Properties

 Label 4598.2.a.cd Level $4598$ Weight $2$ Character orbit 4598.a Self dual yes Analytic conductor $36.715$ Analytic rank $0$ Dimension $10$ 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: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} - 2x^{9} - 19x^{8} + 36x^{7} + 118x^{6} - 220x^{5} - 270x^{4} + 512x^{3} + 176x^{2} - 392x + 44$$ x^10 - 2*x^9 - 19*x^8 + 36*x^7 + 118*x^6 - 220*x^5 - 270*x^4 + 512*x^3 + 176*x^2 - 392*x + 44 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2x^{9} - 19x^{8} + 36x^{7} + 118x^{6} - 220x^{5} - 270x^{4} + 512x^{3} + 176x^{2} - 392x + 44$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 59 \nu^{9} - 21 \nu^{8} - 1216 \nu^{7} + 276 \nu^{6} + 8338 \nu^{5} - 1532 \nu^{4} - 22372 \nu^{3} + 3632 \nu^{2} + 19636 \nu - 2940 ) / 892$$ (59*v^9 - 21*v^8 - 1216*v^7 + 276*v^6 + 8338*v^5 - 1532*v^4 - 22372*v^3 + 3632*v^2 + 19636*v - 2940) / 892 $$\beta_{3}$$ $$=$$ $$( 53 \nu^{9} + 155 \nu^{8} - 858 \nu^{7} - 2738 \nu^{6} + 3340 \nu^{5} + 12412 \nu^{4} - 2272 \nu^{3} - 13640 \nu^{2} - 2000 \nu - 1492 ) / 892$$ (53*v^9 + 155*v^8 - 858*v^7 - 2738*v^6 + 3340*v^5 + 12412*v^4 - 2272*v^3 - 13640*v^2 - 2000*v - 1492) / 892 $$\beta_{4}$$ $$=$$ $$( 97 \nu^{9} - 95 \nu^{8} - 1848 \nu^{7} + 1376 \nu^{6} + 11448 \nu^{5} - 6442 \nu^{4} - 26576 \nu^{3} + 9252 \nu^{2} + 20188 \nu - 2596 ) / 892$$ (97*v^9 - 95*v^8 - 1848*v^7 + 1376*v^6 + 11448*v^5 - 6442*v^4 - 26576*v^3 + 9252*v^2 + 20188*v - 2596) / 892 $$\beta_{5}$$ $$=$$ $$( - 159 \nu^{9} - 19 \nu^{8} + 3020 \nu^{7} + 632 \nu^{6} - 18048 \nu^{5} - 2894 \nu^{4} + 39820 \nu^{3} + 3456 \nu^{2} - 27004 \nu + 16 ) / 892$$ (-159*v^9 - 19*v^8 + 3020*v^7 + 632*v^6 - 18048*v^5 - 2894*v^4 + 39820*v^3 + 3456*v^2 - 27004*v + 16) / 892 $$\beta_{6}$$ $$=$$ $$( - 120 \nu^{9} - 48 \nu^{8} + 2254 \nu^{7} + 1045 \nu^{6} - 13213 \nu^{5} - 4776 \nu^{4} + 29144 \nu^{3} + 6008 \nu^{2} - 21508 \nu - 1368 ) / 446$$ (-120*v^9 - 48*v^8 + 2254*v^7 + 1045*v^6 - 13213*v^5 - 4776*v^4 + 29144*v^3 + 6008*v^2 - 21508*v - 1368) / 446 $$\beta_{7}$$ $$=$$ $$( 120 \nu^{9} + 48 \nu^{8} - 2254 \nu^{7} - 1045 \nu^{6} + 13213 \nu^{5} + 4776 \nu^{4} - 29144 \nu^{3} - 5562 \nu^{2} + 21508 \nu - 862 ) / 446$$ (120*v^9 + 48*v^8 - 2254*v^7 - 1045*v^6 + 13213*v^5 + 4776*v^4 - 29144*v^3 - 5562*v^2 + 21508*v - 862) / 446 $$\beta_{8}$$ $$=$$ $$( 199 \nu^{9} + 35 \nu^{8} - 3697 \nu^{7} - 906 \nu^{6} + 21263 \nu^{5} + 3148 \nu^{4} - 44926 \nu^{3} + 42 \nu^{2} + 31200 \nu - 4912 ) / 446$$ (199*v^9 + 35*v^8 - 3697*v^7 - 906*v^6 + 21263*v^5 + 3148*v^4 - 44926*v^3 + 42*v^2 + 31200*v - 4912) / 446 $$\beta_{9}$$ $$=$$ $$( - 535 \nu^{9} + 9 \nu^{8} + 10142 \nu^{7} + 710 \nu^{6} - 60980 \nu^{5} - 554 \nu^{4} + 138928 \nu^{3} - 11496 \nu^{2} - 104624 \nu + 17316 ) / 892$$ (-535*v^9 + 9*v^8 + 10142*v^7 + 710*v^6 - 60980*v^5 - 554*v^4 + 138928*v^3 - 11496*v^2 - 104624*v + 17316) / 892
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{7} + \beta_{6} + 5$$ b7 + b6 + 5 $$\nu^{3}$$ $$=$$ $$-\beta_{9} - \beta_{8} + \beta_{6} - 2\beta_{5} - \beta_{4} - 2\beta_{2} + 7\beta _1 + 2$$ -b9 - b8 + b6 - 2*b5 - b4 - 2*b2 + 7*b1 + 2 $$\nu^{4}$$ $$=$$ $$-\beta_{9} - \beta_{8} + 11\beta_{7} + 10\beta_{6} - \beta_{5} - 2\beta_{4} - 2\beta_{3} - 4\beta_{2} + 3\beta _1 + 38$$ -b9 - b8 + 11*b7 + 10*b6 - b5 - 2*b4 - 2*b3 - 4*b2 + 3*b1 + 38 $$\nu^{5}$$ $$=$$ $$- 15 \beta_{9} - 15 \beta_{8} + 5 \beta_{7} + 14 \beta_{6} - 21 \beta_{5} - 14 \beta_{4} - 2 \beta_{3} - 30 \beta_{2} + 61 \beta _1 + 36$$ -15*b9 - 15*b8 + 5*b7 + 14*b6 - 21*b5 - 14*b4 - 2*b3 - 30*b2 + 61*b1 + 36 $$\nu^{6}$$ $$=$$ $$- 21 \beta_{9} - 17 \beta_{8} + 103 \beta_{7} + 96 \beta_{6} - 19 \beta_{5} - 38 \beta_{4} - 30 \beta_{3} - 66 \beta_{2} + 59 \beta _1 + 336$$ -21*b9 - 17*b8 + 103*b7 + 96*b6 - 19*b5 - 38*b4 - 30*b3 - 66*b2 + 59*b1 + 336 $$\nu^{7}$$ $$=$$ $$- 175 \beta_{9} - 173 \beta_{8} + 101 \beta_{7} + 174 \beta_{6} - 199 \beta_{5} - 160 \beta_{4} - 38 \beta_{3} - 362 \beta_{2} + 577 \beta _1 + 502$$ -175*b9 - 173*b8 + 101*b7 + 174*b6 - 199*b5 - 160*b4 - 38*b3 - 362*b2 + 577*b1 + 502 $$\nu^{8}$$ $$=$$ $$- 301 \beta_{9} - 235 \beta_{8} + 977 \beta_{7} + 950 \beta_{6} - 275 \beta_{5} - 510 \beta_{4} - 348 \beta_{3} - 844 \beta_{2} + 849 \beta _1 + 3214$$ -301*b9 - 235*b8 + 977*b7 + 950*b6 - 275*b5 - 510*b4 - 348*b3 - 844*b2 + 849*b1 + 3214 $$\nu^{9}$$ $$=$$ $$- 1901 \beta_{9} - 1855 \beta_{8} + 1465 \beta_{7} + 2074 \beta_{6} - 1927 \beta_{5} - 1754 \beta_{4} - 536 \beta_{3} - 4060 \beta_{2} + 5697 \beta _1 + 6318$$ -1901*b9 - 1855*b8 + 1465*b7 + 2074*b6 - 1927*b5 - 1754*b4 - 536*b3 - 4060*b2 + 5697*b1 + 6318

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
 −2.84834 −2.53955 −1.58152 −1.28398 0.120966 1.20026 1.39462 2.10092 2.12868 3.30795
1.00000 −2.84834 1.00000 −3.07094 −2.84834 4.67138 1.00000 5.11302 −3.07094
1.2 1.00000 −2.53955 1.00000 1.81466 −2.53955 0.280173 1.00000 3.44929 1.81466
1.3 1.00000 −1.58152 1.00000 2.60241 −1.58152 1.55057 1.00000 −0.498782 2.60241
1.4 1.00000 −1.28398 1.00000 −1.35936 −1.28398 −3.16828 1.00000 −1.35139 −1.35936
1.5 1.00000 0.120966 1.00000 −2.16907 0.120966 3.98760 1.00000 −2.98537 −2.16907
1.6 1.00000 1.20026 1.00000 −4.16480 1.20026 0.346148 1.00000 −1.55937 −4.16480
1.7 1.00000 1.39462 1.00000 2.16074 1.39462 −1.18224 1.00000 −1.05504 2.16074
1.8 1.00000 2.10092 1.00000 2.46005 2.10092 3.09995 1.00000 1.41386 2.46005
1.9 1.00000 2.12868 1.00000 −0.444747 2.12868 −0.813941 1.00000 1.53127 −0.444747
1.10 1.00000 3.30795 1.00000 −0.828943 3.30795 2.22864 1.00000 7.94251 −0.828943
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 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.cd 10
11.b odd 2 1 4598.2.a.cc 10
11.d odd 10 2 418.2.f.h 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.h 20 11.d odd 10 2
4598.2.a.cc 10 11.b odd 2 1
4598.2.a.cd 10 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4598))$$:

 $$T_{3}^{10} - 2 T_{3}^{9} - 19 T_{3}^{8} + 36 T_{3}^{7} + 118 T_{3}^{6} - 220 T_{3}^{5} - 270 T_{3}^{4} + 512 T_{3}^{3} + 176 T_{3}^{2} - 392 T_{3} + 44$$ T3^10 - 2*T3^9 - 19*T3^8 + 36*T3^7 + 118*T3^6 - 220*T3^5 - 270*T3^4 + 512*T3^3 + 176*T3^2 - 392*T3 + 44 $$T_{5}^{10} + 3 T_{5}^{9} - 23 T_{5}^{8} - 56 T_{5}^{7} + 191 T_{5}^{6} + 369 T_{5}^{5} - 635 T_{5}^{4} - 1062 T_{5}^{3} + 591 T_{5}^{2} + 1191 T_{5} + 349$$ T5^10 + 3*T5^9 - 23*T5^8 - 56*T5^7 + 191*T5^6 + 369*T5^5 - 635*T5^4 - 1062*T5^3 + 591*T5^2 + 1191*T5 + 349 $$T_{7}^{10} - 11 T_{7}^{9} + 27 T_{7}^{8} + 88 T_{7}^{7} - 449 T_{7}^{6} + 307 T_{7}^{5} + 809 T_{7}^{4} - 856 T_{7}^{3} - 329 T_{7}^{2} + 351 T_{7} - 59$$ T7^10 - 11*T7^9 + 27*T7^8 + 88*T7^7 - 449*T7^6 + 307*T7^5 + 809*T7^4 - 856*T7^3 - 329*T7^2 + 351*T7 - 59 $$T_{13}^{10} - 11 T_{13}^{9} - T_{13}^{8} + 356 T_{13}^{7} - 940 T_{13}^{6} - 1480 T_{13}^{5} + 6152 T_{13}^{4} - 576 T_{13}^{3} - 6912 T_{13}^{2} + 1216 T_{13} + 1984$$ T13^10 - 11*T13^9 - T13^8 + 356*T13^7 - 940*T13^6 - 1480*T13^5 + 6152*T13^4 - 576*T13^3 - 6912*T13^2 + 1216*T13 + 1984

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{10}$$
$3$ $$T^{10} - 2 T^{9} - 19 T^{8} + 36 T^{7} + \cdots + 44$$
$5$ $$T^{10} + 3 T^{9} - 23 T^{8} - 56 T^{7} + \cdots + 349$$
$7$ $$T^{10} - 11 T^{9} + 27 T^{8} + 88 T^{7} + \cdots - 59$$
$11$ $$T^{10}$$
$13$ $$T^{10} - 11 T^{9} - T^{8} + 356 T^{7} + \cdots + 1984$$
$17$ $$T^{10} - 12 T^{9} - 11 T^{8} + \cdots - 1381$$
$19$ $$(T - 1)^{10}$$
$23$ $$T^{10} - 14 T^{9} - 3 T^{8} + \cdots - 377581$$
$29$ $$T^{10} - 16 T^{9} - 38 T^{8} + \cdots - 19407424$$
$31$ $$T^{10} - 12 T^{9} - 127 T^{8} + \cdots - 5575484$$
$37$ $$T^{10} + T^{9} - 117 T^{8} + \cdots + 260516$$
$41$ $$T^{10} + 5 T^{9} - 309 T^{8} + \cdots + 301928884$$
$43$ $$T^{10} - 22 T^{9} + 52 T^{8} + \cdots + 350900$$
$47$ $$T^{10} - 8 T^{9} - 185 T^{8} + \cdots - 347771$$
$53$ $$T^{10} - 2 T^{9} - 369 T^{8} + \cdots - 210485900$$
$59$ $$T^{10} + 7 T^{9} - 121 T^{8} + \cdots - 23104$$
$61$ $$T^{10} - 35 T^{9} + 297 T^{8} + \cdots - 342661$$
$67$ $$T^{10} - 9 T^{9} - 335 T^{8} + \cdots + 6908404$$
$71$ $$T^{10} + 4 T^{9} - 429 T^{8} + \cdots - 23572844$$
$73$ $$T^{10} - 5 T^{9} - 387 T^{8} + \cdots + 36593104$$
$79$ $$T^{10} - 18 T^{9} - 145 T^{8} + \cdots + 8477116$$
$83$ $$T^{10} - 7 T^{9} + \cdots - 3201446201$$
$89$ $$T^{10} - 22 T^{9} - 135 T^{8} + \cdots - 64913216$$
$97$ $$T^{10} - 32 T^{9} + 69 T^{8} + \cdots + 86549804$$