Properties

Label 60.6.i.a
Level $60$
Weight $6$
Character orbit 60.i
Analytic conductor $9.623$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 60.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.62302918878\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 2 x^{19} + 2 x^{18} - 382 x^{17} + 117610 x^{16} - 661518 x^{15} + 1160778 x^{14} - 161859774 x^{13} + 12872525097 x^{12} - 60314430972 x^{11} + 238998177000 x^{10} - 23647959230160 x^{9} + 244333990200544 x^{8} + 369253960052800 x^{7} + 17753999958800000 x^{6} - 195554595639827200 x^{5} + 1195920778734496000 x^{4} - 5202205084237824000 x^{3} + 36839424090977280000 x^{2} - 600638984600463360000 x + 4896482487496796160000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{23}\cdot 3^{14}\cdot 5^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{6} q^{3} -\beta_{5} q^{5} + ( 4 + 4 \beta_{1} + \beta_{8} ) q^{7} + ( -8 \beta_{1} - \beta_{15} ) q^{9} +O(q^{10})\) \( q -\beta_{6} q^{3} -\beta_{5} q^{5} + ( 4 + 4 \beta_{1} + \beta_{8} ) q^{7} + ( -8 \beta_{1} - \beta_{15} ) q^{9} + \beta_{17} q^{11} + ( 53 - 53 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{12} - \beta_{14} + \beta_{15} - \beta_{19} ) q^{13} + ( -7 - 66 \beta_{1} - 2 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{16} + \beta_{17} - \beta_{18} ) q^{15} + ( -5 \beta_{2} - \beta_{4} + \beta_{5} + 4 \beta_{7} - \beta_{11} + \beta_{13} - 2 \beta_{14} - 2 \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{17} + ( -120 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} + \beta_{4} - \beta_{5} + 4 \beta_{6} + 5 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - 2 \beta_{15} - \beta_{16} - \beta_{18} ) q^{19} + ( 109 - \beta_{2} + 5 \beta_{3} + 2 \beta_{4} - 10 \beta_{5} - 4 \beta_{6} + 6 \beta_{7} + 5 \beta_{8} - 2 \beta_{9} + 2 \beta_{11} - \beta_{13} - 3 \beta_{14} + 2 \beta_{16} + 3 \beta_{17} - \beta_{19} ) q^{21} + ( -3 + 3 \beta_{1} - 2 \beta_{2} + \beta_{4} - 9 \beta_{5} - 21 \beta_{6} - 9 \beta_{7} + \beta_{9} - \beta_{11} + 3 \beta_{12} + \beta_{13} + 2 \beta_{14} - 2 \beta_{15} + \beta_{16} + 4 \beta_{17} + 4 \beta_{18} - \beta_{19} ) q^{23} + ( 207 - 28 \beta_{1} + 23 \beta_{2} + 8 \beta_{3} - \beta_{4} - 2 \beta_{5} - 15 \beta_{6} - \beta_{7} + 9 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} - \beta_{11} - 3 \beta_{12} - 3 \beta_{13} + 3 \beta_{14} - 9 \beta_{15} - \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{25} + ( 77 + 77 \beta_{1} + 8 \beta_{2} - 2 \beta_{4} - 13 \beta_{5} - 3 \beta_{6} - 22 \beta_{7} + 13 \beta_{8} - 6 \beta_{9} + \beta_{11} + 3 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} - 3 \beta_{16} - 3 \beta_{17} + 3 \beta_{18} - 2 \beta_{19} ) q^{27} + ( 6 \beta_{1} + 23 \beta_{2} - 31 \beta_{5} - 26 \beta_{6} + 15 \beta_{7} + 7 \beta_{9} - 3 \beta_{10} - 3 \beta_{11} + 3 \beta_{12} - \beta_{13} + 2 \beta_{15} - 5 \beta_{18} + \beta_{19} ) q^{29} + ( -252 - 66 \beta_{2} - 10 \beta_{3} + 2 \beta_{4} - 66 \beta_{6} + 3 \beta_{7} - 10 \beta_{8} - 7 \beta_{10} + 3 \beta_{11} - 7 \beta_{12} + \beta_{13} - 10 \beta_{14} + 2 \beta_{16} + 3 \beta_{17} + \beta_{19} ) q^{31} + ( -27 + 27 \beta_{1} + 9 \beta_{2} - 17 \beta_{3} + \beta_{4} + 27 \beta_{5} + \beta_{6} + 47 \beta_{7} - 11 \beta_{9} + 7 \beta_{11} + 9 \beta_{12} + 2 \beta_{13} + 2 \beta_{16} - 2 \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{33} + ( -3 - 21 \beta_{1} - 117 \beta_{2} - 3 \beta_{4} + 7 \beta_{5} + 74 \beta_{6} - 2 \beta_{7} + 15 \beta_{9} + 12 \beta_{10} - 10 \beta_{11} - 9 \beta_{12} - 4 \beta_{13} - 6 \beta_{14} + 8 \beta_{15} - 3 \beta_{16} - 7 \beta_{17} - \beta_{18} + 4 \beta_{19} ) q^{35} + ( -29 - 29 \beta_{1} + 94 \beta_{2} - 5 \beta_{4} + 5 \beta_{5} - 5 \beta_{7} - 43 \beta_{8} - 5 \beta_{9} + 7 \beta_{10} - 5 \beta_{11} + 4 \beta_{13} + 19 \beta_{14} + 19 \beta_{15} - 4 \beta_{16} - 5 \beta_{17} + 5 \beta_{18} - 5 \beta_{19} ) q^{37} + ( 106 \beta_{1} + 16 \beta_{2} + 40 \beta_{3} - \beta_{4} + 81 \beta_{5} - 10 \beta_{6} - 55 \beta_{7} - 40 \beta_{8} - 13 \beta_{9} - 9 \beta_{10} + 11 \beta_{11} + 9 \beta_{12} - 5 \beta_{13} + 6 \beta_{15} + \beta_{16} + 3 \beta_{18} + 5 \beta_{19} ) q^{39} + ( 42 + 182 \beta_{2} + 7 \beta_{4} + 105 \beta_{5} + 216 \beta_{6} - 58 \beta_{7} + 21 \beta_{9} - 21 \beta_{10} - 6 \beta_{11} - 21 \beta_{12} + 14 \beta_{14} + 7 \beta_{16} - 9 \beta_{17} ) q^{41} + ( -1169 + 1169 \beta_{1} + 26 \beta_{3} - 5 \beta_{4} + 5 \beta_{5} + 257 \beta_{6} + 5 \beta_{7} - 5 \beta_{9} + 5 \beta_{11} + 23 \beta_{12} + 5 \beta_{13} - 10 \beta_{14} + 10 \beta_{15} + 5 \beta_{16} + 5 \beta_{17} + 5 \beta_{18} + 5 \beta_{19} ) q^{43} + ( -957 + 48 \beta_{1} + 28 \beta_{2} - 43 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 7 \beta_{6} - 6 \beta_{7} - 34 \beta_{8} - 23 \beta_{9} + 36 \beta_{10} + 14 \beta_{11} - 27 \beta_{12} - 7 \beta_{13} + 7 \beta_{14} - 6 \beta_{15} + 5 \beta_{16} - 9 \beta_{17} - 3 \beta_{18} + 8 \beta_{19} ) q^{45} + ( -21 - 21 \beta_{1} - 207 \beta_{2} + 2 \beta_{4} + 118 \beta_{5} + 24 \beta_{6} + 132 \beta_{7} + 28 \beta_{9} + 21 \beta_{10} - 22 \beta_{11} - 2 \beta_{13} + 4 \beta_{14} + 4 \beta_{15} + 2 \beta_{16} - 17 \beta_{17} + 17 \beta_{18} + 2 \beta_{19} ) q^{47} + ( 789 \beta_{1} - 277 \beta_{2} - 35 \beta_{3} - 11 \beta_{4} + 5 \beta_{5} + 277 \beta_{6} + 35 \beta_{8} - 5 \beta_{9} - 22 \beta_{10} + 22 \beta_{12} - 6 \beta_{13} - 2 \beta_{15} + 11 \beta_{16} + 5 \beta_{18} + 6 \beta_{19} ) q^{49} + ( -1554 + 55 \beta_{2} + 10 \beta_{3} - 9 \beta_{4} - 155 \beta_{5} + \beta_{6} + 116 \beta_{7} + 10 \beta_{8} - 31 \beta_{9} - 63 \beta_{10} + 24 \beta_{11} - 63 \beta_{12} + 10 \beta_{13} + 18 \beta_{14} - 9 \beta_{16} - 16 \beta_{17} + 10 \beta_{19} ) q^{51} + ( -54 + 54 \beta_{1} - 43 \beta_{2} - 3 \beta_{4} - 83 \beta_{5} - 482 \beta_{6} - 218 \beta_{7} + 46 \beta_{9} - 19 \beta_{11} + 54 \beta_{12} - 3 \beta_{13} - 6 \beta_{14} + 6 \beta_{15} - 3 \beta_{16} - 27 \beta_{17} - 27 \beta_{18} + 3 \beta_{19} ) q^{53} + ( 1001 + 549 \beta_{1} + 645 \beta_{2} + 57 \beta_{3} - \beta_{4} + 7 \beta_{5} - 356 \beta_{6} + \beta_{7} - 14 \beta_{8} - 7 \beta_{9} + 59 \beta_{10} + \beta_{11} - 38 \beta_{12} + 17 \beta_{13} + 8 \beta_{14} + 36 \beta_{15} - 9 \beta_{16} + \beta_{17} + 7 \beta_{18} - 5 \beta_{19} ) q^{55} + ( 1663 + 1663 \beta_{1} + 34 \beta_{2} + 4 \beta_{4} - 109 \beta_{5} - 21 \beta_{6} - 226 \beta_{7} + 19 \beta_{8} - 42 \beta_{9} + 63 \beta_{10} + 25 \beta_{11} - 15 \beta_{13} - 3 \beta_{14} - 3 \beta_{15} + 15 \beta_{16} + 6 \beta_{17} - 6 \beta_{18} + 4 \beta_{19} ) q^{57} + ( 102 \beta_{1} + 425 \beta_{2} - 192 \beta_{5} - 429 \beta_{6} + 170 \beta_{7} + 40 \beta_{9} - 51 \beta_{10} - 34 \beta_{11} + 51 \beta_{12} - 2 \beta_{13} + 4 \beta_{15} + 29 \beta_{18} + 2 \beta_{19} ) q^{59} + ( 2824 - 847 \beta_{2} - 35 \beta_{3} - 2 \beta_{4} - 847 \beta_{6} - 11 \beta_{7} - 35 \beta_{8} - 90 \beta_{10} - 11 \beta_{11} - 90 \beta_{12} - 9 \beta_{13} + 26 \beta_{14} - 2 \beta_{16} - 11 \beta_{17} - 9 \beta_{19} ) q^{61} + ( 3352 - 3352 \beta_{1} + 45 \beta_{2} + 36 \beta_{3} + 3 \beta_{4} + 174 \beta_{5} - 54 \beta_{6} + 219 \beta_{7} - 39 \beta_{9} + 30 \beta_{11} + 117 \beta_{12} - 6 \beta_{13} - 16 \beta_{14} + 16 \beta_{15} - 6 \beta_{16} + 21 \beta_{17} + 21 \beta_{18} - 3 \beta_{19} ) q^{63} + ( -48 - 186 \beta_{1} - 1100 \beta_{2} + 17 \beta_{4} - 26 \beta_{5} + 713 \beta_{6} + 4 \beta_{7} + 33 \beta_{9} + 117 \beta_{10} - 9 \beta_{11} - 69 \beta_{12} + 16 \beta_{13} + 34 \beta_{14} - 32 \beta_{15} + 17 \beta_{16} + 43 \beta_{17} - 6 \beta_{18} - 16 \beta_{19} ) q^{65} + ( -3977 - 3977 \beta_{1} + 637 \beta_{2} + 25 \beta_{4} - 25 \beta_{5} + 25 \beta_{7} + 64 \beta_{8} + 25 \beta_{9} + 83 \beta_{10} + 25 \beta_{11} - 15 \beta_{13} - 90 \beta_{14} - 90 \beta_{15} + 15 \beta_{16} + 25 \beta_{17} - 25 \beta_{18} + 25 \beta_{19} ) q^{67} + ( 4261 \beta_{1} + 27 \beta_{2} - 70 \beta_{3} + 10 \beta_{4} + 239 \beta_{5} - 3 \beta_{6} - 95 \beta_{7} + 70 \beta_{8} - 67 \beta_{9} - 108 \beta_{10} + 19 \beta_{11} + 108 \beta_{12} + 34 \beta_{13} - 9 \beta_{15} - 10 \beta_{16} - 34 \beta_{18} - 34 \beta_{19} ) q^{69} + ( 330 + 1371 \beta_{2} - 40 \beta_{4} + 170 \beta_{5} + 1427 \beta_{6} - 150 \beta_{7} + 34 \beta_{9} - 165 \beta_{10} - 62 \beta_{11} - 165 \beta_{12} - 80 \beta_{14} - 40 \beta_{16} + 44 \beta_{17} ) q^{71} + ( -2739 + 2739 \beta_{1} - 62 \beta_{3} + 4 \beta_{4} - 30 \beta_{5} + 1428 \beta_{6} - 30 \beta_{7} + 30 \beta_{9} - 30 \beta_{11} + 166 \beta_{12} - 30 \beta_{13} + 86 \beta_{14} - 86 \beta_{15} - 30 \beta_{16} - 30 \beta_{17} - 30 \beta_{18} - 4 \beta_{19} ) q^{73} + ( -5108 - 4940 \beta_{1} + 32 \beta_{2} + 81 \beta_{3} + 16 \beta_{4} + 11 \beta_{5} - 140 \beta_{6} + 38 \beta_{7} + 118 \beta_{8} - 54 \beta_{9} + 171 \beta_{10} + 37 \beta_{11} - 72 \beta_{12} + 33 \beta_{13} - 48 \beta_{14} + 24 \beta_{15} - 9 \beta_{16} + 18 \beta_{17} + 51 \beta_{18} - 44 \beta_{19} ) q^{75} + ( -144 - 144 \beta_{1} - 1325 \beta_{2} + 4 \beta_{4} - 69 \beta_{5} - 13 \beta_{6} + 229 \beta_{7} + 49 \beta_{9} + 144 \beta_{10} + 17 \beta_{11} - 4 \beta_{13} + 8 \beta_{14} + 8 \beta_{15} + 4 \beta_{16} + 76 \beta_{17} - 76 \beta_{18} + 4 \beta_{19} ) q^{77} + ( -13358 \beta_{1} - 943 \beta_{2} + 80 \beta_{3} + 45 \beta_{4} - 13 \beta_{5} + 943 \beta_{6} - 80 \beta_{8} + 13 \beta_{9} - 144 \beta_{10} + 144 \beta_{12} + 32 \beta_{13} + 38 \beta_{15} - 45 \beta_{16} - 13 \beta_{18} - 32 \beta_{19} ) q^{79} + ( -509 - 52 \beta_{2} - 5 \beta_{3} + 10 \beta_{4} - 215 \beta_{5} - 94 \beta_{6} - 30 \beta_{7} - 5 \beta_{8} - 43 \beta_{9} - 180 \beta_{10} - 26 \beta_{11} - 180 \beta_{12} - 35 \beta_{13} - 32 \beta_{14} + 10 \beta_{16} + 27 \beta_{17} - 35 \beta_{19} ) q^{81} + ( -249 + 249 \beta_{1} + 14 \beta_{2} - \beta_{4} - 261 \beta_{5} - 2193 \beta_{6} + 69 \beta_{7} - 13 \beta_{9} - 53 \beta_{11} + 249 \beta_{12} - \beta_{13} - 2 \beta_{14} + 2 \beta_{15} - \beta_{16} + 66 \beta_{17} + 66 \beta_{18} + \beta_{19} ) q^{83} + ( 11919 + 8155 \beta_{1} + 2003 \beta_{2} - 263 \beta_{3} + 42 \beta_{4} + 2 \beta_{5} - 1577 \beta_{6} + 21 \beta_{7} + 76 \beta_{8} - 2 \beta_{9} + 267 \beta_{10} + 21 \beta_{11} - 144 \beta_{12} - 34 \beta_{13} - 141 \beta_{14} - 7 \beta_{15} + 57 \beta_{16} + 21 \beta_{17} + 2 \beta_{18} - 23 \beta_{19} ) q^{85} + ( 3555 + 3555 \beta_{1} - 17 \beta_{2} + 29 \beta_{4} - 209 \beta_{5} - 36 \beta_{6} + 99 \beta_{7} - 107 \beta_{8} + 43 \beta_{9} + 117 \beta_{10} + 65 \beta_{11} + 17 \beta_{13} - 36 \beta_{14} - 36 \beta_{15} - 17 \beta_{16} + 61 \beta_{17} - 61 \beta_{18} + 29 \beta_{19} ) q^{87} + ( 396 \beta_{1} + 1659 \beta_{2} - 131 \beta_{5} - 1739 \beta_{6} - 225 \beta_{7} - 9 \beta_{9} - 198 \beta_{10} + 45 \beta_{11} + 198 \beta_{12} + 44 \beta_{13} - 88 \beta_{15} - 28 \beta_{18} - 44 \beta_{19} ) q^{89} + ( 10618 - 2625 \beta_{2} + 310 \beta_{3} - 36 \beta_{4} - 2625 \beta_{6} - 12 \beta_{7} + 310 \beta_{8} - 321 \beta_{10} - 12 \beta_{11} - 321 \beta_{12} + 24 \beta_{13} + 96 \beta_{14} - 36 \beta_{16} - 12 \beta_{17} + 24 \beta_{19} ) q^{91} + ( 15388 - 15388 \beta_{1} - 6 \beta_{2} - 53 \beta_{3} - 48 \beta_{4} - 298 \beta_{5} + 85 \beta_{6} - 53 \beta_{7} + 29 \beta_{9} - 78 \beta_{11} + 225 \beta_{12} - 23 \beta_{13} + 114 \beta_{14} - 114 \beta_{15} - 23 \beta_{16} - 57 \beta_{17} - 57 \beta_{18} + 48 \beta_{19} ) q^{93} + ( -150 - 450 \beta_{1} - 2739 \beta_{2} - 30 \beta_{4} + 50 \beta_{5} + 1327 \beta_{6} + 37 \beta_{7} - 66 \beta_{9} + 300 \beta_{10} - 37 \beta_{11} - 150 \beta_{12} + 15 \beta_{13} - 60 \beta_{14} - 30 \beta_{15} - 30 \beta_{16} - 60 \beta_{17} + 55 \beta_{18} - 15 \beta_{19} ) q^{95} + ( -7335 - 7335 \beta_{1} + 1996 \beta_{2} - 20 \beta_{4} + 20 \beta_{5} - 20 \beta_{7} + 282 \beta_{8} - 20 \beta_{9} + 238 \beta_{10} - 20 \beta_{11} - 14 \beta_{13} + 46 \beta_{14} + 46 \beta_{15} + 14 \beta_{16} - 20 \beta_{17} + 20 \beta_{18} - 20 \beta_{19} ) q^{97} + ( 24864 \beta_{1} + 95 \beta_{2} - 175 \beta_{3} - 35 \beta_{4} - 399 \beta_{5} - 173 \beta_{6} - 35 \beta_{7} + 175 \beta_{8} + 115 \beta_{9} - 117 \beta_{10} + 7 \beta_{11} + 117 \beta_{12} - 79 \beta_{13} - 40 \beta_{15} + 35 \beta_{16} + 111 \beta_{18} + 79 \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 2q^{3} + 76q^{7} + O(q^{10}) \) \( 20q + 2q^{3} + 76q^{7} + 1068q^{13} - 130q^{15} + 2180q^{21} + 4060q^{25} + 1454q^{27} - 4720q^{31} - 460q^{33} - 612q^{37} - 24012q^{43} - 18860q^{45} - 31700q^{51} + 19200q^{55} + 33476q^{57} + 59880q^{61} + 67208q^{63} - 80804q^{67} - 56956q^{73} - 102470q^{75} - 9980q^{81} + 239260q^{85} + 71540q^{87} + 218520q^{91} + 307928q^{93} - 151164q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 2 x^{19} + 2 x^{18} - 382 x^{17} + 117610 x^{16} - 661518 x^{15} + 1160778 x^{14} - 161859774 x^{13} + 12872525097 x^{12} - 60314430972 x^{11} + 238998177000 x^{10} - 23647959230160 x^{9} + 244333990200544 x^{8} + 369253960052800 x^{7} + 17753999958800000 x^{6} - 195554595639827200 x^{5} + 1195920778734496000 x^{4} - 5202205084237824000 x^{3} + 36839424090977280000 x^{2} - 600638984600463360000 x + 4896482487496796160000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(\)\(68\!\cdots\!09\)\( \nu^{19} + \)\(55\!\cdots\!00\)\( \nu^{18} - \)\(58\!\cdots\!58\)\( \nu^{17} - \)\(11\!\cdots\!22\)\( \nu^{16} + \)\(81\!\cdots\!34\)\( \nu^{15} + \)\(35\!\cdots\!98\)\( \nu^{14} - \)\(32\!\cdots\!22\)\( \nu^{13} - \)\(88\!\cdots\!42\)\( \nu^{12} + \)\(80\!\cdots\!21\)\( \nu^{11} + \)\(44\!\cdots\!58\)\( \nu^{10} - \)\(21\!\cdots\!76\)\( \nu^{9} - \)\(13\!\cdots\!60\)\( \nu^{8} + \)\(42\!\cdots\!16\)\( \nu^{7} + \)\(16\!\cdots\!92\)\( \nu^{6} + \)\(11\!\cdots\!40\)\( \nu^{5} - \)\(33\!\cdots\!00\)\( \nu^{4} - \)\(66\!\cdots\!00\)\( \nu^{3} + \)\(99\!\cdots\!00\)\( \nu^{2} + \)\(23\!\cdots\!00\)\( \nu - \)\(17\!\cdots\!00\)\(\)\()/ \)\(65\!\cdots\!00\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(32\!\cdots\!37\)\( \nu^{19} + \)\(33\!\cdots\!70\)\( \nu^{18} + \)\(31\!\cdots\!14\)\( \nu^{17} - \)\(26\!\cdots\!34\)\( \nu^{16} - \)\(35\!\cdots\!82\)\( \nu^{15} + \)\(52\!\cdots\!66\)\( \nu^{14} + \)\(23\!\cdots\!86\)\( \nu^{13} - \)\(43\!\cdots\!14\)\( \nu^{12} - \)\(39\!\cdots\!53\)\( \nu^{11} + \)\(47\!\cdots\!56\)\( \nu^{10} + \)\(30\!\cdots\!48\)\( \nu^{9} + \)\(17\!\cdots\!60\)\( \nu^{8} - \)\(75\!\cdots\!88\)\( \nu^{7} - \)\(44\!\cdots\!76\)\( \nu^{6} + \)\(13\!\cdots\!20\)\( \nu^{5} - \)\(32\!\cdots\!00\)\( \nu^{4} - \)\(28\!\cdots\!00\)\( \nu^{3} - \)\(18\!\cdots\!00\)\( \nu^{2} + \)\(12\!\cdots\!00\)\( \nu - \)\(23\!\cdots\!00\)\(\)\()/ \)\(42\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(21\!\cdots\!55\)\( \nu^{19} - \)\(23\!\cdots\!16\)\( \nu^{18} + \)\(30\!\cdots\!02\)\( \nu^{17} + \)\(10\!\cdots\!18\)\( \nu^{16} + \)\(31\!\cdots\!62\)\( \nu^{15} - \)\(29\!\cdots\!50\)\( \nu^{14} + \)\(15\!\cdots\!58\)\( \nu^{13} + \)\(76\!\cdots\!02\)\( \nu^{12} + \)\(55\!\cdots\!99\)\( \nu^{11} - \)\(31\!\cdots\!42\)\( \nu^{10} + \)\(13\!\cdots\!52\)\( \nu^{9} + \)\(33\!\cdots\!80\)\( \nu^{8} + \)\(52\!\cdots\!80\)\( \nu^{7} - \)\(58\!\cdots\!44\)\( \nu^{6} - \)\(68\!\cdots\!80\)\( \nu^{5} - \)\(28\!\cdots\!60\)\( \nu^{4} + \)\(45\!\cdots\!00\)\( \nu^{3} - \)\(51\!\cdots\!00\)\( \nu^{2} + \)\(69\!\cdots\!00\)\( \nu + \)\(20\!\cdots\!00\)\(\)\()/ \)\(26\!\cdots\!40\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(53\!\cdots\!87\)\( \nu^{19} - \)\(19\!\cdots\!02\)\( \nu^{18} + \)\(73\!\cdots\!43\)\( \nu^{17} - \)\(46\!\cdots\!98\)\( \nu^{16} + \)\(10\!\cdots\!92\)\( \nu^{15} - \)\(21\!\cdots\!24\)\( \nu^{14} + \)\(16\!\cdots\!32\)\( \nu^{13} - \)\(84\!\cdots\!60\)\( \nu^{12} + \)\(26\!\cdots\!55\)\( \nu^{11} - \)\(23\!\cdots\!18\)\( \nu^{10} + \)\(15\!\cdots\!77\)\( \nu^{9} - \)\(10\!\cdots\!10\)\( \nu^{8} + \)\(45\!\cdots\!32\)\( \nu^{7} - \)\(43\!\cdots\!44\)\( \nu^{6} + \)\(16\!\cdots\!60\)\( \nu^{5} - \)\(43\!\cdots\!00\)\( \nu^{4} + \)\(25\!\cdots\!00\)\( \nu^{3} - \)\(21\!\cdots\!00\)\( \nu^{2} + \)\(17\!\cdots\!00\)\( \nu - \)\(11\!\cdots\!00\)\(\)\()/ \)\(35\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(10\!\cdots\!69\)\( \nu^{19} + \)\(67\!\cdots\!82\)\( \nu^{18} + \)\(33\!\cdots\!38\)\( \nu^{17} - \)\(30\!\cdots\!98\)\( \nu^{16} + \)\(12\!\cdots\!30\)\( \nu^{15} + \)\(30\!\cdots\!78\)\( \nu^{14} + \)\(35\!\cdots\!22\)\( \nu^{13} - \)\(61\!\cdots\!66\)\( \nu^{12} + \)\(14\!\cdots\!33\)\( \nu^{11} + \)\(30\!\cdots\!12\)\( \nu^{10} + \)\(42\!\cdots\!40\)\( \nu^{9} - \)\(67\!\cdots\!20\)\( \nu^{8} + \)\(23\!\cdots\!36\)\( \nu^{7} - \)\(46\!\cdots\!20\)\( \nu^{6} + \)\(12\!\cdots\!60\)\( \nu^{5} - \)\(24\!\cdots\!00\)\( \nu^{4} + \)\(18\!\cdots\!00\)\( \nu^{3} - \)\(47\!\cdots\!00\)\( \nu^{2} + \)\(28\!\cdots\!00\)\( \nu - \)\(21\!\cdots\!00\)\(\)\()/ \)\(42\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(11\!\cdots\!95\)\( \nu^{19} + \)\(61\!\cdots\!04\)\( \nu^{18} - \)\(65\!\cdots\!38\)\( \nu^{17} + \)\(47\!\cdots\!58\)\( \nu^{16} - \)\(13\!\cdots\!38\)\( \nu^{15} + \)\(12\!\cdots\!90\)\( \nu^{14} - \)\(10\!\cdots\!62\)\( \nu^{13} + \)\(21\!\cdots\!82\)\( \nu^{12} - \)\(14\!\cdots\!11\)\( \nu^{11} + \)\(12\!\cdots\!78\)\( \nu^{10} - \)\(11\!\cdots\!48\)\( \nu^{9} + \)\(29\!\cdots\!40\)\( \nu^{8} - \)\(33\!\cdots\!20\)\( \nu^{7} + \)\(12\!\cdots\!16\)\( \nu^{6} - \)\(24\!\cdots\!80\)\( \nu^{5} + \)\(20\!\cdots\!00\)\( \nu^{4} - \)\(21\!\cdots\!00\)\( \nu^{3} + \)\(86\!\cdots\!00\)\( \nu^{2} - \)\(69\!\cdots\!00\)\( \nu + \)\(67\!\cdots\!00\)\(\)\()/ \)\(42\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(12\!\cdots\!29\)\( \nu^{19} - \)\(18\!\cdots\!63\)\( \nu^{18} - \)\(82\!\cdots\!42\)\( \nu^{17} - \)\(72\!\cdots\!68\)\( \nu^{16} + \)\(14\!\cdots\!80\)\( \nu^{15} - \)\(26\!\cdots\!52\)\( \nu^{14} + \)\(68\!\cdots\!52\)\( \nu^{13} - \)\(22\!\cdots\!56\)\( \nu^{12} + \)\(17\!\cdots\!03\)\( \nu^{11} - \)\(26\!\cdots\!33\)\( \nu^{10} + \)\(67\!\cdots\!90\)\( \nu^{9} - \)\(33\!\cdots\!20\)\( \nu^{8} + \)\(55\!\cdots\!76\)\( \nu^{7} - \)\(77\!\cdots\!20\)\( \nu^{6} + \)\(25\!\cdots\!60\)\( \nu^{5} - \)\(33\!\cdots\!00\)\( \nu^{4} + \)\(24\!\cdots\!00\)\( \nu^{3} - \)\(12\!\cdots\!00\)\( \nu^{2} + \)\(97\!\cdots\!00\)\( \nu + \)\(13\!\cdots\!00\)\(\)\()/ \)\(21\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(23\!\cdots\!79\)\( \nu^{19} + \)\(69\!\cdots\!28\)\( \nu^{18} - \)\(11\!\cdots\!14\)\( \nu^{17} + \)\(31\!\cdots\!14\)\( \nu^{16} + \)\(27\!\cdots\!58\)\( \nu^{15} - \)\(17\!\cdots\!42\)\( \nu^{14} - \)\(15\!\cdots\!06\)\( \nu^{13} - \)\(17\!\cdots\!78\)\( \nu^{12} + \)\(28\!\cdots\!79\)\( \nu^{11} + \)\(20\!\cdots\!14\)\( \nu^{10} - \)\(13\!\cdots\!32\)\( \nu^{9} - \)\(34\!\cdots\!00\)\( \nu^{8} + \)\(30\!\cdots\!76\)\( \nu^{7} + \)\(56\!\cdots\!64\)\( \nu^{6} + \)\(19\!\cdots\!20\)\( \nu^{5} - \)\(16\!\cdots\!60\)\( \nu^{4} - \)\(33\!\cdots\!00\)\( \nu^{3} + \)\(40\!\cdots\!00\)\( \nu^{2} - \)\(23\!\cdots\!00\)\( \nu + \)\(22\!\cdots\!00\)\(\)\()/ \)\(26\!\cdots\!40\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(98\!\cdots\!83\)\( \nu^{19} + \)\(99\!\cdots\!50\)\( \nu^{18} - \)\(19\!\cdots\!39\)\( \nu^{17} + \)\(43\!\cdots\!34\)\( \nu^{16} - \)\(12\!\cdots\!28\)\( \nu^{15} + \)\(15\!\cdots\!44\)\( \nu^{14} - \)\(25\!\cdots\!36\)\( \nu^{13} + \)\(67\!\cdots\!84\)\( \nu^{12} - \)\(16\!\cdots\!57\)\( \nu^{11} + \)\(18\!\cdots\!94\)\( \nu^{10} - \)\(32\!\cdots\!33\)\( \nu^{9} + \)\(76\!\cdots\!90\)\( \nu^{8} - \)\(61\!\cdots\!92\)\( \nu^{7} + \)\(49\!\cdots\!96\)\( \nu^{6} - \)\(10\!\cdots\!20\)\( \nu^{5} + \)\(66\!\cdots\!00\)\( \nu^{4} - \)\(68\!\cdots\!00\)\( \nu^{3} + \)\(55\!\cdots\!00\)\( \nu^{2} - \)\(30\!\cdots\!00\)\( \nu + \)\(21\!\cdots\!00\)\(\)\()/ \)\(96\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(65\!\cdots\!29\)\( \nu^{19} - \)\(15\!\cdots\!10\)\( \nu^{18} + \)\(94\!\cdots\!62\)\( \nu^{17} + \)\(25\!\cdots\!18\)\( \nu^{16} + \)\(73\!\cdots\!54\)\( \nu^{15} - \)\(25\!\cdots\!42\)\( \nu^{14} + \)\(14\!\cdots\!98\)\( \nu^{13} - \)\(52\!\cdots\!02\)\( \nu^{12} + \)\(76\!\cdots\!61\)\( \nu^{11} - \)\(19\!\cdots\!32\)\( \nu^{10} + \)\(16\!\cdots\!44\)\( \nu^{9} - \)\(93\!\cdots\!40\)\( \nu^{8} + \)\(69\!\cdots\!96\)\( \nu^{7} + \)\(10\!\cdots\!12\)\( \nu^{6} + \)\(57\!\cdots\!00\)\( \nu^{5} + \)\(22\!\cdots\!00\)\( \nu^{4} + \)\(81\!\cdots\!00\)\( \nu^{3} + \)\(18\!\cdots\!00\)\( \nu^{2} + \)\(41\!\cdots\!00\)\( \nu + \)\(22\!\cdots\!00\)\(\)\()/ \)\(47\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(60\!\cdots\!25\)\( \nu^{19} - \)\(21\!\cdots\!86\)\( \nu^{18} - \)\(41\!\cdots\!98\)\( \nu^{17} + \)\(10\!\cdots\!18\)\( \nu^{16} + \)\(67\!\cdots\!62\)\( \nu^{15} - \)\(27\!\cdots\!70\)\( \nu^{14} + \)\(40\!\cdots\!98\)\( \nu^{13} - \)\(39\!\cdots\!98\)\( \nu^{12} + \)\(96\!\cdots\!29\)\( \nu^{11} - \)\(27\!\cdots\!12\)\( \nu^{10} + \)\(22\!\cdots\!52\)\( \nu^{9} - \)\(10\!\cdots\!60\)\( \nu^{8} + \)\(46\!\cdots\!60\)\( \nu^{7} - \)\(16\!\cdots\!84\)\( \nu^{6} - \)\(26\!\cdots\!80\)\( \nu^{5} - \)\(24\!\cdots\!00\)\( \nu^{4} + \)\(41\!\cdots\!00\)\( \nu^{3} - \)\(23\!\cdots\!00\)\( \nu^{2} - \)\(15\!\cdots\!00\)\( \nu + \)\(60\!\cdots\!00\)\(\)\()/ \)\(42\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(66\!\cdots\!67\)\( \nu^{19} - \)\(38\!\cdots\!24\)\( \nu^{18} + \)\(40\!\cdots\!22\)\( \nu^{17} + \)\(65\!\cdots\!98\)\( \nu^{16} - \)\(77\!\cdots\!34\)\( \nu^{15} - \)\(13\!\cdots\!54\)\( \nu^{14} + \)\(61\!\cdots\!18\)\( \nu^{13} + \)\(71\!\cdots\!14\)\( \nu^{12} - \)\(78\!\cdots\!87\)\( \nu^{11} - \)\(27\!\cdots\!62\)\( \nu^{10} + \)\(53\!\cdots\!56\)\( \nu^{9} + \)\(11\!\cdots\!40\)\( \nu^{8} - \)\(50\!\cdots\!68\)\( \nu^{7} - \)\(17\!\cdots\!52\)\( \nu^{6} - \)\(76\!\cdots\!40\)\( \nu^{5} + \)\(93\!\cdots\!00\)\( \nu^{4} - \)\(61\!\cdots\!00\)\( \nu^{3} - \)\(33\!\cdots\!00\)\( \nu^{2} + \)\(23\!\cdots\!00\)\( \nu + \)\(52\!\cdots\!00\)\(\)\()/ \)\(47\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(11\!\cdots\!90\)\( \nu^{19} + \)\(28\!\cdots\!03\)\( \nu^{18} + \)\(15\!\cdots\!34\)\( \nu^{17} - \)\(18\!\cdots\!54\)\( \nu^{16} + \)\(12\!\cdots\!34\)\( \nu^{15} + \)\(28\!\cdots\!70\)\( \nu^{14} - \)\(10\!\cdots\!74\)\( \nu^{13} - \)\(36\!\cdots\!46\)\( \nu^{12} + \)\(10\!\cdots\!48\)\( \nu^{11} + \)\(31\!\cdots\!91\)\( \nu^{10} - \)\(50\!\cdots\!36\)\( \nu^{9} - \)\(42\!\cdots\!40\)\( \nu^{8} - \)\(35\!\cdots\!20\)\( \nu^{7} + \)\(54\!\cdots\!52\)\( \nu^{6} + \)\(77\!\cdots\!80\)\( \nu^{5} - \)\(16\!\cdots\!00\)\( \nu^{4} - \)\(41\!\cdots\!00\)\( \nu^{3} - \)\(70\!\cdots\!00\)\( \nu^{2} + \)\(33\!\cdots\!00\)\( \nu - \)\(28\!\cdots\!00\)\(\)\()/ \)\(70\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(14\!\cdots\!07\)\( \nu^{19} - \)\(23\!\cdots\!49\)\( \nu^{18} + \)\(11\!\cdots\!74\)\( \nu^{17} + \)\(15\!\cdots\!76\)\( \nu^{16} + \)\(15\!\cdots\!20\)\( \nu^{15} - \)\(31\!\cdots\!56\)\( \nu^{14} + \)\(23\!\cdots\!76\)\( \nu^{13} - \)\(20\!\cdots\!68\)\( \nu^{12} + \)\(19\!\cdots\!89\)\( \nu^{11} - \)\(33\!\cdots\!59\)\( \nu^{10} + \)\(22\!\cdots\!50\)\( \nu^{9} - \)\(30\!\cdots\!00\)\( \nu^{8} + \)\(55\!\cdots\!08\)\( \nu^{7} - \)\(27\!\cdots\!40\)\( \nu^{6} - \)\(20\!\cdots\!40\)\( \nu^{5} - \)\(10\!\cdots\!00\)\( \nu^{4} + \)\(63\!\cdots\!00\)\( \nu^{3} - \)\(11\!\cdots\!00\)\( \nu^{2} + \)\(50\!\cdots\!00\)\( \nu + \)\(20\!\cdots\!00\)\(\)\()/ \)\(87\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(10\!\cdots\!93\)\( \nu^{19} + \)\(79\!\cdots\!94\)\( \nu^{18} - \)\(80\!\cdots\!34\)\( \nu^{17} + \)\(69\!\cdots\!54\)\( \nu^{16} + \)\(12\!\cdots\!50\)\( \nu^{15} + \)\(49\!\cdots\!46\)\( \nu^{14} - \)\(12\!\cdots\!86\)\( \nu^{13} - \)\(13\!\cdots\!62\)\( \nu^{12} + \)\(12\!\cdots\!21\)\( \nu^{11} + \)\(73\!\cdots\!04\)\( \nu^{10} - \)\(12\!\cdots\!80\)\( \nu^{9} - \)\(82\!\cdots\!60\)\( \nu^{8} - \)\(18\!\cdots\!08\)\( \nu^{7} + \)\(42\!\cdots\!20\)\( \nu^{6} - \)\(72\!\cdots\!40\)\( \nu^{5} + \)\(33\!\cdots\!00\)\( \nu^{4} - \)\(23\!\cdots\!00\)\( \nu^{3} + \)\(19\!\cdots\!00\)\( \nu^{2} - \)\(56\!\cdots\!00\)\( \nu + \)\(15\!\cdots\!00\)\(\)\()/ \)\(38\!\cdots\!00\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(13\!\cdots\!55\)\( \nu^{19} + \)\(26\!\cdots\!28\)\( \nu^{18} - \)\(13\!\cdots\!26\)\( \nu^{17} - \)\(13\!\cdots\!14\)\( \nu^{16} + \)\(14\!\cdots\!14\)\( \nu^{15} - \)\(26\!\cdots\!90\)\( \nu^{14} - \)\(16\!\cdots\!94\)\( \nu^{13} - \)\(11\!\cdots\!26\)\( \nu^{12} + \)\(15\!\cdots\!43\)\( \nu^{11} + \)\(63\!\cdots\!66\)\( \nu^{10} - \)\(16\!\cdots\!56\)\( \nu^{9} - \)\(21\!\cdots\!80\)\( \nu^{8} + \)\(70\!\cdots\!40\)\( \nu^{7} + \)\(45\!\cdots\!72\)\( \nu^{6} - \)\(39\!\cdots\!40\)\( \nu^{5} - \)\(16\!\cdots\!00\)\( \nu^{4} - \)\(26\!\cdots\!00\)\( \nu^{3} + \)\(14\!\cdots\!00\)\( \nu^{2} - \)\(20\!\cdots\!00\)\( \nu - \)\(51\!\cdots\!00\)\(\)\()/ \)\(42\!\cdots\!00\)\( \)
\(\beta_{17}\)\(=\)\((\)\(\)\(30\!\cdots\!63\)\( \nu^{19} - \)\(91\!\cdots\!91\)\( \nu^{18} + \)\(26\!\cdots\!16\)\( \nu^{17} + \)\(52\!\cdots\!84\)\( \nu^{16} + \)\(34\!\cdots\!80\)\( \nu^{15} - \)\(23\!\cdots\!04\)\( \nu^{14} + \)\(33\!\cdots\!84\)\( \nu^{13} - \)\(40\!\cdots\!12\)\( \nu^{12} + \)\(37\!\cdots\!01\)\( \nu^{11} - \)\(23\!\cdots\!81\)\( \nu^{10} + \)\(36\!\cdots\!00\)\( \nu^{9} - \)\(58\!\cdots\!00\)\( \nu^{8} + \)\(56\!\cdots\!72\)\( \nu^{7} - \)\(27\!\cdots\!60\)\( \nu^{6} + \)\(49\!\cdots\!40\)\( \nu^{5} + \)\(28\!\cdots\!00\)\( \nu^{4} + \)\(48\!\cdots\!00\)\( \nu^{3} - \)\(13\!\cdots\!00\)\( \nu^{2} - \)\(61\!\cdots\!00\)\( \nu + \)\(54\!\cdots\!00\)\(\)\()/ \)\(43\!\cdots\!00\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(11\!\cdots\!59\)\( \nu^{19} - \)\(76\!\cdots\!22\)\( \nu^{18} + \)\(87\!\cdots\!42\)\( \nu^{17} - \)\(85\!\cdots\!02\)\( \nu^{16} - \)\(12\!\cdots\!50\)\( \nu^{15} - \)\(39\!\cdots\!98\)\( \nu^{14} + \)\(13\!\cdots\!18\)\( \nu^{13} + \)\(87\!\cdots\!06\)\( \nu^{12} - \)\(12\!\cdots\!23\)\( \nu^{11} - \)\(67\!\cdots\!52\)\( \nu^{10} + \)\(12\!\cdots\!40\)\( \nu^{9} + \)\(15\!\cdots\!80\)\( \nu^{8} + \)\(44\!\cdots\!04\)\( \nu^{7} - \)\(41\!\cdots\!60\)\( \nu^{6} - \)\(63\!\cdots\!80\)\( \nu^{5} - \)\(33\!\cdots\!00\)\( \nu^{4} + \)\(13\!\cdots\!00\)\( \nu^{3} - \)\(30\!\cdots\!00\)\( \nu^{2} + \)\(96\!\cdots\!00\)\( \nu - \)\(61\!\cdots\!00\)\(\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(39\!\cdots\!05\)\( \nu^{19} - \)\(27\!\cdots\!32\)\( \nu^{18} - \)\(56\!\cdots\!46\)\( \nu^{17} + \)\(64\!\cdots\!66\)\( \nu^{16} - \)\(45\!\cdots\!46\)\( \nu^{15} + \)\(14\!\cdots\!10\)\( \nu^{14} - \)\(56\!\cdots\!34\)\( \nu^{13} + \)\(55\!\cdots\!54\)\( \nu^{12} - \)\(48\!\cdots\!37\)\( \nu^{11} + \)\(11\!\cdots\!66\)\( \nu^{10} - \)\(11\!\cdots\!16\)\( \nu^{9} + \)\(82\!\cdots\!40\)\( \nu^{8} - \)\(62\!\cdots\!00\)\( \nu^{7} - \)\(17\!\cdots\!48\)\( \nu^{6} - \)\(73\!\cdots\!80\)\( \nu^{5} + \)\(24\!\cdots\!00\)\( \nu^{4} - \)\(37\!\cdots\!00\)\( \nu^{3} + \)\(10\!\cdots\!00\)\( \nu^{2} - \)\(99\!\cdots\!00\)\( \nu + \)\(40\!\cdots\!00\)\(\)\()/ \)\(42\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-5 \beta_{19} + 5 \beta_{18} - 5 \beta_{17} - 5 \beta_{16} + 5 \beta_{13} + 3 \beta_{11} + 30 \beta_{10} - 7 \beta_{9} - 20 \beta_{8} - 15 \beta_{7} - 8 \beta_{6} - 35 \beta_{5} - 5 \beta_{4} - 148 \beta_{2} + 40 \beta_{1} + 40\)\()/600\)
\(\nu^{2}\)\(=\)\((\)\(-51 \beta_{19} + 48 \beta_{18} + 9 \beta_{16} + 96 \beta_{15} + 51 \beta_{13} + 107 \beta_{12} - 51 \beta_{11} - 107 \beta_{10} + 9 \beta_{9} + 171 \beta_{8} + 255 \beta_{7} - 405 \beta_{6} - 285 \beta_{5} - 9 \beta_{4} - 171 \beta_{3} + 387 \beta_{2} - 39908 \beta_{1}\)\()/360\)
\(\nu^{3}\)\(=\)\((\)\(-183 \beta_{19} - 999 \beta_{18} - 999 \beta_{17} - 927 \beta_{16} - 324 \beta_{15} + 324 \beta_{14} - 927 \beta_{13} + 3676 \beta_{12} - 1329 \beta_{11} + 633 \beta_{9} + 543 \beta_{7} - 42828 \beta_{6} - 2937 \beta_{5} + 183 \beta_{4} - 3006 \beta_{3} + 294 \beta_{2} - 15080 \beta_{1} + 15080\)\()/360\)
\(\nu^{4}\)\(=\)\((\)\(295 \beta_{19} - 1222 \beta_{17} - 5437 \beta_{16} + 5168 \beta_{14} + 295 \beta_{13} - 16181 \beta_{12} + 4969 \beta_{11} - 16181 \beta_{10} + 12913 \beta_{9} + 3643 \beta_{8} + 45413 \beta_{7} + 66203 \beta_{6} + 64565 \beta_{5} - 5437 \beta_{4} + 3643 \beta_{3} + 63401 \beta_{2} - 2782044\)\()/120\)
\(\nu^{5}\)\(=\)\((\)\(90327 \beta_{19} - 103167 \beta_{18} + 103167 \beta_{17} - 72729 \beta_{16} - 122672 \beta_{15} - 122672 \beta_{14} + 72729 \beta_{13} + 96887 \beta_{11} - 267666 \beta_{10} + 69037 \beta_{9} + 425884 \beta_{8} - 16123 \beta_{7} - 6560 \beta_{6} - 123127 \beta_{5} + 90327 \beta_{4} + 1606084 \beta_{2} + 13127632 \beta_{1} + 13127632\)\()/120\)
\(\nu^{6}\)\(=\)\((\)\(2956113 \beta_{19} + 882192 \beta_{18} + 1237581 \beta_{16} + 4194072 \beta_{15} - 2956113 \beta_{13} - 15858557 \beta_{12} + 16137357 \beta_{11} + 15858557 \beta_{10} - 8596599 \beta_{9} - 2107845 \beta_{8} - 80686785 \beta_{7} + 83626887 \beta_{6} + 49857123 \beta_{5} - 1237581 \beta_{4} + 2107845 \beta_{3} - 89449113 \beta_{2} - 75618124 \beta_{1}\)\()/360\)
\(\nu^{7}\)\(=\)\((\)\(95351517 \beta_{19} + 63335793 \beta_{18} + 63335793 \beta_{17} + 70737069 \beta_{16} + 135841332 \beta_{15} - 135841332 \beta_{14} + 70737069 \beta_{13} + 43810460 \beta_{12} - 28722969 \beta_{11} - 110545791 \beta_{9} + 269780679 \beta_{7} - 1732118760 \beta_{6} - 426563121 \beta_{5} - 95351517 \beta_{4} + 352973682 \beta_{3} + 39808722 \beta_{2} - 14582099368 \beta_{1} + 14582099368\)\()/360\)
\(\nu^{8}\)\(=\)\((\)\(-275668645 \beta_{19} - 435594430 \beta_{17} + 41198711 \beta_{16} + 1561769784 \beta_{14} - 275668645 \beta_{13} + 1041337155 \beta_{12} - 1170804687 \beta_{11} + 1041337155 \beta_{10} - 1575206095 \beta_{9} - 132358717 \beta_{8} - 4916143699 \beta_{7} - 9967437081 \beta_{6} - 7876030475 \beta_{5} + 41198711 \beta_{4} - 132358717 \beta_{3} - 9328565739 \beta_{2} - 269553198356\)\()/120\)
\(\nu^{9}\)\(=\)\((\)\(-3675273565 \beta_{19} + 391886377 \beta_{18} - 391886377 \beta_{17} + 9660640931 \beta_{16} + 10984928280 \beta_{15} + 10984928280 \beta_{14} - 9660640931 \beta_{13} - 10034080545 \beta_{11} - 39000159714 \beta_{10} + 15639396997 \beta_{9} - 28387084612 \beta_{8} + 92898079245 \beta_{7} + 6358806980 \beta_{6} + 35469308465 \beta_{5} - 3675273565 \beta_{4} + 501794895688 \beta_{2} - 765857074864 \beta_{1} - 765857074864\)\()/120\)
\(\nu^{10}\)\(=\)\((\)\(184806014037 \beta_{19} - 613510580664 \beta_{18} - 368149758327 \beta_{16} - 2348816151552 \beta_{15} - 184806014037 \beta_{13} - 33877196821 \beta_{12} - 862055521395 \beta_{11} + 33877196821 \beta_{10} + 1050906142929 \beta_{9} + 124212653379 \beta_{8} + 4310277606975 \beta_{7} - 5001856239381 \beta_{6} - 3042707625189 \beta_{5} + 368149758327 \beta_{4} - 124212653379 \beta_{3} + 5365961390211 \beta_{2} + 452353482886300 \beta_{1}\)\()/360\)
\(\nu^{11}\)\(=\)\((\)\(-6401045939703 \beta_{19} + 6476333444697 \beta_{18} + 6476333444697 \beta_{17} + 1871261129361 \beta_{16} - 1814502674436 \beta_{15} + 1814502674436 \beta_{14} + 1871261129361 \beta_{13} - 61012413231572 \beta_{12} + 24916339987791 \beta_{11} + 5283540351657 \beta_{9} - 33902746275729 \beta_{7} + 668845461177636 \beta_{6} + 117096655421511 \beta_{5} + 6401045939703 \beta_{4} - 13022004542382 \beta_{3} - 7154801481018 \beta_{2} - 1049056004214344 \beta_{1} + 1049056004214344\)\()/360\)
\(\nu^{12}\)\(=\)\((\)\(40462377172519 \beta_{19} + 62493239576538 \beta_{17} + 33684102976803 \beta_{16} - 278815812304208 \beta_{14} + 40462377172519 \beta_{13} + 140970526907147 \beta_{12} + 67086177127625 \beta_{11} + 140970526907147 \beta_{10} - 13781290399055 \beta_{9} + 6329173994395 \beta_{8} + 38844965040837 \beta_{7} - 330014759810597 \beta_{6} - 68906451995275 \beta_{5} + 33684102976803 \beta_{4} + 6329173994395 \beta_{3} - 323293772433239 \beta_{2} + 56161376473363876\)\()/120\)
\(\nu^{13}\)\(=\)\((\)\(-835514983998889 \beta_{19} + 1326273498508417 \beta_{18} - 1326273498508417 \beta_{17} - 226539141889945 \beta_{16} + 1091372277829136 \beta_{15} + 1091372277829136 \beta_{14} + 226539141889945 \beta_{13} - 162408965097545 \beta_{11} + 6933603763409070 \beta_{10} - 3021273600044499 \beta_{9} - 391452019633316 \beta_{8} - 11764308064226939 \beta_{7} - 673106018901344 \beta_{6} - 2530015110507831 \beta_{5} - 835514983998889 \beta_{4} - 71848102304412924 \beta_{2} - 350609608074455024 \beta_{1} - 350609608074455024\)\()/120\)
\(\nu^{14}\)\(=\)\((\)\(-33641428610767791 \beta_{19} + 30237551496588048 \beta_{18} + 25807248655445133 \beta_{16} + 217031241768476952 \beta_{15} + 33641428610767791 \beta_{13} + 258781032048781315 \beta_{12} - 120543219446733939 \beta_{11} - 258781032048781315 \beta_{10} + 8068410249072777 \beta_{9} + 10868822037172539 \beta_{8} + 602716097233669695 \beta_{7} - 1314721100180848185 \beta_{6} - 278136760310215581 \beta_{5} - 25807248655445133 \beta_{4} - 10868822037172539 \beta_{3} + 1367747232112296423 \beta_{2} - 45629897881948263628 \beta_{1}\)\()/360\)
\(\nu^{15}\)\(=\)\((\)\(-242255713914931299 \beta_{19} - 1591039486187032527 \beta_{18} - 1591039486187032527 \beta_{17} - 1288485198842706387 \beta_{16} - 2226415237399749132 \beta_{15} + 2226415237399749132 \beta_{14} - 1288485198842706387 \beta_{13} + 4683710104480835996 \beta_{12} - 2260888200225545817 \beta_{11} + 966999078020998977 \beta_{9} + 318945405265830663 \beta_{7} - 48407995275199607784 \beta_{6} - 6150500205756903537 \beta_{5} + 242255713914931299 \beta_{4} - 1949030879953924494 \beta_{3} + 321486120821707410 \beta_{2} + 531021285366184112792 \beta_{1} - 531021285366184112792\)\()/360\)
\(\nu^{16}\)\(=\)\((\)\(159538407963153563 \beta_{19} + 2117889108165341122 \beta_{17} - 2497428689825169161 \beta_{16} + 9858614796421820728 \beta_{14} + 159538407963153563 \beta_{13} - 35468577960050230333 \beta_{12} + 9684354953463014321 \beta_{11} - 35468577960050230333 \beta_{10} + 21327010837445264177 \beta_{9} + 4071663805114700867 \beta_{8} + 57773335894763133997 \beta_{7} + 236477443718188354663 \beta_{6} + 106635054187226320885 \beta_{5} - 2497428689825169161 \beta_{4} + 4071663805114700867 \beta_{3} + 227172678116068120405 \beta_{2} - 2347755204175592764308\)\()/120\)
\(\nu^{17}\)\(=\)\((\)\(156570575301041600931 \beta_{19} - 156817788901609004055 \beta_{18} + 156817788901609004055 \beta_{17} - 63794331403065462429 \beta_{16} - 306420574198260418216 \beta_{15} - 306420574198260418216 \beta_{14} + 63794331403065462429 \beta_{13} + 178483870679779067551 \beta_{11} - 76597161176595137250 \beta_{10} + 65277805377330958277 \beta_{9} + 314636564714607287996 \beta_{8} - 299893274317511612339 \beta_{7} - 21913295378737466620 \beta_{6} - 266137052194728934031 \beta_{5} + 156570575301041600931 \beta_{4} + 1083008358656595110216 \beta_{2} + 64330174987951423548944 \beta_{1} + 64330174987951423548944\)\()/120\)
\(\nu^{18}\)\(=\)\((\)\(551714020835759260245 \beta_{19} + 7689122849622147660744 \beta_{18} + 3832545164761782585801 \beta_{16} + 8007396901119838204416 \beta_{15} - 551714020835759260245 \beta_{13} - 32748874690512806679061 \beta_{12} + 22732298495849829227277 \beta_{11} + 32748874690512806679061 \beta_{10} - 16163528508332959532079 \beta_{9} - 7211816815612554660285 \beta_{8} - 113661492479249146136385 \beta_{7} + 243165140369760841001259 \beta_{6} + 67694317965960704358171 \beta_{5} - 3832545164761782585801 \beta_{4} + 7211816815612554660285 \beta_{3} - 253014741501203734022013 \beta_{2} - 1163980413055784349719012 \beta_{1}\)\()/360\)
\(\nu^{19}\)\(=\)\((\)\(76736757247153198053705 \beta_{19} + 89185861605209666675865 \beta_{18} + 89185861605209666675865 \beta_{17} + 124859789415254563645713 \beta_{16} + 266269569826530429411900 \beta_{15} - 266269569826530429411900 \beta_{14} + 124859789415254563645713 \beta_{13} + 421490781694700207489836 \beta_{12} - 173713127593692517321329 \beta_{11} - 207086331731789484632343 \beta_{9} + 535992500997929168578863 \beta_{7} - 3724692647299510516468956 \beta_{6} - 1368004795629480841189497 \beta_{5} - 76736757247153198053705 \beta_{4} + 313221011267298902333778 \beta_{3} + 82226542316534920986630 \beta_{2} - 50082411852137497839769736 \beta_{1} + 50082411852137497839769736\)\()/360\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(1\) \(\beta_{1}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
14.4102 9.21685i
2.77873 + 6.78762i
9.05032 16.3862i
6.78762 + 2.77873i
−5.69920 7.08001i
−9.21685 + 14.4102i
−4.98882 + 11.3442i
−16.3862 + 9.05032i
11.3442 4.98882i
−7.08001 5.69920i
14.4102 + 9.21685i
2.77873 6.78762i
9.05032 + 16.3862i
6.78762 2.77873i
−5.69920 + 7.08001i
−9.21685 14.4102i
−4.98882 11.3442i
−16.3862 9.05032i
11.3442 + 4.98882i
−7.08001 + 5.69920i
0 −15.0272 4.14525i 0 29.4180 + 47.5351i 0 −98.6056 98.6056i 0 208.634 + 124.583i 0
17.2 0 −12.0403 + 9.90111i 0 −51.4858 + 21.7765i 0 56.0116 + 56.0116i 0 46.9362 238.424i 0
17.3 0 −12.0317 9.91146i 0 −26.7203 49.1022i 0 1.70680 + 1.70680i 0 46.5259 + 238.504i 0
17.4 0 −9.90111 + 12.0403i 0 51.4858 21.7765i 0 56.0116 + 56.0116i 0 −46.9362 238.424i 0
17.5 0 0.876355 15.5638i 0 55.7692 3.84667i 0 151.287 + 151.287i 0 −241.464 27.2788i 0
17.6 0 4.14525 + 15.0272i 0 −29.4180 47.5351i 0 −98.6056 98.6056i 0 −208.634 + 124.583i 0
17.7 0 4.61326 14.8902i 0 −31.2988 + 46.3183i 0 −91.3999 91.3999i 0 −200.436 137.385i 0
17.8 0 9.91146 + 12.0317i 0 26.7203 + 49.1022i 0 1.70680 + 1.70680i 0 −46.5259 + 238.504i 0
17.9 0 14.8902 4.61326i 0 31.2988 46.3183i 0 −91.3999 91.3999i 0 200.436 137.385i 0
17.10 0 15.5638 0.876355i 0 −55.7692 + 3.84667i 0 151.287 + 151.287i 0 241.464 27.2788i 0
53.1 0 −15.0272 + 4.14525i 0 29.4180 47.5351i 0 −98.6056 + 98.6056i 0 208.634 124.583i 0
53.2 0 −12.0403 9.90111i 0 −51.4858 21.7765i 0 56.0116 56.0116i 0 46.9362 + 238.424i 0
53.3 0 −12.0317 + 9.91146i 0 −26.7203 + 49.1022i 0 1.70680 1.70680i 0 46.5259 238.504i 0
53.4 0 −9.90111 12.0403i 0 51.4858 + 21.7765i 0 56.0116 56.0116i 0 −46.9362 + 238.424i 0
53.5 0 0.876355 + 15.5638i 0 55.7692 + 3.84667i 0 151.287 151.287i 0 −241.464 + 27.2788i 0
53.6 0 4.14525 15.0272i 0 −29.4180 + 47.5351i 0 −98.6056 + 98.6056i 0 −208.634 124.583i 0
53.7 0 4.61326 + 14.8902i 0 −31.2988 46.3183i 0 −91.3999 + 91.3999i 0 −200.436 + 137.385i 0
53.8 0 9.91146 12.0317i 0 26.7203 49.1022i 0 1.70680 1.70680i 0 −46.5259 238.504i 0
53.9 0 14.8902 + 4.61326i 0 31.2988 + 46.3183i 0 −91.3999 + 91.3999i 0 200.436 + 137.385i 0
53.10 0 15.5638 + 0.876355i 0 −55.7692 3.84667i 0 151.287 151.287i 0 241.464 + 27.2788i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.6.i.a 20
3.b odd 2 1 inner 60.6.i.a 20
5.b even 2 1 300.6.i.d 20
5.c odd 4 1 inner 60.6.i.a 20
5.c odd 4 1 300.6.i.d 20
15.d odd 2 1 300.6.i.d 20
15.e even 4 1 inner 60.6.i.a 20
15.e even 4 1 300.6.i.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.6.i.a 20 1.a even 1 1 trivial
60.6.i.a 20 3.b odd 2 1 inner
60.6.i.a 20 5.c odd 4 1 inner
60.6.i.a 20 15.e even 4 1 inner
300.6.i.d 20 5.b even 2 1
300.6.i.d 20 5.c odd 4 1
300.6.i.d 20 15.d odd 2 1
300.6.i.d 20 15.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(60, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \)
$3$ \( \)\(71\!\cdots\!49\)\( - \)\(59\!\cdots\!86\)\( T + 24315330918113857602 T^{2} - 24315330918113857602 T^{3} + 713412772707958785 T^{4} - 1552937676622632576 T^{5} + 13168998902797632 T^{6} + 1475495581402368 T^{7} - 67120599719250 T^{8} - 1501871397876 T^{9} + 1631209020732 T^{10} - 6180540732 T^{11} - 1136693250 T^{12} + 102829824 T^{13} + 3776832 T^{14} - 1832832 T^{15} + 3465 T^{16} - 486 T^{17} + 2 T^{18} - 2 T^{19} + T^{20} \)
$5$ \( \)\(88\!\cdots\!25\)\( - \)\(18\!\cdots\!50\)\( T^{2} + \)\(92\!\cdots\!25\)\( T^{4} - \)\(75\!\cdots\!00\)\( T^{6} + \)\(21\!\cdots\!50\)\( T^{8} - 572014068164062500 T^{10} + 223623592968750 T^{12} - 79557300000 T^{14} + 9885225 T^{16} - 2030 T^{18} + T^{20} \)
$7$ \( ( 543712157584108832 - 320394899292050528 T + 94400033234941456 T^{2} - 317014977593600 T^{3} + 350507914896 T^{4} + 51358440016 T^{5} + 1299045768 T^{6} + 2545600 T^{7} + 722 T^{8} - 38 T^{9} + T^{10} )^{2} \)
$11$ \( ( \)\(10\!\cdots\!00\)\( + \)\(17\!\cdots\!00\)\( T^{2} + 8739867847886000 T^{4} + 148426968100 T^{6} + 713200 T^{8} + T^{10} )^{2} \)
$13$ \( ( \)\(18\!\cdots\!68\)\( + \)\(94\!\cdots\!96\)\( T + \)\(24\!\cdots\!56\)\( T^{2} + \)\(25\!\cdots\!00\)\( T^{3} + 102002982524759304 T^{4} + 90570325682088 T^{5} + 859305056868 T^{6} + 738763200 T^{7} + 142578 T^{8} - 534 T^{9} + T^{10} )^{2} \)
$17$ \( \)\(14\!\cdots\!00\)\( + \)\(51\!\cdots\!00\)\( T^{4} + \)\(18\!\cdots\!00\)\( T^{8} + \)\(46\!\cdots\!00\)\( T^{12} + 37399539538700 T^{16} + T^{20} \)
$19$ \( ( \)\(41\!\cdots\!24\)\( + \)\(81\!\cdots\!80\)\( T^{2} + 30003821365046271040 T^{4} + 32974631202160 T^{6} + 10434620 T^{8} + T^{10} )^{2} \)
$23$ \( \)\(58\!\cdots\!00\)\( + \)\(45\!\cdots\!00\)\( T^{4} + \)\(26\!\cdots\!00\)\( T^{8} + \)\(50\!\cdots\!00\)\( T^{12} + 390882458038700 T^{16} + T^{20} \)
$29$ \( ( -\)\(23\!\cdots\!00\)\( + \)\(12\!\cdots\!00\)\( T^{2} - \)\(22\!\cdots\!00\)\( T^{4} + 968689921080100 T^{6} - 62269600 T^{8} + T^{10} )^{2} \)
$31$ \( ( -212642556537831424 + 203322620257280 T + 42049903760 T^{2} - 44043140 T^{3} + 1180 T^{4} + T^{5} )^{4} \)
$37$ \( ( \)\(52\!\cdots\!68\)\( - \)\(26\!\cdots\!44\)\( T + \)\(67\!\cdots\!76\)\( T^{2} - \)\(29\!\cdots\!00\)\( T^{3} + \)\(70\!\cdots\!64\)\( T^{4} - 32464163714746771512 T^{5} + 15797647340229348 T^{6} - 401704866000 T^{7} + 46818 T^{8} + 306 T^{9} + T^{10} )^{2} \)
$41$ \( ( \)\(15\!\cdots\!00\)\( + \)\(77\!\cdots\!00\)\( T^{2} + \)\(23\!\cdots\!00\)\( T^{4} + 225218348874649600 T^{6} + 822509200 T^{8} + T^{10} )^{2} \)
$43$ \( ( \)\(36\!\cdots\!68\)\( + \)\(28\!\cdots\!56\)\( T + \)\(11\!\cdots\!76\)\( T^{2} + \)\(84\!\cdots\!00\)\( T^{3} + \)\(48\!\cdots\!64\)\( T^{4} + \)\(67\!\cdots\!88\)\( T^{5} + 47499919042751748 T^{6} + 630377762400 T^{7} + 72072018 T^{8} + 12006 T^{9} + T^{10} )^{2} \)
$47$ \( \)\(11\!\cdots\!00\)\( + \)\(99\!\cdots\!00\)\( T^{4} + \)\(21\!\cdots\!00\)\( T^{8} + \)\(13\!\cdots\!00\)\( T^{12} + 717527152383950700 T^{16} + T^{20} \)
$53$ \( \)\(29\!\cdots\!00\)\( + \)\(97\!\cdots\!00\)\( T^{4} + \)\(85\!\cdots\!00\)\( T^{8} + \)\(26\!\cdots\!00\)\( T^{12} + 3015871858397206700 T^{16} + T^{20} \)
$59$ \( ( -\)\(79\!\cdots\!00\)\( + \)\(16\!\cdots\!00\)\( T^{2} - \)\(49\!\cdots\!00\)\( T^{4} + 2777807805320652100 T^{6} - 3205986100 T^{8} + T^{10} )^{2} \)
$61$ \( ( \)\(17\!\cdots\!76\)\( + 972926567300152080 T + 10634306567760 T^{2} - 2039434440 T^{3} - 14970 T^{4} + T^{5} )^{4} \)
$67$ \( ( \)\(17\!\cdots\!32\)\( + \)\(39\!\cdots\!32\)\( T + \)\(43\!\cdots\!16\)\( T^{2} + \)\(31\!\cdots\!00\)\( T^{3} + \)\(70\!\cdots\!16\)\( T^{4} + \)\(35\!\cdots\!16\)\( T^{5} + 8761162486743451908 T^{6} + 2202950766400 T^{7} + 816160802 T^{8} + 40402 T^{9} + T^{10} )^{2} \)
$71$ \( ( \)\(56\!\cdots\!00\)\( + \)\(60\!\cdots\!00\)\( T^{2} + \)\(48\!\cdots\!00\)\( T^{4} + \)\(14\!\cdots\!00\)\( T^{6} + 19800130300 T^{8} + T^{10} )^{2} \)
$73$ \( ( \)\(10\!\cdots\!32\)\( + \)\(90\!\cdots\!88\)\( T + \)\(37\!\cdots\!96\)\( T^{2} + \)\(20\!\cdots\!00\)\( T^{3} + \)\(43\!\cdots\!76\)\( T^{4} + \)\(57\!\cdots\!84\)\( T^{5} + 34577272694032990728 T^{6} - 64099440976000 T^{7} + 405498242 T^{8} + 28478 T^{9} + T^{10} )^{2} \)
$79$ \( ( \)\(13\!\cdots\!76\)\( + \)\(29\!\cdots\!80\)\( T^{2} + \)\(24\!\cdots\!60\)\( T^{4} + 95355840030014907360 T^{6} + 16438357380 T^{8} + T^{10} )^{2} \)
$83$ \( \)\(63\!\cdots\!00\)\( + \)\(15\!\cdots\!00\)\( T^{4} + \)\(47\!\cdots\!00\)\( T^{8} + \)\(41\!\cdots\!00\)\( T^{12} + \)\(11\!\cdots\!00\)\( T^{16} + T^{20} \)
$89$ \( ( -\)\(91\!\cdots\!00\)\( + \)\(33\!\cdots\!00\)\( T^{2} - \)\(36\!\cdots\!00\)\( T^{4} + \)\(14\!\cdots\!00\)\( T^{6} - 23917269100 T^{8} + T^{10} )^{2} \)
$97$ \( ( \)\(40\!\cdots\!32\)\( + \)\(39\!\cdots\!52\)\( T + \)\(18\!\cdots\!36\)\( T^{2} - \)\(76\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!56\)\( T^{4} + \)\(47\!\cdots\!16\)\( T^{5} + 73922155197995630088 T^{6} - 185320895356800 T^{7} + 2856319362 T^{8} + 75582 T^{9} + T^{10} )^{2} \)
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