# Properties

 Label 273.2.bd.a Level $273$ Weight $2$ Character orbit 273.bd Analytic conductor $2.180$ Analytic rank $0$ Dimension $16$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(43,273)]

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, 0, 1]))

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

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

chi := DirichletCharacter("273.43");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.bd (of order $$6$$, 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 $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 22x^{14} + 187x^{12} + 774x^{10} + 1619x^{8} + 1618x^{6} + 690x^{4} + 96x^{2} + 1$$ x^16 + 22*x^14 + 187*x^12 + 774*x^10 + 1619*x^8 + 1618*x^6 + 690*x^4 + 96*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 22x^{14} + 187x^{12} + 774x^{10} + 1619x^{8} + 1618x^{6} + 690x^{4} + 96x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{14} + 20\nu^{12} + 147\nu^{10} + 480\nu^{8} + 646\nu^{6} + 183\nu^{4} - 105\nu^{2} + 26\nu - 6 ) / 52$$ (v^14 + 20*v^12 + 147*v^10 + 480*v^8 + 646*v^6 + 183*v^4 - 105*v^2 + 26*v - 6) / 52 $$\beta_{2}$$ $$=$$ $$( -\nu^{14} - 20\nu^{12} - 147\nu^{10} - 480\nu^{8} - 646\nu^{6} - 183\nu^{4} + 105\nu^{2} + 26\nu + 6 ) / 52$$ (-v^14 - 20*v^12 - 147*v^10 - 480*v^8 - 646*v^6 - 183*v^4 + 105*v^2 + 26*v + 6) / 52 $$\beta_{3}$$ $$=$$ $$( -5\nu^{14} - 107\nu^{12} - 870\nu^{10} - 3340\nu^{8} - 6095\nu^{6} - 4633\nu^{4} - 1107\nu^{2} - 23 ) / 104$$ (-5*v^14 - 107*v^12 - 870*v^10 - 3340*v^8 - 6095*v^6 - 4633*v^4 - 1107*v^2 - 23) / 104 $$\beta_{4}$$ $$=$$ $$( -5\nu^{14} - 107\nu^{12} - 870\nu^{10} - 3340\nu^{8} - 6095\nu^{6} - 4633\nu^{4} - 1003\nu^{2} + 289 ) / 104$$ (-5*v^14 - 107*v^12 - 870*v^10 - 3340*v^8 - 6095*v^6 - 4633*v^4 - 1003*v^2 + 289) / 104 $$\beta_{5}$$ $$=$$ $$( 7 \nu^{15} + 5 \nu^{14} + 149 \nu^{13} + 107 \nu^{12} + 1210 \nu^{11} + 870 \nu^{10} + 4680 \nu^{9} + \cdots + 23 ) / 208$$ (7*v^15 + 5*v^14 + 149*v^13 + 107*v^12 + 1210*v^11 + 870*v^10 + 4680*v^9 + 3340*v^8 + 8733*v^7 + 6095*v^6 + 6819*v^5 + 4633*v^4 + 1237*v^3 + 1107*v^2 - 167*v + 23) / 208 $$\beta_{6}$$ $$=$$ $$( 5 \nu^{15} - 13 \nu^{14} + 115 \nu^{13} - 283 \nu^{12} + 1054 \nu^{11} - 2362 \nu^{10} + 4964 \nu^{9} + \cdots - 215 ) / 208$$ (5*v^15 - 13*v^14 + 115*v^13 - 283*v^12 + 1054*v^11 - 2362*v^10 + 4964*v^9 - 9440*v^8 + 12935*v^7 - 18339*v^6 + 18465*v^5 - 15457*v^4 + 12763*v^3 - 4495*v^2 + 2639*v - 215) / 208 $$\beta_{7}$$ $$=$$ $$( 15 \nu^{14} + 323 \nu^{12} + 2656 \nu^{10} + 10400 \nu^{8} + 19631 \nu^{6} + 15823 \nu^{4} + \cdots + 203 ) / 104$$ (15*v^14 + 323*v^12 + 2656*v^10 + 10400*v^8 + 19631*v^6 + 15823*v^4 + 4285*v^2 - 52*v + 203) / 104 $$\beta_{8}$$ $$=$$ $$( 9 \nu^{15} + 197 \nu^{13} + 2 \nu^{12} + 1662 \nu^{11} + 46 \nu^{10} + 6796 \nu^{9} + 406 \nu^{8} + \cdots + 160 ) / 104$$ (9*v^15 + 197*v^13 + 2*v^12 + 1662*v^11 + 46*v^10 + 6796*v^9 + 406*v^8 + 13901*v^7 + 1684*v^6 + 13243*v^5 + 3146*v^4 + 5117*v^3 + 1952*v^2 + 747*v + 160) / 104 $$\beta_{9}$$ $$=$$ $$( 9 \nu^{15} + 197 \nu^{13} - 2 \nu^{12} + 1662 \nu^{11} - 46 \nu^{10} + 6796 \nu^{9} - 406 \nu^{8} + \cdots - 160 ) / 104$$ (9*v^15 + 197*v^13 - 2*v^12 + 1662*v^11 - 46*v^10 + 6796*v^9 - 406*v^8 + 13901*v^7 - 1684*v^6 + 13243*v^5 - 3146*v^4 + 5117*v^3 - 1952*v^2 + 747*v - 160) / 104 $$\beta_{10}$$ $$=$$ $$( - 23 \nu^{15} + 7 \nu^{14} - 501 \nu^{13} + 149 \nu^{12} - 4194 \nu^{11} + 1210 \nu^{10} - 16932 \nu^{9} + \cdots - 167 ) / 208$$ (-23*v^15 + 7*v^14 - 501*v^13 + 149*v^12 - 4194*v^11 + 1210*v^10 - 16932*v^9 + 4680*v^8 - 33897*v^7 + 8733*v^6 - 31119*v^5 + 6819*v^4 - 11237*v^3 + 1237*v^2 - 1101*v - 167) / 208 $$\beta_{11}$$ $$=$$ $$( - 23 \nu^{15} - 7 \nu^{14} - 501 \nu^{13} - 149 \nu^{12} - 4194 \nu^{11} - 1210 \nu^{10} - 16932 \nu^{9} + \cdots + 167 ) / 208$$ (-23*v^15 - 7*v^14 - 501*v^13 - 149*v^12 - 4194*v^11 - 1210*v^10 - 16932*v^9 - 4680*v^8 - 33897*v^7 - 8733*v^6 - 31119*v^5 - 6819*v^4 - 11237*v^3 - 1237*v^2 - 1101*v + 167) / 208 $$\beta_{12}$$ $$=$$ $$( 6\nu^{15} + 133\nu^{13} + 1142\nu^{11} + 4791\nu^{9} + 10194\nu^{7} + 10354\nu^{5} + 4323\nu^{3} + 471\nu + 26 ) / 52$$ (6*v^15 + 133*v^13 + 1142*v^11 + 4791*v^9 + 10194*v^7 + 10354*v^5 + 4323*v^3 + 471*v + 26) / 52 $$\beta_{13}$$ $$=$$ $$( - 41 \nu^{15} + 5 \nu^{14} - 895 \nu^{13} + 119 \nu^{12} - 7518 \nu^{11} + 1094 \nu^{10} - 30524 \nu^{9} + \cdots - 317 ) / 208$$ (-41*v^15 + 5*v^14 - 895*v^13 + 119*v^12 - 7518*v^11 + 1094*v^10 - 30524*v^9 + 4840*v^8 - 61699*v^7 + 10375*v^6 - 57605*v^5 + 9313*v^4 - 21367*v^3 + 2055*v^2 - 2075*v - 317) / 208 $$\beta_{14}$$ $$=$$ $$( - 41 \nu^{15} - 5 \nu^{14} - 895 \nu^{13} - 119 \nu^{12} - 7518 \nu^{11} - 1094 \nu^{10} - 30524 \nu^{9} + \cdots + 317 ) / 208$$ (-41*v^15 - 5*v^14 - 895*v^13 - 119*v^12 - 7518*v^11 - 1094*v^10 - 30524*v^9 - 4840*v^8 - 61699*v^7 - 10375*v^6 - 57605*v^5 - 9313*v^4 - 21367*v^3 - 2055*v^2 - 2075*v + 317) / 208 $$\beta_{15}$$ $$=$$ $$( 59 \nu^{15} + 5 \nu^{14} + 1293 \nu^{13} + 107 \nu^{12} + 10934 \nu^{11} + 870 \nu^{10} + 44928 \nu^{9} + \cdots - 185 ) / 208$$ (59*v^15 + 5*v^14 + 1293*v^13 + 107*v^12 + 10934*v^11 + 870*v^10 + 44928*v^9 + 3340*v^8 + 92869*v^7 + 6095*v^6 + 90383*v^5 + 4633*v^4 + 35401*v^3 + 1003*v^2 + 3577*v - 185) / 208
 $$\nu$$ $$=$$ $$\beta_{2} + \beta_1$$ b2 + b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{3} - 3$$ b4 - b3 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{14} + \beta_{13} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - 5\beta_{2} - 5\beta_1$$ b14 + b13 - b11 - b10 + b9 + b8 - 5*b2 - 5*b1 $$\nu^{4}$$ $$=$$ $$\beta_{11} - \beta_{10} + \beta_{7} - 6\beta_{4} + 8\beta_{3} + \beta _1 + 15$$ b11 - b10 + b7 - 6*b4 + 8*b3 + b1 + 15 $$\nu^{5}$$ $$=$$ $$- 9 \beta_{14} - 9 \beta_{13} - 2 \beta_{12} + 10 \beta_{11} + 10 \beta_{10} - 6 \beta_{9} - 6 \beta_{8} + \cdots + 1$$ -9*b14 - 9*b13 - 2*b12 + 10*b11 + 10*b10 - 6*b9 - 6*b8 - 2*b5 - b3 + 29*b2 + 29*b1 + 1 $$\nu^{6}$$ $$=$$ $$\beta_{14} - \beta_{13} - 8 \beta_{11} + 8 \beta_{10} - 9 \beta_{7} + 35 \beta_{4} - 57 \beta_{3} + \cdots - 84$$ b14 - b13 - 8*b11 + 8*b10 - 9*b7 + 35*b4 - 57*b3 + 2*b2 - 11*b1 - 84 $$\nu^{7}$$ $$=$$ $$- 4 \beta_{15} + 66 \beta_{14} + 66 \beta_{13} + 24 \beta_{12} - 79 \beta_{11} - 79 \beta_{10} + \cdots - 10$$ -4*b15 + 66*b14 + 66*b13 + 24*b12 - 79*b11 - 79*b10 + 35*b9 + 35*b8 + b7 + 2*b6 + 24*b5 - 2*b4 + 12*b3 - 176*b2 - 177*b1 - 10 $$\nu^{8}$$ $$=$$ $$- 13 \beta_{14} + 13 \beta_{13} + 57 \beta_{11} - 57 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + \cdots + 500$$ -13*b14 + 13*b13 + 57*b11 - 57*b10 - 2*b9 + 2*b8 + 66*b7 - 211*b4 + 391*b3 - 28*b2 + 94*b1 + 500 $$\nu^{9}$$ $$=$$ $$60 \beta_{15} - 453 \beta_{14} - 453 \beta_{13} - 218 \beta_{12} + 569 \beta_{11} + 569 \beta_{10} + \cdots + 79$$ 60*b15 - 453*b14 - 453*b13 - 218*b12 + 569*b11 + 569*b10 - 211*b9 - 211*b8 - 13*b7 - 26*b6 - 222*b5 + 30*b4 - 111*b3 + 1098*b2 + 1111*b1 + 79 $$\nu^{10}$$ $$=$$ $$124 \beta_{14} - 124 \beta_{13} - 403 \beta_{11} + 403 \beta_{10} + 30 \beta_{9} - 30 \beta_{8} + \cdots - 3091$$ 124*b14 - 124*b13 - 403*b11 + 403*b10 + 30*b9 - 30*b8 - 453*b7 + 1309*b4 - 2633*b3 + 280*b2 - 733*b1 - 3091 $$\nu^{11}$$ $$=$$ $$- 620 \beta_{15} + 3026 \beta_{14} + 3026 \beta_{13} + 1778 \beta_{12} - 3927 \beta_{11} - 3927 \beta_{10} + \cdots - 579$$ -620*b15 + 3026*b14 + 3026*b13 + 1778*b12 - 3927*b11 - 3927*b10 + 1309*b9 + 1309*b8 + 124*b7 + 248*b6 + 1862*b5 - 310*b4 + 931*b3 - 6973*b2 - 7097*b1 - 579 $$\nu^{12}$$ $$=$$ $$- 1055 \beta_{14} + 1055 \beta_{13} + 2861 \beta_{11} - 2861 \beta_{10} - 310 \beta_{9} + 310 \beta_{8} + \cdots + 19574$$ -1055*b14 + 1055*b13 + 2861*b11 - 2861*b10 - 310*b9 + 310*b8 + 3026*b7 - 8282*b4 + 17572*b3 - 2440*b2 + 5466*b1 + 19574 $$\nu^{13}$$ $$=$$ $$5500 \beta_{15} - 19978 \beta_{14} - 19978 \beta_{13} - 13702 \beta_{12} + 26552 \beta_{11} + \cdots + 4101$$ 5500*b15 - 19978*b14 - 19978*b13 - 13702*b12 + 26552*b11 + 26552*b10 - 8282*b9 - 8282*b8 - 1055*b7 - 2110*b6 - 14822*b5 + 2750*b4 - 7411*b3 + 44808*b2 + 45863*b1 + 4101 $$\nu^{14}$$ $$=$$ $$8466 \beta_{14} - 8466 \beta_{13} - 20354 \beta_{11} + 20354 \beta_{10} + 2750 \beta_{9} - 2750 \beta_{8} + \cdots - 125893$$ 8466*b14 - 8466*b13 - 20354*b11 + 20354*b10 + 2750*b9 - 2750*b8 - 19978*b7 + 53090*b4 - 116816*b3 + 19762*b2 - 39740*b1 - 125893 $$\nu^{15}$$ $$=$$ $$- 45024 \beta_{15} + 131294 \beta_{14} + 131294 \beta_{13} + 102072 \beta_{12} - 177792 \beta_{11} + \cdots - 28524$$ -45024*b15 + 131294*b14 + 131294*b13 + 102072*b12 - 177792*b11 - 177792*b10 + 53090*b9 + 53090*b8 + 8466*b7 + 16932*b6 + 114096*b5 - 22512*b4 + 57048*b3 - 290299*b2 - 298765*b1 - 28524

## Character values

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

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$1$$ $$\beta_{12}$$ $$1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 − 2.60802i − 1.77930i − 0.775848i − 0.485989i 0.106359i 1.10207i 1.98765i 2.45308i 2.60802i 1.77930i 0.775848i 0.485989i − 0.106359i − 1.10207i − 1.98765i − 2.45308i
−2.25861 1.30401i −0.500000 + 0.866025i 2.40088 + 4.15844i 1.50528i 2.25861 1.30401i −0.866025 + 0.500000i 7.30704i −0.500000 0.866025i −1.96290 + 3.39983i
43.2 −1.54092 0.889651i −0.500000 + 0.866025i 0.582956 + 1.00971i 0.681820i 1.54092 0.889651i 0.866025 0.500000i 1.48409i −0.500000 0.866025i −0.606581 + 1.05063i
43.3 −0.671904 0.387924i −0.500000 + 0.866025i −0.699030 1.21076i 3.03444i 0.671904 0.387924i 0.866025 0.500000i 2.63638i −0.500000 0.866025i 1.17713 2.03885i
43.4 −0.420879 0.242995i −0.500000 + 0.866025i −0.881907 1.52751i 1.06536i 0.420879 0.242995i −0.866025 + 0.500000i 1.82917i −0.500000 0.866025i 0.258876 0.448387i
43.5 0.0921099 + 0.0531797i −0.500000 + 0.866025i −0.994344 1.72225i 1.41292i −0.0921099 + 0.0531797i −0.866025 + 0.500000i 0.424234i −0.500000 0.866025i 0.0751388 0.130144i
43.6 0.954423 + 0.551037i −0.500000 + 0.866025i −0.392717 0.680206i 3.28432i −0.954423 + 0.551037i 0.866025 0.500000i 3.06975i −0.500000 0.866025i 1.80978 3.13463i
43.7 1.72135 + 0.993824i −0.500000 + 0.866025i 0.975372 + 1.68939i 2.85284i −1.72135 + 0.993824i −0.866025 + 0.500000i 0.0979034i −0.500000 0.866025i −2.83522 + 4.91075i
43.8 2.12443 + 1.22654i −0.500000 + 0.866025i 2.00879 + 3.47933i 0.0682999i −2.12443 + 1.22654i 0.866025 0.500000i 4.94928i −0.500000 0.866025i 0.0837724 0.145098i
127.1 −2.25861 + 1.30401i −0.500000 0.866025i 2.40088 4.15844i 1.50528i 2.25861 + 1.30401i −0.866025 0.500000i 7.30704i −0.500000 + 0.866025i −1.96290 3.39983i
127.2 −1.54092 + 0.889651i −0.500000 0.866025i 0.582956 1.00971i 0.681820i 1.54092 + 0.889651i 0.866025 + 0.500000i 1.48409i −0.500000 + 0.866025i −0.606581 1.05063i
127.3 −0.671904 + 0.387924i −0.500000 0.866025i −0.699030 + 1.21076i 3.03444i 0.671904 + 0.387924i 0.866025 + 0.500000i 2.63638i −0.500000 + 0.866025i 1.17713 + 2.03885i
127.4 −0.420879 + 0.242995i −0.500000 0.866025i −0.881907 + 1.52751i 1.06536i 0.420879 + 0.242995i −0.866025 0.500000i 1.82917i −0.500000 + 0.866025i 0.258876 + 0.448387i
127.5 0.0921099 0.0531797i −0.500000 0.866025i −0.994344 + 1.72225i 1.41292i −0.0921099 0.0531797i −0.866025 0.500000i 0.424234i −0.500000 + 0.866025i 0.0751388 + 0.130144i
127.6 0.954423 0.551037i −0.500000 0.866025i −0.392717 + 0.680206i 3.28432i −0.954423 0.551037i 0.866025 + 0.500000i 3.06975i −0.500000 + 0.866025i 1.80978 + 3.13463i
127.7 1.72135 0.993824i −0.500000 0.866025i 0.975372 1.68939i 2.85284i −1.72135 0.993824i −0.866025 0.500000i 0.0979034i −0.500000 + 0.866025i −2.83522 4.91075i
127.8 2.12443 1.22654i −0.500000 0.866025i 2.00879 3.47933i 0.0682999i −2.12443 1.22654i 0.866025 + 0.500000i 4.94928i −0.500000 + 0.866025i 0.0837724 + 0.145098i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 43.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.bd.a 16
3.b odd 2 1 819.2.ct.b 16
13.e even 6 1 inner 273.2.bd.a 16
13.f odd 12 1 3549.2.a.bb 8
13.f odd 12 1 3549.2.a.bd 8
39.h odd 6 1 819.2.ct.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bd.a 16 1.a even 1 1 trivial
273.2.bd.a 16 13.e even 6 1 inner
819.2.ct.b 16 3.b odd 2 1
819.2.ct.b 16 39.h odd 6 1
3549.2.a.bb 8 13.f odd 12 1
3549.2.a.bd 8 13.f odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} - 11 T_{2}^{14} + 88 T_{2}^{12} - 6 T_{2}^{11} - 315 T_{2}^{10} + 12 T_{2}^{9} + 824 T_{2}^{8} + \cdots + 1$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 11 T^{14} + \cdots + 1$$
$3$ $$(T^{2} + T + 1)^{8}$$
$5$ $$T^{16} + 34 T^{14} + \cdots + 9$$
$7$ $$(T^{4} - T^{2} + 1)^{4}$$
$11$ $$T^{16} + 12 T^{15} + \cdots + 144$$
$13$ $$T^{16} + \cdots + 815730721$$
$17$ $$T^{16} + \cdots + 2883582601$$
$19$ $$T^{16} + \cdots + 25335725584$$
$23$ $$T^{16} + \cdots + 1446433024$$
$29$ $$T^{16} + \cdots + 2538849769$$
$31$ $$T^{16} + 222 T^{14} + \cdots + 7929856$$
$37$ $$T^{16} + \cdots + 16292735449$$
$41$ $$T^{16} - 18 T^{15} + \cdots + 692224$$
$43$ $$T^{16} + \cdots + 15764309136$$
$47$ $$T^{16} + 222 T^{14} + \cdots + 1336336$$
$53$ $$(T^{8} - 6 T^{7} + \cdots - 76707)^{2}$$
$59$ $$T^{16} + \cdots + 3887273104$$
$61$ $$T^{16} + \cdots + 103866332089$$
$67$ $$T^{16} + \cdots + 5866334464$$
$71$ $$T^{16} + \cdots + 84\!\cdots\!84$$
$73$ $$T^{16} + \cdots + 17309352523401$$
$79$ $$(T^{8} + 2 T^{7} + \cdots + 75240852)^{2}$$
$83$ $$T^{16} + \cdots + 2140872301584$$
$89$ $$T^{16} + \cdots + 72\!\cdots\!16$$
$97$ $$T^{16} + \cdots + 941955655936$$