# Properties

 Label 130.3.k.a Level $130$ Weight $3$ Character orbit 130.k Analytic conductor $3.542$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [130,3,Mod(21,130)]

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([0, 1]))

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

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

chi := DirichletCharacter("130.21");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$130 = 2 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 130.k (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: $$x^{12} + 76x^{10} + 1956x^{8} + 19924x^{6} + 77560x^{4} + 85248x^{2} + 2916$$ x^12 + 76*x^10 + 1956*x^8 + 19924*x^6 + 77560*x^4 + 85248*x^2 + 2916 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{4} - 1) q^{2} - \beta_{2} q^{3} + 2 \beta_{4} q^{4} - \beta_{6} q^{5} + (\beta_{2} - \beta_1) q^{6} + (\beta_{8} + \beta_{5}) q^{7} + ( - 2 \beta_{4} + 2) q^{8} + ( - \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{3} - \beta_{2} + 4) q^{9}+O(q^{10})$$ q + (-b4 - 1) * q^2 - b2 * q^3 + 2*b4 * q^4 - b6 * q^5 + (b2 - b1) * q^6 + (b8 + b5) * q^7 + (-2*b4 + 2) * q^8 + (-b11 - b10 + b9 - b8 + b3 - b2 + 4) * q^9 $$q + ( - \beta_{4} - 1) q^{2} - \beta_{2} q^{3} + 2 \beta_{4} q^{4} - \beta_{6} q^{5} + (\beta_{2} - \beta_1) q^{6} + (\beta_{8} + \beta_{5}) q^{7} + ( - 2 \beta_{4} + 2) q^{8} + ( - \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{3} - \beta_{2} + 4) q^{9} + (\beta_{6} - \beta_{5}) q^{10} + (\beta_{10} + \beta_{7} - \beta_{5} + \beta_{3}) q^{11} + 2 \beta_1 q^{12} + (\beta_{11} + \beta_{9} + \beta_{6} + \beta_{3} + 2 \beta_1 - 1) q^{13} + (\beta_{9} - \beta_{8} - \beta_{6} - \beta_{5}) q^{14} + ( - \beta_{11} + \beta_{9} + 2 \beta_{4} + 2) q^{15} - 4 q^{16} + (\beta_{11} - \beta_{10} + 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_1) q^{17} + (2 \beta_{11} - 2 \beta_{9} + \beta_{7} - 4 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 4) q^{18} + ( - \beta_{11} + 2 \beta_{9} - \beta_{7} - \beta_{6} + 3 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 3) q^{19} + 2 \beta_{5} q^{20} + ( - 2 \beta_{8} + \beta_{7} + 4 \beta_{5} - 4 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 4) q^{21} + ( - \beta_{11} - \beta_{10} + \beta_{6} + \beta_{5} - 2 \beta_{3}) q^{22} + (\beta_{9} + \beta_{8} + 2 \beta_{6} - 2 \beta_{5} - 8 \beta_{4} - 3 \beta_1) q^{23} + ( - 2 \beta_{2} - 2 \beta_1) q^{24} + 5 \beta_{4} q^{25} + ( - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \cdots + 1) q^{26}+ \cdots + (5 \beta_{10} + 4 \beta_{8} - 3 \beta_{7} + 35 \beta_{5} + 5 \beta_{4} - 3 \beta_{3} + \cdots - 5) q^{99}+O(q^{100})$$ q + (-b4 - 1) * q^2 - b2 * q^3 + 2*b4 * q^4 - b6 * q^5 + (b2 - b1) * q^6 + (b8 + b5) * q^7 + (-2*b4 + 2) * q^8 + (-b11 - b10 + b9 - b8 + b3 - b2 + 4) * q^9 + (b6 - b5) * q^10 + (b10 + b7 - b5 + b3) * q^11 + 2*b1 * q^12 + (b11 + b9 + b6 + b3 + 2*b1 - 1) * q^13 + (b9 - b8 - b6 - b5) * q^14 + (-b11 + b9 + 2*b4 + 2) * q^15 - 4 * q^16 + (b11 - b10 + 2*b7 + 2*b6 - 2*b5 + 2*b1) * q^17 + (2*b11 - 2*b9 + b7 - 4*b4 - b3 + b2 - b1 - 4) * q^18 + (-b11 + 2*b9 - b7 - b6 + 3*b4 + b3 + b2 - b1 + 3) * q^19 + 2*b5 * q^20 + (-2*b8 + b7 + 4*b5 - 4*b4 + b3 - b2 - b1 + 4) * q^21 + (-b11 - b10 + b6 + b5 - 2*b3) * q^22 + (b9 + b8 + 2*b6 - 2*b5 - 8*b4 - 3*b1) * q^23 + (-2*b2 - 2*b1) * q^24 + 5*b4 * q^25 + (-b11 + b10 - b9 - b8 + b7 - b6 + b5 + b4 - b3 - 2*b2 - 2*b1 + 1) * q^26 + (-2*b11 - 2*b10 - b9 + b8 - 6*b6 - 6*b5 - 2*b3 - 2*b2 + 14) * q^27 + (-2*b9 + 2*b6) * q^28 + (2*b9 - 2*b8 - 2*b6 - 2*b5 + b3 - b2 - 1) * q^29 + (b11 - b10 - b9 - b8 - 4*b4) * q^30 + (3*b11 - 4*b9 - 2*b7 - 5*b6 - 13*b4 + 2*b3 + b2 - b1 - 13) * q^31 + (4*b4 + 4) * q^32 + (4*b10 + 3*b8 - b7 + 11*b5 + 6*b4 - b3 - 2*b2 - 2*b1 - 6) * q^33 + (2*b10 - 2*b7 + 4*b5 - 2*b3 - 2*b2 - 2*b1) * q^34 + (-b11 - b10 - b6 - b5 + b3 + 3*b2 - 3) * q^35 + (-2*b11 + 2*b10 + 2*b9 + 2*b8 - 2*b7 + 8*b4 + 2*b1) * q^36 + (5*b8 - 2*b7 - 7*b5 + 3*b4 - 2*b3 + 8*b2 + 8*b1 - 3) * q^37 + (b11 - b10 - 2*b9 - 2*b8 + 2*b7 + b6 - b5 - 6*b4 + 2*b1) * q^38 + (4*b11 + 3*b10 - 5*b9 + 3*b8 - b7 + 4*b6 + 3*b5 + 11*b4 - 2*b3 - b1 - 16) * q^39 + (-2*b6 - 2*b5) * q^40 + (b7 + 18*b6 + 3*b4 - b3 + b2 - b1 + 3) * q^41 + (-2*b9 + 2*b8 - 4*b6 - 4*b5 - 2*b3 + 2*b2 - 8) * q^42 + (-2*b11 + 2*b10 - 6*b7 - b6 + b5 + 18*b4 + b1) * q^43 + (2*b11 - 2*b7 - 2*b6 + 2*b3) * q^44 + (-2*b11 - b9 + b7 + b4 - b3 - 3*b2 + 3*b1 + 1) * q^45 + (-2*b8 + 4*b5 + 8*b4 + 3*b2 + 3*b1 - 8) * q^46 + (-8*b10 - b8 - 2*b7 + 13*b5 - 2*b4 - 2*b3 + 6*b2 + 6*b1 + 2) * q^47 + 4*b2 * q^48 + (-3*b11 + 3*b10 - b9 - b8 + 4*b6 - 4*b5 + 7*b4) * q^49 + (-5*b4 + 5) * q^50 + (b11 - b10 - 4*b7 - b6 + b5 + 6*b4 - 10*b1) * q^51 + (-2*b10 + 2*b8 - 2*b7 - 2*b5 - 2*b4 + 4*b2) * q^52 + (-2*b11 - 2*b10 + 8*b9 - 8*b8 - 12*b6 - 12*b5 + 4*b3 - 2*b2 + 2) * q^53 + (4*b11 + 2*b9 - 2*b7 + 12*b6 - 14*b4 + 2*b3 + 2*b2 - 2*b1 - 14) * q^54 + (-2*b9 + 2*b8 - b6 - b5 - 3*b3 + 6*b2 + 1) * q^55 + (2*b9 + 2*b8 - 2*b6 + 2*b5) * q^56 + (2*b11 - 7*b9 + 4*b7 - 7*b6 - 17*b4 - 4*b3 - 6*b2 + 6*b1 - 17) * q^57 + (-4*b9 + b7 + 4*b6 + b4 - b3 + b2 - b1 + 1) * q^58 + (5*b10 + 2*b8 + 2*b7 + 15*b5 - 8*b4 + 2*b3 + 8) * q^59 + (2*b10 + 2*b8 + 4*b4 - 4) * q^60 + (5*b11 + 5*b10 - b9 + b8 - 4*b6 - 4*b5 - b3 + 5*b2 - 1) * q^61 + (-3*b11 + 3*b10 + 4*b9 + 4*b8 + 4*b7 + 5*b6 - 5*b5 + 26*b4 + 2*b1) * q^62 + (6*b10 + b8 - 2*b7 - 11*b5 + 8*b4 - 2*b3 - 7*b2 - 7*b1 - 8) * q^63 - 8*b4 * q^64 + (b11 - 2*b9 + 2*b8 + 3*b7 + b6 - 6*b4 - b3 + 2*b2 - b1 - 5) * q^65 + (-4*b11 - 4*b10 + 3*b9 - 3*b8 - 11*b6 - 11*b5 + 2*b3 + 4*b2 + 12) * q^66 + (-2*b11 - b9 + 27*b6 + 12*b4 - 8*b2 + 8*b1 + 12) * q^67 + (-2*b11 - 2*b10 - 4*b6 - 4*b5 + 4*b3 + 4*b2) * q^68 + (6*b11 - 6*b10 - 8*b9 - 8*b8 + 5*b7 - 4*b6 + 4*b5 - 59*b4 - 13*b1) * q^69 + (2*b11 + b7 + 2*b6 + 3*b4 - b3 - 3*b2 + 3*b1 + 3) * q^70 + (-b11 + 4*b9 + 2*b7 - 9*b6 + 8*b4 - 2*b3 - 8*b2 + 8*b1 + 8) * q^71 + (-4*b10 - 4*b8 + 2*b7 - 8*b4 + 2*b3 - 2*b2 - 2*b1 + 8) * q^72 + (-10*b10 - 11*b8 + 4*b7 + 29*b5 - 3*b4 + 4*b3 - 8*b2 - 8*b1 + 3) * q^73 + (5*b9 - 5*b8 + 7*b6 + 7*b5 + 4*b3 - 16*b2 + 6) * q^74 + 5*b1 * q^75 + (2*b10 + 4*b8 - 2*b7 + 2*b5 + 6*b4 - 2*b3 - 2*b2 - 2*b1 - 6) * q^76 + (-3*b11 + 3*b10 + 3*b9 + 3*b8 + 6*b7 + 3*b6 - 3*b5 - 6*b1) * q^77 + (-7*b11 + b10 + 8*b9 + 2*b8 - b7 - 7*b6 + b5 + 5*b4 + 3*b3 + b2 + b1 + 27) * q^78 + (3*b11 + 3*b10 - 4*b9 + 4*b8 - 11*b6 - 11*b5 + 12*b2 - 16) * q^79 + 4*b6 * q^80 + (-8*b11 - 8*b10 + 8*b9 - 8*b8 - 12*b6 - 12*b5 - 3*b3 - 9*b2 + 56) * q^81 + (-2*b7 - 18*b6 + 18*b5 - 6*b4 + 2*b1) * q^82 + (2*b11 + 9*b9 - 2*b7 - 11*b6 + 12*b4 + 2*b3 + 3*b2 - 3*b1 + 12) * q^83 + (4*b9 - 2*b7 + 8*b6 + 8*b4 + 2*b3 - 2*b2 + 2*b1 + 8) * q^84 + (-2*b10 + 6*b8 - b7 + 2*b5 - 6*b4 - b3 + 2*b2 + 2*b1 + 6) * q^85 + (-4*b10 + 6*b7 - 2*b5 - 18*b4 + 6*b3 - b2 - b1 + 18) * q^86 + (-4*b9 + 4*b8 - 5*b6 - 5*b5 - 4*b3 - 4) * q^87 + (-2*b11 + 2*b10 + 4*b7 + 2*b6 - 2*b5) * q^88 + (-2*b10 - 8*b8 + 6*b7 - 18*b5 + 5*b4 + 6*b3 - 14*b2 - 14*b1 - 5) * q^89 + (2*b11 - 2*b10 + b9 + b8 - 2*b7 - 2*b4 - 6*b1) * q^90 + (-3*b11 + 3*b10 + b9 + 3*b8 - b7 + 23*b6 + 3*b5 - 2*b4 + 4*b3 - 13*b2 - 2*b1 - 35) * q^91 + (-2*b9 + 2*b8 - 4*b6 - 4*b5 - 6*b2 + 16) * q^92 + (6*b11 + 5*b9 - b7 + 43*b6 - 2*b4 + b3 + 8*b2 - 8*b1 - 2) * q^93 + (8*b11 + 8*b10 - b9 + b8 - 13*b6 - 13*b5 + 4*b3 - 12*b2 - 4) * q^94 + (b11 - b10 - 3*b9 - 3*b8 + 3*b7 - 3*b4 + 4*b1) * q^95 + (-4*b2 + 4*b1) * q^96 + (-4*b11 + 6*b9 + 2*b7 - 20*b6 - 14*b4 - 2*b3 + 10*b2 - 10*b1 - 14) * q^97 + (-6*b10 + 2*b8 + 8*b5 - 7*b4 + 7) * q^98 + (5*b10 + 4*b8 - 3*b7 + 35*b5 + 5*b4 - 3*b3 + 13*b2 + 13*b1 - 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 12 q^{2} + 24 q^{8} + 44 q^{9}+O(q^{10})$$ 12 * q - 12 * q^2 + 24 * q^8 + 44 * q^9 $$12 q - 12 q^{2} + 24 q^{8} + 44 q^{9} + 8 q^{11} - 4 q^{13} + 20 q^{15} - 48 q^{16} - 44 q^{18} + 36 q^{19} + 52 q^{21} - 16 q^{22} + 8 q^{26} + 144 q^{27} - 8 q^{29} - 136 q^{31} + 48 q^{32} - 60 q^{33} - 40 q^{35} - 44 q^{37} - 172 q^{39} + 32 q^{41} - 104 q^{42} + 16 q^{44} - 96 q^{46} - 16 q^{47} + 60 q^{50} - 8 q^{52} + 24 q^{53} - 144 q^{54} - 212 q^{57} + 8 q^{58} + 124 q^{59} - 40 q^{60} + 24 q^{61} - 80 q^{63} - 60 q^{65} + 120 q^{66} + 136 q^{67} + 40 q^{70} + 84 q^{71} + 88 q^{72} + 12 q^{73} + 88 q^{74} - 72 q^{76} + 312 q^{78} - 168 q^{79} + 596 q^{81} + 160 q^{83} + 104 q^{84} + 60 q^{85} + 224 q^{86} - 64 q^{87} - 44 q^{89} - 404 q^{91} + 192 q^{92} + 4 q^{93} + 32 q^{94} - 192 q^{97} + 60 q^{98} - 52 q^{99}+O(q^{100})$$ 12 * q - 12 * q^2 + 24 * q^8 + 44 * q^9 + 8 * q^11 - 4 * q^13 + 20 * q^15 - 48 * q^16 - 44 * q^18 + 36 * q^19 + 52 * q^21 - 16 * q^22 + 8 * q^26 + 144 * q^27 - 8 * q^29 - 136 * q^31 + 48 * q^32 - 60 * q^33 - 40 * q^35 - 44 * q^37 - 172 * q^39 + 32 * q^41 - 104 * q^42 + 16 * q^44 - 96 * q^46 - 16 * q^47 + 60 * q^50 - 8 * q^52 + 24 * q^53 - 144 * q^54 - 212 * q^57 + 8 * q^58 + 124 * q^59 - 40 * q^60 + 24 * q^61 - 80 * q^63 - 60 * q^65 + 120 * q^66 + 136 * q^67 + 40 * q^70 + 84 * q^71 + 88 * q^72 + 12 * q^73 + 88 * q^74 - 72 * q^76 + 312 * q^78 - 168 * q^79 + 596 * q^81 + 160 * q^83 + 104 * q^84 + 60 * q^85 + 224 * q^86 - 64 * q^87 - 44 * q^89 - 404 * q^91 + 192 * q^92 + 4 * q^93 + 32 * q^94 - 192 * q^97 + 60 * q^98 - 52 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 76x^{10} + 1956x^{8} + 19924x^{6} + 77560x^{4} + 85248x^{2} + 2916$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{10} + 62\nu^{8} + 1088\nu^{6} + 3626\nu^{4} - 13712\nu^{2} + 4320 ) / 25584$$ (v^10 + 62*v^8 + 1088*v^6 + 3626*v^4 - 13712*v^2 + 4320) / 25584 $$\beta_{3}$$ $$=$$ $$( 381\nu^{10} + 27886\nu^{8} + 666104\nu^{6} + 5594338\nu^{4} + 10074960\nu^{2} - 14932512 ) / 1867632$$ (381*v^10 + 27886*v^8 + 666104*v^6 + 5594338*v^4 + 10074960*v^2 - 14932512) / 1867632 $$\beta_{4}$$ $$=$$ $$( -40\nu^{11} - 3013\nu^{9} - 76566\nu^{7} - 767584\nu^{5} - 3004498\nu^{3} - 3780144\nu ) / 690768$$ (-40*v^11 - 3013*v^9 - 76566*v^7 - 767584*v^5 - 3004498*v^3 - 3780144*v) / 690768 $$\beta_{5}$$ $$=$$ $$( - 7226 \nu^{11} - 4797 \nu^{10} - 545018 \nu^{9} - 350181 \nu^{8} - 13818696 \nu^{7} - 8332389 \nu^{6} - 136050644 \nu^{5} - 71628804 \nu^{4} + \cdots - 86605038 ) / 50426064$$ (-7226*v^11 - 4797*v^10 - 545018*v^9 - 350181*v^8 - 13818696*v^7 - 8332389*v^6 - 136050644*v^5 - 71628804*v^4 - 488498420*v^3 - 197089542*v^2 - 441263880*v - 86605038) / 50426064 $$\beta_{6}$$ $$=$$ $$( 7226 \nu^{11} - 4797 \nu^{10} + 545018 \nu^{9} - 350181 \nu^{8} + 13818696 \nu^{7} - 8332389 \nu^{6} + 136050644 \nu^{5} - 71628804 \nu^{4} + \cdots - 86605038 ) / 50426064$$ (7226*v^11 - 4797*v^10 + 545018*v^9 - 350181*v^8 + 13818696*v^7 - 8332389*v^6 + 136050644*v^5 - 71628804*v^4 + 488498420*v^3 - 197089542*v^2 + 441263880*v - 86605038) / 50426064 $$\beta_{7}$$ $$=$$ $$( -545\nu^{11} - 42851\nu^{9} - 1166468\nu^{7} - 13107342\nu^{5} - 59511202\nu^{3} - 83385324\nu ) / 1867632$$ (-545*v^11 - 42851*v^9 - 1166468*v^7 - 13107342*v^5 - 59511202*v^3 - 83385324*v) / 1867632 $$\beta_{8}$$ $$=$$ $$( - 19079 \nu^{11} - 1422 \nu^{10} - 1462190 \nu^{9} - 116946 \nu^{8} - 38169996 \nu^{7} - 3595455 \nu^{6} - 398348462 \nu^{5} - 51101838 \nu^{4} + \cdots - 316556910 ) / 50426064$$ (-19079*v^11 - 1422*v^10 - 1462190*v^9 - 116946*v^8 - 38169996*v^7 - 3595455*v^6 - 398348462*v^5 - 51101838*v^4 - 1592128640*v^3 - 303685020*v^2 - 1683122616*v - 316556910) / 50426064 $$\beta_{9}$$ $$=$$ $$( - 19079 \nu^{11} + 1422 \nu^{10} - 1462190 \nu^{9} + 116946 \nu^{8} - 38169996 \nu^{7} + 3595455 \nu^{6} - 398348462 \nu^{5} + 51101838 \nu^{4} + \cdots + 316556910 ) / 50426064$$ (-19079*v^11 + 1422*v^10 - 1462190*v^9 + 116946*v^8 - 38169996*v^7 + 3595455*v^6 - 398348462*v^5 + 51101838*v^4 - 1592128640*v^3 + 303685020*v^2 - 1683122616*v + 316556910) / 50426064 $$\beta_{10}$$ $$=$$ $$( 29716 \nu^{11} + 5580 \nu^{10} + 2252269 \nu^{9} + 432306 \nu^{8} + 57681021 \nu^{7} + 11515635 \nu^{6} + 582044530 \nu^{5} + 123051978 \nu^{4} + \cdots + 438480054 ) / 50426064$$ (29716*v^11 + 5580*v^10 + 2252269*v^9 + 432306*v^8 + 57681021*v^7 + 11515635*v^6 + 582044530*v^5 + 123051978*v^4 + 2227874890*v^3 + 478423188*v^2 + 2321628678*v + 438480054) / 50426064 $$\beta_{11}$$ $$=$$ $$( - 29716 \nu^{11} + 5580 \nu^{10} - 2252269 \nu^{9} + 432306 \nu^{8} - 57681021 \nu^{7} + 11515635 \nu^{6} - 582044530 \nu^{5} + 123051978 \nu^{4} + \cdots + 438480054 ) / 50426064$$ (-29716*v^11 + 5580*v^10 - 2252269*v^9 + 432306*v^8 - 57681021*v^7 + 11515635*v^6 - 582044530*v^5 + 123051978*v^4 - 2227874890*v^3 + 478423188*v^2 - 2321628678*v + 438480054) / 50426064
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{3} + \beta_{2} - 13$$ b11 + b10 - b9 + b8 - b3 + b2 - 13 $$\nu^{3}$$ $$=$$ $$2\beta_{11} - 2\beta_{10} + \beta_{9} + \beta_{8} - 2\beta_{7} + 6\beta_{6} - 6\beta_{5} - 14\beta_{4} - 20\beta_1$$ 2*b11 - 2*b10 + b9 + b8 - 2*b7 + 6*b6 - 6*b5 - 14*b4 - 20*b1 $$\nu^{4}$$ $$=$$ $$- 35 \beta_{11} - 35 \beta_{10} + 35 \beta_{9} - 35 \beta_{8} - 12 \beta_{6} - 12 \beta_{5} + 24 \beta_{3} - 36 \beta_{2} + 326$$ -35*b11 - 35*b10 + 35*b9 - 35*b8 - 12*b6 - 12*b5 + 24*b3 - 36*b2 + 326 $$\nu^{5}$$ $$=$$ $$- 94 \beta_{11} + 94 \beta_{10} - 11 \beta_{9} - 11 \beta_{8} + 58 \beta_{7} - 243 \beta_{6} + 243 \beta_{5} + 562 \beta_{4} + 528 \beta_1$$ -94*b11 + 94*b10 - 11*b9 - 11*b8 + 58*b7 - 243*b6 + 243*b5 + 562*b4 + 528*b1 $$\nu^{6}$$ $$=$$ $$1122 \beta_{11} + 1122 \beta_{10} - 1050 \beta_{9} + 1050 \beta_{8} + 600 \beta_{6} + 600 \beta_{5} - 608 \beta_{3} + 1256 \beta_{2} - 9342$$ 1122*b11 + 1122*b10 - 1050*b9 + 1050*b8 + 600*b6 + 600*b5 - 608*b3 + 1256*b2 - 9342 $$\nu^{7}$$ $$=$$ $$3564 \beta_{11} - 3564 \beta_{10} - 342 \beta_{9} - 342 \beta_{8} - 1452 \beta_{7} + 7986 \beta_{6} - 7986 \beta_{5} - 21228 \beta_{4} - 15418 \beta_1$$ 3564*b11 - 3564*b10 - 342*b9 - 342*b8 - 1452*b7 + 7986*b6 - 7986*b5 - 21228*b4 - 15418*b1 $$\nu^{8}$$ $$=$$ $$- 35206 \beta_{11} - 35206 \beta_{10} + 30958 \beta_{9} - 30958 \beta_{8} - 23544 \beta_{6} - 23544 \beta_{5} + 16186 \beta_{3} - 44410 \beta_{2} + 279622$$ -35206*b11 - 35206*b10 + 30958*b9 - 30958*b8 - 23544*b6 - 23544*b5 + 16186*b3 - 44410*b2 + 279622 $$\nu^{9}$$ $$=$$ $$- 126428 \beta_{11} + 126428 \beta_{10} + 29306 \beta_{9} + 29306 \beta_{8} + 33692 \beta_{7} - 251304 \beta_{6} + 251304 \beta_{5} + 776708 \beta_{4} + 467684 \beta_1$$ -126428*b11 + 126428*b10 + 29306*b9 + 29306*b8 + 33692*b7 - 251304*b6 + 251304*b5 + 776708*b4 + 467684*b1 $$\nu^{10}$$ $$=$$ $$1102658 \beta_{11} + 1102658 \beta_{10} - 917618 \beta_{9} + 917618 \beta_{8} + 850440 \beta_{6} + 850440 \beta_{5} - 442764 \beta_{3} + 1556724 \beta_{2} - 8537120$$ 1102658*b11 + 1102658*b10 - 917618*b9 + 917618*b8 + 850440*b6 + 850440*b5 - 442764*b3 + 1556724*b2 - 8537120 $$\nu^{11}$$ $$=$$ $$4354756 \beta_{11} - 4354756 \beta_{10} - 1416862 \beta_{9} - 1416862 \beta_{8} - 721276 \beta_{7} + 7855470 \beta_{6} - 7855470 \beta_{5} - 27622204 \beta_{4} - 14440296 \beta_1$$ 4354756*b11 - 4354756*b10 - 1416862*b9 - 1416862*b8 - 721276*b7 + 7855470*b6 - 7855470*b5 - 27622204*b4 - 14440296*b1

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
21.1
 − 5.08458i − 2.34685i − 1.30814i − 0.187967i 3.23045i 5.69710i 5.08458i 2.34685i 1.30814i 0.187967i − 3.23045i − 5.69710i
−1.00000 1.00000i −5.08458 2.00000i −1.58114 1.58114i 5.08458 + 5.08458i −4.62330 + 4.62330i 2.00000 2.00000i 16.8530 3.16228i
21.2 −1.00000 1.00000i −2.34685 2.00000i 1.58114 + 1.58114i 2.34685 + 2.34685i 4.49093 4.49093i 2.00000 2.00000i −3.49227 3.16228i
21.3 −1.00000 1.00000i −1.30814 2.00000i −1.58114 1.58114i 1.30814 + 1.30814i 2.97945 2.97945i 2.00000 2.00000i −7.28877 3.16228i
21.4 −1.00000 1.00000i −0.187967 2.00000i 1.58114 + 1.58114i 0.187967 + 0.187967i −7.64727 + 7.64727i 2.00000 2.00000i −8.96467 3.16228i
21.5 −1.00000 1.00000i 3.23045 2.00000i −1.58114 1.58114i −3.23045 3.23045i 4.80613 4.80613i 2.00000 2.00000i 1.43578 3.16228i
21.6 −1.00000 1.00000i 5.69710 2.00000i 1.58114 + 1.58114i −5.69710 5.69710i −0.00593756 + 0.00593756i 2.00000 2.00000i 23.4569 3.16228i
31.1 −1.00000 + 1.00000i −5.08458 2.00000i −1.58114 + 1.58114i 5.08458 5.08458i −4.62330 4.62330i 2.00000 + 2.00000i 16.8530 3.16228i
31.2 −1.00000 + 1.00000i −2.34685 2.00000i 1.58114 1.58114i 2.34685 2.34685i 4.49093 + 4.49093i 2.00000 + 2.00000i −3.49227 3.16228i
31.3 −1.00000 + 1.00000i −1.30814 2.00000i −1.58114 + 1.58114i 1.30814 1.30814i 2.97945 + 2.97945i 2.00000 + 2.00000i −7.28877 3.16228i
31.4 −1.00000 + 1.00000i −0.187967 2.00000i 1.58114 1.58114i 0.187967 0.187967i −7.64727 7.64727i 2.00000 + 2.00000i −8.96467 3.16228i
31.5 −1.00000 + 1.00000i 3.23045 2.00000i −1.58114 + 1.58114i −3.23045 + 3.23045i 4.80613 + 4.80613i 2.00000 + 2.00000i 1.43578 3.16228i
31.6 −1.00000 + 1.00000i 5.69710 2.00000i 1.58114 1.58114i −5.69710 + 5.69710i −0.00593756 0.00593756i 2.00000 + 2.00000i 23.4569 3.16228i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 21.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d 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.k.a 12
3.b odd 2 1 1170.3.r.b 12
5.b even 2 1 650.3.k.k 12
5.c odd 4 1 650.3.f.l 12
5.c odd 4 1 650.3.f.m 12
13.d odd 4 1 inner 130.3.k.a 12
39.f even 4 1 1170.3.r.b 12
65.f even 4 1 650.3.f.m 12
65.g odd 4 1 650.3.k.k 12
65.k even 4 1 650.3.f.l 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.3.k.a 12 1.a even 1 1 trivial
130.3.k.a 12 13.d odd 4 1 inner
650.3.f.l 12 5.c odd 4 1
650.3.f.l 12 65.k even 4 1
650.3.f.m 12 5.c odd 4 1
650.3.f.m 12 65.f even 4 1
650.3.k.k 12 5.b even 2 1
650.3.k.k 12 65.g odd 4 1
1170.3.r.b 12 3.b odd 2 1
1170.3.r.b 12 39.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - 38T_{3}^{4} - 24T_{3}^{3} + 256T_{3}^{2} + 336T_{3} + 54$$ acting on $$S_{3}^{\mathrm{new}}(130, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2 T + 2)^{6}$$
$3$ $$(T^{6} - 38 T^{4} - 24 T^{3} + 256 T^{2} + \cdots + 54)^{2}$$
$5$ $$(T^{4} + 25)^{3}$$
$7$ $$T^{12} - 424 T^{9} + 9792 T^{8} + \cdots + 11664$$
$11$ $$T^{12} - 8 T^{11} + \cdots + 231339836484$$
$13$ $$T^{12} + 4 T^{11} + \cdots + 23298085122481$$
$17$ $$T^{12} + 2108 T^{10} + \cdots + 77722561281600$$
$19$ $$T^{12} - 36 T^{11} + \cdots + 172186884$$
$23$ $$T^{12} + 2568 T^{10} + \cdots + 76889744100$$
$29$ $$(T^{6} + 4 T^{5} - 840 T^{4} + \cdots - 5430204)^{2}$$
$31$ $$T^{12} + 136 T^{11} + \cdots + 99\!\cdots\!00$$
$37$ $$T^{12} + 44 T^{11} + \cdots + 50\!\cdots\!44$$
$41$ $$T^{12} - 32 T^{11} + \cdots + 10\!\cdots\!84$$
$43$ $$T^{12} + 17184 T^{10} + \cdots + 42\!\cdots\!00$$
$47$ $$T^{12} + 16 T^{11} + \cdots + 40\!\cdots\!44$$
$53$ $$(T^{6} - 12 T^{5} - 14228 T^{4} + \cdots - 67585859136)^{2}$$
$59$ $$T^{12} - 124 T^{11} + \cdots + 16\!\cdots\!84$$
$61$ $$(T^{6} - 12 T^{5} - 9516 T^{4} + \cdots - 71111196)^{2}$$
$67$ $$T^{12} - 136 T^{11} + \cdots + 30\!\cdots\!84$$
$71$ $$T^{12} - 84 T^{11} + \cdots + 76\!\cdots\!64$$
$73$ $$T^{12} - 12 T^{11} + \cdots + 24\!\cdots\!00$$
$79$ $$(T^{6} + 84 T^{5} - 7092 T^{4} + \cdots + 9524927376)^{2}$$
$83$ $$T^{12} - 160 T^{11} + \cdots + 22\!\cdots\!64$$
$89$ $$T^{12} + 44 T^{11} + \cdots + 17\!\cdots\!04$$
$97$ $$T^{12} + 192 T^{11} + \cdots + 26\!\cdots\!44$$
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