Properties

Label 378.2.u.a
Level 378
Weight 2
Character orbit 378.u
Analytic conductor 3.018
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 378.u (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\)
\(\chi(n)\) \(\zeta_{18}^{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−0.766044 0.642788i
0.939693 + 0.342020i
0.939693 0.342020i
−0.766044 + 0.642788i
−0.173648 + 0.984808i
−0.173648 0.984808i
0.766044 + 0.642788i 1.11334 1.32683i 0.173648 + 0.984808i 2.37939 + 0.866025i 1.70574 0.300767i 0.173648 0.984808i −0.500000 + 0.866025i −0.520945 2.95442i 1.26604 + 2.19285i
85.1 −0.939693 0.342020i 0.592396 1.62760i 0.766044 + 0.642788i 0.152704 0.866025i −1.11334 + 1.32683i 0.766044 0.642788i −0.500000 0.866025i −2.29813 1.92836i −0.439693 + 0.761570i
169.1 −0.939693 + 0.342020i 0.592396 + 1.62760i 0.766044 0.642788i 0.152704 + 0.866025i −1.11334 1.32683i 0.766044 + 0.642788i −0.500000 + 0.866025i −2.29813 + 1.92836i −0.439693 0.761570i
211.1 0.766044 0.642788i 1.11334 + 1.32683i 0.173648 0.984808i 2.37939 0.866025i 1.70574 + 0.300767i 0.173648 + 0.984808i −0.500000 0.866025i −0.520945 + 2.95442i 1.26604 2.19285i
295.1 0.173648 0.984808i −1.70574 0.300767i −0.939693 0.342020i −1.03209 + 0.866025i −0.592396 + 1.62760i −0.939693 + 0.342020i −0.500000 + 0.866025i 2.81908 + 1.02606i 0.673648 + 1.16679i
337.1 0.173648 + 0.984808i −1.70574 + 0.300767i −0.939693 + 0.342020i −1.03209 0.866025i −0.592396 1.62760i −0.939693 0.342020i −0.500000 0.866025i 2.81908 1.02606i 0.673648 1.16679i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.u.a 6
27.e even 9 1 inner 378.2.u.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.u.a 6 1.a even 1 1 trivial
378.2.u.a 6 27.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 3 T_{5}^{5} + 3 T_{5}^{3} + 9 T_{5}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{3} + T^{6} \)
$3$ \( 1 + 9 T^{3} + 27 T^{6} \)
$5$ \( 1 - 3 T + 18 T^{3} - 36 T^{4} - 75 T^{5} + 379 T^{6} - 375 T^{7} - 900 T^{8} + 2250 T^{9} - 9375 T^{11} + 15625 T^{12} \)
$7$ \( 1 + T^{3} + T^{6} \)
$11$ \( 1 - 6 T + 18 T^{2} - 36 T^{3} - 108 T^{4} + 912 T^{5} - 3203 T^{6} + 10032 T^{7} - 13068 T^{8} - 47916 T^{9} + 263538 T^{10} - 966306 T^{11} + 1771561 T^{12} \)
$13$ \( 1 + 3 T + 6 T^{2} + 8 T^{3} - 27 T^{4} - 891 T^{5} - 3483 T^{6} - 11583 T^{7} - 4563 T^{8} + 17576 T^{9} + 171366 T^{10} + 1113879 T^{11} + 4826809 T^{12} \)
$17$ \( 1 - 42 T^{2} + 18 T^{3} + 1050 T^{4} - 378 T^{5} - 20081 T^{6} - 6426 T^{7} + 303450 T^{8} + 88434 T^{9} - 3507882 T^{10} + 24137569 T^{12} \)
$19$ \( 1 - 3 T - 24 T^{2} + 23 T^{3} + 279 T^{4} + 666 T^{5} - 6501 T^{6} + 12654 T^{7} + 100719 T^{8} + 157757 T^{9} - 3127704 T^{10} - 7428297 T^{11} + 47045881 T^{12} \)
$23$ \( 1 - 6 T - 36 T^{2} + 396 T^{3} - 810 T^{4} - 5874 T^{5} + 51733 T^{6} - 135102 T^{7} - 428490 T^{8} + 4818132 T^{9} - 10074276 T^{10} - 38618058 T^{11} + 148035889 T^{12} \)
$29$ \( 1 + 3 T + 108 T^{2} + 90 T^{3} + 4851 T^{4} - 3507 T^{5} + 149293 T^{6} - 101703 T^{7} + 4079691 T^{8} + 2195010 T^{9} + 76386348 T^{10} + 61533447 T^{11} + 594823321 T^{12} \)
$31$ \( 1 + 3 T + 24 T^{2} + 296 T^{3} + 1530 T^{4} + 7119 T^{5} + 55665 T^{6} + 220689 T^{7} + 1470330 T^{8} + 8818136 T^{9} + 22164504 T^{10} + 85887453 T^{11} + 887503681 T^{12} \)
$37$ \( 1 - 12 T + 21 T^{2} + 284 T^{3} + 18 T^{4} - 15444 T^{5} + 118797 T^{6} - 571428 T^{7} + 24642 T^{8} + 14385452 T^{9} + 39357381 T^{10} - 832127484 T^{11} + 2565726409 T^{12} \)
$41$ \( 1 + 12 T + 144 T^{2} + 1044 T^{3} + 9216 T^{4} + 58152 T^{5} + 440263 T^{6} + 2384232 T^{7} + 15492096 T^{8} + 71953524 T^{9} + 406909584 T^{10} + 1390274412 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 + 6 T - 6 T^{2} + 284 T^{3} - 684 T^{4} - 7920 T^{5} + 123837 T^{6} - 340560 T^{7} - 1264716 T^{8} + 22579988 T^{9} - 20512806 T^{10} + 882050658 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 - 24 T + 306 T^{2} - 2844 T^{3} + 25254 T^{4} - 215106 T^{5} + 1621333 T^{6} - 10109982 T^{7} + 55786086 T^{8} - 295272612 T^{9} + 1493182386 T^{10} - 5504280168 T^{11} + 10779215329 T^{12} \)
$53$ \( ( 1 + 42 T^{2} + 153 T^{3} + 2226 T^{4} + 148877 T^{6} )^{2} \)
$59$ \( 1 + 24 T + 252 T^{2} + 1395 T^{3} - 2475 T^{4} - 150699 T^{5} - 1633823 T^{6} - 8891241 T^{7} - 8615475 T^{8} + 286503705 T^{9} + 3053574972 T^{10} + 17158183176 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 + 3 T + 114 T^{2} + 926 T^{3} + 12897 T^{4} + 70029 T^{5} + 1006281 T^{6} + 4271769 T^{7} + 47989737 T^{8} + 210184406 T^{9} + 1578425874 T^{10} + 2533788903 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 - 3 T - 96 T^{2} + 1094 T^{3} - 3735 T^{4} - 48879 T^{5} + 940191 T^{6} - 3274893 T^{7} - 16766415 T^{8} + 329034722 T^{9} - 1934507616 T^{10} - 4050375321 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 + 12 T - 24 T^{2} - 738 T^{3} - 228 T^{4} + 5556 T^{5} - 117857 T^{6} + 394476 T^{7} - 1149348 T^{8} - 264138318 T^{9} - 609880344 T^{10} + 21650752212 T^{11} + 128100283921 T^{12} \)
$73$ \( 1 - 3 T - 96 T^{2} + 635 T^{3} + 1935 T^{4} - 19872 T^{5} + 150873 T^{6} - 1450656 T^{7} + 10311615 T^{8} + 247025795 T^{9} - 2726231136 T^{10} - 6219214779 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 - 33 T + 492 T^{2} - 4402 T^{3} + 17532 T^{4} + 134901 T^{5} - 2452347 T^{6} + 10657179 T^{7} + 109417212 T^{8} - 2170357678 T^{9} + 19163439852 T^{10} - 101542861167 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 + 144 T^{2} + 774 T^{3} + 20916 T^{4} + 72810 T^{5} + 2086741 T^{6} + 6043230 T^{7} + 144090324 T^{8} + 442563138 T^{9} + 6833998224 T^{10} + 326940373369 T^{12} \)
$89$ \( 1 - 21 T + 138 T^{2} + 891 T^{3} - 11325 T^{4} - 132708 T^{5} + 2900905 T^{6} - 11811012 T^{7} - 89705325 T^{8} + 628127379 T^{9} + 8658429258 T^{10} - 117265248429 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 + 15 T + 120 T^{2} + 2138 T^{3} + 14679 T^{4} + 127917 T^{5} + 2485869 T^{6} + 12407949 T^{7} + 138114711 T^{8} + 1951294874 T^{9} + 10623513720 T^{10} + 128810103855 T^{11} + 832972004929 T^{12} \)
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