Properties

Label 74.2.f.a
Level $74$
Weight $2$
Character orbit 74.f
Analytic conductor $0.591$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 74.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.590892974957\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{18}^{4} - \zeta_{18}) q^{2} + ( - \zeta_{18}^{3} - \zeta_{18}) q^{3} - \zeta_{18}^{5} q^{4} + ( - 2 \zeta_{18}^{5} - \zeta_{18}^{3} + 2 \zeta_{18}^{2} + \zeta_{18} + 1) q^{5} + ( - \zeta_{18}^{5} + \zeta_{18}^{2} + \zeta_{18}) q^{6} + (\zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{2} + \zeta_{18} + 1) q^{7} + \zeta_{18}^{3} q^{8} + ( - \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{18}^{4} - \zeta_{18}) q^{2} + ( - \zeta_{18}^{3} - \zeta_{18}) q^{3} - \zeta_{18}^{5} q^{4} + ( - 2 \zeta_{18}^{5} - \zeta_{18}^{3} + 2 \zeta_{18}^{2} + \zeta_{18} + 1) q^{5} + ( - \zeta_{18}^{5} + \zeta_{18}^{2} + \zeta_{18}) q^{6} + (\zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{2} + \zeta_{18} + 1) q^{7} + \zeta_{18}^{3} q^{8} + ( - \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{9} + (\zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{3} - \zeta_{18}^{2} - 2) q^{10} + (\zeta_{18}^{5} - \zeta_{18}^{4} - 3 \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18}) q^{11} + (\zeta_{18}^{5} + \zeta_{18}^{3} - \zeta_{18}^{2} - 1) q^{12} + (3 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - \zeta_{18} - 1) q^{13} + (\zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18} + 1) q^{14} + ( - 3 \zeta_{18}^{2} - \zeta_{18} - 3) q^{15} - \zeta_{18} q^{16} - 3 \zeta_{18}^{4} q^{17} + ( - \zeta_{18}^{4} + \zeta_{18}^{2} - 1) q^{18} + (3 \zeta_{18}^{3} - \zeta_{18}^{2} + 3 \zeta_{18}) q^{19} + ( - 2 \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18}^{2} + 1) q^{20} + (\zeta_{18}^{5} - \zeta_{18}^{3} - 2 \zeta_{18} + 1) q^{21} + (\zeta_{18}^{5} - \zeta_{18}^{3} + 3 \zeta_{18} + 1) q^{22} + (\zeta_{18}^{5} + \zeta_{18}^{4} + 5 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 2 \zeta_{18} - 5) q^{23} + ( - \zeta_{18}^{4} - \zeta_{18}^{3} + 1) q^{24} + ( - 4 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} + \zeta_{18} + 4) q^{25} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - 3 \zeta_{18}^{3} - \zeta_{18}^{2} - \zeta_{18}) q^{26} + (3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - \zeta_{18}^{3} - 3 \zeta_{18} + 1) q^{27} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3}) q^{28} + ( - 4 \zeta_{18}^{5} + 5 \zeta_{18}^{4} + 5 \zeta_{18}^{2} - 4 \zeta_{18}) q^{29} + ( - \zeta_{18}^{5} - 3 \zeta_{18}^{4} + \zeta_{18}^{2} + 3 \zeta_{18} + 3) q^{30} + ( - \zeta_{18}^{5} + \zeta_{18}^{2} + \zeta_{18} - 3) q^{31} + ( - \zeta_{18}^{5} + \zeta_{18}^{2}) q^{32} + (\zeta_{18}^{5} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - \zeta_{18} - 2) q^{33} + 3 \zeta_{18}^{2} q^{34} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - \zeta_{18} + 2) q^{35} + ( - \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 1) q^{36} + (2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 5 \zeta_{18}^{3} - 4 \zeta_{18}^{2} - 2 \zeta_{18} - 1) q^{37} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{2} - 3 \zeta_{18} + 1) q^{38} + ( - 5 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 4 \zeta_{18}^{3} + 4 \zeta_{18}^{2} + 3 \zeta_{18} + 5) q^{39} + (\zeta_{18}^{4} + 2 \zeta_{18}^{2} + 1) q^{40} + (2 \zeta_{18}^{5} - 6 \zeta_{18}^{4} - 6 \zeta_{18}^{3} + 2 \zeta_{18} + 4) q^{41} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} + 2 \zeta_{18}^{2}) q^{42} + ( - 6 \zeta_{18}^{5} + 5 \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} + 2) q^{43} + (3 \zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} - 3 \zeta_{18}^{2}) q^{44} + (\zeta_{18}^{4} + \zeta_{18}^{2}) q^{45} + (2 \zeta_{18}^{5} - 5 \zeta_{18}^{4} - \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 3) q^{46} + ( - 4 \zeta_{18}^{5} - 4 \zeta_{18}^{4} + \zeta_{18}^{3} + 4 \zeta_{18}^{2} - 1) q^{47} + (\zeta_{18}^{4} + \zeta_{18}^{2}) q^{48} + (\zeta_{18}^{5} + 4 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 4 \zeta_{18} - 1) q^{49} + (\zeta_{18}^{5} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{3} - 3 \zeta_{18} - 1) q^{50} + (3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 3 \zeta_{18}) q^{51} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} + 2 \zeta_{18}^{2} + 3 \zeta_{18} + 1) q^{52} + ( - 4 \zeta_{18}^{5} + 4 \zeta_{18}^{3} - \zeta_{18}^{2} + 3 \zeta_{18} - 5) q^{53} + ( - 3 \zeta_{18}^{5} + \zeta_{18}^{4} - 3 \zeta_{18}^{3}) q^{54} + ( - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 5 \zeta_{18}^{2} - 2 \zeta_{18} - 2) q^{55} + (\zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18}) q^{56} + (\zeta_{18}^{5} - 6 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 3) q^{57} + ( - 4 \zeta_{18}^{5} + 4 \zeta_{18}^{3} - \zeta_{18}^{2} - 5) q^{58} + (8 \zeta_{18}^{5} - 8 \zeta_{18}^{3} - 5 \zeta_{18}^{2} - \zeta_{18} + 3) q^{59} + (3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + \zeta_{18}^{3} - 3 \zeta_{18} - 1) q^{60} + (4 \zeta_{18}^{5} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{3} + 4 \zeta_{18}) q^{61} + (\zeta_{18}^{5} - 3 \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} + 3 \zeta_{18} - 1) q^{62} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{3} - \zeta_{18}) q^{63} + (\zeta_{18}^{3} - 1) q^{64} + (4 \zeta_{18}^{5} + 9 \zeta_{18}^{4} + 8 \zeta_{18}^{3} + 4 \zeta_{18}^{2} - 4) q^{65} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{3} - 2 \zeta_{18}^{2} - \zeta_{18}) q^{66} + (7 \zeta_{18}^{5} + \zeta_{18}^{3} - 7 \zeta_{18}^{2} - \zeta_{18} - 1) q^{67} - 3 q^{68} + (\zeta_{18}^{5} - 8 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - \zeta_{18}^{2} + 6 \zeta_{18} + 6) q^{69} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + \zeta_{18} - 3) q^{70} + ( - 4 \zeta_{18}^{4} + 6 \zeta_{18}^{3} + 8 \zeta_{18}^{2} + 6 \zeta_{18} - 4) q^{71} + (\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} - 1) q^{72} + (3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{2} - \zeta_{18} - 6) q^{73} + ( - 2 \zeta_{18}^{5} - \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 4 \zeta_{18} + 4) q^{74} + (4 \zeta_{18}^{5} + \zeta_{18}^{4} - 5 \zeta_{18}^{2} - 5 \zeta_{18} - 5) q^{75} + ( - 3 \zeta_{18}^{5} + \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - \zeta_{18} + 3) q^{76} + ( - 2 \zeta_{18}^{4} - \zeta_{18}^{3} + 2 \zeta_{18}^{2} - \zeta_{18} - 2) q^{77} + (3 \zeta_{18}^{5} + 5 \zeta_{18}^{4} + 5 \zeta_{18}^{3} - \zeta_{18} - 4) q^{78} + ( - \zeta_{18}^{5} - 4 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18}^{2} + 6 \zeta_{18} + 6) q^{79} + (\zeta_{18}^{4} - \zeta_{18}^{2} - \zeta_{18} - 2) q^{80} + ( - 3 \zeta_{18}^{5} + \zeta_{18}^{4} - 6 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 5 \zeta_{18} + 5) q^{81} + (2 \zeta_{18}^{5} + 4 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 4 \zeta_{18}^{2} + 2 \zeta_{18}) q^{82} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 4 \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{83} + (2 \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18} - 2) q^{84} + ( - 3 \zeta_{18}^{5} - 6 \zeta_{18}^{3} - 3 \zeta_{18}) q^{85} + (\zeta_{18}^{5} + 2 \zeta_{18}^{4} + 6 \zeta_{18}^{3} - 6 \zeta_{18}^{2} - 2 \zeta_{18} - 1) q^{86} + ( - 6 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + 5 \zeta_{18} - 4) q^{87} + ( - 3 \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18} + 3) q^{88} + ( - 5 \zeta_{18}^{5} + 5 \zeta_{18}^{3} - 3 \zeta_{18}^{2} - \zeta_{18} - 8) q^{89} + ( - \zeta_{18}^{2} - 1) q^{90} + (3 \zeta_{18}^{5} + 3 \zeta_{18}^{3}) q^{91} + (3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 5 \zeta_{18}^{2} - 2 \zeta_{18} + 3) q^{92} + ( - \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 3 \zeta_{18} - 1) q^{93} + ( - \zeta_{18}^{4} + 4 \zeta_{18}^{3} + 4 \zeta_{18}^{2} - 4) q^{94} + (\zeta_{18}^{5} - \zeta_{18}^{3} + 8 \zeta_{18}^{2} + \zeta_{18} + 9) q^{95} + ( - \zeta_{18}^{2} - 1) q^{96} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 14 \zeta_{18}^{3} + 2 \zeta_{18}^{2} + 14) q^{97} + ( - 4 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18} + 2) q^{98} + ( - 4 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} - 4 \zeta_{18} + 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + 3 q^{5} + 6 q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{3} + 3 q^{5} + 6 q^{7} + 3 q^{8} - 3 q^{9} - 6 q^{10} - 9 q^{11} - 3 q^{12} + 3 q^{14} - 18 q^{15} - 6 q^{18} + 9 q^{19} + 3 q^{20} + 3 q^{21} + 3 q^{22} - 15 q^{23} + 3 q^{24} + 21 q^{25} - 9 q^{26} + 3 q^{27} - 3 q^{28} + 18 q^{30} - 18 q^{31} - 3 q^{33} + 3 q^{35} - 6 q^{36} + 9 q^{37} + 6 q^{38} + 18 q^{39} + 6 q^{40} + 6 q^{41} - 3 q^{42} + 12 q^{43} - 3 q^{44} + 15 q^{46} - 3 q^{47} + 6 q^{50} + 9 q^{52} - 18 q^{53} - 9 q^{54} - 18 q^{55} + 3 q^{56} + 12 q^{57} - 18 q^{58} - 6 q^{59} - 3 q^{60} - 12 q^{61} - 3 q^{62} + 6 q^{63} - 3 q^{64} - 3 q^{66} - 3 q^{67} - 18 q^{68} + 42 q^{69} - 12 q^{70} - 6 q^{71} - 6 q^{72} - 36 q^{73} + 18 q^{74} - 30 q^{75} + 9 q^{76} - 15 q^{77} - 9 q^{78} + 30 q^{79} - 12 q^{80} + 12 q^{81} - 6 q^{82} + 6 q^{83} - 6 q^{84} - 18 q^{85} + 12 q^{86} - 27 q^{87} + 9 q^{88} - 33 q^{89} - 6 q^{90} + 9 q^{91} + 12 q^{92} + 3 q^{93} - 12 q^{94} + 51 q^{95} - 6 q^{96} + 42 q^{97} + 9 q^{98} + 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(-\zeta_{18}^{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−0.173648 + 0.984808i
−0.766044 + 0.642788i
−0.766044 0.642788i
0.939693 + 0.342020i
−0.173648 0.984808i
0.939693 0.342020i
0.939693 0.342020i −0.326352 0.118782i 0.766044 0.642788i −0.0209445 0.118782i −0.347296 0.233956 + 1.32683i 0.500000 0.866025i −2.20574 1.85083i −0.0603074 0.104455i
9.1 −0.173648 0.984808i 0.266044 1.50881i −0.939693 + 0.342020i −1.79813 1.50881i −1.53209 1.93969 + 1.62760i 0.500000 + 0.866025i 0.613341 + 0.223238i −1.17365 + 2.03282i
33.1 −0.173648 + 0.984808i 0.266044 + 1.50881i −0.939693 0.342020i −1.79813 + 1.50881i −1.53209 1.93969 1.62760i 0.500000 0.866025i 0.613341 0.223238i −1.17365 2.03282i
49.1 −0.766044 + 0.642788i −1.43969 1.20805i 0.173648 0.984808i 3.31908 1.20805i 1.87939 0.826352 0.300767i 0.500000 + 0.866025i 0.0923963 + 0.524005i −1.76604 + 3.05888i
53.1 0.939693 + 0.342020i −0.326352 + 0.118782i 0.766044 + 0.642788i −0.0209445 + 0.118782i −0.347296 0.233956 1.32683i 0.500000 + 0.866025i −2.20574 + 1.85083i −0.0603074 + 0.104455i
71.1 −0.766044 0.642788i −1.43969 + 1.20805i 0.173648 + 0.984808i 3.31908 + 1.20805i 1.87939 0.826352 + 0.300767i 0.500000 0.866025i 0.0923963 0.524005i −1.76604 3.05888i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.2.f.a 6
3.b odd 2 1 666.2.x.c 6
4.b odd 2 1 592.2.bc.b 6
37.f even 9 1 inner 74.2.f.a 6
37.f even 9 1 2738.2.a.m 3
37.h even 18 1 2738.2.a.p 3
111.p odd 18 1 666.2.x.c 6
148.p odd 18 1 592.2.bc.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.f.a 6 1.a even 1 1 trivial
74.2.f.a 6 37.f even 9 1 inner
592.2.bc.b 6 4.b odd 2 1
592.2.bc.b 6 148.p odd 18 1
666.2.x.c 6 3.b odd 2 1
666.2.x.c 6 111.p odd 18 1
2738.2.a.m 3 37.f even 9 1
2738.2.a.p 3 37.h even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 3T_{3}^{5} + 6T_{3}^{4} + 8T_{3}^{3} + 12T_{3}^{2} + 6T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(74, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$3$ \( T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} - 3 T^{5} - 6 T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} - 6 T^{5} + 18 T^{4} - 30 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( T^{6} + 9 T^{5} + 57 T^{4} + 182 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$13$ \( T^{6} + 36 T^{4} - 90 T^{3} + 81 T + 81 \) Copy content Toggle raw display
$17$ \( T^{6} - 27T^{3} + 729 \) Copy content Toggle raw display
$19$ \( T^{6} - 9 T^{5} + 63 T^{4} + \cdots + 2809 \) Copy content Toggle raw display
$23$ \( T^{6} + 15 T^{5} + 171 T^{4} + 804 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$29$ \( T^{6} + 63 T^{4} - 342 T^{3} + \cdots + 29241 \) Copy content Toggle raw display
$31$ \( (T^{3} + 9 T^{2} + 24 T + 17)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 9 T^{5} + 54 T^{4} + \cdots + 50653 \) Copy content Toggle raw display
$41$ \( T^{6} - 6 T^{5} + 36 T^{4} + \cdots + 207936 \) Copy content Toggle raw display
$43$ \( (T^{3} - 6 T^{2} - 81 T + 467)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 3 T^{5} + 54 T^{4} - 169 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$53$ \( T^{6} + 18 T^{5} + 144 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$59$ \( T^{6} + 6 T^{5} + 120 T^{4} + \cdots + 687241 \) Copy content Toggle raw display
$61$ \( T^{6} + 12 T^{5} + 48 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$67$ \( T^{6} + 3 T^{5} - 36 T^{4} + \cdots + 103041 \) Copy content Toggle raw display
$71$ \( T^{6} + 6 T^{5} - 144 T^{4} + \cdots + 207936 \) Copy content Toggle raw display
$73$ \( (T^{3} + 18 T^{2} + 87 T + 53)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} - 30 T^{5} + 360 T^{4} + \cdots + 45369 \) Copy content Toggle raw display
$83$ \( T^{6} - 6 T^{5} - 12 T^{4} + \cdots + 32041 \) Copy content Toggle raw display
$89$ \( T^{6} + 33 T^{5} + 516 T^{4} + \cdots + 687241 \) Copy content Toggle raw display
$97$ \( T^{6} - 42 T^{5} + 1188 T^{4} + \cdots + 6594624 \) Copy content Toggle raw display
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