Properties

Label 1134.2.t.e
Level 1134
Weight 2
Character orbit 1134.t
Analytic conductor 9.055
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1134.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{2} + ( 1 - \zeta_{24}^{4} ) q^{4} + ( -\zeta_{24} + 2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{5} + ( -1 + \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{7} -\zeta_{24}^{6} q^{8} +O(q^{10})\) \( q + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{2} + ( 1 - \zeta_{24}^{4} ) q^{4} + ( -\zeta_{24} + 2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{5} + ( -1 + \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{7} -\zeta_{24}^{6} q^{8} + ( 2 - \zeta_{24} - 2 \zeta_{24}^{3} - \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{10} + 3 \zeta_{24}^{6} q^{11} + ( -\zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{13} + ( -\zeta_{24} - \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{14} -\zeta_{24}^{4} q^{16} + ( -\zeta_{24} + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{17} + ( -2 - 2 \zeta_{24} - \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{19} + ( -2 \zeta_{24} + \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{20} + 3 \zeta_{24}^{4} q^{22} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{23} + ( 4 - 6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{25} + ( -2 \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{26} + ( -1 + 2 \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{4} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{28} + ( -3 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{29} + ( 1 + 6 \zeta_{24} + 3 \zeta_{24}^{3} + \zeta_{24}^{4} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{31} -\zeta_{24}^{2} q^{32} + ( 2 + 2 \zeta_{24} + \zeta_{24}^{3} + 2 \zeta_{24}^{4} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{34} + ( -2 \zeta_{24} - 2 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 5 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{35} + ( -4 + 3 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{37} + ( -\zeta_{24} - 4 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{38} + ( 1 + \zeta_{24} - \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{40} + ( -2 \zeta_{24} + 4 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 4 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{41} + ( -4 + 3 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{43} + 3 \zeta_{24}^{2} q^{44} + ( 3 \zeta_{24} + 3 \zeta_{24}^{7} ) q^{46} + ( -\zeta_{24} + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{47} + ( -5 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{49} + ( -6 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{50} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{52} + ( 3 \zeta_{24} + 3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{53} + ( -3 - 3 \zeta_{24} + 3 \zeta_{24}^{3} + 6 \zeta_{24}^{4} - 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{55} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{56} + ( -3 - 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{58} + ( -8 \zeta_{24} + \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{59} + ( -4 + \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{61} + ( 3 \zeta_{24} + 2 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{62} - q^{64} + ( -3 \zeta_{24} + 6 \zeta_{24}^{2} - 6 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{65} + ( 10 - 10 \zeta_{24}^{4} ) q^{67} + ( \zeta_{24} + 4 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{68} + ( -2 + 4 \zeta_{24} + 2 \zeta_{24}^{3} - 5 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{70} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{71} + ( 4 - 2 \zeta_{24} - 4 \zeta_{24}^{3} - 2 \zeta_{24}^{4} + 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{73} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{74} + ( -4 - \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{76} + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{77} + ( -3 \zeta_{24} + 7 \zeta_{24}^{4} - 3 \zeta_{24}^{7} ) q^{79} + ( -\zeta_{24} - \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{80} + ( 4 + 4 \zeta_{24} + 2 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{82} + ( 4 \zeta_{24} + \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{83} + ( -3 \zeta_{24} - 3 \zeta_{24}^{7} ) q^{85} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{86} + 3 q^{88} + ( -6 \zeta_{24}^{2} + 12 \zeta_{24}^{6} ) q^{89} + ( \zeta_{24} + 2 \zeta_{24}^{3} - 6 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{91} + ( 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{92} + ( 2 + 2 \zeta_{24} + \zeta_{24}^{3} + 2 \zeta_{24}^{4} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{94} + ( 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{95} + ( -5 - 4 \zeta_{24} - 2 \zeta_{24}^{3} - 5 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{97} + ( 2 \zeta_{24} - 5 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 5 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{4} - 8q^{7} + O(q^{10}) \) \( 8q + 4q^{4} - 8q^{7} + 12q^{10} - 4q^{16} - 24q^{19} + 12q^{22} + 32q^{25} - 4q^{28} + 12q^{31} + 24q^{34} - 16q^{37} - 16q^{43} - 40q^{49} - 24q^{58} - 24q^{61} - 8q^{64} + 40q^{67} - 36q^{70} + 24q^{73} - 24q^{76} + 28q^{79} + 48q^{82} + 24q^{88} - 24q^{91} + 24q^{94} - 60q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1 - \zeta_{24}^{2}\) \(\zeta_{24}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
593.1
0.258819 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
0.258819 + 0.965926i
−0.258819 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.866025 + 0.500000i 0 0.500000 0.866025i −4.18154 0 −1.00000 2.44949i 1.00000i 0 3.62132 2.09077i
593.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.717439 0 −1.00000 + 2.44949i 1.00000i 0 −0.621320 + 0.358719i
593.3 0.866025 0.500000i 0 0.500000 0.866025i −0.717439 0 −1.00000 + 2.44949i 1.00000i 0 −0.621320 + 0.358719i
593.4 0.866025 0.500000i 0 0.500000 0.866025i 4.18154 0 −1.00000 2.44949i 1.00000i 0 3.62132 2.09077i
1025.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −4.18154 0 −1.00000 + 2.44949i 1.00000i 0 3.62132 + 2.09077i
1025.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.717439 0 −1.00000 2.44949i 1.00000i 0 −0.621320 0.358719i
1025.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.717439 0 −1.00000 2.44949i 1.00000i 0 −0.621320 0.358719i
1025.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 4.18154 0 −1.00000 + 2.44949i 1.00000i 0 3.62132 + 2.09077i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1025.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
63.k odd 6 1 inner
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.t.e 8
3.b odd 2 1 inner 1134.2.t.e 8
7.d odd 6 1 1134.2.l.f 8
9.c even 3 1 126.2.k.a 8
9.c even 3 1 1134.2.l.f 8
9.d odd 6 1 126.2.k.a 8
9.d odd 6 1 1134.2.l.f 8
21.g even 6 1 1134.2.l.f 8
36.f odd 6 1 1008.2.bt.c 8
36.h even 6 1 1008.2.bt.c 8
45.h odd 6 1 3150.2.bf.a 8
45.j even 6 1 3150.2.bf.a 8
45.k odd 12 1 3150.2.bp.b 8
45.k odd 12 1 3150.2.bp.e 8
45.l even 12 1 3150.2.bp.b 8
45.l even 12 1 3150.2.bp.e 8
63.g even 3 1 882.2.d.a 8
63.h even 3 1 882.2.k.a 8
63.i even 6 1 126.2.k.a 8
63.j odd 6 1 882.2.k.a 8
63.k odd 6 1 882.2.d.a 8
63.k odd 6 1 inner 1134.2.t.e 8
63.l odd 6 1 882.2.k.a 8
63.n odd 6 1 882.2.d.a 8
63.o even 6 1 882.2.k.a 8
63.s even 6 1 882.2.d.a 8
63.s even 6 1 inner 1134.2.t.e 8
63.t odd 6 1 126.2.k.a 8
252.n even 6 1 7056.2.k.f 8
252.o even 6 1 7056.2.k.f 8
252.r odd 6 1 1008.2.bt.c 8
252.bj even 6 1 1008.2.bt.c 8
252.bl odd 6 1 7056.2.k.f 8
252.bn odd 6 1 7056.2.k.f 8
315.q odd 6 1 3150.2.bf.a 8
315.bq even 6 1 3150.2.bf.a 8
315.bs even 12 1 3150.2.bp.b 8
315.bs even 12 1 3150.2.bp.e 8
315.bu odd 12 1 3150.2.bp.b 8
315.bu odd 12 1 3150.2.bp.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.k.a 8 9.c even 3 1
126.2.k.a 8 9.d odd 6 1
126.2.k.a 8 63.i even 6 1
126.2.k.a 8 63.t odd 6 1
882.2.d.a 8 63.g even 3 1
882.2.d.a 8 63.k odd 6 1
882.2.d.a 8 63.n odd 6 1
882.2.d.a 8 63.s even 6 1
882.2.k.a 8 63.h even 3 1
882.2.k.a 8 63.j odd 6 1
882.2.k.a 8 63.l odd 6 1
882.2.k.a 8 63.o even 6 1
1008.2.bt.c 8 36.f odd 6 1
1008.2.bt.c 8 36.h even 6 1
1008.2.bt.c 8 252.r odd 6 1
1008.2.bt.c 8 252.bj even 6 1
1134.2.l.f 8 7.d odd 6 1
1134.2.l.f 8 9.c even 3 1
1134.2.l.f 8 9.d odd 6 1
1134.2.l.f 8 21.g even 6 1
1134.2.t.e 8 1.a even 1 1 trivial
1134.2.t.e 8 3.b odd 2 1 inner
1134.2.t.e 8 63.k odd 6 1 inner
1134.2.t.e 8 63.s even 6 1 inner
3150.2.bf.a 8 45.h odd 6 1
3150.2.bf.a 8 45.j even 6 1
3150.2.bf.a 8 315.q odd 6 1
3150.2.bf.a 8 315.bq even 6 1
3150.2.bp.b 8 45.k odd 12 1
3150.2.bp.b 8 45.l even 12 1
3150.2.bp.b 8 315.bs even 12 1
3150.2.bp.b 8 315.bu odd 12 1
3150.2.bp.e 8 45.k odd 12 1
3150.2.bp.e 8 45.l even 12 1
3150.2.bp.e 8 315.bs even 12 1
3150.2.bp.e 8 315.bu odd 12 1
7056.2.k.f 8 252.n even 6 1
7056.2.k.f 8 252.o even 6 1
7056.2.k.f 8 252.bl odd 6 1
7056.2.k.f 8 252.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 \(S_{2}^{\mathrm{new}}(1134, [\chi])\):

\( T_{5}^{4} - 18 T_{5}^{2} + 9 \)
\( T_{11}^{2} + 9 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$3$ \( \)
$5$ \( ( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} )^{2} \)
$7$ \( ( 1 + 2 T + 7 T^{2} )^{4} \)
$11$ \( ( 1 - 13 T^{2} + 121 T^{4} )^{4} \)
$13$ \( ( 1 + 20 T^{2} + 231 T^{4} + 3380 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( 1 - 32 T^{2} + 478 T^{4} + 1024 T^{6} - 81341 T^{8} + 295936 T^{10} + 39923038 T^{12} - 772402208 T^{14} + 6975757441 T^{16} \)
$19$ \( ( 1 + 12 T + 92 T^{2} + 528 T^{3} + 2487 T^{4} + 10032 T^{5} + 33212 T^{6} + 82308 T^{7} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - 28 T^{2} + 529 T^{4} )^{4} \)
$29$ \( 1 + 62 T^{2} + 1849 T^{4} + 19406 T^{6} + 39940 T^{8} + 16320446 T^{10} + 1307762569 T^{12} + 36879045902 T^{14} + 500246412961 T^{16} \)
$31$ \( ( 1 - 6 T + 23 T^{2} - 66 T^{3} - 468 T^{4} - 2046 T^{5} + 22103 T^{6} - 178746 T^{7} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 8 T - 8 T^{2} - 16 T^{3} + 1447 T^{4} - 592 T^{5} - 10952 T^{6} + 405224 T^{7} + 1874161 T^{8} )^{2} \)
$41$ \( 1 - 20 T^{2} + 1546 T^{4} + 90160 T^{6} - 2184845 T^{8} + 151558960 T^{10} + 4368626506 T^{12} - 95002084820 T^{14} + 7984925229121 T^{16} \)
$43$ \( ( 1 + 8 T - 20 T^{2} - 16 T^{3} + 2455 T^{4} - 688 T^{5} - 36980 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2} \)
$47$ \( 1 - 152 T^{2} + 13198 T^{4} - 834176 T^{6} + 42212419 T^{8} - 1842694784 T^{10} + 64402029838 T^{12} - 1638440730008 T^{14} + 23811286661761 T^{16} \)
$53$ \( 1 + 158 T^{2} + 13753 T^{4} + 883694 T^{6} + 47672164 T^{8} + 2482296446 T^{10} + 108517785193 T^{12} + 3501969058382 T^{14} + 62259690411361 T^{16} \)
$59$ \( 1 - 38 T^{2} - 4727 T^{4} + 30058 T^{6} + 20937316 T^{8} + 104631898 T^{10} - 57278765447 T^{12} - 1602860278358 T^{14} + 146830437604321 T^{16} \)
$61$ \( ( 1 + 12 T + 176 T^{2} + 1536 T^{3} + 15591 T^{4} + 93696 T^{5} + 654896 T^{6} + 2723772 T^{7} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 - 10 T + 33 T^{2} - 670 T^{3} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 - 176 T^{2} + 15234 T^{4} - 887216 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 - 12 T + 182 T^{2} - 1608 T^{3} + 16131 T^{4} - 117384 T^{5} + 969878 T^{6} - 4668204 T^{7} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 - 14 T + 7 T^{2} - 434 T^{3} + 13996 T^{4} - 34286 T^{5} + 43687 T^{6} - 6902546 T^{7} + 38950081 T^{8} )^{2} \)
$83$ \( 1 - 278 T^{2} + 44473 T^{4} - 5291174 T^{6} + 496693924 T^{8} - 36450897686 T^{10} + 2110613909833 T^{12} - 90889423796582 T^{14} + 2252292232139041 T^{16} \)
$89$ \( ( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 + 30 T + 545 T^{2} + 7350 T^{3} + 79716 T^{4} + 712950 T^{5} + 5127905 T^{6} + 27380190 T^{7} + 88529281 T^{8} )^{2} \)
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