Newspace parameters
Level: | \( N \) | \(=\) | \( 480 = 2^{5} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 480.m (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.83281929702\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
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: | \( x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{17} \) |
Twist minimal: | no (minimal twist has level 120) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \( x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 \) :
\(\beta_{1}\) | \(=\) | \( ( 3\nu^{13} + 69\nu^{11} + 506\nu^{9} + 1488\nu^{7} + 1638\nu^{5} + 594\nu^{3} + 28\nu ) / 8 \) |
\(\beta_{2}\) | \(=\) | \( ( -7\nu^{14} - 164\nu^{12} - 1250\nu^{10} - 3984\nu^{8} - 5334\nu^{6} - 3024\nu^{4} - 596\nu^{2} - 16 ) / 8 \) |
\(\beta_{3}\) | \(=\) | \( ( 3 \nu^{15} + 22 \nu^{14} + 62 \nu^{13} + 522 \nu^{12} + 344 \nu^{11} + 4080 \nu^{10} + 284 \nu^{9} + 13636 \nu^{8} - 2010 \nu^{7} + 20076 \nu^{6} - 3772 \nu^{5} + 13332 \nu^{4} - 2000 \nu^{3} + \cdots + 344 ) / 16 \) |
\(\beta_{4}\) | \(=\) | \( ( 15\nu^{14} + 352\nu^{12} + 2692\nu^{10} + 8638\nu^{8} + 11726\nu^{6} + 6808\nu^{4} + 1496\nu^{2} + 76 ) / 8 \) |
\(\beta_{5}\) | \(=\) | \( ( - 5 \nu^{15} + 12 \nu^{14} - 110 \nu^{13} + 282 \nu^{12} - 728 \nu^{11} + 2164 \nu^{10} - 1625 \nu^{9} + 7004 \nu^{8} - 118 \nu^{7} + 9752 \nu^{6} + 2276 \nu^{5} + 6092 \nu^{4} + 1648 \nu^{3} + \cdots + 136 ) / 8 \) |
\(\beta_{6}\) | \(=\) | \( ( 16 \nu^{15} + \nu^{14} + 376 \nu^{13} + 20 \nu^{12} + 2885 \nu^{11} + 100 \nu^{10} + 9330 \nu^{9} - 4 \nu^{8} + 12928 \nu^{7} - 918 \nu^{6} + 7872 \nu^{5} - 1440 \nu^{4} + 1834 \nu^{3} - 664 \nu^{2} + \cdots - 72 ) / 16 \) |
\(\beta_{7}\) | \(=\) | \( ( 8 \nu^{15} - 21 \nu^{14} + 196 \nu^{13} - 494 \nu^{12} + 1629 \nu^{11} - 3798 \nu^{10} + 6072 \nu^{9} - 12334 \nu^{8} + 10856 \nu^{7} - 17306 \nu^{6} + 9544 \nu^{5} - 11028 \nu^{4} + 3818 \nu^{3} + \cdots - 252 ) / 16 \) |
\(\beta_{8}\) | \(=\) | \( ( - 14 \nu^{15} + 21 \nu^{14} - 326 \nu^{13} + 494 \nu^{12} - 2455 \nu^{11} + 3798 \nu^{10} - 7652 \nu^{9} + 12334 \nu^{8} - 9812 \nu^{7} + 17306 \nu^{6} - 5276 \nu^{5} + 11028 \nu^{4} + \cdots + 252 ) / 16 \) |
\(\beta_{9}\) | \(=\) | \( ( 6\nu^{14} + 141\nu^{12} + 1082\nu^{10} + 3502\nu^{8} + 4876\nu^{6} + 3046\nu^{4} + 804\nu^{2} + 68 ) / 2 \) |
\(\beta_{10}\) | \(=\) | \( ( - 19 \nu^{15} + 6 \nu^{14} - 454 \nu^{13} + 138 \nu^{12} - 3598 \nu^{11} + 1012 \nu^{10} - 12338 \nu^{9} + 2976 \nu^{8} - 19070 \nu^{7} + 3276 \nu^{6} - 13604 \nu^{5} + 1188 \nu^{4} + \cdots + 16 ) / 16 \) |
\(\beta_{11}\) | \(=\) | \( ( -11\nu^{15} - 261\nu^{13} - 2042\nu^{11} - 6862\nu^{9} - 10334\nu^{7} - 7410\nu^{5} - 2412\nu^{3} - 252\nu ) / 8 \) |
\(\beta_{12}\) | \(=\) | \( ( - 11 \nu^{15} + 12 \nu^{14} - 261 \nu^{13} + 282 \nu^{12} - 2040 \nu^{11} + 2164 \nu^{10} - 6817 \nu^{9} + 7004 \nu^{8} - 10018 \nu^{7} + 9752 \nu^{6} - 6554 \nu^{5} + 6092 \nu^{4} - 1640 \nu^{3} + \cdots + 136 ) / 8 \) |
\(\beta_{13}\) | \(=\) | \( ( 38 \nu^{15} - \nu^{14} + 896 \nu^{13} - 20 \nu^{12} + 6921 \nu^{11} - 100 \nu^{10} + 22674 \nu^{9} + 4 \nu^{8} + 32332 \nu^{7} + 918 \nu^{6} + 20960 \nu^{5} + 1440 \nu^{4} + 5842 \nu^{3} + 664 \nu^{2} + \cdots + 72 ) / 16 \) |
\(\beta_{14}\) | \(=\) | \( ( - 38 \nu^{15} - \nu^{14} - 896 \nu^{13} - 20 \nu^{12} - 6921 \nu^{11} - 100 \nu^{10} - 22674 \nu^{9} + 4 \nu^{8} - 32332 \nu^{7} + 918 \nu^{6} - 20960 \nu^{5} + 1440 \nu^{4} - 5842 \nu^{3} + \cdots + 72 ) / 16 \) |
\(\beta_{15}\) | \(=\) | \( ( -19\nu^{15} - 451\nu^{13} - 3529\nu^{11} - 11832\nu^{9} - 17582\nu^{7} - 11966\nu^{5} - 3498\nu^{3} - 344\nu ) / 8 \) |
\(\nu\) | \(=\) | \( ( \beta_{15} - 2\beta_{12} + \beta_{9} + \beta_{8} + \beta_{7} - 2\beta_1 ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( - \beta_{15} + \beta_{14} + \beta_{13} + \beta_{11} + 2 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{4} - \beta_{2} + \beta _1 - 6 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( - 6 \beta_{15} - 3 \beta_{13} + 8 \beta_{12} - 3 \beta_{11} - 3 \beta_{9} - 4 \beta_{8} - 4 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + 9 \beta_1 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( 14 \beta_{15} - 20 \beta_{14} - 20 \beta_{13} - 12 \beta_{11} - 28 \beta_{10} - 13 \beta_{9} + 16 \beta_{8} - 8 \beta_{7} + 8 \beta_{4} + 8 \beta_{3} + 12 \beta_{2} - 10 \beta _1 + 48 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( 78 \beta_{15} - 10 \beta_{14} + 50 \beta_{13} - 83 \beta_{12} + 55 \beta_{11} + 27 \beta_{9} + 47 \beta_{8} + 47 \beta_{7} + 40 \beta_{6} + 29 \beta_{5} - 95 \beta_1 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( ( - 181 \beta_{15} + 280 \beta_{14} + 280 \beta_{13} + 144 \beta_{11} + 362 \beta_{10} + 161 \beta_{9} - 211 \beta_{8} + 77 \beta_{7} - 74 \beta_{4} - 134 \beta_{3} - 134 \beta_{2} + 114 \beta _1 - 504 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( - 984 \beta_{15} + 175 \beta_{14} - 672 \beta_{13} + 956 \beta_{12} - 756 \beta_{11} - 292 \beta_{9} - 571 \beta_{8} - 571 \beta_{7} - 497 \beta_{6} - 372 \beta_{5} + 1091 \beta_1 ) / 2 \) |
\(\nu^{8}\) | \(=\) | \( 1130 \beta_{15} - 1792 \beta_{14} - 1792 \beta_{13} - 876 \beta_{11} - 2260 \beta_{10} - 990 \beta_{9} + 1326 \beta_{8} - 426 \beta_{7} + 396 \beta_{4} + 900 \beta_{3} + 784 \beta_{2} - 680 \beta _1 + 2910 \) |
\(\nu^{9}\) | \(=\) | \( ( 12191 \beta_{15} - 2388 \beta_{14} + 8496 \beta_{13} - 11446 \beta_{12} + 9624 \beta_{11} + 3407 \beta_{9} + 6971 \beta_{8} + 6971 \beta_{7} + 6108 \beta_{6} + 4632 \beta_{5} - 13034 \beta_1 ) / 2 \) |
\(\nu^{10}\) | \(=\) | \( - 13931 \beta_{15} + 22287 \beta_{14} + 22287 \beta_{13} + 10707 \beta_{11} + 27862 \beta_{10} + 12147 \beta_{9} - 16395 \beta_{8} + 5019 \beta_{7} - 4587 \beta_{4} - 11376 \beta_{3} - 9423 \beta_{2} + \cdots - 34858 \) |
\(\nu^{11}\) | \(=\) | \( - 74968 \beta_{15} + 15180 \beta_{14} - 52613 \beta_{13} + 69508 \beta_{12} - 59741 \beta_{11} - 20483 \beta_{9} - 42622 \beta_{8} - 42622 \beta_{7} - 37433 \beta_{6} - 28542 \beta_{5} + \cdots + 79079 \beta_1 \) |
\(\nu^{12}\) | \(=\) | \( 170986 \beta_{15} - 274428 \beta_{14} - 274428 \beta_{13} - 131044 \beta_{11} - 341972 \beta_{10} - 148879 \beta_{9} + 201492 \beta_{8} - 60596 \beta_{7} + 54976 \beta_{4} + 140896 \beta_{3} + \cdots + 423384 \) |
\(\nu^{13}\) | \(=\) | \( 919486 \beta_{15} - 188422 \beta_{14} + 646906 \beta_{13} - 848621 \beta_{12} + 735189 \beta_{11} + 249125 \beta_{9} + 521589 \beta_{8} + 521589 \beta_{7} + 458484 \beta_{6} + 350371 \beta_{5} + \cdots - 965133 \beta_1 \) |
\(\nu^{14}\) | \(=\) | \( - 2095431 \beta_{15} + 3367136 \beta_{14} + 3367136 \beta_{13} + 1604464 \beta_{11} + 4190862 \beta_{10} + 1823791 \beta_{9} - 2470697 \beta_{8} + 738231 \beta_{7} - 667774 \beta_{4} + \cdots - 5168936 \) |
\(\nu^{15}\) | \(=\) | \( - 11266000 \beta_{15} + 2318785 \beta_{14} - 7933264 \beta_{13} + 10380392 \beta_{12} - 9018824 \beta_{11} - 3042952 \beta_{9} - 6385193 \beta_{8} - 6385193 \beta_{7} + \cdots + 11804009 \beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/480\mathbb{Z}\right)^\times\).
\(n\) | \(31\) | \(97\) | \(161\) | \(421\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
239.1 |
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0 | −1.30656 | − | 1.13705i | 0 | −2.10100 | − | 0.765367i | 0 | −2.27411 | 0 | 0.414214 | + | 2.97127i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.2 | 0 | −1.30656 | − | 1.13705i | 0 | 2.10100 | + | 0.765367i | 0 | 2.27411 | 0 | 0.414214 | + | 2.97127i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.3 | 0 | −1.30656 | + | 1.13705i | 0 | −2.10100 | + | 0.765367i | 0 | −2.27411 | 0 | 0.414214 | − | 2.97127i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.4 | 0 | −1.30656 | + | 1.13705i | 0 | 2.10100 | − | 0.765367i | 0 | 2.27411 | 0 | 0.414214 | − | 2.97127i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.5 | 0 | −0.541196 | − | 1.64533i | 0 | −1.25928 | + | 1.84776i | 0 | 3.29066 | 0 | −2.41421 | + | 1.78089i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.6 | 0 | −0.541196 | − | 1.64533i | 0 | 1.25928 | − | 1.84776i | 0 | −3.29066 | 0 | −2.41421 | + | 1.78089i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.7 | 0 | −0.541196 | + | 1.64533i | 0 | −1.25928 | − | 1.84776i | 0 | 3.29066 | 0 | −2.41421 | − | 1.78089i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.8 | 0 | −0.541196 | + | 1.64533i | 0 | 1.25928 | + | 1.84776i | 0 | −3.29066 | 0 | −2.41421 | − | 1.78089i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.9 | 0 | 0.541196 | − | 1.64533i | 0 | −1.25928 | + | 1.84776i | 0 | −3.29066 | 0 | −2.41421 | − | 1.78089i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.10 | 0 | 0.541196 | − | 1.64533i | 0 | 1.25928 | − | 1.84776i | 0 | 3.29066 | 0 | −2.41421 | − | 1.78089i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.11 | 0 | 0.541196 | + | 1.64533i | 0 | −1.25928 | − | 1.84776i | 0 | −3.29066 | 0 | −2.41421 | + | 1.78089i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.12 | 0 | 0.541196 | + | 1.64533i | 0 | 1.25928 | + | 1.84776i | 0 | 3.29066 | 0 | −2.41421 | + | 1.78089i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.13 | 0 | 1.30656 | − | 1.13705i | 0 | −2.10100 | − | 0.765367i | 0 | 2.27411 | 0 | 0.414214 | − | 2.97127i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.14 | 0 | 1.30656 | − | 1.13705i | 0 | 2.10100 | + | 0.765367i | 0 | −2.27411 | 0 | 0.414214 | − | 2.97127i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.15 | 0 | 1.30656 | + | 1.13705i | 0 | −2.10100 | + | 0.765367i | 0 | 2.27411 | 0 | 0.414214 | + | 2.97127i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.16 | 0 | 1.30656 | + | 1.13705i | 0 | 2.10100 | − | 0.765367i | 0 | −2.27411 | 0 | 0.414214 | + | 2.97127i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
24.f | even | 2 | 1 | inner |
40.e | odd | 2 | 1 | inner |
120.m | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 480.2.m.b | 16 | |
3.b | odd | 2 | 1 | inner | 480.2.m.b | 16 | |
4.b | odd | 2 | 1 | 120.2.m.b | ✓ | 16 | |
5.b | even | 2 | 1 | inner | 480.2.m.b | 16 | |
5.c | odd | 4 | 2 | 2400.2.b.i | 16 | ||
8.b | even | 2 | 1 | 120.2.m.b | ✓ | 16 | |
8.d | odd | 2 | 1 | inner | 480.2.m.b | 16 | |
12.b | even | 2 | 1 | 120.2.m.b | ✓ | 16 | |
15.d | odd | 2 | 1 | inner | 480.2.m.b | 16 | |
15.e | even | 4 | 2 | 2400.2.b.i | 16 | ||
20.d | odd | 2 | 1 | 120.2.m.b | ✓ | 16 | |
20.e | even | 4 | 2 | 600.2.b.i | 16 | ||
24.f | even | 2 | 1 | inner | 480.2.m.b | 16 | |
24.h | odd | 2 | 1 | 120.2.m.b | ✓ | 16 | |
40.e | odd | 2 | 1 | inner | 480.2.m.b | 16 | |
40.f | even | 2 | 1 | 120.2.m.b | ✓ | 16 | |
40.i | odd | 4 | 2 | 600.2.b.i | 16 | ||
40.k | even | 4 | 2 | 2400.2.b.i | 16 | ||
60.h | even | 2 | 1 | 120.2.m.b | ✓ | 16 | |
60.l | odd | 4 | 2 | 600.2.b.i | 16 | ||
120.i | odd | 2 | 1 | 120.2.m.b | ✓ | 16 | |
120.m | even | 2 | 1 | inner | 480.2.m.b | 16 | |
120.q | odd | 4 | 2 | 2400.2.b.i | 16 | ||
120.w | even | 4 | 2 | 600.2.b.i | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
120.2.m.b | ✓ | 16 | 4.b | odd | 2 | 1 | |
120.2.m.b | ✓ | 16 | 8.b | even | 2 | 1 | |
120.2.m.b | ✓ | 16 | 12.b | even | 2 | 1 | |
120.2.m.b | ✓ | 16 | 20.d | odd | 2 | 1 | |
120.2.m.b | ✓ | 16 | 24.h | odd | 2 | 1 | |
120.2.m.b | ✓ | 16 | 40.f | even | 2 | 1 | |
120.2.m.b | ✓ | 16 | 60.h | even | 2 | 1 | |
120.2.m.b | ✓ | 16 | 120.i | odd | 2 | 1 | |
480.2.m.b | 16 | 1.a | even | 1 | 1 | trivial | |
480.2.m.b | 16 | 3.b | odd | 2 | 1 | inner | |
480.2.m.b | 16 | 5.b | even | 2 | 1 | inner | |
480.2.m.b | 16 | 8.d | odd | 2 | 1 | inner | |
480.2.m.b | 16 | 15.d | odd | 2 | 1 | inner | |
480.2.m.b | 16 | 24.f | even | 2 | 1 | inner | |
480.2.m.b | 16 | 40.e | odd | 2 | 1 | inner | |
480.2.m.b | 16 | 120.m | even | 2 | 1 | inner | |
600.2.b.i | 16 | 20.e | even | 4 | 2 | ||
600.2.b.i | 16 | 40.i | odd | 4 | 2 | ||
600.2.b.i | 16 | 60.l | odd | 4 | 2 | ||
600.2.b.i | 16 | 120.w | even | 4 | 2 | ||
2400.2.b.i | 16 | 5.c | odd | 4 | 2 | ||
2400.2.b.i | 16 | 15.e | even | 4 | 2 | ||
2400.2.b.i | 16 | 40.k | even | 4 | 2 | ||
2400.2.b.i | 16 | 120.q | odd | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} - 16T_{7}^{2} + 56 \)
acting on \(S_{2}^{\mathrm{new}}(480, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( (T^{8} + 4 T^{6} + 14 T^{4} + 36 T^{2} + \cdots + 81)^{2} \)
$5$
\( (T^{8} - 4 T^{6} + 22 T^{4} - 100 T^{2} + \cdots + 625)^{2} \)
$7$
\( (T^{4} - 16 T^{2} + 56)^{4} \)
$11$
\( (T^{4} + 24 T^{2} + 112)^{4} \)
$13$
\( (T^{4} - 32 T^{2} + 224)^{4} \)
$17$
\( (T^{4} - 16 T^{2} + 32)^{4} \)
$19$
\( (T^{2} - 4 T - 4)^{8} \)
$23$
\( (T^{4} + 8 T^{2} + 8)^{4} \)
$29$
\( (T^{4} - 40 T^{2} + 112)^{4} \)
$31$
\( (T^{4} + 48 T^{2} + 64)^{4} \)
$37$
\( (T^{4} - 64 T^{2} + 224)^{4} \)
$41$
\( (T^{4} + 80 T^{2} + 448)^{4} \)
$43$
\( (T^{4} + 112 T^{2} + 2744)^{4} \)
$47$
\( (T^{4} + 8 T^{2} + 8)^{4} \)
$53$
\( (T^{4} + 144 T^{2} + 2592)^{4} \)
$59$
\( (T^{4} + 24 T^{2} + 112)^{4} \)
$61$
\( (T^{2} + 72)^{8} \)
$67$
\( (T^{4} + 16 T^{2} + 56)^{4} \)
$71$
\( (T^{4} - 192 T^{2} + 7168)^{4} \)
$73$
\( (T^{4} + 64 T^{2} + 896)^{4} \)
$79$
\( (T^{4} + 272 T^{2} + 64)^{4} \)
$83$
\( (T^{4} - 136 T^{2} + 4232)^{4} \)
$89$
\( (T^{4} + 96 T^{2} + 1792)^{4} \)
$97$
\( (T^{4} + 128 T^{2} + 896)^{4} \)
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