# Properties

 Label 750.2.g.f Level 750 Weight 2 Character orbit 750.g Analytic conductor 5.989 Analytic rank 0 Dimension 16 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$750 = 2 \cdot 3 \cdot 5^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 750.g (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.98878015160$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$5^{6}$$ Twist minimal: no (minimal twist has level 150) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{2} + \beta_{6} q^{3} + \beta_{6} q^{4} -\beta_{3} q^{6} + ( 1 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} ) q^{7} -\beta_{3} q^{8} + ( -1 + \beta_{3} + \beta_{5} - \beta_{6} ) q^{9} +O(q^{10})$$ $$q -\beta_{5} q^{2} + \beta_{6} q^{3} + \beta_{6} q^{4} -\beta_{3} q^{6} + ( 1 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} ) q^{7} -\beta_{3} q^{8} + ( -1 + \beta_{3} + \beta_{5} - \beta_{6} ) q^{9} + ( 1 - \beta_{5} + \beta_{8} + \beta_{11} + \beta_{15} ) q^{11} + ( -1 + \beta_{3} + \beta_{5} - \beta_{6} ) q^{12} + ( \beta_{5} - \beta_{6} + \beta_{13} ) q^{13} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{14} + ( -1 + \beta_{3} + \beta_{5} - \beta_{6} ) q^{16} + ( 1 + \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} ) q^{17} + q^{18} + ( 1 + \beta_{1} - 2 \beta_{3} - \beta_{6} + \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{15} ) q^{19} -\beta_{2} q^{21} + ( -\beta_{3} - \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{22} + ( \beta_{1} - \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{11} - \beta_{15} ) q^{23} + q^{24} + ( \beta_{3} - \beta_{6} + \beta_{11} ) q^{26} -\beta_{5} q^{27} -\beta_{2} q^{28} + ( 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{8} + \beta_{10} + \beta_{12} ) q^{29} + ( -1 + \beta_{1} - 2 \beta_{3} + \beta_{5} + \beta_{10} + \beta_{12} + \beta_{14} ) q^{31} + q^{32} + ( -1 - \beta_{3} + \beta_{5} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{33} + ( -2 + \beta_{1} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{8} - \beta_{14} ) q^{34} -\beta_{5} q^{36} + ( 1 - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{14} + \beta_{15} ) q^{37} + ( -1 + \beta_{3} - \beta_{9} + \beta_{10} + \beta_{12} - \beta_{13} ) q^{38} + ( 1 - \beta_{5} + \beta_{6} - \beta_{7} ) q^{39} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{13} + \beta_{15} ) q^{41} -\beta_{1} q^{42} + ( 1 - 2 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{43} + ( -1 - \beta_{3} + \beta_{5} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{44} + ( \beta_{3} + \beta_{4} + \beta_{5} + \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{46} + ( -\beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} ) q^{47} -\beta_{5} q^{48} + ( 5 - 2 \beta_{3} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{11} + \beta_{13} ) q^{49} + ( 2 - \beta_{1} - \beta_{2} + \beta_{12} ) q^{51} + ( 1 - \beta_{5} + \beta_{6} - \beta_{7} ) q^{52} + ( \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{10} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{53} + \beta_{6} q^{54} -\beta_{1} q^{56} + ( 1 + \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{57} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{58} + ( -\beta_{5} + \beta_{6} - \beta_{9} - \beta_{10} - \beta_{12} + \beta_{13} + \beta_{15} ) q^{59} + ( 2 - \beta_{2} - \beta_{3} - \beta_{4} - 4 \beta_{5} + 3 \beta_{6} - \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{14} - \beta_{15} ) q^{61} + ( -1 + \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - \beta_{9} - \beta_{10} - \beta_{12} ) q^{62} + ( -1 + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{9} ) q^{63} -\beta_{5} q^{64} + ( -1 + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{15} ) q^{66} + ( 1 - \beta_{2} - 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{12} + \beta_{13} + \beta_{15} ) q^{67} + ( 2 - \beta_{1} - \beta_{2} + \beta_{12} ) q^{68} + ( -\beta_{1} - \beta_{2} - \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{69} + ( \beta_{2} - 3 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} + 2 \beta_{14} + \beta_{15} ) q^{71} + \beta_{6} q^{72} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{14} + \beta_{15} ) q^{73} + ( -1 - \beta_{3} + \beta_{6} + \beta_{7} - \beta_{11} - \beta_{13} ) q^{74} + ( 1 + \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{76} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{14} + \beta_{15} ) q^{77} + ( -\beta_{3} + \beta_{7} + \beta_{8} + \beta_{14} + \beta_{15} ) q^{78} + ( 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{10} + \beta_{11} + \beta_{13} + 2 \beta_{14} ) q^{79} -\beta_{3} q^{81} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{6} + \beta_{7} - 2 \beta_{11} + \beta_{12} - \beta_{13} ) q^{82} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{12} - \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{83} + ( -1 + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{9} ) q^{84} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} ) q^{86} + ( -2 - \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{8} + \beta_{14} ) q^{87} + ( -1 + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{15} ) q^{88} + ( -6 - \beta_{2} - \beta_{3} - \beta_{4} - 6 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{14} ) q^{89} + ( -\beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{91} + ( -\beta_{1} - \beta_{2} - \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{92} + ( 1 + 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{8} + \beta_{9} ) q^{93} + ( 1 + \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{12} - \beta_{13} - \beta_{15} ) q^{94} + \beta_{6} q^{96} + ( 2 \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{97} + ( -2 - 2 \beta_{5} - 3 \beta_{6} - \beta_{8} + 2 \beta_{10} + \beta_{11} + \beta_{14} + \beta_{15} ) q^{98} + ( 1 + \beta_{3} - \beta_{6} - \beta_{7} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 4q^{2} - 4q^{3} - 4q^{4} - 4q^{6} + 8q^{7} - 4q^{8} - 4q^{9} + O(q^{10})$$ $$16q - 4q^{2} - 4q^{3} - 4q^{4} - 4q^{6} + 8q^{7} - 4q^{8} - 4q^{9} + 2q^{11} - 4q^{12} + 4q^{13} - 2q^{14} - 4q^{16} - 2q^{17} + 16q^{18} - 2q^{21} - 8q^{22} - 6q^{23} + 16q^{24} + 4q^{26} - 4q^{27} - 2q^{28} + 10q^{29} - 18q^{31} + 16q^{32} - 8q^{33} - 12q^{34} - 4q^{36} - 2q^{37} - 10q^{38} + 4q^{39} + 22q^{41} - 2q^{42} + 4q^{43} - 8q^{44} - 6q^{46} - 2q^{47} - 4q^{48} + 52q^{49} + 28q^{51} + 4q^{52} - 16q^{53} - 4q^{54} - 2q^{56} + 20q^{57} + 10q^{58} - 20q^{59} + 12q^{61} + 2q^{62} - 2q^{63} - 4q^{64} + 2q^{66} + 18q^{67} + 28q^{68} - 6q^{69} - 28q^{71} - 4q^{72} + 24q^{73} - 12q^{74} + 20q^{76} - 14q^{77} - 6q^{78} + 20q^{79} - 4q^{81} + 12q^{82} - 36q^{83} - 2q^{84} - 6q^{86} - 20q^{87} + 2q^{88} - 70q^{89} + 12q^{91} - 6q^{92} + 32q^{93} - 2q^{94} - 4q^{96} + 28q^{97} - 38q^{98} + 12q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-124331836970744504653 \nu^{15} + 74147766037120284798 \nu^{14} + 6575637123195862599235 \nu^{13} - 7361565068046490698696 \nu^{12} - 114171670315622323364112 \nu^{11} + 129297227346064400578539 \nu^{10} + 940839863277170749902944 \nu^{9} - 995606786520726165159631 \nu^{8} - 3722976693962032640661930 \nu^{7} + 3383528419425177424453517 \nu^{6} + 6541910923410674123460038 \nu^{5} - 4124409552879054036847250 \nu^{4} - 3793916280374814064053405 \nu^{3} + 4367880524954808485966775 \nu^{2} + 10909980253880609874290 \nu - 3937464960694376208807150$$$$)/$$$$60\!\cdots\!75$$ $$\beta_{2}$$ $$=$$ $$($$$$209159736194390048391 \nu^{15} - 2183126573404481428433 \nu^{14} - 921402929145846731332 \nu^{13} + 50578789975047200516323 \nu^{12} - 21053103630938467151890 \nu^{11} - 516605312765193244552153 \nu^{10} + 226513676475120458160054 \nu^{9} + 2708996138401524234569682 \nu^{8} - 395430506863010239140326 \nu^{7} - 8305753956702263280322716 \nu^{6} - 520837642160472819150593 \nu^{5} + 12660157620475920634264621 \nu^{4} - 1174098052347100141915960 \nu^{3} - 3854641598501837716805475 \nu^{2} + 11309202599703784457448365 \nu + 7156143218758356286761830$$$$)/$$$$60\!\cdots\!75$$ $$\beta_{3}$$ $$=$$ $$($$$$102309325228650587446 \nu^{15} - 425968439454909023504 \nu^{14} - 2318583371847014159396 \nu^{13} + 9842957759035774370007 \nu^{12} + 23588922138538677715666 \nu^{11} - 96439403899597794479889 \nu^{10} - 132289916480158900957958 \nu^{9} + 469952553315433857842339 \nu^{8} + 521913912002145220106872 \nu^{7} - 1201274203759528179191999 \nu^{6} - 1439503235730439393390400 \nu^{5} + 1561882257402276944726861 \nu^{4} + 1751624753872932949676820 \nu^{3} - 1056349544697974528792295 \nu^{2} + 648738016230841178303300 \nu + 2136427940493114503422100$$$$)/$$$$12\!\cdots\!55$$ $$\beta_{4}$$ $$=$$ $$($$$$-670479263019788745785 \nu^{15} + 5808783625511129416298 \nu^{14} + 2490277702882352021335 \nu^{13} - 124610339454716337184667 \nu^{12} + 108110770501365951462261 \nu^{11} + 1150222220431644145305581 \nu^{10} - 1387265760628987533293941 \nu^{9} - 5213351275070544328036707 \nu^{8} + 5474050671796158148715760 \nu^{7} + 13763120912521780127885204 \nu^{6} - 8085776684625405647461687 \nu^{5} - 19148256425341350974570756 \nu^{4} + 4925945238169235710102170 \nu^{3} + 4133943182196483827525490 \nu^{2} - 17734990834966873730081015 \nu - 9407164594818287670817255$$$$)/$$$$60\!\cdots\!75$$ $$\beta_{5}$$ $$=$$ $$($$$$-161168803239414780910 \nu^{15} + 661637860075799467724 \nu^{14} + 3827536126025598552754 \nu^{13} - 15994665739743758299833 \nu^{12} - 40151303735551147100534 \nu^{11} + 164810455922834007902603 \nu^{10} + 224574784032425709435836 \nu^{9} - 865180285576144364779891 \nu^{8} - 794296728242089068127598 \nu^{7} + 2456515842661058092401641 \nu^{6} + 1829541259881354437224758 \nu^{5} - 3808469691277882506316428 \nu^{4} - 1780293400073412257719180 \nu^{3} + 3317734863222409081306840 \nu^{2} - 1107814885746452854956660 \nu - 2686892356943722466767320$$$$)/$$$$12\!\cdots\!55$$ $$\beta_{6}$$ $$=$$ $$($$$$164194301487618633410 \nu^{15} - 719766109199837227876 \nu^{14} - 3474010950605366952410 \nu^{13} + 16401138606042229504145 \nu^{12} + 31059323892781834514230 \nu^{11} - 157188061964900911061375 \nu^{10} - 130154545341889076617028 \nu^{9} + 733989392081838127266340 \nu^{8} + 317412747752376679716350 \nu^{7} - 1736893910036273955685895 \nu^{6} - 552527714012012839873230 \nu^{5} + 1984064979606023800874943 \nu^{4} + 47852376229771545163160 \nu^{3} - 1332489074800974413015230 \nu^{2} + 2233536035317308402213160 \nu + 1025511991580005661458970$$$$)/$$$$12\!\cdots\!55$$ $$\beta_{7}$$ $$=$$ $$($$$$825056642206646510094 \nu^{15} - 3309109148671381347040 \nu^{14} - 23177084126480302713084 \nu^{13} + 89796640790059785170276 \nu^{12} + 284678894770484789278470 \nu^{11} - 1003793325992617889898975 \nu^{10} - 1895809192339090928409754 \nu^{9} + 5642406357669034386077444 \nu^{8} + 7320601361282167300608548 \nu^{7} - 16118425636653729282281897 \nu^{6} - 16098263057030993567480452 \nu^{5} + 22846655483573752362728564 \nu^{4} + 5489966983986384854079830 \nu^{3} - 18490124851268324162235165 \nu^{2} + 35595526950785121038204610 \nu + 37013945057741259112165345$$$$)/$$$$60\!\cdots\!75$$ $$\beta_{8}$$ $$=$$ $$($$$$104598747224765426 \nu^{15} - 492665369409056577 \nu^{14} - 1860609118043167572 \nu^{13} + 9933253670715682559 \nu^{12} + 13887839469746901817 \nu^{11} - 82481752825489061671 \nu^{10} - 45801221197687274008 \nu^{9} + 310048599187705680258 \nu^{8} + 153197059671127395164 \nu^{7} - 644079276196991518248 \nu^{6} - 490690606807926089402 \nu^{5} + 923340028190882985219 \nu^{4} + 505928391316692432855 \nu^{3} - 397177517625382665270 \nu^{2} - 138919009303557699665 \nu - 1862833702346511294005$$$$)/$$$$75\!\cdots\!75$$ $$\beta_{9}$$ $$=$$ $$($$$$859713236482115609852 \nu^{15} - 5521435521997049744420 \nu^{14} - 12322052446611331480586 \nu^{13} + 125296370582894203734943 \nu^{12} + 49886061591338898032808 \nu^{11} - 1227766370670126017216354 \nu^{10} + 177628563116669271684179 \nu^{9} + 6061236953707159374779795 \nu^{8} - 930510402260991489977198 \nu^{7} - 16997086547094582400276701 \nu^{6} - 228956620702937293280472 \nu^{5} + 25809069667239674846026920 \nu^{4} - 474060716346173434011010 \nu^{3} - 19060768052021310152386215 \nu^{2} + 20901413307917712051094190 \nu + 18643312122164707842914550$$$$)/$$$$60\!\cdots\!75$$ $$\beta_{10}$$ $$=$$ $$($$$$-1299460115152502818714 \nu^{15} + 4487270811728375756266 \nu^{14} + 35341444863509058882955 \nu^{13} - 116807717399885052847095 \nu^{12} - 422002315761784881699595 \nu^{11} + 1316936298960483886112208 \nu^{10} + 2694574233052155278678463 \nu^{9} - 7812655204639800338589865 \nu^{8} - 10080713695248487371901390 \nu^{7} + 25268828066017828142550770 \nu^{6} + 21944052692670850116485037 \nu^{5} - 42553858164431329311678133 \nu^{4} - 23595384347996078052222815 \nu^{3} + 42303268443597858106548325 \nu^{2} - 9053887991424944922843230 \nu - 47144183174991746287899365$$$$)/$$$$60\!\cdots\!75$$ $$\beta_{11}$$ $$=$$ $$($$$$-1574592836099337492490 \nu^{15} + 6900925397189051255252 \nu^{14} + 34052760586501547966220 \nu^{13} - 156344262534607442120813 \nu^{12} - 328474931937103535490956 \nu^{11} + 1495723271670466969311864 \nu^{10} + 1690323491955496542009196 \nu^{9} - 7016503552026061478237508 \nu^{8} - 6153182205258557589646920 \nu^{7} + 17123602090687909484947781 \nu^{6} + 16502279233119671663573202 \nu^{5} - 19653504225630344950830459 \nu^{4} - 24004942108699133343922970 \nu^{3} + 7139685534975283351542585 \nu^{2} + 16902766314303507068679540 \nu - 7059007511747828921024195$$$$)/$$$$60\!\cdots\!75$$ $$\beta_{12}$$ $$=$$ $$($$$$-262837931740539278 \nu^{15} + 1471708972889622906 \nu^{14} + 4022774968630009901 \nu^{13} - 31863082084194063237 \nu^{12} - 17249871320965632266 \nu^{11} + 289468012713592247678 \nu^{10} - 80770827777351194341 \nu^{9} - 1246646574051873818054 \nu^{8} + 732992288018196793613 \nu^{7} + 2766600939541982046314 \nu^{6} - 2091724606383707231214 \nu^{5} - 3031581271759148501187 \nu^{4} + 3978851942084407266110 \nu^{3} + 1163746031625484625835 \nu^{2} - 4675679905232544341130 \nu + 447549436734811744865$$$$)/$$$$75\!\cdots\!75$$ $$\beta_{13}$$ $$=$$ $$($$$$2365643750506328162390 \nu^{15} - 11172455764491969352578 \nu^{14} - 54486762474848731757146 \nu^{13} + 280490443341696305981869 \nu^{12} + 547695740967263618074580 \nu^{11} - 2974115515932565987189328 \nu^{10} - 2893036099906017171709470 \nu^{9} + 15856621799947896145515181 \nu^{8} + 9804946141833297228418482 \nu^{7} - 43571452012557918258947878 \nu^{6} - 24738962377274390845889010 \nu^{5} + 55972537153637957963167105 \nu^{4} + 27756165531065041896261570 \nu^{3} - 23458020504097178603691515 \nu^{2} + 29567004918840064282809300 \nu + 28618714464164239808312900$$$$)/$$$$60\!\cdots\!75$$ $$\beta_{14}$$ $$=$$ $$($$$$4440686169386381053775 \nu^{15} - 19902964987349651471199 \nu^{14} - 93871916386571627516816 \nu^{13} + 454378839163092906914867 \nu^{12} + 870047975964352242942891 \nu^{11} - 4420227892556850128769322 \nu^{10} - 4113043054477141084514633 \nu^{9} + 21409153250151372843474469 \nu^{8} + 13081532680110449770817232 \nu^{7} - 55567324465729717195955414 \nu^{6} - 28289173160041205306722842 \nu^{5} + 74171394571608120776072916 \nu^{4} + 20855257506891577069134660 \nu^{3} - 58426962480746019385967885 \nu^{2} + 47689420538635743669219245 \nu + 65518010360836941583122880$$$$)/$$$$60\!\cdots\!75$$ $$\beta_{15}$$ $$=$$ $$($$$$-4950975914693728326203 \nu^{15} + 24775096076881407709297 \nu^{14} + 98590510294232394702802 \nu^{13} - 575907897989313445655630 \nu^{12} - 837401864414122805737204 \nu^{11} + 5704652211920580391491333 \nu^{10} + 3417436366642788689691063 \nu^{9} - 28289655256498226674975694 \nu^{8} - 9878555739706953941970584 \nu^{7} + 76261374056709458463173010 \nu^{6} + 24574219343129833574456582 \nu^{5} - 110696767774899917556625145 \nu^{4} - 16613144882732870207385855 \nu^{3} + 87542704144517841307017610 \nu^{2} - 75965468295132556936805890 \nu - 75703546008947066945526650$$$$)/$$$$60\!\cdots\!75$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{14} + \beta_{13} + 3 \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - 4 \beta_{6} - \beta_{5} - \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 1$$$$)/5$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{13} - 2 \beta_{12} + 4 \beta_{11} + 4 \beta_{8} - 3 \beta_{7} - 8 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} + 9 \beta_{3} + 2 \beta_{2} + 19$$$$)/5$$ $$\nu^{3}$$ $$=$$ $$($$$$10 \beta_{14} + 9 \beta_{13} - 4 \beta_{12} + 17 \beta_{11} + 9 \beta_{10} - 3 \beta_{9} + 5 \beta_{8} - 9 \beta_{7} - 58 \beta_{6} - 9 \beta_{5} + 25 \beta_{3} - 17 \beta_{2} - 11 \beta_{1} + 16$$$$)/5$$ $$\nu^{4}$$ $$=$$ $$($$$$-14 \beta_{15} - 6 \beta_{14} + 21 \beta_{13} - 32 \beta_{12} + 34 \beta_{11} - 8 \beta_{10} + 2 \beta_{9} + 20 \beta_{8} - 33 \beta_{7} - 124 \beta_{6} + 67 \beta_{5} + 54 \beta_{4} + 160 \beta_{3} + 10 \beta_{2} - 6 \beta_{1} + 71$$$$)/5$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{15} + 95 \beta_{14} + 101 \beta_{13} - 78 \beta_{12} + 133 \beta_{11} + 48 \beta_{10} + 29 \beta_{9} + 70 \beta_{8} - 66 \beta_{7} - 694 \beta_{6} + 94 \beta_{5} + 99 \beta_{4} + 259 \beta_{3} - 142 \beta_{2} - 27 \beta_{1} + 136$$$$)/5$$ $$\nu^{6}$$ $$=$$ $$($$$$-188 \beta_{15} - 44 \beta_{14} + 216 \beta_{13} - 438 \beta_{12} + 305 \beta_{11} - 188 \beta_{10} - 74 \beta_{9} + 146 \beta_{8} - 275 \beta_{7} - 1791 \beta_{6} + 1140 \beta_{5} + 538 \beta_{4} + 1950 \beta_{3} - 62 \beta_{2} - 58 \beta_{1} - 2$$$$)/5$$ $$\nu^{7}$$ $$=$$ $$($$$$-294 \beta_{15} + 785 \beta_{14} + 1093 \beta_{13} - 1027 \beta_{12} + 1065 \beta_{11} - 14 \beta_{10} + 460 \beta_{9} + 895 \beta_{8} - 575 \beta_{7} - 7612 \beta_{6} + 3225 \beta_{5} + 1762 \beta_{4} + 3188 \beta_{3} - 1134 \beta_{2} + 385 \beta_{1} + 998$$$$)/5$$ $$\nu^{8}$$ $$=$$ $$($$$$-2326 \beta_{15} + 154 \beta_{14} + 2443 \beta_{13} - 5008 \beta_{12} + 2768 \beta_{11} - 2612 \beta_{10} - 2034 \beta_{9} + 1584 \beta_{8} - 2001 \beta_{7} - 23351 \beta_{6} + 15294 \beta_{5} + 5230 \beta_{4} + 19654 \beta_{3} - 2046 \beta_{2} + 302 \beta_{1} - 4313$$$$)/5$$ $$\nu^{9}$$ $$=$$ $$($$$$-6366 \beta_{15} + 6142 \beta_{14} + 11172 \beta_{13} - 12226 \beta_{12} + 8125 \beta_{11} - 5168 \beta_{10} + 2171 \beta_{9} + 9946 \beta_{8} - 5220 \beta_{7} - 81375 \beta_{6} + 55205 \beta_{5} + 23432 \beta_{4} + 37656 \beta_{3} - 7670 \beta_{2} + 9832 \beta_{1} + 3989$$$$)/5$$ $$\nu^{10}$$ $$=$$ $$($$$$-29146 \beta_{15} + 10412 \beta_{14} + 27709 \beta_{13} - 51044 \beta_{12} + 22991 \beta_{11} - 29986 \beta_{10} - 33108 \beta_{9} + 18638 \beta_{8} - 12447 \beta_{7} - 280449 \beta_{6} + 193558 \beta_{5} + 55762 \beta_{4} + 175091 \beta_{3} - 27366 \beta_{2} + 21584 \beta_{1} - 69988$$$$)/5$$ $$\nu^{11}$$ $$=$$ $$($$$$-102443 \beta_{15} + 50808 \beta_{14} + 108302 \beta_{13} - 136698 \beta_{12} + 52213 \beta_{11} - 98198 \beta_{10} - 34842 \beta_{9} + 97802 \beta_{8} - 42181 \beta_{7} - 870471 \beta_{6} + 770314 \beta_{5} + 275809 \beta_{4} + 411936 \beta_{3} - 34301 \beta_{2} + 163831 \beta_{1} - 66005$$$$)/5$$ $$\nu^{12}$$ $$=$$ $$($$$$-368948 \beta_{15} + 197614 \beta_{14} + 302610 \beta_{13} - 484604 \beta_{12} + 147042 \beta_{11} - 337056 \beta_{10} - 456660 \beta_{9} + 208408 \beta_{8} - 48619 \beta_{7} - 3155553 \beta_{6} + 2425481 \beta_{5} + 651926 \beta_{4} + 1395453 \beta_{3} - 244322 \beta_{2} + 497230 \beta_{1} - 919477$$$$)/5$$ $$\nu^{13}$$ $$=$$ $$($$$$-1423552 \beta_{15} + 497613 \beta_{14} + 997183 \beta_{13} - 1431438 \beta_{12} + 153682 \beta_{11} - 1393652 \beta_{10} - 1137777 \beta_{9} + 856273 \beta_{8} - 229364 \beta_{7} - 9282256 \beta_{6} + 9741546 \beta_{5} + 3025200 \beta_{4} + 4012368 \beta_{3} + 112272 \beta_{2} + 2441651 \beta_{1} - 2173853$$$$)/5$$ $$\nu^{14}$$ $$=$$ $$($$$$-4656109 \beta_{15} + 2773421 \beta_{14} + 3075782 \beta_{13} - 4332658 \beta_{12} + 171592 \beta_{11} - 3971820 \beta_{10} - 6009522 \beta_{9} + 2114727 \beta_{8} + 354056 \beta_{7} - 33467217 \beta_{6} + 30139656 \beta_{5} + 7837666 \beta_{4} + 9461247 \beta_{3} - 1130570 \beta_{2} + 8628656 \beta_{1} - 11783339$$$$)/5$$ $$\nu^{15}$$ $$=$$ $$($$$$-18116812 \beta_{15} + 5876439 \beta_{14} + 8595411 \beta_{13} - 13723632 \beta_{12} - 3292879 \beta_{11} - 17500853 \beta_{10} - 20676584 \beta_{9} + 6540556 \beta_{8} + 797053 \beta_{7} - 96863665 \beta_{6} + 116294843 \beta_{5} + 31660806 \beta_{4} + 32080639 \beta_{3} + 6166877 \beta_{2} + 34587777 \beta_{1} - 40122453$$$$)/5$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/750\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$251$$ $$\chi(n)$$ $$\beta_{6}$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1
 3.42137 − 0.309017i −0.705457 + 0.309017i −1.80334 − 0.309017i 2.32349 + 0.309017i −2.79002 + 0.809017i 0.543374 − 0.809017i −1.16141 − 0.809017i 2.17199 + 0.809017i −2.79002 − 0.809017i 0.543374 + 0.809017i −1.16141 + 0.809017i 2.17199 − 0.809017i 3.42137 + 0.309017i −0.705457 − 0.309017i −1.80334 + 0.309017i 2.32349 − 0.309017i
0.309017 + 0.951057i −0.809017 + 0.587785i −0.809017 + 0.587785i 0 −0.809017 0.587785i −3.52206 −0.809017 0.587785i 0.309017 0.951057i 0
151.2 0.309017 + 0.951057i −0.809017 + 0.587785i −0.809017 + 0.587785i 0 −0.809017 0.587785i −0.329315 −0.809017 0.587785i 0.309017 0.951057i 0
151.3 0.309017 + 0.951057i −0.809017 + 0.587785i −0.809017 + 0.587785i 0 −0.809017 0.587785i 2.61995 −0.809017 0.587785i 0.309017 0.951057i 0
151.4 0.309017 + 0.951057i −0.809017 + 0.587785i −0.809017 + 0.587785i 0 −0.809017 0.587785i 3.23143 −0.809017 0.587785i 0.309017 0.951057i 0
301.1 −0.809017 + 0.587785i 0.309017 0.951057i 0.309017 0.951057i 0 0.309017 + 0.951057i −4.80694 0.309017 + 0.951057i −0.809017 0.587785i 0
301.2 −0.809017 + 0.587785i 0.309017 0.951057i 0.309017 0.951057i 0 0.309017 + 0.951057i −0.533559 0.309017 + 0.951057i −0.809017 0.587785i 0
301.3 −0.809017 + 0.587785i 0.309017 0.951057i 0.309017 0.951057i 0 0.309017 + 0.951057i 2.70913 0.309017 + 0.951057i −0.809017 0.587785i 0
301.4 −0.809017 + 0.587785i 0.309017 0.951057i 0.309017 0.951057i 0 0.309017 + 0.951057i 4.63137 0.309017 + 0.951057i −0.809017 0.587785i 0
451.1 −0.809017 0.587785i 0.309017 + 0.951057i 0.309017 + 0.951057i 0 0.309017 0.951057i −4.80694 0.309017 0.951057i −0.809017 + 0.587785i 0
451.2 −0.809017 0.587785i 0.309017 + 0.951057i 0.309017 + 0.951057i 0 0.309017 0.951057i −0.533559 0.309017 0.951057i −0.809017 + 0.587785i 0
451.3 −0.809017 0.587785i 0.309017 + 0.951057i 0.309017 + 0.951057i 0 0.309017 0.951057i 2.70913 0.309017 0.951057i −0.809017 + 0.587785i 0
451.4 −0.809017 0.587785i 0.309017 + 0.951057i 0.309017 + 0.951057i 0 0.309017 0.951057i 4.63137 0.309017 0.951057i −0.809017 + 0.587785i 0
601.1 0.309017 0.951057i −0.809017 0.587785i −0.809017 0.587785i 0 −0.809017 + 0.587785i −3.52206 −0.809017 + 0.587785i 0.309017 + 0.951057i 0
601.2 0.309017 0.951057i −0.809017 0.587785i −0.809017 0.587785i 0 −0.809017 + 0.587785i −0.329315 −0.809017 + 0.587785i 0.309017 + 0.951057i 0
601.3 0.309017 0.951057i −0.809017 0.587785i −0.809017 0.587785i 0 −0.809017 + 0.587785i 2.61995 −0.809017 + 0.587785i 0.309017 + 0.951057i 0
601.4 0.309017 0.951057i −0.809017 0.587785i −0.809017 0.587785i 0 −0.809017 + 0.587785i 3.23143 −0.809017 + 0.587785i 0.309017 + 0.951057i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 601.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 750.2.g.f 16
5.b even 2 1 750.2.g.g 16
5.c odd 4 1 150.2.h.b 16
5.c odd 4 1 750.2.h.d 16
15.e even 4 1 450.2.l.c 16
25.d even 5 1 inner 750.2.g.f 16
25.d even 5 1 3750.2.a.v 8
25.e even 10 1 750.2.g.g 16
25.e even 10 1 3750.2.a.u 8
25.f odd 20 1 150.2.h.b 16
25.f odd 20 1 750.2.h.d 16
25.f odd 20 2 3750.2.c.k 16
75.l even 20 1 450.2.l.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.h.b 16 5.c odd 4 1
150.2.h.b 16 25.f odd 20 1
450.2.l.c 16 15.e even 4 1
450.2.l.c 16 75.l even 20 1
750.2.g.f 16 1.a even 1 1 trivial
750.2.g.f 16 25.d even 5 1 inner
750.2.g.g 16 5.b even 2 1
750.2.g.g 16 25.e even 10 1
750.2.h.d 16 5.c odd 4 1
750.2.h.d 16 25.f odd 20 1
3750.2.a.u 8 25.e even 10 1
3750.2.a.v 8 25.d even 5 1
3750.2.c.k 16 25.f odd 20 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(750, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$
$3$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$
$5$ 
$7$ $$( 1 - 4 T + 23 T^{2} - 48 T^{3} + 191 T^{4} - 264 T^{5} + 1285 T^{6} - 2084 T^{7} + 10564 T^{8} - 14588 T^{9} + 62965 T^{10} - 90552 T^{11} + 458591 T^{12} - 806736 T^{13} + 2705927 T^{14} - 3294172 T^{15} + 5764801 T^{16} )^{2}$$
$11$ $$1 - 2 T + 11 T^{2} - 50 T^{3} + 270 T^{4} + 764 T^{5} - 168 T^{6} + 4404 T^{7} - 16480 T^{8} + 222470 T^{9} + 461835 T^{10} - 66610 T^{11} + 2053685 T^{12} + 10233320 T^{13} + 85541270 T^{14} + 166971000 T^{15} - 141281840 T^{16} + 1836681000 T^{17} + 10350493670 T^{18} + 13620548920 T^{19} + 30068002085 T^{20} - 10727607110 T^{21} + 818168874435 T^{22} + 4335310932370 T^{23} - 3532634358880 T^{24} + 10384401631164 T^{25} - 4357487332968 T^{26} + 217978116346804 T^{27} + 847375661714670 T^{28} - 1726135607196550 T^{29} + 4177248169415651 T^{30} - 8354496338831302 T^{31} + 45949729863572161 T^{32}$$
$13$ $$1 - 4 T - 32 T^{2} + 140 T^{3} + 450 T^{4} - 792 T^{5} - 9112 T^{6} - 23836 T^{7} + 215355 T^{8} + 396480 T^{9} - 1875728 T^{10} - 5612928 T^{11} - 11046644 T^{12} + 129713180 T^{13} + 240223360 T^{14} - 1044878996 T^{15} - 1574397811 T^{16} - 13583426948 T^{17} + 40597747840 T^{18} + 284979856460 T^{19} - 315503199284 T^{20} - 2084040875904 T^{21} - 9053780791952 T^{22} + 24878532020160 T^{23} + 175671689420955 T^{24} - 252768847054828 T^{25} - 1256166577728088 T^{26} - 1419391032077304 T^{27} + 10484138305116450 T^{28} + 42402514922915420 T^{29} - 125996044342377248 T^{30} - 204743572056363028 T^{31} + 665416609183179841 T^{32}$$
$17$ $$1 + 2 T - 48 T^{2} - 90 T^{3} + 750 T^{4} - 484 T^{5} - 3528 T^{6} + 71062 T^{7} + 11315 T^{8} - 819420 T^{9} + 506008 T^{10} - 6479544 T^{11} - 20643104 T^{12} + 143754730 T^{13} - 564492280 T^{14} - 443213438 T^{15} + 23546978889 T^{16} - 7534628446 T^{17} - 163138268920 T^{18} + 706266988490 T^{19} - 1724132689184 T^{20} - 9200025905208 T^{21} + 12213803014552 T^{22} - 336239715429660 T^{23} + 78930695444915 T^{24} + 8427091679629814 T^{25} - 7112426480784072 T^{26} - 16587597812894372 T^{27} + 436966677922320750 T^{28} - 891412022961534330 T^{29} - 8082135674851244592 T^{30} + 5724846103019631586 T^{31} + 48661191875666868481 T^{32}$$
$19$ $$1 + 24 T^{2} + 30 T^{3} + 330 T^{4} - 740 T^{5} + 7630 T^{6} - 18810 T^{7} + 160255 T^{8} + 244020 T^{9} + 7105152 T^{10} + 5384250 T^{11} + 141174008 T^{12} + 174364190 T^{13} + 1779959840 T^{14} - 4165821270 T^{15} + 38170716365 T^{16} - 79150604130 T^{17} + 642565502240 T^{18} + 1195963979210 T^{19} + 18397937896568 T^{20} + 13331936040750 T^{21} + 334268135478912 T^{22} + 218122581750780 T^{23} + 2721700895135455 T^{24} - 6069755595222990 T^{25} + 46780035547021630 T^{26} - 86202791584682060 T^{27} + 730393923291833130 T^{28} + 1261589503867711770 T^{29} + 19176160458789218904 T^{30} +$$$$28\!\cdots\!81$$$$T^{32}$$
$23$ $$1 + 6 T + 38 T^{2} - 90 T^{3} - 1030 T^{4} - 11032 T^{5} - 10802 T^{6} + 96694 T^{7} + 2058175 T^{8} + 6403360 T^{9} + 17647902 T^{10} - 190703348 T^{11} - 1079985444 T^{12} - 6006115670 T^{13} + 3603112480 T^{14} + 82638392094 T^{15} + 852025843989 T^{16} + 1900683018162 T^{17} + 1906046501920 T^{18} - 73076409356890 T^{19} - 302224206634404 T^{20} - 1227432158976364 T^{21} + 2612522861554878 T^{22} + 21802323074301920 T^{23} + 161177712130722175 T^{24} + 174160655447503322 T^{25} - 447489174129836498 T^{26} - 10511397249306442664 T^{27} - 22572063164980930630 T^{28} - 45363272574282064470 T^{29} +$$$$44\!\cdots\!42$$$$T^{30} +$$$$15\!\cdots\!42$$$$T^{31} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$1 - 10 T - 36 T^{2} + 270 T^{3} + 3910 T^{4} - 4460 T^{5} - 185440 T^{6} - 333090 T^{7} + 4522435 T^{8} + 38473780 T^{9} - 69131068 T^{10} - 1395128220 T^{11} - 3409890032 T^{12} + 36344819210 T^{13} + 245632726680 T^{14} - 543098432830 T^{15} - 7660398480495 T^{16} - 15749854552070 T^{17} + 206577123137880 T^{18} + 886413795712690 T^{19} - 2411750431722992 T^{20} - 28615682794524780 T^{21} - 41120771452036828 T^{22} + 663667946139678020 T^{23} + 2262331886599280035 T^{24} - 4832185253102205210 T^{25} - 78015949343189273440 T^{26} - 54414273555047997340 T^{27} +$$$$13\!\cdots\!10$$$$T^{28} +$$$$27\!\cdots\!30$$$$T^{29} -$$$$10\!\cdots\!16$$$$T^{30} -$$$$86\!\cdots\!90$$$$T^{31} +$$$$25\!\cdots\!21$$$$T^{32}$$
$31$ $$1 + 18 T + 121 T^{2} + 610 T^{3} + 6060 T^{4} + 54724 T^{5} + 309962 T^{6} + 1456784 T^{7} + 9332270 T^{8} + 64865970 T^{9} + 327433735 T^{10} + 1367357590 T^{11} + 7702756335 T^{12} + 47160385620 T^{13} + 227624335970 T^{14} + 1111476930100 T^{15} + 6249005152660 T^{16} + 34455784833100 T^{17} + 218746986867170 T^{18} + 1404955048005420 T^{19} + 7113657233255535 T^{20} + 39146286915106090 T^{21} + 290598645096078535 T^{22} + 1784632401545702670 T^{23} + 7959409441979521070 T^{24} + 38516818529710942064 T^{25} +$$$$25\!\cdots\!62$$$$T^{26} +$$$$13\!\cdots\!44$$$$T^{27} +$$$$47\!\cdots\!60$$$$T^{28} +$$$$14\!\cdots\!10$$$$T^{29} +$$$$91\!\cdots\!41$$$$T^{30} +$$$$42\!\cdots\!18$$$$T^{31} +$$$$72\!\cdots\!81$$$$T^{32}$$
$37$ $$1 + 2 T - 63 T^{2} - 260 T^{3} + 4140 T^{4} + 9346 T^{5} - 200368 T^{6} - 602598 T^{7} + 10433820 T^{8} + 16405980 T^{9} - 438910957 T^{10} - 556272754 T^{11} + 21598802491 T^{12} + 5413755160 T^{13} - 841023854600 T^{14} - 345750792988 T^{15} + 33745348532144 T^{16} - 12792779340556 T^{17} - 1151361656947400 T^{18} + 274222940119480 T^{19} + 40479633275335051 T^{20} - 38574153933647578 T^{21} - 1126125433574363413 T^{22} + 1557450477606455340 T^{23} + 36648578375910008220 T^{24} - 78314684477033810046 T^{25} -$$$$96\!\cdots\!32$$$$T^{26} +$$$$16\!\cdots\!98$$$$T^{27} +$$$$27\!\cdots\!40$$$$T^{28} -$$$$63\!\cdots\!20$$$$T^{29} -$$$$56\!\cdots\!07$$$$T^{30} +$$$$66\!\cdots\!86$$$$T^{31} +$$$$12\!\cdots\!41$$$$T^{32}$$
$41$ $$1 - 22 T + 316 T^{2} - 3710 T^{3} + 39530 T^{4} - 407436 T^{5} + 3918652 T^{6} - 35382126 T^{7} + 301988495 T^{8} - 2475859380 T^{9} + 19707200260 T^{10} - 150750040860 T^{11} + 1111070917960 T^{12} - 7891095298530 T^{13} + 54497096107120 T^{14} - 366799083339050 T^{15} + 2387443151993985 T^{16} - 15038762416901050 T^{17} + 91609618556068720 T^{18} - 543862179069986130 T^{19} + 3139620868205567560 T^{20} - 17465327034634372860 T^{21} + 93611255533262302660 T^{22} -$$$$48\!\cdots\!80$$$$T^{23} +$$$$24\!\cdots\!95$$$$T^{24} -$$$$11\!\cdots\!86$$$$T^{25} +$$$$52\!\cdots\!52$$$$T^{26} -$$$$22\!\cdots\!76$$$$T^{27} +$$$$89\!\cdots\!30$$$$T^{28} -$$$$34\!\cdots\!10$$$$T^{29} +$$$$11\!\cdots\!76$$$$T^{30} -$$$$34\!\cdots\!22$$$$T^{31} +$$$$63\!\cdots\!41$$$$T^{32}$$
$43$ $$( 1 - 2 T + 182 T^{2} - 166 T^{3} + 15756 T^{4} + 5778 T^{5} + 902810 T^{6} + 1211542 T^{7} + 41601734 T^{8} + 52096306 T^{9} + 1669295690 T^{10} + 459391446 T^{11} + 53866628556 T^{12} - 24403401538 T^{13} + 1150488074918 T^{14} - 543637222214 T^{15} + 11688200277601 T^{16} )^{2}$$
$47$ $$1 + 2 T - 38 T^{2} + 100 T^{3} - 1990 T^{4} + 35816 T^{5} + 199152 T^{6} - 2069788 T^{7} + 8558115 T^{8} - 67009360 T^{9} + 14666138 T^{10} + 6437016726 T^{11} - 54118623424 T^{12} + 182885390860 T^{13} - 36525931520 T^{14} - 14678432881668 T^{15} + 117494617211029 T^{16} - 689886345438396 T^{17} - 80685782727680 T^{18} + 18987709935257780 T^{19} - 264081618468247744 T^{20} + 1476297646083587082 T^{21} + 158089459546829402 T^{22} - 33948491063428533680 T^{23} +$$$$20\!\cdots\!15$$$$T^{24} -$$$$23\!\cdots\!96$$$$T^{25} +$$$$10\!\cdots\!48$$$$T^{26} +$$$$88\!\cdots\!48$$$$T^{27} -$$$$23\!\cdots\!90$$$$T^{28} +$$$$54\!\cdots\!00$$$$T^{29} -$$$$97\!\cdots\!22$$$$T^{30} +$$$$24\!\cdots\!86$$$$T^{31} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 + 16 T - 77 T^{2} - 2870 T^{3} - 940 T^{4} + 339118 T^{5} + 1065708 T^{6} - 31144306 T^{7} - 181262900 T^{8} + 2150574730 T^{9} + 18826444257 T^{10} - 119643607568 T^{11} - 1549694959149 T^{12} + 4868839460340 T^{13} + 105542006347440 T^{14} - 92971675742196 T^{15} - 6013439087391336 T^{16} - 4927498814336388 T^{17} + 296467495829958960 T^{18} + 724858212337038180 T^{19} - 12227838630960960669 T^{20} - 50034417451198291024 T^{21} +$$$$41\!\cdots\!53$$$$T^{22} +$$$$25\!\cdots\!10$$$$T^{23} -$$$$11\!\cdots\!00$$$$T^{24} -$$$$10\!\cdots\!98$$$$T^{25} +$$$$18\!\cdots\!92$$$$T^{26} +$$$$31\!\cdots\!46$$$$T^{27} -$$$$46\!\cdots\!40$$$$T^{28} -$$$$74\!\cdots\!10$$$$T^{29} -$$$$10\!\cdots\!13$$$$T^{30} +$$$$11\!\cdots\!12$$$$T^{31} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$1 + 20 T - 111 T^{2} - 3640 T^{3} + 11660 T^{4} + 317820 T^{5} - 1817980 T^{6} - 11679820 T^{7} + 217773860 T^{8} - 627241760 T^{9} - 13630850673 T^{10} + 133224376740 T^{11} + 186113511463 T^{12} - 9371240514920 T^{13} + 48131664631280 T^{14} + 252534601553160 T^{15} - 4526062111550040 T^{16} + 14899541491636440 T^{17} + 167546324581485680 T^{18} - 1924656005713754680 T^{19} + 2255204605374809143 T^{20} + 95245344150556405260 T^{21} -$$$$57\!\cdots\!93$$$$T^{22} -$$$$15\!\cdots\!40$$$$T^{23} +$$$$31\!\cdots\!60$$$$T^{24} -$$$$10\!\cdots\!80$$$$T^{25} -$$$$92\!\cdots\!80$$$$T^{26} +$$$$95\!\cdots\!80$$$$T^{27} +$$$$20\!\cdots\!60$$$$T^{28} -$$$$38\!\cdots\!60$$$$T^{29} -$$$$68\!\cdots\!71$$$$T^{30} +$$$$73\!\cdots\!80$$$$T^{31} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$1 - 12 T - 104 T^{2} + 1460 T^{3} + 9310 T^{4} - 110696 T^{5} - 1128308 T^{6} + 9605804 T^{7} + 95707895 T^{8} - 545482080 T^{9} - 8568890440 T^{10} + 31012865040 T^{11} + 647686771260 T^{12} - 1172976299580 T^{13} - 45238788397880 T^{14} + 2951178504100 T^{15} + 3155660575885285 T^{16} + 180021888750100 T^{17} - 168333531628511480 T^{18} - 266243333454967980 T^{19} + 8967768052669329660 T^{20} + 26193351096196217040 T^{21} -$$$$44\!\cdots\!40$$$$T^{22} -$$$$17\!\cdots\!80$$$$T^{23} +$$$$18\!\cdots\!95$$$$T^{24} +$$$$11\!\cdots\!64$$$$T^{25} -$$$$80\!\cdots\!08$$$$T^{26} -$$$$48\!\cdots\!56$$$$T^{27} +$$$$24\!\cdots\!10$$$$T^{28} +$$$$23\!\cdots\!60$$$$T^{29} -$$$$10\!\cdots\!64$$$$T^{30} -$$$$72\!\cdots\!12$$$$T^{31} +$$$$36\!\cdots\!61$$$$T^{32}$$
$67$ $$1 - 18 T + 22 T^{2} + 1780 T^{3} - 17550 T^{4} + 18696 T^{5} + 1171492 T^{6} - 10100148 T^{7} - 24274805 T^{8} + 841343440 T^{9} - 2302610202 T^{10} - 43017975694 T^{11} + 410757974896 T^{12} - 505566173460 T^{13} - 15187840052640 T^{14} + 85758790868132 T^{15} - 114408376419531 T^{16} + 5745838988164844 T^{17} - 68178213996300960 T^{18} - 152055599028349980 T^{19} + 8277233653844258416 T^{20} - 58079649036785149258 T^{21} -$$$$20\!\cdots\!38$$$$T^{22} +$$$$50\!\cdots\!20$$$$T^{23} -$$$$98\!\cdots\!05$$$$T^{24} -$$$$27\!\cdots\!56$$$$T^{25} +$$$$21\!\cdots\!08$$$$T^{26} +$$$$22\!\cdots\!68$$$$T^{27} -$$$$14\!\cdots\!50$$$$T^{28} +$$$$97\!\cdots\!60$$$$T^{29} +$$$$80\!\cdots\!38$$$$T^{30} -$$$$44\!\cdots\!74$$$$T^{31} +$$$$16\!\cdots\!81$$$$T^{32}$$
$71$ $$1 + 28 T + 256 T^{2} + 1150 T^{3} + 20770 T^{4} + 328604 T^{5} + 2351902 T^{6} + 21115094 T^{7} + 321977895 T^{8} + 2983992020 T^{9} + 18685321160 T^{10} + 180970630790 T^{11} + 2127756781760 T^{12} + 17298444226470 T^{13} + 123520240637920 T^{14} + 1239239342309850 T^{15} + 12274938369025885 T^{16} + 87985993303999350 T^{17} + 622665533055754720 T^{18} + 6191303471540104170 T^{19} + 54069876583671738560 T^{20} +$$$$32\!\cdots\!90$$$$T^{21} +$$$$23\!\cdots\!60$$$$T^{22} +$$$$27\!\cdots\!20$$$$T^{23} +$$$$20\!\cdots\!95$$$$T^{24} +$$$$96\!\cdots\!14$$$$T^{25} +$$$$76\!\cdots\!02$$$$T^{26} +$$$$75\!\cdots\!84$$$$T^{27} +$$$$34\!\cdots\!70$$$$T^{28} +$$$$13\!\cdots\!50$$$$T^{29} +$$$$21\!\cdots\!36$$$$T^{30} +$$$$16\!\cdots\!28$$$$T^{31} +$$$$41\!\cdots\!21$$$$T^{32}$$
$73$ $$1 - 24 T + 108 T^{2} + 1120 T^{3} - 1630 T^{4} - 255872 T^{5} + 3551748 T^{6} - 13656736 T^{7} - 69280205 T^{8} - 551771520 T^{9} + 23775538892 T^{10} - 240365625128 T^{11} + 1168271575756 T^{12} - 990327853600 T^{13} + 44032383357120 T^{14} - 1148702844107296 T^{15} + 12070885159211549 T^{16} - 83855307619832608 T^{17} + 234648570910092480 T^{18} - 385254370623911200 T^{19} + 33176857761768645196 T^{20} -$$$$49\!\cdots\!04$$$$T^{21} +$$$$35\!\cdots\!88$$$$T^{22} -$$$$60\!\cdots\!40$$$$T^{23} -$$$$55\!\cdots\!05$$$$T^{24} -$$$$80\!\cdots\!68$$$$T^{25} +$$$$15\!\cdots\!52$$$$T^{26} -$$$$80\!\cdots\!44$$$$T^{27} -$$$$37\!\cdots\!30$$$$T^{28} +$$$$18\!\cdots\!60$$$$T^{29} +$$$$13\!\cdots\!72$$$$T^{30} -$$$$21\!\cdots\!68$$$$T^{31} +$$$$65\!\cdots\!61$$$$T^{32}$$
$79$ $$1 - 20 T + 224 T^{2} - 1780 T^{3} + 6070 T^{4} - 5760 T^{5} + 875320 T^{6} - 20333540 T^{7} + 305719495 T^{8} - 3198483920 T^{9} + 22059896192 T^{10} - 116979470440 T^{11} + 660067500148 T^{12} - 7893347694940 T^{13} + 139701887310360 T^{14} - 1828379420794980 T^{15} + 18018691485968485 T^{16} - 144441974242803420 T^{17} + 871879478703956760 T^{18} - 3891728254165522660 T^{19} + 25709682596232111988 T^{20} -$$$$35\!\cdots\!60$$$$T^{21} +$$$$53\!\cdots\!32$$$$T^{22} -$$$$61\!\cdots\!80$$$$T^{23} +$$$$46\!\cdots\!95$$$$T^{24} -$$$$24\!\cdots\!60$$$$T^{25} +$$$$82\!\cdots\!20$$$$T^{26} -$$$$43\!\cdots\!40$$$$T^{27} +$$$$35\!\cdots\!70$$$$T^{28} -$$$$83\!\cdots\!20$$$$T^{29} +$$$$82\!\cdots\!44$$$$T^{30} -$$$$58\!\cdots\!80$$$$T^{31} +$$$$23\!\cdots\!21$$$$T^{32}$$
$83$ $$1 + 36 T + 563 T^{2} + 5840 T^{3} + 65350 T^{4} + 846708 T^{5} + 9529358 T^{6} + 93908524 T^{7} + 968532230 T^{8} + 10602294780 T^{9} + 108997499647 T^{10} + 1021062007352 T^{11} + 9465131610121 T^{12} + 92769329470780 T^{13} + 910324505619660 T^{14} + 8373646504807004 T^{15} + 75200600125807724 T^{16} + 695012659898981332 T^{17} + 6271225519213837740 T^{18} + 53044296590108883860 T^{19} +$$$$44\!\cdots\!41$$$$T^{20} +$$$$40\!\cdots\!36$$$$T^{21} +$$$$35\!\cdots\!43$$$$T^{22} +$$$$28\!\cdots\!60$$$$T^{23} +$$$$21\!\cdots\!30$$$$T^{24} +$$$$17\!\cdots\!72$$$$T^{25} +$$$$14\!\cdots\!42$$$$T^{26} +$$$$10\!\cdots\!36$$$$T^{27} +$$$$69\!\cdots\!50$$$$T^{28} +$$$$51\!\cdots\!20$$$$T^{29} +$$$$41\!\cdots\!27$$$$T^{30} +$$$$22\!\cdots\!52$$$$T^{31} +$$$$50\!\cdots\!81$$$$T^{32}$$
$89$ $$1 + 70 T + 2119 T^{2} + 35330 T^{3} + 343460 T^{4} + 2032360 T^{5} + 10870620 T^{6} + 64684440 T^{7} - 584089940 T^{8} - 17002587830 T^{9} - 99745085123 T^{10} + 366225143690 T^{11} + 5693869232523 T^{12} + 24776539056740 T^{13} + 308063609313680 T^{14} + 3148141734399680 T^{15} + 20769024208268760 T^{16} + 280184614361571520 T^{17} + 2440171849373659280 T^{18} + 17466691962290941060 T^{19} +$$$$35\!\cdots\!43$$$$T^{20} +$$$$20\!\cdots\!10$$$$T^{21} -$$$$49\!\cdots\!03$$$$T^{22} -$$$$75\!\cdots\!70$$$$T^{23} -$$$$22\!\cdots\!40$$$$T^{24} +$$$$22\!\cdots\!60$$$$T^{25} +$$$$33\!\cdots\!20$$$$T^{26} +$$$$56\!\cdots\!40$$$$T^{27} +$$$$84\!\cdots\!60$$$$T^{28} +$$$$77\!\cdots\!70$$$$T^{29} +$$$$41\!\cdots\!79$$$$T^{30} +$$$$12\!\cdots\!30$$$$T^{31} +$$$$15\!\cdots\!61$$$$T^{32}$$
$97$ $$1 - 28 T + 237 T^{2} + 3480 T^{3} - 102760 T^{4} + 823556 T^{5} + 5247032 T^{6} - 171941308 T^{7} + 1253083480 T^{8} + 7160994200 T^{9} - 216456884977 T^{10} + 1333703151276 T^{11} + 12125462474171 T^{12} - 281297168199120 T^{13} + 1492624771812080 T^{14} + 15832426927626952 T^{15} - 314898947215884176 T^{16} + 1535745411979814344 T^{17} + 14044106477979860720 T^{18} -$$$$25\!\cdots\!60$$$$T^{19} +$$$$10\!\cdots\!51$$$$T^{20} +$$$$11\!\cdots\!32$$$$T^{21} -$$$$18\!\cdots\!33$$$$T^{22} +$$$$57\!\cdots\!00$$$$T^{23} +$$$$98\!\cdots\!80$$$$T^{24} -$$$$13\!\cdots\!36$$$$T^{25} +$$$$38\!\cdots\!68$$$$T^{26} +$$$$58\!\cdots\!68$$$$T^{27} -$$$$71\!\cdots\!60$$$$T^{28} +$$$$23\!\cdots\!60$$$$T^{29} +$$$$15\!\cdots\!53$$$$T^{30} -$$$$17\!\cdots\!04$$$$T^{31} +$$$$61\!\cdots\!21$$$$T^{32}$$