Properties

Label 76.2.f.a
Level $76$
Weight $2$
Character orbit 76.f
Analytic conductor $0.607$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 76.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.606863055362\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 3 x^{15} + 6 x^{14} - 9 x^{13} + 12 x^{12} - 9 x^{11} + 3 x^{10} + 6 x^{9} - 10 x^{8} + 12 x^{7} + 12 x^{6} - 72 x^{5} + 192 x^{4} - 288 x^{3} + 384 x^{2} - 384 x + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 3 x^{15} + 6 x^{14} - 9 x^{13} + 12 x^{12} - 9 x^{11} + 3 x^{10} + 6 x^{9} - 10 x^{8} + 12 x^{7} + 12 x^{6} - 72 x^{5} + 192 x^{4} - 288 x^{3} + 384 x^{2} - 384 x + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -26 \nu^{15} - 397 \nu^{14} + 335 \nu^{13} - 666 \nu^{12} + 419 \nu^{11} - 428 \nu^{10} + 625 \nu^{9} - 183 \nu^{8} - 1340 \nu^{7} - 730 \nu^{6} + 1304 \nu^{5} - 4308 \nu^{4} + 17312 \nu^{3} - 29216 \nu^{2} + 15616 \nu - 5056 \)\()/40768\)
\(\beta_{3}\)\(=\)\((\)\( 79 \nu^{15} - 341 \nu^{14} - 1114 \nu^{13} + 629 \nu^{12} - 1716 \nu^{11} + 965 \nu^{10} - 1475 \nu^{9} + 2974 \nu^{8} - 1522 \nu^{7} - 4412 \nu^{6} - 1972 \nu^{5} - 472 \nu^{4} - 2064 \nu^{3} + 46496 \nu^{2} - 86528 \nu + 32128 \)\()/81536\)
\(\beta_{4}\)\(=\)\((\)\( 130 \nu^{15} - 269 \nu^{14} + 1363 \nu^{13} - 2 \nu^{12} - 37 \nu^{11} + 1552 \nu^{10} + 109 \nu^{9} + 5325 \nu^{8} + 7288 \nu^{7} - 270 \nu^{6} + 31896 \nu^{5} - 13740 \nu^{4} + 48288 \nu^{3} + 8880 \nu^{2} + 34816 \nu + 62912 \)\()/81536\)
\(\beta_{5}\)\(=\)\((\)\( 339 \nu^{15} + 2061 \nu^{14} + 1416 \nu^{13} - 2119 \nu^{12} - 1594 \nu^{11} + 3285 \nu^{10} + 10699 \nu^{9} + 6764 \nu^{8} + 8742 \nu^{7} + 1712 \nu^{6} - 2468 \nu^{5} + 37120 \nu^{4} - 24656 \nu^{3} + 17216 \nu^{2} + 174400 \nu + 45056 \)\()/163072\)
\(\beta_{6}\)\(=\)\((\)\( -425 \nu^{15} - 845 \nu^{14} - 666 \nu^{13} - 883 \nu^{12} - 1268 \nu^{11} - 5251 \nu^{10} - 8251 \nu^{9} + 2638 \nu^{8} - 7234 \nu^{7} - 1948 \nu^{6} + 13596 \nu^{5} - 6968 \nu^{4} - 29168 \nu^{3} + 4832 \nu^{2} - 4544 \nu + 51840 \)\()/163072\)
\(\beta_{7}\)\(=\)\((\)\( -230 \nu^{15} - 97 \nu^{14} - 949 \nu^{13} + 290 \nu^{12} + 367 \nu^{11} - 1404 \nu^{10} - 1791 \nu^{9} + 1781 \nu^{8} - 9924 \nu^{7} + 7986 \nu^{6} - 1280 \nu^{5} - 6508 \nu^{4} - 26512 \nu^{3} - 20656 \nu^{2} + 640 \nu - 52928 \)\()/81536\)
\(\beta_{8}\)\(=\)\((\)\( -537 \nu^{15} + 2473 \nu^{14} - 5608 \nu^{13} + 1525 \nu^{12} - 1506 \nu^{11} + 97 \nu^{10} + 7023 \nu^{9} - 7988 \nu^{8} + 4302 \nu^{7} + 5808 \nu^{6} - 39828 \nu^{5} + 22208 \nu^{4} - 122128 \nu^{3} + 90176 \nu^{2} - 960 \nu - 147968 \)\()/163072\)
\(\beta_{9}\)\(=\)\((\)\( -361 \nu^{15} - 550 \nu^{14} + 835 \nu^{13} - 1671 \nu^{12} + 2921 \nu^{11} - 3225 \nu^{10} + 934 \nu^{9} - 1559 \nu^{8} - 11082 \nu^{7} + 10938 \nu^{6} + 3300 \nu^{5} - 14652 \nu^{4} + 10176 \nu^{3} - 44048 \nu^{2} + 36864 \nu - 95168 \)\()/81536\)
\(\beta_{10}\)\(=\)\((\)\( 885 \nu^{15} - 431 \nu^{14} + 114 \nu^{13} + 4223 \nu^{12} - 4660 \nu^{11} + 5903 \nu^{10} + 759 \nu^{9} + 4874 \nu^{8} + 16498 \nu^{7} + 18316 \nu^{6} + 10916 \nu^{5} + 7832 \nu^{4} - 21296 \nu^{3} + 141536 \nu^{2} - 153536 \nu + 273536 \)\()/163072\)
\(\beta_{11}\)\(=\)\((\)\(-621 \nu^{15} + 2879 \nu^{14} - 5566 \nu^{13} + 9309 \nu^{12} - 9892 \nu^{11} + 5725 \nu^{10} - 607 \nu^{9} - 5790 \nu^{8} + 14522 \nu^{7} - 11356 \nu^{6} - 7180 \nu^{5} + 96520 \nu^{4} - 160096 \nu^{3} + 299616 \nu^{2} - 308736 \nu + 352896\)\()/81536\)
\(\beta_{12}\)\(=\)\((\)\( \nu^{15} - 3 \nu^{14} + 6 \nu^{13} - 9 \nu^{12} + 12 \nu^{11} - 9 \nu^{10} + 3 \nu^{9} + 6 \nu^{8} - 10 \nu^{7} + 12 \nu^{6} + 12 \nu^{5} - 72 \nu^{4} + 192 \nu^{3} - 288 \nu^{2} + 384 \nu - 384 \)\()/128\)
\(\beta_{13}\)\(=\)\((\)\( 1441 \nu^{15} + 1063 \nu^{14} + 2720 \nu^{13} - 1949 \nu^{12} + 522 \nu^{11} + 2391 \nu^{10} + 3345 \nu^{9} + 3420 \nu^{8} + 4898 \nu^{7} + 18096 \nu^{6} + 62516 \nu^{5} - 41344 \nu^{4} + 54288 \nu^{3} - 26496 \nu^{2} + 175424 \nu + 125952 \)\()/163072\)
\(\beta_{14}\)\(=\)\((\)\( 475 \nu^{15} - 491 \nu^{14} + 900 \nu^{13} - 731 \nu^{12} + 662 \nu^{11} - 703 \nu^{10} + 27 \nu^{9} + 1600 \nu^{8} + 418 \nu^{7} - 1616 \nu^{6} + 6180 \nu^{5} - 22304 \nu^{4} + 36704 \nu^{3} - 25600 \nu^{2} + 15040 \nu - 6656 \)\()/40768\)
\(\beta_{15}\)\(=\)\((\)\(1346 \nu^{15} - 2357 \nu^{14} + 3471 \nu^{13} - 6350 \nu^{12} + 7867 \nu^{11} - 3172 \nu^{10} + 821 \nu^{9} + 2561 \nu^{8} - 6044 \nu^{7} - 9158 \nu^{6} + 25216 \nu^{5} - 79964 \nu^{4} + 127856 \nu^{3} - 164848 \nu^{2} + 232000 \nu - 249664\)\()/81536\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} - \beta_{7} + \beta_{3} - \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(-\beta_{15} + \beta_{13} + \beta_{12} - \beta_{10} - \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{3} + \beta_{1}\)
\(\nu^{4}\)\(=\)\(-2 \beta_{14} + \beta_{13} + 2 \beta_{12} + \beta_{11} - \beta_{9} - \beta_{8} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} - 1\)
\(\nu^{5}\)\(=\)\(\beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} + \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_{1} - 2\)
\(\nu^{6}\)\(=\)\(-2 \beta_{15} - 2 \beta_{14} + 3 \beta_{13} + 2 \beta_{12} - \beta_{11} + 2 \beta_{10} + \beta_{8} - \beta_{7} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - 1\)
\(\nu^{7}\)\(=\)\(-\beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{11} + 3 \beta_{10} - 4 \beta_{9} + 4 \beta_{8} - 2 \beta_{7} + 4 \beta_{6} - \beta_{5} + 3 \beta_{4} - 5 \beta_{3} - 5 \beta_{1}\)
\(\nu^{8}\)\(=\)\(2 \beta_{14} - 6 \beta_{13} + 2 \beta_{12} - 3 \beta_{9} + 9 \beta_{7} + 6 \beta_{5} + 6 \beta_{4} + \beta_{3} - 5 \beta_{2} - 5 \beta_{1} + 6\)
\(\nu^{9}\)\(=\)\(3 \beta_{15} + 8 \beta_{14} - 7 \beta_{13} - 3 \beta_{12} + 4 \beta_{11} + 3 \beta_{10} + 8 \beta_{9} + 4 \beta_{8} + 5 \beta_{7} - 3 \beta_{6} + 9 \beta_{5} + 4 \beta_{4} - 6 \beta_{3} + 12 \beta_{2} - 3 \beta_{1}\)
\(\nu^{10}\)\(=\)\(4 \beta_{15} - 16 \beta_{14} + 5 \beta_{13} - 8 \beta_{12} - 7 \beta_{11} + 2 \beta_{10} - 5 \beta_{9} - 5 \beta_{8} - 14 \beta_{6} - 4 \beta_{5} - \beta_{4} + 9 \beta_{3} - \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{11}\)\(=\)\(11 \beta_{15} - 13 \beta_{14} - 9 \beta_{13} + 8 \beta_{12} + 5 \beta_{11} + 9 \beta_{7} - 13 \beta_{6} - 4 \beta_{5} + 9 \beta_{4} - 5 \beta_{3} - 24 \beta_{2} - 18 \beta_{1} + 44\)
\(\nu^{12}\)\(=\)\(24 \beta_{14} - 19 \beta_{13} + 11 \beta_{11} - 7 \beta_{8} + 31 \beta_{7} - 20 \beta_{6} - 2 \beta_{5} + 11 \beta_{4} - 30 \beta_{3} + 18 \beta_{1} - 5\)
\(\nu^{13}\)\(=\)\(21 \beta_{14} - 18 \beta_{13} - 21 \beta_{12} - 5 \beta_{11} + 5 \beta_{10} + 24 \beta_{9} - 24 \beta_{8} - 10 \beta_{7} + 2 \beta_{6} + 19 \beta_{5} - 5 \beta_{4} - 29 \beta_{3} - 22 \beta_{2} + 9 \beta_{1} - 26\)
\(\nu^{14}\)\(=\)\(-26 \beta_{15} + 18 \beta_{14} + 28 \beta_{13} + 6 \beta_{12} - 8 \beta_{11} - 52 \beta_{10} - 7 \beta_{9} - 7 \beta_{7} - 22 \beta_{6} - 6 \beta_{5} + 2 \beta_{4} + 13 \beta_{3} - 55 \beta_{2} - 17 \beta_{1} + 24\)
\(\nu^{15}\)\(=\)\(21 \beta_{15} + 2 \beta_{14} + 43 \beta_{13} + 39 \beta_{12} + 32 \beta_{11} + 21 \beta_{10} - 40 \beta_{9} - 20 \beta_{8} + 25 \beta_{7} + \beta_{6} + 5 \beta_{5} - 72 \beta_{4} + 10 \beta_{3} - 29 \beta_{1} - 24\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(1 - \beta_{2}\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1
1.05003 0.947334i
−0.112075 1.40977i
1.34543 0.435684i
−0.835469 1.14105i
1.16486 + 0.801943i
−1.35532 0.403874i
0.570443 + 1.29406i
−0.327894 + 1.37568i
1.05003 + 0.947334i
−0.112075 + 1.40977i
1.34543 + 0.435684i
−0.835469 + 1.14105i
1.16486 0.801943i
−1.35532 + 0.403874i
0.570443 1.29406i
−0.327894 1.37568i
−1.34543 0.435684i 0.982349 1.70148i 1.62036 + 1.17236i −0.349646 + 0.605604i −2.06299 + 1.86122i 3.80025i −1.66930 2.28330i −0.430019 0.744815i 0.734275 0.662462i
27.2 −1.16486 + 0.801943i 0.305055 0.528371i 0.713775 1.86829i 1.59295 2.75907i 0.0683782 + 0.860112i 2.36291i 0.666820 + 2.74870i 1.31388 + 2.27571i 0.357061 + 4.49138i
27.3 −1.05003 0.947334i −0.982349 + 1.70148i 0.205118 + 1.98945i −0.349646 + 0.605604i 2.64336 0.855988i 3.80025i 1.66930 2.28330i −0.430019 0.744815i 0.940847 0.304670i
27.4 −0.570443 + 1.29406i −0.637123 + 1.10353i −1.34919 1.47638i −1.60333 + 2.77705i −1.06459 1.45398i 1.25044i 2.68016 0.903746i 0.688149 + 1.19191i −2.67906 3.65895i
27.5 0.112075 1.40977i −0.305055 + 0.528371i −1.97488 0.316000i 1.59295 2.75907i 0.710690 + 0.489273i 2.36291i −0.666820 + 2.74870i 1.31388 + 2.27571i −3.71111 2.55491i
27.6 0.327894 + 1.37568i 1.42689 2.47144i −1.78497 + 0.902152i −0.139977 + 0.242447i 3.86777 + 1.15256i 1.55280i −1.82635 2.15973i −2.57201 4.45486i −0.379427 0.113066i
27.7 0.835469 1.14105i 0.637123 1.10353i −0.603985 1.90662i −1.60333 + 2.77705i −0.726885 1.64895i 1.25044i −2.68016 0.903746i 0.688149 + 1.19191i 1.82921 + 4.14961i
27.8 1.35532 0.403874i −1.42689 + 2.47144i 1.67377 1.09475i −0.139977 + 0.242447i −0.935735 + 3.92587i 1.55280i 1.82635 2.15973i −2.57201 4.45486i −0.0917953 + 0.385126i
31.1 −1.34543 + 0.435684i 0.982349 + 1.70148i 1.62036 1.17236i −0.349646 0.605604i −2.06299 1.86122i 3.80025i −1.66930 + 2.28330i −0.430019 + 0.744815i 0.734275 + 0.662462i
31.2 −1.16486 0.801943i 0.305055 + 0.528371i 0.713775 + 1.86829i 1.59295 + 2.75907i 0.0683782 0.860112i 2.36291i 0.666820 2.74870i 1.31388 2.27571i 0.357061 4.49138i
31.3 −1.05003 + 0.947334i −0.982349 1.70148i 0.205118 1.98945i −0.349646 0.605604i 2.64336 + 0.855988i 3.80025i 1.66930 + 2.28330i −0.430019 + 0.744815i 0.940847 + 0.304670i
31.4 −0.570443 1.29406i −0.637123 1.10353i −1.34919 + 1.47638i −1.60333 2.77705i −1.06459 + 1.45398i 1.25044i 2.68016 + 0.903746i 0.688149 1.19191i −2.67906 + 3.65895i
31.5 0.112075 + 1.40977i −0.305055 0.528371i −1.97488 + 0.316000i 1.59295 + 2.75907i 0.710690 0.489273i 2.36291i −0.666820 2.74870i 1.31388 2.27571i −3.71111 + 2.55491i
31.6 0.327894 1.37568i 1.42689 + 2.47144i −1.78497 0.902152i −0.139977 0.242447i 3.86777 1.15256i 1.55280i −1.82635 + 2.15973i −2.57201 + 4.45486i −0.379427 + 0.113066i
31.7 0.835469 + 1.14105i 0.637123 + 1.10353i −0.603985 + 1.90662i −1.60333 2.77705i −0.726885 + 1.64895i 1.25044i −2.68016 + 0.903746i 0.688149 1.19191i 1.82921 4.14961i
31.8 1.35532 + 0.403874i −1.42689 2.47144i 1.67377 + 1.09475i −0.139977 0.242447i −0.935735 3.92587i 1.55280i 1.82635 + 2.15973i −2.57201 + 4.45486i −0.0917953 0.385126i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.2.f.a 16
3.b odd 2 1 684.2.r.a 16
4.b odd 2 1 inner 76.2.f.a 16
8.b even 2 1 1216.2.n.f 16
8.d odd 2 1 1216.2.n.f 16
12.b even 2 1 684.2.r.a 16
19.d odd 6 1 inner 76.2.f.a 16
57.f even 6 1 684.2.r.a 16
76.f even 6 1 inner 76.2.f.a 16
152.l odd 6 1 1216.2.n.f 16
152.o even 6 1 1216.2.n.f 16
228.n odd 6 1 684.2.r.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.f.a 16 1.a even 1 1 trivial
76.2.f.a 16 4.b odd 2 1 inner
76.2.f.a 16 19.d odd 6 1 inner
76.2.f.a 16 76.f even 6 1 inner
684.2.r.a 16 3.b odd 2 1
684.2.r.a 16 12.b even 2 1
684.2.r.a 16 57.f even 6 1
684.2.r.a 16 228.n odd 6 1
1216.2.n.f 16 8.b even 2 1
1216.2.n.f 16 8.d odd 2 1
1216.2.n.f 16 152.l odd 6 1
1216.2.n.f 16 152.o even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(76, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 + 384 T + 384 T^{2} + 288 T^{3} + 192 T^{4} + 72 T^{5} + 12 T^{6} - 12 T^{7} - 10 T^{8} - 6 T^{9} + 3 T^{10} + 9 T^{11} + 12 T^{12} + 9 T^{13} + 6 T^{14} + 3 T^{15} + T^{16} \)
$3$ \( 361 + 1330 T^{2} + 3836 T^{4} + 3388 T^{6} + 2137 T^{8} + 644 T^{10} + 140 T^{12} + 14 T^{14} + T^{16} \)
$5$ \( ( 4 + 20 T + 80 T^{2} + 104 T^{3} + 112 T^{4} + 10 T^{5} + 11 T^{6} + T^{7} + T^{8} )^{2} \)
$7$ \( ( 304 + 396 T^{2} + 164 T^{4} + 24 T^{6} + T^{8} )^{2} \)
$11$ \( ( 3724 + 2379 T^{2} + 527 T^{4} + 45 T^{6} + T^{8} )^{2} \)
$13$ \( ( 3136 - 3360 T + 416 T^{2} + 840 T^{3} - 40 T^{4} - 126 T^{5} + 13 T^{6} + 9 T^{7} + T^{8} )^{2} \)
$17$ \( ( 784 + 224 T + 568 T^{2} - 88 T^{3} + 304 T^{4} + 2 T^{5} + 19 T^{6} - T^{7} + T^{8} )^{2} \)
$19$ \( 16983563041 + 1740697597 T^{2} + 145047273 T^{4} + 9041606 T^{6} + 577562 T^{8} + 25046 T^{10} + 1113 T^{12} + 37 T^{14} + T^{16} \)
$23$ \( 757115775376 - 144579803840 T^{2} + 20237455072 T^{4} - 1127527592 T^{6} + 44152900 T^{8} - 1031672 T^{10} + 17449 T^{12} - 161 T^{14} + T^{16} \)
$29$ \( ( 98596 - 175212 T + 76156 T^{2} + 49104 T^{3} + 6872 T^{4} - 264 T^{5} - 85 T^{6} + 3 T^{7} + T^{8} )^{2} \)
$31$ \( ( 7600 - 11396 T^{2} + 4028 T^{4} - 124 T^{6} + T^{8} )^{2} \)
$37$ \( ( 784 + 4364 T^{2} + 2480 T^{4} + 124 T^{6} + T^{8} )^{2} \)
$41$ \( ( 841 - 4524 T + 9098 T^{2} - 5304 T^{3} - 121 T^{4} + 816 T^{5} + 226 T^{6} + 24 T^{7} + T^{8} )^{2} \)
$43$ \( 1478656 - 51889152 T^{2} + 1816439296 T^{4} - 156265536 T^{6} + 8972448 T^{8} - 299796 T^{10} + 7357 T^{12} - 105 T^{14} + T^{16} \)
$47$ \( 37896336 - 30583008 T^{2} + 18254160 T^{4} - 4337064 T^{6} + 740988 T^{8} - 62100 T^{10} + 3717 T^{12} - 69 T^{14} + T^{16} \)
$53$ \( ( 802816 + 86016 T - 54272 T^{2} - 6144 T^{3} + 3104 T^{4} + 192 T^{5} - 61 T^{6} - 3 T^{7} + T^{8} )^{2} \)
$59$ \( 1300683225625 + 372289816150 T^{2} + 87663766556 T^{4} + 4801625812 T^{6} + 186526705 T^{8} + 3754220 T^{10} + 54188 T^{12} + 266 T^{14} + T^{16} \)
$61$ \( ( 209764 + 154804 T + 100504 T^{2} + 22048 T^{3} + 5752 T^{4} + 286 T^{5} + 199 T^{6} + 13 T^{7} + T^{8} )^{2} \)
$67$ \( 361 + 162374 T^{2} + 72986084 T^{4} + 21600260 T^{6} + 5484889 T^{8} + 250876 T^{10} + 8708 T^{12} + 106 T^{14} + T^{16} \)
$71$ \( 3550253056 + 1282247680 T^{2} + 330357248 T^{4} + 38055616 T^{6} + 3118240 T^{8} + 141884 T^{10} + 4661 T^{12} + 83 T^{14} + T^{16} \)
$73$ \( ( 1190281 - 1069180 T + 844754 T^{2} - 121336 T^{3} + 20167 T^{4} - 1112 T^{5} + 170 T^{6} - 8 T^{7} + T^{8} )^{2} \)
$79$ \( 136386521399296 + 10254625669120 T^{2} + 495973302272 T^{4} + 14537668096 T^{6} + 312083200 T^{8} + 4438016 T^{10} + 45617 T^{12} + 263 T^{14} + T^{16} \)
$83$ \( ( 255664 + 127279 T^{2} + 9351 T^{4} + 181 T^{6} + T^{8} )^{2} \)
$89$ \( ( 250000 + 186000 T + 14128 T^{2} - 23808 T^{3} + 2480 T^{4} + 576 T^{5} - 37 T^{6} - 9 T^{7} + T^{8} )^{2} \)
$97$ \( ( 24649 + 78186 T + 98368 T^{2} + 49800 T^{3} + 9161 T^{4} - 600 T^{5} - 88 T^{6} + 6 T^{7} + T^{8} )^{2} \)
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