Newspace parameters
Level: | \( N \) | \(=\) | \( 675 = 3^{3} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 675.q (of order \(12\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.38990213644\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{12})\) |
Coefficient field: | 16.0.9349208943630483456.9 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + 9297 x^{8} - 11276 x^{7} + 11224 x^{6} - 9024 x^{5} + 5736 x^{4} - 2780 x^{3} + \cdots + 25 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{6}\cdot 3^{4} \) |
Twist minimal: | no (minimal twist has level 225) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + 9297 x^{8} - 11276 x^{7} + 11224 x^{6} - 9024 x^{5} + 5736 x^{4} - 2780 x^{3} + \cdots + 25 \) :
\(\beta_{1}\) | \(=\) | \( ( - 3456 \nu^{15} + 25920 \nu^{14} - 151876 \nu^{13} + 594074 \nu^{12} - 1879372 \nu^{11} + 4666596 \nu^{10} - 9554736 \nu^{9} + 15945783 \nu^{8} - 21928484 \nu^{7} + \cdots + 125460 ) / 17095 \) |
\(\beta_{2}\) | \(=\) | \( ( 1757 \nu^{15} + 29034 \nu^{14} - 217536 \nu^{13} + 1408900 \nu^{12} - 5469225 \nu^{11} + 17560899 \nu^{10} - 42890877 \nu^{9} + 87005139 \nu^{8} - 141294613 \nu^{7} + \cdots + 2553985 ) / 17095 \) |
\(\beta_{3}\) | \(=\) | \( ( 1757 \nu^{15} - 55389 \nu^{14} + 373425 \nu^{13} - 2022461 \nu^{12} + 7436448 \nu^{11} - 22510833 \nu^{10} + 53250351 \nu^{9} - 104737641 \nu^{8} + 166646564 \nu^{7} + \cdots - 2875650 ) / 17095 \) |
\(\beta_{4}\) | \(=\) | \( ( - 54 \nu^{14} + 378 \nu^{13} - 2193 \nu^{12} + 8244 \nu^{11} - 25569 \nu^{10} + 61284 \nu^{9} - 122021 \nu^{8} + 195620 \nu^{7} - 258293 \nu^{6} + 273134 \nu^{5} - 230109 \nu^{4} + \cdots - 3308 ) / 13 \) |
\(\beta_{5}\) | \(=\) | \( ( 13659 \nu^{15} - 66806 \nu^{14} + 357125 \nu^{13} - 945718 \nu^{12} + 2245378 \nu^{11} - 2537530 \nu^{10} + 267574 \nu^{9} + 10564902 \nu^{8} - 29075066 \nu^{7} + \cdots + 1260620 ) / 17095 \) |
\(\beta_{6}\) | \(=\) | \( ( - 13659 \nu^{15} + 138079 \nu^{14} - 856036 \nu^{13} + 3832406 \nu^{12} - 13079663 \nu^{11} + 36027161 \nu^{10} - 80292162 \nu^{9} + 148038300 \nu^{8} + \cdots + 2779060 ) / 17095 \) |
\(\beta_{7}\) | \(=\) | \( ( 22230 \nu^{15} - 144633 \nu^{14} + 820486 \nu^{13} - 2915351 \nu^{12} + 8665028 \nu^{11} - 19406977 \nu^{10} + 35943838 \nu^{9} - 51803400 \nu^{8} + 59433982 \nu^{7} + \cdots + 599315 ) / 17095 \) |
\(\beta_{8}\) | \(=\) | \( ( 22230 \nu^{15} - 188817 \nu^{14} + 1129774 \nu^{13} - 4704014 \nu^{12} + 15376262 \nu^{11} - 40133218 \nu^{10} + 85426762 \nu^{9} - 149690685 \nu^{8} + \cdots - 1317955 ) / 17095 \) |
\(\beta_{9}\) | \(=\) | \( ( - 26630 \nu^{15} + 182630 \nu^{14} - 1056740 \nu^{13} + 3923413 \nu^{12} - 12097498 \nu^{11} + 28666706 \nu^{10} - 56590650 \nu^{9} + 89528958 \nu^{8} + \cdots - 218885 ) / 17095 \) |
\(\beta_{10}\) | \(=\) | \( ( - 26630 \nu^{15} + 216820 \nu^{14} - 1296070 \nu^{13} + 5311527 \nu^{12} - 17314892 \nu^{11} + 44845414 \nu^{10} - 95362110 \nu^{9} + 166733397 \nu^{8} + \cdots + 2071845 ) / 17095 \) |
\(\beta_{11}\) | \(=\) | \( ( 63850 \nu^{15} - 464147 \nu^{14} + 2710563 \nu^{13} - 10426206 \nu^{12} + 32745418 \nu^{11} - 80184009 \nu^{10} + 162524611 \nu^{9} - 267170328 \nu^{8} + \cdots - 667700 ) / 17095 \) |
\(\beta_{12}\) | \(=\) | \( ( - 63850 \nu^{15} + 493603 \nu^{14} - 2916755 \nu^{13} + 11625486 \nu^{12} - 37260602 \nu^{11} + 94237414 \nu^{10} - 196316692 \nu^{9} + 334828656 \nu^{8} + \cdots + 3005770 ) / 17095 \) |
\(\beta_{13}\) | \(=\) | \( ( - 97569 \nu^{15} + 712700 \nu^{14} - 4168110 \nu^{13} + 16084516 \nu^{12} - 50637049 \nu^{11} + 124422927 \nu^{10} - 253084785 \nu^{9} + 418004961 \nu^{8} + \cdots + 1608045 ) / 17095 \) |
\(\beta_{14}\) | \(=\) | \( ( 97569 \nu^{15} - 750835 \nu^{14} + 4435055 \nu^{13} - 17639109 \nu^{12} + 56494322 \nu^{11} - 142673023 \nu^{10} + 297005785 \nu^{9} - 505980039 \nu^{8} + \cdots - 4286700 ) / 17095 \) |
\(\beta_{15}\) | \(=\) | \( ( - 162552 \nu^{15} + 1254645 \nu^{14} - 7410690 \nu^{13} + 29505615 \nu^{12} - 94503294 \nu^{11} + 238745115 \nu^{10} - 496891020 \nu^{9} + 846295960 \nu^{8} + \cdots + 7071655 ) / 17095 \) |
\(\nu\) | \(=\) | \( ( 2 \beta_{15} + \beta_{14} - \beta_{13} - 2 \beta_{12} + 2 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} - \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 3 ) / 6 \) |
\(\nu^{2}\) | \(=\) | \( ( 2 \beta_{15} - 2 \beta_{13} + 2 \beta_{12} + 6 \beta_{11} + 6 \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{5} + \beta_{4} + 4 \beta_{3} - 12 ) / 6 \) |
\(\nu^{3}\) | \(=\) | \( ( - 10 \beta_{15} - 6 \beta_{14} + 3 \beta_{13} + 15 \beta_{12} - 3 \beta_{11} - 21 \beta_{10} - 12 \beta_{9} - 6 \beta_{8} - 3 \beta_{7} + 12 \beta_{6} - 9 \beta_{5} - 5 \beta_{4} - 3 \beta_{3} - 9 \beta_{2} - 9 \beta _1 - 24 ) / 6 \) |
\(\nu^{4}\) | \(=\) | \( ( - 11 \beta_{15} - 2 \beta_{14} + 8 \beta_{13} - 2 \beta_{12} - 22 \beta_{11} - 9 \beta_{10} - 27 \beta_{9} - 9 \beta_{7} + 8 \beta_{6} - 14 \beta_{5} - 10 \beta_{4} - 10 \beta_{3} - 4 \beta_{2} - 9 \beta _1 + 18 ) / 3 \) |
\(\nu^{5}\) | \(=\) | \( ( 43 \beta_{15} + 28 \beta_{14} + 7 \beta_{13} - 121 \beta_{12} - 19 \beta_{11} + 138 \beta_{10} + 33 \beta_{9} + 55 \beta_{8} + 5 \beta_{7} - 85 \beta_{6} + 50 \beta_{5} - \beta_{4} - 2 \beta_{3} + 38 \beta_{2} + 75 \beta _1 + 183 ) / 6 \) |
\(\nu^{6}\) | \(=\) | \( ( 185 \beta_{15} + 24 \beta_{14} - 90 \beta_{13} - 96 \beta_{12} + 312 \beta_{11} + 258 \beta_{10} + 438 \beta_{9} + 70 \beta_{8} + 155 \beta_{7} - 222 \beta_{6} + 294 \beta_{5} + 160 \beta_{4} + 102 \beta_{3} + 78 \beta_{2} + 270 \beta _1 - 30 ) / 6 \) |
\(\nu^{7}\) | \(=\) | \( ( - 128 \beta_{15} - 180 \beta_{14} - 177 \beta_{13} + 822 \beta_{12} + 438 \beta_{11} - 801 \beta_{10} + 207 \beta_{9} - 364 \beta_{8} + 112 \beta_{7} + 459 \beta_{6} - 81 \beta_{5} + 251 \beta_{4} + 75 \beta_{3} - 156 \beta_{2} + \cdots - 1299 ) / 6 \) |
\(\nu^{8}\) | \(=\) | \( ( - 714 \beta_{15} - 108 \beta_{14} + 188 \beta_{13} + 784 \beta_{12} - 984 \beta_{11} - 1338 \beta_{10} - 1560 \beta_{9} - 486 \beta_{8} - 564 \beta_{7} + 1116 \beta_{6} - 1220 \beta_{5} - 462 \beta_{4} - 292 \beta_{3} + \cdots - 606 ) / 3 \) |
\(\nu^{9}\) | \(=\) | \( ( - 379 \beta_{15} + 1278 \beta_{14} + 1737 \beta_{13} - 4707 \beta_{12} - 5265 \beta_{11} + 3675 \beta_{10} - 4830 \beta_{9} + 1806 \beta_{8} - 1968 \beta_{7} - 1404 \beta_{6} - 1953 \beta_{5} - 3119 \beta_{4} + \cdots + 8202 ) / 6 \) |
\(\nu^{10}\) | \(=\) | \( ( 10223 \beta_{15} + 2460 \beta_{14} - 586 \beta_{13} - 16730 \beta_{12} + 10062 \beta_{11} + 24006 \beta_{10} + 18930 \beta_{9} + 9581 \beta_{8} + 7024 \beta_{7} - 19002 \beta_{6} + 17128 \beta_{5} + \cdots + 17628 ) / 6 \) |
\(\nu^{11}\) | \(=\) | \( ( 12443 \beta_{15} - 8283 \beta_{14} - 13376 \beta_{13} + 19613 \beta_{12} + 50589 \beta_{11} - 5778 \beta_{10} + 56559 \beta_{9} - 3993 \beta_{8} + 22341 \beta_{7} - 7959 \beta_{6} + 32984 \beta_{5} + \cdots - 43491 ) / 6 \) |
\(\nu^{12}\) | \(=\) | \( ( - 32900 \beta_{15} - 13056 \beta_{14} - 5274 \beta_{13} + 74346 \beta_{12} - 14082 \beta_{11} - 95841 \beta_{10} - 44046 \beta_{9} - 39501 \beta_{8} - 16731 \beta_{7} + 70494 \beta_{6} - 49980 \beta_{5} + \cdots - 89109 ) / 3 \) |
\(\nu^{13}\) | \(=\) | \( ( - 158381 \beta_{15} + 42692 \beta_{14} + 88595 \beta_{13} - 4160 \beta_{12} - 420446 \beta_{11} - 140085 \beta_{10} - 525717 \beta_{9} - 50609 \beta_{8} - 208546 \beta_{7} + 203251 \beta_{6} + \cdots + 151791 ) / 6 \) |
\(\nu^{14}\) | \(=\) | \( ( 352742 \beta_{15} + 244178 \beta_{14} + 172508 \beta_{13} - 1157114 \beta_{12} - 198050 \beta_{11} + 1355046 \beta_{10} + 149892 \beta_{9} + 563485 \beta_{8} + 56225 \beta_{7} + \cdots + 1521186 ) / 6 \) |
\(\nu^{15}\) | \(=\) | \( ( 1567315 \beta_{15} - 107706 \beta_{14} - 494328 \beta_{13} - 1118625 \beta_{12} + 3071523 \beta_{11} + 2418363 \beta_{10} + 4219254 \beta_{9} + 968762 \beta_{8} + 1689964 \beta_{7} + \cdots + 356820 ) / 6 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).
\(n\) | \(326\) | \(352\) |
\(\chi(n)\) | \(1 + \beta_{1}\) | \(-\beta_{9} - \beta_{10}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
143.1 |
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−2.25487 | + | 0.604191i | 0 | 2.98735 | − | 1.72474i | 0 | 0 | 1.04649 | + | 3.90555i | −2.39264 | + | 2.39264i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
143.2 | −0.716682 | + | 0.192034i | 0 | −1.25529 | + | 0.724745i | 0 | 0 | 0.332613 | + | 1.24133i | 1.80977 | − | 1.80977i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
143.3 | 0.716682 | − | 0.192034i | 0 | −1.25529 | + | 0.724745i | 0 | 0 | −0.332613 | − | 1.24133i | −1.80977 | + | 1.80977i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
143.4 | 2.25487 | − | 0.604191i | 0 | 2.98735 | − | 1.72474i | 0 | 0 | −1.04649 | − | 3.90555i | 2.39264 | − | 2.39264i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
332.1 | −0.604191 | − | 2.25487i | 0 | −2.98735 | + | 1.72474i | 0 | 0 | −3.90555 | + | 1.04649i | 2.39264 | + | 2.39264i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
332.2 | −0.192034 | − | 0.716682i | 0 | 1.25529 | − | 0.724745i | 0 | 0 | −1.24133 | + | 0.332613i | −1.80977 | − | 1.80977i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
332.3 | 0.192034 | + | 0.716682i | 0 | 1.25529 | − | 0.724745i | 0 | 0 | 1.24133 | − | 0.332613i | 1.80977 | + | 1.80977i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
332.4 | 0.604191 | + | 2.25487i | 0 | −2.98735 | + | 1.72474i | 0 | 0 | 3.90555 | − | 1.04649i | −2.39264 | − | 2.39264i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
368.1 | −0.604191 | + | 2.25487i | 0 | −2.98735 | − | 1.72474i | 0 | 0 | −3.90555 | − | 1.04649i | 2.39264 | − | 2.39264i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
368.2 | −0.192034 | + | 0.716682i | 0 | 1.25529 | + | 0.724745i | 0 | 0 | −1.24133 | − | 0.332613i | −1.80977 | + | 1.80977i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
368.3 | 0.192034 | − | 0.716682i | 0 | 1.25529 | + | 0.724745i | 0 | 0 | 1.24133 | + | 0.332613i | 1.80977 | − | 1.80977i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
368.4 | 0.604191 | − | 2.25487i | 0 | −2.98735 | − | 1.72474i | 0 | 0 | 3.90555 | + | 1.04649i | −2.39264 | + | 2.39264i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
557.1 | −2.25487 | − | 0.604191i | 0 | 2.98735 | + | 1.72474i | 0 | 0 | 1.04649 | − | 3.90555i | −2.39264 | − | 2.39264i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
557.2 | −0.716682 | − | 0.192034i | 0 | −1.25529 | − | 0.724745i | 0 | 0 | 0.332613 | − | 1.24133i | 1.80977 | + | 1.80977i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
557.3 | 0.716682 | + | 0.192034i | 0 | −1.25529 | − | 0.724745i | 0 | 0 | −0.332613 | + | 1.24133i | −1.80977 | − | 1.80977i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
557.4 | 2.25487 | + | 0.604191i | 0 | 2.98735 | + | 1.72474i | 0 | 0 | −1.04649 | + | 3.90555i | 2.39264 | + | 2.39264i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
5.c | odd | 4 | 2 | inner |
9.d | odd | 6 | 1 | inner |
45.h | odd | 6 | 1 | inner |
45.l | even | 12 | 2 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 675.2.q.b | 16 | |
3.b | odd | 2 | 1 | 225.2.p.a | ✓ | 16 | |
5.b | even | 2 | 1 | inner | 675.2.q.b | 16 | |
5.c | odd | 4 | 2 | inner | 675.2.q.b | 16 | |
9.c | even | 3 | 1 | 225.2.p.a | ✓ | 16 | |
9.d | odd | 6 | 1 | inner | 675.2.q.b | 16 | |
15.d | odd | 2 | 1 | 225.2.p.a | ✓ | 16 | |
15.e | even | 4 | 2 | 225.2.p.a | ✓ | 16 | |
45.h | odd | 6 | 1 | inner | 675.2.q.b | 16 | |
45.j | even | 6 | 1 | 225.2.p.a | ✓ | 16 | |
45.k | odd | 12 | 2 | 225.2.p.a | ✓ | 16 | |
45.l | even | 12 | 2 | inner | 675.2.q.b | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
225.2.p.a | ✓ | 16 | 3.b | odd | 2 | 1 | |
225.2.p.a | ✓ | 16 | 9.c | even | 3 | 1 | |
225.2.p.a | ✓ | 16 | 15.d | odd | 2 | 1 | |
225.2.p.a | ✓ | 16 | 15.e | even | 4 | 2 | |
225.2.p.a | ✓ | 16 | 45.j | even | 6 | 1 | |
225.2.p.a | ✓ | 16 | 45.k | odd | 12 | 2 | |
675.2.q.b | 16 | 1.a | even | 1 | 1 | trivial | |
675.2.q.b | 16 | 5.b | even | 2 | 1 | inner | |
675.2.q.b | 16 | 5.c | odd | 4 | 2 | inner | |
675.2.q.b | 16 | 9.d | odd | 6 | 1 | inner | |
675.2.q.b | 16 | 45.h | odd | 6 | 1 | inner | |
675.2.q.b | 16 | 45.l | even | 12 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} - 30T_{2}^{12} + 891T_{2}^{8} - 270T_{2}^{4} + 81 \)
acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} - 30 T^{12} + 891 T^{8} + \cdots + 81 \)
$3$
\( T^{16} \)
$5$
\( T^{16} \)
$7$
\( T^{16} - 270 T^{12} + 72171 T^{8} + \cdots + 531441 \)
$11$
\( (T^{4} - 18 T^{2} + 324)^{4} \)
$13$
\( T^{16} - 1080 T^{12} + \cdots + 136048896 \)
$17$
\( (T^{8} + 120 T^{4} + 144)^{2} \)
$19$
\( (T^{4} + 20 T^{2} + 4)^{4} \)
$23$
\( T^{16} - 2910 T^{12} + 8468091 T^{8} + \cdots + 81 \)
$29$
\( (T^{4} + 27 T^{2} + 729)^{4} \)
$31$
\( (T^{4} - 4 T^{3} + 18 T^{2} + 8 T + 4)^{4} \)
$37$
\( (T^{8} + 6264 T^{4} + 7290000)^{2} \)
$41$
\( (T^{4} - 18 T^{3} + 117 T^{2} - 162 T + 81)^{4} \)
$43$
\( T^{16} - 1080 T^{12} + \cdots + 136048896 \)
$47$
\( T^{16} - 270 T^{12} + 72171 T^{8} + \cdots + 531441 \)
$53$
\( (T^{8} + 2784 T^{4} + 1440000)^{2} \)
$59$
\( (T^{4} + 72 T^{2} + 5184)^{4} \)
$61$
\( (T^{4} + 10 T^{3} + 81 T^{2} + 190 T + 361)^{4} \)
$67$
\( T^{16} - 2430 T^{12} + \cdots + 3486784401 \)
$71$
\( (T^{4} + 252 T^{2} + 8100)^{4} \)
$73$
\( T^{16} \)
$79$
\( (T^{8} - 200 T^{6} + 31536 T^{4} + \cdots + 71639296)^{2} \)
$83$
\( T^{16} - 7230 T^{12} + \cdots + 6343189807761 \)
$89$
\( (T^{4} - 198 T^{2} + 2025)^{4} \)
$97$
\( T^{16} - 25056 T^{12} + \cdots + 13\!\cdots\!00 \)
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