Properties

Label 201.4.a.c
Level 201
Weight 4
Character orbit 201.a
Self dual yes
Analytic conductor 11.859
Analytic rank 1
Dimension 7
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 201 = 3 \cdot 67 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 201.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.8593839112\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} -3 q^{3} + ( 3 + \beta_{1} + \beta_{2} ) q^{4} + ( 1 - \beta_{2} + \beta_{5} ) q^{5} + 3 \beta_{1} q^{6} + ( -4 + \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{7} + ( -6 - 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{8} + 9 q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} -3 q^{3} + ( 3 + \beta_{1} + \beta_{2} ) q^{4} + ( 1 - \beta_{2} + \beta_{5} ) q^{5} + 3 \beta_{1} q^{6} + ( -4 + \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{7} + ( -6 - 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{8} + 9 q^{9} + ( -7 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{10} + ( -19 + 3 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{11} + ( -9 - 3 \beta_{1} - 3 \beta_{2} ) q^{12} + ( 1 + 8 \beta_{1} + \beta_{2} - \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{13} + ( -3 + 5 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{14} + ( -3 + 3 \beta_{2} - 3 \beta_{5} ) q^{15} + ( 10 + 20 \beta_{1} + 5 \beta_{2} + 7 \beta_{4} - 5 \beta_{5} ) q^{16} + ( 12 + 13 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{17} -9 \beta_{1} q^{18} + ( -20 + 3 \beta_{1} + 5 \beta_{2} + 8 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} ) q^{19} + ( -52 - 3 \beta_{1} - 7 \beta_{2} - 4 \beta_{3} - 5 \beta_{4} + 6 \beta_{5} - 4 \beta_{6} ) q^{20} + ( 12 - 3 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} ) q^{21} + ( -33 + 35 \beta_{1} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{22} + ( -60 + \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 8 \beta_{4} + 6 \beta_{5} - 5 \beta_{6} ) q^{23} + ( 18 + 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 6 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} ) q^{24} + ( -21 - \beta_{1} + 3 \beta_{2} + 7 \beta_{3} - 2 \beta_{4} + 8 \beta_{6} ) q^{25} + ( -87 - 20 \beta_{1} - 7 \beta_{2} - 5 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{26} -27 q^{27} + ( -37 + 7 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{28} + ( -33 - 15 \beta_{1} + 10 \beta_{2} - 13 \beta_{3} - 10 \beta_{4} + \beta_{5} + \beta_{6} ) q^{29} + ( 21 - 6 \beta_{1} + 3 \beta_{2} + 9 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 6 \beta_{6} ) q^{30} + ( -71 + 7 \beta_{1} + 6 \beta_{2} - 10 \beta_{3} - 8 \beta_{4} - 7 \beta_{5} - 5 \beta_{6} ) q^{31} + ( -151 - 43 \beta_{1} - 21 \beta_{2} + 7 \beta_{3} - 3 \beta_{4} + 10 \beta_{5} - 6 \beta_{6} ) q^{32} + ( 57 - 9 \beta_{1} + 6 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} ) q^{33} + ( -110 - 53 \beta_{1} - 11 \beta_{2} - 3 \beta_{3} - 14 \beta_{4} + \beta_{5} + 12 \beta_{6} ) q^{34} + ( -102 - 11 \beta_{1} + 9 \beta_{2} + 13 \beta_{4} - 18 \beta_{5} - 4 \beta_{6} ) q^{35} + ( 27 + 9 \beta_{1} + 9 \beta_{2} ) q^{36} + ( 51 + 11 \beta_{1} + 16 \beta_{2} - \beta_{3} + 11 \beta_{4} - 21 \beta_{5} + 12 \beta_{6} ) q^{37} + ( 22 + 17 \beta_{1} + 25 \beta_{2} + 19 \beta_{3} + 12 \beta_{4} - 39 \beta_{5} + 12 \beta_{6} ) q^{38} + ( -3 - 24 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 12 \beta_{4} - 9 \beta_{5} + 3 \beta_{6} ) q^{39} + ( 28 + 57 \beta_{1} - \beta_{2} + 5 \beta_{3} + 6 \beta_{4} + 10 \beta_{5} - 2 \beta_{6} ) q^{40} + ( -110 - 33 \beta_{1} - 15 \beta_{2} + 14 \beta_{3} + 5 \beta_{4} + 8 \beta_{5} - 14 \beta_{6} ) q^{41} + ( 9 - 15 \beta_{1} + 6 \beta_{2} - 15 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} ) q^{42} + ( -35 - 34 \beta_{1} - 7 \beta_{2} + 9 \beta_{3} - 9 \beta_{4} + 3 \beta_{5} - 16 \beta_{6} ) q^{43} + ( -253 - 45 \beta_{1} - 32 \beta_{2} - 6 \beta_{3} - 15 \beta_{4} + 17 \beta_{5} - 18 \beta_{6} ) q^{44} + ( 9 - 9 \beta_{2} + 9 \beta_{5} ) q^{45} + ( -82 + 25 \beta_{1} - 39 \beta_{2} - 13 \beta_{3} - 17 \beta_{4} + 23 \beta_{5} - 11 \beta_{6} ) q^{46} + ( -138 - 60 \beta_{1} + 8 \beta_{2} + \beta_{3} + 7 \beta_{4} - 16 \beta_{5} + 29 \beta_{6} ) q^{47} + ( -30 - 60 \beta_{1} - 15 \beta_{2} - 21 \beta_{4} + 15 \beta_{5} ) q^{48} + ( 32 - 26 \beta_{1} - 25 \beta_{2} - 5 \beta_{3} - 29 \beta_{4} + 45 \beta_{5} - 12 \beta_{6} ) q^{49} + ( 77 + 39 \beta_{1} + 35 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 33 \beta_{5} + 14 \beta_{6} ) q^{50} + ( -36 - 39 \beta_{1} - 15 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} - 6 \beta_{6} ) q^{51} + ( 175 + 82 \beta_{1} + \beta_{2} - 3 \beta_{3} + 32 \beta_{4} + 5 \beta_{5} - 5 \beta_{6} ) q^{52} + ( 30 - 31 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} + 27 \beta_{4} + 16 \beta_{5} - 24 \beta_{6} ) q^{53} + 27 \beta_{1} q^{54} + ( 165 + \beta_{1} + 24 \beta_{2} + 20 \beta_{3} + 27 \beta_{4} - 41 \beta_{5} + 28 \beta_{6} ) q^{55} + ( -42 - 4 \beta_{1} + 12 \beta_{2} - 39 \beta_{3} - 14 \beta_{4} - 12 \beta_{5} + 11 \beta_{6} ) q^{56} + ( 60 - 9 \beta_{1} - 15 \beta_{2} - 24 \beta_{3} + 6 \beta_{4} + 12 \beta_{5} - 6 \beta_{6} ) q^{57} + ( 225 - 5 \beta_{1} + 10 \beta_{2} - 6 \beta_{3} - 28 \beta_{4} + 32 \beta_{5} + 21 \beta_{6} ) q^{58} + ( -35 - 71 \beta_{1} + 12 \beta_{2} + 14 \beta_{3} - 2 \beta_{4} + 11 \beta_{5} + 20 \beta_{6} ) q^{59} + ( 156 + 9 \beta_{1} + 21 \beta_{2} + 12 \beta_{3} + 15 \beta_{4} - 18 \beta_{5} + 12 \beta_{6} ) q^{60} + ( 61 - 48 \beta_{1} - 35 \beta_{2} - 29 \beta_{3} + 46 \beta_{4} + 15 \beta_{5} - 29 \beta_{6} ) q^{61} + ( -52 + 40 \beta_{1} - 14 \beta_{2} + 15 \beta_{3} - 4 \beta_{4} + 26 \beta_{5} + 7 \beta_{6} ) q^{62} + ( -36 + 9 \beta_{4} - 18 \beta_{5} - 9 \beta_{6} ) q^{63} + ( 251 + 125 \beta_{1} + \beta_{2} - 28 \beta_{3} + 2 \beta_{4} + 30 \beta_{5} - 48 \beta_{6} ) q^{64} + ( 27 - 13 \beta_{1} + 12 \beta_{2} + 18 \beta_{3} - 55 \beta_{4} + 33 \beta_{5} - 2 \beta_{6} ) q^{65} + ( 99 - 105 \beta_{1} + 6 \beta_{3} - 3 \beta_{4} - 9 \beta_{5} + 6 \beta_{6} ) q^{66} + 67 q^{67} + ( 515 + 183 \beta_{1} + 58 \beta_{2} - 12 \beta_{3} + 47 \beta_{4} - 7 \beta_{5} - 26 \beta_{6} ) q^{68} + ( 180 - 3 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} - 24 \beta_{4} - 18 \beta_{5} + 15 \beta_{6} ) q^{69} + ( 156 + 75 \beta_{1} - 5 \beta_{2} + 49 \beta_{3} - 22 \beta_{4} + 4 \beta_{5} + 14 \beta_{6} ) q^{70} + ( -161 + 32 \beta_{1} + 17 \beta_{2} - 45 \beta_{3} - 40 \beta_{4} + 31 \beta_{5} - 23 \beta_{6} ) q^{71} + ( -54 - 18 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} - 18 \beta_{4} - 9 \beta_{5} + 18 \beta_{6} ) q^{72} + ( 136 - 16 \beta_{1} + 55 \beta_{3} + 44 \beta_{4} - 50 \beta_{5} + 16 \beta_{6} ) q^{73} + ( 38 - 82 \beta_{1} + 10 \beta_{2} + 45 \beta_{3} - 47 \beta_{4} - \beta_{5} + 44 \beta_{6} ) q^{74} + ( 63 + 3 \beta_{1} - 9 \beta_{2} - 21 \beta_{3} + 6 \beta_{4} - 24 \beta_{6} ) q^{75} + ( 231 - 65 \beta_{1} + 20 \beta_{2} + 46 \beta_{3} + 7 \beta_{4} - 57 \beta_{5} + 46 \beta_{6} ) q^{76} + ( 35 + 17 \beta_{1} + 22 \beta_{2} - 24 \beta_{3} - 61 \beta_{4} + 41 \beta_{5} + 28 \beta_{6} ) q^{77} + ( 261 + 60 \beta_{1} + 21 \beta_{2} + 15 \beta_{3} - 6 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{78} + ( -326 - 67 \beta_{1} - 21 \beta_{2} - 71 \beta_{3} - 51 \beta_{4} - 16 \beta_{5} - 15 \beta_{6} ) q^{79} + ( -253 - 98 \beta_{1} - 17 \beta_{2} + 18 \beta_{3} + 32 \beta_{4} - 61 \beta_{5} + 28 \beta_{6} ) q^{80} + 81 q^{81} + ( 206 + 171 \beta_{1} + 15 \beta_{2} - 3 \beta_{3} + 54 \beta_{4} - 36 \beta_{5} - 44 \beta_{6} ) q^{82} + ( -396 + 101 \beta_{1} - 13 \beta_{2} + 4 \beta_{3} - 29 \beta_{4} + 32 \beta_{5} - 22 \beta_{6} ) q^{83} + ( 111 - 21 \beta_{1} - 6 \beta_{3} - 12 \beta_{4} + 3 \beta_{5} - 9 \beta_{6} ) q^{84} + ( -83 - 45 \beta_{1} - 22 \beta_{2} - 4 \beta_{3} - 39 \beta_{4} + 31 \beta_{5} + 4 \beta_{6} ) q^{85} + ( 280 + 81 \beta_{1} + 35 \beta_{2} + 12 \beta_{3} + 63 \beta_{4} - 38 \beta_{5} - 30 \beta_{6} ) q^{86} + ( 99 + 45 \beta_{1} - 30 \beta_{2} + 39 \beta_{3} + 30 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{87} + ( 499 + 129 \beta_{1} + 10 \beta_{2} - 38 \beta_{3} + 75 \beta_{4} + 17 \beta_{5} - 66 \beta_{6} ) q^{88} + ( -214 + 7 \beta_{1} + 45 \beta_{2} + 16 \beta_{3} + 52 \beta_{4} - 56 \beta_{5} + 68 \beta_{6} ) q^{89} + ( -63 + 18 \beta_{1} - 9 \beta_{2} - 27 \beta_{3} + 9 \beta_{4} + 9 \beta_{5} - 18 \beta_{6} ) q^{90} + ( -249 - 15 \beta_{1} + 30 \beta_{2} - 36 \beta_{3} + 53 \beta_{4} - 91 \beta_{5} + 26 \beta_{6} ) q^{91} + ( -70 + 202 \beta_{1} - 38 \beta_{2} - 63 \beta_{3} + 10 \beta_{4} + 26 \beta_{5} - 49 \beta_{6} ) q^{92} + ( 213 - 21 \beta_{1} - 18 \beta_{2} + 30 \beta_{3} + 24 \beta_{4} + 21 \beta_{5} + 15 \beta_{6} ) q^{93} + ( 864 + 272 \beta_{1} + 122 \beta_{2} + 12 \beta_{3} - 41 \beta_{4} - 5 \beta_{5} + 45 \beta_{6} ) q^{94} + ( -481 + 187 \beta_{1} - 44 \beta_{2} + 6 \beta_{3} - 33 \beta_{4} + 41 \beta_{5} - 48 \beta_{6} ) q^{95} + ( 453 + 129 \beta_{1} + 63 \beta_{2} - 21 \beta_{3} + 9 \beta_{4} - 30 \beta_{5} + 18 \beta_{6} ) q^{96} + ( -113 + 59 \beta_{1} - 46 \beta_{2} - 6 \beta_{3} - 53 \beta_{4} - 17 \beta_{5} + 82 \beta_{6} ) q^{97} + ( 64 + 26 \beta_{1} + 5 \beta_{2} - 108 \beta_{3} + 65 \beta_{4} + 16 \beta_{5} - 62 \beta_{6} ) q^{98} + ( -171 + 27 \beta_{1} - 18 \beta_{2} + 9 \beta_{4} - 9 \beta_{5} + 18 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - q^{2} - 21q^{3} + 21q^{4} + 11q^{5} + 3q^{6} - 33q^{7} - 45q^{8} + 63q^{9} + O(q^{10}) \) \( 7q - q^{2} - 21q^{3} + 21q^{4} + 11q^{5} + 3q^{6} - 33q^{7} - 45q^{8} + 63q^{9} - 51q^{10} - 130q^{11} - 63q^{12} + 16q^{13} + 5q^{14} - 33q^{15} + 77q^{16} + 90q^{17} - 9q^{18} - 132q^{19} - 359q^{20} + 99q^{21} - 192q^{22} - 399q^{23} + 135q^{24} - 132q^{25} - 638q^{26} - 189q^{27} - 245q^{28} - 302q^{29} + 153q^{30} - 555q^{31} - 1031q^{32} + 390q^{33} - 832q^{34} - 775q^{35} + 189q^{36} + 297q^{37} + 98q^{38} - 48q^{39} + 305q^{40} - 717q^{41} - 15q^{42} - 245q^{43} - 1766q^{44} + 99q^{45} - 497q^{46} - 1072q^{47} - 231q^{48} + 314q^{49} + 454q^{50} - 270q^{51} + 1344q^{52} + 265q^{53} + 27q^{54} + 1096q^{55} - 477q^{56} + 396q^{57} + 1610q^{58} - 255q^{59} + 1077q^{60} + 418q^{61} - 191q^{62} - 297q^{63} + 1889q^{64} + 262q^{65} + 576q^{66} + 469q^{67} + 3720q^{68} + 1197q^{69} + 1309q^{70} - 1194q^{71} - 405q^{72} + 995q^{73} + 259q^{74} + 396q^{75} + 1506q^{76} + 230q^{77} + 1914q^{78} - 2640q^{79} - 1949q^{80} + 567q^{81} + 1535q^{82} - 2579q^{83} + 735q^{84} - 562q^{85} + 1991q^{86} + 906q^{87} + 3624q^{88} - 1604q^{89} - 459q^{90} - 2116q^{91} - 351q^{92} + 1665q^{93} + 6178q^{94} - 3028q^{95} + 3093q^{96} - 808q^{97} + 258q^{98} - 1170q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - x^{6} - 38 x^{5} + 18 x^{4} + 373 x^{3} - 151 x^{2} - 956 x + 498\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 11 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - 47 \nu^{5} + 27 \nu^{4} + 1339 \nu^{3} - 408 \nu^{2} - 6314 \nu + 1966 \)\()/466\)
\(\beta_{4}\)\(=\)\((\)\( -25 \nu^{6} + 10 \nu^{5} + 956 \nu^{4} + 310 \nu^{3} - 9139 \nu^{2} - 5250 \nu + 17954 \)\()/932\)
\(\beta_{5}\)\(=\)\((\)\( -35 \nu^{6} + 14 \nu^{5} + 1152 \nu^{4} + 434 \nu^{3} - 7389 \nu^{2} - 4554 \nu + 4818 \)\()/932\)
\(\beta_{6}\)\(=\)\((\)\( -87 \nu^{6} + 128 \nu^{5} + 3010 \nu^{4} - 2556 \nu^{3} - 23919 \nu^{2} + 13418 \nu + 32134 \)\()/1864\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 11\)
\(\nu^{3}\)\(=\)\(-2 \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + 18 \beta_{1} + 6\)
\(\nu^{4}\)\(=\)\(-5 \beta_{5} + 7 \beta_{4} + 29 \beta_{2} + 44 \beta_{1} + 210\)
\(\nu^{5}\)\(=\)\(-58 \beta_{6} + 22 \beta_{5} + 67 \beta_{4} - 39 \beta_{3} + 53 \beta_{2} + 427 \beta_{1} + 343\)
\(\nu^{6}\)\(=\)\(-48 \beta_{6} - 170 \beta_{5} + 282 \beta_{4} - 28 \beta_{3} + 777 \beta_{2} + 1501 \beta_{1} + 4939\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
5.41567
3.01477
1.85609
0.535294
−2.11972
−3.26131
−4.44080
−5.41567 −3.00000 21.3295 −5.98460 16.2470 0.240383 −72.1883 9.00000 32.4106
1.2 −3.01477 −3.00000 1.08886 14.7190 9.04432 −32.2394 20.8355 9.00000 −44.3744
1.3 −1.85609 −3.00000 −4.55494 −4.35854 5.56826 17.6123 23.3031 9.00000 8.08983
1.4 −0.535294 −3.00000 −7.71346 12.7037 1.60588 −8.67573 8.41132 9.00000 −6.80020
1.5 2.11972 −3.00000 −3.50678 1.76139 −6.35917 22.2326 −24.3912 9.00000 3.73365
1.6 3.26131 −3.00000 2.63612 7.83356 −9.78392 −27.8825 −17.4933 9.00000 25.5476
1.7 4.44080 −3.00000 11.7207 −15.6745 −13.3224 −4.28763 16.5228 9.00000 −69.6071
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.4.a.c 7
3.b odd 2 1 603.4.a.c 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.a.c 7 1.a even 1 1 trivial
603.4.a.c 7 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(67\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} + T_{2}^{6} - 38 T_{2}^{5} - 18 T_{2}^{4} + 373 T_{2}^{3} + 151 T_{2}^{2} - 956 T_{2} - 498 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(201))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 18 T^{2} + 30 T^{3} + 197 T^{4} + 535 T^{5} + 1596 T^{6} + 5246 T^{7} + 12768 T^{8} + 34240 T^{9} + 100864 T^{10} + 122880 T^{11} + 589824 T^{12} + 262144 T^{13} + 2097152 T^{14} \)
$3$ \( ( 1 + 3 T )^{7} \)
$5$ \( 1 - 11 T + 564 T^{2} - 4911 T^{3} + 151982 T^{4} - 1060709 T^{5} + 26199969 T^{6} - 153622402 T^{7} + 3274996125 T^{8} - 16573578125 T^{9} + 296839843750 T^{10} - 1198974609375 T^{11} + 17211914062500 T^{12} - 41961669921875 T^{13} + 476837158203125 T^{14} \)
$7$ \( 1 + 33 T + 1588 T^{2} + 42261 T^{3} + 1234294 T^{4} + 27110135 T^{5} + 630536859 T^{6} + 11313814502 T^{7} + 216274142637 T^{8} + 3189480272615 T^{9} + 49808214998458 T^{10} + 584946638401461 T^{11} + 7539127677789484 T^{12} + 53737648731044817 T^{13} + 558545864083284007 T^{14} \)
$11$ \( 1 + 130 T + 12649 T^{2} + 795976 T^{3} + 43010377 T^{4} + 1823159982 T^{5} + 73998893113 T^{6} + 2649933947120 T^{7} + 98492526733403 T^{8} + 3229839120871902 T^{9} + 101416219136189507 T^{10} + 2498113665588874696 T^{11} + 52838012094938569499 T^{12} + \)\(72\!\cdots\!30\)\( T^{13} + \)\(74\!\cdots\!11\)\( T^{14} \)
$13$ \( 1 - 16 T + 7959 T^{2} - 109108 T^{3} + 31715953 T^{4} - 346791968 T^{5} + 86867713007 T^{6} - 773800401336 T^{7} + 190848365476379 T^{8} - 1673898592270112 T^{9} + 336331803702597469 T^{10} - 2542007471543656948 T^{11} + \)\(40\!\cdots\!63\)\( T^{12} - \)\(17\!\cdots\!64\)\( T^{13} + \)\(24\!\cdots\!13\)\( T^{14} \)
$17$ \( 1 - 90 T + 22744 T^{2} - 1911672 T^{3} + 225354330 T^{4} - 18527392242 T^{5} + 1406248452333 T^{6} - 111062566559448 T^{7} + 6908898646312029 T^{8} - 447206208631339698 T^{9} + 26724291454104182010 T^{10} - \)\(11\!\cdots\!92\)\( T^{11} + \)\(65\!\cdots\!92\)\( T^{12} - \)\(12\!\cdots\!10\)\( T^{13} + \)\(69\!\cdots\!17\)\( T^{14} \)
$19$ \( 1 + 132 T + 27166 T^{2} + 2676198 T^{3} + 306425632 T^{4} + 27877812224 T^{5} + 2477154191679 T^{6} + 218161488799340 T^{7} + 16990800600726261 T^{8} + 1311536236430649344 T^{9} + 98879781730555071328 T^{10} + \)\(59\!\cdots\!78\)\( T^{11} + \)\(41\!\cdots\!34\)\( T^{12} + \)\(13\!\cdots\!12\)\( T^{13} + \)\(71\!\cdots\!19\)\( T^{14} \)
$23$ \( 1 + 399 T + 120929 T^{2} + 26998356 T^{3} + 5056877920 T^{4} + 789200793876 T^{5} + 107618659812076 T^{6} + 12648457670971446 T^{7} + 1309396233933528692 T^{8} + \)\(11\!\cdots\!64\)\( T^{9} + \)\(91\!\cdots\!60\)\( T^{10} + \)\(59\!\cdots\!76\)\( T^{11} + \)\(32\!\cdots\!03\)\( T^{12} + \)\(12\!\cdots\!31\)\( T^{13} + \)\(39\!\cdots\!23\)\( T^{14} \)
$29$ \( 1 + 302 T + 118692 T^{2} + 24523836 T^{3} + 6134239790 T^{4} + 976849670402 T^{5} + 196111334443977 T^{6} + 27082052279613256 T^{7} + 4782959335754155053 T^{8} + \)\(58\!\cdots\!42\)\( T^{9} + \)\(88\!\cdots\!10\)\( T^{10} + \)\(86\!\cdots\!76\)\( T^{11} + \)\(10\!\cdots\!08\)\( T^{12} + \)\(63\!\cdots\!22\)\( T^{13} + \)\(51\!\cdots\!29\)\( T^{14} \)
$31$ \( 1 + 555 T + 271396 T^{2} + 88527427 T^{3} + 25510960062 T^{4} + 5959786423933 T^{5} + 1257751884350603 T^{6} + 227313677390195178 T^{7} + 37469686386688813973 T^{8} + \)\(52\!\cdots\!73\)\( T^{9} + \)\(67\!\cdots\!02\)\( T^{10} + \)\(69\!\cdots\!47\)\( T^{11} + \)\(63\!\cdots\!96\)\( T^{12} + \)\(38\!\cdots\!55\)\( T^{13} + \)\(20\!\cdots\!31\)\( T^{14} \)
$37$ \( 1 - 297 T + 244257 T^{2} - 53082932 T^{3} + 27333286436 T^{4} - 4757108863596 T^{5} + 1945271636872882 T^{6} - 284859005095362286 T^{7} + 98533844222522091946 T^{8} - \)\(12\!\cdots\!64\)\( T^{9} + \)\(35\!\cdots\!72\)\( T^{10} - \)\(34\!\cdots\!92\)\( T^{11} + \)\(81\!\cdots\!01\)\( T^{12} - \)\(50\!\cdots\!13\)\( T^{13} + \)\(85\!\cdots\!37\)\( T^{14} \)
$41$ \( 1 + 717 T + 524798 T^{2} + 217402755 T^{3} + 92867110408 T^{4} + 28152925652091 T^{5} + 9212861250096577 T^{6} + 2301992224371063114 T^{7} + \)\(63\!\cdots\!17\)\( T^{8} + \)\(13\!\cdots\!31\)\( T^{9} + \)\(30\!\cdots\!88\)\( T^{10} + \)\(49\!\cdots\!55\)\( T^{11} + \)\(81\!\cdots\!98\)\( T^{12} + \)\(76\!\cdots\!57\)\( T^{13} + \)\(73\!\cdots\!41\)\( T^{14} \)
$43$ \( 1 + 245 T + 359696 T^{2} + 76388477 T^{3} + 62401534594 T^{4} + 11109706720611 T^{5} + 6942528586006991 T^{6} + 1048659332073804406 T^{7} + \)\(55\!\cdots\!37\)\( T^{8} + \)\(70\!\cdots\!39\)\( T^{9} + \)\(31\!\cdots\!42\)\( T^{10} + \)\(30\!\cdots\!77\)\( T^{11} + \)\(11\!\cdots\!72\)\( T^{12} + \)\(61\!\cdots\!05\)\( T^{13} + \)\(20\!\cdots\!43\)\( T^{14} \)
$47$ \( 1 + 1072 T + 701142 T^{2} + 361323318 T^{3} + 171310164116 T^{4} + 71686508192356 T^{5} + 26832050950548543 T^{6} + 8985939150556985372 T^{7} + \)\(27\!\cdots\!89\)\( T^{8} + \)\(77\!\cdots\!24\)\( T^{9} + \)\(19\!\cdots\!72\)\( T^{10} + \)\(41\!\cdots\!38\)\( T^{11} + \)\(84\!\cdots\!06\)\( T^{12} + \)\(13\!\cdots\!08\)\( T^{13} + \)\(13\!\cdots\!47\)\( T^{14} \)
$53$ \( 1 - 265 T + 755634 T^{2} - 182171295 T^{3} + 266533136548 T^{4} - 56856210837867 T^{5} + 58221326573118101 T^{6} - 10598351489083918522 T^{7} + \)\(86\!\cdots\!77\)\( T^{8} - \)\(12\!\cdots\!43\)\( T^{9} + \)\(87\!\cdots\!84\)\( T^{10} - \)\(89\!\cdots\!95\)\( T^{11} + \)\(55\!\cdots\!38\)\( T^{12} - \)\(28\!\cdots\!85\)\( T^{13} + \)\(16\!\cdots\!53\)\( T^{14} \)
$59$ \( 1 + 255 T + 969971 T^{2} + 205350210 T^{3} + 446647333452 T^{4} + 77109073022682 T^{5} + 130446123069888346 T^{6} + 18786375157850246514 T^{7} + \)\(26\!\cdots\!34\)\( T^{8} + \)\(32\!\cdots\!62\)\( T^{9} + \)\(38\!\cdots\!28\)\( T^{10} + \)\(36\!\cdots\!10\)\( T^{11} + \)\(35\!\cdots\!29\)\( T^{12} + \)\(19\!\cdots\!55\)\( T^{13} + \)\(15\!\cdots\!59\)\( T^{14} \)
$61$ \( 1 - 418 T + 533547 T^{2} - 69919496 T^{3} + 165816862261 T^{4} - 21570768811310 T^{5} + 56110297038126535 T^{6} - 8894859365737545968 T^{7} + \)\(12\!\cdots\!35\)\( T^{8} - \)\(11\!\cdots\!10\)\( T^{9} + \)\(19\!\cdots\!01\)\( T^{10} - \)\(18\!\cdots\!16\)\( T^{11} + \)\(32\!\cdots\!47\)\( T^{12} - \)\(57\!\cdots\!58\)\( T^{13} + \)\(31\!\cdots\!61\)\( T^{14} \)
$67$ \( ( 1 - 67 T )^{7} \)
$71$ \( 1 + 1194 T + 2119729 T^{2} + 2030552208 T^{3} + 2030237493621 T^{4} + 1591046209096086 T^{5} + 1143282411189789429 T^{6} + \)\(72\!\cdots\!92\)\( T^{7} + \)\(40\!\cdots\!19\)\( T^{8} + \)\(20\!\cdots\!06\)\( T^{9} + \)\(93\!\cdots\!51\)\( T^{10} + \)\(33\!\cdots\!28\)\( T^{11} + \)\(12\!\cdots\!79\)\( T^{12} + \)\(25\!\cdots\!34\)\( T^{13} + \)\(75\!\cdots\!71\)\( T^{14} \)
$73$ \( 1 - 995 T + 1995565 T^{2} - 1405191688 T^{3} + 1682422298588 T^{4} - 898691201756752 T^{5} + 862620396675994770 T^{6} - \)\(39\!\cdots\!22\)\( T^{7} + \)\(33\!\cdots\!90\)\( T^{8} - \)\(13\!\cdots\!28\)\( T^{9} + \)\(99\!\cdots\!44\)\( T^{10} - \)\(32\!\cdots\!48\)\( T^{11} + \)\(17\!\cdots\!05\)\( T^{12} - \)\(34\!\cdots\!55\)\( T^{13} + \)\(13\!\cdots\!73\)\( T^{14} \)
$79$ \( 1 + 2640 T + 4147213 T^{2} + 4352043552 T^{3} + 3750155025025 T^{4} + 2811865160939504 T^{5} + 2095389973016555157 T^{6} + \)\(14\!\cdots\!32\)\( T^{7} + \)\(10\!\cdots\!23\)\( T^{8} + \)\(68\!\cdots\!84\)\( T^{9} + \)\(44\!\cdots\!75\)\( T^{10} + \)\(25\!\cdots\!32\)\( T^{11} + \)\(12\!\cdots\!87\)\( T^{12} + \)\(37\!\cdots\!40\)\( T^{13} + \)\(70\!\cdots\!79\)\( T^{14} \)
$83$ \( 1 + 2579 T + 5794900 T^{2} + 8508860117 T^{3} + 11216042833030 T^{4} + 11632488029207865 T^{5} + 11056776765634490023 T^{6} + \)\(87\!\cdots\!70\)\( T^{7} + \)\(63\!\cdots\!01\)\( T^{8} + \)\(38\!\cdots\!85\)\( T^{9} + \)\(20\!\cdots\!90\)\( T^{10} + \)\(90\!\cdots\!37\)\( T^{11} + \)\(35\!\cdots\!00\)\( T^{12} + \)\(90\!\cdots\!11\)\( T^{13} + \)\(19\!\cdots\!83\)\( T^{14} \)
$89$ \( 1 + 1604 T + 3567844 T^{2} + 3909328178 T^{3} + 5592539105542 T^{4} + 4998329615768928 T^{5} + 5685233180415121117 T^{6} + \)\(42\!\cdots\!28\)\( T^{7} + \)\(40\!\cdots\!73\)\( T^{8} + \)\(24\!\cdots\!08\)\( T^{9} + \)\(19\!\cdots\!78\)\( T^{10} + \)\(96\!\cdots\!38\)\( T^{11} + \)\(62\!\cdots\!56\)\( T^{12} + \)\(19\!\cdots\!24\)\( T^{13} + \)\(86\!\cdots\!89\)\( T^{14} \)
$97$ \( 1 + 808 T + 2914363 T^{2} + 2834641908 T^{3} + 5362881471945 T^{4} + 4736280886443000 T^{5} + 7063874922316465019 T^{6} + \)\(50\!\cdots\!92\)\( T^{7} + \)\(64\!\cdots\!87\)\( T^{8} + \)\(39\!\cdots\!00\)\( T^{9} + \)\(40\!\cdots\!65\)\( T^{10} + \)\(19\!\cdots\!28\)\( T^{11} + \)\(18\!\cdots\!59\)\( T^{12} + \)\(46\!\cdots\!12\)\( T^{13} + \)\(52\!\cdots\!97\)\( T^{14} \)
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