Properties

Label 230.6.a.h
Level $230$
Weight $6$
Character orbit 230.a
Self dual yes
Analytic conductor $36.888$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 230.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.8882785570\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - x^{5} - 1168 x^{4} - 2857 x^{3} + 297325 x^{2} + 680040 x - 8930700\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} + ( 2 - \beta_{1} ) q^{3} + 16 q^{4} + 25 q^{5} + ( 8 - 4 \beta_{1} ) q^{6} + ( 61 + 2 \beta_{1} + \beta_{4} ) q^{7} + 64 q^{8} + ( 150 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{9} +O(q^{10})\) \( q + 4 q^{2} + ( 2 - \beta_{1} ) q^{3} + 16 q^{4} + 25 q^{5} + ( 8 - 4 \beta_{1} ) q^{6} + ( 61 + 2 \beta_{1} + \beta_{4} ) q^{7} + 64 q^{8} + ( 150 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{9} + 100 q^{10} + ( 26 - 9 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{11} + ( 32 - 16 \beta_{1} ) q^{12} + ( 78 - 6 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{13} + ( 244 + 8 \beta_{1} + 4 \beta_{4} ) q^{14} + ( 50 - 25 \beta_{1} ) q^{15} + 256 q^{16} + ( 109 - 13 \beta_{1} + \beta_{2} - 3 \beta_{3} - 4 \beta_{4} + \beta_{5} ) q^{17} + ( 600 + 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} ) q^{18} + ( 569 + 17 \beta_{1} + 4 \beta_{2} - \beta_{3} - 4 \beta_{4} + 3 \beta_{5} ) q^{19} + 400 q^{20} + ( -626 - 99 \beta_{1} - 7 \beta_{2} - 5 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{21} + ( 104 - 36 \beta_{1} - 4 \beta_{2} - 4 \beta_{4} - 4 \beta_{5} ) q^{22} + 529 q^{23} + ( 128 - 64 \beta_{1} ) q^{24} + 625 q^{25} + ( 312 - 24 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{26} + ( -546 - 149 \beta_{1} - 18 \beta_{2} + 5 \beta_{3} - 16 \beta_{4} - 6 \beta_{5} ) q^{27} + ( 976 + 32 \beta_{1} + 16 \beta_{4} ) q^{28} + ( 985 + 14 \beta_{1} - 15 \beta_{2} + 7 \beta_{3} - 14 \beta_{4} + \beta_{5} ) q^{29} + ( 200 - 100 \beta_{1} ) q^{30} + ( 1700 + 81 \beta_{1} + 25 \beta_{2} - 8 \beta_{3} - 8 \beta_{4} + 2 \beta_{5} ) q^{31} + 1024 q^{32} + ( 3823 + 145 \beta_{1} + 52 \beta_{2} + 21 \beta_{3} + 22 \beta_{4} - 17 \beta_{5} ) q^{33} + ( 436 - 52 \beta_{1} + 4 \beta_{2} - 12 \beta_{3} - 16 \beta_{4} + 4 \beta_{5} ) q^{34} + ( 1525 + 50 \beta_{1} + 25 \beta_{4} ) q^{35} + ( 2400 + 16 \beta_{1} + 16 \beta_{2} + 16 \beta_{3} + 16 \beta_{4} ) q^{36} + ( 2839 + 122 \beta_{1} - 2 \beta_{2} + 19 \beta_{3} - 13 \beta_{4} - 4 \beta_{5} ) q^{37} + ( 2276 + 68 \beta_{1} + 16 \beta_{2} - 4 \beta_{3} - 16 \beta_{4} + 12 \beta_{5} ) q^{38} + ( 2347 + 116 \beta_{1} + 2 \beta_{2} - 9 \beta_{3} + 12 \beta_{4} + 27 \beta_{5} ) q^{39} + 1600 q^{40} + ( 126 + 90 \beta_{1} - 58 \beta_{2} - 13 \beta_{3} + 59 \beta_{4} - 11 \beta_{5} ) q^{41} + ( -2504 - 396 \beta_{1} - 28 \beta_{2} - 20 \beta_{3} - 12 \beta_{4} - 4 \beta_{5} ) q^{42} + ( 2217 + 195 \beta_{1} - 78 \beta_{2} + 12 \beta_{3} + 54 \beta_{4} + 27 \beta_{5} ) q^{43} + ( 416 - 144 \beta_{1} - 16 \beta_{2} - 16 \beta_{4} - 16 \beta_{5} ) q^{44} + ( 3750 + 25 \beta_{1} + 25 \beta_{2} + 25 \beta_{3} + 25 \beta_{4} ) q^{45} + 2116 q^{46} + ( 4073 + 143 \beta_{1} + 115 \beta_{2} - 22 \beta_{3} - 55 \beta_{4} - 18 \beta_{5} ) q^{47} + ( 512 - 256 \beta_{1} ) q^{48} + ( 4636 + 449 \beta_{1} + 63 \beta_{2} - 51 \beta_{3} + 52 \beta_{4} - 3 \beta_{5} ) q^{49} + 2500 q^{50} + ( 4300 + 420 \beta_{1} - 20 \beta_{2} - 11 \beta_{3} + 38 \beta_{4} + 28 \beta_{5} ) q^{51} + ( 1248 - 96 \beta_{1} - 16 \beta_{2} - 16 \beta_{3} - 16 \beta_{4} + 16 \beta_{5} ) q^{52} + ( 2866 + 185 \beta_{1} - 22 \beta_{2} + \beta_{3} - 57 \beta_{4} - 45 \beta_{5} ) q^{53} + ( -2184 - 596 \beta_{1} - 72 \beta_{2} + 20 \beta_{3} - 64 \beta_{4} - 24 \beta_{5} ) q^{54} + ( 650 - 225 \beta_{1} - 25 \beta_{2} - 25 \beta_{4} - 25 \beta_{5} ) q^{55} + ( 3904 + 128 \beta_{1} + 64 \beta_{4} ) q^{56} + ( -6866 - 719 \beta_{1} - 133 \beta_{2} - 41 \beta_{3} - 41 \beta_{4} + 57 \beta_{5} ) q^{57} + ( 3940 + 56 \beta_{1} - 60 \beta_{2} + 28 \beta_{3} - 56 \beta_{4} + 4 \beta_{5} ) q^{58} + ( 10348 + 657 \beta_{1} + 20 \beta_{2} + 51 \beta_{3} - 49 \beta_{4} - 37 \beta_{5} ) q^{59} + ( 800 - 400 \beta_{1} ) q^{60} + ( 5322 + 855 \beta_{1} - 73 \beta_{2} - 50 \beta_{3} - 37 \beta_{4} + 33 \beta_{5} ) q^{61} + ( 6800 + 324 \beta_{1} + 100 \beta_{2} - 32 \beta_{3} - 32 \beta_{4} + 8 \beta_{5} ) q^{62} + ( 22837 + 1729 \beta_{1} + 229 \beta_{2} + 72 \beta_{3} - 56 \beta_{4} + 13 \beta_{5} ) q^{63} + 4096 q^{64} + ( 1950 - 150 \beta_{1} - 25 \beta_{2} - 25 \beta_{3} - 25 \beta_{4} + 25 \beta_{5} ) q^{65} + ( 15292 + 580 \beta_{1} + 208 \beta_{2} + 84 \beta_{3} + 88 \beta_{4} - 68 \beta_{5} ) q^{66} + ( 8967 + 236 \beta_{1} + 66 \beta_{2} + 103 \beta_{3} - 59 \beta_{4} + 16 \beta_{5} ) q^{67} + ( 1744 - 208 \beta_{1} + 16 \beta_{2} - 48 \beta_{3} - 64 \beta_{4} + 16 \beta_{5} ) q^{68} + ( 1058 - 529 \beta_{1} ) q^{69} + ( 6100 + 200 \beta_{1} + 100 \beta_{4} ) q^{70} + ( -2485 + 257 \beta_{1} - 242 \beta_{2} - 11 \beta_{3} + 11 \beta_{4} - 76 \beta_{5} ) q^{71} + ( 9600 + 64 \beta_{1} + 64 \beta_{2} + 64 \beta_{3} + 64 \beta_{4} ) q^{72} + ( 5376 + 1250 \beta_{1} - 33 \beta_{2} - 90 \beta_{3} + 121 \beta_{4} - 69 \beta_{5} ) q^{73} + ( 11356 + 488 \beta_{1} - 8 \beta_{2} + 76 \beta_{3} - 52 \beta_{4} - 16 \beta_{5} ) q^{74} + ( 1250 - 625 \beta_{1} ) q^{75} + ( 9104 + 272 \beta_{1} + 64 \beta_{2} - 16 \beta_{3} - 64 \beta_{4} + 48 \beta_{5} ) q^{76} + ( -23631 - 2044 \beta_{1} - 289 \beta_{2} - 53 \beta_{3} + 57 \beta_{4} - 30 \beta_{5} ) q^{77} + ( 9388 + 464 \beta_{1} + 8 \beta_{2} - 36 \beta_{3} + 48 \beta_{4} + 108 \beta_{5} ) q^{78} + ( 12417 + 435 \beta_{1} + 104 \beta_{2} + 28 \beta_{3} + 212 \beta_{4} + 141 \beta_{5} ) q^{79} + 6400 q^{80} + ( 24896 + 1732 \beta_{1} + 443 \beta_{2} + 53 \beta_{3} - 25 \beta_{4} - 66 \beta_{5} ) q^{81} + ( 504 + 360 \beta_{1} - 232 \beta_{2} - 52 \beta_{3} + 236 \beta_{4} - 44 \beta_{5} ) q^{82} + ( 10468 + 1805 \beta_{1} - 36 \beta_{2} - 59 \beta_{3} - 305 \beta_{4} + 55 \beta_{5} ) q^{83} + ( -10016 - 1584 \beta_{1} - 112 \beta_{2} - 80 \beta_{3} - 48 \beta_{4} - 16 \beta_{5} ) q^{84} + ( 2725 - 325 \beta_{1} + 25 \beta_{2} - 75 \beta_{3} - 100 \beta_{4} + 25 \beta_{5} ) q^{85} + ( 8868 + 780 \beta_{1} - 312 \beta_{2} + 48 \beta_{3} + 216 \beta_{4} + 108 \beta_{5} ) q^{86} + ( 75 - 1013 \beta_{1} + 325 \beta_{2} + 82 \beta_{3} - 41 \beta_{4} + 66 \beta_{5} ) q^{87} + ( 1664 - 576 \beta_{1} - 64 \beta_{2} - 64 \beta_{4} - 64 \beta_{5} ) q^{88} + ( -10763 - 3059 \beta_{1} + 14 \beta_{2} + 128 \beta_{3} + 52 \beta_{4} - 65 \beta_{5} ) q^{89} + ( 15000 + 100 \beta_{1} + 100 \beta_{2} + 100 \beta_{3} + 100 \beta_{4} ) q^{90} + ( -14696 - 1149 \beta_{1} - 177 \beta_{2} - 44 \beta_{3} + 147 \beta_{4} + \beta_{5} ) q^{91} + 8464 q^{92} + ( -34707 - 1774 \beta_{1} - 548 \beta_{2} - 147 \beta_{3} - 78 \beta_{4} - 9 \beta_{5} ) q^{93} + ( 16292 + 572 \beta_{1} + 460 \beta_{2} - 88 \beta_{3} - 220 \beta_{4} - 72 \beta_{5} ) q^{94} + ( 14225 + 425 \beta_{1} + 100 \beta_{2} - 25 \beta_{3} - 100 \beta_{4} + 75 \beta_{5} ) q^{95} + ( 2048 - 1024 \beta_{1} ) q^{96} + ( -2078 - 65 \beta_{1} + 247 \beta_{2} + 152 \beta_{3} + 333 \beta_{4} - 93 \beta_{5} ) q^{97} + ( 18544 + 1796 \beta_{1} + 252 \beta_{2} - 204 \beta_{3} + 208 \beta_{4} - 12 \beta_{5} ) q^{98} + ( -58266 - 7570 \beta_{1} - 434 \beta_{2} + 131 \beta_{3} - 158 \beta_{4} - 334 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 24 q^{2} + 11 q^{3} + 96 q^{4} + 150 q^{5} + 44 q^{6} + 366 q^{7} + 384 q^{8} + 899 q^{9} + O(q^{10}) \) \( 6 q + 24 q^{2} + 11 q^{3} + 96 q^{4} + 150 q^{5} + 44 q^{6} + 366 q^{7} + 384 q^{8} + 899 q^{9} + 600 q^{10} + 151 q^{11} + 176 q^{12} + 463 q^{13} + 1464 q^{14} + 275 q^{15} + 1536 q^{16} + 644 q^{17} + 3596 q^{18} + 3431 q^{19} + 2400 q^{20} - 3846 q^{21} + 604 q^{22} + 3174 q^{23} + 704 q^{24} + 3750 q^{25} + 1852 q^{26} - 3364 q^{27} + 5856 q^{28} + 5973 q^{29} + 1100 q^{30} + 10262 q^{31} + 6144 q^{32} + 23025 q^{33} + 2576 q^{34} + 9150 q^{35} + 14384 q^{36} + 17207 q^{37} + 13724 q^{38} + 14136 q^{39} + 9600 q^{40} + 784 q^{41} - 15384 q^{42} + 13452 q^{43} + 2416 q^{44} + 22475 q^{45} + 12696 q^{46} + 24572 q^{47} + 2816 q^{48} + 28050 q^{49} + 15000 q^{50} + 26125 q^{51} + 7408 q^{52} + 17563 q^{53} - 13456 q^{54} + 3775 q^{55} + 23424 q^{56} - 41798 q^{57} + 23892 q^{58} + 62911 q^{59} + 4400 q^{60} + 32851 q^{61} + 41048 q^{62} + 138693 q^{63} + 24576 q^{64} + 11575 q^{65} + 92100 q^{66} + 54177 q^{67} + 10304 q^{68} + 5819 q^{69} + 36600 q^{70} - 14368 q^{71} + 57536 q^{72} + 33276 q^{73} + 68828 q^{74} + 6875 q^{75} + 54896 q^{76} - 143678 q^{77} + 56544 q^{78} + 74296 q^{79} + 38400 q^{80} + 150834 q^{81} + 3136 q^{82} + 65145 q^{83} - 61536 q^{84} + 16100 q^{85} + 53808 q^{86} - 790 q^{87} + 9664 q^{88} - 67562 q^{89} + 89900 q^{90} - 89487 q^{91} + 50784 q^{92} - 209450 q^{93} + 98288 q^{94} + 85775 q^{95} + 11264 q^{96} - 13201 q^{97} + 112200 q^{98} - 355951 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 1168 x^{4} - 2857 x^{3} + 297325 x^{2} + 680040 x - 8930700\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 51 \nu^{5} + 1939 \nu^{4} - 51558 \nu^{3} - 1670407 \nu^{2} + 1317025 \nu + 92904390 \)\()/1380040\)
\(\beta_{3}\)\(=\)\((\)\( 49 \nu^{5} - 843 \nu^{4} - 57654 \nu^{3} + 889995 \nu^{2} + 14649059 \nu - 166695750 \)\()/414012\)
\(\beta_{4}\)\(=\)\((\)\( -643 \nu^{5} + 2613 \nu^{4} + 731214 \nu^{3} + 251391 \nu^{2} - 171142265 \nu - 222262350 \)\()/4140120\)
\(\beta_{5}\)\(=\)\((\)\( 416 \nu^{5} - 7861 \nu^{4} - 319078 \nu^{3} + 4409948 \nu^{2} + 34179795 \nu - 336035700 \)\()/1035030\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + \beta_{2} + 5 \beta_{1} + 389\)
\(\nu^{3}\)\(=\)\(6 \beta_{5} + 22 \beta_{4} + \beta_{3} + 24 \beta_{2} + 653 \beta_{1} + 1916\)
\(\nu^{4}\)\(=\)\(-18 \beta_{5} + 856 \beta_{4} + 766 \beta_{3} + 1340 \beta_{2} + 7597 \beta_{1} + 258320\)
\(\nu^{5}\)\(=\)\(6750 \beta_{5} + 22449 \beta_{4} + 4641 \beta_{3} + 33129 \beta_{2} + 509251 \beta_{1} + 3035031\)

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
31.1755
16.9517
4.70878
−7.33442
−19.8456
−24.6560
4.00000 −29.1755 16.0000 25.0000 −116.702 213.932 64.0000 608.209 100.000
1.2 4.00000 −14.9517 16.0000 25.0000 −59.8068 52.9827 64.0000 −19.4468 100.000
1.3 4.00000 −2.70878 16.0000 25.0000 −10.8351 −158.180 64.0000 −235.663 100.000
1.4 4.00000 9.33442 16.0000 25.0000 37.3377 234.539 64.0000 −155.869 100.000
1.5 4.00000 21.8456 16.0000 25.0000 87.3823 7.44621 64.0000 234.229 100.000
1.6 4.00000 26.6560 16.0000 25.0000 106.624 15.2806 64.0000 467.540 100.000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(23\) \(-1\)

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 230.6.a.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.6.a.h 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 11 T_{3}^{5} - 1118 T_{3}^{4} + 12081 T_{3}^{3} + 252311 T_{3}^{2} - 1797792 T_{3} - 6422832 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(230))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -4 + T )^{6} \)
$3$ \( -6422832 - 1797792 T + 252311 T^{2} + 12081 T^{3} - 1118 T^{4} - 11 T^{5} + T^{6} \)
$5$ \( ( -25 + T )^{6} \)
$7$ \( -47846477632 + 10585036248 T - 626498184 T^{2} + 9120101 T^{3} + 2532 T^{4} - 366 T^{5} + T^{6} \)
$11$ \( 2157987582267680 - 26372754820008 T + 67344293300 T^{2} + 158528826 T^{3} - 610989 T^{4} - 151 T^{5} + T^{6} \)
$13$ \( 478562359504036 - 11137252879756 T + 39939428811 T^{2} + 138419329 T^{3} - 472338 T^{4} - 463 T^{5} + T^{6} \)
$17$ \( -272262610938044960 - 121566831742144 T + 1397489332410 T^{2} + 567640143 T^{3} - 2201562 T^{4} - 644 T^{5} + T^{6} \)
$19$ \( -1978462874618267040 - 6154748858246472 T - 557034009444 T^{2} + 8473144050 T^{3} - 345599 T^{4} - 3431 T^{5} + T^{6} \)
$23$ \( ( -529 + T )^{6} \)
$29$ \( \)\(66\!\cdots\!00\)\( - 607539761445239600 T - 112788052438808 T^{2} + 202371060589 T^{3} - 31416361 T^{4} - 5973 T^{5} + T^{6} \)
$31$ \( -\)\(75\!\cdots\!00\)\( - 1877402346007870773 T - 29218706361494 T^{2} + 330872359727 T^{3} - 18716599 T^{4} - 10262 T^{5} + T^{6} \)
$37$ \( -\)\(80\!\cdots\!96\)\( + 16057812080689754528 T - 7878269686079496 T^{2} + 1210659025836 T^{3} + 15824706 T^{4} - 17207 T^{5} + T^{6} \)
$41$ \( \)\(98\!\cdots\!86\)\( + \)\(43\!\cdots\!29\)\( T + 47024476361319342 T^{2} - 2013520936865 T^{3} - 441661173 T^{4} - 784 T^{5} + T^{6} \)
$43$ \( -\)\(10\!\cdots\!00\)\( - \)\(11\!\cdots\!40\)\( T + 121977252449680896 T^{2} + 5128568639712 T^{3} - 664413912 T^{4} - 13452 T^{5} + T^{6} \)
$47$ \( \)\(17\!\cdots\!24\)\( - \)\(39\!\cdots\!36\)\( T + 117036382718481876 T^{2} + 20955382541022 T^{3} - 846942887 T^{4} - 24572 T^{5} + T^{6} \)
$53$ \( \)\(16\!\cdots\!32\)\( - \)\(15\!\cdots\!96\)\( T + 1027877500306144 T^{2} + 19810306468104 T^{3} - 856685182 T^{4} - 17563 T^{5} + T^{6} \)
$59$ \( \)\(17\!\cdots\!00\)\( - \)\(10\!\cdots\!40\)\( T - 664490338288281680 T^{2} + 51198600408652 T^{3} + 194498230 T^{4} - 62911 T^{5} + T^{6} \)
$61$ \( -\)\(20\!\cdots\!76\)\( + \)\(86\!\cdots\!20\)\( T - 1281481551581967148 T^{2} + 78852264811606 T^{3} - 1424276417 T^{4} - 32851 T^{5} + T^{6} \)
$67$ \( -\)\(68\!\cdots\!60\)\( + \)\(31\!\cdots\!20\)\( T - 4156653962960001152 T^{2} + 187405861986548 T^{3} - 1991940256 T^{4} - 54177 T^{5} + T^{6} \)
$71$ \( -\)\(87\!\cdots\!48\)\( - \)\(89\!\cdots\!05\)\( T + 4849502127121286102 T^{2} - 118312698869227 T^{3} - 6038755919 T^{4} + 14368 T^{5} + T^{6} \)
$73$ \( \)\(87\!\cdots\!76\)\( - \)\(14\!\cdots\!08\)\( T + 2667780476203006392 T^{2} + 244298554213774 T^{3} - 7001768859 T^{4} - 33276 T^{5} + T^{6} \)
$79$ \( -\)\(97\!\cdots\!80\)\( + \)\(50\!\cdots\!56\)\( T + 31816946719342495040 T^{2} + 285576278672368 T^{3} - 9088286564 T^{4} - 74296 T^{5} + T^{6} \)
$83$ \( \)\(48\!\cdots\!76\)\( - \)\(64\!\cdots\!56\)\( T + 9732500253030612400 T^{2} + 460204523564044 T^{3} - 7935937420 T^{4} - 65145 T^{5} + T^{6} \)
$89$ \( \)\(17\!\cdots\!20\)\( + \)\(67\!\cdots\!84\)\( T - 25706393357413448512 T^{2} - 1356435038524976 T^{3} - 13356756704 T^{4} + 67562 T^{5} + T^{6} \)
$97$ \( -\)\(23\!\cdots\!24\)\( + \)\(14\!\cdots\!80\)\( T + 55804317905071506212 T^{2} - 301853890580668 T^{3} - 14681458151 T^{4} + 13201 T^{5} + T^{6} \)
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