Properties

Label 847.2.f.q
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + ( - \beta_{3} - \beta_{2}) q^{3} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{4} + (\beta_{7} + \beta_{5} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{7} - \beta_{5} + \beta_{4}) q^{6} - \beta_{3} q^{7} + ( - 2 \beta_{6} - \beta_{5} - 3 \beta_{3} - 3 \beta_{2} + 2) q^{8} + (2 \beta_{6} + \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + ( - \beta_{3} - \beta_{2}) q^{3} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{4} + (\beta_{7} + \beta_{5} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{7} - \beta_{5} + \beta_{4}) q^{6} - \beta_{3} q^{7} + ( - 2 \beta_{6} - \beta_{5} - 3 \beta_{3} - 3 \beta_{2} + 2) q^{8} + (2 \beta_{6} + \beta_{3} - 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_{6} + 2 \beta_{5} + \beta_{4} + 2 \beta_{3} - 2 \beta_1 - 2) q^{13} - \beta_{7} q^{14} + ( - \beta_{7} - \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{15} + ( - \beta_{7} - \beta_{5} + 2 \beta_{4} + \beta_{2} - \beta_1) q^{16} + ( - \beta_{7} + 3 \beta_{6} - \beta_{5} + \beta_{4} + 3 \beta_{3} + \beta_{2}) q^{17} + (\beta_{7} + \beta_{4} - 2 \beta_1) q^{18} + (2 \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} + 2) q^{19} + (2 \beta_{6} + \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - \beta_1 + 2) q^{20} + ( - \beta_{6} - \beta_{2}) q^{21} + ( - 2 \beta_{7} - \beta_{6} - \beta_{2} + 2 \beta_1 - 2) q^{23} + ( - \beta_{6} + \beta_{4} + 3 \beta_{3} - 3) q^{24} + (3 \beta_{7} - 2 \beta_{6} - \beta_{3} - \beta_{2} + 2) q^{25} + (\beta_{3} - 3 \beta_{2} + \beta_1 + 3) q^{26} + ( - 4 \beta_{6} - 4 \beta_{3} - 3 \beta_{2}) q^{27} + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{28} + ( - 3 \beta_{7} - 3 \beta_{4} + 3 \beta_1) q^{29} + ( - 2 \beta_{7} - \beta_{6} - 2 \beta_{5} - 4 \beta_{3} - 4 \beta_{2} + 1) q^{30} + (3 \beta_{6} + \beta_{5} + \beta_{4} + 3 \beta_{3} - \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_{6} - \beta_{4}) q^{35} + ( - \beta_{7} + \beta_{6} + \beta_{5} - 1) q^{36} + (\beta_{7} + \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{37} + (\beta_{7} + 4 \beta_{6} + \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 7 \beta_{2} + \beta_1) q^{38} + ( - \beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_{2} + 2 \beta_1) q^{39} + ( - 3 \beta_{7} - 3 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} + 3) q^{40} + ( - 2 \beta_{6} + 2 \beta_{5} + \beta_{3} + \beta_{2} + 2) q^{41} + ( - \beta_{5} + \beta_{4} + \beta_1) 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} + (2 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} + 4 \beta_{3} + 3 \beta_1 - 4) q^{46} + ( - \beta_{7} - \beta_{6} + 4 \beta_{5} - 3 \beta_{3} - 3 \beta_{2} + 1) q^{47} + (\beta_{7} + \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{48} - \beta_{2} q^{49} + (2 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} - \beta_{4} + 3 \beta_{3} + 9 \beta_{2} - \beta_1) q^{50} + (\beta_{7} + \beta_{4} - 4 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{51} + (6 \beta_{6} + \beta_{5} + 3 \beta_{3} + 3 \beta_{2} - 6) q^{52} + (3 \beta_{6} - 3 \beta_{5} + 4 \beta_{4} - \beta_{3} + 3 \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} + (\beta_{6} + 2 \beta_{5} - 3 \beta_{4} + \beta_{3} - 2 \beta_1 - 1) q^{57} + (3 \beta_{6} - 3 \beta_{5} - 3 \beta_{3} - 3 \beta_{2} - 3) q^{58} + ( - \beta_{7} - \beta_{4} + 6 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{59} + ( - 4 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} + 3 \beta_{4} - 4 \beta_{3} - 6 \beta_{2} + \beta_1) q^{60} + (5 \beta_{7} + 5 \beta_{5} - \beta_{4} - 7 \beta_{2} - 4 \beta_1) q^{61} + (2 \beta_{7} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{62} + ( - 2 \beta_{6} - \beta_{3} - \beta_{2} + 2) q^{63} + ( - 2 \beta_{6} + \beta_{5} - 3 \beta_{4} - \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} + (2 \beta_{6} + \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_1 - 2) q^{68} + (2 \beta_{7} + \beta_{6} + 4 \beta_{3} + 4 \beta_{2} - 1) q^{69} + ( - 3 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{70} + (2 \beta_{7} - 5 \beta_{6} + 2 \beta_{5} - 3 \beta_{4} - 5 \beta_{3} - \beta_{2} + \beta_1) q^{71} + (\beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{72} + (5 \beta_{7} + 5 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 3) q^{73} + (\beta_{7} - 3 \beta_{6} - 2 \beta_{3} - 2 \beta_{2} + 3) q^{74} + (\beta_{6} + 3 \beta_{5} - 3 \beta_{4} + \beta_{3} - 3 \beta_1 - 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_{5} - \beta_{4} - 2 \beta_{3} + 2 \beta_1 + 2) q^{79} + ( - \beta_{7} + \beta_{6} - 4 \beta_{5} - 4 \beta_{3} - 4 \beta_{2} - 1) q^{80} + (4 \beta_{3} + 6 \beta_{2} - 6) q^{81} + (\beta_{7} + 4 \beta_{6} + \beta_{5} - 5 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{82} + (3 \beta_{7} - 7 \beta_{6} + 3 \beta_{5} - 5 \beta_{4} - 7 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{83} + (\beta_{7} + \beta_{4} + 2 \beta_{3} + \beta_{2} - 1) q^{84} + ( - \beta_{7} - 3 \beta_{6} + \beta_{5} - 3 \beta_{3} - 3 \beta_{2} + 3) q^{85} + ( - 6 \beta_{5} + 5 \beta_{4} + 6 \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} + (3 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{90} + (\beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 2) q^{91} + (3 \beta_{7} + 3 \beta_{4} + 6 \beta_{3} + 7 \beta_{2} + 2 \beta_1 - 7) q^{92} + ( - \beta_{4} + 3 \beta_{2} + \beta_1) q^{93} + ( - 4 \beta_{7} + 7 \beta_{6} - 4 \beta_{5} - 2 \beta_{4} + 7 \beta_{3} + \beta_{2} + 6 \beta_1) q^{94} + (3 \beta_{7} + 3 \beta_{4} + 6 \beta_{3} + 5 \beta_{2} - 5) q^{95} + (3 \beta_{6} + 2 \beta_{3} + 2 \beta_{2} - 3) q^{96} + (4 \beta_{6} + 5 \beta_{5} - 2 \beta_{3} - 5 \beta_1 + 2) q^{97} + ( - \beta_{5} + \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} - 4 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} - 2 q^{7} - 3 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} - 4 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} - 2 q^{7} - 3 q^{8} - 2 q^{9} + 28 q^{10} - 14 q^{12} - 5 q^{13} + q^{14} + 6 q^{15} - 3 q^{16} + 11 q^{17} - 4 q^{18} + 9 q^{19} + 21 q^{20} - 4 q^{21} - 16 q^{23} - 21 q^{24} + 5 q^{25} + 21 q^{26} - 22 q^{27} - 7 q^{28} + 9 q^{29} - 14 q^{30} - 11 q^{31} + 20 q^{32} - 24 q^{34} + 3 q^{35} - 2 q^{36} + 6 q^{37} + 35 q^{38} + 5 q^{39} + 16 q^{40} + 22 q^{41} - 3 q^{42} - 16 q^{43} + 18 q^{45} - 29 q^{46} + 7 q^{47} + 4 q^{48} - 2 q^{49} + 34 q^{50} - 3 q^{51} - 21 q^{52} + 2 q^{53} - 4 q^{54} - 18 q^{56} + 3 q^{57} - 39 q^{58} + 25 q^{59} - 38 q^{60} - 7 q^{61} + 5 q^{62} + 8 q^{63} + q^{64} - 24 q^{65} - 30 q^{67} - 8 q^{68} + 8 q^{69} - 2 q^{70} - 14 q^{71} - 3 q^{72} - 3 q^{73} + 9 q^{74} + 5 q^{75} + 52 q^{76} - 18 q^{78} + 9 q^{79} - 33 q^{80} - 28 q^{81} + 31 q^{82} - 23 q^{83} - 4 q^{84} + 10 q^{85} - 17 q^{86} - 12 q^{87} - 34 q^{89} - 2 q^{90} + 5 q^{91} - 34 q^{92} + 8 q^{93} + 30 q^{94} - 24 q^{95} - 10 q^{96} + 30 q^{97} - 4 q^{98}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(365\)
\(\chi(n)\) \(1\) \(-\beta_{3}\)

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
1.43801 1.04478i
−0.628998 + 0.456994i
0.453245 + 1.39494i
−0.762262 2.34600i
1.43801 + 1.04478i
−0.628998 0.456994i
0.453245 1.39494i
−0.762262 + 2.34600i
−0.549273 1.69049i −0.500000 0.363271i −0.938015 + 0.681508i −0.858290 + 2.64154i −0.339469 + 1.04478i −0.809017 + 0.587785i −1.20872 0.878189i −0.809017 2.48990i 4.93693
148.2 0.240256 + 0.739431i −0.500000 0.363271i 1.12900 0.820265i −0.0687611 + 0.211625i 0.148486 0.456994i −0.809017 + 0.587785i 2.13577 + 1.55173i −0.809017 2.48990i −0.173002
323.1 −1.18661 0.862123i −0.500000 + 1.53884i 0.0467549 + 0.143897i −0.377594 + 0.274338i 1.91998 1.39494i 0.309017 + 0.951057i −0.837913 + 2.57883i 0.309017 + 0.224514i 0.684570
323.2 1.99563 + 1.44991i −0.500000 + 1.53884i 1.26226 + 3.88484i 2.80464 2.03769i −3.22899 + 2.34600i 0.309017 + 0.951057i −1.58914 + 4.89086i 0.309017 + 0.224514i 8.55150
372.1 −0.549273 + 1.69049i −0.500000 + 0.363271i −0.938015 0.681508i −0.858290 2.64154i −0.339469 1.04478i −0.809017 0.587785i −1.20872 + 0.878189i −0.809017 + 2.48990i 4.93693
372.2 0.240256 0.739431i −0.500000 + 0.363271i 1.12900 + 0.820265i −0.0687611 0.211625i 0.148486 + 0.456994i −0.809017 0.587785i 2.13577 1.55173i −0.809017 + 2.48990i −0.173002
729.1 −1.18661 + 0.862123i −0.500000 1.53884i 0.0467549 0.143897i −0.377594 0.274338i 1.91998 + 1.39494i 0.309017 0.951057i −0.837913 2.57883i 0.309017 0.224514i 0.684570
729.2 1.99563 1.44991i −0.500000 1.53884i 1.26226 3.88484i 2.80464 + 2.03769i −3.22899 2.34600i 0.309017 0.951057i −1.58914 4.89086i 0.309017 0.224514i 8.55150
\(n\): 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.q 8
11.b odd 2 1 77.2.f.a 8
11.c even 5 1 847.2.a.k 4
11.c even 5 1 inner 847.2.f.q 8
11.c even 5 2 847.2.f.s 8
11.d odd 10 1 77.2.f.a 8
11.d odd 10 1 847.2.a.l 4
11.d odd 10 2 847.2.f.p 8
33.d even 2 1 693.2.m.g 8
33.f even 10 1 693.2.m.g 8
33.f even 10 1 7623.2.a.ch 4
33.h odd 10 1 7623.2.a.co 4
77.b even 2 1 539.2.f.d 8
77.h odd 6 2 539.2.q.c 16
77.i even 6 2 539.2.q.b 16
77.j odd 10 1 5929.2.a.bb 4
77.l even 10 1 539.2.f.d 8
77.l even 10 1 5929.2.a.bi 4
77.n even 30 2 539.2.q.b 16
77.o odd 30 2 539.2.q.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.a 8 11.b odd 2 1
77.2.f.a 8 11.d odd 10 1
539.2.f.d 8 77.b even 2 1
539.2.f.d 8 77.l even 10 1
539.2.q.b 16 77.i even 6 2
539.2.q.b 16 77.n even 30 2
539.2.q.c 16 77.h odd 6 2
539.2.q.c 16 77.o odd 30 2
693.2.m.g 8 33.d even 2 1
693.2.m.g 8 33.f even 10 1
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.d odd 10 2
847.2.f.q 8 1.a even 1 1 trivial
847.2.f.q 8 11.c even 5 1 inner
847.2.f.s 8 11.c even 5 2
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 \(S_{2}^{\mathrm{new}}(847, [\chi])\):

\( T_{2}^{8} - T_{2}^{7} + T_{2}^{6} - T_{2}^{5} + 16T_{2}^{4} + 25T_{2}^{3} + 35T_{2}^{2} + 25 \) Copy content Toggle raw display
\( T_{3}^{4} + 2T_{3}^{3} + 4T_{3}^{2} + 3T_{3} + 1 \) Copy content Toggle raw display
\( T_{13}^{8} + 5T_{13}^{7} + 61T_{13}^{6} + 155T_{13}^{5} + 936T_{13}^{4} + 2815T_{13}^{3} + 4341T_{13}^{2} + 2900T_{13} + 841 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + T^{6} - T^{5} + 16 T^{4} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( (T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 3 T^{7} + 7 T^{6} - 15 T^{5} + \cdots + 1 \) 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
$13$ \( T^{8} + 5 T^{7} + 61 T^{6} + 155 T^{5} + \cdots + 841 \) Copy content Toggle raw display
$17$ \( T^{8} - 11 T^{7} + 73 T^{6} + \cdots + 5041 \) Copy content Toggle raw display
$19$ \( T^{8} - 9 T^{7} + 81 T^{6} + \cdots + 21025 \) Copy content Toggle raw display
$23$ \( (T^{4} + 8 T^{3} - 9 T^{2} - 150 T - 205)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 9 T^{7} + 36 T^{6} + \cdots + 164025 \) Copy content Toggle raw display
$31$ \( T^{8} + 11 T^{7} + 96 T^{6} + 431 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} + 11 T^{6} + 39 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$41$ \( T^{8} - 22 T^{7} + 242 T^{6} + \cdots + 6241 \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T - 41)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} - 7 T^{7} - 6 T^{6} + \cdots + 93025 \) Copy content Toggle raw display
$53$ \( T^{8} - 2 T^{7} - 67 T^{6} + \cdots + 755161 \) Copy content Toggle raw display
$59$ \( T^{8} - 25 T^{7} + 324 T^{6} + \cdots + 1413721 \) Copy content Toggle raw display
$61$ \( T^{8} + 7 T^{7} + 14 T^{6} + \cdots + 990025 \) Copy content Toggle raw display
$67$ \( (T^{4} + 15 T^{3} + 67 T^{2} + 45 T - 199)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 14 T^{7} + 133 T^{6} + \cdots + 982081 \) Copy content Toggle raw display
$73$ \( T^{8} + 3 T^{7} + 144 T^{6} + \cdots + 24750625 \) Copy content Toggle raw display
$79$ \( T^{8} - 9 T^{7} + 83 T^{6} + \cdots + 73441 \) Copy content Toggle raw display
$83$ \( T^{8} + 23 T^{7} + 264 T^{6} + \cdots + 5040025 \) Copy content Toggle raw display
$89$ \( (T^{4} + 17 T^{3} - 44 T^{2} - 1120 T - 755)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 30 T^{7} + 555 T^{6} + \cdots + 17850625 \) Copy content Toggle raw display
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