Properties

Label 33.3.h.a
Level $33$
Weight $3$
Character orbit 33.h
Analytic conductor $0.899$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 33.h (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.899184872389\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \(x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{20} q^{2} + ( 2 + 2 \zeta_{20} - 2 \zeta_{20}^{2} - \zeta_{20}^{3} + 2 \zeta_{20}^{4} + 2 \zeta_{20}^{5} - 2 \zeta_{20}^{6} ) q^{3} -3 \zeta_{20}^{2} q^{4} + ( -5 \zeta_{20} + 6 \zeta_{20}^{3} - 6 \zeta_{20}^{5} + 5 \zeta_{20}^{7} ) q^{5} + ( 2 \zeta_{20} + 2 \zeta_{20}^{2} - 2 \zeta_{20}^{3} - \zeta_{20}^{4} + 2 \zeta_{20}^{5} + 2 \zeta_{20}^{6} - 2 \zeta_{20}^{7} ) q^{6} + ( -3 + 7 \zeta_{20}^{2} - 3 \zeta_{20}^{4} ) q^{7} -7 \zeta_{20}^{3} q^{8} + ( 4 \zeta_{20} + 8 \zeta_{20}^{5} + \zeta_{20}^{6} - 8 \zeta_{20}^{7} ) q^{9} +O(q^{10})\) \( q + \zeta_{20} q^{2} + ( 2 + 2 \zeta_{20} - 2 \zeta_{20}^{2} - \zeta_{20}^{3} + 2 \zeta_{20}^{4} + 2 \zeta_{20}^{5} - 2 \zeta_{20}^{6} ) q^{3} -3 \zeta_{20}^{2} q^{4} + ( -5 \zeta_{20} + 6 \zeta_{20}^{3} - 6 \zeta_{20}^{5} + 5 \zeta_{20}^{7} ) q^{5} + ( 2 \zeta_{20} + 2 \zeta_{20}^{2} - 2 \zeta_{20}^{3} - \zeta_{20}^{4} + 2 \zeta_{20}^{5} + 2 \zeta_{20}^{6} - 2 \zeta_{20}^{7} ) q^{6} + ( -3 + 7 \zeta_{20}^{2} - 3 \zeta_{20}^{4} ) q^{7} -7 \zeta_{20}^{3} q^{8} + ( 4 \zeta_{20} + 8 \zeta_{20}^{5} + \zeta_{20}^{6} - 8 \zeta_{20}^{7} ) q^{9} + ( -5 + \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{10} + ( -12 \zeta_{20} + 6 \zeta_{20}^{3} - 9 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{11} + ( -6 - 6 \zeta_{20}^{3} + 3 \zeta_{20}^{5} - 6 \zeta_{20}^{7} ) q^{12} + ( 2 - 2 \zeta_{20}^{2} + 10 \zeta_{20}^{6} ) q^{13} + ( -3 \zeta_{20} + 7 \zeta_{20}^{3} - 3 \zeta_{20}^{5} ) q^{14} + ( -11 + 2 \zeta_{20} + 8 \zeta_{20}^{2} - 2 \zeta_{20}^{3} - 11 \zeta_{20}^{4} + 10 \zeta_{20}^{7} ) q^{15} + 5 \zeta_{20}^{4} q^{16} + ( 6 \zeta_{20} - 6 \zeta_{20}^{3} + 6 \zeta_{20}^{5} - 6 \zeta_{20}^{7} ) q^{17} + ( 8 - 4 \zeta_{20}^{2} + 8 \zeta_{20}^{4} + \zeta_{20}^{7} ) q^{18} + ( 20 - 2 \zeta_{20}^{2} + 2 \zeta_{20}^{4} - 20 \zeta_{20}^{6} ) q^{19} + ( 15 \zeta_{20} - 3 \zeta_{20}^{5} + 3 \zeta_{20}^{7} ) q^{20} + ( 8 + 11 \zeta_{20}^{3} - 6 \zeta_{20}^{4} - 13 \zeta_{20}^{5} + 6 \zeta_{20}^{6} + 11 \zeta_{20}^{7} ) q^{21} + ( -2 - 10 \zeta_{20}^{2} + 4 \zeta_{20}^{4} - 7 \zeta_{20}^{6} ) q^{22} + ( 4 \zeta_{20}^{3} + 20 \zeta_{20}^{5} + 4 \zeta_{20}^{7} ) q^{23} + ( 14 - 14 \zeta_{20} - 14 \zeta_{20}^{2} - 7 \zeta_{20}^{6} ) q^{24} + ( 1 - 12 \zeta_{20}^{2} + 12 \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{25} + ( 2 \zeta_{20} - 2 \zeta_{20}^{3} + 10 \zeta_{20}^{7} ) q^{26} + ( 7 \zeta_{20} + 7 \zeta_{20}^{3} + 22 \zeta_{20}^{4} - 7 \zeta_{20}^{5} - 7 \zeta_{20}^{7} ) q^{27} + ( 9 \zeta_{20}^{2} - 21 \zeta_{20}^{4} + 9 \zeta_{20}^{6} ) q^{28} + ( 33 \zeta_{20} - 33 \zeta_{20}^{3} - 27 \zeta_{20}^{7} ) q^{29} + ( -10 - 11 \zeta_{20} + 12 \zeta_{20}^{2} + 8 \zeta_{20}^{3} - 12 \zeta_{20}^{4} - 11 \zeta_{20}^{5} + 10 \zeta_{20}^{6} ) q^{30} + ( -11 + 11 \zeta_{20}^{2} + 24 \zeta_{20}^{6} ) q^{31} + 33 \zeta_{20}^{5} q^{32} + ( -5 - 12 \zeta_{20} - 3 \zeta_{20}^{2} + 6 \zeta_{20}^{3} - \zeta_{20}^{4} - 20 \zeta_{20}^{5} - 23 \zeta_{20}^{6} + 24 \zeta_{20}^{7} ) q^{33} + 6 q^{34} + ( -23 \zeta_{20} + 22 \zeta_{20}^{5} - 22 \zeta_{20}^{7} ) q^{35} + ( 3 - 24 \zeta_{20} - 3 \zeta_{20}^{2} + 12 \zeta_{20}^{3} + 3 \zeta_{20}^{4} - 24 \zeta_{20}^{5} - 3 \zeta_{20}^{6} ) q^{36} + ( -22 + 6 \zeta_{20}^{2} - 22 \zeta_{20}^{4} ) q^{37} + ( 20 \zeta_{20} - 2 \zeta_{20}^{3} + 2 \zeta_{20}^{5} - 20 \zeta_{20}^{7} ) q^{38} + ( -6 \zeta_{20} - 4 \zeta_{20}^{2} - 16 \zeta_{20}^{3} + 24 \zeta_{20}^{4} + 16 \zeta_{20}^{5} - 4 \zeta_{20}^{6} + 6 \zeta_{20}^{7} ) q^{39} + ( -7 + 42 \zeta_{20}^{2} - 7 \zeta_{20}^{4} ) q^{40} + ( -22 \zeta_{20} - 8 \zeta_{20}^{3} - 22 \zeta_{20}^{5} ) q^{41} + ( -11 + 8 \zeta_{20} + 11 \zeta_{20}^{2} - 6 \zeta_{20}^{5} - 2 \zeta_{20}^{6} + 6 \zeta_{20}^{7} ) q^{42} + ( -36 - 6 \zeta_{20}^{4} + 6 \zeta_{20}^{6} ) q^{43} + ( 6 \zeta_{20} + 30 \zeta_{20}^{3} - 12 \zeta_{20}^{5} + 21 \zeta_{20}^{7} ) q^{44} + ( -12 + \zeta_{20}^{3} - 44 \zeta_{20}^{4} - 6 \zeta_{20}^{5} + 44 \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{45} + ( -4 + 4 \zeta_{20}^{2} + 24 \zeta_{20}^{6} ) q^{46} + ( 6 \zeta_{20} - 14 \zeta_{20}^{3} + 6 \zeta_{20}^{5} ) q^{47} + ( -10 \zeta_{20} + 10 \zeta_{20}^{2} + 10 \zeta_{20}^{3} + 5 \zeta_{20}^{7} ) q^{48} + ( -33 \zeta_{20}^{2} + 9 \zeta_{20}^{4} - 33 \zeta_{20}^{6} ) q^{49} + ( \zeta_{20} - 12 \zeta_{20}^{3} + 12 \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{50} + ( 12 - 6 \zeta_{20}^{2} + 12 \zeta_{20}^{4} - 12 \zeta_{20}^{7} ) q^{51} + ( 30 - 36 \zeta_{20}^{2} + 36 \zeta_{20}^{4} - 30 \zeta_{20}^{6} ) q^{52} + ( 2 \zeta_{20} + 15 \zeta_{20}^{5} - 15 \zeta_{20}^{7} ) q^{53} + ( 7 + 14 \zeta_{20}^{4} + 22 \zeta_{20}^{5} - 14 \zeta_{20}^{6} ) q^{54} + ( 64 - 43 \zeta_{20}^{2} + 48 \zeta_{20}^{4} - 7 \zeta_{20}^{6} ) q^{55} + ( 21 \zeta_{20}^{3} - 49 \zeta_{20}^{5} + 21 \zeta_{20}^{7} ) q^{56} + ( 36 + 56 \zeta_{20} - 36 \zeta_{20}^{2} + 22 \zeta_{20}^{5} - 40 \zeta_{20}^{6} - 22 \zeta_{20}^{7} ) q^{57} + ( 27 + 6 \zeta_{20}^{2} - 6 \zeta_{20}^{4} - 27 \zeta_{20}^{6} ) q^{58} + ( -5 \zeta_{20} + 5 \zeta_{20}^{3} + 30 \zeta_{20}^{7} ) q^{59} + ( 30 \zeta_{20} + 33 \zeta_{20}^{2} - 36 \zeta_{20}^{3} - 24 \zeta_{20}^{4} + 36 \zeta_{20}^{5} + 33 \zeta_{20}^{6} - 30 \zeta_{20}^{7} ) q^{60} + ( 54 \zeta_{20}^{2} - 60 \zeta_{20}^{4} + 54 \zeta_{20}^{6} ) q^{61} + ( -11 \zeta_{20} + 11 \zeta_{20}^{3} + 24 \zeta_{20}^{7} ) q^{62} + ( -4 + 44 \zeta_{20} + 7 \zeta_{20}^{2} - 52 \zeta_{20}^{3} - 7 \zeta_{20}^{4} + 44 \zeta_{20}^{5} + 4 \zeta_{20}^{6} ) q^{63} + 13 \zeta_{20}^{6} q^{64} + ( 22 \zeta_{20}^{3} - 74 \zeta_{20}^{5} + 22 \zeta_{20}^{7} ) q^{65} + ( -24 - 5 \zeta_{20} + 12 \zeta_{20}^{2} - 3 \zeta_{20}^{3} - 18 \zeta_{20}^{4} - \zeta_{20}^{5} + 4 \zeta_{20}^{6} - 23 \zeta_{20}^{7} ) q^{66} + ( 30 + 66 \zeta_{20}^{4} - 66 \zeta_{20}^{6} ) q^{67} -18 \zeta_{20} q^{68} + ( -32 + 8 \zeta_{20} - 12 \zeta_{20}^{2} + 40 \zeta_{20}^{3} + 12 \zeta_{20}^{4} + 8 \zeta_{20}^{5} + 32 \zeta_{20}^{6} ) q^{69} + ( 22 - 45 \zeta_{20}^{2} + 22 \zeta_{20}^{4} ) q^{70} + ( -82 \zeta_{20} + 72 \zeta_{20}^{3} - 72 \zeta_{20}^{5} + 82 \zeta_{20}^{7} ) q^{71} + ( 7 \zeta_{20} - 56 \zeta_{20}^{2} - 7 \zeta_{20}^{3} + 28 \zeta_{20}^{4} + 7 \zeta_{20}^{5} - 56 \zeta_{20}^{6} - 7 \zeta_{20}^{7} ) q^{72} + ( -21 - 28 \zeta_{20}^{2} - 21 \zeta_{20}^{4} ) q^{73} + ( -22 \zeta_{20} + 6 \zeta_{20}^{3} - 22 \zeta_{20}^{5} ) q^{74} + ( -22 - 21 \zeta_{20} + 22 \zeta_{20}^{2} + 13 \zeta_{20}^{5} - 2 \zeta_{20}^{6} - 13 \zeta_{20}^{7} ) q^{75} + ( -60 - 54 \zeta_{20}^{4} + 54 \zeta_{20}^{6} ) q^{76} + ( \zeta_{20} - 61 \zeta_{20}^{3} + 64 \zeta_{20}^{5} - 46 \zeta_{20}^{7} ) q^{77} + ( -6 - 4 \zeta_{20}^{3} - 22 \zeta_{20}^{4} + 24 \zeta_{20}^{5} + 22 \zeta_{20}^{6} - 4 \zeta_{20}^{7} ) q^{78} + ( -81 + 81 \zeta_{20}^{2} + 13 \zeta_{20}^{6} ) q^{79} + ( 5 \zeta_{20} - 30 \zeta_{20}^{3} + 5 \zeta_{20}^{5} ) q^{80} + ( -16 \zeta_{20} + 79 \zeta_{20}^{2} + 16 \zeta_{20}^{3} + 8 \zeta_{20}^{7} ) q^{81} + ( -22 \zeta_{20}^{2} - 8 \zeta_{20}^{4} - 22 \zeta_{20}^{6} ) q^{82} + ( 49 \zeta_{20} + 50 \zeta_{20}^{3} - 50 \zeta_{20}^{5} - 49 \zeta_{20}^{7} ) q^{83} + ( 18 + 33 \zeta_{20} - 42 \zeta_{20}^{2} - 33 \zeta_{20}^{3} + 18 \zeta_{20}^{4} + 6 \zeta_{20}^{7} ) q^{84} + ( -30 + 36 \zeta_{20}^{2} - 36 \zeta_{20}^{4} + 30 \zeta_{20}^{6} ) q^{85} + ( -36 \zeta_{20} - 6 \zeta_{20}^{5} + 6 \zeta_{20}^{7} ) q^{86} + ( 93 - 66 \zeta_{20}^{3} + 21 \zeta_{20}^{4} + 12 \zeta_{20}^{5} - 21 \zeta_{20}^{6} - 66 \zeta_{20}^{7} ) q^{87} + ( -49 + 63 \zeta_{20}^{2} + 21 \zeta_{20}^{4} + 21 \zeta_{20}^{6} ) q^{88} + ( -48 \zeta_{20}^{3} + 68 \zeta_{20}^{5} - 48 \zeta_{20}^{7} ) q^{89} + ( -1 - 12 \zeta_{20} + \zeta_{20}^{2} - 44 \zeta_{20}^{5} - 5 \zeta_{20}^{6} + 44 \zeta_{20}^{7} ) q^{90} + ( -46 + 90 \zeta_{20}^{2} - 90 \zeta_{20}^{4} + 46 \zeta_{20}^{6} ) q^{91} + ( 12 \zeta_{20} - 12 \zeta_{20}^{3} - 72 \zeta_{20}^{7} ) q^{92} + ( -46 \zeta_{20} + 22 \zeta_{20}^{2} + 9 \zeta_{20}^{3} + 26 \zeta_{20}^{4} - 9 \zeta_{20}^{5} + 22 \zeta_{20}^{6} + 46 \zeta_{20}^{7} ) q^{93} + ( 6 \zeta_{20}^{2} - 14 \zeta_{20}^{4} + 6 \zeta_{20}^{6} ) q^{94} + ( -88 \zeta_{20} + 88 \zeta_{20}^{3} + 82 \zeta_{20}^{7} ) q^{95} + ( -33 - 33 \zeta_{20}^{2} + 66 \zeta_{20}^{3} + 33 \zeta_{20}^{4} + 33 \zeta_{20}^{6} ) q^{96} + ( 99 - 99 \zeta_{20}^{2} - 39 \zeta_{20}^{6} ) q^{97} + ( -33 \zeta_{20}^{3} + 9 \zeta_{20}^{5} - 33 \zeta_{20}^{7} ) q^{98} + ( -32 + 3 \zeta_{20} + 16 \zeta_{20}^{2} + 4 \zeta_{20}^{3} - 112 \zeta_{20}^{4} - 6 \zeta_{20}^{5} + 20 \zeta_{20}^{6} - 6 \zeta_{20}^{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{3} - 6q^{4} + 10q^{6} - 4q^{7} + 2q^{9} + O(q^{10}) \) \( 8q + 4q^{3} - 6q^{4} + 10q^{6} - 4q^{7} + 2q^{9} - 44q^{10} - 48q^{12} + 32q^{13} - 50q^{15} - 10q^{16} + 40q^{18} + 112q^{19} + 88q^{21} - 58q^{22} + 70q^{24} - 42q^{25} - 44q^{27} + 78q^{28} - 12q^{30} - 18q^{31} - 90q^{33} + 48q^{34} + 6q^{36} - 120q^{37} - 64q^{39} + 42q^{40} - 70q^{42} - 264q^{43} + 80q^{45} + 24q^{46} + 20q^{48} - 150q^{49} + 60q^{51} + 36q^{52} + 316q^{55} + 136q^{57} + 186q^{58} + 180q^{60} + 336q^{61} + 4q^{63} + 26q^{64} - 124q^{66} - 24q^{67} - 240q^{69} + 42q^{70} - 280q^{72} - 182q^{73} - 136q^{75} - 264q^{76} + 40q^{78} - 460q^{79} + 158q^{81} - 72q^{82} + 24q^{84} - 36q^{85} + 660q^{87} - 266q^{88} - 16q^{90} + 84q^{91} + 36q^{93} + 52q^{94} - 330q^{96} + 516q^{97} + 40q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(-\zeta_{20}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−0.951057 + 0.309017i
0.951057 0.309017i
−0.587785 0.809017i
0.587785 + 0.809017i
−0.951057 0.309017i
0.951057 + 0.309017i
−0.587785 + 0.809017i
0.587785 0.809017i
−0.951057 + 0.309017i 0.303706 + 2.98459i −2.42705 + 1.76336i 4.16750 + 1.35410i −1.21113 2.74466i 1.73607 1.26133i 4.11450 5.66312i −8.81553 + 1.81288i −4.38197
5.2 0.951057 0.309017i 2.93236 0.633446i −2.42705 + 1.76336i −4.16750 1.35410i 2.59310 1.50859i 1.73607 1.26133i −4.11450 + 5.66312i 8.19749 3.71499i −4.38197
14.1 −0.587785 0.809017i −2.74466 1.21113i 0.927051 2.85317i 3.88998 5.35410i 0.633446 + 2.93236i −2.73607 + 8.42075i −6.65740 + 2.16312i 6.06633 + 6.64828i −6.61803
14.2 0.587785 + 0.809017i 1.50859 2.59310i 0.927051 2.85317i −3.88998 + 5.35410i 2.98459 0.303706i −2.73607 + 8.42075i 6.65740 2.16312i −4.44829 7.82385i −6.61803
20.1 −0.951057 0.309017i 0.303706 2.98459i −2.42705 1.76336i 4.16750 1.35410i −1.21113 + 2.74466i 1.73607 + 1.26133i 4.11450 + 5.66312i −8.81553 1.81288i −4.38197
20.2 0.951057 + 0.309017i 2.93236 + 0.633446i −2.42705 1.76336i −4.16750 + 1.35410i 2.59310 + 1.50859i 1.73607 + 1.26133i −4.11450 5.66312i 8.19749 + 3.71499i −4.38197
26.1 −0.587785 + 0.809017i −2.74466 + 1.21113i 0.927051 + 2.85317i 3.88998 + 5.35410i 0.633446 2.93236i −2.73607 8.42075i −6.65740 2.16312i 6.06633 6.64828i −6.61803
26.2 0.587785 0.809017i 1.50859 + 2.59310i 0.927051 + 2.85317i −3.88998 5.35410i 2.98459 + 0.303706i −2.73607 8.42075i 6.65740 + 2.16312i −4.44829 + 7.82385i −6.61803
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.3.h.a 8
3.b odd 2 1 inner 33.3.h.a 8
11.b odd 2 1 363.3.h.g 8
11.c even 5 1 inner 33.3.h.a 8
11.c even 5 1 363.3.b.f 4
11.c even 5 2 363.3.h.h 8
11.d odd 10 1 363.3.b.g 4
11.d odd 10 1 363.3.h.g 8
11.d odd 10 2 363.3.h.i 8
33.d even 2 1 363.3.h.g 8
33.f even 10 1 363.3.b.g 4
33.f even 10 1 363.3.h.g 8
33.f even 10 2 363.3.h.i 8
33.h odd 10 1 inner 33.3.h.a 8
33.h odd 10 1 363.3.b.f 4
33.h odd 10 2 363.3.h.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.h.a 8 1.a even 1 1 trivial
33.3.h.a 8 3.b odd 2 1 inner
33.3.h.a 8 11.c even 5 1 inner
33.3.h.a 8 33.h odd 10 1 inner
363.3.b.f 4 11.c even 5 1
363.3.b.f 4 33.h odd 10 1
363.3.b.g 4 11.d odd 10 1
363.3.b.g 4 33.f even 10 1
363.3.h.g 8 11.b odd 2 1
363.3.h.g 8 11.d odd 10 1
363.3.h.g 8 33.d even 2 1
363.3.h.g 8 33.f even 10 1
363.3.h.h 8 11.c even 5 2
363.3.h.h 8 33.h odd 10 2
363.3.h.i 8 11.d odd 10 2
363.3.h.i 8 33.f even 10 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - T_{2}^{6} + T_{2}^{4} - T_{2}^{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(33, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 7 T^{2} + 33 T^{4} + 119 T^{6} + 305 T^{8} + 1904 T^{10} + 8448 T^{12} + 28672 T^{14} + 65536 T^{16} \)
$3$ \( 1 - 4 T + 7 T^{2} + 8 T^{3} - 95 T^{4} + 72 T^{5} + 567 T^{6} - 2916 T^{7} + 6561 T^{8} \)
$5$ \( 1 + 46 T^{2} + 171 T^{4} - 24844 T^{6} - 756019 T^{8} - 15527500 T^{10} + 66796875 T^{12} + 11230468750 T^{14} + 152587890625 T^{16} \)
$7$ \( ( 1 + 2 T + 15 T^{2} + 292 T^{3} + 2909 T^{4} + 14308 T^{5} + 36015 T^{6} + 235298 T^{7} + 5764801 T^{8} )^{2} \)
$11$ \( 1 - 316 T^{2} + 49126 T^{4} - 4626556 T^{6} + 214358881 T^{8} \)
$13$ \( ( 1 - 16 T - 33 T^{2} + 3022 T^{3} - 37075 T^{4} + 510718 T^{5} - 942513 T^{6} - 77228944 T^{7} + 815730721 T^{8} )^{2} \)
$17$ \( 1 + 542 T^{2} + 210243 T^{4} + 68683324 T^{6} + 19666656005 T^{8} + 5736499903804 T^{10} + 1466604171668163 T^{12} + 315781252578530462 T^{14} + 48661191875666868481 T^{16} \)
$19$ \( ( 1 - 56 T + 975 T^{2} - 1714 T^{3} - 131251 T^{4} - 618754 T^{5} + 127062975 T^{6} - 2634569336 T^{7} + 16983563041 T^{8} )^{2} \)
$23$ \( ( 1 - 1108 T^{2} + 827878 T^{4} - 310063828 T^{6} + 78310985281 T^{8} )^{2} \)
$29$ \( 1 + 1943 T^{2} + 1137963 T^{4} + 548468221 T^{6} + 624074488400 T^{8} + 387921151817101 T^{10} + 569261908832338443 T^{12} + \)\(68\!\cdots\!63\)\( T^{14} + \)\(25\!\cdots\!21\)\( T^{16} \)
$31$ \( ( 1 + 9 T - 165 T^{2} + 26771 T^{3} + 1096464 T^{4} + 25726931 T^{5} - 152380965 T^{6} + 7987533129 T^{7} + 852891037441 T^{8} )^{2} \)
$37$ \( ( 1 + 60 T + 471 T^{2} + 15310 T^{3} + 2183181 T^{4} + 20959390 T^{5} + 882729831 T^{6} + 153943584540 T^{7} + 3512479453921 T^{8} )^{2} \)
$41$ \( 1 + 4486 T^{2} + 5535915 T^{4} - 12051684736 T^{6} - 42824146054651 T^{8} - 34055180711284096 T^{10} + 44203867349769380715 T^{12} + \)\(10\!\cdots\!66\)\( T^{14} + \)\(63\!\cdots\!41\)\( T^{16} \)
$43$ \( ( 1 + 66 T + 4742 T^{2} + 122034 T^{3} + 3418801 T^{4} )^{4} \)
$47$ \( 1 + 3922 T^{2} + 13265163 T^{4} + 41243684144 T^{6} + 99026272083365 T^{8} + 201256021887478064 T^{10} + \)\(31\!\cdots\!43\)\( T^{12} + \)\(45\!\cdots\!02\)\( T^{14} + \)\(56\!\cdots\!21\)\( T^{16} \)
$53$ \( 1 + 5719 T^{2} + 20370555 T^{4} + 53901639581 T^{6} + 152432594812784 T^{8} + 425309862982728461 T^{10} + \)\(12\!\cdots\!55\)\( T^{12} + \)\(28\!\cdots\!79\)\( T^{14} + \)\(38\!\cdots\!21\)\( T^{16} \)
$59$ \( 1 + 6862 T^{2} + 26290683 T^{4} + 59032874564 T^{6} + 105471828660605 T^{8} + 715322651959705604 T^{10} + \)\(38\!\cdots\!43\)\( T^{12} + \)\(12\!\cdots\!22\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} \)
$61$ \( ( 1 - 168 T + 8303 T^{2} - 15126 T^{3} - 11368595 T^{4} - 56283846 T^{5} + 114962017823 T^{6} - 8655422892648 T^{7} + 191707312997281 T^{8} )^{2} \)
$67$ \( ( 1 + 6 T + 3542 T^{2} + 26934 T^{3} + 20151121 T^{4} )^{4} \)
$71$ \( 1 + 178 T^{2} - 293397 T^{4} - 100251984784 T^{6} + 448183018648805 T^{8} - 2547571456947861904 T^{10} - \)\(18\!\cdots\!17\)\( T^{12} + \)\(29\!\cdots\!98\)\( T^{14} + \)\(41\!\cdots\!21\)\( T^{16} \)
$73$ \( ( 1 + 91 T - 2193 T^{2} - 423367 T^{3} - 13036600 T^{4} - 2256122743 T^{5} - 62277342513 T^{6} + 13771414592299 T^{7} + 806460091894081 T^{8} )^{2} \)
$79$ \( ( 1 + 230 T + 19119 T^{2} + 959620 T^{3} + 61893701 T^{4} + 5988988420 T^{5} + 744686598639 T^{6} + 55910114769830 T^{7} + 1517108809906561 T^{8} )^{2} \)
$83$ \( 1 + 1774 T^{2} - 45091365 T^{4} - 141925594684 T^{6} + 1890623755738109 T^{8} - 6735550430629165564 T^{10} - \)\(10\!\cdots\!65\)\( T^{12} + \)\(18\!\cdots\!14\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} \)
$89$ \( ( 1 - 22052 T^{2} + 224754438 T^{4} - 1383591898532 T^{6} + 3936588805702081 T^{8} )^{2} \)
$97$ \( ( 1 - 258 T + 27455 T^{2} - 2679828 T^{3} + 299189869 T^{4} - 25214501652 T^{5} + 2430571409855 T^{6} - 214906777271682 T^{7} + 7837433594376961 T^{8} )^{2} \)
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