Properties

Label 189.4.f.b
Level $189$
Weight $4$
Character orbit 189.f
Analytic conductor $11.151$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 189.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.1513609911\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 3 x^{15} + 58 x^{14} - 129 x^{13} + 2107 x^{12} - 4455 x^{11} + 42901 x^{10} - 76404 x^{9} + 599392 x^{8} - 1089732 x^{7} + 4808401 x^{6} - 7939134 x^{5} + 26225236 x^{4} - 39450864 x^{3} + 62254768 x^{2} - 39660672 x + 21307456\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{14} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -5 \beta_{3} + \beta_{8} ) q^{4} + ( 4 \beta_{3} + \beta_{12} ) q^{5} + ( 7 - 7 \beta_{3} ) q^{7} + ( 1 - 4 \beta_{2} + \beta_{4} - \beta_{5} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -5 \beta_{3} + \beta_{8} ) q^{4} + ( 4 \beta_{3} + \beta_{12} ) q^{5} + ( 7 - 7 \beta_{3} ) q^{7} + ( 1 - 4 \beta_{2} + \beta_{4} - \beta_{5} ) q^{8} + ( -3 + 6 \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{9} ) q^{10} + ( 2 + \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{12} - \beta_{13} ) q^{11} + ( 7 \beta_{1} - 7 \beta_{2} - 7 \beta_{3} + \beta_{6} - 2 \beta_{8} - \beta_{15} ) q^{13} + ( 7 \beta_{1} - 7 \beta_{2} ) q^{14} + ( -9 - 8 \beta_{1} + 9 \beta_{3} + 3 \beta_{4} - 2 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} - \beta_{10} + 2 \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{16} + ( -21 + 4 \beta_{2} + 2 \beta_{4} + \beta_{5} + 2 \beta_{7} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{17} + ( 17 + 11 \beta_{2} - \beta_{4} + \beta_{5} + 2 \beta_{7} - \beta_{9} + 3 \beta_{10} - \beta_{11} ) q^{19} + ( 42 - 7 \beta_{1} - 42 \beta_{3} - 8 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + 8 \beta_{8} + 3 \beta_{9} - \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{20} + ( 4 \beta_{1} - 4 \beta_{2} - 18 \beta_{3} - 3 \beta_{6} + 2 \beta_{8} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{22} + ( -4 \beta_{1} + 4 \beta_{2} + 25 \beta_{3} - 2 \beta_{6} - 5 \beta_{8} + \beta_{12} - \beta_{13} + \beta_{14} - 4 \beta_{15} ) q^{23} + ( -29 - 11 \beta_{1} + 29 \beta_{3} - \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - 9 \beta_{7} + \beta_{8} - \beta_{9} + 3 \beta_{10} + \beta_{11} + 9 \beta_{12} + \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{25} + ( -94 + 2 \beta_{2} + 14 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} + 2 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{26} + ( -35 + 7 \beta_{4} ) q^{28} + ( 80 - 4 \beta_{1} - 80 \beta_{3} + 5 \beta_{4} - 5 \beta_{7} - 5 \beta_{8} - \beta_{9} - 4 \beta_{10} - \beta_{11} + 5 \beta_{12} + \beta_{13} + \beta_{14} - 4 \beta_{15} ) q^{29} + ( 10 \beta_{1} - 10 \beta_{2} - 3 \beta_{3} - 3 \beta_{6} + 5 \beta_{8} + 3 \beta_{12} + \beta_{13} - 4 \beta_{14} + 3 \beta_{15} ) q^{31} + ( -5 \beta_{1} + 5 \beta_{2} + 94 \beta_{3} - 3 \beta_{6} - 4 \beta_{12} - 4 \beta_{13} - 2 \beta_{14} - 4 \beta_{15} ) q^{32} + ( 49 - 35 \beta_{1} - 49 \beta_{3} - 3 \beta_{4} - 5 \beta_{5} - 5 \beta_{6} + 14 \beta_{7} + 3 \beta_{8} + 6 \beta_{9} + 4 \beta_{11} - 14 \beta_{12} - 6 \beta_{13} - 4 \beta_{14} ) q^{34} + ( 28 + 7 \beta_{7} ) q^{35} + ( 31 - 2 \beta_{2} + 13 \beta_{4} - 5 \beta_{5} - \beta_{7} - 3 \beta_{9} - 3 \beta_{10} - 2 \beta_{11} ) q^{37} + ( 122 + 21 \beta_{1} - 122 \beta_{3} - 6 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + 6 \beta_{8} - 3 \beta_{9} + 6 \beta_{11} - 2 \beta_{12} + 3 \beta_{13} - 6 \beta_{14} ) q^{38} + ( 48 \beta_{1} - 48 \beta_{2} + 63 \beta_{3} - 7 \beta_{8} + 6 \beta_{12} - 9 \beta_{13} - 3 \beta_{15} ) q^{40} + ( 3 \beta_{1} - 3 \beta_{2} + 62 \beta_{3} + 15 \beta_{8} + 5 \beta_{12} + 3 \beta_{13} + 6 \beta_{15} ) q^{41} + ( 7 + 2 \beta_{1} - 7 \beta_{3} + 7 \beta_{4} + 7 \beta_{5} + 7 \beta_{6} + 9 \beta_{7} - 7 \beta_{8} - 7 \beta_{9} - \beta_{10} + 4 \beta_{11} - 9 \beta_{12} + 7 \beta_{13} - 4 \beta_{14} - \beta_{15} ) q^{43} + ( -46 - 28 \beta_{2} - 14 \beta_{4} - 5 \beta_{5} - 14 \beta_{7} - 6 \beta_{9} + \beta_{10} + 4 \beta_{11} ) q^{44} + ( 25 + 60 \beta_{2} - 11 \beta_{4} + 9 \beta_{5} + 2 \beta_{7} - 6 \beta_{9} + 7 \beta_{10} + 4 \beta_{11} ) q^{46} + ( 80 - 33 \beta_{1} - 80 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 8 \beta_{7} + 2 \beta_{8} + 4 \beta_{10} + \beta_{11} + 8 \beta_{12} - \beta_{14} + 4 \beta_{15} ) q^{47} -49 \beta_{3} q^{49} + ( -37 \beta_{1} + 37 \beta_{2} + 135 \beta_{3} - 6 \beta_{6} + 11 \beta_{8} + 16 \beta_{12} + 18 \beta_{13} - 2 \beta_{14} + 8 \beta_{15} ) q^{50} + ( 105 - 131 \beta_{1} - 105 \beta_{3} + 3 \beta_{4} - 13 \beta_{5} - 13 \beta_{6} + 10 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} - 2 \beta_{11} - 10 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 3 \beta_{15} ) q^{52} + ( -89 + 23 \beta_{2} - 25 \beta_{4} + 9 \beta_{5} - 9 \beta_{7} - \beta_{9} + 5 \beta_{10} - 7 \beta_{11} ) q^{53} + ( -131 + 54 \beta_{2} - 24 \beta_{4} + 5 \beta_{5} + 14 \beta_{9} + \beta_{10} + 7 \beta_{11} ) q^{55} + ( 7 - 28 \beta_{1} - 7 \beta_{3} + 7 \beta_{4} - 7 \beta_{5} - 7 \beta_{6} - 7 \beta_{8} ) q^{56} + ( 47 \beta_{1} - 47 \beta_{2} + 23 \beta_{3} - 9 \beta_{6} + 7 \beta_{8} + 14 \beta_{12} + 8 \beta_{13} + 8 \beta_{14} - \beta_{15} ) q^{58} + ( 10 \beta_{1} - 10 \beta_{2} - 16 \beta_{3} + 10 \beta_{6} + 17 \beta_{8} - 13 \beta_{12} - 5 \beta_{13} - 5 \beta_{14} - 4 \beta_{15} ) q^{59} + ( -78 + 5 \beta_{1} + 78 \beta_{3} + 4 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} - 7 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} - 9 \beta_{11} + 7 \beta_{12} + 2 \beta_{13} + 9 \beta_{14} + 4 \beta_{15} ) q^{61} + ( -128 - 32 \beta_{2} - 16 \beta_{5} + 16 \beta_{7} - 11 \beta_{10} + 4 \beta_{11} ) q^{62} + ( -25 + 22 \beta_{2} + 9 \beta_{4} + \beta_{5} - 34 \beta_{7} - 6 \beta_{9} - 3 \beta_{10} + 4 \beta_{11} ) q^{64} + ( -61 + 90 \beta_{1} + 61 \beta_{3} + 35 \beta_{4} + 13 \beta_{5} + 13 \beta_{6} + 27 \beta_{7} - 35 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} - 6 \beta_{11} - 27 \beta_{12} - 3 \beta_{13} + 6 \beta_{14} + 3 \beta_{15} ) q^{65} + ( 4 \beta_{1} - 4 \beta_{2} + 93 \beta_{3} + 19 \beta_{6} - 28 \beta_{8} - 34 \beta_{12} + \beta_{14} + \beta_{15} ) q^{67} + ( 50 \beta_{1} - 50 \beta_{2} + 304 \beta_{3} + 4 \beta_{6} - 58 \beta_{8} - 26 \beta_{12} - 26 \beta_{13} + 4 \beta_{14} - 13 \beta_{15} ) q^{68} + ( -21 + 42 \beta_{1} + 21 \beta_{3} + 7 \beta_{4} + 7 \beta_{5} + 7 \beta_{6} + 14 \beta_{7} - 7 \beta_{8} + 7 \beta_{9} - 14 \beta_{12} - 7 \beta_{13} ) q^{70} + ( -122 + 19 \beta_{2} - 3 \beta_{4} - \beta_{5} + \beta_{7} + 9 \beta_{9} - 11 \beta_{10} - 17 \beta_{11} ) q^{71} + ( 110 + 13 \beta_{2} - 6 \beta_{5} - 8 \beta_{7} - 10 \beta_{9} - 16 \beta_{10} - 21 \beta_{11} ) q^{73} + ( -60 \beta_{1} + 36 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} - 24 \beta_{7} - 36 \beta_{8} - 5 \beta_{10} - 8 \beta_{11} + 24 \beta_{12} + 8 \beta_{14} - 5 \beta_{15} ) q^{74} + ( 113 \beta_{1} - 113 \beta_{2} - 110 \beta_{3} + 18 \beta_{6} + 50 \beta_{8} + 18 \beta_{12} + 5 \beta_{13} + 10 \beta_{14} + 3 \beta_{15} ) q^{76} + ( 7 \beta_{1} - 7 \beta_{2} - 14 \beta_{3} + 7 \beta_{8} + 7 \beta_{12} - 7 \beta_{13} ) q^{77} + ( -38 + 60 \beta_{1} + 38 \beta_{3} - 60 \beta_{4} - 7 \beta_{5} - 7 \beta_{6} - 36 \beta_{7} + 60 \beta_{8} + 2 \beta_{9} - 5 \beta_{10} - 11 \beta_{11} + 36 \beta_{12} - 2 \beta_{13} + 11 \beta_{14} - 5 \beta_{15} ) q^{79} + ( -349 + 86 \beta_{2} - 17 \beta_{4} + 16 \beta_{5} - 8 \beta_{7} + 3 \beta_{9} + 10 \beta_{10} + 4 \beta_{11} ) q^{80} + ( -12 - 24 \beta_{2} + 8 \beta_{4} - 19 \beta_{5} + 22 \beta_{7} + 14 \beta_{9} - 6 \beta_{10} - 6 \beta_{11} ) q^{82} + ( 361 + 20 \beta_{1} - 361 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 30 \beta_{7} - 2 \beta_{8} - 4 \beta_{9} + 15 \beta_{10} - 15 \beta_{11} - 30 \beta_{12} + 4 \beta_{13} + 15 \beta_{14} + 15 \beta_{15} ) q^{83} + ( -193 \beta_{1} + 193 \beta_{2} + 17 \beta_{3} - 3 \beta_{6} - 16 \beta_{8} - 20 \beta_{12} + 26 \beta_{13} + 13 \beta_{15} ) q^{85} + ( 26 \beta_{1} - 26 \beta_{2} + 26 \beta_{3} + 24 \beta_{6} + 46 \beta_{8} - 4 \beta_{12} + 26 \beta_{13} + 24 \beta_{14} + 33 \beta_{15} ) q^{86} + ( -203 + 73 \beta_{1} + 203 \beta_{3} + 3 \beta_{4} + 7 \beta_{5} + 7 \beta_{6} - 46 \beta_{7} - 3 \beta_{8} - 6 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} + 46 \beta_{12} + 6 \beta_{13} + 2 \beta_{14} + 3 \beta_{15} ) q^{88} + ( -538 + 9 \beta_{2} + 31 \beta_{4} + 2 \beta_{5} + 35 \beta_{7} - 25 \beta_{9} + 28 \beta_{10} - 2 \beta_{11} ) q^{89} + ( -49 - 49 \beta_{2} - 14 \beta_{4} - 7 \beta_{5} + 7 \beta_{10} ) q^{91} + ( 506 + 101 \beta_{1} - 506 \beta_{3} - 76 \beta_{4} + 14 \beta_{5} + 14 \beta_{6} - 10 \beta_{7} + 76 \beta_{8} - 26 \beta_{9} + \beta_{10} - 14 \beta_{11} + 10 \beta_{12} + 26 \beta_{13} + 14 \beta_{14} + \beta_{15} ) q^{92} + ( 65 \beta_{1} - 65 \beta_{2} + 415 \beta_{3} - 2 \beta_{6} - 45 \beta_{8} + 20 \beta_{12} + 4 \beta_{13} - 6 \beta_{14} + \beta_{15} ) q^{94} + ( -64 \beta_{1} + 64 \beta_{2} + 114 \beta_{3} - \beta_{6} + 45 \beta_{8} + 13 \beta_{12} + 5 \beta_{13} - 6 \beta_{14} + 15 \beta_{15} ) q^{95} + ( -44 + 164 \beta_{1} + 44 \beta_{3} + 45 \beta_{4} - 18 \beta_{5} - 18 \beta_{6} - \beta_{7} - 45 \beta_{8} - 5 \beta_{9} - 14 \beta_{10} - 21 \beta_{11} + \beta_{12} + 5 \beta_{13} + 21 \beta_{14} - 14 \beta_{15} ) q^{97} -49 \beta_{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 3q^{2} - 43q^{4} + 30q^{5} + 56q^{7} - 12q^{8} + O(q^{10}) \) \( 16q + 3q^{2} - 43q^{4} + 30q^{5} + 56q^{7} - 12q^{8} - 28q^{10} + 24q^{11} - 68q^{13} - 21q^{14} - 103q^{16} - 336q^{17} + 352q^{19} + 330q^{20} - 151q^{22} + 228q^{23} - 244q^{25} - 1590q^{26} - 602q^{28} + 618q^{29} - 72q^{31} + 786q^{32} + 261q^{34} + 420q^{35} + 420q^{37} + 1032q^{38} + 375q^{40} + 420q^{41} + 2q^{43} - 774q^{44} + 804q^{46} + 570q^{47} - 392q^{49} + 1110q^{50} + 431q^{52} - 1056q^{53} - 1676q^{55} - 42q^{56} - 37q^{58} - 150q^{59} - 578q^{61} - 2340q^{62} - 224q^{64} - 366q^{65} + 898q^{67} + 2526q^{68} - 98q^{70} - 1764q^{71} + 1944q^{73} - 222q^{74} - 1423q^{76} - 168q^{77} + 158q^{79} - 4950q^{80} - 422q^{82} + 2958q^{83} + 774q^{85} - 114q^{86} - 1317q^{88} - 8760q^{89} - 952q^{91} + 4629q^{92} + 3234q^{94} + 930q^{95} + 60q^{97} - 294q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 3 x^{15} + 58 x^{14} - 129 x^{13} + 2107 x^{12} - 4455 x^{11} + 42901 x^{10} - 76404 x^{9} + 599392 x^{8} - 1089732 x^{7} + 4808401 x^{6} - 7939134 x^{5} + 26225236 x^{4} - 39450864 x^{3} + 62254768 x^{2} - 39660672 x + 21307456\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-1081453415523969731 \nu^{15} - 10258576004084808911 \nu^{14} - 31433377624576942326 \nu^{13} - 563112173463188181077 \nu^{12} - 1308525829509857362353 \nu^{11} - 19607931782297760198339 \nu^{10} - 14926937550628598170271 \nu^{9} - 359065610897655236169716 \nu^{8} - 338828317369372158410328 \nu^{7} - 4446748427923459524058988 \nu^{6} - 805082242542684152633379 \nu^{5} - 24422079356124957269325774 \nu^{4} - 45221194666110986753154188 \nu^{3} - 98665615226027792616047816 \nu^{2} + 66635913671816055091959360 \nu + 202367040539048711739007360\)\()/ \)\(35\!\cdots\!16\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-5480043342153615460870 \nu^{15} + 15816131405703515847823 \nu^{14} - 323761712199266631472107 \nu^{13} + 688788532248435498730128 \nu^{12} - 11871367046005927356534519 \nu^{11} + 23658573685667169180098169 \nu^{10} - 246413116060118064521225473 \nu^{9} + 410084388547192134528065113 \nu^{8} - 3491874996428086949591717172 \nu^{7} + 5776274652209615948002027584 \nu^{6} - 28916019729366622881044804946 \nu^{5} + 43042265965218272972249226897 \nu^{4} - 157806969726691414058965486918 \nu^{3} + 190099815283061711298509725204 \nu^{2} - 398088886881135987217134517992 \nu + 255791123727576180195754455360\)\()/ \)\(20\!\cdots\!32\)\( \)
\(\beta_{4}\)\(=\)\((\)\(1687867031332089763 \nu^{15} - 3911365059476662759 \nu^{14} + 87827458008222534547 \nu^{13} - 121262064624893357608 \nu^{12} + 3053225843557130668743 \nu^{11} - 3933561928595653407420 \nu^{10} + 55211622207168577437130 \nu^{9} - 38673276033546388324153 \nu^{8} + 703155352666153263370135 \nu^{7} - 549372430264523428234594 \nu^{6} + 4125985367090929146959841 \nu^{5} + 2107477952623602112602834 \nu^{4} + 17666235855524926401730675 \nu^{3} + 28005004456097216819188856 \nu^{2} - 19934483917834105012486016 \nu + 578872191177709113244256834\)\()/ \)\(44\!\cdots\!02\)\( \)
\(\beta_{5}\)\(=\)\((\)\(8783001140284028181 \nu^{15} + 43470149901470719037 \nu^{14} + 332821804139329780724 \nu^{13} + 2573036738066154190169 \nu^{12} + 12649080834663548149251 \nu^{11} + 90172535054297494176855 \nu^{10} + 185057932167480145725615 \nu^{9} + 1717981502421183404200274 \nu^{8} + 3100452292179167318791910 \nu^{7} + 21134997279088250763825752 \nu^{6} + 12277381946895279057086577 \nu^{5} + 126325352685871990571834538 \nu^{4} + 171938049839893255647257886 \nu^{3} + 549338085042333396718616792 \nu^{2} - 373048536194748485484768832 \nu + 235409574861886198715451272\)\()/ \)\(89\!\cdots\!04\)\( \)
\(\beta_{6}\)\(=\)\((\)\(29460956419830751006980 \nu^{15} - 146666044184232704051475 \nu^{14} + 1862302214682745583125477 \nu^{13} - 6909499106332927973864602 \nu^{12} + 67273151099427449143557363 \nu^{11} - 239051194693874552744954649 \nu^{10} + 1430382102375466260188062149 \nu^{9} - 4230292106716474676060775123 \nu^{8} + 19510880324788114783035875668 \nu^{7} - 56882842190615164525482260272 \nu^{6} + 161975086998518761662187758384 \nu^{5} - 380821978469456085459609973215 \nu^{4} + 728685728922575079241140050778 \nu^{3} - 1714754479816438558919396828964 \nu^{2} + 957588221219743771484502226072 \nu - 539874778203584536413805602496\)\()/ \)\(10\!\cdots\!16\)\( \)
\(\beta_{7}\)\(=\)\((\)\(3539754752942150019523 \nu^{15} - 122684183678165230727 \nu^{14} + 174452770437151922132298 \nu^{13} + 144026909845419610167733 \nu^{12} + 6206184692543187159145479 \nu^{11} + 5970463195997448266509557 \nu^{10} + 109662495109947112821687583 \nu^{9} + 167763726599491300132839868 \nu^{8} + 1453858721703117529295747064 \nu^{7} + 2248924998488557613003892484 \nu^{6} + 7755005966247873356147868357 \nu^{5} + 19709283468356252697727458882 \nu^{4} + 32847078045789804696058658638 \nu^{3} + 113765355656410604286498506488 \nu^{2} - 78621694409674998087306087360 \nu + 246119870160029385338081735152\)\()/ \)\(11\!\cdots\!96\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-71240563447997000991310 \nu^{15} + 205609708274145706021699 \nu^{14} - 4208902258590466209137391 \nu^{13} + 8954250919229661483491664 \nu^{12} - 154327771598077055634948747 \nu^{11} + 307561457913673199341276197 \nu^{10} - 3203370508781534838775931149 \nu^{9} + 5331097051113497748864846469 \nu^{8} - 45394374953565130344692323236 \nu^{7} + 75091570478725007324026358592 \nu^{6} - 375908256481766097453582464298 \nu^{5} + 559549457547837548639239949661 \nu^{4} - 2051490606446988382766551329934 \nu^{3} + 2677864510805352460248543352084 \nu^{2} - 5175155529454767833822748733896 \nu + 3325284608458490342544807919680\)\()/ \)\(20\!\cdots\!32\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-2617641835021904687439 \nu^{15} - 3372327952932177795054 \nu^{14} - 125932472913636998540857 \nu^{13} - 286521233422339986132716 \nu^{12} - 4544122869502149854944242 \nu^{11} - 10548575069088617264428929 \nu^{10} - 79786781545579506525770196 \nu^{9} - 231803742121211616794455647 \nu^{8} - 1075375971285375372397604683 \nu^{7} - 2934567040691549008218021194 \nu^{6} - 5398746937938613910571968322 \nu^{5} - 21253520324500359154703761356 \nu^{4} - 20114644835516234343752409504 \nu^{3} - 108438132615112983231707785104 \nu^{2} + 74418143780426672332885099328 \nu - 221045516789505940659354953552\)\()/ \)\(55\!\cdots\!48\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-3809862470040652283741 \nu^{15} - 14769415057371219629753 \nu^{14} - 172328119360261931923704 \nu^{13} - 909621437509063763774861 \nu^{12} - 6406200475966836318029319 \nu^{11} - 32739844754691313323928983 \nu^{10} - 109879003718808001091047415 \nu^{9} - 642010859435203696943474534 \nu^{8} - 1547488464000946140245119866 \nu^{7} - 8170019990431308401539977368 \nu^{6} - 6802759803441621335354456589 \nu^{5} - 49698655346260753623587537442 \nu^{4} - 18187269795968750353398884354 \nu^{3} - 225597487047461019176333633528 \nu^{2} + 153661530081349823189504832576 \nu - 204383968954270213143532500320\)\()/ \)\(55\!\cdots\!48\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-16592390587936102543113 \nu^{15} - 22245915187769923423837 \nu^{14} - 838627098561879404524434 \nu^{13} - 1801995220427686941413183 \nu^{12} - 30256941523477787117007651 \nu^{11} - 69865177894596342184100265 \nu^{10} - 547355686106949911166448029 \nu^{9} - 1536892823637354853152643084 \nu^{8} - 7159711910494780372983639480 \nu^{7} - 20796371320500948446466869060 \nu^{6} - 35964276420328560529690244697 \nu^{5} - 141104917239183063441863736138 \nu^{4} - 78404574561279509257324541364 \nu^{3} - 720517779594544382537815231192 \nu^{2} + 494495938406601078878400115392 \nu - 928459961735252745979723573760\)\()/ \)\(22\!\cdots\!92\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-5474231123656222072887489 \nu^{15} + 18318056950062213203751045 \nu^{14} - 319355137885079791561710338 \nu^{13} + 797424233016352632647016935 \nu^{12} - 11552044685709304655329441059 \nu^{11} + 27600828637521099459638233647 \nu^{10} - 234945797128772158752503051043 \nu^{9} + 473336819183249885674760955900 \nu^{8} - 3250573583318933748338329629374 \nu^{7} + 6712542559546595652097450811390 \nu^{6} - 25839053516139413861421518322069 \nu^{5} + 47619810469827187410919620053502 \nu^{4} - 136878829936228523653248481107588 \nu^{3} + 243868423746525923021168549316948 \nu^{2} - 294867591484777323253525477117808 \nu + 185101199218695590295877788750848\)\()/ \)\(64\!\cdots\!92\)\( \)
\(\beta_{13}\)\(=\)\((\)\(739384269402108045261284 \nu^{15} - 2450926269331313704910840 \nu^{14} + 42880997647968135595922657 \nu^{13} - 108000532350344066163407207 \nu^{12} + 1547933425754127653577132621 \nu^{11} - 3743139188841429540343527765 \nu^{10} + 31266490745281530427680871838 \nu^{9} - 64803651962444011651602785261 \nu^{8} + 430655832943973911084820085797 \nu^{7} - 910994197054063026404301521555 \nu^{6} + 3377502581931716502770141530950 \nu^{5} - 6430369281384919278267793375263 \nu^{4} + 17940055819560613837739330767874 \nu^{3} - 31243630661672669129909192047580 \nu^{2} + 37707884925294121168102052403512 \nu - 23574875317009746283265948433152\)\()/ \)\(45\!\cdots\!28\)\( \)
\(\beta_{14}\)\(=\)\((\)\(27500148910237919496179767 \nu^{15} - 88300862898779656319608205 \nu^{14} + 1531995634088142305990462830 \nu^{13} - 3731337023038496624039217615 \nu^{12} + 54469454075990557086088842813 \nu^{11} - 130221170460690171109098837273 \nu^{10} + 1059671422215455552444584648867 \nu^{9} - 2210517275553785466829549027964 \nu^{8} + 14286933270430834727048925893992 \nu^{7} - 31971808815936801988031130111444 \nu^{6} + 105547422805888389595816642180119 \nu^{5} - 222861263658948728669692032830874 \nu^{4} + 572245472357422578248561959100668 \nu^{3} - 1086112217099221855624368057535352 \nu^{2} + 1112053465314347599071773278201360 \nu - 685572092018790295891744167781248\)\()/ \)\(12\!\cdots\!84\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-7445479490958153507104347 \nu^{15} + 26732036690555565160044174 \nu^{14} - 424855555625552569168518941 \nu^{13} + 1182715762133630881265375961 \nu^{12} - 15156830072926200217513130730 \nu^{11} + 41208273751508774474184288540 \nu^{10} - 298944186966088153513189940254 \nu^{9} + 714525189834034177189697996769 \nu^{8} - 4020700186637257679634353295896 \nu^{7} + 10019135132131701934450566692640 \nu^{6} - 29802486891611854668938635978335 \nu^{5} + 68579134357990951915309720423719 \nu^{4} - 155658302257860579162933656705482 \nu^{3} + 311345059781046098987981175043488 \nu^{2} - 264305572911128359627730623812056 \nu + 158697333991145125016941606877376\)\()/ \)\(32\!\cdots\!96\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} - 13 \beta_{3}\)
\(\nu^{3}\)\(=\)\(-\beta_{5} + \beta_{4} - 20 \beta_{2} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{15} - 2 \beta_{13} + 2 \beta_{12} - \beta_{10} + 2 \beta_{9} - 27 \beta_{8} - 2 \beta_{7} + 27 \beta_{4} + 257 \beta_{3} - 8 \beta_{1} - 257\)
\(\nu^{5}\)\(=\)\(-4 \beta_{15} - 2 \beta_{14} - 4 \beta_{13} - 4 \beta_{12} - 32 \beta_{8} - 35 \beta_{6} + 62 \beta_{3} + 453 \beta_{2} - 453 \beta_{1}\)
\(\nu^{6}\)\(=\)\(4 \beta_{11} + 37 \beta_{10} - 86 \beta_{9} + 46 \beta_{7} + \beta_{5} - 687 \beta_{4} + 342 \beta_{2} + 5775\)
\(\nu^{7}\)\(=\)\(185 \beta_{15} + 106 \beta_{14} + 214 \beta_{13} + 250 \beta_{12} - 106 \beta_{11} + 185 \beta_{10} - 214 \beta_{9} + 972 \beta_{8} - 250 \beta_{7} + 998 \beta_{6} + 998 \beta_{5} - 972 \beta_{4} - 3250 \beta_{3} + 10837 \beta_{1} + 3250\)
\(\nu^{8}\)\(=\)\(1108 \beta_{15} - 154 \beta_{14} + 2888 \beta_{13} - 640 \beta_{12} + 17494 \beta_{8} + 154 \beta_{6} - 137466 \beta_{3} - 11319 \beta_{2} + 11319 \beta_{1}\)
\(\nu^{9}\)\(=\)\(3868 \beta_{11} - 6392 \beta_{10} + 8332 \beta_{9} + 9964 \beta_{7} - 26952 \beta_{5} + 28969 \beta_{4} - 267314 \beta_{2} - 115437\)
\(\nu^{10}\)\(=\)\(-32012 \beta_{15} + 3856 \beta_{14} - 88864 \beta_{13} + 648 \beta_{12} - 3856 \beta_{11} - 32012 \beta_{10} + 88864 \beta_{9} - 449379 \beta_{8} - 648 \beta_{7} - 9313 \beta_{6} - 9313 \beta_{5} + 449379 \beta_{4} + 3377929 \beta_{3} - 346412 \beta_{1} - 3377929\)
\(\nu^{11}\)\(=\)\(-198609 \beta_{15} - 121416 \beta_{14} - 284570 \beta_{13} - 335534 \beta_{12} - 851561 \beta_{8} - 715018 \beta_{6} + 3670779 \beta_{3} + 6725628 \beta_{2} - 6725628 \beta_{1}\)
\(\nu^{12}\)\(=\)\(70910 \beta_{11} + 919910 \beta_{10} - 2619280 \beta_{9} - 379312 \beta_{7} + 411887 \beta_{5} - 11648742 \beta_{4} + 10284309 \beta_{2} + 84708632\)
\(\nu^{13}\)\(=\)\(5863177 \beta_{15} + 3540560 \beta_{14} + 9060926 \beta_{13} + 10411970 \beta_{12} - 3540560 \beta_{11} + 5863177 \beta_{10} - 9060926 \beta_{9} + 24785055 \beta_{8} - 10411970 \beta_{7} + 18857203 \beta_{6} + 18857203 \beta_{5} - 24785055 \beta_{4} - 111743867 \beta_{3} + 171625816 \beta_{1} + 111743867\)
\(\nu^{14}\)\(=\)\(26357375 \beta_{15} - 685622 \beta_{14} + 75309154 \beta_{13} + 19353238 \beta_{12} + 304356334 \beta_{8} + 15617540 \beta_{6} - 2155099748 \beta_{3} - 301101285 \beta_{2} + 301101285 \beta_{1}\)
\(\nu^{15}\)\(=\)\(99274802 \beta_{11} - 168292714 \beta_{10} + 276515780 \beta_{9} + 308708012 \beta_{7} - 496684262 \beta_{5} + 716124370 \beta_{4} - 4427034893 \beta_{2} - 3331785418\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(136\)
\(\chi(n)\) \(-\beta_{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
−2.47591 4.28840i
−1.96709 3.40709i
−1.46974 2.54566i
0.403686 + 0.699204i
0.797492 + 1.38130i
1.30789 + 2.26533i
2.28179 + 3.95218i
2.62188 + 4.54123i
−2.47591 + 4.28840i
−1.96709 + 3.40709i
−1.46974 + 2.54566i
0.403686 0.699204i
0.797492 1.38130i
1.30789 2.26533i
2.28179 3.95218i
2.62188 4.54123i
−2.47591 4.28840i 0 −8.26023 + 14.3071i 8.06998 13.9776i 0 3.50000 + 6.06218i 42.1917 0 −79.9221
64.2 −1.96709 3.40709i 0 −3.73885 + 6.47588i −1.21571 + 2.10567i 0 3.50000 + 6.06218i −2.05480 0 9.56561
64.3 −1.46974 2.54566i 0 −0.320267 + 0.554718i −1.28443 + 2.22469i 0 3.50000 + 6.06218i −21.6330 0 7.55109
64.4 0.403686 + 0.699204i 0 3.67408 6.36369i 9.11444 15.7867i 0 3.50000 + 6.06218i 12.3917 0 14.7175
64.5 0.797492 + 1.38130i 0 2.72801 4.72505i −1.27816 + 2.21384i 0 3.50000 + 6.06218i 21.4622 0 −4.07730
64.6 1.30789 + 2.26533i 0 0.578868 1.00263i −6.77153 + 11.7286i 0 3.50000 + 6.06218i 23.9546 0 −35.4255
64.7 2.28179 + 3.95218i 0 −6.41313 + 11.1079i 10.3955 18.0055i 0 3.50000 + 6.06218i −22.0250 0 94.8813
64.8 2.62188 + 4.54123i 0 −9.74848 + 16.8849i −2.03009 + 3.51621i 0 3.50000 + 6.06218i −60.2873 0 −21.2906
127.1 −2.47591 + 4.28840i 0 −8.26023 14.3071i 8.06998 + 13.9776i 0 3.50000 6.06218i 42.1917 0 −79.9221
127.2 −1.96709 + 3.40709i 0 −3.73885 6.47588i −1.21571 2.10567i 0 3.50000 6.06218i −2.05480 0 9.56561
127.3 −1.46974 + 2.54566i 0 −0.320267 0.554718i −1.28443 2.22469i 0 3.50000 6.06218i −21.6330 0 7.55109
127.4 0.403686 0.699204i 0 3.67408 + 6.36369i 9.11444 + 15.7867i 0 3.50000 6.06218i 12.3917 0 14.7175
127.5 0.797492 1.38130i 0 2.72801 + 4.72505i −1.27816 2.21384i 0 3.50000 6.06218i 21.4622 0 −4.07730
127.6 1.30789 2.26533i 0 0.578868 + 1.00263i −6.77153 11.7286i 0 3.50000 6.06218i 23.9546 0 −35.4255
127.7 2.28179 3.95218i 0 −6.41313 11.1079i 10.3955 + 18.0055i 0 3.50000 6.06218i −22.0250 0 94.8813
127.8 2.62188 4.54123i 0 −9.74848 16.8849i −2.03009 3.51621i 0 3.50000 6.06218i −60.2873 0 −21.2906
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.4.f.b 16
3.b odd 2 1 63.4.f.b 16
9.c even 3 1 inner 189.4.f.b 16
9.c even 3 1 567.4.a.g 8
9.d odd 6 1 63.4.f.b 16
9.d odd 6 1 567.4.a.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.f.b 16 3.b odd 2 1
63.4.f.b 16 9.d odd 6 1
189.4.f.b 16 1.a even 1 1 trivial
189.4.f.b 16 9.c even 3 1 inner
567.4.a.g 8 9.c even 3 1
567.4.a.i 8 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{16} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(189, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 21307456 - 39660672 T + 62254768 T^{2} - 39450864 T^{3} + 26225236 T^{4} - 7939134 T^{5} + 4808401 T^{6} - 1089732 T^{7} + 599392 T^{8} - 76404 T^{9} + 42901 T^{10} - 4455 T^{11} + 2107 T^{12} - 129 T^{13} + 58 T^{14} - 3 T^{15} + T^{16} \)
$3$ \( T^{16} \)
$5$ \( 28841980243729 + 38849239074834 T + 34081386376861 T^{2} + 18503393913870 T^{3} + 7528104076330 T^{4} + 2078095242474 T^{5} + 431614721467 T^{6} + 51994843578 T^{7} + 4920908542 T^{8} + 20676930 T^{9} + 52776610 T^{10} - 1206468 T^{11} + 273883 T^{12} - 12060 T^{13} + 1072 T^{14} - 30 T^{15} + T^{16} \)
$7$ \( ( 49 - 7 T + T^{2} )^{8} \)
$11$ \( 19352068738871482384 + 35849465685845921328 T + 70022073886875510460 T^{2} - 6126106677903291300 T^{3} + 1180370289224013085 T^{4} - 5974651611808092 T^{5} + 6354194039535937 T^{6} - 123362981984280 T^{7} + 15045119406877 T^{8} - 147232944726 T^{9} + 17620195678 T^{10} - 209410236 T^{11} + 12555637 T^{12} - 79428 T^{13} + 4336 T^{14} - 24 T^{15} + T^{16} \)
$13$ \( \)\(64\!\cdots\!64\)\( + \)\(37\!\cdots\!48\)\( T + \)\(31\!\cdots\!56\)\( T^{2} + \)\(11\!\cdots\!68\)\( T^{3} + \)\(66\!\cdots\!89\)\( T^{4} + 20910710007378924876 T^{5} + 886830899435150467 T^{6} + 20768687397798344 T^{7} + 653974912972458 T^{8} + 12443182657628 T^{9} + 321160744711 T^{10} + 4528688904 T^{11} + 83171594 T^{12} + 699772 T^{13} + 11511 T^{14} + 68 T^{15} + T^{16} \)
$17$ \( ( 2073513126204 - 1554538753260 T + 250373208411 T^{2} + 8466840468 T^{3} - 32700159 T^{4} - 2480508 T^{5} - 8910 T^{6} + 168 T^{7} + T^{8} )^{2} \)
$19$ \( ( -604815137888177 + 18413307722920 T + 473899377403 T^{2} - 15172506716 T^{3} - 63882233 T^{4} + 3616846 T^{5} - 12980 T^{6} - 176 T^{7} + T^{8} )^{2} \)
$23$ \( \)\(21\!\cdots\!25\)\( - \)\(22\!\cdots\!70\)\( T + \)\(29\!\cdots\!74\)\( T^{2} - \)\(10\!\cdots\!80\)\( T^{3} + \)\(52\!\cdots\!59\)\( T^{4} - \)\(79\!\cdots\!98\)\( T^{5} + \)\(47\!\cdots\!08\)\( T^{6} - 3517723859432587740 T^{7} + 152181350553640662 T^{8} - 1548626050062468 T^{9} + 32495897803311 T^{10} - 187084830924 T^{11} + 1900227492 T^{12} - 7913088 T^{13} + 71811 T^{14} - 228 T^{15} + T^{16} \)
$29$ \( \)\(24\!\cdots\!00\)\( - \)\(55\!\cdots\!40\)\( T + \)\(10\!\cdots\!56\)\( T^{2} - \)\(50\!\cdots\!84\)\( T^{3} + \)\(22\!\cdots\!25\)\( T^{4} - \)\(12\!\cdots\!60\)\( T^{5} + \)\(74\!\cdots\!66\)\( T^{6} - 70465987056078523104 T^{7} + 1458128461714497559 T^{8} - 12124179768658290 T^{9} + 179453796065908 T^{10} - 1582604233902 T^{11} + 13305242827 T^{12} - 67672590 T^{13} + 273127 T^{14} - 618 T^{15} + T^{16} \)
$31$ \( \)\(45\!\cdots\!64\)\( - \)\(18\!\cdots\!28\)\( T + \)\(68\!\cdots\!40\)\( T^{2} - \)\(25\!\cdots\!60\)\( T^{3} + \)\(85\!\cdots\!73\)\( T^{4} - \)\(64\!\cdots\!46\)\( T^{5} + \)\(10\!\cdots\!38\)\( T^{6} - \)\(27\!\cdots\!32\)\( T^{7} + 8636930654645597013 T^{8} - 11092690384226676 T^{9} + 335166439923216 T^{10} - 26374414860 T^{11} + 9147853935 T^{12} + 110484 T^{13} + 115983 T^{14} + 72 T^{15} + T^{16} \)
$37$ \( ( 71324346320230788 - 1595546991527796 T - 52771182253767 T^{2} + 124534542204 T^{3} + 6138784098 T^{4} + 12199410 T^{5} - 131643 T^{6} - 210 T^{7} + T^{8} )^{2} \)
$41$ \( \)\(21\!\cdots\!96\)\( + \)\(49\!\cdots\!00\)\( T + \)\(11\!\cdots\!68\)\( T^{2} + \)\(10\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!61\)\( T^{4} + \)\(33\!\cdots\!36\)\( T^{5} + \)\(82\!\cdots\!29\)\( T^{6} - \)\(31\!\cdots\!84\)\( T^{7} + 49314133653098744023 T^{8} - 201906961420668972 T^{9} + 1916269491754600 T^{10} - 5615393885100 T^{11} + 28486127767 T^{12} - 62292312 T^{13} + 263530 T^{14} - 420 T^{15} + T^{16} \)
$43$ \( \)\(43\!\cdots\!56\)\( + \)\(39\!\cdots\!04\)\( T + \)\(65\!\cdots\!80\)\( T^{2} + \)\(19\!\cdots\!28\)\( T^{3} + \)\(81\!\cdots\!93\)\( T^{4} + \)\(34\!\cdots\!58\)\( T^{5} + \)\(19\!\cdots\!74\)\( T^{6} - \)\(71\!\cdots\!80\)\( T^{7} + \)\(30\!\cdots\!85\)\( T^{8} - 1017173615702195738 T^{9} + 24829751960735392 T^{10} - 6200975997630 T^{11} + 145631819591 T^{12} - 19080886 T^{13} + 461133 T^{14} - 2 T^{15} + T^{16} \)
$47$ \( \)\(70\!\cdots\!16\)\( - \)\(98\!\cdots\!84\)\( T + \)\(33\!\cdots\!28\)\( T^{2} - \)\(18\!\cdots\!20\)\( T^{3} + \)\(74\!\cdots\!09\)\( T^{4} - \)\(46\!\cdots\!58\)\( T^{5} + \)\(82\!\cdots\!16\)\( T^{6} - \)\(30\!\cdots\!48\)\( T^{7} + 46608343810770753639 T^{8} - 200539340662390434 T^{9} + 1543211389982358 T^{10} - 5116276763784 T^{11} + 28446777261 T^{12} - 85943106 T^{13} + 326565 T^{14} - 570 T^{15} + T^{16} \)
$53$ \( ( 86158340642449528308 + 2650393862685934092 T + 24282335794603149 T^{2} + 81691443067554 T^{3} + 17633836809 T^{4} - 437892894 T^{5} - 587934 T^{6} + 528 T^{7} + T^{8} )^{2} \)
$59$ \( \)\(12\!\cdots\!04\)\( + \)\(27\!\cdots\!44\)\( T + \)\(66\!\cdots\!48\)\( T^{2} - \)\(59\!\cdots\!08\)\( T^{3} + \)\(58\!\cdots\!85\)\( T^{4} - \)\(18\!\cdots\!36\)\( T^{5} + \)\(10\!\cdots\!68\)\( T^{6} - \)\(16\!\cdots\!44\)\( T^{7} + \)\(13\!\cdots\!49\)\( T^{8} - 10902094912961533638 T^{9} + 77902055977792626 T^{10} - 14914159463124 T^{11} + 326995394229 T^{12} - 15347970 T^{13} + 696915 T^{14} + 150 T^{15} + T^{16} \)
$61$ \( \)\(63\!\cdots\!25\)\( - \)\(27\!\cdots\!70\)\( T + \)\(96\!\cdots\!64\)\( T^{2} - \)\(15\!\cdots\!20\)\( T^{3} + \)\(22\!\cdots\!23\)\( T^{4} - \)\(86\!\cdots\!44\)\( T^{5} + \)\(13\!\cdots\!62\)\( T^{6} + \)\(42\!\cdots\!94\)\( T^{7} + \)\(16\!\cdots\!62\)\( T^{8} + 8730419005915000562 T^{9} + 47477314905361183 T^{10} + 113392302812838 T^{11} + 276580191338 T^{12} + 285723382 T^{13} + 670113 T^{14} + 578 T^{15} + T^{16} \)
$67$ \( \)\(26\!\cdots\!24\)\( + \)\(66\!\cdots\!88\)\( T + \)\(16\!\cdots\!76\)\( T^{2} - \)\(14\!\cdots\!32\)\( T^{3} + \)\(46\!\cdots\!29\)\( T^{4} - \)\(79\!\cdots\!94\)\( T^{5} + \)\(10\!\cdots\!25\)\( T^{6} - \)\(15\!\cdots\!18\)\( T^{7} + \)\(86\!\cdots\!67\)\( T^{8} - 67401351317987780362 T^{9} + 442845833216678812 T^{10} - 349068470579910 T^{11} + 1076951523329 T^{12} - 573468188 T^{13} + 1614654 T^{14} - 898 T^{15} + T^{16} \)
$71$ \( ( \)\(36\!\cdots\!25\)\( - 41762863933845347934 T - 84625618285280943 T^{2} + 370893752864550 T^{3} + 541546496631 T^{4} - 1037538936 T^{5} - 1297368 T^{6} + 882 T^{7} + T^{8} )^{2} \)
$73$ \( ( \)\(45\!\cdots\!76\)\( + 20102375258878654764 T + 7761474434502279 T^{2} - 334588931457660 T^{3} + 98139802230 T^{4} + 1566589032 T^{5} - 1393785 T^{6} - 972 T^{7} + T^{8} )^{2} \)
$79$ \( \)\(18\!\cdots\!81\)\( + \)\(48\!\cdots\!24\)\( T + \)\(66\!\cdots\!71\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!64\)\( T^{4} - \)\(61\!\cdots\!12\)\( T^{5} + \)\(60\!\cdots\!11\)\( T^{6} - \)\(11\!\cdots\!92\)\( T^{7} + \)\(18\!\cdots\!06\)\( T^{8} - \)\(11\!\cdots\!08\)\( T^{9} + 3035745212081161276 T^{10} - 188665356763566 T^{11} + 3616369662683 T^{12} - 140819368 T^{13} + 2312334 T^{14} - 158 T^{15} + T^{16} \)
$83$ \( \)\(19\!\cdots\!00\)\( - \)\(17\!\cdots\!80\)\( T + \)\(12\!\cdots\!24\)\( T^{2} - \)\(23\!\cdots\!52\)\( T^{3} + \)\(30\!\cdots\!17\)\( T^{4} - \)\(18\!\cdots\!20\)\( T^{5} + \)\(80\!\cdots\!95\)\( T^{6} - \)\(22\!\cdots\!88\)\( T^{7} + \)\(48\!\cdots\!11\)\( T^{8} - \)\(75\!\cdots\!68\)\( T^{9} + 10000025916310895868 T^{10} - 10530605267612676 T^{11} + 10919355505209 T^{12} - 8956255896 T^{13} + 6620490 T^{14} - 2958 T^{15} + T^{16} \)
$89$ \( ( \)\(44\!\cdots\!84\)\( + \)\(56\!\cdots\!32\)\( T + 1426979281925511527 T^{2} - 1461374513042748 T^{3} - 5942862637383 T^{4} - 1699594764 T^{5} + 5183942 T^{6} + 4380 T^{7} + T^{8} )^{2} \)
$97$ \( \)\(48\!\cdots\!00\)\( + \)\(42\!\cdots\!00\)\( T + \)\(34\!\cdots\!84\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(54\!\cdots\!85\)\( T^{4} + \)\(15\!\cdots\!00\)\( T^{5} + \)\(57\!\cdots\!66\)\( T^{6} + \)\(10\!\cdots\!72\)\( T^{7} + \)\(21\!\cdots\!95\)\( T^{8} + \)\(14\!\cdots\!32\)\( T^{9} + 19460632456966806684 T^{10} + 7559097291028692 T^{11} + 13126774498371 T^{12} + 2499642396 T^{13} + 4191849 T^{14} - 60 T^{15} + T^{16} \)
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