Properties

Label 1859.4.a.c
Level $1859$
Weight $4$
Character orbit 1859.a
Self dual yes
Analytic conductor $109.685$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1859.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.684550701\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 34 x^{4} - 26 x^{3} + 249 x^{2} + 274 x - 200\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 143)
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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} + ( -1 - \beta_{4} ) q^{3} + ( 4 - \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{1} + \beta_{3} ) q^{5} + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{6} + ( 9 + 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{7} + ( 6 - \beta_{1} + \beta_{2} - 4 \beta_{4} - 2 \beta_{5} ) q^{8} + ( -\beta_{1} + \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} + ( -1 - \beta_{4} ) q^{3} + ( 4 - \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{1} + \beta_{3} ) q^{5} + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{6} + ( 9 + 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{7} + ( 6 - \beta_{1} + \beta_{2} - 4 \beta_{4} - 2 \beta_{5} ) q^{8} + ( -\beta_{1} + \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{9} + ( -9 - 3 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} ) q^{10} -11 q^{11} + ( -3 - 9 \beta_{1} + \beta_{2} - \beta_{3} - 6 \beta_{4} - \beta_{5} ) q^{12} + ( -18 - 13 \beta_{1} - \beta_{2} - 2 \beta_{3} - 8 \beta_{4} - 6 \beta_{5} ) q^{14} + ( 4 + 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{15} + ( 6 - 5 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} - 12 \beta_{4} - 4 \beta_{5} ) q^{16} + ( -23 - 3 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} + 5 \beta_{5} ) q^{17} + ( 7 - 4 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} + 12 \beta_{4} - 9 \beta_{5} ) q^{18} + ( 9 - 11 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 11 \beta_{4} + 9 \beta_{5} ) q^{19} + ( 3 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{20} + ( -1 - 18 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} - 6 \beta_{4} - 7 \beta_{5} ) q^{21} + ( -11 + 11 \beta_{1} ) q^{22} + ( -31 - 4 \beta_{1} + 7 \beta_{2} + 7 \beta_{3} + 5 \beta_{4} ) q^{23} + ( 101 - 9 \beta_{1} + 5 \beta_{2} + \beta_{3} - 2 \beta_{4} - 9 \beta_{5} ) q^{24} + ( -36 + 17 \beta_{1} - 8 \beta_{2} + \beta_{3} - 12 \beta_{4} - 4 \beta_{5} ) q^{25} + ( -90 + 16 \beta_{1} + 6 \beta_{2} - 7 \beta_{3} - 6 \beta_{4} + 11 \beta_{5} ) q^{27} + ( 106 - 3 \beta_{1} + 25 \beta_{2} - 6 \beta_{3} - 8 \beta_{4} - 4 \beta_{5} ) q^{28} + ( -21 - 14 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} - 10 \beta_{4} - 21 \beta_{5} ) q^{29} + ( -36 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 8 \beta_{4} + 7 \beta_{5} ) q^{30} + ( 9 - 50 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{31} + ( 60 - 5 \beta_{1} + 19 \beta_{2} - 16 \beta_{3} - 16 \beta_{4} - 2 \beta_{5} ) q^{32} + ( 11 + 11 \beta_{4} ) q^{33} + ( -13 + 3 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + 19 \beta_{5} ) q^{34} + ( -36 + 29 \beta_{1} - \beta_{2} + 8 \beta_{3} + 4 \beta_{4} - 8 \beta_{5} ) q^{35} + ( 69 + 16 \beta_{1} - 8 \beta_{2} - 5 \beta_{3} + 14 \beta_{4} - 5 \beta_{5} ) q^{36} + ( 122 + 3 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} - 14 \beta_{4} - 11 \beta_{5} ) q^{37} + ( 112 - 16 \beta_{1} + 22 \beta_{2} - 2 \beta_{3} - 42 \beta_{4} + 26 \beta_{5} ) q^{38} + ( 6 + 51 \beta_{1} - 19 \beta_{2} + 8 \beta_{3} - 4 \beta_{4} - 10 \beta_{5} ) q^{40} + ( 30 + 22 \beta_{1} + 18 \beta_{2} + 8 \beta_{3} - 23 \beta_{4} ) q^{41} + ( 257 + 15 \beta_{1} + 39 \beta_{2} - 13 \beta_{3} + 12 \beta_{4} - \beta_{5} ) q^{42} + ( -66 - 29 \beta_{1} + 21 \beta_{2} - 16 \beta_{3} + 10 \beta_{4} - 4 \beta_{5} ) q^{43} + ( -44 + 11 \beta_{1} - 11 \beta_{2} ) q^{44} + ( 27 - 6 \beta_{2} - 19 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} ) q^{45} + ( -2 - 18 \beta_{1} + \beta_{2} + 5 \beta_{3} - 4 \beta_{4} - 5 \beta_{5} ) q^{46} + ( 23 - 29 \beta_{1} + 9 \beta_{2} + 23 \beta_{3} - 26 \beta_{4} + 3 \beta_{5} ) q^{47} + ( 271 - 65 \beta_{1} + 15 \beta_{2} - 3 \beta_{3} + 26 \beta_{4} - 25 \beta_{5} ) q^{48} + ( 66 + 20 \beta_{1} + 45 \beta_{2} - 10 \beta_{3} - 3 \beta_{4} + 25 \beta_{5} ) q^{49} + ( -158 + 70 \beta_{1} + 12 \beta_{2} - 16 \beta_{3} + 10 \beta_{4} + 18 \beta_{5} ) q^{50} + ( 61 - 7 \beta_{1} - 7 \beta_{2} + 7 \beta_{3} + 32 \beta_{4} - 3 \beta_{5} ) q^{51} + ( -19 + 61 \beta_{1} - 6 \beta_{2} + 16 \beta_{3} - 39 \beta_{4} + 29 \beta_{5} ) q^{53} + ( -316 + 85 \beta_{1} - 22 \beta_{2} + 5 \beta_{3} - 50 \beta_{4} + 13 \beta_{5} ) q^{54} + ( -11 - 11 \beta_{1} - 11 \beta_{3} ) q^{55} + ( 296 - 119 \beta_{1} + 25 \beta_{2} + 4 \beta_{3} - 64 \beta_{4} - 18 \beta_{5} ) q^{56} + ( 277 - 70 \beta_{1} + 7 \beta_{3} + 69 \beta_{4} - 19 \beta_{5} ) q^{57} + ( 277 + 10 \beta_{1} + 60 \beta_{2} - 31 \beta_{3} + 6 \beta_{4} - 35 \beta_{5} ) q^{58} + ( -284 + 76 \beta_{1} - 15 \beta_{2} + 4 \beta_{3} - 36 \beta_{4} - 9 \beta_{5} ) q^{59} + ( -143 - \beta_{1} - 11 \beta_{2} - \beta_{3} + 18 \beta_{4} + 11 \beta_{5} ) q^{60} + ( 61 + 2 \beta_{1} - 16 \beta_{2} + 31 \beta_{3} - 68 \beta_{4} + 25 \beta_{5} ) q^{61} + ( 543 + 10 \beta_{1} + 47 \beta_{2} + 6 \beta_{3} + 34 \beta_{4} + 30 \beta_{5} ) q^{62} + ( 1 + 7 \beta_{1} + 28 \beta_{2} + 10 \beta_{3} + 43 \beta_{5} ) q^{63} + ( 90 - 85 \beta_{1} - \beta_{2} + 30 \beta_{3} - 44 \beta_{4} - 28 \beta_{5} ) q^{64} + ( -22 - 11 \beta_{1} - 22 \beta_{2} + 11 \beta_{3} + 22 \beta_{4} - 11 \beta_{5} ) q^{66} + ( 118 + 94 \beta_{1} - \beta_{2} + 30 \beta_{3} + 36 \beta_{4} - 63 \beta_{5} ) q^{67} + ( 51 + 31 \beta_{1} - 37 \beta_{2} - 23 \beta_{3} - 6 \beta_{4} + 23 \beta_{5} ) q^{68} + ( -45 - 19 \beta_{1} + \beta_{2} - 2 \beta_{3} - 14 \beta_{4} ) q^{69} + ( -318 + 17 \beta_{1} - 21 \beta_{2} - 4 \beta_{3} + 28 \beta_{4} - 10 \beta_{5} ) q^{70} + ( -110 + 17 \beta_{1} + 56 \beta_{2} + 8 \beta_{3} + 4 \beta_{4} + 87 \beta_{5} ) q^{71} + ( -177 + 8 \beta_{1} + 4 \beta_{2} - 15 \beta_{3} - 46 \beta_{4} + 49 \beta_{5} ) q^{72} + ( 370 - 67 \beta_{1} + 32 \beta_{2} + 7 \beta_{3} - 3 \beta_{4} + 42 \beta_{5} ) q^{73} + ( 186 - 189 \beta_{1} + 44 \beta_{2} - 25 \beta_{3} - 44 \beta_{4} - 19 \beta_{5} ) q^{74} + ( 325 + 65 \beta_{1} - 18 \beta_{2} + 26 \beta_{3} + 87 \beta_{4} - 25 \beta_{5} ) q^{75} + ( 188 - 104 \beta_{1} + 40 \beta_{2} - 88 \beta_{4} ) q^{76} + ( -99 - 33 \beta_{1} - 11 \beta_{2} + 11 \beta_{3} + 11 \beta_{4} + 11 \beta_{5} ) q^{77} + ( -190 + 276 \beta_{1} - 34 \beta_{2} - 4 \beta_{3} - 48 \beta_{4} - 18 \beta_{5} ) q^{79} + ( -466 + 39 \beta_{1} - \beta_{2} + 18 \beta_{3} + 52 \beta_{4} - 4 \beta_{5} ) q^{80} + ( 119 - 119 \beta_{1} - 60 \beta_{2} - \beta_{3} + 58 \beta_{4} + 14 \beta_{5} ) q^{81} + ( -153 - 136 \beta_{1} + 32 \beta_{2} - 23 \beta_{3} - 102 \beta_{4} + 3 \beta_{5} ) q^{82} + ( 255 + 66 \beta_{1} - 2 \beta_{2} + 37 \beta_{3} + 101 \beta_{4} + \beta_{5} ) q^{83} + ( 17 - 283 \beta_{1} - 11 \beta_{2} - 5 \beta_{3} - 110 \beta_{4} - 63 \beta_{5} ) q^{84} + ( 395 - 21 \beta_{1} - 29 \beta_{2} - 47 \beta_{3} - 38 \beta_{4} + 27 \beta_{5} ) q^{85} + ( 206 - 11 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} - 96 \beta_{4} - 96 \beta_{5} ) q^{86} + ( 392 + 116 \beta_{1} + 74 \beta_{2} - 20 \beta_{3} - 64 \beta_{4} + 2 \beta_{5} ) q^{87} + ( -66 + 11 \beta_{1} - 11 \beta_{2} + 44 \beta_{4} + 22 \beta_{5} ) q^{88} + ( 105 + 26 \beta_{1} - 97 \beta_{2} + 21 \beta_{3} + 28 \beta_{4} - 36 \beta_{5} ) q^{89} + ( 33 + 36 \beta_{1} - 18 \beta_{2} - 3 \beta_{3} - 10 \beta_{4} - 43 \beta_{5} ) q^{90} + ( 485 + 14 \beta_{1} - 20 \beta_{2} - 65 \beta_{3} - 42 \beta_{4} - 3 \beta_{5} ) q^{92} + ( 135 + 124 \beta_{1} + 97 \beta_{2} - 43 \beta_{3} - 52 \beta_{4} + 64 \beta_{5} ) q^{93} + ( 419 - 111 \beta_{1} + 101 \beta_{2} - 23 \beta_{3} - 42 \beta_{4} + 63 \beta_{5} ) q^{94} + ( -106 - 80 \beta_{1} + 18 \beta_{2} + 14 \beta_{3} + 28 \beta_{4} + 68 \beta_{5} ) q^{95} + ( 207 - 293 \beta_{1} - 5 \beta_{2} - 7 \beta_{3} + 2 \beta_{4} - 65 \beta_{5} ) q^{96} + ( 423 - 2 \beta_{1} - 7 \beta_{2} + 47 \beta_{3} + 128 \beta_{4} + 64 \beta_{5} ) q^{97} + ( -325 - 246 \beta_{1} - 49 \beta_{2} + 22 \beta_{3} - 206 \beta_{4} - 32 \beta_{5} ) q^{98} + ( 11 \beta_{1} - 11 \beta_{3} - 55 \beta_{4} + 22 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} - 6 q^{3} + 26 q^{4} + 8 q^{5} + 15 q^{6} + 53 q^{7} + 36 q^{8} + O(q^{10}) \) \( 6 q + 6 q^{2} - 6 q^{3} + 26 q^{4} + 8 q^{5} + 15 q^{6} + 53 q^{7} + 36 q^{8} - 52 q^{10} - 66 q^{11} - 19 q^{12} - 120 q^{14} + 23 q^{15} + 26 q^{16} - 117 q^{17} + 27 q^{18} + 67 q^{19} - 10 q^{20} - 19 q^{21} - 66 q^{22} - 158 q^{23} + 609 q^{24} - 234 q^{25} - 531 q^{27} + 670 q^{28} - 145 q^{29} - 211 q^{30} + 58 q^{31} + 364 q^{32} + 66 q^{33} - 43 q^{34} - 210 q^{35} + 383 q^{36} + 753 q^{37} + 738 q^{38} + 4 q^{40} + 232 q^{41} + 1593 q^{42} - 390 q^{43} - 286 q^{44} + 107 q^{45} - 5 q^{46} + 205 q^{47} + 1625 q^{48} + 491 q^{49} - 938 q^{50} + 363 q^{51} - 65 q^{53} - 1917 q^{54} - 88 q^{55} + 1816 q^{56} + 1657 q^{57} + 1685 q^{58} - 1735 q^{59} - 871 q^{60} + 421 q^{61} + 3394 q^{62} + 125 q^{63} + 570 q^{64} - 165 q^{66} + 703 q^{67} + 209 q^{68} - 272 q^{69} - 1968 q^{70} - 445 q^{71} - 1035 q^{72} + 2340 q^{73} + 1135 q^{74} + 1941 q^{75} + 1208 q^{76} - 583 q^{77} - 1234 q^{79} - 2766 q^{80} + 606 q^{81} - 897 q^{82} + 1601 q^{83} + 7 q^{84} + 2245 q^{85} + 1146 q^{86} + 2462 q^{87} - 396 q^{88} + 442 q^{89} + 113 q^{90} + 2737 q^{92} + 982 q^{93} + 2733 q^{94} - 504 q^{95} + 1153 q^{96} + 2682 q^{97} - 2036 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 34 x^{4} - 26 x^{3} + 249 x^{2} + 274 x - 200\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 11 \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} + 2 \nu^{4} + 29 \nu^{3} - 32 \nu^{2} - 158 \nu + 56 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{5} - 7 \nu^{4} - 85 \nu^{3} + 121 \nu^{2} + 452 \nu - 256 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{5} + 7 \nu^{4} + 86 \nu^{3} - 123 \nu^{2} - 467 \nu + 266 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 11\)
\(\nu^{3}\)\(=\)\(2 \beta_{5} + 4 \beta_{4} + 2 \beta_{2} + 17 \beta_{1} + 12\)
\(\nu^{4}\)\(=\)\(4 \beta_{5} + 4 \beta_{4} - 6 \beta_{3} + 29 \beta_{2} + 37 \beta_{1} + 211\)
\(\nu^{5}\)\(=\)\(66 \beta_{5} + 124 \beta_{4} - 14 \beta_{3} + 84 \beta_{2} + 377 \beta_{1} + 474\)

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
5.34903
3.11199
0.512502
−2.30499
−2.36134
−4.30719
−4.34903 −6.31201 10.9140 2.31865 27.4511 33.8589 −12.6732 12.8415 −10.0839
1.2 −2.11199 4.05013 −3.53950 16.1679 −8.55384 9.75779 24.3713 −10.5964 −34.1465
1.3 0.487498 0.0966010 −7.76235 −13.1745 0.0470928 11.9172 −7.68411 −26.9907 −6.42254
1.4 3.30499 0.708355 2.92295 6.96953 2.34111 −22.1312 −16.7796 −26.4982 23.0342
1.5 3.36134 −9.16081 3.29862 0.852032 −30.7926 −5.75009 −15.8029 56.9204 2.86397
1.6 5.30719 4.61773 20.1662 −5.13365 24.5072 25.3474 64.5685 −5.67656 −27.2453
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(1\)
\(13\) \(1\)

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 1859.4.a.c 6
13.b even 2 1 143.4.a.b 6
39.d odd 2 1 1287.4.a.f 6
52.b odd 2 1 2288.4.a.m 6
143.d odd 2 1 1573.4.a.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.a.b 6 13.b even 2 1
1287.4.a.f 6 39.d odd 2 1
1573.4.a.d 6 143.d odd 2 1
1859.4.a.c 6 1.a even 1 1 trivial
2288.4.a.m 6 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 6 T_{2}^{5} - 19 T_{2}^{4} + 142 T_{2}^{3} - 18 T_{2}^{2} - 564 T_{2} + 264 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1859))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 264 - 564 T - 18 T^{2} + 142 T^{3} - 19 T^{4} - 6 T^{5} + T^{6} \)
$3$ \( 74 - 885 T + 1248 T^{2} - 165 T^{3} - 63 T^{4} + 6 T^{5} + T^{6} \)
$5$ \( 15056 - 23180 T + 5561 T^{2} + 1260 T^{3} - 226 T^{4} - 8 T^{5} + T^{6} \)
$7$ \( 12700172 - 460832 T - 323489 T^{2} + 27171 T^{3} + 130 T^{4} - 53 T^{5} + T^{6} \)
$11$ \( ( 11 + T )^{6} \)
$13$ \( T^{6} \)
$17$ \( 64149376 - 157625536 T - 22448136 T^{2} - 794704 T^{3} - 5164 T^{4} + 117 T^{5} + T^{6} \)
$19$ \( -49335856960 - 589693504 T + 76596244 T^{2} + 867528 T^{3} - 21599 T^{4} - 67 T^{5} + T^{6} \)
$23$ \( 158607016 - 51818057 T + 2352898 T^{2} + 189307 T^{3} - 12859 T^{4} + 158 T^{5} + T^{6} \)
$29$ \( -14715687201280 + 119074831168 T + 2277465152 T^{2} - 10732860 T^{3} - 95224 T^{4} + 145 T^{5} + T^{6} \)
$31$ \( -2334074232384 - 86254699608 T + 891421992 T^{2} + 11109836 T^{3} - 97807 T^{4} - 58 T^{5} + T^{6} \)
$37$ \( -1475360000224 + 103934262392 T - 1645497258 T^{2} - 1748943 T^{3} + 156065 T^{4} - 753 T^{5} + T^{6} \)
$41$ \( 19515694713824 - 910183130164 T + 3296886452 T^{2} + 29057067 T^{3} - 120352 T^{4} - 232 T^{5} + T^{6} \)
$43$ \( -4882557023552 + 631154555136 T + 3718734996 T^{2} - 44017642 T^{3} - 145665 T^{4} + 390 T^{5} + T^{6} \)
$47$ \( -359796596534272 - 2125147528896 T + 16886548384 T^{2} + 46170564 T^{3} - 246756 T^{4} - 205 T^{5} + T^{6} \)
$53$ \( -33677222445392 - 248966794992 T + 64547895656 T^{2} - 10899448 T^{3} - 516849 T^{4} + 65 T^{5} + T^{6} \)
$59$ \( -1424934900572200 - 19485743810312 T - 60724065976 T^{2} + 127053105 T^{3} + 981455 T^{4} + 1735 T^{5} + T^{6} \)
$61$ \( 703896062106752 + 24146584784304 T + 218212216264 T^{2} + 138065472 T^{3} - 867714 T^{4} - 421 T^{5} + T^{6} \)
$67$ \( 7632911822856224 - 117416217721980 T + 118138713346 T^{2} + 640787379 T^{3} - 848187 T^{4} - 703 T^{5} + T^{6} \)
$71$ \( 73522818265301472 + 386520749704044 T + 386950360446 T^{2} - 866398753 T^{3} - 1391111 T^{4} + 445 T^{5} + T^{6} \)
$73$ \( -260628470656 - 434654013728 T + 58758647484 T^{2} - 575587107 T^{3} + 1844048 T^{4} - 2340 T^{5} + T^{6} \)
$79$ \( -308455474338688000 + 738459184377088 T + 1128895136224 T^{2} - 2048250216 T^{3} - 1892108 T^{4} + 1234 T^{5} + T^{6} \)
$83$ \( 24040793615106048 - 96495047900160 T - 215268704544 T^{2} + 1001426576 T^{3} - 112857 T^{4} - 1601 T^{5} + T^{6} \)
$89$ \( -7197734921201920 - 481209895272256 T + 849107749888 T^{2} + 937002900 T^{3} - 1826759 T^{4} - 442 T^{5} + T^{6} \)
$97$ \( 46775041266827264 + 38000372296192 T - 2162558674944 T^{2} + 3186447168 T^{3} + 583721 T^{4} - 2682 T^{5} + T^{6} \)
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