# Properties

 Label 45.2.l.a Level $45$ Weight $2$ Character orbit 45.l Analytic conductor $0.359$ Analytic rank $0$ Dimension $16$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,2,Mod(2,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("45.2");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 45.l (of order $$12$$, degree $$4$$, 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: $$0.359326809096$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} + \cdots + 1$$ x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 34*x^12 + 18*x^11 - 72*x^10 + 132*x^9 - 93*x^8 - 102*x^7 + 144*x^6 - 432*x^5 + 502*x^4 + 288*x^3 + 72*x^2 + 12*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} + \cdots + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 80029143512 \nu^{15} + 385788744870 \nu^{14} - 820783926284 \nu^{13} + 848040618120 \nu^{12} + \cdots + 33432180594 ) / 3707507912227$$ (-80029143512*v^15 + 385788744870*v^14 - 820783926284*v^13 + 848040618120*v^12 + 1720995499366*v^11 - 6827412737484*v^10 + 6402779061545*v^9 - 3349022853273*v^8 - 9000312527194*v^7 + 24987667997140*v^6 - 8831452106135*v^5 + 17087672692002*v^4 + 9515651467064*v^3 - 97367215891956*v^2 + 394928996361*v + 33432180594) / 3707507912227 $$\beta_{3}$$ $$=$$ $$( 94386116202 \nu^{15} - 619740656932 \nu^{14} + 2033008548312 \nu^{13} - 4441986378774 \nu^{12} + \cdots - 80029143512 ) / 3707507912227$$ (94386116202*v^15 - 619740656932*v^14 + 2033008548312*v^13 - 4441986378774*v^12 + 5386888154268*v^11 - 640680728681*v^10 - 7214824090311*v^9 + 16443022873810*v^8 - 16824695358916*v^7 - 2692744559593*v^6 + 17484917305182*v^5 - 49690281510088*v^4 + 74318822560500*v^3 - 6157027329225*v^2 - 993781902738*v - 80029143512) / 3707507912227 $$\beta_{4}$$ $$=$$ $$( 140020985001 \nu^{15} - 843322184609 \nu^{14} + 2535236829090 \nu^{13} + \cdots - 5622211270526 ) / 3707507912227$$ (140020985001*v^15 - 843322184609*v^14 + 2535236829090*v^13 - 5074237635644*v^12 + 4819083120574*v^11 + 2468033098841*v^10 - 10010107392923*v^9 + 18125438747586*v^8 - 12670141167362*v^7 - 14375839504346*v^6 + 19963910433530*v^5 - 58749535677614*v^4 + 69548698425511*v^3 + 39778718221026*v^2 + 6229211681286*v - 5622211270526) / 3707507912227 $$\beta_{5}$$ $$=$$ $$( - 140094553942 \nu^{15} + 840493754711 \nu^{14} - 2524530400854 \nu^{13} + \cdots - 1652709999986 ) / 3707507912227$$ (-140094553942*v^15 + 840493754711*v^14 - 2524530400854*v^13 + 5054110370148*v^12 - 4783342099524*v^11 - 2485960949906*v^10 + 10068880032759*v^9 - 18433708480508*v^8 + 12720523783684*v^7 + 14340027118406*v^6 - 20209428153588*v^5 + 60275329582886*v^4 - 68801672173612*v^3 - 39600205283397*v^2 - 9908294946195*v - 1652709999986) / 3707507912227 $$\beta_{6}$$ $$=$$ $$( 190424165289 \nu^{15} - 1059314239720 \nu^{14} + 2941649532255 \nu^{13} + \cdots + 7166890770427 ) / 3707507912227$$ (190424165289*v^15 - 1059314239720*v^14 + 2941649532255*v^13 - 5449473018186*v^12 + 3809266070290*v^11 + 5471229679434*v^10 - 11091715879228*v^9 + 18612405885154*v^8 - 7831377865421*v^7 - 23866374631831*v^6 + 15235400765019*v^5 - 70138530799181*v^4 + 64517669865637*v^3 + 87537780396607*v^2 + 51260000406122*v + 7166890770427) / 3707507912227 $$\beta_{7}$$ $$=$$ $$( - 250381984552 \nu^{15} + 1889212496921 \nu^{14} - 6927613311171 \nu^{13} + \cdots + 5660844178894 ) / 3707507912227$$ (-250381984552*v^15 + 1889212496921*v^14 - 6927613311171*v^13 + 16582911879478*v^12 - 24294968776849*v^11 + 12446807908885*v^10 + 21076350180684*v^9 - 61892560799590*v^8 + 81094540846877*v^7 - 23942148678807*v^6 - 64342714660344*v^5 + 171068416087064*v^4 - 307914761564074*v^3 + 166080601083280*v^2 + 37629440983980*v + 5660844178894) / 3707507912227 $$\beta_{8}$$ $$=$$ $$( - 254444403226 \nu^{15} + 1391318146672 \nu^{14} - 3674576400880 \nu^{13} + \cdots + 146893504700 ) / 3707507912227$$ (-254444403226*v^15 + 1391318146672*v^14 - 3674576400880*v^13 + 6138067615014*v^12 - 1944897155536*v^11 - 13014144746287*v^10 + 20020382213401*v^9 - 23141068580356*v^8 - 1175929695472*v^7 + 52668080553873*v^6 - 35147821517452*v^5 + 83865626894092*v^4 - 55287519626372*v^3 - 184869896542460*v^2 - 1923868016767*v + 146893504700) / 3707507912227 $$\beta_{9}$$ $$=$$ $$( 196858530 \nu^{15} - 1212076232 \nu^{14} + 3734556994 \nu^{13} - 7676123862 \nu^{12} + \cdots + 785074438 ) / 1973128213$$ (196858530*v^15 - 1212076232*v^14 + 3734556994*v^13 - 7676123862*v^12 + 7906540260*v^11 + 2288455604*v^10 - 14534291253*v^9 + 28282811704*v^8 - 22763827488*v^7 - 16547104520*v^6 + 31379154942*v^5 - 90050852224*v^4 + 112985327352*v^3 + 38916218628*v^2 + 6077223668*v + 785074438) / 1973128213 $$\beta_{10}$$ $$=$$ $$( - 677106342206 \nu^{15} + 3890533925017 \nu^{14} - 11114551602021 \nu^{13} + \cdots - 11893459432082 ) / 3707507912227$$ (-677106342206*v^15 + 3890533925017*v^14 - 11114551602021*v^13 + 21019011866538*v^12 - 16017341587322*v^11 - 19694741515785*v^10 + 47355950801010*v^9 - 76380921929866*v^8 + 36855540630045*v^7 + 91030841771699*v^6 - 84926350980924*v^5 + 264557518559700*v^4 - 259338049254711*v^3 - 298633135730430*v^2 - 72960092263182*v - 11893459432082) / 3707507912227 $$\beta_{11}$$ $$=$$ $$( - 785074438 \nu^{15} + 4907305158 \nu^{14} - 15343416116 \nu^{13} + 31997236762 \nu^{12} + \cdots - 3343669588 ) / 1973128213$$ (-785074438*v^15 + 4907305158*v^14 - 15343416116*v^13 + 31997236762*v^12 - 34368654754*v^11 - 6224799624*v^10 + 58813815140*v^9 - 118164117069*v^8 + 101294734438*v^7 + 57313765188*v^6 - 129597823592*v^5 + 370531312158*v^4 - 484158220100*v^3 - 113116110792*v^2 - 17609140908*v - 3343669588) / 1973128213 $$\beta_{12}$$ $$=$$ $$( - 1616321538 \nu^{15} + 9938916556 \nu^{14} - 30594542475 \nu^{13} + 62871582510 \nu^{12} + \cdots - 10233974547 ) / 3810388399$$ (-1616321538*v^15 + 9938916556*v^14 - 30594542475*v^13 + 62871582510*v^12 - 64714538406*v^11 - 18641341380*v^10 + 118271636061*v^9 - 231117756606*v^8 + 186162436800*v^7 + 134499720676*v^6 - 250255719435*v^5 + 736991238906*v^4 - 924410484114*v^3 - 318408624800*v^2 - 82003950372*v - 10233974547) / 3810388399 $$\beta_{13}$$ $$=$$ $$( - 2029780580195 \nu^{15} + 12360646755715 \nu^{14} - 37626995630777 \nu^{13} + \cdots - 14351633608601 ) / 3707507912227$$ (-2029780580195*v^15 + 12360646755715*v^14 - 37626995630777*v^13 + 76333636000087*v^12 - 75496074288529*v^11 - 30557686749160*v^10 + 149878719509936*v^9 - 281602693598733*v^8 + 213102612708737*v^7 + 190394270523687*v^6 - 312115024344538*v^5 + 903298035710624*v^4 - 1096523230221273*v^3 - 494201049779995*v^2 - 88708185746586*v - 14351633608601) / 3707507912227 $$\beta_{14}$$ $$=$$ $$( 2802573231023 \nu^{15} - 17485025240258 \nu^{14} + 54572756591878 \nu^{13} + \cdots + 12411058785072 ) / 3707507912227$$ (2802573231023*v^15 - 17485025240258*v^14 + 54572756591878*v^13 - 113605741903926*v^12 + 121395144569897*v^11 + 23632381842971*v^10 - 209882210310499*v^9 + 419879212252284*v^8 - 356956659982399*v^7 - 208763256058877*v^6 + 460873742162788*v^5 - 1317355191257852*v^4 + 1711060513528000*v^3 + 423279276478650*v^2 + 69635354161975*v + 12411058785072) / 3707507912227 $$\beta_{15}$$ $$=$$ $$( - 2852772870096 \nu^{15} + 17521813816267 \nu^{14} - 53872302373806 \nu^{13} + \cdots - 17972736434118 ) / 3707507912227$$ (-2852772870096*v^15 + 17521813816267*v^14 - 53872302373806*v^13 + 110572117714578*v^12 - 113406120362540*v^11 - 33753433646508*v^10 + 208555923938660*v^9 - 406505628692916*v^8 + 326089380120540*v^7 + 239007446234759*v^6 - 439893314072684*v^5 + 1297608293014434*v^4 - 1622375494927184*v^3 - 574674020284387*v^2 - 143994055202790*v - 17972736434118) / 3707507912227
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{8} + \beta_{3} - 2\beta_{2} + \beta_1$$ b8 + b3 - 2*b2 + b1 $$\nu^{3}$$ $$=$$ $$\beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} + \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + 4\beta_{3} - \beta_{2} + 1$$ b15 + b14 - b12 + b11 + b8 + b7 + b5 + b4 + 4*b3 - b2 + 1 $$\nu^{4}$$ $$=$$ $$- \beta_{15} + 5 \beta_{14} + 2 \beta_{13} + \beta_{12} + 8 \beta_{11} - \beta_{10} + \beta_{7} + \cdots + 8$$ -b15 + 5*b14 + 2*b13 + b12 + 8*b11 - b10 + b7 - b6 + 5*b5 + 4*b4 + 5*b3 + 8 $$\nu^{5}$$ $$=$$ $$- 16 \beta_{15} + 8 \beta_{13} + 14 \beta_{12} + 7 \beta_{11} + 3 \beta_{9} - 8 \beta_{8} - 8 \beta_{6} + \cdots + 14$$ -16*b15 + 8*b13 + 14*b12 + 7*b11 + 3*b9 - 8*b8 - 8*b6 - 8*b5 + 8*b3 + 7*b2 + 11*b1 + 14 $$\nu^{6}$$ $$=$$ $$- 35 \beta_{15} - 16 \beta_{14} + 8 \beta_{13} + 30 \beta_{12} - 8 \beta_{11} + 16 \beta_{10} + \cdots + 16$$ -35*b15 - 16*b14 + 8*b13 + 30*b12 - 8*b11 + 16*b10 - b9 - 16*b8 + 8*b7 - 16*b6 - 57*b5 - 8*b4 + 34*b3 + 16*b2 + 25*b1 + 16 $$\nu^{7}$$ $$=$$ $$\beta_{15} - \beta_{14} - 10 \beta_{12} - 10 \beta_{11} + 51 \beta_{10} - \beta_{8} + 51 \beta_{7} + \cdots + 10$$ b15 - b14 - 10*b12 - 10*b11 + 51*b10 - b8 + 51*b7 - 98*b5 + b4 + 97*b3 + 10*b2 + 10 $$\nu^{8}$$ $$=$$ $$50 \beta_{15} + 148 \beta_{14} + 50 \beta_{13} - 50 \beta_{12} + 145 \beta_{11} + 50 \beta_{10} + \cdots + 50$$ 50*b15 + 148*b14 + 50*b13 - 50*b12 + 145*b11 + 50*b10 - 10*b9 + 100*b7 + 50*b6 + 10*b5 + 50*b4 + 128*b3 - 138*b1 + 50 $$\nu^{9}$$ $$=$$ $$- 288 \beta_{15} + 278 \beta_{14} + 298 \beta_{13} + 228 \beta_{12} + 456 \beta_{11} - 55 \beta_{9} + \cdots + 228$$ -288*b15 + 278*b14 + 298*b13 + 228*b12 + 456*b11 - 55*b9 - 278*b8 - 10*b4 - 10*b3 + 158*b2 - 278*b1 + 228 $$\nu^{10}$$ $$=$$ $$- 1385 \beta_{15} - 288 \beta_{14} + 576 \beta_{13} + 1032 \beta_{12} + 288 \beta_{11} + 288 \beta_{10} + \cdots + 288$$ -1385*b15 - 288*b14 + 576*b13 + 1032*b12 + 288*b11 + 288*b10 - 140*b9 - 1097*b8 - 288*b7 - 288*b6 - 1315*b5 - 576*b4 - 218*b3 + 744*b2 - 70*b1 + 288 $$\nu^{11}$$ $$=$$ $$- 1533 \beta_{15} - 1603 \beta_{14} + 817 \beta_{12} - 1245 \beta_{11} + 1673 \beta_{10} - 1603 \beta_{8} + \cdots - 817$$ -1533*b15 - 1603*b14 + 817*b12 - 1245*b11 + 1673*b10 - 1603*b8 - 4331*b5 - 1533*b4 + 1245*b2 - 817 $$\nu^{12}$$ $$=$$ $$3206 \beta_{15} - 1603 \beta_{13} - 3206 \beta_{12} - 1603 \beta_{11} + 3206 \beta_{10} + 428 \beta_{9} + \cdots - 3923$$ 3206*b15 - 1603*b13 - 3206*b12 - 1603*b11 + 3206*b10 + 428*b9 + 1603*b7 + 3206*b6 - 3545*b5 - 1195*b4 - 428*b3 - 3545*b1 - 3923 $$\nu^{13}$$ $$=$$ $$8782 \beta_{15} + 8782 \beta_{14} - 6751 \beta_{12} + 6751 \beta_{11} + 428 \beta_{8} + 9210 \beta_{6} + \cdots - 6751$$ 8782*b15 + 8782*b14 - 6751*b12 + 6751*b11 + 428*b8 + 9210*b6 + 9210*b5 + 428*b4 - 8782*b3 - 2459*b2 - 14219*b1 - 6751 $$\nu^{14}$$ $$=$$ $$- 8782 \beta_{15} + 8782 \beta_{14} + 8782 \beta_{13} + 8782 \beta_{12} + 17564 \beta_{11} + \cdots - 8782$$ -8782*b15 + 8782*b14 + 8782*b13 + 8782*b12 + 17564*b11 - 8782*b10 - 2459*b9 - 15075*b8 - 17564*b7 + 8782*b6 + 15105*b5 - 8782*b4 - 36503*b3 + 3376*b2 - 21398*b1 - 8782 $$\nu^{15}$$ $$=$$ $$- 47744 \beta_{15} - 45285 \beta_{14} + 36503 \beta_{12} - 22803 \beta_{11} - 45285 \beta_{8} + \cdots - 36503$$ -47744*b15 - 45285*b14 + 36503*b12 - 22803*b11 - 45285*b8 - 50203*b7 - 45285*b5 - 47744*b4 - 78841*b3 + 22803*b2 - 36503

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$37$$ $$\chi(n)$$ $$1 + \beta_{11}$$ $$-\beta_{12}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 0.601150 − 2.24352i 0.430324 − 1.60599i −0.0499037 + 0.186243i −0.347596 + 1.29724i 0.601150 + 2.24352i 0.430324 + 1.60599i −0.0499037 − 0.186243i −0.347596 − 1.29724i 2.24352 − 0.601150i 1.60599 − 0.430324i −0.186243 + 0.0499037i −1.29724 + 0.347596i 2.24352 + 0.601150i 1.60599 + 0.430324i −0.186243 − 0.0499037i −1.29724 − 0.347596i
−0.601150 2.24352i −1.72336 0.173261i −2.93996 + 1.69739i 1.70912 1.44185i 0.647285 + 3.97056i 0.751454 0.201351i 2.29074 + 2.29074i 2.93996 + 0.597183i −4.26225 2.96768i
2.2 −0.430324 1.60599i 1.35314 + 1.08121i −0.661975 + 0.382191i −2.23073 + 0.154373i 1.15412 2.63840i −1.73749 + 0.465559i −1.45267 1.45267i 0.661975 + 2.92605i 1.20786 + 3.51610i
2.3 0.0499037 + 0.186243i −0.806271 1.53295i 1.69985 0.981412i −0.250705 + 2.22197i 0.245265 0.226662i −2.35868 + 0.632007i 0.540289 + 0.540289i −1.69985 + 2.47194i −0.426337 + 0.0641924i
2.4 0.347596 + 1.29724i −1.18953 + 1.25897i 0.170031 0.0981673i −1.59371 1.56847i −2.04667 1.10550i 1.97869 0.530190i 2.08575 + 2.08575i −0.170031 2.99518i 1.48073 2.61262i
23.1 −0.601150 + 2.24352i −1.72336 + 0.173261i −2.93996 1.69739i 1.70912 + 1.44185i 0.647285 3.97056i 0.751454 + 0.201351i 2.29074 2.29074i 2.93996 0.597183i −4.26225 + 2.96768i
23.2 −0.430324 + 1.60599i 1.35314 1.08121i −0.661975 0.382191i −2.23073 0.154373i 1.15412 + 2.63840i −1.73749 0.465559i −1.45267 + 1.45267i 0.661975 2.92605i 1.20786 3.51610i
23.3 0.0499037 0.186243i −0.806271 + 1.53295i 1.69985 + 0.981412i −0.250705 2.22197i 0.245265 + 0.226662i −2.35868 0.632007i 0.540289 0.540289i −1.69985 2.47194i −0.426337 0.0641924i
23.4 0.347596 1.29724i −1.18953 1.25897i 0.170031 + 0.0981673i −1.59371 + 1.56847i −2.04667 + 1.10550i 1.97869 + 0.530190i 2.08575 2.08575i −0.170031 + 2.99518i 1.48073 + 2.61262i
32.1 −2.24352 0.601150i 0.173261 + 1.72336i 2.93996 + 1.69739i 2.10323 + 0.759216i 0.647285 3.97056i −0.201351 + 0.751454i −2.29074 2.29074i −2.93996 + 0.597183i −4.26225 2.96768i
32.2 −1.60599 0.430324i −1.08121 1.35314i 0.661975 + 0.382191i −1.24906 1.85468i 1.15412 + 2.63840i 0.465559 1.73749i 1.45267 + 1.45267i −0.661975 + 2.92605i 1.20786 + 3.51610i
32.3 0.186243 + 0.0499037i 1.53295 + 0.806271i −1.69985 0.981412i −2.04963 + 0.893868i 0.245265 + 0.226662i 0.632007 2.35868i −0.540289 0.540289i 1.69985 + 2.47194i −0.426337 + 0.0641924i
32.4 1.29724 + 0.347596i −1.25897 + 1.18953i −0.170031 0.0981673i 0.561484 2.16443i −2.04667 + 1.10550i −0.530190 + 1.97869i −2.08575 2.08575i 0.170031 2.99518i 1.48073 2.61262i
38.1 −2.24352 + 0.601150i 0.173261 1.72336i 2.93996 1.69739i 2.10323 0.759216i 0.647285 + 3.97056i −0.201351 0.751454i −2.29074 + 2.29074i −2.93996 0.597183i −4.26225 + 2.96768i
38.2 −1.60599 + 0.430324i −1.08121 + 1.35314i 0.661975 0.382191i −1.24906 + 1.85468i 1.15412 2.63840i 0.465559 + 1.73749i 1.45267 1.45267i −0.661975 2.92605i 1.20786 3.51610i
38.3 0.186243 0.0499037i 1.53295 0.806271i −1.69985 + 0.981412i −2.04963 0.893868i 0.245265 0.226662i 0.632007 + 2.35868i −0.540289 + 0.540289i 1.69985 2.47194i −0.426337 0.0641924i
38.4 1.29724 0.347596i −1.25897 1.18953i −0.170031 + 0.0981673i 0.561484 + 2.16443i −2.04667 1.10550i −0.530190 1.97869i −2.08575 + 2.08575i 0.170031 + 2.99518i 1.48073 + 2.61262i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
9.d odd 6 1 inner
45.l even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.2.l.a 16
3.b odd 2 1 135.2.m.a 16
4.b odd 2 1 720.2.cu.c 16
5.b even 2 1 225.2.p.b 16
5.c odd 4 1 inner 45.2.l.a 16
5.c odd 4 1 225.2.p.b 16
9.c even 3 1 135.2.m.a 16
9.c even 3 1 405.2.f.a 16
9.d odd 6 1 inner 45.2.l.a 16
9.d odd 6 1 405.2.f.a 16
15.d odd 2 1 675.2.q.a 16
15.e even 4 1 135.2.m.a 16
15.e even 4 1 675.2.q.a 16
20.e even 4 1 720.2.cu.c 16
36.h even 6 1 720.2.cu.c 16
45.h odd 6 1 225.2.p.b 16
45.j even 6 1 675.2.q.a 16
45.k odd 12 1 135.2.m.a 16
45.k odd 12 1 405.2.f.a 16
45.k odd 12 1 675.2.q.a 16
45.l even 12 1 inner 45.2.l.a 16
45.l even 12 1 225.2.p.b 16
45.l even 12 1 405.2.f.a 16
180.v odd 12 1 720.2.cu.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.l.a 16 1.a even 1 1 trivial
45.2.l.a 16 5.c odd 4 1 inner
45.2.l.a 16 9.d odd 6 1 inner
45.2.l.a 16 45.l even 12 1 inner
135.2.m.a 16 3.b odd 2 1
135.2.m.a 16 9.c even 3 1
135.2.m.a 16 15.e even 4 1
135.2.m.a 16 45.k odd 12 1
225.2.p.b 16 5.b even 2 1
225.2.p.b 16 5.c odd 4 1
225.2.p.b 16 45.h odd 6 1
225.2.p.b 16 45.l even 12 1
405.2.f.a 16 9.c even 3 1
405.2.f.a 16 9.d odd 6 1
405.2.f.a 16 45.k odd 12 1
405.2.f.a 16 45.l even 12 1
675.2.q.a 16 15.d odd 2 1
675.2.q.a 16 15.e even 4 1
675.2.q.a 16 45.j even 6 1
675.2.q.a 16 45.k odd 12 1
720.2.cu.c 16 4.b odd 2 1
720.2.cu.c 16 20.e even 4 1
720.2.cu.c 16 36.h even 6 1
720.2.cu.c 16 180.v odd 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(45, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 6 T^{15} + \cdots + 1$$
$3$ $$T^{16} + 6 T^{15} + \cdots + 6561$$
$5$ $$T^{16} + 6 T^{15} + \cdots + 390625$$
$7$ $$T^{16} + 2 T^{15} + \cdots + 2401$$
$11$ $$(T^{8} - 10 T^{6} + 102 T^{4} + \cdots + 4)^{2}$$
$13$ $$T^{16} + 2 T^{15} + \cdots + 4477456$$
$17$ $$T^{16} + 964 T^{12} + \cdots + 16$$
$19$ $$(T^{8} + 60 T^{6} + \cdots + 324)^{2}$$
$23$ $$T^{16} - 18 T^{15} + \cdots + 62742241$$
$29$ $$T^{16} + \cdots + 981506241$$
$31$ $$(T^{8} + 2 T^{7} + \cdots + 676)^{2}$$
$37$ $$(T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 4)^{2}$$
$41$ $$(T^{8} + 12 T^{7} + \cdots + 32761)^{2}$$
$43$ $$T^{16} + 2 T^{15} + \cdots + 11316496$$
$47$ $$T^{16} + \cdots + 33243864241$$
$53$ $$T^{16} + \cdots + 409600000000$$
$59$ $$T^{16} + \cdots + 592240896$$
$61$ $$(T^{8} - 4 T^{7} + \cdots + 11449)^{2}$$
$67$ $$T^{16} + \cdots + 539415333601$$
$71$ $$(T^{8} + 116 T^{6} + \cdots + 128164)^{2}$$
$73$ $$(T^{8} + 4 T^{7} + \cdots + 270400)^{2}$$
$79$ $$T^{16} + \cdots + 17\!\cdots\!96$$
$83$ $$T^{16} + \cdots + 13841287201$$
$89$ $$(T^{8} - 300 T^{6} + \cdots + 3969)^{2}$$
$97$ $$T^{16} + \cdots + 6146560000$$