# Properties

 Label 130.2.l.b Level $130$ Weight $2$ Character orbit 130.l Analytic conductor $1.038$ Analytic rank $0$ Dimension $8$ 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(101,130)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(130, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5]))

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

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

chi := DirichletCharacter("130.101");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$130 = 2 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 130.l (of order $$6$$, degree $$2$$, 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.22581504.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16$$ x^8 - 4*x^7 + 5*x^6 + 2*x^5 - 11*x^4 + 4*x^3 + 20*x^2 - 32*x + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{7} + 2\nu^{6} - \nu^{5} - 4\nu^{4} + 3\nu^{3} + 2\nu^{2} - 8\nu + 8 ) / 8$$ (-v^7 + 2*v^6 - v^5 - 4*v^4 + 3*v^3 + 2*v^2 - 8*v + 8) / 8 $$\beta_{2}$$ $$=$$ $$( -\nu^{7} + 3\nu^{6} - \nu^{5} - 3\nu^{4} + 5\nu^{3} + 3\nu^{2} - 12\nu + 12 ) / 4$$ (-v^7 + 3*v^6 - v^5 - 3*v^4 + 5*v^3 + 3*v^2 - 12*v + 12) / 4 $$\beta_{3}$$ $$=$$ $$( \nu^{7} - 2\nu^{6} + 4\nu^{4} - 2\nu^{3} - 6\nu^{2} + 11\nu - 4 ) / 2$$ (v^7 - 2*v^6 + 4*v^4 - 2*v^3 - 6*v^2 + 11*v - 4) / 2 $$\beta_{4}$$ $$=$$ $$( -3\nu^{7} + 7\nu^{6} - 3\nu^{5} - 11\nu^{4} + 15\nu^{3} + 11\nu^{2} - 40\nu + 32 ) / 4$$ (-3*v^7 + 7*v^6 - 3*v^5 - 11*v^4 + 15*v^3 + 11*v^2 - 40*v + 32) / 4 $$\beta_{5}$$ $$=$$ $$( -2\nu^{7} + 5\nu^{6} - 3\nu^{5} - 7\nu^{4} + 11\nu^{3} + 7\nu^{2} - 27\nu + 22 ) / 2$$ (-2*v^7 + 5*v^6 - 3*v^5 - 7*v^4 + 11*v^3 + 7*v^2 - 27*v + 22) / 2 $$\beta_{6}$$ $$=$$ $$( 7\nu^{7} - 20\nu^{6} + 11\nu^{5} + 30\nu^{4} - 45\nu^{3} - 28\nu^{2} + 116\nu - 88 ) / 8$$ (7*v^7 - 20*v^6 + 11*v^5 + 30*v^4 - 45*v^3 - 28*v^2 + 116*v - 88) / 8 $$\beta_{7}$$ $$=$$ $$( 9\nu^{7} - 22\nu^{6} + 13\nu^{5} + 32\nu^{4} - 47\nu^{3} - 30\nu^{2} + 132\nu - 104 ) / 8$$ (9*v^7 - 22*v^6 + 13*v^5 + 32*v^4 - 47*v^3 - 30*v^2 + 132*v - 104) / 8
 $$\nu$$ $$=$$ $$( \beta_{7} + \beta_{5} + \beta _1 + 1 ) / 2$$ (b7 + b5 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{7} + 2\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 3\beta _1 + 3 ) / 2$$ (b7 + 2*b5 - b4 - b3 - b2 - 3*b1 + 3) / 2 $$\nu^{3}$$ $$=$$ $$( 2\beta_{7} + \beta_{5} + 3\beta_{4} + \beta_{3} - \beta_{2} - 2\beta _1 - 2 ) / 2$$ (2*b7 + b5 + 3*b4 + b3 - b2 - 2*b1 - 2) / 2 $$\nu^{4}$$ $$=$$ $$( -\beta_{7} + 2\beta_{6} + \beta_{5} + 2\beta_{2} - 7\beta _1 - 1 ) / 2$$ (-b7 + 2*b6 + b5 + 2*b2 - 7*b1 - 1) / 2 $$\nu^{5}$$ $$=$$ $$( \beta_{7} - 4\beta_{5} + 7\beta_{4} + \beta_{3} + 3\beta_{2} - 3\beta _1 - 3 ) / 2$$ (b7 - 4*b5 + 7*b4 + b3 + 3*b2 - 3*b1 - 3) / 2 $$\nu^{6}$$ $$=$$ $$( -2\beta_{6} - \beta_{5} - 5\beta_{4} - \beta_{3} + 9\beta_{2} + 2\beta _1 - 2 ) / 2$$ (-2*b6 - b5 - 5*b4 - b3 + 9*b2 + 2*b1 - 2) / 2 $$\nu^{7}$$ $$=$$ $$( 3\beta_{7} - 12\beta_{6} - 3\beta_{5} - 10\beta_{4} - 2\beta_{3} + 2\beta_{2} - \beta _1 + 11 ) / 2$$ (3*b7 - 12*b6 - 3*b5 - 10*b4 - 2*b3 + 2*b2 - b1 + 11) / 2

## Character values

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

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$1$$ $$1 - \beta_{4}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 1.20036 + 0.747754i 0.665665 − 1.24775i −1.27597 + 0.609843i 1.40994 − 0.109843i 1.20036 − 0.747754i 0.665665 + 1.24775i −1.27597 − 0.609843i 1.40994 + 0.109843i
−0.866025 + 0.500000i −0.547394 0.948114i 0.500000 0.866025i 1.00000i 0.948114 + 0.547394i −3.45632 1.99551i 1.00000i 0.900720 1.56009i 0.500000 + 0.866025i
101.2 −0.866025 + 0.500000i 0.913419 + 1.58209i 0.500000 0.866025i 1.00000i −1.58209 0.913419i 3.45632 + 1.99551i 1.00000i −0.168669 + 0.292144i 0.500000 + 0.866025i
101.3 0.866025 0.500000i −1.66612 2.88581i 0.500000 0.866025i 1.00000i −2.88581 1.66612i −1.24653 0.719687i 1.00000i −4.05193 + 7.01815i 0.500000 + 0.866025i
101.4 0.866025 0.500000i 0.300098 + 0.519785i 0.500000 0.866025i 1.00000i 0.519785 + 0.300098i 1.24653 + 0.719687i 1.00000i 1.31988 2.28610i 0.500000 + 0.866025i
121.1 −0.866025 0.500000i −0.547394 + 0.948114i 0.500000 + 0.866025i 1.00000i 0.948114 0.547394i −3.45632 + 1.99551i 1.00000i 0.900720 + 1.56009i 0.500000 0.866025i
121.2 −0.866025 0.500000i 0.913419 1.58209i 0.500000 + 0.866025i 1.00000i −1.58209 + 0.913419i 3.45632 1.99551i 1.00000i −0.168669 0.292144i 0.500000 0.866025i
121.3 0.866025 + 0.500000i −1.66612 + 2.88581i 0.500000 + 0.866025i 1.00000i −2.88581 + 1.66612i −1.24653 + 0.719687i 1.00000i −4.05193 7.01815i 0.500000 0.866025i
121.4 0.866025 + 0.500000i 0.300098 0.519785i 0.500000 + 0.866025i 1.00000i 0.519785 0.300098i 1.24653 0.719687i 1.00000i 1.31988 + 2.28610i 0.500000 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 101.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.2.l.b 8
3.b odd 2 1 1170.2.bs.g 8
4.b odd 2 1 1040.2.da.d 8
5.b even 2 1 650.2.m.c 8
5.c odd 4 1 650.2.n.d 8
5.c odd 4 1 650.2.n.e 8
13.b even 2 1 1690.2.l.j 8
13.c even 3 1 1690.2.d.k 8
13.c even 3 1 1690.2.l.j 8
13.d odd 4 1 1690.2.e.s 8
13.d odd 4 1 1690.2.e.t 8
13.e even 6 1 inner 130.2.l.b 8
13.e even 6 1 1690.2.d.k 8
13.f odd 12 1 1690.2.a.t 4
13.f odd 12 1 1690.2.a.u 4
13.f odd 12 1 1690.2.e.s 8
13.f odd 12 1 1690.2.e.t 8
39.h odd 6 1 1170.2.bs.g 8
52.i odd 6 1 1040.2.da.d 8
65.l even 6 1 650.2.m.c 8
65.r odd 12 1 650.2.n.d 8
65.r odd 12 1 650.2.n.e 8
65.s odd 12 1 8450.2.a.ci 4
65.s odd 12 1 8450.2.a.cm 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.l.b 8 1.a even 1 1 trivial
130.2.l.b 8 13.e even 6 1 inner
650.2.m.c 8 5.b even 2 1
650.2.m.c 8 65.l even 6 1
650.2.n.d 8 5.c odd 4 1
650.2.n.d 8 65.r odd 12 1
650.2.n.e 8 5.c odd 4 1
650.2.n.e 8 65.r odd 12 1
1040.2.da.d 8 4.b odd 2 1
1040.2.da.d 8 52.i odd 6 1
1170.2.bs.g 8 3.b odd 2 1
1170.2.bs.g 8 39.h odd 6 1
1690.2.a.t 4 13.f odd 12 1
1690.2.a.u 4 13.f odd 12 1
1690.2.d.k 8 13.c even 3 1
1690.2.d.k 8 13.e even 6 1
1690.2.e.s 8 13.d odd 4 1
1690.2.e.s 8 13.f odd 12 1
1690.2.e.t 8 13.d odd 4 1
1690.2.e.t 8 13.f odd 12 1
1690.2.l.j 8 13.b even 2 1
1690.2.l.j 8 13.c even 3 1
8450.2.a.ci 4 65.s odd 12 1
8450.2.a.cm 4 65.s odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 2T_{3}^{7} + 10T_{3}^{6} - 4T_{3}^{5} + 40T_{3}^{4} + 8T_{3}^{3} + 40T_{3}^{2} - 16T_{3} + 16$$ acting on $$S_{2}^{\mathrm{new}}(130, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{2}$$
$3$ $$T^{8} + 2 T^{7} + 10 T^{6} - 4 T^{5} + \cdots + 16$$
$5$ $$(T^{2} + 1)^{4}$$
$7$ $$T^{8} - 18 T^{6} + 291 T^{4} + \cdots + 1089$$
$11$ $$T^{8} + 6 T^{7} - 6 T^{6} - 108 T^{5} + \cdots + 81$$
$13$ $$T^{8} + 2 T^{7} - 2 T^{6} + \cdots + 28561$$
$17$ $$T^{8} - 6 T^{7} + 54 T^{6} + \cdots + 11664$$
$19$ $$T^{8} + 6 T^{7} + 6 T^{6} - 36 T^{5} + \cdots + 9$$
$23$ $$T^{8} - 12 T^{7} + 108 T^{6} + \cdots + 2304$$
$29$ $$T^{8} + 96 T^{6} + 7104 T^{4} + \cdots + 4460544$$
$31$ $$T^{8} + 144 T^{6} + 3324 T^{4} + \cdots + 144$$
$37$ $$T^{8} + 30 T^{7} + 390 T^{6} + \cdots + 9801$$
$41$ $$T^{8} - 12 T^{7} + 12 T^{6} + \cdots + 20736$$
$43$ $$T^{8} - 4 T^{7} + 136 T^{6} + \cdots + 135424$$
$47$ $$T^{8} + 228 T^{6} + 11718 T^{4} + \cdots + 408321$$
$53$ $$(T^{4} - 30 T^{3} + 294 T^{2} - 882 T - 579)^{2}$$
$59$ $$(T^{2} + 6 T + 12)^{4}$$
$61$ $$T^{8} - 26 T^{7} + 454 T^{6} + \cdots + 394384$$
$67$ $$T^{8} - 24 T^{7} + 96 T^{6} + \cdots + 9437184$$
$71$ $$(T^{4} - 36 T^{2} + 1296)^{2}$$
$73$ $$T^{8} + 264 T^{6} + 15228 T^{4} + \cdots + 1296$$
$79$ $$(T^{4} - 10 T^{3} - 54 T^{2} + 260 T - 188)^{2}$$
$83$ $$T^{8} + 480 T^{6} + 67932 T^{4} + \cdots + 4511376$$
$89$ $$T^{8} + 24 T^{7} + 210 T^{6} + \cdots + 42849$$
$97$ $$T^{8} + 6 T^{7} - 306 T^{6} + \cdots + 18558864$$