# Properties

 Label 130.2.m.a Level $130$ Weight $2$ Character orbit 130.m 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(49,130)]

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([3, 5]))

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

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

chi := DirichletCharacter("130.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$130 = 2 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 130.m (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: 8.0.50027374224.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 20x^{6} + 132x^{4} + 332x^{2} + 256$$ x^8 + 20*x^6 + 132*x^4 + 332*x^2 + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 20x^{6} + 132x^{4} + 332x^{2} + 256$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{4} + 10\nu^{2} + 2\nu + 16 ) / 4$$ (v^4 + 10*v^2 + 2*v + 16) / 4 $$\beta_{3}$$ $$=$$ $$( \nu^{7} + 20\nu^{5} + 116\nu^{3} + 172\nu + 32 ) / 64$$ (v^7 + 20*v^5 + 116*v^3 + 172*v + 32) / 64 $$\beta_{4}$$ $$=$$ $$( \nu^{7} + 16\nu^{5} + 68\nu^{3} + 8\nu^{2} + 60\nu + 48 ) / 16$$ (v^7 + 16*v^5 + 68*v^3 + 8*v^2 + 60*v + 48) / 16 $$\beta_{5}$$ $$=$$ $$( -\nu^{7} - 16\nu^{5} - 68\nu^{3} + 8\nu^{2} - 60\nu + 32 ) / 16$$ (-v^7 - 16*v^5 - 68*v^3 + 8*v^2 - 60*v + 32) / 16 $$\beta_{6}$$ $$=$$ $$( \nu^{7} + 8\nu^{6} + 20\nu^{5} + 128\nu^{4} + 132\nu^{3} + 544\nu^{2} + 268\nu + 544 ) / 64$$ (v^7 + 8*v^6 + 20*v^5 + 128*v^4 + 132*v^3 + 544*v^2 + 268*v + 544) / 64 $$\beta_{7}$$ $$=$$ $$( \nu^{7} - 8\nu^{6} + 20\nu^{5} - 128\nu^{4} + 132\nu^{3} - 544\nu^{2} + 268\nu - 544 ) / 64$$ (v^7 - 8*v^6 + 20*v^5 - 128*v^4 + 132*v^3 - 544*v^2 + 268*v - 544) / 64
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4} - 5$$ b5 + b4 - 5 $$\nu^{3}$$ $$=$$ $$2\beta_{7} + 2\beta_{6} - 4\beta_{3} - 6\beta _1 + 2$$ 2*b7 + 2*b6 - 4*b3 - 6*b1 + 2 $$\nu^{4}$$ $$=$$ $$-10\beta_{5} - 10\beta_{4} + 4\beta_{2} - 2\beta _1 + 34$$ -10*b5 - 10*b4 + 4*b2 - 2*b1 + 34 $$\nu^{5}$$ $$=$$ $$-24\beta_{7} - 24\beta_{6} + 2\beta_{5} - 2\beta_{4} + 64\beta_{3} + 44\beta _1 - 30$$ -24*b7 - 24*b6 + 2*b5 - 2*b4 + 64*b3 + 44*b1 - 30 $$\nu^{6}$$ $$=$$ $$-4\beta_{7} + 4\beta_{6} + 92\beta_{5} + 92\beta_{4} - 64\beta_{2} + 32\beta _1 - 272$$ -4*b7 + 4*b6 + 92*b5 + 92*b4 - 64*b2 + 32*b1 - 272 $$\nu^{7}$$ $$=$$ $$248\beta_{7} + 248\beta_{6} - 40\beta_{5} + 40\beta_{4} - 752\beta_{3} - 356\beta _1 + 336$$ 248*b7 + 248*b6 - 40*b5 + 40*b4 - 752*b3 - 356*b1 + 336

## Character values

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

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$-1$$ $$\beta_{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 3.07108i − 1.17644i 1.83766i 2.40987i 3.07108i 1.17644i − 1.83766i − 2.40987i
−0.500000 0.866025i −2.65963 + 1.53554i −0.500000 + 0.866025i 1.36822 1.76861i 2.65963 + 1.53554i 1.84755 3.20005i 1.00000 3.21577 5.56988i −2.21577 0.300609i
49.2 −0.500000 0.866025i −1.01883 + 0.588222i −0.500000 + 0.866025i 0.235468 + 2.22364i 1.01883 + 0.588222i −1.04346 + 1.80732i 1.00000 −0.807991 + 1.39948i 1.80799 1.31574i
49.3 −0.500000 0.866025i 1.59146 0.918829i −0.500000 + 0.866025i −2.21022 0.339024i −1.59146 0.918829i 2.39871 4.15469i 1.00000 0.188495 0.326482i 0.811505 + 2.08362i
49.4 −0.500000 0.866025i 2.08700 1.20493i −0.500000 + 0.866025i 2.10653 + 0.750022i −2.08700 1.20493i −0.702803 + 1.21729i 1.00000 1.40373 2.43133i −0.403726 2.19932i
69.1 −0.500000 + 0.866025i −2.65963 1.53554i −0.500000 0.866025i 1.36822 + 1.76861i 2.65963 1.53554i 1.84755 + 3.20005i 1.00000 3.21577 + 5.56988i −2.21577 + 0.300609i
69.2 −0.500000 + 0.866025i −1.01883 0.588222i −0.500000 0.866025i 0.235468 2.22364i 1.01883 0.588222i −1.04346 1.80732i 1.00000 −0.807991 1.39948i 1.80799 + 1.31574i
69.3 −0.500000 + 0.866025i 1.59146 + 0.918829i −0.500000 0.866025i −2.21022 + 0.339024i −1.59146 + 0.918829i 2.39871 + 4.15469i 1.00000 0.188495 + 0.326482i 0.811505 2.08362i
69.4 −0.500000 + 0.866025i 2.08700 + 1.20493i −0.500000 0.866025i 2.10653 0.750022i −2.08700 + 1.20493i −0.702803 1.21729i 1.00000 1.40373 + 2.43133i −0.403726 + 2.19932i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.l 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.m.a 8
3.b odd 2 1 1170.2.bj.b 8
4.b odd 2 1 1040.2.df.c 8
5.b even 2 1 130.2.m.b yes 8
5.c odd 4 2 650.2.m.e 16
13.c even 3 1 1690.2.c.f 8
13.e even 6 1 130.2.m.b yes 8
13.e even 6 1 1690.2.c.e 8
13.f odd 12 2 1690.2.b.e 16
15.d odd 2 1 1170.2.bj.a 8
20.d odd 2 1 1040.2.df.a 8
39.h odd 6 1 1170.2.bj.a 8
52.i odd 6 1 1040.2.df.a 8
65.l even 6 1 inner 130.2.m.a 8
65.l even 6 1 1690.2.c.f 8
65.n even 6 1 1690.2.c.e 8
65.o even 12 2 8450.2.a.cs 8
65.r odd 12 2 650.2.m.e 16
65.s odd 12 2 1690.2.b.e 16
65.t even 12 2 8450.2.a.cr 8
195.y odd 6 1 1170.2.bj.b 8
260.w odd 6 1 1040.2.df.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.m.a 8 1.a even 1 1 trivial
130.2.m.a 8 65.l even 6 1 inner
130.2.m.b yes 8 5.b even 2 1
130.2.m.b yes 8 13.e even 6 1
650.2.m.e 16 5.c odd 4 2
650.2.m.e 16 65.r odd 12 2
1040.2.df.a 8 20.d odd 2 1
1040.2.df.a 8 52.i odd 6 1
1040.2.df.c 8 4.b odd 2 1
1040.2.df.c 8 260.w odd 6 1
1170.2.bj.a 8 15.d odd 2 1
1170.2.bj.a 8 39.h odd 6 1
1170.2.bj.b 8 3.b odd 2 1
1170.2.bj.b 8 195.y odd 6 1
1690.2.b.e 16 13.f odd 12 2
1690.2.b.e 16 65.s odd 12 2
1690.2.c.e 8 13.e even 6 1
1690.2.c.e 8 65.n even 6 1
1690.2.c.f 8 13.c even 3 1
1690.2.c.f 8 65.l even 6 1
8450.2.a.cr 8 65.t even 12 2
8450.2.a.cs 8 65.o even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 10T_{3}^{6} + 84T_{3}^{4} - 60T_{3}^{3} - 148T_{3}^{2} + 96T_{3} + 256$$ acting on $$S_{2}^{\mathrm{new}}(130, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{4}$$
$3$ $$T^{8} - 10 T^{6} + 84 T^{4} + \cdots + 256$$
$5$ $$T^{8} - 3 T^{7} + 2 T^{6} + 15 T^{5} + \cdots + 625$$
$7$ $$T^{8} - 5 T^{7} + 34 T^{6} + \cdots + 2704$$
$11$ $$T^{8} + 3 T^{7} - 16 T^{6} - 57 T^{5} + \cdots + 784$$
$13$ $$T^{8} - 4 T^{7} + 28 T^{6} + \cdots + 28561$$
$17$ $$T^{8} + 15 T^{7} + 83 T^{6} + 120 T^{5} + \cdots + 4$$
$19$ $$T^{8} - 9 T^{7} + 18 T^{6} + \cdots + 9216$$
$23$ $$T^{8} + 6 T^{7} - 4 T^{6} - 96 T^{5} + \cdots + 256$$
$29$ $$T^{8} + 3 T^{7} + 27 T^{6} + 66 T^{5} + \cdots + 576$$
$31$ $$T^{8} + 60 T^{6} + 1176 T^{4} + \cdots + 576$$
$37$ $$T^{8} - 20 T^{7} + 280 T^{6} + \cdots + 11881$$
$41$ $$T^{8} - 21 T^{7} + 119 T^{6} + \cdots + 1106704$$
$43$ $$T^{8} - 18 T^{7} + 72 T^{6} + \cdots + 2359296$$
$47$ $$(T^{4} - 3 T^{3} - 117 T^{2} + 267 T - 96)^{2}$$
$53$ $$T^{8} + 356 T^{6} + \cdots + 17783089$$
$59$ $$T^{8} - 30 T^{7} + 344 T^{6} + \cdots + 369664$$
$61$ $$T^{8} + 5 T^{7} + 271 T^{6} + \cdots + 28751044$$
$67$ $$(T^{2} + 4 T + 16)^{4}$$
$71$ $$T^{8} - 160 T^{6} + \cdots + 16777216$$
$73$ $$(T^{4} - 13 T^{3} - 114 T^{2} + 1298 T - 1406)^{2}$$
$79$ $$(T^{4} - 2 T^{3} - 216 T^{2} + 502 T + 1384)^{2}$$
$83$ $$(T^{4} + 24 T^{3} + 78 T^{2} - 1602 T - 9744)^{2}$$
$89$ $$T^{8} + 39 T^{7} + 548 T^{6} + \cdots + 59474944$$
$97$ $$T^{8} + 4 T^{7} + 76 T^{6} + \cdots + 327184$$