Properties

Label 315.2.m
Level 315
Weight 2
Character orbit m
Rep. character \(\chi_{315}(8,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 24
Newform subspaces 2
Sturm bound 96
Trace bound 5

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.m (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 15 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(96\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(315, [\chi])\).

Total New Old
Modular forms 112 24 88
Cusp forms 80 24 56
Eisenstein series 32 0 32

Trace form

\( 24q + O(q^{10}) \) \( 24q + 32q^{10} - 8q^{13} - 40q^{16} - 16q^{22} - 16q^{25} + 8q^{37} - 56q^{40} + 80q^{43} + 32q^{46} + 72q^{52} - 32q^{55} - 56q^{58} - 64q^{61} - 96q^{67} + 16q^{70} - 40q^{73} + 32q^{76} + 40q^{82} + 24q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(315, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
315.2.m.a \(12\) \(2.515\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(-4\) \(0\) \(q+\beta _{2}q^{2}+(\beta _{5}+\beta _{6}-\beta _{8})q^{4}+(-\beta _{3}+\cdots)q^{5}+\cdots\)
315.2.m.b \(12\) \(2.515\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(4\) \(0\) \(q+(\beta _{1}+\beta _{8}+\beta _{9})q^{2}+(-\beta _{5}-\beta _{6}+\beta _{8}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(315, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(315, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 8 T^{3} - 5 T^{4} + 32 T^{6} + 32 T^{7} - 5 T^{8} - 80 T^{9} - 96 T^{10} + 72 T^{11} + 193 T^{12} + 144 T^{13} - 384 T^{14} - 640 T^{15} - 80 T^{16} + 1024 T^{17} + 2048 T^{18} - 1280 T^{20} - 4096 T^{21} + 4096 T^{24} \))(\( 1 + 8 T^{3} - 5 T^{4} + 32 T^{6} - 32 T^{7} - 5 T^{8} + 80 T^{9} - 96 T^{10} - 72 T^{11} + 193 T^{12} - 144 T^{13} - 384 T^{14} + 640 T^{15} - 80 T^{16} - 1024 T^{17} + 2048 T^{18} - 1280 T^{20} + 4096 T^{21} + 4096 T^{24} \))
$3$ 1
$5$ (\( 1 + 4 T + 12 T^{2} + 12 T^{3} - 5 T^{4} - 160 T^{5} - 408 T^{6} - 800 T^{7} - 125 T^{8} + 1500 T^{9} + 7500 T^{10} + 12500 T^{11} + 15625 T^{12} \))(\( 1 - 4 T + 12 T^{2} - 12 T^{3} - 5 T^{4} + 160 T^{5} - 408 T^{6} + 800 T^{7} - 125 T^{8} - 1500 T^{9} + 7500 T^{10} - 12500 T^{11} + 15625 T^{12} \))
$7$ (\( ( 1 + T^{4} )^{3} \))(\( ( 1 + T^{4} )^{3} \))
$11$ (\( 1 - 52 T^{2} + 1458 T^{4} - 30084 T^{6} + 494911 T^{8} - 6805160 T^{10} + 80542716 T^{12} - 823424360 T^{14} + 7245991951 T^{16} - 53295641124 T^{18} + 312535248498 T^{20} - 1348746079252 T^{22} + 3138428376721 T^{24} \))(\( 1 - 52 T^{2} + 1458 T^{4} - 30084 T^{6} + 494911 T^{8} - 6805160 T^{10} + 80542716 T^{12} - 823424360 T^{14} + 7245991951 T^{16} - 53295641124 T^{18} + 312535248498 T^{20} - 1348746079252 T^{22} + 3138428376721 T^{24} \))
$13$ (\( 1 + 4 T + 8 T^{2} + 108 T^{3} + 298 T^{4} + 204 T^{5} + 4264 T^{6} + 17924 T^{7} + 42687 T^{8} + 200904 T^{9} + 1148048 T^{10} + 3595224 T^{11} + 7337420 T^{12} + 46737912 T^{13} + 194020112 T^{14} + 441386088 T^{15} + 1219183407 T^{16} + 6655055732 T^{17} + 20581513576 T^{18} + 12800697468 T^{19} + 243087754858 T^{20} + 1145285932284 T^{21} + 1102867934792 T^{22} + 7168641576148 T^{23} + 23298085122481 T^{24} \))(\( 1 + 4 T + 8 T^{2} + 108 T^{3} + 298 T^{4} + 204 T^{5} + 4264 T^{6} + 17924 T^{7} + 42687 T^{8} + 200904 T^{9} + 1148048 T^{10} + 3595224 T^{11} + 7337420 T^{12} + 46737912 T^{13} + 194020112 T^{14} + 441386088 T^{15} + 1219183407 T^{16} + 6655055732 T^{17} + 20581513576 T^{18} + 12800697468 T^{19} + 243087754858 T^{20} + 1145285932284 T^{21} + 1102867934792 T^{22} + 7168641576148 T^{23} + 23298085122481 T^{24} \))
$17$ (\( 1 - 8 T + 32 T^{2} - 232 T^{3} + 1238 T^{4} - 3432 T^{5} + 14752 T^{6} - 67976 T^{7} + 254479 T^{8} - 1447664 T^{9} + 6834752 T^{10} - 33133872 T^{11} + 165039860 T^{12} - 563275824 T^{13} + 1975243328 T^{14} - 7112373232 T^{15} + 21254340559 T^{16} - 96516199432 T^{17} + 356077417888 T^{18} - 1408282325736 T^{19} + 8635987711958 T^{20} - 27512387347304 T^{21} + 64511804814368 T^{22} - 274175170461064 T^{23} + 582622237229761 T^{24} \))(\( 1 + 8 T + 32 T^{2} + 232 T^{3} + 1238 T^{4} + 3432 T^{5} + 14752 T^{6} + 67976 T^{7} + 254479 T^{8} + 1447664 T^{9} + 6834752 T^{10} + 33133872 T^{11} + 165039860 T^{12} + 563275824 T^{13} + 1975243328 T^{14} + 7112373232 T^{15} + 21254340559 T^{16} + 96516199432 T^{17} + 356077417888 T^{18} + 1408282325736 T^{19} + 8635987711958 T^{20} + 27512387347304 T^{21} + 64511804814368 T^{22} + 274175170461064 T^{23} + 582622237229761 T^{24} \))
$19$ (\( 1 - 84 T^{2} + 4642 T^{4} - 179332 T^{6} + 5560223 T^{8} - 138383432 T^{10} + 2894445596 T^{12} - 49956418952 T^{14} + 724613821583 T^{16} - 8436831931492 T^{18} + 78837699636322 T^{20} - 515009565655284 T^{22} + 2213314919066161 T^{24} \))(\( 1 - 84 T^{2} + 4642 T^{4} - 179332 T^{6} + 5560223 T^{8} - 138383432 T^{10} + 2894445596 T^{12} - 49956418952 T^{14} + 724613821583 T^{16} - 8436831931492 T^{18} + 78837699636322 T^{20} - 515009565655284 T^{22} + 2213314919066161 T^{24} \))
$23$ (\( 1 - 8 T + 32 T^{2} - 184 T^{3} + 518 T^{4} + 1752 T^{5} - 13664 T^{6} + 145640 T^{7} - 745937 T^{8} - 382288 T^{9} + 8743232 T^{10} - 77776560 T^{11} + 600339476 T^{12} - 1788860880 T^{13} + 4625169728 T^{14} - 4651298096 T^{15} - 208743756017 T^{16} + 937388994520 T^{17} - 2022762387296 T^{18} + 5965254183144 T^{19} + 40565090375558 T^{20} - 331412089709192 T^{21} + 1325648358836768 T^{22} - 7622478063311416 T^{23} + 21914624432020321 T^{24} \))(\( 1 + 8 T + 32 T^{2} + 184 T^{3} + 518 T^{4} - 1752 T^{5} - 13664 T^{6} - 145640 T^{7} - 745937 T^{8} + 382288 T^{9} + 8743232 T^{10} + 77776560 T^{11} + 600339476 T^{12} + 1788860880 T^{13} + 4625169728 T^{14} + 4651298096 T^{15} - 208743756017 T^{16} - 937388994520 T^{17} - 2022762387296 T^{18} - 5965254183144 T^{19} + 40565090375558 T^{20} + 331412089709192 T^{21} + 1325648358836768 T^{22} + 7622478063311416 T^{23} + 21914624432020321 T^{24} \))
$29$ (\( ( 1 + 16 T + 216 T^{2} + 1904 T^{3} + 15483 T^{4} + 98368 T^{5} + 589488 T^{6} + 2852672 T^{7} + 13021203 T^{8} + 46436656 T^{9} + 152772696 T^{10} + 328178384 T^{11} + 594823321 T^{12} )^{2} \))(\( ( 1 - 16 T + 216 T^{2} - 1904 T^{3} + 15483 T^{4} - 98368 T^{5} + 589488 T^{6} - 2852672 T^{7} + 13021203 T^{8} - 46436656 T^{9} + 152772696 T^{10} - 328178384 T^{11} + 594823321 T^{12} )^{2} \))
$31$ (\( ( 1 + 114 T^{2} - 168 T^{3} + 6583 T^{4} - 10856 T^{5} + 253628 T^{6} - 336536 T^{7} + 6326263 T^{8} - 5004888 T^{9} + 105281394 T^{10} + 887503681 T^{12} )^{2} \))(\( ( 1 + 114 T^{2} - 168 T^{3} + 6583 T^{4} - 10856 T^{5} + 253628 T^{6} - 336536 T^{7} + 6326263 T^{8} - 5004888 T^{9} + 105281394 T^{10} + 887503681 T^{12} )^{2} \))
$37$ (\( 1 - 4 T + 8 T^{2} - 28 T^{3} + 122 T^{4} - 124 T^{5} - 88 T^{6} + 102780 T^{7} - 827937 T^{8} + 2005016 T^{9} - 4255984 T^{10} + 201457960 T^{11} - 3018818132 T^{12} + 7453944520 T^{13} - 5826442096 T^{14} + 101560075448 T^{15} - 1551687235857 T^{16} + 7127171900460 T^{17} - 225783923992 T^{18} - 11771552764492 T^{19} + 428522493378362 T^{20} - 3638928714262156 T^{21} + 38468674979342792 T^{22} - 711670487117841652 T^{23} + 6582952005840035281 T^{24} \))(\( 1 - 4 T + 8 T^{2} - 28 T^{3} + 122 T^{4} - 124 T^{5} - 88 T^{6} + 102780 T^{7} - 827937 T^{8} + 2005016 T^{9} - 4255984 T^{10} + 201457960 T^{11} - 3018818132 T^{12} + 7453944520 T^{13} - 5826442096 T^{14} + 101560075448 T^{15} - 1551687235857 T^{16} + 7127171900460 T^{17} - 225783923992 T^{18} - 11771552764492 T^{19} + 428522493378362 T^{20} - 3638928714262156 T^{21} + 38468674979342792 T^{22} - 711670487117841652 T^{23} + 6582952005840035281 T^{24} \))
$41$ (\( 1 - 232 T^{2} + 30646 T^{4} - 2820520 T^{6} + 198407727 T^{8} - 11090030832 T^{10} + 503431026356 T^{12} - 18642341828592 T^{14} + 560652817055247 T^{16} - 13397764013825320 T^{18} + 244706018571642166 T^{20} - 3114056959955357032 T^{22} + 22563490300366186081 T^{24} \))(\( 1 - 232 T^{2} + 30646 T^{4} - 2820520 T^{6} + 198407727 T^{8} - 11090030832 T^{10} + 503431026356 T^{12} - 18642341828592 T^{14} + 560652817055247 T^{16} - 13397764013825320 T^{18} + 244706018571642166 T^{20} - 3114056959955357032 T^{22} + 22563490300366186081 T^{24} \))
$43$ (\( 1 - 40 T + 800 T^{2} - 11608 T^{3} + 144630 T^{4} - 1609192 T^{5} + 16036512 T^{6} - 146487000 T^{7} + 1250458847 T^{8} - 9986195856 T^{9} + 74890568512 T^{10} - 532487979120 T^{11} + 3591291202036 T^{12} - 22896983102160 T^{13} + 138472661178688 T^{14} - 793972473922992 T^{15} + 4275069956582447 T^{16} - 21534825789741000 T^{17} + 101372614391645088 T^{18} - 437408334444495544 T^{19} + 1690464406149432630 T^{20} - 5834095039362873544 T^{21} + 17289185850627399200 T^{22} - 37171749578848908280 T^{23} + 39959630797262576401 T^{24} \))(\( 1 - 40 T + 800 T^{2} - 11608 T^{3} + 144630 T^{4} - 1609192 T^{5} + 16036512 T^{6} - 146487000 T^{7} + 1250458847 T^{8} - 9986195856 T^{9} + 74890568512 T^{10} - 532487979120 T^{11} + 3591291202036 T^{12} - 22896983102160 T^{13} + 138472661178688 T^{14} - 793972473922992 T^{15} + 4275069956582447 T^{16} - 21534825789741000 T^{17} + 101372614391645088 T^{18} - 437408334444495544 T^{19} + 1690464406149432630 T^{20} - 5834095039362873544 T^{21} + 17289185850627399200 T^{22} - 37171749578848908280 T^{23} + 39959630797262576401 T^{24} \))
$47$ (\( 1 - 24 T + 288 T^{2} - 2920 T^{3} + 30374 T^{4} - 283512 T^{5} + 2319776 T^{6} - 18683912 T^{7} + 148359759 T^{8} - 1083670000 T^{9} + 7566143296 T^{10} - 53830728976 T^{11} + 378456487636 T^{12} - 2530044261872 T^{13} + 16713610540864 T^{14} - 112509870410000 T^{15} + 723948297156879 T^{16} - 4285061928427384 T^{17} + 25005365019046304 T^{18} - 143633734128706056 T^{19} + 723244021064328614 T^{20} - 3267860981460079640 T^{21} + 15148550083919054112 T^{22} - 59331821162016295272 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 + 24 T + 288 T^{2} + 2920 T^{3} + 30374 T^{4} + 283512 T^{5} + 2319776 T^{6} + 18683912 T^{7} + 148359759 T^{8} + 1083670000 T^{9} + 7566143296 T^{10} + 53830728976 T^{11} + 378456487636 T^{12} + 2530044261872 T^{13} + 16713610540864 T^{14} + 112509870410000 T^{15} + 723948297156879 T^{16} + 4285061928427384 T^{17} + 25005365019046304 T^{18} + 143633734128706056 T^{19} + 723244021064328614 T^{20} + 3267860981460079640 T^{21} + 15148550083919054112 T^{22} + 59331821162016295272 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))
$53$ (\( 1 + 40 T + 800 T^{2} + 11880 T^{3} + 147222 T^{4} + 1477736 T^{5} + 11899040 T^{6} + 74456936 T^{7} + 249326879 T^{8} - 1557249200 T^{9} - 37151695552 T^{10} - 419908594096 T^{11} - 3501062916044 T^{12} - 22255155487088 T^{13} - 104359112805568 T^{14} - 231838589148400 T^{15} + 1967309001538799 T^{16} + 31137555057789448 T^{17} + 263734619648416160 T^{18} + 1735912940938169032 T^{19} + 9165996141741389142 T^{20} + 39201191470609340040 T^{21} + \)\(13\!\cdots\!00\)\( T^{22} + \)\(37\!\cdots\!80\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 - 40 T + 800 T^{2} - 11880 T^{3} + 147222 T^{4} - 1477736 T^{5} + 11899040 T^{6} - 74456936 T^{7} + 249326879 T^{8} + 1557249200 T^{9} - 37151695552 T^{10} + 419908594096 T^{11} - 3501062916044 T^{12} + 22255155487088 T^{13} - 104359112805568 T^{14} + 231838589148400 T^{15} + 1967309001538799 T^{16} - 31137555057789448 T^{17} + 263734619648416160 T^{18} - 1735912940938169032 T^{19} + 9165996141741389142 T^{20} - 39201191470609340040 T^{21} + \)\(13\!\cdots\!00\)\( T^{22} - \)\(37\!\cdots\!80\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))
$59$ (\( ( 1 + 40 T + 986 T^{2} + 16824 T^{3} + 222455 T^{4} + 2325552 T^{5} + 19830604 T^{6} + 137207568 T^{7} + 774365855 T^{8} + 3455296296 T^{9} + 11947717946 T^{10} + 28596971960 T^{11} + 42180533641 T^{12} )^{2} \))(\( ( 1 - 40 T + 986 T^{2} - 16824 T^{3} + 222455 T^{4} - 2325552 T^{5} + 19830604 T^{6} - 137207568 T^{7} + 774365855 T^{8} - 3455296296 T^{9} + 11947717946 T^{10} - 28596971960 T^{11} + 42180533641 T^{12} )^{2} \))
$61$ (\( ( 1 + 16 T + 254 T^{2} + 2800 T^{3} + 31591 T^{4} + 270400 T^{5} + 2316548 T^{6} + 16494400 T^{7} + 117550111 T^{8} + 635546800 T^{9} + 3516843614 T^{10} + 13513540816 T^{11} + 51520374361 T^{12} )^{2} \))(\( ( 1 + 16 T + 254 T^{2} + 2800 T^{3} + 31591 T^{4} + 270400 T^{5} + 2316548 T^{6} + 16494400 T^{7} + 117550111 T^{8} + 635546800 T^{9} + 3516843614 T^{10} + 13513540816 T^{11} + 51520374361 T^{12} )^{2} \))
$67$ (\( 1 + 48 T + 1152 T^{2} + 18992 T^{3} + 248566 T^{4} + 2816976 T^{5} + 29214848 T^{6} + 287145808 T^{7} + 2747657919 T^{8} + 25947593440 T^{9} + 239514732288 T^{10} + 2122341535712 T^{11} + 17856126120308 T^{12} + 142196882892704 T^{13} + 1075181633240832 T^{14} + 7804076045794720 T^{15} + 55368387192377199 T^{16} + 387682764750601456 T^{17} + 2642727885393245312 T^{18} + 17072879135116363248 T^{19} + \)\(10\!\cdots\!06\)\( T^{20} + \)\(51\!\cdots\!24\)\( T^{21} + \)\(20\!\cdots\!48\)\( T^{22} + \)\(58\!\cdots\!84\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( 1 + 48 T + 1152 T^{2} + 18992 T^{3} + 248566 T^{4} + 2816976 T^{5} + 29214848 T^{6} + 287145808 T^{7} + 2747657919 T^{8} + 25947593440 T^{9} + 239514732288 T^{10} + 2122341535712 T^{11} + 17856126120308 T^{12} + 142196882892704 T^{13} + 1075181633240832 T^{14} + 7804076045794720 T^{15} + 55368387192377199 T^{16} + 387682764750601456 T^{17} + 2642727885393245312 T^{18} + 17072879135116363248 T^{19} + \)\(10\!\cdots\!06\)\( T^{20} + \)\(51\!\cdots\!24\)\( T^{21} + \)\(20\!\cdots\!48\)\( T^{22} + \)\(58\!\cdots\!84\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))
$71$ (\( 1 - 404 T^{2} + 75490 T^{4} - 8307268 T^{6} + 571593487 T^{8} - 25299911144 T^{10} + 1141503138780 T^{12} - 127536852076904 T^{14} + 14525151353321647 T^{16} - 1064163389407837828 T^{18} + 48747934073742497890 T^{20} - \)\(13\!\cdots\!04\)\( T^{22} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( 1 - 404 T^{2} + 75490 T^{4} - 8307268 T^{6} + 571593487 T^{8} - 25299911144 T^{10} + 1141503138780 T^{12} - 127536852076904 T^{14} + 14525151353321647 T^{16} - 1064163389407837828 T^{18} + 48747934073742497890 T^{20} - \)\(13\!\cdots\!04\)\( T^{22} + \)\(16\!\cdots\!41\)\( T^{24} \))
$73$ (\( 1 + 20 T + 200 T^{2} + 1452 T^{3} + 7482 T^{4} + 35804 T^{5} + 273832 T^{6} + 2000484 T^{7} + 27883951 T^{8} + 188107816 T^{9} - 317778928 T^{10} - 34324729064 T^{11} - 473910414484 T^{12} - 2505705221672 T^{13} - 1693443907312 T^{14} + 73177138256872 T^{15} + 791855160530191 T^{16} + 4147146552651012 T^{17} + 41440153853169448 T^{18} + 395541056577748988 T^{19} + 6033934407551514042 T^{20} + 85481543900405009676 T^{21} + \)\(85\!\cdots\!00\)\( T^{22} + \)\(62\!\cdots\!40\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( 1 + 20 T + 200 T^{2} + 1452 T^{3} + 7482 T^{4} + 35804 T^{5} + 273832 T^{6} + 2000484 T^{7} + 27883951 T^{8} + 188107816 T^{9} - 317778928 T^{10} - 34324729064 T^{11} - 473910414484 T^{12} - 2505705221672 T^{13} - 1693443907312 T^{14} + 73177138256872 T^{15} + 791855160530191 T^{16} + 4147146552651012 T^{17} + 41440153853169448 T^{18} + 395541056577748988 T^{19} + 6033934407551514042 T^{20} + 85481543900405009676 T^{21} + \)\(85\!\cdots\!00\)\( T^{22} + \)\(62\!\cdots\!40\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))
$79$ (\( 1 - 372 T^{2} + 73090 T^{4} - 9878564 T^{6} + 1031952175 T^{8} - 90900376040 T^{10} + 7353042693660 T^{12} - 567309246865640 T^{14} + 40194620804376175 T^{16} - 2401354986961351844 T^{18} + \)\(11\!\cdots\!90\)\( T^{20} - \)\(35\!\cdots\!72\)\( T^{22} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( 1 - 372 T^{2} + 73090 T^{4} - 9878564 T^{6} + 1031952175 T^{8} - 90900376040 T^{10} + 7353042693660 T^{12} - 567309246865640 T^{14} + 40194620804376175 T^{16} - 2401354986961351844 T^{18} + \)\(11\!\cdots\!90\)\( T^{20} - \)\(35\!\cdots\!72\)\( T^{22} + \)\(59\!\cdots\!41\)\( T^{24} \))
$83$ (\( 1 - 24 T + 288 T^{2} - 2760 T^{3} + 26518 T^{4} - 273208 T^{5} + 2728608 T^{6} - 24159272 T^{7} + 240213375 T^{8} - 2803489520 T^{9} + 29745965888 T^{10} - 283235701200 T^{11} + 2619914616372 T^{12} - 23508563199600 T^{13} + 204919959002432 T^{14} - 1602998862172240 T^{15} + 11400123459243375 T^{16} - 95164354313291896 T^{17} + 892092118297640352 T^{18} - 7413786218774013416 T^{19} + 59726285411863089238 T^{20} - \)\(51\!\cdots\!80\)\( T^{21} + \)\(44\!\cdots\!12\)\( T^{22} - \)\(30\!\cdots\!08\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( 1 + 24 T + 288 T^{2} + 2760 T^{3} + 26518 T^{4} + 273208 T^{5} + 2728608 T^{6} + 24159272 T^{7} + 240213375 T^{8} + 2803489520 T^{9} + 29745965888 T^{10} + 283235701200 T^{11} + 2619914616372 T^{12} + 23508563199600 T^{13} + 204919959002432 T^{14} + 1602998862172240 T^{15} + 11400123459243375 T^{16} + 95164354313291896 T^{17} + 892092118297640352 T^{18} + 7413786218774013416 T^{19} + 59726285411863089238 T^{20} + \)\(51\!\cdots\!80\)\( T^{21} + \)\(44\!\cdots\!12\)\( T^{22} + \)\(30\!\cdots\!08\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))
$89$ (\( ( 1 + 28 T + 684 T^{2} + 11188 T^{3} + 161827 T^{4} + 1876496 T^{5} + 19427016 T^{6} + 167008144 T^{7} + 1281831667 T^{8} + 7887193172 T^{9} + 42915692844 T^{10} + 156353664572 T^{11} + 496981290961 T^{12} )^{2} \))(\( ( 1 - 28 T + 684 T^{2} - 11188 T^{3} + 161827 T^{4} - 1876496 T^{5} + 19427016 T^{6} - 167008144 T^{7} + 1281831667 T^{8} - 7887193172 T^{9} + 42915692844 T^{10} - 156353664572 T^{11} + 496981290961 T^{12} )^{2} \))
$97$ (\( 1 - 12 T + 72 T^{2} - 884 T^{3} + 15450 T^{4} - 102628 T^{5} + 509864 T^{6} - 1529948 T^{7} - 62272049 T^{8} + 410787880 T^{9} - 1704538608 T^{10} + 69640900504 T^{11} - 1520682376276 T^{12} + 6755167348888 T^{13} - 16038003762672 T^{14} + 374915006803240 T^{15} - 5512899724366769 T^{16} - 13138184051516636 T^{17} + 424702438321119656 T^{18} - 8292166339419780964 T^{19} + \)\(12\!\cdots\!50\)\( T^{20} - \)\(67\!\cdots\!28\)\( T^{21} + \)\(53\!\cdots\!28\)\( T^{22} - \)\(85\!\cdots\!36\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( 1 - 12 T + 72 T^{2} - 884 T^{3} + 15450 T^{4} - 102628 T^{5} + 509864 T^{6} - 1529948 T^{7} - 62272049 T^{8} + 410787880 T^{9} - 1704538608 T^{10} + 69640900504 T^{11} - 1520682376276 T^{12} + 6755167348888 T^{13} - 16038003762672 T^{14} + 374915006803240 T^{15} - 5512899724366769 T^{16} - 13138184051516636 T^{17} + 424702438321119656 T^{18} - 8292166339419780964 T^{19} + \)\(12\!\cdots\!50\)\( T^{20} - \)\(67\!\cdots\!28\)\( T^{21} + \)\(53\!\cdots\!28\)\( T^{22} - \)\(85\!\cdots\!36\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))
show more
show less