# Properties

 Label 495.2.n.h 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: $$x^{16} - 2 x^{15} + 5 x^{14} - 8 x^{13} + 47 x^{12} + 32 x^{11} + 171 x^{10} + 26 x^{9} + 360 x^{8} - 172 x^{7} + 471 x^{6} - 430 x^{5} + 383 x^{4} + 70 x^{3} + 17 x^{2} + 4 x + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{14} + \beta_{10} - \beta_{5} - \beta_{3} + \beta_1 - 1) q^{2} + (\beta_{12} - \beta_{9} + \beta_{6} - \beta_{4} - \beta_{2} + \beta_1 + 1) q^{4} + \beta_{8} q^{5} + (\beta_{11} + \beta_{9} - \beta_{7} + \beta_{5} + \beta_{3} - \beta_{2}) q^{7} + ( - 2 \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} - \beta_{10} - 2 \beta_{8} + \beta_{6} + \beta_{5} + \cdots + 2) q^{8}+O(q^{10})$$ q + (b14 + b10 - b5 - b3 + b1 - 1) * q^2 + (b12 - b9 + b6 - b4 - b2 + b1 + 1) * q^4 + b8 * q^5 + (b11 + b9 - b7 + b5 + b3 - b2) * q^7 + (-2*b15 - b14 - b13 + b12 - b10 - 2*b8 + b6 + b5 - 2*b4 - b2 - b1 + 2) * q^8 $$q + (\beta_{14} + \beta_{10} - \beta_{5} - \beta_{3} + \beta_1 - 1) q^{2} + (\beta_{12} - \beta_{9} + \beta_{6} - \beta_{4} - \beta_{2} + \beta_1 + 1) q^{4} + \beta_{8} q^{5} + (\beta_{11} + \beta_{9} - \beta_{7} + \beta_{5} + \beta_{3} - \beta_{2}) q^{7} + ( - 2 \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} - \beta_{10} - 2 \beta_{8} + \beta_{6} + \beta_{5} + \cdots + 2) q^{8}+ \cdots + (8 \beta_{14} - 3 \beta_{13} + 2 \beta_{12} + 2 \beta_{9} + 5 \beta_{8} - 3 \beta_{7} + \cdots - 4) q^{98}+O(q^{100})$$ q + (b14 + b10 - b5 - b3 + b1 - 1) * q^2 + (b12 - b9 + b6 - b4 - b2 + b1 + 1) * q^4 + b8 * q^5 + (b11 + b9 - b7 + b5 + b3 - b2) * q^7 + (-2*b15 - b14 - b13 + b12 - b10 - 2*b8 + b6 + b5 - 2*b4 - b2 - b1 + 2) * q^8 - b3 * q^10 + (-b15 - b14 + b13 + b11 + b9 - b8 - b6 - b2 - 1) * q^11 + (b15 - b14 - b12 + b11 + b9 - 2*b7 - b6 + 2*b4 - 1) * q^13 + (-3*b14 - 2*b10 - 2*b8 - b6 - 2*b2 + 3) * q^14 + (b15 + 2*b14 - b12 + 2*b11 + b10 + 3*b9 - 2*b8 - 2*b7 - 3*b6 - 2*b5 + 3*b4 - b3 + 2*b2 - 4) * q^16 + (b13 - b12 + 2*b10 + 2*b8 + b7 - b5 - 2*b3 - 2) * q^17 + (-b15 - b14 + 2*b11 - b10 - b8 + 2*b5 - b4 - b2 - 2*b1 + 1) * q^19 + (b15 + b14 - b12 + b11 + b10 + b9 - b6 - b5 + b4 - b3 + b1 - 2) * q^20 + (b15 + 2*b11 - b8 - b7 - b6 - b5 + b4 + b3 - b1 - 1) * q^22 + (2*b14 - 2*b13 + b12 + b10 + 2*b9 - 2*b7 + b6 + 2*b2 + b1 - 2) * q^23 + (b14 + b8 + b2 - 1) * q^25 + (2*b12 - 3*b11 - 4*b9 + b8 + b7 + 3*b6 - b4 + b2 + 3) * q^26 + (b15 + b14 - b12 + 2*b11 + 3*b9 - 4*b8 - 2*b7 - 3*b6 - 2*b5 + 3*b4 + b2 - 3) * q^28 + (-b12 + 2*b9 - b8 - 2*b6 + 2*b4 + 2*b2 - 1) * q^29 + (-b15 - 2*b13 + b12 - b11 - b10 - b9 - 2*b7 + 3*b6 + b5 - b4 + b3 + 3*b2 - b1 + 1) * q^31 + (2*b15 + 4*b14 + b13 + b12 + b10 - 2*b9 + 5*b8 + b7 + b6 + 2*b3 - b2 + b1 - 2) * q^32 + (3*b15 + 5*b14 + b10 + 5*b8 - b3 + b1 - 2) * q^34 + (2*b14 - b13 + b10 + b6 - b5 - b4 + b2) * q^35 + (-b11 - b9 - 2*b8 + b6 - 2*b5 - 2*b3 + 3*b1 + 1) * q^37 + (4*b14 - 2*b13 - b12 + 2*b11 - b10 + 2*b9 - 2*b8 - b7 - 2*b6 - b5 + 2*b4 + b3 + 4*b2 - 1) * q^38 + (b12 - 2*b9 + b7 + b6 + b5 - 2*b4 + b3 - 2*b2 + 3) * q^40 + (b15 - 3*b14 + 2*b13 - 2*b12 - 2*b11 - 2*b10 - b8 - 2*b6 + 2*b5 + b4 - 3*b2 - 2*b1 + 1) * q^41 + (-2*b15 - b14 - 2*b13 + b12 - 2*b10 + 3*b9 - 3*b8 - 2*b7 + b6 + 4*b3 + 2*b2 - 2*b1 + 3) * q^43 + (-b15 - 2*b14 + 2*b12 - b11 - 4*b10 - 3*b9 + b8 - b7 + 3*b6 + 4*b5 - 2*b4 + 4*b3 - 2*b1 + 2) * q^44 + (2*b15 - 2*b14 + 2*b13 - 2*b12 + 2*b11 + 2*b9 - 4*b6 + 3*b4 - b3 - 4*b2 + b1 - 1) * q^46 + (2*b15 - 4*b14 + 3*b13 - 3*b12 + 2*b11 - 3*b6 + 2*b4 - 3*b2 + 1) * q^47 + (-3*b15 + 2*b14 + 2*b12 - b11 + 3*b10 - 4*b9 + b8 + 2*b7 + 4*b6 - b5 - 4*b4 - 3*b3 + 2*b2 + 1) * q^49 + (b8 + b5) * q^50 + (-2*b14 - b13 + b12 - 2*b11 - 2*b10 + b8 + 3*b5 + 2*b2 - 3*b1 + 3) * q^52 + (2*b15 - 2*b14 - 2*b13 - 2*b12 + 2*b11 - b10 + 2*b9 - b7 + b5 + 3*b3 + 2*b2 - 3*b1 + 1) * q^53 + (-b15 - b14 + b12 - b9 - b8 + b7 + 2*b6 - 2*b4 - b2 + 2) * q^55 + (b15 - 3*b14 + 3*b13 + b12 + 2*b10 - 6*b9 + 3*b7 + b6 - 3*b2 + 2*b1 + 1) * q^56 + (2*b15 + 3*b14 + 2*b13 - 2*b12 + b11 + 6*b10 + 4*b8 - 2*b6 - 2*b5 + 2*b4 + 2*b2 + 2*b1 - 5) * q^58 + (2*b12 - b11 - 2*b9 - b8 + 2*b6 + 2*b5 - b4 + 2*b3 + 2*b2 - 4*b1 + 5) * q^59 + (-7*b14 + 2*b13 - b12 - b11 - 2*b10 - b9 + 2*b7 + b6 + b5 - b4 + 2*b3 - 7*b2 + 3) * q^61 + (-b12 - 2*b11 + 2*b9 + b8 - b7 - b6 + 4*b4 + 6*b2 - 2*b1 - 3) * q^62 + (-b15 - 8*b14 + 2*b13 + b12 - b11 - 7*b10 - b9 + 2*b7 - b6 + 7*b5 - 3*b4 + b3 - 7*b2 - b1 + 7) * q^64 + (-b15 + 2*b14 - 2*b13 + b12 + b9 - 2*b7 + b6 + 2*b2 - 2) * q^65 + (-3*b15 + 2*b14 - b12 + 2*b10 + b9 + 2*b8 - b6 - 2*b3 + 2*b1 - 4) * q^67 + (-b14 - 2*b13 - 2*b10 + b7 + 2*b6 + 2*b5 - 4*b4 - 2*b3 - 6*b2 + 2*b1 + 6) * q^68 + (b12 + b8 - 2*b2 + 2*b1 - 1) * q^70 + (2*b15 + 3*b14 + 3*b13 - 2*b12 - b11 - 2*b10 + b9 + 2*b8 + b7 - b6 - 2*b5 + b4 + 2*b3 + 3*b2 + 1) * q^71 + (-b12 - 2*b11 + b8 + b7 - b6 + b5 + 2*b4 + b3 - 3*b2 + 2*b1) * q^73 + (-b15 + 2*b10 - 5*b8 + b6 - 2*b5 - b4 - 5*b2 + 2*b1) * q^74 + (b15 + b10 - 2*b9 + 5*b3 + b1 + 1) * q^76 + (-2*b15 + b14 + 3*b12 - 4*b11 - b10 - 6*b9 + 2*b7 + 4*b6 - b5 - 4*b4 - b3 + 4*b2 + 2*b1 + 1) * q^77 + (-2*b15 + 4*b14 - 2*b13 + 2*b12 - 2*b11 + 4*b10 - 2*b9 + 4*b7 + 4*b6 - 4*b5 - 2*b4 - 6*b3 + b2 + 6*b1 - 3) * q^79 + (-2*b15 - 2*b14 - 2*b13 + 2*b12 - b11 - 2*b10 - 2*b8 + 2*b6 + b5 - 2*b4 - b1 + 4) * q^80 + (4*b15 + 4*b14 + 2*b13 - 4*b12 - b11 + 3*b10 + 3*b9 + 2*b8 + b7 - 3*b6 - 5*b5 + 3*b4 - 3*b3 + 4*b2 - 6) * q^82 + (2*b15 + 3*b14 - 4*b13 - 2*b12 + 2*b11 + b10 + 4*b9 + 6*b8 - 2*b7 - 4*b6 - b5 + 4*b4 - b3 + 3*b2 - 5) * q^83 + (b13 - b12 - b10 + 2*b8 + 2*b5 + b2 - 2*b1 - 1) * q^85 + (3*b15 - 2*b14 + 3*b13 - 3*b12 + 3*b11 + 4*b10 + 3*b9 - 2*b7 - 6*b6 - 4*b5 + 9*b4 + 6*b2 - 12) * q^86 + (-9*b14 - b12 - 3*b11 - 6*b10 + b9 - 5*b8 + b7 - 2*b6 + b5 + b4 + b3 + 2*b2 - 5*b1 + 6) * q^88 + (b15 + 3*b14 + b13 - b12 + 2*b10 + 2*b9 + 4*b8 + b7 - b6 - 2*b3 - b2 + 2*b1 - 9) * q^89 + (-5*b15 + 7*b14 - 4*b13 + 4*b12 - 3*b11 + 3*b10 + b8 + 6*b6 + b5 - 5*b4 + 5*b2 - b1 - 3) * q^91 + (3*b12 - 3*b8 + b5 + b3 - 5*b2 + 2*b1 + 4) * q^92 + (-2*b15 + b14 - 2*b13 + 5*b12 - 5*b11 - 7*b9 + 2*b7 + 7*b6 - b5 - 7*b4 + b2 + 7) * q^94 + (-b12 + 2*b11 + b9 - b6 + 2*b5 - b4 + 2*b3 - b2 - b1 + 1) * q^95 + (b15 - b14 + b13 - b12 + b11 + 2*b10 + b9 + 3*b7 - 2*b6 - 2*b5 + 4*b4 + b3 + b2 - b1 - 2) * q^97 + (8*b14 - 3*b13 + 2*b12 + 2*b9 + 5*b8 - 3*b7 + 2*b6 + 3*b2 - 4) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 2 q^{2} - 8 q^{4} + 4 q^{5} - 4 q^{7} + 6 q^{8}+O(q^{10})$$ 16 * q + 2 * q^2 - 8 * q^4 + 4 * q^5 - 4 * q^7 + 6 * q^8 $$16 q + 2 q^{2} - 8 q^{4} + 4 q^{5} - 4 q^{7} + 6 q^{8} + 8 q^{10} - 4 q^{11} + 2 q^{13} + 22 q^{14} + 8 q^{16} + 4 q^{17} - 4 q^{19} - 2 q^{20} - 28 q^{22} - 8 q^{23} - 4 q^{25} - 6 q^{26} - 2 q^{28} + 26 q^{29} - 10 q^{31} - 56 q^{32} - 4 q^{34} + 4 q^{35} + 22 q^{37} + 30 q^{38} - 6 q^{40} + 6 q^{41} + 28 q^{43} - 68 q^{44} + 16 q^{46} + 20 q^{47} + 10 q^{49} + 2 q^{50} + 30 q^{52} - 14 q^{53} - 6 q^{55} - 68 q^{56} - 6 q^{58} + 16 q^{59} - 38 q^{61} + 20 q^{62} + 10 q^{64} - 12 q^{65} + 20 q^{67} + 48 q^{68} - 22 q^{70} + 54 q^{71} + 2 q^{73} - 28 q^{74} - 44 q^{76} - 34 q^{77} - 12 q^{79} + 22 q^{80} + 30 q^{82} + 28 q^{83} - 4 q^{85} - 74 q^{86} + 46 q^{88} - 76 q^{89} - 34 q^{91} + 8 q^{92} - 10 q^{94} + 4 q^{95} - 18 q^{97} - 8 q^{98}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 274909709246406 \nu^{15} - 784278699533661 \nu^{14} + 143020561508636 \nu^{13} + \cdots + 46\!\cdots\!04 ) / 12\!\cdots\!44$$ (-274909709246406*v^15 - 784278699533661*v^14 + 143020561508636*v^13 - 2005995543669531*v^12 - 8519106391972926*v^11 - 61146526572552240*v^10 - 145336767693588186*v^9 - 264117958519829524*v^8 - 331573544358140394*v^7 - 446597103764880540*v^6 - 319729317918190560*v^5 - 231641863194072003*v^4 - 90506767183986926*v^3 + 93570035844837543*v^2 - 568591142787875238*v + 4684863033388304) / 124366022234909444 $$\beta_{3}$$ $$=$$ $$( 553668747953762 \nu^{15} + \cdots - 30\!\cdots\!65 ) / 24\!\cdots\!88$$ (553668747953762*v^15 - 1294038871869219*v^14 + 2678375186769114*v^13 - 4724034435676990*v^12 + 25734959323311664*v^11 + 10614112411878989*v^10 + 69986082534625206*v^9 - 46186287420472496*v^8 + 104166886954478224*v^7 - 245911872417401744*v^6 + 55312393060713098*v^5 - 394772810552973845*v^4 + 140885354640541754*v^3 + 27555273511115606*v^2 - 3822684831751662*v - 301927752492954565) / 248732044469818888 $$\beta_{4}$$ $$=$$ $$( 14\!\cdots\!63 \nu^{15} + \cdots + 41\!\cdots\!35 ) / 24\!\cdots\!88$$ (1439074005313563*v^15 + 1851897765207855*v^14 + 2324766414796124*v^13 + 2360640185706480*v^12 + 53528410268329273*v^11 + 228500108198224671*v^10 + 616925067838708542*v^9 + 967946523561484984*v^8 + 1386230620986522104*v^7 + 1425871732223573700*v^6 + 1350563990196551169*v^5 + 508765947296826397*v^4 + 596242561123071756*v^3 - 337614021434331534*v^2 + 1967072679306622413*v + 413981182405604135) / 248732044469818888 $$\beta_{5}$$ $$=$$ $$( - 13\!\cdots\!73 \nu^{15} + \cdots + 274909709246406 ) / 12\!\cdots\!44$$ (-1334098118026473*v^15 + 1517569107740666*v^14 - 4205273217640779*v^13 + 4401649942608156*v^12 - 52349415876667248*v^11 - 98327207412452760*v^10 - 256970306079422968*v^9 - 232606049029434234*v^8 - 493881573755262372*v^7 - 190246844863133334*v^6 - 349853038170026583*v^5 + 14783651457386572*v^4 + 112813715492085963*v^3 - 563917677730686336*v^2 + 5784501870373928*v + 274909709246406) / 124366022234909444 $$\beta_{6}$$ $$=$$ $$( 49\!\cdots\!38 \nu^{15} + \cdots - 55\!\cdots\!07 ) / 24\!\cdots\!88$$ (4950341416425938*v^15 + 8565337944679831*v^14 - 13364241040070848*v^13 + 53906232696629134*v^12 + 80585581972894742*v^11 + 1030149467982352343*v^10 + 1391133956348731186*v^9 + 3191521592275583908*v^8 + 1992929875410893800*v^7 + 5482691994217503452*v^6 - 1446990931651661606*v^5 + 6219461329136756829*v^4 - 6598860258546821788*v^3 + 7194317225999645966*v^2 + 1483100099170413112*v - 55763223898352207) / 248732044469818888 $$\beta_{7}$$ $$=$$ $$( - 76\!\cdots\!39 \nu^{15} + \cdots - 37\!\cdots\!23 ) / 24\!\cdots\!88$$ (-7689574809486539*v^15 + 14417915972893703*v^14 - 40719039631172776*v^13 + 64673902756106082*v^12 - 373412049536210877*v^11 - 261321357649369131*v^10 - 1538010628569371690*v^9 - 519517240851217752*v^8 - 3509882114878012604*v^7 + 791986271093937708*v^6 - 5007977547533482581*v^5 + 3295679700257036373*v^4 - 4507955070755752244*v^3 + 484965482191497092*v^2 - 1796302457038402649*v - 376917382847082123) / 248732044469818888 $$\beta_{8}$$ $$=$$ $$( - 79\!\cdots\!85 \nu^{15} + \cdots - 244926581437456 ) / 24\!\cdots\!88$$ (-7996357615436185*v^15 + 7113165089580568*v^14 - 19707199958346938*v^13 + 14633853633104868*v^12 - 292468610266020577*v^11 - 692989384122531926*v^10 - 1533967646866839956*v^9 - 1641929205864583504*v^8 - 2679309963838852172*v^7 - 1751713015381200100*v^6 - 1355347926840362115*v^5 - 1186243882333216876*v^4 + 1857024347168333810*v^3 - 5185130631661255590*v^2 + 90224891838242869*v - 244926581437456) / 248732044469818888 $$\beta_{9}$$ $$=$$ $$( - 99\!\cdots\!70 \nu^{15} + \cdots - 62\!\cdots\!55 ) / 24\!\cdots\!88$$ (-9923394814730370*v^15 + 19483671586929623*v^14 - 51095562919719476*v^13 + 79967884092907126*v^12 - 470124075549151302*v^11 - 327483559332676297*v^10 - 1794502293098529654*v^9 - 488100125702895684*v^8 - 4005504188033716152*v^7 + 1194357667186697532*v^6 - 5361647578042256546*v^5 + 3784296383430534165*v^4 - 4192774164604126624*v^3 - 864449632553452018*v^2 + 31677413386224412*v - 624001392880083455) / 248732044469818888 $$\beta_{10}$$ $$=$$ $$( - 54\!\cdots\!93 \nu^{15} + \cdots - 22\!\cdots\!40 ) / 12\!\cdots\!44$$ (-5497038814695493*v^15 + 11007843731223050*v^14 - 27534589269448528*v^13 + 43719731554509720*v^12 - 258034906429298419*v^11 - 176315398520139426*v^10 - 938989218743731712*v^9 - 156009803880583268*v^8 - 2005341392896135220*v^7 + 892265473943206304*v^6 - 2634535670125058247*v^5 + 2277211903121095982*v^4 - 2129436406478004480*v^3 - 437058555324799526*v^2 - 104622610840257705*v - 22324765494603340) / 124366022234909444 $$\beta_{11}$$ $$=$$ $$( - 14\!\cdots\!53 \nu^{15} + \cdots + 18\!\cdots\!54 ) / 24\!\cdots\!88$$ (-14237456455843353*v^15 + 8144045718544856*v^14 - 30137086783330880*v^13 + 12727897036830450*v^12 - 507211526823956967*v^11 - 1409219186344899196*v^10 - 3074760077481916412*v^9 - 3779964285440740540*v^8 - 5546470109830386496*v^7 - 4703855168870146888*v^6 - 3133211800080577119*v^5 - 3257452744531745816*v^4 + 3137595733462195724*v^3 - 8606178666465003832*v^2 - 1801341363730993457*v + 18253448300139254) / 248732044469818888 $$\beta_{12}$$ $$=$$ $$( - 14\!\cdots\!74 \nu^{15} + \cdots - 38\!\cdots\!23 ) / 24\!\cdots\!88$$ (-14347599686866674*v^15 + 9723085162322699*v^14 - 34539788766771066*v^13 + 20685781237821498*v^12 - 524154385298967080*v^11 - 1349027752041544833*v^10 - 3089476577515634734*v^9 - 3672993166187436608*v^8 - 5787816772121294744*v^7 - 4443385422641245576*v^6 - 3686314678933949826*v^5 - 2781796675359935187*v^4 + 2651503125175466950*v^3 - 8008865683061113730*v^2 - 1454572150144297626*v - 383695910164439023) / 248732044469818888 $$\beta_{13}$$ $$=$$ $$( 18\!\cdots\!06 \nu^{15} + \cdots - 11\!\cdots\!58 ) / 12\!\cdots\!44$$ (18531899779261006*v^15 - 33938549156435524*v^14 + 84524592027254251*v^13 - 129058061625899722*v^12 + 837482018154361001*v^11 + 752962612570292994*v^10 + 3185365486979204256*v^9 + 942372174626117730*v^8 + 6447269279178828544*v^7 - 2125661015087147782*v^6 + 7619595794373178122*v^5 - 6140010126413766874*v^4 + 5195494791680033913*v^3 + 3253089239364610664*v^2 + 241366565868854323*v - 11257512017065858) / 124366022234909444 $$\beta_{14}$$ $$=$$ $$( - 22\!\cdots\!40 \nu^{15} + \cdots + 15\!\cdots\!45 ) / 12\!\cdots\!44$$ (-22324765494603340*v^15 + 50146569803902173*v^14 - 122631671204239750*v^13 + 206132713226275248*v^12 - 1092983709800866700*v^11 - 456357589398008461*v^10 - 3641219501057031714*v^9 + 358545315884044872*v^8 - 7880905774176619132*v^7 + 5845201057967909700*v^6 - 11407230021901379444*v^5 + 12234184832804494447*v^4 - 10827597087554175202*v^3 + 566702821855770680*v^2 + 57537541916542746*v + 15323548861844345) / 124366022234909444 $$\beta_{15}$$ $$=$$ $$( 91\!\cdots\!22 \nu^{15} + \cdots - 49\!\cdots\!25 ) / 24\!\cdots\!88$$ (91442776575316422*v^15 - 186572448315471337*v^14 + 459651261004779450*v^13 - 740928671285460332*v^12 + 4303795152282388204*v^11 + 2790485160591147685*v^10 + 15289009782018226266*v^9 + 1556631325320829296*v^8 + 31927969909310058828*v^7 - 17289921029964539848*v^6 + 41897475129387418778*v^5 - 40367508360363927983*v^4 + 34255519869345816022*v^3 + 7009973887022542852*v^2 - 356067839358896986*v - 49589969014520525) / 248732044469818888
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{12} + \beta_{11} + \beta_{9} - 2\beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} - 1$$ -b12 + b11 + b9 - 2*b8 - b6 + b5 + b4 - 1 $$\nu^{3}$$ $$=$$ $$\beta_{15} + 2 \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} - 5 \beta_{10} + \beta_{9} + 5 \beta_{5} + \beta_{4} + 4 \beta_{3} + \beta_{2} - 4 \beta _1 + 3$$ b15 + 2*b14 - b13 - b12 + b11 - 5*b10 + b9 + 5*b5 + b4 + 4*b3 + b2 - 4*b1 + 3 $$\nu^{4}$$ $$=$$ $$6 \beta_{15} + 11 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + \beta_{11} - 10 \beta_{10} + 7 \beta_{8} + 2 \beta_{6} + 3 \beta_{5} + 6 \beta_{4} + 9 \beta_{2} - 3 \beta _1 - 9$$ 6*b15 + 11*b14 - 2*b13 + 2*b12 + b11 - 10*b10 + 7*b8 + 2*b6 + 3*b5 + 6*b4 + 9*b2 - 3*b1 - 9 $$\nu^{5}$$ $$=$$ $$3 \beta_{15} + 6 \beta_{14} - \beta_{13} + 21 \beta_{12} - 14 \beta_{10} - 12 \beta_{9} + 5 \beta_{8} - \beta_{7} + 21 \beta_{6} - 22 \beta_{3} + \beta_{2} - 14 \beta _1 - 24$$ 3*b15 + 6*b14 - b13 + 21*b12 - 14*b10 - 12*b9 + 5*b8 - b7 + 21*b6 - 22*b3 + b2 - 14*b1 - 24 $$\nu^{6}$$ $$=$$ $$83 \beta_{12} - 15 \beta_{11} - 59 \beta_{9} + 25 \beta_{8} - 20 \beta_{7} + 79 \beta_{6} - 43 \beta_{5} - 44 \beta_{4} - 43 \beta_{3} - 35 \beta_{2} - 51 \beta _1 - 9$$ 83*b12 - 15*b11 - 59*b9 + 25*b8 - 20*b7 + 79*b6 - 43*b5 - 44*b4 - 43*b3 - 35*b2 - 51*b1 - 9 $$\nu^{7}$$ $$=$$ $$- 47 \beta_{15} - 90 \beta_{14} + 19 \beta_{13} + 225 \beta_{12} - 114 \beta_{11} + 151 \beta_{10} - 161 \beta_{9} + 73 \beta_{8} - 64 \beta_{7} + 161 \beta_{6} - 301 \beta_{5} - 161 \beta_{4} - 151 \beta_{3} - 90 \beta_{2} + 10$$ -47*b15 - 90*b14 + 19*b13 + 225*b12 - 114*b11 + 151*b10 - 161*b9 + 73*b8 - 64*b7 + 161*b6 - 301*b5 - 161*b4 - 151*b3 - 90*b2 + 10 $$\nu^{8}$$ $$=$$ $$- 365 \beta_{15} - 666 \beta_{14} + 178 \beta_{13} + 365 \beta_{12} - 365 \beta_{11} + 867 \beta_{10} - 365 \beta_{9} - 66 \beta_{7} + 187 \beta_{6} - 867 \beta_{5} - 535 \beta_{4} - 401 \beta_{3} - 433 \beta_{2} + 401 \beta _1 + 219$$ -365*b15 - 666*b14 + 178*b13 + 365*b12 - 365*b11 + 867*b10 - 365*b9 - 66*b7 + 187*b6 - 867*b5 - 535*b4 - 401*b3 - 433*b2 + 401*b1 + 219 $$\nu^{9}$$ $$=$$ $$- 1045 \beta_{15} - 1912 \beta_{14} + 543 \beta_{13} - 543 \beta_{12} - 532 \beta_{11} + 2677 \beta_{10} - 607 \beta_{8} - 779 \beta_{6} - 1500 \beta_{5} - 1045 \beta_{4} - 1150 \beta_{2} + 1500 \beta _1 + 1369$$ -1045*b15 - 1912*b14 + 543*b13 - 543*b12 - 532*b11 + 2677*b10 - 607*b8 - 779*b6 - 1500*b5 - 1045*b4 - 1150*b2 + 1500*b1 + 1369 $$\nu^{10}$$ $$=$$ $$- 1736 \beta_{15} - 3236 \beta_{14} + 768 \beta_{13} - 5576 \beta_{12} + 4618 \beta_{10} + 3220 \beta_{9} - 2468 \beta_{8} + 768 \beta_{7} - 5576 \beta_{6} + 3372 \beta_{3} - 768 \beta_{2} + 4618 \beta _1 + 4461$$ -1736*b15 - 3236*b14 + 768*b13 - 5576*b12 + 4618*b10 + 3220*b9 - 2468*b8 + 768*b7 - 5576*b6 + 3372*b3 - 768*b2 + 4618*b1 + 4461 $$\nu^{11}$$ $$=$$ $$- 22276 \beta_{12} + 5386 \beta_{11} + 14964 \beta_{9} - 7500 \beta_{8} + 4808 \beta_{7} - 19772 \beta_{6} + 14398 \beta_{5} + 9578 \beta_{4} + 14398 \beta_{3} + 5260 \beta_{2} + 10021 \beta _1 + 6934$$ -22276*b12 + 5386*b11 + 14964*b9 - 7500*b8 + 4808*b7 - 19772*b6 + 14398*b5 + 9578*b4 + 14398*b3 + 5260*b2 + 10021*b1 + 6934 $$\nu^{12}$$ $$=$$ $$16902 \beta_{15} + 31300 \beta_{14} - 7890 \beta_{13} - 60515 \beta_{12} + 29227 \beta_{11} - 44174 \beta_{10} + 46129 \beta_{9} - 15850 \beta_{8} + 14386 \beta_{7} - 46129 \beta_{6} + 73841 \beta_{5} + \cdots - 1955$$ 16902*b15 + 31300*b14 - 7890*b13 - 60515*b12 + 29227*b11 - 44174*b10 + 46129*b9 - 15850*b8 + 14386*b7 - 46129*b6 + 73841*b5 + 46129*b4 + 44174*b3 + 31300*b2 - 1955 $$\nu^{13}$$ $$=$$ $$88227 \beta_{15} + 162068 \beta_{14} - 43613 \beta_{13} - 88227 \beta_{12} + 88227 \beta_{11} - 225119 \beta_{10} + 88227 \beta_{9} + 24792 \beta_{7} - 44614 \beta_{6} + 225119 \beta_{5} + \cdots - 70543$$ 88227*b15 + 162068*b14 - 43613*b13 - 88227*b12 + 88227*b11 - 225119*b10 + 88227*b9 + 24792*b7 - 44614*b6 + 225119*b5 + 140291*b4 + 89130*b3 + 115059*b2 - 89130*b1 - 70543 $$\nu^{14}$$ $$=$$ $$268732 \beta_{15} + 493851 \beta_{14} - 131840 \beta_{13} + 131840 \beta_{12} + 160781 \beta_{11} - 684110 \beta_{10} + 142097 \beta_{8} + 208696 \beta_{6} + 416131 \beta_{5} + 268732 \beta_{4} + \cdots - 362011$$ 268732*b15 + 493851*b14 - 131840*b13 + 131840*b12 + 160781*b11 - 684110*b10 + 142097*b8 + 208696*b6 + 416131*b5 + 268732*b4 + 273937*b2 - 416131*b1 - 362011 $$\nu^{15}$$ $$=$$ $$492987 \beta_{15} + 909118 \beta_{14} - 237637 \beta_{13} + 1454159 \beta_{12} - 1274254 \beta_{10} - 815950 \beta_{9} + 671481 \beta_{8} - 237637 \beta_{7} + 1454159 \beta_{6} + \cdots - 1094186$$ 492987*b15 + 909118*b14 - 237637*b13 + 1454159*b12 - 1274254*b10 - 815950*b9 + 671481*b8 - 237637*b7 + 1454159*b6 - 810648*b3 + 237637*b2 - 1274254*b1 - 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
 −0.659965 + 2.03116i −0.458960 + 1.41253i 0.0698401 − 0.214946i 0.431051 − 1.32664i −0.659965 − 2.03116i −0.458960 − 1.41253i 0.0698401 + 0.214946i 0.431051 + 1.32664i −1.41763 + 1.02997i −0.166559 + 0.121012i 0.735494 − 0.534368i 2.46673 − 1.79218i −1.41763 − 1.02997i −0.166559 − 0.121012i 0.735494 + 0.534368i 2.46673 + 1.79218i
−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
91.2 −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
91.3 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
91.4 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
136.1 −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
136.2 −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
136.3 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
136.4 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
181.1 −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
181.2 −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
181.3 −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
181.4 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
361.1 −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
361.2 −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
361.3 −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
361.4 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
 $$n$$: 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.h yes 16
3.b odd 2 1 495.2.n.g 16
11.c even 5 1 inner 495.2.n.h yes 16
11.c even 5 1 5445.2.a.ca 8
11.d odd 10 1 5445.2.a.cc 8
33.f even 10 1 5445.2.a.cb 8
33.h odd 10 1 495.2.n.g 16
33.h odd 10 1 5445.2.a.cd 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} - 2 T_{2}^{15} + 10 T_{2}^{14} - 22 T_{2}^{13} + 61 T_{2}^{12} - 46 T_{2}^{11} + 113 T_{2}^{10} + 84 T_{2}^{9} + 252 T_{2}^{8} + 18 T_{2}^{7} + 111 T_{2}^{6} + 44 T_{2}^{5} + 507 T_{2}^{4} + 416 T_{2}^{3} + 147 T_{2}^{2} + 10 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(495, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 2 T^{15} + 10 T^{14} - 22 T^{13} + \cdots + 1$$
$3$ $$T^{16}$$
$5$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{4}$$
$7$ $$T^{16} + 4 T^{15} + 17 T^{14} + \cdots + 10201$$
$11$ $$T^{16} + 4 T^{15} + 5 T^{14} + \cdots + 214358881$$
$13$ $$T^{16} - 2 T^{15} + 22 T^{14} + \cdots + 1771561$$
$17$ $$T^{16} - 4 T^{15} + 26 T^{14} + \cdots + 17901361$$
$19$ $$T^{16} + 4 T^{15} + 82 T^{14} + \cdots + 35153041$$
$23$ $$(T^{8} + 4 T^{7} - 65 T^{6} - 46 T^{5} + \cdots - 1)^{2}$$
$29$ $$T^{16} - 26 T^{15} + 324 T^{14} + \cdots + 3575881$$
$31$ $$T^{16} + 10 T^{15} + \cdots + 532732561$$
$37$ $$T^{16} - 22 T^{15} + \cdots + 15099740161$$
$41$ $$T^{16} - 6 T^{15} + \cdots + 7009547478025$$
$43$ $$(T^{8} - 14 T^{7} - 172 T^{6} + \cdots - 2417279)^{2}$$
$47$ $$T^{16} - 20 T^{15} + 173 T^{14} + \cdots + 346921$$
$53$ $$T^{16} + 14 T^{15} + \cdots + 154209363025$$
$59$ $$T^{16} - 16 T^{15} + \cdots + 7826763961$$
$61$ $$T^{16} + 38 T^{15} + \cdots + 213905325001$$
$67$ $$(T^{8} - 10 T^{7} - 173 T^{6} + \cdots - 597971)^{2}$$
$71$ $$T^{16} - 54 T^{15} + \cdots + 81898970538025$$
$73$ $$T^{16} - 2 T^{15} + \cdots + 9273553653001$$
$79$ $$T^{16} + 12 T^{15} + \cdots + 32898311847025$$
$83$ $$T^{16} - 28 T^{15} + \cdots + 17\!\cdots\!21$$
$89$ $$(T^{8} + 38 T^{7} + 438 T^{6} + \cdots - 5696725)^{2}$$
$97$ $$T^{16} + 18 T^{15} + \cdots + 57744177883681$$