Properties

Label 2151.2.a.k
Level $2151$
Weight $2$
Character orbit 2151.a
Self dual yes
Analytic conductor $17.176$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1758214748\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + 229 x^{12} - 14020 x^{11} + 3356 x^{10} + 23404 x^{9} - 9429 x^{8} - 21252 x^{7} + 10479 x^{6} + 9108 x^{5} - 4844 x^{4} - 1184 x^{3} + 640 x^{2} - 56 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( 1 + \beta_{9} ) q^{5} -\beta_{12} q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( 1 + \beta_{9} ) q^{5} -\beta_{12} q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{7} + \beta_{9} ) q^{10} + ( 1 + \beta_{8} ) q^{11} + ( \beta_{1} - \beta_{4} ) q^{13} + ( 1 + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{12} - \beta_{13} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{14} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} - \beta_{13} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{16} + ( 2 - \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{12} + \beta_{13} ) q^{17} + ( -\beta_{2} - \beta_{5} + \beta_{13} ) q^{19} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{6} - \beta_{8} + \beta_{10} ) q^{20} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{16} ) q^{22} + ( 1 - \beta_{1} - \beta_{5} - \beta_{9} - \beta_{12} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{23} + ( 2 + 2 \beta_{1} - \beta_{4} + \beta_{7} + 2 \beta_{9} + \beta_{14} + \beta_{16} - \beta_{17} - \beta_{19} ) q^{25} + ( 3 - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{11} - \beta_{16} - \beta_{17} ) q^{26} + ( -3 + 2 \beta_{1} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} - 2 \beta_{11} - \beta_{13} - \beta_{14} - \beta_{15} + \beta_{17} + \beta_{19} ) q^{28} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{29} + ( -\beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{12} + \beta_{13} + 2 \beta_{15} - \beta_{17} + \beta_{18} ) q^{31} + ( -1 + \beta_{1} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} - 2 \beta_{15} - \beta_{16} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{32} + ( 1 - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{14} - \beta_{16} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{34} + ( \beta_{1} - \beta_{4} - \beta_{6} - \beta_{8} - \beta_{10} - 2 \beta_{11} - \beta_{12} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{35} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{16} - \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{37} + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{7} + \beta_{8} - \beta_{10} + 2 \beta_{12} + \beta_{15} ) q^{38} + ( -1 + 3 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} - \beta_{15} - \beta_{19} ) q^{40} + ( 4 + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} + 2 \beta_{16} - \beta_{18} ) q^{41} + ( -2 + 3 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} - 2 \beta_{15} - \beta_{16} + 2 \beta_{17} - \beta_{18} - \beta_{19} ) q^{43} + ( 1 - 4 \beta_{1} - \beta_{2} + \beta_{3} + 5 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + 4 \beta_{12} + \beta_{13} - \beta_{14} - 2 \beta_{16} + 2 \beta_{18} + 2 \beta_{19} ) q^{44} + ( 1 + 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{8} - 3 \beta_{9} + \beta_{10} + \beta_{11} - 3 \beta_{12} + 2 \beta_{14} + \beta_{15} + 2 \beta_{16} - \beta_{18} - \beta_{19} ) q^{46} + ( 2 + 2 \beta_{4} + \beta_{5} - \beta_{7} + 3 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{16} - \beta_{17} ) q^{47} + ( 2 + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{12} + \beta_{13} + \beta_{16} + \beta_{17} - 2 \beta_{18} ) q^{49} + ( 3 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 3 \beta_{7} + 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} - 2 \beta_{17} - 2 \beta_{19} ) q^{50} + ( -\beta_{1} + 2 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{11} - \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} - \beta_{16} - 2 \beta_{17} + 2 \beta_{18} ) q^{52} + ( 1 - \beta_{1} - \beta_{3} - \beta_{7} - \beta_{9} + \beta_{12} - \beta_{13} + \beta_{18} + \beta_{19} ) q^{53} + ( 1 - \beta_{2} + \beta_{4} + 3 \beta_{8} + \beta_{9} + \beta_{11} - \beta_{13} - \beta_{15} - 2 \beta_{16} - \beta_{17} + \beta_{19} ) q^{55} + ( 2 + \beta_{1} + 4 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - \beta_{7} - 2 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{14} + 3 \beta_{16} + 3 \beta_{17} - 4 \beta_{18} - \beta_{19} ) q^{56} + ( 3 + 3 \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + 3 \beta_{7} - \beta_{9} + \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + \beta_{13} + 2 \beta_{14} + 2 \beta_{15} + \beta_{16} - 3 \beta_{17} - 2 \beta_{19} ) q^{58} + ( 2 - 2 \beta_{1} - \beta_{3} - \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{17} - \beta_{18} ) q^{59} + ( -1 - 5 \beta_{1} + 4 \beta_{4} + 3 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} + \beta_{11} + 3 \beta_{12} - \beta_{14} - \beta_{16} + \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{61} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - \beta_{16} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{62} + ( 2 \beta_{1} - \beta_{3} + 3 \beta_{5} - 2 \beta_{6} - \beta_{9} - \beta_{11} - 2 \beta_{14} + \beta_{16} + 2 \beta_{17} - 2 \beta_{18} ) q^{64} + ( 2 + 4 \beta_{1} - \beta_{2} - 2 \beta_{4} + 2 \beta_{8} + \beta_{9} + \beta_{11} - 2 \beta_{13} + \beta_{14} - 3 \beta_{15} - \beta_{16} - \beta_{18} - \beta_{19} ) q^{65} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} + \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{67} + ( \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{8} - \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{16} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{68} + ( 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{15} + 4 \beta_{16} + 3 \beta_{17} - 4 \beta_{18} - \beta_{19} ) q^{70} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{71} + ( 2 + 2 \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} - \beta_{14} + \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{73} + ( 3 - 4 \beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} - \beta_{16} - 2 \beta_{17} + 2 \beta_{18} + 2 \beta_{19} ) q^{74} + ( -4 - 3 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + 3 \beta_{8} - 2 \beta_{10} + 3 \beta_{12} - \beta_{14} - \beta_{15} - 3 \beta_{16} - \beta_{17} + 3 \beta_{18} + 2 \beta_{19} ) q^{76} + ( 1 - 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{13} - \beta_{14} - 2 \beta_{16} + 2 \beta_{19} ) q^{77} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{7} + 3 \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{16} - \beta_{17} ) q^{79} + ( 4 + 6 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{13} + 3 \beta_{16} + \beta_{17} - 3 \beta_{18} - 3 \beta_{19} ) q^{80} + ( -3 + 6 \beta_{1} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} + \beta_{16} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{82} + ( -2 + 2 \beta_{1} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} - 2 \beta_{15} + \beta_{16} + 3 \beta_{17} - \beta_{18} - \beta_{19} ) q^{83} + ( -\beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} - 3 \beta_{8} - \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{85} + ( -2 + 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - 3 \beta_{11} + \beta_{12} - \beta_{13} - 3 \beta_{14} - \beta_{15} + \beta_{16} + 2 \beta_{17} - \beta_{18} ) q^{86} + ( -2 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + 4 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + 3 \beta_{12} + 2 \beta_{13} - \beta_{14} - 2 \beta_{16} + \beta_{17} + 2 \beta_{18} + 3 \beta_{19} ) q^{88} + ( 4 + 4 \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{8} + 2 \beta_{9} - 2 \beta_{11} + 3 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{89} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{8} + \beta_{11} - 2 \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{15} + 2 \beta_{16} + \beta_{17} - 3 \beta_{18} - \beta_{19} ) q^{91} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} - 3 \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} - 4 \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{92} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{14} - \beta_{15} + \beta_{16} + 3 \beta_{17} - \beta_{18} ) q^{94} + ( 1 - \beta_{1} - 4 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{7} + \beta_{8} + \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} - 2 \beta_{17} + 2 \beta_{18} + 2 \beta_{19} ) q^{95} + ( \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{12} - 2 \beta_{13} + \beta_{14} + 2 \beta_{16} + 2 \beta_{17} - \beta_{18} ) q^{97} + ( -1 + 2 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{14} - 2 \beta_{15} - 3 \beta_{16} - \beta_{17} + 3 \beta_{18} + \beta_{19} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + O(q^{10}) \) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + 4q^{10} + 12q^{11} - 4q^{13} + 20q^{14} + 22q^{16} + 24q^{17} - 4q^{19} + 40q^{20} - 6q^{22} + 12q^{23} + 22q^{25} + 30q^{26} - 12q^{28} + 24q^{29} - 4q^{31} + 28q^{32} + 8q^{34} + 20q^{35} - 10q^{37} + 26q^{38} + 6q^{40} + 66q^{41} + 8q^{43} + 36q^{44} - 12q^{46} + 28q^{47} + 18q^{49} + 28q^{50} - 18q^{52} + 28q^{53} - 4q^{55} + 60q^{56} + 54q^{59} - 4q^{61} + 20q^{62} + 22q^{64} + 42q^{65} + 12q^{67} + 12q^{68} + 20q^{70} + 36q^{71} + 14q^{73} - 50q^{76} + 8q^{77} - 12q^{79} + 88q^{80} - 8q^{82} + 20q^{83} + 4q^{85} + 18q^{86} - 10q^{88} + 130q^{89} - 6q^{91} - 46q^{92} - 26q^{94} - 2q^{97} + 12q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + 229 x^{12} - 14020 x^{11} + 3356 x^{10} + 23404 x^{9} - 9429 x^{8} - 21252 x^{7} + 10479 x^{6} + 9108 x^{5} - 4844 x^{4} - 1184 x^{3} + 640 x^{2} - 56 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\((\)\(-8259046 \nu^{19} + 25715994 \nu^{18} + 170049029 \nu^{17} - 526870261 \nu^{16} - 1318767164 \nu^{15} + 3879963697 \nu^{14} + 4637406235 \nu^{13} - 10641470398 \nu^{12} - 5690004117 \nu^{11} - 8255925475 \nu^{10} - 9466063919 \nu^{9} + 110440481462 \nu^{8} + 44748634260 \nu^{7} - 233246301320 \nu^{6} - 75991923150 \nu^{5} + 196387146324 \nu^{4} + 62363096099 \nu^{3} - 59487470803 \nu^{2} - 16518890109 \nu + 2324646198\)\()/ 615206542 \)
\(\beta_{5}\)\(=\)\((\)\(-14412353 \nu^{19} + 53211591 \nu^{18} + 303862297 \nu^{17} - 1266785187 \nu^{16} - 2385590232 \nu^{15} + 12309098593 \nu^{14} + 7821689951 \nu^{13} - 63597294912 \nu^{12} - 1856788512 \nu^{11} + 190290241568 \nu^{10} - 59687153306 \nu^{9} - 335996303347 \nu^{8} + 168783410560 \nu^{7} + 340443884242 \nu^{6} - 189016862749 \nu^{5} - 182021298167 \nu^{4} + 78826196104 \nu^{3} + 42654573730 \nu^{2} - 3285022906 \nu - 1169865603\)\()/ 615206542 \)
\(\beta_{6}\)\(=\)\((\)\(-25152759 \nu^{19} + 155801032 \nu^{18} + 337358422 \nu^{17} - 3732397225 \nu^{16} + 554113853 \nu^{15} + 36468511186 \nu^{14} - 33169211667 \nu^{13} - 189004335507 \nu^{12} + 240555769633 \nu^{11} + 564343295095 \nu^{10} - 821292093320 \nu^{9} - 984891173929 \nu^{8} + 1487550147038 \nu^{7} + 968327695840 \nu^{6} - 1389923753527 \nu^{5} - 476065715156 \nu^{4} + 573846563225 \nu^{3} + 80270642859 \nu^{2} - 63595522316 \nu + 4109545279\)\()/ 615206542 \)
\(\beta_{7}\)\(=\)\((\)\(-18194582 \nu^{19} + 67002479 \nu^{18} + 389694649 \nu^{17} - 1608799145 \nu^{16} - 3118198456 \nu^{15} + 15676712010 \nu^{14} + 10579309020 \nu^{13} - 80281652938 \nu^{12} - 4793202241 \nu^{11} + 233192405803 \nu^{10} - 69698642148 \nu^{9} - 385701840578 \nu^{8} + 204676228239 \nu^{7} + 343079978087 \nu^{6} - 238050299893 \nu^{5} - 138330331238 \nu^{4} + 114948803841 \nu^{3} + 13748437164 \nu^{2} - 15016860419 \nu + 435415269\)\()/ 307603271 \)
\(\beta_{8}\)\(=\)\((\)\(79748961 \nu^{19} - 301609179 \nu^{18} - 1707908090 \nu^{17} + 7153614242 \nu^{16} + 14002650970 \nu^{15} - 68713614510 \nu^{14} - 53462767366 \nu^{13} + 346201298514 \nu^{12} + 79045302167 \nu^{11} - 988002485813 \nu^{10} + 70776349317 \nu^{9} + 1607226024841 \nu^{8} - 406229547534 \nu^{7} - 1418565745122 \nu^{6} + 545824666009 \nu^{5} + 586935486325 \nu^{4} - 312635355213 \nu^{3} - 68965179349 \nu^{2} + 61187604393 \nu - 5283572531\)\()/ 1230413084 \)
\(\beta_{9}\)\(=\)\((\)\(-32909587 \nu^{19} + 80534179 \nu^{18} + 838637985 \nu^{17} - 1930987718 \nu^{16} - 8936959131 \nu^{15} + 18748215845 \nu^{14} + 52196104835 \nu^{13} - 95322242093 \nu^{12} - 183140190454 \nu^{11} + 273459017045 \nu^{10} + 396206848876 \nu^{9} - 443707767420 \nu^{8} - 519105112175 \nu^{7} + 384965658988 \nu^{6} + 383186074902 \nu^{5} - 154604743387 \nu^{4} - 133521035197 \nu^{3} + 20700322923 \nu^{2} + 13079021136 \nu - 1325315495\)\()/ 307603271 \)
\(\beta_{10}\)\(=\)\((\)\(165811103 \nu^{19} - 636904335 \nu^{18} - 3626875370 \nu^{17} + 15370014952 \nu^{16} + 30755798996 \nu^{15} - 150919713262 \nu^{14} - 125091051332 \nu^{13} + 781865754296 \nu^{12} + 223739180405 \nu^{11} - 2312306020795 \nu^{10} + 10235198061 \nu^{9} + 3941510138311 \nu^{8} - 660569594262 \nu^{7} - 3712320960866 \nu^{6} + 950270360355 \nu^{5} + 1712651972849 \nu^{4} - 497624917207 \nu^{3} - 278185798363 \nu^{2} + 73661197513 \nu - 234896285\)\()/ 1230413084 \)
\(\beta_{11}\)\(=\)\((\)\(167543794 \nu^{19} - 625032345 \nu^{18} - 3630490016 \nu^{17} + 14940203236 \nu^{16} + 30200134326 \nu^{15} - 144695688234 \nu^{14} - 117401664390 \nu^{13} + 734184616508 \nu^{12} + 178571636242 \nu^{11} - 2099123586481 \nu^{10} + 148889232756 \nu^{9} + 3370944216337 \nu^{8} - 891165421092 \nu^{7} - 2829691525926 \nu^{6} + 1145957832622 \nu^{5} + 1006992862467 \nu^{4} - 563447135410 \nu^{3} - 54991098473 \nu^{2} + 75649845204 \nu - 9584403627\)\()/ 1230413084 \)
\(\beta_{12}\)\(=\)\((\)\(-191187961 \nu^{19} + 812153314 \nu^{18} + 3922186244 \nu^{17} - 19617511978 \nu^{16} - 29167126050 \nu^{15} + 193013398266 \nu^{14} + 82628335904 \nu^{13} - 1003246916152 \nu^{12} + 55963406827 \nu^{11} + 2979330609416 \nu^{10} - 907371831707 \nu^{9} - 5092905104298 \nu^{8} + 2194123680594 \nu^{7} + 4777975720548 \nu^{6} - 2310123795985 \nu^{5} - 2148992184846 \nu^{4} + 1041865182955 \nu^{3} + 310811596208 \nu^{2} - 137345767347 \nu + 8319810742\)\()/ 1230413084 \)
\(\beta_{13}\)\(=\)\((\)\(-230262066 \nu^{19} + 792118541 \nu^{18} + 5255031164 \nu^{17} - 19043970058 \nu^{16} - 47988197224 \nu^{15} + 185896992694 \nu^{14} + 225813453148 \nu^{13} - 953933229802 \nu^{12} - 583595480738 \nu^{11} + 2776224670379 \nu^{10} + 806330690734 \nu^{9} - 4601849814699 \nu^{8} - 505963606680 \nu^{7} + 4128749475290 \nu^{6} + 28588928198 \nu^{5} - 1761383572467 \nu^{4} + 90722687714 \nu^{3} + 265089961095 \nu^{2} - 20820501810 \nu - 6936694175\)\()/ 1230413084 \)
\(\beta_{14}\)\(=\)\((\)\(-148218686 \nu^{19} + 540557051 \nu^{18} + 3269848589 \nu^{17} - 12943039301 \nu^{16} - 28156991606 \nu^{15} + 125696279845 \nu^{14} + 118362276789 \nu^{13} - 640863691862 \nu^{12} - 234690327751 \nu^{11} + 1849614742124 \nu^{10} + 98257492451 \nu^{9} - 3030189040325 \nu^{8} + 396576234042 \nu^{7} + 2663213045448 \nu^{6} - 658526239498 \nu^{5} - 1073350264885 \nu^{4} + 355842728817 \nu^{3} + 119016332276 \nu^{2} - 51667771349 \nu + 5574045831\)\()/ 615206542 \)
\(\beta_{15}\)\(=\)\((\)\(349773159 \nu^{19} - 1538748210 \nu^{18} - 6932521542 \nu^{17} + 36955819636 \nu^{16} + 47527928106 \nu^{15} - 361128470164 \nu^{14} - 94162854506 \nu^{13} + 1862951048504 \nu^{12} - 396106182055 \nu^{11} - 5493115468418 \nu^{10} + 2517671450063 \nu^{9} + 9357507914978 \nu^{8} - 5421598473634 \nu^{7} - 8855184929684 \nu^{6} + 5417893882787 \nu^{5} + 4162998154950 \nu^{4} - 2308907122843 \nu^{3} - 714054374770 \nu^{2} + 262319755187 \nu - 7156850618\)\()/ 1230413084 \)
\(\beta_{16}\)\(=\)\((\)\(-409558402 \nu^{19} + 1641156191 \nu^{18} + 8685638574 \nu^{17} - 39676028714 \nu^{16} - 69029091930 \nu^{15} + 390583010652 \nu^{14} + 237254998916 \nu^{13} - 2030486662212 \nu^{12} - 163978080660 \nu^{11} + 6030009661713 \nu^{10} - 1211519056514 \nu^{9} - 10319684222695 \nu^{8} + 3665469555820 \nu^{7} + 9743414760590 \nu^{6} - 4170396163194 \nu^{5} - 4499164209241 \nu^{4} + 1929328806792 \nu^{3} + 731216944173 \nu^{2} - 243115237582 \nu + 7398565297\)\()/ 1230413084 \)
\(\beta_{17}\)\(=\)\((\)\(598257155 \nu^{19} - 2412992855 \nu^{18} - 12540986368 \nu^{17} + 57939652746 \nu^{16} + 97829257148 \nu^{15} - 565548619752 \nu^{14} - 323375601322 \nu^{13} + 2909343190140 \nu^{12} + 158040286339 \nu^{11} - 8528444005017 \nu^{10} + 1879234764699 \nu^{9} + 14360787887307 \nu^{8} - 5299261526434 \nu^{7} - 13279130648006 \nu^{6} + 5831232262335 \nu^{5} + 5948526383965 \nu^{4} - 2641800836793 \nu^{3} - 909185235221 \nu^{2} + 330179570559 \nu - 14477391829\)\()/ 1230413084 \)
\(\beta_{18}\)\(=\)\((\)\(339515531 \nu^{19} - 1337024169 \nu^{18} - 7181512587 \nu^{17} + 32022014849 \nu^{16} + 57068773546 \nu^{15} - 311403994141 \nu^{14} - 198680269865 \nu^{13} + 1593298102484 \nu^{12} + 167872279748 \nu^{11} - 4633920382372 \nu^{10} + 836812332066 \nu^{9} + 7712530360549 \nu^{8} - 2622318172054 \nu^{7} - 7002436929558 \nu^{6} + 2968266256213 \nu^{5} + 3031788732943 \nu^{4} - 1375942998806 \nu^{3} - 421428522916 \nu^{2} + 184808475650 \nu - 10645201535\)\()/ 615206542 \)
\(\beta_{19}\)\(=\)\((\)\(-796840493 \nu^{19} + 3030854544 \nu^{18} + 17248139170 \nu^{17} - 72932486780 \nu^{16} - 142898221962 \nu^{15} + 713538822384 \nu^{14} + 547600302602 \nu^{13} - 3679266043972 \nu^{12} - 772152880431 \nu^{11} + 10809765842264 \nu^{10} - 998078437201 \nu^{9} - 18235080824624 \nu^{8} + 4797063610670 \nu^{7} + 16858150511980 \nu^{6} - 5950105648445 \nu^{5} - 7497534043164 \nu^{4} + 2848228155609 \nu^{3} + 1111002194104 \nu^{2} - 373123885017 \nu + 17183676468\)\()/ 1230413084 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{18} + \beta_{17} + \beta_{16} - \beta_{13} + \beta_{10} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + 7 \beta_{2} + \beta_{1} + 14\)
\(\nu^{5}\)\(=\)\(\beta_{19} + \beta_{18} + \beta_{17} - \beta_{16} - 2 \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} - 2 \beta_{11} - \beta_{10} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{4} + 8 \beta_{3} + 29 \beta_{1} - 1\)
\(\nu^{6}\)\(=\)\(-12 \beta_{18} + 12 \beta_{17} + 11 \beta_{16} - 2 \beta_{14} - 10 \beta_{13} - \beta_{11} + 10 \beta_{10} - \beta_{9} - 10 \beta_{7} - 12 \beta_{6} + 13 \beta_{5} - 11 \beta_{3} + 46 \beta_{2} + 12 \beta_{1} + 76\)
\(\nu^{7}\)\(=\)\(13 \beta_{19} + 11 \beta_{18} + 15 \beta_{17} - 12 \beta_{16} - 25 \beta_{15} - 15 \beta_{14} - 12 \beta_{13} + 14 \beta_{12} - 26 \beta_{11} - 13 \beta_{10} + 13 \beta_{9} - 15 \beta_{7} - 12 \beta_{6} + \beta_{5} + 14 \beta_{4} + 55 \beta_{3} - \beta_{2} + 179 \beta_{1} - 16\)
\(\nu^{8}\)\(=\)\(-110 \beta_{18} + 113 \beta_{17} + 94 \beta_{16} - 3 \beta_{15} - 29 \beta_{14} - 84 \beta_{13} + 4 \beta_{12} - 17 \beta_{11} + 77 \beta_{10} - 10 \beta_{9} + 3 \beta_{8} - 84 \beta_{7} - 109 \beta_{6} + 124 \beta_{5} + 2 \beta_{4} - 95 \beta_{3} + 303 \beta_{2} + 107 \beta_{1} + 442\)
\(\nu^{9}\)\(=\)\(122 \beta_{19} + 88 \beta_{18} + 156 \beta_{17} - 106 \beta_{16} - 230 \beta_{15} - 156 \beta_{14} - 111 \beta_{13} + 140 \beta_{12} - 247 \beta_{11} - 124 \beta_{10} + 124 \beta_{9} + 6 \beta_{8} - 155 \beta_{7} - 111 \beta_{6} + 22 \beta_{5} + 140 \beta_{4} + 368 \beta_{3} - 20 \beta_{2} + 1152 \beta_{1} - 179\)
\(\nu^{10}\)\(=\)\(3 \beta_{19} - 906 \beta_{18} + 962 \beta_{17} + 735 \beta_{16} - 54 \beta_{15} - 297 \beta_{14} - 663 \beta_{13} + 70 \beta_{12} - 191 \beta_{11} + 547 \beta_{10} - 70 \beta_{9} + 50 \beta_{8} - 670 \beta_{7} - 891 \beta_{6} + 1047 \beta_{5} + 39 \beta_{4} - 756 \beta_{3} + 2018 \beta_{2} + 843 \beta_{1} + 2674\)
\(\nu^{11}\)\(=\)\(1017 \beta_{19} + 631 \beta_{18} + 1401 \beta_{17} - 842 \beta_{16} - 1891 \beta_{15} - 1400 \beta_{14} - 930 \beta_{13} + 1232 \beta_{12} - 2080 \beta_{11} - 1051 \beta_{10} + 1050 \beta_{9} + 112 \beta_{8} - 1382 \beta_{7} - 932 \beta_{6} + 289 \beta_{5} + 1233 \beta_{4} + 2461 \beta_{3} - 256 \beta_{2} + 7625 \beta_{1} - 1719\)
\(\nu^{12}\)\(=\)\(71 \beta_{19} - 7057 \beta_{18} + 7741 \beta_{17} + 5503 \beta_{16} - 638 \beta_{15} - 2651 \beta_{14} - 5060 \beta_{13} + 825 \beta_{12} - 1805 \beta_{11} + 3769 \beta_{10} - 414 \beta_{9} + 568 \beta_{8} - 5208 \beta_{7} - 6912 \beta_{6} + 8315 \beta_{5} + 502 \beta_{4} - 5785 \beta_{3} + 13581 \beta_{2} + 6234 \beta_{1} + 16610\)
\(\nu^{13}\)\(=\)\(8020 \beta_{19} + 4328 \beta_{18} + 11664 \beta_{17} - 6391 \beta_{16} - 14735 \beta_{15} - 11634 \beta_{14} - 7390 \beta_{13} + 10167 \beta_{12} - 16483 \beta_{11} - 8396 \beta_{10} + 8373 \beta_{9} + 1370 \beta_{8} - 11433 \beta_{7} - 7432 \beta_{6} + 3031 \beta_{5} + 10187 \beta_{4} + 16564 \beta_{3} - 2683 \beta_{2} + 51412 \beta_{1} - 15143\)
\(\nu^{14}\)\(=\)\(1031 \beta_{19} - 53193 \beta_{18} + 60120 \beta_{17} + 40187 \beta_{16} - 6269 \beta_{15} - 22040 \beta_{14} - 37808 \beta_{13} + 8224 \beta_{12} - 15535 \beta_{11} + 25655 \beta_{10} - 2126 \beta_{9} + 5527 \beta_{8} - 39798 \beta_{7} - 52035 \beta_{6} + 63771 \beta_{5} + 5379 \beta_{4} - 43266 \beta_{3} + 92166 \beta_{2} + 44509 \beta_{1} + 105213\)
\(\nu^{15}\)\(=\)\(61387 \beta_{19} + 29207 \beta_{18} + 92831 \beta_{17} - 47505 \beta_{16} - 111466 \beta_{15} - 92352 \beta_{14} - 56804 \beta_{13} + 80819 \beta_{12} - 126089 \beta_{11} - 64836 \beta_{10} + 64496 \beta_{9} + 13967 \beta_{8} - 90569 \beta_{7} - 57335 \beta_{6} + 28122 \beta_{5} + 81094 \beta_{4} + 112385 \beta_{3} - 25133 \beta_{2} + 350875 \beta_{1} - 126191\)
\(\nu^{16}\)\(=\)\(11935 \beta_{19} - 392736 \beta_{18} + 455945 \beta_{17} + 288926 \beta_{16} - 55638 \beta_{15} - 175615 \beta_{14} - 278448 \beta_{13} + 74825 \beta_{12} - 126220 \beta_{11} + 173857 \beta_{10} - 8875 \beta_{9} + 49622 \beta_{8} - 300273 \beta_{7} - 384390 \beta_{6} + 478633 \beta_{5} + 51948 \beta_{4} - 318763 \beta_{3} + 629515 \beta_{2} + 311402 \beta_{1} + 676831\)
\(\nu^{17}\)\(=\)\(461694 \beta_{19} + 196752 \beta_{18} + 717438 \beta_{17} - 349721 \beta_{16} - 828105 \beta_{15} - 711768 \beta_{14} - 426864 \beta_{13} + 627260 \beta_{12} - 943132 \beta_{11} - 490244 \beta_{10} + 486148 \beta_{9} + 129009 \beta_{8} - 698133 \beta_{7} - 432103 \beta_{6} + 242137 \beta_{5} + 630520 \beta_{4} + 768609 \beta_{3} - 219439 \beta_{2} + 2413799 \beta_{1} - 1012104\)
\(\nu^{18}\)\(=\)\(121329 \beta_{19} - 2860243 \beta_{18} + 3400864 \beta_{17} + 2055926 \beta_{16} - 463661 \beta_{15} - 1360816 \beta_{14} - 2029634 \beta_{13} + 643242 \beta_{12} - 986936 \beta_{11} + 1177131 \beta_{10} - 19724 \beta_{9} + 424437 \beta_{8} - 2243051 \beta_{7} - 2803667 \beta_{6} + 3541678 \beta_{5} + 469347 \beta_{4} - 2323830 \beta_{3} + 4321105 \beta_{2} + 2152885 \beta_{1} + 4409800\)
\(\nu^{19}\)\(=\)\(3434226 \beta_{19} + 1333646 \beta_{18} + 5432860 \beta_{17} - 2563761 \beta_{16} - 6079625 \beta_{15} - 5376569 \beta_{14} - 3156382 \beta_{13} + 4789799 \beta_{12} - 6948307 \beta_{11} - 3655836 \beta_{10} + 3612101 \beta_{9} + 1122042 \beta_{8} - 5284162 \beta_{7} - 3200668 \beta_{6} + 1985377 \beta_{5} + 4825302 \beta_{4} + 5294928 \beta_{3} - 1827712 \beta_{2} + 16693786 \beta_{1} - 7897579\)

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
−2.66148
−2.12763
−2.02621
−1.88454
−1.38158
−1.25296
−0.921026
−0.510429
0.0242456
0.0888996
0.244320
0.854685
0.946263
1.22231
1.61701
1.66953
2.39589
2.41299
2.61567
2.67404
−2.66148 0 5.08345 2.46126 0 −5.08484 −8.20653 0 −6.55057
1.2 −2.12763 0 2.52683 2.10198 0 0.789580 −1.12089 0 −4.47224
1.3 −2.02621 0 2.10552 3.48073 0 1.37004 −0.213806 0 −7.05267
1.4 −1.88454 0 1.55150 −1.57704 0 −2.47195 0.845212 0 2.97201
1.5 −1.38158 0 −0.0912347 −1.35155 0 −1.50510 2.88921 0 1.86728
1.6 −1.25296 0 −0.430099 −1.67805 0 −0.437804 3.04481 0 2.10252
1.7 −0.921026 0 −1.15171 1.07676 0 3.99910 2.90281 0 −0.991725
1.8 −0.510429 0 −1.73946 1.97336 0 −1.30841 1.90873 0 −1.00726
1.9 0.0242456 0 −1.99941 −2.24441 0 −4.21532 −0.0969680 0 −0.0544170
1.10 0.0888996 0 −1.99210 0.674348 0 0.687902 −0.354896 0 0.0599493
1.11 0.244320 0 −1.94031 4.22460 0 0.0748971 −0.962697 0 1.03216
1.12 0.854685 0 −1.26951 −2.82017 0 3.33105 −2.79440 0 −2.41036
1.13 0.946263 0 −1.10459 4.23503 0 2.19590 −2.93776 0 4.00746
1.14 1.22231 0 −0.505952 0.403117 0 −4.45690 −3.06306 0 0.492736
1.15 1.61701 0 0.614711 1.46591 0 3.83563 −2.24002 0 2.37038
1.16 1.66953 0 0.787326 −3.19534 0 −2.48662 −2.02459 0 −5.33471
1.17 2.39589 0 3.74030 4.10699 0 −3.62464 4.16958 0 9.83991
1.18 2.41299 0 3.82251 −0.666388 0 0.549662 4.39769 0 −1.60799
1.19 2.61567 0 4.84173 2.80971 0 3.73688 7.43303 0 7.34928
1.20 2.67404 0 5.15050 0.519163 0 1.02094 8.42455 0 1.38826
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(239\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2151.2.a.k yes 20
3.b odd 2 1 2151.2.a.j 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2151.2.a.j 20 3.b odd 2 1
2151.2.a.k yes 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2151))\):

\(T_{2}^{20} - \cdots\)
\(T_{5}^{20} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 56 T + 640 T^{2} - 1184 T^{3} - 4844 T^{4} + 9108 T^{5} + 10479 T^{6} - 21252 T^{7} - 9429 T^{8} + 23404 T^{9} + 3356 T^{10} - 14020 T^{11} + 229 T^{12} + 4804 T^{13} - 540 T^{14} - 936 T^{15} + 164 T^{16} + 96 T^{17} - 21 T^{18} - 4 T^{19} + T^{20} \)
$3$ \( T^{20} \)
$5$ \( -78784 + 446336 T - 539584 T^{2} - 1119728 T^{3} + 2771933 T^{4} - 315580 T^{5} - 3183445 T^{6} + 1814628 T^{7} + 1367627 T^{8} - 1358932 T^{9} - 125829 T^{10} + 434624 T^{11} - 72922 T^{12} - 62284 T^{13} + 22471 T^{14} + 2548 T^{15} - 2271 T^{16} + 212 T^{17} + 67 T^{18} - 16 T^{19} + T^{20} \)
$7$ \( -24064 + 377344 T - 623424 T^{2} - 2062112 T^{3} + 5283336 T^{4} + 267016 T^{5} - 7678796 T^{6} + 2298084 T^{7} + 4793126 T^{8} - 1782552 T^{9} - 1582428 T^{10} + 540512 T^{11} + 298773 T^{12} - 81170 T^{13} - 32927 T^{14} + 6402 T^{15} + 2091 T^{16} - 254 T^{17} - 71 T^{18} + 4 T^{19} + T^{20} \)
$11$ \( -735404 + 5927504 T + 2158580 T^{2} - 121705004 T^{3} + 333492181 T^{4} - 274997872 T^{5} - 74716698 T^{6} + 200093180 T^{7} - 50715478 T^{8} - 41211180 T^{9} + 21550057 T^{10} + 1669232 T^{11} - 2882328 T^{12} + 337008 T^{13} + 143450 T^{14} - 34204 T^{15} - 1868 T^{16} + 1108 T^{17} - 45 T^{18} - 12 T^{19} + T^{20} \)
$13$ \( -55588096 - 671521536 T - 804061312 T^{2} + 1604441280 T^{3} + 1740089840 T^{4} - 1603958528 T^{5} - 1026634744 T^{6} + 630443800 T^{7} + 287015328 T^{8} - 122982896 T^{9} - 44926398 T^{10} + 13314386 T^{11} + 4199567 T^{12} - 831078 T^{13} - 237453 T^{14} + 29422 T^{15} + 7871 T^{16} - 542 T^{17} - 139 T^{18} + 4 T^{19} + T^{20} \)
$17$ \( -31861696 + 154334720 T + 393118128 T^{2} - 1292123688 T^{3} - 298480871 T^{4} + 2363839072 T^{5} - 1342679605 T^{6} - 510325904 T^{7} + 677447169 T^{8} - 117945840 T^{9} - 82810353 T^{10} + 37525452 T^{11} - 994894 T^{12} - 2385764 T^{13} + 452067 T^{14} + 27420 T^{15} - 16669 T^{16} + 1260 T^{17} + 131 T^{18} - 24 T^{19} + T^{20} \)
$19$ \( 3041439616 + 3958838656 T - 7998798016 T^{2} - 9000652608 T^{3} + 8908750168 T^{4} + 7048661608 T^{5} - 5385758264 T^{6} - 2120067084 T^{7} + 1710089242 T^{8} + 165961092 T^{9} - 228546670 T^{10} + 365236 T^{11} + 15858313 T^{12} - 592532 T^{13} - 630877 T^{14} + 28738 T^{15} + 14652 T^{16} - 560 T^{17} - 186 T^{18} + 4 T^{19} + T^{20} \)
$23$ \( -83221504 + 373662720 T + 143640576 T^{2} - 1575570304 T^{3} + 355010576 T^{4} + 2067147808 T^{5} - 571994784 T^{6} - 1234452120 T^{7} + 256273092 T^{8} + 356125644 T^{9} - 45624008 T^{10} - 48498150 T^{11} + 4165703 T^{12} + 3245784 T^{13} - 230685 T^{14} - 110566 T^{15} + 7786 T^{16} + 1846 T^{17} - 140 T^{18} - 12 T^{19} + T^{20} \)
$29$ \( 3203296 - 68652192 T + 236646608 T^{2} + 154913412 T^{3} - 1799466551 T^{4} + 2236072788 T^{5} + 1618436518 T^{6} - 6169430988 T^{7} + 5852742391 T^{8} - 2362511208 T^{9} + 167611702 T^{10} + 178891332 T^{11} - 49070298 T^{12} - 1137348 T^{13} + 1908452 T^{14} - 155580 T^{15} - 23975 T^{16} + 3712 T^{17} + 16 T^{18} - 24 T^{19} + T^{20} \)
$31$ \( 10178867008 + 103828470656 T + 247610176096 T^{2} + 140850923184 T^{3} - 130559397275 T^{4} - 140540619078 T^{5} + 9779602561 T^{6} + 40076139720 T^{7} + 4222457408 T^{8} - 4902250930 T^{9} - 848611006 T^{10} + 284378616 T^{11} + 58396830 T^{12} - 8664670 T^{13} - 1940327 T^{14} + 142476 T^{15} + 33636 T^{16} - 1194 T^{17} - 292 T^{18} + 4 T^{19} + T^{20} \)
$37$ \( 1484607744 + 26409003264 T + 128132050944 T^{2} + 72892591808 T^{3} - 239435924080 T^{4} - 235596064768 T^{5} + 46294717904 T^{6} + 87819349480 T^{7} + 3534003572 T^{8} - 11499558076 T^{9} - 1120419296 T^{10} + 691457922 T^{11} + 77716373 T^{12} - 21596282 T^{13} - 2456945 T^{14} + 360386 T^{15} + 39694 T^{16} - 3028 T^{17} - 318 T^{18} + 10 T^{19} + T^{20} \)
$41$ \( -14710729952 + 502667338560 T - 1227836157616 T^{2} - 688471986480 T^{3} + 2683318808688 T^{4} - 1439503749312 T^{5} - 417608514348 T^{6} + 667550649220 T^{7} - 226345289334 T^{8} + 4550456168 T^{9} + 16470733266 T^{10} - 4276951376 T^{11} + 209246965 T^{12} + 95076954 T^{13} - 20317586 T^{14} + 1276244 T^{15} + 96923 T^{16} - 23148 T^{17} + 1776 T^{18} - 66 T^{19} + T^{20} \)
$43$ \( -14800745004544 + 38987773290496 T + 5268506801920 T^{2} - 43867327505408 T^{3} + 10131909678080 T^{4} + 9868972483328 T^{5} - 2609686871248 T^{6} - 1028407678112 T^{7} + 261762187646 T^{8} + 59618367880 T^{9} - 13798507102 T^{10} - 2043876658 T^{11} + 420293697 T^{12} + 41889264 T^{13} - 7581773 T^{14} - 498610 T^{15} + 79226 T^{16} + 3142 T^{17} - 440 T^{18} - 8 T^{19} + T^{20} \)
$47$ \( 168178221056 + 7201797439488 T - 23763653779456 T^{2} + 18779811930112 T^{3} + 6989788740672 T^{4} - 12244236337344 T^{5} + 2500474927968 T^{6} + 1125993932160 T^{7} - 398200411148 T^{8} - 29526116748 T^{9} + 22268285088 T^{10} - 554548410 T^{11} - 605754753 T^{12} + 46048242 T^{13} + 8152697 T^{14} - 976324 T^{15} - 41555 T^{16} + 8824 T^{17} - 91 T^{18} - 28 T^{19} + T^{20} \)
$53$ \( -1686381404704 + 8634725122720 T - 7283403266480 T^{2} - 24585563325536 T^{3} + 52572503908784 T^{4} - 33582396681192 T^{5} + 3721516294652 T^{6} + 3138556960816 T^{7} - 863600599618 T^{8} - 66482135528 T^{9} + 42861820344 T^{10} - 1329475378 T^{11} - 950863863 T^{12} + 77760164 T^{13} + 10122067 T^{14} - 1301822 T^{15} - 38560 T^{16} + 9754 T^{17} - 122 T^{18} - 28 T^{19} + T^{20} \)
$59$ \( -599715727156736 + 899163528456704 T + 34405864328192 T^{2} - 597212647574208 T^{3} + 243326203694512 T^{4} + 44595742432176 T^{5} - 41181687417024 T^{6} + 2752259463160 T^{7} + 2442245248468 T^{8} - 428133345628 T^{9} - 51740149188 T^{10} + 17583622504 T^{11} - 262309857 T^{12} - 291456466 T^{13} + 23616644 T^{14} + 1516380 T^{15} - 271847 T^{16} + 5976 T^{17} + 810 T^{18} - 54 T^{19} + T^{20} \)
$61$ \( 2509733584926884 + 2300207570935064 T - 1278438835270480 T^{2} - 1253775537839700 T^{3} + 312695223577341 T^{4} + 246307886678642 T^{5} - 48888987781879 T^{6} - 21228605797416 T^{7} + 4346420215136 T^{8} + 786112610286 T^{9} - 185654476790 T^{10} - 11972725072 T^{11} + 3782033982 T^{12} + 70710782 T^{13} - 40767775 T^{14} + 424 T^{15} + 242840 T^{16} - 1494 T^{17} - 764 T^{18} + 4 T^{19} + T^{20} \)
$67$ \( 78953104580608 + 273178398023680 T - 1929455344896000 T^{2} + 314423726103552 T^{3} + 705591752248064 T^{4} - 64543482476032 T^{5} - 95034020721856 T^{6} + 1558485976896 T^{7} + 5688248024064 T^{8} + 124728079072 T^{9} - 176038383664 T^{10} - 7039329612 T^{11} + 3069053289 T^{12} + 140566160 T^{13} - 31661212 T^{14} - 1360520 T^{15} + 194494 T^{16} + 6448 T^{17} - 668 T^{18} - 12 T^{19} + T^{20} \)
$71$ \( -49968319344987136 + 7359183101873152 T + 22379742326361600 T^{2} - 1030774350011136 T^{3} - 3614543458982336 T^{4} + 7030079929792 T^{5} + 277385457716096 T^{6} + 181488851648 T^{7} - 12046269842052 T^{8} + 183160832448 T^{9} + 313870646256 T^{10} - 12016261860 T^{11} - 4786940725 T^{12} + 303063272 T^{13} + 38263987 T^{14} - 3534964 T^{15} - 116633 T^{16} + 18676 T^{17} - 137 T^{18} - 36 T^{19} + T^{20} \)
$73$ \( -8242928417788672 + 18774218617403648 T + 7460842411691776 T^{2} - 7395707546662656 T^{3} - 1650677474587760 T^{4} + 961843178818800 T^{5} + 102502986340424 T^{6} - 58317233666272 T^{7} - 2344124231384 T^{8} + 1930120690996 T^{9} - 7480406 T^{10} - 37297557716 T^{11} + 964609847 T^{12} + 427263706 T^{13} - 18480569 T^{14} - 2822566 T^{15} + 156356 T^{16} + 9854 T^{17} - 634 T^{18} - 14 T^{19} + T^{20} \)
$79$ \( -27457749050074528 - 7851847385408800 T + 18131336822453968 T^{2} + 6477683637286496 T^{3} - 2682274644016464 T^{4} - 1084107327251288 T^{5} + 154827237453188 T^{6} + 78971531969304 T^{7} - 3130052916946 T^{8} - 2975337541052 T^{9} - 36492734274 T^{10} + 60923339714 T^{11} + 2525523953 T^{12} - 676746176 T^{13} - 39826020 T^{14} + 3924654 T^{15} + 273454 T^{16} - 11092 T^{17} - 857 T^{18} + 12 T^{19} + T^{20} \)
$83$ \( -979929064644044 + 19464110561091776 T + 17120104151475908 T^{2} - 4612145663978540 T^{3} - 5720387030676003 T^{4} + 219281505290800 T^{5} + 591099246591190 T^{6} - 23553376573232 T^{7} - 29278447650556 T^{8} + 2306320464008 T^{9} + 674877665741 T^{10} - 84887338400 T^{11} - 5200432244 T^{12} + 1121988624 T^{13} - 234558 T^{14} - 6731560 T^{15} + 175566 T^{16} + 18848 T^{17} - 761 T^{18} - 20 T^{19} + T^{20} \)
$89$ \( \)\(11\!\cdots\!36\)\( - 48967818399886188544 T - 8725745830857481216 T^{2} + 6881271015718381056 T^{3} - 532775467255492096 T^{4} - 255988533757039232 T^{5} + 49952106256557296 T^{6} + 1560889528776840 T^{7} - 1223026116172422 T^{8} + 85208764043820 T^{9} + 9504492062604 T^{10} - 1711568000022 T^{11} + 44804366803 T^{12} + 9278763260 T^{13} - 913504348 T^{14} + 17489892 T^{15} + 2279292 T^{16} - 194348 T^{17} + 7045 T^{18} - 130 T^{19} + T^{20} \)
$97$ \( -3450021631132928 + 14734390439709440 T - 12141215702952576 T^{2} - 1963235496962432 T^{3} + 3135632893459120 T^{4} + 39372759005328 T^{5} - 304913741155056 T^{6} + 4590712407256 T^{7} + 14672741234180 T^{8} - 307455906932 T^{9} - 391747297476 T^{10} + 7733865120 T^{11} + 6157821585 T^{12} - 96011310 T^{13} - 57796780 T^{14} + 604804 T^{15} + 313509 T^{16} - 1784 T^{17} - 888 T^{18} + 2 T^{19} + T^{20} \)
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