Properties

Label 8001.2.a.r
Level 8001
Weight 2
Character orbit 8001.a
Self dual Yes
Analytic conductor 63.888
Analytic rank 1
Dimension 16
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8001.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut -\mathstrut \) \(5\) \(x^{15}\mathstrut -\mathstrut \) \(13\) \(x^{14}\mathstrut +\mathstrut \) \(98\) \(x^{13}\mathstrut +\mathstrut \) \(9\) \(x^{12}\mathstrut -\mathstrut \) \(712\) \(x^{11}\mathstrut +\mathstrut \) \(565\) \(x^{10}\mathstrut +\mathstrut \) \(2282\) \(x^{9}\mathstrut -\mathstrut \) \(3082\) \(x^{8}\mathstrut -\mathstrut \) \(2747\) \(x^{7}\mathstrut +\mathstrut \) \(5821\) \(x^{6}\mathstrut -\mathstrut \) \(158\) \(x^{5}\mathstrut -\mathstrut \) \(3341\) \(x^{4}\mathstrut +\mathstrut \) \(1002\) \(x^{3}\mathstrut +\mathstrut \) \(416\) \(x^{2}\mathstrut -\mathstrut \) \(148\) \(x\mathstrut +\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 5 \nu + 3 \)
\(\beta_{4}\)\(=\)\((\)\(34171\) \(\nu^{15}\mathstrut -\mathstrut \) \(45717\) \(\nu^{14}\mathstrut -\mathstrut \) \(952649\) \(\nu^{13}\mathstrut +\mathstrut \) \(996036\) \(\nu^{12}\mathstrut +\mathstrut \) \(10901947\) \(\nu^{11}\mathstrut -\mathstrut \) \(8612786\) \(\nu^{10}\mathstrut -\mathstrut \) \(64480493\) \(\nu^{9}\mathstrut +\mathstrut \) \(37846618\) \(\nu^{8}\mathstrut +\mathstrut \) \(203131982\) \(\nu^{7}\mathstrut -\mathstrut \) \(90034741\) \(\nu^{6}\mathstrut -\mathstrut \) \(313228957\) \(\nu^{5}\mathstrut +\mathstrut \) \(112849336\) \(\nu^{4}\mathstrut +\mathstrut \) \(182483997\) \(\nu^{3}\mathstrut -\mathstrut \) \(61371742\) \(\nu^{2}\mathstrut -\mathstrut \) \(23762490\) \(\nu\mathstrut +\mathstrut \) \(10476522\)\()/3260350\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(95749\) \(\nu^{15}\mathstrut +\mathstrut \) \(241013\) \(\nu^{14}\mathstrut +\mathstrut \) \(1943846\) \(\nu^{13}\mathstrut -\mathstrut \) \(4685549\) \(\nu^{12}\mathstrut -\mathstrut \) \(14893598\) \(\nu^{11}\mathstrut +\mathstrut \) \(33446774\) \(\nu^{10}\mathstrut +\mathstrut \) \(52754722\) \(\nu^{9}\mathstrut -\mathstrut \) \(102849072\) \(\nu^{8}\mathstrut -\mathstrut \) \(85522528\) \(\nu^{7}\mathstrut +\mathstrut \) \(106565099\) \(\nu^{6}\mathstrut +\mathstrut \) \(65698528\) \(\nu^{5}\mathstrut +\mathstrut \) \(47018671\) \(\nu^{4}\mathstrut -\mathstrut \) \(57366413\) \(\nu^{3}\mathstrut -\mathstrut \) \(63764457\) \(\nu^{2}\mathstrut +\mathstrut \) \(11672715\) \(\nu\mathstrut +\mathstrut \) \(5521822\)\()/1630175\)
\(\beta_{6}\)\(=\)\((\)\(138556\) \(\nu^{15}\mathstrut -\mathstrut \) \(479387\) \(\nu^{14}\mathstrut -\mathstrut \) \(2447289\) \(\nu^{13}\mathstrut +\mathstrut \) \(9459921\) \(\nu^{12}\mathstrut +\mathstrut \) \(14196617\) \(\nu^{11}\mathstrut -\mathstrut \) \(69724171\) \(\nu^{10}\mathstrut -\mathstrut \) \(19774673\) \(\nu^{9}\mathstrut +\mathstrut \) \(231708473\) \(\nu^{8}\mathstrut -\mathstrut \) \(83578798\) \(\nu^{7}\mathstrut -\mathstrut \) \(316169501\) \(\nu^{6}\mathstrut +\mathstrut \) \(271416573\) \(\nu^{5}\mathstrut +\mathstrut \) \(75977871\) \(\nu^{4}\mathstrut -\mathstrut \) \(188741583\) \(\nu^{3}\mathstrut +\mathstrut \) \(54036313\) \(\nu^{2}\mathstrut +\mathstrut \) \(31200060\) \(\nu\mathstrut -\mathstrut \) \(10011933\)\()/1630175\)
\(\beta_{7}\)\(=\)\((\)\(312927\) \(\nu^{15}\mathstrut -\mathstrut \) \(1689159\) \(\nu^{14}\mathstrut -\mathstrut \) \(4050143\) \(\nu^{13}\mathstrut +\mathstrut \) \(33947762\) \(\nu^{12}\mathstrut +\mathstrut \) \(2433199\) \(\nu^{11}\mathstrut -\mathstrut \) \(256582162\) \(\nu^{10}\mathstrut +\mathstrut \) \(180181899\) \(\nu^{9}\mathstrut +\mathstrut \) \(886177826\) \(\nu^{8}\mathstrut -\mathstrut \) \(982488526\) \(\nu^{7}\mathstrut -\mathstrut \) \(1312231707\) \(\nu^{6}\mathstrut +\mathstrut \) \(1885163751\) \(\nu^{5}\mathstrut +\mathstrut \) \(532378722\) \(\nu^{4}\mathstrut -\mathstrut \) \(1161023271\) \(\nu^{3}\mathstrut +\mathstrut \) \(13012206\) \(\nu^{2}\mathstrut +\mathstrut \) \(194829910\) \(\nu\mathstrut -\mathstrut \) \(7559716\)\()/3260350\)
\(\beta_{8}\)\(=\)\((\)\(174111\) \(\nu^{15}\mathstrut -\mathstrut \) \(416897\) \(\nu^{14}\mathstrut -\mathstrut \) \(3848034\) \(\nu^{13}\mathstrut +\mathstrut \) \(8756876\) \(\nu^{12}\mathstrut +\mathstrut \) \(33610177\) \(\nu^{11}\mathstrut -\mathstrut \) \(71442151\) \(\nu^{10}\mathstrut -\mathstrut \) \(147009638\) \(\nu^{9}\mathstrut +\mathstrut \) \(284665963\) \(\nu^{8}\mathstrut +\mathstrut \) \(336839837\) \(\nu^{7}\mathstrut -\mathstrut \) \(569728931\) \(\nu^{6}\mathstrut -\mathstrut \) \(387906862\) \(\nu^{5}\mathstrut +\mathstrut \) \(524853251\) \(\nu^{4}\mathstrut +\mathstrut \) \(199086302\) \(\nu^{3}\mathstrut -\mathstrut \) \(178089422\) \(\nu^{2}\mathstrut -\mathstrut \) \(29637740\) \(\nu\mathstrut +\mathstrut \) \(12678877\)\()/1630175\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(62553\) \(\nu^{15}\mathstrut +\mathstrut \) \(185357\) \(\nu^{14}\mathstrut +\mathstrut \) \(1244303\) \(\nu^{13}\mathstrut -\mathstrut \) \(3782409\) \(\nu^{12}\mathstrut -\mathstrut \) \(9231198\) \(\nu^{11}\mathstrut +\mathstrut \) \(29429019\) \(\nu^{10}\mathstrut +\mathstrut \) \(30643366\) \(\nu^{9}\mathstrut -\mathstrut \) \(108077121\) \(\nu^{8}\mathstrut -\mathstrut \) \(40734444\) \(\nu^{7}\mathstrut +\mathstrut \) \(186611071\) \(\nu^{6}\mathstrut +\mathstrut \) \(9095064\) \(\nu^{5}\mathstrut -\mathstrut \) \(131435121\) \(\nu^{4}\mathstrut -\mathstrut \) \(543294\) \(\nu^{3}\mathstrut +\mathstrut \) \(37570814\) \(\nu^{2}\mathstrut +\mathstrut \) \(3158402\) \(\nu\mathstrut -\mathstrut \) \(3490290\)\()/326035\)
\(\beta_{10}\)\(=\)\((\)\(322256\) \(\nu^{15}\mathstrut -\mathstrut \) \(1375567\) \(\nu^{14}\mathstrut -\mathstrut \) \(5035219\) \(\nu^{13}\mathstrut +\mathstrut \) \(27365426\) \(\nu^{12}\mathstrut +\mathstrut \) \(19868527\) \(\nu^{11}\mathstrut -\mathstrut \) \(204238826\) \(\nu^{10}\mathstrut +\mathstrut \) \(53766867\) \(\nu^{9}\mathstrut +\mathstrut \) \(693632208\) \(\nu^{8}\mathstrut -\mathstrut \) \(546168183\) \(\nu^{7}\mathstrut -\mathstrut \) \(997385891\) \(\nu^{6}\mathstrut +\mathstrut \) \(1188437858\) \(\nu^{5}\mathstrut +\mathstrut \) \(353563286\) \(\nu^{4}\mathstrut -\mathstrut \) \(736318918\) \(\nu^{3}\mathstrut +\mathstrut \) \(56736448\) \(\nu^{2}\mathstrut +\mathstrut \) \(98969000\) \(\nu\mathstrut -\mathstrut \) \(11457663\)\()/1630175\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(387602\) \(\nu^{15}\mathstrut +\mathstrut \) \(1093459\) \(\nu^{14}\mathstrut +\mathstrut \) \(7892793\) \(\nu^{13}\mathstrut -\mathstrut \) \(22401737\) \(\nu^{12}\mathstrut -\mathstrut \) \(60887074\) \(\nu^{11}\mathstrut +\mathstrut \) \(175479737\) \(\nu^{10}\mathstrut +\mathstrut \) \(217918476\) \(\nu^{9}\mathstrut -\mathstrut \) \(652127926\) \(\nu^{8}\mathstrut -\mathstrut \) \(350064674\) \(\nu^{7}\mathstrut +\mathstrut \) \(1148743407\) \(\nu^{6}\mathstrut +\mathstrut \) \(201090249\) \(\nu^{5}\mathstrut -\mathstrut \) \(822768247\) \(\nu^{4}\mathstrut -\mathstrut \) \(52865479\) \(\nu^{3}\mathstrut +\mathstrut \) \(185190694\) \(\nu^{2}\mathstrut +\mathstrut \) \(5524390\) \(\nu\mathstrut +\mathstrut \) \(2433991\)\()/1630175\)
\(\beta_{12}\)\(=\)\((\)\(397191\) \(\nu^{15}\mathstrut -\mathstrut \) \(1572742\) \(\nu^{14}\mathstrut -\mathstrut \) \(6632689\) \(\nu^{13}\mathstrut +\mathstrut \) \(31550366\) \(\nu^{12}\mathstrut +\mathstrut \) \(32948807\) \(\nu^{11}\mathstrut -\mathstrut \) \(238552241\) \(\nu^{10}\mathstrut +\mathstrut \) \(2632702\) \(\nu^{9}\mathstrut +\mathstrut \) \(829385698\) \(\nu^{8}\mathstrut -\mathstrut \) \(449364548\) \(\nu^{7}\mathstrut -\mathstrut \) \(1260230866\) \(\nu^{6}\mathstrut +\mathstrut \) \(1103837898\) \(\nu^{5}\mathstrut +\mathstrut \) \(577130586\) \(\nu^{4}\mathstrut -\mathstrut \) \(695428333\) \(\nu^{3}\mathstrut -\mathstrut \) \(15812362\) \(\nu^{2}\mathstrut +\mathstrut \) \(100033165\) \(\nu\mathstrut -\mathstrut \) \(6777548\)\()/1630175\)
\(\beta_{13}\)\(=\)\((\)\(443728\) \(\nu^{15}\mathstrut -\mathstrut \) \(1561421\) \(\nu^{14}\mathstrut -\mathstrut \) \(8006447\) \(\nu^{13}\mathstrut +\mathstrut \) \(31434138\) \(\nu^{12}\mathstrut +\mathstrut \) \(49072551\) \(\nu^{11}\mathstrut -\mathstrut \) \(239584888\) \(\nu^{10}\mathstrut -\mathstrut \) \(92916154\) \(\nu^{9}\mathstrut +\mathstrut \) \(848762304\) \(\nu^{8}\mathstrut -\mathstrut \) \(151345429\) \(\nu^{7}\mathstrut -\mathstrut \) \(1357041408\) \(\nu^{6}\mathstrut +\mathstrut \) \(646884804\) \(\nu^{5}\mathstrut +\mathstrut \) \(763592393\) \(\nu^{4}\mathstrut -\mathstrut \) \(420987859\) \(\nu^{3}\mathstrut -\mathstrut \) \(120963051\) \(\nu^{2}\mathstrut +\mathstrut \) \(53266175\) \(\nu\mathstrut +\mathstrut \) \(3556906\)\()/1630175\)
\(\beta_{14}\)\(=\)\((\)\(509784\) \(\nu^{15}\mathstrut -\mathstrut \) \(2052553\) \(\nu^{14}\mathstrut -\mathstrut \) \(8419656\) \(\nu^{13}\mathstrut +\mathstrut \) \(41075579\) \(\nu^{12}\mathstrut +\mathstrut \) \(40710358\) \(\nu^{11}\mathstrut -\mathstrut \) \(309487504\) \(\nu^{10}\mathstrut +\mathstrut \) \(12346983\) \(\nu^{9}\mathstrut +\mathstrut \) \(1070084317\) \(\nu^{8}\mathstrut -\mathstrut \) \(592856117\) \(\nu^{7}\mathstrut -\mathstrut \) \(1609704994\) \(\nu^{6}\mathstrut +\mathstrut \) \(1400337842\) \(\nu^{5}\mathstrut +\mathstrut \) \(719985124\) \(\nu^{4}\mathstrut -\mathstrut \) \(824771807\) \(\nu^{3}\mathstrut -\mathstrut \) \(26429048\) \(\nu^{2}\mathstrut +\mathstrut \) \(92347345\) \(\nu\mathstrut -\mathstrut \) \(4486597\)\()/1630175\)
\(\beta_{15}\)\(=\)\((\)\(540201\) \(\nu^{15}\mathstrut -\mathstrut \) \(2114267\) \(\nu^{14}\mathstrut -\mathstrut \) \(9125559\) \(\nu^{13}\mathstrut +\mathstrut \) \(42440381\) \(\nu^{12}\mathstrut +\mathstrut \) \(47185962\) \(\nu^{11}\mathstrut -\mathstrut \) \(321313406\) \(\nu^{10}\mathstrut -\mathstrut \) \(17132488\) \(\nu^{9}\mathstrut +\mathstrut \) \(1120057338\) \(\nu^{8}\mathstrut -\mathstrut \) \(525465538\) \(\nu^{7}\mathstrut -\mathstrut \) \(1711378566\) \(\nu^{6}\mathstrut +\mathstrut \) \(1336320913\) \(\nu^{5}\mathstrut +\mathstrut \) \(794016061\) \(\nu^{4}\mathstrut -\mathstrut \) \(824333673\) \(\nu^{3}\mathstrut -\mathstrut \) \(8611797\) \(\nu^{2}\mathstrut +\mathstrut \) \(109330955\) \(\nu\mathstrut -\mathstrut \) \(15755608\)\()/1630175\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(10\) \(\beta_{2}\mathstrut +\mathstrut \) \(28\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{6}\)\(=\)\(\beta_{14}\mathstrut -\mathstrut \) \(2\) \(\beta_{13}\mathstrut +\mathstrut \) \(10\) \(\beta_{12}\mathstrut +\mathstrut \) \(11\) \(\beta_{11}\mathstrut -\mathstrut \) \(10\) \(\beta_{10}\mathstrut +\mathstrut \) \(13\) \(\beta_{8}\mathstrut +\mathstrut \) \(12\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(57\) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(85\)
\(\nu^{7}\)\(=\)\(-\)\(11\) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(8\) \(\beta_{13}\mathstrut +\mathstrut \) \(3\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(10\) \(\beta_{9}\mathstrut +\mathstrut \) \(17\) \(\beta_{8}\mathstrut +\mathstrut \) \(11\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(12\) \(\beta_{5}\mathstrut -\mathstrut \) \(15\) \(\beta_{4}\mathstrut +\mathstrut \) \(65\) \(\beta_{3}\mathstrut +\mathstrut \) \(84\) \(\beta_{2}\mathstrut +\mathstrut \) \(167\) \(\beta_{1}\mathstrut +\mathstrut \) \(40\)
\(\nu^{8}\)\(=\)\(-\)\(3\) \(\beta_{15}\mathstrut +\mathstrut \) \(15\) \(\beta_{14}\mathstrut -\mathstrut \) \(28\) \(\beta_{13}\mathstrut +\mathstrut \) \(76\) \(\beta_{12}\mathstrut +\mathstrut \) \(94\) \(\beta_{11}\mathstrut -\mathstrut \) \(76\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(125\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(108\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(33\) \(\beta_{4}\mathstrut +\mathstrut \) \(65\) \(\beta_{3}\mathstrut +\mathstrut \) \(398\) \(\beta_{2}\mathstrut +\mathstrut \) \(33\) \(\beta_{1}\mathstrut +\mathstrut \) \(514\)
\(\nu^{9}\)\(=\)\(-\)\(90\) \(\beta_{15}\mathstrut +\mathstrut \) \(13\) \(\beta_{14}\mathstrut +\mathstrut \) \(41\) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(56\) \(\beta_{11}\mathstrut +\mathstrut \) \(16\) \(\beta_{10}\mathstrut +\mathstrut \) \(72\) \(\beta_{9}\mathstrut +\mathstrut \) \(198\) \(\beta_{8}\mathstrut +\mathstrut \) \(94\) \(\beta_{7}\mathstrut +\mathstrut \) \(41\) \(\beta_{6}\mathstrut -\mathstrut \) \(108\) \(\beta_{5}\mathstrut -\mathstrut \) \(160\) \(\beta_{4}\mathstrut +\mathstrut \) \(439\) \(\beta_{3}\mathstrut +\mathstrut \) \(671\) \(\beta_{2}\mathstrut +\mathstrut \) \(1040\) \(\beta_{1}\mathstrut +\mathstrut \) \(386\)
\(\nu^{10}\)\(=\)\(-\)\(47\) \(\beta_{15}\mathstrut +\mathstrut \) \(152\) \(\beta_{14}\mathstrut -\mathstrut \) \(283\) \(\beta_{13}\mathstrut +\mathstrut \) \(524\) \(\beta_{12}\mathstrut +\mathstrut \) \(745\) \(\beta_{11}\mathstrut -\mathstrut \) \(524\) \(\beta_{10}\mathstrut -\mathstrut \) \(21\) \(\beta_{9}\mathstrut +\mathstrut \) \(1081\) \(\beta_{8}\mathstrut +\mathstrut \) \(56\) \(\beta_{7}\mathstrut +\mathstrut \) \(880\) \(\beta_{6}\mathstrut -\mathstrut \) \(41\) \(\beta_{5}\mathstrut -\mathstrut \) \(376\) \(\beta_{4}\mathstrut +\mathstrut \) \(446\) \(\beta_{3}\mathstrut +\mathstrut \) \(2789\) \(\beta_{2}\mathstrut +\mathstrut \) \(373\) \(\beta_{1}\mathstrut +\mathstrut \) \(3243\)
\(\nu^{11}\)\(=\)\(-\)\(659\) \(\beta_{15}\mathstrut +\mathstrut \) \(124\) \(\beta_{14}\mathstrut +\mathstrut \) \(113\) \(\beta_{13}\mathstrut +\mathstrut \) \(11\) \(\beta_{12}\mathstrut +\mathstrut \) \(693\) \(\beta_{11}\mathstrut +\mathstrut \) \(168\) \(\beta_{10}\mathstrut +\mathstrut \) \(447\) \(\beta_{9}\mathstrut +\mathstrut \) \(1967\) \(\beta_{8}\mathstrut +\mathstrut \) \(745\) \(\beta_{7}\mathstrut +\mathstrut \) \(540\) \(\beta_{6}\mathstrut -\mathstrut \) \(880\) \(\beta_{5}\mathstrut -\mathstrut \) \(1497\) \(\beta_{4}\mathstrut +\mathstrut \) \(2892\) \(\beta_{3}\mathstrut +\mathstrut \) \(5242\) \(\beta_{2}\mathstrut +\mathstrut \) \(6685\) \(\beta_{1}\mathstrut +\mathstrut \) \(3289\)
\(\nu^{12}\)\(=\)\(-\)\(488\) \(\beta_{15}\mathstrut +\mathstrut \) \(1308\) \(\beta_{14}\mathstrut -\mathstrut \) \(2518\) \(\beta_{13}\mathstrut +\mathstrut \) \(3447\) \(\beta_{12}\mathstrut +\mathstrut \) \(5755\) \(\beta_{11}\mathstrut -\mathstrut \) \(3453\) \(\beta_{10}\mathstrut -\mathstrut \) \(287\) \(\beta_{9}\mathstrut +\mathstrut \) \(8905\) \(\beta_{8}\mathstrut +\mathstrut \) \(693\) \(\beta_{7}\mathstrut +\mathstrut \) \(6864\) \(\beta_{6}\mathstrut -\mathstrut \) \(540\) \(\beta_{5}\mathstrut -\mathstrut \) \(3687\) \(\beta_{4}\mathstrut +\mathstrut \) \(3041\) \(\beta_{3}\mathstrut +\mathstrut \) \(19751\) \(\beta_{2}\mathstrut +\mathstrut \) \(3616\) \(\beta_{1}\mathstrut +\mathstrut \) \(21084\)
\(\nu^{13}\)\(=\)\(-\)\(4564\) \(\beta_{15}\mathstrut +\mathstrut \) \(1055\) \(\beta_{14}\mathstrut -\mathstrut \) \(621\) \(\beta_{13}\mathstrut +\mathstrut \) \(40\) \(\beta_{12}\mathstrut +\mathstrut \) \(7185\) \(\beta_{11}\mathstrut +\mathstrut \) \(1480\) \(\beta_{10}\mathstrut +\mathstrut \) \(2467\) \(\beta_{9}\mathstrut +\mathstrut \) \(17932\) \(\beta_{8}\mathstrut +\mathstrut \) \(5755\) \(\beta_{7}\mathstrut +\mathstrut \) \(5856\) \(\beta_{6}\mathstrut -\mathstrut \) \(6864\) \(\beta_{5}\mathstrut -\mathstrut \) \(13129\) \(\beta_{4}\mathstrut +\mathstrut \) \(18896\) \(\beta_{3}\mathstrut +\mathstrut \) \(40463\) \(\beta_{2}\mathstrut +\mathstrut \) \(44042\) \(\beta_{1}\mathstrut +\mathstrut \) \(26378\)
\(\nu^{14}\)\(=\)\(-\)\(4226\) \(\beta_{15}\mathstrut +\mathstrut \) \(10308\) \(\beta_{14}\mathstrut -\mathstrut \) \(20979\) \(\beta_{13}\mathstrut +\mathstrut \) \(22018\) \(\beta_{12}\mathstrut +\mathstrut \) \(44116\) \(\beta_{11}\mathstrut -\mathstrut \) \(22192\) \(\beta_{10}\mathstrut -\mathstrut \) \(3248\) \(\beta_{9}\mathstrut +\mathstrut \) \(71487\) \(\beta_{8}\mathstrut +\mathstrut \) \(7185\) \(\beta_{7}\mathstrut +\mathstrut \) \(52445\) \(\beta_{6}\mathstrut -\mathstrut \) \(5856\) \(\beta_{5}\mathstrut -\mathstrut \) \(33443\) \(\beta_{4}\mathstrut +\mathstrut \) \(20897\) \(\beta_{3}\mathstrut +\mathstrut \) \(141527\) \(\beta_{2}\mathstrut +\mathstrut \) \(32386\) \(\beta_{1}\mathstrut +\mathstrut \) \(140173\)
\(\nu^{15}\)\(=\)\(-\)\(30550\) \(\beta_{15}\mathstrut +\mathstrut \) \(8435\) \(\beta_{14}\mathstrut -\mathstrut \) \(14647\) \(\beta_{13}\mathstrut -\mathstrut \) \(580\) \(\beta_{12}\mathstrut +\mathstrut \) \(67671\) \(\beta_{11}\mathstrut +\mathstrut \) \(11924\) \(\beta_{10}\mathstrut +\mathstrut \) \(11585\) \(\beta_{9}\mathstrut +\mathstrut \) \(155148\) \(\beta_{8}\mathstrut +\mathstrut \) \(44116\) \(\beta_{7}\mathstrut +\mathstrut \) \(57046\) \(\beta_{6}\mathstrut -\mathstrut \) \(52445\) \(\beta_{5}\mathstrut -\mathstrut \) \(111002\) \(\beta_{4}\mathstrut +\mathstrut \) \(123435\) \(\beta_{3}\mathstrut +\mathstrut \) \(310155\) \(\beta_{2}\mathstrut +\mathstrut \) \(296102\) \(\beta_{1}\mathstrut +\mathstrut \) \(204740\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.76723
2.62936
2.41798
1.84800
1.68977
1.55853
1.09349
0.639731
0.298113
0.0296587
−0.415886
−0.823041
−1.82937
−2.04658
−2.36717
−2.48981
−2.76723 0 5.65757 3.65468 0 −1.00000 −10.1213 0 −10.1133
1.2 −2.62936 0 4.91351 0.281520 0 −1.00000 −7.66067 0 −0.740216
1.3 −2.41798 0 3.84663 −2.86533 0 −1.00000 −4.46511 0 6.92831
1.4 −1.84800 0 1.41512 3.49192 0 −1.00000 1.08086 0 −6.45308
1.5 −1.68977 0 0.855315 −2.75353 0 −1.00000 1.93425 0 4.65282
1.6 −1.55853 0 0.429028 1.78044 0 −1.00000 2.44841 0 −2.77487
1.7 −1.09349 0 −0.804288 −1.42245 0 −1.00000 3.06645 0 1.55543
1.8 −0.639731 0 −1.59074 2.86363 0 −1.00000 2.29711 0 −1.83195
1.9 −0.298113 0 −1.91113 −2.54722 0 −1.00000 1.16596 0 0.759360
1.10 −0.0296587 0 −1.99912 −3.07938 0 −1.00000 0.118609 0 0.0913304
1.11 0.415886 0 −1.82704 0.233776 0 −1.00000 −1.59161 0 0.0972242
1.12 0.823041 0 −1.32260 3.74037 0 −1.00000 −2.73464 0 3.07848
1.13 1.82937 0 1.34661 1.45514 0 −1.00000 −1.19530 0 2.66200
1.14 2.04658 0 2.18847 2.05615 0 −1.00000 0.385720 0 4.20807
1.15 2.36717 0 3.60352 −4.44142 0 −1.00000 3.79580 0 −10.5136
1.16 2.48981 0 4.19916 −1.44828 0 −1.00000 5.47548 0 −3.60594
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(127\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{16} + \cdots\)
\(T_{5}^{16} - \cdots\)