Properties

Label 150.2.g.c
Level 150
Weight 2
Character orbit 150.g
Analytic conductor 1.198
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 150.g (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.19775603032\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.1064390625.3
Defining polynomial: \(x^{8} - 2 x^{7} - 3 x^{6} - 5 x^{5} + 36 x^{4} - 35 x^{3} + 23 x^{2} - 171 x + 361\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 3 x^{6} - 5 x^{5} + 36 x^{4} - 35 x^{3} + 23 x^{2} - 171 x + 361\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -27571 \nu^{7} + 156 \nu^{6} - 50800 \nu^{5} + 197116 \nu^{4} - 261151 \nu^{3} + 1281772 \nu^{2} - 1105295 \nu + 4276691 \)\()/4238558\)
\(\beta_{3}\)\(=\)\((\)\( -45697 \nu^{7} - 18958 \nu^{6} + 87368 \nu^{5} + 389890 \nu^{4} - 351515 \nu^{3} + 1275844 \nu^{2} - 687257 \nu + 5880747 \)\()/4238558\)
\(\beta_{4}\)\(=\)\((\)\( 2894 \nu^{7} + 7027 \nu^{6} - 3119 \nu^{5} - 38495 \nu^{4} - 16673 \nu^{3} + 24798 \nu^{2} + 23050 \nu - 523849 \)\()/223082\)
\(\beta_{5}\)\(=\)\((\)\( 60697 \nu^{7} + 29865 \nu^{6} - 107003 \nu^{5} - 736723 \nu^{4} + 989612 \nu^{3} + 38090 \nu^{2} + 939005 \nu - 13281190 \)\()/4238558\)
\(\beta_{6}\)\(=\)\((\)\( -72436 \nu^{7} - 26109 \nu^{6} + 188067 \nu^{5} + 265983 \nu^{4} - 1082509 \nu^{3} + 501044 \nu^{2} + 3422970 \nu + 7026371 \)\()/4238558\)
\(\beta_{7}\)\(=\)\((\)\( 5808 \nu^{7} + 2617 \nu^{6} - 8495 \nu^{5} - 68083 \nu^{4} + 17029 \nu^{3} - 19146 \nu^{2} + 101760 \nu - 868243 \)\()/223082\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} - \beta_{6} + 4 \beta_{3} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(-3 \beta_{7} + 5 \beta_{5} - \beta_{2} + 5\)
\(\nu^{4}\)\(=\)\(-8 \beta_{6} - 7 \beta_{5} + \beta_{4} + 7 \beta_{3} - 4 \beta_{2} + 8 \beta_{1} - 12\)
\(\nu^{5}\)\(=\)\(2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} + 39 \beta_{3} - 39 \beta_{2} - 4 \beta_{1} + 12\)
\(\nu^{6}\)\(=\)\(-39 \beta_{7} + 16 \beta_{5} + 39 \beta_{4} - 26 \beta_{3} + 6 \beta_{1} + 26\)
\(\nu^{7}\)\(=\)\(71 \beta_{7} - 100 \beta_{6} - 101 \beta_{5} + 164 \beta_{3} - 101 \beta_{2} + 71 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(\beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−0.815575 1.64827i
1.31557 + 1.28500i
−1.86886 1.45788i
2.36886 0.0809628i
−1.86886 + 1.45788i
2.36886 + 0.0809628i
−0.815575 + 1.64827i
1.31557 1.28500i
−0.309017 0.951057i −0.809017 + 0.587785i −0.809017 + 0.587785i −2.12459 0.697217i 0.809017 + 0.587785i −4.25729 0.809017 + 0.587785i 0.309017 0.951057i −0.00655751 + 2.23606i
31.2 −0.309017 0.951057i −0.809017 + 0.587785i −0.809017 + 0.587785i 0.00655751 + 2.23606i 0.809017 + 0.587785i 2.63925 0.809017 + 0.587785i 0.309017 0.951057i 2.12459 0.697217i
61.1 0.809017 0.587785i 0.309017 0.951057i 0.309017 0.951057i −2.05984 0.870094i −0.309017 0.951057i 2.92807 −0.309017 0.951057i −0.809017 0.587785i −2.17787 + 0.506822i
61.2 0.809017 0.587785i 0.309017 0.951057i 0.309017 0.951057i 2.17787 + 0.506822i −0.309017 0.951057i −2.31003 −0.309017 0.951057i −0.809017 0.587785i 2.05984 0.870094i
91.1 0.809017 + 0.587785i 0.309017 + 0.951057i 0.309017 + 0.951057i −2.05984 + 0.870094i −0.309017 + 0.951057i 2.92807 −0.309017 + 0.951057i −0.809017 + 0.587785i −2.17787 0.506822i
91.2 0.809017 + 0.587785i 0.309017 + 0.951057i 0.309017 + 0.951057i 2.17787 0.506822i −0.309017 + 0.951057i −2.31003 −0.309017 + 0.951057i −0.809017 + 0.587785i 2.05984 + 0.870094i
121.1 −0.309017 + 0.951057i −0.809017 0.587785i −0.809017 0.587785i −2.12459 + 0.697217i 0.809017 0.587785i −4.25729 0.809017 0.587785i 0.309017 + 0.951057i −0.00655751 2.23606i
121.2 −0.309017 + 0.951057i −0.809017 0.587785i −0.809017 0.587785i 0.00655751 2.23606i 0.809017 0.587785i 2.63925 0.809017 0.587785i 0.309017 + 0.951057i 2.12459 + 0.697217i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.2.g.c 8
3.b odd 2 1 450.2.h.d 8
5.b even 2 1 750.2.g.d 8
5.c odd 4 2 750.2.h.e 16
25.d even 5 1 inner 150.2.g.c 8
25.d even 5 1 3750.2.a.l 4
25.e even 10 1 750.2.g.d 8
25.e even 10 1 3750.2.a.q 4
25.f odd 20 2 750.2.h.e 16
25.f odd 20 2 3750.2.c.h 8
75.j odd 10 1 450.2.h.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.g.c 8 1.a even 1 1 trivial
150.2.g.c 8 25.d even 5 1 inner
450.2.h.d 8 3.b odd 2 1
450.2.h.d 8 75.j odd 10 1
750.2.g.d 8 5.b even 2 1
750.2.g.d 8 25.e even 10 1
750.2.h.e 16 5.c odd 4 2
750.2.h.e 16 25.f odd 20 2
3750.2.a.l 4 25.d even 5 1
3750.2.a.q 4 25.e even 10 1
3750.2.c.h 8 25.f odd 20 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + T_{7}^{3} - 19 T_{7}^{2} - 4 T_{7} + 76 \) acting on \(S_{2}^{\mathrm{new}}(150, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$3$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$5$ \( 1 + 4 T + T^{2} - 16 T^{3} - 39 T^{4} - 80 T^{5} + 25 T^{6} + 500 T^{7} + 625 T^{8} \)
$7$ \( ( 1 + T + 9 T^{2} + 17 T^{3} + 104 T^{4} + 119 T^{5} + 441 T^{6} + 343 T^{7} + 2401 T^{8} )^{2} \)
$11$ \( 1 + 5 T + 38 T^{2} + 210 T^{3} + 1048 T^{4} + 4455 T^{5} + 19301 T^{6} + 69050 T^{7} + 243280 T^{8} + 759550 T^{9} + 2335421 T^{10} + 5929605 T^{11} + 15343768 T^{12} + 33820710 T^{13} + 67319318 T^{14} + 97435855 T^{15} + 214358881 T^{16} \)
$13$ \( 1 - 6 T + 6 T^{2} + 26 T^{3} + 3 T^{4} - 1366 T^{5} + 5284 T^{6} - 2112 T^{7} - 15227 T^{8} - 27456 T^{9} + 892996 T^{10} - 3001102 T^{11} + 85683 T^{12} + 9653618 T^{13} + 28960854 T^{14} - 376491102 T^{15} + 815730721 T^{16} \)
$17$ \( 1 + 2 T + 44 T^{2} + 12 T^{3} + 1273 T^{4} - 258 T^{5} + 28916 T^{6} - 13984 T^{7} + 544693 T^{8} - 237728 T^{9} + 8356724 T^{10} - 1267554 T^{11} + 106322233 T^{12} + 17038284 T^{13} + 1062053036 T^{14} + 820677346 T^{15} + 6975757441 T^{16} \)
$19$ \( ( 1 - 4 T - 3 T^{2} - 62 T^{3} + 605 T^{4} - 1178 T^{5} - 1083 T^{6} - 27436 T^{7} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 + 10 T + 37 T^{2} + 200 T^{3} + 1389 T^{4} + 4600 T^{5} + 19573 T^{6} + 121670 T^{7} + 279841 T^{8} )^{2} \)
$29$ \( 1 + 18 T + 160 T^{2} + 1050 T^{3} + 5925 T^{4} + 26514 T^{5} + 92732 T^{6} + 300180 T^{7} + 1280525 T^{8} + 8705220 T^{9} + 77987612 T^{10} + 646649946 T^{11} + 4190639925 T^{12} + 21536706450 T^{13} + 95171731360 T^{14} + 310497773562 T^{15} + 500246412961 T^{16} \)
$31$ \( 1 - 9 T - 20 T^{2} + 300 T^{3} + 690 T^{4} - 8217 T^{5} - 2797 T^{6} + 139770 T^{7} - 752500 T^{8} + 4332870 T^{9} - 2687917 T^{10} - 244792647 T^{11} + 637229490 T^{12} + 8588745300 T^{13} - 17750073620 T^{14} - 247613526999 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 - 21 T + 258 T^{2} - 2462 T^{3} + 19746 T^{4} - 143741 T^{5} + 980785 T^{6} - 6308436 T^{7} + 38864404 T^{8} - 233412132 T^{9} + 1342694665 T^{10} - 7280912873 T^{11} + 37007183106 T^{12} - 170724822134 T^{13} + 661957413522 T^{14} - 1993569419793 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 - 2 T + 36 T^{2} - 168 T^{3} + 2201 T^{4} - 17286 T^{5} + 84044 T^{6} - 959864 T^{7} + 2420157 T^{8} - 39354424 T^{9} + 141277964 T^{10} - 1191368406 T^{11} + 6219499961 T^{12} - 19463841768 T^{13} + 171003752676 T^{14} - 389508547762 T^{15} + 7984925229121 T^{16} \)
$43$ \( ( 1 + 16 T + 168 T^{2} + 1280 T^{3} + 9326 T^{4} + 55040 T^{5} + 310632 T^{6} + 1272112 T^{7} + 3418801 T^{8} )^{2} \)
$47$ \( 1 + 10 T + 66 T^{2} + 210 T^{3} + 1787 T^{4} - 6510 T^{5} - 94072 T^{6} - 974600 T^{7} - 2764935 T^{8} - 45806200 T^{9} - 207805048 T^{10} - 675887730 T^{11} + 8719989947 T^{12} + 48162451470 T^{13} + 711428211714 T^{14} + 5066231204630 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 - 7 T + 62 T^{2} - 816 T^{3} + 3586 T^{4} + 23943 T^{5} - 77365 T^{6} + 2388872 T^{7} - 30975296 T^{8} + 126610216 T^{9} - 217318285 T^{10} + 3564562011 T^{11} + 28295264866 T^{12} - 341247522288 T^{13} + 1374190389998 T^{14} - 8222977978859 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 + 25 T + 212 T^{2} + 150 T^{3} - 12692 T^{4} - 128475 T^{5} - 252121 T^{6} + 7858000 T^{7} + 98067400 T^{8} + 463622000 T^{9} - 877633201 T^{10} - 26386067025 T^{11} - 153793545812 T^{12} + 107238644850 T^{13} + 8942273131892 T^{14} + 62216287120475 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 - 10 T - 62 T^{2} + 700 T^{3} + 5543 T^{4} - 26410 T^{5} - 544444 T^{6} + 851800 T^{7} + 32679625 T^{8} + 51959800 T^{9} - 2025876124 T^{10} - 5994568210 T^{11} + 76747496663 T^{12} + 591217410700 T^{13} - 3194263210382 T^{14} - 31427428360210 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 + 2 T + 34 T^{2} + 232 T^{3} - 637 T^{4} - 10798 T^{5} + 66956 T^{6} + 2488096 T^{7} - 7257587 T^{8} + 166702432 T^{9} + 300565484 T^{10} - 3247638874 T^{11} - 12836264077 T^{12} + 313229024824 T^{13} + 3075584993746 T^{14} + 12121423210646 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 - 182 T^{2} - 750 T^{3} + 7983 T^{4} + 134400 T^{5} + 952256 T^{6} - 5830800 T^{7} - 134438695 T^{8} - 413986800 T^{9} + 4800322496 T^{10} + 48103238400 T^{11} + 202861449423 T^{12} - 1353172013250 T^{13} - 23314251673622 T^{14} + 645753531245761 T^{16} \)
$73$ \( 1 + 24 T + 226 T^{2} + 1116 T^{3} + 5643 T^{4} + 16584 T^{5} - 622036 T^{6} - 12920832 T^{7} - 137738827 T^{8} - 943220736 T^{9} - 3314829844 T^{10} + 6451457928 T^{11} + 160251273963 T^{12} + 2313547897788 T^{13} + 34201535141314 T^{14} + 265137564458328 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 + 6 T + 34 T^{2} + 336 T^{3} + 4761 T^{4} - 34122 T^{5} + 60506 T^{6} - 838512 T^{7} + 1917227 T^{8} - 66242448 T^{9} + 377617946 T^{10} - 16823476758 T^{11} + 185441335641 T^{12} + 1033890950064 T^{13} + 8264973487714 T^{14} + 115223453916954 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 - 11 T - 4 T^{2} - 264 T^{3} + 5188 T^{4} + 21759 T^{5} + 513329 T^{6} - 7685372 T^{7} + 27478708 T^{8} - 637885876 T^{9} + 3536323481 T^{10} + 12441513333 T^{11} + 246213769348 T^{12} - 1039906729752 T^{13} - 1307761493476 T^{14} - 298496560885897 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 - 9 T - 166 T^{2} + 1356 T^{3} + 12786 T^{4} - 119727 T^{5} + 426031 T^{6} + 2824668 T^{7} - 98110168 T^{8} + 251395452 T^{9} + 3374591551 T^{10} - 84403823463 T^{11} + 802222293426 T^{12} + 7571984612844 T^{13} - 82498894299526 T^{14} - 398082014059761 T^{15} + 3936588805702081 T^{16} \)
$97$ \( 1 - T + 28 T^{2} - 632 T^{3} + 5996 T^{4} - 78691 T^{5} + 707075 T^{6} - 3586376 T^{7} + 72919024 T^{8} - 347878472 T^{9} + 6652868675 T^{10} - 71819151043 T^{11} + 530821568876 T^{12} - 5427199042424 T^{13} + 23323216138012 T^{14} - 80798284478113 T^{15} + 7837433594376961 T^{16} \)
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