Properties

Label 43.4.a.b
Level $43$
Weight $4$
Character orbit 43.a
Self dual yes
Analytic conductor $2.537$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.53708213025\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 32 x^{4} - 16 x^{3} + 251 x^{2} + 276 x + 60\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 1 - \beta_{3} ) q^{3} + ( 4 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{4} + ( 7 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{5} + ( -1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{6} + ( 2 + 2 \beta_{1} + \beta_{2} + 3 \beta_{5} ) q^{7} + ( 10 - \beta_{2} + 4 \beta_{3} - 3 \beta_{4} ) q^{8} + ( 12 + 4 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - 5 \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 1 - \beta_{3} ) q^{3} + ( 4 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{4} + ( 7 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{5} + ( -1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{6} + ( 2 + 2 \beta_{1} + \beta_{2} + 3 \beta_{5} ) q^{7} + ( 10 - \beta_{2} + 4 \beta_{3} - 3 \beta_{4} ) q^{8} + ( 12 + 4 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - 5 \beta_{5} ) q^{9} + ( 9 - 5 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} + \beta_{4} - \beta_{5} ) q^{10} + ( -3 + 6 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + 4 \beta_{5} ) q^{11} + ( -25 + 4 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} ) q^{12} + ( 7 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 8 \beta_{5} ) q^{13} + ( -30 - \beta_{1} + 5 \beta_{2} - 5 \beta_{3} + 2 \beta_{4} + 6 \beta_{5} ) q^{14} + ( -22 + 2 \beta_{1} + \beta_{2} - 8 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} ) q^{15} + ( -10 - 14 \beta_{1} - 7 \beta_{2} + 6 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{16} + ( 4 + 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} ) q^{17} + ( -18 - 9 \beta_{1} - 8 \beta_{2} - \beta_{3} + 7 \beta_{4} - 11 \beta_{5} ) q^{18} + ( -11 - 10 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{19} + ( 23 - 6 \beta_{1} - 12 \beta_{2} + 19 \beta_{3} - 7 \beta_{4} - 11 \beta_{5} ) q^{20} + ( 4 - 6 \beta_{1} - 4 \beta_{2} + 16 \beta_{3} - 14 \beta_{4} + 16 \beta_{5} ) q^{21} + ( -83 - 5 \beta_{1} + 19 \beta_{2} - 14 \beta_{3} + 5 \beta_{4} + 6 \beta_{5} ) q^{22} + ( 24 + 6 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} - 10 \beta_{4} - 7 \beta_{5} ) q^{23} + ( -89 + 18 \beta_{1} + 11 \beta_{2} - 11 \beta_{3} - 12 \beta_{4} + 21 \beta_{5} ) q^{24} + ( 18 - 20 \beta_{1} - 9 \beta_{2} + 13 \beta_{3} + 12 \beta_{4} + 11 \beta_{5} ) q^{25} + ( 3 - 3 \beta_{1} - 9 \beta_{2} - 6 \beta_{3} + \beta_{4} - 10 \beta_{5} ) q^{26} + ( 30 + 4 \beta_{1} - 11 \beta_{2} - 8 \beta_{3} + 13 \beta_{4} - 26 \beta_{5} ) q^{27} + ( -70 + 24 \beta_{1} + 16 \beta_{2} - 26 \beta_{3} - 6 \beta_{5} ) q^{28} + ( 81 + 24 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} + 18 \beta_{4} - 8 \beta_{5} ) q^{29} + ( -70 + 44 \beta_{1} + 17 \beta_{2} - 24 \beta_{3} + 17 \beta_{4} ) q^{30} + ( 42 - 18 \beta_{1} - 6 \beta_{2} + \beta_{3} - 21 \beta_{4} - \beta_{5} ) q^{31} + ( 86 + 16 \beta_{1} - 23 \beta_{2} + 12 \beta_{3} + 17 \beta_{4} - 14 \beta_{5} ) q^{32} + ( 104 + 10 \beta_{1} - 11 \beta_{2} + 32 \beta_{3} - 40 \beta_{4} + 23 \beta_{5} ) q^{33} + ( -14 - 17 \beta_{1} - 9 \beta_{3} - 17 \beta_{4} + 15 \beta_{5} ) q^{34} + ( 32 + 18 \beta_{1} + 8 \beta_{2} - 16 \beta_{3} + 28 \beta_{4} + 30 \beta_{5} ) q^{35} + ( 17 + 7 \beta_{1} - 16 \beta_{2} + 32 \beta_{3} - 2 \beta_{4} + 11 \beta_{5} ) q^{36} + ( 47 - 18 \beta_{1} + 31 \beta_{2} - 3 \beta_{3} + 6 \beta_{4} - 9 \beta_{5} ) q^{37} + ( 95 + 2 \beta_{1} + 2 \beta_{2} + 11 \beta_{3} - 19 \beta_{4} + 11 \beta_{5} ) q^{38} + ( 26 - 26 \beta_{1} - 11 \beta_{2} - 46 \beta_{3} + 32 \beta_{4} - 45 \beta_{5} ) q^{39} + ( 107 - 22 \beta_{1} - 26 \beta_{2} + 55 \beta_{3} - 27 \beta_{4} - 27 \beta_{5} ) q^{40} + ( 78 - 28 \beta_{1} + 27 \beta_{2} - 17 \beta_{3} + 16 \beta_{4} + 3 \beta_{5} ) q^{41} + ( 72 - 40 \beta_{1} - 10 \beta_{2} + 64 \beta_{3} - 10 \beta_{4} + 6 \beta_{5} ) q^{42} -43 q^{43} + ( -74 + 41 \beta_{1} + 43 \beta_{2} - 69 \beta_{3} + 12 \beta_{4} + 22 \beta_{5} ) q^{44} + ( 56 + 2 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} - 21 \beta_{4} - 31 \beta_{5} ) q^{45} + ( -6 - 71 \beta_{1} + 3 \beta_{2} - \beta_{3} - 26 \beta_{4} + 3 \beta_{5} ) q^{46} + ( 79 - 44 \beta_{1} - 9 \beta_{2} + 5 \beta_{3} - 40 \beta_{4} + 23 \beta_{5} ) q^{47} + ( -171 + 54 \beta_{1} + 37 \beta_{2} - 21 \beta_{3} + 44 \beta_{4} - 9 \beta_{5} ) q^{48} + ( 43 - 30 \beta_{1} - 32 \beta_{2} - 6 \beta_{3} + 46 \beta_{4} - 8 \beta_{5} ) q^{49} + ( 228 + 9 \beta_{1} - 45 \beta_{2} + 61 \beta_{3} - 8 \beta_{4} + 9 \beta_{5} ) q^{50} + ( -203 - 76 \beta_{1} + \beta_{2} - 3 \beta_{3} - 5 \beta_{4} + 20 \beta_{5} ) q^{51} + ( 6 + 3 \beta_{1} - 9 \beta_{2} + 33 \beta_{3} + 2 \beta_{4} + 22 \beta_{5} ) q^{52} + ( 67 + 20 \beta_{1} - 37 \beta_{2} - 30 \beta_{3} + 7 \beta_{4} + 26 \beta_{5} ) q^{53} + ( 46 + 8 \beta_{1} - 45 \beta_{2} + 27 \beta_{4} - 54 \beta_{5} ) q^{54} + ( -290 + 78 \beta_{1} + 27 \beta_{2} - 40 \beta_{3} - 6 \beta_{4} - \beta_{5} ) q^{55} + ( -144 + 92 \beta_{1} + 38 \beta_{2} - 84 \beta_{3} + 6 \beta_{4} - 32 \beta_{5} ) q^{56} + ( -156 - 12 \beta_{1} + 27 \beta_{2} + 18 \beta_{3} + 3 \beta_{4} + 22 \beta_{5} ) q^{57} + ( -189 - 47 \beta_{1} - 12 \beta_{2} - 58 \beta_{3} + 20 \beta_{4} + 11 \beta_{5} ) q^{58} + ( 62 + 42 \beta_{1} - 8 \beta_{2} + 30 \beta_{3} - 4 \beta_{4} - 10 \beta_{5} ) q^{59} + ( -460 + 102 \beta_{1} + 57 \beta_{2} - 96 \beta_{3} + 17 \beta_{4} + 28 \beta_{5} ) q^{60} + ( -208 + 46 \beta_{1} + 25 \beta_{2} - 4 \beta_{3} - 46 \beta_{4} + 13 \beta_{5} ) q^{61} + ( 272 - 75 \beta_{1} + 6 \beta_{2} + 59 \beta_{3} - 3 \beta_{4} - 39 \beta_{5} ) q^{62} + ( -340 + 86 \beta_{1} + 31 \beta_{2} + 66 \beta_{3} - 90 \beta_{4} + 63 \beta_{5} ) q^{63} + ( 62 + 68 \beta_{1} - 45 \beta_{2} + 12 \beta_{3} + 23 \beta_{4} - 38 \beta_{5} ) q^{64} + ( -12 + 18 \beta_{1} + 9 \beta_{2} + 6 \beta_{3} - 28 \beta_{4} - 53 \beta_{5} ) q^{65} + ( 40 - 203 \beta_{1} - 23 \beta_{2} + 127 \beta_{3} + 2 \beta_{4} - 18 \beta_{5} ) q^{66} + ( -113 + 14 \beta_{1} - 69 \beta_{2} + 78 \beta_{3} - \beta_{4} - 56 \beta_{5} ) q^{67} + ( 95 - 3 \beta_{1} + 18 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} - 3 \beta_{5} ) q^{68} + ( -186 - 42 \beta_{1} + 8 \beta_{2} - 48 \beta_{3} - 31 \beta_{4} + 23 \beta_{5} ) q^{69} + ( -324 + 70 \beta_{1} + 50 \beta_{2} - 102 \beta_{3} + 56 \beta_{4} + 66 \beta_{5} ) q^{70} + ( -62 - 56 \beta_{1} - 38 \beta_{2} - 28 \beta_{3} + 116 \beta_{4} - 46 \beta_{5} ) q^{71} + ( 133 + 30 \beta_{1} - 19 \beta_{2} + 115 \beta_{3} - 54 \beta_{4} + 81 \beta_{5} ) q^{72} + ( 148 + 146 \beta_{1} - 51 \beta_{2} + 92 \beta_{3} + 26 \beta_{4} + 9 \beta_{5} ) q^{73} + ( 229 - 100 \beta_{1} + 53 \beta_{2} - 87 \beta_{3} - 114 \beta_{4} + 87 \beta_{5} ) q^{74} + ( -402 + 34 \beta_{1} + 54 \beta_{2} + 8 \beta_{3} + 39 \beta_{4} + 59 \beta_{5} ) q^{75} + ( 167 - 68 \beta_{1} - 4 \beta_{2} + 9 \beta_{3} - 7 \beta_{4} - 15 \beta_{5} ) q^{76} + ( 434 - 14 \beta_{1} - 55 \beta_{2} + 24 \beta_{3} - 50 \beta_{4} - 75 \beta_{5} ) q^{77} + ( 334 + 107 \beta_{1} - 7 \beta_{2} - 65 \beta_{3} + 54 \beta_{4} - 92 \beta_{5} ) q^{78} + ( -297 + 96 \beta_{1} + 14 \beta_{2} - 53 \beta_{3} + 95 \beta_{4} - 35 \beta_{5} ) q^{79} + ( 409 - 198 \beta_{1} - 66 \beta_{2} + 85 \beta_{3} - 25 \beta_{4} + 11 \beta_{5} ) q^{80} + ( -61 - 106 \beta_{1} + 17 \beta_{2} - 140 \beta_{3} + 101 \beta_{4} - 40 \beta_{5} ) q^{81} + ( 300 - 63 \beta_{1} + 75 \beta_{2} - 103 \beta_{3} - 72 \beta_{4} + 83 \beta_{5} ) q^{82} + ( -79 - 136 \beta_{1} + 37 \beta_{2} + 102 \beta_{3} - 5 \beta_{4} + 64 \beta_{5} ) q^{83} + ( 610 - 146 \beta_{1} - 100 \beta_{2} + 80 \beta_{3} - 20 \beta_{4} - 98 \beta_{5} ) q^{84} + ( 14 + 74 \beta_{1} + 8 \beta_{2} + 26 \beta_{3} + 7 \beta_{4} - 27 \beta_{5} ) q^{85} + ( -43 + 43 \beta_{1} ) q^{86} + ( 33 - 180 \beta_{1} - 76 \beta_{2} - 95 \beta_{3} + 33 \beta_{4} - 47 \beta_{5} ) q^{87} + ( -120 + 212 \beta_{1} + 82 \beta_{2} - 208 \beta_{3} + 32 \beta_{4} + 46 \beta_{5} ) q^{88} + ( 576 - 2 \beta_{1} + 89 \beta_{2} - 62 \beta_{3} - 34 \beta_{4} + 17 \beta_{5} ) q^{89} + ( 160 - 153 \beta_{1} - 18 \beta_{2} + 23 \beta_{3} - 57 \beta_{4} - 32 \beta_{5} ) q^{90} + ( -594 + 66 \beta_{1} + 51 \beta_{2} + 108 \beta_{3} - 114 \beta_{4} + 63 \beta_{5} ) q^{91} + ( 595 - 95 \beta_{1} - 3 \beta_{2} + 46 \beta_{3} + 5 \beta_{4} + 41 \beta_{5} ) q^{92} + ( 313 + 236 \beta_{1} + 59 \beta_{2} - 71 \beta_{3} - 19 \beta_{4} - 8 \beta_{5} ) q^{93} + ( 553 - 128 \beta_{1} + 35 \beta_{2} + 121 \beta_{3} - 4 \beta_{4} - 39 \beta_{5} ) q^{94} + ( -13 - 30 \beta_{1} - 33 \beta_{2} + 37 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{95} + ( -149 + 74 \beta_{1} - 25 \beta_{2} - 163 \beta_{3} + 72 \beta_{4} - 43 \beta_{5} ) q^{96} + ( 4 + 102 \beta_{1} - 3 \beta_{2} - 29 \beta_{3} + 22 \beta_{4} + 133 \beta_{5} ) q^{97} + ( 371 + 117 \beta_{1} - 106 \beta_{2} + 68 \beta_{3} + 70 \beta_{4} - 64 \beta_{5} ) q^{98} + ( -213 + 192 \beta_{1} + 3 \beta_{2} + 64 \beta_{3} - 95 \beta_{4} + 172 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} + 7q^{3} + 22q^{4} + 43q^{5} - 3q^{6} + 8q^{7} + 54q^{8} + 81q^{9} + O(q^{10}) \) \( 6q + 6q^{2} + 7q^{3} + 22q^{4} + 43q^{5} - 3q^{6} + 8q^{7} + 54q^{8} + 81q^{9} + 57q^{10} - 28q^{11} - 157q^{12} + 56q^{13} - 184q^{14} - 124q^{15} - 54q^{16} + 19q^{17} - 81q^{18} - 75q^{19} + 135q^{20} - 18q^{21} - 504q^{22} + 131q^{23} - 567q^{24} + 105q^{25} + 44q^{26} + 238q^{27} - 404q^{28} + 515q^{29} - 396q^{30} + 237q^{31} + 558q^{32} + 540q^{33} - 107q^{34} + 198q^{35} + 73q^{36} + 269q^{37} + 527q^{38} + 290q^{39} + 613q^{40} + 471q^{41} + 362q^{42} - 258q^{43} - 428q^{44} + 334q^{45} - 67q^{46} + 415q^{47} - 989q^{48} + 350q^{49} + 1335q^{50} - 1241q^{51} - 8q^{52} + 450q^{53} + 402q^{54} - 1732q^{55} - 780q^{56} - 1000q^{57} - 1055q^{58} + 356q^{59} - 2732q^{60} - 1328q^{61} + 1603q^{62} - 2290q^{63} + 466q^{64} - 62q^{65} + 156q^{66} - 632q^{67} + 571q^{68} - 1130q^{69} - 1902q^{70} - 144q^{71} + 567q^{72} + 864q^{73} + 1207q^{74} - 2494q^{75} + 1005q^{76} + 2660q^{77} + 2222q^{78} - 1613q^{79} + 2399q^{80} - 102q^{81} + 1673q^{82} - 682q^{83} + 3758q^{84} + 84q^{85} - 258q^{86} + 449q^{87} - 608q^{88} + 3378q^{89} + 930q^{90} - 3900q^{91} + 3491q^{92} + 1879q^{93} + 3197q^{94} - 79q^{95} - 591q^{96} - 55q^{97} + 2398q^{98} - 1612q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 32 x^{4} - 16 x^{3} + 251 x^{2} + 276 x + 60\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 2 \nu^{4} - 16 \nu^{3} - 36 \nu^{2} - 25 \nu - 34 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} + 2 \nu^{4} - 24 \nu^{3} - 36 \nu^{2} + 103 \nu + 30 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} + 2 \nu^{4} - 24 \nu^{3} - 44 \nu^{2} + 111 \nu + 118 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 36 \nu^{3} + 24 \nu^{2} + 331 \nu + 182 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{4} + \beta_{3} + \beta_{1} + 11\)
\(\nu^{3}\)\(=\)\(-\beta_{3} + \beta_{2} + 16 \beta_{1} + 8\)
\(\nu^{4}\)\(=\)\(-2 \beta_{5} - 15 \beta_{4} + 20 \beta_{3} - 3 \beta_{2} + 24 \beta_{1} + 179\)
\(\nu^{5}\)\(=\)\(4 \beta_{5} - 6 \beta_{4} - 20 \beta_{3} + 30 \beta_{2} + 269 \beta_{1} + 200\)

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
4.31455
4.15653
−0.299707
−0.847740
−3.17112
−4.15251
−3.31455 −7.10409 2.98627 8.51910 23.5469 9.84502 16.6183 23.4681 −28.2370
1.2 −3.15653 7.20925 1.96369 1.36370 −22.7562 13.0131 19.0538 24.9733 −4.30455
1.3 1.29971 1.43046 −6.31076 20.4116 1.85918 29.9522 −18.5998 −24.9538 26.5291
1.4 1.84774 9.49653 −4.58586 2.98245 17.5471 −26.0720 −23.2554 63.1842 5.51080
1.5 4.17112 2.46717 9.39827 −7.54340 10.2909 4.58222 5.83236 −20.9131 −31.4645
1.6 5.15251 −6.49933 18.5484 17.2665 −33.4879 −23.3206 54.3507 15.2413 88.9661
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(43\) \(1\)

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 43.4.a.b 6
3.b odd 2 1 387.4.a.h 6
4.b odd 2 1 688.4.a.i 6
5.b even 2 1 1075.4.a.b 6
7.b odd 2 1 2107.4.a.c 6
43.b odd 2 1 1849.4.a.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.4.a.b 6 1.a even 1 1 trivial
387.4.a.h 6 3.b odd 2 1
688.4.a.i 6 4.b odd 2 1
1075.4.a.b 6 5.b even 2 1
1849.4.a.c 6 43.b odd 2 1
2107.4.a.c 6 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 6 T_{2}^{5} - 17 T_{2}^{4} + 124 T_{2}^{3} + 26 T_{2}^{2} - 608 T_{2} + 540 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(43))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 540 - 608 T + 26 T^{2} + 124 T^{3} - 17 T^{4} - 6 T^{5} + T^{6} \)
$3$ \( 11156 - 11756 T + 2140 T^{2} + 588 T^{3} - 97 T^{4} - 7 T^{5} + T^{6} \)
$5$ \( -92116 + 106884 T - 30356 T^{2} + 508 T^{3} + 497 T^{4} - 43 T^{5} + T^{6} \)
$7$ \( 10690992 - 3729072 T + 285404 T^{2} + 9640 T^{3} - 1172 T^{4} - 8 T^{5} + T^{6} \)
$11$ \( -53187648 + 87126072 T + 3182929 T^{2} - 145260 T^{3} - 4842 T^{4} + 28 T^{5} + T^{6} \)
$13$ \( -458957340 - 3215268 T + 3791673 T^{2} + 84276 T^{3} - 4406 T^{4} - 56 T^{5} + T^{6} \)
$17$ \( 181639863 + 65687277 T + 3967071 T^{2} - 76658 T^{3} - 6335 T^{4} - 19 T^{5} + T^{6} \)
$19$ \( -673814000 + 113047600 T + 3291160 T^{2} - 206344 T^{3} - 2895 T^{4} + 75 T^{5} + T^{6} \)
$23$ \( -9170218345 + 4069168837 T - 233173145 T^{2} + 4291590 T^{3} - 20807 T^{4} - 131 T^{5} + T^{6} \)
$29$ \( 331483322700 - 16936584300 T - 704330628 T^{2} + 4732868 T^{3} + 58249 T^{4} - 515 T^{5} + T^{6} \)
$31$ \( 8546933895145 - 204439793629 T + 368947627 T^{2} + 14285746 T^{3} - 53373 T^{4} - 237 T^{5} + T^{6} \)
$37$ \( -15081152424000 - 80718597120 T + 5676973984 T^{2} + 37379592 T^{3} - 177607 T^{4} - 269 T^{5} + T^{6} \)
$41$ \( 84893308383715 - 827086183663 T - 3213994857 T^{2} + 49708910 T^{3} - 63763 T^{4} - 471 T^{5} + T^{6} \)
$43$ \( ( 43 + T )^{6} \)
$47$ \( -68794630166960 - 2018901432752 T - 675059768 T^{2} + 98566408 T^{3} - 201307 T^{4} - 415 T^{5} + T^{6} \)
$53$ \( -2796076662325740 - 11537303514238 T + 85831493161 T^{2} + 184990872 T^{3} - 571558 T^{4} - 450 T^{5} + T^{6} \)
$59$ \( -18355561214400 - 181544427840 T + 2268932784 T^{2} + 17967392 T^{3} - 78324 T^{4} - 356 T^{5} + T^{6} \)
$61$ \( 51013136843120 + 779382380976 T - 30113941508 T^{2} - 52724216 T^{3} + 434108 T^{4} + 1328 T^{5} + T^{6} \)
$67$ \( 119387526352596 - 5235646534404 T - 122185236567 T^{2} - 697381148 T^{3} - 876950 T^{4} + 632 T^{5} + T^{6} \)
$71$ \( -1133110269466432 + 76457292519680 T + 392378097392 T^{2} - 184951424 T^{3} - 1284236 T^{4} + 144 T^{5} + T^{6} \)
$73$ \( 75431034605642544 - 350187317967184 T + 138693949724 T^{2} + 1240846952 T^{3} - 1253444 T^{4} - 864 T^{5} + T^{6} \)
$79$ \( -24008230108815760 + 129229651676016 T + 70634693192 T^{2} - 845482784 T^{3} - 227153 T^{4} + 1613 T^{5} + T^{6} \)
$83$ \( 108732474040310676 - 794994150872130 T + 1672505022177 T^{2} - 166505888 T^{3} - 2310718 T^{4} + 682 T^{5} + T^{6} \)
$89$ \( -17611748593784080 + 198773575263992 T - 298079697748 T^{2} - 1148334440 T^{3} + 3622036 T^{4} - 3378 T^{5} + T^{6} \)
$97$ \( 162742133524474039 + 1097497984768263 T + 1699324914399 T^{2} - 1191302086 T^{3} - 2979167 T^{4} + 55 T^{5} + T^{6} \)
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