Properties

 Label 273.2.l.b Level $273$ Weight $2$ Character orbit 273.l Analytic conductor $2.180$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(16,273)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(273, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("273.16");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.l (of order $$3$$, degree $$2$$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 11 x^{14} - 4 x^{13} + 87 x^{12} - 35 x^{11} + 326 x^{10} - 205 x^{9} + 895 x^{8} - 481 x^{7} + 1005 x^{6} - 544 x^{5} + 811 x^{4} - 312 x^{3} + 195 x^{2} + 13 x + 1$$ x^16 + 11*x^14 - 4*x^13 + 87*x^12 - 35*x^11 + 326*x^10 - 205*x^9 + 895*x^8 - 481*x^7 + 1005*x^6 - 544*x^5 + 811*x^4 - 312*x^3 + 195*x^2 + 13*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 11 x^{14} - 4 x^{13} + 87 x^{12} - 35 x^{11} + 326 x^{10} - 205 x^{9} + 895 x^{8} - 481 x^{7} + 1005 x^{6} - 544 x^{5} + 811 x^{4} - 312 x^{3} + 195 x^{2} + 13 x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 850102445244 \nu^{15} - 602738721141 \nu^{14} - 9256957678228 \nu^{13} - 3074932976676 \nu^{12} - 70844833728257 \nu^{11} + \cdots - 13972229088464 ) / 200652098581830$$ (-850102445244*v^15 - 602738721141*v^14 - 9256957678228*v^13 - 3074932976676*v^12 - 70844833728257*v^11 - 21460338818148*v^10 - 251702286046476*v^9 - 11860911508749*v^8 - 637132974569996*v^7 - 98337615982400*v^6 - 589939014330730*v^5 - 26247556294384*v^4 - 577279503454940*v^3 - 108368220879957*v^2 - 208029345221468*v - 13972229088464) / 200652098581830 $$\beta_{3}$$ $$=$$ $$( 945658852056 \nu^{15} + 5453415441339 \nu^{14} + 13517776750957 \nu^{13} + 52870958416314 \nu^{12} + \cdots - 611267573740024 ) / 200652098581830$$ (945658852056*v^15 + 5453415441339*v^14 + 13517776750957*v^13 + 52870958416314*v^12 + 88760072093213*v^11 + 389923212586737*v^10 + 338939264634954*v^9 + 1187742608006541*v^8 + 384615393503879*v^7 + 2746301731062440*v^6 + 187320602242210*v^5 + 1195526676592186*v^4 - 1373541222944320*v^3 + 455677150451133*v^2 + 30629905626407*v - 611267573740024) / 200652098581830 $$\beta_{4}$$ $$=$$ $$( 1548397573197 \nu^{15} + 5359246221883 \nu^{14} + 19993119508609 \nu^{13} + 49756879408343 \nu^{12} + \cdots - 10161380439778 ) / 200652098581830$$ (1548397573197*v^15 + 5359246221883*v^14 + 19993119508609*v^13 + 49756879408343*v^12 + 139973996494901*v^11 + 364492101483669*v^10 + 525071177418723*v^9 + 1064033894083157*v^8 + 891852285648643*v^7 + 2481887787922950*v^6 + 676023888749330*v^5 + 1083373096954242*v^4 - 999941039148235*v^3 + 497936518850021*v^2 + 33550802926699*v - 10161380439778) / 200652098581830 $$\beta_{5}$$ $$=$$ $$( - 3810848648686 \nu^{15} + 2398500018441 \nu^{14} - 35957350192522 \nu^{13} + 44493471781581 \nu^{12} + \cdots - 8612982866581 ) / 200652098581830$$ (-3810848648686*v^15 + 2398500018441*v^14 - 35957350192522*v^13 + 44493471781581*v^12 - 278712020050663*v^11 + 344198532927168*v^10 - 856384219169819*v^9 + 1557997436445829*v^8 - 2334814734982064*v^7 + 3362003460236605*v^6 - 1249677488024080*v^5 + 3339064567965244*v^4 - 1980977600835720*v^3 + 766323242696737*v^2 - 136810746763792*v - 8612982866581) / 200652098581830 $$\beta_{6}$$ $$=$$ $$( 22240023182504 \nu^{15} + 33095488971866 \nu^{14} + 219131096234958 \nu^{13} + 229934387339641 \nu^{12} + \cdots - 24\!\cdots\!91 ) / 601956295745490$$ (22240023182504*v^15 + 33095488971866*v^14 + 219131096234958*v^13 + 229934387339641*v^12 + 1505862005807122*v^11 + 1718479268465178*v^10 + 3895919201439001*v^9 + 3400337677941704*v^8 + 5141508670414336*v^7 + 11075617584988655*v^6 - 11233975942651870*v^5 + 399705759957354*v^4 - 15416083486559320*v^3 + 6039092233128032*v^2 - 10163858567310982*v - 2454346762052891) / 601956295745490 $$\beta_{7}$$ $$=$$ $$( - 23960698558757 \nu^{15} + 80722315132867 \nu^{14} - 248272416035934 \nu^{13} + 945357317741162 \nu^{12} + \cdots - 15\!\cdots\!87 ) / 601956295745490$$ (-23960698558757*v^15 + 80722315132867*v^14 - 248272416035934*v^13 + 945357317741162*v^12 - 2278547346806461*v^11 + 7369609606825686*v^10 - 9561210057401608*v^9 + 27440315802224893*v^8 - 34791569261503438*v^7 + 68922263214064465*v^6 - 51995180624536250*v^5 + 58806711807132708*v^4 - 56537190221514755*v^3 + 39140928712744339*v^2 - 18100020514647914*v - 1527844957759087) / 601956295745490 $$\beta_{8}$$ $$=$$ $$( - 1778153235146 \nu^{15} + 3078673249492 \nu^{14} - 23056845757699 \nu^{13} + 36844479871649 \nu^{12} + \cdots + 13468531477239 ) / 40130419716366$$ (-1778153235146*v^15 + 3078673249492*v^14 - 23056845757699*v^13 + 36844479871649*v^12 - 206484663504642*v^11 + 300268949144295*v^10 - 984482545211503*v^9 + 1154069266885630*v^8 - 3294396074768529*v^7 + 3159979917916907*v^6 - 5797955983317930*v^5 + 2782805001313508*v^4 - 5401830167261236*v^3 + 2544041682468796*v^2 - 2398400014950427*v + 13468531477239) / 40130419716366 $$\beta_{9}$$ $$=$$ $$( 13972229088464 \nu^{15} - 850102445244 \nu^{14} + 153091781251963 \nu^{13} - 65145874032084 \nu^{12} + \cdots + 174261731510394 ) / 200652098581830$$ (13972229088464*v^15 - 850102445244*v^14 + 153091781251963*v^13 - 65145874032084*v^12 + 1212508997719692*v^11 - 559872851824497*v^10 + 4533486344021116*v^9 - 3116009249181596*v^8 + 12493284122666531*v^7 - 7357775166121180*v^6 + 13943752617923920*v^5 - 8190831638455146*v^4 + 11305230234449920*v^3 - 4936614979055708*v^2 + 2616216451370523*v + 174261731510394) / 200652098581830 $$\beta_{10}$$ $$=$$ $$( - 78334216662179 \nu^{15} + 42305988951004 \nu^{14} - 796493782907553 \nu^{13} + 800506448243009 \nu^{12} + \cdots - 15\!\cdots\!24 ) / 601956295745490$$ (-78334216662179*v^15 + 42305988951004*v^14 - 796493782907553*v^13 + 800506448243009*v^12 - 6287052669219262*v^11 + 6369277178514687*v^10 - 21667564253989861*v^9 + 29202146424854866*v^8 - 60056970145125211*v^7 + 67366680956630260*v^6 - 51818201522647850*v^5 + 68348199564129876*v^4 - 46941064229416025*v^3 + 32439278838552808*v^2 - 2452125901935503*v - 1561887871205524) / 601956295745490 $$\beta_{11}$$ $$=$$ $$( 40582993882149 \nu^{15} + 45466225490261 \nu^{14} + 457612339093228 \nu^{13} + 320116699841551 \nu^{12} + \cdots + 541126055004554 ) / 200652098581830$$ (40582993882149*v^15 + 45466225490261*v^14 + 457612339093228*v^13 + 320116699841551*v^12 + 3451826421856817*v^11 + 2289567725966748*v^10 + 12469185555804981*v^9 + 4596778428280669*v^8 + 29626192865873556*v^7 + 13380711590415220*v^6 + 26853299881601610*v^5 + 5262660243488864*v^4 + 13462802422151435*v^3 + 6376257529840357*v^2 + 432729870487758*v + 541126055004554) / 200652098581830 $$\beta_{12}$$ $$=$$ $$( - 24469220650027 \nu^{15} + 15238766802188 \nu^{14} - 263471571349674 \nu^{13} + 261485413803346 \nu^{12} + \cdots - 273843217222007 ) / 120391259149098$$ (-24469220650027*v^15 + 15238766802188*v^14 - 263471571349674*v^13 + 261485413803346*v^12 - 2133804675429128*v^11 + 2112117989573874*v^10 - 8069708080242212*v^9 + 9430098598560608*v^8 - 23566610847554492*v^7 + 22912974402719057*v^6 - 27864296349778072*v^5 + 22807330510966992*v^4 - 25099987842740767*v^3 + 13749352221620660*v^2 - 6383574622903984*v - 273843217222007) / 120391259149098 $$\beta_{13}$$ $$=$$ $$( - 131547920533543 \nu^{15} + 66691456016783 \nu^{14} + \cdots + 20950999529842 ) / 601956295745490$$ (-131547920533543*v^15 + 66691456016783*v^14 - 1384852130591721*v^13 + 1273004752646143*v^12 - 11058467161934039*v^11 + 10269404382982179*v^10 - 40207214376187487*v^9 + 47455714076985947*v^8 - 114175121032835207*v^7 + 112908211013582750*v^6 - 120446275827543100*v^5 + 117538497500186772*v^4 - 107879081650263685*v^3 + 65375118976888151*v^2 - 20971641574908571*v + 20950999529842) / 601956295745490 $$\beta_{14}$$ $$=$$ $$( 46673194766134 \nu^{15} + 504608087166 \nu^{14} + 508750470699368 \nu^{13} - 187060135461519 \nu^{12} + \cdots - 79869396342261 ) / 200652098581830$$ (46673194766134*v^15 + 504608087166*v^14 + 508750470699368*v^13 - 187060135461519*v^12 + 4004999085302952*v^11 - 1633893875813922*v^10 + 14795782515868121*v^9 - 9718282575984076*v^8 + 40199282496485536*v^7 - 22689027227537705*v^6 + 43268255944038050*v^5 - 26716032806738496*v^4 + 34523127081241160*v^3 - 14919838930830898*v^2 + 8016090006501768*v - 79869396342261) / 200652098581830 $$\beta_{15}$$ $$=$$ $$( - 140712016641263 \nu^{15} - 74541657903737 \nu^{14} + \cdots + 431512462062377 ) / 601956295745490$$ (-140712016641263*v^15 - 74541657903737*v^14 - 1577680530611646*v^13 - 227311479417322*v^12 - 12231961189578409*v^11 - 1099348556784906*v^10 - 45544873806547732*v^9 + 8153767607643457*v^8 - 118092162473133742*v^7 + 16417888475821585*v^6 - 126589546885231730*v^5 + 37652742907899012*v^4 - 88243990870548815*v^3 + 19109720516331511*v^2 - 19457791048108706*v + 431512462062377) / 601956295745490
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{14} - 3\beta_{9} + \beta_{5} - \beta_{3}$$ b14 - 3*b9 + b5 - b3 $$\nu^{3}$$ $$=$$ $$-\beta_{13} + \beta_{12} - \beta_{6} + \beta_{3} - 4\beta_{2} - 3\beta _1 + 1$$ -b13 + b12 - b6 + b3 - 4*b2 - 3*b1 + 1 $$\nu^{4}$$ $$=$$ $$- \beta_{15} - 7 \beta_{14} - \beta_{13} + \beta_{12} - \beta_{10} + 15 \beta_{9} - \beta_{8} - 6 \beta_{5} + 6 \beta_{4} + \beta _1 - 15$$ -b15 - 7*b14 - b13 + b12 - b10 + 15*b9 - b8 - 6*b5 + 6*b4 + b1 - 15 $$\nu^{5}$$ $$=$$ $$- \beta_{15} + 8 \beta_{14} + 9 \beta_{13} - \beta_{12} - \beta_{11} - 8 \beta_{9} + \beta_{8} - 8 \beta_{7} + 8 \beta_{6} + \beta_{5} - 8 \beta_{3} + 20 \beta_{2} - 8 \beta_1$$ -b15 + 8*b14 + 9*b13 - b12 - b11 - 8*b9 + b8 - 8*b7 + 8*b6 + b5 - 8*b3 + 20*b2 - 8*b1 $$\nu^{6}$$ $$=$$ $$11 \beta_{15} - \beta_{13} - \beta_{12} + 10 \beta_{11} + 2 \beta_{10} - 9 \beta_{7} + \beta_{6} - 34 \beta_{4} + 45 \beta_{3} - 10 \beta_{2} - 11 \beta _1 + 84$$ 11*b15 - b13 - b12 + 10*b11 + 2*b10 - 9*b7 + b6 - 34*b4 + 45*b3 - 10*b2 - 11*b1 + 84 $$\nu^{7}$$ $$=$$ $$- \beta_{15} - 56 \beta_{14} - \beta_{13} - 41 \beta_{12} - 13 \beta_{10} + 56 \beta_{9} - 12 \beta_{8} + 54 \beta_{7} - 10 \beta_{5} + 10 \beta_{4} + 111 \beta _1 - 56$$ -b15 - 56*b14 - b13 - 41*b12 - 13*b10 + 56*b9 - 12*b8 + 54*b7 - 10*b5 + 10*b4 + 111*b1 - 56 $$\nu^{8}$$ $$=$$ $$- 24 \beta_{15} + 286 \beta_{14} + 81 \beta_{13} - 78 \beta_{12} - 78 \beta_{11} + 65 \beta_{10} - 494 \beta_{9} + 78 \beta_{8} + 62 \beta_{7} - 8 \beta_{6} + 198 \beta_{5} - 286 \beta_{3} + 77 \beta_{2} + 8 \beta_1$$ -24*b15 + 286*b14 + 81*b13 - 78*b12 - 78*b11 + 65*b10 - 494*b9 + 78*b8 + 62*b7 - 8*b6 + 198*b5 - 286*b3 + 77*b2 + 8*b1 $$\nu^{9}$$ $$=$$ $$118 \beta_{15} - 437 \beta_{13} + 335 \beta_{12} + 105 \beta_{11} + 102 \beta_{10} - 16 \beta_{7} - 335 \beta_{6} - 79 \beta_{4} + 382 \beta_{3} - 651 \beta_{2} - 316 \beta _1 + 382$$ 118*b15 - 437*b13 + 335*b12 + 105*b11 + 102*b10 - 16*b7 - 335*b6 - 79*b4 + 382*b3 - 651*b2 - 316*b1 + 382 $$\nu^{10}$$ $$=$$ $$- 437 \beta_{15} - 1818 \beta_{14} - 437 \beta_{13} + 594 \beta_{12} - 644 \beta_{10} + 2986 \beta_{9} - 555 \beta_{8} + 50 \beta_{7} - 1184 \beta_{5} + 1184 \beta_{4} + 547 \beta _1 - 2986$$ -437*b15 - 1818*b14 - 437*b13 + 594*b12 - 644*b10 + 2986*b9 - 555*b8 + 50*b7 - 1184*b5 + 1184*b4 + 547*b1 - 2986 $$\nu^{11}$$ $$=$$ $$- 762 \beta_{15} + 2583 \beta_{14} + 3017 \beta_{13} - 812 \beta_{12} - 812 \beta_{11} + 168 \beta_{10} - 2596 \beta_{9} + 812 \beta_{8} - 2037 \beta_{7} + 2087 \beta_{6} + 587 \beta_{5} - 2583 \beta_{3} + \cdots - 2087 \beta_1$$ -762*b15 + 2583*b14 + 3017*b13 - 812*b12 - 812*b11 + 168*b10 - 2596*b9 + 812*b8 - 2037*b7 + 2087*b6 + 587*b5 - 2583*b3 + 3940*b2 - 2087*b1 $$\nu^{12}$$ $$=$$ $$4423 \beta_{15} - 1476 \beta_{13} - 98 \beta_{12} + 3779 \beta_{11} + 1574 \beta_{10} - 2849 \beta_{7} + 98 \beta_{6} - 7211 \beta_{4} + 11577 \beta_{3} - 3777 \beta_{2} - 3875 \beta _1 + 18359$$ 4423*b15 - 1476*b13 - 98*b12 + 3779*b11 + 1574*b10 - 2849*b7 + 98*b6 - 7211*b4 + 11577*b3 - 3777*b2 - 3875*b1 + 18359 $$\nu^{13}$$ $$=$$ $$- 1476 \beta_{15} - 17376 \beta_{14} - 1476 \beta_{13} - 7051 \beta_{12} - 6829 \beta_{10} + 17658 \beta_{9} - 5899 \beta_{8} + 13880 \beta_{7} - 4266 \beta_{5} + 4266 \beta_{4} + 24295 \beta _1 - 17658$$ -1476*b15 - 17376*b14 - 1476*b13 - 7051*b12 - 6829*b10 + 17658*b9 - 5899*b8 + 13880*b7 - 4266*b5 + 4266*b4 + 24295*b1 - 17658 $$\nu^{14}$$ $$=$$ $$- 11252 \beta_{15} + 73854 \beta_{14} + 30104 \beta_{13} - 25132 \beta_{12} - 25132 \beta_{11} + 18303 \beta_{10} - 114139 \beta_{9} + 25132 \beta_{8} + 13331 \beta_{7} + 549 \beta_{6} + 44456 \beta_{5} + \cdots - 549 \beta_1$$ -11252*b15 + 73854*b14 + 30104*b13 - 25132*b12 - 25132*b11 + 18303*b10 - 114139*b9 + 25132*b8 + 13331*b7 + 549*b6 + 44456*b5 - 73854*b3 + 25801*b2 - 549*b1 $$\nu^{15}$$ $$=$$ $$48185 \beta_{15} - 116740 \beta_{13} + 80356 \beta_{12} + 41356 \beta_{11} + 36384 \beta_{10} - 11801 \beta_{7} - 80356 \beta_{6} - 30616 \beta_{4} + 116428 \beta_{3} - 151544 \beta_{2} + \cdots + 120172$$ 48185*b15 - 116740*b13 + 80356*b12 + 41356*b11 + 36384*b10 - 11801*b7 - 80356*b6 - 30616*b4 + 116428*b3 - 151544*b2 - 71188*b1 + 120172

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$1$$ $$-\beta_{9}$$ $$-\beta_{9}$$

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 1.21707 + 2.10803i 0.857510 + 1.48525i 0.415625 + 0.719884i 0.379240 + 0.656863i −0.0340180 − 0.0589209i −0.532778 − 0.922798i −1.02737 − 1.77946i −1.27528 − 2.20885i 1.21707 − 2.10803i 0.857510 − 1.48525i 0.415625 − 0.719884i 0.379240 − 0.656863i −0.0340180 + 0.0589209i −0.532778 + 0.922798i −1.02737 + 1.77946i −1.27528 + 2.20885i
−2.43414 0.500000 0.866025i 3.92506 −0.613891 + 1.06329i −1.21707 + 2.10803i 2.20121 + 1.46788i −4.68588 −0.500000 0.866025i 1.49430 2.58820i
16.2 −1.71502 0.500000 0.866025i 0.941295 1.22863 2.12806i −0.857510 + 1.48525i −2.38702 1.14112i 1.81570 −0.500000 0.866025i −2.10713 + 3.64966i
16.3 −0.831251 0.500000 0.866025i −1.30902 −1.30847 + 2.26634i −0.415625 + 0.719884i 1.78280 1.95490i 2.75063 −0.500000 0.866025i 1.08767 1.88389i
16.4 −0.758480 0.500000 0.866025i −1.42471 −0.357869 + 0.619848i −0.379240 + 0.656863i −1.32400 + 2.29064i 2.59757 −0.500000 0.866025i 0.271437 0.470142i
16.5 0.0680360 0.500000 0.866025i −1.99537 1.52954 2.64923i 0.0340180 0.0589209i 0.910236 + 2.48424i −0.271829 −0.500000 0.866025i 0.104063 0.180243i
16.6 1.06556 0.500000 0.866025i −0.864591 1.19023 2.06154i 0.532778 0.922798i −0.813611 2.51755i −3.05238 −0.500000 0.866025i 1.26826 2.19668i
16.7 2.05474 0.500000 0.866025i 2.22196 −0.274662 + 0.475728i 1.02737 1.77946i 2.59269 0.527227i 0.456078 −0.500000 0.866025i −0.564359 + 0.977499i
16.8 2.55056 0.500000 0.866025i 4.50537 −1.39351 + 2.41363i 1.27528 2.20885i −2.46231 0.968004i 6.39011 −0.500000 0.866025i −3.55423 + 6.15611i
256.1 −2.43414 0.500000 + 0.866025i 3.92506 −0.613891 1.06329i −1.21707 2.10803i 2.20121 1.46788i −4.68588 −0.500000 + 0.866025i 1.49430 + 2.58820i
256.2 −1.71502 0.500000 + 0.866025i 0.941295 1.22863 + 2.12806i −0.857510 1.48525i −2.38702 + 1.14112i 1.81570 −0.500000 + 0.866025i −2.10713 3.64966i
256.3 −0.831251 0.500000 + 0.866025i −1.30902 −1.30847 2.26634i −0.415625 0.719884i 1.78280 + 1.95490i 2.75063 −0.500000 + 0.866025i 1.08767 + 1.88389i
256.4 −0.758480 0.500000 + 0.866025i −1.42471 −0.357869 0.619848i −0.379240 0.656863i −1.32400 2.29064i 2.59757 −0.500000 + 0.866025i 0.271437 + 0.470142i
256.5 0.0680360 0.500000 + 0.866025i −1.99537 1.52954 + 2.64923i 0.0340180 + 0.0589209i 0.910236 2.48424i −0.271829 −0.500000 + 0.866025i 0.104063 + 0.180243i
256.6 1.06556 0.500000 + 0.866025i −0.864591 1.19023 + 2.06154i 0.532778 + 0.922798i −0.813611 + 2.51755i −3.05238 −0.500000 + 0.866025i 1.26826 + 2.19668i
256.7 2.05474 0.500000 + 0.866025i 2.22196 −0.274662 0.475728i 1.02737 + 1.77946i 2.59269 + 0.527227i 0.456078 −0.500000 + 0.866025i −0.564359 0.977499i
256.8 2.55056 0.500000 + 0.866025i 4.50537 −1.39351 2.41363i 1.27528 + 2.20885i −2.46231 + 0.968004i 6.39011 −0.500000 + 0.866025i −3.55423 6.15611i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 16.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.l.b yes 16
3.b odd 2 1 819.2.s.e 16
7.c even 3 1 273.2.j.b 16
13.c even 3 1 273.2.j.b 16
21.h odd 6 1 819.2.n.e 16
39.i odd 6 1 819.2.n.e 16
91.h even 3 1 inner 273.2.l.b yes 16
273.s odd 6 1 819.2.s.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.j.b 16 7.c even 3 1
273.2.j.b 16 13.c even 3 1
273.2.l.b yes 16 1.a even 1 1 trivial
273.2.l.b yes 16 91.h even 3 1 inner
819.2.n.e 16 21.h odd 6 1
819.2.n.e 16 39.i odd 6 1
819.2.s.e 16 3.b odd 2 1
819.2.s.e 16 273.s odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 11T_{2}^{6} - 2T_{2}^{5} + 34T_{2}^{4} + 13T_{2}^{3} - 26T_{2}^{2} - 13T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{8} - 11 T^{6} - 2 T^{5} + 34 T^{4} + \cdots + 1)^{2}$$
$3$ $$(T^{2} - T + 1)^{8}$$
$5$ $$T^{16} + 19 T^{14} + 10 T^{13} + \cdots + 3969$$
$7$ $$T^{16} - T^{15} - 3 T^{14} + \cdots + 5764801$$
$11$ $$T^{16} + 2 T^{15} + 45 T^{14} + \cdots + 1290496$$
$13$ $$T^{16} - 5 T^{15} + 6 T^{14} + \cdots + 815730721$$
$17$ $$(T^{8} - 2 T^{7} - 68 T^{6} + 92 T^{5} + \cdots + 6257)^{2}$$
$19$ $$T^{16} + 11 T^{15} + 133 T^{14} + \cdots + 37161216$$
$23$ $$(T^{8} + 4 T^{7} - 50 T^{6} - 158 T^{5} + \cdots - 153)^{2}$$
$29$ $$T^{16} - 15 T^{15} + \cdots + 539772289$$
$31$ $$T^{16} - 3 T^{15} + \cdots + 1771652281$$
$37$ $$(T^{8} + 4 T^{7} - 119 T^{6} - 333 T^{5} + \cdots - 1999)^{2}$$
$41$ $$T^{16} - 19 T^{15} + \cdots + 8083371138129$$
$43$ $$T^{16} - 11 T^{15} + 209 T^{14} + \cdots + 99740169$$
$47$ $$T^{16} - 5 T^{15} + \cdots + 67950412929$$
$53$ $$T^{16} - 36 T^{15} + \cdots + 9921384931329$$
$59$ $$(T^{8} - 17 T^{7} - 66 T^{6} + \cdots + 441561)^{2}$$
$61$ $$T^{16} + 22 T^{15} + \cdots + 124397290000$$
$67$ $$T^{16} - 26 T^{15} + \cdots + 2264374886656$$
$71$ $$T^{16} - 9 T^{15} + \cdots + 83773776969$$
$73$ $$T^{16} + \cdots + 160073179912009$$
$79$ $$T^{16} - 16 T^{15} + \cdots + 131635449856$$
$83$ $$(T^{8} - 18 T^{7} - 196 T^{6} + \cdots - 5580400)^{2}$$
$89$ $$(T^{8} + 20 T^{7} - 82 T^{6} + \cdots + 683431)^{2}$$
$97$ $$T^{16} + \cdots + 575206497472369$$