^{5} + 327194836 T^{6} - 74625980 T^{7} + 23818625 T^{8} - 4789757 T^{9} + 585984 T^{10} - 24387 T^{11} + 3165 T^{12} - 965 T^{13} + 186 T^{14} - 17 T^{15} + T^{16} \)
$61$ \( 253446400 - 543190400 T + 431878400 T^{2} + 14292000 T^{3} - 34066000 T^{4} - 62609400 T^{5} + 50920960 T^{6} + 23644410 T^{7} + 20094055 T^{8} + 3416695 T^{9} + 1144190 T^{10} + 171715 T^{11} + 29831 T^{12} + 2843 T^{13} + 224 T^{14} + 7 T^{15} + T^{16} \)
$67$ \( ( -27395 - 35255 T + 5380 T^{2} + 12660 T^{3} - 255 T^{4} - 1160 T^{5} - 16 T^{6} + 19 T^{7} + T^{8} )^{2} \)
$71$ \( 379119841 + 2290763150 T + 29448788941 T^{2} + 21101907378 T^{3} + 5632223589 T^{4} + 26342786 T^{5} + 368390466 T^{6} + 176908812 T^{7} + 46858071 T^{8} + 11881588 T^{9} + 3773584 T^{10} + 650792 T^{11} + 69296 T^{12} + 4668 T^{13} + 424 T^{14} + 14 T^{15} + T^{16} \)
$73$ \( 105069332736 + 261490854528 T + 295833229056 T^{2} + 133897989024 T^{3} + 215013584176 T^{4} + 70926929544 T^{5} + 12085086536 T^{6} + 1087296874 T^{7} + 147153271 T^{8} + 38702959 T^{9} + 9580124 T^{10} + 1540823 T^{11} + 180449 T^{12} + 14721 T^{13} + 914 T^{14} + 35 T^{15} + T^{16} \)
$79$ \( 15858514175625 + 3558361826250 T + 370304355375 T^{2} - 42305420625 T^{3} + 166705429150 T^{4} - 38045770625 T^{5} + 6491197800 T^{6} - 286563275 T^{7} + 99898775 T^{8} - 22354075 T^{9} + 5985185 T^{10} - 796750 T^{11} + 87835 T^{12} - 6050 T^{13} + 410 T^{14} - 15 T^{15} + T^{16} \)
$83$ \( 756470016 + 1315571328 T + 1763125056 T^{2} + 2163259104 T^{3} + 2105559616 T^{4} + 1476165304 T^{5} + 750623116 T^{6} + 282797224 T^{7} + 80756581 T^{8} + 17485189 T^{9} + 2830604 T^{10} + 321583 T^{11} + 23949 T^{12} + 421 T^{13} + 14 T^{14} - 5 T^{15} + T^{16} \)
$89$ \( ( 952400 - 1415520 T + 512284 T^{2} + 45228 T^{3} - 34845 T^{4} + 2150 T^{5} + 320 T^{6} - 37 T^{7} + T^{8} )^{2} \)
$97$ \( 4647025244416 - 1516351057536 T + 624201528704 T^{2} + 57600817888 T^{3} + 5668873616 T^{4} - 2128609208 T^{5} + 10342618584 T^{6} + 770043138 T^{7} + 186574551 T^{8} + 7455377 T^{9} + 3268781 T^{10} - 194579 T^{11} + 45834 T^{12} - 4337 T^{13} + 511 T^{14} - 20 T^{15} + T^{16} \)
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