# Properties

 Label 37.4.e.a Level $37$ Weight $4$ Character orbit 37.e Analytic conductor $2.183$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,4,Mod(11,37)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("37.11");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$37$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 37.e (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: $$2.18307067021$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 82 x^{14} + 2679 x^{12} + 44392 x^{10} + 392767 x^{8} + 1779258 x^{6} + 3438825 x^{4} + 1208748 x^{2} + 82944$$ x^16 + 82*x^14 + 2679*x^12 + 44392*x^10 + 392767*x^8 + 1779258*x^6 + 3438825*x^4 + 1208748*x^2 + 82944 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{12}\cdot 3^{3}$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} - \beta_{6} q^{3} + (\beta_{5} - 2 \beta_{4} - \beta_{3} + 2) q^{4} + ( - \beta_{11} + \beta_{4} + 1) q^{5} + ( - \beta_{9} + 3 \beta_{4} - \beta_{2} - \beta_1 - 1) q^{6} + (\beta_{15} - \beta_{5} - 3 \beta_{4} + \beta_{3} + 3) q^{7} + ( - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} + 4 \beta_{4} + \beta_{2} + \beta_1 - 2) q^{8} + (\beta_{15} + \beta_{14} - 2 \beta_{13} + 2 \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{5} - 6 \beta_{4} + \cdots + \beta_1) q^{9}+O(q^{10})$$ q + b2 * q^2 - b6 * q^3 + (b5 - 2*b4 - b3 + 2) * q^4 + (-b11 + b4 + 1) * q^5 + (-b9 + 3*b4 - b2 - b1 - 1) * q^6 + (b15 - b5 - 3*b4 + b3 + 3) * q^7 + (-b14 + b13 + b12 - b11 + 4*b4 + b2 + b1 - 2) * q^8 + (b15 + b14 - 2*b13 + 2*b8 + b7 + 2*b6 - b5 - 6*b4 + b1) * q^9 $$q + \beta_{2} q^{2} - \beta_{6} q^{3} + (\beta_{5} - 2 \beta_{4} - \beta_{3} + 2) q^{4} + ( - \beta_{11} + \beta_{4} + 1) q^{5} + ( - \beta_{9} + 3 \beta_{4} - \beta_{2} - \beta_1 - 1) q^{6} + (\beta_{15} - \beta_{5} - 3 \beta_{4} + \beta_{3} + 3) q^{7} + ( - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} + 4 \beta_{4} + \beta_{2} + \beta_1 - 2) q^{8} + (\beta_{15} + \beta_{14} - 2 \beta_{13} + 2 \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{5} - 6 \beta_{4} + \cdots + \beta_1) q^{9}+ \cdots + (29 \beta_{15} + 14 \beta_{14} - 28 \beta_{13} - 21 \beta_{12} + 42 \beta_{11} + \cdots + 438 \beta_1) q^{99}+O(q^{100})$$ q + b2 * q^2 - b6 * q^3 + (b5 - 2*b4 - b3 + 2) * q^4 + (-b11 + b4 + 1) * q^5 + (-b9 + 3*b4 - b2 - b1 - 1) * q^6 + (b15 - b5 - 3*b4 + b3 + 3) * q^7 + (-b14 + b13 + b12 - b11 + 4*b4 + b2 + b1 - 2) * q^8 + (b15 + b14 - 2*b13 + 2*b8 + b7 + 2*b6 - b5 - 6*b4 + b1) * q^9 + (b12 + b11 - b8 - b7 + 2*b3 - 2*b1 - 1) * q^10 + (b14 + b13 - 2*b10 + b9 + b8 + b7 - b4 + 2) * q^11 + (-b15 + b10 + b9 - b8 - b7 - b6 + b4) * q^12 + (b13 - b11 + b10 - 2*b8 - b6 - b4 + 3*b1 - 1) * q^13 + (-2*b15 + b14 - b13 - 3*b12 + 3*b11 + b9 + b8 - b7 + 2*b6 + 2*b5 - b4 - b3 - b2 - 2*b1) * q^14 + (-b15 - b14 - 2*b12 + b10 - b9 + b7 + b5 + 5*b4 - 2*b3 - 3*b2 + b1 - 8) * q^15 + (-2*b15 - b14 + 2*b13 - b12 + 2*b11 - 2*b7 - b5 + 10*b4 + b1) * q^16 + (-b15 - 3*b14 - b12 - b8 + b7 + b6 + 8*b4 + 3*b2 - 16) * q^17 + (-4*b13 + 3*b11 + 3*b10 - 3*b5 - 7*b4 - 3*b3 + 7*b1 - 7) * q^18 + (-b13 + 6*b11 - 4*b10 - 2*b8 - b6 + 2*b5 - 6*b4 + 2*b3 - 12*b1 - 6) * q^19 + (-2*b15 + 2*b14 + b12 - 2*b10 + 2*b9 + 2*b7 + 3*b5 + 2*b4 - 6*b3 - 8*b2 + 3*b1 - 8) * q^20 + (-2*b14 + 4*b13 + 3*b12 - 6*b11 - 7*b8 - 7*b6 - 3*b5 + 10*b4 - 3*b2 - 3*b1) * q^21 + (3*b15 + 3*b14 + 2*b12 + 5*b10 - 5*b9 - 7*b8 - 3*b7 + 7*b6 - 6*b5 + 7*b4 + 12*b3 - 8*b2 - 6*b1 - 4) * q^22 + (4*b15 - b14 + b13 - 4*b12 + 4*b11 + 4*b9 + 2*b7 + 2*b5 - 40*b4 - b3 + 5*b2 + 4*b1 + 18) * q^23 + (2*b15 - 2*b11 + 3*b10 + 16*b8 + 4*b7 + 8*b6 - 2*b5 - b4 - 2*b3 + 3*b1 - 1) * q^24 + (-2*b15 + 2*b14 - 4*b13 + 3*b12 - 6*b11 - b10 - b9 + 8*b8 - 2*b7 + 8*b6 + 7*b5 - 8*b4 + 8*b2 + 9*b1) * q^25 + (b14 + b13 + b12 + b11 - 4*b10 + 2*b9 + 9*b8 - b7 - 2*b4 + 8*b3 - b2 - 7*b1 - 15) * q^26 + (3*b14 + 3*b13 - 3*b12 - 3*b11 - 6*b10 + 3*b9 - 8*b8 - 5*b7 - 3*b4 + 2*b3 - 2*b1 + 70) * q^27 + (6*b12 - 12*b11 - b10 - b9 + 8*b8 + 8*b6 - 7*b5 + 38*b4 + 7*b1) * q^28 + (4*b15 + 3*b14 - 3*b13 - 7*b12 + 7*b11 - 4*b9 - 9*b8 + 2*b7 - 18*b6 - 14*b5 + 22*b4 + 7*b3 + 11*b2 + 18*b1 - 9) * q^29 + (3*b15 - 4*b14 + 2*b13 + 10*b12 - 5*b11 - 9*b6 - 5*b5 + 29*b4 + 5*b3 - 29) * q^30 + (-b14 + b13 + 4*b12 - 4*b11 + 3*b9 + 20*b5 - 17*b4 - 10*b3 + 19*b2 + 9*b1 + 7) * q^31 + (2*b15 - 6*b13 + 2*b11 - 2*b10 + 4*b7 - 16*b4 - 31*b1 - 16) * q^32 + (-b15 + 6*b10 - 12*b9 - 15*b6 + 14*b5 - 21*b4 - 14*b3 - 42*b2 - 21*b1 + 33) * q^33 + (b15 - 6*b14 + 3*b13 + 6*b12 - 3*b11 + b6 + 10*b5 - 23*b4 - 10*b3 - 16*b2 - 8*b1 + 23) * q^34 + (-b15 - 4*b14 - 11*b12 + 5*b10 - 5*b9 - 8*b8 + b7 + 8*b6 - 8*b5 - 3*b4 + 16*b3 + 24*b2 - 8*b1 + 16) * q^35 + (b14 + b13 - 6*b10 + 3*b9 + 10*b8 + 2*b7 - 3*b4 - 16*b3 + 24*b2 - 8*b1 + 27) * q^36 + (4*b15 + b12 + 10*b11 + 7*b10 - 4*b9 + 2*b8 - 3*b7 - 15*b6 + 4*b5 - 70*b4 - 7*b3 + 24*b2 + 22*b1 + 37) * q^37 + (b14 + b13 - 8*b12 - 8*b11 + 6*b10 - 3*b9 - 31*b8 + 3*b7 + 3*b4 - 20*b3 - 16*b2 + 36*b1 + 64) * q^38 + (-3*b15 + b14 - 5*b12 - b10 + b9 + 8*b8 + 3*b7 - 8*b6 + 25*b4 + 30*b2 - 52) * q^39 + (b15 - 2*b14 + b13 - 4*b12 + 2*b11 + 4*b10 - 8*b9 + b6 - 6*b5 + 91*b4 + 6*b3 + 16*b2 + 8*b1 - 83) * q^40 + (-3*b15 + 6*b12 - 3*b11 - 5*b10 + 10*b9 + 7*b6 + 11*b5 - 77*b4 - 11*b3 - 6*b2 - 3*b1 + 67) * q^41 + (-5*b15 + 3*b13 + 12*b11 - 7*b10 - 2*b8 - 10*b7 - b6 + 11*b5 + 38*b4 + 11*b3 - 45*b1 + 38) * q^42 + (-8*b15 + 3*b14 - 3*b13 + 10*b12 - 10*b11 + 4*b9 - 16*b8 - 4*b7 - 32*b6 - 16*b5 + 24*b4 + 8*b3 - 6*b2 + 2*b1 - 14) * q^43 + (-5*b15 + 2*b14 - b13 - 28*b12 + 14*b11 - 7*b10 + 14*b9 - 33*b6 - 2*b5 - 14*b4 + 2*b3 - 48*b2 - 24*b1) * q^44 + (-2*b15 + 7*b14 - 7*b13 + 11*b12 - 11*b11 + 6*b9 + 16*b8 - b7 + 32*b6 + 2*b5 + 30*b4 - b3 + 30*b2 + 29*b1 - 18) * q^45 + (b15 - 2*b14 + 4*b13 + b12 - 2*b11 - 2*b10 - 2*b9 + 47*b8 + b7 + 47*b6 + 3*b5 + 11*b4 + 24*b2 + 45*b1) * q^46 + (11*b12 + 11*b11 - 2*b10 + b9 - 30*b8 - 2*b7 - b4 + 34*b3 - 21*b2 - 13*b1 - 57) * q^47 + (-2*b14 - 2*b13 + 8*b12 + 8*b11 - 2*b10 + b9 + 39*b8 + 3*b7 - b4 + 6*b3 - 6*b1 - 8) * q^48 + (b15 - 2*b14 + 4*b13 + 3*b12 - 6*b11 + 6*b10 + 6*b9 - 8*b8 + b7 - 8*b6 - 13*b5 + 36*b4 - 24*b2 - 35*b1) * q^49 + (b13 - 2*b11 + 9*b10 - 32*b8 - 16*b6 + 5*b5 - 99*b4 + 5*b3 + 56*b1 - 99) * q^50 + (4*b15 - b14 + b13 - 11*b12 + 11*b11 + 2*b9 + 31*b8 + 2*b7 + 62*b6 + 4*b5 + 66*b4 - 2*b3 - 6*b2 - 8*b1 - 34) * q^51 + (b15 + 3*b14 - 5*b12 + 6*b10 - 6*b9 - 9*b8 - b7 + 9*b6 - 4*b5 + 13*b4 + 8*b3 - 59*b2 - 4*b1 - 14) * q^52 + (5*b15 + 3*b14 - 6*b13 - 2*b12 + 4*b11 - 2*b10 - 2*b9 + 30*b8 + 5*b7 + 30*b6 - 43*b5 - 31*b4 + 43*b1) * q^53 + (4*b15 + 11*b14 - 3*b12 - 4*b10 + 4*b9 - 32*b8 - 4*b7 + 32*b6 - 16*b5 + 6*b4 + 32*b3 + 52*b2 - 16*b1 - 20) * q^54 + (4*b15 + 16*b13 + 7*b11 - 11*b10 - 16*b8 + 8*b7 - 8*b6 - 24*b5 - 11*b4 - 24*b3 + 67*b1 - 11) * q^55 + (b15 + b13 + b11 + 3*b10 - 46*b8 + 2*b7 - 23*b6 + 5*b5 - 36*b4 + 5*b3 - 59*b1 - 36) * q^56 + (8*b15 - 14*b14 + 24*b12 + 13*b10 - 13*b9 - 7*b8 - 8*b7 + 7*b6 + 6*b5 + 52*b4 - 12*b3 - 101*b2 + 6*b1 - 78) * q^57 + (10*b14 - 20*b13 - 4*b12 + 8*b11 - 3*b10 - 3*b9 - 26*b8 - 26*b6 - 5*b5 - 150*b4 - 67*b2 - 129*b1) * q^58 + (-13*b15 - 12*b14 - 4*b12 - 13*b10 + 13*b9 - 10*b8 + 13*b7 + 10*b6 - 4*b5 + 61*b4 + 8*b3 - 40*b2 - 4*b1 - 148) * q^59 + (-4*b15 + 7*b14 - 7*b13 - 10*b12 + 10*b11 + 2*b9 - 2*b7 + 26*b5 - 46*b4 - 13*b3 - 19*b2 - 32*b1 + 22) * q^60 + (-7*b15 + b13 + 3*b11 + 3*b10 + 66*b8 - 14*b7 + 33*b6 + 13*b5 + 77*b4 + 13*b3 - 77*b1 + 77) * q^61 + (-2*b15 - 11*b14 + 22*b13 + 6*b12 - 12*b11 - 3*b10 - 3*b9 + 24*b8 - 2*b7 + 24*b6 + 28*b5 - 102*b4 + 79*b2 + 130*b1) * q^62 + (-8*b14 - 8*b13 - 6*b12 - 6*b11 + 18*b10 - 9*b9 + 54*b8 - 2*b7 + 9*b4 + 5*b3 - 6*b2 + b1 - 189) * q^63 + (-14*b14 - 14*b13 - 6*b12 - 6*b11 + 8*b10 - 4*b9 - 16*b8 + 16*b7 + 4*b4 - 43*b3 - 16*b2 + 59*b1 + 198) * q^64 + (-4*b15 - 10*b12 + 20*b11 - 5*b10 - 5*b9 + 11*b8 - 4*b7 + 11*b6 + 66*b4 + 5*b2 + 10*b1) * q^65 + (-10*b15 - 14*b14 + 14*b13 + 16*b12 - 16*b11 + 2*b9 - 55*b8 - 5*b7 - 110*b6 - 32*b5 + 404*b4 + 16*b3 + 131*b2 + 147*b1 - 203) * q^66 + (-8*b15 + 16*b14 - 8*b13 - 18*b12 + 9*b11 - 45*b6 + 14*b5 + 120*b4 - 14*b3 - 32*b2 - 16*b1 - 120) * q^67 + (2*b15 + 5*b14 - 5*b13 + 7*b12 - 7*b11 + 2*b9 - 7*b8 + b7 - 14*b6 + 22*b5 + 96*b4 - 11*b3 + 91*b2 + 80*b1 - 49) * q^68 + (-6*b15 + 33*b13 - 22*b11 - 5*b10 - 16*b8 - 12*b7 - 8*b6 + 9*b5 + 73*b4 + 9*b3 - 164*b1 + 73) * q^69 + (5*b15 + 8*b14 - 4*b13 + 10*b12 - 5*b11 - 12*b10 + 24*b9 - 33*b6 + 20*b5 - 351*b4 - 20*b3 - 90*b2 - 45*b1 + 327) * q^70 + (5*b15 + 22*b14 - 11*b13 - 18*b12 + 9*b11 + 2*b10 - 4*b9 + 30*b6 + 5*b5 - 239*b4 - 5*b3 - 112*b2 - 56*b1 + 243) * q^71 + (6*b15 + 19*b14 - 4*b12 - 3*b10 + 3*b9 - 24*b8 - 6*b7 + 24*b6 - 9*b5 - 242*b4 + 18*b3 + 119*b2 - 9*b1 + 478) * q^72 + (-2*b14 - 2*b13 - 5*b12 - 5*b11 + 10*b10 - 5*b9 + 53*b8 + b7 + 5*b4 - 27*b3 + 109*b2 - 82*b1 - 2) * q^73 + (-16*b15 - 7*b14 + 4*b13 - 19*b12 - 5*b11 - 11*b10 + b9 + 38*b8 + 12*b7 - 26*b6 + 41*b5 - 198*b4 - 44*b3 + 113*b2 + 94*b1 + 58) * q^74 + (-5*b14 - 5*b13 + 3*b12 + 3*b11 - 8*b10 + 4*b9 - 66*b8 - 10*b7 - 4*b4 + 2*b3 - 40*b2 + 38*b1 + 212) * q^75 + (9*b15 - 9*b14 + 2*b12 - 5*b10 + 5*b9 + 31*b8 - 9*b7 - 31*b6 - 20*b5 + 89*b4 + 40*b3 + 144*b2 - 20*b1 - 188) * q^76 + (15*b15 + 16*b14 - 8*b13 + 36*b12 - 18*b11 - 4*b10 + 8*b9 + 41*b6 - 8*b5 + 281*b4 + 8*b3 + 138*b2 + 69*b1 - 289) * q^77 + (16*b15 + 2*b14 - b13 + 22*b12 - 11*b11 + 12*b10 - 24*b9 + 4*b6 + 6*b5 - 262*b4 - 6*b3 - 100*b2 - 50*b1 + 286) * q^78 + (17*b15 - 9*b13 + 6*b11 + 14*b10 - 44*b8 + 34*b7 - 22*b6 + 79*b4 - 71*b1 + 79) * q^79 + (24*b15 - 13*b14 + 13*b13 - 10*b12 + 10*b11 - 2*b9 - 32*b8 + 12*b7 - 64*b6 - 22*b5 - 294*b4 + 11*b3 - 48*b2 - 37*b1 + 148) * q^80 + (-2*b15 - 6*b14 + 3*b13 + 36*b12 - 18*b11 + 15*b10 - 30*b9 - 94*b6 + 11*b5 + 260*b4 - 11*b3 - 192*b2 - 96*b1 - 230) * q^81 + (10*b15 - 11*b14 + 11*b13 + 8*b12 - 8*b11 - 11*b9 + 39*b8 + 5*b7 + 78*b6 - 10*b5 + 163*b4 + 5*b3 + 136*b2 + 141*b1 - 76) * q^82 + (-14*b15 - 7*b14 + 14*b13 + 7*b12 - 14*b11 + 6*b10 + 6*b9 + 43*b8 - 14*b7 + 43*b6 + 24*b5 + 80*b4 + 88*b2 + 152*b1) * q^83 + (-2*b14 - 2*b13 - 9*b12 - 9*b11 + 2*b10 - b9 - 7*b8 + 17*b7 + b4 - 19*b3 - 45*b2 + 64*b1 + 160) * q^84 + (-8*b14 - 8*b13 - 5*b12 - 5*b11 + 8*b10 - 4*b9 + 53*b8 - 23*b7 + 4*b4 + 34*b3 - 21*b2 - 13*b1 + 212) * q^85 + (9*b15 + 11*b14 - 22*b13 - 10*b12 + 20*b11 - 24*b10 - 24*b9 + 11*b8 + 9*b7 + 11*b6 - 14*b5 + 67*b4 - 95*b2 - 176*b1) * q^86 + (4*b15 - 27*b13 + 8*b11 - 34*b10 - 8*b8 + 8*b7 - 4*b6 - 17*b5 - 278*b4 - 17*b3 + 101*b1 - 278) * q^87 + (6*b15 - 19*b14 + 19*b13 + 34*b12 - 34*b11 - 19*b9 + 15*b8 + 3*b7 + 30*b6 - 40*b5 + 635*b4 + 20*b3 - 64*b2 - 44*b1 - 308) * q^88 + (10*b15 + 6*b14 + 13*b12 - 29*b10 + 29*b9 - 30*b8 - 10*b7 + 30*b6 - 11*b5 - 15*b4 + 22*b3 - 62*b2 - 11*b1 - 28) * q^89 + (5*b15 + 6*b14 - 12*b13 - 8*b12 + 16*b11 + 9*b10 + 9*b9 + 33*b8 + 5*b7 + 33*b6 + 15*b5 - 355*b4 - 15*b1) * q^90 + (-7*b15 + 3*b14 - 17*b12 + 11*b10 - 11*b9 - 28*b8 + 7*b7 + 28*b6 + 10*b5 + 81*b4 - 20*b3 + 36*b2 + 10*b1 - 140) * q^91 + (-22*b15 + b13 - 30*b11 + 22*b10 - 32*b8 - 44*b7 - 16*b6 + 29*b5 - 142*b4 + 29*b3 - 21*b1 - 142) * q^92 + (-5*b15 + 9*b13 - 17*b11 + 12*b10 + 18*b8 - 10*b7 + 9*b6 + 4*b5 + 43*b4 + 4*b3 - 99*b1 + 43) * q^93 + (-12*b15 + 34*b14 - 71*b12 - 29*b10 + 29*b9 + 12*b7 + 2*b5 + 84*b4 - 4*b3 - 229*b2 + 2*b1 - 226) * q^94 + (3*b15 - 14*b14 + 28*b13 + 5*b12 - 10*b11 + 12*b10 + 12*b9 - 80*b8 + 3*b7 - 80*b6 - 22*b5 - 523*b4 - 67*b2 - 112*b1) * q^95 + (10*b15 - 8*b12 + 21*b10 - 21*b9 - 64*b8 - 10*b7 + 64*b6 + 4*b5 + 44*b4 - 8*b3 - 79*b2 + 4*b1 - 46) * q^96 + (-36*b15 - 17*b14 + 17*b13 + 26*b12 - 26*b11 - 24*b9 - b8 - 18*b7 - 2*b6 + 34*b5 + 354*b4 - 17*b3 + 5*b2 - 12*b1 - 165) * q^97 + (-7*b13 + 24*b11 - 25*b10 + 108*b8 + 54*b6 - 15*b5 + 279*b4 - 15*b3 - 111*b1 + 279) * q^98 + (29*b15 + 14*b14 - 28*b13 - 21*b12 + 42*b11 + 19*b10 + 19*b9 + 23*b8 + 29*b7 + 23*b6 - 66*b5 - 615*b4 + 186*b2 + 438*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 6 q^{2} - 3 q^{3} + 18 q^{4} + 18 q^{5} + 23 q^{7} - 53 q^{9}+O(q^{10})$$ 16 * q - 6 * q^2 - 3 * q^3 + 18 * q^4 + 18 * q^5 + 23 * q^7 - 53 * q^9 $$16 q - 6 q^{2} - 3 q^{3} + 18 q^{4} + 18 q^{5} + 23 q^{7} - 53 q^{9} - 4 q^{10} + 36 q^{11} + 14 q^{12} - 9 q^{13} - 93 q^{15} + 90 q^{16} - 210 q^{17} - 144 q^{18} - 135 q^{19} - 18 q^{20} + 71 q^{21} + 18 q^{22} - 126 q^{24} - 72 q^{25} - 276 q^{26} + 1170 q^{27} + 256 q^{28} - 236 q^{30} - 552 q^{32} + 336 q^{33} + 274 q^{34} - 27 q^{35} + 180 q^{36} - 33 q^{37} + 1344 q^{38} - 909 q^{39} - 756 q^{40} + 642 q^{41} + 846 q^{42} - 6 q^{44} + 74 q^{46} - 468 q^{47} - 284 q^{48} + 187 q^{49} - 1932 q^{50} + 180 q^{52} - 249 q^{53} - 342 q^{54} + 162 q^{55} - 996 q^{56} - 141 q^{57} - 1496 q^{58} - 1455 q^{59} + 1188 q^{61} - 510 q^{62} - 3472 q^{63} + 3476 q^{64} + 579 q^{65} - 1033 q^{67} + 810 q^{69} + 2934 q^{70} + 2319 q^{71} + 5196 q^{72} - 1672 q^{73} - 1110 q^{74} + 4364 q^{75} - 3450 q^{76} - 2472 q^{77} + 2622 q^{78} + 1569 q^{79} - 1508 q^{81} + 975 q^{83} + 3064 q^{84} + 3128 q^{85} - 36 q^{86} - 5892 q^{87} + 522 q^{89} - 2908 q^{90} - 1773 q^{91} - 3462 q^{92} + 222 q^{93} - 1614 q^{94} - 4311 q^{95} + 378 q^{96} + 5748 q^{98} - 3606 q^{99}+O(q^{100})$$ 16 * q - 6 * q^2 - 3 * q^3 + 18 * q^4 + 18 * q^5 + 23 * q^7 - 53 * q^9 - 4 * q^10 + 36 * q^11 + 14 * q^12 - 9 * q^13 - 93 * q^15 + 90 * q^16 - 210 * q^17 - 144 * q^18 - 135 * q^19 - 18 * q^20 + 71 * q^21 + 18 * q^22 - 126 * q^24 - 72 * q^25 - 276 * q^26 + 1170 * q^27 + 256 * q^28 - 236 * q^30 - 552 * q^32 + 336 * q^33 + 274 * q^34 - 27 * q^35 + 180 * q^36 - 33 * q^37 + 1344 * q^38 - 909 * q^39 - 756 * q^40 + 642 * q^41 + 846 * q^42 - 6 * q^44 + 74 * q^46 - 468 * q^47 - 284 * q^48 + 187 * q^49 - 1932 * q^50 + 180 * q^52 - 249 * q^53 - 342 * q^54 + 162 * q^55 - 996 * q^56 - 141 * q^57 - 1496 * q^58 - 1455 * q^59 + 1188 * q^61 - 510 * q^62 - 3472 * q^63 + 3476 * q^64 + 579 * q^65 - 1033 * q^67 + 810 * q^69 + 2934 * q^70 + 2319 * q^71 + 5196 * q^72 - 1672 * q^73 - 1110 * q^74 + 4364 * q^75 - 3450 * q^76 - 2472 * q^77 + 2622 * q^78 + 1569 * q^79 - 1508 * q^81 + 975 * q^83 + 3064 * q^84 + 3128 * q^85 - 36 * q^86 - 5892 * q^87 + 522 * q^89 - 2908 * q^90 - 1773 * q^91 - 3462 * q^92 + 222 * q^93 - 1614 * q^94 - 4311 * q^95 + 378 * q^96 + 5748 * q^98 - 3606 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 82 x^{14} + 2679 x^{12} + 44392 x^{10} + 392767 x^{8} + 1779258 x^{6} + 3438825 x^{4} + 1208748 x^{2} + 82944$$ :

 $$\beta_{1}$$ $$=$$ $$( - \nu^{14} + 33 \nu^{12} + 2850 \nu^{10} + 38114 \nu^{8} - 57933 \nu^{6} - 2426979 \nu^{4} - 5725236 \nu^{2} + 2386944 \nu + 746496 ) / 4773888$$ (-v^14 + 33*v^12 + 2850*v^10 + 38114*v^8 - 57933*v^6 - 2426979*v^4 - 5725236*v^2 + 2386944*v + 746496) / 4773888 $$\beta_{2}$$ $$=$$ $$( \nu^{14} - 33 \nu^{12} - 2850 \nu^{10} - 38114 \nu^{8} + 57933 \nu^{6} + 2426979 \nu^{4} + 5725236 \nu^{2} + 2386944 \nu - 746496 ) / 4773888$$ (v^14 - 33*v^12 - 2850*v^10 - 38114*v^8 + 57933*v^6 + 2426979*v^4 + 5725236*v^2 + 2386944*v - 746496) / 4773888 $$\beta_{3}$$ $$=$$ $$( - \nu^{14} + 33 \nu^{12} + 2850 \nu^{10} + 38114 \nu^{8} - 57933 \nu^{6} - 2426979 \nu^{4} - 951348 \nu^{2} + 2386944 \nu + 48485376 ) / 4773888$$ (-v^14 + 33*v^12 + 2850*v^10 + 38114*v^8 - 57933*v^6 - 2426979*v^4 - 951348*v^2 + 2386944*v + 48485376) / 4773888 $$\beta_{4}$$ $$=$$ $$( 9 \nu^{15} + 739 \nu^{13} + 24078 \nu^{11} + 396678 \nu^{9} + 3496789 \nu^{7} + 16071255 \nu^{5} + 33376404 \nu^{3} + 16603968 \nu + 2386944 ) / 4773888$$ (9*v^15 + 739*v^13 + 24078*v^11 + 396678*v^9 + 3496789*v^7 + 16071255*v^5 + 33376404*v^3 + 16603968*v + 2386944) / 4773888 $$\beta_{5}$$ $$=$$ $$( 91 \nu^{15} - \nu^{14} + 7357 \nu^{13} + 33 \nu^{12} + 237930 \nu^{11} + 2850 \nu^{10} + 3928666 \nu^{9} + 38114 \nu^{8} + 35025823 \nu^{7} - 57933 \nu^{6} + 163139529 \nu^{5} + \cdots + 24615936 ) / 4773888$$ (91*v^15 - v^14 + 7357*v^13 + 33*v^12 + 237930*v^11 + 2850*v^10 + 3928666*v^9 + 38114*v^8 + 35025823*v^7 - 57933*v^6 + 163139529*v^5 - 2426979*v^4 + 339489276*v^3 - 3338292*v^2 + 167680128*v + 24615936) / 4773888 $$\beta_{6}$$ $$=$$ $$( 17 \nu^{15} - 256 \nu^{14} + 2029 \nu^{13} - 19524 \nu^{12} + 94518 \nu^{11} - 575760 \nu^{10} + 2208314 \nu^{9} - 8200840 \nu^{8} + 27220525 \nu^{7} - 57273696 \nu^{6} + \cdots - 18948096 ) / 9547776$$ (17*v^15 - 256*v^14 + 2029*v^13 - 19524*v^12 + 94518*v^11 - 575760*v^10 + 2208314*v^9 - 8200840*v^8 + 27220525*v^7 - 57273696*v^6 + 166876233*v^5 - 174043668*v^4 + 401322492*v^3 - 152430960*v^2 + 72244992*v - 18948096) / 9547776 $$\beta_{7}$$ $$=$$ $$( - 199 \nu^{14} - 16743 \nu^{12} - 570378 \nu^{10} - 10065646 \nu^{8} - 97272171 \nu^{6} - 490768347 \nu^{4} - 1018240500 \nu^{2} - 128332800 ) / 4773888$$ (-199*v^14 - 16743*v^12 - 570378*v^10 - 10065646*v^8 - 97272171*v^6 - 490768347*v^4 - 1018240500*v^2 - 128332800) / 4773888 $$\beta_{8}$$ $$=$$ $$( 64 \nu^{14} + 4881 \nu^{12} + 143940 \nu^{10} + 2050210 \nu^{8} + 14318424 \nu^{6} + 43510917 \nu^{4} + 38107740 \nu^{2} + 4737024 ) / 1193472$$ (64*v^14 + 4881*v^12 + 143940*v^10 + 2050210*v^8 + 14318424*v^6 + 43510917*v^4 + 38107740*v^2 + 4737024) / 1193472 $$\beta_{9}$$ $$=$$ $$( - 229 \nu^{15} - 17307 \nu^{13} - 503526 \nu^{11} - 7010806 \nu^{9} - 46783329 \nu^{7} - 125829903 \nu^{5} - 52301748 \nu^{3} + 26089920 \nu + 2386944 ) / 4773888$$ (-229*v^15 - 17307*v^13 - 503526*v^11 - 7010806*v^9 - 46783329*v^7 - 125829903*v^5 - 52301748*v^3 + 26089920*v + 2386944) / 4773888 $$\beta_{10}$$ $$=$$ $$( - 34 \nu^{15} + 91 \nu^{14} - 2578 \nu^{13} + 6987 \nu^{12} - 75372 \nu^{11} + 206850 \nu^{10} - 1058212 \nu^{9} + 2923894 \nu^{8} - 7182874 \nu^{7} + 19534959 \nu^{6} + \cdots + 1631232 ) / 1363968$$ (-34*v^15 + 91*v^14 - 2578*v^13 + 6987*v^12 - 75372*v^11 + 206850*v^10 - 1058212*v^9 + 2923894*v^8 - 7182874*v^7 + 19534959*v^6 - 20271594*v^5 + 49673775*v^4 - 12239736*v^3 + 9020964*v^2 + 1355136*v + 1631232) / 1363968 $$\beta_{11}$$ $$=$$ $$( - 632 \nu^{15} - 101 \nu^{14} - 51664 \nu^{13} - 10653 \nu^{12} - 1679760 \nu^{11} - 439422 \nu^{10} - 27616640 \nu^{9} - 9082874 \nu^{8} - 241086904 \nu^{7} + \cdots - 254002176 ) / 9547776$$ (-632*v^15 - 101*v^14 - 51664*v^13 - 10653*v^12 - 1679760*v^11 - 439422*v^10 - 27616640*v^9 - 9082874*v^8 - 241086904*v^7 - 98904753*v^6 - 1065468912*v^5 - 536700345*v^4 - 1948507776*v^3 - 1168513980*v^2 - 459321600*v - 254002176) / 9547776 $$\beta_{12}$$ $$=$$ $$( 632 \nu^{15} - 101 \nu^{14} + 51664 \nu^{13} - 10653 \nu^{12} + 1679760 \nu^{11} - 439422 \nu^{10} + 27616640 \nu^{9} - 9082874 \nu^{8} + 241086904 \nu^{7} + \cdots - 254002176 ) / 9547776$$ (632*v^15 - 101*v^14 + 51664*v^13 - 10653*v^12 + 1679760*v^11 - 439422*v^10 + 27616640*v^9 - 9082874*v^8 + 241086904*v^7 - 98904753*v^6 + 1065468912*v^5 - 536700345*v^4 + 1948507776*v^3 - 1168513980*v^2 + 459321600*v - 254002176) / 9547776 $$\beta_{13}$$ $$=$$ $$( - 668 \nu^{15} - 297 \nu^{14} - 54620 \nu^{13} - 22833 \nu^{12} - 1776072 \nu^{11} - 701334 \nu^{10} - 29203352 \nu^{9} - 11048418 \nu^{8} - 255074060 \nu^{7} + \cdots - 308192256 ) / 9547776$$ (-668*v^15 - 297*v^14 - 54620*v^13 - 22833*v^12 - 1776072*v^11 - 701334*v^10 - 29203352*v^9 - 11048418*v^8 - 255074060*v^7 - 95639589*v^6 - 1129753932*v^5 - 449610237*v^4 - 2086787280*v^3 - 977766444*v^2 - 606893568*v - 308192256) / 9547776 $$\beta_{14}$$ $$=$$ $$( 668 \nu^{15} - 297 \nu^{14} + 54620 \nu^{13} - 22833 \nu^{12} + 1776072 \nu^{11} - 701334 \nu^{10} + 29203352 \nu^{9} - 11048418 \nu^{8} + 255074060 \nu^{7} + \cdots - 308192256 ) / 9547776$$ (668*v^15 - 297*v^14 + 54620*v^13 - 22833*v^12 + 1776072*v^11 - 701334*v^10 + 29203352*v^9 - 11048418*v^8 + 255074060*v^7 - 95639589*v^6 + 1129753932*v^5 - 449610237*v^4 + 2086787280*v^3 - 977766444*v^2 + 606893568*v - 308192256) / 9547776 $$\beta_{15}$$ $$=$$ $$( - 1466 \nu^{15} + 199 \nu^{14} - 118418 \nu^{13} + 16743 \nu^{12} - 3790812 \nu^{11} + 570378 \nu^{10} - 61054052 \nu^{9} + 10065646 \nu^{8} - 518466770 \nu^{7} + \cdots + 128332800 ) / 9547776$$ (-1466*v^15 + 199*v^14 - 118418*v^13 + 16743*v^12 - 3790812*v^11 + 570378*v^10 - 61054052*v^9 + 10065646*v^8 - 518466770*v^7 + 97272171*v^6 - 2206167018*v^5 + 490768347*v^4 - 3787127544*v^3 + 1018240500*v^2 - 553921152*v + 128332800) / 9547776
 $$\nu$$ $$=$$ $$\beta_{2} + \beta_1$$ b2 + b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - \beta _1 - 10$$ b3 - b1 - 10 $$\nu^{3}$$ $$=$$ $$\beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} - 4\beta_{4} - 17\beta_{2} - 17\beta _1 + 2$$ b14 - b13 - b12 + b11 - 4*b4 - 17*b2 - 17*b1 + 2 $$\nu^{4}$$ $$=$$ $$-\beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} + 2\beta_{7} - 23\beta_{3} + 23\beta _1 + 166$$ -b14 - b13 - b12 - b11 + 2*b7 - 23*b3 + 23*b1 + 166 $$\nu^{5}$$ $$=$$ $$4 \beta_{15} - 26 \beta_{14} + 26 \beta_{13} + 30 \beta_{12} - 30 \beta_{11} - 2 \beta_{9} + 2 \beta_{7} + 98 \beta_{4} + 321 \beta_{2} + 321 \beta _1 - 48$$ 4*b15 - 26*b14 + 26*b13 + 30*b12 - 30*b11 - 2*b9 + 2*b7 + 98*b4 + 321*b2 + 321*b1 - 48 $$\nu^{6}$$ $$=$$ $$26 \beta_{14} + 26 \beta_{13} + 34 \beta_{12} + 34 \beta_{11} + 8 \beta_{10} - 4 \beta_{9} - 16 \beta_{8} - 64 \beta_{7} + 4 \beta_{4} + 493 \beta_{3} - 16 \beta_{2} - 477 \beta _1 - 3114$$ 26*b14 + 26*b13 + 34*b12 + 34*b11 + 8*b10 - 4*b9 - 16*b8 - 64*b7 + 4*b4 + 493*b3 - 16*b2 - 477*b1 - 3114 $$\nu^{7}$$ $$=$$ $$- 152 \beta_{15} + 571 \beta_{14} - 571 \beta_{13} - 723 \beta_{12} + 723 \beta_{11} + 92 \beta_{9} - 32 \beta_{8} - 76 \beta_{7} - 64 \beta_{6} + 32 \beta_{5} - 1912 \beta_{4} - 16 \beta_{3} - 6321 \beta_{2} - 6337 \beta _1 + 910$$ -152*b15 + 571*b14 - 571*b13 - 723*b12 + 723*b11 + 92*b9 - 32*b8 - 76*b7 - 64*b6 + 32*b5 - 1912*b4 - 16*b3 - 6321*b2 - 6337*b1 + 910 $$\nu^{8}$$ $$=$$ $$- 555 \beta_{14} - 555 \beta_{13} - 891 \beta_{12} - 891 \beta_{11} - 400 \beta_{10} + 200 \beta_{9} + 752 \beta_{8} + 1614 \beta_{7} - 200 \beta_{4} - 10371 \beta_{3} + 624 \beta_{2} + 9747 \beta _1 + 61550$$ -555*b14 - 555*b13 - 891*b12 - 891*b11 - 400*b10 + 200*b9 + 752*b8 + 1614*b7 - 200*b4 - 10371*b3 + 624*b2 + 9747*b1 + 61550 $$\nu^{9}$$ $$=$$ $$4300 \beta_{15} - 12036 \beta_{14} + 12036 \beta_{13} + 16272 \beta_{12} - 16272 \beta_{11} - 2966 \beta_{9} + 1632 \beta_{8} + 2150 \beta_{7} + 3264 \beta_{6} - 1504 \beta_{5} + 37550 \beta_{4} + \cdots - 17292$$ 4300*b15 - 12036*b14 + 12036*b13 + 16272*b12 - 16272*b11 - 2966*b9 + 1632*b8 + 2150*b7 + 3264*b6 - 1504*b5 + 37550*b4 + 752*b3 + 126945*b2 + 127697*b1 - 17292 $$\nu^{10}$$ $$=$$ $$11284 \beta_{14} + 11284 \beta_{13} + 21324 \beta_{12} + 21324 \beta_{11} + 13496 \beta_{10} - 6748 \beta_{9} - 24480 \beta_{8} - 37724 \beta_{7} + 6748 \beta_{4} + 217049 \beta_{3} - 16800 \beta_{2} + \cdots - 1245778$$ 11284*b14 + 11284*b13 + 21324*b12 + 21324*b11 + 13496*b10 - 6748*b9 - 24480*b8 - 37724*b7 + 6748*b4 + 217049*b3 - 16800*b2 - 200249*b1 - 1245778 $$\nu^{11}$$ $$=$$ $$- 109024 \beta_{15} + 250901 \beta_{14} - 250901 \beta_{13} - 356469 \beta_{12} + 356469 \beta_{11} + 82448 \beta_{9} - 55616 \beta_{8} - 54512 \beta_{7} - 111232 \beta_{6} + 50368 \beta_{5} + \cdots + 346858$$ -109024*b15 + 250901*b14 - 250901*b13 - 356469*b12 + 356469*b11 + 82448*b9 - 55616*b8 - 54512*b7 - 111232*b6 + 50368*b5 - 776164*b4 - 25184*b3 - 2577105*b2 - 2602289*b1 + 346858 $$\nu^{12}$$ $$=$$ $$- 225717 \beta_{14} - 225717 \beta_{13} - 490677 \beta_{12} - 490677 \beta_{11} - 385152 \beta_{10} + 192576 \beta_{9} + 684064 \beta_{8} + 853354 \beta_{7} - 192576 \beta_{4} + \cdots + 25523478$$ -225717*b14 - 225717*b13 - 490677*b12 - 490677*b11 - 385152*b10 + 192576*b9 + 684064*b8 + 853354*b7 - 192576*b4 - 4539631*b3 + 385824*b2 + 4153807*b1 + 25523478 $$\nu^{13}$$ $$=$$ $$2621780 \beta_{15} - 5216782 \beta_{14} + 5216782 \beta_{13} + 7718370 \beta_{12} - 7718370 \beta_{11} - 2115146 \beta_{9} + 1596224 \beta_{8} + 1310890 \beta_{7} + 3192448 \beta_{6} + \cdots - 7358248$$ 2621780*b15 - 5216782*b14 + 5216782*b13 + 7718370*b12 - 7718370*b11 - 2115146*b9 + 1596224*b8 + 1310890*b7 + 3192448*b6 - 1477184*b5 + 16831642*b4 + 738592*b3 + 52684161*b2 + 53422753*b1 - 7358248 $$\nu^{14}$$ $$=$$ $$4478190 \beta_{14} + 4478190 \beta_{13} + 11078742 \beta_{12} + 11078742 \beta_{11} + 10044520 \beta_{10} - 5022260 \beta_{9} - 17605232 \beta_{8} - 18982968 \beta_{7} + \cdots - 526752122$$ 4478190*b14 + 4478190*b13 + 11078742*b12 + 11078742*b11 + 10044520*b10 - 5022260*b9 - 17605232*b8 - 18982968*b7 + 5022260*b4 + 95035845*b3 - 8050800*b2 - 86985045*b1 - 526752122 $$\nu^{15}$$ $$=$$ $$- 61211560 \beta_{15} + 108470415 \beta_{14} - 108470415 \beta_{13} - 166238007 \beta_{12} + 166238007 \beta_{11} + 51654980 \beta_{9} - 41774304 \beta_{8} + \cdots + 162850758$$ -61211560*b15 + 108470415*b14 - 108470415*b13 - 166238007*b12 + 166238007*b11 + 51654980*b9 - 41774304*b8 - 30605780*b7 - 83548608*b6 + 40398304*b5 - 377356496*b4 - 20199152*b3 - 1082580465*b2 - 1102779617*b1 + 162850758

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/37\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$1 - \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 4.65090i 4.17312i 3.17259i 0.575927i 0.301955i − 2.31004i − 2.56981i − 4.53054i − 4.65090i − 4.17312i − 3.17259i − 0.575927i − 0.301955i 2.31004i 2.56981i 4.53054i
−4.02780 + 2.32545i 1.78194 3.08642i 6.81543 11.8047i 11.2971 + 6.52240i 16.5753i −10.9405 + 18.9495i 26.1886i 7.14934 + 12.3830i −60.6701
11.2 −3.61403 + 2.08656i −3.72941 + 6.45953i 4.70747 8.15358i −2.02566 1.16952i 31.1266i 3.33293 5.77281i 5.90472i −14.3170 24.7978i 9.76108
11.3 −2.74754 + 1.58629i 2.85325 4.94198i 1.03266 1.78861i −13.1878 7.61398i 18.1044i 14.3541 24.8621i 18.8283i −2.78209 4.81872i 48.3120
11.4 −0.498768 + 0.287964i −1.51417 + 2.62262i −3.83415 + 6.64095i −6.67348 3.85293i 1.74410i −10.3028 + 17.8450i 9.02381i 8.91459 + 15.4405i 4.43802
11.5 −0.261501 + 0.150978i 0.675960 1.17080i −3.95441 + 6.84924i 17.4040 + 10.0482i 0.408219i 10.5914 18.3449i 4.80375i 12.5862 + 21.7999i −6.06822
11.6 2.00056 1.15502i 3.80485 6.59020i −1.33185 + 2.30684i −1.68208 0.971152i 17.5787i −1.95249 + 3.38181i 24.6336i −15.4538 26.7667i −4.48680
11.7 2.22552 1.28490i −4.96296 + 8.59609i −0.698050 + 1.20906i 5.97182 + 3.44783i 25.5077i 8.67107 15.0187i 24.1461i −35.7619 61.9414i 17.7205
11.8 3.92356 2.26527i −0.409472 + 0.709227i 6.26291 10.8477i −2.10394 1.21471i 3.71026i −2.25379 + 3.90368i 20.5044i 13.1647 + 22.8019i −11.0066
27.1 −4.02780 2.32545i 1.78194 + 3.08642i 6.81543 + 11.8047i 11.2971 6.52240i 16.5753i −10.9405 18.9495i 26.1886i 7.14934 12.3830i −60.6701
27.2 −3.61403 2.08656i −3.72941 6.45953i 4.70747 + 8.15358i −2.02566 + 1.16952i 31.1266i 3.33293 + 5.77281i 5.90472i −14.3170 + 24.7978i 9.76108
27.3 −2.74754 1.58629i 2.85325 + 4.94198i 1.03266 + 1.78861i −13.1878 + 7.61398i 18.1044i 14.3541 + 24.8621i 18.8283i −2.78209 + 4.81872i 48.3120
27.4 −0.498768 0.287964i −1.51417 2.62262i −3.83415 6.64095i −6.67348 + 3.85293i 1.74410i −10.3028 17.8450i 9.02381i 8.91459 15.4405i 4.43802
27.5 −0.261501 0.150978i 0.675960 + 1.17080i −3.95441 6.84924i 17.4040 10.0482i 0.408219i 10.5914 + 18.3449i 4.80375i 12.5862 21.7999i −6.06822
27.6 2.00056 + 1.15502i 3.80485 + 6.59020i −1.33185 2.30684i −1.68208 + 0.971152i 17.5787i −1.95249 3.38181i 24.6336i −15.4538 + 26.7667i −4.48680
27.7 2.22552 + 1.28490i −4.96296 8.59609i −0.698050 1.20906i 5.97182 3.44783i 25.5077i 8.67107 + 15.0187i 24.1461i −35.7619 + 61.9414i 17.7205
27.8 3.92356 + 2.26527i −0.409472 0.709227i 6.26291 + 10.8477i −2.10394 + 1.21471i 3.71026i −2.25379 3.90368i 20.5044i 13.1647 22.8019i −11.0066
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 27.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.4.e.a 16
3.b odd 2 1 333.4.s.c 16
37.e even 6 1 inner 37.4.e.a 16
37.g odd 12 2 1369.4.a.g 16
111.h odd 6 1 333.4.s.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.4.e.a 16 1.a even 1 1 trivial
37.4.e.a 16 37.e even 6 1 inner
333.4.s.c 16 3.b odd 2 1
333.4.s.c 16 111.h odd 6 1
1369.4.a.g 16 37.g odd 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(37, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 6 T^{15} - 23 T^{14} + \cdots + 82944$$
$3$ $$T^{16} + 3 T^{15} + \cdots + 1475789056$$
$5$ $$T^{16} - 18 T^{15} + \cdots + 5481540530361$$
$7$ $$T^{16} - 23 T^{15} + \cdots + 31\!\cdots\!04$$
$11$ $$(T^{8} - 18 T^{7} + \cdots - 233130203136)^{2}$$
$13$ $$T^{16} + 9 T^{15} + \cdots + 16\!\cdots\!76$$
$17$ $$T^{16} + 210 T^{15} + \cdots + 13\!\cdots\!81$$
$19$ $$T^{16} + 135 T^{15} + \cdots + 14\!\cdots\!96$$
$23$ $$T^{16} + 121024 T^{14} + \cdots + 77\!\cdots\!64$$
$29$ $$T^{16} + 228973 T^{14} + \cdots + 26\!\cdots\!04$$
$31$ $$T^{16} + 193236 T^{14} + \cdots + 91\!\cdots\!56$$
$37$ $$T^{16} + 33 T^{15} + \cdots + 43\!\cdots\!61$$
$41$ $$T^{16} - 642 T^{15} + \cdots + 11\!\cdots\!09$$
$43$ $$T^{16} + 636216 T^{14} + \cdots + 44\!\cdots\!44$$
$47$ $$(T^{8} + 234 T^{7} + \cdots + 18\!\cdots\!44)^{2}$$
$53$ $$T^{16} + 249 T^{15} + \cdots + 29\!\cdots\!56$$
$59$ $$T^{16} + 1455 T^{15} + \cdots + 12\!\cdots\!84$$
$61$ $$T^{16} - 1188 T^{15} + \cdots + 21\!\cdots\!69$$
$67$ $$T^{16} + 1033 T^{15} + \cdots + 87\!\cdots\!04$$
$71$ $$T^{16} - 2319 T^{15} + \cdots + 35\!\cdots\!44$$
$73$ $$(T^{8} + 836 T^{7} + \cdots + 72\!\cdots\!36)^{2}$$
$79$ $$T^{16} - 1569 T^{15} + \cdots + 11\!\cdots\!36$$
$83$ $$T^{16} - 975 T^{15} + \cdots + 56\!\cdots\!04$$
$89$ $$T^{16} - 522 T^{15} + \cdots + 46\!\cdots\!81$$
$97$ $$T^{16} + 5602893 T^{14} + \cdots + 42\!\cdots\!64$$