Properties

Label 378.2.l.a
Level 378
Weight 2
Character orbit 378.l
Analytic conductor 3.018
Analytic rank 0
Dimension 16
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 378.l (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + 2785 x^{8} - 2640 x^{7} - 2601 x^{6} + 10260 x^{5} - 10611 x^{4} - 1944 x^{3} + 16767 x^{2} - 17496 x + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(50 \nu^{15} + 1352 \nu^{14} - 6827 \nu^{13} + 7676 \nu^{12} + 27422 \nu^{11} - 107246 \nu^{10} + 107467 \nu^{9} + 206194 \nu^{8} - 757363 \nu^{7} + 724572 \nu^{6} + 756198 \nu^{5} - 2730942 \nu^{4} + 2372247 \nu^{3} + 1318518 \nu^{2} - 4444713 \nu + 2825604\)\()/142155\)
\(\beta_{2}\)\(=\)\((\)\(169 \nu^{15} - 866 \nu^{14} + 1319 \nu^{13} + 2308 \nu^{12} - 13199 \nu^{11} + 19055 \nu^{10} + 12131 \nu^{9} - 90010 \nu^{8} + 128722 \nu^{7} + 13734 \nu^{6} - 313347 \nu^{5} + 432351 \nu^{4} - 55431 \nu^{3} - 524799 \nu^{2} + 629856 \nu - 244944\)\()/47385\)
\(\beta_{3}\)\(=\)\((\)\(154 \nu^{15} - 1325 \nu^{14} + 3608 \nu^{13} - 224 \nu^{12} - 22478 \nu^{11} + 55022 \nu^{10} - 23518 \nu^{9} - 159688 \nu^{8} + 382978 \nu^{7} - 226785 \nu^{6} - 566733 \nu^{5} + 1355805 \nu^{4} - 871830 \nu^{3} - 949725 \nu^{2} + 2135727 \nu - 1285227\)\()/47385\)
\(\beta_{4}\)\(=\)\((\)\(1445 \nu^{15} - 9836 \nu^{14} + 21081 \nu^{13} + 15627 \nu^{12} - 172766 \nu^{11} + 334353 \nu^{10} + 13019 \nu^{9} - 1283242 \nu^{8} + 2419149 \nu^{7} - 693301 \nu^{6} - 4779069 \nu^{5} + 8827731 \nu^{4} - 3909951 \nu^{3} - 8165529 \nu^{2} + 13935564 \nu - 7117227\)\()/47385\)
\(\beta_{5}\)\(=\)\((\)\(2858 \nu^{15} - 19265 \nu^{14} + 40866 \nu^{13} + 31392 \nu^{12} - 338066 \nu^{11} + 649239 \nu^{10} + 37829 \nu^{9} - 2521996 \nu^{8} + 4717386 \nu^{7} - 1293175 \nu^{6} - 9430341 \nu^{5} + 17293545 \nu^{4} - 7519770 \nu^{3} - 16197165 \nu^{2} + 27464589 \nu - 14041269\)\()/47385\)
\(\beta_{6}\)\(=\)\((\)\(-11192 \nu^{15} + 70123 \nu^{14} - 136087 \nu^{13} - 145859 \nu^{12} + 1215277 \nu^{11} - 2154880 \nu^{10} - 494788 \nu^{9} + 9027170 \nu^{8} - 15633236 \nu^{7} + 2544783 \nu^{6} + 33710346 \nu^{5} - 57047193 \nu^{4} + 20576268 \nu^{3} + 57644217 \nu^{2} - 89475273 \nu + 42998607\)\()/142155\)
\(\beta_{7}\)\(=\)\((\)\(16898 \nu^{15} - 108472 \nu^{14} + 217033 \nu^{13} + 208526 \nu^{12} - 1883968 \nu^{11} + 3436840 \nu^{10} + 565132 \nu^{9} - 13978340 \nu^{8} + 24892859 \nu^{7} - 5100702 \nu^{6} - 52089894 \nu^{5} + 90692622 \nu^{4} - 35165097 \nu^{3} - 88915158 \nu^{2} + 142069707 \nu - 69445998\)\()/142155\)
\(\beta_{8}\)\(=\)\((\)\(-4120 \nu^{15} + 25571 \nu^{14} - 48788 \nu^{13} - 55006 \nu^{12} + 441224 \nu^{11} - 771188 \nu^{10} - 200900 \nu^{9} + 3269857 \nu^{8} - 5585140 \nu^{7} + 796053 \nu^{6} + 12187116 \nu^{5} - 20322468 \nu^{4} + 7028856 \nu^{3} + 20790351 \nu^{2} - 31653180 \nu + 14963454\)\()/28431\)
\(\beta_{9}\)\(=\)\((\)\(8586 \nu^{15} - 53254 \nu^{14} + 101261 \nu^{13} + 115567 \nu^{12} - 919066 \nu^{11} + 1601225 \nu^{10} + 428149 \nu^{9} - 6806990 \nu^{8} + 11591968 \nu^{7} - 1602394 \nu^{6} - 25345758 \nu^{5} + 42144849 \nu^{4} - 14468679 \nu^{3} - 43148781 \nu^{2} + 65494089 \nu - 30865131\)\()/47385\)
\(\beta_{10}\)\(=\)\((\)\(-26068 \nu^{15} + 165707 \nu^{14} - 325403 \nu^{13} - 334861 \nu^{12} + 2873363 \nu^{11} - 5145350 \nu^{10} - 1063457 \nu^{9} + 21324130 \nu^{8} - 37270684 \nu^{7} + 6581337 \nu^{6} + 79507809 \nu^{5} - 135722682 \nu^{4} + 50026572 \nu^{3} + 135776493 \nu^{2} - 212064642 \nu + 102032298\)\()/142155\)
\(\beta_{11}\)\(=\)\((\)\(-31130 \nu^{15} + 194263 \nu^{14} - 374983 \nu^{13} - 405716 \nu^{12} + 3354958 \nu^{11} - 5936959 \nu^{10} - 1366837 \nu^{9} + 24841391 \nu^{8} - 43003982 \nu^{7} + 7063563 \nu^{6} + 92428227 \nu^{5} - 156501153 \nu^{4} + 56669868 \nu^{3} + 157282722 \nu^{2} - 244082322 \nu + 117273501\)\()/142155\)
\(\beta_{12}\)\(=\)\((\)\(33311 \nu^{15} - 207064 \nu^{14} + 396466 \nu^{13} + 441842 \nu^{12} - 3574291 \nu^{11} + 6268915 \nu^{10} + 1580329 \nu^{9} - 26474600 \nu^{8} + 45384563 \nu^{7} - 6735609 \nu^{6} - 98586783 \nu^{5} + 165029724 \nu^{4} - 57791394 \nu^{3} - 167924421 \nu^{2} + 256798269 \nu - 121811526\)\()/142155\)
\(\beta_{13}\)\(=\)\((\)\(5368 \nu^{15} - 32959 \nu^{14} + 62025 \nu^{13} + 72930 \nu^{12} - 567820 \nu^{11} + 981351 \nu^{10} + 280045 \nu^{9} - 4204439 \nu^{8} + 7108452 \nu^{7} - 906329 \nu^{6} - 15657912 \nu^{5} + 25851399 \nu^{4} - 8698959 \nu^{3} - 26672436 \nu^{2} + 40175190 \nu - 18826182\)\()/15795\)
\(\beta_{14}\)\(=\)\((\)\(-62668 \nu^{15} + 397070 \nu^{14} - 778856 \nu^{13} - 802747 \nu^{12} + 6878696 \nu^{11} - 12322064 \nu^{10} - 2527469 \nu^{9} + 51016561 \nu^{8} - 89230126 \nu^{7} + 15905550 \nu^{6} + 190088901 \nu^{5} - 324842130 \nu^{4} + 120144465 \nu^{3} + 324340605 \nu^{2} - 507326409 \nu + 244346949\)\()/142155\)
\(\beta_{15}\)\(=\)\((\)\(5105 \nu^{15} - 31804 \nu^{14} + 61039 \nu^{13} + 67583 \nu^{12} - 549514 \nu^{11} + 965497 \nu^{10} + 239686 \nu^{9} - 4072523 \nu^{8} + 6992981 \nu^{7} - 1054329 \nu^{6} - 15172956 \nu^{5} + 25447554 \nu^{4} - 8949609 \nu^{3} - 25866621 \nu^{2} + 39638646 \nu - 18816948\)\()/10935\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{15} + 2 \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} - \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{14} - 2 \beta_{13} - \beta_{11} + \beta_{10} + 3 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - \beta_{6} - 2 \beta_{4} - \beta_{3} - 2 \beta_{1} + 3\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{13} + \beta_{12} + 2 \beta_{11} + 3 \beta_{10} + 6 \beta_{9} + 2 \beta_{8} + 5 \beta_{7} + 2 \beta_{5} - 3 \beta_{4} - 5 \beta_{3} + 8 \beta_{2} + \beta_{1} - 4\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(5 \beta_{15} - 3 \beta_{13} - \beta_{12} + 3 \beta_{11} + 6 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} + \beta_{7} + 7 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 11 \beta_{1} + 8\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-10 \beta_{14} + 3 \beta_{13} + 3 \beta_{12} + 9 \beta_{11} + 5 \beta_{10} + 6 \beta_{9} + 30 \beta_{8} - 3 \beta_{7} - 2 \beta_{6} + 6 \beta_{5} - 7 \beta_{4} + \beta_{3} + 16 \beta_{2} + 21 \beta_{1} - 1\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(-7 \beta_{15} - 2 \beta_{14} + 7 \beta_{13} - 2 \beta_{12} + 11 \beta_{11} - 2 \beta_{10} + 8 \beta_{9} + 15 \beta_{8} + 30 \beta_{7} - 3 \beta_{6} - 4 \beta_{5} - 12 \beta_{4} + 32 \beta_{3} - 21 \beta_{2} + 51 \beta_{1} + 20\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-36 \beta_{15} + 3 \beta_{14} + 6 \beta_{13} + 37 \beta_{12} + 24 \beta_{11} - 21 \beta_{10} + 33 \beta_{9} + 49 \beta_{8} + 93 \beta_{7} - \beta_{5} - 39 \beta_{4} - 17 \beta_{3} + 46 \beta_{2} + 90 \beta_{1} - 3\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(-42 \beta_{15} + 68 \beta_{14} + 10 \beta_{13} + 66 \beta_{12} + 50 \beta_{11} - 25 \beta_{10} + 42 \beta_{9} - 71 \beta_{8} + 179 \beta_{7} + 61 \beta_{6} + 18 \beta_{5} + 50 \beta_{4} + 31 \beta_{3} - 16 \beta_{2} + 163 \beta_{1} + 94\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(-44 \beta_{15} + 32 \beta_{14} + 216 \beta_{12} + 72 \beta_{11} + 14 \beta_{10} + 40 \beta_{9} + 50 \beta_{8} - 19 \beta_{7} + 66 \beta_{6} + 126 \beta_{5} + 66 \beta_{4} - 64 \beta_{3} + 110 \beta_{2} + 67 \beta_{1} + 168\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(-58 \beta_{15} - 18 \beta_{14} + 90 \beta_{13} + 172 \beta_{12} + 126 \beta_{11} + 162 \beta_{10} + 62 \beta_{9} + 14 \beta_{8} - 161 \beta_{7} - 8 \beta_{6} + 167 \beta_{5} + 158 \beta_{4} + 153 \beta_{3} - 144 \beta_{2} - 61 \beta_{1} + 188\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(-126 \beta_{15} - 351 \beta_{14} - 100 \beta_{13} + 231 \beta_{12} + 13 \beta_{11} + 333 \beta_{10} + 48 \beta_{9} + 80 \beta_{8} - 482 \beta_{7} - 81 \beta_{6} + 141 \beta_{5} - 219 \beta_{4} - 138 \beta_{3} + 86 \beta_{2} - 46 \beta_{1} + 217\)\()/3\)
\(\nu^{12}\)\(=\)\((\)\(-209 \beta_{15} - 418 \beta_{14} - 251 \beta_{13} - 170 \beta_{12} + 68 \beta_{11} + 416 \beta_{10} + 301 \beta_{9} - 320 \beta_{8} + 325 \beta_{7} - 36 \beta_{6} - 298 \beta_{5} + 153 \beta_{4} - 357 \beta_{3} + 376 \beta_{2} + 1031 \beta_{1} - 97\)\()/3\)
\(\nu^{13}\)\(=\)\((\)\(385 \beta_{15} - 609 \beta_{14} - 1440 \beta_{13} + 402 \beta_{12} - 186 \beta_{11} + 414 \beta_{10} - 155 \beta_{9} - 478 \beta_{8} - 220 \beta_{7} + 98 \beta_{6} - 597 \beta_{5} + 277 \beta_{4} - 1199 \beta_{3} + 1540 \beta_{2} + 2320 \beta_{1} + 1243\)\()/3\)
\(\nu^{14}\)\(=\)\((\)\(1281 \beta_{15} - 511 \beta_{14} - 835 \beta_{13} + 960 \beta_{12} + 1042 \beta_{11} + 812 \beta_{10} - 27 \beta_{9} + 3809 \beta_{8} + 541 \beta_{7} - 1481 \beta_{6} - 882 \beta_{5} + 1178 \beta_{4} - 1454 \beta_{3} + 2715 \beta_{2} + 2486 \beta_{1} - 1641\)\()/3\)
\(\nu^{15}\)\(=\)\((\)\(2826 \beta_{15} - 1179 \beta_{14} - 1012 \beta_{13} + 2828 \beta_{12} + 1522 \beta_{11} + 2838 \beta_{10} - 2730 \beta_{9} + 6361 \beta_{8} - 11 \beta_{7} - 2559 \beta_{6} - 431 \beta_{5} - 2124 \beta_{4} + 809 \beta_{3} + 757 \beta_{2} + 1787 \beta_{1} + 208\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\)
\(\chi(n)\) \(\beta_{8}\) \(\beta_{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
1.73109 0.0577511i
−1.68301 + 0.409224i
0.320287 + 1.70218i
0.765614 1.55365i
1.27866 1.16834i
−1.70672 0.295146i
1.58110 + 0.707199i
1.71298 + 0.256290i
1.27866 + 1.16834i
−1.70672 + 0.295146i
1.58110 0.707199i
1.71298 0.256290i
1.73109 + 0.0577511i
−1.68301 0.409224i
0.320287 1.70218i
0.765614 + 1.55365i
1.00000i 0 −1.00000 −1.14095 + 1.97618i 0 1.42337 2.23025i 1.00000i 0 1.97618 + 1.14095i
143.2 1.00000i 0 −1.00000 −0.714925 + 1.23829i 0 0.327442 + 2.62541i 1.00000i 0 1.23829 + 0.714925i
143.3 1.00000i 0 −1.00000 0.0338034 0.0585493i 0 1.19767 + 2.35915i 1.00000i 0 −0.0585493 0.0338034i
143.4 1.00000i 0 −1.00000 1.82207 3.15592i 0 −1.58246 2.12034i 1.00000i 0 −3.15592 1.82207i
143.5 1.00000i 0 −1.00000 −1.77612 + 3.07634i 0 2.63804 + 0.201867i 1.00000i 0 −3.07634 1.77612i
143.6 1.00000i 0 −1.00000 −0.483662 + 0.837727i 0 −2.16249 1.52435i 1.00000i 0 −0.837727 0.483662i
143.7 1.00000i 0 −1.00000 0.450129 0.779646i 0 1.57151 2.12847i 1.00000i 0 0.779646 + 0.450129i
143.8 1.00000i 0 −1.00000 1.80966 3.13442i 0 −2.41308 + 1.08492i 1.00000i 0 3.13442 + 1.80966i
341.1 1.00000i 0 −1.00000 −1.77612 3.07634i 0 2.63804 0.201867i 1.00000i 0 −3.07634 + 1.77612i
341.2 1.00000i 0 −1.00000 −0.483662 0.837727i 0 −2.16249 + 1.52435i 1.00000i 0 −0.837727 + 0.483662i
341.3 1.00000i 0 −1.00000 0.450129 + 0.779646i 0 1.57151 + 2.12847i 1.00000i 0 0.779646 0.450129i
341.4 1.00000i 0 −1.00000 1.80966 + 3.13442i 0 −2.41308 1.08492i 1.00000i 0 3.13442 1.80966i
341.5 1.00000i 0 −1.00000 −1.14095 1.97618i 0 1.42337 + 2.23025i 1.00000i 0 1.97618 1.14095i
341.6 1.00000i 0 −1.00000 −0.714925 1.23829i 0 0.327442 2.62541i 1.00000i 0 1.23829 0.714925i
341.7 1.00000i 0 −1.00000 0.0338034 + 0.0585493i 0 1.19767 2.35915i 1.00000i 0 −0.0585493 + 0.0338034i
341.8 1.00000i 0 −1.00000 1.82207 + 3.15592i 0 −1.58246 + 2.12034i 1.00000i 0 −3.15592 + 1.82207i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 341.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.l.a 16
3.b odd 2 1 126.2.l.a 16
4.b odd 2 1 3024.2.ca.c 16
7.b odd 2 1 2646.2.l.a 16
7.c even 3 1 2646.2.m.b 16
7.c even 3 1 2646.2.t.b 16
7.d odd 6 1 378.2.t.a 16
7.d odd 6 1 2646.2.m.a 16
9.c even 3 1 126.2.t.a yes 16
9.c even 3 1 1134.2.k.b 16
9.d odd 6 1 378.2.t.a 16
9.d odd 6 1 1134.2.k.a 16
12.b even 2 1 1008.2.ca.c 16
21.c even 2 1 882.2.l.b 16
21.g even 6 1 126.2.t.a yes 16
21.g even 6 1 882.2.m.a 16
21.h odd 6 1 882.2.m.b 16
21.h odd 6 1 882.2.t.a 16
28.f even 6 1 3024.2.df.c 16
36.f odd 6 1 1008.2.df.c 16
36.h even 6 1 3024.2.df.c 16
63.g even 3 1 882.2.m.a 16
63.h even 3 1 882.2.l.b 16
63.i even 6 1 inner 378.2.l.a 16
63.j odd 6 1 2646.2.l.a 16
63.k odd 6 1 882.2.m.b 16
63.k odd 6 1 1134.2.k.a 16
63.l odd 6 1 882.2.t.a 16
63.n odd 6 1 2646.2.m.a 16
63.o even 6 1 2646.2.t.b 16
63.s even 6 1 1134.2.k.b 16
63.s even 6 1 2646.2.m.b 16
63.t odd 6 1 126.2.l.a 16
84.j odd 6 1 1008.2.df.c 16
252.r odd 6 1 3024.2.ca.c 16
252.bj even 6 1 1008.2.ca.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.l.a 16 3.b odd 2 1
126.2.l.a 16 63.t odd 6 1
126.2.t.a yes 16 9.c even 3 1
126.2.t.a yes 16 21.g even 6 1
378.2.l.a 16 1.a even 1 1 trivial
378.2.l.a 16 63.i even 6 1 inner
378.2.t.a 16 7.d odd 6 1
378.2.t.a 16 9.d odd 6 1
882.2.l.b 16 21.c even 2 1
882.2.l.b 16 63.h even 3 1
882.2.m.a 16 21.g even 6 1
882.2.m.a 16 63.g even 3 1
882.2.m.b 16 21.h odd 6 1
882.2.m.b 16 63.k odd 6 1
882.2.t.a 16 21.h odd 6 1
882.2.t.a 16 63.l odd 6 1
1008.2.ca.c 16 12.b even 2 1
1008.2.ca.c 16 252.bj even 6 1
1008.2.df.c 16 36.f odd 6 1
1008.2.df.c 16 84.j odd 6 1
1134.2.k.a 16 9.d odd 6 1
1134.2.k.a 16 63.k odd 6 1
1134.2.k.b 16 9.c even 3 1
1134.2.k.b 16 63.s even 6 1
2646.2.l.a 16 7.b odd 2 1
2646.2.l.a 16 63.j odd 6 1
2646.2.m.a 16 7.d odd 6 1
2646.2.m.a 16 63.n odd 6 1
2646.2.m.b 16 7.c even 3 1
2646.2.m.b 16 63.s even 6 1
2646.2.t.b 16 7.c even 3 1
2646.2.t.b 16 63.o even 6 1
3024.2.ca.c 16 4.b odd 2 1
3024.2.ca.c 16 252.r odd 6 1
3024.2.df.c 16 28.f even 6 1
3024.2.df.c 16 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(378, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{8} \)
$3$ \( \)
$5$ \( 1 - 16 T^{2} + 24 T^{3} + 123 T^{4} - 330 T^{5} - 98 T^{6} + 2124 T^{7} - 4594 T^{8} + 102 T^{9} + 38946 T^{10} - 84396 T^{11} - 70916 T^{12} + 641196 T^{13} - 735151 T^{14} - 1444464 T^{15} + 6891966 T^{16} - 7222320 T^{17} - 18378775 T^{18} + 80149500 T^{19} - 44322500 T^{20} - 263737500 T^{21} + 608531250 T^{22} + 7968750 T^{23} - 1794531250 T^{24} + 4148437500 T^{25} - 957031250 T^{26} - 16113281250 T^{27} + 30029296875 T^{28} + 29296875000 T^{29} - 97656250000 T^{30} + 152587890625 T^{32} \)
$7$ \( 1 - 2 T + 6 T^{2} + 8 T^{3} + 23 T^{4} - 210 T^{5} + 463 T^{6} - 1148 T^{7} - 1350 T^{8} - 8036 T^{9} + 22687 T^{10} - 72030 T^{11} + 55223 T^{12} + 134456 T^{13} + 705894 T^{14} - 1647086 T^{15} + 5764801 T^{16} \)
$11$ \( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3735 T^{4} + 17808 T^{5} + 77432 T^{6} + 326142 T^{7} + 1351880 T^{8} + 5458080 T^{9} + 21295098 T^{10} + 79655676 T^{11} + 287397262 T^{12} + 1023482544 T^{13} + 3603683689 T^{14} + 12488798178 T^{15} + 42242848866 T^{16} + 137376779958 T^{17} + 436045726369 T^{18} + 1362255266064 T^{19} + 4207783312942 T^{20} + 12828626275476 T^{21} + 37725565107978 T^{22} + 106362538291680 T^{23} + 289787484046280 T^{24} + 769025775838122 T^{25} + 2008386661704632 T^{26} + 5080830230240688 T^{27} + 11722029987052935 T^{28} + 24027807652175976 T^{29} + 40253482359823546 T^{30} + 50126978032987812 T^{31} + 45949729863572161 T^{32} \)
$13$ \( 1 - 6 T + 47 T^{2} - 210 T^{3} + 888 T^{4} - 2658 T^{5} + 9181 T^{6} - 12564 T^{7} + 21641 T^{8} + 214770 T^{9} - 1221396 T^{10} + 6263526 T^{11} - 19011605 T^{12} + 72302586 T^{13} - 179114935 T^{14} + 742674348 T^{15} - 2320143696 T^{16} + 9654766524 T^{17} - 30270424015 T^{18} + 158848781442 T^{19} - 542990450405 T^{20} + 2325603359118 T^{21} - 5895445205364 T^{22} + 13476498996090 T^{23} + 17653228533161 T^{24} - 133234930122372 T^{25} + 1265678813665669 T^{26} - 4763562327350346 T^{27} + 20688699588763128 T^{28} - 63603772384373130 T^{29} + 185056690127866583 T^{30} - 307115358084544542 T^{31} + 665416609183179841 T^{32} \)
$17$ \( 1 - 18 T + 95 T^{2} + 42 T^{3} - 849 T^{4} - 7584 T^{5} + 29152 T^{6} + 139356 T^{7} - 154873 T^{8} - 2564070 T^{9} - 13186653 T^{10} + 89037906 T^{11} + 222290986 T^{12} - 933503742 T^{13} - 5890959001 T^{14} + 7644891330 T^{15} + 98422426836 T^{16} + 129963152610 T^{17} - 1702487151289 T^{18} - 4586303884446 T^{19} + 18565965441706 T^{20} + 126421094099442 T^{21} - 318293746666557 T^{22} - 1052137081279110 T^{23} - 1080356482159993 T^{24} + 16525932117115932 T^{25} + 58770254185889248 T^{26} - 259918061597088672 T^{27} - 494646279408067089 T^{28} + 415992277382049354 T^{29} + 15995893523143088255 T^{30} - 51523614927176684274 T^{31} + 48661191875666868481 T^{32} \)
$19$ \( 1 + 80 T^{2} + 3483 T^{4} - 882 T^{5} + 107770 T^{6} - 26190 T^{7} + 2596658 T^{8} + 477594 T^{9} + 51326154 T^{10} + 58243698 T^{11} + 858124540 T^{12} + 2331197604 T^{13} + 13169269781 T^{14} + 60976736712 T^{15} + 222780924306 T^{16} + 1158557997528 T^{17} + 4754106390941 T^{18} + 15989684365836 T^{19} + 111831648177340 T^{20} + 144217162374102 T^{21} + 2414684133271674 T^{22} + 426907779315966 T^{23} + 44100504838916978 T^{24} - 8451190804832010 T^{25} + 660745010603213770 T^{26} - 102744408348229158 T^{27} + 7708975863107438763 T^{28} + 63920534862630729680 T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \)
$23$ \( 1 - 6 T + 130 T^{2} - 708 T^{3} + 8559 T^{4} - 45012 T^{5} + 399452 T^{6} - 2115900 T^{7} + 15122540 T^{8} - 81277422 T^{9} + 486352566 T^{10} - 2645846004 T^{11} + 13727139058 T^{12} - 75456395706 T^{13} + 352786829089 T^{14} - 1928667059838 T^{15} + 8415100540206 T^{16} - 44359342376274 T^{17} + 186624232588081 T^{18} - 918077966554902 T^{19} + 3841416321129778 T^{20} - 17029572406923372 T^{21} + 71997634475241174 T^{22} - 276735434692157634 T^{23} + 1184261007351333740 T^{24} - 3811058916389561700 T^{25} + 16547902757314520348 T^{26} - 42887872823221682124 T^{27} + \)\(18\!\cdots\!39\)\( T^{28} - \)\(35\!\cdots\!64\)\( T^{29} + \)\(15\!\cdots\!70\)\( T^{30} - \)\(15\!\cdots\!42\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \)
$29$ \( 1 + 6 T + 196 T^{2} + 1104 T^{3} + 19989 T^{4} + 108858 T^{5} + 1435478 T^{6} + 7594740 T^{7} + 81322766 T^{8} + 416245488 T^{9} + 3844091772 T^{10} + 18872529852 T^{11} + 155981060614 T^{12} + 727660406082 T^{13} + 5509975202215 T^{14} + 24244157784798 T^{15} + 170557455776958 T^{16} + 703080575759142 T^{17} + 4633889145062815 T^{18} + 17746909643933898 T^{19} + 110322440532130534 T^{20} + 387097271801319948 T^{21} + 2286555434049814812 T^{22} + 7180183182179343792 T^{23} + 40681421983566770126 T^{24} + \)\(11\!\cdots\!60\)\( T^{25} + \)\(60\!\cdots\!78\)\( T^{26} + \)\(13\!\cdots\!82\)\( T^{27} + \)\(70\!\cdots\!49\)\( T^{28} + \)\(11\!\cdots\!56\)\( T^{29} + \)\(58\!\cdots\!76\)\( T^{30} + \)\(51\!\cdots\!94\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \)
$31$ \( 1 - 292 T^{2} + 41022 T^{4} - 3694910 T^{6} + 240356405 T^{8} - 12117919608 T^{10} + 500572972675 T^{12} - 17863857948586 T^{14} + 576631021225626 T^{16} - 17167167488591146 T^{18} + 462289652297788675 T^{20} - 10754698258162077048 T^{22} + \)\(20\!\cdots\!05\)\( T^{24} - \)\(30\!\cdots\!10\)\( T^{26} + \)\(32\!\cdots\!42\)\( T^{28} - \)\(22\!\cdots\!32\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \)
$37$ \( 1 + 2 T - 116 T^{2} - 656 T^{3} + 3854 T^{4} + 48538 T^{5} + 98460 T^{6} - 747702 T^{7} - 5251023 T^{8} - 50514234 T^{9} - 417738792 T^{10} + 801423570 T^{11} + 29966075382 T^{12} + 116541321036 T^{13} - 322717550496 T^{14} - 3343790405826 T^{15} - 14327838321804 T^{16} - 123720245015562 T^{17} - 441800326629024 T^{18} + 5903167534436508 T^{19} + 56161249804004502 T^{20} + 55573881576866490 T^{21} - 1071803450698157928 T^{22} - 4795411055555611122 T^{23} - 18444110399566611183 T^{24} - 97172652768258663054 T^{25} + \)\(47\!\cdots\!40\)\( T^{26} + \)\(86\!\cdots\!94\)\( T^{27} + \)\(25\!\cdots\!74\)\( T^{28} - \)\(15\!\cdots\!32\)\( T^{29} - \)\(10\!\cdots\!24\)\( T^{30} + \)\(66\!\cdots\!86\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \)
$41$ \( 1 - 6 T - 223 T^{2} + 1686 T^{3} + 25980 T^{4} - 231654 T^{5} - 1971341 T^{6} + 20408106 T^{7} + 109216031 T^{8} - 1268388768 T^{9} - 4836103872 T^{10} + 57438167556 T^{11} + 192545389345 T^{12} - 1824668193534 T^{13} - 7677147470143 T^{14} + 27964729912410 T^{15} + 313424888729076 T^{16} + 1146553926408810 T^{17} - 12905284897310383 T^{18} - 125757956566556814 T^{19} + 544087251940916545 T^{20} + 6654567885439614756 T^{21} - 22971997512303721152 T^{22} - \)\(24\!\cdots\!08\)\( T^{23} + \)\(87\!\cdots\!51\)\( T^{24} + \)\(66\!\cdots\!66\)\( T^{25} - \)\(26\!\cdots\!41\)\( T^{26} - \)\(12\!\cdots\!14\)\( T^{27} + \)\(58\!\cdots\!80\)\( T^{28} + \)\(15\!\cdots\!06\)\( T^{29} - \)\(84\!\cdots\!03\)\( T^{30} - \)\(93\!\cdots\!06\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \)
$43$ \( 1 + 2 T - 209 T^{2} - 602 T^{3} + 20774 T^{4} + 72052 T^{5} - 1457073 T^{6} - 4933350 T^{7} + 91147851 T^{8} + 235722522 T^{9} - 5466637218 T^{10} - 9144255228 T^{11} + 301177025103 T^{12} + 290829011802 T^{13} - 14773623661707 T^{14} - 4808085837138 T^{15} + 658882369500660 T^{16} - 206747690996934 T^{17} - 27316430150496243 T^{18} + 23122942241341614 T^{19} + 1029664314599161503 T^{20} - 1344282723462890004 T^{21} - 34556598512153357682 T^{22} + 64073768536679251854 T^{23} + \)\(10\!\cdots\!51\)\( T^{24} - \)\(24\!\cdots\!50\)\( T^{25} - \)\(31\!\cdots\!77\)\( T^{26} + \)\(66\!\cdots\!64\)\( T^{27} + \)\(83\!\cdots\!74\)\( T^{28} - \)\(10\!\cdots\!86\)\( T^{29} - \)\(15\!\cdots\!41\)\( T^{30} + \)\(63\!\cdots\!14\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \)
$47$ \( ( 1 - 18 T + 379 T^{2} - 4272 T^{3} + 53788 T^{4} - 467070 T^{5} + 4528654 T^{6} - 32587152 T^{7} + 257763508 T^{8} - 1531596144 T^{9} + 10003796686 T^{10} - 48492608610 T^{11} + 262468281628 T^{12} - 979761869904 T^{13} + 4085322609691 T^{14} - 9119216168334 T^{15} + 23811286661761 T^{16} )^{2} \)
$53$ \( 1 - 36 T + 766 T^{2} - 12024 T^{3} + 153999 T^{4} - 1662408 T^{5} + 15312632 T^{6} - 121481640 T^{7} + 835010600 T^{8} - 5038052724 T^{9} + 28358007246 T^{10} - 182105031078 T^{11} + 1622318702806 T^{12} - 17289317216124 T^{13} + 177226593270661 T^{14} - 1605696832118286 T^{15} + 12594715934262750 T^{16} - 85101932102269158 T^{17} + 497829500497286749 T^{18} - 2573981679184892748 T^{19} + 12800874900435389686 T^{20} - 76155503249444531454 T^{21} + \)\(62\!\cdots\!34\)\( T^{22} - \)\(59\!\cdots\!88\)\( T^{23} + \)\(51\!\cdots\!00\)\( T^{24} - \)\(40\!\cdots\!20\)\( T^{25} + \)\(26\!\cdots\!68\)\( T^{26} - \)\(15\!\cdots\!76\)\( T^{27} + \)\(75\!\cdots\!59\)\( T^{28} - \)\(31\!\cdots\!52\)\( T^{29} + \)\(10\!\cdots\!54\)\( T^{30} - \)\(26\!\cdots\!52\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \)
$59$ \( ( 1 + 30 T + 700 T^{2} + 11136 T^{3} + 154015 T^{4} + 1732056 T^{5} + 17715805 T^{6} + 156334302 T^{7} + 1281973738 T^{8} + 9223723818 T^{9} + 61668717205 T^{10} + 355727929224 T^{11} + 1866255354415 T^{12} + 7961396993664 T^{13} + 29526373548700 T^{14} + 74659544544570 T^{15} + 146830437604321 T^{16} )^{2} \)
$61$ \( 1 - 472 T^{2} + 107634 T^{4} - 15742946 T^{6} + 1661711189 T^{8} - 136358974992 T^{10} + 9312516091879 T^{12} - 573351459745642 T^{14} + 34616863644604218 T^{16} - 2133440781713533882 T^{18} + \)\(12\!\cdots\!39\)\( T^{20} - \)\(70\!\cdots\!12\)\( T^{22} + \)\(31\!\cdots\!09\)\( T^{24} - \)\(11\!\cdots\!46\)\( T^{26} + \)\(28\!\cdots\!14\)\( T^{28} - \)\(46\!\cdots\!52\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \)
$67$ \( ( 1 + 14 T + 435 T^{2} + 5596 T^{3} + 90428 T^{4} + 1008078 T^{5} + 11375926 T^{6} + 106594220 T^{7} + 933924096 T^{8} + 7141812740 T^{9} + 51066531814 T^{10} + 303192563514 T^{11} + 1822225569788 T^{12} + 7555300098772 T^{13} + 39349396243515 T^{14} + 84849962474522 T^{15} + 406067677556641 T^{16} )^{2} \)
$71$ \( 1 - 650 T^{2} + 199389 T^{4} - 38930632 T^{6} + 5566228364 T^{8} - 640511863116 T^{10} + 63163988645884 T^{12} - 5475157404521894 T^{14} + 416179213677825948 T^{16} - 27600268476194867654 T^{18} + \)\(16\!\cdots\!04\)\( T^{20} - \)\(82\!\cdots\!36\)\( T^{22} + \)\(35\!\cdots\!04\)\( T^{24} - \)\(12\!\cdots\!32\)\( T^{26} + \)\(32\!\cdots\!49\)\( T^{28} - \)\(53\!\cdots\!50\)\( T^{30} + \)\(41\!\cdots\!21\)\( T^{32} \)
$73$ \( 1 + 434 T^{2} + 101253 T^{4} + 70380 T^{5} + 16361620 T^{6} + 29833866 T^{7} + 2027081504 T^{8} + 6817588704 T^{9} + 204710173188 T^{10} + 1064159524062 T^{11} + 17706469884892 T^{12} + 124668495857988 T^{13} + 1383495353685575 T^{14} + 11427374402737884 T^{15} + 102528286806790386 T^{16} + 834198331399865532 T^{17} + 7372646739790429175 T^{18} + 48498164253186917796 T^{19} + \)\(50\!\cdots\!72\)\( T^{20} + \)\(22\!\cdots\!66\)\( T^{21} + \)\(30\!\cdots\!32\)\( T^{22} + \)\(75\!\cdots\!88\)\( T^{23} + \)\(16\!\cdots\!24\)\( T^{24} + \)\(17\!\cdots\!58\)\( T^{25} + \)\(70\!\cdots\!80\)\( T^{26} + \)\(22\!\cdots\!60\)\( T^{27} + \)\(23\!\cdots\!13\)\( T^{28} + \)\(52\!\cdots\!06\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \)
$79$ \( ( 1 - 16 T + 483 T^{2} - 6542 T^{3} + 113126 T^{4} - 1278384 T^{5} + 16380706 T^{6} - 152801626 T^{7} + 1573503228 T^{8} - 12071328454 T^{9} + 102231986146 T^{10} - 630293168976 T^{11} + 4406266863206 T^{12} - 20130102962258 T^{13} + 117411241016643 T^{14} - 307262543778544 T^{15} + 1517108809906561 T^{16} )^{2} \)
$83$ \( 1 - 487 T^{2} - 312 T^{3} + 123774 T^{4} + 132990 T^{5} - 22183883 T^{6} - 29138634 T^{7} + 3170469341 T^{8} + 4110229572 T^{9} - 386467088226 T^{10} - 393115428402 T^{11} + 41656592194789 T^{12} + 25282823866380 T^{13} - 4030130568645907 T^{14} - 780079655467782 T^{15} + 351851707607703156 T^{16} - 64746611403825906 T^{17} - 27763569487401653323 T^{18} + 14456390010085821060 T^{19} + \)\(19\!\cdots\!69\)\( T^{20} - \)\(15\!\cdots\!86\)\( T^{21} - \)\(12\!\cdots\!94\)\( T^{22} + \)\(11\!\cdots\!44\)\( T^{23} + \)\(71\!\cdots\!81\)\( T^{24} - \)\(54\!\cdots\!02\)\( T^{25} - \)\(34\!\cdots\!67\)\( T^{26} + \)\(17\!\cdots\!30\)\( T^{27} + \)\(13\!\cdots\!14\)\( T^{28} - \)\(27\!\cdots\!56\)\( T^{29} - \)\(35\!\cdots\!23\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( 1 - 24 T - 10 T^{2} + 3636 T^{3} + 1197 T^{4} - 543420 T^{5} + 1792912 T^{6} + 58468350 T^{7} - 544062388 T^{8} - 3510821196 T^{9} + 61278252588 T^{10} + 296095488138 T^{11} - 6613804791944 T^{12} - 16297764313308 T^{13} + 597103648531739 T^{14} - 218011613471172 T^{15} - 39896652829458150 T^{16} - 19403033598934308 T^{17} + 4729658000019904619 T^{18} - 11489418610188427452 T^{19} - \)\(41\!\cdots\!04\)\( T^{20} + \)\(16\!\cdots\!62\)\( T^{21} + \)\(30\!\cdots\!68\)\( T^{22} - \)\(15\!\cdots\!84\)\( T^{23} - \)\(21\!\cdots\!28\)\( T^{24} + \)\(20\!\cdots\!50\)\( T^{25} + \)\(55\!\cdots\!12\)\( T^{26} - \)\(15\!\cdots\!80\)\( T^{27} + \)\(29\!\cdots\!37\)\( T^{28} + \)\(79\!\cdots\!84\)\( T^{29} - \)\(19\!\cdots\!10\)\( T^{30} - \)\(41\!\cdots\!76\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \)
$97$ \( 1 - 6 T + 395 T^{2} - 2298 T^{3} + 68379 T^{4} - 571872 T^{5} + 8922340 T^{6} - 109039644 T^{7} + 1233764039 T^{8} - 14418445794 T^{9} + 171203907675 T^{10} - 1605167799666 T^{11} + 21002714173786 T^{12} - 194363495836338 T^{13} + 2133943680044999 T^{14} - 22669145285986746 T^{15} + 200280216945639852 T^{16} - 2198907092740714362 T^{17} + 20078276085543395591 T^{18} - \)\(17\!\cdots\!74\)\( T^{19} + \)\(18\!\cdots\!66\)\( T^{20} - \)\(13\!\cdots\!62\)\( T^{21} + \)\(14\!\cdots\!75\)\( T^{22} - \)\(11\!\cdots\!22\)\( T^{23} + \)\(96\!\cdots\!79\)\( T^{24} - \)\(82\!\cdots\!48\)\( T^{25} + \)\(65\!\cdots\!60\)\( T^{26} - \)\(40\!\cdots\!16\)\( T^{27} + \)\(47\!\cdots\!39\)\( T^{28} - \)\(15\!\cdots\!46\)\( T^{29} + \)\(25\!\cdots\!55\)\( T^{30} - \)\(37\!\cdots\!58\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \)
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