Properties

Label 495.2.n.e
Level $495$
Weight $2$
Character orbit 495.n
Analytic conductor $3.953$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.n (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.13140625.1
Defining polynomial: \(x^{8} - 3 x^{7} + 5 x^{6} - 3 x^{5} + 4 x^{4} + 3 x^{3} + 5 x^{2} + 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + ( -\beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} ) q^{4} + ( -1 + \beta_{3} - \beta_{4} - \beta_{7} ) q^{5} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{7} + ( -1 + 2 \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{8} +O(q^{10})\) \( q + \beta_{6} q^{2} + ( -\beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} ) q^{4} + ( -1 + \beta_{3} - \beta_{4} - \beta_{7} ) q^{5} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{7} + ( -1 + 2 \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{8} + \beta_{5} q^{10} + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{11} + ( -2 - 2 \beta_{4} + 3 \beta_{6} - 2 \beta_{7} ) q^{13} + ( 2 + 2 \beta_{1} - \beta_{3} + 2 \beta_{4} ) q^{14} + ( \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{16} + ( 1 - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{17} + ( 3 \beta_{1} + 5 \beta_{3} - \beta_{5} - \beta_{6} ) q^{19} + ( -\beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{20} + ( 1 - \beta_{2} - 3 \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{22} + ( -2 + \beta_{1} + \beta_{2} ) q^{23} -\beta_{3} q^{25} + ( -\beta_{1} - \beta_{2} + \beta_{6} + 3 \beta_{7} ) q^{26} + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + \beta_{5} + \beta_{6} + 4 \beta_{7} ) q^{28} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{6} + 3 \beta_{7} ) q^{29} -5 \beta_{6} q^{31} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{5} + 4 \beta_{7} ) q^{32} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{7} ) q^{34} + ( 1 + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{35} + ( -3 \beta_{1} - 3 \beta_{2} + 7 \beta_{3} - 7 \beta_{4} + 3 \beta_{6} - 4 \beta_{7} ) q^{37} + ( 3 - 4 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 6 \beta_{6} + 2 \beta_{7} ) q^{38} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{40} + ( -1 + 4 \beta_{1} - \beta_{3} - \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{41} + ( -6 + 2 \beta_{1} + 2 \beta_{2} - \beta_{5} ) q^{43} + ( -\beta_{1} - \beta_{2} + \beta_{3} - 5 \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{44} + ( 1 - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{46} + ( 2 \beta_{1} + 3 \beta_{3} - 3 \beta_{5} - 3 \beta_{6} ) q^{47} + ( 1 - 4 \beta_{1} - \beta_{2} - \beta_{3} + 4 \beta_{5} + \beta_{6} ) q^{49} + ( \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} ) q^{50} + ( -5 + 2 \beta_{1} - 5 \beta_{4} - \beta_{5} - \beta_{6} ) q^{52} + ( -1 + 5 \beta_{2} + 5 \beta_{4} - 5 \beta_{5} - \beta_{6} - \beta_{7} ) q^{53} + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{55} + ( -3 - 3 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} + 2 \beta_{5} - 4 \beta_{7} ) q^{56} + ( 1 - 4 \beta_{1} + \beta_{3} + \beta_{4} + 4 \beta_{5} + 4 \beta_{6} ) q^{58} + ( -4 \beta_{1} - \beta_{2} + 5 \beta_{3} - 5 \beta_{4} + 4 \beta_{6} - 7 \beta_{7} ) q^{59} + ( 2 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{61} + ( 5 \beta_{1} + 5 \beta_{2} - 5 \beta_{6} - 5 \beta_{7} ) q^{62} + ( -4 - 2 \beta_{4} - 3 \beta_{6} - 4 \beta_{7} ) q^{64} + ( -2 \beta_{3} + 3 \beta_{5} + 2 \beta_{7} ) q^{65} + ( -5 - 4 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} + \beta_{5} - 6 \beta_{7} ) q^{67} + ( -4 - \beta_{2} - 4 \beta_{4} + \beta_{5} + 3 \beta_{6} - 4 \beta_{7} ) q^{68} + ( 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{7} ) q^{70} + ( 3 + 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{71} + ( 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{6} - \beta_{7} ) q^{73} + ( -3 - 6 \beta_{1} - 3 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} ) q^{74} + ( 6 - 7 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} + \beta_{5} + 2 \beta_{7} ) q^{76} + ( -2 - 3 \beta_{1} - \beta_{2} + \beta_{3} + 6 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{77} + ( 6 + 3 \beta_{2} - 3 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} + 6 \beta_{7} ) q^{79} + ( -1 - \beta_{1} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{80} + ( 4 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + \beta_{4} - 4 \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{82} + ( 1 - 4 \beta_{1} - \beta_{2} - \beta_{3} + 7 \beta_{4} + 4 \beta_{5} + \beta_{6} + 7 \beta_{7} ) q^{83} + ( -3 + \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} ) q^{85} + ( 2 - \beta_{2} + \beta_{4} + \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{86} + ( -5 - \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{88} + ( -1 + 6 \beta_{1} + 6 \beta_{2} - 6 \beta_{5} ) q^{89} + ( 8 + 2 \beta_{1} - \beta_{3} + 8 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{91} + ( \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} + 4 \beta_{7} ) q^{92} + ( 2 + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{94} + ( -\beta_{1} + 2 \beta_{2} + \beta_{6} - 5 \beta_{7} ) q^{95} + ( 8 + 2 \beta_{2} + 13 \beta_{4} - 2 \beta_{5} + 8 \beta_{7} ) q^{97} + ( -4 + 3 \beta_{3} + 2 \beta_{5} - 3 \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{2} - 2q^{4} - 2q^{5} - q^{7} - 4q^{8} + O(q^{10}) \) \( 8q + 2q^{2} - 2q^{4} - 2q^{5} - q^{7} - 4q^{8} + 2q^{10} - 3q^{11} - 2q^{13} + 16q^{14} + 4q^{16} + 13q^{17} + 15q^{19} + 3q^{20} - 7q^{22} - 10q^{23} - 2q^{25} - 10q^{26} - 6q^{28} + 9q^{29} - 10q^{31} - 16q^{32} + 4q^{34} + 4q^{35} + 24q^{37} - 4q^{40} - 8q^{41} - 38q^{43} + 12q^{44} + 3q^{46} + q^{49} + 2q^{50} - 28q^{52} - 13q^{53} + 7q^{55} - 22q^{56} + 12q^{58} + 27q^{59} + 6q^{61} + 30q^{62} - 26q^{64} - 2q^{65} - 38q^{67} - 11q^{68} + 16q^{70} + 20q^{71} + 13q^{73} - 20q^{74} - 34q^{77} + 37q^{79} - q^{80} + 28q^{82} - 27q^{83} - 12q^{85} + 3q^{86} - 36q^{88} + 16q^{89} + 44q^{91} - 11q^{92} + 17q^{94} + 15q^{95} + 24q^{97} - 16q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} + 5 x^{6} - 3 x^{5} + 4 x^{4} + 3 x^{3} + 5 x^{2} + 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{6} - 3 \nu^{5} - 4 \nu^{3} - 7 \nu^{2} - 12 \nu - 7 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - 7 \nu^{5} + 20 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} + 6 \nu + 9 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{6} - 9 \nu^{5} + 12 \nu^{4} - 16 \nu^{3} + 13 \nu^{2} - 10 \nu - 1 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} + 10 \nu^{6} - 17 \nu^{5} + 8 \nu^{4} - 4 \nu^{3} - 13 \nu^{2} - 8 \nu - 5 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{7} - 12 \nu^{6} + 23 \nu^{5} - 20 \nu^{4} + 16 \nu^{3} + \nu^{2} + 6 \nu - 1 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{7} + 18 \nu^{6} - 35 \nu^{5} + 32 \nu^{4} - 28 \nu^{3} - 11 \nu^{2} - 12 \nu - 7 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{5} + \beta_{4} - \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \beta_{4} - 3 \beta_{2} - 2 \beta_{1}\)
\(\nu^{4}\)\(=\)\(3 \beta_{7} + 4 \beta_{6} + \beta_{4} - \beta_{3} - 5 \beta_{2} - 4 \beta_{1}\)
\(\nu^{5}\)\(=\)\(4 \beta_{7} - 6 \beta_{5} - 4 \beta_{3} - 6 \beta_{2} - 6 \beta_{1} - 1\)
\(\nu^{6}\)\(=\)\(-16 \beta_{6} - 16 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} - 7 \beta_{1} - 6\)
\(\nu^{7}\)\(=\)\(-16 \beta_{7} - 51 \beta_{6} - 29 \beta_{5} - 23 \beta_{4} + 29 \beta_{2} - 16\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(397\)
\(\chi(n)\) \(\beta_{7}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
−0.227943 0.701538i
0.418926 + 1.28932i
−0.227943 + 0.701538i
0.418926 1.28932i
−0.386111 0.280526i
1.69513 + 1.23158i
−0.386111 + 0.280526i
1.69513 1.23158i
−0.596764 + 0.433574i 0 −0.449894 + 1.38463i −0.809017 0.587785i 0 0.318714 0.980901i −0.787747 2.42443i 0 0.737640
91.2 1.09676 0.796845i 0 −0.0501062 + 0.154211i −0.809017 0.587785i 0 −1.12773 + 3.47080i 0.905781 + 2.78771i 0 −1.35567
136.1 −0.596764 0.433574i 0 −0.449894 1.38463i −0.809017 + 0.587785i 0 0.318714 + 0.980901i −0.787747 + 2.42443i 0 0.737640
136.2 1.09676 + 0.796845i 0 −0.0501062 0.154211i −0.809017 + 0.587785i 0 −1.12773 3.47080i 0.905781 2.78771i 0 −1.35567
181.1 −0.147481 0.453901i 0 1.43376 1.04169i 0.309017 0.951057i 0 −2.17239 + 1.57833i −1.45650 1.05821i 0 −0.477260
181.2 0.647481 + 1.99274i 0 −1.93376 + 1.40496i 0.309017 0.951057i 0 2.48141 1.80285i −0.661536 0.480634i 0 2.09529
361.1 −0.147481 + 0.453901i 0 1.43376 + 1.04169i 0.309017 + 0.951057i 0 −2.17239 1.57833i −1.45650 + 1.05821i 0 −0.477260
361.2 0.647481 1.99274i 0 −1.93376 1.40496i 0.309017 + 0.951057i 0 2.48141 + 1.80285i −0.661536 + 0.480634i 0 2.09529
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.n.e 8
3.b odd 2 1 55.2.g.b 8
11.c even 5 1 inner 495.2.n.e 8
11.c even 5 1 5445.2.a.bp 4
11.d odd 10 1 5445.2.a.bi 4
12.b even 2 1 880.2.bo.h 8
15.d odd 2 1 275.2.h.a 8
15.e even 4 2 275.2.z.a 16
33.d even 2 1 605.2.g.k 8
33.f even 10 1 605.2.a.k 4
33.f even 10 2 605.2.g.e 8
33.f even 10 1 605.2.g.k 8
33.h odd 10 1 55.2.g.b 8
33.h odd 10 1 605.2.a.j 4
33.h odd 10 2 605.2.g.m 8
132.n odd 10 1 9680.2.a.cm 4
132.o even 10 1 880.2.bo.h 8
132.o even 10 1 9680.2.a.cn 4
165.o odd 10 1 275.2.h.a 8
165.o odd 10 1 3025.2.a.bd 4
165.r even 10 1 3025.2.a.w 4
165.v even 20 2 275.2.z.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.g.b 8 3.b odd 2 1
55.2.g.b 8 33.h odd 10 1
275.2.h.a 8 15.d odd 2 1
275.2.h.a 8 165.o odd 10 1
275.2.z.a 16 15.e even 4 2
275.2.z.a 16 165.v even 20 2
495.2.n.e 8 1.a even 1 1 trivial
495.2.n.e 8 11.c even 5 1 inner
605.2.a.j 4 33.h odd 10 1
605.2.a.k 4 33.f even 10 1
605.2.g.e 8 33.f even 10 2
605.2.g.k 8 33.d even 2 1
605.2.g.k 8 33.f even 10 1
605.2.g.m 8 33.h odd 10 2
880.2.bo.h 8 12.b even 2 1
880.2.bo.h 8 132.o even 10 1
3025.2.a.w 4 165.r even 10 1
3025.2.a.bd 4 165.o odd 10 1
5445.2.a.bi 4 11.d odd 10 1
5445.2.a.bp 4 11.c even 5 1
9680.2.a.cm 4 132.n odd 10 1
9680.2.a.cn 4 132.o even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 5 T^{2} + 2 T^{3} - T^{4} - 2 T^{5} + 5 T^{6} - 2 T^{7} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$7$ \( 961 - 341 T + 777 T^{2} + 197 T^{3} + 30 T^{4} - 17 T^{5} + 7 T^{6} + T^{7} + T^{8} \)
$11$ \( 14641 + 3993 T + 2178 T^{2} - 99 T^{3} + 75 T^{4} - 9 T^{5} + 18 T^{6} + 3 T^{7} + T^{8} \)
$13$ \( 19321 - 3336 T + 1439 T^{2} + 128 T^{3} + 129 T^{4} - 4 T^{5} + 21 T^{6} + 2 T^{7} + T^{8} \)
$17$ \( 361 - 931 T + 2544 T^{2} - 2537 T^{3} + 1379 T^{4} - 449 T^{5} + 96 T^{6} - 13 T^{7} + T^{8} \)
$19$ \( 625 + 3750 T + 67125 T^{2} - 33875 T^{3} + 7950 T^{4} - 1025 T^{5} + 135 T^{6} - 15 T^{7} + T^{8} \)
$23$ \( ( -11 - 10 T + 4 T^{2} + 5 T^{3} + T^{4} )^{2} \)
$29$ \( 203401 - 18942 T - 2329 T^{2} + 429 T^{3} + 1384 T^{4} - 363 T^{5} + 99 T^{6} - 9 T^{7} + T^{8} \)
$31$ \( 390625 - 156250 T + 78125 T^{2} - 6250 T^{3} - 625 T^{4} + 250 T^{5} + 125 T^{6} + 10 T^{7} + T^{8} \)
$37$ \( 1324801 + 740093 T + 124041 T^{2} - 46831 T^{3} + 19344 T^{4} - 3133 T^{5} + 359 T^{6} - 24 T^{7} + T^{8} \)
$41$ \( 101761 + 120263 T + 69513 T^{2} + 23171 T^{3} + 5430 T^{4} + 831 T^{5} + 93 T^{6} + 8 T^{7} + T^{8} \)
$43$ \( ( 211 + 289 T + 121 T^{2} + 19 T^{3} + T^{4} )^{2} \)
$47$ \( 28561 + 3887 T^{2} + 1170 T^{3} + 259 T^{4} - 90 T^{5} + 23 T^{6} + T^{8} \)
$53$ \( 885481 - 313353 T + 81665 T^{2} - 13753 T^{3} + 5824 T^{4} + 133 T^{5} + 85 T^{6} + 13 T^{7} + T^{8} \)
$59$ \( 687241 - 395433 T + 157155 T^{2} - 54243 T^{3} + 18994 T^{4} - 3177 T^{5} + 385 T^{6} - 27 T^{7} + T^{8} \)
$61$ \( 28561 + 26364 T + 17745 T^{2} + 6396 T^{3} + 1504 T^{4} + 96 T^{5} + 10 T^{6} - 6 T^{7} + T^{8} \)
$67$ \( ( -4079 - 1014 T + 22 T^{2} + 19 T^{3} + T^{4} )^{2} \)
$71$ \( 17161 - 9825 T + 11477 T^{2} - 10445 T^{3} + 5634 T^{4} - 1325 T^{5} + 213 T^{6} - 20 T^{7} + T^{8} \)
$73$ \( 121 + 319 T + 584 T^{2} + 633 T^{3} + 479 T^{4} + 201 T^{5} + 56 T^{6} - 13 T^{7} + T^{8} \)
$79$ \( 45954841 - 17171207 T + 6346418 T^{2} - 1002329 T^{3} + 104405 T^{4} - 8839 T^{5} + 698 T^{6} - 37 T^{7} + T^{8} \)
$83$ \( 2886601 - 2101663 T + 461353 T^{2} + 174129 T^{3} + 42080 T^{4} + 5379 T^{5} + 493 T^{6} + 27 T^{7} + T^{8} \)
$89$ \( ( 1861 + 472 T - 102 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$97$ \( 9066121 + 1951128 T - 356419 T^{2} - 209856 T^{3} + 96589 T^{4} - 7968 T^{5} + 749 T^{6} - 24 T^{7} + T^{8} \)
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