Properties

Label 63.3.j.a
Level $63$
Weight $3$
Character orbit 63.j
Analytic conductor $1.717$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 63.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.71662566547\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.63369648.1
Defining polynomial: \( x^{6} - x^{5} + 12x^{4} + 17x^{3} + 118x^{2} + 33x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{4} - 1) q^{2} + 3 q^{3} + ( - \beta_1 - 4) q^{4} + ( - \beta_{4} - 2 \beta_{2} - 4) q^{5} + ( - 3 \beta_{5} - 3 \beta_{4} - 3) q^{6} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2}) q^{7} + (3 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{4} - 1) q^{2} + 3 q^{3} + ( - \beta_1 - 4) q^{4} + ( - \beta_{4} - 2 \beta_{2} - 4) q^{5} + ( - 3 \beta_{5} - 3 \beta_{4} - 3) q^{6} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2}) q^{7} + (3 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{8} + 9 q^{9} + (4 \beta_{5} + 2 \beta_{4} + \beta_{3} + 8 \beta_{2} - \beta_1 + 4) q^{10} + (2 \beta_{5} - \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{11} + ( - 3 \beta_1 - 12) q^{12} + (\beta_{5} + 2 \beta_{4} - \beta_{3} + 5 \beta_{2} + 6) q^{13} + ( - 4 \beta_{5} - \beta_{3} - 15 \beta_{2} - \beta_1 - 13) q^{14} + ( - 3 \beta_{4} - 6 \beta_{2} - 12) q^{15} + (4 \beta_{5} - 4 \beta_{4} + 2 \beta_1 + 15) q^{16} + (6 \beta_{4} + 2 \beta_{3} + 5 \beta_{2} + 2 \beta_1 + 10) q^{17} + ( - 9 \beta_{5} - 9 \beta_{4} - 9) q^{18} + (2 \beta_{3} - 7 \beta_{2} - 7) q^{19} + (7 \beta_{4} + \beta_{3} + 7 \beta_{2} + \beta_1 + 14) q^{20} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 3 \beta_{2}) q^{21} + ( - \beta_{5} - 2 \beta_{4} + 5 \beta_{3} + 19 \beta_{2} + 18) q^{22} + ( - 3 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{23} + (9 \beta_{5} + 9 \beta_{4} - 6 \beta_{3} - 6 \beta_{2} + 3 \beta_1 + 6) q^{24} + (4 \beta_{5} + 8 \beta_{4} + \beta_{3} - 5 \beta_{2} - 1) q^{25} + ( - 2 \beta_{5} - 2 \beta_{3} - 9 \beta_{2} + 4 \beta_1 + 7) q^{26} + 27 q^{27} + (11 \beta_{5} + \beta_{4} - 3 \beta_{3} - 31 \beta_{2} + 3 \beta_1 - 17) q^{28} + ( - \beta_{4} - 6 \beta_{2} - 12) q^{29} + (12 \beta_{5} + 6 \beta_{4} + 3 \beta_{3} + 24 \beta_{2} - 3 \beta_1 + 12) q^{30} + ( - 2 \beta_{5} + 2 \beta_{4} - 4 \beta_1 - 14) q^{31} + ( - 5 \beta_{5} - 5 \beta_{4} + 4 \beta_{3} + 60 \beta_{2} - 2 \beta_1 + 25) q^{32} + (6 \beta_{5} - 3 \beta_{3} + 6 \beta_{2} + 6 \beta_1) q^{33} + ( - 22 \beta_{5} - 11 \beta_{4} - 42 \beta_{2} - 22) q^{34} + ( - 4 \beta_{5} - 10 \beta_{4} - 3 \beta_{3} - 10 \beta_{2} - 21) q^{35} + ( - 9 \beta_1 - 36) q^{36} + ( - 7 \beta_{5} - 14 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 10) q^{37} + (\beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 1) q^{38} + (3 \beta_{5} + 6 \beta_{4} - 3 \beta_{3} + 15 \beta_{2} + 18) q^{39} + ( - 4 \beta_{5} - 2 \beta_{4} - 21 \beta_{2} - 4) q^{40} + (10 \beta_{5} + 4 \beta_{3} + 5 \beta_{2} - 8 \beta_1 + 5) q^{41} + ( - 12 \beta_{5} - 3 \beta_{3} - 45 \beta_{2} - 3 \beta_1 - 39) q^{42} + (8 \beta_{5} + 4 \beta_{4} - 3 \beta_{3} + 38 \beta_{2} + 3 \beta_1 + 8) q^{43} + ( - 26 \beta_{5} + 2 \beta_{3} + 21 \beta_{2} - 4 \beta_1 - 47) q^{44} + ( - 9 \beta_{4} - 18 \beta_{2} - 36) q^{45} + (4 \beta_{5} + 2 \beta_{4} + 21 \beta_{2} + 4) q^{46} + (18 \beta_{5} + 18 \beta_{4} + 4 \beta_{3} - 24 \beta_{2} - 2 \beta_1 + 6) q^{47} + (12 \beta_{5} - 12 \beta_{4} + 6 \beta_1 + 45) q^{48} + (12 \beta_{4} - 6 \beta_{3} + 22 \beta_{2} + 3 \beta_1 + 18) q^{49} + (2 \beta_{5} - 3 \beta_{3} - 31 \beta_{2} + 6 \beta_1 + 33) q^{50} + (18 \beta_{4} + 6 \beta_{3} + 15 \beta_{2} + 6 \beta_1 + 30) q^{51} + ( - 11 \beta_{5} - 22 \beta_{4} + 10 \beta_{2} - 1) q^{52} + ( - 15 \beta_{4} - 3 \beta_{3} + \beta_{2} - 3 \beta_1 + 2) q^{53} + ( - 27 \beta_{5} - 27 \beta_{4} - 27) q^{54} + ( - 3 \beta_{5} + 3 \beta_{4} - \beta_1 + 28) q^{55} + (12 \beta_{5} - 12 \beta_{4} + 9 \beta_{3} + 23 \beta_{2} + 70) q^{56} + (6 \beta_{3} - 21 \beta_{2} - 21) q^{57} + (12 \beta_{5} + 6 \beta_{4} + \beta_{3} + 8 \beta_{2} - \beta_1 + 12) q^{58} + ( - 6 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} - 34 \beta_{2} + 3 \beta_1 - 23) q^{59} + (21 \beta_{4} + 3 \beta_{3} + 21 \beta_{2} + 3 \beta_1 + 42) q^{60} + ( - 6 \beta_{5} + 6 \beta_{4} - 8 \beta_1 + 10) q^{61} + (24 \beta_{5} + 24 \beta_{4} - 12 \beta_{3} - 40 \beta_{2} + 6 \beta_1 + 4) q^{62} + ( - 9 \beta_{5} + 9 \beta_{4} + 9 \beta_{3} + 9 \beta_{2}) q^{63} + ( - 20 \beta_{5} + 20 \beta_{4} - 3 \beta_1 - 22) q^{64} + ( - 8 \beta_{5} - 8 \beta_{4} - 38 \beta_{2} - 27) q^{65} + ( - 3 \beta_{5} - 6 \beta_{4} + 15 \beta_{3} + 57 \beta_{2} + 54) q^{66} + (6 \beta_{5} - 6 \beta_{4} + 7 \beta_1 - 35) q^{67} + ( - 18 \beta_{4} - 3 \beta_{3} - 68 \beta_{2} - 3 \beta_1 - 136) q^{68} + ( - 9 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 6) q^{69} + (26 \beta_{5} + 7 \beta_{4} + 3 \beta_{3} + 45 \beta_{2} - 4 \beta_1 - 3) q^{70} + ( - 6 \beta_{5} - 6 \beta_{4} + 4 \beta_{3} + 32 \beta_{2} - 2 \beta_1 + 10) q^{71} + (27 \beta_{5} + 27 \beta_{4} - 18 \beta_{3} - 18 \beta_{2} + 9 \beta_1 + 18) q^{72} + ( - 36 \beta_{5} - 18 \beta_{4} - 7 \beta_{3} + 18 \beta_{2} + 7 \beta_1 - 36) q^{73} + (18 \beta_{5} + 2 \beta_{3} + 51 \beta_{2} - 4 \beta_1 - 33) q^{74} + (12 \beta_{5} + 24 \beta_{4} + 3 \beta_{3} - 15 \beta_{2} - 3) q^{75} + (8 \beta_{5} + 16 \beta_{4} + 3 \beta_{3} - 26 \beta_{2} - 18) q^{76} + ( - 3 \beta_{5} - 20 \beta_{4} + 4 \beta_{3} + 46 \beta_{2} - 11 \beta_1 + 32) q^{77} + ( - 6 \beta_{5} - 6 \beta_{3} - 27 \beta_{2} + 12 \beta_1 + 21) q^{78} + (2 \beta_{5} - 2 \beta_{4} - 3 \beta_1 + 31) q^{79} + (7 \beta_{4} + 2 \beta_{3} + 12 \beta_{2} + 2 \beta_1 + 24) q^{80} + 81 q^{81} + (17 \beta_{5} + 34 \beta_{4} - 2 \beta_{3} + 68 \beta_{2} + 85) q^{82} + (9 \beta_{3} + 8 \beta_{2} + 9 \beta_1 + 16) q^{83} + (33 \beta_{5} + 3 \beta_{4} - 9 \beta_{3} - 93 \beta_{2} + 9 \beta_1 - 51) q^{84} + ( - 23 \beta_{5} - 46 \beta_{4} - 12 \beta_{3} - 72 \beta_{2} - 95) q^{85} + (29 \beta_{4} + 7 \beta_{3} + 35 \beta_{2} + 7 \beta_1 + 70) q^{86} + ( - 3 \beta_{4} - 18 \beta_{2} - 36) q^{87} + (23 \beta_{5} + 46 \beta_{4} - 12 \beta_{3} - 138 \beta_{2} - 115) q^{88} + (8 \beta_{5} + 5 \beta_{3} + 28 \beta_{2} - 10 \beta_1 - 20) q^{89} + (36 \beta_{5} + 18 \beta_{4} + 9 \beta_{3} + 72 \beta_{2} - 9 \beta_1 + 36) q^{90} + (18 \beta_{5} + 12 \beta_{4} + 9 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 45) q^{91} + (9 \beta_{4} - 2 \beta_{3} + 20 \beta_{2} - 2 \beta_1 + 40) q^{92} + ( - 6 \beta_{5} + 6 \beta_{4} - 12 \beta_1 - 42) q^{93} + (6 \beta_{5} - 6 \beta_{4} + 12 \beta_1 + 144) q^{94} + (\beta_{5} + \beta_{4} - 4 \beta_{3} + 32 \beta_{2} + 2 \beta_1 + 17) q^{95} + ( - 15 \beta_{5} - 15 \beta_{4} + 12 \beta_{3} + 180 \beta_{2} - 6 \beta_1 + 75) q^{96} + ( - 24 \beta_{5} - 12 \beta_{4} - \beta_{3} + 42 \beta_{2} + \beta_1 - 24) q^{97} + ( - 9 \beta_{5} - 5 \beta_{4} - 12 \beta_{3} - 96 \beta_{2} + 21 \beta_1) q^{98} + (18 \beta_{5} - 9 \beta_{3} + 18 \beta_{2} + 18 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 18 q^{3} - 26 q^{4} - 15 q^{5} - 2 q^{7} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 18 q^{3} - 26 q^{4} - 15 q^{5} - 2 q^{7} + 54 q^{9} - 19 q^{10} - 9 q^{11} - 78 q^{12} + 11 q^{13} - 24 q^{14} - 45 q^{15} + 94 q^{16} + 33 q^{17} - 19 q^{19} + 45 q^{20} - 6 q^{21} + 65 q^{22} + 15 q^{23} - 26 q^{25} + 81 q^{26} + 162 q^{27} - 42 q^{28} - 51 q^{29} - 57 q^{30} - 92 q^{31} - 27 q^{33} + 93 q^{34} - 57 q^{35} - 234 q^{36} + 7 q^{37} - 21 q^{38} + 33 q^{39} + 57 q^{40} - 27 q^{41} - 72 q^{42} - 99 q^{43} - 273 q^{44} - 135 q^{45} - 57 q^{46} + 282 q^{48} + 6 q^{49} + 294 q^{50} + 99 q^{51} + 63 q^{52} + 45 q^{53} + 166 q^{55} + 360 q^{56} - 57 q^{57} - 7 q^{58} + 135 q^{60} + 44 q^{61} - 18 q^{63} - 138 q^{64} + 195 q^{66} - 196 q^{67} - 567 q^{68} + 45 q^{69} - 257 q^{70} - 101 q^{73} - 411 q^{74} - 78 q^{75} - 99 q^{76} + 105 q^{77} + 243 q^{78} + 180 q^{79} + 93 q^{80} + 486 q^{81} + 151 q^{82} + 99 q^{83} - 126 q^{84} - 159 q^{85} + 249 q^{86} - 153 q^{87} - 495 q^{88} - 243 q^{89} - 171 q^{90} + 177 q^{91} + 147 q^{92} - 276 q^{93} + 888 q^{94} - 161 q^{97} + 360 q^{98} - 81 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 12x^{4} + 17x^{3} + 118x^{2} + 33x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{5} - 24\nu^{4} + 288\nu^{3} - 236\nu^{2} - 66\nu + 2241 ) / 1449 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 44\nu^{5} - 45\nu^{4} + 540\nu^{3} + 604\nu^{2} + 5310\nu + 36 ) / 1449 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{5} - 3\nu^{4} + 36\nu^{3} + 16\nu^{2} + 354\nu + 99 ) / 63 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 83\nu^{5} - 30\nu^{4} + 843\nu^{3} + 2281\nu^{2} + 9819\nu + 5337 ) / 1449 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 32\nu^{5} - 62\nu^{4} + 422\nu^{3} + 249\nu^{2} + 3130\nu - 1335 ) / 483 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{4} + \beta_{3} - 7\beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{5} + 2\beta_{4} + 13\beta _1 - 33 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -24\beta_{5} - 12\beta_{4} - 19\beta_{3} + 94\beta_{2} + 19\beta _1 - 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -52\beta_{5} - 104\beta_{4} - 187\beta_{3} + 571\beta_{2} + 519 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1 + \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−0.140998 + 0.244215i
1.98253 3.43384i
−1.34153 + 2.32360i
−1.34153 2.32360i
1.98253 + 3.43384i
−0.140998 0.244215i
3.09257i 3.00000 −5.56399 −5.67824 3.27834i 9.27771i 6.63848 + 2.22048i 4.83675i 9.00000 −10.1385 + 17.5604i
11.2 1.03435i 3.00000 2.93011 −2.10422 1.21487i 3.10306i −4.75661 5.13563i 7.16819i 9.00000 1.25661 2.17651i
11.3 3.79027i 3.00000 −10.3661 0.282467 + 0.163082i 11.3708i −2.88187 + 6.37925i 24.1293i 9.00000 −0.618126 + 1.07063i
23.1 3.79027i 3.00000 −10.3661 0.282467 0.163082i 11.3708i −2.88187 6.37925i 24.1293i 9.00000 −0.618126 1.07063i
23.2 1.03435i 3.00000 2.93011 −2.10422 + 1.21487i 3.10306i −4.75661 + 5.13563i 7.16819i 9.00000 1.25661 + 2.17651i
23.3 3.09257i 3.00000 −5.56399 −5.67824 + 3.27834i 9.27771i 6.63848 2.22048i 4.83675i 9.00000 −10.1385 17.5604i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.3.j.a 6
3.b odd 2 1 189.3.j.a 6
7.b odd 2 1 441.3.j.c 6
7.c even 3 1 63.3.n.a yes 6
7.c even 3 1 441.3.r.b 6
7.d odd 6 1 441.3.n.c 6
7.d odd 6 1 441.3.r.c 6
9.c even 3 1 189.3.n.a 6
9.d odd 6 1 63.3.n.a yes 6
21.h odd 6 1 189.3.n.a 6
63.h even 3 1 189.3.j.a 6
63.i even 6 1 441.3.j.c 6
63.j odd 6 1 inner 63.3.j.a 6
63.n odd 6 1 441.3.r.b 6
63.o even 6 1 441.3.n.c 6
63.s even 6 1 441.3.r.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.j.a 6 1.a even 1 1 trivial
63.3.j.a 6 63.j odd 6 1 inner
63.3.n.a yes 6 7.c even 3 1
63.3.n.a yes 6 9.d odd 6 1
189.3.j.a 6 3.b odd 2 1
189.3.j.a 6 63.h even 3 1
189.3.n.a 6 9.c even 3 1
189.3.n.a 6 21.h odd 6 1
441.3.j.c 6 7.b odd 2 1
441.3.j.c 6 63.i even 6 1
441.3.n.c 6 7.d odd 6 1
441.3.n.c 6 63.o even 6 1
441.3.r.b 6 7.c even 3 1
441.3.r.b 6 63.n odd 6 1
441.3.r.c 6 7.d odd 6 1
441.3.r.c 6 63.s even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 25T_{2}^{4} + 163T_{2}^{2} + 147 \) acting on \(S_{3}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 25 T^{4} + 163 T^{2} + \cdots + 147 \) Copy content Toggle raw display
$3$ \( (T - 3)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 15 T^{5} + 88 T^{4} + 195 T^{3} + \cdots + 27 \) Copy content Toggle raw display
$7$ \( T^{6} + 2 T^{5} - T^{4} - 532 T^{3} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{6} + 9 T^{5} - 188 T^{4} + \cdots + 1982907 \) Copy content Toggle raw display
$13$ \( T^{6} - 11 T^{5} + 182 T^{4} + \cdots + 499849 \) Copy content Toggle raw display
$17$ \( T^{6} - 33 T^{5} - 252 T^{4} + \cdots + 31434507 \) Copy content Toggle raw display
$19$ \( T^{6} + 19 T^{5} + 422 T^{4} + \cdots + 214369 \) Copy content Toggle raw display
$23$ \( T^{6} - 15 T^{5} - 84 T^{4} + \cdots + 128547 \) Copy content Toggle raw display
$29$ \( T^{6} + 51 T^{5} + 1144 T^{4} + \cdots + 694083 \) Copy content Toggle raw display
$31$ \( (T^{3} + 46 T^{2} - 324 T - 4632)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 7 T^{5} + 2230 T^{4} + \cdots + 564110001 \) Copy content Toggle raw display
$41$ \( T^{6} + 27 T^{5} + \cdots + 1387137027 \) Copy content Toggle raw display
$43$ \( T^{6} + 99 T^{5} + \cdots + 432265681 \) Copy content Toggle raw display
$47$ \( T^{6} + 10128 T^{4} + \cdots + 29274835968 \) Copy content Toggle raw display
$53$ \( T^{6} - 45 T^{5} + \cdots + 5141134827 \) Copy content Toggle raw display
$59$ \( T^{6} + 4896 T^{4} + \cdots + 813189888 \) Copy content Toggle raw display
$61$ \( (T^{3} - 22 T^{2} - 4996 T + 160696)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 98 T^{2} - 1156 T - 210392)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + 5568 T^{4} + \cdots + 109734912 \) Copy content Toggle raw display
$73$ \( T^{6} + 101 T^{5} + \cdots + 653628591729 \) Copy content Toggle raw display
$79$ \( (T^{3} - 90 T^{2} + 2268 T - 16712)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} - 99 T^{5} + \cdots + 165842832483 \) Copy content Toggle raw display
$89$ \( T^{6} + 243 T^{5} + \cdots + 15857178627 \) Copy content Toggle raw display
$97$ \( T^{6} + 161 T^{5} + \cdots + 36908941689 \) Copy content Toggle raw display
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