Properties

Label 99.2.e.d
Level 99
Weight 2
Character orbit 99.e
Analytic conductor 0.791
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 99.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.790518980011\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{18}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1
−0.173648 + 0.984808i
0.939693 0.342020i
−0.766044 0.642788i
−0.173648 0.984808i
0.939693 + 0.342020i
−0.766044 + 0.642788i
−0.766044 + 1.32683i 0.592396 + 1.62760i −0.173648 0.300767i 0.266044 + 0.460802i −2.61334 0.460802i 1.43969 2.49362i −2.53209 −2.29813 + 1.92836i −0.815207
34.2 −0.173648 + 0.300767i 1.11334 1.32683i 0.939693 + 1.62760i −0.326352 0.565258i 0.205737 + 0.565258i −0.266044 + 0.460802i −1.34730 −0.520945 2.95442i 0.226682
34.3 0.939693 1.62760i −1.70574 0.300767i −0.766044 1.32683i −1.43969 2.49362i −2.09240 + 2.49362i 0.326352 0.565258i 0.879385 2.81908 + 1.02606i −5.41147
67.1 −0.766044 1.32683i 0.592396 1.62760i −0.173648 + 0.300767i 0.266044 0.460802i −2.61334 + 0.460802i 1.43969 + 2.49362i −2.53209 −2.29813 1.92836i −0.815207
67.2 −0.173648 0.300767i 1.11334 + 1.32683i 0.939693 1.62760i −0.326352 + 0.565258i 0.205737 0.565258i −0.266044 0.460802i −1.34730 −0.520945 + 2.95442i 0.226682
67.3 0.939693 + 1.62760i −1.70574 + 0.300767i −0.766044 + 1.32683i −1.43969 + 2.49362i −2.09240 2.49362i 0.326352 + 0.565258i 0.879385 2.81908 1.02606i −5.41147
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.3
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 99.2.e.d 6
3.b odd 2 1 297.2.e.d 6
9.c even 3 1 inner 99.2.e.d 6
9.c even 3 1 891.2.a.l 3
9.d odd 6 1 297.2.e.d 6
9.d odd 6 1 891.2.a.k 3
11.b odd 2 1 1089.2.e.h 6
99.g even 6 1 9801.2.a.bd 3
99.h odd 6 1 1089.2.e.h 6
99.h odd 6 1 9801.2.a.be 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.e.d 6 1.a even 1 1 trivial
99.2.e.d 6 9.c even 3 1 inner
297.2.e.d 6 3.b odd 2 1
297.2.e.d 6 9.d odd 6 1
891.2.a.k 3 9.d odd 6 1
891.2.a.l 3 9.c even 3 1
1089.2.e.h 6 11.b odd 2 1
1089.2.e.h 6 99.h odd 6 1
9801.2.a.bd 3 99.g even 6 1
9801.2.a.be 3 99.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 3 T_{2}^{4} + 2 T_{2}^{3} + 9 T_{2}^{2} + 3 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(99, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T^{2} + 2 T^{3} + 3 T^{4} - 3 T^{5} - T^{6} - 6 T^{7} + 12 T^{8} + 16 T^{9} - 48 T^{10} + 64 T^{12} \)
$3$ \( 1 + 9 T^{3} + 27 T^{6} \)
$5$ \( 1 + 3 T - 6 T^{2} - 13 T^{3} + 63 T^{4} + 60 T^{5} - 259 T^{6} + 300 T^{7} + 1575 T^{8} - 1625 T^{9} - 3750 T^{10} + 9375 T^{11} + 15625 T^{12} \)
$7$ \( 1 - 3 T - 12 T^{2} + 19 T^{3} + 171 T^{4} - 126 T^{5} - 1161 T^{6} - 882 T^{7} + 8379 T^{8} + 6517 T^{9} - 28812 T^{10} - 50421 T^{11} + 117649 T^{12} \)
$11$ \( ( 1 - T + T^{2} )^{3} \)
$13$ \( 1 - 9 T + 27 T^{2} - 20 T^{3} + 117 T^{4} - 1467 T^{5} + 7086 T^{6} - 19071 T^{7} + 19773 T^{8} - 43940 T^{9} + 771147 T^{10} - 3341637 T^{11} + 4826809 T^{12} \)
$17$ \( ( 1 + 3 T + 45 T^{2} + 103 T^{3} + 765 T^{4} + 867 T^{5} + 4913 T^{6} )^{2} \)
$19$ \( ( 1 + 9 T + 75 T^{2} + 333 T^{3} + 1425 T^{4} + 3249 T^{5} + 6859 T^{6} )^{2} \)
$23$ \( 1 - 6 T - 36 T^{2} + 82 T^{3} + 1986 T^{4} - 1806 T^{5} - 47893 T^{6} - 41538 T^{7} + 1050594 T^{8} + 997694 T^{9} - 10074276 T^{10} - 38618058 T^{11} + 148035889 T^{12} \)
$29$ \( 1 - 6 T - 6 T^{2} + 18 T^{3} - 264 T^{4} + 3900 T^{5} - 14429 T^{6} + 113100 T^{7} - 222024 T^{8} + 439002 T^{9} - 4243686 T^{10} - 123066894 T^{11} + 594823321 T^{12} \)
$31$ \( 1 - 9 T - 18 T^{2} + 187 T^{3} + 1881 T^{4} - 6768 T^{5} - 31569 T^{6} - 209808 T^{7} + 1807641 T^{8} + 5570917 T^{9} - 16623378 T^{10} - 257662359 T^{11} + 887503681 T^{12} \)
$37$ \( ( 1 + 12 T + 138 T^{2} + 885 T^{3} + 5106 T^{4} + 16428 T^{5} + 50653 T^{6} )^{2} \)
$41$ \( 1 - 6 T - 6 T^{2} - 202 T^{3} - 108 T^{4} + 9528 T^{5} + 5627 T^{6} + 390648 T^{7} - 181548 T^{8} - 13922042 T^{9} - 16954566 T^{10} - 695137206 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 + 3 T + 24 T^{2} + 525 T^{3} + 1725 T^{4} + 13692 T^{5} + 213923 T^{6} + 588756 T^{7} + 3189525 T^{8} + 41741175 T^{9} + 82051224 T^{10} + 441025329 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 + 12 T + 18 T^{2} - 386 T^{3} - 1434 T^{4} + 3954 T^{5} + 43979 T^{6} + 185838 T^{7} - 3167706 T^{8} - 40075678 T^{9} + 87834258 T^{10} + 2752140084 T^{11} + 10779215329 T^{12} \)
$53$ \( ( 1 - 3 T + 33 T^{2} - 261 T^{3} + 1749 T^{4} - 8427 T^{5} + 148877 T^{6} )^{2} \)
$59$ \( 1 + 21 T + 198 T^{2} + 1081 T^{3} + 2481 T^{4} - 40380 T^{5} - 564133 T^{6} - 2382420 T^{7} + 8636361 T^{8} + 222014699 T^{9} + 2399237478 T^{10} + 15013410279 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 - 21 T + 114 T^{2} - 1101 T^{3} + 26439 T^{4} - 183108 T^{5} + 602021 T^{6} - 11169588 T^{7} + 98379519 T^{8} - 249906081 T^{9} + 1578425874 T^{10} - 17736522321 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 + 3 T - 66 T^{2} + 575 T^{3} + 1125 T^{4} - 29808 T^{5} + 214875 T^{6} - 1997136 T^{7} + 5050125 T^{8} + 172938725 T^{9} - 1329973986 T^{10} + 4050375321 T^{11} + 90458382169 T^{12} \)
$71$ \( ( 1 - 12 T + 222 T^{2} - 1593 T^{3} + 15762 T^{4} - 60492 T^{5} + 357911 T^{6} )^{2} \)
$73$ \( ( 1 + 6 T + 114 T^{2} + 173 T^{3} + 8322 T^{4} + 31974 T^{5} + 389017 T^{6} )^{2} \)
$79$ \( 1 - 9 T - 180 T^{2} + 533 T^{3} + 34533 T^{4} - 61128 T^{5} - 2824521 T^{6} - 4829112 T^{7} + 215520453 T^{8} + 262789787 T^{9} - 7011014580 T^{10} - 27693507591 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 - 222 T^{2} - 54 T^{3} + 30858 T^{4} + 5994 T^{5} - 2946269 T^{6} + 497502 T^{7} + 212580762 T^{8} - 30876498 T^{9} - 10535747262 T^{10} + 326940373369 T^{12} \)
$89$ \( ( 1 + 12 T + 159 T^{2} + 1728 T^{3} + 14151 T^{4} + 95052 T^{5} + 704969 T^{6} )^{2} \)
$97$ \( 1 - 3 T - 141 T^{2} - 724 T^{3} + 8361 T^{4} + 79623 T^{5} - 423066 T^{6} + 7723431 T^{7} + 78668649 T^{8} - 660775252 T^{9} - 12482628621 T^{10} - 25762020771 T^{11} + 832972004929 T^{12} \)
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