Properties

 Label 726.2.b.c Level $726$ Weight $2$ Character orbit 726.b Analytic conductor $5.797$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [726,2,Mod(725,726)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(726, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("726.725");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$726 = 2 \cdot 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 726.b (of order $$2$$, degree $$1$$, 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: $$5.79713918674$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.185640625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 3x^{7} + x^{6} + x^{5} + 4x^{4} + 3x^{3} + 9x^{2} - 81x + 81$$ x^8 - 3*x^7 + x^6 + x^5 + 4*x^4 + 3*x^3 + 9*x^2 - 81*x + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 66) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} + x^{6} + x^{5} + 4x^{4} + 3x^{3} + 9x^{2} - 81x + 81$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{7} - \nu^{5} - 4\nu^{4} - 16\nu^{3} + 57\nu^{2} - 54\nu + 27 ) / 108$$ (-v^7 - v^5 - 4*v^4 - 16*v^3 + 57*v^2 - 54*v + 27) / 108 $$\beta_{3}$$ $$=$$ $$( \nu^{7} - 3\nu^{6} + \nu^{5} + \nu^{4} + 4\nu^{3} + 3\nu^{2} + 9\nu - 81 ) / 27$$ (v^7 - 3*v^6 + v^5 + v^4 + 4*v^3 + 3*v^2 + 9*v - 81) / 27 $$\beta_{4}$$ $$=$$ $$( -3\nu^{7} + 5\nu^{6} + 6\nu^{5} + 2\nu^{4} - 10\nu^{3} - 19\nu^{2} - 69\nu + 144 ) / 36$$ (-3*v^7 + 5*v^6 + 6*v^5 + 2*v^4 - 10*v^3 - 19*v^2 - 69*v + 144) / 36 $$\beta_{5}$$ $$=$$ $$( 10\nu^{7} - 15\nu^{6} - 17\nu^{5} - 2\nu^{4} + 46\nu^{3} + 108\nu^{2} + 153\nu - 567 ) / 108$$ (10*v^7 - 15*v^6 - 17*v^5 - 2*v^4 + 46*v^3 + 108*v^2 + 153*v - 567) / 108 $$\beta_{6}$$ $$=$$ $$( 11\nu^{7} - 18\nu^{6} - 7\nu^{5} - 28\nu^{4} + 32\nu^{3} + 93\nu^{2} + 252\nu - 567 ) / 108$$ (11*v^7 - 18*v^6 - 7*v^5 - 28*v^4 + 32*v^3 + 93*v^2 + 252*v - 567) / 108 $$\beta_{7}$$ $$=$$ $$( -20\nu^{7} + 21\nu^{6} + 25\nu^{5} + 22\nu^{4} - 2\nu^{3} - 126\nu^{2} - 423\nu + 783 ) / 108$$ (-20*v^7 + 21*v^6 + 25*v^5 + 22*v^4 - 2*v^3 - 126*v^2 - 423*v + 783) / 108
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{2} + \beta _1 + 1$$ b5 + b4 + b2 + b1 + 1 $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{6} + 2\beta_{5} + \beta_{4} - \beta_{3} - 2\beta_{2} + 2$$ b7 + b6 + 2*b5 + b4 - b3 - 2*b2 + 2 $$\nu^{4}$$ $$=$$ $$-\beta_{7} - 4\beta_{6} + 3\beta_{5} + 2\beta_{4} + 3\beta_{3} + 4\beta _1 + 3$$ -b7 - 4*b6 + 3*b5 + 2*b4 + 3*b3 + 4*b1 + 3 $$\nu^{5}$$ $$=$$ $$-\beta_{7} + 2\beta_{6} + 3\beta_{5} + 10\beta_{4} + 4\beta_{3} - 2\beta_{2} + 4\beta _1 + 6$$ -b7 + 2*b6 + 3*b5 + 10*b4 + 4*b3 - 2*b2 + 4*b1 + 6 $$\nu^{6}$$ $$=$$ $$-3\beta_{7} + 9\beta_{5} + 14\beta_{4} - 8\beta_{3} - 8\beta_{2} + \beta _1 - 9$$ -3*b7 + 9*b5 + 14*b4 - 8*b3 - 8*b2 + b1 - 9 $$\nu^{7}$$ $$=$$ $$-11\beta_{7} - 2\beta_{6} + 10\beta_{5} + 23\beta_{4} - 17\beta_{2} - 17\beta _1 + 34$$ -11*b7 - 2*b6 + 10*b5 + 23*b4 - 17*b2 - 17*b1 + 34

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/726\mathbb{Z}\right)^\times$$.

 $$n$$ $$485$$ $$607$$ $$\chi(n)$$ $$-1$$ $$-1$$

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
725.1
 1.71634 + 0.232753i 1.71634 − 0.232753i 1.55470 + 0.763481i 1.55470 − 0.763481i −0.245684 + 1.71454i −0.245684 − 1.71454i −1.52536 + 0.820539i −1.52536 − 0.820539i
−1.00000 −1.71634 0.232753i 1.00000 2.65532i 1.71634 + 0.232753i 1.64108i −1.00000 2.89165 + 0.798968i 2.65532i
725.2 −1.00000 −1.71634 + 0.232753i 1.00000 2.65532i 1.71634 0.232753i 1.64108i −1.00000 2.89165 0.798968i 2.65532i
725.3 −1.00000 −1.55470 0.763481i 1.00000 2.11929i 1.55470 + 0.763481i 3.42908i −1.00000 1.83419 + 2.37397i 2.11929i
725.4 −1.00000 −1.55470 + 0.763481i 1.00000 2.11929i 1.55470 0.763481i 3.42908i −1.00000 1.83419 2.37397i 2.11929i
725.5 −1.00000 0.245684 1.71454i 1.00000 0.943715i −0.245684 + 1.71454i 1.52696i −1.00000 −2.87928 0.842471i 0.943715i
725.6 −1.00000 0.245684 + 1.71454i 1.00000 0.943715i −0.245684 1.71454i 1.52696i −1.00000 −2.87928 + 0.842471i 0.943715i
725.7 −1.00000 1.52536 0.820539i 1.00000 0.753205i −1.52536 + 0.820539i 0.465507i −1.00000 1.65343 2.50323i 0.753205i
725.8 −1.00000 1.52536 + 0.820539i 1.00000 0.753205i −1.52536 0.820539i 0.465507i −1.00000 1.65343 + 2.50323i 0.753205i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 725.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 726.2.b.c 8
3.b odd 2 1 726.2.b.e 8
11.b odd 2 1 726.2.b.e 8
11.c even 5 1 66.2.h.b yes 8
11.c even 5 1 726.2.h.f 8
11.c even 5 1 726.2.h.h 8
11.c even 5 1 726.2.h.j 8
11.d odd 10 1 66.2.h.a 8
11.d odd 10 1 726.2.h.a 8
11.d odd 10 1 726.2.h.c 8
11.d odd 10 1 726.2.h.d 8
33.d even 2 1 inner 726.2.b.c 8
33.f even 10 1 66.2.h.b yes 8
33.f even 10 1 726.2.h.f 8
33.f even 10 1 726.2.h.h 8
33.f even 10 1 726.2.h.j 8
33.h odd 10 1 66.2.h.a 8
33.h odd 10 1 726.2.h.a 8
33.h odd 10 1 726.2.h.c 8
33.h odd 10 1 726.2.h.d 8
44.g even 10 1 528.2.bn.b 8
44.h odd 10 1 528.2.bn.a 8
132.n odd 10 1 528.2.bn.a 8
132.o even 10 1 528.2.bn.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.h.a 8 11.d odd 10 1
66.2.h.a 8 33.h odd 10 1
66.2.h.b yes 8 11.c even 5 1
66.2.h.b yes 8 33.f even 10 1
528.2.bn.a 8 44.h odd 10 1
528.2.bn.a 8 132.n odd 10 1
528.2.bn.b 8 44.g even 10 1
528.2.bn.b 8 132.o even 10 1
726.2.b.c 8 1.a even 1 1 trivial
726.2.b.c 8 33.d even 2 1 inner
726.2.b.e 8 3.b odd 2 1
726.2.b.e 8 11.b odd 2 1
726.2.h.a 8 11.d odd 10 1
726.2.h.a 8 33.h odd 10 1
726.2.h.c 8 11.d odd 10 1
726.2.h.c 8 33.h odd 10 1
726.2.h.d 8 11.d odd 10 1
726.2.h.d 8 33.h odd 10 1
726.2.h.f 8 11.c even 5 1
726.2.h.f 8 33.f even 10 1
726.2.h.h 8 11.c even 5 1
726.2.h.h 8 33.f even 10 1
726.2.h.j 8 11.c even 5 1
726.2.h.j 8 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(726, [\chi])$$:

 $$T_{5}^{8} + 13T_{5}^{6} + 49T_{5}^{4} + 52T_{5}^{2} + 16$$ T5^8 + 13*T5^6 + 49*T5^4 + 52*T5^2 + 16 $$T_{17}^{4} + 5T_{17}^{3} - 35T_{17}^{2} - 250T_{17} - 380$$ T17^4 + 5*T17^3 - 35*T17^2 - 250*T17 - 380

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{8}$$
$3$ $$T^{8} + 3 T^{7} + T^{6} - T^{5} + 4 T^{4} + \cdots + 81$$
$5$ $$T^{8} + 13 T^{6} + 49 T^{4} + 52 T^{2} + \cdots + 16$$
$7$ $$T^{8} + 17 T^{6} + 69 T^{4} + 88 T^{2} + \cdots + 16$$
$11$ $$T^{8}$$
$13$ $$T^{8} + 52 T^{6} + 784 T^{4} + \cdots + 4096$$
$17$ $$(T^{4} + 5 T^{3} - 35 T^{2} - 250 T - 380)^{2}$$
$19$ $$T^{8} + 85 T^{6} + 1885 T^{4} + \cdots + 400$$
$23$ $$T^{8} + 88 T^{6} + 2544 T^{4} + \cdots + 30976$$
$29$ $$(T^{4} + T^{3} - 79 T^{2} - 304 T - 124)^{2}$$
$31$ $$(T^{4} + 11 T^{3} - 19 T^{2} - 464 T - 1084)^{2}$$
$37$ $$(T^{4} + 12 T^{3} + 4 T^{2} - 192 T + 176)^{2}$$
$41$ $$(T^{4} - T^{3} - 59 T^{2} + 94 T - 4)^{2}$$
$43$ $$T^{8} + 113 T^{6} + 4549 T^{4} + \cdots + 430336$$
$47$ $$T^{8} + 212 T^{6} + 13264 T^{4} + \cdots + 1048576$$
$53$ $$T^{8} + 185 T^{6} + 11085 T^{4} + \cdots + 2310400$$
$59$ $$T^{8} + 202 T^{6} + 10299 T^{4} + \cdots + 844561$$
$61$ $$T^{8} + 212 T^{6} + 3664 T^{4} + \cdots + 4096$$
$67$ $$(T^{4} + T^{3} - 149 T^{2} - 284 T + 3076)^{2}$$
$71$ $$(T^{4} + 100 T^{2} + 80)^{2}$$
$73$ $$T^{8} + 90 T^{6} + 2035 T^{4} + \cdots + 24025$$
$79$ $$T^{8} + 157 T^{6} + 3909 T^{4} + \cdots + 55696$$
$83$ $$(T^{4} - 12 T^{3} - 31 T^{2} + 552 T - 859)^{2}$$
$89$ $$T^{8} + 513 T^{6} + \cdots + 12702096$$
$97$ $$(T^{4} + 8 T^{3} - 131 T^{2} - 928 T - 1439)^{2}$$