Properties

Label 175.2.e
Level 175
Weight 2
Character orbit e
Rep. character \(\chi_{175}(51,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 20
Newform subspaces 5
Sturm bound 40
Trace bound 2

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Defining parameters

Level: \( N \) = \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 175.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 5 \)
Sturm bound: \(40\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(175, [\chi])\).

Total New Old
Modular forms 52 32 20
Cusp forms 28 20 8
Eisenstein series 24 12 12

Trace form

\( 20q + 2q^{2} + 2q^{3} - 6q^{4} - 16q^{6} - 2q^{7} - 12q^{8} - 4q^{9} + O(q^{10}) \) \( 20q + 2q^{2} + 2q^{3} - 6q^{4} - 16q^{6} - 2q^{7} - 12q^{8} - 4q^{9} - 6q^{12} + 8q^{13} + 22q^{14} + 6q^{16} + 4q^{17} - 8q^{18} - 4q^{19} - 18q^{21} + 8q^{22} - 2q^{23} + 12q^{24} - 22q^{26} - 4q^{27} + 22q^{28} - 8q^{29} - 2q^{31} - 6q^{32} + 12q^{33} - 16q^{34} + 36q^{36} + 8q^{38} - 24q^{39} - 4q^{41} + 2q^{42} - 20q^{43} + 24q^{46} + 4q^{47} - 12q^{48} + 8q^{49} - 14q^{51} - 20q^{52} - 8q^{53} + 38q^{54} - 18q^{56} + 16q^{57} - 2q^{58} - 2q^{59} - 18q^{61} + 24q^{62} - 64q^{64} - 48q^{66} + 22q^{67} + 20q^{68} + 48q^{69} + 4q^{71} - 8q^{72} + 4q^{73} + 46q^{74} + 68q^{76} - 28q^{77} - 8q^{78} - 6q^{79} + 26q^{81} - 18q^{82} - 4q^{83} - 22q^{84} + 40q^{86} - 2q^{87} + 4q^{88} - 38q^{91} - 12q^{92} - 12q^{93} + 30q^{94} - 14q^{96} - 24q^{97} + 14q^{98} - 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(175, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
175.2.e.a \(2\) \(1.397\) \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(0\) \(-1\) \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
175.2.e.b \(2\) \(1.397\) \(\Q(\sqrt{-3}) \) None \(1\) \(-1\) \(0\) \(1\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
175.2.e.c \(4\) \(1.397\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(2\) \(2\) \(0\) \(-2\) \(q+(1+\beta _{1}+\beta _{2})q^{2}+(\beta _{1}-\beta _{2}+\beta _{3})q^{3}+\cdots\)
175.2.e.d \(6\) \(1.397\) 6.0.1783323.2 None \(-1\) \(3\) \(0\) \(2\) \(q-\beta _{1}q^{2}+(\beta _{3}+\beta _{4}-\beta _{5})q^{3}+(-\beta _{3}+\cdots)q^{4}+\cdots\)
175.2.e.e \(6\) \(1.397\) 6.0.1783323.2 None \(1\) \(-3\) \(0\) \(-2\) \(q+\beta _{1}q^{2}+(-\beta _{3}-\beta _{4}+\beta _{5})q^{3}+(-\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(175, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(175, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 - T - T^{2} - 2 T^{3} + 4 T^{4} \))(\( 1 - 2 T + T^{2} + 2 T^{3} - 3 T^{4} + 4 T^{5} + 4 T^{6} - 16 T^{7} + 16 T^{8} \))(\( 1 + T - T^{2} - T^{4} - 2 T^{5} + 5 T^{6} - 4 T^{7} - 4 T^{8} - 16 T^{10} + 32 T^{11} + 64 T^{12} \))(\( 1 - T - T^{2} - T^{4} + 2 T^{5} + 5 T^{6} + 4 T^{7} - 4 T^{8} - 16 T^{10} - 32 T^{11} + 64 T^{12} \))
$3$ (\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 6 T^{5} - 9 T^{6} - 54 T^{7} + 81 T^{8} \))(\( 1 - 3 T + 2 T^{2} + T^{3} - 5 T^{4} + 22 T^{5} - 53 T^{6} + 66 T^{7} - 45 T^{8} + 27 T^{9} + 162 T^{10} - 729 T^{11} + 729 T^{12} \))(\( 1 + 3 T + 2 T^{2} - T^{3} - 5 T^{4} - 22 T^{5} - 53 T^{6} - 66 T^{7} - 45 T^{8} - 27 T^{9} + 162 T^{10} + 729 T^{11} + 729 T^{12} \))
$5$ 1
$7$ (\( 1 + T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \))(\( 1 - 2 T - 4 T^{2} + 13 T^{3} - 28 T^{4} - 98 T^{5} + 343 T^{6} \))(\( 1 + 2 T - 4 T^{2} - 13 T^{3} - 28 T^{4} + 98 T^{5} + 343 T^{6} \))
$11$ (\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 + 4 T - 2 T^{2} - 16 T^{3} + 27 T^{4} - 176 T^{5} - 242 T^{6} + 5324 T^{7} + 14641 T^{8} \))(\( 1 - 2 T - 10 T^{2} - 30 T^{3} + 44 T^{4} + 388 T^{5} + 23 T^{6} + 4268 T^{7} + 5324 T^{8} - 39930 T^{9} - 146410 T^{10} - 322102 T^{11} + 1771561 T^{12} \))(\( 1 - 2 T - 10 T^{2} - 30 T^{3} + 44 T^{4} + 388 T^{5} + 23 T^{6} + 4268 T^{7} + 5324 T^{8} - 39930 T^{9} - 146410 T^{10} - 322102 T^{11} + 1771561 T^{12} \))
$13$ (\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 4 T + 22 T^{2} - 52 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 4 T + 24 T^{2} + 111 T^{3} + 312 T^{4} + 676 T^{5} + 2197 T^{6} )^{2} \))(\( ( 1 - 4 T + 24 T^{2} - 111 T^{3} + 312 T^{4} - 676 T^{5} + 2197 T^{6} )^{2} \))
$17$ (\( 1 + 2 T - 13 T^{2} + 34 T^{3} + 289 T^{4} \))(\( 1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4} \))(\( 1 - 4 T - 14 T^{2} + 16 T^{3} + 339 T^{4} + 272 T^{5} - 4046 T^{6} - 19652 T^{7} + 83521 T^{8} \))(\( 1 - 10 T + 20 T^{2} - 78 T^{3} + 1430 T^{4} - 4570 T^{5} + 2447 T^{6} - 77690 T^{7} + 413270 T^{8} - 383214 T^{9} + 1670420 T^{10} - 14198570 T^{11} + 24137569 T^{12} \))(\( 1 + 10 T + 20 T^{2} + 78 T^{3} + 1430 T^{4} + 4570 T^{5} + 2447 T^{6} + 77690 T^{7} + 413270 T^{8} + 383214 T^{9} + 1670420 T^{10} + 14198570 T^{11} + 24137569 T^{12} \))
$19$ (\( 1 + 6 T + 17 T^{2} + 114 T^{3} + 361 T^{4} \))(\( 1 + 6 T + 17 T^{2} + 114 T^{3} + 361 T^{4} \))(\( 1 - 30 T^{2} + 539 T^{4} - 10830 T^{6} + 130321 T^{8} \))(\( 1 - 4 T - 24 T^{2} + 158 T^{3} + 204 T^{4} - 1724 T^{5} + 2823 T^{6} - 32756 T^{7} + 73644 T^{8} + 1083722 T^{9} - 3127704 T^{10} - 9904396 T^{11} + 47045881 T^{12} \))(\( 1 - 4 T - 24 T^{2} + 158 T^{3} + 204 T^{4} - 1724 T^{5} + 2823 T^{6} - 32756 T^{7} + 73644 T^{8} + 1083722 T^{9} - 3127704 T^{10} - 9904396 T^{11} + 47045881 T^{12} \))
$23$ (\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 + 2 T - 41 T^{2} - 2 T^{3} + 1404 T^{4} - 46 T^{5} - 21689 T^{6} + 24334 T^{7} + 279841 T^{8} \))(\( 1 - T - 58 T^{2} + 51 T^{3} + 2069 T^{4} - 1060 T^{5} - 53233 T^{6} - 24380 T^{7} + 1094501 T^{8} + 620517 T^{9} - 16230778 T^{10} - 6436343 T^{11} + 148035889 T^{12} \))(\( 1 + T - 58 T^{2} - 51 T^{3} + 2069 T^{4} + 1060 T^{5} - 53233 T^{6} + 24380 T^{7} + 1094501 T^{8} - 620517 T^{9} - 16230778 T^{10} + 6436343 T^{11} + 148035889 T^{12} \))
$29$ (\( ( 1 - 7 T + 29 T^{2} )^{2} \))(\( ( 1 - 7 T + 29 T^{2} )^{2} \))(\( ( 1 + T + 29 T^{2} )^{4} \))(\( ( 1 + 8 T + 82 T^{2} + 389 T^{3} + 2378 T^{4} + 6728 T^{5} + 24389 T^{6} )^{2} \))(\( ( 1 + 8 T + 82 T^{2} + 389 T^{3} + 2378 T^{4} + 6728 T^{5} + 24389 T^{6} )^{2} \))
$31$ (\( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} \))(\( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} \))(\( ( 1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4} )^{2} \))(\( 1 + 5 T - 50 T^{2} - 119 T^{3} + 2065 T^{4} + 80 T^{5} - 77809 T^{6} + 2480 T^{7} + 1984465 T^{8} - 3545129 T^{9} - 46176050 T^{10} + 143145755 T^{11} + 887503681 T^{12} \))(\( 1 + 5 T - 50 T^{2} - 119 T^{3} + 2065 T^{4} + 80 T^{5} - 77809 T^{6} + 2480 T^{7} + 1984465 T^{8} - 3545129 T^{9} - 46176050 T^{10} + 143145755 T^{11} + 887503681 T^{12} \))
$37$ (\( 1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4} \))(\( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} \))(\( ( 1 - 37 T^{2} + 1369 T^{4} )^{2} \))(\( 1 + 10 T - 20 T^{2} - 118 T^{3} + 3370 T^{4} + 3970 T^{5} - 134113 T^{6} + 146890 T^{7} + 4613530 T^{8} - 5977054 T^{9} - 37483220 T^{10} + 693439570 T^{11} + 2565726409 T^{12} \))(\( 1 - 10 T - 20 T^{2} + 118 T^{3} + 3370 T^{4} - 3970 T^{5} - 134113 T^{6} - 146890 T^{7} + 4613530 T^{8} + 5977054 T^{9} - 37483220 T^{10} - 693439570 T^{11} + 2565726409 T^{12} \))
$41$ (\( ( 1 - 5 T + 41 T^{2} )^{2} \))(\( ( 1 - 5 T + 41 T^{2} )^{2} \))(\( ( 1 + 10 T + 99 T^{2} + 410 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + T + 77 T^{2} + 187 T^{3} + 3157 T^{4} + 1681 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 + T + 77 T^{2} + 187 T^{3} + 3157 T^{4} + 1681 T^{5} + 68921 T^{6} )^{2} \))
$43$ (\( ( 1 - 7 T + 43 T^{2} )^{2} \))(\( ( 1 + 7 T + 43 T^{2} )^{2} \))(\( ( 1 + 10 T + 109 T^{2} + 430 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 9 T + 109 T^{2} + 721 T^{3} + 4687 T^{4} + 16641 T^{5} + 79507 T^{6} )^{2} \))(\( ( 1 - 9 T + 109 T^{2} - 721 T^{3} + 4687 T^{4} - 16641 T^{5} + 79507 T^{6} )^{2} \))
$47$ (\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( ( 1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4} )^{2} \))(\( 1 - 28 T + 390 T^{2} - 4298 T^{3} + 42442 T^{4} - 351442 T^{5} + 2516387 T^{6} - 16517774 T^{7} + 93754378 T^{8} - 446231254 T^{9} + 1903075590 T^{10} - 6421660196 T^{11} + 10779215329 T^{12} \))(\( 1 + 28 T + 390 T^{2} + 4298 T^{3} + 42442 T^{4} + 351442 T^{5} + 2516387 T^{6} + 16517774 T^{7} + 93754378 T^{8} + 446231254 T^{9} + 1903075590 T^{10} + 6421660196 T^{11} + 10779215329 T^{12} \))
$53$ (\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 + 8 T - 50 T^{2} + 64 T^{3} + 6795 T^{4} + 3392 T^{5} - 140450 T^{6} + 1191016 T^{7} + 7890481 T^{8} \))(\( 1 - 10 T - 40 T^{2} + 474 T^{3} + 2810 T^{4} - 10870 T^{5} - 143977 T^{6} - 576110 T^{7} + 7893290 T^{8} + 70567698 T^{9} - 315619240 T^{10} - 4181954930 T^{11} + 22164361129 T^{12} \))(\( 1 + 10 T - 40 T^{2} - 474 T^{3} + 2810 T^{4} + 10870 T^{5} - 143977 T^{6} + 576110 T^{7} + 7893290 T^{8} - 70567698 T^{9} - 315619240 T^{10} + 4181954930 T^{11} + 22164361129 T^{12} \))
$59$ (\( 1 + 10 T + 41 T^{2} + 590 T^{3} + 3481 T^{4} \))(\( 1 + 10 T + 41 T^{2} + 590 T^{3} + 3481 T^{4} \))(\( 1 - 8 T + 2 T^{2} + 448 T^{3} - 3413 T^{4} + 26432 T^{5} + 6962 T^{6} - 1643032 T^{7} + 12117361 T^{8} \))(\( 1 - 5 T - 90 T^{2} + 815 T^{3} + 2965 T^{4} - 29480 T^{5} + 26483 T^{6} - 1739320 T^{7} + 10321165 T^{8} + 167383885 T^{9} - 1090562490 T^{10} - 3574621495 T^{11} + 42180533641 T^{12} \))(\( 1 - 5 T - 90 T^{2} + 815 T^{3} + 2965 T^{4} - 29480 T^{5} + 26483 T^{6} - 1739320 T^{7} + 10321165 T^{8} + 167383885 T^{9} - 1090562490 T^{10} - 3574621495 T^{11} + 42180533641 T^{12} \))
$61$ (\( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} \))(\( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} \))(\( 1 - 6 T - 23 T^{2} + 378 T^{3} - 2436 T^{4} + 23058 T^{5} - 85583 T^{6} - 1361886 T^{7} + 13845841 T^{8} \))(\( 1 + 5 T - 140 T^{2} - 269 T^{3} + 14425 T^{4} + 8210 T^{5} - 1000579 T^{6} + 500810 T^{7} + 53675425 T^{8} - 61057889 T^{9} - 1938417740 T^{10} + 4222981505 T^{11} + 51520374361 T^{12} \))(\( 1 + 5 T - 140 T^{2} - 269 T^{3} + 14425 T^{4} + 8210 T^{5} - 1000579 T^{6} + 500810 T^{7} + 53675425 T^{8} - 61057889 T^{9} - 1938417740 T^{10} + 4222981505 T^{11} + 51520374361 T^{12} \))
$67$ (\( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 - 22 T + 231 T^{2} - 2618 T^{3} + 27092 T^{4} - 175406 T^{5} + 1036959 T^{6} - 6616786 T^{7} + 20151121 T^{8} \))(\( 1 + 12 T - 85 T^{2} - 388 T^{3} + 18282 T^{4} + 44708 T^{5} - 1099813 T^{6} + 2995436 T^{7} + 82067898 T^{8} - 116696044 T^{9} - 1712845285 T^{10} + 16201501284 T^{11} + 90458382169 T^{12} \))(\( 1 - 12 T - 85 T^{2} + 388 T^{3} + 18282 T^{4} - 44708 T^{5} - 1099813 T^{6} - 2995436 T^{7} + 82067898 T^{8} + 116696044 T^{9} - 1712845285 T^{10} - 16201501284 T^{11} + 90458382169 T^{12} \))
$71$ (\( ( 1 + 2 T + 71 T^{2} )^{2} \))(\( ( 1 + 2 T + 71 T^{2} )^{2} \))(\( ( 1 + 8 T + 86 T^{2} + 568 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 7 T + 124 T^{2} - 571 T^{3} + 8804 T^{4} - 35287 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 - 7 T + 124 T^{2} - 571 T^{3} + 8804 T^{4} - 35287 T^{5} + 357911 T^{6} )^{2} \))
$73$ (\( 1 - 6 T - 37 T^{2} - 438 T^{3} + 5329 T^{4} \))(\( 1 + 6 T - 37 T^{2} + 438 T^{3} + 5329 T^{4} \))(\( 1 - 4 T - 126 T^{2} + 16 T^{3} + 13667 T^{4} + 1168 T^{5} - 671454 T^{6} - 1556068 T^{7} + 28398241 T^{8} \))(\( 1 + 12 T - 43 T^{2} - 364 T^{3} + 5670 T^{4} - 19828 T^{5} - 841663 T^{6} - 1447444 T^{7} + 30215430 T^{8} - 141602188 T^{9} - 1221124363 T^{10} + 24876859116 T^{11} + 151334226289 T^{12} \))(\( 1 - 12 T - 43 T^{2} + 364 T^{3} + 5670 T^{4} + 19828 T^{5} - 841663 T^{6} + 1447444 T^{7} + 30215430 T^{8} + 141602188 T^{9} - 1221124363 T^{10} - 24876859116 T^{11} + 151334226289 T^{12} \))
$79$ (\( 1 - 2 T - 75 T^{2} - 158 T^{3} + 6241 T^{4} \))(\( 1 - 2 T - 75 T^{2} - 158 T^{3} + 6241 T^{4} \))(\( 1 + 24 T + 282 T^{2} + 3264 T^{3} + 34691 T^{4} + 257856 T^{5} + 1759962 T^{6} + 11832936 T^{7} + 38950081 T^{8} \))(\( 1 - 7 T - 96 T^{2} + 1499 T^{3} + 771 T^{4} - 65582 T^{5} + 599343 T^{6} - 5180978 T^{7} + 4811811 T^{8} + 739065461 T^{9} - 3739207776 T^{10} - 21539394793 T^{11} + 243087455521 T^{12} \))(\( 1 - 7 T - 96 T^{2} + 1499 T^{3} + 771 T^{4} - 65582 T^{5} + 599343 T^{6} - 5180978 T^{7} + 4811811 T^{8} + 739065461 T^{9} - 3739207776 T^{10} - 21539394793 T^{11} + 243087455521 T^{12} \))
$83$ (\( ( 1 + 11 T + 83 T^{2} )^{2} \))(\( ( 1 - 11 T + 83 T^{2} )^{2} \))(\( ( 1 + 2 T + 5 T^{2} + 166 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 26 T + 448 T^{2} + 4715 T^{3} + 37184 T^{4} + 179114 T^{5} + 571787 T^{6} )^{2} \))(\( ( 1 - 26 T + 448 T^{2} - 4715 T^{3} + 37184 T^{4} - 179114 T^{5} + 571787 T^{6} )^{2} \))
$89$ (\( 1 + 9 T - 8 T^{2} + 801 T^{3} + 7921 T^{4} \))(\( 1 + 9 T - 8 T^{2} + 801 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 119 T^{2} + 138 T^{3} + 12900 T^{4} + 12282 T^{5} - 942599 T^{6} - 4229814 T^{7} + 62742241 T^{8} \))(\( 1 - 6 T - 94 T^{2} - 198 T^{3} + 3584 T^{4} + 53484 T^{5} - 294637 T^{6} + 4760076 T^{7} + 28388864 T^{8} - 139583862 T^{9} - 5897770654 T^{10} - 33504356694 T^{11} + 496981290961 T^{12} \))(\( 1 - 6 T - 94 T^{2} - 198 T^{3} + 3584 T^{4} + 53484 T^{5} - 294637 T^{6} + 4760076 T^{7} + 28388864 T^{8} - 139583862 T^{9} - 5897770654 T^{10} - 33504356694 T^{11} + 496981290961 T^{12} \))
$97$ (\( ( 1 + 16 T + 97 T^{2} )^{2} \))(\( ( 1 - 16 T + 97 T^{2} )^{2} \))(\( ( 1 + 12 T + 198 T^{2} + 1164 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 7 T + 181 T^{2} - 1617 T^{3} + 17557 T^{4} - 65863 T^{5} + 912673 T^{6} )^{2} \))(\( ( 1 + 7 T + 181 T^{2} + 1617 T^{3} + 17557 T^{4} + 65863 T^{5} + 912673 T^{6} )^{2} \))
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