Properties

Label 418.2.f.g
Level $418$
Weight $2$
Character orbit 418.f
Analytic conductor $3.338$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.f (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.33774680449\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - x^{15} + 7 x^{14} - 13 x^{13} + 51 x^{12} - 74 x^{11} + 332 x^{10} - 614 x^{9} + 1832 x^{8} - 2960 x^{7} + 5348 x^{6} - 6872 x^{5} + 8232 x^{4} - 6344 x^{3} + 3984 x^{2} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + (\beta_{10} - \beta_{8} + \beta_{6} + 1) q^{2} + \beta_1 q^{3} + \beta_{6} q^{4} + ( - \beta_{15} + \beta_{12} + \beta_{11} + \beta_{9} - \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{5} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{6} + ( - \beta_{15} - \beta_{13} - \beta_{10} - \beta_{7} + \beta_{6} + \beta_{4} - 2) q^{7} + \beta_{8} q^{8} + (\beta_{13} - \beta_{12} - \beta_{11} + \beta_{6} - \beta_{5} + \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{10} - \beta_{8} + \beta_{6} + 1) q^{2} + \beta_1 q^{3} + \beta_{6} q^{4} + ( - \beta_{15} + \beta_{12} + \beta_{11} + \beta_{9} - \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{5} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{6} + ( - \beta_{15} - \beta_{13} - \beta_{10} - \beta_{7} + \beta_{6} + \beta_{4} - 2) q^{7} + \beta_{8} q^{8} + (\beta_{13} - \beta_{12} - \beta_{11} + \beta_{6} - \beta_{5} + \beta_{4}) q^{9} + ( - \beta_{15} + \beta_{14} + \beta_{12} + \beta_{11} + \beta_{9} + \beta_{7} - \beta_{6} - \beta_{3} - \beta_1) q^{10} + ( - 2 \beta_{9} - \beta_{7} + \beta_1 - 1) q^{11} + \beta_{3} q^{12} + ( - 2 \beta_{15} + \beta_{14} + \beta_{13} + \beta_{9} + 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \cdots - \beta_1) q^{13}+ \cdots + (\beta_{15} + \beta_{12} + \beta_{11} - 2 \beta_{10} + \beta_{9} + 4 \beta_{8} - \beta_{7} - 4 \beta_{6} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} + q^{3} - 4 q^{4} + q^{5} - q^{6} - 12 q^{7} + 4 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} + q^{3} - 4 q^{4} + q^{5} - q^{6} - 12 q^{7} + 4 q^{8} - q^{9} + 4 q^{10} + 4 q^{11} - 4 q^{12} - 12 q^{13} + 12 q^{14} - 10 q^{15} - 4 q^{16} + 6 q^{17} + q^{18} - 4 q^{19} + q^{20} + 14 q^{21} - 9 q^{22} - 26 q^{23} - q^{24} + 33 q^{25} - 3 q^{26} + 13 q^{27} - 2 q^{28} + 10 q^{30} + 11 q^{31} - 16 q^{32} - 31 q^{33} - 6 q^{34} - 4 q^{35} - q^{36} + 4 q^{38} + 12 q^{39} - q^{40} - 7 q^{41} + 6 q^{42} + 66 q^{43} - 11 q^{44} - 42 q^{45} - 24 q^{46} + 47 q^{47} + q^{48} + 24 q^{49} + 37 q^{50} - 41 q^{51} + 3 q^{52} + 15 q^{53} - 8 q^{54} - 9 q^{55} - 28 q^{56} + q^{57} - 18 q^{59} + 5 q^{60} - 43 q^{61} + 9 q^{62} - 37 q^{63} - 4 q^{64} + 52 q^{65} - 39 q^{66} - 50 q^{67} + 6 q^{68} + 81 q^{69} - 11 q^{70} - 5 q^{71} + q^{72} - 13 q^{73} + 39 q^{75} + 16 q^{76} - 10 q^{77} - 22 q^{78} + 5 q^{79} + q^{80} + 2 q^{82} + 7 q^{83} - q^{84} - 29 q^{85} + 29 q^{86} + 4 q^{87} - 4 q^{88} - 28 q^{89} - 18 q^{90} - 11 q^{92} - 5 q^{93} + 28 q^{94} + q^{95} - q^{96} - 11 q^{97} - 24 q^{98} + 43 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + 7 x^{14} - 13 x^{13} + 51 x^{12} - 74 x^{11} + 332 x^{10} - 614 x^{9} + 1832 x^{8} - 2960 x^{7} + 5348 x^{6} - 6872 x^{5} + 8232 x^{4} - 6344 x^{3} + 3984 x^{2} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 54\!\cdots\!43 \nu^{15} + \cdots + 19\!\cdots\!04 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 16\!\cdots\!51 \nu^{15} + \cdots - 32\!\cdots\!04 ) / 49\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 22\!\cdots\!11 \nu^{15} + \cdots + 54\!\cdots\!20 ) / 49\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 26\!\cdots\!92 \nu^{15} + \cdots + 83\!\cdots\!12 ) / 49\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20\!\cdots\!94 \nu^{15} + \cdots - 85\!\cdots\!08 ) / 49\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 20\!\cdots\!39 \nu^{15} + \cdots - 93\!\cdots\!80 ) / 49\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12\!\cdots\!44 \nu^{15} + \cdots - 29\!\cdots\!88 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 26\!\cdots\!97 \nu^{15} + \cdots + 16\!\cdots\!00 ) / 49\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 34\!\cdots\!70 \nu^{15} + \cdots + 88\!\cdots\!16 ) / 49\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 43\!\cdots\!42 \nu^{15} + \cdots - 18\!\cdots\!72 ) / 49\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 44\!\cdots\!39 \nu^{15} + \cdots - 59\!\cdots\!68 ) / 49\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 23\!\cdots\!43 \nu^{15} + \cdots + 46\!\cdots\!16 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 60\!\cdots\!98 \nu^{15} + \cdots - 67\!\cdots\!88 ) / 49\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 81\!\cdots\!10 \nu^{15} + \cdots - 95\!\cdots\!20 ) / 49\!\cdots\!40 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} - \beta_{12} - \beta_{11} - 3\beta_{10} + 3\beta_{8} - 2\beta_{6} - \beta_{5} + \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{13} + \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} - 5 \beta_{4} - 4 \beta_{3} - 5 \beta_{2} - 6 \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 7 \beta_{15} + \beta_{14} - 7 \beta_{13} - 3 \beta_{10} + 6 \beta_{9} + 12 \beta_{6} + 6 \beta_{5} + 2 \beta_{4} + 8 \beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{15} + 2 \beta_{13} - 9 \beta_{12} - 2 \beta_{11} - 9 \beta_{9} + 9 \beta_{8} - 7 \beta_{7} - 7 \beta_{6} - 2 \beta_{5} + 27 \beta_{3} - 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{15} - 43 \beta_{14} - 36 \beta_{13} + 34 \beta_{12} + 43 \beta_{11} + 27 \beta_{10} + 2 \beta_{9} - 83 \beta_{8} - 34 \beta_{7} - 16 \beta_{6} + 34 \beta_{5} - 43 \beta_{4} - 9 \beta_{3} - 52 \beta_{2} - 9 \beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 63 \beta_{15} + 22 \beta_{14} + 2 \beta_{13} + 39 \beta_{12} - 2 \beta_{11} - 69 \beta_{10} + 22 \beta_{9} + 69 \beta_{8} + 63 \beta_{7} + 39 \beta_{5} + 24 \beta_{4} + 175 \beta_{2} - 22 \beta _1 + 39 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 194 \beta_{15} + 22 \beta_{14} + 39 \beta_{13} - 39 \beta_{11} + 483 \beta_{10} - 255 \beta_{9} - 191 \beta_{8} - 22 \beta_{7} + 39 \beta_{6} + 39 \beta_{5} - 73 \beta_{4} + 190 \beta_{3} - 95 \beta_{2} + 290 \beta _1 + 147 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 199 \beta_{15} - 407 \beta_{14} + 199 \beta_{13} - 467 \beta_{10} + 34 \beta_{9} - 174 \beta_{7} + 34 \beta_{6} + 34 \beta_{5} + 660 \beta_{4} + 34 \beta_{2} + 407 \beta _1 - 234 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1126 \beta_{14} + 1126 \beta_{13} + 199 \beta_{12} + 199 \beta_{9} + 1255 \beta_{8} + 1499 \beta_{7} - 1255 \beta_{6} - 1300 \beta_{5} - 874 \beta_{4} - 1257 \beta_{3} - 2174 \beta _1 + 2005 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 955 \beta_{14} - 1592 \beta_{13} + 370 \beta_{12} - 955 \beta_{11} + 3151 \beta_{10} - 3529 \beta_{8} - 370 \beta_{7} + 4106 \beta_{6} + 1592 \beta_{5} + 955 \beta_{4} - 1333 \beta_{3} - 378 \beta_{2} + 5305 \beta _1 - 1325 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 947 \beta_{15} + 8 \beta_{14} + 7860 \beta_{13} - 8815 \beta_{12} - 6638 \beta_{11} - 17531 \beta_{10} + 8 \beta_{9} + 17531 \beta_{8} - 947 \beta_{7} - 2858 \beta_{6} - 7593 \beta_{5} + 12336 \beta_{4} + 5690 \beta_{3} + \cdots - 10451 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3314 \beta_{15} + 8 \beta_{14} - 7593 \beta_{13} + 8134 \beta_{12} + 15727 \beta_{11} + 24341 \beta_{10} + 4271 \beta_{9} - 20867 \beta_{8} - 8 \beta_{7} - 15727 \beta_{6} - 7593 \beta_{5} - 40163 \beta_{4} + \cdots + 20851 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 52035 \beta_{15} + 4087 \beta_{14} - 43901 \beta_{13} + 8134 \beta_{12} - 8134 \beta_{11} - 51023 \beta_{10} + 39622 \beta_{9} - 192 \beta_{7} + 64678 \beta_{6} + 47756 \beta_{5} + 43390 \beta_{4} + \cdots - 55302 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 52678 \beta_{15} + 26698 \beta_{14} + 79376 \beta_{13} - 96579 \beta_{12} - 52678 \beta_{11} - 96579 \beta_{9} + 136591 \beta_{8} - 17011 \beta_{7} - 83913 \beta_{6} - 79568 \beta_{5} - 2392 \beta_{4} + \cdots - 15019 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/418\mathbb{Z}\right)^\times\).

\(n\) \(287\) \(343\)
\(\chi(n)\) \(1\) \(\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
115.1
−2.04714 1.48734i
0.0560397 + 0.0407152i
0.956027 + 0.694595i
1.84409 + 1.33981i
−0.744204 + 2.29042i
−0.494109 + 1.52071i
0.388450 1.19553i
0.540846 1.66455i
−2.04714 + 1.48734i
0.0560397 0.0407152i
0.956027 0.694595i
1.84409 1.33981i
−0.744204 2.29042i
−0.494109 1.52071i
0.388450 + 1.19553i
0.540846 + 1.66455i
−0.309017 0.951057i −2.04714 1.48734i −0.809017 + 0.587785i 0.0815989 0.251136i −0.781939 + 2.40656i −3.52166 + 2.55863i 0.809017 + 0.587785i 1.05158 + 3.23642i −0.264060
115.2 −0.309017 0.951057i 0.0560397 + 0.0407152i −0.809017 + 0.587785i 0.340306 1.04736i 0.0214053 0.0658786i −1.61489 + 1.17328i 0.809017 + 0.587785i −0.925568 2.84861i −1.10125
115.3 −0.309017 0.951057i 0.956027 + 0.694595i −0.809017 + 0.587785i −0.0364140 + 0.112071i 0.365170 1.12388i 1.41067 1.02491i 0.809017 + 0.587785i −0.495524 1.52507i 0.117838
115.4 −0.309017 0.951057i 1.84409 + 1.33981i −0.809017 + 0.587785i −0.694508 + 2.13748i 0.704381 2.16786i −3.74627 + 2.72182i 0.809017 + 0.587785i 0.678534 + 2.08831i 2.24748
191.1 0.809017 + 0.587785i −0.744204 + 2.29042i 0.309017 + 0.951057i 2.77774 2.01814i −1.94835 + 1.41556i 0.326530 + 1.00496i −0.309017 + 0.951057i −2.26515 1.64573i 3.43347
191.2 0.809017 + 0.587785i −0.494109 + 1.52071i 0.309017 + 0.951057i −3.03200 + 2.20287i −1.29359 + 0.939851i −0.259070 0.797336i −0.309017 + 0.951057i 0.358634 + 0.260563i −3.74775
191.3 0.809017 + 0.587785i 0.388450 1.19553i 0.309017 + 0.951057i −1.36174 + 0.989363i 1.01697 0.738875i 1.33687 + 4.11446i −0.309017 + 0.951057i 1.14866 + 0.834553i −1.68320
191.4 0.809017 + 0.587785i 0.540846 1.66455i 0.309017 + 0.951057i 2.42502 1.76188i 1.41595 1.02875i 0.0678059 + 0.208685i −0.309017 + 0.951057i −0.0511674 0.0371753i 2.99749
229.1 −0.309017 + 0.951057i −2.04714 + 1.48734i −0.809017 0.587785i 0.0815989 + 0.251136i −0.781939 2.40656i −3.52166 2.55863i 0.809017 0.587785i 1.05158 3.23642i −0.264060
229.2 −0.309017 + 0.951057i 0.0560397 0.0407152i −0.809017 0.587785i 0.340306 + 1.04736i 0.0214053 + 0.0658786i −1.61489 1.17328i 0.809017 0.587785i −0.925568 + 2.84861i −1.10125
229.3 −0.309017 + 0.951057i 0.956027 0.694595i −0.809017 0.587785i −0.0364140 0.112071i 0.365170 + 1.12388i 1.41067 + 1.02491i 0.809017 0.587785i −0.495524 + 1.52507i 0.117838
229.4 −0.309017 + 0.951057i 1.84409 1.33981i −0.809017 0.587785i −0.694508 2.13748i 0.704381 + 2.16786i −3.74627 2.72182i 0.809017 0.587785i 0.678534 2.08831i 2.24748
267.1 0.809017 0.587785i −0.744204 2.29042i 0.309017 0.951057i 2.77774 + 2.01814i −1.94835 1.41556i 0.326530 1.00496i −0.309017 0.951057i −2.26515 + 1.64573i 3.43347
267.2 0.809017 0.587785i −0.494109 1.52071i 0.309017 0.951057i −3.03200 2.20287i −1.29359 0.939851i −0.259070 + 0.797336i −0.309017 0.951057i 0.358634 0.260563i −3.74775
267.3 0.809017 0.587785i 0.388450 + 1.19553i 0.309017 0.951057i −1.36174 0.989363i 1.01697 + 0.738875i 1.33687 4.11446i −0.309017 0.951057i 1.14866 0.834553i −1.68320
267.4 0.809017 0.587785i 0.540846 + 1.66455i 0.309017 0.951057i 2.42502 + 1.76188i 1.41595 + 1.02875i 0.0678059 0.208685i −0.309017 0.951057i −0.0511674 + 0.0371753i 2.99749
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 267.4
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 418.2.f.g 16
11.c even 5 1 inner 418.2.f.g 16
11.c even 5 1 4598.2.a.bw 8
11.d odd 10 1 4598.2.a.bz 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.g 16 1.a even 1 1 trivial
418.2.f.g 16 11.c even 5 1 inner
4598.2.a.bw 8 11.c even 5 1
4598.2.a.bz 8 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - T_{3}^{15} + 7 T_{3}^{14} - 13 T_{3}^{13} + 51 T_{3}^{12} - 74 T_{3}^{11} + 332 T_{3}^{10} - 614 T_{3}^{9} + 1832 T_{3}^{8} - 2960 T_{3}^{7} + 5348 T_{3}^{6} - 6872 T_{3}^{5} + 8232 T_{3}^{4} - 6344 T_{3}^{3} + 3984 T_{3}^{2} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} - T^{15} + 7 T^{14} - 13 T^{13} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{16} - T^{15} - 6 T^{14} + 3 T^{13} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{16} + 12 T^{15} + 74 T^{14} + \cdots + 3481 \) Copy content Toggle raw display
$11$ \( T^{16} - 4 T^{15} - 4 T^{14} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{16} + 12 T^{15} + 120 T^{14} + \cdots + 102400 \) Copy content Toggle raw display
$17$ \( T^{16} - 6 T^{15} + \cdots + 14059082041 \) Copy content Toggle raw display
$19$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 13 T^{7} - 47 T^{6} + \cdots - 142921)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + 74 T^{14} + \cdots + 1124663296 \) Copy content Toggle raw display
$31$ \( T^{16} - 11 T^{15} + \cdots + 528816016 \) Copy content Toggle raw display
$37$ \( T^{16} + 100 T^{14} + 367 T^{13} + \cdots + 6370576 \) Copy content Toggle raw display
$41$ \( T^{16} + 7 T^{15} + 106 T^{14} + \cdots + 146410000 \) Copy content Toggle raw display
$43$ \( (T^{8} - 33 T^{7} + 336 T^{6} + \cdots - 1634476)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} - 47 T^{15} + \cdots + 6705277197025 \) Copy content Toggle raw display
$53$ \( T^{16} - 15 T^{15} + \cdots + 10973819536 \) Copy content Toggle raw display
$59$ \( T^{16} + 18 T^{15} + 338 T^{14} + \cdots + 2560000 \) Copy content Toggle raw display
$61$ \( T^{16} + 43 T^{15} + \cdots + 49999196025 \) Copy content Toggle raw display
$67$ \( (T^{8} + 25 T^{7} - 165 T^{6} + \cdots - 72329500)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + 5 T^{15} + \cdots + 5035015405456 \) Copy content Toggle raw display
$73$ \( T^{16} + 13 T^{15} + \cdots + 7776423813376 \) Copy content Toggle raw display
$79$ \( T^{16} - 5 T^{15} + \cdots + 7299831312400 \) Copy content Toggle raw display
$83$ \( T^{16} - 7 T^{15} + \cdots + 2600869023841 \) Copy content Toggle raw display
$89$ \( (T^{8} + 14 T^{7} - 233 T^{6} + \cdots - 481856)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + 11 T^{15} + \cdots + 281430250000 \) Copy content Toggle raw display
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