Properties

 Label 4598.2.a.bw Level $4598$ Weight $2$ Character orbit 4598.a Self dual yes Analytic conductor $36.715$ Analytic rank $0$ Dimension $8$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4598,2,Mod(1,4598)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4598, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4598.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4598 = 2 \cdot 11^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4598.a (trivial)

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: yes Analytic conductor: $$36.7152148494$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} - 11x^{6} + 22x^{5} + 34x^{4} - 68x^{3} - 28x^{2} + 60x - 4$$ x^8 - 2*x^7 - 11*x^6 + 22*x^5 + 34*x^4 - 68*x^3 - 28*x^2 + 60*x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 418) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} - 11x^{6} + 22x^{5} + 34x^{4} - 68x^{3} - 28x^{2} + 60x - 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 9\nu^{3} - 18\nu ) / 2$$ (-v^5 + 9*v^3 - 18*v) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{7} - 9\nu^{5} + 2\nu^{4} + 20\nu^{3} - 10\nu^{2} - 12\nu + 8 ) / 4$$ (v^7 - 9*v^5 + 2*v^4 + 20*v^3 - 10*v^2 - 12*v + 8) / 4 $$\beta_{4}$$ $$=$$ $$( \nu^{7} - 11\nu^{5} + 36\nu^{3} + 2\nu^{2} - 36\nu ) / 4$$ (v^7 - 11*v^5 + 36*v^3 + 2*v^2 - 36*v) / 4 $$\beta_{5}$$ $$=$$ $$( -\nu^{7} + 11\nu^{5} - \nu^{4} - 35\nu^{3} + 4\nu^{2} + 32\nu - 2 ) / 2$$ (-v^7 + 11*v^5 - v^4 - 35*v^3 + 4*v^2 + 32*v - 2) / 2 $$\beta_{6}$$ $$=$$ $$( -\nu^{7} + 11\nu^{5} - \nu^{4} - 35\nu^{3} + 6\nu^{2} + 30\nu - 8 ) / 2$$ (-v^7 + 11*v^5 - v^4 - 35*v^3 + 6*v^2 + 30*v - 8) / 2 $$\beta_{7}$$ $$=$$ $$( -\nu^{7} + \nu^{6} + 11\nu^{5} - 9\nu^{4} - 34\nu^{3} + 16\nu^{2} + 28\nu ) / 2$$ (-v^7 + v^6 + 11*v^5 - 9*v^4 - 34*v^3 + 16*v^2 + 28*v) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} - \beta_{5} + \beta _1 + 3$$ b6 - b5 + b1 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 5\beta _1 - 1$$ b5 + b4 + b3 + b2 + 5*b1 - 1 $$\nu^{4}$$ $$=$$ $$6\beta_{6} - 7\beta_{5} - 3\beta_{4} + \beta_{3} + \beta_{2} + 7\beta _1 + 15$$ 6*b6 - 7*b5 - 3*b4 + b3 + b2 + 7*b1 + 15 $$\nu^{5}$$ $$=$$ $$9\beta_{5} + 9\beta_{4} + 9\beta_{3} + 7\beta_{2} + 27\beta _1 - 9$$ 9*b5 + 9*b4 + 9*b3 + 7*b2 + 27*b1 - 9 $$\nu^{6}$$ $$=$$ $$2\beta_{7} + 36\beta_{6} - 47\beta_{5} - 25\beta_{4} + 7\beta_{3} + 7\beta_{2} + 43\beta _1 + 83$$ 2*b7 + 36*b6 - 47*b5 - 25*b4 + 7*b3 + 7*b2 + 43*b1 + 83 $$\nu^{7}$$ $$=$$ $$-2\beta_{6} + 65\beta_{5} + 67\beta_{4} + 63\beta_{3} + 41\beta_{2} + 151\beta _1 - 69$$ -2*b6 + 65*b5 + 67*b4 + 63*b3 + 41*b2 + 151*b1 - 69

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.40829 2.27943 1.59897 1.18171 0.0692689 −1.25705 −1.75021 −2.53041
−1.00000 −2.40829 1.00000 −3.43347 2.40829 1.05667 −1.00000 2.79988 3.43347
1.2 −1.00000 −2.27943 1.00000 −2.24748 2.27943 4.63064 −1.00000 2.19578 2.24748
1.3 −1.00000 −1.59897 1.00000 3.74775 1.59897 −0.838369 −1.00000 −0.443297 −3.74775
1.4 −1.00000 −1.18171 1.00000 −0.117838 1.18171 −1.74369 −1.00000 −1.60355 0.117838
1.5 −1.00000 −0.0692689 1.00000 1.10125 0.0692689 1.99611 −1.00000 −2.99520 −1.10125
1.6 −1.00000 1.25705 1.00000 1.68320 −1.25705 4.32620 −1.00000 −1.41983 −1.68320
1.7 −1.00000 1.75021 1.00000 −2.99749 −1.75021 0.219425 −1.00000 0.0632464 2.99749
1.8 −1.00000 2.53041 1.00000 0.264060 −2.53041 4.35301 −1.00000 3.40297 −0.264060
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$11$$ $$1$$
$$19$$ $$-1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4598.2.a.bw 8
11.b odd 2 1 4598.2.a.bz 8
11.c even 5 2 418.2.f.g 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.g 16 11.c even 5 2
4598.2.a.bw 8 1.a even 1 1 trivial
4598.2.a.bz 8 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4598))$$:

 $$T_{3}^{8} + 2T_{3}^{7} - 11T_{3}^{6} - 22T_{3}^{5} + 34T_{3}^{4} + 68T_{3}^{3} - 28T_{3}^{2} - 60T_{3} - 4$$ T3^8 + 2*T3^7 - 11*T3^6 - 22*T3^5 + 34*T3^4 + 68*T3^3 - 28*T3^2 - 60*T3 - 4 $$T_{5}^{8} + 2T_{5}^{7} - 20T_{5}^{6} - 36T_{5}^{5} + 99T_{5}^{4} + 100T_{5}^{3} - 180T_{5}^{2} + 20T_{5} + 5$$ T5^8 + 2*T5^7 - 20*T5^6 - 36*T5^5 + 99*T5^4 + 100*T5^3 - 180*T5^2 + 20*T5 + 5 $$T_{7}^{8} - 14T_{7}^{7} + 64T_{7}^{6} - 70T_{7}^{5} - 211T_{7}^{4} + 442T_{7}^{3} + 10T_{7}^{2} - 290T_{7} + 59$$ T7^8 - 14*T7^7 + 64*T7^6 - 70*T7^5 - 211*T7^4 + 442*T7^3 + 10*T7^2 - 290*T7 + 59 $$T_{13}^{8} - 9T_{13}^{7} - 23T_{13}^{6} + 394T_{13}^{5} - 744T_{13}^{4} - 1240T_{13}^{3} + 2080T_{13}^{2} + 1760T_{13} + 320$$ T13^8 - 9*T13^7 - 23*T13^6 + 394*T13^5 - 744*T13^4 - 1240*T13^3 + 2080*T13^2 + 1760*T13 + 320

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{8}$$
$3$ $$T^{8} + 2 T^{7} - 11 T^{6} - 22 T^{5} + \cdots - 4$$
$5$ $$T^{8} + 2 T^{7} - 20 T^{6} - 36 T^{5} + \cdots + 5$$
$7$ $$T^{8} - 14 T^{7} + 64 T^{6} - 70 T^{5} + \cdots + 59$$
$11$ $$T^{8}$$
$13$ $$T^{8} - 9 T^{7} - 23 T^{6} + 394 T^{5} + \cdots + 320$$
$17$ $$T^{8} - 3 T^{7} - 89 T^{6} + \cdots - 118571$$
$19$ $$(T - 1)^{8}$$
$23$ $$T^{8} + 13 T^{7} - 47 T^{6} + \cdots - 142921$$
$29$ $$T^{8} - 10 T^{7} - 28 T^{6} + \cdots + 33536$$
$31$ $$T^{8} + 2 T^{7} - 71 T^{6} + \cdots + 22996$$
$37$ $$T^{8} - 15 T^{7} + 35 T^{6} + \cdots + 2524$$
$41$ $$T^{8} - 9 T^{7} - 75 T^{6} + \cdots + 12100$$
$43$ $$T^{8} - 33 T^{7} + 336 T^{6} + \cdots - 1634476$$
$47$ $$T^{8} + 19 T^{7} - 129 T^{6} + \cdots - 2589455$$
$53$ $$T^{8} - 10 T^{7} - 237 T^{6} + \cdots + 104756$$
$59$ $$T^{8} - T^{7} - 149 T^{6} + 794 T^{5} + \cdots + 1600$$
$61$ $$T^{8} - 26 T^{7} + 152 T^{6} + \cdots - 223605$$
$67$ $$T^{8} + 25 T^{7} - 165 T^{6} + \cdots - 72329500$$
$71$ $$T^{8} + 10 T^{7} - 213 T^{6} + \cdots - 2243884$$
$73$ $$T^{8} - 11 T^{7} - 183 T^{6} + \cdots + 2788624$$
$79$ $$T^{8} - 10 T^{7} - 249 T^{6} + \cdots - 2701820$$
$83$ $$T^{8} - 16 T^{7} - 98 T^{6} + \cdots - 1612721$$
$89$ $$T^{8} + 14 T^{7} - 233 T^{6} + \cdots - 481856$$
$97$ $$T^{8} - 22 T^{7} - 11 T^{6} + \cdots + 530500$$