Properties

 Label 130.2.p.a Level $130$ Weight $2$ Character orbit 130.p Analytic conductor $1.038$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [130,2,Mod(7,130)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(130, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([3, 11]))

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

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

chi := DirichletCharacter("130.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$130 = 2 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 130.p (of order $$12$$, degree $$4$$, 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: $$1.03805522628$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 24x^{10} + 192x^{8} + 680x^{6} + 1104x^{4} + 672x^{2} + 4$$ x^12 + 24*x^10 + 192*x^8 + 680*x^6 + 1104*x^4 + 672*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} + (\beta_{11} + \beta_{8} + \cdots + \beta_{5}) q^{3}+ \cdots + (\beta_{11} - 2 \beta_{9} + \beta_{8} + \cdots - 4) q^{9}+O(q^{10})$$ q + b6 * q^2 + (b11 + b8 + b7 + b5) * q^3 + b3 * q^4 + (-b11 + b10 + b7 - b4 + b3 + b1 - 1) * q^5 - b8 * q^6 + (-b7 + b2 - b1) * q^7 + (b6 + b4) * q^8 + (b11 - 2*b9 + b8 + b7 - 2*b4 + 2*b3 + 2*b1 - 4) * q^9 $$q + \beta_{6} q^{2} + (\beta_{11} + \beta_{8} + \cdots + \beta_{5}) q^{3}+ \cdots + (3 \beta_{11} - \beta_{10} + 2 \beta_{8} + \cdots + 3) q^{99}+O(q^{100})$$ q + b6 * q^2 + (b11 + b8 + b7 + b5) * q^3 + b3 * q^4 + (-b11 + b10 + b7 - b4 + b3 + b1 - 1) * q^5 - b8 * q^6 + (-b7 + b2 - b1) * q^7 + (b6 + b4) * q^8 + (b11 - 2*b9 + b8 + b7 - 2*b4 + 2*b3 + 2*b1 - 4) * q^9 + (b11 + b7 + b2) * q^10 + (-b11 + b10 - b7 + b6 - b5 - b4 + b1) * q^11 + (b8 + b7 + b5 - b2 + b1) * q^12 + (2*b11 - b10 - b9 + b8 + b7 - b6 + b5 - b2 - 1) * q^13 + (-b10 - b8 - b7 + b4 - b3 + b2 - 2*b1 + 1) * q^14 + (2*b10 - b9 + b8 + b7 + 3*b6 - b4 - 2*b3 - b2 + 3*b1 - 1) * q^15 + (b3 - 1) * q^16 + (b11 - b10 + b9 - b7 - b6 + 2*b4 - 2*b1 + 2) * q^17 + (-b11 - b10 + 2*b9 - b8 - 2*b7 - 2*b6 - b5 + 3*b4 - b3 - 3*b1 + 3) * q^18 + (b11 - b9 + b8 - 3*b6 + b5 - b4 - b3 - b2 + b1 + 3) * q^19 + (-b11 + b9 - b8) * q^20 + (b11 - b10 - 2*b6 + b5 - b4 - b3 - 1) * q^21 + (-b10 - b7 - b5 + b2 - b1 + 1) * q^22 + (-4*b11 + b10 + 2*b9 - 2*b8 - b7 - b6 - 3*b5 + 2*b2 - 3*b1 + 1) * q^23 + b11 * q^24 + (-b11 + 2*b9 - b7 - 2*b6 - b5 + 2*b4 - 2*b3 - b2 - b1 + 1) * q^25 + (-b11 + b10 - b6 - 2*b2 + b1) * q^26 + (-4*b11 - 2*b8 - 2*b7 + b6 - 2*b5 + b4 - 2*b2 + 2*b1 - 1) * q^27 + (b5 - b1) * q^28 + (-2*b11 - 2*b10 + 2*b9 - 2*b8 - 2*b7 + b6 + 2*b4 - b3 - 2*b1 + 3) * q^29 + (-b10 - 3*b6 - b5 - 3*b4 + 2*b3) * q^30 + (b11 + b9 + b8 + 3*b6 - b5 - 3*b3 + b2) * q^31 + b4 * q^32 + (b11 + b9 - 2*b8 + 3*b4 + 2*b2 - 2*b1) * q^33 + (-b11 + b10 - b9 + 2*b6 - b2 + 2*b1 - 2) * q^34 + (b11 + b7 + 3*b5 - b4 - 2*b3 - b2 - b1) * q^35 + (2*b10 - 2*b9 + b8 + b7 + 2*b6 - 2*b4 + 3*b1 - 4) * q^36 + (2*b11 - 2*b10 - 2*b9 - b8 - b6 + b2 - b1) * q^37 + (-b10 - b7 + 2*b6 - 4*b3 - b2 - b1 + 2) * q^38 + (-3*b11 - b10 - b9 - b8 - b7 - b6 - b5 - 2*b4 + 3*b3 - b2 - 1) * q^39 + (b11 + b8 + b7 + b5 + b1) * q^40 + (-2*b11 - b10 + 2*b9 - b8 - 2*b7 + b6 + 4*b4 - 3*b1 + 3) * q^41 + (-b10 + b9 - b7 - 2*b6 + b5 + b4 - 3*b3 - b2 - b1 + 3) * q^42 + (-2*b11 + 2*b10 - b9 - b8 - 2*b7 - 3*b5 - b4 + b3 - 2) * q^43 + (-b11 + b9 - b8 - 2*b7 + b6 - b5 + b4 + b2 - 2*b1 + 1) * q^44 + (2*b11 - 3*b8 - 3*b7 - b5 + b4 + 2*b3 + 2*b2 - 2*b1) * q^45 + (b11 + 3*b10 - 2*b9 + 3*b7 + b6 - b5 - 4*b4 + 2*b3 + 4*b2 + 2*b1 - 4) * q^46 + (2*b11 + 3*b7 + b6 + 3*b5 - b4 - b2 + 1) * q^47 + (-b11 - b2 + b1) * q^48 + (-2*b11 + 2*b10 + 2*b9 - b8 + b7 + 2*b6 - 3*b5 + 2*b4 + 2*b2 + 2*b1) * q^49 + (2*b10 - 2*b9 + b8 + b7 - b6 - b5 - 4*b4 + 4*b1 - 4) * q^50 + (3*b11 - 2*b10 + 4*b8 + 2*b7 - 4*b6 + 2*b5 - 2*b4 + 4*b3 - b1 - 1) * q^51 + (b11 + b10 - 2*b9 + 2*b8 + 2*b7 - 2*b4 - b2 + 3*b1 - 2) * q^52 + (2*b11 - 3*b9 + 3*b8 + 5*b7 + 2*b5 - 2*b2 + 5*b1) * q^53 + (2*b11 + 4*b8 - b6 + b3 - 1) * q^54 + (-b10 + 2*b9 - b8 - b7 - b6 + b4 + b3 + 3*b2 - 5*b1 - 1) * q^55 + (b11 - b9 + b7 + b5) * q^56 + (2*b11 - b10 + 3*b8 + b7 + 2*b5 + b4 + b3 + 2*b2 - 3*b1) * q^57 + (2*b11 + 2*b8 + 2*b6 + 2*b5 + b4 + b3) * q^58 + (b11 - 2*b10 + 2*b9 - b8 + 2*b6 + 2*b5 + b4 - b3 + b2 - b1 + 4) * q^59 + (-b11 + b10 + b9 - b7 + 2*b6 - b5 + 2*b4 - 2*b3 + 3) * q^60 + (b11 - 2*b10 + b9 + b7 + 2*b5 + 2*b4 + 2*b3 - 3*b2 + b1 + 2) * q^61 + (-2*b11 + b9 - 2*b8 - 2*b7 - 3*b6 - 2*b5 - 3*b4 + 3*b3 + b2) * q^62 + (-2*b11 + 2*b10 - 2*b9 + 2*b7 - 2*b4 + 3*b3 - 2*b2 + 4*b1 - 1) * q^63 - q^64 + (b11 + b10 - 3*b9 + 5*b6 - b5 + 3*b3 - b2 + 4*b1 - 1) * q^65 + (-b11 + 2*b7 + 2*b5 - b2 + 2*b1 - 3) * q^66 + (2*b10 - 2*b9 + b8 + b7 - 4*b6 - b5 - 2*b4 + 2*b3 + 2*b1 - 2) * q^67 + (b11 - b10 + b8 - 2*b6 + b3 - b1) * q^68 + (5*b11 + 4*b10 - 2*b9 + 7*b8 + 6*b7 + 2*b6 + 4*b5 - 3*b4 - 5*b3 - 3*b2 + 7*b1 - 4) * q^69 + (2*b11 - 2*b9 + b8 + 3*b7 - 2*b6 + 3*b5 - 2*b4 - 2*b2 + 2*b1 + 1) * q^70 + (b9 - 3*b8 + 2*b6 + b5 + 2) * q^71 + (-2*b10 + b9 - b8 - b7 - 4*b6 - b5 + b2 - 3*b1 + 2) * q^72 + (2*b11 - b10 + 3*b8 - b7 + b6 + 4*b5 + 2*b4 - b3 - 4*b2 + b1 + 1) * q^73 + (-2*b11 + 2*b9 - b7 + b5 + 2*b4 - b3 - 2*b2 - 2*b1 + 2) * q^74 + (4*b11 + b10 - b9 + 4*b8 + b7 + b6 - b5 - b4 + 4*b3 + 3*b2 - b1 - 4) * q^75 + (b10 - b9 + b8 - 2*b6 - 4*b4 + 3*b3 - b2 + b1) * q^76 + (-2*b11 + 2*b10 - b8 - b7 + 4*b6 - 3*b5 - b4 + 5*b3 + b2 - b1 - 1) * q^77 + (2*b11 + b10 + 2*b8 + 2*b6 + 3*b4 + b2 - 2*b1 + 2) * q^78 + (-b11 - 3*b10 - 5*b8 - 2*b7 - 2*b6 - 3*b5 + b4 - 5*b3 + 3*b2 - 4*b1 + 4) * q^79 + (-b10 + b9 - b8 - b7 + b4 - b3 - b1 + 1) * q^80 + (-b11 + b10 + b9 - 3*b8 + 4*b6 - b5 + 7*b4 - 5*b3 + 3*b2 - 2*b1 + 5) * q^81 + (2*b11 + b10 - 2*b9 + b8 + b7 + 3*b6 + b5 + 2*b3 + b2 + b1 - 4) * q^82 + (-2*b11 - b10 + 2*b9 - b8 + 3*b6 + b5 - 2*b4 - b3 + b2 + b1 - 2) * q^83 + (b11 - b9 + b8 + b5 - 2*b4 - 2*b3 - b2 + b1) * q^84 + (-2*b11 - b9 + b7 + 4*b6 + b5 - b4 - b2 + 3*b1 - 2) * q^85 + (-b11 + b10 - 2*b9 + b8 + b7 - b6 - 2*b5 - 2*b4 + b3 + b2 + b1 - 2) * q^86 + (-b11 + 2*b10 + 2*b9 + b8 + 6*b6 + 2*b5 + 6*b4 - 6*b3 - b2 + b1 + 4) * q^87 + (-b9 + b6 + b3 + b2 - 1) * q^88 + (b10 - 2*b8 - b7 + 3*b6 - 3*b5 + b4 + 4*b3 + 4*b2 - 3*b1 - 2) * q^89 + (-3*b11 - 2*b10 + b8 + 2*b6 + 2*b5 + 4*b4 - 2*b3 - 3*b2 + b1 + 1) * q^90 + (-3*b10 - 3*b7 - 5*b6 - 6*b3 - b1 + 4) * q^91 + (-2*b11 - 2*b10 + 3*b9 - 4*b8 - 2*b6 - b5 + b4 - b3 + 3*b2 - 4*b1 + 3) * q^92 + (b11 - b10 - b9 - 6*b8 - b7 - 3*b6 - 4*b5 - 5*b4 + 7*b3 + 5*b2 - 2*b1 - 7) * q^93 + (b11 + b10 - b9 + b8 + 4*b7 + b6 + 3*b5 - b4 + 2*b3 - 3*b2 + 4*b1) * q^94 + (-5*b11 + 4*b10 - b9 + 3*b6 - 5*b4 + 6*b3 - 2*b2 + 2*b1 - 3) * q^95 + (b11 + b8) * q^96 + (-b9 + b8 - b4 + 6*b3 - b2 + b1 - 12) * q^97 + (-b11 + 2*b9 - b8 - 2*b5 - 2*b4 + 2*b3 + 3*b2 + b1 - 4) * q^98 + (3*b11 - b10 + 2*b8 - 7*b6 + b5 - b4 + 4*b3 - 2*b1 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 6 q^{4} - 24 q^{9}+O(q^{10})$$ 12 * q + 6 * q^4 - 24 * q^9 $$12 q + 6 q^{4} - 24 q^{9} + 6 q^{11} - 12 q^{13} - 6 q^{15} - 6 q^{16} + 12 q^{17} + 12 q^{18} + 36 q^{19} - 6 q^{20} - 24 q^{21} + 6 q^{22} + 6 q^{23} - 12 q^{25} + 6 q^{26} - 12 q^{27} + 6 q^{29} + 6 q^{30} - 24 q^{31} - 6 q^{33} - 12 q^{34} - 12 q^{35} - 24 q^{36} - 6 q^{38} + 6 q^{39} + 18 q^{41} + 6 q^{42} + 6 q^{44} + 12 q^{45} - 6 q^{46} + 12 q^{47} - 24 q^{50} - 6 q^{52} + 18 q^{53} - 6 q^{54} - 24 q^{55} + 6 q^{56} + 6 q^{58} + 18 q^{59} + 24 q^{60} + 18 q^{61} + 12 q^{62} + 30 q^{63} - 12 q^{64} + 30 q^{65} - 36 q^{66} + 12 q^{67} - 42 q^{69} + 24 q^{70} + 18 q^{71} + 6 q^{72} + 6 q^{74} - 12 q^{75} + 30 q^{76} + 30 q^{77} + 30 q^{78} - 6 q^{80} + 30 q^{81} - 18 q^{82} - 48 q^{83} - 6 q^{84} - 18 q^{85} + 12 q^{87} + 6 q^{89} - 12 q^{90} - 6 q^{91} - 42 q^{93} + 24 q^{94} + 30 q^{95} - 102 q^{97} - 48 q^{98} + 54 q^{99}+O(q^{100})$$ 12 * q + 6 * q^4 - 24 * q^9 + 6 * q^11 - 12 * q^13 - 6 * q^15 - 6 * q^16 + 12 * q^17 + 12 * q^18 + 36 * q^19 - 6 * q^20 - 24 * q^21 + 6 * q^22 + 6 * q^23 - 12 * q^25 + 6 * q^26 - 12 * q^27 + 6 * q^29 + 6 * q^30 - 24 * q^31 - 6 * q^33 - 12 * q^34 - 12 * q^35 - 24 * q^36 - 6 * q^38 + 6 * q^39 + 18 * q^41 + 6 * q^42 + 6 * q^44 + 12 * q^45 - 6 * q^46 + 12 * q^47 - 24 * q^50 - 6 * q^52 + 18 * q^53 - 6 * q^54 - 24 * q^55 + 6 * q^56 + 6 * q^58 + 18 * q^59 + 24 * q^60 + 18 * q^61 + 12 * q^62 + 30 * q^63 - 12 * q^64 + 30 * q^65 - 36 * q^66 + 12 * q^67 - 42 * q^69 + 24 * q^70 + 18 * q^71 + 6 * q^72 + 6 * q^74 - 12 * q^75 + 30 * q^76 + 30 * q^77 + 30 * q^78 - 6 * q^80 + 30 * q^81 - 18 * q^82 - 48 * q^83 - 6 * q^84 - 18 * q^85 + 12 * q^87 + 6 * q^89 - 12 * q^90 - 6 * q^91 - 42 * q^93 + 24 * q^94 + 30 * q^95 - 102 * q^97 - 48 * q^98 + 54 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 24x^{10} + 192x^{8} + 680x^{6} + 1104x^{4} + 672x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$( 7\nu^{10} + 136\nu^{8} + 668\nu^{6} + 702\nu^{4} + 176\nu^{2} + 380\nu + 1728 ) / 760$$ (7*v^10 + 136*v^8 + 668*v^6 + 702*v^4 + 176*v^2 + 380*v + 1728) / 760 $$\beta_{2}$$ $$=$$ $$( 7\nu^{10} + 136\nu^{8} + 668\nu^{6} + 702\nu^{4} + 176\nu^{2} - 380\nu + 1728 ) / 760$$ (7*v^10 + 136*v^8 + 668*v^6 + 702*v^4 + 176*v^2 - 380*v + 1728) / 760 $$\beta_{3}$$ $$=$$ $$( 4\nu^{11} + 87\nu^{9} + 576\nu^{7} + 1504\nu^{5} + 1542\nu^{3} + 456\nu + 20 ) / 40$$ (4*v^11 + 87*v^9 + 576*v^7 + 1504*v^5 + 1542*v^3 + 456*v + 20) / 40 $$\beta_{4}$$ $$=$$ $$( - 60 \nu^{11} + 36 \nu^{10} - 1315 \nu^{9} + 808 \nu^{8} - 8820 \nu^{7} + 5634 \nu^{6} - 23280 \nu^{5} + \cdots + 744 ) / 760$$ (-60*v^11 + 36*v^10 - 1315*v^9 + 808*v^8 - 8820*v^7 + 5634*v^6 - 23280*v^5 + 15336*v^4 - 23250*v^3 + 14368*v^2 - 5040*v + 744) / 760 $$\beta_{5}$$ $$=$$ $$( - 11 \nu^{11} + 89 \nu^{10} - 173 \nu^{9} + 1892 \nu^{8} - 154 \nu^{7} + 11886 \nu^{6} + 5194 \nu^{5} + \cdots + 1016 ) / 760$$ (-11*v^11 + 89*v^10 - 173*v^9 + 1892*v^8 - 154*v^7 + 11886*v^6 + 5194*v^5 + 27654*v^4 + 19402*v^3 + 21292*v^2 + 19216*v + 1016) / 760 $$\beta_{6}$$ $$=$$ $$( - 60 \nu^{11} - 36 \nu^{10} - 1315 \nu^{9} - 808 \nu^{8} - 8820 \nu^{7} - 5634 \nu^{6} - 23280 \nu^{5} + \cdots - 744 ) / 760$$ (-60*v^11 - 36*v^10 - 1315*v^9 - 808*v^8 - 8820*v^7 - 5634*v^6 - 23280*v^5 - 15336*v^4 - 23250*v^3 - 14368*v^2 - 5040*v - 744) / 760 $$\beta_{7}$$ $$=$$ $$( 11 \nu^{11} + 82 \nu^{10} + 173 \nu^{9} + 1756 \nu^{8} + 154 \nu^{7} + 11218 \nu^{6} - 5194 \nu^{5} + \cdots - 712 ) / 760$$ (11*v^11 + 82*v^10 + 173*v^9 + 1756*v^8 + 154*v^7 + 11218*v^6 - 5194*v^5 + 26952*v^4 - 19402*v^3 + 21116*v^2 - 19596*v - 712) / 760 $$\beta_{8}$$ $$=$$ $$( - 36 \nu^{11} + 125 \nu^{10} - 808 \nu^{9} + 2700 \nu^{8} - 5634 \nu^{7} + 17520 \nu^{6} - 15336 \nu^{5} + \cdots + 240 ) / 760$$ (-36*v^11 + 125*v^10 - 808*v^9 + 2700*v^8 - 5634*v^7 + 17520*v^6 - 15336*v^5 + 42990*v^4 - 14368*v^3 + 35280*v^2 - 1504*v + 240) / 760 $$\beta_{9}$$ $$=$$ $$( 3 \nu^{11} - 134 \nu^{10} + 99 \nu^{9} - 2902 \nu^{8} + 1182 \nu^{7} - 18976 \nu^{6} + 6218 \nu^{5} + \cdots - 1756 ) / 760$$ (3*v^11 - 134*v^10 + 99*v^9 - 2902*v^8 + 1182*v^7 - 18976*v^6 + 6218*v^5 - 47584*v^4 + 14054*v^3 - 41152*v^2 + 10892*v - 1756) / 760 $$\beta_{10}$$ $$=$$ $$( - 105 \nu^{11} - 178 \nu^{10} - 2135 \nu^{9} - 3784 \nu^{8} - 11920 \nu^{7} - 23772 \nu^{6} + \cdots - 132 ) / 760$$ (-105*v^11 - 178*v^10 - 2135*v^9 - 3784*v^8 - 11920*v^7 - 23772*v^6 - 18890*v^5 - 55308*v^4 + 9330*v^3 - 42204*v^2 + 28040*v - 132) / 760 $$\beta_{11}$$ $$=$$ $$( -421\nu^{10} - 9048\nu^{8} - 58144\nu^{6} - 140586\nu^{4} - 112968\nu^{2} + 380\nu - 784 ) / 760$$ (-421*v^10 - 9048*v^8 - 58144*v^6 - 140586*v^4 - 112968*v^2 + 380*v - 784) / 760
 $$\nu$$ $$=$$ $$-\beta_{2} + \beta_1$$ -b2 + b1 $$\nu^{2}$$ $$=$$ $$\beta_{11} + 2\beta_{7} - \beta_{6} + 2\beta_{5} + \beta_{4} + \beta _1 - 4$$ b11 + 2*b7 - b6 + 2*b5 + b4 + b1 - 4 $$\nu^{3}$$ $$=$$ $$- 2 \beta_{11} - 4 \beta_{8} - 2 \beta_{7} + 5 \beta_{6} - 2 \beta_{5} + 5 \beta_{4} + 6 \beta_{3} + \cdots - 3$$ -2*b11 - 4*b8 - 2*b7 + 5*b6 - 2*b5 + 5*b4 + 6*b3 + 10*b2 - 10*b1 - 3 $$\nu^{4}$$ $$=$$ $$- 13 \beta_{11} + 2 \beta_{10} - 4 \beta_{9} + 2 \beta_{8} - 27 \beta_{7} + 14 \beta_{6} - 29 \beta_{5} + \cdots + 28$$ -13*b11 + 2*b10 - 4*b9 + 2*b8 - 27*b7 + 14*b6 - 29*b5 - 16*b4 + 2*b3 + 3*b2 - 7*b1 + 28 $$\nu^{5}$$ $$=$$ $$32 \beta_{11} - 5 \beta_{10} + 59 \beta_{8} + 24 \beta_{7} - 77 \beta_{6} + 35 \beta_{5} - 72 \beta_{4} + \cdots + 50$$ 32*b11 - 5*b10 + 59*b8 + 24*b7 - 77*b6 + 35*b5 - 72*b4 - 95*b3 - 118*b2 + 110*b1 + 50 $$\nu^{6}$$ $$=$$ $$168 \beta_{11} - 24 \beta_{10} + 48 \beta_{9} - 24 \beta_{8} + 352 \beta_{7} - 174 \beta_{6} + \cdots - 292$$ 168*b11 - 24*b10 + 48*b9 - 24*b8 + 352*b7 - 174*b6 + 376*b5 + 198*b4 - 24*b3 - 56*b2 + 80*b1 - 292 $$\nu^{7}$$ $$=$$ $$- 422 \beta_{11} + 96 \beta_{10} - 748 \beta_{8} - 270 \beta_{7} + 1008 \beta_{6} - 478 \beta_{5} + \cdots - 680$$ -422*b11 + 96*b10 - 748*b8 - 270*b7 + 1008*b6 - 478*b5 + 912*b4 + 1264*b3 + 1440*b2 - 1288*b1 - 680 $$\nu^{8}$$ $$=$$ $$- 2122 \beta_{11} + 270 \beta_{10} - 540 \beta_{9} + 270 \beta_{8} - 4460 \beta_{7} + 2132 \beta_{6} + \cdots + 3460$$ -2122*b11 + 270*b10 - 540*b9 + 270*b8 - 4460*b7 + 2132*b6 - 4730*b5 - 2402*b4 + 270*b3 + 792*b2 - 1006*b1 + 3460 $$\nu^{9}$$ $$=$$ $$5316 \beta_{11} - 1368 \beta_{10} + 9264 \beta_{8} + 3156 \beta_{7} - 12698 \beta_{6} + 6108 \beta_{5} + \cdots + 8706$$ 5316*b11 - 1368*b10 + 9264*b8 + 3156*b7 - 12698*b6 + 6108*b5 - 11330*b4 - 16044*b3 - 17724*b2 + 15564*b1 + 8706 $$\nu^{10}$$ $$=$$ $$26474 \beta_{11} - 3156 \beta_{10} + 6312 \beta_{9} - 3156 \beta_{8} + 55718 \beta_{7} - 26196 \beta_{6} + \cdots - 42312$$ 26474*b11 - 3156*b10 + 6312*b9 - 3156*b8 + 55718*b7 - 26196*b6 + 58874*b5 + 29352*b4 - 3156*b3 - 10290*b2 + 12642*b1 - 42312 $$\nu^{11}$$ $$=$$ $$- 66116 \beta_{11} + 17810 \beta_{10} - 114422 \beta_{8} - 38016 \beta_{7} + 158054 \beta_{6} + \cdots - 109084$$ -66116*b11 + 17810*b10 - 114422*b8 - 38016*b7 + 158054*b6 - 76406*b5 + 140244*b4 + 200358*b3 + 218764*b2 - 190664*b1 - 109084

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/130\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$-\beta_{4} - \beta_{6}$$ $$\beta_{4}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 1.55227i 0.0775341i − 1.62980i 3.51623i − 1.32083i − 2.19540i − 1.55227i − 0.0775341i 1.62980i − 3.51623i 1.32083i 2.19540i
−0.866025 0.500000i −0.776134 2.89657i 0.500000 + 0.866025i 2.12044 0.709753i −0.776134 + 2.89657i −0.638961 1.10671i 1.00000i −5.18966 + 2.99625i −2.19123 0.445554i
7.2 −0.866025 0.500000i −0.0387670 0.144681i 0.500000 + 0.866025i 0.105914 + 2.23356i −0.0387670 + 0.144681i 1.10259 + 1.90974i 1.00000i 2.57865 1.48878i 1.02506 1.98727i
7.3 −0.866025 0.500000i 0.814901 + 3.04125i 0.500000 + 0.866025i −2.22635 + 0.208246i 0.814901 3.04125i 0.402397 + 0.696972i 1.00000i −5.98707 + 3.45663i 2.03220 + 0.932829i
37.1 0.866025 0.500000i −1.75811 0.471085i 0.500000 0.866025i −1.28703 1.82854i −1.75811 + 0.471085i 1.48736 2.57618i 1.00000i 0.270964 + 0.156441i −2.02887 0.940048i
37.2 0.866025 0.500000i 0.660414 + 0.176957i 0.500000 0.866025i 0.483457 + 2.18318i 0.660414 0.176957i 0.189447 0.328132i 1.00000i −2.19324 1.26627i 1.51028 + 1.64896i
37.3 0.866025 0.500000i 1.09770 + 0.294128i 0.500000 0.866025i 0.803571 2.08669i 1.09770 0.294128i −2.54283 + 4.40431i 1.00000i −1.47964 0.854273i −0.347432 2.20891i
93.1 −0.866025 + 0.500000i −0.776134 + 2.89657i 0.500000 0.866025i 2.12044 + 0.709753i −0.776134 2.89657i −0.638961 + 1.10671i 1.00000i −5.18966 2.99625i −2.19123 + 0.445554i
93.2 −0.866025 + 0.500000i −0.0387670 + 0.144681i 0.500000 0.866025i 0.105914 2.23356i −0.0387670 0.144681i 1.10259 1.90974i 1.00000i 2.57865 + 1.48878i 1.02506 + 1.98727i
93.3 −0.866025 + 0.500000i 0.814901 3.04125i 0.500000 0.866025i −2.22635 0.208246i 0.814901 + 3.04125i 0.402397 0.696972i 1.00000i −5.98707 3.45663i 2.03220 0.932829i
123.1 0.866025 + 0.500000i −1.75811 + 0.471085i 0.500000 + 0.866025i −1.28703 + 1.82854i −1.75811 0.471085i 1.48736 + 2.57618i 1.00000i 0.270964 0.156441i −2.02887 + 0.940048i
123.2 0.866025 + 0.500000i 0.660414 0.176957i 0.500000 + 0.866025i 0.483457 2.18318i 0.660414 + 0.176957i 0.189447 + 0.328132i 1.00000i −2.19324 + 1.26627i 1.51028 1.64896i
123.3 0.866025 + 0.500000i 1.09770 0.294128i 0.500000 + 0.866025i 0.803571 + 2.08669i 1.09770 + 0.294128i −2.54283 4.40431i 1.00000i −1.47964 + 0.854273i −0.347432 + 2.20891i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.t even 12 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.2.p.a 12
5.b even 2 1 650.2.t.e 12
5.c odd 4 1 130.2.s.a yes 12
5.c odd 4 1 650.2.w.e 12
13.f odd 12 1 130.2.s.a yes 12
65.o even 12 1 650.2.t.e 12
65.s odd 12 1 650.2.w.e 12
65.t even 12 1 inner 130.2.p.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.p.a 12 1.a even 1 1 trivial
130.2.p.a 12 65.t even 12 1 inner
130.2.s.a yes 12 5.c odd 4 1
130.2.s.a yes 12 13.f odd 12 1
650.2.t.e 12 5.b even 2 1
650.2.t.e 12 65.o even 12 1
650.2.w.e 12 5.c odd 4 1
650.2.w.e 12 65.s odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + 12 T_{3}^{10} + 4 T_{3}^{9} + 24 T_{3}^{8} + 24 T_{3}^{7} - 280 T_{3}^{6} + 48 T_{3}^{5} + \cdots + 4$$ acting on $$S_{2}^{\mathrm{new}}(130, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{3}$$
$3$ $$T^{12} + 12 T^{10} + \cdots + 4$$
$5$ $$T^{12} + 6 T^{10} + \cdots + 15625$$
$7$ $$T^{12} + 21 T^{10} + \cdots + 169$$
$11$ $$T^{12} - 6 T^{11} + \cdots + 6889$$
$13$ $$T^{12} + 12 T^{11} + \cdots + 4826809$$
$17$ $$T^{12} - 12 T^{11} + \cdots + 4$$
$19$ $$T^{12} - 36 T^{11} + \cdots + 221841$$
$23$ $$T^{12} - 6 T^{11} + \cdots + 23078416$$
$29$ $$T^{12} - 6 T^{11} + \cdots + 8248384$$
$31$ $$T^{12} + 24 T^{11} + \cdots + 26152996$$
$37$ $$T^{12} + 93 T^{10} + \cdots + 769129$$
$41$ $$T^{12} - 18 T^{11} + \cdots + 1690000$$
$43$ $$T^{12} - 36 T^{10} + \cdots + 22886656$$
$47$ $$(T^{6} - 6 T^{5} + \cdots - 6047)^{2}$$
$53$ $$T^{12} - 18 T^{11} + \cdots + 36881329$$
$59$ $$T^{12} - 18 T^{11} + \cdots + 576$$
$61$ $$T^{12} + \cdots + 307721764$$
$67$ $$T^{12} - 12 T^{11} + \cdots + 2304$$
$71$ $$T^{12} + \cdots + 880427584$$
$73$ $$T^{12} + \cdots + 704902500$$
$79$ $$T^{12} + \cdots + 1427177284$$
$83$ $$(T^{6} + 24 T^{5} + \cdots - 31050)^{2}$$
$89$ $$T^{12} - 6 T^{11} + \cdots + 39828721$$
$97$ $$T^{12} + \cdots + 553217613796$$