Properties

Label 1323.2.g.e
Level $1323$
Weight $2$
Character orbit 1323.g
Analytic conductor $10.564$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
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.

\(f(q)\) \(=\) \( q + ( -\zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{2} + ( -1 + 2 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{4} + ( 1 + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{5} + ( -2 - \zeta_{18} - \zeta_{18}^{2} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{8} +O(q^{10})\) \( q + ( -\zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{2} + ( -1 + 2 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{4} + ( 1 + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{5} + ( -2 - \zeta_{18} - \zeta_{18}^{2} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{8} + ( -2 \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{10} + ( -2 + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{11} + ( -4 \zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{13} + ( -3 \zeta_{18} + 3 \zeta_{18}^{2} - \zeta_{18}^{3} + 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{16} + ( \zeta_{18} - 2 \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{17} + ( -1 - 2 \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{19} + ( 2 - \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{3} ) q^{20} + ( -2 \zeta_{18} + 4 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 4 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{22} + ( -4 - 3 \zeta_{18} - 3 \zeta_{18}^{2} + 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{23} + ( -2 - \zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{25} + ( 1 - 7 \zeta_{18} + 6 \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{26} + ( 3 + 5 \zeta_{18} - 4 \zeta_{18}^{2} - 3 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{29} + ( -1 + 3 \zeta_{18} + 3 \zeta_{18}^{2} + \zeta_{18}^{3} - 6 \zeta_{18}^{4} - 6 \zeta_{18}^{5} ) q^{31} + ( -3 \zeta_{18} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{32} + ( 3 - \zeta_{18} - \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{34} + ( 1 - 3 \zeta_{18} - 3 \zeta_{18}^{2} - \zeta_{18}^{3} + 6 \zeta_{18}^{4} + 6 \zeta_{18}^{5} ) q^{37} + ( 2 + \zeta_{18} + \zeta_{18}^{2} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{38} + ( 3 - 4 \zeta_{18} - 4 \zeta_{18}^{2} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{40} + ( -\zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{41} + ( 1 + \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{43} + ( 5 - 7 \zeta_{18} + 4 \zeta_{18}^{2} - 5 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{44} + ( -\zeta_{18} + 5 \zeta_{18}^{2} + 5 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{46} + ( -3 \zeta_{18} - 2 \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{47} + ( -3 \zeta_{18} + 5 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 5 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{50} + ( 7 - 5 \zeta_{18} - 5 \zeta_{18}^{2} - 5 \zeta_{18}^{4} + 10 \zeta_{18}^{5} ) q^{52} + ( 2 \zeta_{18} - 3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{53} + ( -\zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{55} + ( -3 - 3 \zeta_{18} - 3 \zeta_{18}^{2} + 9 \zeta_{18}^{4} - 6 \zeta_{18}^{5} ) q^{58} + ( 1 + 5 \zeta_{18} - \zeta_{18}^{3} - 5 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{59} + ( 3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 3 \zeta_{18}^{4} ) q^{61} + ( -10 + \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{5} ) q^{62} + ( 4 + 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{64} + ( -5 \zeta_{18} - \zeta_{18}^{2} + 5 \zeta_{18}^{3} - \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{65} + ( 4 - 3 \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{67} + ( 2 - \zeta_{18} - \zeta_{18}^{2} - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{68} + ( -3 + 3 \zeta_{18} + 3 \zeta_{18}^{2} - 6 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{71} + ( \zeta_{18} + 4 \zeta_{18}^{2} - 7 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{73} + ( 10 - \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{5} ) q^{74} + ( 3 \zeta_{18} - 3 \zeta_{18}^{2} + 5 \zeta_{18}^{3} - 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{76} + ( -3 \zeta_{18} + 7 \zeta_{18}^{3} - 3 \zeta_{18}^{5} ) q^{79} + ( -2 \zeta_{18} - \zeta_{18}^{2} + 5 \zeta_{18}^{3} - \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{80} + ( -3 + 3 \zeta_{18}^{3} ) q^{82} + ( 6 + 4 \zeta_{18} + \zeta_{18}^{2} - 6 \zeta_{18}^{3} - 5 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{83} + ( 4 \zeta_{18} - 2 \zeta_{18}^{2} - 3 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{85} + ( -2 - \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{5} ) q^{86} + ( 9 - \zeta_{18} - \zeta_{18}^{2} - 7 \zeta_{18}^{4} + 8 \zeta_{18}^{5} ) q^{88} + ( -4 + 3 \zeta_{18} - 7 \zeta_{18}^{2} + 4 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{89} + ( 1 - 8 \zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{3} + 6 \zeta_{18}^{4} + 6 \zeta_{18}^{5} ) q^{92} + ( -6 - 2 \zeta_{18} + \zeta_{18}^{2} + 6 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{94} + ( -4 - \zeta_{18} + \zeta_{18}^{2} + 4 \zeta_{18}^{3} ) q^{95} + ( -1 - 7 \zeta_{18} + 8 \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 3q^{2} - 3q^{4} + 6q^{5} - 12q^{8} + O(q^{10}) \) \( 6q + 3q^{2} - 3q^{4} + 6q^{5} - 12q^{8} - 12q^{11} - 3q^{13} - 3q^{16} - 6q^{17} - 3q^{19} + 6q^{20} - 9q^{22} - 24q^{23} - 12q^{25} + 3q^{26} + 9q^{29} - 3q^{31} + 9q^{34} + 3q^{37} + 12q^{38} + 18q^{40} + 3q^{43} + 15q^{44} - 3q^{47} - 6q^{50} + 42q^{52} + 6q^{53} - 18q^{58} + 3q^{59} + 6q^{61} - 60q^{62} + 24q^{64} + 15q^{65} + 12q^{67} + 12q^{68} - 18q^{71} - 21q^{73} + 60q^{74} + 15q^{76} + 21q^{79} + 15q^{80} - 9q^{82} + 18q^{83} - 9q^{85} - 12q^{86} + 54q^{88} - 12q^{89} + 3q^{92} - 18q^{94} - 12q^{95} - 3q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-\zeta_{18}\) \(-1 + \zeta_{18}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.939693 0.342020i
−0.173648 + 0.984808i
−0.766044 0.642788i
0.939693 + 0.342020i
−0.173648 0.984808i
−0.766044 + 0.642788i
−0.439693 + 0.761570i 0 0.613341 + 1.06234i 1.34730 0 0 −2.83750 0 −0.592396 + 1.02606i
361.2 0.673648 1.16679i 0 0.0923963 + 0.160035i 2.53209 0 0 2.94356 0 1.70574 2.95442i
361.3 1.26604 2.19285i 0 −2.20574 3.82045i −0.879385 0 0 −6.10607 0 −1.11334 + 1.92836i
667.1 −0.439693 0.761570i 0 0.613341 1.06234i 1.34730 0 0 −2.83750 0 −0.592396 1.02606i
667.2 0.673648 + 1.16679i 0 0.0923963 0.160035i 2.53209 0 0 2.94356 0 1.70574 + 2.95442i
667.3 1.26604 + 2.19285i 0 −2.20574 + 3.82045i −0.879385 0 0 −6.10607 0 −1.11334 1.92836i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 667.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.g.e 6
3.b odd 2 1 441.2.g.b 6
7.b odd 2 1 1323.2.g.d 6
7.c even 3 1 1323.2.f.d 6
7.c even 3 1 1323.2.h.b 6
7.d odd 6 1 189.2.f.b 6
7.d odd 6 1 1323.2.h.c 6
9.c even 3 1 1323.2.h.b 6
9.d odd 6 1 441.2.h.e 6
21.c even 2 1 441.2.g.c 6
21.g even 6 1 63.2.f.a 6
21.g even 6 1 441.2.h.d 6
21.h odd 6 1 441.2.f.c 6
21.h odd 6 1 441.2.h.e 6
28.f even 6 1 3024.2.r.k 6
63.g even 3 1 inner 1323.2.g.e 6
63.g even 3 1 3969.2.a.l 3
63.h even 3 1 1323.2.f.d 6
63.i even 6 1 63.2.f.a 6
63.j odd 6 1 441.2.f.c 6
63.k odd 6 1 567.2.a.c 3
63.k odd 6 1 1323.2.g.d 6
63.l odd 6 1 1323.2.h.c 6
63.n odd 6 1 441.2.g.b 6
63.n odd 6 1 3969.2.a.q 3
63.o even 6 1 441.2.h.d 6
63.s even 6 1 441.2.g.c 6
63.s even 6 1 567.2.a.h 3
63.t odd 6 1 189.2.f.b 6
84.j odd 6 1 1008.2.r.h 6
252.n even 6 1 9072.2.a.bs 3
252.r odd 6 1 1008.2.r.h 6
252.bj even 6 1 3024.2.r.k 6
252.bn odd 6 1 9072.2.a.ca 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.a 6 21.g even 6 1
63.2.f.a 6 63.i even 6 1
189.2.f.b 6 7.d odd 6 1
189.2.f.b 6 63.t odd 6 1
441.2.f.c 6 21.h odd 6 1
441.2.f.c 6 63.j odd 6 1
441.2.g.b 6 3.b odd 2 1
441.2.g.b 6 63.n odd 6 1
441.2.g.c 6 21.c even 2 1
441.2.g.c 6 63.s even 6 1
441.2.h.d 6 21.g even 6 1
441.2.h.d 6 63.o even 6 1
441.2.h.e 6 9.d odd 6 1
441.2.h.e 6 21.h odd 6 1
567.2.a.c 3 63.k odd 6 1
567.2.a.h 3 63.s even 6 1
1008.2.r.h 6 84.j odd 6 1
1008.2.r.h 6 252.r odd 6 1
1323.2.f.d 6 7.c even 3 1
1323.2.f.d 6 63.h even 3 1
1323.2.g.d 6 7.b odd 2 1
1323.2.g.d 6 63.k odd 6 1
1323.2.g.e 6 1.a even 1 1 trivial
1323.2.g.e 6 63.g even 3 1 inner
1323.2.h.b 6 7.c even 3 1
1323.2.h.b 6 9.c even 3 1
1323.2.h.c 6 7.d odd 6 1
1323.2.h.c 6 63.l odd 6 1
3024.2.r.k 6 28.f even 6 1
3024.2.r.k 6 252.bj even 6 1
3969.2.a.l 3 63.g even 3 1
3969.2.a.q 3 63.n odd 6 1
9072.2.a.bs 3 252.n even 6 1
9072.2.a.ca 3 252.bn odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1323, [\chi])\):

\( T_{2}^{6} - 3 T_{2}^{5} + 9 T_{2}^{4} - 6 T_{2}^{3} + 9 T_{2}^{2} + 9 \)
\( T_{5}^{3} - 3 T_{5}^{2} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 3 T^{2} - 3 T^{4} + 6 T^{5} - 11 T^{6} + 12 T^{7} - 12 T^{8} + 48 T^{10} - 96 T^{11} + 64 T^{12} \)
$3$ 1
$5$ \( ( 1 - 3 T + 15 T^{2} - 27 T^{3} + 75 T^{4} - 75 T^{5} + 125 T^{6} )^{2} \)
$7$ 1
$11$ \( ( 1 + 6 T + 42 T^{2} + 135 T^{3} + 462 T^{4} + 726 T^{5} + 1331 T^{6} )^{2} \)
$13$ \( 1 + 3 T + 3 T^{2} + 76 T^{3} + 45 T^{4} - 135 T^{5} + 3246 T^{6} - 1755 T^{7} + 7605 T^{8} + 166972 T^{9} + 85683 T^{10} + 1113879 T^{11} + 4826809 T^{12} \)
$17$ \( 1 + 6 T - 24 T^{2} - 54 T^{3} + 1338 T^{4} + 1914 T^{5} - 18929 T^{6} + 32538 T^{7} + 386682 T^{8} - 265302 T^{9} - 2004504 T^{10} + 8519142 T^{11} + 24137569 T^{12} \)
$19$ \( 1 + 3 T - 42 T^{2} - 41 T^{3} + 1341 T^{4} + 216 T^{5} - 29541 T^{6} + 4104 T^{7} + 484101 T^{8} - 281219 T^{9} - 5473482 T^{10} + 7428297 T^{11} + 47045881 T^{12} \)
$23$ \( ( 1 + 12 T + 96 T^{2} + 549 T^{3} + 2208 T^{4} + 6348 T^{5} + 12167 T^{6} )^{2} \)
$29$ \( 1 - 9 T + 30 T^{2} - 81 T^{3} - 579 T^{4} + 9414 T^{5} - 59051 T^{6} + 273006 T^{7} - 486939 T^{8} - 1975509 T^{9} + 21218430 T^{10} - 184600341 T^{11} + 594823321 T^{12} \)
$31$ \( 1 + 3 T - 6 T^{2} + 319 T^{3} + 171 T^{4} - 1962 T^{5} + 62727 T^{6} - 60822 T^{7} + 164331 T^{8} + 9503329 T^{9} - 5541126 T^{10} + 85887453 T^{11} + 887503681 T^{12} \)
$37$ \( 1 - 3 T - 24 T^{2} - 301 T^{3} + 171 T^{4} + 6552 T^{5} + 58893 T^{6} + 242424 T^{7} + 234099 T^{8} - 15246553 T^{9} - 44979864 T^{10} - 208031871 T^{11} + 2565726409 T^{12} \)
$41$ \( 1 - 114 T^{2} - 18 T^{3} + 8322 T^{4} + 1026 T^{5} - 394913 T^{6} + 42066 T^{7} + 13989282 T^{8} - 1240578 T^{9} - 322136754 T^{10} + 4750104241 T^{12} \)
$43$ \( 1 - 3 T - 114 T^{2} + 149 T^{3} + 9063 T^{4} - 5670 T^{5} - 441093 T^{6} - 243810 T^{7} + 16757487 T^{8} + 11846543 T^{9} - 389743314 T^{10} - 441025329 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 + 3 T - 78 T^{2} - 405 T^{3} + 2481 T^{4} + 11064 T^{5} - 57089 T^{6} + 520008 T^{7} + 5480529 T^{8} - 42048315 T^{9} - 380615118 T^{10} + 688035021 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 - 6 T - 114 T^{2} + 378 T^{3} + 10716 T^{4} - 17304 T^{5} - 587549 T^{6} - 917112 T^{7} + 30101244 T^{8} + 56275506 T^{9} - 899514834 T^{10} - 2509172958 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 - 3 T - 96 T^{2} + 495 T^{3} + 3615 T^{4} - 15798 T^{5} - 107021 T^{6} - 932082 T^{7} + 12583815 T^{8} + 101662605 T^{9} - 1163266656 T^{10} - 2144772897 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 - 6 T - 132 T^{2} + 418 T^{3} + 13698 T^{4} - 19134 T^{5} - 893289 T^{6} - 1167174 T^{7} + 50970258 T^{8} + 94878058 T^{9} - 1827651012 T^{10} - 5067577806 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 - 12 T - 78 T^{2} + 518 T^{3} + 15318 T^{4} - 50094 T^{5} - 815637 T^{6} - 3356298 T^{7} + 68762502 T^{8} + 155795234 T^{9} - 1571787438 T^{10} - 16201501284 T^{11} + 90458382169 T^{12} \)
$71$ \( ( 1 + 9 T + 159 T^{2} + 1305 T^{3} + 11289 T^{4} + 45369 T^{5} + 357911 T^{6} )^{2} \)
$73$ \( 1 + 21 T + 138 T^{2} + 769 T^{3} + 10953 T^{4} + 30402 T^{5} - 450903 T^{6} + 2219346 T^{7} + 58368537 T^{8} + 299154073 T^{9} + 3918957258 T^{10} + 43534503453 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 - 21 T + 84 T^{2} - 499 T^{3} + 25767 T^{4} - 195678 T^{5} + 408327 T^{6} - 15458562 T^{7} + 160811847 T^{8} - 246026461 T^{9} + 3271806804 T^{10} - 64618184379 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 - 18 T + 30 T^{2} + 702 T^{3} + 8088 T^{4} - 126648 T^{5} + 719359 T^{6} - 10511784 T^{7} + 55718232 T^{8} + 401394474 T^{9} + 1423749630 T^{10} - 70902731574 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 + 12 T - 60 T^{2} - 198 T^{3} + 7584 T^{4} - 70800 T^{5} - 1684181 T^{6} - 6301200 T^{7} + 60072864 T^{8} - 139583862 T^{9} - 3764534460 T^{10} + 67008713388 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 + 3 T - 114 T^{2} - 149 T^{3} + 2421 T^{4} - 11502 T^{5} + 340233 T^{6} - 1115694 T^{7} + 22779189 T^{8} - 135988277 T^{9} - 10092338034 T^{10} + 25762020771 T^{11} + 832972004929 T^{12} \)
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