# Properties

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

# Learn more

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: $$x^{8} - 16x^{6} - 4x^{5} + 75x^{4} + 32x^{3} - 90x^{2} - 28x - 2$$ x^8 - 16*x^6 - 4*x^5 + 75*x^4 + 32*x^3 - 90*x^2 - 28*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 16x^{6} - 4x^{5} + 75x^{4} + 32x^{3} - 90x^{2} - 28x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{7} - \nu^{6} - 15\nu^{5} + 11\nu^{4} + 64\nu^{3} - 24\nu^{2} - 66\nu - 2 ) / 8$$ (v^7 - v^6 - 15*v^5 + 11*v^4 + 64*v^3 - 24*v^2 - 66*v - 2) / 8 $$\beta_{3}$$ $$=$$ $$( \nu^{6} - 2\nu^{5} - 13\nu^{4} + 20\nu^{3} + 44\nu^{2} - 40\nu - 18 ) / 4$$ (v^6 - 2*v^5 - 13*v^4 + 20*v^3 + 44*v^2 - 40*v - 18) / 4 $$\beta_{4}$$ $$=$$ $$( \nu^{7} + \nu^{6} - 19\nu^{5} - 15\nu^{4} + 108\nu^{3} + 56\nu^{2} - 166\nu - 18 ) / 4$$ (v^7 + v^6 - 19*v^5 - 15*v^4 + 108*v^3 + 56*v^2 - 166*v - 18) / 4 $$\beta_{5}$$ $$=$$ $$( 3\nu^{7} - 2\nu^{6} - 45\nu^{5} + 14\nu^{4} + 196\nu^{3} + 8\nu^{2} - 218\nu - 32 ) / 4$$ (3*v^7 - 2*v^6 - 45*v^5 + 14*v^4 + 196*v^3 + 8*v^2 - 218*v - 32) / 4 $$\beta_{6}$$ $$=$$ $$( 5\nu^{7} + 3\nu^{6} - 87\nu^{5} - 53\nu^{4} + 448\nu^{3} + 232\nu^{2} - 610\nu - 98 ) / 8$$ (5*v^7 + 3*v^6 - 87*v^5 - 53*v^4 + 448*v^3 + 232*v^2 - 610*v - 98) / 8 $$\beta_{7}$$ $$=$$ $$( 7\nu^{7} - 3\nu^{6} - 109\nu^{5} + 13\nu^{4} + 496\nu^{3} + 72\nu^{2} - 582\nu - 78 ) / 8$$ (7*v^7 - 3*v^6 - 109*v^5 + 13*v^4 + 496*v^3 + 72*v^2 - 582*v - 78) / 8
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{7} + \beta_{5} + \beta_{3} + \beta_{2} + 3$$ -b7 + b5 + b3 + b2 + 3 $$\nu^{3}$$ $$=$$ $$-2\beta_{7} + 2\beta_{5} + \beta_{4} + 5\beta _1 + 1$$ -2*b7 + 2*b5 + b4 + 5*b1 + 1 $$\nu^{4}$$ $$=$$ $$-9\beta_{7} + \beta_{6} + 8\beta_{5} + 6\beta_{3} + 10\beta_{2} + 18$$ -9*b7 + b6 + 8*b5 + 6*b3 + 10*b2 + 18 $$\nu^{5}$$ $$=$$ $$-25\beta_{7} + 3\beta_{6} + 24\beta_{5} + 8\beta_{4} - 2\beta_{3} + 30\beta _1 + 12$$ -25*b7 + 3*b6 + 24*b5 + 8*b4 - 2*b3 + 30*b1 + 12 $$\nu^{6}$$ $$=$$ $$-83\beta_{7} + 19\beta_{6} + 68\beta_{5} - 4\beta_{4} + 34\beta_{3} + 86\beta_{2} + 124$$ -83*b7 + 19*b6 + 68*b5 - 4*b4 + 34*b3 + 86*b2 + 124 $$\nu^{7}$$ $$=$$ $$-255\beta_{7} + 53\beta_{6} + 236\beta_{5} + 52\beta_{4} - 38\beta_{3} + 8\beta_{2} + 196\beta _1 + 116$$ -255*b7 + 53*b6 + 236*b5 + 52*b4 - 38*b3 + 8*b2 + 196*b1 + 116

## 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
 −1.93433 −0.115899 2.55131 3.02984 −1.84109 1.24609 −0.182716 −2.75320
1.00000 −2.48823 1.00000 3.25915 −2.48823 4.58972 1.00000 3.19127 3.25915
1.2 1.00000 −2.11129 1.00000 −2.75070 −2.11129 3.66381 1.00000 1.45756 −2.75070
1.3 1.00000 −0.103249 1.00000 −3.91798 −0.103249 −4.45623 1.00000 −2.98934 −3.91798
1.4 1.00000 1.55594 1.00000 2.24033 1.55594 −2.27752 1.00000 −0.579065 2.24033
1.5 1.00000 1.67352 1.00000 −0.566489 1.67352 3.40033 1.00000 −0.199325 −0.566489
1.6 1.00000 2.79123 1.00000 4.02325 2.79123 0.274917 1.00000 4.79095 4.02325
1.7 1.00000 3.30349 1.00000 1.88260 3.30349 2.41197 1.00000 7.91304 1.88260
1.8 1.00000 3.37860 1.00000 −4.17017 3.37860 −3.60700 1.00000 8.41492 −4.17017
 $$n$$: 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.cb yes 8
11.b odd 2 1 4598.2.a.by 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4598.2.a.by 8 11.b odd 2 1
4598.2.a.cb yes 8 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{3}^{8} - 8T_{3}^{7} + 9T_{3}^{6} + 72T_{3}^{5} - 183T_{3}^{4} - 80T_{3}^{3} + 557T_{3}^{2} - 368T_{3} - 44$$ T3^8 - 8*T3^7 + 9*T3^6 + 72*T3^5 - 183*T3^4 - 80*T3^3 + 557*T3^2 - 368*T3 - 44 $$T_{5}^{8} - 38T_{5}^{6} + 12T_{5}^{5} + 470T_{5}^{4} - 288T_{5}^{3} - 1984T_{5}^{2} + 1536T_{5} + 1408$$ T5^8 - 38*T5^6 + 12*T5^5 + 470*T5^4 - 288*T5^3 - 1984*T5^2 + 1536*T5 + 1408 $$T_{7}^{8} - 4T_{7}^{7} - 37T_{7}^{6} + 152T_{7}^{5} + 387T_{7}^{4} - 1716T_{7}^{3} - 867T_{7}^{2} + 5408T_{7} - 1388$$ T7^8 - 4*T7^7 - 37*T7^6 + 152*T7^5 + 387*T7^4 - 1716*T7^3 - 867*T7^2 + 5408*T7 - 1388 $$T_{13}^{8} + 12T_{13}^{7} + 34T_{13}^{6} - 84T_{13}^{5} - 451T_{13}^{4} - 72T_{13}^{3} + 1088T_{13}^{2} + 96T_{13} - 368$$ T13^8 + 12*T13^7 + 34*T13^6 - 84*T13^5 - 451*T13^4 - 72*T13^3 + 1088*T13^2 + 96*T13 - 368

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{8}$$
$3$ $$T^{8} - 8 T^{7} + 9 T^{6} + 72 T^{5} + \cdots - 44$$
$5$ $$T^{8} - 38 T^{6} + 12 T^{5} + \cdots + 1408$$
$7$ $$T^{8} - 4 T^{7} - 37 T^{6} + \cdots - 1388$$
$11$ $$T^{8}$$
$13$ $$T^{8} + 12 T^{7} + 34 T^{6} + \cdots - 368$$
$17$ $$T^{8} + 4 T^{7} - 46 T^{6} - 128 T^{5} + \cdots - 92$$
$19$ $$(T + 1)^{8}$$
$23$ $$T^{8} - 14 T^{7} - 29 T^{6} + \cdots + 35296$$
$29$ $$T^{8} + 2 T^{7} - 123 T^{6} + \cdots + 964$$
$31$ $$T^{8} - 132 T^{6} - 368 T^{5} + \cdots - 9344$$
$37$ $$T^{8} - 24 T^{7} + 134 T^{6} + \cdots + 29569$$
$41$ $$T^{8} - 8 T^{7} - 130 T^{6} + \cdots - 3872$$
$43$ $$T^{8} - 8 T^{7} - 96 T^{6} + \cdots - 67328$$
$47$ $$T^{8} + 16 T^{7} - 172 T^{6} + \cdots - 2659283$$
$53$ $$T^{8} - 36 T^{7} + 413 T^{6} + \cdots + 2770816$$
$59$ $$T^{8} + 24 T^{7} + T^{6} + \cdots - 20032604$$
$61$ $$T^{8} - 12 T^{7} - 194 T^{6} + \cdots + 16192$$
$67$ $$T^{8} - 16 T^{7} - 126 T^{6} + \cdots + 457168$$
$71$ $$T^{8} - 4 T^{7} - 282 T^{6} + \cdots + 3806464$$
$73$ $$T^{8} + 20 T^{7} - 270 T^{6} + \cdots - 37382204$$
$79$ $$T^{8} + 12 T^{7} - 78 T^{6} + \cdots + 51808$$
$83$ $$T^{8} - 20 T^{7} - 234 T^{6} + \cdots - 251648$$
$89$ $$T^{8} - 8 T^{7} - 210 T^{6} + \cdots - 320672$$
$97$ $$T^{8} - 4 T^{7} - 452 T^{6} + \cdots - 1921664$$
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