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: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 4 x^{15} - 24 x^{14} + 94 x^{13} + 262 x^{12} - 936 x^{11} - 1584 x^{10} + 4642 x^{9} + 6259 x^{8} - 11958 x^{7} - 15752 x^{6} + 14670 x^{5} + 18271 x^{4} - 10440 x^{3} + 1135 x^{2} + 21080 x + 11105\)
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

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$3$ \( T^{16} \)
$5$ \( 390625 + 312500 T - 62500 T^{2} - 18750 T^{3} + 100625 T^{4} + 32500 T^{5} - 14500 T^{6} + 2600 T^{7} + 7205 T^{8} + 520 T^{9} - 580 T^{10} + 260 T^{11} + 161 T^{12} - 6 T^{13} - 4 T^{14} + 4 T^{15} + T^{16} \)
$7$ \( 99856 + 1326272 T^{2} + 3943448 T^{4} + 1927704 T^{6} + 407105 T^{8} + 44874 T^{10} + 2683 T^{12} + 82 T^{14} + T^{16} \)
$11$ \( 524176 + 660288 T + 5337520 T^{2} + 18282200 T^{3} + 30332320 T^{4} + 26865184 T^{5} + 13990042 T^{6} + 4331970 T^{7} + 981825 T^{8} + 189020 T^{9} + 38552 T^{10} + 4076 T^{11} + 1480 T^{12} + 50 T^{13} + 55 T^{14} + 2 T^{15} + T^{16} \)
$13$ \( 4096 - 10240 T + 50688 T^{2} + 785920 T^{3} + 1912128 T^{4} + 889600 T^{5} - 1261864 T^{6} - 271020 T^{7} + 751065 T^{8} - 411280 T^{9} + 117681 T^{10} - 21960 T^{11} + 4478 T^{12} - 1060 T^{13} + 188 T^{14} - 20 T^{15} + T^{16} \)
$17$ \( 3748096 - 10222080 T + 7960832 T^{2} + 5089920 T^{3} - 12660192 T^{4} + 5209920 T^{5} + 5488964 T^{6} - 6822760 T^{7} + 3013765 T^{8} - 790760 T^{9} + 246619 T^{10} - 82980 T^{11} + 21558 T^{12} - 3780 T^{13} + 432 T^{14} - 30 T^{15} + T^{16} \)
$19$ \( 2560000 + 15360000 T + 87680000 T^{2} + 192640000 T^{3} + 164608000 T^{4} - 128224000 T^{5} + 36992000 T^{6} + 2408000 T^{7} + 2344400 T^{8} - 98800 T^{9} + 84200 T^{10} - 8000 T^{11} + 4320 T^{12} - 220 T^{13} + 100 T^{14} + T^{16} \)
$23$ \( 20533743616 - 7978721280 T - 793377792 T^{2} - 3091287040 T^{3} + 2328182528 T^{4} - 223645440 T^{5} - 20284864 T^{6} + 2033280 T^{7} + 5589040 T^{8} - 218480 T^{9} - 212024 T^{10} - 8800 T^{11} + 5148 T^{12} + 480 T^{13} - 62 T^{14} - 10 T^{15} + T^{16} \)
$29$ \( 40960000 - 286720000 T + 826880000 T^{2} - 629760000 T^{3} + 407328000 T^{4} - 204992000 T^{5} + 89374000 T^{6} - 29294000 T^{7} + 7936025 T^{8} - 1497150 T^{9} + 227375 T^{10} - 20700 T^{11} + 2170 T^{12} - 310 T^{13} + 80 T^{14} - 10 T^{15} + T^{16} \)
$31$ \( 205176976 + 167705392 T + 215927560 T^{2} + 223722720 T^{3} + 174024880 T^{4} + 46838996 T^{5} + 16443812 T^{6} + 4114340 T^{7} + 4864705 T^{8} + 2692320 T^{9} + 920972 T^{10} + 183994 T^{11} + 28690 T^{12} + 2780 T^{13} + 245 T^{14} + 18 T^{15} + T^{16} \)
$37$ \( 4096 + 71680 T + 778752 T^{2} + 2629120 T^{3} + 4014848 T^{4} + 2963360 T^{5} + 882864 T^{6} - 38560 T^{7} - 37935 T^{8} + 29040 T^{9} + 15644 T^{10} - 8340 T^{11} + 438 T^{12} - 80 T^{13} + 117 T^{14} - 20 T^{15} + T^{16} \)
$41$ \( 78050949376 + 143389173248 T + 115932473600 T^{2} + 44589326080 T^{3} + 7925375520 T^{4} + 1047324864 T^{5} + 1014402852 T^{6} + 420300600 T^{7} + 147587005 T^{8} + 30973970 T^{9} + 5360007 T^{10} + 673116 T^{11} + 73150 T^{12} + 6170 T^{13} + 480 T^{14} + 22 T^{15} + T^{16} \)
$43$ \( 15083769856 + 14436538368 T^{2} + 4906686208 T^{4} + 770631296 T^{6} + 59235280 T^{8} + 2205376 T^{10} + 39548 T^{12} + 328 T^{14} + T^{16} \)
$47$ \( 172199901270016 - 40068754186240 T - 10064501357568 T^{2} + 6907001753600 T^{3} - 738130995712 T^{4} - 318740764160 T^{5} + 128575435264 T^{6} - 21291263360 T^{7} + 1634548240 T^{8} + 26371040 T^{9} - 18529776 T^{10} + 1580680 T^{11} + 1008 T^{12} - 11560 T^{13} + 1102 T^{14} - 50 T^{15} + T^{16} \)
$53$ \( 111534721 + 332565890 T + 605305833 T^{2} + 379694600 T^{3} + 85146708 T^{4} - 67381290 T^{5} - 37207539 T^{6} - 1943930 T^{7} + 4916665 T^{8} + 1201780 T^{9} - 207569 T^{10} - 96790 T^{11} - 2852 T^{12} + 2680 T^{13} + 443 T^{14} + 30 T^{15} + T^{16} \)
$59$ \( 1600000000 - 3200000000 T + 21200000000 T^{2} + 2500000000 T^{3} + 15884000000 T^{4} + 13070000000 T^{5} + 6565625000 T^{6} + 2161637500 T^{7} + 499400625 T^{8} + 80447500 T^{9} + 8962500 T^{10} + 614000 T^{11} + 18150 T^{12} - 100 T^{13} + 125 T^{14} + 20 T^{15} + T^{16} \)
$61$ \( 10137856 - 131537408 T + 391080000 T^{2} + 5456405760 T^{3} + 17593011360 T^{4} - 2661983904 T^{5} + 3192436112 T^{6} - 357951000 T^{7} + 215705265 T^{8} - 18107960 T^{9} + 5810137 T^{10} - 181456 T^{11} + 11750 T^{12} + 240 T^{13} + 140 T^{14} - 12 T^{15} + T^{16} \)
$67$ \( 201983672713216 + 120791446323200 T + 32731352727552 T^{2} + 493948436480 T^{3} - 1485531308032 T^{4} - 246479728640 T^{5} + 10106302464 T^{6} + 7429202560 T^{7} + 754081040 T^{8} - 32153440 T^{9} - 13694576 T^{10} - 1052840 T^{11} + 32288 T^{12} + 12680 T^{13} + 1122 T^{14} + 50 T^{15} + T^{16} \)
$71$ \( 3398330023936 - 2682656161792 T + 1330632499200 T^{2} - 425924648960 T^{3} + 99450024960 T^{4} - 17766322176 T^{5} + 3207030912 T^{6} - 725636800 T^{7} + 185863440 T^{8} - 39399440 T^{9} + 6474872 T^{10} - 821344 T^{11} + 86800 T^{12} - 7540 T^{13} + 540 T^{14} - 28 T^{15} + T^{16} \)
$73$ \( 4778526048256 + 509246832640 T - 294239997952 T^{2} + 483201126400 T^{3} + 181700284928 T^{4} - 18177437440 T^{5} - 4569035264 T^{6} + 1613600120 T^{7} + 10209465 T^{8} - 43529070 T^{9} + 5643141 T^{10} - 49240 T^{11} - 48062 T^{12} + 3530 T^{13} + 88 T^{14} - 20 T^{15} + T^{16} \)
$79$ \( 110872476160000 + 83874581760000 T + 191851535040000 T^{2} - 14144237440000 T^{3} - 330659872000 T^{4} + 170478716000 T^{5} + 75421422000 T^{6} - 23885000 T^{7} + 533087025 T^{8} - 104550 T^{9} + 7574125 T^{10} + 547700 T^{11} + 85070 T^{12} + 5730 T^{13} + 540 T^{14} + 20 T^{15} + T^{16} \)
$83$ \( 25573127056 + 339562436080 T + 1110319587208 T^{2} - 1096003177360 T^{3} + 458530144868 T^{4} - 91903706160 T^{5} + 9475604786 T^{6} - 3232904770 T^{7} + 1075044065 T^{8} - 51058980 T^{9} + 4855216 T^{10} - 1899040 T^{11} + 160248 T^{12} - 3290 T^{13} + 203 T^{14} - 30 T^{15} + T^{16} \)
$89$ \( 36100000000 + 127300000000 T + 260925000000 T^{2} + 324645000000 T^{3} + 269750250000 T^{4} + 115122750000 T^{5} + 42047312500 T^{6} + 12861537500 T^{7} + 3211500625 T^{8} + 621983750 T^{9} + 93500000 T^{10} + 10577250 T^{11} + 896150 T^{12} + 55800 T^{13} + 2475 T^{14} + 70 T^{15} + T^{16} \)
$97$ \( 91700905971481 - 25807000202050 T - 7674896673923 T^{2} - 411721888950 T^{3} + 1028722650748 T^{4} + 161774443900 T^{5} - 9940762061 T^{6} - 7943899040 T^{7} - 646205785 T^{8} + 102052160 T^{9} + 23851659 T^{10} + 1567060 T^{11} - 24112 T^{12} - 8290 T^{13} - 183 T^{14} + 10 T^{15} + T^{16} \)
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