Properties

Label 315.2.i.c
Level 315
Weight 2
Character orbit 315.i
Analytic conductor 2.515
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.142635249.1
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} + 3 x^{6} + 3 x^{5} - 11 x^{4} + 6 x^{3} + 12 x^{2} - 24 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + \nu^{6} - 5 \nu^{5} + 3 \nu^{4} + 5 \nu^{3} - 10 \nu^{2} + 8 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - \nu^{6} - 3 \nu^{5} + \nu^{4} + 3 \nu^{3} - 8 \nu^{2} + 8 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 3 \nu^{6} + \nu^{5} - 7 \nu^{4} + 7 \nu^{3} + 6 \nu^{2} - 16 \nu + 8 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} + 3 \nu^{5} + \nu^{4} - 7 \nu^{3} + 7 \nu^{2} + 6 \nu - 16 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} + \nu^{5} - 3 \nu^{4} + 3 \nu^{3} + 6 \nu^{2} - 6 \nu + 4 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{7} - 7 \nu^{6} + 3 \nu^{5} + 15 \nu^{4} - 19 \nu^{3} - 4 \nu^{2} + 40 \nu - 32 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - 2\)
\(\nu^{4}\)\(=\)\(\beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{3} + \beta_{2} - 2 \beta_{1}\)
\(\nu^{5}\)\(=\)\(2 \beta_{7} - \beta_{6} - 3 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} - \beta_{1} - 1\)
\(\nu^{6}\)\(=\)\(-2 \beta_{6} + 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} + \beta_{1} + 2\)
\(\nu^{7}\)\(=\)\(2 \beta_{7} - \beta_{6} + 3 \beta_{5} + 8 \beta_{4} + 5 \beta_{3} + 3 \beta_{2} + 5\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
106.1
0.818235 + 1.15347i
0.620769 1.27069i
−1.32841 0.485097i
1.38941 0.263711i
0.818235 1.15347i
0.620769 + 1.27069i
−1.32841 + 0.485097i
1.38941 + 0.263711i
−0.875400 + 1.51624i −1.16098 1.28535i −0.532651 0.922579i −0.500000 0.866025i 2.96522 0.635135i −0.500000 + 0.866025i −1.63647 −0.304233 + 2.98453i 1.75080
106.2 −0.159229 + 0.275793i −1.72929 + 0.0977414i 0.949292 + 1.64422i −0.500000 0.866025i 0.248398 0.492490i −0.500000 + 0.866025i −1.24154 2.98089 0.338047i 0.318459
106.3 0.392631 0.680056i 1.02936 + 1.39299i 0.691682 + 1.19803i −0.500000 0.866025i 1.35147 0.153094i −0.500000 + 0.866025i 2.65683 −0.880830 + 2.86778i −0.785261
106.4 1.14200 1.97800i 1.36091 1.07141i −1.60832 2.78570i −0.500000 0.866025i −0.565083 3.91543i −0.500000 + 0.866025i −2.77882 0.704170 2.91619i −2.28400
211.1 −0.875400 1.51624i −1.16098 + 1.28535i −0.532651 + 0.922579i −0.500000 + 0.866025i 2.96522 + 0.635135i −0.500000 0.866025i −1.63647 −0.304233 2.98453i 1.75080
211.2 −0.159229 0.275793i −1.72929 0.0977414i 0.949292 1.64422i −0.500000 + 0.866025i 0.248398 + 0.492490i −0.500000 0.866025i −1.24154 2.98089 + 0.338047i 0.318459
211.3 0.392631 + 0.680056i 1.02936 1.39299i 0.691682 1.19803i −0.500000 + 0.866025i 1.35147 + 0.153094i −0.500000 0.866025i 2.65683 −0.880830 2.86778i −0.785261
211.4 1.14200 + 1.97800i 1.36091 + 1.07141i −1.60832 + 2.78570i −0.500000 + 0.866025i −0.565083 + 3.91543i −0.500000 0.866025i −2.77882 0.704170 + 2.91619i −2.28400
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.i.c 8
3.b odd 2 1 945.2.i.d 8
9.c even 3 1 inner 315.2.i.c 8
9.c even 3 1 2835.2.a.m 4
9.d odd 6 1 945.2.i.d 8
9.d odd 6 1 2835.2.a.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.i.c 8 1.a even 1 1 trivial
315.2.i.c 8 9.c even 3 1 inner
945.2.i.d 8 3.b odd 2 1
945.2.i.d 8 9.d odd 6 1
2835.2.a.m 4 9.c even 3 1
2835.2.a.p 4 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - T_{2}^{7} + 5 T_{2}^{6} + 17 T_{2}^{4} - 6 T_{2}^{3} + 8 T_{2}^{2} + 2 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T - 3 T^{2} + 4 T^{3} + 3 T^{4} - 6 T^{5} + 4 T^{6} + 4 T^{7} - 15 T^{8} + 8 T^{9} + 16 T^{10} - 48 T^{11} + 48 T^{12} + 128 T^{13} - 192 T^{14} - 128 T^{15} + 256 T^{16} \)
$3$ \( 1 + T - 2 T^{2} + 3 T^{3} + 15 T^{4} + 9 T^{5} - 18 T^{6} + 27 T^{7} + 81 T^{8} \)
$5$ \( ( 1 + T + T^{2} )^{4} \)
$7$ \( ( 1 + T + T^{2} )^{4} \)
$11$ \( 1 - T - 9 T^{2} + 80 T^{3} - 100 T^{4} - 635 T^{5} + 2400 T^{6} + 384 T^{7} - 22329 T^{8} + 4224 T^{9} + 290400 T^{10} - 845185 T^{11} - 1464100 T^{12} + 12884080 T^{13} - 15944049 T^{14} - 19487171 T^{15} + 214358881 T^{16} \)
$13$ \( 1 - 12 T + 46 T^{2} - 114 T^{3} + 995 T^{4} - 5253 T^{5} + 13023 T^{6} - 59967 T^{7} + 319775 T^{8} - 779571 T^{9} + 2200887 T^{10} - 11540841 T^{11} + 28418195 T^{12} - 42327402 T^{13} + 222033214 T^{14} - 752982204 T^{15} + 815730721 T^{16} \)
$17$ \( ( 1 - 5 T + 14 T^{2} + 61 T^{3} - 359 T^{4} + 1037 T^{5} + 4046 T^{6} - 24565 T^{7} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 + 9 T + 75 T^{2} + 394 T^{3} + 1947 T^{4} + 7486 T^{5} + 27075 T^{6} + 61731 T^{7} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} )^{4} \)
$29$ \( 1 - 15 T + 39 T^{2} + 24 T^{3} + 4348 T^{4} - 28215 T^{5} - 20142 T^{6} - 397710 T^{7} + 7357161 T^{8} - 11533590 T^{9} - 16939422 T^{10} - 688135635 T^{11} + 3075257788 T^{12} + 492267576 T^{13} + 23198109519 T^{14} - 258748144635 T^{15} + 500246412961 T^{16} \)
$31$ \( 1 - 16 T + 64 T^{2} - 106 T^{3} + 3435 T^{4} - 20277 T^{5} - 29847 T^{6} - 161799 T^{7} + 4834609 T^{8} - 5015769 T^{9} - 28682967 T^{10} - 604072107 T^{11} + 3172294635 T^{12} - 3034690006 T^{13} + 56800235584 T^{14} - 440201825776 T^{15} + 852891037441 T^{16} \)
$37$ \( ( 1 + 15 T + 213 T^{2} + 1714 T^{3} + 12885 T^{4} + 63418 T^{5} + 291597 T^{6} + 759795 T^{7} + 1874161 T^{8} )^{2} \)
$41$ \( 1 - 2 T - 44 T^{2} - 426 T^{3} + 1985 T^{4} + 17281 T^{5} + 170979 T^{6} - 841573 T^{7} - 6842441 T^{8} - 34504493 T^{9} + 287415699 T^{10} + 1191023801 T^{11} + 5609135585 T^{12} - 49354741626 T^{13} - 209004586604 T^{14} - 389508547762 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 - 7 T + 31 T^{2} - 88 T^{3} - 1116 T^{4} + 17139 T^{5} + 37680 T^{6} - 784944 T^{7} + 5466535 T^{8} - 33752592 T^{9} + 69670320 T^{10} + 1362670473 T^{11} - 3815381916 T^{12} - 12936742984 T^{13} + 195962254519 T^{14} - 1902730277749 T^{15} + 11688200277601 T^{16} \)
$47$ \( 1 - 10 T - 86 T^{2} + 618 T^{3} + 9423 T^{4} - 33027 T^{5} - 603099 T^{6} + 677569 T^{7} + 30716757 T^{8} + 31845743 T^{9} - 1332245691 T^{10} - 3428962221 T^{11} + 45981234063 T^{12} + 141735214326 T^{13} - 927012518294 T^{14} - 5066231204630 T^{15} + 23811286661761 T^{16} \)
$53$ \( ( 1 + 112 T^{2} - 119 T^{3} + 7405 T^{4} - 6307 T^{5} + 314608 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( 1 - 13 T - 39 T^{2} + 934 T^{3} + 1620 T^{4} - 17751 T^{5} - 381692 T^{6} - 173558 T^{7} + 35373867 T^{8} - 10239922 T^{9} - 1328669852 T^{10} - 3645682629 T^{11} + 19630124820 T^{12} + 667739295266 T^{13} - 1645040811999 T^{14} - 32352469302647 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 - 32 T + 429 T^{2} - 4324 T^{3} + 51554 T^{4} - 537888 T^{5} + 4331413 T^{6} - 37090382 T^{7} + 322144443 T^{8} - 2262513302 T^{9} + 16117187773 T^{10} - 122090356128 T^{11} + 713808486914 T^{12} - 3652034405524 T^{13} + 22102240600869 T^{14} - 100567770752672 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 - 209 T^{2} + 72 T^{3} + 24028 T^{4} - 9936 T^{5} - 2229779 T^{6} + 300708 T^{7} + 169918447 T^{8} + 20147436 T^{9} - 10009477931 T^{10} - 2988381168 T^{11} + 484191135388 T^{12} + 97209007704 T^{13} - 18905801873321 T^{14} + 406067677556641 T^{16} \)
$71$ \( ( 1 - 13 T + 268 T^{2} - 2163 T^{3} + 26345 T^{4} - 153573 T^{5} + 1350988 T^{6} - 4652843 T^{7} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 + 2 T + 68 T^{2} + 507 T^{3} + 5887 T^{4} + 37011 T^{5} + 362372 T^{6} + 778034 T^{7} + 28398241 T^{8} )^{2} \)
$79$ \( 1 - 177 T^{2} - 8 T^{3} + 11052 T^{4} + 1024 T^{5} - 1379699 T^{6} - 30756 T^{7} + 176681095 T^{8} - 2429724 T^{9} - 8610701459 T^{10} + 504871936 T^{11} + 430476295212 T^{12} - 24616451192 T^{13} - 43026479627217 T^{14} + 1517108809906561 T^{16} \)
$83$ \( 1 + 17 T - 38 T^{2} - 1593 T^{3} + 7218 T^{4} + 142914 T^{5} - 479067 T^{6} - 6921758 T^{7} - 296163 T^{8} - 574505914 T^{9} - 3300292563 T^{10} + 81716367318 T^{11} + 342554160978 T^{12} - 6274891744299 T^{13} - 12423734188022 T^{14} + 461312866823659 T^{15} + 2252292232139041 T^{16} \)
$89$ \( ( 1 + 17 T + 309 T^{2} + 2514 T^{3} + 30601 T^{4} + 223746 T^{5} + 2447589 T^{6} + 11984473 T^{7} + 62742241 T^{8} )^{2} \)
$97$ \( 1 - 4 T - 167 T^{2} + 888 T^{3} + 5042 T^{4} - 25144 T^{5} - 827367 T^{6} - 2199130 T^{7} + 197329403 T^{8} - 213315610 T^{9} - 7784696103 T^{10} - 22948249912 T^{11} + 446364634802 T^{12} + 7625558148216 T^{13} - 139106324823143 T^{14} - 323193137912452 T^{15} + 7837433594376961 T^{16} \)
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