Properties

 Label 197.2.a.c Level $197$ Weight $2$ Character orbit 197.a Self dual yes Analytic conductor $1.573$ 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] = [197,2,Mod(1,197)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(197, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("197.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$197$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 197.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: $$1.57305291982$$ 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: $$x^{10} - 15x^{8} - x^{7} + 78x^{6} + 7x^{5} - 165x^{4} - 15x^{3} + 123x^{2} + 9x - 26$$ x^10 - 15*x^8 - x^7 + 78*x^6 + 7*x^5 - 165*x^4 - 15*x^3 + 123*x^2 + 9*x - 26 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{3} + 1) q^{3} + (\beta_{2} + 1) q^{4} - \beta_{4} q^{5} + (\beta_{9} + \beta_{6} - \beta_1) q^{6} + (\beta_{8} + 1) q^{7} + ( - \beta_{9} - \beta_{8} + \cdots - \beta_{2}) q^{8}+ \cdots + ( - \beta_{9} - \beta_{8} - \beta_{6} + \cdots + 1) q^{9}+O(q^{10})$$ q - b1 * q^2 + (b3 + 1) * q^3 + (b2 + 1) * q^4 - b4 * q^5 + (b9 + b6 - b1) * q^6 + (b8 + 1) * q^7 + (-b9 - b8 - b7 - b6 + b4 - b3 - b2) * q^8 + (-b9 - b8 - b6 - b5 + b4 - b2 + b1 + 1) * q^9 $$q - \beta_1 q^{2} + (\beta_{3} + 1) q^{3} + (\beta_{2} + 1) q^{4} - \beta_{4} q^{5} + (\beta_{9} + \beta_{6} - \beta_1) q^{6} + (\beta_{8} + 1) q^{7} + ( - \beta_{9} - \beta_{8} + \cdots - \beta_{2}) q^{8}+ \cdots + (\beta_{9} + 2 \beta_{8} + 5 \beta_{7} + \cdots - 1) q^{99}+O(q^{100})$$ q - b1 * q^2 + (b3 + 1) * q^3 + (b2 + 1) * q^4 - b4 * q^5 + (b9 + b6 - b1) * q^6 + (b8 + 1) * q^7 + (-b9 - b8 - b7 - b6 + b4 - b3 - b2) * q^8 + (-b9 - b8 - b6 - b5 + b4 - b2 + b1 + 1) * q^9 + (b8 + 2*b7 + b5 - 1) * q^10 + (b7 + b6 + b5) * q^11 + (-b8 - b7 - b6 + b4 + b1 + 1) * q^12 + (b9 + b8 + 1) * q^13 + (-b9 - b7 - b6 - b5 - b3 - b2 - b1 - 1) * q^14 + (-b8 - b7 - b6 - b5 - b4 - b3 - b2 + b1) * q^15 + (b9 + b8 + 2*b6 - b4 + b2) * q^16 + (2*b6 + b5 + b2) * q^17 + (b9 + b8 + b7 + 3*b6 - b4 + b2 - b1 - 1) * q^18 + (-b9 - b8 - b7 - b6 + b4 - b3 + b1 + 2) * q^19 + (b7 - b6 - b5 - b4 - b3 + b1 - 1) * q^20 + (b9 + 2*b8 + b5 + 2*b3) * q^21 + (-b8 - b6 - b5 - b4 + b1) * q^22 + (2*b9 - b4 + b2) * q^23 + (-b7 + b5 + b3 + b2 - 1) * q^24 + (-b9 - b8 - b7 - b6 + b5 + b4 + b3 + b1 + 1) * q^25 + (-b7 - b6 + b3 - b2 - b1 - 1) * q^26 + (-2*b9 - 2*b8 + b7 - b6 - 2*b5 + b4 - b3 - b2 + 2*b1 + 1) * q^27 + (-b9 + b7 + b6 - b3 + 2*b2 + b1 + 1) * q^28 + (b9 + b8 + b7 + b5 + b3 + b2 - 1) * q^29 + (b9 + 3*b8 + 4*b7 + 2*b6 + 3*b5 + 2*b3 + b2 - 3) * q^30 + (-b8 - b5 - b3 + b1 + 2) * q^31 + (-b9 - 2*b6 - b5 + b4 + 2*b3 - 2*b2 + b1 - 2) * q^32 + (b8 + 3*b7 + 3*b6 + 2*b5 + b3 + b2 - b1 - 1) * q^33 + (-2*b9 - 3*b8 - 3*b7 - 3*b6 - 2*b5 + 2*b4 - b3 - 2*b2 + b1) * q^34 + (-b7 + b6 - b5 - b4 - b3 - b1 - 1) * q^35 + (-2*b9 - b8 - 2*b7 - 3*b6 + b4 - 2*b2 + b1 - 1) * q^36 + (-b8 - b7 + 2*b6 - b5 + b4 - b3 + b2 + 2) * q^37 + (-b7 + b6 - b3 + b2 - 3*b1 - 1) * q^38 + (b9 + 3*b8 + b5 - b4 + 2*b3 + b2 - 2*b1 - 1) * q^39 + (-b9 - b4 - b2 - 2) * q^40 + (b9 - b7 - b5 + b4 + b3 - b2 - 2*b1 - 1) * q^41 + (2*b9 - 2*b7 - b5 - b4 - b2 - 2) * q^42 + (-2*b8 - 2*b7 - 2*b6 - b5 + b4 - 2*b3 - 2*b2 + 2) * q^43 + (b9 + 2*b8 + 2*b7 + b5 + b3 - b1 - 3) * q^44 + (-b9 - 2*b8 - 3*b7 - 3*b6 - 3*b5 - b3 - b2 + b1 - 1) * q^45 + (b9 + b7 - b6 + 3*b5 + b4 + 3*b3 - b2 - b1 - 1) * q^46 + (-b9 + 3*b7 - b6 + b5 - b3 + b1 - 1) * q^47 + (b9 + b8 + 2*b6 - b4 - b3 - 2*b1 - 2) * q^48 + (b9 + 4*b8 + 2*b7 - b4 + 2*b3 - 2*b1 - 1) * q^49 + (3*b9 - b7 + 3*b6 - b4 - b3 + 2*b2 - 2*b1 - 1) * q^50 + (b7 + 3*b6 + 2*b5 + b4 - b3 + b2 - b1 - 1) * q^51 + (b9 + b7 + 3*b6 + b5 - b4 + b3 + 3*b2 + b1 + 1) * q^52 + (b8 + 3*b7 - b5 - b3 - 2) * q^53 + (-b9 + b8 + 3*b7 + 3*b6 - b4 - b3 - b1 - 3) * q^54 + (2*b9 + b8 + 4*b6 + 2*b5 - 2*b4 + b2 - 4*b1 - 1) * q^55 + (-3*b9 - 3*b8 - 2*b6 + 2*b4 - 2*b3 - 2*b2 - 1) * q^56 + (-b9 - b8 - 2*b7 - 2*b6 + b5 + b4 + 2*b3 + b2 + 2*b1 + 1) * q^57 + (b9 - b8 - b7 - b6 - b4 - b2 - 1) * q^58 + (-b9 - b8 - b7 + b6 - b4 - b3 + 3*b2 - b1) * q^59 + (-b8 - 2*b6 - 2*b5 - 2*b4 - b2 + 2*b1 - 3) * q^60 + (-2*b9 - 2*b8 - 3*b7 - 2*b6 - 2*b5 + b4 - 3*b3 - 3*b2 + 1) * q^61 + (-b9 + b7 + b5 + b4 + b3 - b2 - 2*b1 - 2) * q^62 + (b8 - 2*b6 + 2*b5 + b4 - 1) * q^63 + (2*b9 + b8 + 2*b7 + 2*b6 - b4 + 2*b1 - 2) * q^64 + (b9 + b8 + b7 + b6 - b4 + b3 + b2 - b1 - 2) * q^65 + (-2*b9 - 4*b8 - 2*b7 - 4*b6 - 4*b5 - b4 - 2*b3 - 2*b2 + 3*b1 + 2) * q^66 + (b8 + 3*b7 - b6 - b4 + b1 + 3) * q^67 + (3*b9 + 3*b8 + b7 + 3*b6 + b3 + 3*b2 - b1 + 2) * q^68 + (-2*b7 - 2*b6 - b5 - 2*b4 - 2*b3 + b2 - 2*b1 - 2) * q^69 + (-3*b9 - 2*b6 + 3*b4 - b2 + 2*b1 + 2) * q^70 + (-b9 + 3*b8 - b7 + b6 - 2*b4 - b2 - b1 - 1) * q^71 + (2*b9 + 2*b8 + b5 - b4 - b3 + 3*b2 + 2*b1 + 1) * q^72 + (-b9 - 3*b8 + b7 - 3*b6 - 2*b5 + 2*b4 - 3*b3 - b2 + b1 + 2) * q^73 + (-4*b9 - 4*b8 - 5*b7 - 3*b6 - 2*b5 + 5*b4 - 3*b2 - b1 + 2) * q^74 + (-3*b9 - 2*b8 - 2*b7 - 4*b6 + b5 + 3*b4 - b3 + 4*b1 + 5) * q^75 + (-b9 - b7 - b6 - b5 + b4 + b3 + b2 - b1 + 5) * q^76 + (-2*b9 - 2*b8 - b7 - b6 + b5 + 2*b4 - b3 + 3*b1 - 1) * q^77 + (-2*b7 - 2*b6 - b5 - 2*b3 - b2 + 2) * q^78 + (3*b9 + 2*b8 + b7 + 3*b6 + b5 - 3*b4 + b2 - 2*b1) * q^79 + (2*b8 + b7 + 3*b6 + 2*b5 + b4 + b3 + b2 + b1 + 1) * q^80 + (-2*b8 + 3*b7 + 3*b6 - b5 - 2*b4 + 3*b3 + b2 + b1 + 1) * q^81 + (2*b9 - 2*b7 + 2*b6 + b4 + 3*b3 + 2*b2 + 2*b1 + 7) * q^82 + (-b8 - 3*b7 - 3*b6 - 2*b5 + 2*b4 - b3 - 3*b2 - b1 + 4) * q^83 + (-2*b8 + b7 + b6 + b5 + 2*b4 + b3 + 3*b1 - 1) * q^84 + (b9 + 2*b8 + b7 + 5*b6 + 2*b5 - 2*b4 - b3 - b1 - 3) * q^85 + (3*b9 + 3*b8 + 2*b7 + 4*b6 + 3*b5 - b4 + 4*b3 + 5*b2 - 2*b1 + 3) * q^86 + (2*b8 + b7 - b6 + 2*b5 + b4 - b3 + b2 - 1) * q^87 + (b9 + 2*b8 + b6 + b5 - b4 + b1 + 1) * q^88 + (b9 - 2*b8 - 2*b6 + b5 + b4 + 2*b3) * q^89 + (b9 + 4*b8 + 3*b7 + 5*b6 + 4*b5 + 2*b4 + b3 + 2*b2 - b1 - 1) * q^90 + (3*b9 + 4*b8 + b7 + b6 - b5 - b4 + 3*b3 - b2 - b1 + 5) * q^91 + (5*b9 + b8 + b7 + 5*b6 + b5 - 4*b4 + 3*b3 + 4*b2 + b1 + 4) * q^92 + (-b9 - 2*b8 - 2*b5 - b4 + 2*b3 + 1) * q^93 + (b8 + 4*b7 - 5*b4 - 2*b3 + b2 - 3) * q^94 + (-b9 - b8 - b7 - b6 - 2*b5 - 3*b4 - b3 - b2 + b1 - 2) * q^95 + (-3*b9 - b8 + b7 - 4*b6 - 3*b5 + 2*b4 - b3 - 3*b2 + 4*b1 + 6) * q^96 + (-b8 - b7 + 3*b5 + b3 - b2 + 2*b1 + 1) * q^97 + (-b9 + b8 - 2*b6 - 2*b5 - 2*b4 - 2*b3 - 2*b2 + b1 + 1) * q^98 + (b9 + 2*b8 + 5*b7 + 5*b6 + b5 + 2*b4 + b3 + b2 - 2*b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + 10 q^{3} + 10 q^{4} + 2 q^{5} - 4 q^{6} + 11 q^{7} - 3 q^{8} + 12 q^{9}+O(q^{10})$$ 10 * q + 10 * q^3 + 10 * q^4 + 2 * q^5 - 4 * q^6 + 11 * q^7 - 3 * q^8 + 12 * q^9 $$10 q + 10 q^{3} + 10 q^{4} + 2 q^{5} - 4 q^{6} + 11 q^{7} - 3 q^{8} + 12 q^{9} - 2 q^{10} + 2 q^{11} + 4 q^{12} + 8 q^{13} - 9 q^{14} - q^{15} - 2 q^{16} - 3 q^{17} - 9 q^{18} + 17 q^{19} - 2 q^{20} - 2 q^{21} + 3 q^{22} - 4 q^{23} - 15 q^{24} + 6 q^{25} - 13 q^{26} + 19 q^{27} + 16 q^{28} - 9 q^{29} - 19 q^{30} + 20 q^{31} - 16 q^{32} - 2 q^{33} - 8 q^{34} - 12 q^{35} - 12 q^{36} + 12 q^{37} - 15 q^{38} - 9 q^{39} - 15 q^{40} - 18 q^{41} - 31 q^{42} + 11 q^{43} - 24 q^{44} - 15 q^{45} - 13 q^{46} + 5 q^{47} - 22 q^{48} + q^{49} - 24 q^{50} - 13 q^{51} + 9 q^{52} - 6 q^{53} - 15 q^{54} - 17 q^{55} - 6 q^{56} + 3 q^{57} - 15 q^{58} - q^{59} - 23 q^{60} + 4 q^{61} - 16 q^{62} - 11 q^{63} - 17 q^{64} - 17 q^{65} + 24 q^{66} + 46 q^{67} + 15 q^{68} - 21 q^{69} + 25 q^{70} - 5 q^{71} + 7 q^{72} + 25 q^{73} + 3 q^{74} + 46 q^{75} + 49 q^{76} - 14 q^{77} + 15 q^{78} - q^{79} + 9 q^{80} + 22 q^{81} + 52 q^{82} + 28 q^{83} - 14 q^{84} - 30 q^{85} + 27 q^{86} - 7 q^{87} + 9 q^{88} - 6 q^{89} - 10 q^{90} + 51 q^{91} + 32 q^{92} + 15 q^{93} - 3 q^{94} - 13 q^{95} + 75 q^{96} + 2 q^{97} + 22 q^{98} - q^{99}+O(q^{100})$$ 10 * q + 10 * q^3 + 10 * q^4 + 2 * q^5 - 4 * q^6 + 11 * q^7 - 3 * q^8 + 12 * q^9 - 2 * q^10 + 2 * q^11 + 4 * q^12 + 8 * q^13 - 9 * q^14 - q^15 - 2 * q^16 - 3 * q^17 - 9 * q^18 + 17 * q^19 - 2 * q^20 - 2 * q^21 + 3 * q^22 - 4 * q^23 - 15 * q^24 + 6 * q^25 - 13 * q^26 + 19 * q^27 + 16 * q^28 - 9 * q^29 - 19 * q^30 + 20 * q^31 - 16 * q^32 - 2 * q^33 - 8 * q^34 - 12 * q^35 - 12 * q^36 + 12 * q^37 - 15 * q^38 - 9 * q^39 - 15 * q^40 - 18 * q^41 - 31 * q^42 + 11 * q^43 - 24 * q^44 - 15 * q^45 - 13 * q^46 + 5 * q^47 - 22 * q^48 + q^49 - 24 * q^50 - 13 * q^51 + 9 * q^52 - 6 * q^53 - 15 * q^54 - 17 * q^55 - 6 * q^56 + 3 * q^57 - 15 * q^58 - q^59 - 23 * q^60 + 4 * q^61 - 16 * q^62 - 11 * q^63 - 17 * q^64 - 17 * q^65 + 24 * q^66 + 46 * q^67 + 15 * q^68 - 21 * q^69 + 25 * q^70 - 5 * q^71 + 7 * q^72 + 25 * q^73 + 3 * q^74 + 46 * q^75 + 49 * q^76 - 14 * q^77 + 15 * q^78 - q^79 + 9 * q^80 + 22 * q^81 + 52 * q^82 + 28 * q^83 - 14 * q^84 - 30 * q^85 + 27 * q^86 - 7 * q^87 + 9 * q^88 - 6 * q^89 - 10 * q^90 + 51 * q^91 + 32 * q^92 + 15 * q^93 - 3 * q^94 - 13 * q^95 + 75 * q^96 + 2 * q^97 + 22 * q^98 - q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 15x^{8} - x^{7} + 78x^{6} + 7x^{5} - 165x^{4} - 15x^{3} + 123x^{2} + 9x - 26$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$( \nu^{8} - 2\nu^{7} - 10\nu^{6} + 17\nu^{5} + 30\nu^{4} - 36\nu^{3} - 27\nu^{2} + 7\nu + 6 ) / 4$$ (v^8 - 2*v^7 - 10*v^6 + 17*v^5 + 30*v^4 - 36*v^3 - 27*v^2 + 7*v + 6) / 4 $$\beta_{4}$$ $$=$$ $$( \nu^{8} - 10\nu^{6} - 3\nu^{5} + 26\nu^{4} + 14\nu^{3} - 11\nu^{2} - 9\nu - 2 ) / 2$$ (v^8 - 10*v^6 - 3*v^5 + 26*v^4 + 14*v^3 - 11*v^2 - 9*v - 2) / 2 $$\beta_{5}$$ $$=$$ $$( \nu^{8} - 12\nu^{6} - \nu^{5} + 44\nu^{4} + 2\nu^{3} - 53\nu^{2} + 5\nu + 16 ) / 2$$ (v^8 - 12*v^6 - v^5 + 44*v^4 + 2*v^3 - 53*v^2 + 5*v + 16) / 2 $$\beta_{6}$$ $$=$$ $$( \nu^{9} - 14\nu^{7} + \nu^{6} + 64\nu^{5} - 12\nu^{4} - 107\nu^{3} + 29\nu^{2} + 44\nu - 8 ) / 4$$ (v^9 - 14*v^7 + v^6 + 64*v^5 - 12*v^4 - 107*v^3 + 29*v^2 + 44*v - 8) / 4 $$\beta_{7}$$ $$=$$ $$( \nu^{9} - \nu^{8} - 12\nu^{7} + 11\nu^{6} + 47\nu^{5} - 46\nu^{4} - 67\nu^{3} + 80\nu^{2} + 21\nu - 30 ) / 4$$ (v^9 - v^8 - 12*v^7 + 11*v^6 + 47*v^5 - 46*v^4 - 67*v^3 + 80*v^2 + 21*v - 30) / 4 $$\beta_{8}$$ $$=$$ $$\nu^{7} - \nu^{6} - 10\nu^{5} + 8\nu^{4} + 27\nu^{3} - 18\nu^{2} - 14\nu + 8$$ v^7 - v^6 - 10*v^5 + 8*v^4 + 27*v^3 - 18*v^2 - 14*v + 8 $$\beta_{9}$$ $$=$$ $$( -\nu^{9} + \nu^{8} + 12\nu^{7} - 9\nu^{6} - 47\nu^{5} + 24\nu^{4} + 67\nu^{3} - 18\nu^{2} - 25\nu + 4 ) / 2$$ (-v^9 + v^8 + 12*v^7 - 9*v^6 - 47*v^5 + 24*v^4 + 67*v^3 - 18*v^2 - 25*v + 4) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + 4\beta_1$$ b9 + b8 + b7 + b6 - b4 + b3 + b2 + 4*b1 $$\nu^{4}$$ $$=$$ $$\beta_{9} + \beta_{8} + 2\beta_{6} - \beta_{4} + 7\beta_{2} + 14$$ b9 + b8 + 2*b6 - b4 + 7*b2 + 14 $$\nu^{5}$$ $$=$$ $$9\beta_{9} + 8\beta_{8} + 8\beta_{7} + 10\beta_{6} + \beta_{5} - 9\beta_{4} + 6\beta_{3} + 10\beta_{2} + 19\beta _1 + 2$$ 9*b9 + 8*b8 + 8*b7 + 10*b6 + b5 - 9*b4 + 6*b3 + 10*b2 + 19*b1 + 2 $$\nu^{6}$$ $$=$$ $$12\beta_{9} + 11\beta_{8} + 2\beta_{7} + 22\beta_{6} - 11\beta_{4} + 46\beta_{2} + 2\beta _1 + 74$$ 12*b9 + 11*b8 + 2*b7 + 22*b6 - 11*b4 + 46*b2 + 2*b1 + 74 $$\nu^{7}$$ $$=$$ $$67 \beta_{9} + 57 \beta_{8} + 55 \beta_{7} + 79 \beta_{6} + 10 \beta_{5} - 66 \beta_{4} + 33 \beta_{3} + \cdots + 28$$ 67*b9 + 57*b8 + 55*b7 + 79*b6 + 10*b5 - 66*b4 + 33*b3 + 81*b2 + 98*b1 + 28 $$\nu^{8}$$ $$=$$ $$107 \beta_{9} + 94 \beta_{8} + 30 \beta_{7} + 184 \beta_{6} + 3 \beta_{5} - 95 \beta_{4} + 4 \beta_{3} + \cdots + 417$$ 107*b9 + 94*b8 + 30*b7 + 184*b6 + 3*b5 - 95*b4 + 4*b3 + 305*b2 + 30*b1 + 417 $$\nu^{9}$$ $$=$$ $$469 \beta_{9} + 394 \beta_{8} + 363 \beta_{7} + 579 \beta_{6} + 76 \beta_{5} - 456 \beta_{4} + \cdots + 279$$ 469*b9 + 394*b8 + 363*b7 + 579*b6 + 76*b5 - 456*b4 + 185*b3 + 610*b2 + 538*b1 + 279

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
 2.62911 1.94873 1.85218 0.931586 0.530276 −0.669007 −0.896375 −1.82964 −2.22583 −2.27104
−2.62911 0.0119479 4.91223 0.817206 −0.0314124 2.54267 −7.65657 −2.99986 −2.14853
1.2 −1.94873 2.99027 1.79756 3.37189 −5.82723 −0.532209 0.394492 5.94168 −6.57091
1.3 −1.85218 2.60828 1.43059 −3.95640 −4.83102 3.45456 1.05466 3.80315 7.32799
1.4 −0.931586 −2.17005 −1.13215 −1.44818 2.02159 1.12960 2.91786 1.70911 1.34911
1.5 −0.530276 0.899248 −1.71881 3.03207 −0.476849 0.743459 1.97199 −2.19135 −1.60783
1.6 0.669007 1.75170 −1.55243 0.170148 1.17190 5.01834 −2.37660 0.0684478 0.113830
1.7 0.896375 3.41221 −1.19651 −0.448431 3.05862 −2.39168 −2.86527 8.64317 −0.401962
1.8 1.82964 0.661864 1.34760 2.39480 1.21098 −2.47846 −1.19366 −2.56194 4.38163
1.9 2.22583 −1.75215 2.95430 1.17649 −3.89998 3.66330 2.12410 0.0700264 2.61867
1.10 2.27104 1.58668 3.15762 −3.10959 3.60342 −0.149594 2.62899 −0.482441 −7.06199
 $$n$$: 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
$$197$$ $$-1$$

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 197.2.a.c 10
3.b odd 2 1 1773.2.a.f 10
4.b odd 2 1 3152.2.a.m 10
5.b even 2 1 4925.2.a.i 10
7.b odd 2 1 9653.2.a.j 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
197.2.a.c 10 1.a even 1 1 trivial
1773.2.a.f 10 3.b odd 2 1
3152.2.a.m 10 4.b odd 2 1
4925.2.a.i 10 5.b even 2 1
9653.2.a.j 10 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} - 15T_{2}^{8} + T_{2}^{7} + 78T_{2}^{6} - 7T_{2}^{5} - 165T_{2}^{4} + 15T_{2}^{3} + 123T_{2}^{2} - 9T_{2} - 26$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(197))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} - 15 T^{8} + \cdots - 26$$
$3$ $$T^{10} - 10 T^{9} + \cdots + 2$$
$5$ $$T^{10} - 2 T^{9} + \cdots + 32$$
$7$ $$T^{10} - 11 T^{9} + \cdots + 64$$
$11$ $$T^{10} - 2 T^{9} + \cdots - 5906$$
$13$ $$T^{10} - 8 T^{9} + \cdots + 448$$
$17$ $$T^{10} + 3 T^{9} + \cdots - 11008$$
$19$ $$T^{10} - 17 T^{9} + \cdots - 26944$$
$23$ $$T^{10} + 4 T^{9} + \cdots - 55696$$
$29$ $$T^{10} + 9 T^{9} + \cdots - 1849$$
$31$ $$T^{10} - 20 T^{9} + \cdots - 1018$$
$37$ $$T^{10} - 12 T^{9} + \cdots - 1031837$$
$41$ $$T^{10} + 18 T^{9} + \cdots - 12249251$$
$43$ $$T^{10} - 11 T^{9} + \cdots + 958064$$
$47$ $$T^{10} - 5 T^{9} + \cdots - 6076144$$
$53$ $$T^{10} + 6 T^{9} + \cdots + 24986$$
$59$ $$T^{10} + T^{9} + \cdots - 2663552$$
$61$ $$T^{10} - 4 T^{9} + \cdots + 7550167$$
$67$ $$T^{10} + \cdots + 142552394$$
$71$ $$T^{10} + \cdots - 112062458$$
$73$ $$T^{10} - 25 T^{9} + \cdots + 28387712$$
$79$ $$T^{10} + T^{9} + \cdots - 29837000$$
$83$ $$T^{10} - 28 T^{9} + \cdots + 303296$$
$89$ $$T^{10} + 6 T^{9} + \cdots + 57477472$$
$97$ $$T^{10} + \cdots - 151216646$$