# Properties

 Label 495.2.n.g Level $495$ Weight $2$ Character orbit 495.n Analytic conductor $3.953$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$495 = 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 495.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.95259490005$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 2 x^{15} + 5 x^{14} - 8 x^{13} + 47 x^{12} + 32 x^{11} + 171 x^{10} + 26 x^{9} + 360 x^{8} - 172 x^{7} + 471 x^{6} - 430 x^{5} + 383 x^{4} + 70 x^{3} + 17 x^{2} + 4 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} + \beta_{3} + \beta_{5} - \beta_{10} - \beta_{14} ) q^{2} + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} - \beta_{9} + \beta_{12} ) q^{4} -\beta_{8} q^{5} + ( -\beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} + \beta_{9} + \beta_{11} ) q^{7} + ( -2 + \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{8} + \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{8} +O(q^{10})$$ $$q + ( 1 - \beta_{1} + \beta_{3} + \beta_{5} - \beta_{10} - \beta_{14} ) q^{2} + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} - \beta_{9} + \beta_{12} ) q^{4} -\beta_{8} q^{5} + ( -\beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} + \beta_{9} + \beta_{11} ) q^{7} + ( -2 + \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{8} + \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{8} -\beta_{3} q^{10} + ( 1 + \beta_{2} + \beta_{6} + \beta_{8} - \beta_{9} - \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{11} + ( -1 + 2 \beta_{4} - \beta_{6} - 2 \beta_{7} + \beta_{9} + \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} ) q^{13} + ( -3 + 2 \beta_{2} + \beta_{6} + 2 \beta_{8} + 2 \beta_{10} + 3 \beta_{14} ) q^{14} + ( -4 + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{12} + 2 \beta_{14} + \beta_{15} ) q^{16} + ( 2 + 2 \beta_{3} + \beta_{5} - \beta_{7} - 2 \beta_{8} - 2 \beta_{10} + \beta_{12} - \beta_{13} ) q^{17} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{8} - \beta_{10} + 2 \beta_{11} - \beta_{14} - \beta_{15} ) q^{19} + ( 2 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{20} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{11} + \beta_{15} ) q^{22} + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - \beta_{10} - \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{23} + ( -1 + \beta_{2} + \beta_{8} + \beta_{14} ) q^{25} + ( -3 - \beta_{2} + \beta_{4} - 3 \beta_{6} - \beta_{7} - \beta_{8} + 4 \beta_{9} + 3 \beta_{11} - 2 \beta_{12} ) q^{26} + ( -3 + \beta_{2} + 3 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} + 3 \beta_{9} + 2 \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{28} + ( 1 - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} + \beta_{8} - 2 \beta_{9} + \beta_{12} ) q^{29} + ( 1 - \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{31} + ( 2 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{6} - \beta_{7} - 5 \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{12} - \beta_{13} - 4 \beta_{14} - 2 \beta_{15} ) q^{32} + ( -2 + \beta_{1} - \beta_{3} + 5 \beta_{8} + \beta_{10} + 5 \beta_{14} + 3 \beta_{15} ) q^{34} + ( -\beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{10} + \beta_{13} - 2 \beta_{14} ) q^{35} + ( 1 + 3 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{9} - \beta_{11} ) q^{37} + ( 1 - 4 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} + 2 \beta_{13} - 4 \beta_{14} ) q^{38} + ( 3 - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{12} ) q^{40} + ( -1 + 2 \beta_{1} + 3 \beta_{2} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{8} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} - \beta_{15} ) q^{41} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} + \beta_{12} - 2 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{43} + ( -2 + 2 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{9} + 4 \beta_{10} + \beta_{11} - 2 \beta_{12} + 2 \beta_{14} + \beta_{15} ) q^{44} + ( -1 + \beta_{1} - 4 \beta_{2} - \beta_{3} + 3 \beta_{4} - 4 \beta_{6} + 2 \beta_{9} + 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{46} + ( -1 + 3 \beta_{2} - 2 \beta_{4} + 3 \beta_{6} - 2 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} + 4 \beta_{14} - 2 \beta_{15} ) q^{47} + ( 1 + 2 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} - \beta_{5} + 4 \beta_{6} + 2 \beta_{7} + \beta_{8} - 4 \beta_{9} + 3 \beta_{10} - \beta_{11} + 2 \beta_{12} + 2 \beta_{14} - 3 \beta_{15} ) q^{49} + ( -\beta_{5} - \beta_{8} ) q^{50} + ( 3 - 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{5} + \beta_{8} - 2 \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{52} + ( -1 + 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{5} + \beta_{7} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{53} + ( 2 - \beta_{2} - 2 \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{55} + ( -1 - 2 \beta_{1} + 3 \beta_{2} - \beta_{6} - 3 \beta_{7} + 6 \beta_{9} - 2 \beta_{10} - \beta_{12} - 3 \beta_{13} + 3 \beta_{14} - \beta_{15} ) q^{56} + ( -5 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 4 \beta_{8} + 6 \beta_{10} + \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + 3 \beta_{14} + 2 \beta_{15} ) q^{58} + ( -5 + 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{8} + 2 \beta_{9} + \beta_{11} - 2 \beta_{12} ) q^{59} + ( 3 - 7 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} - 7 \beta_{14} ) q^{61} + ( 3 + 2 \beta_{1} - 6 \beta_{2} - 4 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{11} + \beta_{12} ) q^{62} + ( 7 - \beta_{1} - 7 \beta_{2} + \beta_{3} - 3 \beta_{4} + 7 \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{9} - 7 \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{13} - 8 \beta_{14} - \beta_{15} ) q^{64} + ( 2 - 2 \beta_{2} - \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{65} + ( -4 + 2 \beta_{1} - 2 \beta_{3} - \beta_{6} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{12} + 2 \beta_{14} - 3 \beta_{15} ) q^{67} + ( -6 - 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{10} + 2 \beta_{13} + \beta_{14} ) q^{68} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{8} + \beta_{12} ) q^{70} + ( -1 - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} + 2 \beta_{12} - 3 \beta_{13} - 3 \beta_{14} - 2 \beta_{15} ) q^{71} + ( 2 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{11} - \beta_{12} ) q^{73} + ( -2 \beta_{1} + 5 \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{6} + 5 \beta_{8} - 2 \beta_{10} + \beta_{15} ) q^{74} + ( 1 + \beta_{1} + 5 \beta_{3} - 2 \beta_{9} + \beta_{10} + \beta_{15} ) q^{76} + ( -1 - 2 \beta_{1} - 4 \beta_{2} + \beta_{3} + 4 \beta_{4} + \beta_{5} - 4 \beta_{6} - 2 \beta_{7} + 6 \beta_{9} + \beta_{10} + 4 \beta_{11} - 3 \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{77} + ( -3 + 6 \beta_{1} + \beta_{2} - 6 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} - 2 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + 4 \beta_{14} - 2 \beta_{15} ) q^{79} + ( -4 + \beta_{1} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{10} + \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{80} + ( -6 + 4 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} - 3 \beta_{6} + \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} - \beta_{11} - 4 \beta_{12} + 2 \beta_{13} + 4 \beta_{14} + 4 \beta_{15} ) q^{82} + ( 5 - 3 \beta_{2} + \beta_{3} - 4 \beta_{4} + \beta_{5} + 4 \beta_{6} + 2 \beta_{7} - 6 \beta_{8} - 4 \beta_{9} - \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + 4 \beta_{13} - 3 \beta_{14} - 2 \beta_{15} ) q^{83} + ( -1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{5} + 2 \beta_{8} - \beta_{10} - \beta_{12} + \beta_{13} ) q^{85} + ( 12 - 6 \beta_{2} - 9 \beta_{4} + 4 \beta_{5} + 6 \beta_{6} + 2 \beta_{7} - 3 \beta_{9} - 4 \beta_{10} - 3 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} + 2 \beta_{14} - 3 \beta_{15} ) q^{86} + ( 6 - 5 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} - 5 \beta_{8} + \beta_{9} - 6 \beta_{10} - 3 \beta_{11} - \beta_{12} - 9 \beta_{14} ) q^{88} + ( 9 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{6} - \beta_{7} - 4 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{12} - \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{89} + ( -3 - \beta_{1} + 5 \beta_{2} - 5 \beta_{4} + \beta_{5} + 6 \beta_{6} + \beta_{8} + 3 \beta_{10} - 3 \beta_{11} + 4 \beta_{12} - 4 \beta_{13} + 7 \beta_{14} - 5 \beta_{15} ) q^{91} + ( -4 - 2 \beta_{1} + 5 \beta_{2} - \beta_{3} - \beta_{5} + 3 \beta_{8} - 3 \beta_{12} ) q^{92} + ( 7 + \beta_{2} - 7 \beta_{4} - \beta_{5} + 7 \beta_{6} + 2 \beta_{7} - 7 \beta_{9} - 5 \beta_{11} + 5 \beta_{12} - 2 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{94} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{9} - 2 \beta_{11} + \beta_{12} ) q^{95} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{97} + ( 4 - 3 \beta_{2} - 2 \beta_{6} + 3 \beta_{7} - 5 \beta_{8} - 2 \beta_{9} - 2 \beta_{12} + 3 \beta_{13} - 8 \beta_{14} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 2 q^{2} - 8 q^{4} - 4 q^{5} - 4 q^{7} - 6 q^{8} + O(q^{10})$$ $$16 q - 2 q^{2} - 8 q^{4} - 4 q^{5} - 4 q^{7} - 6 q^{8} + 8 q^{10} + 4 q^{11} + 2 q^{13} - 22 q^{14} + 8 q^{16} - 4 q^{17} - 4 q^{19} + 2 q^{20} - 28 q^{22} + 8 q^{23} - 4 q^{25} + 6 q^{26} - 2 q^{28} - 26 q^{29} - 10 q^{31} + 56 q^{32} - 4 q^{34} - 4 q^{35} + 22 q^{37} - 30 q^{38} - 6 q^{40} - 6 q^{41} + 28 q^{43} + 68 q^{44} + 16 q^{46} - 20 q^{47} + 10 q^{49} - 2 q^{50} + 30 q^{52} + 14 q^{53} - 6 q^{55} + 68 q^{56} - 6 q^{58} - 16 q^{59} - 38 q^{61} - 20 q^{62} + 10 q^{64} + 12 q^{65} + 20 q^{67} - 48 q^{68} - 22 q^{70} - 54 q^{71} + 2 q^{73} + 28 q^{74} - 44 q^{76} + 34 q^{77} - 12 q^{79} - 22 q^{80} + 30 q^{82} - 28 q^{83} - 4 q^{85} + 74 q^{86} + 46 q^{88} + 76 q^{89} - 34 q^{91} - 8 q^{92} - 10 q^{94} - 4 q^{95} - 18 q^{97} + 8 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{15} + 5 x^{14} - 8 x^{13} + 47 x^{12} + 32 x^{11} + 171 x^{10} + 26 x^{9} + 360 x^{8} - 172 x^{7} + 471 x^{6} - 430 x^{5} + 383 x^{4} + 70 x^{3} + 17 x^{2} + 4 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-274909709246406 \nu^{15} - 784278699533661 \nu^{14} + 143020561508636 \nu^{13} - 2005995543669531 \nu^{12} - 8519106391972926 \nu^{11} - 61146526572552240 \nu^{10} - 145336767693588186 \nu^{9} - 264117958519829524 \nu^{8} - 331573544358140394 \nu^{7} - 446597103764880540 \nu^{6} - 319729317918190560 \nu^{5} - 231641863194072003 \nu^{4} - 90506767183986926 \nu^{3} + 93570035844837543 \nu^{2} - 568591142787875238 \nu + 4684863033388304$$$$)/ 124366022234909444$$ $$\beta_{3}$$ $$=$$ $$($$$$553668747953762 \nu^{15} - 1294038871869219 \nu^{14} + 2678375186769114 \nu^{13} - 4724034435676990 \nu^{12} + 25734959323311664 \nu^{11} + 10614112411878989 \nu^{10} + 69986082534625206 \nu^{9} - 46186287420472496 \nu^{8} + 104166886954478224 \nu^{7} - 245911872417401744 \nu^{6} + 55312393060713098 \nu^{5} - 394772810552973845 \nu^{4} + 140885354640541754 \nu^{3} + 27555273511115606 \nu^{2} - 3822684831751662 \nu - 301927752492954565$$$$)/ 248732044469818888$$ $$\beta_{4}$$ $$=$$ $$($$$$1439074005313563 \nu^{15} + 1851897765207855 \nu^{14} + 2324766414796124 \nu^{13} + 2360640185706480 \nu^{12} + 53528410268329273 \nu^{11} + 228500108198224671 \nu^{10} + 616925067838708542 \nu^{9} + 967946523561484984 \nu^{8} + 1386230620986522104 \nu^{7} + 1425871732223573700 \nu^{6} + 1350563990196551169 \nu^{5} + 508765947296826397 \nu^{4} + 596242561123071756 \nu^{3} - 337614021434331534 \nu^{2} + 1967072679306622413 \nu + 413981182405604135$$$$)/ 248732044469818888$$ $$\beta_{5}$$ $$=$$ $$($$$$-1334098118026473 \nu^{15} + 1517569107740666 \nu^{14} - 4205273217640779 \nu^{13} + 4401649942608156 \nu^{12} - 52349415876667248 \nu^{11} - 98327207412452760 \nu^{10} - 256970306079422968 \nu^{9} - 232606049029434234 \nu^{8} - 493881573755262372 \nu^{7} - 190246844863133334 \nu^{6} - 349853038170026583 \nu^{5} + 14783651457386572 \nu^{4} + 112813715492085963 \nu^{3} - 563917677730686336 \nu^{2} + 5784501870373928 \nu + 274909709246406$$$$)/ 124366022234909444$$ $$\beta_{6}$$ $$=$$ $$($$$$4950341416425938 \nu^{15} + 8565337944679831 \nu^{14} - 13364241040070848 \nu^{13} + 53906232696629134 \nu^{12} + 80585581972894742 \nu^{11} + 1030149467982352343 \nu^{10} + 1391133956348731186 \nu^{9} + 3191521592275583908 \nu^{8} + 1992929875410893800 \nu^{7} + 5482691994217503452 \nu^{6} - 1446990931651661606 \nu^{5} + 6219461329136756829 \nu^{4} - 6598860258546821788 \nu^{3} + 7194317225999645966 \nu^{2} + 1483100099170413112 \nu - 55763223898352207$$$$)/ 248732044469818888$$ $$\beta_{7}$$ $$=$$ $$($$$$-7689574809486539 \nu^{15} + 14417915972893703 \nu^{14} - 40719039631172776 \nu^{13} + 64673902756106082 \nu^{12} - 373412049536210877 \nu^{11} - 261321357649369131 \nu^{10} - 1538010628569371690 \nu^{9} - 519517240851217752 \nu^{8} - 3509882114878012604 \nu^{7} + 791986271093937708 \nu^{6} - 5007977547533482581 \nu^{5} + 3295679700257036373 \nu^{4} - 4507955070755752244 \nu^{3} + 484965482191497092 \nu^{2} - 1796302457038402649 \nu - 376917382847082123$$$$)/ 248732044469818888$$ $$\beta_{8}$$ $$=$$ $$($$$$-7996357615436185 \nu^{15} + 7113165089580568 \nu^{14} - 19707199958346938 \nu^{13} + 14633853633104868 \nu^{12} - 292468610266020577 \nu^{11} - 692989384122531926 \nu^{10} - 1533967646866839956 \nu^{9} - 1641929205864583504 \nu^{8} - 2679309963838852172 \nu^{7} - 1751713015381200100 \nu^{6} - 1355347926840362115 \nu^{5} - 1186243882333216876 \nu^{4} + 1857024347168333810 \nu^{3} - 5185130631661255590 \nu^{2} + 90224891838242869 \nu - 244926581437456$$$$)/ 248732044469818888$$ $$\beta_{9}$$ $$=$$ $$($$$$-9923394814730370 \nu^{15} + 19483671586929623 \nu^{14} - 51095562919719476 \nu^{13} + 79967884092907126 \nu^{12} - 470124075549151302 \nu^{11} - 327483559332676297 \nu^{10} - 1794502293098529654 \nu^{9} - 488100125702895684 \nu^{8} - 4005504188033716152 \nu^{7} + 1194357667186697532 \nu^{6} - 5361647578042256546 \nu^{5} + 3784296383430534165 \nu^{4} - 4192774164604126624 \nu^{3} - 864449632553452018 \nu^{2} + 31677413386224412 \nu - 624001392880083455$$$$)/ 248732044469818888$$ $$\beta_{10}$$ $$=$$ $$($$$$-5497038814695493 \nu^{15} + 11007843731223050 \nu^{14} - 27534589269448528 \nu^{13} + 43719731554509720 \nu^{12} - 258034906429298419 \nu^{11} - 176315398520139426 \nu^{10} - 938989218743731712 \nu^{9} - 156009803880583268 \nu^{8} - 2005341392896135220 \nu^{7} + 892265473943206304 \nu^{6} - 2634535670125058247 \nu^{5} + 2277211903121095982 \nu^{4} - 2129436406478004480 \nu^{3} - 437058555324799526 \nu^{2} - 104622610840257705 \nu - 22324765494603340$$$$)/ 124366022234909444$$ $$\beta_{11}$$ $$=$$ $$($$$$-14237456455843353 \nu^{15} + 8144045718544856 \nu^{14} - 30137086783330880 \nu^{13} + 12727897036830450 \nu^{12} - 507211526823956967 \nu^{11} - 1409219186344899196 \nu^{10} - 3074760077481916412 \nu^{9} - 3779964285440740540 \nu^{8} - 5546470109830386496 \nu^{7} - 4703855168870146888 \nu^{6} - 3133211800080577119 \nu^{5} - 3257452744531745816 \nu^{4} + 3137595733462195724 \nu^{3} - 8606178666465003832 \nu^{2} - 1801341363730993457 \nu + 18253448300139254$$$$)/ 248732044469818888$$ $$\beta_{12}$$ $$=$$ $$($$$$-14347599686866674 \nu^{15} + 9723085162322699 \nu^{14} - 34539788766771066 \nu^{13} + 20685781237821498 \nu^{12} - 524154385298967080 \nu^{11} - 1349027752041544833 \nu^{10} - 3089476577515634734 \nu^{9} - 3672993166187436608 \nu^{8} - 5787816772121294744 \nu^{7} - 4443385422641245576 \nu^{6} - 3686314678933949826 \nu^{5} - 2781796675359935187 \nu^{4} + 2651503125175466950 \nu^{3} - 8008865683061113730 \nu^{2} - 1454572150144297626 \nu - 383695910164439023$$$$)/ 248732044469818888$$ $$\beta_{13}$$ $$=$$ $$($$$$18531899779261006 \nu^{15} - 33938549156435524 \nu^{14} + 84524592027254251 \nu^{13} - 129058061625899722 \nu^{12} + 837482018154361001 \nu^{11} + 752962612570292994 \nu^{10} + 3185365486979204256 \nu^{9} + 942372174626117730 \nu^{8} + 6447269279178828544 \nu^{7} - 2125661015087147782 \nu^{6} + 7619595794373178122 \nu^{5} - 6140010126413766874 \nu^{4} + 5195494791680033913 \nu^{3} + 3253089239364610664 \nu^{2} + 241366565868854323 \nu - 11257512017065858$$$$)/ 124366022234909444$$ $$\beta_{14}$$ $$=$$ $$($$$$-22324765494603340 \nu^{15} + 50146569803902173 \nu^{14} - 122631671204239750 \nu^{13} + 206132713226275248 \nu^{12} - 1092983709800866700 \nu^{11} - 456357589398008461 \nu^{10} - 3641219501057031714 \nu^{9} + 358545315884044872 \nu^{8} - 7880905774176619132 \nu^{7} + 5845201057967909700 \nu^{6} - 11407230021901379444 \nu^{5} + 12234184832804494447 \nu^{4} - 10827597087554175202 \nu^{3} + 566702821855770680 \nu^{2} + 57537541916542746 \nu + 15323548861844345$$$$)/ 124366022234909444$$ $$\beta_{15}$$ $$=$$ $$($$$$91442776575316422 \nu^{15} - 186572448315471337 \nu^{14} + 459651261004779450 \nu^{13} - 740928671285460332 \nu^{12} + 4303795152282388204 \nu^{11} + 2790485160591147685 \nu^{10} + 15289009782018226266 \nu^{9} + 1556631325320829296 \nu^{8} + 31927969909310058828 \nu^{7} - 17289921029964539848 \nu^{6} + 41897475129387418778 \nu^{5} - 40367508360363927983 \nu^{4} + 34255519869345816022 \nu^{3} + 7009973887022542852 \nu^{2} - 356067839358896986 \nu - 49589969014520525$$$$)/ 248732044469818888$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{12} + \beta_{11} + \beta_{9} - 2 \beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{15} + 2 \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} - 5 \beta_{10} + \beta_{9} + 5 \beta_{5} + \beta_{4} + 4 \beta_{3} + \beta_{2} - 4 \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$6 \beta_{15} + 11 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + \beta_{11} - 10 \beta_{10} + 7 \beta_{8} + 2 \beta_{6} + 3 \beta_{5} + 6 \beta_{4} + 9 \beta_{2} - 3 \beta_{1} - 9$$ $$\nu^{5}$$ $$=$$ $$3 \beta_{15} + 6 \beta_{14} - \beta_{13} + 21 \beta_{12} - 14 \beta_{10} - 12 \beta_{9} + 5 \beta_{8} - \beta_{7} + 21 \beta_{6} - 22 \beta_{3} + \beta_{2} - 14 \beta_{1} - 24$$ $$\nu^{6}$$ $$=$$ $$83 \beta_{12} - 15 \beta_{11} - 59 \beta_{9} + 25 \beta_{8} - 20 \beta_{7} + 79 \beta_{6} - 43 \beta_{5} - 44 \beta_{4} - 43 \beta_{3} - 35 \beta_{2} - 51 \beta_{1} - 9$$ $$\nu^{7}$$ $$=$$ $$-47 \beta_{15} - 90 \beta_{14} + 19 \beta_{13} + 225 \beta_{12} - 114 \beta_{11} + 151 \beta_{10} - 161 \beta_{9} + 73 \beta_{8} - 64 \beta_{7} + 161 \beta_{6} - 301 \beta_{5} - 161 \beta_{4} - 151 \beta_{3} - 90 \beta_{2} + 10$$ $$\nu^{8}$$ $$=$$ $$-365 \beta_{15} - 666 \beta_{14} + 178 \beta_{13} + 365 \beta_{12} - 365 \beta_{11} + 867 \beta_{10} - 365 \beta_{9} - 66 \beta_{7} + 187 \beta_{6} - 867 \beta_{5} - 535 \beta_{4} - 401 \beta_{3} - 433 \beta_{2} + 401 \beta_{1} + 219$$ $$\nu^{9}$$ $$=$$ $$-1045 \beta_{15} - 1912 \beta_{14} + 543 \beta_{13} - 543 \beta_{12} - 532 \beta_{11} + 2677 \beta_{10} - 607 \beta_{8} - 779 \beta_{6} - 1500 \beta_{5} - 1045 \beta_{4} - 1150 \beta_{2} + 1500 \beta_{1} + 1369$$ $$\nu^{10}$$ $$=$$ $$-1736 \beta_{15} - 3236 \beta_{14} + 768 \beta_{13} - 5576 \beta_{12} + 4618 \beta_{10} + 3220 \beta_{9} - 2468 \beta_{8} + 768 \beta_{7} - 5576 \beta_{6} + 3372 \beta_{3} - 768 \beta_{2} + 4618 \beta_{1} + 4461$$ $$\nu^{11}$$ $$=$$ $$-22276 \beta_{12} + 5386 \beta_{11} + 14964 \beta_{9} - 7500 \beta_{8} + 4808 \beta_{7} - 19772 \beta_{6} + 14398 \beta_{5} + 9578 \beta_{4} + 14398 \beta_{3} + 5260 \beta_{2} + 10021 \beta_{1} + 6934$$ $$\nu^{12}$$ $$=$$ $$16902 \beta_{15} + 31300 \beta_{14} - 7890 \beta_{13} - 60515 \beta_{12} + 29227 \beta_{11} - 44174 \beta_{10} + 46129 \beta_{9} - 15850 \beta_{8} + 14386 \beta_{7} - 46129 \beta_{6} + 73841 \beta_{5} + 46129 \beta_{4} + 44174 \beta_{3} + 31300 \beta_{2} - 1955$$ $$\nu^{13}$$ $$=$$ $$88227 \beta_{15} + 162068 \beta_{14} - 43613 \beta_{13} - 88227 \beta_{12} + 88227 \beta_{11} - 225119 \beta_{10} + 88227 \beta_{9} + 24792 \beta_{7} - 44614 \beta_{6} + 225119 \beta_{5} + 140291 \beta_{4} + 89130 \beta_{3} + 115059 \beta_{2} - 89130 \beta_{1} - 70543$$ $$\nu^{14}$$ $$=$$ $$268732 \beta_{15} + 493851 \beta_{14} - 131840 \beta_{13} + 131840 \beta_{12} + 160781 \beta_{11} - 684110 \beta_{10} + 142097 \beta_{8} + 208696 \beta_{6} + 416131 \beta_{5} + 268732 \beta_{4} + 273937 \beta_{2} - 416131 \beta_{1} - 362011$$ $$\nu^{15}$$ $$=$$ $$492987 \beta_{15} + 909118 \beta_{14} - 237637 \beta_{13} + 1454159 \beta_{12} - 1274254 \beta_{10} - 815950 \beta_{9} + 671481 \beta_{8} - 237637 \beta_{7} + 1454159 \beta_{6} - 810648 \beta_{3} + 237637 \beta_{2} - 1274254 \beta_{1} - 1094186$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/495\mathbb{Z}\right)^\times$$.

 $$n$$ $$46$$ $$56$$ $$397$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
91.1
 0.431051 − 1.32664i 0.0698401 − 0.214946i −0.458960 + 1.41253i −0.659965 + 2.03116i 0.431051 + 1.32664i 0.0698401 + 0.214946i −0.458960 − 1.41253i −0.659965 − 2.03116i 2.46673 − 1.79218i 0.735494 − 0.534368i −0.166559 + 0.121012i −1.41763 + 1.02997i 2.46673 + 1.79218i 0.735494 + 0.534368i −0.166559 − 0.121012i −1.41763 − 1.02997i
−1.93752 + 1.40769i 0 1.15436 3.55277i −0.809017 0.587785i 0 0.608715 1.87343i 1.28446 + 3.95317i 0 2.39491
91.2 −0.991861 + 0.720629i 0 −0.153553 + 0.472586i −0.809017 0.587785i 0 −0.139581 + 0.429587i −0.945971 2.91140i 0 1.22601
91.3 0.392557 0.285209i 0 −0.545277 + 1.67819i −0.809017 0.587785i 0 −0.500445 + 1.54021i 0.564470 + 1.73726i 0 −0.485227
91.4 0.918793 0.667542i 0 −0.219466 + 0.675446i −0.809017 0.587785i 0 1.26738 3.90059i 0.951141 + 2.92731i 0 −1.13569
136.1 −1.93752 1.40769i 0 1.15436 + 3.55277i −0.809017 + 0.587785i 0 0.608715 + 1.87343i 1.28446 3.95317i 0 2.39491
136.2 −0.991861 0.720629i 0 −0.153553 0.472586i −0.809017 + 0.587785i 0 −0.139581 0.429587i −0.945971 + 2.91140i 0 1.22601
136.3 0.392557 + 0.285209i 0 −0.545277 1.67819i −0.809017 + 0.587785i 0 −0.500445 1.54021i 0.564470 1.73726i 0 −0.485227
136.4 0.918793 + 0.667542i 0 −0.219466 0.675446i −0.809017 + 0.587785i 0 1.26738 + 3.90059i 0.951141 2.92731i 0 −1.13569
181.1 −0.633189 1.94876i 0 −1.77869 + 1.29229i 0.309017 0.951057i 0 0.477268 0.346756i 0.329192 + 0.239172i 0 −2.04904
181.2 0.0280832 + 0.0864312i 0 1.61135 1.17072i 0.309017 0.951057i 0 1.98801 1.44438i 0.293484 + 0.213228i 0 0.0908791
181.3 0.372637 + 1.14686i 0 0.441609 0.320848i 0.309017 0.951057i 0 −3.49122 + 2.53652i 2.48368 + 1.80450i 0 1.20588
181.4 0.850504 + 2.61758i 0 −4.51034 + 3.27695i 0.309017 0.951057i 0 −2.21013 + 1.60575i −7.96046 5.78361i 0 2.75229
361.1 −0.633189 + 1.94876i 0 −1.77869 1.29229i 0.309017 + 0.951057i 0 0.477268 + 0.346756i 0.329192 0.239172i 0 −2.04904
361.2 0.0280832 0.0864312i 0 1.61135 + 1.17072i 0.309017 + 0.951057i 0 1.98801 + 1.44438i 0.293484 0.213228i 0 0.0908791
361.3 0.372637 1.14686i 0 0.441609 + 0.320848i 0.309017 + 0.951057i 0 −3.49122 2.53652i 2.48368 1.80450i 0 1.20588
361.4 0.850504 2.61758i 0 −4.51034 3.27695i 0.309017 + 0.951057i 0 −2.21013 1.60575i −7.96046 + 5.78361i 0 2.75229
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.n.g 16
3.b odd 2 1 495.2.n.h yes 16
11.c even 5 1 inner 495.2.n.g 16
11.c even 5 1 5445.2.a.cd 8
11.d odd 10 1 5445.2.a.cb 8
33.f even 10 1 5445.2.a.cc 8
33.h odd 10 1 495.2.n.h yes 16
33.h odd 10 1 5445.2.a.ca 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.n.g 16 1.a even 1 1 trivial
495.2.n.g 16 11.c even 5 1 inner
495.2.n.h yes 16 3.b odd 2 1
495.2.n.h yes 16 33.h odd 10 1
5445.2.a.ca 8 33.h odd 10 1
5445.2.a.cb 8 11.d odd 10 1
5445.2.a.cc 8 33.f even 10 1
5445.2.a.cd 8 11.c even 5 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(495, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 10 T + 147 T^{2} - 416 T^{3} + 507 T^{4} - 44 T^{5} + 111 T^{6} - 18 T^{7} + 252 T^{8} - 84 T^{9} + 113 T^{10} + 46 T^{11} + 61 T^{12} + 22 T^{13} + 10 T^{14} + 2 T^{15} + T^{16}$$
$3$ $$T^{16}$$
$5$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$
$7$ $$10201 - 11716 T + 42770 T^{2} - 93918 T^{3} + 141613 T^{4} - 12940 T^{5} + 67089 T^{6} + 236 T^{7} + 11278 T^{8} + 2470 T^{9} + 1023 T^{10} + 56 T^{11} + 339 T^{12} + 68 T^{13} + 17 T^{14} + 4 T^{15} + T^{16}$$
$11$ $$214358881 - 77948684 T + 8857805 T^{2} + 4831530 T^{3} - 3221020 T^{4} + 822558 T^{5} - 69817 T^{6} - 67540 T^{7} + 39185 T^{8} - 6140 T^{9} - 577 T^{10} + 618 T^{11} - 220 T^{12} + 30 T^{13} + 5 T^{14} - 4 T^{15} + T^{16}$$
$13$ $$1771561 - 4187326 T + 19282197 T^{2} - 9306352 T^{3} + 8028471 T^{4} - 698808 T^{5} + 516503 T^{6} - 162496 T^{7} + 138174 T^{8} + 23204 T^{9} + 18817 T^{10} - 2222 T^{11} + 285 T^{12} + 72 T^{13} + 22 T^{14} - 2 T^{15} + T^{16}$$
$17$ $$17901361 - 19885700 T + 25132150 T^{2} - 29222956 T^{3} + 43480689 T^{4} + 11075740 T^{5} + 6538046 T^{6} + 1341152 T^{7} + 627811 T^{8} + 52584 T^{9} + 33542 T^{10} + 52 T^{11} + 1325 T^{12} + 104 T^{13} + 26 T^{14} + 4 T^{15} + T^{16}$$
$19$ $$35153041 + 55697026 T + 30042243 T^{2} - 29607732 T^{3} + 109056655 T^{4} - 80540488 T^{5} + 34790545 T^{6} - 6114572 T^{7} + 2963354 T^{8} - 412384 T^{9} + 64751 T^{10} + 184 T^{11} + 2941 T^{12} + 170 T^{13} + 82 T^{14} + 4 T^{15} + T^{16}$$
$23$ $$( -1 + 144 T + 2139 T^{2} + 3042 T^{3} + 1309 T^{4} + 46 T^{5} - 65 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$29$ $$3575881 - 7582910 T + 34737783 T^{2} - 5121864 T^{3} - 7295667 T^{4} + 3409876 T^{5} + 13217257 T^{6} + 3724640 T^{7} + 2812424 T^{8} + 729336 T^{9} + 271317 T^{10} + 61422 T^{11} + 12793 T^{12} + 2332 T^{13} + 324 T^{14} + 26 T^{15} + T^{16}$$
$31$ $$532732561 - 232148698 T + 861928051 T^{2} - 55894584 T^{3} + 415883995 T^{4} - 75969308 T^{5} + 56721005 T^{6} - 20139130 T^{7} + 5119880 T^{8} - 66838 T^{9} - 8463 T^{10} + 8328 T^{11} + 2595 T^{12} - 212 T^{13} + 23 T^{14} + 10 T^{15} + T^{16}$$
$37$ $$15099740161 - 32785142324 T + 34844222737 T^{2} - 23499419304 T^{3} + 11331012975 T^{4} - 4198160720 T^{5} + 1263079891 T^{6} - 319523350 T^{7} + 69603096 T^{8} - 13170076 T^{9} + 2173163 T^{10} - 309060 T^{11} + 38079 T^{12} - 3976 T^{13} + 345 T^{14} - 22 T^{15} + T^{16}$$
$41$ $$7009547478025 + 3408541733650 T + 1268271619940 T^{2} + 390587874660 T^{3} + 112367424641 T^{4} + 22539995470 T^{5} + 4196786057 T^{6} + 650757888 T^{7} + 93030264 T^{8} + 8643852 T^{9} + 1035637 T^{10} + 81556 T^{11} + 14605 T^{12} + 1328 T^{13} + 131 T^{14} + 6 T^{15} + T^{16}$$
$43$ $$( -2417279 + 230960 T + 588171 T^{2} - 163146 T^{3} + 1462 T^{4} + 3012 T^{5} - 172 T^{6} - 14 T^{7} + T^{8} )^{2}$$
$47$ $$346921 - 8402674 T + 482524928 T^{2} - 696940520 T^{3} + 609417097 T^{4} - 304759058 T^{5} + 108986459 T^{6} - 11871812 T^{7} + 3204788 T^{8} + 575426 T^{9} + 206219 T^{10} - 6396 T^{11} + 1885 T^{12} + 204 T^{13} + 173 T^{14} + 20 T^{15} + T^{16}$$
$53$ $$154209363025 + 113127575600 T + 480485102535 T^{2} - 188678579310 T^{3} + 90024582606 T^{4} - 37894574890 T^{5} + 11993414307 T^{6} - 2211877732 T^{7} + 304597269 T^{8} - 28175348 T^{9} + 2317017 T^{10} - 198694 T^{11} + 21190 T^{12} - 1312 T^{13} + 131 T^{14} - 14 T^{15} + T^{16}$$
$59$ $$7826763961 + 31640583974 T + 50875341559 T^{2} + 7206884342 T^{3} + 23281535617 T^{4} + 6721433890 T^{5} + 1232074643 T^{6} + 123881672 T^{7} + 24221454 T^{8} + 5330994 T^{9} + 1339291 T^{10} + 211154 T^{11} + 28999 T^{12} + 2886 T^{13} + 283 T^{14} + 16 T^{15} + T^{16}$$
$61$ $$213905325001 + 373785216814 T + 511455808625 T^{2} + 379483580754 T^{3} + 174006451512 T^{4} + 52111822156 T^{5} + 10949075039 T^{6} + 1647981972 T^{7} + 197830815 T^{8} + 23969764 T^{9} + 4658079 T^{10} + 842900 T^{11} + 128108 T^{12} + 12650 T^{13} + 873 T^{14} + 38 T^{15} + T^{16}$$
$67$ $$( -597971 + 641474 T - 125028 T^{2} - 43982 T^{3} + 9702 T^{4} + 1238 T^{5} - 173 T^{6} - 10 T^{7} + T^{8} )^{2}$$
$71$ $$81898970538025 + 88060394525200 T + 89903665448335 T^{2} + 41442238296660 T^{3} + 12688050289111 T^{4} + 2722370918340 T^{5} + 435531417337 T^{6} + 51688617562 T^{7} + 4732563154 T^{8} + 340827008 T^{9} + 25001127 T^{10} + 2411564 T^{11} + 265165 T^{12} + 23292 T^{13} + 1441 T^{14} + 54 T^{15} + T^{16}$$
$73$ $$9273553653001 + 9080676590414 T + 6195758698607 T^{2} + 2689813879378 T^{3} + 852991217576 T^{4} + 198752348572 T^{5} + 35382249093 T^{6} + 4878227924 T^{7} + 587193759 T^{8} + 62108674 T^{9} + 5389797 T^{10} + 310088 T^{11} + 22340 T^{12} + 1382 T^{13} + 47 T^{14} - 2 T^{15} + T^{16}$$
$79$ $$32898311847025 - 45865106176100 T + 29793809632440 T^{2} - 10670996133940 T^{3} + 3023950506036 T^{4} - 832370672176 T^{5} + 191922480702 T^{6} - 9326792896 T^{7} + 2458571126 T^{8} - 73620152 T^{9} + 14857516 T^{10} - 148356 T^{11} + 56213 T^{12} + 948 T^{13} + 302 T^{14} + 12 T^{15} + T^{16}$$
$83$ $$1719577524548521 + 980184080862360 T + 412575498858237 T^{2} + 103588090069398 T^{3} + 19454525419563 T^{4} + 2345915231342 T^{5} + 226771390063 T^{6} + 15394976940 T^{7} + 2991568514 T^{8} + 297421532 T^{9} + 27540273 T^{10} + 2560466 T^{11} + 231273 T^{12} + 12254 T^{13} + 611 T^{14} + 28 T^{15} + T^{16}$$
$89$ $$( -5696725 + 6461770 T - 2873896 T^{2} + 629456 T^{3} - 64783 T^{4} + 888 T^{5} + 438 T^{6} - 38 T^{7} + T^{8} )^{2}$$
$97$ $$57744177883681 + 14222697997776 T + 27638886070115 T^{2} + 19008150901302 T^{3} + 7016936587964 T^{4} + 1612371435030 T^{5} + 262417855317 T^{6} + 31190419902 T^{7} + 2908626391 T^{8} + 223335780 T^{9} + 16959189 T^{10} + 952230 T^{11} + 53280 T^{12} + 3732 T^{13} + 403 T^{14} + 18 T^{15} + T^{16}$$