# Properties

 Label 65.4.e.a Level $65$ Weight $4$ Character orbit 65.e Analytic conductor $3.835$ Analytic rank $0$ Dimension $14$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [65,4,Mod(16,65)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(65, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("65.16");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$65 = 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 65.e (of order $$3$$, 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: $$3.83512415037$$ Analytic rank: $$0$$ Dimension: $$14$$ Relative dimension: $$7$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{14} - 2 x^{13} + 45 x^{12} - 52 x^{11} + 1311 x^{10} - 1336 x^{9} + 20343 x^{8} - 11166 x^{7} + 216447 x^{6} - 107836 x^{5} + 1201133 x^{4} + 4392 x^{3} + \cdots + 1157776$$ x^14 - 2*x^13 + 45*x^12 - 52*x^11 + 1311*x^10 - 1336*x^9 + 20343*x^8 - 11166*x^7 + 216447*x^6 - 107836*x^5 + 1201133*x^4 + 4392*x^3 + 3890980*x^2 - 1936800*x + 1157776 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{11} - \beta_{6}) q^{3} + (\beta_{13} - 4 \beta_{6} - \beta_{5} - 4) q^{4} - 5 q^{5} + (\beta_{13} - \beta_{11} + \beta_{7} - 3 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 3) q^{6} + (2 \beta_{12} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{7} + (\beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} - \beta_{2} + 2) q^{8} + ( - \beta_{13} + 2 \beta_{12} + 3 \beta_{11} + \beta_{9} - 3 \beta_{7} - 12 \beta_{6} + \cdots - 12) q^{9}+O(q^{10})$$ q - b1 * q^2 + (b11 - b6) * q^3 + (b13 - 4*b6 - b5 - 4) * q^4 - 5 * q^5 + (b13 - b11 + b7 - 3*b6 - b5 - b4 + b3 - 2*b2 - 2*b1 - 3) * q^6 + (2*b12 - b10 + b9 + b8 + b4 - b3 + b2 + b1) * q^7 + (b9 + b8 - b7 + b5 - b2 + 2) * q^8 + (-b13 + 2*b12 + 3*b11 + b9 - 3*b7 - 12*b6 + b5 + 4*b2 + 4*b1 - 12) * q^9 $$q - \beta_1 q^{2} + (\beta_{11} - \beta_{6}) q^{3} + (\beta_{13} - 4 \beta_{6} - \beta_{5} - 4) q^{4} - 5 q^{5} + (\beta_{13} - \beta_{11} + \beta_{7} - 3 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 3) q^{6} + (2 \beta_{12} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{7} + (\beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} - \beta_{2} + 2) q^{8} + ( - \beta_{13} + 2 \beta_{12} + 3 \beta_{11} + \beta_{9} - 3 \beta_{7} - 12 \beta_{6} + \cdots - 12) q^{9}+ \cdots + ( - 17 \beta_{12} + 29 \beta_{9} - 7 \beta_{8} + 79 \beta_{7} + 40 \beta_{5} + \cdots + 853) q^{99}+O(q^{100})$$ q - b1 * q^2 + (b11 - b6) * q^3 + (b13 - 4*b6 - b5 - 4) * q^4 - 5 * q^5 + (b13 - b11 + b7 - 3*b6 - b5 - b4 + b3 - 2*b2 - 2*b1 - 3) * q^6 + (2*b12 - b10 + b9 + b8 + b4 - b3 + b2 + b1) * q^7 + (b9 + b8 - b7 + b5 - b2 + 2) * q^8 + (-b13 + 2*b12 + 3*b11 + b9 - 3*b7 - 12*b6 + b5 + 4*b2 + 4*b1 - 12) * q^9 + 5*b1 * q^10 + (b13 - 2*b11 - b10 - 2*b9 + 12*b6 - b3 - 5*b1) * q^11 + (b12 + b9 + b8 - 5*b7 + 5*b5 + b4 - 3*b2 - 15) * q^12 + (b13 - 2*b12 - b11 + 2*b10 - b8 + 2*b7 + 15*b6 + 2*b5 - b3 + 3*b1 + 15) * q^13 + (-2*b12 + 2*b9 + 2*b8 + 8*b7 - 4*b5 - 2*b4 - 9*b2 + 10) * q^14 + (-5*b11 + 5*b6) * q^15 + (-b13 - b12 + 8*b11 + b10 - b9 - 24*b6 - b1) * q^16 + (-3*b13 + 2*b12 + 5*b11 - b10 + b9 + b8 - 5*b7 + 17*b6 + 3*b5 - 4*b4 + 4*b3 + 8*b2 + 8*b1 + 17) * q^17 + (-2*b12 - 5*b9 - b8 - 5*b7 + 6*b5 - 6*b4 - 10*b2 + 22) * q^18 + (8*b13 - 2*b12 - 4*b11 + 4*b10 - b9 - 4*b8 + 4*b7 - 34*b6 - 8*b5 + b4 - b3 - 7*b2 - 7*b1 - 34) * q^19 + (-5*b13 + 20*b6 + 5*b5 + 20) * q^20 + (b12 + 5*b9 - 3*b8 - 3*b7 - 8*b5 + 9*b4 + 17*b2) * q^21 + (13*b13 - 14*b11 - 3*b10 + 3*b8 + 14*b7 - 39*b6 - 13*b5 + 3*b4 - 3*b3 - 39) * q^22 + (-7*b13 + 5*b11 + 3*b10 - 2*b9 - 14*b6 - 5*b3 + 5*b1) * q^23 + (-10*b13 + 2*b12 + 10*b11 - 3*b10 + 2*b9 + 39*b6 + b3 - 12*b1) * q^24 + 25 * q^25 + (-10*b13 - 5*b12 + 3*b11 - b10 - 4*b9 - 3*b8 + 3*b7 + 24*b6 + 8*b5 + b4 + 4*b3 + 17*b2 - 15*b1 + 37) * q^26 + (3*b12 - 4*b9 - 4*b8 - 9*b7 + 2*b5 + 3*b4 + 5*b2 - 64) * q^27 + (-11*b13 - 4*b11 + 8*b9 + 36*b6 + 8*b3 + 12*b1) * q^28 + (16*b13 - 6*b12 - 7*b11 - b10 - 9*b9 - 9*b6 + 4*b3 + 2*b1) * q^29 + (-5*b13 + 5*b11 - 5*b7 + 15*b6 + 5*b5 + 5*b4 - 5*b3 + 10*b2 + 10*b1 + 15) * q^30 + (-2*b9 - 7*b8 + 10*b7 - 17*b5 + 5*b4 - 3*b2 + 79) * q^31 + (2*b13 + 2*b12 - 4*b11 - 5*b10 + b9 + 5*b8 + 4*b7 - 16*b6 - 2*b5 - 6*b4 + 6*b3 - 30*b2 - 30*b1 - 16) * q^32 + (11*b13 - 2*b12 - 33*b11 - b10 - b9 + b8 + 33*b7 + 47*b6 - 11*b5 + 4*b4 - 4*b3 + 12*b2 + 12*b1 + 47) * q^33 + (3*b12 - 11*b9 - b8 - 39*b7 + 20*b5 - 7*b4 + 38*b2 + 53) * q^34 + (-10*b12 + 5*b10 - 5*b9 - 5*b8 - 5*b4 + 5*b3 - 5*b2 - 5*b1) * q^35 + (-5*b13 - b12 - 19*b11 - 8*b10 - 10*b9 + 54*b6 - 44*b1) * q^36 + (b13 - b12 + 8*b11 + 8*b10 + 2*b9 + 75*b6 - 4*b3 - 38*b1) * q^37 + (-3*b12 + 6*b9 - 16*b7 + 18*b5 + 3*b4 - 54*b2 - 73) * q^38 + (-20*b13 + 14*b12 + 21*b11 - 5*b10 + 11*b9 + 9*b8 + 12*b7 - 47*b6 + 13*b5 + 3*b4 - 8*b3 + 23*b2 + 8*b1 + 61) * q^39 + (-5*b9 - 5*b8 + 5*b7 - 5*b5 + 5*b2 - 10) * q^40 + (-17*b13 - 5*b12 - 7*b11 + 6*b10 - 6*b9 + 23*b6 - 2*b3 + 22*b1) * q^41 + (b13 + 14*b12 + 67*b11 + 2*b10 + 21*b9 - 205*b6 - 9*b3 + 64*b1) * q^42 + (4*b13 - 6*b12 + 25*b11 + 6*b10 - 3*b9 - 6*b8 - 25*b7 + 46*b6 - 4*b5 + 3*b4 - 3*b3 + 65*b2 + 65*b1 + 46) * q^43 + (20*b9 + 11*b8 + 8*b7 + 2*b5 + 9*b4 - 77*b2 + 11) * q^44 + (5*b13 - 10*b12 - 15*b11 - 5*b9 + 15*b7 + 60*b6 - 5*b5 - 20*b2 - 20*b1 + 60) * q^45 + (14*b13 - 16*b12 - 49*b11 + 13*b10 - 8*b9 - 13*b8 + 49*b7 + 24*b6 - 14*b5 - 8*b4 + 8*b3 + 11*b2 + 11*b1 + 24) * q^46 + (-10*b12 + 9*b9 + 11*b8 - 9*b7 - 20*b5 - 12*b4 - 38*b2 + 52) * q^47 + (-7*b13 + 16*b12 + 31*b11 - 4*b10 + 8*b9 + 4*b8 - 31*b7 - 321*b6 + 7*b5 - 5*b4 + 5*b3 + 44*b2 + 44*b1 - 321) * q^48 + (16*b13 - 4*b12 - 40*b11 + 4*b10 + 8*b9 + 254*b6 + 12*b3 - 28*b1) * q^49 - 25*b1 * q^50 + (-7*b12 - 21*b9 - 16*b8 - 25*b7 + 9*b5 - 12*b4 - 70*b2 - 99) * q^51 + (33*b13 + 8*b12 - 8*b11 + 2*b10 - 4*b9 + 13*b7 - 245*b6 - 10*b5 - 3*b3 + 58*b2 - 9*b1 - 166) * q^52 + (7*b12 - 13*b9 + 9*b8 + 31*b7 - 14*b5 - 15*b4 - 15*b2 - 65) * q^53 + (21*b13 + 13*b12 - 7*b11 - 10*b10 - 6*b9 + 40*b6 - 22*b3 + 55*b1) * q^54 + (-5*b13 + 10*b11 + 5*b10 + 10*b9 - 60*b6 + 5*b3 + 25*b1) * q^55 + (-13*b13 - 16*b12 + 17*b11 - 5*b10 - 8*b9 + 5*b8 - 17*b7 + 182*b6 + 13*b5 - 12*b4 + 12*b3 + 55*b2 + 55*b1 + 182) * q^56 + (13*b12 + 2*b9 + 21*b8 - 39*b7 + 45*b5 - 6*b4 - 38*b2 + 37) * q^57 + (-3*b13 + 34*b12 + 7*b11 - 18*b10 + 17*b9 + 18*b8 - 7*b7 + 160*b6 + 3*b5 + 26*b4 - 26*b3 - 71*b2 - 71*b1 + 160) * q^58 + (-38*b13 + 20*b12 + 25*b11 - 9*b10 + 10*b9 + 9*b8 - 25*b7 - 229*b6 + 38*b5 - 6*b4 + 6*b3 - 54*b2 - 54*b1 - 229) * q^59 + (-5*b12 - 5*b9 - 5*b8 + 25*b7 - 25*b5 - 5*b4 + 15*b2 + 75) * q^60 + (-8*b13 - 6*b12 + 7*b11 + 7*b10 - 3*b9 - 7*b8 - 7*b7 - 58*b6 + 8*b5 + 13*b4 - 13*b3 + 37*b2 + 37*b1 - 58) * q^61 + (41*b13 + 12*b12 - 14*b11 + 3*b10 + 30*b9 + 15*b6 + 3*b3 - 19*b1) * q^62 + (39*b13 - 9*b12 - 25*b11 + 2*b10 + 18*b9 + 142*b6 + 34*b3 - 190*b1) * q^63 + (17*b12 + 9*b9 + 20*b8 + 13*b7 - 26*b5 + 6*b4 - 15*b2 - 146) * q^64 + (-5*b13 + 10*b12 + 5*b11 - 10*b10 + 5*b8 - 10*b7 - 75*b6 - 10*b5 + 5*b3 - 15*b1 - 75) * q^65 + (-b12 + 51*b9 + 13*b8 + 15*b7 + 24*b5 + 37*b4 + 4*b2 + 305) * q^66 + (-28*b13 - 3*b12 + 47*b11 - 10*b10 - 19*b9 + 186*b6 - 3*b3 + 61*b1) * q^67 + (14*b13 + 8*b12 - 11*b11 - 30*b10 - 31*b9 - 81*b6 - 17*b3 - 83*b1) * q^68 + (30*b13 + 28*b12 + 21*b11 - 9*b10 + 14*b9 + 9*b8 - 21*b7 - 99*b6 - 30*b5 + 14*b4 - 14*b3 + 160*b2 + 160*b1 - 99) * q^69 + (10*b12 - 10*b9 - 10*b8 - 40*b7 + 20*b5 + 10*b4 + 45*b2 - 50) * q^70 + (-34*b13 + 4*b12 + 11*b11 + 5*b10 + 2*b9 - 5*b8 - 11*b7 + 63*b6 + 34*b5 + 10*b4 - 10*b3 - 22*b2 - 22*b1 + 63) * q^71 + (36*b13 - 12*b12 - 3*b11 - 3*b10 - 6*b9 + 3*b8 + 3*b7 - 218*b6 - 36*b5 - 18*b4 + 18*b3 - 22*b2 - 22*b1 - 218) * q^72 + (18*b12 - 16*b9 - 3*b8 - 26*b7 + 5*b5 + 5*b4 + 29*b2 + 373) * q^73 + (23*b13 - 20*b12 - 12*b11 + 15*b10 - 10*b9 - 15*b8 + 12*b7 - 519*b6 - 23*b5 - 13*b4 + 13*b3 + 86*b2 + 86*b1 - 519) * q^74 + (25*b11 - 25*b6) * q^75 + (-57*b13 - 11*b12 + 62*b11 + 14*b10 - 23*b9 + 403*b6 - 15*b3 - 45*b1) * q^76 + (-9*b12 - 7*b9 + 2*b8 - 2*b7 - 39*b5 - 18*b4 - 164*b2 - 435) * q^77 + (36*b13 - 36*b12 - 41*b11 + 15*b10 - b9 - 10*b8 + 81*b7 - 359*b6 - 51*b5 - 42*b4 + 47*b3 - 25*b2 - 106*b1 - 60) * q^78 + (-24*b12 - 16*b9 - 27*b8 + 18*b7 - 31*b5 - 13*b4 - 81*b2 + 41) * q^79 + (5*b13 + 5*b12 - 40*b11 - 5*b10 + 5*b9 + 120*b6 + 5*b1) * q^80 + (-15*b13 + 3*b12 - 66*b11 + 3*b10 + 15*b9 - 23*b6 + 6*b3 + 60*b1) * q^81 + (-8*b13 + 4*b12 - 41*b11 + 29*b10 + 2*b9 - 29*b8 + 41*b7 + 280*b6 + 8*b5 + 16*b4 - 16*b3 + 137*b2 + 137*b1 + 280) * q^82 + (-15*b12 + 22*b9 - 20*b8 + 46*b7 - 76*b5 + 27*b4 + 13*b2 + 95) * q^83 + (15*b13 - 62*b12 - 3*b11 + 27*b10 - 31*b9 - 27*b8 + 3*b7 + 377*b6 - 15*b5 - 39*b4 + 39*b3 - 167*b2 - 167*b1 + 377) * q^84 + (15*b13 - 10*b12 - 25*b11 + 5*b10 - 5*b9 - 5*b8 + 25*b7 - 85*b6 - 15*b5 + 20*b4 - 20*b3 - 40*b2 - 40*b1 - 85) * q^85 + (-3*b12 - 27*b9 - 8*b8 + 45*b7 + 47*b5 - 22*b4 + 19*b2 + 716) * q^86 + (-86*b13 - 14*b12 - 65*b11 - 24*b10 - 7*b9 + 24*b8 + 65*b7 + 20*b6 + 86*b5 + b4 - b3 - 61*b2 - 61*b1 + 20) * q^87 + (-78*b13 - 2*b12 + 47*b11 - 4*b10 - 13*b9 + 457*b6 - 5*b3 + 62*b1) * q^88 + (31*b13 + 29*b12 + 4*b11 - 15*b10 + 9*b9 + 79*b6 - 34*b3 - 40*b1) * q^89 + (10*b12 + 25*b9 + 5*b8 + 25*b7 - 30*b5 + 30*b4 + 50*b2 - 110) * q^90 + (2*b13 - 20*b12 - 10*b11 - 2*b10 - 34*b9 - 8*b8 - 91*b7 - 484*b6 - 5*b5 + 14*b4 - 43*b3 + 140*b2 + 43*b1 - 223) * q^91 + (11*b12 + 21*b9 + 12*b8 - 117*b7 + 88*b5 + 20*b4 + 60*b2 + 390) * q^92 + (87*b13 - 31*b12 + 54*b11 + 41*b10 - 3*b9 - 320*b6 + 18*b3 + 16*b1) * q^93 + (-93*b13 - 43*b12 + 13*b11 + 42*b10 - 12*b9 + 242*b6 + 32*b3 + 64*b1) * q^94 + (-40*b13 + 10*b12 + 20*b11 - 20*b10 + 5*b9 + 20*b8 - 20*b7 + 170*b6 + 40*b5 - 5*b4 + 5*b3 + 35*b2 + 35*b1 + 170) * q^95 + (-23*b12 - 43*b9 - 23*b8 + 15*b7 - 25*b5 - 43*b4 - 221*b2 + 5) * q^96 + (-61*b13 + 26*b12 + b11 + 17*b10 + 13*b9 - 17*b8 - b7 + 271*b6 + 61*b5 + 10*b4 - 10*b3 + 42*b2 + 42*b1 + 271) * q^97 + (-88*b13 + 32*b12 + 176*b11 - 8*b10 + 16*b9 + 8*b8 - 176*b7 - 208*b6 + 88*b5 + 48*b4 - 48*b3 + 282*b2 + 282*b1 - 208) * q^98 + (-17*b12 + 29*b9 - 7*b8 + 79*b7 + 40*b5 + 19*b4 + 123*b2 + 853) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14 q - 2 q^{2} + 4 q^{3} - 30 q^{4} - 70 q^{5} - 23 q^{6} - 7 q^{7} + 42 q^{8} - 87 q^{9}+O(q^{10})$$ 14 * q - 2 * q^2 + 4 * q^3 - 30 * q^4 - 70 * q^5 - 23 * q^6 - 7 * q^7 + 42 * q^8 - 87 * q^9 $$14 q - 2 q^{2} + 4 q^{3} - 30 q^{4} - 70 q^{5} - 23 q^{6} - 7 q^{7} + 42 q^{8} - 87 q^{9} + 10 q^{10} - 87 q^{11} - 158 q^{12} + 123 q^{13} + 132 q^{14} - 20 q^{15} + 134 q^{16} + 114 q^{17} + 414 q^{18} - 245 q^{19} + 150 q^{20} - 76 q^{21} - 338 q^{22} + 74 q^{23} - 334 q^{24} + 350 q^{25} + 243 q^{26} - 884 q^{27} - 230 q^{28} + 88 q^{29} + 115 q^{30} + 1000 q^{31} - 80 q^{32} + 194 q^{33} + 854 q^{34} + 35 q^{35} - 425 q^{36} - 633 q^{37} - 596 q^{38} + 970 q^{39} - 210 q^{40} - 162 q^{41} + 1439 q^{42} + 280 q^{43} + 440 q^{44} + 435 q^{45} + 11 q^{46} + 950 q^{47} - 2281 q^{48} - 1694 q^{49} - 50 q^{50} - 860 q^{51} - 956 q^{52} - 1206 q^{53} - 51 q^{54} + 435 q^{55} + 1277 q^{56} + 916 q^{57} + 1213 q^{58} - 1410 q^{59} + 790 q^{60} - 412 q^{61} + 56 q^{62} - 1241 q^{63} - 2358 q^{64} - 615 q^{65} + 4346 q^{66} - 1398 q^{67} + 493 q^{68} - 1080 q^{69} - 660 q^{70} + 584 q^{71} - 1545 q^{72} + 5076 q^{73} - 3840 q^{74} + 100 q^{75} - 3292 q^{76} - 5506 q^{77} + 1179 q^{78} + 928 q^{79} - 670 q^{80} + 473 q^{81} + 1583 q^{82} + 932 q^{83} + 3081 q^{84} - 570 q^{85} + 9858 q^{86} + 282 q^{87} - 3389 q^{88} - 443 q^{89} - 2070 q^{90} + 487 q^{91} + 6182 q^{92} + 2116 q^{93} - 2017 q^{94} + 1225 q^{95} + 954 q^{96} + 1870 q^{97} - 1364 q^{98} + 11378 q^{99}+O(q^{100})$$ 14 * q - 2 * q^2 + 4 * q^3 - 30 * q^4 - 70 * q^5 - 23 * q^6 - 7 * q^7 + 42 * q^8 - 87 * q^9 + 10 * q^10 - 87 * q^11 - 158 * q^12 + 123 * q^13 + 132 * q^14 - 20 * q^15 + 134 * q^16 + 114 * q^17 + 414 * q^18 - 245 * q^19 + 150 * q^20 - 76 * q^21 - 338 * q^22 + 74 * q^23 - 334 * q^24 + 350 * q^25 + 243 * q^26 - 884 * q^27 - 230 * q^28 + 88 * q^29 + 115 * q^30 + 1000 * q^31 - 80 * q^32 + 194 * q^33 + 854 * q^34 + 35 * q^35 - 425 * q^36 - 633 * q^37 - 596 * q^38 + 970 * q^39 - 210 * q^40 - 162 * q^41 + 1439 * q^42 + 280 * q^43 + 440 * q^44 + 435 * q^45 + 11 * q^46 + 950 * q^47 - 2281 * q^48 - 1694 * q^49 - 50 * q^50 - 860 * q^51 - 956 * q^52 - 1206 * q^53 - 51 * q^54 + 435 * q^55 + 1277 * q^56 + 916 * q^57 + 1213 * q^58 - 1410 * q^59 + 790 * q^60 - 412 * q^61 + 56 * q^62 - 1241 * q^63 - 2358 * q^64 - 615 * q^65 + 4346 * q^66 - 1398 * q^67 + 493 * q^68 - 1080 * q^69 - 660 * q^70 + 584 * q^71 - 1545 * q^72 + 5076 * q^73 - 3840 * q^74 + 100 * q^75 - 3292 * q^76 - 5506 * q^77 + 1179 * q^78 + 928 * q^79 - 670 * q^80 + 473 * q^81 + 1583 * q^82 + 932 * q^83 + 3081 * q^84 - 570 * q^85 + 9858 * q^86 + 282 * q^87 - 3389 * q^88 - 443 * q^89 - 2070 * q^90 + 487 * q^91 + 6182 * q^92 + 2116 * q^93 - 2017 * q^94 + 1225 * q^95 + 954 * q^96 + 1870 * q^97 - 1364 * q^98 + 11378 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} - 2 x^{13} + 45 x^{12} - 52 x^{11} + 1311 x^{10} - 1336 x^{9} + 20343 x^{8} - 11166 x^{7} + 216447 x^{6} - 107836 x^{5} + 1201133 x^{4} + 4392 x^{3} + \cdots + 1157776$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 3326902201366 \nu^{13} + \cdots - 34\!\cdots\!32 ) / 56\!\cdots\!80$$ (-3326902201366*v^13 - 2766228314997701*v^12 + 11090467347398791*v^11 - 121266915258692077*v^10 + 339767666647636262*v^9 - 3327251392008764616*v^8 + 9765865833206795238*v^7 - 47894626257442780887*v^6 + 107338183474008449073*v^5 - 448488882766637910911*v^4 + 1428291819822049017266*v^3 - 1996292904611150474188*v^2 + 1025300300389103849008*v - 3457172445310064178032) / 5667671277177588202680 $$\beta_{3}$$ $$=$$ $$( 52\!\cdots\!09 \nu^{13} + \cdots - 26\!\cdots\!84 ) / 30\!\cdots\!40$$ (523182621589451209*v^13 - 9656850387029169079*v^12 - 51548294721017382733*v^11 - 482638851525362726780*v^10 - 1797569657940990701054*v^9 - 16562698242066108904416*v^8 - 56408785023659441536929*v^7 - 291010723711943929819965*v^6 - 543445008878086388369499*v^5 - 3249542083993666821809920*v^4 - 4973187773049461746164962*v^3 - 17451225567745373351681468*v^2 + 2058286116109970969931872*v - 26798037593785203502237984) / 3049207147121542453041840 $$\beta_{4}$$ $$=$$ $$( - 11\!\cdots\!27 \nu^{13} + \cdots - 84\!\cdots\!28 ) / 30\!\cdots\!40$$ (-1125081546642793027*v^13 - 2408288576668246169*v^12 - 147803903983998124325*v^11 - 111800433850915779496*v^10 - 5662303956159972477064*v^9 - 5144956502746555452936*v^8 - 137251651130763906028389*v^7 - 106332974031561851807523*v^6 - 1756776106496933063184075*v^5 - 1356335139590250309306188*v^4 - 13483802757577222030935952*v^3 - 8565335904671014832634268*v^2 - 33899923377346925117832824*v - 8403439655074630684471328) / 3049207147121542453041840 $$\beta_{5}$$ $$=$$ $$( 27\!\cdots\!33 \nu^{13} + \cdots - 68\!\cdots\!76 ) / 56\!\cdots\!80$$ (2772882119400433*v^13 - 11240177946460261*v^12 + 121439914173163109*v^11 - 344129235433627088*v^10 + 3331696133349789592*v^9 - 9833545004689183776*v^8 + 47931774447423233643*v^7 - 108058281474787515675*v^6 + 448847642592424414887*v^5 - 1432287871843882364944*v^4 + 1996278292856682074716*v^3 - 6705916487494163130368*v^2 + 3463615989493669846832*v - 68015907133654147154176) / 5667671277177588202680 $$\beta_{6}$$ $$=$$ $$( - 80\!\cdots\!83 \nu^{13} + \cdots - 24\!\cdots\!72 ) / 15\!\cdots\!20$$ (-803246385992115283*v^13 + 1607387708676398020*v^12 - 35401971952910806166*v^11 + 38785476355139719937*v^10 - 1020435211831074967300*v^9 + 981739669357251863610*v^8 - 15445410605787243520365*v^7 + 6342031236855331330956*v^6 - 160976616045583268600898*v^5 + 57744905925337470856951*v^4 - 844162231881641808180580*v^3 - 387738357659408555967490*v^2 - 2588412831627201246290768*v - 244681753975911281789672) / 1524603573560771226520920 $$\beta_{7}$$ $$=$$ $$( 21\!\cdots\!61 \nu^{13} + \cdots - 31\!\cdots\!24 ) / 37\!\cdots\!20$$ (2131050886361861*v^13 - 7078708615767206*v^12 + 82657666589744074*v^11 - 148894721129722153*v^10 + 2177054950642700894*v^9 - 4385741761573758984*v^8 + 27506221241217534687*v^7 - 32929279100056183086*v^6 + 242485382589670238322*v^5 - 673122166986874185779*v^4 + 790854353619076791662*v^3 - 3249857420262978259612*v^2 + 1684054104950570125312*v - 31161600099767064168224) / 3778447518118392135120 $$\beta_{8}$$ $$=$$ $$( - 65\!\cdots\!50 \nu^{13} + \cdots - 18\!\cdots\!20 ) / 10\!\cdots\!80$$ (-656489828737648250*v^13 - 15352922279544514753*v^12 + 2390413038873043949*v^11 - 695250316842788423345*v^10 - 130749819332753701280*v^9 - 19731288856958630125344*v^8 + 5324034591068163772638*v^7 - 297376033000661613485319*v^6 - 16840725158958478622313*v^5 - 2870465209056811524797095*v^4 + 480993845726616190651900*v^3 - 13588277499547702362985232*v^2 - 5789238438512979317812936*v - 18451347378483204641213920) / 1016402382373847484347280 $$\beta_{9}$$ $$=$$ $$( 24\!\cdots\!43 \nu^{13} + \cdots + 32\!\cdots\!92 ) / 33\!\cdots\!60$$ (240776569303982243*v^13 + 2343726727992895121*v^12 + 10625804261716880635*v^11 + 115736359142663681074*v^10 + 384910595842668353006*v^9 + 3390442105610720729944*v^8 + 7750747891805967635701*v^7 + 53960586050122612238467*v^6 + 109604763729477362022805*v^5 + 526319732408929498401722*v^4 + 903910004622752144621258*v^3 + 2610209525021017906740372*v^2 + 2915635347027244163333016*v + 3231275045977176936718592) / 338800794124615828115760 $$\beta_{10}$$ $$=$$ $$( - 32\!\cdots\!23 \nu^{13} + \cdots - 30\!\cdots\!96 ) / 30\!\cdots\!40$$ (-3206920903483164323*v^13 - 25352804090212209904*v^12 - 119042349862537272604*v^11 - 1221240433616532962591*v^10 - 4295784337961621550614*v^9 - 35874303342450747049176*v^8 - 74366871983261753595105*v^7 - 561663449340413167929924*v^6 - 1188949524137806073667312*v^5 - 5510335792829984711922253*v^4 - 9529386527201435498846822*v^3 - 25949441155773831694799468*v^2 - 66579934269117511588600288*v - 30026951185982206529185696) / 3049207147121542453041840 $$\beta_{11}$$ $$=$$ $$( - 12\!\cdots\!32 \nu^{13} + \cdots - 41\!\cdots\!12 ) / 30\!\cdots\!40$$ (-12549454344631442332*v^13 + 23638077185578916455*v^12 - 558547498249194857831*v^11 + 574265948056030994669*v^10 - 16150452414931348474120*v^9 + 13544629763724394804320*v^8 - 246253259925534708512376*v^7 + 74104987360288600787481*v^6 - 2570858950447925959958493*v^5 + 328734238420555078632067*v^4 - 13589108299332344834997940*v^3 - 6829718777778506466822760*v^2 - 41294792334358961377974200*v - 4114000637808287291206112) / 3049207147121542453041840 $$\beta_{12}$$ $$=$$ $$( 15\!\cdots\!52 \nu^{13} + \cdots - 36\!\cdots\!60 ) / 26\!\cdots\!20$$ (150294597506125552*v^13 - 236229705238513059*v^12 + 6529683510611703899*v^11 - 6814505435024938365*v^10 + 190701968826187170316*v^9 - 189042715739597659136*v^8 + 2822650191673616078284*v^7 - 2303343473883607283237*v^6 + 29197080469760985712897*v^5 - 25224409684057507559115*v^4 + 133376584392658326929608*v^3 - 107410320449157270249088*v^2 + 383493444892260233186584*v - 367797214494660663420960) / 26061599548047371393520 $$\beta_{13}$$ $$=$$ $$( - 88\!\cdots\!19 \nu^{13} + \cdots - 29\!\cdots\!68 ) / 15\!\cdots\!20$$ (-8893051341786666919*v^13 + 16265044636518966031*v^12 - 392156326522348797671*v^11 + 372854951930030952572*v^10 - 11348996282101806207352*v^9 + 9135652426025631927576*v^8 - 172451279943090072394413*v^7 + 47036697125546134254897*v^6 - 1810979376689637055606173*v^5 + 307653433578045294113476*v^4 - 9592947921801254220068356*v^3 - 4932148253488061327157952*v^2 - 30129241278352617766691408*v - 2937217183934646247698368) / 1524603573560771226520920
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{13} - 12\beta_{6} - \beta_{5} - 12$$ b13 - 12*b6 - b5 - 12 $$\nu^{3}$$ $$=$$ $$-\beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + 17\beta_{2} - 2$$ -b9 - b8 + b7 - b5 + 17*b2 - 2 $$\nu^{4}$$ $$=$$ $$-25\beta_{13} - \beta_{12} + 8\beta_{11} + \beta_{10} - \beta_{9} + 200\beta_{6} - \beta_1$$ -25*b13 - b12 + 8*b11 + b10 - b9 + 200*b6 - b1 $$\nu^{5}$$ $$=$$ $$- 34 \beta_{13} - 2 \beta_{12} + 36 \beta_{11} - 27 \beta_{10} - \beta_{9} + 27 \beta_{8} - 36 \beta_{7} + 80 \beta_{6} + 34 \beta_{5} + 6 \beta_{4} - 6 \beta_{3} - 322 \beta_{2} - 322 \beta _1 + 80$$ -34*b13 - 2*b12 + 36*b11 - 27*b10 - b9 + 27*b8 - 36*b7 + 80*b6 + 34*b5 + 6*b4 - 6*b3 - 322*b2 - 322*b1 + 80 $$\nu^{6}$$ $$=$$ $$-23\beta_{12} + 9\beta_{9} - 20\beta_{8} - 307\beta_{7} + 590\beta_{5} + 6\beta_{4} - 55\beta_{2} + 3758$$ -23*b12 + 9*b9 - 20*b8 - 307*b7 + 590*b5 + 6*b4 - 55*b2 + 3758 $$\nu^{7}$$ $$=$$ $$1000 \beta_{13} + 20 \beta_{12} - 1050 \beta_{11} + 630 \beta_{10} + 928 \beta_{9} - 2572 \beta_{6} + 258 \beta_{3} + 6477 \beta_1$$ 1000*b13 + 20*b12 - 1050*b11 + 630*b10 + 928*b9 - 2572*b6 + 258*b3 + 6477*b1 $$\nu^{8}$$ $$=$$ $$13717 \beta_{13} + 824 \beta_{12} - 8848 \beta_{11} - 260 \beta_{10} + 412 \beta_{9} + 260 \beta_{8} + 8848 \beta_{7} - 75872 \beta_{6} - 13717 \beta_{5} - 360 \beta_{4} + 360 \beta_{3} + 2340 \beta_{2} + \cdots - 75872$$ 13717*b13 + 824*b12 - 8848*b11 - 260*b10 + 412*b9 + 260*b8 + 8848*b7 - 75872*b6 - 13717*b5 - 360*b4 + 360*b3 + 2340*b2 + 2340*b1 - 75872 $$\nu^{9}$$ $$=$$ $$204 \beta_{12} - 22313 \beta_{9} - 14237 \beta_{8} + 28673 \beta_{7} - 27617 \beta_{5} - 7872 \beta_{4} + 135469 \beta_{2} - 76778$$ 204*b12 - 22313*b9 - 14237*b8 + 28673*b7 - 27617*b5 - 7872*b4 + 135469*b2 - 76778 $$\nu^{10}$$ $$=$$ $$- 317765 \beta_{13} - 6773 \beta_{12} + 230140 \beta_{11} + 857 \beta_{10} - 26513 \beta_{9} + 1604200 \beta_{6} - 13824 \beta_{3} - 83517 \beta_1$$ -317765*b13 - 6773*b12 + 230140*b11 + 857*b10 - 26513*b9 + 1604200*b6 - 13824*b3 - 83517*b1 $$\nu^{11}$$ $$=$$ $$- 736038 \beta_{13} + 2270 \beta_{12} + 755784 \beta_{11} - 319479 \beta_{10} + 1135 \beta_{9} + 319479 \beta_{8} - 755784 \beta_{7} + 2178976 \beta_{6} + 736038 \beta_{5} + 210678 \beta_{4} + \cdots + 2178976$$ -736038*b13 + 2270*b12 + 755784*b11 - 319479*b10 + 1135*b9 + 319479*b8 - 755784*b7 + 2178976*b6 + 736038*b5 + 210678*b4 - 210678*b3 - 2911874*b2 - 2911874*b1 + 2178976 $$\nu^{12}$$ $$=$$ $$- 106531 \beta_{12} + 643321 \beta_{9} + 97080 \beta_{8} - 5715939 \beta_{7} + 7367150 \beta_{5} + 439710 \beta_{4} - 2663275 \beta_{2} + 34980918$$ -106531*b12 + 643321*b9 + 97080*b8 - 5715939*b7 + 7367150*b5 + 439710*b4 - 2663275*b2 + 34980918 $$\nu^{13}$$ $$=$$ $$19166580 \beta_{13} - 129568 \beta_{12} - 19461158 \beta_{11} + 7172990 \beta_{10} + 12213120 \beta_{9} - 59610892 \beta_{6} + 5299266 \beta_{3} + 63856489 \beta_1$$ 19166580*b13 - 129568*b12 - 19461158*b11 + 7172990*b10 + 12213120*b9 - 59610892*b6 + 5299266*b3 + 63856489*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/65\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 2.45261 + 4.24804i 1.92689 + 3.33746i 1.48434 + 2.57096i 0.272699 + 0.472328i −1.08655 − 1.88197i −1.78789 − 3.09671i −2.26209 − 3.91806i 2.45261 − 4.24804i 1.92689 − 3.33746i 1.48434 − 2.57096i 0.272699 − 0.472328i −1.08655 + 1.88197i −1.78789 + 3.09671i −2.26209 + 3.91806i
−2.45261 4.24804i −3.39607 5.88216i −8.03055 + 13.9093i −5.00000 −16.6584 + 28.8532i 2.60104 4.50514i 39.5414 −9.56652 + 16.5697i 12.2630 + 21.2402i
16.2 −1.92689 3.33746i 4.40164 + 7.62386i −3.42577 + 5.93361i −5.00000 16.9629 29.3806i −17.9862 + 31.1531i −4.42588 −25.2488 + 43.7322i 9.63443 + 16.6873i
16.3 −1.48434 2.57096i 1.47367 + 2.55247i −0.406551 + 0.704167i −5.00000 4.37486 7.57749i 13.2000 22.8630i −21.3357 9.15660 15.8597i 7.42172 + 12.8548i
16.4 −0.272699 0.472328i −3.51122 6.08161i 3.85127 6.67060i −5.00000 −1.91501 + 3.31689i −11.4358 + 19.8074i −8.56412 −11.1573 + 19.3250i 1.36349 + 2.36164i
16.5 1.08655 + 1.88197i −1.99894 3.46226i 1.63880 2.83848i −5.00000 4.34391 7.52387i 10.9938 19.0418i 24.5074 5.50850 9.54100i −5.43277 9.40984i
16.6 1.78789 + 3.09671i 4.37841 + 7.58363i −2.39307 + 4.14492i −5.00000 −15.6562 + 27.1173i 11.1225 19.2648i 11.4920 −24.8410 + 43.0258i −8.93943 15.4835i
16.7 2.26209 + 3.91806i 0.652503 + 1.13017i −6.23413 + 10.7978i −5.00000 −2.95204 + 5.11309i −11.9953 + 20.7765i −20.2152 12.6485 21.9078i −11.3105 19.5903i
61.1 −2.45261 + 4.24804i −3.39607 + 5.88216i −8.03055 13.9093i −5.00000 −16.6584 28.8532i 2.60104 + 4.50514i 39.5414 −9.56652 16.5697i 12.2630 21.2402i
61.2 −1.92689 + 3.33746i 4.40164 7.62386i −3.42577 5.93361i −5.00000 16.9629 + 29.3806i −17.9862 31.1531i −4.42588 −25.2488 43.7322i 9.63443 16.6873i
61.3 −1.48434 + 2.57096i 1.47367 2.55247i −0.406551 0.704167i −5.00000 4.37486 + 7.57749i 13.2000 + 22.8630i −21.3357 9.15660 + 15.8597i 7.42172 12.8548i
61.4 −0.272699 + 0.472328i −3.51122 + 6.08161i 3.85127 + 6.67060i −5.00000 −1.91501 3.31689i −11.4358 19.8074i −8.56412 −11.1573 19.3250i 1.36349 2.36164i
61.5 1.08655 1.88197i −1.99894 + 3.46226i 1.63880 + 2.83848i −5.00000 4.34391 + 7.52387i 10.9938 + 19.0418i 24.5074 5.50850 + 9.54100i −5.43277 + 9.40984i
61.6 1.78789 3.09671i 4.37841 7.58363i −2.39307 4.14492i −5.00000 −15.6562 27.1173i 11.1225 + 19.2648i 11.4920 −24.8410 43.0258i −8.93943 + 15.4835i
61.7 2.26209 3.91806i 0.652503 1.13017i −6.23413 10.7978i −5.00000 −2.95204 5.11309i −11.9953 20.7765i −20.2152 12.6485 + 21.9078i −11.3105 + 19.5903i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 61.7 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.4.e.a 14
13.c even 3 1 inner 65.4.e.a 14
13.c even 3 1 845.4.a.k 7
13.e even 6 1 845.4.a.h 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.4.e.a 14 1.a even 1 1 trivial
65.4.e.a 14 13.c even 3 1 inner
845.4.a.h 7 13.e even 6 1
845.4.a.k 7 13.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{14} + 2 T_{2}^{13} + 45 T_{2}^{12} + 52 T_{2}^{11} + 1311 T_{2}^{10} + 1336 T_{2}^{9} + 20343 T_{2}^{8} + 11166 T_{2}^{7} + 216447 T_{2}^{6} + 107836 T_{2}^{5} + 1201133 T_{2}^{4} - 4392 T_{2}^{3} + \cdots + 1157776$$ acting on $$S_{4}^{\mathrm{new}}(65, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{14} + 2 T^{13} + 45 T^{12} + \cdots + 1157776$$
$3$ $$T^{14} - 4 T^{13} + \cdots + 3196771600$$
$5$ $$(T + 5)^{14}$$
$7$ $$T^{14} + 7 T^{13} + \cdots + 17\!\cdots\!25$$
$11$ $$T^{14} + 87 T^{13} + \cdots + 59\!\cdots\!89$$
$13$ $$T^{14} - 123 T^{13} + \cdots + 24\!\cdots\!13$$
$17$ $$T^{14} - 114 T^{13} + \cdots + 41\!\cdots\!44$$
$19$ $$T^{14} + 245 T^{13} + \cdots + 65\!\cdots\!25$$
$23$ $$T^{14} - 74 T^{13} + \cdots + 95\!\cdots\!64$$
$29$ $$T^{14} - 88 T^{13} + \cdots + 86\!\cdots\!76$$
$31$ $$(T^{7} - 500 T^{6} + \cdots - 274963230720000)^{2}$$
$37$ $$T^{14} + 633 T^{13} + \cdots + 70\!\cdots\!69$$
$41$ $$T^{14} + 162 T^{13} + \cdots + 67\!\cdots\!00$$
$43$ $$T^{14} - 280 T^{13} + \cdots + 24\!\cdots\!36$$
$47$ $$(T^{7} - 475 T^{6} + \cdots - 35\!\cdots\!00)^{2}$$
$53$ $$(T^{7} + 603 T^{6} + \cdots - 88\!\cdots\!00)^{2}$$
$59$ $$T^{14} + 1410 T^{13} + \cdots + 33\!\cdots\!24$$
$61$ $$T^{14} + 412 T^{13} + \cdots + 46\!\cdots\!00$$
$67$ $$T^{14} + 1398 T^{13} + \cdots + 13\!\cdots\!44$$
$71$ $$T^{14} - 584 T^{13} + \cdots + 85\!\cdots\!36$$
$73$ $$(T^{7} - 2538 T^{6} + \cdots - 16\!\cdots\!00)^{2}$$
$79$ $$(T^{7} - 464 T^{6} + \cdots - 13\!\cdots\!00)^{2}$$
$83$ $$(T^{7} - 466 T^{6} + \cdots - 21\!\cdots\!00)^{2}$$
$89$ $$T^{14} + 443 T^{13} + \cdots + 11\!\cdots\!29$$
$97$ $$T^{14} - 1870 T^{13} + \cdots + 13\!\cdots\!04$$