Properties

Label 4235.2.a.bp
Level $4235$
Weight $2$
Character orbit 4235.a
Self dual yes
Analytic conductor $33.817$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4235.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.8166452560\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 2 x^{17} - 28 x^{16} + 54 x^{15} + 317 x^{14} - 580 x^{13} - 1874 x^{12} + 3158 x^{11} + 6290 x^{10} - 9228 x^{9} - 12411 x^{8} + 14224 x^{7} + 14618 x^{6} - 10744 x^{5} - 9817 x^{4} + 3146 x^{3} + 3048 x^{2} - 16 x - 176\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 385)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{7} q^{3} + ( 1 + \beta_{2} ) q^{4} + q^{5} + ( -\beta_{1} - \beta_{4} + \beta_{5} - \beta_{11} - \beta_{14} + \beta_{16} ) q^{6} + q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} + ( 2 + \beta_{15} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{7} q^{3} + ( 1 + \beta_{2} ) q^{4} + q^{5} + ( -\beta_{1} - \beta_{4} + \beta_{5} - \beta_{11} - \beta_{14} + \beta_{16} ) q^{6} + q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} + ( 2 + \beta_{15} ) q^{9} + \beta_{1} q^{10} + ( 1 - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{10} + \beta_{14} ) q^{12} -\beta_{6} q^{13} + \beta_{1} q^{14} + \beta_{7} q^{15} + ( 3 + \beta_{1} + \beta_{2} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{11} - \beta_{13} ) q^{16} + ( 2 \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{11} - \beta_{16} ) q^{17} + ( -1 + \beta_{1} + \beta_{6} + \beta_{7} - 3 \beta_{8} - \beta_{10} - 2 \beta_{11} + 2 \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{18} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{12} + \beta_{13} ) q^{19} + ( 1 + \beta_{2} ) q^{20} + \beta_{7} q^{21} + ( -1 - \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} + \beta_{14} - \beta_{15} - \beta_{16} ) q^{23} + ( 1 - \beta_{1} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} - 2 \beta_{14} + \beta_{15} + \beta_{16} ) q^{24} + q^{25} + ( \beta_{1} + \beta_{4} - \beta_{8} + \beta_{9} + \beta_{11} - \beta_{13} + \beta_{14} ) q^{26} + ( \beta_{1} + \beta_{3} - \beta_{5} + 2 \beta_{7} + \beta_{8} + \beta_{12} - \beta_{13} + \beta_{14} + \beta_{17} ) q^{27} + ( 1 + \beta_{2} ) q^{28} + ( \beta_{1} - \beta_{2} - \beta_{5} - \beta_{8} + \beta_{11} - \beta_{12} ) q^{29} + ( -\beta_{1} - \beta_{4} + \beta_{5} - \beta_{11} - \beta_{14} + \beta_{16} ) q^{30} + ( 3 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{12} - \beta_{14} + \beta_{15} + \beta_{17} ) q^{31} + ( 1 + 4 \beta_{1} + \beta_{3} - \beta_{5} + 3 \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} ) q^{32} + ( 4 - \beta_{1} - 2 \beta_{7} + \beta_{9} + \beta_{11} - \beta_{12} - \beta_{14} ) q^{34} + q^{35} + ( 4 - 5 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + 3 \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{36} + ( 2 - \beta_{1} - \beta_{3} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{14} - \beta_{17} ) q^{37} + ( -2 + \beta_{1} + \beta_{5} - \beta_{7} - 2 \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{17} ) q^{38} + ( -3 - \beta_{1} - \beta_{2} + 2 \beta_{5} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} ) q^{39} + ( 2 \beta_{1} + \beta_{3} ) q^{40} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{8} - \beta_{9} + \beta_{11} + 2 \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} ) q^{41} + ( -\beta_{1} - \beta_{4} + \beta_{5} - \beta_{11} - \beta_{14} + \beta_{16} ) q^{42} + ( \beta_{1} + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{17} ) q^{43} + ( 2 + \beta_{15} ) q^{45} + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{7} - \beta_{8} - 2 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{16} + \beta_{17} ) q^{46} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{5} + 2 \beta_{7} - \beta_{11} + \beta_{13} - \beta_{15} + \beta_{16} ) q^{47} + ( -2 + 2 \beta_{1} + \beta_{4} - 3 \beta_{5} + \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{12} + 3 \beta_{13} + \beta_{15} - \beta_{16} ) q^{48} + q^{49} + \beta_{1} q^{50} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{14} + 2 \beta_{17} ) q^{51} + ( \beta_{1} + \beta_{2} + \beta_{4} + 3 \beta_{5} - 2 \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{52} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} - \beta_{11} + \beta_{12} - \beta_{14} - \beta_{17} ) q^{53} + ( 1 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{6} + 3 \beta_{8} - \beta_{9} - 3 \beta_{11} + \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - \beta_{15} + 3 \beta_{16} - \beta_{17} ) q^{54} + ( 2 \beta_{1} + \beta_{3} ) q^{56} + ( 1 - 2 \beta_{1} - \beta_{3} + \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - 3 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{57} + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} - \beta_{14} + \beta_{17} ) q^{58} + ( 1 - 2 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} ) q^{59} + ( 1 - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{10} + \beta_{14} ) q^{60} + ( 2 \beta_{2} - \beta_{5} - \beta_{8} - \beta_{9} + \beta_{13} - \beta_{16} ) q^{61} + ( 4 - \beta_{2} - 2 \beta_{4} - \beta_{6} - \beta_{9} + \beta_{10} - 3 \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{17} ) q^{62} + ( 2 + \beta_{15} ) q^{63} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} + 3 \beta_{8} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{64} -\beta_{6} q^{65} + ( 2 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{16} ) q^{67} + ( -3 + 5 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} + 2 \beta_{14} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{68} + ( 4 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - \beta_{16} ) q^{69} + \beta_{1} q^{70} + ( -1 - 2 \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{8} - \beta_{11} - \beta_{14} ) q^{71} + ( -6 + 6 \beta_{1} - 2 \beta_{2} + \beta_{3} - 3 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} - 4 \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{11} + 2 \beta_{13} - \beta_{14} + 2 \beta_{15} - \beta_{17} ) q^{72} + ( \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} + 3 \beta_{13} - 2 \beta_{14} - \beta_{17} ) q^{73} + ( -2 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{74} + \beta_{7} q^{75} + ( 3 + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{76} + ( -2 - 6 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - 3 \beta_{13} + \beta_{14} - 2 \beta_{15} + \beta_{16} + \beta_{17} ) q^{78} + ( -3 \beta_{1} + 3 \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} - \beta_{17} ) q^{79} + ( 3 + \beta_{1} + \beta_{2} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{11} - \beta_{13} ) q^{80} + ( 4 + \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - \beta_{7} - 3 \beta_{8} - \beta_{11} + \beta_{13} - \beta_{14} + 3 \beta_{15} - \beta_{16} ) q^{81} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{17} ) q^{82} + ( 2 \beta_{1} + \beta_{4} + 2 \beta_{6} - 2 \beta_{8} - \beta_{10} + 2 \beta_{12} - 2 \beta_{13} + \beta_{15} - 2 \beta_{17} ) q^{83} + ( 1 - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{10} + \beta_{14} ) q^{84} + ( 2 \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{11} - \beta_{16} ) q^{85} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + 3 \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} + \beta_{17} ) q^{86} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{87} + ( 3 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + \beta_{15} + \beta_{16} ) q^{89} + ( -1 + \beta_{1} + \beta_{6} + \beta_{7} - 3 \beta_{8} - \beta_{10} - 2 \beta_{11} + 2 \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{90} -\beta_{6} q^{91} + ( -3 - 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - \beta_{4} + 4 \beta_{5} + 2 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + 3 \beta_{14} - \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{92} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} + 2 \beta_{13} - 3 \beta_{14} + \beta_{15} + \beta_{16} + 2 \beta_{17} ) q^{93} + ( -2 - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{10} - 2 \beta_{13} + \beta_{16} + \beta_{17} ) q^{94} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{12} + \beta_{13} ) q^{95} + ( 3 - 6 \beta_{1} - 3 \beta_{4} + 4 \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{10} - 2 \beta_{11} - \beta_{12} + 5 \beta_{13} - 3 \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} ) q^{96} + ( 3 + \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{97} + \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 2q^{2} + 5q^{3} + 24q^{4} + 18q^{5} - q^{6} + 18q^{7} + 6q^{8} + 37q^{9} + O(q^{10}) \) \( 18q + 2q^{2} + 5q^{3} + 24q^{4} + 18q^{5} - q^{6} + 18q^{7} + 6q^{8} + 37q^{9} + 2q^{10} + 15q^{12} + 8q^{13} + 2q^{14} + 5q^{15} + 44q^{16} - 5q^{17} + 2q^{18} + 15q^{19} + 24q^{20} + 5q^{21} + 4q^{23} + 8q^{24} + 18q^{25} + 14q^{26} + 20q^{27} + 24q^{28} - 6q^{29} - q^{30} + 22q^{31} - 6q^{32} + 44q^{34} + 18q^{35} + 83q^{36} + 26q^{37} - 11q^{38} - 38q^{39} + 6q^{40} + 7q^{41} - q^{42} + 10q^{43} + 37q^{45} - 40q^{46} - q^{47} - 15q^{48} + 18q^{49} + 2q^{50} - 11q^{51} + 18q^{52} + 23q^{53} + 13q^{54} + 6q^{56} - 16q^{57} + 2q^{58} + 30q^{59} + 15q^{60} + 17q^{61} + 57q^{62} + 37q^{63} + 64q^{64} + 8q^{65} + 29q^{67} - 66q^{68} + 54q^{69} + 2q^{70} - 2q^{71} - 77q^{72} + 3q^{73} - 48q^{74} + 5q^{75} + 47q^{76} + 10q^{78} - 18q^{79} + 44q^{80} + 110q^{81} + 56q^{82} + 9q^{83} + 15q^{84} - 5q^{85} + 25q^{86} + 23q^{87} + 59q^{89} + 2q^{90} + 8q^{91} - 74q^{92} + 33q^{93} - 19q^{94} + 15q^{95} + 14q^{96} + 30q^{97} + 2q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} - 2 x^{17} - 28 x^{16} + 54 x^{15} + 317 x^{14} - 580 x^{13} - 1874 x^{12} + 3158 x^{11} + 6290 x^{10} - 9228 x^{9} - 12411 x^{8} + 14224 x^{7} + 14618 x^{6} - 10744 x^{5} - 9817 x^{4} + 3146 x^{3} + 3048 x^{2} - 16 x - 176\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 6 \nu \)
\(\beta_{4}\)\(=\)\((\)\(-816960 \nu^{17} + 2255867 \nu^{16} + 21055270 \nu^{15} - 60082628 \nu^{14} - 210456368 \nu^{13} + 632416299 \nu^{12} + 1019559172 \nu^{11} - 3339503298 \nu^{10} - 2433071454 \nu^{9} + 9309964908 \nu^{8} + 2576263422 \nu^{7} - 13389191277 \nu^{6} - 1022378476 \nu^{5} + 9357207550 \nu^{4} + 351411224 \nu^{3} - 2761902073 \nu^{2} - 216452570 \nu + 168084952\)\()/2227868\)
\(\beta_{5}\)\(=\)\((\)\(1319945 \nu^{17} - 3626264 \nu^{16} - 34314866 \nu^{15} + 96973182 \nu^{14} + 347707617 \nu^{13} - 1026800452 \nu^{12} - 1724705334 \nu^{11} + 5471648118 \nu^{10} + 4312537074 \nu^{9} - 15476616276 \nu^{8} - 5105784911 \nu^{7} + 22779563346 \nu^{6} + 2734933636 \nu^{5} - 16449014832 \nu^{4} - 1056764613 \nu^{3} + 5031635418 \nu^{2} + 411389508 \nu - 323928792\)\()/2227868\)
\(\beta_{6}\)\(=\)\((\)\(-1404970 \nu^{17} + 3780947 \nu^{16} + 36540544 \nu^{15} - 100751078 \nu^{14} - 370573528 \nu^{13} + 1061364933 \nu^{12} + 1841735098 \nu^{11} - 5612196832 \nu^{10} - 4629724958 \nu^{9} + 15678706052 \nu^{8} + 5576925138 \nu^{7} - 22610713437 \nu^{6} - 3168374124 \nu^{5} + 15830872676 \nu^{4} + 1297181256 \nu^{3} - 4677606553 \nu^{2} - 434813104 \nu + 298271312\)\()/2227868\)
\(\beta_{7}\)\(=\)\((\)\(-3494855 \nu^{17} + 9764098 \nu^{16} + 90446516 \nu^{15} - 260823938 \nu^{14} - 910154271 \nu^{13} + 2757128768 \nu^{12} + 4462271110 \nu^{11} - 14653546126 \nu^{10} - 10911924870 \nu^{9} + 41268166936 \nu^{8} + 12287191969 \nu^{7} - 60304065576 \nu^{6} - 5863959586 \nu^{5} + 43073936872 \nu^{4} + 2211117527 \nu^{3} - 13004973606 \nu^{2} - 1016134956 \nu + 817140288\)\()/4455736\)
\(\beta_{8}\)\(=\)\((\)\(3681009 \nu^{17} - 10001908 \nu^{16} - 95815724 \nu^{15} + 267404218 \nu^{14} + 972933489 \nu^{13} - 2830400454 \nu^{12} - 4844609962 \nu^{11} + 15074037090 \nu^{10} + 12210250374 \nu^{9} - 42593425200 \nu^{8} - 14731770147 \nu^{7} + 62570241838 \nu^{6} + 8249862870 \nu^{5} - 45018627968 \nu^{4} - 3238435689 \nu^{3} + 13693983540 \nu^{2} + 1156444596 \nu - 881675160\)\()/4455736\)
\(\beta_{9}\)\(=\)\((\)\(-2019375 \nu^{17} + 5620788 \nu^{16} + 52264236 \nu^{15} - 150234226 \nu^{14} - 525947833 \nu^{13} + 1589551438 \nu^{12} + 2578526366 \nu^{11} - 8460573146 \nu^{10} - 6304189942 \nu^{9} + 23887170908 \nu^{8} + 7093748799 \nu^{7} - 35063523236 \nu^{6} - 3380420346 \nu^{5} + 25242671100 \nu^{4} + 1279478681 \nu^{3} - 7708927626 \nu^{2} - 588831410 \nu + 487365336\)\()/2227868\)
\(\beta_{10}\)\(=\)\((\)\(-5620963 \nu^{17} + 15618254 \nu^{16} + 145277224 \nu^{15} - 416660350 \nu^{14} - 1458787503 \nu^{13} + 4396529884 \nu^{12} + 7125121230 \nu^{11} - 23305214606 \nu^{10} - 17292234094 \nu^{9} + 65367889636 \nu^{8} + 19126905513 \nu^{7} - 94914129924 \nu^{6} - 8744938822 \nu^{5} + 67202957572 \nu^{4} + 3323305787 \nu^{3} - 20135817862 \nu^{2} - 1621011268 \nu + 1262317704\)\()/4455736\)
\(\beta_{11}\)\(=\)\((\)\(3688663 \nu^{17} - 10111498 \nu^{16} - 95822274 \nu^{15} + 270214862 \nu^{14} + 969897915 \nu^{13} - 2858362608 \nu^{12} - 4802559124 \nu^{11} + 15209105388 \nu^{10} + 11969811462 \nu^{9} - 42917345898 \nu^{8} - 14066376001 \nu^{7} + 62927500708 \nu^{6} + 7382847570 \nu^{5} - 45188553084 \nu^{4} - 2767323217 \nu^{3} + 13732699798 \nu^{2} + 1074734650 \nu - 876816348\)\()/2227868\)
\(\beta_{12}\)\(=\)\((\)\(-9070279 \nu^{17} + 24718536 \nu^{16} + 236054868 \nu^{15} - 660782758 \nu^{14} - 2396335763 \nu^{13} + 6993458170 \nu^{12} + 11927306270 \nu^{11} - 37242277318 \nu^{10} - 30038423490 \nu^{9} + 105228170632 \nu^{8} + 36185979109 \nu^{7} - 154590680766 \nu^{6} - 20214537106 \nu^{5} + 111253613000 \nu^{4} + 7957130171 \nu^{3} - 33858344916 \nu^{2} - 2863714444 \nu + 2180404840\)\()/4455736\)
\(\beta_{13}\)\(=\)\((\)\(5956650 \nu^{17} - 16368237 \nu^{16} - 154638528 \nu^{15} + 437355758 \nu^{14} + 1563734178 \nu^{13} - 4625326767 \nu^{12} - 7731294326 \nu^{11} + 24601248878 \nu^{10} + 19218362010 \nu^{9} - 69371525690 \nu^{8} - 22470072148 \nu^{7} + 101585091069 \nu^{6} + 11704825162 \nu^{5} - 72788048540 \nu^{4} - 4435335212 \nu^{3} + 22066138029 \nu^{2} + 1751862786 \nu - 1411206752\)\()/2227868\)
\(\beta_{14}\)\(=\)\((\)\(11947639 \nu^{17} - 32671464 \nu^{16} - 310321848 \nu^{15} + 872715794 \nu^{14} + 3140449751 \nu^{13} - 9225906882 \nu^{12} - 15547648298 \nu^{11} + 49043404274 \nu^{10} + 38754588478 \nu^{9} - 138175741352 \nu^{8} - 45626346569 \nu^{7} + 202051896914 \nu^{6} + 24227665706 \nu^{5} - 144426557712 \nu^{4} - 9321097623 \nu^{3} + 43607448488 \nu^{2} + 3565229508 \nu - 2766807104\)\()/4455736\)
\(\beta_{15}\)\(=\)\((\)\(-7559885 \nu^{17} + 20794276 \nu^{16} + 196202226 \nu^{15} - 555649790 \nu^{14} - 1983049725 \nu^{13} + 5877133890 \nu^{12} + 9795396730 \nu^{11} - 31267758290 \nu^{10} - 24302139180 \nu^{9} + 88216405924 \nu^{8} + 28279515887 \nu^{7} - 129311316056 \nu^{6} - 14560176524 \nu^{5} + 92821313642 \nu^{4} + 5509444921 \nu^{3} - 28219452272 \nu^{2} - 2235461278 \nu + 1807951924\)\()/2227868\)
\(\beta_{16}\)\(=\)\((\)\(17825543 \nu^{17} - 48539622 \nu^{16} - 463327892 \nu^{15} + 1296748662 \nu^{14} + 4694030479 \nu^{13} - 13711285756 \nu^{12} - 23281031570 \nu^{11} + 72910025298 \nu^{10} + 58220639342 \nu^{9} - 205524724216 \nu^{8} - 68988249961 \nu^{7} + 300793219888 \nu^{6} + 37003951374 \nu^{5} - 215289093124 \nu^{4} - 14049552159 \nu^{3} + 65121956186 \nu^{2} + 5224692996 \nu - 4151506792\)\()/4455736\)
\(\beta_{17}\)\(=\)\((\)\(-23245789 \nu^{17} + 63356008 \nu^{16} + 604662944 \nu^{15} - 1693449166 \nu^{14} - 6133289141 \nu^{13} + 17919217902 \nu^{12} + 30484202262 \nu^{11} - 95394232990 \nu^{10} - 76561491406 \nu^{9} + 269397508516 \nu^{8} + 91650814599 \nu^{7} - 395473480546 \nu^{6} - 50457333038 \nu^{5} + 284382610852 \nu^{4} + 19687976769 \nu^{3} - 86511099796 \nu^{2} - 7215473472 \nu + 5546657016\)\()/4455736\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{13} + \beta_{11} + \beta_{8} - \beta_{7} - \beta_{5} + 7 \beta_{2} + \beta_{1} + 17\)
\(\nu^{5}\)\(=\)\(-\beta_{16} - \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} + \beta_{9} + 3 \beta_{8} - \beta_{5} + 9 \beta_{3} + 40 \beta_{1} + 1\)
\(\nu^{6}\)\(=\)\(-\beta_{15} - \beta_{14} - 11 \beta_{13} + \beta_{12} + 11 \beta_{11} + 13 \beta_{8} - 11 \beta_{7} - \beta_{6} - 12 \beta_{5} + \beta_{4} - \beta_{3} + 48 \beta_{2} + 12 \beta_{1} + 110\)
\(\nu^{7}\)\(=\)\(-14 \beta_{16} - 14 \beta_{15} + 16 \beta_{14} - 14 \beta_{13} + 16 \beta_{12} + 15 \beta_{11} + \beta_{10} + 12 \beta_{9} + 40 \beta_{8} + \beta_{7} - \beta_{6} - 15 \beta_{5} + 3 \beta_{4} + 68 \beta_{3} + 2 \beta_{2} + 278 \beta_{1} + 14\)
\(\nu^{8}\)\(=\)\(-2 \beta_{17} - 15 \beta_{15} - 11 \beta_{14} - 99 \beta_{13} + 19 \beta_{12} + 95 \beta_{11} + 2 \beta_{9} + 123 \beta_{8} - 99 \beta_{7} - 13 \beta_{6} - 110 \beta_{5} + 17 \beta_{4} - 14 \beta_{3} + 336 \beta_{2} + 113 \beta_{1} + 742\)
\(\nu^{9}\)\(=\)\(-4 \beta_{17} - 142 \beta_{16} - 140 \beta_{15} + 176 \beta_{14} - 145 \beta_{13} + 182 \beta_{12} + 160 \beta_{11} + 11 \beta_{10} + 110 \beta_{9} + 393 \beta_{8} + 16 \beta_{7} - 15 \beta_{6} - 158 \beta_{5} + 57 \beta_{4} + 488 \beta_{3} + 40 \beta_{2} + 1971 \beta_{1} + 142\)
\(\nu^{10}\)\(=\)\(-42 \beta_{17} - 5 \beta_{16} - 162 \beta_{15} - 80 \beta_{14} - 832 \beta_{13} + 246 \beta_{12} + 762 \beta_{11} + 37 \beta_{9} + 1036 \beta_{8} - 825 \beta_{7} - 125 \beta_{6} - 919 \beta_{5} + 203 \beta_{4} - 142 \beta_{3} + 2396 \beta_{2} + 983 \beta_{1} + 5103\)
\(\nu^{11}\)\(=\)\(-84 \beta_{17} - 1270 \beta_{16} - 1235 \beta_{15} + 1663 \beta_{14} - 1337 \beta_{13} + 1795 \beta_{12} + 1494 \beta_{11} + 77 \beta_{10} + 924 \beta_{9} + 3441 \beta_{8} + 168 \beta_{7} - 156 \beta_{6} - 1459 \beta_{5} + 716 \beta_{4} + 3429 \beta_{3} + 523 \beta_{2} + 14140 \beta_{1} + 1285\)
\(\nu^{12}\)\(=\)\(-560 \beta_{17} - 120 \beta_{16} - 1541 \beta_{15} - 435 \beta_{14} - 6761 \beta_{13} + 2681 \beta_{12} + 5948 \beta_{11} - 5 \beta_{10} + 464 \beta_{9} + 8319 \beta_{8} - 6598 \beta_{7} - 1080 \beta_{6} - 7369 \beta_{5} + 2090 \beta_{4} - 1271 \beta_{3} + 17316 \beta_{2} + 8261 \beta_{1} + 35540\)
\(\nu^{13}\)\(=\)\(-1140 \beta_{17} - 10664 \beta_{16} - 10267 \beta_{15} + 14527 \beta_{14} - 11627 \beta_{13} + 16377 \beta_{12} + 13038 \beta_{11} + 395 \beta_{10} + 7468 \beta_{9} + 28491 \beta_{8} + 1444 \beta_{7} - 1414 \beta_{6} - 12677 \beta_{5} + 7520 \beta_{4} + 23862 \beta_{3} + 5693 \beta_{2} + 102283 \beta_{1} + 11055\)
\(\nu^{14}\)\(=\)\(-6110 \beta_{17} - 1816 \beta_{16} - 13747 \beta_{15} - 1325 \beta_{14} - 53898 \beta_{13} + 26495 \beta_{12} + 45967 \beta_{11} - 125 \beta_{10} + 4928 \beta_{9} + 65534 \beta_{8} - 51479 \beta_{7} - 8878 \beta_{6} - 57904 \beta_{5} + 19832 \beta_{4} - 10655 \beta_{3} + 126340 \beta_{2} + 68212 \beta_{1} + 249958\)
\(\nu^{15}\)\(=\)\(-12748 \beta_{17} - 86383 \beta_{16} - 82692 \beta_{15} + 121208 \beta_{14} - 97756 \beta_{13} + 142416 \beta_{12} + 109391 \beta_{11} + 1099 \beta_{10} + 59139 \beta_{9} + 228820 \beta_{8} + 10886 \beta_{7} - 12012 \beta_{6} - 106650 \beta_{5} + 71630 \beta_{4} + 165328 \beta_{3} + 56101 \beta_{2} + 744647 \beta_{1} + 92746\)
\(\nu^{16}\)\(=\)\(-59724 \beta_{17} - 22228 \beta_{16} - 118122 \beta_{15} + 7852 \beta_{14} - 424628 \beta_{13} + 246086 \beta_{12} + 354147 \beta_{11} - 1941 \beta_{10} + 47768 \beta_{9} + 512776 \beta_{8} - 395361 \beta_{7} - 71045 \beta_{6} - 450229 \beta_{5} + 178767 \beta_{4} - 85784 \beta_{3} + 928247 \beta_{2} + 557517 \beta_{1} + 1772555\)
\(\nu^{17}\)\(=\)\(-127964 \beta_{17} - 684398 \beta_{16} - 653941 \beta_{15} + 982463 \beta_{14} - 804613 \beta_{13} + 1199801 \beta_{12} + 895273 \beta_{11} - 6980 \beta_{10} + 462688 \beta_{9} + 1805367 \beta_{8} + 73441 \beta_{7} - 98571 \beta_{6} - 880104 \beta_{5} + 642663 \beta_{4} + 1143633 \beta_{3} + 520211 \beta_{2} + 5450103 \beta_{1} + 767629\)

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
−2.67006
−2.64654
−2.35862
−1.95691
−1.29163
−0.817277
−0.740902
−0.729335
−0.371973
0.238980
0.940613
1.11352
1.38970
1.97890
2.05277
2.52971
2.56132
2.77774
−2.67006 −3.09834 5.12921 1.00000 8.27276 1.00000 −8.35518 6.59974 −2.67006
1.2 −2.64654 2.16636 5.00419 1.00000 −5.73337 1.00000 −7.95071 1.69313 −2.64654
1.3 −2.35862 3.12567 3.56309 1.00000 −7.37227 1.00000 −3.68672 6.76983 −2.35862
1.4 −1.95691 0.270998 1.82949 1.00000 −0.530318 1.00000 0.333679 −2.92656 −1.95691
1.5 −1.29163 −0.466181 −0.331697 1.00000 0.602132 1.00000 3.01169 −2.78268 −1.29163
1.6 −0.817277 −1.30837 −1.33206 1.00000 1.06930 1.00000 2.72321 −1.28816 −0.817277
1.7 −0.740902 3.32552 −1.45106 1.00000 −2.46389 1.00000 2.55690 8.05911 −0.740902
1.8 −0.729335 1.69348 −1.46807 1.00000 −1.23511 1.00000 2.52938 −0.132123 −0.729335
1.9 −0.371973 −1.74007 −1.86164 1.00000 0.647257 1.00000 1.43642 0.0278278 −0.371973
1.10 0.238980 −2.82984 −1.94289 1.00000 −0.676275 1.00000 −0.942271 5.00799 0.238980
1.11 0.940613 0.854982 −1.11525 1.00000 0.804208 1.00000 −2.93024 −2.26901 0.940613
1.12 1.11352 2.27506 −0.760083 1.00000 2.53332 1.00000 −3.07340 2.17591 1.11352
1.13 1.38970 −1.33908 −0.0687332 1.00000 −1.86092 1.00000 −2.87492 −1.20687 1.38970
1.14 1.97890 2.99507 1.91606 1.00000 5.92696 1.00000 −0.166100 5.97045 1.97890
1.15 2.05277 −3.41466 2.21386 1.00000 −7.00951 1.00000 0.439000 8.65991 2.05277
1.16 2.52971 3.17620 4.39941 1.00000 8.03485 1.00000 6.06980 7.08825 2.52971
1.17 2.56132 0.468192 4.56034 1.00000 1.19919 1.00000 6.55784 −2.78080 2.56132
1.18 2.77774 −1.15501 5.71583 1.00000 −3.20831 1.00000 10.3216 −1.66596 2.77774
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 4235.2.a.bp 18
11.b odd 2 1 4235.2.a.bo 18
11.d odd 10 2 385.2.n.f 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.n.f 36 11.d odd 10 2
4235.2.a.bo 18 11.b odd 2 1
4235.2.a.bp 18 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4235))\):

\(T_{2}^{18} - \cdots\)
\(T_{3}^{18} - \cdots\)
\(T_{13}^{18} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -176 - 16 T + 3048 T^{2} + 3146 T^{3} - 9817 T^{4} - 10744 T^{5} + 14618 T^{6} + 14224 T^{7} - 12411 T^{8} - 9228 T^{9} + 6290 T^{10} + 3158 T^{11} - 1874 T^{12} - 580 T^{13} + 317 T^{14} + 54 T^{15} - 28 T^{16} - 2 T^{17} + T^{18} \)
$3$ \( 4400 - 16240 T - 31284 T^{2} + 115244 T^{3} + 62219 T^{4} - 224597 T^{5} - 57876 T^{6} + 191761 T^{7} + 21138 T^{8} - 85889 T^{9} - 132 T^{10} + 21207 T^{11} - 1687 T^{12} - 2832 T^{13} + 384 T^{14} + 190 T^{15} - 33 T^{16} - 5 T^{17} + T^{18} \)
$5$ \( ( -1 + T )^{18} \)
$7$ \( ( -1 + T )^{18} \)
$11$ \( T^{18} \)
$13$ \( 156496 - 2810576 T - 7005504 T^{2} + 36595538 T^{3} + 67479449 T^{4} - 63191848 T^{5} - 82015293 T^{6} + 55611044 T^{7} + 13201425 T^{8} - 11951605 T^{9} - 323926 T^{10} + 1076987 T^{11} - 56740 T^{12} - 46981 T^{13} + 4291 T^{14} + 986 T^{15} - 111 T^{16} - 8 T^{17} + T^{18} \)
$17$ \( -122446139024 + 69183081504 T + 111363376840 T^{2} - 41616043110 T^{3} - 32593479383 T^{4} + 9346421146 T^{5} + 4815898154 T^{6} - 1081521632 T^{7} - 418252806 T^{8} + 72364761 T^{9} + 22695142 T^{10} - 2912221 T^{11} - 779832 T^{12} + 69574 T^{13} + 16497 T^{14} - 909 T^{15} - 196 T^{16} + 5 T^{17} + T^{18} \)
$19$ \( -23658496 + 130744320 T + 604119040 T^{2} - 3797703680 T^{3} + 4400762624 T^{4} - 320193536 T^{5} - 1534964608 T^{6} + 377253024 T^{7} + 186008816 T^{8} - 58109312 T^{9} - 10034968 T^{10} + 3921588 T^{11} + 215296 T^{12} - 134146 T^{13} + 809 T^{14} + 2269 T^{15} - 97 T^{16} - 15 T^{17} + T^{18} \)
$23$ \( -53798725376 + 70329940352 T + 135467086656 T^{2} - 54886455776 T^{3} - 74681680784 T^{4} + 8382019136 T^{5} + 14867513152 T^{6} - 151984292 T^{7} - 1372424404 T^{8} - 37516616 T^{9} + 67655505 T^{10} + 2607341 T^{11} - 1909199 T^{12} - 70133 T^{13} + 31102 T^{14} + 861 T^{15} - 273 T^{16} - 4 T^{17} + T^{18} \)
$29$ \( 2868305 + 264255970 T - 1932521414 T^{2} - 3069369862 T^{3} + 527256367 T^{4} + 2003625913 T^{5} + 123778152 T^{6} - 499971438 T^{7} - 58906153 T^{8} + 56827797 T^{9} + 8134343 T^{10} - 3114094 T^{11} - 483732 T^{12} + 86429 T^{13} + 13957 T^{14} - 1166 T^{15} - 192 T^{16} + 6 T^{17} + T^{18} \)
$31$ \( 1619504742400 + 364622848000 T - 1480775782400 T^{2} - 77555916800 T^{3} + 439104122880 T^{4} - 9787950080 T^{5} - 59584428800 T^{6} + 4331054720 T^{7} + 4124412416 T^{8} - 468233056 T^{9} - 145629952 T^{10} + 22253448 T^{11} + 2378952 T^{12} - 506062 T^{13} - 10975 T^{14} + 5419 T^{15} - 114 T^{16} - 22 T^{17} + T^{18} \)
$37$ \( 281306189824 + 477516103680 T - 1376198596608 T^{2} - 279936310848 T^{3} + 612978482320 T^{4} + 28120241120 T^{5} - 99921306752 T^{6} + 3593238060 T^{7} + 7002217712 T^{8} - 554945460 T^{9} - 240295667 T^{10} + 26734904 T^{11} + 4046049 T^{12} - 600340 T^{13} - 26310 T^{14} + 6398 T^{15} - 57 T^{16} - 26 T^{17} + T^{18} \)
$41$ \( -1359292436480 - 487828295680 T + 2517899121664 T^{2} + 1230835542016 T^{3} - 1173133966336 T^{4} - 651266113792 T^{5} + 121039691520 T^{6} + 70269571264 T^{7} - 8698300976 T^{8} - 2984878944 T^{9} + 349082424 T^{10} + 61692124 T^{11} - 7383480 T^{12} - 657428 T^{13} + 82361 T^{14} + 3448 T^{15} - 458 T^{16} - 7 T^{17} + T^{18} \)
$43$ \( -55605780736 + 123891889408 T + 16290001856 T^{2} - 128400592896 T^{3} + 41308073968 T^{4} + 34162071456 T^{5} - 21445470672 T^{6} + 567740828 T^{7} + 2093518784 T^{8} - 399909206 T^{9} - 52455329 T^{10} + 20199688 T^{11} - 203279 T^{12} - 398636 T^{13} + 23258 T^{14} + 3400 T^{15} - 283 T^{16} - 10 T^{17} + T^{18} \)
$47$ \( 311511344 - 3637101624 T + 8982601096 T^{2} + 3981276408 T^{3} - 40952034087 T^{4} + 58640106830 T^{5} - 36297968530 T^{6} + 8850056640 T^{7} + 797228532 T^{8} - 786636219 T^{9} + 79743570 T^{10} + 19385925 T^{11} - 3482292 T^{12} - 183580 T^{13} + 54099 T^{14} + 487 T^{15} - 376 T^{16} + T^{17} + T^{18} \)
$53$ \( -4047305752576 - 8169931792384 T - 3051990935552 T^{2} + 2810623389184 T^{3} + 1543819748352 T^{4} - 414110821184 T^{5} - 221104397264 T^{6} + 39851753352 T^{7} + 13315216372 T^{8} - 2390743662 T^{9} - 339828367 T^{10} + 71691335 T^{11} + 3463870 T^{12} - 1084264 T^{13} - 707 T^{14} + 8004 T^{15} - 212 T^{16} - 23 T^{17} + T^{18} \)
$59$ \( -1255676285317120 + 739101047562240 T + 170734668207104 T^{2} - 173950586635264 T^{3} + 8046726798592 T^{4} + 13835711693824 T^{5} - 2172232084544 T^{6} - 405831305216 T^{7} + 112347110832 T^{8} + 1461395040 T^{9} - 2330227408 T^{10} + 118107844 T^{11} + 21433472 T^{12} - 2063714 T^{13} - 66927 T^{14} + 13209 T^{15} - 160 T^{16} - 30 T^{17} + T^{18} \)
$61$ \( -1920738463744 + 5283989884928 T + 1498244870144 T^{2} - 4442032171008 T^{3} + 1249227653376 T^{4} + 421583078400 T^{5} - 221436944320 T^{6} - 274425248 T^{7} + 13172485360 T^{8} - 1352475216 T^{9} - 326560400 T^{10} + 59149212 T^{11} + 2540460 T^{12} - 994050 T^{13} + 18231 T^{14} + 6947 T^{15} - 311 T^{16} - 17 T^{17} + T^{18} \)
$67$ \( -32144763136 - 210265178880 T + 938846298496 T^{2} - 879420885856 T^{3} - 207814974864 T^{4} + 482027814992 T^{5} - 47771751704 T^{6} - 62976197804 T^{7} + 14058906080 T^{8} + 1871060782 T^{9} - 755014969 T^{10} + 15200036 T^{11} + 13952737 T^{12} - 1162801 T^{13} - 71328 T^{14} + 11443 T^{15} - 135 T^{16} - 29 T^{17} + T^{18} \)
$71$ \( 34405546365455 + 16462900740235 T - 139900092825894 T^{2} + 102063169645328 T^{3} - 13985120919705 T^{4} - 7559088475795 T^{5} + 1986125586490 T^{6} + 206688729889 T^{7} - 86078437868 T^{8} - 2363179805 T^{9} + 1905463195 T^{10} + 4505845 T^{11} - 24007703 T^{12} + 127736 T^{13} + 172805 T^{14} - 975 T^{15} - 655 T^{16} + 2 T^{17} + T^{18} \)
$73$ \( 3761805788656 + 23644584575688 T - 37764281934488 T^{2} - 8435169767096 T^{3} + 22483448898989 T^{4} - 5557407702146 T^{5} - 1314001905952 T^{6} + 557516017013 T^{7} + 2886544320 T^{8} - 18301836731 T^{9} + 1140188563 T^{10} + 246998125 T^{11} - 24057858 T^{12} - 1491493 T^{13} + 196407 T^{14} + 3835 T^{15} - 721 T^{16} - 3 T^{17} + T^{18} \)
$79$ \( 16304004109 - 360390902606 T + 539627425734 T^{2} + 679098105558 T^{3} - 781165656377 T^{4} - 747580346217 T^{5} + 205410997888 T^{6} + 347475576858 T^{7} + 107399161513 T^{8} + 7088648403 T^{9} - 1908623221 T^{10} - 296369678 T^{11} + 5218898 T^{12} + 2995759 T^{13} + 77297 T^{14} - 12270 T^{15} - 546 T^{16} + 18 T^{17} + T^{18} \)
$83$ \( 105847641630896 + 278444930116056 T - 824279468981508 T^{2} + 286756456653506 T^{3} + 84383559993173 T^{4} - 41628822083088 T^{5} - 1135092123530 T^{6} + 1888105780981 T^{7} - 74571572836 T^{8} - 39286512915 T^{9} + 2687519145 T^{10} + 425568045 T^{11} - 36146374 T^{12} - 2484881 T^{13} + 239457 T^{14} + 7437 T^{15} - 781 T^{16} - 9 T^{17} + T^{18} \)
$89$ \( -656130284154880 - 3790427199221760 T - 3225351946776576 T^{2} + 5134760345625600 T^{3} - 1611632962756352 T^{4} + 31721187347200 T^{5} + 64389714423488 T^{6} - 9076102714592 T^{7} - 497799607152 T^{8} + 193505690160 T^{9} - 7884119096 T^{10} - 1384374124 T^{11} + 137227320 T^{12} + 1119690 T^{13} - 648795 T^{14} + 22090 T^{15} + 736 T^{16} - 59 T^{17} + T^{18} \)
$97$ \( -2821637141144320 + 2399352977393280 T + 3298035038641664 T^{2} + 667045421665760 T^{3} - 199308700219568 T^{4} - 66031080766088 T^{5} + 4353380143272 T^{6} + 2490606328158 T^{7} - 36041485817 T^{8} - 50903853353 T^{9} + 79825296 T^{10} + 619625128 T^{11} - 2428032 T^{12} - 4499624 T^{13} + 57358 T^{14} + 17935 T^{15} - 419 T^{16} - 30 T^{17} + T^{18} \)
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