Properties

Label 201.2.e.b
Level 201
Weight 2
Character orbit 201.e
Analytic conductor 1.605
Analytic rank 0
Dimension 10
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 201 = 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 201.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.60499308063\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} - q^{3} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} ) q^{4} -\beta_{5} q^{5} + \beta_{1} q^{6} + ( -\beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{7} + ( 1 - \beta_{3} + \beta_{5} - \beta_{6} ) q^{8} + q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} - q^{3} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} ) q^{4} -\beta_{5} q^{5} + \beta_{1} q^{6} + ( -\beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{7} + ( 1 - \beta_{3} + \beta_{5} - \beta_{6} ) q^{8} + q^{9} + ( -2 + \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{10} + ( \beta_{7} + \beta_{9} ) q^{11} + ( -\beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{8} ) q^{12} + ( \beta_{1} + \beta_{2} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{13} + ( -3 - \beta_{3} - 2 \beta_{6} + \beta_{9} ) q^{14} + \beta_{5} q^{15} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{16} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{6} + \beta_{8} ) q^{17} -\beta_{1} q^{18} + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} ) q^{19} + ( 3 \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{20} + ( \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{21} + ( 1 + \beta_{6} - \beta_{9} ) q^{22} + ( -1 + 2 \beta_{1} + \beta_{4} ) q^{23} + ( -1 + \beta_{3} - \beta_{5} + \beta_{6} ) q^{24} + ( \beta_{3} + \beta_{6} + \beta_{9} ) q^{25} + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} + 2 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} ) q^{26} - q^{27} + ( -2 - \beta_{2} + 2 \beta_{4} + \beta_{6} + \beta_{7} - 3 \beta_{8} ) q^{28} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{8} ) q^{29} + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{6} ) q^{30} + ( -2 \beta_{4} - \beta_{7} - \beta_{9} ) q^{31} + ( \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{32} + ( -\beta_{7} - \beta_{9} ) q^{33} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{34} + ( \beta_{1} + \beta_{2} + \beta_{3} + 5 \beta_{4} + 2 \beta_{5} - \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{35} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} ) q^{36} + ( 4 + 3 \beta_{1} - \beta_{2} - 4 \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{37} + ( \beta_{1} + \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - \beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{38} + ( -\beta_{1} - \beta_{2} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{39} + ( -1 + 2 \beta_{3} - \beta_{9} ) q^{40} + ( -3 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{41} + ( 3 + \beta_{3} + 2 \beta_{6} - \beta_{9} ) q^{42} + ( 1 - 3 \beta_{3} + 3 \beta_{5} + \beta_{6} - \beta_{9} ) q^{43} + ( -1 + \beta_{2} + \beta_{4} - \beta_{6} + 2 \beta_{8} ) q^{44} -\beta_{5} q^{45} + ( -\beta_{1} - \beta_{3} + 6 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} ) q^{46} + ( \beta_{1} - \beta_{2} + \beta_{3} - 6 \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{47} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{48} + ( -6 + \beta_{1} + 3 \beta_{2} + 6 \beta_{4} - 3 \beta_{6} - 2 \beta_{7} ) q^{49} + ( 4 + 2 \beta_{1} - \beta_{2} - 4 \beta_{4} + \beta_{6} + 3 \beta_{8} ) q^{50} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} - \beta_{8} ) q^{51} + ( -1 + 6 \beta_{3} - 5 \beta_{5} + 3 \beta_{6} - 2 \beta_{9} ) q^{52} + ( -3 + \beta_{3} - 3 \beta_{5} + \beta_{6} + \beta_{9} ) q^{53} + \beta_{1} q^{54} + ( -2 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{55} + ( 4 \beta_{1} + 4 \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{8} ) q^{56} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{6} ) q^{57} + ( 4 - 2 \beta_{3} + 6 \beta_{5} - \beta_{6} + 2 \beta_{9} ) q^{58} + ( 4 + 3 \beta_{3} - 2 \beta_{5} + \beta_{6} - 2 \beta_{9} ) q^{59} + ( -3 \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{60} + ( -2 \beta_{2} + 2 \beta_{6} + \beta_{7} - 3 \beta_{8} ) q^{61} + ( -1 - 2 \beta_{3} - \beta_{6} + \beta_{9} ) q^{62} + ( -\beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{63} + ( -1 - \beta_{5} - 2 \beta_{6} ) q^{64} + ( -1 - 2 \beta_{1} + \beta_{4} - \beta_{7} - \beta_{8} ) q^{65} + ( -1 - \beta_{6} + \beta_{9} ) q^{66} + ( -\beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{8} ) q^{67} + ( -1 + \beta_{3} - 2 \beta_{5} - 2 \beta_{9} ) q^{68} + ( 1 - 2 \beta_{1} - \beta_{4} ) q^{69} + ( 6 + 5 \beta_{3} + 3 \beta_{5} + \beta_{6} + 2 \beta_{9} ) q^{70} + ( -2 \beta_{1} - 2 \beta_{3} - \beta_{4} + 4 \beta_{5} + 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} ) q^{71} + ( 1 - \beta_{3} + \beta_{5} - \beta_{6} ) q^{72} + ( 3 + \beta_{1} + 3 \beta_{2} - 3 \beta_{4} - 3 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{73} + ( -7 \beta_{1} - 2 \beta_{2} - 7 \beta_{3} + 6 \beta_{4} + \beta_{5} - \beta_{8} ) q^{74} + ( -\beta_{3} - \beta_{6} - \beta_{9} ) q^{75} + ( -2 - 5 \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{9} ) q^{76} + ( 8 - 2 \beta_{2} - 8 \beta_{4} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{77} + ( 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} - 2 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} ) q^{78} + ( -3 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{79} + ( -1 + 3 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{80} + q^{81} + ( -8 + \beta_{3} - \beta_{5} + 2 \beta_{9} ) q^{82} + ( 3 - \beta_{1} - \beta_{2} - 3 \beta_{4} + \beta_{6} + \beta_{8} ) q^{83} + ( 2 + \beta_{2} - 2 \beta_{4} - \beta_{6} - \beta_{7} + 3 \beta_{8} ) q^{84} + ( -5 + 2 \beta_{2} + 5 \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{85} + ( -4 - 6 \beta_{1} - 2 \beta_{2} + 4 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{86} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{8} ) q^{87} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{88} + ( 1 - 2 \beta_{3} + \beta_{5} - \beta_{9} ) q^{89} + ( -2 + \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{90} + ( -7 + 4 \beta_{3} + \beta_{5} - 2 \beta_{6} + 2 \beta_{9} ) q^{91} + ( -1 + 5 \beta_{3} - \beta_{5} + 2 \beta_{6} ) q^{92} + ( 2 \beta_{4} + \beta_{7} + \beta_{9} ) q^{93} + ( -7 \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{94} + ( -4 + 3 \beta_{1} + \beta_{2} + 4 \beta_{4} - \beta_{6} - 2 \beta_{8} ) q^{95} + ( -\beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{96} + ( -2 - 2 \beta_{1} + 2 \beta_{4} - \beta_{7} - 3 \beta_{8} ) q^{97} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 7 \beta_{5} + 5 \beta_{7} - 7 \beta_{8} + 5 \beta_{9} ) q^{98} + ( \beta_{7} + \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 2q^{2} - 10q^{3} - 6q^{4} + 2q^{5} + 2q^{6} - q^{7} + 12q^{8} + 10q^{9} + O(q^{10}) \) \( 10q - 2q^{2} - 10q^{3} - 6q^{4} + 2q^{5} + 2q^{6} - q^{7} + 12q^{8} + 10q^{9} - 8q^{10} + 2q^{11} + 6q^{12} + 3q^{13} - 22q^{14} - 2q^{15} + 2q^{17} - 2q^{18} + 3q^{19} + 16q^{20} + q^{21} + 6q^{22} - q^{23} - 12q^{24} + 7q^{26} - 10q^{27} - 9q^{28} + 12q^{29} + 8q^{30} - 12q^{31} - 9q^{32} - 2q^{33} - 20q^{34} + 19q^{35} - 6q^{36} + 27q^{37} - 16q^{38} - 3q^{39} - 22q^{40} - 7q^{41} + 22q^{42} + 12q^{43} - 7q^{44} + 2q^{45} + 30q^{46} - 33q^{47} - 24q^{49} + 21q^{50} - 2q^{51} - 32q^{52} - 24q^{53} + 2q^{54} + 6q^{55} - q^{56} - 3q^{57} + 44q^{58} + 24q^{59} - 16q^{60} + q^{61} + 2q^{62} - q^{63} - 8q^{64} - 6q^{65} - 6q^{66} + 2q^{67} - 18q^{68} + q^{69} + 42q^{70} - q^{71} + 12q^{72} + 12q^{73} + 43q^{74} + 2q^{76} + 40q^{77} - 7q^{78} + 7q^{79} - q^{80} + 10q^{81} - 74q^{82} + 12q^{83} + 9q^{84} - 27q^{85} - 27q^{86} - 12q^{87} - 10q^{88} + 12q^{89} - 8q^{90} - 80q^{91} - 28q^{92} + 12q^{93} + 30q^{94} - 12q^{95} + 9q^{96} - 9q^{97} + 4q^{98} + 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} + 10 x^{8} - 8 x^{7} + 49 x^{6} - 39 x^{5} + 128 x^{4} - 14 x^{3} + 119 x^{2} - 49 x + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -24307 \nu^{9} + 181995 \nu^{8} - 604465 \nu^{7} + 1626651 \nu^{6} - 3301924 \nu^{5} + 7319233 \nu^{4} - 12172941 \nu^{3} + 14158031 \nu^{2} - 13105288 \nu + 6437963 \)\()/4414074\)
\(\beta_{3}\)\(=\)\((\)\( 32653 \nu^{9} - 100089 \nu^{8} + 396649 \nu^{7} - 539103 \nu^{6} + 1764532 \nu^{5} - 2509639 \nu^{4} + 5937753 \nu^{3} - 2828441 \nu^{2} + 1453144 \nu - 2618021 \)\()/4414074\)
\(\beta_{4}\)\(=\)\((\)\( -53429 \nu^{9} + 74205 \nu^{8} - 434201 \nu^{7} + 30783 \nu^{6} - 2078918 \nu^{5} + 319199 \nu^{4} - 4329273 \nu^{3} - 5189747 \nu^{2} - 3529610 \nu + 1164877 \)\()/4414074\)
\(\beta_{5}\)\(=\)\((\)\( 33718 \nu^{9} - 85104 \nu^{8} + 337264 \nu^{7} - 351819 \nu^{6} + 1500352 \nu^{5} - 2133904 \nu^{4} + 4154526 \nu^{3} - 2404976 \nu^{2} + 1235584 \nu - 7130123 \)\()/2207037\)
\(\beta_{6}\)\(=\)\((\)\( -31943 \nu^{9} + 110079 \nu^{8} - 436239 \nu^{7} + 663959 \nu^{6} - 1940652 \nu^{5} + 2760129 \nu^{4} - 5655213 \nu^{3} + 3110751 \nu^{2} - 1598184 \nu + 1081311 \)\()/1471358\)
\(\beta_{7}\)\(=\)\((\)\( -112429 \nu^{9} - 45435 \nu^{8} - 824203 \nu^{7} - 1270707 \nu^{6} - 5458108 \nu^{5} - 6300671 \nu^{4} - 13242393 \nu^{3} - 19664257 \nu^{2} - 25007374 \nu - 11599567 \)\()/4414074\)
\(\beta_{8}\)\(=\)\((\)\( -62752 \nu^{9} + 76248 \nu^{8} - 512362 \nu^{7} - 36093 \nu^{6} - 2500291 \nu^{5} - 400286 \nu^{4} - 5308260 \nu^{3} - 6568339 \nu^{2} - 6992440 \nu - 4073797 \)\()/2207037\)
\(\beta_{9}\)\(=\)\((\)\( -117800 \nu^{9} + 300861 \nu^{8} - 1192301 \nu^{7} + 1489527 \nu^{6} - 5304068 \nu^{5} + 7543811 \nu^{4} - 12249042 \nu^{3} + 8502109 \nu^{2} - 4368056 \nu + 12355714 \)\()/2207037\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} - \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{6} - \beta_{5} + 5 \beta_{3} - 1\)
\(\nu^{4}\)\(=\)\(-7 \beta_{8} + \beta_{7} + \beta_{6} + 13 \beta_{4} - \beta_{2} - 8 \beta_{1} - 13\)
\(\nu^{5}\)\(=\)\(2 \beta_{9} - 10 \beta_{8} + 2 \beta_{7} + 10 \beta_{5} + 9 \beta_{4} - 29 \beta_{3} - 8 \beta_{2} - 29 \beta_{1}\)
\(\nu^{6}\)\(=\)\(10 \beta_{9} - 12 \beta_{6} + 45 \beta_{5} - 56 \beta_{3} + 65\)
\(\nu^{7}\)\(=\)\(80 \beta_{8} - 22 \beta_{7} - 55 \beta_{6} - 68 \beta_{4} + 55 \beta_{2} + 178 \beta_{1} + 68\)
\(\nu^{8}\)\(=\)\(-77 \beta_{9} + 288 \beta_{8} - 77 \beta_{7} - 288 \beta_{5} - 352 \beta_{4} + 381 \beta_{3} + 102 \beta_{2} + 381 \beta_{1}\)
\(\nu^{9}\)\(=\)\(-179 \beta_{9} + 365 \beta_{6} - 585 \beta_{5} + 1123 \beta_{3} - 490\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\) \(68\) \(136\)
\(\chi(n)\) \(1\) \(-\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
1.28367 + 2.22339i
0.947050 + 1.64034i
0.330147 + 0.571831i
−0.527336 0.913372i
−1.03354 1.79014i
1.28367 2.22339i
0.947050 1.64034i
0.330147 0.571831i
−0.527336 + 0.913372i
−1.03354 + 1.79014i
−1.28367 2.22339i −1.00000 −2.29564 + 3.97616i −1.02393 1.28367 + 2.22339i −0.850574 + 1.47324i 6.65272 1.00000 1.31439 + 2.27659i
37.2 −0.947050 1.64034i −1.00000 −0.793808 + 1.37492i 1.30648 0.947050 + 1.64034i 2.53710 4.39438i −0.781096 1.00000 −1.23731 2.14308i
37.3 −0.330147 0.571831i −1.00000 0.782006 1.35447i 3.22431 0.330147 + 0.571831i −1.02143 + 1.76916i −2.35330 1.00000 −1.06450 1.84376i
37.4 0.527336 + 0.913372i −1.00000 0.443834 0.768743i 0.832996 −0.527336 0.913372i 1.13029 1.95772i 3.04554 1.00000 0.439269 + 0.760836i
37.5 1.03354 + 1.79014i −1.00000 −1.13639 + 1.96829i −3.33986 −1.03354 1.79014i −2.29539 + 3.97573i −0.563865 1.00000 −3.45186 5.97880i
163.1 −1.28367 + 2.22339i −1.00000 −2.29564 3.97616i −1.02393 1.28367 2.22339i −0.850574 1.47324i 6.65272 1.00000 1.31439 2.27659i
163.2 −0.947050 + 1.64034i −1.00000 −0.793808 1.37492i 1.30648 0.947050 1.64034i 2.53710 + 4.39438i −0.781096 1.00000 −1.23731 + 2.14308i
163.3 −0.330147 + 0.571831i −1.00000 0.782006 + 1.35447i 3.22431 0.330147 0.571831i −1.02143 1.76916i −2.35330 1.00000 −1.06450 + 1.84376i
163.4 0.527336 0.913372i −1.00000 0.443834 + 0.768743i 0.832996 −0.527336 + 0.913372i 1.13029 + 1.95772i 3.04554 1.00000 0.439269 0.760836i
163.5 1.03354 1.79014i −1.00000 −1.13639 1.96829i −3.33986 −1.03354 + 1.79014i −2.29539 3.97573i −0.563865 1.00000 −3.45186 + 5.97880i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.2.e.b 10
3.b odd 2 1 603.2.g.g 10
67.c even 3 1 inner 201.2.e.b 10
201.g odd 6 1 603.2.g.g 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.e.b 10 1.a even 1 1 trivial
201.2.e.b 10 67.c even 3 1 inner
603.2.g.g 10 3.b odd 2 1
603.2.g.g 10 201.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{10} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(201, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T - 4 T^{3} - 7 T^{4} - 5 T^{5} + 6 T^{6} + 14 T^{7} + 7 T^{8} - 3 T^{9} - T^{10} - 6 T^{11} + 28 T^{12} + 112 T^{13} + 96 T^{14} - 160 T^{15} - 448 T^{16} - 512 T^{17} + 1024 T^{19} + 1024 T^{20} \)
$3$ \( ( 1 + T )^{10} \)
$5$ \( ( 1 - T + 13 T^{2} - 7 T^{3} + 82 T^{4} - 32 T^{5} + 410 T^{6} - 175 T^{7} + 1625 T^{8} - 625 T^{9} + 3125 T^{10} )^{2} \)
$7$ \( 1 + T - 5 T^{2} + 30 T^{3} + 47 T^{4} - 157 T^{5} + 235 T^{6} + 403 T^{7} - 4271 T^{8} - 6187 T^{9} + 5314 T^{10} - 43309 T^{11} - 209279 T^{12} + 138229 T^{13} + 564235 T^{14} - 2638699 T^{15} + 5529503 T^{16} + 24706290 T^{17} - 28824005 T^{18} + 40353607 T^{19} + 282475249 T^{20} \)
$11$ \( 1 - 2 T - 30 T^{2} + 80 T^{3} + 455 T^{4} - 1472 T^{5} - 3496 T^{6} + 17382 T^{7} + 3117 T^{8} - 82000 T^{9} + 195386 T^{10} - 902000 T^{11} + 377157 T^{12} + 23135442 T^{13} - 51184936 T^{14} - 237067072 T^{15} + 806060255 T^{16} + 1558973680 T^{17} - 6430766430 T^{18} - 4715895382 T^{19} + 25937424601 T^{20} \)
$13$ \( 1 - 3 T - 23 T^{2} + 60 T^{3} + 401 T^{4} - 1189 T^{5} + 669 T^{6} + 7033 T^{7} - 81821 T^{8} - 71981 T^{9} + 1930474 T^{10} - 935753 T^{11} - 13827749 T^{12} + 15451501 T^{13} + 19107309 T^{14} - 441467377 T^{15} + 1935550409 T^{16} + 3764911020 T^{17} - 18761806583 T^{18} - 31813498119 T^{19} + 137858491849 T^{20} \)
$17$ \( 1 - 2 T - 45 T^{2} + 52 T^{3} + 896 T^{4} - 118 T^{5} - 18426 T^{6} + 13204 T^{7} + 413548 T^{8} - 278580 T^{9} - 7528996 T^{10} - 4735860 T^{11} + 119515372 T^{12} + 64871252 T^{13} - 1538957946 T^{14} - 167543126 T^{15} + 21627261824 T^{16} + 21337610996 T^{17} - 313909084845 T^{18} - 237175752994 T^{19} + 2015993900449 T^{20} \)
$19$ \( 1 - 3 T - 63 T^{2} + 188 T^{3} + 2217 T^{4} - 5955 T^{5} - 56201 T^{6} + 118667 T^{7} + 1161603 T^{8} - 994263 T^{9} - 22074290 T^{10} - 18890997 T^{11} + 419338683 T^{12} + 813936953 T^{13} - 7324170521 T^{14} - 14745169545 T^{15} + 104300718177 T^{16} + 168047886932 T^{17} - 1069964471583 T^{18} - 968063093337 T^{19} + 6131066257801 T^{20} \)
$23$ \( 1 + T - 84 T^{2} - 87 T^{3} + 3730 T^{4} + 3331 T^{5} - 128394 T^{6} - 65555 T^{7} + 3769325 T^{8} + 533650 T^{9} - 94045060 T^{10} + 12273950 T^{11} + 1993972925 T^{12} - 797607685 T^{13} - 35929905354 T^{14} + 21439458533 T^{15} + 552173865970 T^{16} - 296219813889 T^{17} - 6578122763604 T^{18} + 1801152661463 T^{19} + 41426511213649 T^{20} \)
$29$ \( 1 - 12 T + 35 T^{2} + 42 T^{3} - 562 T^{4} + 7172 T^{5} - 47688 T^{6} + 109166 T^{7} + 246850 T^{8} - 8769548 T^{9} + 78548584 T^{10} - 254316892 T^{11} + 207600850 T^{12} + 2662449574 T^{13} - 33728816328 T^{14} + 147105960628 T^{15} - 334290706402 T^{16} + 724494804978 T^{17} + 17508624453635 T^{18} - 174085751710428 T^{19} + 420707233300201 T^{20} \)
$31$ \( 1 + 12 T - 46 T^{2} - 672 T^{3} + 5651 T^{4} + 43136 T^{5} - 272368 T^{6} - 1029508 T^{7} + 14657629 T^{8} + 20680480 T^{9} - 457112294 T^{10} + 641094880 T^{11} + 14085981469 T^{12} - 30670072828 T^{13} - 251537567728 T^{14} + 1234947057536 T^{15} + 5015283301331 T^{16} - 18488476682592 T^{17} - 39232987722286 T^{18} + 317275465928052 T^{19} + 819628286980801 T^{20} \)
$37$ \( 1 - 27 T + 335 T^{2} - 2834 T^{3} + 19821 T^{4} - 109389 T^{5} + 333119 T^{6} + 1221661 T^{7} - 28104657 T^{8} + 266964541 T^{9} - 1848541638 T^{10} + 9877688017 T^{11} - 38475275433 T^{12} + 61880794633 T^{13} + 624318638159 T^{14} - 7585466112273 T^{15} + 50855263152789 T^{16} - 269036939794922 T^{17} + 1176680617063535 T^{18} - 3508966974467079 T^{19} + 4808584372417849 T^{20} \)
$41$ \( 1 + 7 T - 65 T^{2} - 282 T^{3} + 2255 T^{4} - 7849 T^{5} - 127467 T^{6} + 270227 T^{7} + 5626873 T^{8} + 5908829 T^{9} - 153257794 T^{10} + 242261989 T^{11} + 9458773513 T^{12} + 18624315067 T^{13} - 360191277387 T^{14} - 909355321649 T^{15} + 10711485063455 T^{16} - 54920705234442 T^{17} - 519020139892865 T^{18} + 2291673540757727 T^{19} + 13422659310152401 T^{20} \)
$43$ \( ( 1 - 6 T + 44 T^{2} + 85 T^{3} + 2623 T^{4} - 11734 T^{5} + 112789 T^{6} + 157165 T^{7} + 3498308 T^{8} - 20512806 T^{9} + 147008443 T^{10} )^{2} \)
$47$ \( 1 + 33 T + 467 T^{2} + 4526 T^{3} + 44547 T^{4} + 406231 T^{5} + 2822455 T^{6} + 18880275 T^{7} + 144571805 T^{8} + 930825353 T^{9} + 5559199434 T^{10} + 43748791591 T^{11} + 319359117245 T^{12} + 1960206791325 T^{13} + 13772680036855 T^{14} + 93167051538617 T^{15} + 480181705260963 T^{16} + 2292976243215538 T^{17} + 11119870871042387 T^{18} + 36931305612391311 T^{19} + 52599132235830049 T^{20} \)
$53$ \( ( 1 + 12 T + 176 T^{2} + 799 T^{3} + 6435 T^{4} + 7802 T^{5} + 341055 T^{6} + 2244391 T^{7} + 26202352 T^{8} + 94685772 T^{9} + 418195493 T^{10} )^{2} \)
$59$ \( ( 1 - 12 T + 228 T^{2} - 1793 T^{3} + 21391 T^{4} - 131822 T^{5} + 1262069 T^{6} - 6241433 T^{7} + 46826412 T^{8} - 145408332 T^{9} + 714924299 T^{10} )^{2} \)
$61$ \( 1 - T - 155 T^{2} + 972 T^{3} + 9803 T^{4} - 108505 T^{5} - 205309 T^{6} + 4289939 T^{7} + 3661333 T^{8} - 50850637 T^{9} - 549294066 T^{10} - 3101888857 T^{11} + 13623820093 T^{12} + 973734644159 T^{13} - 2842675769869 T^{14} - 91642921640005 T^{15} + 505054229860883 T^{16} + 3054746036612412 T^{17} - 29714633514578555 T^{18} - 11694146092834141 T^{19} + 713342911662882601 T^{20} \)
$67$ \( 1 - 2 T - 84 T^{2} + 293 T^{3} + 3191 T^{4} - 37614 T^{5} + 213797 T^{6} + 1315277 T^{7} - 25264092 T^{8} - 40302242 T^{9} + 1350125107 T^{10} \)
$71$ \( 1 + T - 156 T^{2} + 193 T^{3} + 6698 T^{4} - 31933 T^{5} - 276042 T^{6} - 1283315 T^{7} + 66333973 T^{8} + 159797802 T^{9} - 7065147508 T^{10} + 11345643942 T^{11} + 334389557893 T^{12} - 459312554965 T^{13} - 7014691246602 T^{14} - 57614455865483 T^{15} + 858015701702858 T^{16} + 1755358190569463 T^{17} - 100737550874338716 T^{18} + 45848500718449031 T^{19} + 3255243551009881201 T^{20} \)
$73$ \( 1 - 12 T + 5 T^{2} + 918 T^{3} - 10402 T^{4} + 39590 T^{5} + 311486 T^{6} - 3886402 T^{7} + 11681842 T^{8} + 28585534 T^{9} + 25423120 T^{10} + 2086743982 T^{11} + 62252536018 T^{12} - 1511876446834 T^{13} + 8845654496126 T^{14} + 82072904366870 T^{15} - 1574178621858178 T^{16} + 10141511840531046 T^{17} + 4032300459470405 T^{18} - 706459040499214956 T^{19} + 4297625829703557649 T^{20} \)
$79$ \( 1 - 7 T - 287 T^{2} + 1980 T^{3} + 48845 T^{4} - 297751 T^{5} - 5791633 T^{6} + 24523715 T^{7} + 564939799 T^{8} - 863580187 T^{9} - 47031915594 T^{10} - 68222834773 T^{11} + 3525789285559 T^{12} + 12091147919885 T^{13} - 225584574472273 T^{14} - 916196619858649 T^{15} + 11873606764923245 T^{16} + 38023739792594820 T^{17} - 435410228443183007 T^{18} - 838961171878328233 T^{19} + 9468276082626847201 T^{20} \)
$83$ \( 1 - 12 T - 277 T^{2} + 2862 T^{3} + 59360 T^{4} - 467894 T^{5} - 8191686 T^{6} + 37927936 T^{7} + 964403038 T^{8} - 1586803864 T^{9} - 85681157216 T^{10} - 131704720712 T^{11} + 6643772528782 T^{12} + 21686700741632 T^{13} - 388763663719206 T^{14} - 1843053482615842 T^{15} + 19407180563183840 T^{16} + 77663377932312474 T^{17} - 623884948302514357 T^{18} - 2243283063210484836 T^{19} + 15516041187205853449 T^{20} \)
$89$ \( ( 1 - 6 T + 396 T^{2} - 1909 T^{3} + 66559 T^{4} - 244594 T^{5} + 5923751 T^{6} - 15121189 T^{7} + 279167724 T^{8} - 376453446 T^{9} + 5584059449 T^{10} )^{2} \)
$97$ \( 1 + 9 T - 293 T^{2} - 978 T^{3} + 60983 T^{4} + 1013 T^{5} - 8276571 T^{6} + 8984413 T^{7} + 837449737 T^{8} - 638925773 T^{9} - 80446237178 T^{10} - 61975799981 T^{11} + 7879564575433 T^{12} + 8199831165949 T^{13} - 732718879775451 T^{14} + 8698975680341 T^{15} + 50797131776585207 T^{16} - 79020722219594514 T^{17} - 2296368043152449573 T^{18} + 6842079527891086953 T^{19} + 73742412689492826049 T^{20} \)
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