Properties

Label 735.2.p.c
Level 735
Weight 2
Character orbit 735.p
Analytic conductor 5.869
Analytic rank 0
Dimension 8
CM no
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 735.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 105)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(\zeta_{24}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
374.1
−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 1.50000i −0.724745 1.57313i −0.500000 + 0.866025i −2.09077 0.792893i −1.73205 + 2.44949i 0 −1.73205 −1.94949 + 2.28024i 0.621320 + 3.82282i
374.2 −0.866025 1.50000i 1.72474 0.158919i −0.500000 + 0.866025i 0.358719 2.20711i −1.73205 2.44949i 0 −1.73205 2.94949 0.548188i −3.62132 + 1.37333i
374.3 0.866025 + 1.50000i −0.724745 1.57313i −0.500000 + 0.866025i −0.358719 + 2.20711i 1.73205 2.44949i 0 1.73205 −1.94949 + 2.28024i −3.62132 + 1.37333i
374.4 0.866025 + 1.50000i 1.72474 0.158919i −0.500000 + 0.866025i 2.09077 + 0.792893i 1.73205 + 2.44949i 0 1.73205 2.94949 0.548188i 0.621320 + 3.82282i
509.1 −0.866025 + 1.50000i −0.724745 + 1.57313i −0.500000 0.866025i −2.09077 + 0.792893i −1.73205 2.44949i 0 −1.73205 −1.94949 2.28024i 0.621320 3.82282i
509.2 −0.866025 + 1.50000i 1.72474 + 0.158919i −0.500000 0.866025i 0.358719 + 2.20711i −1.73205 + 2.44949i 0 −1.73205 2.94949 + 0.548188i −3.62132 1.37333i
509.3 0.866025 1.50000i −0.724745 + 1.57313i −0.500000 0.866025i −0.358719 2.20711i 1.73205 + 2.44949i 0 1.73205 −1.94949 2.28024i −3.62132 1.37333i
509.4 0.866025 1.50000i 1.72474 + 0.158919i −0.500000 0.866025i 2.09077 0.792893i 1.73205 2.44949i 0 1.73205 2.94949 + 0.548188i 0.621320 3.82282i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 509.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner
35.c odd 2 1 inner
35.i odd 6 1 inner
105.g even 2 1 inner
105.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.p.c 8
3.b odd 2 1 inner 735.2.p.c 8
5.b even 2 1 735.2.p.a 8
7.b odd 2 1 735.2.p.a 8
7.c even 3 1 105.2.g.a 4
7.c even 3 1 inner 735.2.p.c 8
7.d odd 6 1 105.2.g.c yes 4
7.d odd 6 1 735.2.p.a 8
15.d odd 2 1 735.2.p.a 8
21.c even 2 1 735.2.p.a 8
21.g even 6 1 105.2.g.c yes 4
21.g even 6 1 735.2.p.a 8
21.h odd 6 1 105.2.g.a 4
21.h odd 6 1 inner 735.2.p.c 8
28.f even 6 1 1680.2.k.a 4
28.g odd 6 1 1680.2.k.c 4
35.c odd 2 1 inner 735.2.p.c 8
35.i odd 6 1 105.2.g.a 4
35.i odd 6 1 inner 735.2.p.c 8
35.j even 6 1 105.2.g.c yes 4
35.j even 6 1 735.2.p.a 8
35.k even 12 2 525.2.b.j 8
35.l odd 12 2 525.2.b.j 8
84.j odd 6 1 1680.2.k.a 4
84.n even 6 1 1680.2.k.c 4
105.g even 2 1 inner 735.2.p.c 8
105.o odd 6 1 105.2.g.c yes 4
105.o odd 6 1 735.2.p.a 8
105.p even 6 1 105.2.g.a 4
105.p even 6 1 inner 735.2.p.c 8
105.w odd 12 2 525.2.b.j 8
105.x even 12 2 525.2.b.j 8
140.p odd 6 1 1680.2.k.a 4
140.s even 6 1 1680.2.k.c 4
420.ba even 6 1 1680.2.k.a 4
420.be odd 6 1 1680.2.k.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.g.a 4 7.c even 3 1
105.2.g.a 4 21.h odd 6 1
105.2.g.a 4 35.i odd 6 1
105.2.g.a 4 105.p even 6 1
105.2.g.c yes 4 7.d odd 6 1
105.2.g.c yes 4 21.g even 6 1
105.2.g.c yes 4 35.j even 6 1
105.2.g.c yes 4 105.o odd 6 1
525.2.b.j 8 35.k even 12 2
525.2.b.j 8 35.l odd 12 2
525.2.b.j 8 105.w odd 12 2
525.2.b.j 8 105.x even 12 2
735.2.p.a 8 5.b even 2 1
735.2.p.a 8 7.b odd 2 1
735.2.p.a 8 7.d odd 6 1
735.2.p.a 8 15.d odd 2 1
735.2.p.a 8 21.c even 2 1
735.2.p.a 8 21.g even 6 1
735.2.p.a 8 35.j even 6 1
735.2.p.a 8 105.o odd 6 1
735.2.p.c 8 1.a even 1 1 trivial
735.2.p.c 8 3.b odd 2 1 inner
735.2.p.c 8 7.c even 3 1 inner
735.2.p.c 8 21.h odd 6 1 inner
735.2.p.c 8 35.c odd 2 1 inner
735.2.p.c 8 35.i odd 6 1 inner
735.2.p.c 8 105.g even 2 1 inner
735.2.p.c 8 105.p even 6 1 inner
1680.2.k.a 4 28.f even 6 1
1680.2.k.a 4 84.j odd 6 1
1680.2.k.a 4 140.p odd 6 1
1680.2.k.a 4 420.ba even 6 1
1680.2.k.c 4 28.g odd 6 1
1680.2.k.c 4 84.n even 6 1
1680.2.k.c 4 140.s even 6 1
1680.2.k.c 4 420.be odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\):

\( T_{2}^{4} + 3 T_{2}^{2} + 9 \)
\( T_{13} + 4 \)
\( T_{257}^{4} - 200 T_{257}^{2} + 40000 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} - 3 T^{4} - 4 T^{6} + 16 T^{8} )^{2} \)
$3$ \( ( 1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4} )^{2} \)
$5$ \( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} \)
$7$ 1
$11$ \( ( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{2}( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} )^{2} \)
$13$ \( ( 1 + 4 T + 13 T^{2} )^{8} \)
$17$ \( ( 1 + 26 T^{2} + 387 T^{4} + 7514 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 + 19 T^{2} + 361 T^{4} )^{4} \)
$23$ \( ( 1 - 34 T^{2} + 627 T^{4} - 17986 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 26 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 - 34 T^{2} + 195 T^{4} - 32674 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 37 T^{2} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 + 70 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 - 62 T^{2} + 1849 T^{4} )^{4} \)
$47$ \( ( 1 + 86 T^{2} + 5187 T^{4} + 189974 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 - 53 T^{2} + 2809 T^{4} )^{4} \)
$59$ \( ( 1 - 70 T^{2} + 1419 T^{4} - 243670 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 + 26 T^{2} - 3045 T^{4} + 96746 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 + 110 T^{2} + 7611 T^{4} + 493790 T^{6} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 - 134 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 + 8 T - 9 T^{2} + 584 T^{3} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 - 18 T + 83 T^{2} )^{4}( 1 + 18 T + 83 T^{2} )^{4} \)
$89$ \( ( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 8 T + 97 T^{2} )^{8} \)
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