Properties

Label 57.2.i.a
Level $57$
Weight $2$
Character orbit 57.i
Analytic conductor $0.455$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 57.i (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.455147291521\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(40\)
\(\chi(n)\) \(1\) \(\zeta_{18}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.939693 + 0.342020i
−0.766044 0.642788i
−0.766044 + 0.642788i
−0.173648 0.984808i
0.939693 0.342020i
−0.173648 + 0.984808i
1.43969 + 0.524005i −0.173648 0.984808i 0.266044 + 0.223238i −1.93969 + 1.62760i 0.266044 1.50881i −0.266044 + 0.460802i −1.26604 2.19285i −0.939693 + 0.342020i −3.64543 + 1.32683i
16.1 −0.266044 0.223238i 0.939693 0.342020i −0.326352 1.85083i −0.233956 + 1.32683i −0.326352 0.118782i 0.326352 + 0.565258i −0.673648 + 1.16679i 0.766044 0.642788i 0.358441 0.300767i
25.1 −0.266044 + 0.223238i 0.939693 + 0.342020i −0.326352 + 1.85083i −0.233956 1.32683i −0.326352 + 0.118782i 0.326352 0.565258i −0.673648 1.16679i 0.766044 + 0.642788i 0.358441 + 0.300767i
28.1 0.326352 + 1.85083i −0.766044 + 0.642788i −1.43969 + 0.524005i −0.826352 0.300767i −1.43969 1.20805i 1.43969 2.49362i 0.439693 + 0.761570i 0.173648 0.984808i 0.286989 1.62760i
43.1 1.43969 0.524005i −0.173648 + 0.984808i 0.266044 0.223238i −1.93969 1.62760i 0.266044 + 1.50881i −0.266044 0.460802i −1.26604 + 2.19285i −0.939693 0.342020i −3.64543 1.32683i
55.1 0.326352 1.85083i −0.766044 0.642788i −1.43969 0.524005i −0.826352 + 0.300767i −1.43969 + 1.20805i 1.43969 + 2.49362i 0.439693 0.761570i 0.173648 + 0.984808i 0.286989 + 1.62760i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.2.i.a 6
3.b odd 2 1 171.2.u.a 6
4.b odd 2 1 912.2.bo.b 6
19.e even 9 1 inner 57.2.i.a 6
19.e even 9 1 1083.2.a.m 3
19.f odd 18 1 1083.2.a.n 3
57.j even 18 1 3249.2.a.x 3
57.l odd 18 1 171.2.u.a 6
57.l odd 18 1 3249.2.a.w 3
76.l odd 18 1 912.2.bo.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.i.a 6 1.a even 1 1 trivial
57.2.i.a 6 19.e even 9 1 inner
171.2.u.a 6 3.b odd 2 1
171.2.u.a 6 57.l odd 18 1
912.2.bo.b 6 4.b odd 2 1
912.2.bo.b 6 76.l odd 18 1
1083.2.a.m 3 19.e even 9 1
1083.2.a.n 3 19.f odd 18 1
3249.2.a.w 3 57.l odd 18 1
3249.2.a.x 3 57.j even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 3T_{2}^{5} + 6T_{2}^{4} - 8T_{2}^{3} + 3T_{2}^{2} + 3T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(57, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$5$ \( T^{6} + 6 T^{5} + 18 T^{4} + 30 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} - 3 T^{5} + 9 T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} - 3 T^{5} + 27 T^{4} + \cdots + 1369 \) Copy content Toggle raw display
$13$ \( T^{6} + 6 T^{5} + 30 T^{4} + 125 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$17$ \( T^{6} + 12 T^{5} + 114 T^{4} + \cdots + 2809 \) Copy content Toggle raw display
$19$ \( T^{6} - 18 T^{5} + 162 T^{4} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} + 9 T^{5} + 18 T^{4} + \cdots + 5329 \) Copy content Toggle raw display
$29$ \( T^{6} - 12 T^{5} + 60 T^{4} + \cdots + 2809 \) Copy content Toggle raw display
$31$ \( T^{6} + 24 T^{5} + 387 T^{4} + \cdots + 239121 \) Copy content Toggle raw display
$37$ \( (T^{3} - 6 T^{2} - 9 T + 51)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 6 T^{5} + 81 T^{4} + \cdots + 2601 \) Copy content Toggle raw display
$43$ \( T^{6} - 18 T^{5} + 144 T^{4} + \cdots + 361 \) Copy content Toggle raw display
$47$ \( T^{6} + 33 T^{5} + 477 T^{4} + \cdots + 45369 \) Copy content Toggle raw display
$53$ \( T^{6} + 24 T^{5} + 375 T^{4} + \cdots + 83521 \) Copy content Toggle raw display
$59$ \( T^{6} + 54 T^{4} - 224 T^{3} + \cdots + 7921 \) Copy content Toggle raw display
$61$ \( T^{6} + 21 T^{5} + 204 T^{4} + \cdots + 1369 \) Copy content Toggle raw display
$67$ \( T^{6} - 30 T^{5} + 372 T^{4} + \cdots + 87616 \) Copy content Toggle raw display
$71$ \( T^{6} - 18 T^{5} + 144 T^{4} + \cdots + 23409 \) Copy content Toggle raw display
$73$ \( T^{6} + 27 T^{5} + 324 T^{4} + \cdots + 11881 \) Copy content Toggle raw display
$79$ \( T^{6} - 12 T^{5} + 168 T^{4} + \cdots + 2809 \) Copy content Toggle raw display
$83$ \( T^{6} - 15 T^{5} + 189 T^{4} + \cdots + 2601 \) Copy content Toggle raw display
$89$ \( T^{6} - 45 T^{5} + 900 T^{4} + \cdots + 1265625 \) Copy content Toggle raw display
$97$ \( T^{6} - 21 T^{5} + 210 T^{4} + \cdots + 72361 \) Copy content Toggle raw display
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