Properties

 Label 290.2.b.b Level $290$ Weight $2$ Character orbit 290.b Analytic conductor $2.316$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [290,2,Mod(59,290)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(290, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("290.59");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$290 = 2 \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 290.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: $$2.31566165862$$ 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: $$x^{10} + 24x^{8} + 152x^{6} + 377x^{4} + 352x^{2} + 64$$ x^10 + 24*x^8 + 152*x^6 + 377*x^4 + 352*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 24x^{8} + 152x^{6} + 377x^{4} + 352x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 11 \nu^{9} + 200 \nu^{8} - 224 \nu^{7} + 4128 \nu^{6} - 968 \nu^{5} + 16384 \nu^{4} - 3059 \nu^{3} + 17480 \nu^{2} - 5240 \nu + 1696 ) / 1216$$ (-11*v^9 + 200*v^8 - 224*v^7 + 4128*v^6 - 968*v^5 + 16384*v^4 - 3059*v^3 + 17480*v^2 - 5240*v + 1696) / 1216 $$\beta_{3}$$ $$=$$ $$( 11 \nu^{9} + 200 \nu^{8} + 224 \nu^{7} + 4128 \nu^{6} + 968 \nu^{5} + 16384 \nu^{4} + 3059 \nu^{3} + 17480 \nu^{2} + 5240 \nu + 1696 ) / 1216$$ (11*v^9 + 200*v^8 + 224*v^7 + 4128*v^6 + 968*v^5 + 16384*v^4 + 3059*v^3 + 17480*v^2 + 5240*v + 1696) / 1216 $$\beta_{4}$$ $$=$$ $$( 17\nu^{8} + 360\nu^{6} + 1572\nu^{4} + 2033\nu^{2} + 360 ) / 76$$ (17*v^8 + 360*v^6 + 1572*v^4 + 2033*v^2 + 360) / 76 $$\beta_{5}$$ $$=$$ $$( 45\nu^{9} + 944\nu^{7} + 3960\nu^{5} + 4389\nu^{3} - 424\nu ) / 608$$ (45*v^9 + 944*v^7 + 3960*v^5 + 4389*v^3 - 424*v) / 608 $$\beta_{6}$$ $$=$$ $$( 55\nu^{9} + 1120\nu^{7} + 4232\nu^{5} + 4351\nu^{3} + 1272\nu ) / 608$$ (55*v^9 + 1120*v^7 + 4232*v^5 + 4351*v^3 + 1272*v) / 608 $$\beta_{7}$$ $$=$$ $$( - 117 \nu^{9} + 20 \nu^{8} - 2576 \nu^{7} + 352 \nu^{6} - 12728 \nu^{5} + 544 \nu^{4} - 20045 \nu^{3} - 76 \nu^{2} - 7288 \nu + 2176 ) / 1216$$ (-117*v^9 + 20*v^8 - 2576*v^7 + 352*v^6 - 12728*v^5 + 544*v^4 - 20045*v^3 - 76*v^2 - 7288*v + 2176) / 1216 $$\beta_{8}$$ $$=$$ $$( - 117 \nu^{9} - 20 \nu^{8} - 2576 \nu^{7} - 352 \nu^{6} - 12728 \nu^{5} - 544 \nu^{4} - 20045 \nu^{3} + 76 \nu^{2} - 7288 \nu - 2176 ) / 1216$$ (-117*v^9 - 20*v^8 - 2576*v^7 - 352*v^6 - 12728*v^5 - 544*v^4 - 20045*v^3 + 76*v^2 - 7288*v - 2176) / 1216 $$\beta_{9}$$ $$=$$ $$( - 32 \nu^{9} - 81 \nu^{8} - 700 \nu^{7} - 1760 \nu^{6} - 3424 \nu^{5} - 8344 \nu^{4} - 5776 \nu^{3} - 12217 \nu^{2} - 2828 \nu - 3128 ) / 304$$ (-32*v^9 - 81*v^8 - 700*v^7 - 1760*v^6 - 3424*v^5 - 8344*v^4 - 5776*v^3 - 12217*v^2 - 2828*v - 3128) / 304
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{9} + \beta_{7} - 2\beta_{4} + \beta_{2} + \beta _1 - 4$$ -b9 + b7 - 2*b4 + b2 + b1 - 4 $$\nu^{3}$$ $$=$$ $$-\beta_{8} - \beta_{7} + 4\beta_{6} - 7\beta_{5} - 2\beta_{3} + 2\beta_{2} - 8\beta_1$$ -b8 - b7 + 4*b6 - 7*b5 - 2*b3 + 2*b2 - 8*b1 $$\nu^{4}$$ $$=$$ $$18\beta_{9} - 6\beta_{8} - 12\beta_{7} + 41\beta_{4} - 4\beta_{3} - 22\beta_{2} - 18\beta _1 + 38$$ 18*b9 - 6*b8 - 12*b7 + 41*b4 - 4*b3 - 22*b2 - 18*b1 + 38 $$\nu^{5}$$ $$=$$ $$18\beta_{8} + 18\beta_{7} - 73\beta_{6} + 126\beta_{5} + 41\beta_{3} - 41\beta_{2} + 103\beta_1$$ 18*b8 + 18*b7 - 73*b6 + 126*b5 + 41*b3 - 41*b2 + 103*b1 $$\nu^{6}$$ $$=$$ $$-294\beta_{9} + 118\beta_{8} + 176\beta_{7} - 678\beta_{4} + 73\beta_{3} + 367\beta_{2} + 294\beta _1 - 531$$ -294*b9 + 118*b8 + 176*b7 - 678*b4 + 73*b3 + 367*b2 + 294*b1 - 531 $$\nu^{7}$$ $$=$$ $$-294\beta_{8} - 294\beta_{7} + 1176\beta_{6} - 2036\beta_{5} - 678\beta_{3} + 678\beta_{2} - 1561\beta_1$$ -294*b8 - 294*b7 + 1176*b6 - 2036*b5 - 678*b3 + 678*b2 - 1561*b1 $$\nu^{8}$$ $$=$$ $$4681 \beta_{9} - 1944 \beta_{8} - 2737 \beta_{7} + 10810 \beta_{4} - 1176 \beta_{3} - 5857 \beta_{2} - 4681 \beta _1 + 8188$$ 4681*b9 - 1944*b8 - 2737*b7 + 10810*b4 - 1176*b3 - 5857*b2 - 4681*b1 + 8188 $$\nu^{9}$$ $$=$$ $$4681\beta_{8} + 4681\beta_{7} - 18636\beta_{6} + 32319\beta_{5} + 10810\beta_{3} - 10810\beta_{2} + 24472\beta_1$$ 4681*b8 + 4681*b7 - 18636*b6 + 32319*b5 + 10810*b3 - 10810*b2 + 24472*b1

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/290\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$117$$ $$\chi(n)$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
59.1
 3.97530i 1.68193i − 0.485804i − 1.35864i − 1.81278i 1.81278i 1.35864i 0.485804i − 1.68193i − 3.97530i
1.00000i 2.97530i −1.00000 −0.639550 + 2.14266i −2.97530 3.57331i 1.00000i −5.85241 2.14266 + 0.639550i
59.2 1.00000i 0.681929i −1.00000 2.23558 0.0464742i −0.681929 0.936197i 1.00000i 2.53497 −0.0464742 2.23558i
59.3 1.00000i 1.48580i −1.00000 −1.29150 1.82539i 1.48580 4.37538i 1.00000i 0.792385 −1.82539 + 1.29150i
59.4 1.00000i 2.35864i −1.00000 1.32739 1.79945i 2.35864 4.21797i 1.00000i −2.56319 −1.79945 1.32739i
59.5 1.00000i 2.81278i −1.00000 −1.63193 + 1.52866i 2.81278 0.647892i 1.00000i −4.91176 1.52866 + 1.63193i
59.6 1.00000i 2.81278i −1.00000 −1.63193 1.52866i 2.81278 0.647892i 1.00000i −4.91176 1.52866 1.63193i
59.7 1.00000i 2.35864i −1.00000 1.32739 + 1.79945i 2.35864 4.21797i 1.00000i −2.56319 −1.79945 + 1.32739i
59.8 1.00000i 1.48580i −1.00000 −1.29150 + 1.82539i 1.48580 4.37538i 1.00000i 0.792385 −1.82539 1.29150i
59.9 1.00000i 0.681929i −1.00000 2.23558 + 0.0464742i −0.681929 0.936197i 1.00000i 2.53497 −0.0464742 + 2.23558i
59.10 1.00000i 2.97530i −1.00000 −0.639550 2.14266i −2.97530 3.57331i 1.00000i −5.85241 2.14266 0.639550i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 59.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 290.2.b.b 10
3.b odd 2 1 2610.2.e.i 10
4.b odd 2 1 2320.2.d.h 10
5.b even 2 1 inner 290.2.b.b 10
5.c odd 4 1 1450.2.a.t 5
5.c odd 4 1 1450.2.a.u 5
15.d odd 2 1 2610.2.e.i 10
20.d odd 2 1 2320.2.d.h 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
290.2.b.b 10 1.a even 1 1 trivial
290.2.b.b 10 5.b even 2 1 inner
1450.2.a.t 5 5.c odd 4 1
1450.2.a.u 5 5.c odd 4 1
2320.2.d.h 10 4.b odd 2 1
2320.2.d.h 10 20.d odd 2 1
2610.2.e.i 10 3.b odd 2 1
2610.2.e.i 10 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{10} + 25T_{3}^{8} + 224T_{3}^{6} + 849T_{3}^{4} + 1209T_{3}^{2} + 400$$ acting on $$S_{2}^{\mathrm{new}}(290, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{5}$$
$3$ $$T^{10} + 25 T^{8} + 224 T^{6} + \cdots + 400$$
$5$ $$T^{10} + 2 T^{8} - 18 T^{7} + \cdots + 3125$$
$7$ $$T^{10} + 51 T^{8} + 877 T^{6} + \cdots + 1600$$
$11$ $$(T^{5} - 27 T^{3} + 4 T^{2} + 172 T - 16)^{2}$$
$13$ $$T^{10} + 53 T^{8} + 356 T^{6} + \cdots + 64$$
$17$ $$T^{10} + 119 T^{8} + 5325 T^{6} + \cdots + 2930944$$
$19$ $$(T^{5} - 4 T^{4} - 56 T^{3} + 144 T^{2} + \cdots - 640)^{2}$$
$23$ $$T^{10} + 167 T^{8} + 9813 T^{6} + \cdots + 2611456$$
$29$ $$(T + 1)^{10}$$
$31$ $$(T^{5} - 5 T^{4} - 22 T^{3} + 175 T^{2} + \cdots + 184)^{2}$$
$37$ $$T^{10} + 212 T^{8} + 12608 T^{6} + \cdots + 262144$$
$41$ $$(T^{5} + 4 T^{4} - 76 T^{3} - 504 T^{2} + \cdots + 608)^{2}$$
$43$ $$T^{10} + 205 T^{8} + 11856 T^{6} + \cdots + 440896$$
$47$ $$T^{10} + 194 T^{8} + 11057 T^{6} + \cdots + 4096$$
$53$ $$T^{10} + 109 T^{8} + 1084 T^{6} + \cdots + 16$$
$59$ $$(T^{5} - 9 T^{4} - 79 T^{3} + 448 T^{2} + \cdots - 4000)^{2}$$
$61$ $$(T^{5} - 5 T^{4} - 169 T^{3} + 998 T^{2} + \cdots - 9808)^{2}$$
$67$ $$T^{10} + 564 T^{8} + 112832 T^{6} + \cdots + 1048576$$
$71$ $$(T^{5} + 6 T^{4} - 208 T^{3} - 1192 T^{2} + \cdots + 62464)^{2}$$
$73$ $$T^{10} + 543 T^{8} + \cdots + 3008303104$$
$79$ $$(T^{5} - 17 T^{4} + 48 T^{3} + 471 T^{2} + \cdots + 3020)^{2}$$
$83$ $$T^{10} + 684 T^{8} + \cdots + 3399356416$$
$89$ $$(T^{5} + 10 T^{4} - 96 T^{3} - 1088 T^{2} + \cdots - 160)^{2}$$
$97$ $$T^{10} + 623 T^{8} + \cdots + 3351946816$$