Properties

Label 847.2.f.w
Level $847$
Weight $2$
Character orbit 847.f
Analytic conductor $6.763$
Analytic rank $0$
Dimension $16$
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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 3 x^{15} + 14 x^{14} - 32 x^{13} + 86 x^{12} - 145 x^{11} + 245 x^{10} - 245 x^{9} + 640 x^{8} - 1175 x^{7} + 2135 x^{6} - 2300 x^{5} + 1850 x^{4} - 925 x^{3} + 700 x^{2} + \cdots + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + ( - \beta_{15} - \beta_{14} + \beta_{11} - \beta_{10}) q^{3} + (\beta_{10} + \beta_{9} + \beta_{7}) q^{4} + (\beta_{15} + 2 \beta_{10} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{3} - 1) q^{5} + ( - \beta_{15} + \beta_{12} - 2 \beta_{10} - \beta_{9} + \beta_{8} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \cdots - 1) q^{6}+ \cdots + ( - \beta_{15} + \beta_{12} - \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + ( - \beta_{15} - \beta_{14} + \beta_{11} - \beta_{10}) q^{3} + (\beta_{10} + \beta_{9} + \beta_{7}) q^{4} + (\beta_{15} + 2 \beta_{10} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{3} - 1) q^{5} + ( - \beta_{15} + \beta_{12} - 2 \beta_{10} - \beta_{9} + \beta_{8} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \cdots - 1) q^{6}+ \cdots + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 5 q^{5} - 2 q^{6} - 4 q^{7} + 5 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 5 q^{5} - 2 q^{6} - 4 q^{7} + 5 q^{8} - 2 q^{9} + 12 q^{10} + 18 q^{12} + 13 q^{13} - 3 q^{14} + 7 q^{15} - 18 q^{16} + 10 q^{17} - 19 q^{18} - 6 q^{19} - 24 q^{20} + 8 q^{21} + 32 q^{23} + 45 q^{24} - 23 q^{25} + 33 q^{26} - 20 q^{27} - 11 q^{28} - 12 q^{29} + 38 q^{30} - 2 q^{31} + 32 q^{32} - 24 q^{34} - 5 q^{35} - 38 q^{36} - 11 q^{37} + 15 q^{38} - 24 q^{39} + 5 q^{40} + 20 q^{41} - 2 q^{42} - 8 q^{43} + 70 q^{45} + 38 q^{46} + 7 q^{47} + 39 q^{48} - 4 q^{49} - 58 q^{50} + 16 q^{51} + 8 q^{52} - 41 q^{53} + 60 q^{54} + 9 q^{57} - 5 q^{58} - 18 q^{59} + 25 q^{60} - 12 q^{61} - 61 q^{62} - 12 q^{63} - 3 q^{64} - 8 q^{65} - 38 q^{67} - 7 q^{68} - 30 q^{69} + 12 q^{70} + q^{71} + 35 q^{72} + 60 q^{73} - 4 q^{74} + 4 q^{75} + 52 q^{76} - 58 q^{78} - 15 q^{79} + 83 q^{80} + 6 q^{81} - 6 q^{82} + 20 q^{83} - 17 q^{84} - 9 q^{85} + 48 q^{86} - 72 q^{87} + 74 q^{89} + 16 q^{90} - 7 q^{91} + 20 q^{92} - 53 q^{93} + 66 q^{94} - 53 q^{95} + 48 q^{96} - 35 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 14 x^{14} - 32 x^{13} + 86 x^{12} - 145 x^{11} + 245 x^{10} - 245 x^{9} + 640 x^{8} - 1175 x^{7} + 2135 x^{6} - 2300 x^{5} + 1850 x^{4} - 925 x^{3} + 700 x^{2} + \cdots + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 33722118880975 \nu^{15} - 215936836813390 \nu^{14} + 154302737507927 \nu^{13} + \cdots + 88\!\cdots\!50 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 686277383720070 \nu^{15} + \cdots - 15\!\cdots\!35 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 874936745197272 \nu^{15} - 514706296938285 \nu^{14} + \cdots - 92\!\cdots\!35 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15\!\cdots\!39 \nu^{15} + \cdots + 95\!\cdots\!50 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\!\cdots\!90 \nu^{15} + \cdots - 18\!\cdots\!25 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16\!\cdots\!90 \nu^{15} + \cdots + 18\!\cdots\!25 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 35\!\cdots\!14 \nu^{15} + \cdots + 82\!\cdots\!05 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 38\!\cdots\!02 \nu^{15} + \cdots + 22\!\cdots\!25 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 41\!\cdots\!60 \nu^{15} + \cdots - 29\!\cdots\!75 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 73\!\cdots\!73 \nu^{15} + \cdots - 84\!\cdots\!55 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 54\!\cdots\!19 \nu^{15} + \cdots + 86\!\cdots\!50 ) / 22\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11\!\cdots\!43 \nu^{15} + \cdots + 59\!\cdots\!75 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 16\!\cdots\!01 \nu^{15} + \cdots + 16\!\cdots\!55 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 16\!\cdots\!14 \nu^{15} + \cdots - 10\!\cdots\!35 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 31\!\cdots\!64 \nu^{15} + \cdots + 34\!\cdots\!95 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{6} + \beta_{5} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + 2\beta_{10} - 2\beta_{8} + 3\beta_{7} + \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + \beta_{13} + \beta_{12} - \beta_{10} + \beta_{8} - 5\beta_{6} - \beta_{5} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} - \beta_{14} - \beta_{11} - 8 \beta_{10} - 6 \beta_{9} - 15 \beta_{7} + \beta_{5} + \beta_{4} - 6 \beta_{2} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 7 \beta_{14} - 7 \beta_{13} - 7 \beta_{12} + 7 \beta_{11} - 8 \beta_{8} + 8 \beta_{7} + 10 \beta_{6} - 10 \beta_{4} - \beta_{3} + 8 \beta_{2} - 18 \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7 \beta_{15} + \beta_{14} - 7 \beta_{13} + 11 \beta_{12} + 54 \beta_{10} + 35 \beta_{9} + 40 \beta_{8} + 47 \beta_{7} - 10 \beta_{6} - 10 \beta_{5} - 2 \beta_{4} + 6 \beta_{3} + 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 42 \beta_{15} + 54 \beta_{14} + 13 \beta_{12} - 45 \beta_{11} + 12 \beta_{10} - 13 \beta_{9} + 30 \beta_{8} - 86 \beta_{7} + 76 \beta_{5} + 165 \beta_{4} + 42 \beta_{3} - 58 \beta_{2} + 28 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 9 \beta_{15} + 38 \beta_{13} - 210 \beta_{12} + 89 \beta_{11} - 234 \beta_{10} - 89 \beta_{9} - 308 \beta_{8} + 80 \beta_{6} + 28 \beta_{5} + 210 \beta_{2} - 28 \beta _1 - 98 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 346 \beta_{15} - 346 \beta_{14} + 248 \beta_{13} + 117 \beta_{11} + 38 \beta_{10} + 290 \beta_{9} + 614 \beta_{7} - 524 \beta_{6} - 1000 \beta_{5} - 1000 \beta_{4} - 248 \beta_{3} + 290 \beta_{2} + 476 \beta _1 - 94 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 56 \beta_{14} - 56 \beta_{13} + 1290 \beta_{12} - 1290 \beta_{11} + 1992 \beta_{8} - 1992 \beta_{7} - 272 \beta_{6} + 272 \beta_{4} - 131 \beta_{3} - 1931 \beta_{2} + 328 \beta _1 + 1406 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2174 \beta_{15} + 1477 \beta_{14} - 2174 \beta_{13} - 913 \beta_{12} - 447 \beta_{10} - 1890 \beta_{9} - 1625 \beta_{8} - 2621 \beta_{7} + 6160 \beta_{6} + 6160 \beta_{5} + 3461 \beta_{4} + 697 \beta_{3} + \cdots - 3461 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 284 \beta_{15} + 848 \beta_{14} - 4374 \beta_{12} + 8050 \beta_{11} + 8579 \beta_{10} + 4374 \beta_{9} - 8295 \beta_{8} + 21093 \beta_{7} - 2273 \beta_{5} - 4375 \beta_{4} + 284 \beta_{3} + 12424 \beta_{2} + \cdots - 8669 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 8898 \beta_{15} + 13556 \beta_{13} + 12425 \beta_{12} - 6647 \beta_{11} + 2237 \beta_{10} + 6647 \beta_{9} + 17819 \beta_{8} - 38325 \beta_{6} - 22394 \beta_{5} - 12425 \beta_{2} + 22394 \beta _1 + 5394 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 3382 \beta_{15} - 3382 \beta_{14} + 1131 \beta_{13} - 29041 \beta_{11} - 85364 \beta_{10} - 50750 \beta_{9} - 133987 \beta_{7} + 17557 \beta_{6} + 31350 \beta_{5} + 31350 \beta_{4} - 1131 \beta_{3} + \cdots + 31232 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 54132 \beta_{14} - 54132 \beta_{13} - 82100 \beta_{12} + 82100 \beta_{11} - 120596 \beta_{8} + 120596 \beta_{7} + 143446 \beta_{6} - 143446 \beta_{4} - 30172 \beta_{3} + 128698 \beta_{2} + \cdots - 67213 \) Copy content Toggle raw display

Character values

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

\(n\) \(122\) \(365\)
\(\chi(n)\) \(1\) \(\beta_{8}\)

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
−1.38112 + 1.00344i
0.183009 0.132964i
0.901622 0.655067i
1.60551 1.16647i
−0.788594 2.42704i
−0.206962 0.636964i
0.435488 + 1.34029i
0.751051 + 2.31150i
−1.38112 1.00344i
0.183009 + 0.132964i
0.901622 + 0.655067i
1.60551 + 1.16647i
−0.788594 + 2.42704i
−0.206962 + 0.636964i
0.435488 1.34029i
0.751051 2.31150i
−0.527541 1.62360i −1.85391 1.34694i −0.739758 + 0.537466i 1.25658 3.86735i −1.20889 + 3.72058i −0.809017 + 0.587785i −1.49936 1.08935i 0.695665 + 2.14104i −6.94194
148.2 0.0699031 + 0.215140i 0.177280 + 0.128801i 1.57664 1.14549i −0.771159 + 2.37338i −0.0153178 + 0.0471435i −0.809017 + 0.587785i 0.722670 + 0.525051i −0.912213 2.80750i −0.564516
148.3 0.344389 + 1.05992i 2.31283 + 1.68037i 0.613206 0.445520i 1.06799 3.28693i −0.984546 + 3.03012i −0.809017 + 0.587785i 2.48664 + 1.80665i 1.59850 + 4.91966i 3.85168
148.4 0.613249 + 1.88739i −2.25424 1.63780i −1.56812 + 1.13930i −0.00832008 + 0.0256066i 1.70875 5.25900i −0.809017 + 0.587785i 0.0990607 + 0.0719718i 1.47215 + 4.53082i −0.0534317
323.1 −2.06456 1.49999i −0.126882 + 0.390502i 1.39441 + 4.29156i −2.77410 + 2.01550i 0.847707 0.615895i 0.309017 + 0.951057i 1.98127 6.09772i 2.29066 + 1.66426i 8.75055
323.2 −0.541834 0.393666i 0.970243 2.98610i −0.479422 1.47551i −1.73435 + 1.26008i −1.70124 + 1.23602i 0.309017 + 0.951057i −0.735015 + 2.26214i −5.54838 4.03113i 1.43578
323.3 1.14012 + 0.828347i −0.668522 + 2.05750i −0.00431527 0.0132810i 1.48162 1.07646i −2.46652 + 1.79203i 0.309017 + 0.951057i 0.877057 2.69930i −1.35932 0.987607i 2.58091
323.4 1.96628 + 1.42858i 0.443194 1.36401i 1.20736 + 3.71587i −1.01825 + 0.739805i 2.82005 2.04888i 0.309017 + 0.951057i −1.43233 + 4.40826i 0.762946 + 0.554312i −3.05904
372.1 −0.527541 + 1.62360i −1.85391 + 1.34694i −0.739758 0.537466i 1.25658 + 3.86735i −1.20889 3.72058i −0.809017 0.587785i −1.49936 + 1.08935i 0.695665 2.14104i −6.94194
372.2 0.0699031 0.215140i 0.177280 0.128801i 1.57664 + 1.14549i −0.771159 2.37338i −0.0153178 0.0471435i −0.809017 0.587785i 0.722670 0.525051i −0.912213 + 2.80750i −0.564516
372.3 0.344389 1.05992i 2.31283 1.68037i 0.613206 + 0.445520i 1.06799 + 3.28693i −0.984546 3.03012i −0.809017 0.587785i 2.48664 1.80665i 1.59850 4.91966i 3.85168
372.4 0.613249 1.88739i −2.25424 + 1.63780i −1.56812 1.13930i −0.00832008 0.0256066i 1.70875 + 5.25900i −0.809017 0.587785i 0.0990607 0.0719718i 1.47215 4.53082i −0.0534317
729.1 −2.06456 + 1.49999i −0.126882 0.390502i 1.39441 4.29156i −2.77410 2.01550i 0.847707 + 0.615895i 0.309017 0.951057i 1.98127 + 6.09772i 2.29066 1.66426i 8.75055
729.2 −0.541834 + 0.393666i 0.970243 + 2.98610i −0.479422 + 1.47551i −1.73435 1.26008i −1.70124 1.23602i 0.309017 0.951057i −0.735015 2.26214i −5.54838 + 4.03113i 1.43578
729.3 1.14012 0.828347i −0.668522 2.05750i −0.00431527 + 0.0132810i 1.48162 + 1.07646i −2.46652 1.79203i 0.309017 0.951057i 0.877057 + 2.69930i −1.35932 + 0.987607i 2.58091
729.4 1.96628 1.42858i 0.443194 + 1.36401i 1.20736 3.71587i −1.01825 0.739805i 2.82005 + 2.04888i 0.309017 0.951057i −1.43233 4.40826i 0.762946 0.554312i −3.05904
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 729.4
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.w 16
11.b odd 2 1 847.2.f.v 16
11.c even 5 2 77.2.f.b 16
11.c even 5 1 847.2.a.p 8
11.c even 5 1 inner 847.2.f.w 16
11.d odd 10 1 847.2.a.o 8
11.d odd 10 1 847.2.f.v 16
11.d odd 10 2 847.2.f.x 16
33.f even 10 1 7623.2.a.cw 8
33.h odd 10 2 693.2.m.i 16
33.h odd 10 1 7623.2.a.ct 8
77.j odd 10 2 539.2.f.e 16
77.j odd 10 1 5929.2.a.bt 8
77.l even 10 1 5929.2.a.bs 8
77.m even 15 4 539.2.q.g 32
77.p odd 30 4 539.2.q.f 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.b 16 11.c even 5 2
539.2.f.e 16 77.j odd 10 2
539.2.q.f 32 77.p odd 30 4
539.2.q.g 32 77.m even 15 4
693.2.m.i 16 33.h odd 10 2
847.2.a.o 8 11.d odd 10 1
847.2.a.p 8 11.c even 5 1
847.2.f.v 16 11.b odd 2 1
847.2.f.v 16 11.d odd 10 1
847.2.f.w 16 1.a even 1 1 trivial
847.2.f.w 16 11.c even 5 1 inner
847.2.f.x 16 11.d odd 10 2
5929.2.a.bs 8 77.l even 10 1
5929.2.a.bt 8 77.j odd 10 1
7623.2.a.ct 8 33.h odd 10 1
7623.2.a.cw 8 33.f even 10 1

Hecke kernels

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

\( T_{2}^{16} - 2 T_{2}^{15} + 4 T_{2}^{14} - 3 T_{2}^{13} + 31 T_{2}^{12} - 70 T_{2}^{11} + 265 T_{2}^{10} - 385 T_{2}^{9} + 840 T_{2}^{8} - 975 T_{2}^{7} + 1085 T_{2}^{6} - 200 T_{2}^{5} + 275 T_{2}^{4} + 325 T_{2}^{3} + 450 T_{2}^{2} - 50 T_{2} + 25 \) Copy content Toggle raw display
\( T_{3}^{16} + 2 T_{3}^{15} + 9 T_{3}^{14} + 26 T_{3}^{13} + 104 T_{3}^{12} + 120 T_{3}^{11} + 682 T_{3}^{10} + 1762 T_{3}^{9} + 6213 T_{3}^{8} + 10062 T_{3}^{7} + 18112 T_{3}^{6} + 11160 T_{3}^{5} + 32304 T_{3}^{4} - 2464 T_{3}^{3} + \cdots + 256 \) Copy content Toggle raw display
\( T_{13}^{16} - 13 T_{13}^{15} + 124 T_{13}^{14} - 829 T_{13}^{13} + 4689 T_{13}^{12} - 21415 T_{13}^{11} + 84482 T_{13}^{10} - 254273 T_{13}^{9} + 648843 T_{13}^{8} - 1416018 T_{13}^{7} + 3693312 T_{13}^{6} + \cdots + 188897536 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 2 T^{15} + 4 T^{14} - 3 T^{13} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( T^{16} + 2 T^{15} + 9 T^{14} + 26 T^{13} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{16} + 5 T^{15} + 34 T^{14} + 169 T^{13} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} - 13 T^{15} + \cdots + 188897536 \) Copy content Toggle raw display
$17$ \( T^{16} - 10 T^{15} + 31 T^{14} + \cdots + 2611456 \) Copy content Toggle raw display
$19$ \( T^{16} + 6 T^{15} + 79 T^{14} + \cdots + 62726400 \) Copy content Toggle raw display
$23$ \( (T^{8} - 16 T^{7} + 40 T^{6} + 342 T^{5} + \cdots + 859)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + 12 T^{15} + 46 T^{14} + \cdots + 245025 \) Copy content Toggle raw display
$31$ \( T^{16} + 2 T^{15} + \cdots + 2629638400 \) Copy content Toggle raw display
$37$ \( T^{16} + 11 T^{15} + 74 T^{14} + \cdots + 212521 \) Copy content Toggle raw display
$41$ \( T^{16} - 20 T^{15} + 171 T^{14} + \cdots + 13424896 \) Copy content Toggle raw display
$43$ \( (T^{8} + 4 T^{7} - 105 T^{6} - 268 T^{5} + \cdots - 971)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} - 7 T^{15} + \cdots + 5345713926400 \) Copy content Toggle raw display
$53$ \( T^{16} + 41 T^{15} + \cdots + 310840815961 \) Copy content Toggle raw display
$59$ \( T^{16} + 18 T^{15} + \cdots + 187142400 \) Copy content Toggle raw display
$61$ \( T^{16} + 12 T^{15} + \cdots + 253446400 \) Copy content Toggle raw display
$67$ \( (T^{8} + 19 T^{7} - 16 T^{6} - 1160 T^{5} + \cdots - 27395)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} - T^{15} - 71 T^{14} + \cdots + 379119841 \) Copy content Toggle raw display
$73$ \( T^{16} - 60 T^{15} + \cdots + 105069332736 \) Copy content Toggle raw display
$79$ \( T^{16} + 15 T^{15} + \cdots + 15858514175625 \) Copy content Toggle raw display
$83$ \( T^{16} - 20 T^{15} + \cdots + 756470016 \) Copy content Toggle raw display
$89$ \( (T^{8} - 37 T^{7} + 320 T^{6} + \cdots + 952400)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + 35 T^{15} + \cdots + 4647025244416 \) Copy content Toggle raw display
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