# Properties

 Label 37.2.f.a Level $37$ Weight $2$ Character orbit 37.f Analytic conductor $0.295$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,2,Mod(7,37)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([16]))

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

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

chi := DirichletCharacter("37.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 37.f (of order $$9$$, degree $$6$$, 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: $$0.295446487479$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-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 primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-\zeta_{18}^{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −0.173648 + 0.984808i −0.766044 + 0.642788i 0.939693 + 0.342020i −0.173648 − 0.984808i −0.766044 − 0.642788i 0.939693 − 0.342020i
−1.76604 + 0.642788i 1.26604 + 0.460802i 1.17365 0.984808i 0.532089 + 3.01763i −2.53209 −0.613341 3.47843i 0.439693 0.761570i −0.907604 0.761570i −2.87939 4.98724i
9.1 −0.0603074 0.342020i −0.439693 + 2.49362i 1.76604 0.642788i −2.87939 2.41609i 0.879385 −0.0923963 0.0775297i −0.673648 1.16679i −3.20574 1.16679i −0.652704 + 1.13052i
12.1 −1.17365 + 0.984808i 0.673648 + 0.565258i 0.0603074 0.342020i −0.652704 + 0.237565i −1.34730 2.20574 0.802823i −1.26604 2.19285i −0.386659 2.19285i 0.532089 0.921605i
16.1 −1.76604 0.642788i 1.26604 0.460802i 1.17365 + 0.984808i 0.532089 3.01763i −2.53209 −0.613341 + 3.47843i 0.439693 + 0.761570i −0.907604 + 0.761570i −2.87939 + 4.98724i
33.1 −0.0603074 + 0.342020i −0.439693 2.49362i 1.76604 + 0.642788i −2.87939 + 2.41609i 0.879385 −0.0923963 + 0.0775297i −0.673648 + 1.16679i −3.20574 + 1.16679i −0.652704 1.13052i
34.1 −1.17365 0.984808i 0.673648 0.565258i 0.0603074 + 0.342020i −0.652704 0.237565i −1.34730 2.20574 + 0.802823i −1.26604 + 2.19285i −0.386659 + 2.19285i 0.532089 + 0.921605i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 34.1 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.f even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.2.f.a 6
3.b odd 2 1 333.2.x.c 6
4.b odd 2 1 592.2.bc.a 6
5.b even 2 1 925.2.p.b 6
5.c odd 4 2 925.2.bc.a 12
37.f even 9 1 inner 37.2.f.a 6
37.f even 9 1 1369.2.a.j 3
37.h even 18 1 1369.2.a.k 3
37.i odd 36 2 1369.2.b.f 6
111.p odd 18 1 333.2.x.c 6
148.p odd 18 1 592.2.bc.a 6
185.x even 18 1 925.2.p.b 6
185.bd odd 36 2 925.2.bc.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.f.a 6 1.a even 1 1 trivial
37.2.f.a 6 37.f even 9 1 inner
333.2.x.c 6 3.b odd 2 1
333.2.x.c 6 111.p odd 18 1
592.2.bc.a 6 4.b odd 2 1
592.2.bc.a 6 148.p odd 18 1
925.2.p.b 6 5.b even 2 1
925.2.p.b 6 185.x even 18 1
925.2.bc.a 12 5.c odd 4 2
925.2.bc.a 12 185.bd odd 36 2
1369.2.a.j 3 37.f even 9 1
1369.2.a.k 3 37.h even 18 1
1369.2.b.f 6 37.i odd 36 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 6T_{2}^{5} + 15T_{2}^{4} + 19T_{2}^{3} + 12T_{2}^{2} + 3T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(37, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 6 T^{5} + 15 T^{4} + 19 T^{3} + \cdots + 1$$
$3$ $$T^{6} - 3 T^{5} + 9 T^{4} - 24 T^{3} + \cdots + 9$$
$5$ $$T^{6} + 6 T^{5} + 24 T^{4} + 64 T^{3} + \cdots + 64$$
$7$ $$T^{6} - 3 T^{5} + 12 T^{4} - 46 T^{3} + \cdots + 1$$
$11$ $$T^{6} + 3 T^{4} - 2 T^{3} + 9 T^{2} + \cdots + 1$$
$13$ $$T^{6} + 9 T^{5} + 27 T^{4} + 28 T^{3} + \cdots + 1$$
$17$ $$T^{6} - 3 T^{5} + 36 T^{4} + \cdots + 47961$$
$19$ $$T^{6} - 3 T^{5} - 36 T^{4} + \cdots + 3249$$
$23$ $$T^{6} + 3 T^{5} + 9 T^{4} + 2 T^{3} + \cdots + 1$$
$29$ $$T^{6} + 15 T^{5} + 162 T^{4} + \cdots + 3249$$
$31$ $$(T^{3} - 3 T^{2} - 9 T + 19)^{2}$$
$37$ $$T^{6} + 6 T^{5} + 96 T^{4} + \cdots + 50653$$
$41$ $$T^{6} + 9 T^{4} - 9 T^{3} + 81 T + 81$$
$43$ $$(T^{3} + 3 T^{2} - 60 T - 71)^{2}$$
$47$ $$T^{6} - 15 T^{5} + 162 T^{4} + \cdots + 3249$$
$53$ $$T^{6} - 24 T^{5} + 228 T^{4} + \cdots + 289$$
$59$ $$T^{6} - 6 T^{5} - 12 T^{4} + \cdots + 32041$$
$61$ $$T^{6} + 27 T^{5} + 324 T^{4} + \cdots + 104329$$
$67$ $$T^{6} + 18 T^{5} + 144 T^{4} + \cdots + 2809$$
$71$ $$T^{6} - 33 T^{5} + 459 T^{4} + \cdots + 567009$$
$73$ $$(T^{3} - 6 T^{2} - 51 T + 109)^{2}$$
$79$ $$T^{6} - 33 T^{5} + 516 T^{4} + \cdots + 687241$$
$83$ $$T^{6} - 21 T^{5} + 144 T^{4} + \cdots + 2601$$
$89$ $$T^{6} - 12 T^{5} + 159 T^{4} + \cdots + 5329$$
$97$ $$T^{6} - 18 T^{5} + 255 T^{4} + \cdots + 5041$$