Properties

Label 69.5.d.a
Level $69$
Weight $5$
Character orbit 69.d
Analytic conductor $7.133$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 69.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.13252745279\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 5598 x^{14} + 11369517 x^{12} + 11272666128 x^{10} + 5958872960073 x^{8} + 1661250441139200 x^{6} + 215215956313867368 x^{4} + 8236967436040727664 x^{2} + 13861812915684254052\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{9} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{2} ) q^{2} + \beta_{3} q^{3} + ( 9 + \beta_{2} + \beta_{5} ) q^{4} + \beta_{10} q^{5} + ( -2 + \beta_{3} - \beta_{4} ) q^{6} + \beta_{11} q^{7} + ( 24 + \beta_{1} + 6 \beta_{2} - 4 \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{8} + 27 q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{2} ) q^{2} + \beta_{3} q^{3} + ( 9 + \beta_{2} + \beta_{5} ) q^{4} + \beta_{10} q^{5} + ( -2 + \beta_{3} - \beta_{4} ) q^{6} + \beta_{11} q^{7} + ( 24 + \beta_{1} + 6 \beta_{2} - 4 \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{8} + 27 q^{9} + ( \beta_{9} + 3 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} ) q^{10} + ( -\beta_{8} - \beta_{10} + \beta_{12} ) q^{11} + ( -1 - 3 \beta_{1} - 6 \beta_{2} + 9 \beta_{3} - 2 \beta_{4} ) q^{12} + ( 9 + 6 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} + \beta_{7} ) q^{13} + ( \beta_{8} + \beta_{10} + 2 \beta_{11} - \beta_{13} - \beta_{14} ) q^{14} + ( \beta_{9} - \beta_{10} - \beta_{11} - \beta_{14} ) q^{15} + ( 46 + 2 \beta_{1} + 26 \beta_{2} - 5 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{16} + ( -3 \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{17} + ( 27 + 27 \beta_{2} ) q^{18} + ( 2 \beta_{8} + 3 \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{19} + ( \beta_{8} + 21 \beta_{10} - 3 \beta_{11} - 2 \beta_{13} + 4 \beta_{14} - 2 \beta_{15} ) q^{20} + ( \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{13} + \beta_{14} ) q^{21} + ( -4 \beta_{8} + \beta_{9} - 9 \beta_{10} - 5 \beta_{11} - \beta_{12} - 4 \beta_{14} ) q^{22} + ( -48 + \beta_{1} - 3 \beta_{2} - 26 \beta_{3} + 8 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{10} + 2 \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{23} + ( -107 - 6 \beta_{1} + 26 \beta_{3} - 7 \beta_{4} - 9 \beta_{5} - 3 \beta_{7} ) q^{24} + ( -191 - 16 \beta_{1} - 43 \beta_{2} + 18 \beta_{3} - 11 \beta_{4} - 21 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} ) q^{25} + ( 101 + 12 \beta_{1} - 51 \beta_{2} - 19 \beta_{3} - \beta_{4} + 2 \beta_{5} - 5 \beta_{6} + \beta_{7} ) q^{26} + 27 \beta_{3} q^{27} + ( 4 \beta_{8} - 2 \beta_{9} - 12 \beta_{10} + 7 \beta_{11} - \beta_{12} + 3 \beta_{15} ) q^{28} + ( -219 + 8 \beta_{1} - 19 \beta_{2} + 21 \beta_{3} - \beta_{4} - 24 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{29} + ( -4 \beta_{8} - 2 \beta_{9} - 16 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} + 5 \beta_{13} - 3 \beta_{14} + 3 \beta_{15} ) q^{30} + ( -23 - 10 \beta_{1} + 7 \beta_{2} - 23 \beta_{3} - \beta_{4} + 10 \beta_{5} - 3 \beta_{6} + 7 \beta_{7} ) q^{31} + ( 321 + 15 \beta_{1} + 11 \beta_{2} - 61 \beta_{3} + 11 \beta_{4} + 38 \beta_{5} - 8 \beta_{6} + 3 \beta_{7} ) q^{32} + ( -\beta_{8} + 2 \beta_{9} + 7 \beta_{10} - 2 \beta_{11} - 3 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 3 \beta_{15} ) q^{33} + ( -6 \beta_{8} - 10 \beta_{9} - 28 \beta_{10} + 6 \beta_{11} + 4 \beta_{12} + 4 \beta_{13} - \beta_{14} + 8 \beta_{15} ) q^{34} + ( 37 - 5 \beta_{1} - 35 \beta_{2} + 30 \beta_{3} + 4 \beta_{4} + 30 \beta_{5} + 9 \beta_{6} - 2 \beta_{7} ) q^{35} + ( 243 + 27 \beta_{2} + 27 \beta_{5} ) q^{36} + ( \beta_{9} + 23 \beta_{10} - 21 \beta_{11} - \beta_{12} - 8 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{37} + ( 4 \beta_{8} - 12 \beta_{9} + 6 \beta_{10} + 16 \beta_{11} + \beta_{12} + 7 \beta_{13} + 7 \beta_{14} + 6 \beta_{15} ) q^{38} + ( 125 + 15 \beta_{1} - 15 \beta_{2} + 8 \beta_{3} - 2 \beta_{4} - 9 \beta_{6} ) q^{39} + ( 4 \beta_{8} - 5 \beta_{9} + 59 \beta_{10} - 6 \beta_{11} - 10 \beta_{12} - 2 \beta_{13} + 24 \beta_{14} - \beta_{15} ) q^{40} + ( 103 + 6 \beta_{1} + 173 \beta_{2} - 69 \beta_{3} - 9 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} - 13 \beta_{7} ) q^{41} + ( 9 \beta_{8} + \beta_{9} + 8 \beta_{10} - \beta_{11} - \beta_{14} ) q^{42} + ( 4 \beta_{8} - 3 \beta_{9} - 41 \beta_{10} + 8 \beta_{11} - 9 \beta_{12} - 10 \beta_{13} - 10 \beta_{14} - 2 \beta_{15} ) q^{43} + ( -19 \beta_{8} - 12 \beta_{9} - 39 \beta_{10} - 30 \beta_{11} + 5 \beta_{12} + 10 \beta_{13} - 20 \beta_{14} + 4 \beta_{15} ) q^{44} + 27 \beta_{10} q^{45} + ( -102 + 44 \beta_{1} - 61 \beta_{2} - 193 \beta_{3} + 36 \beta_{4} + 22 \beta_{5} + 3 \beta_{7} + 10 \beta_{9} - 38 \beta_{10} - \beta_{11} - 7 \beta_{12} + 8 \beta_{13} - 2 \beta_{14} + 7 \beta_{15} ) q^{46} + ( -476 - 3 \beta_{1} + 218 \beta_{2} + 93 \beta_{3} - 9 \beta_{4} - 24 \beta_{5} - 6 \beta_{6} + 5 \beta_{7} ) q^{47} + ( -146 - 15 \beta_{1} - 129 \beta_{2} + 44 \beta_{3} - 31 \beta_{4} - 18 \beta_{5} - 9 \beta_{6} - 9 \beta_{7} ) q^{48} + ( 400 + 32 \beta_{1} - 178 \beta_{2} - 165 \beta_{3} + 16 \beta_{4} + 21 \beta_{5} + 6 \beta_{6} + 11 \beta_{7} ) q^{49} + ( -1255 - 98 \beta_{1} - 463 \beta_{2} + 276 \beta_{3} - 110 \beta_{4} - 48 \beta_{5} - 18 \beta_{6} - 20 \beta_{7} ) q^{50} + ( \beta_{8} - 3 \beta_{9} + 12 \beta_{10} - 6 \beta_{11} + 12 \beta_{12} - 2 \beta_{13} + 8 \beta_{14} - 6 \beta_{15} ) q^{51} + ( -1315 + 10 \beta_{1} + 13 \beta_{2} - 29 \beta_{3} + 51 \beta_{4} - 16 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{52} + ( 3 \beta_{8} + 24 \beta_{9} - 36 \beta_{10} - 7 \beta_{11} + 5 \beta_{12} + 9 \beta_{13} + 3 \beta_{14} + 4 \beta_{15} ) q^{53} + ( -54 + 27 \beta_{3} - 27 \beta_{4} ) q^{54} + ( 453 + 34 \beta_{1} + 223 \beta_{2} - 177 \beta_{3} + 101 \beta_{4} + 27 \beta_{6} + 13 \beta_{7} ) q^{55} + ( 18 \beta_{8} - 6 \beta_{9} - 46 \beta_{10} + \beta_{11} + 9 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 8 \beta_{15} ) q^{56} + ( 2 \beta_{8} - 18 \beta_{9} + 9 \beta_{10} + 12 \beta_{11} + 6 \beta_{12} + 2 \beta_{13} - 8 \beta_{14} - 3 \beta_{15} ) q^{57} + ( -945 - 60 \beta_{1} - 535 \beta_{2} + 163 \beta_{3} + 13 \beta_{4} - 74 \beta_{5} - 27 \beta_{6} + 5 \beta_{7} ) q^{58} + ( 1285 - 87 \beta_{1} + 19 \beta_{2} + 248 \beta_{3} + 2 \beta_{4} - 82 \beta_{5} - 5 \beta_{6} + 8 \beta_{7} ) q^{59} + ( -8 \beta_{8} - 3 \beta_{9} - 87 \beta_{10} - 24 \beta_{11} + 12 \beta_{12} - 2 \beta_{13} - 28 \beta_{14} - 6 \beta_{15} ) q^{60} + ( -22 \beta_{8} - 9 \beta_{9} - \beta_{10} + 3 \beta_{11} + 9 \beta_{12} - 8 \beta_{13} + 22 \beta_{14} - 16 \beta_{15} ) q^{61} + ( 299 + 84 \beta_{1} + 127 \beta_{2} - 67 \beta_{3} + 5 \beta_{4} - 22 \beta_{5} + 7 \beta_{6} - 3 \beta_{7} ) q^{62} + 27 \beta_{11} q^{63} + ( 63 + 72 \beta_{1} + 429 \beta_{2} - 326 \beta_{3} + 82 \beta_{4} - 109 \beta_{5} - 18 \beta_{6} + 2 \beta_{7} ) q^{64} + ( 9 \beta_{8} + 42 \beta_{9} + 41 \beta_{10} + 31 \beta_{11} - 25 \beta_{12} + 7 \beta_{13} - 11 \beta_{14} ) q^{65} + ( -12 \beta_{8} - \beta_{9} + 73 \beta_{10} - 41 \beta_{11} + 3 \beta_{12} + 6 \beta_{13} + 16 \beta_{14} + 3 \beta_{15} ) q^{66} + ( 12 \beta_{8} + 13 \beta_{9} + 67 \beta_{10} + 35 \beta_{11} - 4 \beta_{12} - 34 \beta_{13} + 7 \beta_{14} - 5 \beta_{15} ) q^{67} + ( 8 \beta_{8} + 18 \beta_{9} - 54 \beta_{10} - 65 \beta_{11} + 13 \beta_{12} - 22 \beta_{13} - 46 \beta_{14} - 2 \beta_{15} ) q^{68} + ( -666 + 6 \beta_{1} - 171 \beta_{2} - 55 \beta_{3} + 3 \beta_{4} - 9 \beta_{5} - 18 \beta_{6} - 6 \beta_{7} + 17 \beta_{9} + 37 \beta_{10} + 19 \beta_{11} - 9 \beta_{13} + \beta_{14} - 9 \beta_{15} ) q^{69} + ( -579 + 20 \beta_{1} + 419 \beta_{2} - 201 \beta_{3} - 77 \beta_{4} + 96 \beta_{5} + 39 \beta_{6} - 7 \beta_{7} ) q^{70} + ( 2551 - 30 \beta_{1} + 165 \beta_{2} + 345 \beta_{3} + 9 \beta_{4} + 42 \beta_{5} + 15 \beta_{6} + 13 \beta_{7} ) q^{71} + ( 648 + 27 \beta_{1} + 162 \beta_{2} - 108 \beta_{3} + 54 \beta_{5} + 27 \beta_{6} ) q^{72} + ( -623 - 108 \beta_{1} - 97 \beta_{2} + 337 \beta_{3} - 89 \beta_{4} + 40 \beta_{5} + 63 \beta_{6} + 11 \beta_{7} ) q^{73} + ( -31 \beta_{8} + 57 \beta_{10} + 40 \beta_{11} - 17 \beta_{12} + 14 \beta_{13} + 44 \beta_{14} - 2 \beta_{15} ) q^{74} + ( 480 + 63 \beta_{1} + 252 \beta_{2} - 194 \beta_{3} + 33 \beta_{4} + 144 \beta_{5} + 36 \beta_{6} - 9 \beta_{7} ) q^{75} + ( 50 \beta_{8} + 5 \beta_{9} + 103 \beta_{10} + 20 \beta_{11} - 32 \beta_{12} - 22 \beta_{13} - 12 \beta_{14} - 23 \beta_{15} ) q^{76} + ( 266 - 82 \beta_{1} + 332 \beta_{2} + 426 \beta_{3} - 196 \beta_{4} + 186 \beta_{5} + 30 \beta_{6} + 14 \beta_{7} ) q^{77} + ( -389 - 6 \beta_{1} + 117 \beta_{2} + 113 \beta_{3} + 77 \beta_{4} - 108 \beta_{5} - 9 \beta_{6} + 15 \beta_{7} ) q^{78} + ( -16 \beta_{8} + 10 \beta_{9} + 82 \beta_{10} + 48 \beta_{11} + 17 \beta_{12} - 4 \beta_{13} - \beta_{14} + 31 \beta_{15} ) q^{79} + ( 37 \beta_{8} + 36 \beta_{9} + 251 \beta_{10} - 17 \beta_{11} - 62 \beta_{12} - 26 \beta_{13} + 76 \beta_{14} - 24 \beta_{15} ) q^{80} + 729 q^{81} + ( 4457 - 164 \beta_{1} + 151 \beta_{2} + 107 \beta_{3} + 49 \beta_{4} + 208 \beta_{5} + 9 \beta_{6} - 7 \beta_{7} ) q^{82} + ( -23 \beta_{8} + 54 \beta_{9} + 71 \beta_{10} + 66 \beta_{11} - 43 \beta_{12} + 12 \beta_{13} + 18 \beta_{14} - 6 \beta_{15} ) q^{83} + ( 25 \beta_{8} - 9 \beta_{9} - 18 \beta_{10} + 42 \beta_{11} - 15 \beta_{12} - 2 \beta_{13} + 8 \beta_{14} + 12 \beta_{15} ) q^{84} + ( 2841 + 252 \beta_{1} + 678 \beta_{2} - 321 \beta_{3} - 36 \beta_{4} + 45 \beta_{5} - 6 \beta_{6} + 3 \beta_{7} ) q^{85} + ( -12 \beta_{8} - 90 \beta_{9} - 170 \beta_{10} + 82 \beta_{11} + 71 \beta_{12} - 5 \beta_{13} - 35 \beta_{14} + 18 \beta_{15} ) q^{86} + ( 571 + 72 \beta_{1} + 207 \beta_{2} - 221 \beta_{3} + 65 \beta_{4} - 72 \beta_{5} + 27 \beta_{6} + 9 \beta_{7} ) q^{87} + ( -66 \beta_{8} + 5 \beta_{9} - 271 \beta_{10} - 89 \beta_{11} + 67 \beta_{12} + 34 \beta_{13} - 34 \beta_{14} + 2 \beta_{15} ) q^{88} + ( -27 \beta_{8} - 42 \beta_{9} - 60 \beta_{10} + 106 \beta_{11} + 37 \beta_{12} + 30 \beta_{13} + 26 \beta_{15} ) q^{89} + ( 27 \beta_{9} + 81 \beta_{10} + 27 \beta_{11} - 27 \beta_{12} + 27 \beta_{14} ) q^{90} + ( -16 \beta_{8} + 30 \beta_{9} + 70 \beta_{10} - 17 \beta_{11} - 9 \beta_{12} + 20 \beta_{13} - 11 \beta_{14} + \beta_{15} ) q^{91} + ( -633 + 147 \beta_{1} + 235 \beta_{2} - 497 \beta_{3} + 283 \beta_{4} + 40 \beta_{5} - 10 \beta_{6} + 15 \beta_{7} - 24 \beta_{9} - 66 \beta_{10} - 85 \beta_{11} + 39 \beta_{12} - 20 \beta_{13} - 50 \beta_{14} + 2 \beta_{15} ) q^{92} + ( -637 - 30 \beta_{1} - 69 \beta_{2} - 13 \beta_{3} - 41 \beta_{4} + 90 \beta_{5} - 63 \beta_{6} + 9 \beta_{7} ) q^{93} + ( 4406 + 4 \beta_{1} - 800 \beta_{2} + 398 \beta_{3} - 86 \beta_{4} + 124 \beta_{5} - 30 \beta_{6} + 2 \beta_{7} ) q^{94} + ( 2094 + 33 \beta_{1} - 336 \beta_{2} - 411 \beta_{3} - 285 \beta_{4} - 108 \beta_{5} - 6 \beta_{6} - 15 \beta_{7} ) q^{95} + ( -1606 - 114 \beta_{1} - 387 \beta_{2} + 315 \beta_{3} - 14 \beta_{4} - 135 \beta_{5} - 27 \beta_{6} + 24 \beta_{7} ) q^{96} + ( 54 \beta_{8} - 32 \beta_{9} + 168 \beta_{10} - 2 \beta_{11} - 28 \beta_{12} - 20 \beta_{14} - 6 \beta_{15} ) q^{97} + ( -3610 + 190 \beta_{1} + 578 \beta_{2} - 417 \beta_{3} + 313 \beta_{4} - 42 \beta_{5} + 27 \beta_{6} + 43 \beta_{7} ) q^{98} + ( -27 \beta_{8} - 27 \beta_{10} + 27 \beta_{12} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 12q^{2} + 144q^{4} - 36q^{6} + 372q^{8} + 432q^{9} + O(q^{10}) \) \( 16q + 12q^{2} + 144q^{4} - 36q^{6} + 372q^{8} + 432q^{9} + 104q^{13} + 680q^{16} + 324q^{18} - 732q^{23} - 1764q^{24} - 2984q^{25} + 1800q^{26} - 3528q^{29} - 400q^{31} + 5244q^{32} + 912q^{35} + 3888q^{36} + 2016q^{39} + 1008q^{41} - 1168q^{46} - 8664q^{47} - 2016q^{48} + 7240q^{49} - 18852q^{50} - 20952q^{52} - 972q^{54} + 6816q^{55} - 13352q^{58} + 20112q^{59} + 4248q^{62} - 896q^{64} - 10044q^{69} - 10680q^{70} + 40368q^{71} + 10044q^{72} - 9568q^{73} + 7560q^{75} + 2952q^{77} - 6912q^{78} + 11664q^{81} + 71800q^{82} + 42744q^{85} + 8352q^{87} - 9876q^{92} - 10008q^{93} + 73720q^{94} + 33312q^{95} - 24948q^{96} - 59052q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 5598 x^{14} + 11369517 x^{12} + 11272666128 x^{10} + 5958872960073 x^{8} + 1661250441139200 x^{6} + 215215956313867368 x^{4} + 8236967436040727664 x^{2} + 13861812915684254052\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-2675203270570026020971652895601675753060283 \nu^{14} - 14238637016034482634824278640516987841374339728 \nu^{12} - 26192365673580586995322259887262118770953428077661 \nu^{10} - 21507212716206070573154316958148524759401132663745398 \nu^{8} - 7819349055663772414439095167381626700214510482651169065 \nu^{6} - 880397130743833011536105597947209156825066402346330865106 \nu^{4} + 58645928039717728795111981034405682378647959592927901735258 \nu^{2} + 3145932605498921257485716032829348712845304905358713480865180\)\()/ \)\(79\!\cdots\!60\)\( \)
\(\beta_{2}\)\(=\)\((\)\(74979708817815580100568529133684064367621 \nu^{14} + 391843880960998010680471559981681220247187065 \nu^{12} + 706424583175334031335446741907549350688776134381 \nu^{10} + 581066345025075725649194768358279464989500003380169 \nu^{8} + 228603537156351314925059100504702947000381969945646339 \nu^{6} + 38208426360372215308296878791107875393565702466009011537 \nu^{4} + 1521051582181854269250176698390092598873076801076607787082 \nu^{2} - 6191538260228464584033023346407458222907767384728136986774\)\()/ \)\(14\!\cdots\!60\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-15349746200334540353527 \nu^{14} - 80195850603628591399624332 \nu^{12} - 144554955968766536897123659545 \nu^{10} - 118967751857615522580039139146114 \nu^{8} - 46903401016840871659401309379922109 \nu^{6} - 7890490487914281036403836803891817390 \nu^{4} - 334193936090047349047209936759267141102 \nu^{2} - 2018443559520599225719154364687153455340\)\()/ \)\(27\!\cdots\!84\)\( \)
\(\beta_{4}\)\(=\)\((\)\(2179648048325358332557275810315347932491617 \nu^{14} + 11393928820647357951521331812263405745694980926 \nu^{12} + 20572661832518073936486312009296773623626666771043 \nu^{10} + 17000509616141889802914365455481342060323438406374620 \nu^{8} + 6740914472364304541900529123083637799038400110588306939 \nu^{6} + 1126481078035330995348201518538110472288156258281854057584 \nu^{4} + 38543745350463364576722104748492782203004005105129448234586 \nu^{2} - 837471246847621107247896231420246830542953592229450765748264\)\()/ \)\(26\!\cdots\!20\)\( \)
\(\beta_{5}\)\(=\)\((\)\(300380652793977411495478681125002739014913 \nu^{14} + 1579381946801409666660137930751001570620070795 \nu^{12} + 2876973569863693789695587983396483161636654247293 \nu^{10} + 2402959732457566676791019601057192694771740016853667 \nu^{8} + 967203834398592207473672492846820073463179387096607967 \nu^{6} + 168627360023514863043429239195827041096056758458415566051 \nu^{4} + 7951121515744760196508547547809607912672357800688537994626 \nu^{2} + 25158321495096119254264914742136200236813506084290989715698\)\()/ \)\(14\!\cdots\!60\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-39425043684255471411746565346178505583166889 \nu^{14} - 203747142201501541899931708207246404772006231696 \nu^{12} - 361142526800360236976938950984228539267845022663295 \nu^{10} - 291646089738079061019199029183680693673448125188724498 \nu^{8} - 113553160503602997021328443919006209958367987042405806947 \nu^{6} - 19468853646276654715660526924001804424214588921232313336998 \nu^{4} - 992698070195202451224942980950897904912224349123570869511970 \nu^{2} - 4601897721448289549116825434263716900744726176389336531342028\)\()/ \)\(79\!\cdots\!60\)\( \)
\(\beta_{7}\)\(=\)\((\)\(97988703565348196873196557650394308600292337 \nu^{14} + 506085408501380280391948685028597535162306656482 \nu^{12} + 894836673766688582037280638717872451952813122449499 \nu^{10} + 716762784231273306753143184871750729964000735725728072 \nu^{8} + 272437910144993581027305748685681283423094667550495865995 \nu^{6} + 43334002397429942915873857321401017752636905914128547142204 \nu^{4} + 1519755429808124465173473148319148789598801020508281177762378 \nu^{2} - 2037873200019207075683850526436648420244770867141739578707680\)\()/ \)\(79\!\cdots\!60\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(14\!\cdots\!69\)\( \nu^{15} + \)\(71\!\cdots\!94\)\( \nu^{13} + \)\(11\!\cdots\!03\)\( \nu^{11} + \)\(83\!\cdots\!24\)\( \nu^{9} + \)\(31\!\cdots\!95\)\( \nu^{7} + \)\(68\!\cdots\!88\)\( \nu^{5} + \)\(75\!\cdots\!06\)\( \nu^{3} + \)\(22\!\cdots\!60\)\( \nu\)\()/ \)\(43\!\cdots\!80\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(18\!\cdots\!21\)\( \nu^{15} - \)\(13\!\cdots\!85\)\( \nu^{13} - \)\(33\!\cdots\!61\)\( \nu^{11} - \)\(42\!\cdots\!09\)\( \nu^{9} - \)\(27\!\cdots\!39\)\( \nu^{7} - \)\(86\!\cdots\!17\)\( \nu^{5} - \)\(11\!\cdots\!02\)\( \nu^{3} - \)\(36\!\cdots\!66\)\( \nu\)\()/ \)\(43\!\cdots\!80\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(68\!\cdots\!23\)\( \nu^{15} + \)\(36\!\cdots\!61\)\( \nu^{13} + \)\(66\!\cdots\!99\)\( \nu^{11} + \)\(57\!\cdots\!09\)\( \nu^{9} + \)\(24\!\cdots\!73\)\( \nu^{7} + \)\(51\!\cdots\!81\)\( \nu^{5} + \)\(41\!\cdots\!18\)\( \nu^{3} + \)\(10\!\cdots\!22\)\( \nu\)\()/ \)\(43\!\cdots\!80\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(69\!\cdots\!26\)\( \nu^{15} + \)\(35\!\cdots\!11\)\( \nu^{13} + \)\(63\!\cdots\!12\)\( \nu^{11} + \)\(51\!\cdots\!21\)\( \nu^{9} + \)\(20\!\cdots\!30\)\( \nu^{7} + \)\(35\!\cdots\!77\)\( \nu^{5} + \)\(19\!\cdots\!04\)\( \nu^{3} + \)\(22\!\cdots\!50\)\( \nu\)\()/ \)\(39\!\cdots\!80\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(87\!\cdots\!72\)\( \nu^{15} + \)\(46\!\cdots\!53\)\( \nu^{13} + \)\(88\!\cdots\!30\)\( \nu^{11} + \)\(80\!\cdots\!99\)\( \nu^{9} + \)\(38\!\cdots\!56\)\( \nu^{7} + \)\(97\!\cdots\!99\)\( \nu^{5} + \)\(12\!\cdots\!60\)\( \nu^{3} + \)\(55\!\cdots\!94\)\( \nu\)\()/ \)\(43\!\cdots\!80\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(15\!\cdots\!00\)\( \nu^{15} + \)\(80\!\cdots\!17\)\( \nu^{13} + \)\(14\!\cdots\!82\)\( \nu^{11} + \)\(12\!\cdots\!79\)\( \nu^{9} + \)\(57\!\cdots\!12\)\( \nu^{7} + \)\(12\!\cdots\!55\)\( \nu^{5} + \)\(11\!\cdots\!24\)\( \nu^{3} + \)\(24\!\cdots\!58\)\( \nu\)\()/ \)\(43\!\cdots\!80\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(85\!\cdots\!55\)\( \nu^{15} + \)\(44\!\cdots\!92\)\( \nu^{13} + \)\(81\!\cdots\!77\)\( \nu^{11} + \)\(67\!\cdots\!14\)\( \nu^{9} + \)\(26\!\cdots\!37\)\( \nu^{7} + \)\(44\!\cdots\!50\)\( \nu^{5} + \)\(17\!\cdots\!34\)\( \nu^{3} + \)\(19\!\cdots\!08\)\( \nu\)\()/ \)\(14\!\cdots\!60\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(51\!\cdots\!04\)\( \nu^{15} - \)\(26\!\cdots\!45\)\( \nu^{13} - \)\(47\!\cdots\!94\)\( \nu^{11} - \)\(38\!\cdots\!51\)\( \nu^{9} - \)\(14\!\cdots\!36\)\( \nu^{7} - \)\(21\!\cdots\!63\)\( \nu^{5} - \)\(21\!\cdots\!68\)\( \nu^{3} + \)\(57\!\cdots\!46\)\( \nu\)\()/ \)\(72\!\cdots\!80\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{13} - 8 \beta_{11} + 7 \beta_{10} + 2 \beta_{9} + \beta_{8}\)\()/18\)
\(\nu^{2}\)\(=\)\((\)\(21 \beta_{7} + 45 \beta_{6} + 84 \beta_{5} - 55 \beta_{4} + 27 \beta_{3} - 237 \beta_{2} + 84 \beta_{1} - 4271\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(426 \beta_{15} - 1024 \beta_{14} - 3632 \beta_{13} - 582 \beta_{12} + 13185 \beta_{11} - 102 \beta_{10} - 2373 \beta_{9} - 2861 \beta_{8}\)\()/18\)
\(\nu^{4}\)\(=\)\((\)\(-20567 \beta_{7} - 39345 \beta_{6} - 94716 \beta_{5} + 37929 \beta_{4} - 202535 \beta_{3} + 176381 \beta_{2} - 71064 \beta_{1} + 2212629\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-141975 \beta_{15} + 1370418 \beta_{14} + 3389218 \beta_{13} + 455706 \beta_{12} - 9569255 \beta_{11} - 3378380 \beta_{10} + 1552625 \beta_{9} + 2485297 \beta_{8}\)\()/6\)
\(\nu^{6}\)\(=\)\(27288554 \beta_{7} + 49257471 \beta_{6} + 133548777 \beta_{5} - 45147776 \beta_{4} + 328431057 \beta_{3} - 244782895 \beta_{2} + 86186138 \beta_{1} - 2377067838\)
\(\nu^{7}\)\(=\)\((\)\(8017965 \beta_{15} - 4002931408 \beta_{14} - 9004377080 \beta_{13} - 952949592 \beta_{12} + 23071420920 \beta_{11} + 11012188122 \beta_{10} - 3686332536 \beta_{9} - 6317250662 \beta_{8}\)\()/6\)
\(\nu^{8}\)\(=\)\((\)\(-141281647023 \beta_{7} - 248827903245 \beta_{6} - 707598369318 \beta_{5} + 222677637249 \beta_{4} - 1789794086967 \beta_{3} + 1327054200987 \beta_{2} - 427503272388 \beta_{1} + 11476285426371\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(133141138584 \beta_{15} + 3574305718014 \beta_{14} + 7799201267100 \beta_{13} + 714774643596 \beta_{12} - 19225389141721 \beta_{11} - 10012991871164 \beta_{10} + 3078213708641 \beta_{9} + 5341875598057 \beta_{8}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(362655908564613 \beta_{7} + 632394156823797 \beta_{6} + 1832708651916054 \beta_{5} - 558579514451499 \beta_{4} + 4667337296434389 \beta_{3} - 3488679532222689 \beta_{2} + 1077607177700994 \beta_{1} - 28752066766690149\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-492599294360709 \beta_{15} - 9295679084727778 \beta_{14} - 20065838970641540 \beta_{13} - 1715828923401606 \beta_{12} + 48698725956035817 \beta_{11} + 26144232397052352 \beta_{10} - 7819352797931001 \beta_{9} - 13585256196522497 \beta_{8}\)\()/2\)
\(\nu^{12}\)\(=\)\(-463995512749714542 \beta_{7} - 805869709295106402 \beta_{6} - 2353095227760965019 \beta_{5} + 707177159730023154 \beta_{4} - 6003681557644354926 \beta_{3} + 4512015210452426559 \beta_{2} - 1368550571266462080 \beta_{1} + 36459891591524222037\)
\(\nu^{13}\)\(=\)\((\)\(1419420363414608193 \beta_{15} + 23909359230389950968 \beta_{14} + 51402800380852730760 \beta_{13} + 4264282606505528454 \beta_{12} - 123968407985302024812 \beta_{11} - 67315429573092805374 \beta_{10} + 19936712477866245798 \beta_{9} + 34620564679963963464 \beta_{8}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(2371724755280485993947 \beta_{7} + 4112533990055524813605 \beta_{6} + 12044591981251656791868 \beta_{5} - 3598062847325349544017 \beta_{4} + 30747557643598040589405 \beta_{3} - 23170647922118394079893 \beta_{2} + 6974468790581537604246 \beta_{1} - 185730759614406029597787\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(-3793627134844744602360 \beta_{15} - 61232554896919874776194 \beta_{14} - 131437281476437175214900 \beta_{13} - 10766265303112818040830 \beta_{12} + 316179990051934359126039 \beta_{11} + 172451078111320418644446 \beta_{10} - 50885904977670996430779 \beta_{9} - 88326154249652795877849 \beta_{8}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
32.2151i
32.2151i
1.32778i
1.32778i
19.7839i
19.7839i
7.67355i
7.67355i
23.3661i
23.3661i
50.5339i
50.5339i
25.3025i
25.3025i
19.1905i
19.1905i
−6.46704 5.19615 25.8226 30.6703i −33.6037 37.9803i −63.5230 27.0000 198.346i
22.2 −6.46704 5.19615 25.8226 30.6703i −33.6037 37.9803i −63.5230 27.0000 198.346i
22.3 −5.20161 −5.19615 11.0568 30.5542i 27.0284 6.50343i 25.7128 27.0000 158.931i
22.4 −5.20161 −5.19615 11.0568 30.5542i 27.0284 6.50343i 25.7128 27.0000 158.931i
22.5 −2.02014 5.19615 −11.9191 11.8600i −10.4969 49.3562i 56.4003 27.0000 23.9589i
22.6 −2.02014 5.19615 −11.9191 11.8600i −10.4969 49.3562i 56.4003 27.0000 23.9589i
22.7 −1.44280 −5.19615 −13.9183 11.4538i 7.49700 12.7987i 43.1661 27.0000 16.5255i
22.8 −1.44280 −5.19615 −13.9183 11.4538i 7.49700 12.7987i 43.1661 27.0000 16.5255i
22.9 3.07267 5.19615 −6.55870 40.4877i 15.9661 40.5328i −69.3154 27.0000 124.405i
22.10 3.07267 5.19615 −6.55870 40.4877i 15.9661 40.5328i −69.3154 27.0000 124.405i
22.11 3.93471 −5.19615 −0.518045 11.5477i −20.4454 67.1686i −64.9937 27.0000 45.4369i
22.12 3.93471 −5.19615 −0.518045 11.5477i −20.4454 67.1686i −64.9937 27.0000 45.4369i
22.13 6.68245 5.19615 28.6552 12.7157i 34.7230 72.2771i 84.5676 27.0000 84.9718i
22.14 6.68245 5.19615 28.6552 12.7157i 34.7230 72.2771i 84.5676 27.0000 84.9718i
22.15 7.44175 −5.19615 39.3796 49.1084i −38.6685 11.1740i 173.985 27.0000 365.452i
22.16 7.44175 −5.19615 39.3796 49.1084i −38.6685 11.1740i 173.985 27.0000 365.452i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.5.d.a 16
3.b odd 2 1 207.5.d.c 16
4.b odd 2 1 1104.5.c.c 16
23.b odd 2 1 inner 69.5.d.a 16
69.c even 2 1 207.5.d.c 16
92.b even 2 1 1104.5.c.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.5.d.a 16 1.a even 1 1 trivial
69.5.d.a 16 23.b odd 2 1 inner
207.5.d.c 16 3.b odd 2 1
207.5.d.c 16 69.c even 2 1
1104.5.c.c 16 4.b odd 2 1
1104.5.c.c 16 92.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(69, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 58948 + 39576 T - 16108 T^{2} - 8736 T^{3} + 2037 T^{4} + 438 T^{5} - 82 T^{6} - 6 T^{7} + T^{8} )^{2} \)
$3$ \( ( -27 + T^{2} )^{8} \)
$5$ \( \)\(13\!\cdots\!68\)\( + 43612925227643919168 T^{2} + 546292304197024608 T^{4} + 3429272086394544 T^{6} + 11249509308948 T^{8} + 18733188384 T^{10} + 15902820 T^{12} + 6492 T^{14} + T^{16} \)
$7$ \( \)\(11\!\cdots\!12\)\( + \)\(46\!\cdots\!76\)\( T^{2} + 54678896334705098448 T^{4} + 239438440649798352 T^{6} + 368900302989924 T^{8} + 261514405008 T^{10} + 92291328 T^{12} + 15588 T^{14} + T^{16} \)
$11$ \( \)\(21\!\cdots\!32\)\( + \)\(27\!\cdots\!80\)\( T^{2} + \)\(20\!\cdots\!04\)\( T^{4} + \)\(55\!\cdots\!84\)\( T^{6} + 6249120446446598160 T^{8} + 355203921384432 T^{10} + 10779977988 T^{12} + 165168 T^{14} + T^{16} \)
$13$ \( ( 4394524804672 - 3732129932800 T + 473106325072 T^{2} + 62636036624 T^{3} + 1441593988 T^{4} + 648032 T^{5} - 81608 T^{6} - 52 T^{7} + T^{8} )^{2} \)
$17$ \( \)\(16\!\cdots\!12\)\( + \)\(16\!\cdots\!52\)\( T^{2} + \)\(12\!\cdots\!12\)\( T^{4} + \)\(38\!\cdots\!20\)\( T^{6} + \)\(61\!\cdots\!92\)\( T^{8} + 56643640073680176 T^{10} + 300924141984 T^{12} + 853452 T^{14} + T^{16} \)
$19$ \( \)\(10\!\cdots\!12\)\( + \)\(23\!\cdots\!72\)\( T^{2} + \)\(19\!\cdots\!32\)\( T^{4} + \)\(73\!\cdots\!76\)\( T^{6} + \)\(13\!\cdots\!84\)\( T^{8} + 126833221469012256 T^{10} + 581622510852 T^{12} + 1253844 T^{14} + T^{16} \)
$23$ \( \)\(37\!\cdots\!21\)\( + \)\(98\!\cdots\!92\)\( T + \)\(41\!\cdots\!12\)\( T^{2} + \)\(71\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!08\)\( T^{4} + \)\(33\!\cdots\!32\)\( T^{5} + \)\(70\!\cdots\!68\)\( T^{6} + \)\(20\!\cdots\!88\)\( T^{7} + \)\(34\!\cdots\!78\)\( T^{8} + 74188023829300207668 T^{9} + 90165168024204328 T^{10} + 154519526053692 T^{11} + 261170649628 T^{12} + 417031860 T^{13} + 865432 T^{14} + 732 T^{15} + T^{16} \)
$29$ \( ( \)\(58\!\cdots\!96\)\( - 47814679869995804928 T + 28230051061209152 T^{2} + 446442759765312 T^{3} - 252904089792 T^{4} - 1711093584 T^{5} - 332764 T^{6} + 1764 T^{7} + T^{8} )^{2} \)
$31$ \( ( \)\(11\!\cdots\!76\)\( + 41343730281031826432 T - 213167666787009536 T^{2} - 454361209445632 T^{3} + 1292616030352 T^{4} + 1027271024 T^{5} - 2869820 T^{6} + 200 T^{7} + T^{8} )^{2} \)
$37$ \( \)\(20\!\cdots\!28\)\( + \)\(36\!\cdots\!44\)\( T^{2} + \)\(26\!\cdots\!48\)\( T^{4} + \)\(64\!\cdots\!96\)\( T^{6} + \)\(70\!\cdots\!60\)\( T^{8} + \)\(39\!\cdots\!72\)\( T^{10} + 118292730210756 T^{12} + 17675712 T^{14} + T^{16} \)
$41$ \( ( -\)\(23\!\cdots\!96\)\( + \)\(67\!\cdots\!56\)\( T - 36140980923363659200 T^{2} - 47424708048980736 T^{3} + 40054879989504 T^{4} + 9738664224 T^{5} - 11447068 T^{6} - 504 T^{7} + T^{8} )^{2} \)
$43$ \( \)\(16\!\cdots\!52\)\( + \)\(52\!\cdots\!20\)\( T^{2} + \)\(98\!\cdots\!60\)\( T^{4} + \)\(71\!\cdots\!76\)\( T^{6} + \)\(26\!\cdots\!92\)\( T^{8} + \)\(54\!\cdots\!48\)\( T^{10} + 638091140184612 T^{12} + 39542100 T^{14} + T^{16} \)
$47$ \( ( \)\(66\!\cdots\!56\)\( + \)\(53\!\cdots\!24\)\( T + 11715885615784223648 T^{2} - 659166986051568 T^{3} - 25559567250588 T^{4} - 19921408320 T^{5} + 438812 T^{6} + 4332 T^{7} + T^{8} )^{2} \)
$53$ \( \)\(77\!\cdots\!52\)\( + \)\(48\!\cdots\!48\)\( T^{2} + \)\(73\!\cdots\!36\)\( T^{4} + \)\(46\!\cdots\!28\)\( T^{6} + \)\(14\!\cdots\!88\)\( T^{8} + \)\(22\!\cdots\!44\)\( T^{10} + 1780474781310948 T^{12} + 68163804 T^{14} + T^{16} \)
$59$ \( ( -\)\(21\!\cdots\!68\)\( - \)\(59\!\cdots\!64\)\( T + \)\(69\!\cdots\!32\)\( T^{2} - 1189685333022424704 T^{3} - 777636292460412 T^{4} + 276579769680 T^{5} + 1627040 T^{6} - 10056 T^{7} + T^{8} )^{2} \)
$61$ \( \)\(70\!\cdots\!68\)\( + \)\(74\!\cdots\!28\)\( T^{2} + \)\(18\!\cdots\!32\)\( T^{4} + \)\(14\!\cdots\!40\)\( T^{6} + \)\(28\!\cdots\!36\)\( T^{8} + \)\(23\!\cdots\!88\)\( T^{10} + 8850677579288388 T^{12} + 153434208 T^{14} + T^{16} \)
$67$ \( \)\(75\!\cdots\!52\)\( + \)\(54\!\cdots\!00\)\( T^{2} + \)\(14\!\cdots\!44\)\( T^{4} + \)\(46\!\cdots\!20\)\( T^{6} + \)\(59\!\cdots\!12\)\( T^{8} + \)\(35\!\cdots\!20\)\( T^{10} + 11099153592423108 T^{12} + 169460340 T^{14} + T^{16} \)
$71$ \( ( \)\(53\!\cdots\!08\)\( - \)\(18\!\cdots\!00\)\( T - \)\(53\!\cdots\!84\)\( T^{2} + 3962152689270481920 T^{3} - 461836762083072 T^{4} - 368826084240 T^{5} + 145305164 T^{6} - 20184 T^{7} + T^{8} )^{2} \)
$73$ \( ( -\)\(12\!\cdots\!56\)\( - \)\(14\!\cdots\!04\)\( T + \)\(18\!\cdots\!56\)\( T^{2} + 30001103101443513344 T^{3} + 3544278551932816 T^{4} - 753518077696 T^{5} - 123565220 T^{6} + 4784 T^{7} + T^{8} )^{2} \)
$79$ \( \)\(13\!\cdots\!88\)\( + \)\(28\!\cdots\!84\)\( T^{2} + \)\(17\!\cdots\!44\)\( T^{4} + \)\(24\!\cdots\!32\)\( T^{6} + \)\(13\!\cdots\!64\)\( T^{8} + \)\(38\!\cdots\!64\)\( T^{10} + 54299196895089024 T^{12} + 374178612 T^{14} + T^{16} \)
$83$ \( \)\(17\!\cdots\!52\)\( + \)\(41\!\cdots\!60\)\( T^{2} + \)\(39\!\cdots\!28\)\( T^{4} + \)\(19\!\cdots\!16\)\( T^{6} + \)\(55\!\cdots\!00\)\( T^{8} + \)\(91\!\cdots\!08\)\( T^{10} + 88854184408664772 T^{12} + 462287472 T^{14} + T^{16} \)
$89$ \( \)\(37\!\cdots\!32\)\( + \)\(50\!\cdots\!80\)\( T^{2} + \)\(96\!\cdots\!40\)\( T^{4} + \)\(77\!\cdots\!24\)\( T^{6} + \)\(31\!\cdots\!40\)\( T^{8} + \)\(71\!\cdots\!80\)\( T^{10} + 84583916839236096 T^{12} + 479187324 T^{14} + T^{16} \)
$97$ \( \)\(21\!\cdots\!08\)\( + \)\(77\!\cdots\!16\)\( T^{2} + \)\(10\!\cdots\!88\)\( T^{4} + \)\(62\!\cdots\!16\)\( T^{6} + \)\(17\!\cdots\!84\)\( T^{8} + \)\(25\!\cdots\!60\)\( T^{10} + 189473573625782976 T^{12} + 702957984 T^{14} + T^{16} \)
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