Properties

 Label 130.3.i.b Level $130$ Weight $3$ Character orbit 130.i Analytic conductor $3.542$ 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] = [130,3,Mod(27,130)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(130, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("130.27");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$130 = 2 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 130.i (of order $$4$$, 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: $$3.54224343668$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 4 x^{11} + 34 x^{10} + 12 x^{9} + 435 x^{8} + 2024 x^{7} + 10862 x^{6} + 44868 x^{5} + 149289 x^{4} + 340080 x^{3} + 619204 x^{2} + 735420 x + 354025$$ x^12 - 4*x^11 + 34*x^10 + 12*x^9 + 435*x^8 + 2024*x^7 + 10862*x^6 + 44868*x^5 + 149289*x^4 + 340080*x^3 + 619204*x^2 + 735420*x + 354025 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 5$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + 1) q^{2} + \beta_{2} q^{3} - 2 \beta_{3} q^{4} + (\beta_{10} + \beta_{8} + 1) q^{5} + (\beta_{2} + \beta_1) q^{6} + (\beta_{11} + \beta_{7} - \beta_{3} + 1) q^{7} + ( - 2 \beta_{3} - 2) q^{8} + ( - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + \cdots + 1) q^{9}+O(q^{10})$$ q + (-b3 + 1) * q^2 + b2 * q^3 - 2*b3 * q^4 + (b10 + b8 + 1) * q^5 + (b2 + b1) * q^6 + (b11 + b7 - b3 + 1) * q^7 + (-2*b3 - 2) * q^8 + (-b10 + b9 + b8 - b7 + b6 - b5 + 2*b4 + 4*b3 + 2*b2 + 1) * q^9 $$q + ( - \beta_{3} + 1) q^{2} + \beta_{2} q^{3} - 2 \beta_{3} q^{4} + (\beta_{10} + \beta_{8} + 1) q^{5} + (\beta_{2} + \beta_1) q^{6} + (\beta_{11} + \beta_{7} - \beta_{3} + 1) q^{7} + ( - 2 \beta_{3} - 2) q^{8} + ( - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + \cdots + 1) q^{9}+ \cdots + (9 \beta_{11} + 6 \beta_{10} + 6 \beta_{9} + 6 \beta_{8} + 6 \beta_{7} - 6 \beta_{6} + \cdots + 6) q^{99}+O(q^{100})$$ q + (-b3 + 1) * q^2 + b2 * q^3 - 2*b3 * q^4 + (b10 + b8 + 1) * q^5 + (b2 + b1) * q^6 + (b11 + b7 - b3 + 1) * q^7 + (-2*b3 - 2) * q^8 + (-b10 + b9 + b8 - b7 + b6 - b5 + 2*b4 + 4*b3 + 2*b2 + 1) * q^9 + (b11 + b10 + b8 + b5 - b3 + 1) * q^10 + (b11 + b9 - b8 + b2 - 1) * q^11 + 2*b1 * q^12 + b6 * q^13 + (b11 - b10 + b7 - b6 - 2*b3) * q^14 + (-2*b10 + 2*b9 - 3*b7 - 4*b6 - b5 - b4 + 2*b3 + 2*b2 - b1 - 3) * q^15 - 4 * q^16 + (-b11 + b10 + b9 + 2*b8 - b4 + 2*b3 - 2*b1) * q^17 + (-b11 - b10 - b9 + 2*b8 + 2*b6 + 3*b4 + 3*b3 + 2*b2 + 2*b1 + 5) * q^18 + (-b10 - 2*b9 - 2*b8 + b7 - b6 - b5 - b4 - 3*b3 - b1 - 2) * q^19 + (2*b11 + 2*b5 - 2*b3) * q^20 + (-2*b10 - 2*b8 + b7 + b6 + 2*b5 + 2*b4 - 2*b3 + b2 + b1 + 4) * q^21 + (b11 - b10 + b9 - b8 - b5 + b4 + b3 + b2 + b1 - 1) * q^22 + (-b11 + 3*b10 - b9 - b8 + 2*b6 - 3*b5 - 3*b4 - 2*b3 - 4*b2 - b1 - 6) * q^23 + (-2*b2 + 2*b1) * q^24 + (-2*b11 + 2*b10 - 3*b9 - 2*b8 - b7 - 3*b6 + 2*b5 - 3*b4 - 5*b3 - 5*b2 + b1 + 2) * q^25 + (b7 + b6) * q^26 + (2*b10 + 3*b8 + 4*b7 + b5 - 2*b4 + 2*b3 - b2 - 3*b1) * q^27 + (-2*b10 - 2*b6 - 2*b3 - 2) * q^28 + (-5*b11 + 4*b10 - 3*b9 - 3*b8 - b5 - 2*b4 - 5*b3 - 5*b2 + 3*b1 - 3) * q^29 + (-2*b11 - 2*b10 + 3*b9 + b8 - 7*b7 - b6 - b5 + b4 + 5*b3 + 3*b2 + b1 - 1) * q^30 + (4*b11 + 3*b10 - b9 - b8 + 3*b7 + 3*b6 + 2*b5 + 2*b4 - 2*b3 - 5*b2 - 4*b1 + 5) * q^31 + (4*b3 - 4) * q^32 + (-b11 - b9 + 3*b8 + 9*b6 + b5 + 5*b4 + 11*b3 - 3*b2 + 3*b1 + 15) * q^33 + (2*b10 + 2*b9 + 2*b8 + 2*b5 + 2*b3 + 2*b2 - 2*b1 + 2) * q^34 + (b10 - 3*b7 + b6 + 3*b4 + b3 + 3*b2 - 3*b1 + 7) * q^35 + (-2*b11 - 4*b9 + 2*b8 + 2*b7 + 2*b6 + 2*b5 + 2*b4 - 2*b3 + 4*b1 + 8) * q^36 + (-4*b11 + b10 - 5*b9 - b8 + 7*b7 + 3*b5 - b4 - 3*b3 - 3*b2 + b1 - 1) * q^37 + (-b11 - b10 - b9 - b8 - 2*b6 - 3*b5 - 3*b4 - b3 + b2 - b1 - 5) * q^38 + (b11 + b10 + b9 + b8 + 2*b5 - b4 - b1 + 1) * q^39 + (2*b11 - 2*b10 - 2*b8 + 2*b5 - 2*b3 - 2) * q^40 + (-3*b11 - 7*b10 + 6*b9 - 4*b8 - 10*b7 - 10*b6 - 2*b5 - 2*b4 + 2*b3 + 3*b2 - 3*b1 - 4) * q^41 + (-2*b11 - 2*b10 - 2*b9 - 4*b8 + 2*b7 + 2*b4 - 6*b3 + 2*b1 + 2) * q^42 + (2*b11 - 2*b10 + 2*b9 - 5*b8 - 2*b6 - b5 - 8*b4 - 2*b3 - 5*b1 - 8) * q^43 + (-2*b10 - 2*b5 + 2*b4 + 2*b3 + 2*b1) * q^44 + (-b11 - b10 - 7*b9 + 2*b8 - 2*b7 - b6 - 5*b5 + b4 + 9*b3 - 7*b2 + b1) * q^45 + (2*b11 + 4*b10 + 2*b9 + 2*b8 + 2*b7 + 2*b6 - 4*b5 - 4*b4 + 4*b3 - 3*b2 - 5*b1 - 8) * q^46 + (3*b11 + 4*b9 + 2*b8 + 11*b7 - 2*b5 + 5*b3 + 2*b2 - 6*b1 - 1) * q^47 - 4*b2 * q^48 + (-b11 - b10 - 2*b9 - 2*b8 - b7 + b6 - 2*b5 + 25*b3 - b2 + b1 - 2) * q^49 + (4*b10 - 4*b8 - 4*b7 - 2*b6 - 6*b4 - 7*b3 - 6*b2 - 4*b1 - 3) * q^50 + (-2*b10 + 8*b9 - 2*b8 - 12*b7 - 12*b6 - 6*b5 - 6*b4 + 6*b3 + 2*b2 - 6*b1 - 28) * q^51 + 2*b7 * q^52 + (3*b11 + b10 + 3*b9 - 2*b8 + 12*b6 + 4*b5 - b4 + b3 + 4*b2 - 2*b1 + 3) * q^53 + (2*b11 + 2*b10 + 2*b9 + 2*b8 + 4*b7 - 4*b6 + 4*b5 - 2*b4 + 2*b3 + 2*b2 - 4*b1 + 2) * q^54 + (-b10 + 6*b9 - b8 - 7*b7 - b6 - 3*b5 + 5*b4 + 17*b3 + 4*b2 - b1 - 13) * q^55 + (-2*b11 - 2*b10 - 2*b7 - 2*b6 - 4) * q^56 + (4*b11 - 3*b10 - 3*b9 - 6*b8 + 11*b7 + 3*b4 + 8*b3 + 4*b1 - 14) * q^57 + (-b11 + 9*b10 - b9 - 2*b8 - 4*b5 - 5*b4 - 2*b3 - 8*b2 - 2*b1 - 8) * q^58 + (b11 - 2*b10 + 4*b9 + 4*b8 - b5 + 5*b4 + 7*b3 + 12*b2 - 7*b1 + 4) * q^59 + (-4*b11 + 2*b9 + 2*b8 - 8*b7 + 6*b6 + 4*b4 + 6*b3 + 2*b2 + 4*b1 + 4) * q^60 + (-7*b11 - 6*b9 + 7*b8 - 5*b7 - 5*b6 - b5 - b4 + b3 - 4*b2 + 2*b1 - 11) * q^61 + (7*b11 - b10 - 3*b9 - 3*b8 + 6*b7 + b5 + b4 - 7*b3 - b2 - 9*b1 + 3) * q^62 + (b11 - 2*b10 + b9 + 3*b8 + 9*b6 + 5*b5 + 7*b4 - 6*b3 + 7*b2 + 3*b1 + 2) * q^63 + 8*b3 * q^64 + (b11 - 2*b10 + 2*b9 + b8 + b4 - 5*b3 - b2 - 3*b1 + 1) * q^65 + (-b11 + b10 - 6*b9 + 2*b8 + 9*b7 + 9*b6 + 4*b5 + 4*b4 - 4*b3 - 6*b2 + 26) * q^66 + (-16*b11 - b10 + 9*b9 + 3*b8 - 11*b7 - 5*b5 + b4 + 2*b3 + 5*b2 + 5*b1 + 6) * q^67 + (2*b11 + 2*b10 + 2*b9 + 4*b5 + 2*b4 + 4*b2 + 4) * q^68 + (9*b11 - 2*b10 - 3*b9 - 3*b8 + 12*b7 - 12*b6 + 7*b5 - 10*b4 - 27*b3 - 15*b2 + 5*b1 - 3) * q^69 + (b11 + b10 - 3*b9 - 2*b7 + 4*b6 + 3*b4 - 6*b3 + 6*b2 + 8) * q^70 + (5*b11 + 20*b10 - 19*b9 + 15*b8 + 7*b7 + 7*b6 + 4*b5 + 4*b4 - 4*b3 - 3*b2 + 16*b1 + 55) * q^71 + (-2*b11 + 2*b10 - 6*b9 + 4*b7 + 4*b5 - 2*b4 - 10*b3 - 4*b2 + 4*b1 + 6) * q^72 + (5*b11 - 12*b10 + 5*b9 + b8 + 11*b6 + 11*b5 + 7*b4 - b3 - 7*b2 + b1 + 11) * q^73 + (-3*b11 + 5*b10 - 4*b9 - 4*b8 + 7*b7 - 7*b6 + 2*b5 - 6*b4 - 2*b3 - 4*b2 - 2*b1 - 4) * q^74 + (-b11 + b10 - 5*b9 - 2*b8 - 4*b7 - 2*b6 - 6*b5 - b4 - 16*b3 + 4*b2 + 21*b1 - 6) * q^75 + (-2*b11 + 2*b9 + 2*b8 - 2*b7 - 2*b6 - 4*b5 - 4*b4 + 4*b3 + 2*b2 - 6) * q^76 + (b11 - b10 - b9 - 2*b8 + b4 - 17*b3 + 4*b1 + 15) * q^77 + (2*b11 + 2*b9 - b8 + 3*b5 - b3 + b2 - b1 + 1) * q^78 + (2*b11 - 2*b10 + 4*b9 + 4*b8 - 2*b7 + 2*b6 + 4*b4 - 18*b3 - 6*b2 + 10*b1 + 4) * q^79 + (-4*b10 - 4*b8 - 4) * q^80 + (-6*b11 - 9*b10 - 2*b9 - 3*b8 - 10*b7 - 10*b6 + 5*b5 + 5*b4 - 5*b3 + 9*b2 + 11*b1 - 10) * q^81 + (-10*b11 - 4*b10 + 8*b9 - 2*b8 - 20*b7 - 6*b5 + 4*b4 + 6*b3 + 6*b2 - 2) * q^82 + (-3*b11 + 8*b10 - 3*b9 + 3*b8 + 15*b6 - 3*b5 + 3*b4 + 6*b3 + 15*b2 + 3*b1 + 6) * q^83 + (-4*b11 - 4*b9 - 4*b8 + 2*b7 - 2*b6 - 4*b5 - 8*b3 - 2*b2 + 2*b1 - 4) * q^84 + (-5*b11 - b10 - 8*b9 - 4*b8 + 7*b7 - 9*b6 + 4*b5 - 6*b4 - 48*b3 - 8*b2 + 4*b1 - 22) * q^85 + (-4*b10 + 10*b9 - 4*b8 - 2*b7 - 2*b6 - 6*b5 - 6*b4 + 6*b3 + 5*b2 - 5*b1 - 10) * q^86 + (11*b11 + 3*b10 - 7*b9 + b8 + 12*b7 + 5*b5 - 3*b4 - 24*b3 - 5*b2 + 21*b1 + 20) * q^87 + (-2*b11 - 2*b10 - 2*b9 + 2*b8 - 2*b5 + 2*b4 + 2*b3 - 2*b2 + 2*b1 + 2) * q^88 + (b11 - 7*b10 + 8*b9 + 8*b8 + 3*b7 - 3*b6 - 6*b5 + 14*b4 + 4*b3 + 22*b2 - 8*b1 + 8) * q^89 + (-2*b11 - 8*b9 + 7*b8 - 3*b7 + b6 - 3*b5 - 6*b4 + 9*b3 - 8*b2 - 6*b1 + 9) * q^90 + (-2*b11 - 3*b10 - b8 + b5 + b4 - b3 + 2*b2 + 2*b1 + 7) * q^91 + (6*b11 + 2*b10 + 6*b9 + 6*b8 + 4*b7 - 2*b5 - 2*b4 + 12*b3 + 2*b2 - 8*b1 - 4) * q^92 + (11*b11 + 2*b10 + 11*b9 - 7*b8 + 9*b6 + 15*b5 - 3*b4 - 55*b3 - 3*b2 - 7*b1 - 47) * q^93 + (3*b11 - 3*b10 + 4*b9 + 4*b8 + 11*b7 - 11*b6 + 4*b4 + 6*b3 + 8*b2 - 4*b1 + 4) * q^94 + (8*b11 - 7*b10 + 6*b9 + 2*b8 + 5*b7 - 5*b6 + 3*b4 + 15*b3 + 17*b2 - 4*b1 + 37) * q^95 + (-4*b2 - 4*b1) * q^96 + (20*b11 - 2*b10 - 8*b9 - 7*b8 - 2*b7 + 3*b5 + 2*b4 + 23*b3 - 3*b2 + 9*b1 - 33) * q^97 + (-2*b11 - 2*b9 + 2*b6 - 4*b5 - 2*b4 + 27*b3 - 2*b2 + 23) * q^98 + (9*b11 + 6*b10 + 6*b9 + 6*b8 + 6*b7 - 6*b6 + 15*b5 - 9*b4 - 61*b3 + 12*b2 - 21*b1 + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 12 q^{2} + 4 q^{5} + 8 q^{7} - 24 q^{8}+O(q^{10})$$ 12 * q + 12 * q^2 + 4 * q^5 + 8 * q^7 - 24 * q^8 $$12 q + 12 q^{2} + 4 q^{5} + 8 q^{7} - 24 q^{8} - 16 q^{11} - 32 q^{15} - 48 q^{16} - 8 q^{17} + 52 q^{18} - 8 q^{20} + 56 q^{21} - 16 q^{22} - 60 q^{23} + 56 q^{25} - 12 q^{27} - 16 q^{28} - 16 q^{30} + 32 q^{31} - 48 q^{32} + 156 q^{33} + 68 q^{35} + 104 q^{36} + 28 q^{37} - 32 q^{38} - 16 q^{40} - 8 q^{41} + 56 q^{42} - 52 q^{43} + 24 q^{45} - 120 q^{46} - 48 q^{47} - 12 q^{50} - 328 q^{51} + 20 q^{53} - 192 q^{55} - 32 q^{56} - 148 q^{57} - 96 q^{58} + 32 q^{60} - 104 q^{61} + 32 q^{62} - 16 q^{63} + 312 q^{66} + 88 q^{67} + 16 q^{68} + 88 q^{70} + 560 q^{71} + 104 q^{72} + 108 q^{73} - 40 q^{75} - 64 q^{76} + 188 q^{77} - 16 q^{80} - 60 q^{81} - 8 q^{82} + 40 q^{83} - 168 q^{85} - 104 q^{86} + 220 q^{87} + 32 q^{88} + 144 q^{90} + 104 q^{91} - 120 q^{92} - 620 q^{93} + 396 q^{95} - 416 q^{97} + 300 q^{98}+O(q^{100})$$ 12 * q + 12 * q^2 + 4 * q^5 + 8 * q^7 - 24 * q^8 - 16 * q^11 - 32 * q^15 - 48 * q^16 - 8 * q^17 + 52 * q^18 - 8 * q^20 + 56 * q^21 - 16 * q^22 - 60 * q^23 + 56 * q^25 - 12 * q^27 - 16 * q^28 - 16 * q^30 + 32 * q^31 - 48 * q^32 + 156 * q^33 + 68 * q^35 + 104 * q^36 + 28 * q^37 - 32 * q^38 - 16 * q^40 - 8 * q^41 + 56 * q^42 - 52 * q^43 + 24 * q^45 - 120 * q^46 - 48 * q^47 - 12 * q^50 - 328 * q^51 + 20 * q^53 - 192 * q^55 - 32 * q^56 - 148 * q^57 - 96 * q^58 + 32 * q^60 - 104 * q^61 + 32 * q^62 - 16 * q^63 + 312 * q^66 + 88 * q^67 + 16 * q^68 + 88 * q^70 + 560 * q^71 + 104 * q^72 + 108 * q^73 - 40 * q^75 - 64 * q^76 + 188 * q^77 - 16 * q^80 - 60 * q^81 - 8 * q^82 + 40 * q^83 - 168 * q^85 - 104 * q^86 + 220 * q^87 + 32 * q^88 + 144 * q^90 + 104 * q^91 - 120 * q^92 - 620 * q^93 + 396 * q^95 - 416 * q^97 + 300 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{11} + 34 x^{10} + 12 x^{9} + 435 x^{8} + 2024 x^{7} + 10862 x^{6} + 44868 x^{5} + 149289 x^{4} + 340080 x^{3} + 619204 x^{2} + 735420 x + 354025$$ :

 $$\beta_{1}$$ $$=$$ $$( - 19\!\cdots\!78 \nu^{11} + \cdots - 54\!\cdots\!00 ) / 15\!\cdots\!75$$ (-196798616351355678*v^11 + 1027534402721622748*v^10 - 7829482578836192678*v^9 + 6625999114569840275*v^8 - 90438555512064136855*v^7 - 282692549372532675637*v^6 - 1787657589629997366067*v^5 - 6429138448347996184100*v^4 - 20663547023555466857617*v^3 - 39024436068774979434736*v^2 - 67609698814266662280055*v - 54338567011845131139600) / 1596086174445927626875 $$\beta_{2}$$ $$=$$ $$( 19\!\cdots\!48 \nu^{11} + \cdots + 54\!\cdots\!00 ) / 15\!\cdots\!75$$ (197711861069400248*v^11 - 989951143531948593*v^10 + 7666354153826994748*v^9 - 5403975354826072375*v^8 + 91701748557105721955*v^7 + 295489463176626202817*v^6 + 1841587422407747116297*v^5 + 6867665969073152724200*v^4 + 21786573895562020376447*v^3 + 41510195229121058217151*v^2 + 71576292286515188731005*v + 54009288004587487528600) / 1596086174445927626875 $$\beta_{3}$$ $$=$$ $$( - 861103744344 \nu^{11} + 4543005521154 \nu^{10} - 35270655980094 \nu^{9} + 35262839612400 \nu^{8} + \cdots - 25\!\cdots\!00 ) / 62\!\cdots\!75$$ (-861103744344*v^11 + 4543005521154*v^10 - 35270655980094*v^9 + 35262839612400*v^8 - 423742303166340*v^7 - 1217256904448701*v^6 - 7845850939485166*v^5 - 28997493736991625*v^4 - 93764202692912766*v^3 - 180615403770389703*v^2 - 323348585401865890*v - 255695644105593300) / 6268773587916875 $$\beta_{4}$$ $$=$$ $$( - 64\!\cdots\!23 \nu^{11} + \cdots - 14\!\cdots\!25 ) / 31\!\cdots\!75$$ (-64888255189173523*v^11 + 354430065287404443*v^10 - 2654784016879457223*v^9 + 2714298617032094600*v^8 - 29630073809053077005*v^7 - 89234649128802670467*v^6 - 554514081144107047547*v^5 - 1999314108994501499175*v^4 - 6261815000157635172247*v^3 - 11255185592637288805851*v^2 - 19369762768989163938655*v - 14276602693538131923725) / 319217234889185525375 $$\beta_{5}$$ $$=$$ $$( - 59\!\cdots\!63 \nu^{11} + \cdots - 17\!\cdots\!00 ) / 15\!\cdots\!75$$ (-598089922663512963*v^11 + 3192274388964567458*v^10 - 24583112814900456763*v^9 + 24755191271871408400*v^8 - 287402171180819335555*v^7 - 858551140023450004052*v^6 - 5322072443723131852382*v^5 - 19847547877523533591225*v^4 - 63814938734734501642132*v^3 - 122498485842563614119156*v^2 - 220358605517845898390655*v - 179062056164871257569100) / 1596086174445927626875 $$\beta_{6}$$ $$=$$ $$( 60\!\cdots\!41 \nu^{11} + \cdots + 16\!\cdots\!00 ) / 15\!\cdots\!75$$ (603006665253056241*v^11 - 3156185770358005456*v^10 + 24226424272684226941*v^9 - 22024430264921268125*v^8 + 285006200209595612260*v^7 + 860591372065128593339*v^6 + 5446576370218054866574*v^5 + 20006952749246072252275*v^4 + 63746049483157999517924*v^3 + 120181318497907779191742*v^2 + 209645186322759493863210*v + 160605617053970320988700) / 1596086174445927626875 $$\beta_{7}$$ $$=$$ $$( - 60\!\cdots\!79 \nu^{11} + \cdots - 17\!\cdots\!00 ) / 15\!\cdots\!75$$ (-607324407367248779*v^11 + 3118573508248027514*v^10 - 24090972381849770504*v^9 + 19931920034041932500*v^8 - 286153506341041605040*v^7 - 886968425969049226766*v^6 - 5593707281777208063656*v^5 - 20614662501836313930100*v^4 - 66092845101361232919506*v^3 - 127099531821460916146123*v^2 - 219424224783737420927365*v - 173600933484161443822800) / 1596086174445927626875 $$\beta_{8}$$ $$=$$ $$( 12\!\cdots\!18 \nu^{11} + \cdots + 35\!\cdots\!50 ) / 31\!\cdots\!75$$ (129887138120349418*v^11 - 692198065981594783*v^10 + 5327753827979516538*v^9 - 5418874820887540335*v^8 + 63057299389922841385*v^7 + 181281321391815784647*v^6 + 1165693407698491077132*v^5 + 4285952747368504768660*v^4 + 13670625553440033177097*v^3 + 26203024732906918932136*v^2 + 46317751572081759939630*v + 35520648638543620447350) / 319217234889185525375 $$\beta_{9}$$ $$=$$ $$( - 88\!\cdots\!69 \nu^{11} + \cdots - 25\!\cdots\!75 ) / 15\!\cdots\!75$$ (-889718240561715669*v^11 + 4539170135011551329*v^10 - 35258589926312913294*v^9 + 28893588146864636375*v^8 - 423061508984462089615*v^7 - 1308278714961808435026*v^6 - 8245095339879536318091*v^5 - 30630390729971154344350*v^4 - 98194373872673482978791*v^3 - 189531363670835907630428*v^2 - 327322532690081369325390*v - 254270322412066950185175) / 1596086174445927626875 $$\beta_{10}$$ $$=$$ $$( - 52\!\cdots\!72 \nu^{11} + \cdots - 14\!\cdots\!50 ) / 63\!\cdots\!75$$ (-52759536391463572*v^11 + 280729251577070306*v^10 - 2161795126376976784*v^9 + 2202137668680454818*v^8 - 25726976724833548471*v^7 - 73027132413851789221*v^6 - 475685981390901516601*v^5 - 1736147417724883906668*v^4 - 5565019303898365694432*v^3 - 10567111694020792264810*v^2 - 18601488185277263309640*v - 14307187934467709849750) / 63843446977837105075 $$\beta_{11}$$ $$=$$ $$( 13\!\cdots\!72 \nu^{11} + \cdots + 39\!\cdots\!75 ) / 15\!\cdots\!75$$ (1354893240402459972*v^11 - 7033379785623543277*v^10 + 54452769295317028247*v^9 - 48809727797325558400*v^8 + 648363840089660907895*v^7 + 1967540085783580605238*v^6 + 12357810133780961396133*v^5 + 46087184165669049008325*v^4 + 147103999702055363057608*v^3 + 284419598094046338994164*v^2 + 496133010055912739443695*v + 393089107780700819007275) / 1596086174445927626875
 $$\nu$$ $$=$$ $$( - 4 \beta_{11} + 2 \beta_{10} + \beta_{9} + 6 \beta_{8} - 5 \beta_{7} + 5 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 6 \beta _1 + 6 ) / 10$$ (-4*b11 + 2*b10 + b9 + 6*b8 - 5*b7 + 5*b6 - 2*b5 + 3*b4 + 2*b3 + 2*b2 + 6*b1 + 6) / 10 $$\nu^{2}$$ $$=$$ $$( \beta_{11} + 2 \beta_{10} + 6 \beta_{9} + \beta_{8} - 15 \beta_{7} + 3 \beta_{5} - 2 \beta_{4} - 8 \beta_{3} - 8 \beta_{2} + 11 \beta _1 - 19 ) / 5$$ (b11 + 2*b10 + 6*b9 + b8 - 15*b7 + 3*b5 - 2*b4 - 8*b3 - 8*b2 + 11*b1 - 19) / 5 $$\nu^{3}$$ $$=$$ $$( 62 \beta_{11} + 34 \beta_{10} + 67 \beta_{9} - 78 \beta_{8} - 25 \beta_{7} + 25 \beta_{6} + 86 \beta_{5} - 59 \beta_{4} - 386 \beta_{3} - 76 \beta_{2} - 18 \beta _1 - 308 ) / 10$$ (62*b11 + 34*b10 + 67*b9 - 78*b8 - 25*b7 + 25*b6 + 86*b5 - 59*b4 - 386*b3 - 76*b2 - 18*b1 - 308) / 10 $$\nu^{4}$$ $$=$$ $$47 \beta_{11} + 18 \beta_{10} + 17 \beta_{9} - 29 \beta_{8} + 62 \beta_{7} + 32 \beta_{6} + 49 \beta_{5} - 10 \beta_{4} - 145 \beta_{3} - 45 \beta_{2} - 79 \beta _1 - 114$$ 47*b11 + 18*b10 + 17*b9 - 29*b8 + 62*b7 + 32*b6 + 49*b5 - 10*b4 - 145*b3 - 45*b2 - 79*b1 - 114 $$\nu^{5}$$ $$=$$ $$( 1186 \beta_{11} - 418 \beta_{10} - 619 \beta_{9} - 1134 \beta_{8} + 4065 \beta_{7} + 1945 \beta_{6} + 128 \beta_{5} + 1543 \beta_{4} + 2902 \beta_{3} - 1768 \beta_{2} - 4504 \beta _1 - 3544 ) / 10$$ (1186*b11 - 418*b10 - 619*b9 - 1134*b8 + 4065*b7 + 1945*b6 + 128*b5 + 1543*b4 + 2902*b3 - 1768*b2 - 4504*b1 - 3544) / 10 $$\nu^{6}$$ $$=$$ $$( - 414 \beta_{11} - 5678 \beta_{10} - 2499 \beta_{9} - 3084 \beta_{8} + 6265 \beta_{7} + 210 \beta_{6} - 7002 \beta_{5} + 6808 \beta_{4} + 33392 \beta_{3} + 3937 \beta_{2} - 5554 \beta _1 - 6469 ) / 5$$ (-414*b11 - 5678*b10 - 2499*b9 - 3084*b8 + 6265*b7 + 210*b6 - 7002*b5 + 6808*b4 + 33392*b3 + 3937*b2 - 5554*b1 - 6469) / 5 $$\nu^{7}$$ $$=$$ $$( - 29278 \beta_{11} - 75196 \beta_{10} - 17313 \beta_{9} - 20638 \beta_{8} - 2545 \beta_{7} - 89615 \beta_{6} - 94644 \beta_{5} + 40351 \beta_{4} + 360414 \beta_{3} + 143594 \beta_{2} + \cdots - 3058 ) / 10$$ (-29278*b11 - 75196*b10 - 17313*b9 - 20638*b8 - 2545*b7 - 89615*b6 - 94644*b5 + 40351*b4 + 360414*b3 + 143594*b2 + 9692*b1 - 3058) / 10 $$\nu^{8}$$ $$=$$ $$- 24245 \beta_{11} - 17251 \beta_{10} - 13322 \beta_{9} + 13618 \beta_{8} - 23689 \beta_{7} - 70925 \beta_{6} - 28330 \beta_{5} - 14852 \beta_{4} + 63030 \beta_{3} + 79881 \beta_{2} + \cdots + 83641$$ -24245*b11 - 17251*b10 - 13322*b9 + 13618*b8 - 23689*b7 - 70925*b6 - 28330*b5 - 14852*b4 + 63030*b3 + 79881*b2 + 30269*b1 + 83641 $$\nu^{9}$$ $$=$$ $$( - 1550674 \beta_{11} + 992112 \beta_{10} - 1065839 \beta_{9} + 2303826 \beta_{8} - 1863065 \beta_{7} - 2150215 \beta_{6} + 261858 \beta_{5} - 1974517 \beta_{4} - 4290148 \beta_{3} + \cdots + 9280456 ) / 10$$ (-1550674*b11 + 992112*b10 - 1065839*b9 + 2303826*b8 - 1863065*b7 - 2150215*b6 + 261858*b5 - 1974517*b4 - 4290148*b3 + 1283412*b2 + 2785236*b1 + 9280456) / 10 $$\nu^{10}$$ $$=$$ $$( - 3202329 \beta_{11} + 5925642 \beta_{10} - 1358279 \beta_{9} + 7095731 \beta_{8} - 6403680 \beta_{7} + 3151345 \beta_{6} + 4094943 \beta_{5} - 4354957 \beta_{4} + \cdots + 23256581 ) / 5$$ (-3202329*b11 + 5925642*b10 - 1358279*b9 + 7095731*b8 - 6403680*b7 + 3151345*b6 + 4094943*b5 - 4354957*b4 - 21606708*b3 - 6707043*b2 + 9280106*b1 + 23256581) / 5 $$\nu^{11}$$ $$=$$ $$( 3073282 \beta_{11} + 60334874 \beta_{10} + 29181107 \beta_{9} + 30877252 \beta_{8} - 64443015 \beta_{7} + 102687595 \beta_{6} + 59711976 \beta_{5} - 14816339 \beta_{4} + \cdots + 26279292 ) / 10$$ (3073282*b11 + 60334874*b10 + 29181107*b9 + 30877252*b8 - 64443015*b7 + 102687595*b6 + 59711976*b5 - 14816339*b4 - 231262586*b3 - 130275486*b2 + 69898652*b1 + 26279292) / 10

Character values

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

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

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
27.1
 1.61554 − 4.65786i −1.96771 + 1.30974i −1.24767 + 0.0362500i −0.272159 + 2.46318i 4.21538 + 3.75303i −0.343384 − 2.90434i 1.61554 + 4.65786i −1.96771 − 1.30974i −1.24767 − 0.0362500i −0.272159 − 2.46318i 4.21538 − 3.75303i −0.343384 + 2.90434i
1.00000 + 1.00000i −3.71323 + 3.71323i 2.00000i 3.82477 + 3.22042i −7.42646 0.429967 + 0.429967i −2.00000 + 2.00000i 18.5761i 0.604347 + 7.04519i
27.2 1.00000 + 1.00000i −2.22000 + 2.22000i 2.00000i −4.90152 0.987478i −4.44001 −4.73561 4.73561i −2.00000 + 2.00000i 0.856825i −3.91404 5.88900i
27.3 1.00000 + 1.00000i −0.827569 + 0.827569i 2.00000i 4.04912 2.93336i −1.65514 4.16739 + 4.16739i −2.00000 + 2.00000i 7.63026i 6.98248 + 1.11576i
27.4 1.00000 + 1.00000i 0.370523 0.370523i 2.00000i −2.42963 + 4.37000i 0.741047 2.03473 + 2.03473i −2.00000 + 2.00000i 8.72542i −6.79963 + 1.94038i
27.5 1.00000 + 1.00000i 3.04757 3.04757i 2.00000i 4.40191 + 2.37133i 6.09514 −2.53080 2.53080i −2.00000 + 2.00000i 9.57539i 2.03058 + 6.77324i
27.6 1.00000 + 1.00000i 3.34270 3.34270i 2.00000i −2.94465 4.04092i 6.68541 4.63433 + 4.63433i −2.00000 + 2.00000i 13.3473i 1.09626 6.98557i
53.1 1.00000 1.00000i −3.71323 3.71323i 2.00000i 3.82477 3.22042i −7.42646 0.429967 0.429967i −2.00000 2.00000i 18.5761i 0.604347 7.04519i
53.2 1.00000 1.00000i −2.22000 2.22000i 2.00000i −4.90152 + 0.987478i −4.44001 −4.73561 + 4.73561i −2.00000 2.00000i 0.856825i −3.91404 + 5.88900i
53.3 1.00000 1.00000i −0.827569 0.827569i 2.00000i 4.04912 + 2.93336i −1.65514 4.16739 4.16739i −2.00000 2.00000i 7.63026i 6.98248 1.11576i
53.4 1.00000 1.00000i 0.370523 + 0.370523i 2.00000i −2.42963 4.37000i 0.741047 2.03473 2.03473i −2.00000 2.00000i 8.72542i −6.79963 1.94038i
53.5 1.00000 1.00000i 3.04757 + 3.04757i 2.00000i 4.40191 2.37133i 6.09514 −2.53080 + 2.53080i −2.00000 2.00000i 9.57539i 2.03058 6.77324i
53.6 1.00000 1.00000i 3.34270 + 3.34270i 2.00000i −2.94465 + 4.04092i 6.68541 4.63433 4.63433i −2.00000 2.00000i 13.3473i 1.09626 + 6.98557i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 27.6 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.c odd 4 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.3.i.b 12
5.b even 2 1 650.3.i.c 12
5.c odd 4 1 inner 130.3.i.b 12
5.c odd 4 1 650.3.i.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.3.i.b 12 1.a even 1 1 trivial
130.3.i.b 12 5.c odd 4 1 inner
650.3.i.c 12 5.b even 2 1
650.3.i.c 12 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + 4 T_{3}^{9} + 852 T_{3}^{8} - 128 T_{3}^{7} + 8 T_{3}^{6} + 10024 T_{3}^{5} + 123396 T_{3}^{4} + 106888 T_{3}^{3} + 41472 T_{3}^{2} - 59328 T_{3} + 42436$$ acting on $$S_{3}^{\mathrm{new}}(130, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2 T + 2)^{6}$$
$3$ $$T^{12} + 4 T^{9} + 852 T^{8} + \cdots + 42436$$
$5$ $$T^{12} - 4 T^{11} - 20 T^{10} + \cdots + 244140625$$
$7$ $$T^{12} - 8 T^{11} + 32 T^{10} + \cdots + 2624400$$
$11$ $$(T^{6} + 8 T^{5} - 208 T^{4} - 248 T^{3} + \cdots - 74138)^{2}$$
$13$ $$(T^{4} + 169)^{3}$$
$17$ $$T^{12} + 8 T^{11} + \cdots + 3113728459776$$
$19$ $$T^{12} + 1276 T^{10} + \cdots + 23687202302500$$
$23$ $$T^{12} + \cdots + 630951601562500$$
$29$ $$T^{12} + 5520 T^{10} + \cdots + 60\!\cdots\!00$$
$31$ $$(T^{6} - 16 T^{5} - 3172 T^{4} + \cdots + 39973270)^{2}$$
$37$ $$T^{12} - 28 T^{11} + \cdots + 56\!\cdots\!44$$
$41$ $$(T^{6} + 4 T^{5} - 7802 T^{4} + \cdots - 1381745000)^{2}$$
$43$ $$T^{12} + \cdots + 542737815066756$$
$47$ $$T^{12} + 48 T^{11} + \cdots + 53\!\cdots\!44$$
$53$ $$T^{12} - 20 T^{11} + \cdots + 20\!\cdots\!00$$
$59$ $$T^{12} + 12260 T^{10} + \cdots + 54\!\cdots\!00$$
$61$ $$(T^{6} + 52 T^{5} - 9512 T^{4} + \cdots - 8632277404)^{2}$$
$67$ $$T^{12} - 88 T^{11} + \cdots + 70\!\cdots\!00$$
$71$ $$(T^{6} - 280 T^{5} + \cdots + 2676310808950)^{2}$$
$73$ $$T^{12} - 108 T^{11} + \cdots + 98\!\cdots\!00$$
$79$ $$T^{12} + 23384 T^{10} + \cdots + 83\!\cdots\!00$$
$83$ $$T^{12} - 40 T^{11} + \cdots + 54\!\cdots\!00$$
$89$ $$T^{12} + 43840 T^{10} + \cdots + 18\!\cdots\!00$$
$97$ $$T^{12} + 416 T^{11} + \cdots + 28\!\cdots\!00$$