Newspace parameters
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.i (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.51528766367\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} - 2 x^{15} + 8 x^{14} - 10 x^{13} + 40 x^{12} - 45 x^{11} + 159 x^{10} - 180 x^{9} + 576 x^{8} - 540 x^{7} + 1431 x^{6} - 1215 x^{5} + 3240 x^{4} - 2430 x^{3} + 5832 x^{2} + \cdots + 6561 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 3^{4} \) |
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 8 x^{14} - 10 x^{13} + 40 x^{12} - 45 x^{11} + 159 x^{10} - 180 x^{9} + 576 x^{8} - 540 x^{7} + 1431 x^{6} - 1215 x^{5} + 3240 x^{4} - 2430 x^{3} + 5832 x^{2} + \cdots + 6561 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( - \nu^{15} - 54 \nu^{14} - 13 \nu^{13} - 285 \nu^{12} - 92 \nu^{11} - 1376 \nu^{10} - 564 \nu^{9} - 5133 \nu^{8} - 729 \nu^{7} - 15138 \nu^{6} - 5130 \nu^{5} - 35586 \nu^{4} - 6399 \nu^{3} + \cdots - 105705 ) / 5832 \) |
\(\beta_{3}\) | \(=\) | \( ( - 331 \nu^{15} - 384 \nu^{14} - 1549 \nu^{13} - 3459 \nu^{12} - 7034 \nu^{11} - 20030 \nu^{10} - 31770 \nu^{9} - 69753 \nu^{8} - 77643 \nu^{7} - 230976 \nu^{6} + \cdots - 2009853 ) / 763992 \) |
\(\beta_{4}\) | \(=\) | \( ( 23 \nu^{15} + 98 \nu^{14} + 139 \nu^{13} + 571 \nu^{12} + 668 \nu^{11} + 2808 \nu^{10} + 3036 \nu^{9} + 9171 \nu^{8} + 7335 \nu^{7} + 29214 \nu^{6} + 21006 \nu^{5} + 66582 \nu^{4} + \cdots + 185895 ) / 17496 \) |
\(\beta_{5}\) | \(=\) | \( ( 1046 \nu^{15} - 1099 \nu^{14} + 6769 \nu^{13} - 6206 \nu^{12} + 34925 \nu^{11} - 20859 \nu^{10} + 129333 \nu^{9} - 113013 \nu^{8} + 409716 \nu^{7} - 254097 \nu^{6} + \cdots - 2171691 ) / 763992 \) |
\(\beta_{6}\) | \(=\) | \( ( 3181 \nu^{15} + 19087 \nu^{14} + 19856 \nu^{13} + 110357 \nu^{12} + 111133 \nu^{11} + 545253 \nu^{10} + 450105 \nu^{9} + 1921554 \nu^{8} + 1030491 \nu^{7} + \cdots + 38027556 ) / 2291976 \) |
\(\beta_{7}\) | \(=\) | \( ( 4417 \nu^{15} - 21707 \nu^{14} + 18134 \nu^{13} - 111667 \nu^{12} + 85891 \nu^{11} - 556257 \nu^{10} + 222711 \nu^{9} - 2202156 \nu^{8} + 1032759 \nu^{7} + \cdots - 48914442 ) / 2291976 \) |
\(\beta_{8}\) | \(=\) | \( ( 5084 \nu^{15} - 28999 \nu^{14} + 24811 \nu^{13} - 145040 \nu^{12} + 102185 \nu^{11} - 728211 \nu^{10} + 259437 \nu^{9} - 2833947 \nu^{8} + 1367514 \nu^{7} + \cdots - 64820493 ) / 2291976 \) |
\(\beta_{9}\) | \(=\) | \( ( - 7037 \nu^{15} - 4493 \nu^{14} - 37522 \nu^{13} - 33481 \nu^{12} - 196979 \nu^{11} - 182583 \nu^{10} - 785487 \nu^{9} - 500112 \nu^{8} - 2386251 \nu^{7} + \cdots - 11822922 ) / 2291976 \) |
\(\beta_{10}\) | \(=\) | \( ( 7210 \nu^{15} - 8453 \nu^{14} + 41831 \nu^{13} - 44614 \nu^{12} + 178159 \nu^{11} - 172845 \nu^{10} + 627531 \nu^{9} - 856971 \nu^{8} + 1982160 \nu^{7} + \cdots - 19925757 ) / 2291976 \) |
\(\beta_{11}\) | \(=\) | \( ( - 9662 \nu^{15} + 17977 \nu^{14} - 46603 \nu^{13} + 104798 \nu^{12} - 193703 \nu^{11} + 470901 \nu^{10} - 641559 \nu^{9} + 2068767 \nu^{8} - 2112984 \nu^{7} + \cdots + 46493433 ) / 2291976 \) |
\(\beta_{12}\) | \(=\) | \( ( 3765 \nu^{15} - 343 \nu^{14} + 21146 \nu^{13} - 1619 \nu^{12} + 99331 \nu^{11} + 9839 \nu^{10} + 355491 \nu^{9} - 51720 \nu^{8} + 1071351 \nu^{7} + 70947 \nu^{6} + \cdots - 1106622 ) / 763992 \) |
\(\beta_{13}\) | \(=\) | \( ( - 12086 \nu^{15} + 23761 \nu^{14} - 60991 \nu^{13} + 125582 \nu^{12} - 286595 \nu^{11} + 566589 \nu^{10} - 937479 \nu^{9} + 2445795 \nu^{8} - 3348720 \nu^{7} + \cdots + 58198257 ) / 2291976 \) |
\(\beta_{14}\) | \(=\) | \( ( - 5017 \nu^{15} + 6145 \nu^{14} - 26580 \nu^{13} + 31991 \nu^{12} - 129525 \nu^{11} + 144563 \nu^{10} - 454785 \nu^{9} + 650802 \nu^{8} - 1497951 \nu^{7} + \cdots + 14521680 ) / 763992 \) |
\(\beta_{15}\) | \(=\) | \( ( - 16105 \nu^{15} + 1514 \nu^{14} - 86681 \nu^{13} + 4519 \nu^{12} - 420040 \nu^{11} - 14028 \nu^{10} - 1510836 \nu^{9} + 340047 \nu^{8} - 4533309 \nu^{7} + \cdots + 12984219 ) / 2291976 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{15} + \beta_{12} + \beta_{7} - \beta_{2} - 1 \) |
\(\nu^{3}\) | \(=\) | \( -\beta_{14} - \beta_{10} + \beta_{9} - \beta_{7} + \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 1 \) |
\(\nu^{4}\) | \(=\) | \( - 3 \beta_{14} + 2 \beta_{13} - 3 \beta_{12} - \beta_{11} + 2 \beta_{10} - \beta_{9} - \beta_{8} - 2 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - \beta_{4} + 3 \beta_{3} + 4 \beta_{2} - \beta _1 - 1 \) |
\(\nu^{5}\) | \(=\) | \( 4 \beta_{14} + \beta_{13} - 3 \beta_{11} - 4 \beta_{9} + 3 \beta_{8} + 2 \beta_{6} + 3 \beta_{5} - 5 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 5 \beta _1 + 3 \) |
\(\nu^{6}\) | \(=\) | \( - 7 \beta_{15} + 10 \beta_{14} - \beta_{13} + 7 \beta_{12} + 2 \beta_{11} - 7 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} - 9 \beta_{6} + 4 \beta_{5} + 3 \beta_{4} + 2 \beta_{2} - 5 \) |
\(\nu^{7}\) | \(=\) | \( 4 \beta_{15} - 5 \beta_{14} - 11 \beta_{13} - 4 \beta_{12} + 9 \beta_{11} + 10 \beta_{10} - \beta_{9} - 6 \beta_{8} - \beta_{7} - 4 \beta_{6} - 15 \beta_{5} + 5 \beta_{4} + 12 \beta_{3} - 2 \beta_{2} + \beta _1 + 20 \) |
\(\nu^{8}\) | \(=\) | \( - 11 \beta_{15} - 5 \beta_{14} - 12 \beta_{13} - 13 \beta_{12} + 12 \beta_{11} + 5 \beta_{10} + 14 \beta_{9} - 9 \beta_{7} - 12 \beta_{6} - 6 \beta_{5} - 6 \beta_{4} - 37 \beta_{3} - 15 \beta_{2} + 25 \beta_1 \) |
\(\nu^{9}\) | \(=\) | \( - 7 \beta_{15} - 6 \beta_{14} + 23 \beta_{13} + 13 \beta_{12} + 2 \beta_{11} - 17 \beta_{10} + 17 \beta_{9} - 7 \beta_{8} + 23 \beta_{7} - 36 \beta_{6} + 28 \beta_{5} + 28 \beta_{4} - 64 \beta_{3} - 6 \beta_{2} + 23 \beta _1 - 117 \) |
\(\nu^{10}\) | \(=\) | \( 79 \beta_{15} - 26 \beta_{14} - 48 \beta_{13} - 19 \beta_{12} - 22 \beta_{11} + 15 \beta_{10} + 34 \beta_{9} - 22 \beta_{8} + 21 \beta_{7} + 65 \beta_{6} + 40 \beta_{5} + 22 \beta_{4} - 45 \beta_{3} - 25 \beta_{2} - 93 \beta _1 + 46 \) |
\(\nu^{11}\) | \(=\) | \( - 20 \beta_{15} - 58 \beta_{14} + 34 \beta_{13} - 94 \beta_{12} - 21 \beta_{11} + 32 \beta_{10} + 10 \beta_{9} - 45 \beta_{8} - 3 \beta_{7} + 194 \beta_{6} + 9 \beta_{5} - 104 \beta_{4} + 127 \beta_{3} + 126 \beta_{2} + 5 \beta _1 + 165 \) |
\(\nu^{12}\) | \(=\) | \( - 8 \beta_{15} + 109 \beta_{14} + 232 \beta_{13} + 104 \beta_{12} - 68 \beta_{11} - 15 \beta_{10} - 189 \beta_{9} + 193 \beta_{8} + 119 \beta_{7} + 87 \beta_{6} - 22 \beta_{5} - 62 \beta_{4} + 275 \beta_{3} - 19 \beta_{2} + 137 \beta _1 + 37 \) |
\(\nu^{13}\) | \(=\) | \( - 124 \beta_{15} + 458 \beta_{14} + \beta_{13} + 508 \beta_{12} + 65 \beta_{11} - 137 \beta_{10} + 45 \beta_{9} + 419 \beta_{8} - 181 \beta_{7} - 425 \beta_{6} + 163 \beta_{5} + 192 \beta_{4} + 51 \beta_{3} - 216 \beta_{2} + 124 \beta _1 + 111 \) |
\(\nu^{14}\) | \(=\) | \( - 112 \beta_{15} - 393 \beta_{14} - 134 \beta_{13} - 71 \beta_{12} + 48 \beta_{11} + 4 \beta_{10} + 96 \beta_{9} - 501 \beta_{8} - 276 \beta_{7} - 424 \beta_{6} - 531 \beta_{5} + 292 \beta_{4} + 1464 \beta_{3} + 180 \beta_{2} + \cdots + 438 \) |
\(\nu^{15}\) | \(=\) | \( 522 \beta_{15} - 734 \beta_{14} - 498 \beta_{13} - 528 \beta_{12} - 438 \beta_{11} + 82 \beta_{10} - 346 \beta_{9} - 744 \beta_{8} + 232 \beta_{7} + 378 \beta_{6} - 1872 \beta_{5} - 514 \beta_{4} - 1208 \beta_{3} - 298 \beta_{2} + \cdots - 113 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(136\) | \(281\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-1 - \beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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106.1 |
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−1.32864 | + | 2.30128i | −0.649093 | − | 1.60583i | −2.53058 | − | 4.38310i | −0.500000 | − | 0.866025i | 4.55786 | + | 0.639826i | 0.500000 | − | 0.866025i | 8.13440 | −2.15736 | + | 2.08466i | 2.65729 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
106.2 | −0.978244 | + | 1.69437i | −1.57186 | + | 0.727495i | −0.913922 | − | 1.58296i | −0.500000 | − | 0.866025i | 0.305020 | − | 3.37498i | 0.500000 | − | 0.866025i | −0.336819 | 1.94150 | − | 2.28704i | 1.95649 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
106.3 | −0.522039 | + | 0.904198i | 0.392487 | + | 1.68700i | 0.454951 | + | 0.787999i | −0.500000 | − | 0.866025i | −1.73027 | − | 0.525791i | 0.500000 | − | 0.866025i | −3.03816 | −2.69191 | + | 1.32425i | 1.04408 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
106.4 | −0.219523 | + | 0.380224i | 1.57891 | − | 0.712071i | 0.903620 | + | 1.56512i | −0.500000 | − | 0.866025i | −0.0758596 | + | 0.756655i | 0.500000 | − | 0.866025i | −1.67155 | 1.98591 | − | 2.24859i | 0.439045 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
106.5 | 0.441371 | − | 0.764477i | −1.59660 | − | 0.671465i | 0.610383 | + | 1.05721i | −0.500000 | − | 0.866025i | −1.21801 | + | 0.924200i | 0.500000 | − | 0.866025i | 2.84311 | 2.09827 | + | 2.14412i | −0.882742 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
106.6 | 0.627726 | − | 1.08725i | 1.73059 | + | 0.0710311i | 0.211920 | + | 0.367057i | −0.500000 | − | 0.866025i | 1.16357 | − | 1.83701i | 0.500000 | − | 0.866025i | 3.04302 | 2.98991 | + | 0.245852i | −1.25545 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
106.7 | 1.09035 | − | 1.88855i | −0.551686 | − | 1.64184i | −1.37775 | − | 2.38633i | −0.500000 | − | 0.866025i | −3.70223 | − | 0.748302i | 0.500000 | − | 0.866025i | −1.64751 | −2.39128 | + | 1.81156i | −2.18071 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
106.8 | 1.38900 | − | 2.40581i | 1.16725 | + | 1.27966i | −2.85862 | − | 4.95128i | −0.500000 | − | 0.866025i | 4.69992 | − | 1.03075i | 0.500000 | − | 0.866025i | −10.3265 | −0.275041 | + | 2.98737i | −2.77799 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
211.1 | −1.32864 | − | 2.30128i | −0.649093 | + | 1.60583i | −2.53058 | + | 4.38310i | −0.500000 | + | 0.866025i | 4.55786 | − | 0.639826i | 0.500000 | + | 0.866025i | 8.13440 | −2.15736 | − | 2.08466i | 2.65729 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
211.2 | −0.978244 | − | 1.69437i | −1.57186 | − | 0.727495i | −0.913922 | + | 1.58296i | −0.500000 | + | 0.866025i | 0.305020 | + | 3.37498i | 0.500000 | + | 0.866025i | −0.336819 | 1.94150 | + | 2.28704i | 1.95649 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
211.3 | −0.522039 | − | 0.904198i | 0.392487 | − | 1.68700i | 0.454951 | − | 0.787999i | −0.500000 | + | 0.866025i | −1.73027 | + | 0.525791i | 0.500000 | + | 0.866025i | −3.03816 | −2.69191 | − | 1.32425i | 1.04408 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
211.4 | −0.219523 | − | 0.380224i | 1.57891 | + | 0.712071i | 0.903620 | − | 1.56512i | −0.500000 | + | 0.866025i | −0.0758596 | − | 0.756655i | 0.500000 | + | 0.866025i | −1.67155 | 1.98591 | + | 2.24859i | 0.439045 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
211.5 | 0.441371 | + | 0.764477i | −1.59660 | + | 0.671465i | 0.610383 | − | 1.05721i | −0.500000 | + | 0.866025i | −1.21801 | − | 0.924200i | 0.500000 | + | 0.866025i | 2.84311 | 2.09827 | − | 2.14412i | −0.882742 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
211.6 | 0.627726 | + | 1.08725i | 1.73059 | − | 0.0710311i | 0.211920 | − | 0.367057i | −0.500000 | + | 0.866025i | 1.16357 | + | 1.83701i | 0.500000 | + | 0.866025i | 3.04302 | 2.98991 | − | 0.245852i | −1.25545 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
211.7 | 1.09035 | + | 1.88855i | −0.551686 | + | 1.64184i | −1.37775 | + | 2.38633i | −0.500000 | + | 0.866025i | −3.70223 | + | 0.748302i | 0.500000 | + | 0.866025i | −1.64751 | −2.39128 | − | 1.81156i | −2.18071 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
211.8 | 1.38900 | + | 2.40581i | 1.16725 | − | 1.27966i | −2.85862 | + | 4.95128i | −0.500000 | + | 0.866025i | 4.69992 | + | 1.03075i | 0.500000 | + | 0.866025i | −10.3265 | −0.275041 | − | 2.98737i | −2.77799 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 315.2.i.f | ✓ | 16 |
3.b | odd | 2 | 1 | 945.2.i.f | 16 | ||
9.c | even | 3 | 1 | inner | 315.2.i.f | ✓ | 16 |
9.c | even | 3 | 1 | 2835.2.a.x | 8 | ||
9.d | odd | 6 | 1 | 945.2.i.f | 16 | ||
9.d | odd | 6 | 1 | 2835.2.a.y | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.2.i.f | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
315.2.i.f | ✓ | 16 | 9.c | even | 3 | 1 | inner |
945.2.i.f | 16 | 3.b | odd | 2 | 1 | ||
945.2.i.f | 16 | 9.d | odd | 6 | 1 | ||
2835.2.a.x | 8 | 9.c | even | 3 | 1 | ||
2835.2.a.y | 8 | 9.d | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} - T_{2}^{15} + 14 T_{2}^{14} - 9 T_{2}^{13} + 131 T_{2}^{12} - 77 T_{2}^{11} + 616 T_{2}^{10} - 216 T_{2}^{9} + 1978 T_{2}^{8} - 660 T_{2}^{7} + 3061 T_{2}^{6} - 220 T_{2}^{5} + 2945 T_{2}^{4} - 186 T_{2}^{3} + 1364 T_{2}^{2} + \cdots + 256 \)
acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} - T^{15} + 14 T^{14} - 9 T^{13} + \cdots + 256 \)
$3$
\( T^{16} - T^{15} - T^{14} - 2 T^{13} + \cdots + 6561 \)
$5$
\( (T^{2} + T + 1)^{8} \)
$7$
\( (T^{2} - T + 1)^{8} \)
$11$
\( T^{16} + 4 T^{15} + 80 T^{14} + \cdots + 241367296 \)
$13$
\( T^{16} + 5 T^{15} + 80 T^{14} + \cdots + 3118756 \)
$17$
\( (T^{8} - 4 T^{7} - 88 T^{6} + 416 T^{5} + \cdots - 13682)^{2} \)
$19$
\( (T^{8} - 3 T^{7} - 101 T^{6} + 243 T^{5} + \cdots + 96256)^{2} \)
$23$
\( T^{16} - 8 T^{15} + 128 T^{14} + \cdots + 358875136 \)
$29$
\( T^{16} + 19 T^{15} + \cdots + 181104718096 \)
$31$
\( T^{16} + 108 T^{14} + \cdots + 6879707136 \)
$37$
\( (T^{8} - 21 T^{7} + 105 T^{6} + \cdots - 82944)^{2} \)
$41$
\( T^{16} + 20 T^{15} + \cdots + 836829184 \)
$43$
\( T^{16} + 13 T^{15} + \cdots + 345330171904 \)
$47$
\( T^{16} - 11 T^{15} + 192 T^{14} + \cdots + 4096 \)
$53$
\( (T^{8} + 8 T^{7} - 118 T^{6} + \cdots - 165056)^{2} \)
$59$
\( T^{16} + 7 T^{15} + \cdots + 63471748096 \)
$61$
\( T^{16} + 24 T^{15} + \cdots + 27518828544 \)
$67$
\( T^{16} + 16 T^{15} + 233 T^{14} + \cdots + 5456896 \)
$71$
\( (T^{8} + 5 T^{7} - 290 T^{6} + \cdots - 1249607)^{2} \)
$73$
\( (T^{8} - 10 T^{7} - 88 T^{6} + \cdots + 124561)^{2} \)
$79$
\( T^{16} + 27 T^{15} + 497 T^{14} + \cdots + 16384 \)
$83$
\( T^{16} + 5 T^{15} + \cdots + 6791950062769 \)
$89$
\( (T^{8} - 27 T^{7} - 53 T^{6} + \cdots - 6305216)^{2} \)
$97$
\( T^{16} + 27 T^{15} + \cdots + 4365349992964 \)
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