Properties

Label 670.2.e.j
Level 670
Weight 2
Character orbit 670.e
Analytic conductor 5.350
Analytic rank 0
Dimension 12
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 670 = 2 \cdot 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 670.e (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(5.34997693543\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{5} q^{2} -\beta_{2} q^{3} + ( -1 - \beta_{5} ) q^{4} - q^{5} + ( \beta_{1} - \beta_{2} ) q^{6} + ( -\beta_{6} - \beta_{9} ) q^{7} - q^{8} + ( 2 - \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{5} q^{2} -\beta_{2} q^{3} + ( -1 - \beta_{5} ) q^{4} - q^{5} + ( \beta_{1} - \beta_{2} ) q^{6} + ( -\beta_{6} - \beta_{9} ) q^{7} - q^{8} + ( 2 - \beta_{3} ) q^{9} + \beta_{5} q^{10} + \beta_{11} q^{11} + \beta_{1} q^{12} + ( \beta_{7} + \beta_{10} ) q^{13} + ( 1 + \beta_{4} - \beta_{8} ) q^{14} + \beta_{2} q^{15} + \beta_{5} q^{16} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{11} ) q^{17} + ( -\beta_{3} - 2 \beta_{5} + \beta_{11} ) q^{18} + ( 1 + \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{19} + ( 1 + \beta_{5} ) q^{20} + ( 2 - 2 \beta_{1} + 2 \beta_{5} - \beta_{6} + \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{21} + \beta_{3} q^{22} + ( -1 - \beta_{4} - \beta_{6} - \beta_{8} + \beta_{9} ) q^{23} + \beta_{2} q^{24} + q^{25} + \beta_{10} q^{26} + ( -\beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{27} + ( 1 + \beta_{4} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{28} + ( -1 - \beta_{1} - \beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} ) q^{29} + ( -\beta_{1} + \beta_{2} ) q^{30} + ( 3 - 2 \beta_{1} + 3 \beta_{5} + \beta_{9} + \beta_{10} ) q^{31} + ( 1 + \beta_{5} ) q^{32} + ( 2 + 2 \beta_{1} + 2 \beta_{5} + 2 \beta_{6} - \beta_{11} ) q^{33} + ( -3 + \beta_{1} - 3 \beta_{5} - \beta_{6} + \beta_{11} ) q^{34} + ( \beta_{6} + \beta_{9} ) q^{35} + ( -2 - 2 \beta_{5} + \beta_{11} ) q^{36} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{37} + ( 2 + 2 \beta_{5} + \beta_{6} ) q^{38} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{39} + q^{40} + ( 3 + 3 \beta_{5} + \beta_{6} + 2 \beta_{9} + \beta_{11} ) q^{41} + ( 3 - 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{7} + \beta_{8} ) q^{42} + ( -1 + \beta_{2} - 2 \beta_{4} + \beta_{7} - \beta_{8} ) q^{43} + ( \beta_{3} - \beta_{11} ) q^{44} + ( -2 + \beta_{3} ) q^{45} + ( -\beta_{6} + \beta_{9} ) q^{46} + ( -2 + 2 \beta_{1} - 2 \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} ) q^{47} + ( -\beta_{1} + \beta_{2} ) q^{48} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + 8 \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{49} -\beta_{5} q^{50} + ( 1 + 3 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{51} -\beta_{7} q^{52} + ( -4 - \beta_{2} + \beta_{3} + \beta_{7} + \beta_{8} ) q^{53} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{11} ) q^{54} -\beta_{11} q^{55} + ( \beta_{6} + \beta_{9} ) q^{56} + ( 1 - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{10} + \beta_{11} ) q^{57} + ( -\beta_{2} + \beta_{4} - \beta_{7} + \beta_{8} ) q^{58} + ( 3 - \beta_{3} - 3 \beta_{4} ) q^{59} -\beta_{1} q^{60} + ( -2 - 3 \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{61} + ( 3 - 2 \beta_{2} - \beta_{7} + \beta_{8} ) q^{62} + ( -2 - 2 \beta_{1} - 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{9} - 2 \beta_{10} ) q^{63} + q^{64} + ( -\beta_{7} - \beta_{10} ) q^{65} + ( 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{66} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} ) q^{67} + ( -2 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{68} + ( -1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{69} + ( -1 - \beta_{4} + \beta_{8} ) q^{70} + ( -1 - \beta_{5} - \beta_{9} + \beta_{11} ) q^{71} + ( -2 + \beta_{3} ) q^{72} + ( -1 - 3 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{11} ) q^{73} + ( -\beta_{1} + \beta_{9} + \beta_{11} ) q^{74} -\beta_{2} q^{75} + ( 1 - \beta_{4} ) q^{76} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{10} ) q^{77} + ( -4 + 2 \beta_{1} - 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{78} + ( 2 \beta_{1} + 2 \beta_{6} - 2 \beta_{10} ) q^{79} -\beta_{5} q^{80} + ( 1 - \beta_{3} - 2 \beta_{7} ) q^{81} + ( 2 + \beta_{3} - \beta_{4} + 2 \beta_{8} ) q^{82} + ( 1 - 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + 4 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{10} - 2 \beta_{11} ) q^{83} + ( 1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{84} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{11} ) q^{85} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{86} + ( -1 + \beta_{1} - \beta_{5} - 3 \beta_{6} - 3 \beta_{9} - \beta_{10} - \beta_{11} ) q^{87} -\beta_{11} q^{88} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{8} ) q^{89} + ( \beta_{3} + 2 \beta_{5} - \beta_{11} ) q^{90} + ( 4 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} ) q^{91} + ( 1 + \beta_{4} + \beta_{8} ) q^{92} + ( 4 - \beta_{1} + 4 \beta_{5} - 2 \beta_{6} - 3 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{93} + ( -3 + 2 \beta_{2} - \beta_{4} - \beta_{7} - \beta_{8} ) q^{94} + ( -1 - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{95} -\beta_{1} q^{96} + ( -1 - \beta_{4} - 4 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{97} + ( 8 - \beta_{1} + 8 \beta_{5} + \beta_{6} - \beta_{9} - 2 \beta_{11} ) q^{98} + ( -12 - 12 \beta_{5} - 2 \beta_{10} + 2 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 6q^{2} - 4q^{3} - 6q^{4} - 12q^{5} - 2q^{6} + 3q^{7} - 12q^{8} + 24q^{9} + O(q^{10}) \) \( 12q + 6q^{2} - 4q^{3} - 6q^{4} - 12q^{5} - 2q^{6} + 3q^{7} - 12q^{8} + 24q^{9} - 6q^{10} + 2q^{12} + 6q^{14} + 4q^{15} - 6q^{16} + 13q^{17} + 12q^{18} - 9q^{19} + 6q^{20} + 11q^{21} - 3q^{23} + 4q^{24} + 12q^{25} - 16q^{27} + 3q^{28} - 5q^{29} + 2q^{30} + 14q^{31} + 6q^{32} + 10q^{33} - 13q^{34} - 3q^{35} - 12q^{36} + 2q^{37} + 9q^{38} + 14q^{39} + 12q^{40} + 15q^{41} + 22q^{42} + 4q^{43} - 24q^{45} + 3q^{46} - 11q^{47} + 2q^{48} - 43q^{49} + 6q^{50} + 15q^{51} - 52q^{53} - 8q^{54} - 3q^{56} + 3q^{57} - 10q^{58} + 54q^{59} - 2q^{60} - 6q^{61} + 28q^{62} - 4q^{63} + 12q^{64} + 20q^{66} - 5q^{67} - 26q^{68} + 13q^{69} - 6q^{70} - 6q^{71} - 24q^{72} - 15q^{73} - 2q^{74} - 4q^{75} + 18q^{76} - 10q^{77} - 14q^{78} - 2q^{79} + 6q^{80} + 12q^{81} + 30q^{82} - 15q^{83} + 11q^{84} - 13q^{85} + 2q^{86} + 5q^{87} - 14q^{89} - 12q^{90} + 4q^{91} + 6q^{92} + 28q^{93} - 22q^{94} + 9q^{95} - 2q^{96} + 21q^{97} + 43q^{98} - 72q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 2 x^{11} + 17 x^{10} - 18 x^{9} + 172 x^{8} - 170 x^{7} + 887 x^{6} - 312 x^{5} + 2516 x^{4} - 1220 x^{3} + 3001 x^{2} + 816 x + 289\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-348796 \nu^{11} - 222710 \nu^{10} - 4440773 \nu^{9} - 6874888 \nu^{8} - 54113692 \nu^{7} - 67546606 \nu^{6} - 245833032 \nu^{5} - 415113608 \nu^{4} - 1416606628 \nu^{3} - 870936982 \nu^{2} - 247252488 \nu - 683228368\)\()/ 2189184197 \)
\(\beta_{3}\)\(=\)\((\)\(920302 \nu^{11} - 1488759 \nu^{10} + 13153216 \nu^{9} - 5879220 \nu^{8} + 126841926 \nu^{7} - 63549020 \nu^{6} + 523937960 \nu^{5} + 539035892 \nu^{4} + 1296468102 \nu^{3} + 1389699889 \nu^{2} + 398610832 \nu + 10845118941\)\()/ 2189184197 \)
\(\beta_{4}\)\(=\)\((\)\(-3361874 \nu^{11} - 70211 \nu^{10} - 44238627 \nu^{9} - 42244996 \nu^{8} - 505637770 \nu^{7} - 409277222 \nu^{6} - 2244769184 \nu^{5} - 3444831148 \nu^{4} - 9023530301 \nu^{3} - 7486258763 \nu^{2} - 2129378248 \nu - 15627717517\)\()/ 4378368394 \)
\(\beta_{5}\)\(=\)\((\)\(-2364112 \nu^{11} + 5077020 \nu^{10} - 39967194 \nu^{9} + 46994789 \nu^{8} - 399752376 \nu^{7} + 456012732 \nu^{6} - 2029420738 \nu^{5} + 983435976 \nu^{4} - 5532992184 \nu^{3} + 4300823268 \nu^{2} - 6223763130 \nu - 1681862904\)\()/ 2189184197 \)
\(\beta_{6}\)\(=\)\((\)\(-6523879 \nu^{11} + 16234219 \nu^{10} - 117434582 \nu^{9} + 166555165 \nu^{8} - 1165317522 \nu^{7} + 1596859090 \nu^{6} - 6390494370 \nu^{5} + 4508355550 \nu^{4} - 17814976779 \nu^{3} + 14466200966 \nu^{2} - 32745199461 \nu + 1687129113\)\()/ 4378368394 \)
\(\beta_{7}\)\(=\)\((\)\(-7414792 \nu^{11} + 10182739 \nu^{10} - 106415352 \nu^{9} + 26407220 \nu^{8} - 1035862996 \nu^{7} + 121406599 \nu^{6} - 4330143510 \nu^{5} - 4828401032 \nu^{4} - 11030716450 \nu^{3} - 11989343569 \nu^{2} - 3433035522 \nu - 27349690916\)\()/ 4378368394 \)
\(\beta_{8}\)\(=\)\((\)\(-17099073 \nu^{11} + 8561797 \nu^{10} - 243882439 \nu^{9} - 111695356 \nu^{8} - 2503302607 \nu^{7} - 1295616041 \nu^{6} - 10881668329 \nu^{5} - 15131705802 \nu^{4} - 32764605035 \nu^{3} - 34174990581 \nu^{2} - 9740351807 \nu - 56769709220\)\()/ 8756736788 \)
\(\beta_{9}\)\(=\)\((\)\(36337419 \nu^{11} - 96041698 \nu^{10} + 676341606 \nu^{9} - 1042500477 \nu^{8} + 6852283813 \nu^{7} - 9948243066 \nu^{6} + 38054929314 \nu^{5} - 29582509997 \nu^{4} + 107156512575 \nu^{3} - 88664338686 \nu^{2} + 160818864128 \nu - 10291250985\)\()/ 8756736788 \)
\(\beta_{10}\)\(=\)\((\)\(-21530436 \nu^{11} + 51113670 \nu^{10} - 388041362 \nu^{9} + 552140057 \nu^{8} - 3928361628 \nu^{7} + 5292675316 \nu^{6} - 21143089994 \nu^{5} + 15985842411 \nu^{4} - 58964217372 \nu^{3} + 47915859404 \nu^{2} - 76922812370 \nu + 5587156369\)\()/ 4378368394 \)
\(\beta_{11}\)\(=\)\((\)\(-10900258 \nu^{11} + 23896341 \nu^{10} - 186682754 \nu^{9} + 229094725 \nu^{8} - 1871919954 \nu^{7} + 2216514640 \nu^{6} - 9623165730 \nu^{5} + 5456215772 \nu^{4} - 26368492818 \nu^{3} + 20704632032 \nu^{2} - 30720204818 \nu + 2435804421\)\()/ 2189184197 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{11} + 5 \beta_{5} + \beta_{3}\)
\(\nu^{3}\)\(=\)\(2 \beta_{4} + \beta_{3} - 7 \beta_{2}\)
\(\nu^{4}\)\(=\)\(10 \beta_{11} - 2 \beta_{10} - 37 \beta_{5} - 37\)
\(\nu^{5}\)\(=\)\(12 \beta_{11} - 2 \beta_{10} - 4 \beta_{9} + 4 \beta_{8} - 2 \beta_{7} - 24 \beta_{6} - 28 \beta_{5} - 24 \beta_{4} - 12 \beta_{3} + 53 \beta_{2} - 53 \beta_{1} - 24\)
\(\nu^{6}\)\(=\)\(8 \beta_{8} - 30 \beta_{7} - 4 \beta_{4} - 91 \beta_{3} + 301\)
\(\nu^{7}\)\(=\)\(-125 \beta_{11} + 42 \beta_{10} + 68 \beta_{9} + 238 \beta_{6} + 318 \beta_{5} + 419 \beta_{1} + 318\)
\(\nu^{8}\)\(=\)\(-824 \beta_{11} + 348 \beta_{10} + 152 \beta_{9} - 152 \beta_{8} + 348 \beta_{7} + 96 \beta_{6} + 2649 \beta_{5} + 96 \beta_{4} + 824 \beta_{3} - 8 \beta_{2} + 8 \beta_{1} + 96\)
\(\nu^{9}\)\(=\)\(-848 \beta_{8} + 596 \beta_{7} + 2248 \beta_{4} + 1276 \beta_{3} - 3409 \beta_{2} - 1136\)
\(\nu^{10}\)\(=\)\(7529 \beta_{11} - 3692 \beta_{10} - 2040 \beta_{9} - 1496 \beta_{6} - 23677 \beta_{5} - 248 \beta_{1} - 23677\)
\(\nu^{11}\)\(=\)\(12965 \beta_{11} - 7228 \beta_{10} - 9424 \beta_{9} + 9424 \beta_{8} - 7228 \beta_{7} - 20946 \beta_{6} - 35146 \beta_{5} - 20946 \beta_{4} - 12965 \beta_{3} + 28359 \beta_{2} - 28359 \beta_{1} - 20946\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/670\mathbb{Z}\right)^\times\).

\(n\) \(471\) \(537\)
\(\chi(n)\) \(-1 - \beta_{5}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
171.1
1.56111 2.70392i
1.29203 2.23786i
0.665228 1.15221i
−0.146964 + 0.254549i
−0.943122 + 1.63354i
−1.42828 + 2.47386i
1.56111 + 2.70392i
1.29203 + 2.23786i
0.665228 + 1.15221i
−0.146964 0.254549i
−0.943122 1.63354i
−1.42828 2.47386i
0.500000 + 0.866025i −3.12222 −0.500000 + 0.866025i −1.00000 −1.56111 2.70392i −2.17160 + 3.76132i −1.00000 6.74824 −0.500000 0.866025i
171.2 0.500000 + 0.866025i −2.58406 −0.500000 + 0.866025i −1.00000 −1.29203 2.23786i 1.94629 3.37108i −1.00000 3.67738 −0.500000 0.866025i
171.3 0.500000 + 0.866025i −1.33046 −0.500000 + 0.866025i −1.00000 −0.665228 1.15221i −0.407937 + 0.706567i −1.00000 −1.22989 −0.500000 0.866025i
171.4 0.500000 + 0.866025i 0.293928 −0.500000 + 0.866025i −1.00000 0.146964 + 0.254549i 1.88267 3.26088i −1.00000 −2.91361 −0.500000 0.866025i
171.5 0.500000 + 0.866025i 1.88624 −0.500000 + 0.866025i −1.00000 0.943122 + 1.63354i −1.99648 + 3.45801i −1.00000 0.557917 −0.500000 0.866025i
171.6 0.500000 + 0.866025i 2.85656 −0.500000 + 0.866025i −1.00000 1.42828 + 2.47386i 2.24706 3.89203i −1.00000 5.15996 −0.500000 0.866025i
431.1 0.500000 0.866025i −3.12222 −0.500000 0.866025i −1.00000 −1.56111 + 2.70392i −2.17160 3.76132i −1.00000 6.74824 −0.500000 + 0.866025i
431.2 0.500000 0.866025i −2.58406 −0.500000 0.866025i −1.00000 −1.29203 + 2.23786i 1.94629 + 3.37108i −1.00000 3.67738 −0.500000 + 0.866025i
431.3 0.500000 0.866025i −1.33046 −0.500000 0.866025i −1.00000 −0.665228 + 1.15221i −0.407937 0.706567i −1.00000 −1.22989 −0.500000 + 0.866025i
431.4 0.500000 0.866025i 0.293928 −0.500000 0.866025i −1.00000 0.146964 0.254549i 1.88267 + 3.26088i −1.00000 −2.91361 −0.500000 + 0.866025i
431.5 0.500000 0.866025i 1.88624 −0.500000 0.866025i −1.00000 0.943122 1.63354i −1.99648 3.45801i −1.00000 0.557917 −0.500000 + 0.866025i
431.6 0.500000 0.866025i 2.85656 −0.500000 0.866025i −1.00000 1.42828 2.47386i 2.24706 + 3.89203i −1.00000 5.15996 −0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 431.6
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
67.c Even 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(670, [\chi])\):

\( T_{3}^{6} + 2 T_{3}^{5} - 13 T_{3}^{4} - 22 T_{3}^{3} + 41 T_{3}^{2} + 48 T_{3} - 17 \)
\(T_{7}^{12} - \cdots\)
\(T_{11}^{12} + \cdots\)