# Properties

 Label 99.2.g.b Level $99$ Weight $2$ Character orbit 99.g Analytic conductor $0.791$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 99.g (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.790518980011$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 15x^{14} + 150x^{12} + 837x^{10} + 3372x^{8} + 8010x^{6} + 13761x^{4} + 13392x^{2} + 8649$$ x^16 + 15*x^14 + 150*x^12 + 837*x^10 + 3372*x^8 + 8010*x^6 + 13761*x^4 + 13392*x^2 + 8649 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{14} + \beta_{13} + \beta_{12} + \beta_{9} - \beta_{7} + \beta_{6}) q^{3} + ( - \beta_{12} + \beta_{9} + \beta_{7} - \beta_{2} - 1) q^{4} + (\beta_{7} - \beta_{6}) q^{5} + (\beta_{15} + \beta_{11} - \beta_{10} - \beta_{8} - \beta_{3}) q^{6} + (\beta_{10} - \beta_{4}) q^{7} + ( - \beta_{10} + \beta_{5} + \beta_{4} + \beta_{3}) q^{8} + ( - 2 \beta_{13} - \beta_{9} + 2 \beta_{2} + 1) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b14 + b13 + b12 + b9 - b7 + b6) * q^3 + (-b12 + b9 + b7 - b2 - 1) * q^4 + (b7 - b6) * q^5 + (b15 + b11 - b10 - b8 - b3) * q^6 + (b10 - b4) * q^7 + (-b10 + b5 + b4 + b3) * q^8 + (-2*b13 - b9 + 2*b2 + 1) * q^9 $$q + \beta_1 q^{2} + ( - \beta_{14} + \beta_{13} + \beta_{12} + \beta_{9} - \beta_{7} + \beta_{6}) q^{3} + ( - \beta_{12} + \beta_{9} + \beta_{7} - \beta_{2} - 1) q^{4} + (\beta_{7} - \beta_{6}) q^{5} + (\beta_{15} + \beta_{11} - \beta_{10} - \beta_{8} - \beta_{3}) q^{6} + (\beta_{10} - \beta_{4}) q^{7} + ( - \beta_{10} + \beta_{5} + \beta_{4} + \beta_{3}) q^{8} + ( - 2 \beta_{13} - \beta_{9} + 2 \beta_{2} + 1) q^{9} + ( - \beta_{11} + \beta_{10} - \beta_1) q^{10} + (\beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4}) q^{11} + (3 \beta_{14} + \beta_{13} - 2 \beta_{9} - 2 \beta_{7} - \beta_{6} - 2 \beta_{2} - 1) q^{12} + ( - \beta_{10} - \beta_{8} - 2 \beta_{5} - \beta_{4}) q^{13} + ( - 4 \beta_{13} - 3 \beta_{12} - 3 \beta_{9} + \beta_{7} - 2 \beta_{6} + 2 \beta_{2} + \cdots - 2) q^{14}+ \cdots + ( - \beta_{15} - \beta_{14} + \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + \cdots - 1) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b14 + b13 + b12 + b9 - b7 + b6) * q^3 + (-b12 + b9 + b7 - b2 - 1) * q^4 + (b7 - b6) * q^5 + (b15 + b11 - b10 - b8 - b3) * q^6 + (b10 - b4) * q^7 + (-b10 + b5 + b4 + b3) * q^8 + (-2*b13 - b9 + 2*b2 + 1) * q^9 + (-b11 + b10 - b1) * q^10 + (b10 + b9 + b8 + b7 + b5 + b4) * q^11 + (3*b14 + b13 - 2*b9 - 2*b7 - b6 - 2*b2 - 1) * q^12 + (-b10 - b8 - 2*b5 - b4) * q^13 + (-4*b13 - 3*b12 - 3*b9 + b7 - 2*b6 + 2*b2 - 2) * q^14 + (-b12 - b9 - b6 - 3) * q^15 + (b14 + 3*b13 + 2*b12 - b7 + b6 - 2*b2) * q^16 + (-2*b15 - b11 + b10 + b8 - b4 + b3 - b1) * q^17 + (-b10 - 2*b5 - b3 + b1) * q^18 + (b11 + b5 + b4 - b3 - b1) * q^19 + (-2*b14 + 2*b12 + b9 - b7 + 2*b6 + 2*b2 + 4) * q^20 + (b11 - b10 + b8 + b5 + 2*b4 + b3 + b1) * q^21 + (-2*b14 - 2*b13 + b12 - b10 + b9 - b8 + 2*b7 + b6 + b3 + 3*b2 - b1 + 3) * q^22 + (2*b13 - b7 + 3*b6 - b2 + 1) * q^23 + (-b15 - b11 + 2*b10 + b8 - b5 - 3*b4 - b3 - b1) * q^24 + (b14 + 2*b13 + b12 + b9 - 2*b7 + b6) * q^25 + (b14 + 2*b13 - b12 + 2*b9 - 2*b7 - 2*b2 - 1) * q^26 + (-2*b14 + b13 + 2*b12 + 2*b9 + 2*b7 + b6 + 3) * q^27 + (-2*b11 + b10 + 2*b8 - b5 + b4 + 2*b3 + 2*b1) * q^28 + (-b10 - b8 - b5 - b4) * q^29 + (-b11 + 2*b10 + 2*b8 + b5 + b4 - 2*b1) * q^30 + (b13 + b12 - b9 - 2*b7 - 2*b2 - 2) * q^31 + (-b15 + b11 + b5 - b4 - b3 - b1) * q^32 + (-b15 + b14 - b11 - 2*b9 + b8 - b7 - b6 + b5 - 2*b2 - 2*b1 - 1) * q^33 + (-4*b14 - 4*b13 + b9 + 3*b7 - 4*b6 + 3*b2) * q^34 + (2*b15 + b11 - b8 - b5 - b3 + b1) * q^35 + (-b14 - 2*b12 + 3*b9 + b7 + 2*b6 - 4*b2 - 2) * q^36 + (-b14 - b13 - b12 - 3*b9 - 3*b7 + 2*b6 - 4) * q^37 + (3*b14 - 3*b13 - 3*b9 - 3*b6 + 2*b2 + 4) * q^38 + (b15 + b11 - 2*b8 + b4 + 2*b1) * q^39 + (-b10 - b8 - b3 + b1) * q^40 + (b15 - b11 + b10 - b8 - b5) * q^41 + (2*b14 + 3*b13 + b12 + 3*b9 + 4*b7 - b6 + 2*b2 + 1) * q^42 + (b15 - b10 + b8 + b5 + b4 + b3 + b1) * q^43 + (2*b15 + 2*b14 + 2*b13 + 2*b12 + b11 - 2*b10 - 2*b9 - 2*b8 - 2*b7 - b5 + b4 - b3 + 2*b2 + b1 + 1) * q^44 + (3*b14 - 2*b13 - 3*b12 - b9 + 3*b7 - 3*b6 + 2*b2 - 2) * q^45 + (3*b11 - b10 - 2*b8 + 2*b5 - b3 + b1) * q^46 + (2*b14 + b13 - 3*b12 - b9 + b7 - 2*b6 - 2*b2 - 4) * q^47 + (-2*b14 - 3*b13 - 4*b12 - b9) * q^48 + (4*b14 - 3*b13 - 2*b12 - 2*b9 + 3*b7 - 2*b6 + 2*b2 + 2) * q^49 + (-b15 + b11 + b5 - 2*b4 - 2*b3 - 2*b1) * q^50 + (b10 + 2*b5 + 4*b3 + 2*b1) * q^51 + (-b15 + b8 + b5 + 3*b3 + b1) * q^52 + (-b14 - 2*b13 - b12 + 2*b7 - 4*b2 - 2) * q^53 + (2*b15 + b11 - 2*b10 - 5*b8 - b5 - b3) * q^54 + (b12 + 2*b11 - b10 + b9 + b5 + b4 + 2*b1 + 2) * q^55 + (-7*b14 + 4*b13 + 3*b12 + 6*b9 - b7 + 7*b6 - 3*b2 - 6) * q^56 + (-b15 - 2*b11 + b8 + b3 + b1) * q^57 + (2*b14 + 2*b13 - b12 - b9 - 2*b7 - b6 - 3*b2 - 3) * q^58 + (2*b13 + 4*b12 + 4*b9 - b7 - b6) * q^59 + (-5*b14 + 4*b12 + 2*b9 - b7 + 3*b6 + 4*b2 + 5) * q^60 + (2*b15 + b10 - b4 - 6*b3 - 2*b1) * q^61 + (2*b10 + b8 - b4 - b3) * q^62 + (-2*b15 - b11 - b8 - 3*b4 - b3 - b1) * q^63 + (3*b14 - b13 - 4*b12 - 2*b9 + 5*b7 - 6*b6 + 1) * q^64 + (-b15 - 2*b11 + b10 + b8 - b5 - b4 + b3 - 3*b1) * q^65 + (-b15 - 3*b14 + b13 + 3*b12 - b11 + 3*b10 - b9 + 2*b8 - b4 + 2*b2 + b1 + 4) * q^66 + (-2*b14 + 3*b13 - b12 + 3*b9 - b7 + b6 + b2 + 1) * q^67 + (2*b15 - 2*b11 - b10 + b8 - 2*b5 + 4*b4 + 2*b3 + 2*b1) * q^68 + (-b14 + 2*b9 - 2*b7 + b6 + 2*b2 + 7) * q^69 + (3*b14 - 3*b12 - b9 - 2*b7 + 3*b6) * q^70 + (-b14 - b13 + b12 - b9 + b7) * q^71 + (b15 + 2*b11 - 3*b10 - b8 + 3*b4 + 5*b3 + 2*b1) * q^72 + (-b11 + 3*b8 - b5 + 2*b4 - b3 - 3*b1) * q^73 + (b15 + 2*b11 + 2*b8 + 2*b4 - b3 + 2*b1) * q^74 + (-3*b13 - 3*b2) * q^75 + (-b15 - b11 + 3*b10 + 3*b8 - b5 - b4 - 2*b3 + 2*b1) * q^76 + (b15 + b13 + 3*b12 - b11 + 3*b9 - b7 - b6 - b5 + 2*b3 - 4*b2 + 2*b1 + 4) * q^77 + (4*b14 + 2*b12 - 4*b7 + b6 - 5*b2 - 4) * q^78 + (-3*b15 + 2*b8 + 2*b5 + b3 - b1) * q^79 + (-2*b14 + 2*b12 - 2*b2 - 1) * q^80 + (-3*b14 - 3*b9 - 3*b7 + 3*b2) * q^81 + (-4*b14 + 2*b13 + 5*b12 + 3*b9 - 6*b7 + 8*b6 + 3) * q^82 + (-3*b10 - 3*b8 - 3*b5 - 3*b4 - 2*b1) * q^83 + (b11 + b10 - 2*b8 + 2*b5 - b4 - 3*b3 - 7*b1) * q^84 + (b15 + b11 - 4*b10 - 4*b8 - 3*b5 - b4 - 2*b3 + 2*b1) * q^85 + (7*b13 + b12 + b9 + 2*b7 + 4*b6 + b2 - 1) * q^86 + (b15 + b11 - b8 - b5 + 2*b1) * q^87 + (-2*b15 + 3*b14 + b13 - 2*b12 + 2*b10 - 3*b9 + 3*b6 - 2*b4 - 4*b3 - 3*b1) * q^88 + (-2*b14 - 4*b13 - 2*b9 + 4*b7 + 8*b2 + 4) * q^89 + (-3*b15 - 3*b11 + 2*b10 + 3*b8 + b5 + 2*b3 - 5*b1) * q^90 + (b14 - b13 - 3*b12 - 3*b9 + b7 - 2*b6 - 3) * q^91 + (8*b14 - 3*b13 - 5*b12 - 9*b9 - b7 - 8*b6 - 3*b2 - 6) * q^92 + (-2*b13 - 3*b12 + 2*b7 + b6 + 3*b2) * q^93 + (-2*b15 - 2*b11 + 4*b10 + 4*b8 + 4*b5 + b4 + 3*b3 - 3*b1) * q^94 + (-b15 + b11 - b10 + b8 + b5 + 2*b4) * q^95 + (-2*b10 - b5 + 4*b3 + 5*b1) * q^96 + (-2*b14 + 2*b13 + 4*b12 + 4*b9 - 2*b7 - 2*b6 - b2) * q^97 + (-4*b15 - 2*b11 + b10 + b8 - b5 - 2*b4 + b3 - 2*b1) * q^98 + (-b15 - b14 + b13 - 2*b12 - 2*b11 + 2*b10 + 2*b9 + b8 + b7 + 2*b6 - 2*b5 - 3*b4 + 3*b3 + b2 - b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 6 q^{3} - 14 q^{4} + 6 q^{9}+O(q^{10})$$ 16 * q - 6 * q^3 - 14 * q^4 + 6 * q^9 $$16 q - 6 q^{3} - 14 q^{4} + 6 q^{9} - 12 q^{11} + 12 q^{12} - 6 q^{14} - 30 q^{15} - 2 q^{16} + 36 q^{20} + 6 q^{22} + 12 q^{23} - 12 q^{25} + 18 q^{27} - 4 q^{31} + 18 q^{33} - 18 q^{36} - 28 q^{37} + 66 q^{38} - 54 q^{42} - 42 q^{45} - 30 q^{47} + 42 q^{48} + 10 q^{49} + 20 q^{55} - 120 q^{56} - 6 q^{58} - 36 q^{59} + 30 q^{60} + 40 q^{64} + 54 q^{66} + 8 q^{67} + 96 q^{69} + 24 q^{75} + 72 q^{77} - 42 q^{78} + 30 q^{81} + 12 q^{82} - 72 q^{86} - 6 q^{88} - 12 q^{91} + 18 q^{92} - 24 q^{93} - 4 q^{97} - 36 q^{99}+O(q^{100})$$ 16 * q - 6 * q^3 - 14 * q^4 + 6 * q^9 - 12 * q^11 + 12 * q^12 - 6 * q^14 - 30 * q^15 - 2 * q^16 + 36 * q^20 + 6 * q^22 + 12 * q^23 - 12 * q^25 + 18 * q^27 - 4 * q^31 + 18 * q^33 - 18 * q^36 - 28 * q^37 + 66 * q^38 - 54 * q^42 - 42 * q^45 - 30 * q^47 + 42 * q^48 + 10 * q^49 + 20 * q^55 - 120 * q^56 - 6 * q^58 - 36 * q^59 + 30 * q^60 + 40 * q^64 + 54 * q^66 + 8 * q^67 + 96 * q^69 + 24 * q^75 + 72 * q^77 - 42 * q^78 + 30 * q^81 + 12 * q^82 - 72 * q^86 - 6 * q^88 - 12 * q^91 + 18 * q^92 - 24 * q^93 - 4 * q^97 - 36 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 15x^{14} + 150x^{12} + 837x^{10} + 3372x^{8} + 8010x^{6} + 13761x^{4} + 13392x^{2} + 8649$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 2126395 \nu^{14} - 29523650 \nu^{12} - 288594750 \nu^{10} - 1489331264 \nu^{8} - 5823700650 \nu^{6} - 11995771800 \nu^{4} + \cdots - 22396351233 ) / 13131109983$$ (-2126395*v^14 - 29523650*v^12 - 288594750*v^10 - 1489331264*v^8 - 5823700650*v^6 - 11995771800*v^4 - 23155939764*v^2 - 22396351233) / 13131109983 $$\beta_{3}$$ $$=$$ $$( 2126395 \nu^{15} + 29523650 \nu^{13} + 288594750 \nu^{11} + 1489331264 \nu^{9} + 5823700650 \nu^{7} + 11995771800 \nu^{5} + \cdots + 9265241250 \nu ) / 13131109983$$ (2126395*v^15 + 29523650*v^13 + 288594750*v^11 + 1489331264*v^9 + 5823700650*v^7 + 11995771800*v^5 + 23155939764*v^3 + 9265241250*v) / 13131109983 $$\beta_{4}$$ $$=$$ $$( - 4873266 \nu^{15} - 160857665 \nu^{13} - 1890026425 \nu^{11} - 14824095809 \nu^{9} - 66244476882 \nu^{7} - 206255555451 \nu^{5} + \cdots - 290195282991 \nu ) / 13131109983$$ (-4873266*v^15 - 160857665*v^13 - 1890026425*v^11 - 14824095809*v^9 - 66244476882*v^7 - 206255555451*v^5 - 292920249753*v^3 - 290195282991*v) / 13131109983 $$\beta_{5}$$ $$=$$ $$( 47500907 \nu^{15} + 812018893 \nu^{13} + 8261503794 \nu^{11} + 49665627558 \nu^{9} + 195642483900 \nu^{7} + 477061297716 \nu^{5} + \cdots + 551339033751 \nu ) / 39393329949$$ (47500907*v^15 + 812018893*v^13 + 8261503794*v^11 + 49665627558*v^9 + 195642483900*v^7 + 477061297716*v^5 + 607209996627*v^3 + 551339033751*v) / 39393329949 $$\beta_{6}$$ $$=$$ $$( - 49032332 \nu^{14} - 864786042 \nu^{12} - 8872595664 \nu^{10} - 54770706318 \nu^{8} - 220344752319 \nu^{6} - 565719472953 \nu^{4} + \cdots - 671327583345 ) / 39393329949$$ (-49032332*v^14 - 864786042*v^12 - 8872595664*v^10 - 54770706318*v^8 - 220344752319*v^6 - 565719472953*v^4 - 758542460580*v^2 - 671327583345) / 39393329949 $$\beta_{7}$$ $$=$$ $$( - 54617732 \nu^{14} - 903112393 \nu^{12} - 9132887847 \nu^{10} - 53705137428 \nu^{8} - 210752440350 \nu^{6} - 495377443209 \nu^{4} + \cdots - 448939285020 ) / 39393329949$$ (-54617732*v^14 - 903112393*v^12 - 9132887847*v^10 - 53705137428*v^8 - 210752440350*v^6 - 495377443209*v^4 - 625450988448*v^2 - 448939285020) / 39393329949 $$\beta_{8}$$ $$=$$ $$( 54617732 \nu^{15} + 903112393 \nu^{13} + 9132887847 \nu^{11} + 53705137428 \nu^{9} + 210752440350 \nu^{7} + 495377443209 \nu^{5} + \cdots + 409545955071 \nu ) / 39393329949$$ (54617732*v^15 + 903112393*v^13 + 9132887847*v^11 + 53705137428*v^9 + 210752440350*v^7 + 495377443209*v^5 + 625450988448*v^3 + 409545955071*v) / 39393329949 $$\beta_{9}$$ $$=$$ $$( - 58397849 \nu^{14} - 683729698 \nu^{12} - 6054561519 \nu^{10} - 23065315299 \nu^{8} - 66793461054 \nu^{6} - 2243892963 \nu^{4} + \cdots + 208063920222 ) / 39393329949$$ (-58397849*v^14 - 683729698*v^12 - 6054561519*v^10 - 23065315299*v^8 - 66793461054*v^6 - 2243892963*v^4 + 33072805413*v^2 + 208063920222) / 39393329949 $$\beta_{10}$$ $$=$$ $$( 64777034 \nu^{15} + 772300648 \nu^{13} + 6920345769 \nu^{11} + 27533309091 \nu^{9} + 84264563004 \nu^{7} + 38231208363 \nu^{5} + \cdots - 180268196472 \nu ) / 39393329949$$ (64777034*v^15 + 772300648*v^13 + 6920345769*v^11 + 27533309091*v^9 + 84264563004*v^7 + 38231208363*v^5 + 36395013879*v^3 - 180268196472*v) / 39393329949 $$\beta_{11}$$ $$=$$ $$( 70362434 \nu^{15} + 810626999 \nu^{13} + 7180637952 \nu^{11} + 26467740201 \nu^{9} + 74672251035 \nu^{7} - 32110821381 \nu^{5} + \cdots - 442049824746 \nu ) / 39393329949$$ (70362434*v^15 + 810626999*v^13 + 7180637952*v^11 + 26467740201*v^9 + 74672251035*v^7 - 32110821381*v^5 - 96696458253*v^3 - 442049824746*v) / 39393329949 $$\beta_{12}$$ $$=$$ $$( - 93878026 \nu^{14} - 1321129241 \nu^{12} - 12590096616 \nu^{10} - 63366471351 \nu^{8} - 225132595554 \nu^{6} - 389659389972 \nu^{4} + \cdots - 157488193548 ) / 39393329949$$ (-93878026*v^14 - 1321129241*v^12 - 12590096616*v^10 - 63366471351*v^8 - 225132595554*v^6 - 389659389972*v^4 - 423368055108*v^2 - 157488193548) / 39393329949 $$\beta_{13}$$ $$=$$ $$( 105898756 \nu^{14} + 1495748591 \nu^{12} + 14316065313 \nu^{10} + 72730942857 \nu^{8} + 262435944954 \nu^{6} + 479305190679 \nu^{4} + \cdots + 303881783580 ) / 39393329949$$ (105898756*v^14 + 1495748591*v^12 + 14316065313*v^10 + 72730942857*v^8 + 262435944954*v^6 + 479305190679*v^4 + 574137191214*v^2 + 303881783580) / 39393329949 $$\beta_{14}$$ $$=$$ $$( - 131099776 \nu^{14} - 1868178342 \nu^{12} - 17994696072 \nu^{10} - 92965220511 \nu^{8} - 341617204785 \nu^{6} + \cdots - 446373561123 ) / 39393329949$$ (-131099776*v^14 - 1868178342*v^12 - 17994696072*v^10 - 92965220511*v^8 - 341617204785*v^6 - 654888451842*v^4 - 813584976426*v^2 - 446373561123) / 39393329949 $$\beta_{15}$$ $$=$$ $$( 145719574 \nu^{15} + 2350751337 \nu^{13} + 23664775347 \nu^{11} + 137437507938 \nu^{9} + 540350635431 \nu^{7} + \cdots + 1277566080147 \nu ) / 39393329949$$ (145719574*v^15 + 2350751337*v^13 + 23664775347*v^11 + 137437507938*v^9 + 540350635431*v^7 + 1273655118195*v^5 + 1692345725685*v^3 + 1277566080147*v) / 39393329949
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{12} + \beta_{9} + \beta_{7} - 3\beta_{2} - 3$$ -b12 + b9 + b7 - 3*b2 - 3 $$\nu^{3}$$ $$=$$ $$-\beta_{10} + \beta_{5} + \beta_{4} + 5\beta_{3}$$ -b10 + b5 + b4 + 5*b3 $$\nu^{4}$$ $$=$$ $$\beta_{14} + 9\beta_{13} + 8\beta_{12} - \beta_{7} + \beta_{6} + 12\beta_{2}$$ b14 + 9*b13 + 8*b12 - b7 + b6 + 12*b2 $$\nu^{5}$$ $$=$$ $$-\beta_{15} + \beta_{11} + 8\beta_{10} - 8\beta_{8} + \beta_{5} - 9\beta_{4} - 29\beta_{3} - 29\beta_1$$ -b15 + b11 + 8*b10 - 8*b8 + b5 - 9*b4 - 29*b3 - 29*b1 $$\nu^{6}$$ $$=$$ $$13\beta_{14} - 67\beta_{13} - 14\beta_{12} - 68\beta_{9} - 41\beta_{7} - 26\beta_{6} + 57$$ 13*b14 - 67*b13 - 14*b12 - 68*b9 - 41*b7 - 26*b6 + 57 $$\nu^{7}$$ $$=$$ $$-13\beta_{15} - 26\beta_{11} + 27\beta_{10} + 81\beta_{8} - 53\beta_{5} + \beta_{4} + 13\beta_{3} + 166\beta_1$$ -13*b15 - 26*b11 + 27*b10 + 81*b8 - 53*b5 + b4 + 13*b3 + 166*b1 $$\nu^{8}$$ $$=$$ $$-240\beta_{14} - 15\beta_{13} - 234\beta_{12} + 474\beta_{9} + 369\beta_{7} + 120\beta_{6} - 294\beta_{2} - 294$$ -240*b14 - 15*b13 - 234*b12 + 474*b9 + 369*b7 + 120*b6 - 294*b2 - 294 $$\nu^{9}$$ $$=$$ $$240 \beta_{15} + 120 \beta_{11} - 609 \beta_{10} - 255 \beta_{8} + 219 \beta_{5} + 474 \beta_{4} + 1017 \beta_{3} + 120 \beta_1$$ 240*b15 + 120*b11 - 609*b10 - 255*b8 + 219*b5 + 474*b4 + 1017*b3 + 120*b1 $$\nu^{10}$$ $$=$$ $$969\beta_{14} + 3438\beta_{13} + 2469\beta_{12} + 150\beta_{9} - 1119\beta_{7} + 969\beta_{6} + 1584\beta_{2}$$ 969*b14 + 3438*b13 + 2469*b12 + 150*b9 - 1119*b7 + 969*b6 + 1584*b2 $$\nu^{11}$$ $$=$$ $$- 969 \beta_{15} + 969 \beta_{11} + 2319 \beta_{10} - 2319 \beta_{8} + 969 \beta_{5} - 3438 \beta_{4} - 7341 \beta_{3} - 7341 \beta_1$$ -969*b15 + 969*b11 + 2319*b10 - 2319*b8 + 969*b5 - 3438*b4 - 7341*b3 - 7341*b1 $$\nu^{12}$$ $$=$$ $$7314\beta_{14} - 22581\beta_{13} - 8583\beta_{12} - 23850\beta_{9} - 7953\beta_{7} - 14628\beta_{6} + 8802$$ 7314*b14 - 22581*b13 - 8583*b12 - 23850*b9 - 7953*b7 - 14628*b6 + 8802 $$\nu^{13}$$ $$=$$ $$- 7314 \beta_{15} - 14628 \beta_{11} + 15897 \beta_{10} + 31164 \beta_{8} - 13998 \beta_{5} + 1269 \beta_{4} + 7314 \beta_{3} + 40605 \beta_1$$ -7314*b15 - 14628*b11 + 15897*b10 + 31164*b8 - 13998*b5 + 1269*b4 + 7314*b3 + 40605*b1 $$\nu^{14}$$ $$=$$ $$- 106212 \beta_{14} - 9852 \beta_{13} - 47928 \beta_{12} + 154140 \beta_{9} + 110886 \beta_{7} + 53106 \beta_{6} - 50265 \beta_{2} - 50265$$ -106212*b14 - 9852*b13 - 47928*b12 + 154140*b9 + 110886*b7 + 53106*b6 - 50265*b2 - 50265 $$\nu^{15}$$ $$=$$ $$106212 \beta_{15} + 53106 \beta_{11} - 217098 \beta_{10} - 116064 \beta_{8} + 38076 \beta_{5} + 154140 \beta_{4} + 262185 \beta_{3} + 53106 \beta_1$$ 106212*b15 + 53106*b11 - 217098*b10 - 116064*b8 + 38076*b5 + 154140*b4 + 262185*b3 + 53106*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/99\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1
 −1.29716 − 2.24675i −1.10617 − 1.91594i −0.679041 − 1.17613i −0.618600 − 1.07145i 0.618600 + 1.07145i 0.679041 + 1.17613i 1.10617 + 1.91594i 1.29716 + 2.24675i −1.29716 + 2.24675i −1.10617 + 1.91594i −0.679041 + 1.17613i −0.618600 + 1.07145i 0.618600 − 1.07145i 0.679041 − 1.17613i 1.10617 − 1.91594i 1.29716 − 2.24675i
−1.29716 2.24675i −1.72785 + 0.120512i −2.36526 + 4.09675i −0.137404 0.0793301i 2.51207 + 3.72573i −2.91814 + 1.68479i 7.08384 2.97095 0.416454i 0.411616i
32.2 −1.10617 1.91594i 1.59999 0.663339i −1.44722 + 2.50665i −2.54721 1.47063i −3.04078 2.33173i 1.72189 0.994132i 1.97879 2.11996 2.12268i 6.50707i
32.3 −0.679041 1.17613i −0.323333 1.70160i 0.0778064 0.134765i 0.901139 + 0.520273i −1.78176 + 1.53574i 0.600962 0.346965i −2.92750 −2.79091 + 1.10037i 1.41315i
32.4 −0.618600 1.07145i −1.04881 + 1.37841i 0.234668 0.406456i 1.78348 + 1.02969i 2.12568 + 0.271060i 3.59283 2.07432i −3.05506 −0.800005 2.89137i 2.54787i
32.5 0.618600 + 1.07145i −1.04881 + 1.37841i 0.234668 0.406456i 1.78348 + 1.02969i −2.12568 0.271060i −3.59283 + 2.07432i 3.05506 −0.800005 2.89137i 2.54787i
32.6 0.679041 + 1.17613i −0.323333 1.70160i 0.0778064 0.134765i 0.901139 + 0.520273i 1.78176 1.53574i −0.600962 + 0.346965i 2.92750 −2.79091 + 1.10037i 1.41315i
32.7 1.10617 + 1.91594i 1.59999 0.663339i −1.44722 + 2.50665i −2.54721 1.47063i 3.04078 + 2.33173i −1.72189 + 0.994132i −1.97879 2.11996 2.12268i 6.50707i
32.8 1.29716 + 2.24675i −1.72785 + 0.120512i −2.36526 + 4.09675i −0.137404 0.0793301i −2.51207 3.72573i 2.91814 1.68479i −7.08384 2.97095 0.416454i 0.411616i
65.1 −1.29716 + 2.24675i −1.72785 0.120512i −2.36526 4.09675i −0.137404 + 0.0793301i 2.51207 3.72573i −2.91814 1.68479i 7.08384 2.97095 + 0.416454i 0.411616i
65.2 −1.10617 + 1.91594i 1.59999 + 0.663339i −1.44722 2.50665i −2.54721 + 1.47063i −3.04078 + 2.33173i 1.72189 + 0.994132i 1.97879 2.11996 + 2.12268i 6.50707i
65.3 −0.679041 + 1.17613i −0.323333 + 1.70160i 0.0778064 + 0.134765i 0.901139 0.520273i −1.78176 1.53574i 0.600962 + 0.346965i −2.92750 −2.79091 1.10037i 1.41315i
65.4 −0.618600 + 1.07145i −1.04881 1.37841i 0.234668 + 0.406456i 1.78348 1.02969i 2.12568 0.271060i 3.59283 + 2.07432i −3.05506 −0.800005 + 2.89137i 2.54787i
65.5 0.618600 1.07145i −1.04881 1.37841i 0.234668 + 0.406456i 1.78348 1.02969i −2.12568 + 0.271060i −3.59283 2.07432i 3.05506 −0.800005 + 2.89137i 2.54787i
65.6 0.679041 1.17613i −0.323333 + 1.70160i 0.0778064 + 0.134765i 0.901139 0.520273i 1.78176 + 1.53574i −0.600962 0.346965i 2.92750 −2.79091 1.10037i 1.41315i
65.7 1.10617 1.91594i 1.59999 + 0.663339i −1.44722 2.50665i −2.54721 + 1.47063i 3.04078 2.33173i −1.72189 0.994132i −1.97879 2.11996 + 2.12268i 6.50707i
65.8 1.29716 2.24675i −1.72785 0.120512i −2.36526 4.09675i −0.137404 + 0.0793301i −2.51207 + 3.72573i 2.91814 + 1.68479i −7.08384 2.97095 + 0.416454i 0.411616i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 65.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
11.b odd 2 1 inner
99.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.2.g.b 16
3.b odd 2 1 297.2.g.b 16
9.c even 3 1 297.2.g.b 16
9.c even 3 1 891.2.d.b 16
9.d odd 6 1 inner 99.2.g.b 16
9.d odd 6 1 891.2.d.b 16
11.b odd 2 1 inner 99.2.g.b 16
33.d even 2 1 297.2.g.b 16
99.g even 6 1 inner 99.2.g.b 16
99.g even 6 1 891.2.d.b 16
99.h odd 6 1 297.2.g.b 16
99.h odd 6 1 891.2.d.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.g.b 16 1.a even 1 1 trivial
99.2.g.b 16 9.d odd 6 1 inner
99.2.g.b 16 11.b odd 2 1 inner
99.2.g.b 16 99.g even 6 1 inner
297.2.g.b 16 3.b odd 2 1
297.2.g.b 16 9.c even 3 1
297.2.g.b 16 33.d even 2 1
297.2.g.b 16 99.h odd 6 1
891.2.d.b 16 9.c even 3 1
891.2.d.b 16 9.d odd 6 1
891.2.d.b 16 99.g even 6 1
891.2.d.b 16 99.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} + 15T_{2}^{14} + 150T_{2}^{12} + 837T_{2}^{10} + 3372T_{2}^{8} + 8010T_{2}^{6} + 13761T_{2}^{4} + 13392T_{2}^{2} + 8649$$ acting on $$S_{2}^{\mathrm{new}}(99, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 15 T^{14} + 150 T^{12} + \cdots + 8649$$
$3$ $$(T^{8} + 3 T^{7} + 3 T^{6} - 3 T^{5} - 15 T^{4} + \cdots + 81)^{2}$$
$5$ $$(T^{8} - 7 T^{6} + 48 T^{4} - 63 T^{3} + \cdots + 1)^{2}$$
$7$ $$T^{16} - 33 T^{14} + 765 T^{12} + \cdots + 138384$$
$11$ $$T^{16} + 12 T^{15} + \cdots + 214358881$$
$13$ $$T^{16} - 51 T^{14} + 2046 T^{12} + \cdots + 138384$$
$17$ $$(T^{8} - 93 T^{6} + 3078 T^{4} + \cdots + 214272)^{2}$$
$19$ $$(T^{8} + 63 T^{6} + 1152 T^{4} + \cdots + 3348)^{2}$$
$23$ $$(T^{8} - 6 T^{7} - 25 T^{6} + 222 T^{5} + \cdots + 119716)^{2}$$
$29$ $$T^{16} + 30 T^{14} + 585 T^{12} + \cdots + 2214144$$
$31$ $$(T^{8} + 2 T^{7} + 19 T^{6} - 16 T^{5} + \cdots + 1)^{2}$$
$37$ $$(T^{4} + 7 T^{3} - 51 T^{2} + 52 T + 31)^{4}$$
$41$ $$T^{16} + 132 T^{14} + \cdots + 18034341264$$
$43$ $$T^{16} - 183 T^{14} + \cdots + 54056250000$$
$47$ $$(T^{8} + 15 T^{7} + 50 T^{6} - 375 T^{5} + \cdots + 6889)^{2}$$
$53$ $$(T^{8} + 146 T^{6} + 5619 T^{4} + \cdots + 105625)^{2}$$
$59$ $$(T^{8} + 18 T^{7} + 71 T^{6} + \cdots + 265225)^{2}$$
$61$ $$T^{16} - 477 T^{14} + \cdots + 16\!\cdots\!44$$
$67$ $$(T^{8} - 4 T^{7} + 103 T^{6} + 794 T^{5} + \cdots + 18769)^{2}$$
$71$ $$(T^{8} + 53 T^{6} + 735 T^{4} + 1382 T^{2} + \cdots + 361)^{2}$$
$73$ $$(T^{8} + 408 T^{6} + 58167 T^{4} + \cdots + 64686708)^{2}$$
$79$ $$T^{16} - 279 T^{14} + \cdots + 9069133824$$
$83$ $$T^{16} + \cdots + 162096184890000$$
$89$ $$(T^{8} + 308 T^{6} + 23664 T^{4} + \cdots + 1893376)^{2}$$
$97$ $$(T^{8} + 2 T^{7} + 124 T^{6} + \cdots + 564001)^{2}$$