Properties

Label 722.2.e.m
Level $722$
Weight $2$
Character orbit 722.e
Analytic conductor $5.765$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.e (of order \(9\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.76519902594\)
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: no (minimal twist has level 38)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/722\mathbb{Z}\right)^\times\).

\(n\) \(363\)
\(\chi(n)\) \(-\zeta_{18}^{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
−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 0.0603074 + 0.342020i 0.766044 + 0.642788i 1.53209 1.28558i 0.0603074 0.342020i 0.879385 1.52314i −0.500000 0.866025i 2.70574 0.984808i −1.87939 + 0.684040i
245.1 0.173648 0.984808i 1.17365 + 0.984808i −0.939693 0.342020i −1.87939 + 0.684040i 1.17365 0.984808i −1.34730 2.33359i −0.500000 + 0.866025i −0.113341 0.642788i 0.347296 + 1.96962i
389.1 0.173648 + 0.984808i 1.17365 0.984808i −0.939693 + 0.342020i −1.87939 0.684040i 1.17365 + 0.984808i −1.34730 + 2.33359i −0.500000 0.866025i −0.113341 + 0.642788i 0.347296 1.96962i
415.1 0.766044 + 0.642788i 1.76604 0.642788i 0.173648 + 0.984808i 0.347296 1.96962i 1.76604 + 0.642788i −2.53209 4.38571i −0.500000 + 0.866025i 0.407604 0.342020i 1.53209 1.28558i
423.1 −0.939693 + 0.342020i 0.0603074 0.342020i 0.766044 0.642788i 1.53209 + 1.28558i 0.0603074 + 0.342020i 0.879385 + 1.52314i −0.500000 + 0.866025i 2.70574 + 0.984808i −1.87939 0.684040i
595.1 0.766044 0.642788i 1.76604 + 0.642788i 0.173648 0.984808i 0.347296 + 1.96962i 1.76604 0.642788i −2.53209 + 4.38571i −0.500000 0.866025i 0.407604 + 0.342020i 1.53209 + 1.28558i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 595.1
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 722.2.e.m 6
19.b odd 2 1 722.2.e.a 6
19.c even 3 1 38.2.e.a 6
19.c even 3 1 722.2.e.b 6
19.d odd 6 1 722.2.e.k 6
19.d odd 6 1 722.2.e.l 6
19.e even 9 1 38.2.e.a 6
19.e even 9 1 722.2.a.l 3
19.e even 9 2 722.2.c.k 6
19.e even 9 1 722.2.e.b 6
19.e even 9 1 inner 722.2.e.m 6
19.f odd 18 1 722.2.a.k 3
19.f odd 18 2 722.2.c.l 6
19.f odd 18 1 722.2.e.a 6
19.f odd 18 1 722.2.e.k 6
19.f odd 18 1 722.2.e.l 6
57.h odd 6 1 342.2.u.c 6
57.j even 18 1 6498.2.a.bq 3
57.l odd 18 1 342.2.u.c 6
57.l odd 18 1 6498.2.a.bl 3
76.g odd 6 1 304.2.u.c 6
76.k even 18 1 5776.2.a.bo 3
76.l odd 18 1 304.2.u.c 6
76.l odd 18 1 5776.2.a.bn 3
95.i even 6 1 950.2.l.d 6
95.m odd 12 2 950.2.u.b 12
95.p even 18 1 950.2.l.d 6
95.q odd 36 2 950.2.u.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.e.a 6 19.c even 3 1
38.2.e.a 6 19.e even 9 1
304.2.u.c 6 76.g odd 6 1
304.2.u.c 6 76.l odd 18 1
342.2.u.c 6 57.h odd 6 1
342.2.u.c 6 57.l odd 18 1
722.2.a.k 3 19.f odd 18 1
722.2.a.l 3 19.e even 9 1
722.2.c.k 6 19.e even 9 2
722.2.c.l 6 19.f odd 18 2
722.2.e.a 6 19.b odd 2 1
722.2.e.a 6 19.f odd 18 1
722.2.e.b 6 19.c even 3 1
722.2.e.b 6 19.e even 9 1
722.2.e.k 6 19.d odd 6 1
722.2.e.k 6 19.f odd 18 1
722.2.e.l 6 19.d odd 6 1
722.2.e.l 6 19.f odd 18 1
722.2.e.m 6 1.a even 1 1 trivial
722.2.e.m 6 19.e even 9 1 inner
950.2.l.d 6 95.i even 6 1
950.2.l.d 6 95.p even 18 1
950.2.u.b 12 95.m odd 12 2
950.2.u.b 12 95.q odd 36 2
5776.2.a.bn 3 76.l odd 18 1
5776.2.a.bo 3 76.k even 18 1
6498.2.a.bl 3 57.l odd 18 1
6498.2.a.bq 3 57.j even 18 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(722, [\chi])\):

\( T_{3}^{6} - 6T_{3}^{5} + 15T_{3}^{4} - 19T_{3}^{3} + 12T_{3}^{2} - 3T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{6} + 8T_{5}^{3} + 64 \) Copy content Toggle raw display
\( T_{7}^{6} + 6T_{7}^{5} + 36T_{7}^{4} + 48T_{7}^{3} + 144T_{7}^{2} + 576 \) Copy content Toggle raw display
\( T_{13}^{6} + 6T_{13}^{5} + 12T_{13}^{4} - 64T_{13}^{3} + 96T_{13}^{2} - 96T_{13} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$3$ \( T^{6} - 6 T^{5} + 15 T^{4} - 19 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} + 8T^{3} + 64 \) Copy content Toggle raw display
$7$ \( T^{6} + 6 T^{5} + 36 T^{4} + 48 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$11$ \( T^{6} + 6 T^{5} + 33 T^{4} + 56 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$13$ \( T^{6} + 6 T^{5} + 12 T^{4} - 64 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{6} + 12 T^{5} + 72 T^{4} + \cdots + 12321 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 6 T^{5} + 24 T^{4} - 64 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( T^{6} - 36 T^{4} + 280 T^{3} + \cdots + 23104 \) Copy content Toggle raw display
$31$ \( T^{6} - 6 T^{5} + 60 T^{4} + 160 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$37$ \( (T^{3} + 6 T^{2} - 24 T - 136)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{6} + 6 T^{5} - 12 T^{4} - 53 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$47$ \( T^{6} + 6 T^{5} + 48 T^{4} + \cdots + 87616 \) Copy content Toggle raw display
$53$ \( T^{6} + 30 T^{5} + 384 T^{4} + \cdots + 18496 \) Copy content Toggle raw display
$59$ \( T^{6} - 15 T^{5} + 72 T^{4} - 84 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$61$ \( T^{6} - 6 T^{5} + 96 T^{4} + \cdots + 23104 \) Copy content Toggle raw display
$67$ \( T^{6} - 18 T^{5} + 81 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$71$ \( T^{6} - 18 T^{5} + 144 T^{4} + \cdots + 23104 \) Copy content Toggle raw display
$73$ \( T^{6} - 33 T^{5} + 486 T^{4} + \cdots + 3249 \) Copy content Toggle raw display
$79$ \( T^{6} + 12 T^{5} + 96 T^{4} + \cdots + 18496 \) Copy content Toggle raw display
$83$ \( T^{6} + 6 T^{5} + 63 T^{4} + \cdots + 2601 \) Copy content Toggle raw display
$89$ \( T^{6} - 36 T^{5} + 684 T^{4} + \cdots + 962361 \) Copy content Toggle raw display
$97$ \( T^{6} + 6 T^{5} + 15 T^{4} + 19 T^{3} + \cdots + 1 \) Copy content Toggle raw display
show more
show less