# Properties

 Label 72.2.l.b Level $72$ Weight $2$ Character orbit 72.l Analytic conductor $0.575$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,2,Mod(11,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 1]))

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

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

chi := DirichletCharacter("72.11");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 72.l (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: $$0.574922894553$$ 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: $$x^{16} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 16 x^{12} - 12 x^{11} - 8 x^{10} + 36 x^{9} - 68 x^{8} + 72 x^{7} - 32 x^{6} - 96 x^{5} + 256 x^{4} - 384 x^{3} + 448 x^{2} - 384 x + 256$$ x^16 - 3*x^15 + 7*x^14 - 12*x^13 + 16*x^12 - 12*x^11 - 8*x^10 + 36*x^9 - 68*x^8 + 72*x^7 - 32*x^6 - 96*x^5 + 256*x^4 - 384*x^3 + 448*x^2 - 384*x + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{11} q^{2} + (\beta_{6} - \beta_{3}) q^{3} + (\beta_{15} + 2 \beta_{11} - \beta_{9} - \beta_{8} + \beta_{6} - \beta_{4} + \beta_{2} - \beta_1 - 1) q^{4} + (\beta_{13} + \beta_{12} - \beta_{10}) q^{5} + ( - \beta_{13} + \beta_{11} + \beta_{10} - \beta_{8} + \beta_{3} - \beta_{2}) q^{6} + (\beta_{14} + \beta_{10} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_1 + 1) q^{7} + (\beta_{15} - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} + \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} + \cdots - 1) q^{8}+ \cdots + (2 \beta_{15} - \beta_{13} + 2 \beta_{11} + 2 \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 - 2) q^{9}+O(q^{10})$$ q - b11 * q^2 + (b6 - b3) * q^3 + (b15 + 2*b11 - b9 - b8 + b6 - b4 + b2 - b1 - 1) * q^4 + (b13 + b12 - b10) * q^5 + (-b13 + b11 + b10 - b8 + b3 - b2) * q^6 + (b14 + b10 - b7 - b6 - b5 - b4 - 2*b1 + 1) * q^7 + (b15 - b13 - b12 + b11 - b10 + b7 + b6 + 2*b5 + b4 - b3 - 1) * q^8 + (2*b15 - b13 + 2*b11 + 2*b5 - b4 + b2 - b1 - 2) * q^9 $$q - \beta_{11} q^{2} + (\beta_{6} - \beta_{3}) q^{3} + (\beta_{15} + 2 \beta_{11} - \beta_{9} - \beta_{8} + \beta_{6} - \beta_{4} + \beta_{2} - \beta_1 - 1) q^{4} + (\beta_{13} + \beta_{12} - \beta_{10}) q^{5} + ( - \beta_{13} + \beta_{11} + \beta_{10} - \beta_{8} + \beta_{3} - \beta_{2}) q^{6} + (\beta_{14} + \beta_{10} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_1 + 1) q^{7} + (\beta_{15} - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} + \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} + \cdots - 1) q^{8}+ \cdots + (2 \beta_{15} - \beta_{13} + 2 \beta_{11} + 3 \beta_{6} + 2 \beta_{5} - \beta_{4} + 7 \beta_{2} + \cdots - 5) q^{99}+O(q^{100})$$ q - b11 * q^2 + (b6 - b3) * q^3 + (b15 + 2*b11 - b9 - b8 + b6 - b4 + b2 - b1 - 1) * q^4 + (b13 + b12 - b10) * q^5 + (-b13 + b11 + b10 - b8 + b3 - b2) * q^6 + (b14 + b10 - b7 - b6 - b5 - b4 - 2*b1 + 1) * q^7 + (b15 - b13 - b12 + b11 - b10 + b7 + b6 + 2*b5 + b4 - b3 - 1) * q^8 + (2*b15 - b13 + 2*b11 + 2*b5 - b4 + b2 - b1 - 2) * q^9 + (-2*b15 - b14 + 2*b13 - b11 + b10 - b6 - 2*b5 - b2 + 2) * q^10 + (-b15 + b13 - b11 - b6 - b5 - b2 + 2) * q^11 + (-b15 - 2*b11 - b10 + b9 + b8 + b7 + 2*b4 + b3 + b2 + 3*b1 - 2) * q^12 + (-b15 - b12 - 2*b11 - b10 + 2*b9 + 2*b8 - b6 + b4 + b2 + b1 - 1) * q^13 + (-b15 - 2*b13 - b9 + b8 + b6 + b4 - b2 + 2*b1 - 1) * q^14 + (-3*b15 - 2*b14 + b12 - 3*b11 + b10 + 2*b8 - 2*b6 - b5 + b4 + 2*b3 - 2*b2 + 4*b1 + 1) * q^15 + (b15 + 2*b13 + b12 - b7 - 2*b6 - b4 - b1 + 1) * q^16 + (-b15 - 2*b11 - b6 - 2*b5 + b4 + b3 + 2*b1) * q^17 + (b15 + 2*b14 + b13 - b12 - 2*b3 + 2*b2 - 2*b1) * q^18 + (-2*b13 + b11 + b6 + b5 + b4 + b1 - 1) * q^19 + (b15 - 2*b13 - b12 - b11 + 2*b9 + b8 - b7 - b6 + 2*b5 - b4 + 2*b3 + b2 - 1) * q^20 + (4*b15 + b14 - b13 - 2*b12 + 4*b11 - b10 - 2*b9 - 2*b8 + b7 + 4*b6 + 3*b5 - b4 - 3*b3 + 2*b2 - b1 - 2) * q^21 + (-b15 - b14 - 2*b11 + b9 + b8 - b7 - b6 + b3 + b1) * q^22 + (-b12 + b10 + 2*b9 - 2*b8 + b4 + b3 - b1) * q^23 + (-2*b15 - b14 + 3*b13 + 2*b12 - b11 - 2*b6 - 4*b5 + b3 - 3*b2 + 4) * q^24 + (-b15 + b13 + b11 + b5 + b4 + b3 - b2 - 2*b1) * q^25 + (b15 + b13 + b12 + b10 - b7 - b6 - 2*b5 - b4 - b3 - 4*b2 - b1 + 3) * q^26 + (3*b13 - 3*b11 - 2*b6 - 3*b5 + b3 - 3*b2) * q^27 + (-3*b13 + b12 + 2*b11 - b10 - b9 + b7 + b6 + 2*b5 - b3 + 2*b1 - 2) * q^28 + (b15 - b14 + b13 + 2*b12 + 5*b11 + b10 - 4*b9 - 2*b8 + b7 + 2*b6 - 2*b5 - b4 - 2*b2 + 1) * q^29 + (2*b15 + 2*b14 - 2*b13 - b12 - b10 + b9 + b6 + 2*b5 - 3*b4 - b3 + 6*b2 - 2*b1 - 3) * q^30 + (4*b15 + 2*b14 - 2*b13 - b12 + 2*b11 - b10 - 2*b9 - 2*b8 + 2*b7 + 4*b6 + 4*b5 - b4 - 3*b3 + 2*b2 - b1 - 2) * q^31 + (-b15 + b14 + 2*b13 - b12 + b10 - b7 + b6 - b4 - 2*b3 + 3*b2 - 4*b1 + 3) * q^32 + (-b15 + b6 - b4 - b3 + 3*b2) * q^33 + (-b15 - b14 + b13 + b12 + b11 - b10 + b6 - b2 - b1) * q^34 + (3*b11 + 2*b6 + 3*b5 - 2*b4 + 6*b2 - 3*b1 - 3) * q^35 + (-3*b15 - b14 + 2*b12 + b10 - b9 + b8 - 2*b7 - 2*b6 - 2*b5 + b4 + 2*b3 - 4*b2 + 5) * q^36 + (-3*b15 - 2*b14 - 2*b13 - 3*b11 + 2*b10 + 2*b9 - 2*b6 - b5 + 2*b4 + 3*b3 - 2*b2 + b1 + 1) * q^37 + (-b15 + b12 + b7 + b4 + b1 - 1) * q^38 + (-2*b14 - b13 + b12 + 3*b11 - b10 - 2*b9 + b6 - b5 - b2 - b1) * q^39 + (5*b15 + 2*b14 - b12 + 2*b11 - b10 - 2*b9 - 2*b8 + 2*b7 + 5*b6 - 5*b3 + 2*b2 - b1 - 2) * q^40 + (b15 - b13 - b6 - b4 - 2*b2 + b1 - 2) * q^41 + (2*b15 - b13 + 3*b11 - b10 - 2*b8 + b6 + 4*b5 + 2*b4 - b3 - 2*b1 + 2) * q^42 + (b15 - 2*b13 - 2*b11 - 2*b6 - 2*b5 + b4 + b3 - 2*b2 + 4*b1) * q^43 + (4*b15 + b14 - b13 - b12 + 4*b11 - b9 - 2*b8 + b7 + 4*b6 + 2*b5 - 2*b4 - 3*b3 + 3*b2 - 3*b1 - 2) * q^44 + (b15 + 2*b14 - 2*b13 - b12 - 2*b11 + b10 + 2*b9 - 2*b8 - 2*b7 - b6 + 2*b5 - b4 + b2 - 5*b1 + 1) * q^45 + (-b14 + 3*b13 + b12 - b11 + b9 + b7 - 2*b5 + 2*b4 + 3*b3 - b2 + 2*b1) * q^46 + (2*b15 + b14 - 2*b12 + 2*b11 - b10 - b7 + b6 + b5 - b4 - 2*b3 + 2*b2 - 2*b1 - 1) * q^47 + (-3*b15 - b14 + 2*b13 - b12 - 6*b11 + b10 + 2*b9 + 2*b8 - b7 - 3*b6 - 2*b5 - b4 + 2*b3 - 3*b2 + 2*b1 + 1) * q^48 + (2*b13 - 4*b11 - 4*b5 - b4 + b3 - 2*b2 + 2*b1 + 2) * q^49 + (b15 - b14 + b12 - b10 - 2*b9 + 2*b8 + b7 - b6 - b4 + b2 + 4*b1 + 1) * q^50 + (-2*b15 - b13 + 2*b11 + 2*b6 + 2*b5 + b4 + b2 - b1 + 1) * q^51 + (-b15 + 2*b14 - b12 - b11 + 2*b10 - b8 - b7 + b6 - b4 - b2 + 1) * q^52 + (-5*b15 + 2*b13 + 2*b12 - 7*b11 + 2*b10 + 2*b9 + 4*b8 - 2*b7 - 6*b6 - 3*b5 + 3*b3 - 2*b2 + 3*b1 + 3) * q^53 + (2*b15 + 2*b13 + 4*b11 - b10 - 2*b9 - b8 - b7 + 2*b6 - 3*b4 - b3 - b2 - b1 + 3) * q^54 + (2*b15 + b14 + 3*b13 + b12 + 3*b11 - 2*b10 - 2*b9 + b7 + 2*b6 - b4 - 2*b3 + b2 + b1 - 1) * q^55 + (-6*b15 - b14 + b13 - 5*b11 + b10 + 4*b9 + 2*b8 - b7 - 5*b6 - 2*b5 + 3*b4 + 2*b3 + b2 + 2*b1 - 3) * q^56 + (b15 + b13 + b11 + b5 + b4 - b3 - b2 - 2*b1 + 2) * q^57 + (-2*b15 + b14 - 2*b13 - 2*b12 - 4*b11 - 2*b10 + 2*b9 + 2*b8 + b7 - 2*b6 + 4*b5 + b4 + b3 + b2 + 2*b1 - 1) * q^58 + (3*b15 + 2*b13 - 3*b6 - 2*b4 + b3 - 2*b1) * q^59 + (-b15 - b14 - b13 - b12 + b10 - b8 - 4*b5 + 4*b4 - 4*b2 - 2) * q^60 + (-b15 - b14 + b13 - 3*b11 - b10 + 2*b8 + b7 + 2*b5 + b4 + 6*b1 - 1) * q^61 + (-2*b15 - b14 + b13 + b12 + b11 - b9 - 2*b8 - b7 - 2*b6 - 2*b5 + 2*b4 + 3*b3 + b2) * q^62 + (-4*b15 - b14 + 5*b13 + 2*b12 - 3*b11 - b10 + 4*b9 + 2*b8 - b7 - 4*b6 - 4*b5 + 2*b4 + 3*b3 - b2 + 2*b1 + 1) * q^63 + (2*b15 - b14 - 6*b13 + 2*b11 + b10 - 2*b9 + b6 + 2*b5 + 2*b3 - b2 + b1) * q^64 + (-b15 + 3*b6 + 4*b4 - 4*b3 - b2 + 2) * q^65 + (-b15 - b14 - 3*b13 - b12 - b11 + b10 + 2*b9 - b6 + 2*b4 + 2*b3 - b2 - 2) * q^66 + (2*b15 - 3*b13 + 6*b11 + 2*b6 + 6*b5 - b4 - b3 + 5*b2 - 3*b1 - 5) * q^67 + (b15 - b14 - b12 + b10 + b7 - b6 - b4 - b2 + 2*b1 - 1) * q^68 + (2*b14 - b12 + b11 + b10 + 2*b8 - b6 - 3*b5 - b4 - b3 + b2 - 2*b1) * q^69 + (2*b15 - 4*b13 - 2*b12 + b8 + 2*b7 - b6 + 2*b4 + b2 + b1 - 2) * q^70 + (-b15 - 2*b13 - 2*b12 - 5*b11 - 2*b10 + 2*b9 + 4*b8 + 2*b7 + 3*b5 + 2*b4 - b3 + 2*b2 + 5*b1 - 3) * q^71 + (-3*b13 - b11 - b10 - 2*b9 + 2*b8 + 2*b7 + 3*b6 + 4*b5 - 2*b4 - 3*b3 + 4*b2 + 3*b1 - 4) * q^72 + (-3*b15 + b6 + b4 - 3*b3 + 2) * q^73 + (4*b15 + b14 + b13 + b11 - b10 + b7 + b6 - 3*b4 + 2*b3 + 5*b2 - 9) * q^74 + (-b15 - b13 - 4*b11 - 3*b6 - 4*b5 + 2*b4 + b3 - 5*b2 + 5*b1 + 7) * q^75 + (-5*b15 - b14 + 2*b13 + b12 - 4*b11 + b10 + 2*b9 + 2*b8 - b7 - 5*b6 - 4*b5 + b4 + 4*b3 - b2 + 2*b1 + 1) * q^76 + (-3*b13 - b12 + b10 - 2*b1) * q^77 + (5*b15 + b14 - b13 - 2*b12 + 3*b11 - b10 - b9 + b7 + b6 + 6*b5 - 4*b4 - 4*b3 + b2 - 3*b1 - 6) * q^78 + (-b14 + b13 + 2*b12 + 3*b11 - b10 - 2*b8 - b7 - 2*b5 + b3 - b2 - 4*b1 + 1) * q^79 + (-2*b15 + 2*b13 + 2*b12 + 4*b11 + 2*b10 - 2*b9 - 4*b8 - 2*b7 - 2*b6 - 2*b5 + 4*b3 - 8*b2 - 4*b1 + 6) * q^80 + (-3*b15 - 3*b13 - 3*b6 + 3*b3 - 3*b2 + 3*b1) * q^81 + (-b15 + b14 + b13 - b12 + 2*b9 - b7 + b4 - b3 + b2 - 1) * q^82 + (2*b15 - 5*b13 + 5*b11 + 4*b6 + 5*b5 + 2*b4 - 2*b3 - b2 + 2) * q^83 + (-b15 - b14 + 6*b13 + 2*b12 - b11 + b10 - b9 - b6 - b4 + 5*b2 - 3*b1 + 1) * q^84 + (-b15 - 2*b14 + 3*b13 + 2*b12 + 6*b11 + 2*b10 - 2*b9 - 2*b8 - 2*b7 - b6 - 6*b5 - b4 + 2*b3 - 3*b2 - 3*b1 + 3) * q^85 + (-2*b15 + b14 + 4*b13 + b12 - b10 + b9 - b8 - b7 + 2*b6 - 2*b3 - 4*b1) * q^86 + (b15 + 3*b14 + b13 - 6*b11 - 3*b10 + 2*b9 + 2*b8 + b7 + 5*b5 - 2*b3 + 3*b2 + 3*b1 - 2) * q^87 + (b15 + b14 - b13 - b12 + b11 + b10 - 2*b8 - b6 + 2*b4 + 2*b3 - 3*b2 - b1) * q^88 + (b15 + b11 + 4*b6 + b5 - 4*b4 - b3 + 2*b2 - b1 - 1) * q^89 + (b15 - 2*b14 + b13 + b12 + 6*b11 + b10 - 4*b9 - 2*b8 + b7 + 5*b6 - 2*b5 + b4 + b3 + 4*b2 + b1 - 7) * q^90 + (b15 + 8*b13 - 4*b11 - 4*b6 - 4*b5 - 4*b4 + b3 - 4*b1) * q^91 + (b15 + b14 + 5*b13 + b12 - 2*b11 - b10 - 2*b9 - b8 + 2*b7 - 4*b5 - 2*b4 - 4*b2 + 8) * q^92 + (-2*b15 + 6*b13 + 3*b12 + b11 + 3*b10 - 2*b8 - 2*b7 - 3*b6 - 5*b5 - b4 + 3*b3 - 3*b2 + 4) * q^93 + (3*b15 - 2*b13 + 4*b11 - b9 - b8 + 3*b6 + 4*b5 + b4 - 4*b3 - 3*b2 - 2*b1 + 3) * q^94 + (-2*b13 - 2*b1) * q^95 + (6*b15 + b14 - 4*b13 - 2*b12 + 4*b11 - b10 - 2*b8 + 5*b6 + 2*b5 - 2*b4 - 2*b3 + 3*b2 + b1 - 2) * q^96 + (4*b15 + 3*b13 + 3*b11 + b6 + 3*b5 - 5*b4 - 5*b3 + 4*b2 - 6*b1) * q^97 + (3*b15 - b13 - b12 + 3*b11 - b10 - 2*b9 - 4*b8 + b7 + 5*b6 - b4 - b3 - 4*b2 + 1) * q^98 + (2*b15 - b13 + 2*b11 + 3*b6 + 2*b5 - b4 + 7*b2 - b1 - 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 3 q^{2} - 6 q^{3} - 5 q^{4} - 3 q^{6} - 6 q^{9}+O(q^{10})$$ 16 * q - 3 * q^2 - 6 * q^3 - 5 * q^4 - 3 * q^6 - 6 * q^9 $$16 q - 3 q^{2} - 6 q^{3} - 5 q^{4} - 3 q^{6} - 6 q^{9} + 12 q^{11} - 6 q^{12} - 18 q^{14} + 7 q^{16} - 4 q^{19} + 18 q^{20} - q^{22} + 21 q^{24} - 14 q^{25} - 36 q^{27} - 12 q^{28} + 12 q^{30} + 27 q^{32} + 12 q^{33} - 13 q^{34} + 27 q^{36} - 15 q^{38} - 12 q^{40} - 36 q^{41} + 42 q^{42} + 8 q^{43} + 12 q^{46} - 27 q^{48} + 10 q^{49} + 51 q^{50} + 18 q^{51} - 18 q^{52} + 39 q^{54} - 66 q^{56} + 18 q^{57} + 12 q^{58} + 12 q^{59} - 72 q^{60} + 34 q^{64} - 6 q^{65} - 24 q^{66} - 16 q^{67} - 9 q^{68} + 18 q^{70} - 21 q^{72} - 4 q^{73} - 60 q^{74} + 78 q^{75} - 7 q^{76} - 72 q^{78} - 6 q^{81} - 22 q^{82} + 54 q^{83} + 12 q^{84} - 51 q^{86} - 13 q^{88} - 66 q^{90} - 36 q^{91} + 84 q^{92} + 24 q^{94} + 42 q^{96} + 8 q^{97} - 6 q^{99}+O(q^{100})$$ 16 * q - 3 * q^2 - 6 * q^3 - 5 * q^4 - 3 * q^6 - 6 * q^9 + 12 * q^11 - 6 * q^12 - 18 * q^14 + 7 * q^16 - 4 * q^19 + 18 * q^20 - q^22 + 21 * q^24 - 14 * q^25 - 36 * q^27 - 12 * q^28 + 12 * q^30 + 27 * q^32 + 12 * q^33 - 13 * q^34 + 27 * q^36 - 15 * q^38 - 12 * q^40 - 36 * q^41 + 42 * q^42 + 8 * q^43 + 12 * q^46 - 27 * q^48 + 10 * q^49 + 51 * q^50 + 18 * q^51 - 18 * q^52 + 39 * q^54 - 66 * q^56 + 18 * q^57 + 12 * q^58 + 12 * q^59 - 72 * q^60 + 34 * q^64 - 6 * q^65 - 24 * q^66 - 16 * q^67 - 9 * q^68 + 18 * q^70 - 21 * q^72 - 4 * q^73 - 60 * q^74 + 78 * q^75 - 7 * q^76 - 72 * q^78 - 6 * q^81 - 22 * q^82 + 54 * q^83 + 12 * q^84 - 51 * q^86 - 13 * q^88 - 66 * q^90 - 36 * q^91 + 84 * q^92 + 24 * q^94 + 42 * q^96 + 8 * q^97 - 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 16 x^{12} - 12 x^{11} - 8 x^{10} + 36 x^{9} - 68 x^{8} + 72 x^{7} - 32 x^{6} - 96 x^{5} + 256 x^{4} - 384 x^{3} + 448 x^{2} - 384 x + 256$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{15} - 13 \nu^{14} + 11 \nu^{13} + 4 \nu^{12} - 38 \nu^{11} + 60 \nu^{10} - 104 \nu^{9} + 68 \nu^{8} + 148 \nu^{7} - 344 \nu^{6} + 440 \nu^{5} - 240 \nu^{4} - 32 \nu^{3} + 608 \nu^{2} - 1152 \nu + 1280 ) / 896$$ (v^15 - 13*v^14 + 11*v^13 + 4*v^12 - 38*v^11 + 60*v^10 - 104*v^9 + 68*v^8 + 148*v^7 - 344*v^6 + 440*v^5 - 240*v^4 - 32*v^3 + 608*v^2 - 1152*v + 1280) / 896 $$\beta_{3}$$ $$=$$ $$( \nu^{15} + \nu^{14} + 25 \nu^{13} - 38 \nu^{12} + 46 \nu^{11} - 24 \nu^{10} + 8 \nu^{9} + 68 \nu^{8} - 244 \nu^{7} + 272 \nu^{6} - 8 \nu^{5} - 128 \nu^{4} + 416 \nu^{3} - 1184 \nu^{2} + 1088 \nu - 512 ) / 896$$ (v^15 + v^14 + 25*v^13 - 38*v^12 + 46*v^11 - 24*v^10 + 8*v^9 + 68*v^8 - 244*v^7 + 272*v^6 - 8*v^5 - 128*v^4 + 416*v^3 - 1184*v^2 + 1088*v - 512) / 896 $$\beta_{4}$$ $$=$$ $$( - \nu^{15} + 6 \nu^{14} - 11 \nu^{13} + 24 \nu^{12} - 39 \nu^{11} + 10 \nu^{10} + 20 \nu^{9} - 124 \nu^{8} + 160 \nu^{7} - 160 \nu^{6} - 20 \nu^{5} + 240 \nu^{4} - 528 \nu^{3} + 848 \nu^{2} - 640 \nu + 512 ) / 448$$ (-v^15 + 6*v^14 - 11*v^13 + 24*v^12 - 39*v^11 + 10*v^10 + 20*v^9 - 124*v^8 + 160*v^7 - 160*v^6 - 20*v^5 + 240*v^4 - 528*v^3 + 848*v^2 - 640*v + 512) / 448 $$\beta_{5}$$ $$=$$ $$( 3 \nu^{15} - 11 \nu^{14} + 47 \nu^{13} - 58 \nu^{12} + 40 \nu^{11} + 40 \nu^{10} - 144 \nu^{9} + 316 \nu^{8} - 340 \nu^{7} - 80 \nu^{6} + 592 \nu^{5} - 1168 \nu^{4} + 1248 \nu^{3} - 1088 \nu^{2} + \cdots + 1152 ) / 896$$ (3*v^15 - 11*v^14 + 47*v^13 - 58*v^12 + 40*v^11 + 40*v^10 - 144*v^9 + 316*v^8 - 340*v^7 - 80*v^6 + 592*v^5 - 1168*v^4 + 1248*v^3 - 1088*v^2 + 128*v + 1152) / 896 $$\beta_{6}$$ $$=$$ $$( - 5 \nu^{15} + 23 \nu^{14} - 13 \nu^{13} - 6 \nu^{12} + 50 \nu^{11} - 132 \nu^{10} + 240 \nu^{9} - 116 \nu^{8} - 236 \nu^{7} + 656 \nu^{6} - 856 \nu^{5} + 752 \nu^{4} - 288 \nu^{3} - 1696 \nu^{2} + \cdots - 1920 ) / 896$$ (-5*v^15 + 23*v^14 - 13*v^13 - 6*v^12 + 50*v^11 - 132*v^10 + 240*v^9 - 116*v^8 - 236*v^7 + 656*v^6 - 856*v^5 + 752*v^4 - 288*v^3 - 1696*v^2 + 2176*v - 1920) / 896 $$\beta_{7}$$ $$=$$ $$( - 3 \nu^{15} + 4 \nu^{14} - 5 \nu^{13} + 2 \nu^{12} + 23 \nu^{11} - 40 \nu^{10} + 32 \nu^{9} + 20 \nu^{8} - 24 \nu^{7} + 80 \nu^{6} - 60 \nu^{5} + 104 \nu^{4} + 208 \nu^{3} - 144 \nu^{2} - 352 \nu + 192 ) / 448$$ (-3*v^15 + 4*v^14 - 5*v^13 + 2*v^12 + 23*v^11 - 40*v^10 + 32*v^9 + 20*v^8 - 24*v^7 + 80*v^6 - 60*v^5 + 104*v^4 + 208*v^3 - 144*v^2 - 352*v + 192) / 448 $$\beta_{8}$$ $$=$$ $$( 3 \nu^{15} + 3 \nu^{14} - 9 \nu^{13} + 26 \nu^{12} - 30 \nu^{11} + 12 \nu^{10} + 52 \nu^{9} - 132 \nu^{8} + 164 \nu^{7} - 80 \nu^{6} - 136 \nu^{5} + 400 \nu^{4} - 656 \nu^{3} + 704 \nu^{2} - 320 \nu - 192 ) / 448$$ (3*v^15 + 3*v^14 - 9*v^13 + 26*v^12 - 30*v^11 + 12*v^10 + 52*v^9 - 132*v^8 + 164*v^7 - 80*v^6 - 136*v^5 + 400*v^4 - 656*v^3 + 704*v^2 - 320*v - 192) / 448 $$\beta_{9}$$ $$=$$ $$( - 4 \nu^{15} + 17 \nu^{14} - 37 \nu^{13} + 19 \nu^{12} + 12 \nu^{11} - 86 \nu^{10} + 192 \nu^{9} - 216 \nu^{8} + 52 \nu^{7} + 340 \nu^{6} - 584 \nu^{5} + 792 \nu^{4} - 544 \nu^{3} - 192 \nu^{2} + \cdots - 1088 ) / 448$$ (-4*v^15 + 17*v^14 - 37*v^13 + 19*v^12 + 12*v^11 - 86*v^10 + 192*v^9 - 216*v^8 + 52*v^7 + 340*v^6 - 584*v^5 + 792*v^4 - 544*v^3 - 192*v^2 + 1248*v - 1088) / 448 $$\beta_{10}$$ $$=$$ $$( 5 \nu^{15} - 9 \nu^{14} + 27 \nu^{13} - 36 \nu^{12} + 34 \nu^{11} - 8 \nu^{10} - 72 \nu^{9} + 172 \nu^{8} - 156 \nu^{7} + 72 \nu^{6} + 184 \nu^{5} - 640 \nu^{4} + 736 \nu^{3} - 768 \nu^{2} + 512 \nu - 320 ) / 448$$ (5*v^15 - 9*v^14 + 27*v^13 - 36*v^12 + 34*v^11 - 8*v^10 - 72*v^9 + 172*v^8 - 156*v^7 + 72*v^6 + 184*v^5 - 640*v^4 + 736*v^3 - 768*v^2 + 512*v - 320) / 448 $$\beta_{11}$$ $$=$$ $$( 5 \nu^{15} - 2 \nu^{14} - 8 \nu^{13} + 27 \nu^{12} - 36 \nu^{11} + 48 \nu^{10} - 16 \nu^{9} - 108 \nu^{8} + 208 \nu^{7} - 236 \nu^{6} + 72 \nu^{5} + 144 \nu^{4} - 496 \nu^{3} + 800 \nu^{2} - 384 \nu + 128 ) / 448$$ (5*v^15 - 2*v^14 - 8*v^13 + 27*v^12 - 36*v^11 + 48*v^10 - 16*v^9 - 108*v^8 + 208*v^7 - 236*v^6 + 72*v^5 + 144*v^4 - 496*v^3 + 800*v^2 - 384*v + 128) / 448 $$\beta_{12}$$ $$=$$ $$( - 6 \nu^{15} + 15 \nu^{14} - 24 \nu^{13} + 18 \nu^{12} - 3 \nu^{11} - 66 \nu^{10} + 120 \nu^{9} - 128 \nu^{8} - 20 \nu^{7} + 160 \nu^{6} - 260 \nu^{5} + 432 \nu^{4} - 480 \nu^{3} + 48 \nu^{2} + 192 \nu + 384 ) / 448$$ (-6*v^15 + 15*v^14 - 24*v^13 + 18*v^12 - 3*v^11 - 66*v^10 + 120*v^9 - 128*v^8 - 20*v^7 + 160*v^6 - 260*v^5 + 432*v^4 - 480*v^3 + 48*v^2 + 192*v + 384) / 448 $$\beta_{13}$$ $$=$$ $$( 5 \nu^{15} - 16 \nu^{14} + 48 \nu^{13} - 71 \nu^{12} + 76 \nu^{11} - 22 \nu^{10} - 100 \nu^{9} + 284 \nu^{8} - 408 \nu^{7} + 212 \nu^{6} + 184 \nu^{5} - 920 \nu^{4} + 1520 \nu^{3} - 1888 \nu^{2} + \cdots - 768 ) / 448$$ (5*v^15 - 16*v^14 + 48*v^13 - 71*v^12 + 76*v^11 - 22*v^10 - 100*v^9 + 284*v^8 - 408*v^7 + 212*v^6 + 184*v^5 - 920*v^4 + 1520*v^3 - 1888*v^2 + 1632*v - 768) / 448 $$\beta_{14}$$ $$=$$ $$( 3 \nu^{14} - 7 \nu^{13} + 15 \nu^{12} - 22 \nu^{11} + 16 \nu^{10} + 4 \nu^{9} - 56 \nu^{8} + 92 \nu^{7} - 116 \nu^{6} + 32 \nu^{5} + 144 \nu^{4} - 352 \nu^{3} + 512 \nu^{2} - 448 \nu + 384 ) / 64$$ (3*v^14 - 7*v^13 + 15*v^12 - 22*v^11 + 16*v^10 + 4*v^9 - 56*v^8 + 92*v^7 - 116*v^6 + 32*v^5 + 144*v^4 - 352*v^3 + 512*v^2 - 448*v + 384) / 64 $$\beta_{15}$$ $$=$$ $$( 3 \nu^{15} - 18 \nu^{14} + 47 \nu^{13} - 100 \nu^{12} + 117 \nu^{11} - 100 \nu^{10} - 60 \nu^{9} + 316 \nu^{8} - 592 \nu^{7} + 648 \nu^{6} - 52 \nu^{5} - 888 \nu^{4} + 2144 \nu^{3} - 2768 \nu^{2} + \cdots - 1536 ) / 448$$ (3*v^15 - 18*v^14 + 47*v^13 - 100*v^12 + 117*v^11 - 100*v^10 - 60*v^9 + 316*v^8 - 592*v^7 + 648*v^6 - 52*v^5 - 888*v^4 + 2144*v^3 - 2768*v^2 + 3040*v - 1536) / 448
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{11} + \beta_{8} - \beta_{6} - \beta_{2} + \beta_1$$ -b11 + b8 - b6 - b2 + b1 $$\nu^{3}$$ $$=$$ $$\beta_{15} - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} + \beta_{7} + \beta_{6} + 2\beta_{5} + \beta_{4} - \beta_{3} - 1$$ b15 - b13 - b12 + b11 - b10 + b7 + b6 + 2*b5 + b4 - b3 - 1 $$\nu^{4}$$ $$=$$ $$\beta_{15} + \beta_{14} - \beta_{12} - \beta_{10} + \beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} - 1$$ b15 + b14 - b12 - b10 + b7 + b6 - b4 + b2 - 1 $$\nu^{5}$$ $$=$$ $$\beta_{15} - 2 \beta_{13} + \beta_{12} + 2 \beta_{11} + \beta_{7} + 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta _1 - 5$$ b15 - 2*b13 + b12 + 2*b11 + b7 + 2*b5 - b4 + 2*b3 + 2*b2 + b1 - 5 $$\nu^{6}$$ $$=$$ $$2 \beta_{15} - \beta_{14} - 6 \beta_{13} + 2 \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{6} + 2 \beta_{5} + 2 \beta_{3} - \beta_{2} + \beta_1$$ 2*b15 - b14 - 6*b13 + 2*b11 + b10 - 2*b9 + b6 + 2*b5 + 2*b3 - b2 + b1 $$\nu^{7}$$ $$=$$ $$- \beta_{15} - \beta_{14} - 2 \beta_{13} - 3 \beta_{12} + 3 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{6} + 3 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 2 \beta _1 + 3$$ -b15 - b14 - 2*b13 - 3*b12 + 3*b10 + 2*b9 - 2*b8 + b7 + b6 + 3*b4 + 2*b3 + 3*b2 + 2*b1 + 3 $$\nu^{8}$$ $$=$$ $$\beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 3 \beta_{12} - 4 \beta_{11} + 2 \beta_{10} + 2 \beta_{8} + \beta_{7} + 4 \beta_{6} + 2 \beta_{5} - 5 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} + 5 \beta _1 - 1$$ b15 + 2*b14 - 2*b13 - 3*b12 - 4*b11 + 2*b10 + 2*b8 + b7 + 4*b6 + 2*b5 - 5*b4 - 6*b3 + 2*b2 + 5*b1 - 1 $$\nu^{9}$$ $$=$$ $$- 2 \beta_{15} - 3 \beta_{14} - 4 \beta_{11} - 3 \beta_{10} + 6 \beta_{9} + 12 \beta_{8} - 3 \beta_{6} + 14 \beta_{5} - 4 \beta_{3} + 5 \beta_{2} + \beta _1 - 4$$ -2*b15 - 3*b14 - 4*b11 - 3*b10 + 6*b9 + 12*b8 - 3*b6 + 14*b5 - 4*b3 + 5*b2 + b1 - 4 $$\nu^{10}$$ $$=$$ $$- \beta_{15} - 3 \beta_{14} - 6 \beta_{13} - 7 \beta_{12} - 7 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 3 \beta_{7} - \beta_{6} + 12 \beta_{5} - \beta_{4} + 2 \beta_{3} - 7 \beta_{2} + 7$$ -b15 - 3*b14 - 6*b13 - 7*b12 - 7*b10 + 2*b9 + 2*b8 - 3*b7 - b6 + 12*b5 - b4 + 2*b3 - 7*b2 + 7 $$\nu^{11}$$ $$=$$ $$- 17 \beta_{15} - 4 \beta_{14} + 18 \beta_{13} + 7 \beta_{12} - 8 \beta_{11} + 4 \beta_{10} + 4 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} - 26 \beta_{6} - 22 \beta_{5} - 7 \beta_{4} + 22 \beta_{3} - 16 \beta_{2} + 9 \beta _1 + 21$$ -17*b15 - 4*b14 + 18*b13 + 7*b12 - 8*b11 + 4*b10 + 4*b9 + 2*b8 + 3*b7 - 26*b6 - 22*b5 - 7*b4 + 22*b3 - 16*b2 + 9*b1 + 21 $$\nu^{12}$$ $$=$$ $$8 \beta_{15} - 11 \beta_{14} - 24 \beta_{13} + 18 \beta_{12} + 40 \beta_{11} - 7 \beta_{10} - 34 \beta_{9} + 18 \beta_{7} + 21 \beta_{6} + 10 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - 11 \beta_{2} + 35 \beta _1 - 22$$ 8*b15 - 11*b14 - 24*b13 + 18*b12 + 40*b11 - 7*b10 - 34*b9 + 18*b7 + 21*b6 + 10*b5 - 2*b4 - 4*b3 - 11*b2 + 35*b1 - 22 $$\nu^{13}$$ $$=$$ $$- 5 \beta_{15} - 13 \beta_{14} - 6 \beta_{13} + \beta_{12} - \beta_{10} - 18 \beta_{9} + 18 \beta_{8} + 13 \beta_{7} + 5 \beta_{6} + 31 \beta_{4} + 26 \beta_{3} - \beta_{2} + 38 \beta _1 - 1$$ -5*b15 - 13*b14 - 6*b13 + b12 - b10 - 18*b9 + 18*b8 + 13*b7 + 5*b6 + 31*b4 + 26*b3 - b2 + 38*b1 - 1 $$\nu^{14}$$ $$=$$ $$17 \beta_{15} + 18 \beta_{14} - 34 \beta_{13} - 43 \beta_{12} + 16 \beta_{11} + 18 \beta_{10} - 42 \beta_{8} + 25 \beta_{7} + 48 \beta_{6} + 34 \beta_{5} + 27 \beta_{4} + 2 \beta_{3} - 26 \beta_{2} + 17 \beta _1 - 25$$ 17*b15 + 18*b14 - 34*b13 - 43*b12 + 16*b11 + 18*b10 - 42*b8 + 25*b7 + 48*b6 + 34*b5 + 27*b4 + 2*b3 - 26*b2 + 17*b1 - 25 $$\nu^{15}$$ $$=$$ $$50 \beta_{15} + 45 \beta_{14} - 20 \beta_{13} - 20 \beta_{12} + 48 \beta_{11} + 25 \beta_{10} + 6 \beta_{9} + 12 \beta_{8} + 20 \beta_{7} + 49 \beta_{6} + 38 \beta_{5} - 60 \beta_{4} - 56 \beta_{3} - 11 \beta_{2} - 79 \beta _1 + 8$$ 50*b15 + 45*b14 - 20*b13 - 20*b12 + 48*b11 + 25*b10 + 6*b9 + 12*b8 + 20*b7 + 49*b6 + 38*b5 - 60*b4 - 56*b3 - 11*b2 - 79*b1 + 8

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$55$$ $$65$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1 - \beta_{2}$$

## 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
 0.608741 − 1.27649i −0.186766 − 1.40183i −0.533474 − 1.30973i 1.40985 + 0.111062i 1.12063 + 0.862658i −1.37702 − 0.322193i 0.867527 + 1.11687i −0.409484 + 1.35363i 0.608741 + 1.27649i −0.186766 + 1.40183i −0.533474 + 1.30973i 1.40985 − 0.111062i 1.12063 − 0.862658i −1.37702 + 0.322193i 0.867527 − 1.11687i −0.409484 − 1.35363i
−1.40985 + 0.111062i −1.71646 0.231865i 1.97533 0.313160i 1.74322 + 3.01934i 2.44570 + 0.136260i 1.80802 + 1.04386i −2.75013 + 0.660890i 2.89248 + 0.795973i −2.79300 4.06320i
11.2 −1.12063 + 0.862658i 0.418594 1.68071i 0.511643 1.93345i −1.60936 2.78750i 0.980785 + 2.24456i 1.82223 + 1.05206i 1.09454 + 2.60806i −2.64956 1.40707i 4.20817 + 1.73544i
11.3 −0.867527 + 1.11687i 0.925606 + 1.46399i −0.494795 1.93783i 0.895377 + 1.55084i −2.43807 0.236266i −2.08793 1.20546i 2.59355 + 1.12850i −1.28651 + 2.71015i −2.50885 0.345375i
11.4 −0.608741 1.27649i −1.71646 0.231865i −1.25887 + 1.55411i −1.74322 3.01934i 0.748906 + 2.33220i −1.80802 1.04386i 2.75013 + 0.660890i 2.89248 + 0.795973i −2.79300 + 4.06320i
11.5 0.186766 1.40183i 0.418594 1.68071i −1.93024 0.523628i 1.60936 + 2.78750i −2.27788 0.900696i −1.82223 1.05206i −1.09454 + 2.60806i −2.64956 1.40707i 4.20817 1.73544i
11.6 0.409484 + 1.35363i −1.12774 + 1.31461i −1.66465 + 1.10858i −0.565188 0.978934i −2.24129 0.988231i 3.71499 + 2.14485i −2.18226 1.79937i −0.456412 2.96508i 1.09368 1.16591i
11.7 0.533474 1.30973i 0.925606 + 1.46399i −1.43081 1.39742i −0.895377 1.55084i 2.41122 0.431300i 2.08793 + 1.20546i −2.59355 + 1.12850i −1.28651 + 2.71015i −2.50885 + 0.345375i
11.8 1.37702 0.322193i −1.12774 + 1.31461i 1.79238 0.887333i 0.565188 + 0.978934i −1.12936 + 2.17360i −3.71499 2.14485i 2.18226 1.79937i −0.456412 2.96508i 1.09368 + 1.16591i
59.1 −1.40985 0.111062i −1.71646 + 0.231865i 1.97533 + 0.313160i 1.74322 3.01934i 2.44570 0.136260i 1.80802 1.04386i −2.75013 0.660890i 2.89248 0.795973i −2.79300 + 4.06320i
59.2 −1.12063 0.862658i 0.418594 + 1.68071i 0.511643 + 1.93345i −1.60936 + 2.78750i 0.980785 2.24456i 1.82223 1.05206i 1.09454 2.60806i −2.64956 + 1.40707i 4.20817 1.73544i
59.3 −0.867527 1.11687i 0.925606 1.46399i −0.494795 + 1.93783i 0.895377 1.55084i −2.43807 + 0.236266i −2.08793 + 1.20546i 2.59355 1.12850i −1.28651 2.71015i −2.50885 + 0.345375i
59.4 −0.608741 + 1.27649i −1.71646 + 0.231865i −1.25887 1.55411i −1.74322 + 3.01934i 0.748906 2.33220i −1.80802 + 1.04386i 2.75013 0.660890i 2.89248 0.795973i −2.79300 4.06320i
59.5 0.186766 + 1.40183i 0.418594 + 1.68071i −1.93024 + 0.523628i 1.60936 2.78750i −2.27788 + 0.900696i −1.82223 + 1.05206i −1.09454 2.60806i −2.64956 + 1.40707i 4.20817 + 1.73544i
59.6 0.409484 1.35363i −1.12774 1.31461i −1.66465 1.10858i −0.565188 + 0.978934i −2.24129 + 0.988231i 3.71499 2.14485i −2.18226 + 1.79937i −0.456412 + 2.96508i 1.09368 + 1.16591i
59.7 0.533474 + 1.30973i 0.925606 1.46399i −1.43081 + 1.39742i −0.895377 + 1.55084i 2.41122 + 0.431300i 2.08793 1.20546i −2.59355 1.12850i −1.28651 2.71015i −2.50885 0.345375i
59.8 1.37702 + 0.322193i −1.12774 1.31461i 1.79238 + 0.887333i 0.565188 0.978934i −1.12936 2.17360i −3.71499 + 2.14485i 2.18226 + 1.79937i −0.456412 + 2.96508i 1.09368 1.16591i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
9.d odd 6 1 inner
72.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.2.l.b 16
3.b odd 2 1 216.2.l.b 16
4.b odd 2 1 288.2.p.b 16
8.b even 2 1 288.2.p.b 16
8.d odd 2 1 inner 72.2.l.b 16
9.c even 3 1 216.2.l.b 16
9.c even 3 1 648.2.f.b 16
9.d odd 6 1 inner 72.2.l.b 16
9.d odd 6 1 648.2.f.b 16
12.b even 2 1 864.2.p.b 16
24.f even 2 1 216.2.l.b 16
24.h odd 2 1 864.2.p.b 16
36.f odd 6 1 864.2.p.b 16
36.f odd 6 1 2592.2.f.b 16
36.h even 6 1 288.2.p.b 16
36.h even 6 1 2592.2.f.b 16
72.j odd 6 1 288.2.p.b 16
72.j odd 6 1 2592.2.f.b 16
72.l even 6 1 inner 72.2.l.b 16
72.l even 6 1 648.2.f.b 16
72.n even 6 1 864.2.p.b 16
72.n even 6 1 2592.2.f.b 16
72.p odd 6 1 216.2.l.b 16
72.p odd 6 1 648.2.f.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.l.b 16 1.a even 1 1 trivial
72.2.l.b 16 8.d odd 2 1 inner
72.2.l.b 16 9.d odd 6 1 inner
72.2.l.b 16 72.l even 6 1 inner
216.2.l.b 16 3.b odd 2 1
216.2.l.b 16 9.c even 3 1
216.2.l.b 16 24.f even 2 1
216.2.l.b 16 72.p odd 6 1
288.2.p.b 16 4.b odd 2 1
288.2.p.b 16 8.b even 2 1
288.2.p.b 16 36.h even 6 1
288.2.p.b 16 72.j odd 6 1
648.2.f.b 16 9.c even 3 1
648.2.f.b 16 9.d odd 6 1
648.2.f.b 16 72.l even 6 1
648.2.f.b 16 72.p odd 6 1
864.2.p.b 16 12.b even 2 1
864.2.p.b 16 24.h odd 2 1
864.2.p.b 16 36.f odd 6 1
864.2.p.b 16 72.n even 6 1
2592.2.f.b 16 36.f odd 6 1
2592.2.f.b 16 36.h even 6 1
2592.2.f.b 16 72.j odd 6 1
2592.2.f.b 16 72.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{16} + 27 T_{5}^{14} + 498 T_{5}^{12} + 4923 T_{5}^{10} + 35106 T_{5}^{8} + 123903 T_{5}^{6} + 312453 T_{5}^{4} + 339012 T_{5}^{2} + 266256$$ acting on $$S_{2}^{\mathrm{new}}(72, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 3 T^{15} + 7 T^{14} + 12 T^{13} + \cdots + 256$$
$3$ $$(T^{8} + 3 T^{7} + 6 T^{6} + 15 T^{5} + \cdots + 81)^{2}$$
$5$ $$T^{16} + 27 T^{14} + 498 T^{12} + \cdots + 266256$$
$7$ $$T^{16} - 33 T^{14} + 750 T^{12} + \cdots + 4260096$$
$11$ $$(T^{8} - 6 T^{7} + 8 T^{6} + 24 T^{5} - 9 T^{4} + \cdots + 1)^{2}$$
$13$ $$T^{16} - 57 T^{14} + 2442 T^{12} + \cdots + 266256$$
$17$ $$(T^{8} + 35 T^{6} + 360 T^{4} + 992 T^{2} + \cdots + 784)^{2}$$
$19$ $$(T^{4} + T^{3} - 12 T^{2} - 8 T + 16)^{4}$$
$23$ $$T^{16} + 99 T^{14} + \cdots + 639280656$$
$29$ $$T^{16} + 135 T^{14} + \cdots + 17449353216$$
$31$ $$T^{16} - 117 T^{14} + \cdots + 279189651456$$
$37$ $$(T^{8} + 156 T^{6} + 4896 T^{4} + \cdots + 74304)^{2}$$
$41$ $$(T^{8} + 18 T^{7} + 128 T^{6} + 360 T^{5} + \cdots + 7921)^{2}$$
$43$ $$(T^{8} - 4 T^{7} + 76 T^{6} + 524 T^{5} + \cdots + 6889)^{2}$$
$47$ $$T^{16} + 111 T^{14} + 10818 T^{12} + \cdots + 266256$$
$53$ $$(T^{8} - 228 T^{6} + 15408 T^{4} + \cdots + 297216)^{2}$$
$59$ $$(T^{8} - 6 T^{7} - 58 T^{6} + 420 T^{5} + \cdots + 528529)^{2}$$
$61$ $$T^{16} - 189 T^{14} + \cdots + 1192149524736$$
$67$ $$(T^{8} + 8 T^{7} + 130 T^{6} + \cdots + 582169)^{2}$$
$71$ $$(T^{8} - 168 T^{6} + 3744 T^{4} + \cdots + 74304)^{2}$$
$73$ $$(T^{4} + T^{3} - 78 T^{2} - 224 T + 172)^{4}$$
$79$ $$T^{16} - 249 T^{14} + \cdots + 74509345296$$
$83$ $$(T^{8} - 27 T^{7} + 212 T^{6} + \cdots + 432964)^{2}$$
$89$ $$(T^{8} + 272 T^{6} + 23808 T^{4} + \cdots + 891136)^{2}$$
$97$ $$(T^{8} - 4 T^{7} + 190 T^{6} + \cdots + 1018081)^{2}$$