Properties

Label 21.4.g.a
Level 21
Weight 4
Character orbit 21.g
Analytic conductor 1.239
Analytic rank 0
Dimension 12
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.23904011012\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - x^{11} - 29 x^{9} + 6 x^{8} - 49 x^{7} + 1564 x^{6} - 441 x^{5} + 486 x^{4} - 21141 x^{3} - 59049 x + 531441\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{2} - \beta_{6} ) q^{2} + ( \beta_{7} + \beta_{8} ) q^{3} + ( -2 \beta_{4} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{4} + \beta_{5} q^{5} + ( 2 - \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} ) q^{6} + ( -5 + 3 \beta_{3} - \beta_{4} - \beta_{7} - \beta_{8} - \beta_{10} + 3 \beta_{11} ) q^{7} + ( \beta_{1} - 2 \beta_{2} - \beta_{5} - \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{11} ) q^{8} + ( \beta_{1} - 3 \beta_{3} + \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{9} +O(q^{10})\) \( q + ( \beta_{2} - \beta_{6} ) q^{2} + ( \beta_{7} + \beta_{8} ) q^{3} + ( -2 \beta_{4} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{4} + \beta_{5} q^{5} + ( 2 - \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} ) q^{6} + ( -5 + 3 \beta_{3} - \beta_{4} - \beta_{7} - \beta_{8} - \beta_{10} + 3 \beta_{11} ) q^{7} + ( \beta_{1} - 2 \beta_{2} - \beta_{5} - \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{11} ) q^{8} + ( \beta_{1} - 3 \beta_{3} + \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{9} + ( -6 \beta_{1} - 2 \beta_{3} - 5 \beta_{7} - 4 \beta_{8} + \beta_{10} - \beta_{11} ) q^{10} + ( 2 \beta_{1} - 2 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - 4 \beta_{11} ) q^{11} + ( -10 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 10 \beta_{4} + \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - \beta_{8} - 4 \beta_{10} + 5 \beta_{11} ) q^{12} + ( -9 + 2 \beta_{1} + 3 \beta_{3} - 18 \beta_{4} + 5 \beta_{7} + 3 \beta_{10} - 2 \beta_{11} ) q^{13} + ( -4 \beta_{1} - 10 \beta_{2} + \beta_{5} + 9 \beta_{6} + 4 \beta_{7} - \beta_{8} + 2 \beta_{9} + 5 \beta_{11} ) q^{14} + ( 3 + 4 \beta_{1} + 12 \beta_{2} + \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} + 4 \beta_{11} ) q^{15} + ( 24 + 3 \beta_{1} - \beta_{3} + 24 \beta_{4} + 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{11} ) q^{16} + ( -9 \beta_{1} + 8 \beta_{2} - 4 \beta_{6} + 9 \beta_{7} + 9 \beta_{8} ) q^{17} + ( 6 \beta_{1} - 6 \beta_{4} - 3 \beta_{6} + 6 \beta_{7} + 6 \beta_{10} - 18 \beta_{11} ) q^{18} + ( 17 + \beta_{1} + \beta_{3} - 17 \beta_{4} - \beta_{7} - 2 \beta_{10} + 2 \beta_{11} ) q^{19} + ( 3 \beta_{1} + 12 \beta_{2} + 3 \beta_{5} - 24 \beta_{6} - 3 \beta_{7} - 3 \beta_{9} - 3 \beta_{11} ) q^{20} + ( 13 - 16 \beta_{1} + 16 \beta_{2} - 5 \beta_{3} - 31 \beta_{4} - 3 \beta_{5} - 20 \beta_{6} - \beta_{7} + \beta_{9} + 4 \beta_{10} + 8 \beta_{11} ) q^{21} + ( -16 + 3 \beta_{1} - 5 \beta_{3} + 8 \beta_{7} + 16 \beta_{8} + 5 \beta_{10} + 3 \beta_{11} ) q^{22} + ( 10 \beta_{1} + 4 \beta_{2} - 4 \beta_{6} - 10 \beta_{7} - 5 \beta_{8} - 5 \beta_{11} ) q^{23} + ( 40 - 10 \beta_{1} - 20 \beta_{2} + 10 \beta_{3} + 20 \beta_{4} + 10 \beta_{6} - 3 \beta_{7} - 8 \beta_{8} - 2 \beta_{9} - 5 \beta_{10} + 5 \beta_{11} ) q^{24} + ( 11 \beta_{1} + 13 \beta_{4} - 4 \beta_{7} - 11 \beta_{8} - 15 \beta_{10} - 7 \beta_{11} ) q^{25} + ( 19 \beta_{2} - 7 \beta_{5} + 19 \beta_{6} - 9 \beta_{8} + 9 \beta_{11} ) q^{26} + ( 21 - 24 \beta_{2} + 3 \beta_{3} + 42 \beta_{4} + 3 \beta_{5} + 48 \beta_{6} - 3 \beta_{7} - 3 \beta_{9} + 3 \beta_{10} ) q^{27} + ( -6 - 7 \beta_{1} - 9 \beta_{3} + 80 \beta_{4} - 11 \beta_{7} + 10 \beta_{8} - 4 \beta_{10} + 12 \beta_{11} ) q^{28} + ( 7 \beta_{1} - 52 \beta_{2} + 5 \beta_{5} - 7 \beta_{7} - 14 \beta_{8} + 5 \beta_{9} + 7 \beta_{11} ) q^{29} + ( -138 + 4 \beta_{1} - 12 \beta_{2} + 18 \beta_{3} - 138 \beta_{4} - 5 \beta_{5} + 12 \beta_{6} + 4 \beta_{7} - 7 \beta_{8} + 10 \beta_{9} + 7 \beta_{11} ) q^{30} + ( -110 - 12 \beta_{1} - 4 \beta_{3} - 55 \beta_{4} - 10 \beta_{7} - 8 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} ) q^{31} + ( 9 \beta_{1} + 10 \beta_{5} - 9 \beta_{7} + 9 \beta_{8} - 5 \beta_{9} - 18 \beta_{11} ) q^{32} + ( -49 - 10 \beta_{1} - 16 \beta_{2} - 10 \beta_{3} + 49 \beta_{4} - 5 \beta_{5} - 16 \beta_{6} + 10 \beta_{7} + 7 \beta_{8} + 20 \beta_{10} - 16 \beta_{11} ) q^{33} + ( -4 + 4 \beta_{1} - 14 \beta_{3} - 8 \beta_{4} - 10 \beta_{7} - 14 \beta_{10} - 4 \beta_{11} ) q^{34} + ( -9 \beta_{1} - 12 \beta_{2} - 3 \beta_{5} + 36 \beta_{6} + 9 \beta_{7} + 3 \beta_{8} - 13 \beta_{9} + 6 \beta_{11} ) q^{35} + ( 66 - \beta_{1} + 48 \beta_{2} + 3 \beta_{3} - 4 \beta_{5} - 8 \beta_{7} - 16 \beta_{8} - 4 \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{36} + ( 135 + 19 \beta_{1} + 11 \beta_{3} + 135 \beta_{4} + 19 \beta_{7} + 4 \beta_{8} - 4 \beta_{11} ) q^{37} + ( -3 \beta_{1} + 26 \beta_{2} - 13 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} ) q^{38} + ( -5 \beta_{1} + 63 \beta_{4} + 4 \beta_{5} - 36 \beta_{6} - 16 \beta_{7} + \beta_{8} - 2 \beta_{9} - 15 \beta_{10} + 25 \beta_{11} ) q^{39} + ( 132 - 13 \beta_{1} - 13 \beta_{3} - 132 \beta_{4} + 13 \beta_{7} + 26 \beta_{8} + 26 \beta_{10} ) q^{40} + ( -6 \beta_{1} + 32 \beta_{2} - 16 \beta_{5} - 64 \beta_{6} + 6 \beta_{7} + 16 \beta_{9} + 6 \beta_{11} ) q^{41} + ( 14 + 30 \beta_{1} + 47 \beta_{2} + 14 \beta_{3} - 140 \beta_{4} + 17 \beta_{5} - 43 \beta_{6} + 5 \beta_{7} - 3 \beta_{8} - 8 \beta_{9} - 7 \beta_{10} - 6 \beta_{11} ) q^{42} + ( -87 + 18 \beta_{1} + 31 \beta_{3} - 13 \beta_{7} - 26 \beta_{8} - 31 \beta_{10} + 18 \beta_{11} ) q^{43} + ( -22 \beta_{1} - 12 \beta_{2} - 3 \beta_{5} + 12 \beta_{6} + 22 \beta_{7} + 11 \beta_{8} + 6 \beta_{9} + 11 \beta_{11} ) q^{44} + ( 270 - 30 \beta_{3} + 135 \beta_{4} + 6 \beta_{7} + 21 \beta_{8} + 12 \beta_{9} + 15 \beta_{10} - 15 \beta_{11} ) q^{45} + ( -10 \beta_{1} - 100 \beta_{4} + 24 \beta_{7} + 10 \beta_{8} + 34 \beta_{10} - 14 \beta_{11} ) q^{46} + ( 20 \beta_{2} + 14 \beta_{5} + 20 \beta_{6} + 27 \beta_{8} - 27 \beta_{11} ) q^{47} + ( 44 + 9 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 88 \beta_{4} + 5 \beta_{5} + 8 \beta_{6} + 19 \beta_{7} - 5 \beta_{9} - 4 \beta_{10} - 9 \beta_{11} ) q^{48} + ( 42 - 21 \beta_{1} - 7 \beta_{3} + 175 \beta_{4} + 14 \beta_{7} - 21 \beta_{8} + 35 \beta_{10} - 42 \beta_{11} ) q^{49} + ( -15 \beta_{1} - 7 \beta_{2} - 7 \beta_{5} + 15 \beta_{7} + 30 \beta_{8} - 7 \beta_{9} - 15 \beta_{11} ) q^{50} + ( -219 + 5 \beta_{1} - 12 \beta_{2} - 39 \beta_{3} - 219 \beta_{4} + 5 \beta_{5} + 12 \beta_{6} - 13 \beta_{7} + 13 \beta_{8} - 10 \beta_{9} - 22 \beta_{11} ) q^{51} + ( -308 + 58 \beta_{1} + 40 \beta_{3} - 154 \beta_{4} + 38 \beta_{7} + 18 \beta_{8} - 20 \beta_{10} + 20 \beta_{11} ) q^{52} + ( -28 \beta_{1} - 6 \beta_{5} + 40 \beta_{6} + 28 \beta_{7} - 28 \beta_{8} + 3 \beta_{9} + 56 \beta_{11} ) q^{53} + ( -228 + 15 \beta_{1} - 21 \beta_{2} + 15 \beta_{3} + 228 \beta_{4} + 3 \beta_{5} - 21 \beta_{6} - 15 \beta_{7} - 15 \beta_{8} - 30 \beta_{10} + 33 \beta_{11} ) q^{54} + ( -150 - 28 \beta_{1} + 19 \beta_{3} - 300 \beta_{4} - 9 \beta_{7} + 19 \beta_{10} + 28 \beta_{11} ) q^{55} + ( 45 \beta_{1} - 66 \beta_{2} - 6 \beta_{5} + 2 \beta_{6} - 45 \beta_{7} - 15 \beta_{8} + 23 \beta_{9} - 30 \beta_{11} ) q^{56} + ( 30 - 7 \beta_{1} + 12 \beta_{2} - 6 \beta_{3} - \beta_{5} + 20 \beta_{7} + 40 \beta_{8} - \beta_{9} + 6 \beta_{10} - 7 \beta_{11} ) q^{57} + ( 436 - 93 \beta_{1} - 25 \beta_{3} + 436 \beta_{4} - 93 \beta_{7} - 34 \beta_{8} + 34 \beta_{11} ) q^{58} + ( 54 \beta_{1} - 32 \beta_{2} + 16 \beta_{6} - 54 \beta_{7} - 54 \beta_{8} - 19 \beta_{9} ) q^{59} + ( -6 \beta_{1} + 144 \beta_{4} - 24 \beta_{5} + 72 \beta_{6} - 21 \beta_{7} - 6 \beta_{8} + 12 \beta_{9} - 27 \beta_{10} + 39 \beta_{11} ) q^{60} + ( 114 + 38 \beta_{1} + 38 \beta_{3} - 114 \beta_{4} - 38 \beta_{7} - 89 \beta_{8} - 76 \beta_{10} - 13 \beta_{11} ) q^{61} + ( 6 \beta_{1} - 31 \beta_{2} + 22 \beta_{5} + 62 \beta_{6} - 6 \beta_{7} - 22 \beta_{9} - 6 \beta_{11} ) q^{62} + ( -45 + 23 \beta_{1} - 72 \beta_{2} + 6 \beta_{3} - 282 \beta_{4} - 25 \beta_{5} + 48 \beta_{6} - 11 \beta_{7} + 23 \beta_{8} + 20 \beta_{9} - 30 \beta_{10} - 7 \beta_{11} ) q^{63} + ( -84 - 79 \beta_{1} - 61 \beta_{3} - 18 \beta_{7} - 36 \beta_{8} + 61 \beta_{10} - 79 \beta_{11} ) q^{64} + ( 12 \beta_{1} + 84 \beta_{2} + 14 \beta_{5} - 84 \beta_{6} - 12 \beta_{7} - 6 \beta_{8} - 28 \beta_{9} - 6 \beta_{11} ) q^{65} + ( 332 + 25 \beta_{1} - 40 \beta_{2} - 28 \beta_{3} + 166 \beta_{4} + 20 \beta_{6} - 21 \beta_{7} - 7 \beta_{8} - 19 \beta_{9} + 14 \beta_{10} - 14 \beta_{11} ) q^{66} + ( -62 \beta_{1} - 181 \beta_{4} - 23 \beta_{7} + 62 \beta_{8} + 39 \beta_{10} + 85 \beta_{11} ) q^{67} + ( -48 \beta_{2} + 6 \beta_{5} - 48 \beta_{6} - 30 \beta_{8} + 30 \beta_{11} ) q^{68} + ( 143 - 15 \beta_{1} - 4 \beta_{2} + 11 \beta_{3} + 286 \beta_{4} - 19 \beta_{5} + 8 \beta_{6} + 11 \beta_{7} + 19 \beta_{9} + 11 \beta_{10} + 15 \beta_{11} ) q^{69} + ( -252 + 119 \beta_{1} + 49 \beta_{3} + 168 \beta_{4} + 63 \beta_{7} - 56 \beta_{10} - 14 \beta_{11} ) q^{70} + ( -18 \beta_{1} + 68 \beta_{2} + 4 \beta_{5} + 18 \beta_{7} + 36 \beta_{8} + 4 \beta_{9} - 18 \beta_{11} ) q^{71} + ( -456 - 54 \beta_{1} + 66 \beta_{2} + 24 \beta_{3} - 456 \beta_{4} + 9 \beta_{5} - 66 \beta_{6} + 54 \beta_{7} + 15 \beta_{8} - 18 \beta_{9} + 39 \beta_{11} ) q^{72} + ( -290 + 38 \beta_{1} - 22 \beta_{3} - 145 \beta_{4} + 49 \beta_{7} + 60 \beta_{8} + 11 \beta_{10} - 11 \beta_{11} ) q^{73} + ( -11 \beta_{1} - 38 \beta_{5} - 67 \beta_{6} + 11 \beta_{7} - 11 \beta_{8} + 19 \beta_{9} + 22 \beta_{11} ) q^{74} + ( -147 + 3 \beta_{1} + 60 \beta_{2} + 3 \beta_{3} + 147 \beta_{4} + 18 \beta_{5} + 60 \beta_{6} - 3 \beta_{7} - \beta_{8} - 6 \beta_{10} - 75 \beta_{11} ) q^{75} + ( 18 + 6 \beta_{1} + 12 \beta_{3} + 36 \beta_{4} + 18 \beta_{7} + 12 \beta_{10} - 6 \beta_{11} ) q^{76} + ( -15 \beta_{1} + 148 \beta_{2} + 23 \beta_{5} - 52 \beta_{6} + 15 \beta_{7} + 54 \beta_{8} + 18 \beta_{9} - 39 \beta_{11} ) q^{77} + ( 336 - 57 \beta_{1} - 141 \beta_{2} - 30 \beta_{3} + 21 \beta_{5} + 3 \beta_{7} + 6 \beta_{8} + 21 \beta_{9} + 30 \beta_{10} - 57 \beta_{11} ) q^{78} + ( 325 + 88 \beta_{1} - 12 \beta_{3} + 325 \beta_{4} + 88 \beta_{7} + 50 \beta_{8} - 50 \beta_{11} ) q^{79} + ( 15 \beta_{1} + 72 \beta_{2} - 36 \beta_{6} - 15 \beta_{7} - 15 \beta_{8} + 37 \beta_{9} ) q^{80} + ( -57 \beta_{1} - 144 \beta_{4} + 42 \beta_{5} - 72 \beta_{6} + 75 \beta_{7} + 6 \beta_{8} - 21 \beta_{9} + 81 \beta_{10} + 33 \beta_{11} ) q^{81} + ( 296 - 4 \beta_{1} - 4 \beta_{3} - 296 \beta_{4} + 4 \beta_{7} + 52 \beta_{8} + 8 \beta_{10} + 44 \beta_{11} ) q^{82} + ( -27 \beta_{1} - 56 \beta_{2} + 13 \beta_{5} + 112 \beta_{6} + 27 \beta_{7} - 13 \beta_{9} + 27 \beta_{11} ) q^{83} + ( 220 + 13 \beta_{1} - 20 \beta_{2} - 20 \beta_{3} - 166 \beta_{4} - 19 \beta_{5} + 88 \beta_{6} + 17 \beta_{7} - 56 \beta_{8} - 3 \beta_{9} + 58 \beta_{10} - 59 \beta_{11} ) q^{84} + ( 54 + 17 \beta_{1} - 12 \beta_{3} + 29 \beta_{7} + 58 \beta_{8} + 12 \beta_{10} + 17 \beta_{11} ) q^{85} + ( -62 \beta_{1} - 133 \beta_{2} - 5 \beta_{5} + 133 \beta_{6} + 62 \beta_{7} + 31 \beta_{8} + 10 \beta_{9} + 31 \beta_{11} ) q^{86} + ( 200 + 112 \beta_{1} + 224 \beta_{2} + 146 \beta_{3} + 100 \beta_{4} - 112 \beta_{6} + 15 \beta_{7} - 58 \beta_{8} - 16 \beta_{9} - 73 \beta_{10} + 73 \beta_{11} ) q^{87} + ( 74 \beta_{1} + 124 \beta_{4} - 53 \beta_{7} - 74 \beta_{8} - 127 \beta_{10} - 21 \beta_{11} ) q^{88} + ( -104 \beta_{2} - 48 \beta_{5} - 104 \beta_{6} + 51 \beta_{8} - 51 \beta_{11} ) q^{89} + ( -18 - 15 \beta_{1} + 144 \beta_{2} - 3 \beta_{3} - 36 \beta_{4} - 6 \beta_{5} - 288 \beta_{6} - 108 \beta_{7} + 6 \beta_{9} - 3 \beta_{10} + 15 \beta_{11} ) q^{90} + ( 143 - 77 \beta_{1} - 55 \beta_{3} + 499 \beta_{4} - 75 \beta_{7} + 9 \beta_{8} + 2 \beta_{10} + 29 \beta_{11} ) q^{91} + ( -6 \beta_{1} + 144 \beta_{2} - 14 \beta_{5} + 6 \beta_{7} + 12 \beta_{8} - 14 \beta_{9} - 6 \beta_{11} ) q^{92} + ( -276 + 8 \beta_{1} - 24 \beta_{2} + 36 \beta_{3} - 276 \beta_{4} - 10 \beta_{5} + 24 \beta_{6} - 102 \beta_{7} - 69 \beta_{8} + 20 \beta_{9} + 14 \beta_{11} ) q^{93} + ( -184 - 98 \beta_{1} - 96 \beta_{3} - 92 \beta_{4} - 50 \beta_{7} - 2 \beta_{8} + 48 \beta_{10} - 48 \beta_{11} ) q^{94} + ( -3 \beta_{1} + 32 \beta_{5} + 36 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - 16 \beta_{9} + 6 \beta_{11} ) q^{95} + ( -228 - 27 \beta_{1} + 60 \beta_{2} - 27 \beta_{3} + 228 \beta_{4} - 12 \beta_{5} + 60 \beta_{6} + 27 \beta_{7} + 42 \beta_{8} + 54 \beta_{10} + 60 \beta_{11} ) q^{96} + ( -156 + 37 \beta_{1} - 81 \beta_{3} - 312 \beta_{4} - 44 \beta_{7} - 81 \beta_{10} - 37 \beta_{11} ) q^{97} + ( 42 \beta_{1} + 63 \beta_{2} + 7 \beta_{5} - 140 \beta_{6} - 42 \beta_{7} - 63 \beta_{8} - 84 \beta_{9} + 21 \beta_{11} ) q^{98} + ( -303 + 75 \beta_{1} - 132 \beta_{2} + 87 \beta_{3} - 18 \beta_{5} - 93 \beta_{7} - 186 \beta_{8} - 18 \beta_{9} - 87 \beta_{10} + 75 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 3q^{3} + 14q^{4} - 56q^{7} - 3q^{9} + O(q^{10}) \) \( 12q - 3q^{3} + 14q^{4} - 56q^{7} - 3q^{9} + 30q^{10} - 192q^{12} + 6q^{15} + 134q^{16} + 66q^{18} + 300q^{19} + 357q^{21} - 268q^{22} + 414q^{24} - 42q^{25} - 602q^{28} - 822q^{30} - 930q^{31} - 855q^{33} + 852q^{36} + 764q^{37} - 426q^{39} + 2298q^{40} + 966q^{42} - 1012q^{43} + 2367q^{45} + 608q^{46} - 336q^{49} - 1341q^{51} - 3000q^{52} - 4158q^{54} + 270q^{57} + 2870q^{58} - 918q^{60} + 2358q^{61} + 1071q^{63} - 548q^{64} + 2934q^{66} + 792q^{67} - 4242q^{70} - 2712q^{72} - 2904q^{73} - 2418q^{75} + 4296q^{78} + 1674q^{79} + 837q^{81} + 5040q^{82} + 3864q^{84} + 348q^{85} + 1638q^{87} - 554q^{88} - 1218q^{91} - 1479q^{93} - 1356q^{94} - 4410q^{96} - 3354q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - x^{11} - 29 x^{9} + 6 x^{8} - 49 x^{7} + 1564 x^{6} - 441 x^{5} + 486 x^{4} - 21141 x^{3} - 59049 x + 531441\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-538 \nu^{11} - 22601 \nu^{10} + 146502 \nu^{9} - 1327 \nu^{8} + 161148 \nu^{7} - 632573 \nu^{6} + 6980468 \nu^{5} - 17528769 \nu^{4} + 61874280 \nu^{3} - 81309015 \nu^{2} - 145536102 \nu + 142839531\)\()/ 489398112 \)
\(\beta_{2}\)\(=\)\((\)\(-661 \nu^{11} - 17114 \nu^{10} - 13815 \nu^{9} + 226448 \nu^{8} + 617943 \nu^{7} + 441628 \nu^{6} - 2790145 \nu^{5} - 21396420 \nu^{4} - 82640331 \nu^{3} + 112284954 \nu^{2} + 410108427 \nu + 575491554\)\()/ 489398112 \)
\(\beta_{3}\)\(=\)\((\)\( 109 \nu^{11} - 3322 \nu^{10} - 26109 \nu^{9} + 69172 \nu^{8} + 83625 \nu^{7} - 33772 \nu^{6} + 26593 \nu^{5} - 1141056 \nu^{4} - 12229137 \nu^{3} + 61499898 \nu^{2} - 15136227 \nu - 271271106 \)\()/69914016\)
\(\beta_{4}\)\(=\)\((\)\( -857 \nu^{11} + 5426 \nu^{10} - 627 \nu^{9} + 6088 \nu^{8} - 24405 \nu^{7} + 289700 \nu^{6} - 658001 \nu^{5} + 2301324 \nu^{4} - 3432699 \nu^{3} - 5390226 \nu^{2} + 4113747 \nu + 173722158 \)\()/ 163132704 \)
\(\beta_{5}\)\(=\)\((\)\(-4301 \nu^{11} + 145853 \nu^{10} - 231543 \nu^{9} - 1385435 \nu^{8} - 5417499 \nu^{7} + 15945431 \nu^{6} + 12125077 \nu^{5} + 198180063 \nu^{4} - 245384073 \nu^{3} - 938815677 \nu^{2} - 3646715337 \nu + 8693961417\)\()/ 489398112 \)
\(\beta_{6}\)\(=\)\((\)\(-4345 \nu^{11} + 19933 \nu^{10} + 136449 \nu^{9} + 345029 \nu^{8} - 1557771 \nu^{7} - 3453101 \nu^{6} - 8432491 \nu^{5} + 16879851 \nu^{4} + 90039843 \nu^{3} + 303484887 \nu^{2} - 575773677 \nu - 1180448559\)\()/ 489398112 \)
\(\beta_{7}\)\(=\)\((\)\( -1523 \nu^{11} + 209 \nu^{10} + 6255 \nu^{9} + 6421 \nu^{8} - 82569 \nu^{7} - 227449 \nu^{6} - 641129 \nu^{5} + 1005399 \nu^{4} + 7836021 \nu^{3} - 1371249 \nu^{2} + 13338513 \nu - 151814979 \)\()/54377568\)
\(\beta_{8}\)\(=\)\((\)\( -1523 \nu^{11} + 209 \nu^{10} + 6255 \nu^{9} + 6421 \nu^{8} - 82569 \nu^{7} - 227449 \nu^{6} - 641129 \nu^{5} + 1005399 \nu^{4} + 7836021 \nu^{3} - 1371249 \nu^{2} - 149794191 \nu - 151814979 \)\()/54377568\)
\(\beta_{9}\)\(=\)\((\)\(22376 \nu^{11} + 116845 \nu^{10} - 61380 \nu^{9} - 2138089 \nu^{8} - 3679350 \nu^{7} + 8057845 \nu^{6} + 34199906 \nu^{5} + 126399249 \nu^{4} - 75013614 \nu^{3} - 784745901 \nu^{2} - 2180220300 \nu + 5512873689\)\()/ 489398112 \)
\(\beta_{10}\)\(=\)\((\)\( 635 \nu^{11} - 221 \nu^{10} + 4041 \nu^{9} + 4103 \nu^{8} + 30441 \nu^{7} - 164387 \nu^{6} + 335465 \nu^{5} + 2637 \nu^{4} + 2622699 \nu^{3} + 8961597 \nu^{2} + 22720743 \nu - 66193929 \)\()/13226976\)
\(\beta_{11}\)\(=\)\((\)\(-12970 \nu^{11} - 10169 \nu^{10} + 146502 \nu^{9} + 359201 \nu^{8} + 86556 \nu^{7} - 23405 \nu^{6} - 12463180 \nu^{5} - 12046257 \nu^{4} + 55832328 \nu^{3} + 181515897 \nu^{2} - 145536102 \nu + 876936699\)\()/ 244699056 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{8} + \beta_{7}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{11} + 2 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + 3 \beta_{3} - \beta_{1}\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{11} - 3 \beta_{10} - \beta_{9} - 6 \beta_{8} - 3 \beta_{7} - \beta_{5} + 3 \beta_{3} - 24 \beta_{2} + 2 \beta_{1} + 21\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(11 \beta_{11} + 27 \beta_{10} - 7 \beta_{9} + 2 \beta_{8} + 25 \beta_{7} - 24 \beta_{6} + 14 \beta_{5} - 48 \beta_{4} - 19 \beta_{1}\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-33 \beta_{11} + 12 \beta_{9} - 10 \beta_{8} + 52 \beta_{7} - 6 \beta_{5} + 216 \beta_{4} + 72 \beta_{3} + 138 \beta_{1} + 216\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(134 \beta_{11} - 114 \beta_{10} + 8 \beta_{9} - 280 \beta_{8} - 140 \beta_{7} + 8 \beta_{5} + 114 \beta_{3} - 360 \beta_{2} + 134 \beta_{1} - 1629\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(260 \beta_{11} + 552 \beta_{10} - 10 \beta_{9} + 681 \beta_{8} - 129 \beta_{7} - 912 \beta_{6} + 20 \beta_{5} + 984 \beta_{4} - 406 \beta_{1}\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(-1282 \beta_{11} - 1582 \beta_{9} - 493 \beta_{8} - 1013 \beta_{7} + 1176 \beta_{6} + 791 \beta_{5} + 3648 \beta_{4} - 27 \beta_{3} - 1176 \beta_{2} + 2537 \beta_{1} + 3648\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(1824 \beta_{11} + 2709 \beta_{10} + 681 \beta_{9} + 974 \beta_{8} + 487 \beta_{7} + 681 \beta_{5} - 2709 \beta_{3} + 8424 \beta_{2} + 1824 \beta_{1} - 29619\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(-10351 \beta_{11} - 4467 \beta_{10} + 3905 \beta_{9} + 8606 \beta_{8} - 13073 \beta_{7} + 11088 \beta_{6} - 7810 \beta_{5} + 114192 \beta_{4} + 7409 \beta_{1}\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(-3799 \beta_{11} - 20608 \beta_{9} - 25200 \beta_{8} - 96240 \beta_{7} + 40080 \beta_{6} + 10304 \beta_{5} - 144912 \beta_{4} - 45840 \beta_{3} - 40080 \beta_{2} - 38242 \beta_{1} - 144912\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(1 + \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
2.70662 + 1.29391i
0.00299931 3.00000i
−2.23014 + 2.00661i
2.85284 0.928053i
−2.59957 1.49740i
−0.232749 + 2.99096i
2.70662 1.29391i
0.00299931 + 3.00000i
−2.23014 2.00661i
2.85284 + 0.928053i
−2.59957 + 1.49740i
−0.232749 2.99096i
−3.93653 2.27276i −2.24112 + 4.68800i 6.33084 + 10.9653i −5.80193 + 10.0492i 19.4769 13.3609i −18.4018 2.09174i 21.1897i −16.9548 21.0128i 45.6790 26.3728i
5.2 −2.24076 1.29370i 5.19615 + 0.00519496i −0.652660 1.13044i 8.05907 13.9587i −11.6366 6.73392i −5.67909 + 17.6280i 24.0767i 26.9999 + 0.0539876i −36.1169 + 20.8521i
5.3 −1.65310 0.954416i −3.47555 3.86271i −2.17818 3.77272i 0.623706 1.08029i 2.05878 + 9.70256i 10.0808 15.5363i 23.5862i −2.84113 + 26.8501i −2.06209 + 1.19055i
5.4 1.65310 + 0.954416i 1.60743 + 4.94127i −2.17818 3.77272i −0.623706 + 1.08029i −2.05878 + 9.70256i 10.0808 15.5363i 23.5862i −21.8323 + 15.8855i −2.06209 + 1.19055i
5.5 2.24076 + 1.29370i 2.59358 4.50260i −0.652660 1.13044i −8.05907 + 13.9587i 11.6366 6.73392i −5.67909 + 17.6280i 24.0767i −13.5467 23.3556i −36.1169 + 20.8521i
5.6 3.93653 + 2.27276i −5.18049 0.403134i 6.33084 + 10.9653i 5.80193 10.0492i −19.4769 13.3609i −18.4018 2.09174i 21.1897i 26.6750 + 4.17686i 45.6790 26.3728i
17.1 −3.93653 + 2.27276i −2.24112 4.68800i 6.33084 10.9653i −5.80193 10.0492i 19.4769 + 13.3609i −18.4018 + 2.09174i 21.1897i −16.9548 + 21.0128i 45.6790 + 26.3728i
17.2 −2.24076 + 1.29370i 5.19615 0.00519496i −0.652660 + 1.13044i 8.05907 + 13.9587i −11.6366 + 6.73392i −5.67909 17.6280i 24.0767i 26.9999 0.0539876i −36.1169 20.8521i
17.3 −1.65310 + 0.954416i −3.47555 + 3.86271i −2.17818 + 3.77272i 0.623706 + 1.08029i 2.05878 9.70256i 10.0808 + 15.5363i 23.5862i −2.84113 26.8501i −2.06209 1.19055i
17.4 1.65310 0.954416i 1.60743 4.94127i −2.17818 + 3.77272i −0.623706 1.08029i −2.05878 9.70256i 10.0808 + 15.5363i 23.5862i −21.8323 15.8855i −2.06209 1.19055i
17.5 2.24076 1.29370i 2.59358 + 4.50260i −0.652660 + 1.13044i −8.05907 13.9587i 11.6366 + 6.73392i −5.67909 17.6280i 24.0767i −13.5467 + 23.3556i −36.1169 20.8521i
17.6 3.93653 2.27276i −5.18049 + 0.403134i 6.33084 10.9653i 5.80193 + 10.0492i −19.4769 + 13.3609i −18.4018 + 2.09174i 21.1897i 26.6750 4.17686i 45.6790 + 26.3728i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.4.g.a 12
3.b odd 2 1 inner 21.4.g.a 12
4.b odd 2 1 336.4.bc.d 12
7.b odd 2 1 147.4.g.d 12
7.c even 3 1 147.4.c.a 12
7.c even 3 1 147.4.g.d 12
7.d odd 6 1 inner 21.4.g.a 12
7.d odd 6 1 147.4.c.a 12
12.b even 2 1 336.4.bc.d 12
21.c even 2 1 147.4.g.d 12
21.g even 6 1 inner 21.4.g.a 12
21.g even 6 1 147.4.c.a 12
21.h odd 6 1 147.4.c.a 12
21.h odd 6 1 147.4.g.d 12
28.f even 6 1 336.4.bc.d 12
84.j odd 6 1 336.4.bc.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.g.a 12 1.a even 1 1 trivial
21.4.g.a 12 3.b odd 2 1 inner
21.4.g.a 12 7.d odd 6 1 inner
21.4.g.a 12 21.g even 6 1 inner
147.4.c.a 12 7.c even 3 1
147.4.c.a 12 7.d odd 6 1
147.4.c.a 12 21.g even 6 1
147.4.c.a 12 21.h odd 6 1
147.4.g.d 12 7.b odd 2 1
147.4.g.d 12 7.c even 3 1
147.4.g.d 12 21.c even 2 1
147.4.g.d 12 21.h odd 6 1
336.4.bc.d 12 4.b odd 2 1
336.4.bc.d 12 12.b even 2 1
336.4.bc.d 12 28.f even 6 1
336.4.bc.d 12 84.j odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 17 T^{2} + 83 T^{4} + 222 T^{6} + 1372 T^{8} - 40784 T^{10} - 685760 T^{12} - 2610176 T^{14} + 5619712 T^{16} + 58195968 T^{18} + 1392508928 T^{20} + 18253611008 T^{22} + 68719476736 T^{24} \)
$3$ \( 1 + 3 T + 6 T^{2} + 9 T^{3} - 198 T^{4} - 2565 T^{5} - 36018 T^{6} - 69255 T^{7} - 144342 T^{8} + 177147 T^{9} + 3188646 T^{10} + 43046721 T^{11} + 387420489 T^{12} \)
$5$ \( 1 - 354 T^{2} + 53346 T^{4} - 3913744 T^{6} + 238386078 T^{8} - 66225097710 T^{10} + 12349753291374 T^{12} - 1034767151718750 T^{14} + 58199726074218750 T^{16} - 14929748535156250000 T^{18} + \)\(31\!\cdots\!50\)\( T^{20} - \)\(32\!\cdots\!50\)\( T^{22} + \)\(14\!\cdots\!25\)\( T^{24} \)
$7$ \( ( 1 + 28 T + 476 T^{2} + 10780 T^{3} + 163268 T^{4} + 3294172 T^{5} + 40353607 T^{6} )^{2} \)
$11$ \( 1 + 4742 T^{2} + 10147682 T^{4} + 18336975648 T^{6} + 35136366245806 T^{8} + 55422032379705850 T^{10} + 74318354148055569358 T^{12} + \)\(98\!\cdots\!50\)\( T^{14} + \)\(11\!\cdots\!26\)\( T^{16} + \)\(10\!\cdots\!88\)\( T^{18} + \)\(99\!\cdots\!62\)\( T^{20} + \)\(82\!\cdots\!42\)\( T^{22} + \)\(30\!\cdots\!61\)\( T^{24} \)
$13$ \( ( 1 - 8847 T^{2} + 36037359 T^{4} - 94069474274 T^{6} + 173945448757431 T^{8} - 206118159078589407 T^{10} + \)\(11\!\cdots\!29\)\( T^{12} )^{2} \)
$17$ \( 1 - 17217 T^{2} + 160729722 T^{4} - 865949822635 T^{6} + 2480649769543932 T^{8} + 2994987661331912811 T^{10} - \)\(48\!\cdots\!48\)\( T^{12} + \)\(72\!\cdots\!59\)\( T^{14} + \)\(14\!\cdots\!52\)\( T^{16} - \)\(12\!\cdots\!15\)\( T^{18} + \)\(54\!\cdots\!62\)\( T^{20} - \)\(14\!\cdots\!33\)\( T^{22} + \)\(19\!\cdots\!81\)\( T^{24} \)
$19$ \( ( 1 - 150 T + 30330 T^{2} - 3424500 T^{3} + 468855630 T^{4} - 40538051670 T^{5} + 4025485422214 T^{6} - 278050496404530 T^{7} + 22057726175160030 T^{8} - 1105044021044185500 T^{9} + 67129841495276663130 T^{10} - \)\(22\!\cdots\!50\)\( T^{11} + \)\(10\!\cdots\!41\)\( T^{12} )^{2} \)
$23$ \( 1 + 58691 T^{2} + 1855192226 T^{4} + 43669490183457 T^{6} + 835010896549582396 T^{8} + \)\(13\!\cdots\!91\)\( T^{10} + \)\(17\!\cdots\!88\)\( T^{12} + \)\(19\!\cdots\!99\)\( T^{14} + \)\(18\!\cdots\!16\)\( T^{16} + \)\(14\!\cdots\!33\)\( T^{18} + \)\(89\!\cdots\!66\)\( T^{20} + \)\(41\!\cdots\!59\)\( T^{22} + \)\(10\!\cdots\!61\)\( T^{24} \)
$29$ \( ( 1 - 26333 T^{2} + 912806819 T^{4} - 27528483699758 T^{6} + 542958783509025899 T^{8} - \)\(93\!\cdots\!53\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \)
$31$ \( ( 1 + 465 T + 176469 T^{2} + 48543210 T^{3} + 12033403527 T^{4} + 2446560037821 T^{5} + 457872227799046 T^{6} + 72885470086725411 T^{7} + 10679689925170882887 T^{8} + \)\(12\!\cdots\!10\)\( T^{9} + \)\(13\!\cdots\!09\)\( T^{10} + \)\(10\!\cdots\!15\)\( T^{11} + \)\(69\!\cdots\!41\)\( T^{12} )^{2} \)
$37$ \( ( 1 - 382 T - 32782 T^{2} + 5434056 T^{3} + 8807187550 T^{4} - 649676877530 T^{5} - 374210188787594 T^{6} - 32908082877527090 T^{7} + 22596833686051007950 T^{8} + \)\(70\!\cdots\!12\)\( T^{9} - \)\(21\!\cdots\!42\)\( T^{10} - \)\(12\!\cdots\!26\)\( T^{11} + \)\(16\!\cdots\!29\)\( T^{12} )^{2} \)
$41$ \( ( 1 + 240738 T^{2} + 28558186959 T^{4} + 2299629172416892 T^{6} + \)\(13\!\cdots\!19\)\( T^{8} + \)\(54\!\cdots\!78\)\( T^{10} + \)\(10\!\cdots\!21\)\( T^{12} )^{2} \)
$43$ \( ( 1 + 253 T + 215237 T^{2} + 33567598 T^{3} + 17112848159 T^{4} + 1599304851397 T^{5} + 502592611936843 T^{6} )^{4} \)
$47$ \( 1 - 437385 T^{2} + 98505882894 T^{4} - 16642251751296271 T^{6} + \)\(23\!\cdots\!92\)\( T^{8} - \)\(28\!\cdots\!97\)\( T^{10} + \)\(30\!\cdots\!08\)\( T^{12} - \)\(30\!\cdots\!13\)\( T^{14} + \)\(27\!\cdots\!72\)\( T^{16} - \)\(20\!\cdots\!19\)\( T^{18} + \)\(13\!\cdots\!14\)\( T^{20} - \)\(63\!\cdots\!65\)\( T^{22} + \)\(15\!\cdots\!21\)\( T^{24} \)
$53$ \( 1 + 362162 T^{2} + 31342470242 T^{4} + 2991238457561520 T^{6} + \)\(17\!\cdots\!22\)\( T^{8} + \)\(23\!\cdots\!54\)\( T^{10} + \)\(11\!\cdots\!62\)\( T^{12} + \)\(51\!\cdots\!66\)\( T^{14} + \)\(85\!\cdots\!02\)\( T^{16} + \)\(32\!\cdots\!80\)\( T^{18} + \)\(75\!\cdots\!02\)\( T^{20} + \)\(19\!\cdots\!38\)\( T^{22} + \)\(11\!\cdots\!21\)\( T^{24} \)
$59$ \( 1 - 661854 T^{2} + 202275596298 T^{4} - 40619516236130848 T^{6} + \)\(74\!\cdots\!38\)\( T^{8} - \)\(16\!\cdots\!86\)\( T^{10} + \)\(35\!\cdots\!06\)\( T^{12} - \)\(68\!\cdots\!26\)\( T^{14} + \)\(13\!\cdots\!78\)\( T^{16} - \)\(30\!\cdots\!08\)\( T^{18} + \)\(64\!\cdots\!78\)\( T^{20} - \)\(88\!\cdots\!54\)\( T^{22} + \)\(56\!\cdots\!41\)\( T^{24} \)
$61$ \( ( 1 - 1179 T + 941982 T^{2} - 564310665 T^{3} + 276007099296 T^{4} - 128710086478551 T^{5} + 56963412393227764 T^{6} - 29214744138987984531 T^{7} + \)\(14\!\cdots\!56\)\( T^{8} - \)\(65\!\cdots\!65\)\( T^{9} + \)\(25\!\cdots\!22\)\( T^{10} - \)\(71\!\cdots\!79\)\( T^{11} + \)\(13\!\cdots\!81\)\( T^{12} )^{2} \)
$67$ \( ( 1 - 396 T - 208818 T^{2} + 528240452 T^{3} - 109541196954 T^{4} - 71767532435832 T^{5} + 113262131816538126 T^{6} - 21585018357998139816 T^{7} - \)\(99\!\cdots\!26\)\( T^{8} + \)\(14\!\cdots\!44\)\( T^{9} - \)\(17\!\cdots\!98\)\( T^{10} - \)\(97\!\cdots\!28\)\( T^{11} + \)\(74\!\cdots\!09\)\( T^{12} )^{2} \)
$71$ \( ( 1 - 1922318 T^{2} + 1602890240687 T^{4} - 746575415906526884 T^{6} + \)\(20\!\cdots\!27\)\( T^{8} - \)\(31\!\cdots\!38\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \)
$73$ \( ( 1 + 1452 T + 1911888 T^{2} + 1755642240 T^{3} + 1485047372688 T^{4} + 1012976615051412 T^{5} + 691104732915920110 T^{6} + \)\(39\!\cdots\!04\)\( T^{7} + \)\(22\!\cdots\!32\)\( T^{8} + \)\(10\!\cdots\!20\)\( T^{9} + \)\(43\!\cdots\!48\)\( T^{10} + \)\(12\!\cdots\!64\)\( T^{11} + \)\(34\!\cdots\!69\)\( T^{12} )^{2} \)
$79$ \( ( 1 - 837 T - 518715 T^{2} + 592038158 T^{3} + 210634653159 T^{4} - 180227265033825 T^{5} - 13471835399034906 T^{6} - 88859070525012044175 T^{7} + \)\(51\!\cdots\!39\)\( T^{8} + \)\(70\!\cdots\!02\)\( T^{9} - \)\(30\!\cdots\!15\)\( T^{10} - \)\(24\!\cdots\!63\)\( T^{11} + \)\(14\!\cdots\!61\)\( T^{12} )^{2} \)
$83$ \( ( 1 + 2862735 T^{2} + 3680795850915 T^{4} + 2710866692924348218 T^{6} + \)\(12\!\cdots\!35\)\( T^{8} + \)\(30\!\cdots\!35\)\( T^{10} + \)\(34\!\cdots\!09\)\( T^{12} )^{2} \)
$89$ \( 1 - 1635561 T^{2} + 1787180462010 T^{4} - 817237014874197619 T^{6} - \)\(17\!\cdots\!36\)\( T^{8} + \)\(75\!\cdots\!95\)\( T^{10} - \)\(70\!\cdots\!84\)\( T^{12} + \)\(37\!\cdots\!95\)\( T^{14} - \)\(42\!\cdots\!56\)\( T^{16} - \)\(10\!\cdots\!39\)\( T^{18} + \)\(10\!\cdots\!10\)\( T^{20} - \)\(49\!\cdots\!61\)\( T^{22} + \)\(15\!\cdots\!61\)\( T^{24} \)
$97$ \( ( 1 - 3316347 T^{2} + 4922505333747 T^{4} - 4971002050523297522 T^{6} + \)\(41\!\cdots\!63\)\( T^{8} - \)\(23\!\cdots\!27\)\( T^{10} + \)\(57\!\cdots\!89\)\( T^{12} )^{2} \)
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