Properties

Label 450.2.l.c
Level 450
Weight 2
Character orbit 450.l
Analytic conductor 3.593
Analytic rank 0
Dimension 16
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 450.l (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 150)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 4 x^{15} - 24 x^{14} + 94 x^{13} + 262 x^{12} - 936 x^{11} - 1584 x^{10} + 4642 x^{9} + 6259 x^{8} - 11958 x^{7} - 15752 x^{6} + 14670 x^{5} + 18271 x^{4} - 10440 x^{3} + 1135 x^{2} + 21080 x + 11105\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-216062652991087486517 \nu^{15} - 222689963418511649230 \nu^{14} + 11692850809057402771308 \nu^{13} - 4830492818371367785816 \nu^{12} - 200158344597704877408533 \nu^{11} + 142199510139739938929725 \nu^{10} + 1696861859405690275607183 \nu^{9} - 1203647771351280118822772 \nu^{8} - 7616568797732981879968331 \nu^{7} + 3146071202213920505926562 \nu^{6} + 19090952137993148860203137 \nu^{5} - 132407938274590801116617 \nu^{4} - 24907817546557276804454900 \nu^{3} + 3697181661562006774932655 \nu^{2} - 2518379054720136093673150 \nu - 36766787356071608007667585\)\()/ \)\(30\!\cdots\!75\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-265732875567395073657 \nu^{15} + 1990174463411413335455 \nu^{14} + 15780868218717044920838 \nu^{13} - 82635447635890840435206 \nu^{12} - 310570846147264249920748 \nu^{11} + 1241691800858697588974155 \nu^{10} + 3010621887147134394227693 \nu^{9} - 9299193782108641569228897 \nu^{8} - 15187750120575825142764691 \nu^{7} + 37039168692149977669870742 \nu^{6} + 41504619394117749813534187 \nu^{5} - 78953875227996269190814922 \nu^{4} - 49901182956943689204688600 \nu^{3} + 104941588005445229750025955 \nu^{2} - 24344868121453920674255425 \nu - 91429957070907564671351610\)\()/ \)\(30\!\cdots\!75\)\( \)
\(\beta_{3}\)\(=\)\((\)\(702375526475979478783 \nu^{15} - 3304044074082752777785 \nu^{14} - 25795447997562939235427 \nu^{13} + 114953003304995515650009 \nu^{12} + 368633707984163431786287 \nu^{11} - 1482042781994674721659060 \nu^{10} - 2812846670822251548242917 \nu^{9} + 9176301900507627049359998 \nu^{8} + 12537511618422589653976589 \nu^{7} - 26170693115741185316682238 \nu^{6} - 40039221089263675036570378 \nu^{5} + 32685303511358818643459718 \nu^{4} + 72705390025681627618678875 \nu^{3} - 35152535417255483308930345 \nu^{2} - 30211511665347684930563925 \nu + 49504201727591742141925840\)\()/ \)\(30\!\cdots\!75\)\( \)
\(\beta_{4}\)\(=\)\((\)\(2296292574754742295778 \nu^{15} - 18777167771030970954085 \nu^{14} - 5742569515547694072792 \nu^{13} + 376031136320790011131049 \nu^{12} - 424999022122650057110578 \nu^{11} - 3188418937056165399714235 \nu^{10} + 5262312369001650000411893 \nu^{9} + 12651496691912944253088318 \nu^{8} - 21294724748266832493330656 \nu^{7} - 31019540923068154312589818 \nu^{6} + 37963741339531390922508492 \nu^{5} + 52805059704158952082659513 \nu^{4} - 32920117107035471964771150 \nu^{3} - 38527298141204740926854120 \nu^{2} + 44218902932902401592517525 \nu - 10609792203969280509060435\)\()/ \)\(30\!\cdots\!75\)\( \)
\(\beta_{5}\)\(=\)\((\)\(102309325228650587446 \nu^{15} - 425968439454909023504 \nu^{14} - 2318583371847014159396 \nu^{13} + 9842957759035774370007 \nu^{12} + 23588922138538677715666 \nu^{11} - 96439403899597794479889 \nu^{10} - 132289916480158900957958 \nu^{9} + 469952553315433857842339 \nu^{8} + 521913912002145220106872 \nu^{7} - 1201274203759528179191999 \nu^{6} - 1439503235730439393390400 \nu^{5} + 1561882257402276944726861 \nu^{4} + 1751624753872932949676820 \nu^{3} - 1056349544697974528792295 \nu^{2} + 648738016230841178303300 \nu + 2136427940493114503422100\)\()/ \)\(12\!\cdots\!55\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-3180944021456305418003 \nu^{15} + 12108798860040862926030 \nu^{14} + 76658914050049775735647 \nu^{13} - 279795475163391518337309 \nu^{12} - 843282840396799929838402 \nu^{11} + 2772410277587660664967595 \nu^{10} + 5120683136367422331041022 \nu^{9} - 13952896772257730298480513 \nu^{8} - 20136486077065159637670929 \nu^{7} + 38669929503840247602021988 \nu^{6} + 48847780672259798172146818 \nu^{5} - 57387686031561159003451668 \nu^{4} - 56558593741097731772595025 \nu^{3} + 52324310540791207677909920 \nu^{2} - 9360398580013835322450250 \nu - 54250932322302201472419340\)\()/ \)\(30\!\cdots\!75\)\( \)
\(\beta_{7}\)\(=\)\((\)\(3754084957894051064894 \nu^{15} - 15083470458469574064070 \nu^{14} - 89761163865409232604666 \nu^{13} + 347746317243990280362907 \nu^{12} + 998563896155868111731741 \nu^{11} - 3429976210567685569153495 \nu^{10} - 6202656224864865816953881 \nu^{9} + 17064490132080048666621894 \nu^{8} + 24890198037032388014893387 \nu^{7} - 46215997398926518335783349 \nu^{6} - 60890899573543787233277629 \nu^{5} + 60090136619662887556863749 \nu^{4} + 77529726343056299821308475 \nu^{3} - 40267892959128470588816910 \nu^{2} - 36825642304869580862835650 \nu + 68909263042569036125855870\)\()/ \)\(30\!\cdots\!75\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-3973936680890235791447 \nu^{15} + 19337803181486775124430 \nu^{14} + 73342780711804225674553 \nu^{13} - 416649754235775402919971 \nu^{12} - 558606750221222446608433 \nu^{11} + 3759928827840084662619945 \nu^{10} + 1783092343379774194213858 \nu^{9} - 16064794943644897425418542 \nu^{8} - 4220420776903807591816446 \nu^{7} + 35426011517516138530176972 \nu^{6} + 10251508388168254787967127 \nu^{5} - 38968246371604410909723242 \nu^{4} - 1287569082167687474870500 \nu^{3} + 19285624254963832677634030 \nu^{2} - 37529598314145175899342950 \nu - 3543808081464618883649710\)\()/ \)\(30\!\cdots\!75\)\( \)
\(\beta_{9}\)\(=\)\((\)\(161168803239414780910 \nu^{15} - 661637860075799467724 \nu^{14} - 3827536126025598552754 \nu^{13} + 15994665739743758299833 \nu^{12} + 40151303735551147100534 \nu^{11} - 164810455922834007902603 \nu^{10} - 224574784032425709435836 \nu^{9} + 865180285576144364779891 \nu^{8} + 794296728242089068127598 \nu^{7} - 2456515842661058092401641 \nu^{6} - 1829541259881354437224758 \nu^{5} + 3808469691277882506316428 \nu^{4} + 1780293400073412257719180 \nu^{3} - 3317734863222409081306840 \nu^{2} + 1107814885746452854956660 \nu + 2686892356943722466767320\)\()/ \)\(12\!\cdots\!55\)\( \)
\(\beta_{10}\)\(=\)\((\)\(164194301487618633410 \nu^{15} - 719766109199837227876 \nu^{14} - 3474010950605366952410 \nu^{13} + 16401138606042229504145 \nu^{12} + 31059323892781834514230 \nu^{11} - 157188061964900911061375 \nu^{10} - 130154545341889076617028 \nu^{9} + 733989392081838127266340 \nu^{8} + 317412747752376679716350 \nu^{7} - 1736893910036273955685895 \nu^{6} - 552527714012012839873230 \nu^{5} + 1984064979606023800874943 \nu^{4} + 47852376229771545163160 \nu^{3} - 1332489074800974413015230 \nu^{2} + 2233536035317308402213160 \nu + 1025511991580005661458970\)\()/ \)\(12\!\cdots\!55\)\( \)
\(\beta_{11}\)\(=\)\((\)\(838230261608330797746 \nu^{15} - 4786382082628259634787 \nu^{14} - 13869542625980282893647 \nu^{13} + 106085643722406510377459 \nu^{12} + 90720060109888652320392 \nu^{11} - 1002540084989697030577461 \nu^{10} - 234297600527699664967938 \nu^{9} + 4699509630594445307398133 \nu^{8} + 1109399877527339581698789 \nu^{7} - 12618721201525299962308298 \nu^{6} - 6217938413303324609066387 \nu^{5} + 20000504884661427693942799 \nu^{4} + 10638060647839174881601105 \nu^{3} - 18066278410794456581746620 \nu^{2} + 3148595090397016687536785 \nu + 17988954213833427432880420\)\()/ \)\(60\!\cdots\!75\)\( \)
\(\beta_{12}\)\(=\)\((\)\(908669311439016539378 \nu^{15} - 5174938240432228690153 \nu^{14} - 15283113254819993188609 \nu^{13} + 120512535486778277787298 \nu^{12} + 82750556881744744497304 \nu^{11} - 1186706887477185726218359 \nu^{10} + 90637788457715278790657 \nu^{9} + 5770412096942933836755541 \nu^{8} - 1561116277429088140696022 \nu^{7} - 14822422913804415887485051 \nu^{6} + 1820874604516396099156933 \nu^{5} + 20462537822506322798652590 \nu^{4} + 1296008565761348543038545 \nu^{3} - 11744901364273461227406290 \nu^{2} + 12987872084176881105529190 \nu + 4308719570906786186270775\)\()/ \)\(60\!\cdots\!75\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-6533138943972814245933 \nu^{15} + 31619240120523939425865 \nu^{14} + 125897227911835136338677 \nu^{13} - 708040835635361844344624 \nu^{12} - 1007374201550559641412517 \nu^{11} + 6724682741769758012533880 \nu^{10} + 3470026494541648226365932 \nu^{9} - 31333706626178340309002363 \nu^{8} - 7246181075197585096864139 \nu^{7} + 77592234545423436266661168 \nu^{6} + 11636953568620512039028188 \nu^{5} - 98305813076082701994022623 \nu^{4} + 10820382804042389176367050 \nu^{3} + 67691544638516502492240320 \nu^{2} - 84912508264192029168955175 \nu - 61955069294991105282252615\)\()/ \)\(30\!\cdots\!75\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-9361640845971301396397 \nu^{15} + 52366780732480415686465 \nu^{14} + 159762414534900794176883 \nu^{13} - 1190582299946926644582166 \nu^{12} - 1024978252277545106715408 \nu^{11} + 11575346968422089033871410 \nu^{10} + 1598877636542195838733293 \nu^{9} - 55828465749972912641212972 \nu^{8} + 613642773665136143529419 \nu^{7} + 145008784517773349507323362 \nu^{6} + 9914427702507185104901877 \nu^{5} - 183985152384127705242957177 \nu^{4} - 15715753986137640451935575 \nu^{3} + 81809532792011322933042280 \nu^{2} - 134474994846987328679889500 \nu - 86034467893737742078614385\)\()/ \)\(30\!\cdots\!75\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-1965198321477020977800 \nu^{15} + 9926479283735303812942 \nu^{14} + 40235447863223328373118 \nu^{13} - 237493265324739844946336 \nu^{12} - 348172097985448217050033 \nu^{11} + 2410564961185097493334441 \nu^{10} + 1429316307775823908404854 \nu^{9} - 12215850222197537282508032 \nu^{8} - 3751634380400785138202611 \nu^{7} + 32442715594977735558622037 \nu^{6} + 8620102019274457312300996 \nu^{5} - 42222846117366993415189833 \nu^{4} - 4541857374012864958803405 \nu^{3} + 21432392262237651257289005 \nu^{2} - 32435795750138591078265085 \nu - 25977835089457219631295215\)\()/ \)\(60\!\cdots\!75\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{15} - \beta_{14} - \beta_{13} + 4 \beta_{12} - 3 \beta_{11} - 3 \beta_{10} - 2 \beta_{9} - 4 \beta_{8} + \beta_{7} + 2 \beta_{6} + 4 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} - 3 \beta_{1} - 1\)\()/5\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - 7 \beta_{11} - 7 \beta_{10} - 3 \beta_{9} - 6 \beta_{8} - \beta_{7} - 7 \beta_{6} + 6 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - \beta_{2} + 13 \beta_{1} + 16\)\()/5\)
\(\nu^{3}\)\(=\)\((\)\(17 \beta_{15} - 6 \beta_{14} - 26 \beta_{13} + 29 \beta_{12} - 23 \beta_{11} - 53 \beta_{10} - 2 \beta_{9} - 34 \beta_{8} + 6 \beta_{7} + 12 \beta_{6} + 19 \beta_{5} - 17 \beta_{4} - 8 \beta_{3} - 19 \beta_{2} - 13 \beta_{1} + 4\)\()/5\)
\(\nu^{4}\)\(=\)\((\)\(-12 \beta_{15} + 31 \beta_{14} - 89 \beta_{13} + 21 \beta_{12} - 77 \beta_{11} - 107 \beta_{10} - 78 \beta_{9} - 6 \beta_{8} - 11 \beta_{7} - 77 \beta_{6} + 111 \beta_{5} + 22 \beta_{4} + 53 \beta_{3} - 21 \beta_{2} + 153 \beta_{1} + 66\)\()/5\)
\(\nu^{5}\)\(=\)\((\)\(102 \beta_{15} - 31 \beta_{14} - 356 \beta_{13} + 179 \beta_{12} - 158 \beta_{11} - 593 \beta_{10} - 97 \beta_{9} - 244 \beta_{8} - 74 \beta_{7} - 108 \beta_{6} + 159 \beta_{5} - 177 \beta_{4} + 32 \beta_{3} - 99 \beta_{2} + 112 \beta_{1} + 69\)\()/5\)
\(\nu^{6}\)\(=\)\((\)\(-67 \beta_{15} + 396 \beta_{14} - 1679 \beta_{13} + 411 \beta_{12} - 757 \beta_{11} - 1647 \beta_{10} - 1303 \beta_{9} + 284 \beta_{8} - 86 \beta_{7} - 877 \beta_{6} + 1561 \beta_{5} - 3 \beta_{4} + 508 \beta_{3} - 276 \beta_{2} + 1658 \beta_{1} - 104\)\()/5\)
\(\nu^{7}\)\(=\)\((\)\(322 \beta_{15} + 209 \beta_{14} - 4661 \beta_{13} + 1249 \beta_{12} - 1198 \beta_{11} - 6368 \beta_{10} - 2737 \beta_{9} - 1044 \beta_{8} - 1629 \beta_{7} - 3763 \beta_{6} + 1859 \beta_{5} - 1652 \beta_{4} + 1272 \beta_{3} - 399 \beta_{2} + 3612 \beta_{1} + 659\)\()/5\)
\(\nu^{8}\)\(=\)\((\)\(-722 \beta_{15} + 4821 \beta_{14} - 22869 \beta_{13} + 6081 \beta_{12} - 6637 \beta_{11} - 22292 \beta_{10} - 17653 \beta_{9} + 5054 \beta_{8} - 1361 \beta_{7} - 11897 \beta_{6} + 17361 \beta_{5} - 2488 \beta_{4} + 4463 \beta_{3} - 2791 \beta_{2} + 17423 \beta_{1} - 6464\)\()/5\)
\(\nu^{9}\)\(=\)\((\)\(-4648 \beta_{15} + 10089 \beta_{14} - 61441 \beta_{13} + 10719 \beta_{12} - 10283 \beta_{11} - 69933 \beta_{10} - 49987 \beta_{9} + 6701 \beta_{8} - 21274 \beta_{7} - 64498 \beta_{6} + 23899 \beta_{5} - 13622 \beta_{4} + 20527 \beta_{3} + 561 \beta_{2} + 58037 \beta_{1} + 1354\)\()/5\)
\(\nu^{10}\)\(=\)\((\)\(-17042 \beta_{15} + 61161 \beta_{14} - 277069 \beta_{13} + 72921 \beta_{12} - 50587 \beta_{11} - 273867 \beta_{10} - 221948 \beta_{9} + 71349 \beta_{8} - 28166 \beta_{7} - 175417 \beta_{6} + 166606 \beta_{5} - 42628 \beta_{4} + 45218 \beta_{3} - 18106 \beta_{2} + 182258 \beta_{1} - 101659\)\()/5\)
\(\nu^{11}\)\(=\)\((\)\(-134263 \beta_{15} + 202349 \beta_{14} - 797386 \beta_{13} + 111324 \beta_{12} - 86558 \beta_{11} - 792838 \beta_{10} - 749067 \beta_{9} + 261221 \beta_{8} - 241464 \beta_{7} - 907323 \beta_{6} + 302154 \beta_{5} - 100422 \beta_{4} + 265777 \beta_{3} + 51356 \beta_{2} + 747912 \beta_{1} - 106706\)\()/5\)
\(\nu^{12}\)\(=\)\((\)\(-378617 \beta_{15} + 805976 \beta_{14} - 3193009 \beta_{13} + 749601 \beta_{12} - 291027 \beta_{11} - 3144842 \beta_{10} - 2726573 \beta_{9} + 1034699 \beta_{8} - 507941 \beta_{7} - 2581267 \beta_{6} + 1434546 \beta_{5} - 498313 \beta_{4} + 553503 \beta_{3} + 29859 \beta_{2} + 1963313 \beta_{1} - 1309799\)\()/5\)
\(\nu^{13}\)\(=\)\((\)\(-2303643 \beta_{15} + 3165794 \beta_{14} - 9996001 \beta_{13} + 1254869 \beta_{12} - 527138 \beta_{11} - 9072163 \beta_{10} - 10035822 \beta_{9} + 4755751 \beta_{8} - 2651744 \beta_{7} - 11712898 \beta_{6} + 3497229 \beta_{5} - 655547 \beta_{4} + 3100882 \beta_{3} + 1093781 \beta_{2} + 8564457 \beta_{1} - 2885261\)\()/5\)
\(\nu^{14}\)\(=\)\((\)\(-6955377 \beta_{15} + 10771406 \beta_{14} - 35942739 \beta_{13} + 6786036 \beta_{12} - 208367 \beta_{11} - 34294982 \beta_{10} - 33244988 \beta_{9} + 15468999 \beta_{8} - 7800891 \beta_{7} - 36336197 \beta_{6} + 11168716 \beta_{5} - 4560408 \beta_{4} + 7312018 \beta_{3} + 3446899 \beta_{2} + 21728638 \beta_{1} - 16212534\)\()/5\)
\(\nu^{15}\)\(=\)\((\)\(-33676153 \beta_{15} + 43741979 \beta_{14} - 119837181 \beta_{13} + 13712654 \beta_{12} + 1164252 \beta_{11} - 101919263 \beta_{10} - 125083137 \beta_{9} + 70977541 \beta_{8} - 29532179 \beta_{7} - 145033883 \beta_{6} + 35437464 \beta_{5} - 3225587 \beta_{4} + 34156632 \beta_{3} + 18966791 \beta_{2} + 90702602 \beta_{1} - 51277636\)\()/5\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-\beta_{10}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.705457 + 0.309017i
2.32349 + 0.309017i
3.42137 0.309017i
−1.80334 0.309017i
−1.16141 + 0.809017i
0.543374 + 0.809017i
−2.79002 0.809017i
2.17199 0.809017i
−1.16141 0.809017i
0.543374 0.809017i
−2.79002 + 0.809017i
2.17199 + 0.809017i
−0.705457 0.309017i
2.32349 0.309017i
3.42137 + 0.309017i
−1.80334 + 0.309017i
−0.951057 + 0.309017i 0 0.809017 0.587785i −1.97959 1.03982i 0 0.329315i −0.587785 + 0.809017i 0 2.20402 + 0.377200i
19.2 −0.951057 + 0.309017i 0 0.809017 0.587785i 1.47959 + 1.67655i 0 3.23143i −0.587785 + 0.809017i 0 −1.92526 1.13727i
19.3 0.951057 0.309017i 0 0.809017 0.587785i −2.23558 + 0.0466062i 0 3.52206i 0.587785 0.809017i 0 −2.11176 + 0.735158i
19.4 0.951057 0.309017i 0 0.809017 0.587785i 1.73558 1.40988i 0 2.61995i 0.587785 0.809017i 0 1.21496 1.87720i
109.1 −0.587785 + 0.809017i 0 −0.309017 0.951057i −1.86682 1.23085i 0 2.70913i 0.951057 + 0.309017i 0 2.09307 0.786811i
109.2 −0.587785 + 0.809017i 0 −0.309017 0.951057i 1.36682 + 1.76969i 0 0.533559i 0.951057 + 0.309017i 0 −2.23511 + 0.0655797i
109.3 0.587785 0.809017i 0 −0.309017 0.951057i −2.03938 + 0.917020i 0 4.80694i −0.951057 0.309017i 0 −0.456833 + 2.18890i
109.4 0.587785 0.809017i 0 −0.309017 0.951057i 1.53938 + 1.62182i 0 4.63137i −0.951057 0.309017i 0 2.21691 0.292102i
289.1 −0.587785 0.809017i 0 −0.309017 + 0.951057i −1.86682 + 1.23085i 0 2.70913i 0.951057 0.309017i 0 2.09307 + 0.786811i
289.2 −0.587785 0.809017i 0 −0.309017 + 0.951057i 1.36682 1.76969i 0 0.533559i 0.951057 0.309017i 0 −2.23511 0.0655797i
289.3 0.587785 + 0.809017i 0 −0.309017 + 0.951057i −2.03938 0.917020i 0 4.80694i −0.951057 + 0.309017i 0 −0.456833 2.18890i
289.4 0.587785 + 0.809017i 0 −0.309017 + 0.951057i 1.53938 1.62182i 0 4.63137i −0.951057 + 0.309017i 0 2.21691 + 0.292102i
379.1 −0.951057 0.309017i 0 0.809017 + 0.587785i −1.97959 + 1.03982i 0 0.329315i −0.587785 0.809017i 0 2.20402 0.377200i
379.2 −0.951057 0.309017i 0 0.809017 + 0.587785i 1.47959 1.67655i 0 3.23143i −0.587785 0.809017i 0 −1.92526 + 1.13727i
379.3 0.951057 + 0.309017i 0 0.809017 + 0.587785i −2.23558 0.0466062i 0 3.52206i 0.587785 + 0.809017i 0 −2.11176 0.735158i
379.4 0.951057 + 0.309017i 0 0.809017 + 0.587785i 1.73558 + 1.40988i 0 2.61995i 0.587785 + 0.809017i 0 1.21496 + 1.87720i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 379.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.l.c 16
3.b odd 2 1 150.2.h.b 16
15.d odd 2 1 750.2.h.d 16
15.e even 4 1 750.2.g.f 16
15.e even 4 1 750.2.g.g 16
25.e even 10 1 inner 450.2.l.c 16
75.h odd 10 1 150.2.h.b 16
75.h odd 10 1 3750.2.c.k 16
75.j odd 10 1 750.2.h.d 16
75.j odd 10 1 3750.2.c.k 16
75.l even 20 1 750.2.g.f 16
75.l even 20 1 750.2.g.g 16
75.l even 20 1 3750.2.a.u 8
75.l even 20 1 3750.2.a.v 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.h.b 16 3.b odd 2 1
150.2.h.b 16 75.h odd 10 1
450.2.l.c 16 1.a even 1 1 trivial
450.2.l.c 16 25.e even 10 1 inner
750.2.g.f 16 15.e even 4 1
750.2.g.f 16 75.l even 20 1
750.2.g.g 16 15.e even 4 1
750.2.g.g 16 75.l even 20 1
750.2.h.d 16 15.d odd 2 1
750.2.h.d 16 75.j odd 10 1
3750.2.a.u 8 75.l even 20 1
3750.2.a.v 8 75.l even 20 1
3750.2.c.k 16 75.h odd 10 1
3750.2.c.k 16 75.j odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{16} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(450, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$3$ \( \)
$5$ \( 1 + 4 T - 4 T^{2} - 6 T^{3} + 161 T^{4} + 260 T^{5} - 580 T^{6} + 520 T^{7} + 7205 T^{8} + 2600 T^{9} - 14500 T^{10} + 32500 T^{11} + 100625 T^{12} - 18750 T^{13} - 62500 T^{14} + 312500 T^{15} + 390625 T^{16} \)
$7$ \( 1 - 30 T^{2} + 527 T^{4} - 6940 T^{6} + 74703 T^{8} - 716280 T^{10} + 6275029 T^{12} - 50516750 T^{14} + 372187880 T^{16} - 2475320750 T^{18} + 15066344629 T^{20} - 84269625720 T^{22} + 430647929103 T^{24} - 1960378228060 T^{26} + 7294358354927 T^{28} - 20346692185470 T^{30} + 33232930569601 T^{32} \)
$11$ \( 1 + 2 T + 11 T^{2} + 50 T^{3} + 270 T^{4} - 764 T^{5} - 168 T^{6} - 4404 T^{7} - 16480 T^{8} - 222470 T^{9} + 461835 T^{10} + 66610 T^{11} + 2053685 T^{12} - 10233320 T^{13} + 85541270 T^{14} - 166971000 T^{15} - 141281840 T^{16} - 1836681000 T^{17} + 10350493670 T^{18} - 13620548920 T^{19} + 30068002085 T^{20} + 10727607110 T^{21} + 818168874435 T^{22} - 4335310932370 T^{23} - 3532634358880 T^{24} - 10384401631164 T^{25} - 4357487332968 T^{26} - 217978116346804 T^{27} + 847375661714670 T^{28} + 1726135607196550 T^{29} + 4177248169415651 T^{30} + 8354496338831302 T^{31} + 45949729863572161 T^{32} \)
$13$ \( 1 - 20 T + 240 T^{2} - 2100 T^{3} + 14722 T^{4} - 86440 T^{5} + 437000 T^{6} - 1922660 T^{7} + 7352283 T^{8} - 23888120 T^{9} + 61481440 T^{10} - 91205000 T^{11} - 205909876 T^{12} + 2578228580 T^{13} - 14615222400 T^{14} + 65003714260 T^{15} - 249510986035 T^{16} + 845048285380 T^{17} - 2469972585600 T^{18} + 5664368190260 T^{19} - 5880991968436 T^{20} - 33863778065000 T^{21} + 296759167924960 T^{22} - 1498944103918040 T^{23} + 5997483112586043 T^{24} - 20388846764492180 T^{25} + 60244160938013000 T^{26} - 154914344460558280 T^{27} + 342994409173165282 T^{28} - 636037723843731300 T^{29} + 944970332567829360 T^{30} - 1023717860281815140 T^{31} + 665416609183179841 T^{32} \)
$17$ \( 1 - 30 T + 500 T^{2} - 5990 T^{3} + 56782 T^{4} - 447460 T^{5} + 3015460 T^{6} - 17669550 T^{7} + 90820363 T^{8} - 409992380 T^{9} + 1611336300 T^{10} - 5357847740 T^{11} + 13820507184 T^{12} - 18014105810 T^{13} - 71676889400 T^{14} + 711604275870 T^{15} - 3544790874535 T^{16} + 12097272689790 T^{17} - 20714621036600 T^{18} - 88503301844530 T^{19} + 1154302580514864 T^{20} - 7607377618573180 T^{21} + 38893741123454700 T^{22} - 168235729149311740 T^{23} + 633540822991571083 T^{24} - 2095394413157566350 T^{25} + 6079148967047941540 T^{26} - 15335302721813462180 T^{27} + 33082455874380289102 T^{28} - 59328422417106562630 T^{29} + 84188913279700464500 T^{30} - 85872691545294473790 T^{31} + 48661191875666868481 T^{32} \)
$19$ \( 1 + 24 T^{2} - 30 T^{3} + 330 T^{4} + 740 T^{5} + 7630 T^{6} + 18810 T^{7} + 160255 T^{8} - 244020 T^{9} + 7105152 T^{10} - 5384250 T^{11} + 141174008 T^{12} - 174364190 T^{13} + 1779959840 T^{14} + 4165821270 T^{15} + 38170716365 T^{16} + 79150604130 T^{17} + 642565502240 T^{18} - 1195963979210 T^{19} + 18397937896568 T^{20} - 13331936040750 T^{21} + 334268135478912 T^{22} - 218122581750780 T^{23} + 2721700895135455 T^{24} + 6069755595222990 T^{25} + 46780035547021630 T^{26} + 86202791584682060 T^{27} + 730393923291833130 T^{28} - 1261589503867711770 T^{29} + 19176160458789218904 T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \)
$23$ \( 1 - 10 T + 30 T^{2} + 250 T^{3} - 2718 T^{4} + 8680 T^{5} + 13790 T^{6} - 227450 T^{7} + 942143 T^{8} - 3042820 T^{9} + 4039430 T^{10} + 83513580 T^{11} - 909320196 T^{12} + 3918445730 T^{13} - 578679600 T^{14} - 92103435490 T^{15} + 633292773765 T^{16} - 2118379016270 T^{17} - 306121508400 T^{18} + 47675729196910 T^{19} - 254465072968836 T^{20} + 537522046037940 T^{21} + 597980611103270 T^{22} - 10360270966640540 T^{23} + 73780146605597183 T^{24} - 409672172849759350 T^{25} + 571271589636219710 T^{26} + 8270388698692886360 T^{27} - 59563949206231232478 T^{28} + \)\(12\!\cdots\!50\)\( T^{29} + \)\(34\!\cdots\!70\)\( T^{30} - \)\(26\!\cdots\!70\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \)
$29$ \( 1 - 10 T - 36 T^{2} + 270 T^{3} + 3910 T^{4} - 4460 T^{5} - 185440 T^{6} - 333090 T^{7} + 4522435 T^{8} + 38473780 T^{9} - 69131068 T^{10} - 1395128220 T^{11} - 3409890032 T^{12} + 36344819210 T^{13} + 245632726680 T^{14} - 543098432830 T^{15} - 7660398480495 T^{16} - 15749854552070 T^{17} + 206577123137880 T^{18} + 886413795712690 T^{19} - 2411750431722992 T^{20} - 28615682794524780 T^{21} - 41120771452036828 T^{22} + 663667946139678020 T^{23} + 2262331886599280035 T^{24} - 4832185253102205210 T^{25} - 78015949343189273440 T^{26} - 54414273555047997340 T^{27} + \)\(13\!\cdots\!10\)\( T^{28} + \)\(27\!\cdots\!30\)\( T^{29} - \)\(10\!\cdots\!16\)\( T^{30} - \)\(86\!\cdots\!90\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \)
$31$ \( 1 + 18 T + 121 T^{2} + 610 T^{3} + 6060 T^{4} + 54724 T^{5} + 309962 T^{6} + 1456784 T^{7} + 9332270 T^{8} + 64865970 T^{9} + 327433735 T^{10} + 1367357590 T^{11} + 7702756335 T^{12} + 47160385620 T^{13} + 227624335970 T^{14} + 1111476930100 T^{15} + 6249005152660 T^{16} + 34455784833100 T^{17} + 218746986867170 T^{18} + 1404955048005420 T^{19} + 7113657233255535 T^{20} + 39146286915106090 T^{21} + 290598645096078535 T^{22} + 1784632401545702670 T^{23} + 7959409441979521070 T^{24} + 38516818529710942064 T^{25} + \)\(25\!\cdots\!62\)\( T^{26} + \)\(13\!\cdots\!44\)\( T^{27} + \)\(47\!\cdots\!60\)\( T^{28} + \)\(14\!\cdots\!10\)\( T^{29} + \)\(91\!\cdots\!41\)\( T^{30} + \)\(42\!\cdots\!18\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \)
$37$ \( 1 - 20 T + 265 T^{2} - 2300 T^{3} + 17532 T^{4} - 115640 T^{5} + 825500 T^{6} - 5506160 T^{7} + 38914148 T^{8} - 253492260 T^{9} + 1794329215 T^{10} - 11470149820 T^{11} + 74232183599 T^{12} - 424648096560 T^{13} + 2693507120800 T^{14} - 16225080394680 T^{15} + 105486161388440 T^{16} - 600327974603160 T^{17} + 3687411248375200 T^{18} - 21509700035053680 T^{19} + 139123063446085439 T^{20} - 795385575901637740 T^{21} + 4603757853365738935 T^{22} - 24064496080486490580 T^{23} + \)\(13\!\cdots\!08\)\( T^{24} - \)\(71\!\cdots\!20\)\( T^{25} + \)\(39\!\cdots\!00\)\( T^{26} - \)\(20\!\cdots\!20\)\( T^{27} + \)\(11\!\cdots\!92\)\( T^{28} - \)\(56\!\cdots\!00\)\( T^{29} + \)\(23\!\cdots\!85\)\( T^{30} - \)\(66\!\cdots\!60\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \)
$41$ \( 1 + 22 T + 316 T^{2} + 3710 T^{3} + 39530 T^{4} + 407436 T^{5} + 3918652 T^{6} + 35382126 T^{7} + 301988495 T^{8} + 2475859380 T^{9} + 19707200260 T^{10} + 150750040860 T^{11} + 1111070917960 T^{12} + 7891095298530 T^{13} + 54497096107120 T^{14} + 366799083339050 T^{15} + 2387443151993985 T^{16} + 15038762416901050 T^{17} + 91609618556068720 T^{18} + 543862179069986130 T^{19} + 3139620868205567560 T^{20} + 17465327034634372860 T^{21} + 93611255533262302660 T^{22} + \)\(48\!\cdots\!80\)\( T^{23} + \)\(24\!\cdots\!95\)\( T^{24} + \)\(11\!\cdots\!86\)\( T^{25} + \)\(52\!\cdots\!52\)\( T^{26} + \)\(22\!\cdots\!76\)\( T^{27} + \)\(89\!\cdots\!30\)\( T^{28} + \)\(34\!\cdots\!10\)\( T^{29} + \)\(11\!\cdots\!76\)\( T^{30} + \)\(34\!\cdots\!22\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \)
$43$ \( 1 - 360 T^{2} + 63972 T^{4} - 7536360 T^{6} + 666842308 T^{8} - 47508203160 T^{10} + 2846509586204 T^{12} - 147613243096600 T^{14} + 6742598283189430 T^{16} - 272936886485613400 T^{18} + 9731649819823821404 T^{20} - \)\(30\!\cdots\!40\)\( T^{22} + \)\(77\!\cdots\!08\)\( T^{24} - \)\(16\!\cdots\!40\)\( T^{26} + \)\(25\!\cdots\!72\)\( T^{28} - \)\(26\!\cdots\!40\)\( T^{30} + \)\(13\!\cdots\!01\)\( T^{32} \)
$47$ \( 1 - 50 T + 1290 T^{2} - 23780 T^{3} + 355482 T^{4} - 4545300 T^{5} + 51239280 T^{6} - 520209820 T^{7} + 4818856003 T^{8} - 41087515780 T^{9} + 324882626890 T^{10} - 2399419588970 T^{11} + 16710816340744 T^{12} - 111327617454640 T^{13} + 724257830956800 T^{14} - 4728319081214120 T^{15} + 31780196573425365 T^{16} - 222230996817063640 T^{17} + 1599885548583571200 T^{18} - 11558367226993088720 T^{19} + 81543452992418022664 T^{20} - \)\(55\!\cdots\!90\)\( T^{21} + \)\(35\!\cdots\!10\)\( T^{22} - \)\(20\!\cdots\!40\)\( T^{23} + \)\(11\!\cdots\!83\)\( T^{24} - \)\(58\!\cdots\!40\)\( T^{25} + \)\(26\!\cdots\!20\)\( T^{26} - \)\(11\!\cdots\!00\)\( T^{27} + \)\(41\!\cdots\!62\)\( T^{28} - \)\(12\!\cdots\!60\)\( T^{29} + \)\(33\!\cdots\!10\)\( T^{30} - \)\(60\!\cdots\!50\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( 1 + 30 T + 655 T^{2} + 10630 T^{3} + 152332 T^{4} + 1919860 T^{5} + 22723040 T^{6} + 248684220 T^{7} + 2591225028 T^{8} + 25379094630 T^{9} + 238630725505 T^{10} + 2134501287330 T^{11} + 18442776242119 T^{12} + 152410720886420 T^{13} + 1218238235823200 T^{14} + 9336287074370240 T^{15} + 69349166231444240 T^{16} + 494823214941622720 T^{17} + 3422031204427368800 T^{18} + 22690450893407550340 T^{19} + \)\(14\!\cdots\!39\)\( T^{20} + \)\(89\!\cdots\!90\)\( T^{21} + \)\(52\!\cdots\!45\)\( T^{22} + \)\(29\!\cdots\!10\)\( T^{23} + \)\(16\!\cdots\!08\)\( T^{24} + \)\(82\!\cdots\!60\)\( T^{25} + \)\(39\!\cdots\!60\)\( T^{26} + \)\(17\!\cdots\!20\)\( T^{27} + \)\(74\!\cdots\!12\)\( T^{28} + \)\(27\!\cdots\!90\)\( T^{29} + \)\(90\!\cdots\!95\)\( T^{30} + \)\(21\!\cdots\!10\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \)
$59$ \( 1 + 20 T - 111 T^{2} - 3640 T^{3} + 11660 T^{4} + 317820 T^{5} - 1817980 T^{6} - 11679820 T^{7} + 217773860 T^{8} - 627241760 T^{9} - 13630850673 T^{10} + 133224376740 T^{11} + 186113511463 T^{12} - 9371240514920 T^{13} + 48131664631280 T^{14} + 252534601553160 T^{15} - 4526062111550040 T^{16} + 14899541491636440 T^{17} + 167546324581485680 T^{18} - 1924656005713754680 T^{19} + 2255204605374809143 T^{20} + 95245344150556405260 T^{21} - \)\(57\!\cdots\!93\)\( T^{22} - \)\(15\!\cdots\!40\)\( T^{23} + \)\(31\!\cdots\!60\)\( T^{24} - \)\(10\!\cdots\!80\)\( T^{25} - \)\(92\!\cdots\!80\)\( T^{26} + \)\(95\!\cdots\!80\)\( T^{27} + \)\(20\!\cdots\!60\)\( T^{28} - \)\(38\!\cdots\!60\)\( T^{29} - \)\(68\!\cdots\!71\)\( T^{30} + \)\(73\!\cdots\!80\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( 1 - 12 T - 104 T^{2} + 1460 T^{3} + 9310 T^{4} - 110696 T^{5} - 1128308 T^{6} + 9605804 T^{7} + 95707895 T^{8} - 545482080 T^{9} - 8568890440 T^{10} + 31012865040 T^{11} + 647686771260 T^{12} - 1172976299580 T^{13} - 45238788397880 T^{14} + 2951178504100 T^{15} + 3155660575885285 T^{16} + 180021888750100 T^{17} - 168333531628511480 T^{18} - 266243333454967980 T^{19} + 8967768052669329660 T^{20} + 26193351096196217040 T^{21} - \)\(44\!\cdots\!40\)\( T^{22} - \)\(17\!\cdots\!80\)\( T^{23} + \)\(18\!\cdots\!95\)\( T^{24} + \)\(11\!\cdots\!64\)\( T^{25} - \)\(80\!\cdots\!08\)\( T^{26} - \)\(48\!\cdots\!56\)\( T^{27} + \)\(24\!\cdots\!10\)\( T^{28} + \)\(23\!\cdots\!60\)\( T^{29} - \)\(10\!\cdots\!64\)\( T^{30} - \)\(72\!\cdots\!12\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \)
$67$ \( 1 + 50 T + 1390 T^{2} + 30100 T^{3} + 553682 T^{4} + 8918100 T^{5} + 129754300 T^{6} + 1739003620 T^{7} + 21685053163 T^{8} + 253853201940 T^{9} + 2808527300190 T^{10} + 29485013121570 T^{11} + 294663718701784 T^{12} + 2810511152550240 T^{13} + 25630455288876000 T^{14} + 223725991474439120 T^{15} + 1870714073010261365 T^{16} + 14989641428787421040 T^{17} + \)\(11\!\cdots\!00\)\( T^{18} + \)\(84\!\cdots\!20\)\( T^{19} + \)\(59\!\cdots\!64\)\( T^{20} + \)\(39\!\cdots\!90\)\( T^{21} + \)\(25\!\cdots\!10\)\( T^{22} + \)\(15\!\cdots\!20\)\( T^{23} + \)\(88\!\cdots\!83\)\( T^{24} + \)\(47\!\cdots\!40\)\( T^{25} + \)\(23\!\cdots\!00\)\( T^{26} + \)\(10\!\cdots\!00\)\( T^{27} + \)\(45\!\cdots\!02\)\( T^{28} + \)\(16\!\cdots\!00\)\( T^{29} + \)\(51\!\cdots\!10\)\( T^{30} + \)\(12\!\cdots\!50\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \)
$71$ \( 1 - 28 T + 256 T^{2} - 1150 T^{3} + 20770 T^{4} - 328604 T^{5} + 2351902 T^{6} - 21115094 T^{7} + 321977895 T^{8} - 2983992020 T^{9} + 18685321160 T^{10} - 180970630790 T^{11} + 2127756781760 T^{12} - 17298444226470 T^{13} + 123520240637920 T^{14} - 1239239342309850 T^{15} + 12274938369025885 T^{16} - 87985993303999350 T^{17} + 622665533055754720 T^{18} - 6191303471540104170 T^{19} + 54069876583671738560 T^{20} - \)\(32\!\cdots\!90\)\( T^{21} + \)\(23\!\cdots\!60\)\( T^{22} - \)\(27\!\cdots\!20\)\( T^{23} + \)\(20\!\cdots\!95\)\( T^{24} - \)\(96\!\cdots\!14\)\( T^{25} + \)\(76\!\cdots\!02\)\( T^{26} - \)\(75\!\cdots\!84\)\( T^{27} + \)\(34\!\cdots\!70\)\( T^{28} - \)\(13\!\cdots\!50\)\( T^{29} + \)\(21\!\cdots\!36\)\( T^{30} - \)\(16\!\cdots\!28\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \)
$73$ \( 1 - 20 T + 380 T^{2} - 7420 T^{3} + 107282 T^{4} - 1509240 T^{5} + 20338260 T^{6} - 240254580 T^{7} + 2794851843 T^{8} - 31155740720 T^{9} + 320911861980 T^{10} - 3274927660920 T^{11} + 32181004976604 T^{12} - 299417039187980 T^{13} + 2775262759264800 T^{14} - 24764197420905260 T^{15} + 211758644546106765 T^{16} - 1807786411726083980 T^{17} + 14789375244122119200 T^{18} - \)\(11\!\cdots\!60\)\( T^{19} + \)\(91\!\cdots\!64\)\( T^{20} - \)\(67\!\cdots\!60\)\( T^{21} + \)\(48\!\cdots\!20\)\( T^{22} - \)\(34\!\cdots\!40\)\( T^{23} + \)\(22\!\cdots\!83\)\( T^{24} - \)\(14\!\cdots\!40\)\( T^{25} + \)\(87\!\cdots\!40\)\( T^{26} - \)\(47\!\cdots\!80\)\( T^{27} + \)\(24\!\cdots\!22\)\( T^{28} - \)\(12\!\cdots\!60\)\( T^{29} + \)\(46\!\cdots\!20\)\( T^{30} - \)\(17\!\cdots\!40\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \)
$79$ \( 1 + 20 T + 224 T^{2} + 1780 T^{3} + 6070 T^{4} + 5760 T^{5} + 875320 T^{6} + 20333540 T^{7} + 305719495 T^{8} + 3198483920 T^{9} + 22059896192 T^{10} + 116979470440 T^{11} + 660067500148 T^{12} + 7893347694940 T^{13} + 139701887310360 T^{14} + 1828379420794980 T^{15} + 18018691485968485 T^{16} + 144441974242803420 T^{17} + 871879478703956760 T^{18} + 3891728254165522660 T^{19} + 25709682596232111988 T^{20} + \)\(35\!\cdots\!60\)\( T^{21} + \)\(53\!\cdots\!32\)\( T^{22} + \)\(61\!\cdots\!80\)\( T^{23} + \)\(46\!\cdots\!95\)\( T^{24} + \)\(24\!\cdots\!60\)\( T^{25} + \)\(82\!\cdots\!20\)\( T^{26} + \)\(43\!\cdots\!40\)\( T^{27} + \)\(35\!\cdots\!70\)\( T^{28} + \)\(83\!\cdots\!20\)\( T^{29} + \)\(82\!\cdots\!44\)\( T^{30} + \)\(58\!\cdots\!80\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \)
$83$ \( 1 - 30 T + 535 T^{2} - 5780 T^{3} + 43882 T^{4} - 205010 T^{5} + 590510 T^{6} - 2468290 T^{7} + 123659138 T^{8} - 2415976440 T^{9} + 27231081235 T^{10} - 158815212750 T^{11} + 405339323209 T^{12} - 3244201552980 T^{13} + 126271405899100 T^{14} - 2149314211865960 T^{15} + 23236430083058540 T^{16} - 178393079584874680 T^{17} + 869883715238899900 T^{18} - 1854992273373775260 T^{19} + 19236723714775472089 T^{20} - \)\(62\!\cdots\!50\)\( T^{21} + \)\(89\!\cdots\!15\)\( T^{22} - \)\(65\!\cdots\!80\)\( T^{23} + \)\(27\!\cdots\!58\)\( T^{24} - \)\(46\!\cdots\!70\)\( T^{25} + \)\(91\!\cdots\!90\)\( T^{26} - \)\(26\!\cdots\!70\)\( T^{27} + \)\(46\!\cdots\!02\)\( T^{28} - \)\(51\!\cdots\!40\)\( T^{29} + \)\(39\!\cdots\!15\)\( T^{30} - \)\(18\!\cdots\!10\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( 1 + 70 T + 2119 T^{2} + 35330 T^{3} + 343460 T^{4} + 2032360 T^{5} + 10870620 T^{6} + 64684440 T^{7} - 584089940 T^{8} - 17002587830 T^{9} - 99745085123 T^{10} + 366225143690 T^{11} + 5693869232523 T^{12} + 24776539056740 T^{13} + 308063609313680 T^{14} + 3148141734399680 T^{15} + 20769024208268760 T^{16} + 280184614361571520 T^{17} + 2440171849373659280 T^{18} + 17466691962290941060 T^{19} + \)\(35\!\cdots\!43\)\( T^{20} + \)\(20\!\cdots\!10\)\( T^{21} - \)\(49\!\cdots\!03\)\( T^{22} - \)\(75\!\cdots\!70\)\( T^{23} - \)\(22\!\cdots\!40\)\( T^{24} + \)\(22\!\cdots\!60\)\( T^{25} + \)\(33\!\cdots\!20\)\( T^{26} + \)\(56\!\cdots\!40\)\( T^{27} + \)\(84\!\cdots\!60\)\( T^{28} + \)\(77\!\cdots\!70\)\( T^{29} + \)\(41\!\cdots\!79\)\( T^{30} + \)\(12\!\cdots\!30\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \)
$97$ \( 1 + 10 T + 205 T^{2} + 4320 T^{3} + 44952 T^{4} + 667870 T^{5} + 8897460 T^{6} + 68756910 T^{7} + 834554408 T^{8} + 7477618800 T^{9} + 22708693755 T^{10} + 118605463990 T^{11} - 2491804851081 T^{12} - 84263627884560 T^{13} - 895266623177600 T^{14} - 11056342779615180 T^{15} - 138673183182585960 T^{16} - 1072465249622672460 T^{17} - 8423563657478038400 T^{18} - 76905138052285028880 T^{19} - \)\(22\!\cdots\!61\)\( T^{20} + \)\(10\!\cdots\!30\)\( T^{21} + \)\(18\!\cdots\!95\)\( T^{22} + \)\(60\!\cdots\!00\)\( T^{23} + \)\(65\!\cdots\!88\)\( T^{24} + \)\(52\!\cdots\!70\)\( T^{25} + \)\(65\!\cdots\!40\)\( T^{26} + \)\(47\!\cdots\!10\)\( T^{27} + \)\(31\!\cdots\!32\)\( T^{28} + \)\(29\!\cdots\!40\)\( T^{29} + \)\(13\!\cdots\!45\)\( T^{30} + \)\(63\!\cdots\!30\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \)
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