Properties

Label 162.2.e.a
Level $162$
Weight $2$
Character orbit 162.e
Analytic conductor $1.294$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 162.e (of order \(9\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.29357651274\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 54)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(-\zeta_{18}^{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.173648 + 0.984808i
0.939693 + 0.342020i
−0.766044 0.642788i
−0.766044 + 0.642788i
0.939693 0.342020i
−0.173648 0.984808i
−0.939693 + 0.342020i 0 0.766044 0.642788i −0.439693 2.49362i 0 −1.79813 1.50881i −0.500000 + 0.866025i 0 1.26604 + 2.19285i
37.1 0.766044 0.642788i 0 0.173648 0.984808i 1.26604 0.460802i 0 −0.0209445 0.118782i −0.500000 0.866025i 0 0.673648 1.16679i
73.1 0.173648 0.984808i 0 −0.939693 0.342020i 0.673648 0.565258i 0 3.31908 1.20805i −0.500000 + 0.866025i 0 −0.439693 0.761570i
91.1 0.173648 + 0.984808i 0 −0.939693 + 0.342020i 0.673648 + 0.565258i 0 3.31908 + 1.20805i −0.500000 0.866025i 0 −0.439693 + 0.761570i
127.1 0.766044 + 0.642788i 0 0.173648 + 0.984808i 1.26604 + 0.460802i 0 −0.0209445 + 0.118782i −0.500000 + 0.866025i 0 0.673648 + 1.16679i
145.1 −0.939693 0.342020i 0 0.766044 + 0.642788i −0.439693 + 2.49362i 0 −1.79813 + 1.50881i −0.500000 0.866025i 0 1.26604 2.19285i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.1
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 162.2.e.a 6
3.b odd 2 1 54.2.e.a 6
9.c even 3 1 486.2.e.a 6
9.c even 3 1 486.2.e.c 6
9.d odd 6 1 486.2.e.b 6
9.d odd 6 1 486.2.e.d 6
12.b even 2 1 432.2.u.a 6
27.e even 9 1 inner 162.2.e.a 6
27.e even 9 1 486.2.e.a 6
27.e even 9 1 486.2.e.c 6
27.e even 9 1 1458.2.a.d 3
27.e even 9 2 1458.2.c.a 6
27.f odd 18 1 54.2.e.a 6
27.f odd 18 1 486.2.e.b 6
27.f odd 18 1 486.2.e.d 6
27.f odd 18 1 1458.2.a.a 3
27.f odd 18 2 1458.2.c.d 6
108.l even 18 1 432.2.u.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.e.a 6 3.b odd 2 1
54.2.e.a 6 27.f odd 18 1
162.2.e.a 6 1.a even 1 1 trivial
162.2.e.a 6 27.e even 9 1 inner
432.2.u.a 6 12.b even 2 1
432.2.u.a 6 108.l even 18 1
486.2.e.a 6 9.c even 3 1
486.2.e.a 6 27.e even 9 1
486.2.e.b 6 9.d odd 6 1
486.2.e.b 6 27.f odd 18 1
486.2.e.c 6 9.c even 3 1
486.2.e.c 6 27.e even 9 1
486.2.e.d 6 9.d odd 6 1
486.2.e.d 6 27.f odd 18 1
1458.2.a.a 3 27.f odd 18 1
1458.2.a.d 3 27.e even 9 1
1458.2.c.a 6 27.e even 9 2
1458.2.c.d 6 27.f odd 18 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 3 T_{5}^{5} + 9 T_{5}^{4} - 24 T_{5}^{3} + 36 T_{5}^{2} - 27 T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(162, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{3} + T^{6} \)
$3$ 1
$5$ \( 1 - 3 T + 9 T^{2} - 9 T^{3} + 36 T^{4} - 12 T^{5} + 109 T^{6} - 60 T^{7} + 900 T^{8} - 1125 T^{9} + 5625 T^{10} - 9375 T^{11} + 15625 T^{12} \)
$7$ \( 1 - 3 T - 6 T^{2} + 50 T^{3} - 99 T^{4} - 207 T^{5} + 1401 T^{6} - 1449 T^{7} - 4851 T^{8} + 17150 T^{9} - 14406 T^{10} - 50421 T^{11} + 117649 T^{12} \)
$11$ \( 1 - 3 T + 9 T^{2} + 9 T^{3} - 18 T^{4} + 114 T^{5} + 1225 T^{6} + 1254 T^{7} - 2178 T^{8} + 11979 T^{9} + 131769 T^{10} - 483153 T^{11} + 1771561 T^{12} \)
$13$ \( 1 + 12 T + 78 T^{2} + 386 T^{3} + 1566 T^{4} + 5886 T^{5} + 21843 T^{6} + 76518 T^{7} + 264654 T^{8} + 848042 T^{9} + 2227758 T^{10} + 4455516 T^{11} + 4826809 T^{12} \)
$17$ \( 1 - 6 T + 12 T^{2} - 54 T^{3} - 102 T^{4} + 2082 T^{5} - 8345 T^{6} + 35394 T^{7} - 29478 T^{8} - 265302 T^{9} + 1002252 T^{10} - 8519142 T^{11} + 24137569 T^{12} \)
$19$ \( 1 - 9 T + 36 T^{2} - 79 T^{3} - 297 T^{4} + 4806 T^{5} - 27429 T^{6} + 91314 T^{7} - 107217 T^{8} - 541861 T^{9} + 4691556 T^{10} - 22284891 T^{11} + 47045881 T^{12} \)
$23$ \( 1 - 6 T + 36 T^{2} - 180 T^{3} + 1386 T^{4} - 6954 T^{5} + 33589 T^{6} - 159942 T^{7} + 733194 T^{8} - 2190060 T^{9} + 10074276 T^{10} - 38618058 T^{11} + 148035889 T^{12} \)
$29$ \( 1 + 15 T + 99 T^{2} + 387 T^{3} - 162 T^{4} - 17112 T^{5} - 132695 T^{6} - 496248 T^{7} - 136242 T^{8} + 9438543 T^{9} + 70020819 T^{10} + 307667235 T^{11} + 594823321 T^{12} \)
$31$ \( 1 + 18 T + 171 T^{2} + 1253 T^{3} + 7263 T^{4} + 37719 T^{5} + 206634 T^{6} + 1169289 T^{7} + 6979743 T^{8} + 37328123 T^{9} + 157922091 T^{10} + 515324718 T^{11} + 887503681 T^{12} \)
$37$ \( 1 - 15 T + 60 T^{2} - 289 T^{3} + 4725 T^{4} - 17730 T^{5} - 19395 T^{6} - 656010 T^{7} + 6468525 T^{8} - 14638717 T^{9} + 112449660 T^{10} - 1040159355 T^{11} + 2565726409 T^{12} \)
$41$ \( 1 - 3 T + 36 T^{2} + 72 T^{3} + 738 T^{4} - 1119 T^{5} + 93799 T^{6} - 45879 T^{7} + 1240578 T^{8} + 4962312 T^{9} + 101727396 T^{10} - 347568603 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 + 18 T + 144 T^{2} + 740 T^{3} + 432 T^{4} - 23706 T^{5} - 185739 T^{6} - 1019358 T^{7} + 798768 T^{8} + 58835180 T^{9} + 492307344 T^{10} + 2646151974 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 + 9 T + 9 T^{2} - 495 T^{3} - 3222 T^{4} + 3726 T^{5} + 123409 T^{6} + 175122 T^{7} - 7117398 T^{8} - 51392385 T^{9} + 43917129 T^{10} + 2064105063 T^{11} + 10779215329 T^{12} \)
$53$ \( ( 1 - 6 T + 150 T^{2} - 639 T^{3} + 7950 T^{4} - 16854 T^{5} + 148877 T^{6} )^{2} \)
$59$ \( 1 - 6 T + 36 T^{2} - 261 T^{3} - 639 T^{4} + 33681 T^{5} - 161243 T^{6} + 1987179 T^{7} - 2224359 T^{8} - 53603919 T^{9} + 436224996 T^{10} - 4289545794 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 - 18 T + 153 T^{2} - 745 T^{3} - 3915 T^{4} + 107703 T^{5} - 1034862 T^{6} + 6569883 T^{7} - 14567715 T^{8} - 169100845 T^{9} + 2118413673 T^{10} - 15202733418 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 + 9 T + 45 T^{2} + 281 T^{3} - 1836 T^{4} - 68094 T^{5} - 564675 T^{6} - 4562298 T^{7} - 8241804 T^{8} + 84514403 T^{9} + 906800445 T^{10} + 12151125963 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 + 12 T - 24 T^{2} - 738 T^{3} - 228 T^{4} + 5556 T^{5} - 117857 T^{6} + 394476 T^{7} - 1149348 T^{8} - 264138318 T^{9} - 609880344 T^{10} + 21650752212 T^{11} + 128100283921 T^{12} \)
$73$ \( 1 - 3 T - 96 T^{2} + 23 T^{3} + 2853 T^{4} + 12258 T^{5} - 46191 T^{6} + 894834 T^{7} + 15203637 T^{8} + 8947391 T^{9} - 2726231136 T^{10} - 6219214779 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 - 33 T + 510 T^{2} - 4168 T^{3} + 3429 T^{4} + 380187 T^{5} - 5136507 T^{6} + 30034773 T^{7} + 21400389 T^{8} - 2054986552 T^{9} + 19864541310 T^{10} - 101542861167 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 - 18 T + 144 T^{2} - 720 T^{3} + 5580 T^{4} - 58968 T^{5} + 392545 T^{6} - 4894344 T^{7} + 38440620 T^{8} - 411686640 T^{9} + 6833998224 T^{10} - 70902731574 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 - 15 T - 78 T^{2} + 477 T^{3} + 27177 T^{4} - 70638 T^{5} - 2238167 T^{6} - 6286782 T^{7} + 215269017 T^{8} + 336270213 T^{9} - 4893894798 T^{10} - 83760891735 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 + 12 T + 51 T^{2} + 1277 T^{3} + 801 T^{4} - 56169 T^{5} + 617238 T^{6} - 5448393 T^{7} + 7536609 T^{8} + 1165483421 T^{9} + 4514993331 T^{10} + 103048083084 T^{11} + 832972004929 T^{12} \)
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