Properties

Label 847.2.f.s
Level $847$
Weight $2$
Character orbit 847.f
Analytic conductor $6.763$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.f (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.159390625.1
Defining polynomial: \( x^{8} - x^{7} + 6x^{6} - 11x^{5} + 21x^{4} - 5x^{3} + 10x^{2} + 25x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} + 1) q^{3} + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{4} + ( - \beta_{7} - \beta_{6} - \beta_{3} - \beta_{2} + 1) q^{5} + (\beta_{7} + \beta_{5}) q^{6} - \beta_{2} q^{7} + (2 \beta_{6} + \beta_{5} - \beta_{3} - \beta_1 + 1) q^{8} + (\beta_{3} - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} + 1) q^{3} + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{4} + ( - \beta_{7} - \beta_{6} - \beta_{3} - \beta_{2} + 1) q^{5} + (\beta_{7} + \beta_{5}) q^{6} - \beta_{2} q^{7} + (2 \beta_{6} + \beta_{5} - \beta_{3} - \beta_1 + 1) q^{8} + (\beta_{3} - \beta_{2} + 1) q^{9} + (\beta_{6} + 2 \beta_{5} - 2 \beta_{4} + \beta_{2} + 2) q^{10} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 - 1) q^{12} + (2 \beta_{7} + 2 \beta_{4} - 2 \beta_{2} - 3 \beta_1 + 2) q^{13} - \beta_{4} q^{14} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}) q^{15} + (\beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_{2} + 1) q^{16} + (\beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} - 2) q^{17} + ( - \beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_1) q^{18} + (2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_1 - 1) q^{19} + (\beta_{7} + \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{20} + ( - \beta_{6} - \beta_{2}) q^{21} + ( - 2 \beta_{7} - \beta_{6} - \beta_{2} + 2 \beta_1 - 2) q^{23} + ( - 4 \beta_{3} - 3 \beta_{2} - \beta_1 + 3) q^{24} + (2 \beta_{6} + 3 \beta_{4} + \beta_{3} - 1) q^{25} + (\beta_{7} - 3 \beta_{6} + \beta_{5} - 3 \beta_{3} + \beta_{2} - \beta_1) q^{26} + ( - \beta_{6} + 3 \beta_{3} + 3 \beta_{2} + 1) q^{27} + (\beta_{7} + \beta_{3} + \beta_{2}) q^{28} + (3 \beta_{4} - 3 \beta_1) q^{29} + (\beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 2 \beta_1 + 3) q^{30} + (\beta_{7} + \beta_{4} - 3 \beta_{2} - 2 \beta_1 + 3) q^{31} + ( - 3 \beta_{6} - 3 \beta_{2} + 4) q^{32} + (2 \beta_{7} - 3 \beta_{6} - 3 \beta_{2} - 2 \beta_1 - 1) q^{34} + (\beta_{3} + \beta_1) q^{35} + ( - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1 + 1) q^{36} + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{37} + ( - \beta_{7} - 3 \beta_{6} - 2 \beta_{5} - 7 \beta_{3} - 7 \beta_{2} + 3) q^{38} + (\beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{3} - 2 \beta_{2} + 2) q^{39} + ( - 3 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} - 4 \beta_{2}) q^{40} + (2 \beta_{6} - 2 \beta_{5} + 3 \beta_{3} + 2 \beta_1 - 3) q^{41} + ( - \beta_{7} - \beta_{4}) q^{42} + ( - 6 \beta_{6} - 6 \beta_{2} + 1) q^{43} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 + 2) q^{45} + ( - 3 \beta_{7} - 3 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 4) q^{46} + (\beta_{6} - 4 \beta_{5} - \beta_{4} - 2 \beta_{3} + 4 \beta_1 + 2) q^{47} + (3 \beta_{7} - \beta_{6} + 3 \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1) q^{48} + (\beta_{6} + \beta_{3} + \beta_{2} - 1) q^{49} + ( - 2 \beta_{7} - 6 \beta_{6} - \beta_{5} - 9 \beta_{3} - 9 \beta_{2} + 6) q^{50} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} - 4 \beta_{2} - \beta_1) q^{51} + ( - 6 \beta_{6} - \beta_{5} - 3 \beta_{3} + \beta_1 + 3) q^{52} + ( - 3 \beta_{7} - 3 \beta_{4} + 4 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{53} + ( - 4 \beta_{7} - 3 \beta_{5} + 3 \beta_{4} + 4 \beta_1) q^{54} + ( - \beta_{7} - \beta_{6} - \beta_{2} + \beta_1 - 2) q^{56} + (2 \beta_{7} + 2 \beta_{4} - \beta_{2} + \beta_1 + 1) q^{57} + ( - 3 \beta_{6} + 3 \beta_{5} - 6 \beta_{3} - 3 \beta_1 + 6) q^{58} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + 6 \beta_{2} + \beta_1) q^{59} + (4 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + 6 \beta_{3} + 6 \beta_{2} - 2) q^{60} + ( - 5 \beta_{7} + 7 \beta_{6} - \beta_{5} + 7 \beta_{3} + 7 \beta_{2} - 7) q^{61} + (\beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{62} + (2 \beta_{6} + \beta_{3} - 1) q^{63} + (\beta_{7} + \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{64} + (2 \beta_{7} + \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{2} - 2 \beta_1 - 2) q^{65} + (\beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{2} - \beta_1 - 2) q^{67} + (\beta_{7} + \beta_{4} - 2 \beta_{2} - 3 \beta_1 + 2) q^{68} + ( - \beta_{6} + 2 \beta_{4} + 3 \beta_{3} - 3) q^{69} + ( - 2 \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{70} + ( - 2 \beta_{7} - 4 \beta_{6} - 3 \beta_{5} + \beta_{3} + \beta_{2} + 4) q^{71} + ( - \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 3) q^{72} + (2 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} - 5 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + 3 \beta_1) q^{73} + (3 \beta_{6} + \beta_{4} + \beta_{3} - 1) q^{74} + (3 \beta_{7} + 3 \beta_{4} - \beta_{2} + 1) q^{75} + (4 \beta_{7} + 3 \beta_{6} + 4 \beta_{5} - 4 \beta_{4} + 3 \beta_{2} - 4 \beta_1 + 4) q^{76} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 - 3) q^{78} + ( - 2 \beta_{7} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 2) q^{79} + ( - \beta_{6} + 4 \beta_{5} - \beta_{4} - 5 \beta_{3} - 4 \beta_1 + 5) q^{80} + (6 \beta_{6} + 6 \beta_{3} + 4 \beta_{2}) q^{81} + ( - \beta_{7} + 2 \beta_{6} - 5 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 2) q^{82} + ( - 3 \beta_{7} - 3 \beta_{6} - 5 \beta_{5} + 4 \beta_{3} + 4 \beta_{2} + 3) q^{83} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2}) q^{84} + (3 \beta_{6} - \beta_{5} - \beta_{4} + \beta_1) q^{85} + ( - 6 \beta_{7} - 6 \beta_{4} + \beta_1) q^{86} + ( - 3 \beta_{5} + 3 \beta_{4}) q^{87} + (5 \beta_{7} - 3 \beta_{6} - \beta_{5} + \beta_{4} - 3 \beta_{2} - 5 \beta_1 - 1) q^{89} + ( - 2 \beta_{7} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 4 \beta_1 + 1) q^{90} + (2 \beta_{6} + 2 \beta_{5} + \beta_{4} + 2 \beta_{3} - 2 \beta_1 - 2) q^{91} + (5 \beta_{7} + 7 \beta_{6} + 5 \beta_{5} - 3 \beta_{4} + 7 \beta_{3} + 6 \beta_{2} - 2 \beta_1) q^{92} + ( - 3 \beta_{6} - \beta_{5} - 3 \beta_{3} - 3 \beta_{2} + 3) q^{93} + (4 \beta_{7} + 6 \beta_{6} - 2 \beta_{5} - \beta_{3} - \beta_{2} - 6) q^{94} + (3 \beta_{7} + 5 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 5 \beta_{3} + 6 \beta_{2}) q^{95} + ( - 3 \beta_{6} - \beta_{3} + 1) q^{96} + (5 \beta_{7} + 5 \beta_{4} + 6 \beta_{3} + 2 \beta_{2} - 5 \beta_1 - 2) q^{97} + ( - \beta_{5} + \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 6 q^{3} - 7 q^{4} + 3 q^{5} + 2 q^{6} - 2 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} + 6 q^{3} - 7 q^{4} + 3 q^{5} + 2 q^{6} - 2 q^{7} + 12 q^{8} + 8 q^{9} + 28 q^{10} - 14 q^{12} + 5 q^{13} + q^{14} - 9 q^{15} + 7 q^{16} - 14 q^{17} + q^{18} - 6 q^{19} - 4 q^{20} - 4 q^{21} - 16 q^{23} + 9 q^{24} - 5 q^{25} - 9 q^{26} + 18 q^{27} + 3 q^{28} - 6 q^{29} + 26 q^{30} + 14 q^{31} + 20 q^{32} - 24 q^{34} + 3 q^{35} + 3 q^{36} + q^{37} - 15 q^{38} - 29 q^{40} - 18 q^{41} + 2 q^{42} - 16 q^{43} + 18 q^{45} + 26 q^{46} + 7 q^{47} - q^{48} - 2 q^{49} - q^{50} - 8 q^{51} + 4 q^{52} + 7 q^{53} - 4 q^{54} - 18 q^{56} + 3 q^{57} + 36 q^{58} + 17 q^{60} - 12 q^{61} + 5 q^{62} - 2 q^{63} - 4 q^{64} - 24 q^{65} - 30 q^{67} + 7 q^{68} - 22 q^{69} - 12 q^{70} + 21 q^{71} - 3 q^{72} - 8 q^{73} - q^{74} + 52 q^{76} - 18 q^{78} - q^{79} + 37 q^{80} + 32 q^{81} - 34 q^{82} + 22 q^{83} + 11 q^{84} + 5 q^{85} + 13 q^{86} - 12 q^{87} - 34 q^{89} + 18 q^{90} - 5 q^{91} + 51 q^{92} + 3 q^{93} - 50 q^{94} + 41 q^{95} - 15 q^{97} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 6x^{6} - 11x^{5} + 21x^{4} - 5x^{3} + 10x^{2} + 25x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 555\nu^{7} - 2159\nu^{6} + 7489\nu^{5} - 18164\nu^{4} + 40069\nu^{3} - 84434\nu^{2} + 43855\nu + 375 ) / 94655 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -970\nu^{7} - 1002\nu^{6} - 6608\nu^{5} + 9063\nu^{4} - 14943\nu^{3} + 27673\nu^{2} - 68120\nu + 35160 ) / 94655 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -1604\nu^{7} + 4159\nu^{6} - 12059\nu^{5} + 28414\nu^{4} - 81659\nu^{3} + 38305\nu^{2} - 13500\nu - 13875 ) / 94655 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2052\nu^{7} + 2252\nu^{6} - 19912\nu^{5} + 21007\nu^{4} - 82042\nu^{3} + 35785\nu^{2} - 19395\nu - 90925 ) / 94655 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2667\nu^{7} + 6691\nu^{6} - 17466\nu^{5} + 50856\nu^{4} - 82441\nu^{3} + 72554\nu^{2} - 4035\nu - 12035 ) / 94655 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4024\nu^{7} - 1464\nu^{6} + 21519\nu^{5} - 26434\nu^{4} + 59219\nu^{3} + 22635\nu^{2} + 54640\nu + 66675 ) / 94655 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - 3\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{6} + \beta_{5} - 4\beta_{4} - \beta_{3} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{7} + 7\beta_{6} + 2\beta_{5} + 13\beta_{3} + 13\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{7} - 11\beta_{6} - 20\beta_{5} + 20\beta_{4} - 11\beta_{2} + 8\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -19\beta_{7} - 19\beta_{4} - 68\beta_{3} - 36\beta_{2} - 24\beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 111\beta_{7} + 81\beta_{6} + 111\beta_{5} - 55\beta_{4} + 81\beta_{3} + 148\beta_{2} - 56\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(365\)
\(\chi(n)\) \(1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
148.1
−0.762262 2.34600i
0.453245 + 1.39494i
−0.628998 0.456994i
1.43801 + 1.04478i
−0.762262 + 2.34600i
0.453245 1.39494i
−0.628998 + 0.456994i
1.43801 1.04478i
−0.762262 2.34600i 1.30902 + 0.951057i −3.30464 + 2.40097i −1.07128 + 3.29706i 1.23337 3.79591i −0.809017 + 0.587785i 4.16042 + 3.02272i −0.118034 0.363271i 8.55150
148.2 0.453245 + 1.39494i 1.30902 + 0.951057i −0.122406 + 0.0889332i 0.144228 0.443888i −0.733366 + 2.25707i −0.809017 + 0.587785i 2.19369 + 1.59381i −0.118034 0.363271i 0.684570
323.1 −0.628998 0.456994i 0.190983 0.587785i −0.431239 1.32722i 0.180019 0.130791i −0.388742 + 0.282438i 0.309017 + 0.951057i −0.815793 + 2.51075i 2.11803 + 1.53884i −0.173002
323.2 1.43801 + 1.04478i 0.190983 0.587785i 0.358290 + 1.10270i 2.24703 1.63256i 0.888742 0.645709i 0.309017 + 0.951057i 0.461691 1.42094i 2.11803 + 1.53884i 4.93693
372.1 −0.762262 + 2.34600i 1.30902 0.951057i −3.30464 2.40097i −1.07128 3.29706i 1.23337 + 3.79591i −0.809017 0.587785i 4.16042 3.02272i −0.118034 + 0.363271i 8.55150
372.2 0.453245 1.39494i 1.30902 0.951057i −0.122406 0.0889332i 0.144228 + 0.443888i −0.733366 2.25707i −0.809017 0.587785i 2.19369 1.59381i −0.118034 + 0.363271i 0.684570
729.1 −0.628998 + 0.456994i 0.190983 + 0.587785i −0.431239 + 1.32722i 0.180019 + 0.130791i −0.388742 0.282438i 0.309017 0.951057i −0.815793 2.51075i 2.11803 1.53884i −0.173002
729.2 1.43801 1.04478i 0.190983 + 0.587785i 0.358290 1.10270i 2.24703 + 1.63256i 0.888742 + 0.645709i 0.309017 0.951057i 0.461691 + 1.42094i 2.11803 1.53884i 4.93693
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 729.2
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 847.2.f.s 8
11.b odd 2 1 847.2.f.p 8
11.c even 5 1 847.2.a.k 4
11.c even 5 2 847.2.f.q 8
11.c even 5 1 inner 847.2.f.s 8
11.d odd 10 2 77.2.f.a 8
11.d odd 10 1 847.2.a.l 4
11.d odd 10 1 847.2.f.p 8
33.f even 10 2 693.2.m.g 8
33.f even 10 1 7623.2.a.ch 4
33.h odd 10 1 7623.2.a.co 4
77.j odd 10 1 5929.2.a.bb 4
77.l even 10 2 539.2.f.d 8
77.l even 10 1 5929.2.a.bi 4
77.n even 30 4 539.2.q.b 16
77.o odd 30 4 539.2.q.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.a 8 11.d odd 10 2
539.2.f.d 8 77.l even 10 2
539.2.q.b 16 77.n even 30 4
539.2.q.c 16 77.o odd 30 4
693.2.m.g 8 33.f even 10 2
847.2.a.k 4 11.c even 5 1
847.2.a.l 4 11.d odd 10 1
847.2.f.p 8 11.b odd 2 1
847.2.f.p 8 11.d odd 10 1
847.2.f.q 8 11.c even 5 2
847.2.f.s 8 1.a even 1 1 trivial
847.2.f.s 8 11.c even 5 1 inner
5929.2.a.bb 4 77.j odd 10 1
5929.2.a.bi 4 77.l even 10 1
7623.2.a.ch 4 33.f even 10 1
7623.2.a.co 4 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\):

\( T_{2}^{8} - T_{2}^{7} + 6T_{2}^{6} - 11T_{2}^{5} + 21T_{2}^{4} - 5T_{2}^{3} + 10T_{2}^{2} + 25T_{2} + 25 \) Copy content Toggle raw display
\( T_{3}^{4} - 3T_{3}^{3} + 4T_{3}^{2} - 2T_{3} + 1 \) Copy content Toggle raw display
\( T_{13}^{8} - 5T_{13}^{7} - 4T_{13}^{6} + 105T_{13}^{5} + 611T_{13}^{4} - 735T_{13}^{3} + 2456T_{13}^{2} - 1015T_{13} + 841 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + 6 T^{6} - 11 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( (T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 3 T^{7} + 12 T^{6} - 45 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 5 T^{7} - 4 T^{6} + 105 T^{5} + \cdots + 841 \) Copy content Toggle raw display
$17$ \( T^{8} + 14 T^{7} + 113 T^{6} + \cdots + 5041 \) Copy content Toggle raw display
$19$ \( T^{8} + 6 T^{7} + 11 T^{6} + \cdots + 21025 \) Copy content Toggle raw display
$23$ \( (T^{4} + 8 T^{3} - 9 T^{2} - 150 T - 205)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 6 T^{7} + 81 T^{6} + \cdots + 164025 \) Copy content Toggle raw display
$31$ \( T^{8} - 14 T^{7} + 91 T^{6} - 279 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$37$ \( T^{8} - T^{7} + 26 T^{6} - 121 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$41$ \( T^{8} + 18 T^{7} + 227 T^{6} + \cdots + 6241 \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T - 41)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} - 7 T^{7} + 179 T^{6} + \cdots + 93025 \) Copy content Toggle raw display
$53$ \( T^{8} - 7 T^{7} + 238 T^{6} + \cdots + 755161 \) Copy content Toggle raw display
$59$ \( T^{8} + 109 T^{6} - 675 T^{5} + \cdots + 1413721 \) Copy content Toggle raw display
$61$ \( T^{8} + 12 T^{7} + 269 T^{6} + \cdots + 990025 \) Copy content Toggle raw display
$67$ \( (T^{4} + 15 T^{3} + 67 T^{2} + 45 T - 199)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 21 T^{7} + 293 T^{6} + \cdots + 982081 \) Copy content Toggle raw display
$73$ \( T^{8} + 8 T^{7} + 69 T^{6} + \cdots + 24750625 \) Copy content Toggle raw display
$79$ \( T^{8} + T^{7} - 2 T^{6} - 5 T^{5} + \cdots + 73441 \) Copy content Toggle raw display
$83$ \( T^{8} - 22 T^{7} + 479 T^{6} + \cdots + 5040025 \) Copy content Toggle raw display
$89$ \( (T^{4} + 17 T^{3} - 44 T^{2} - 1120 T - 755)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 15 T^{7} + 230 T^{6} + \cdots + 17850625 \) Copy content Toggle raw display
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