# Properties

 Label 441.2.f.d Level $441$ Weight $2$ Character orbit 441.f Analytic conductor $3.521$ 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] = [441,2,Mod(148,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("441.148");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.f (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.309123.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$$ x^6 - 3*x^5 + 10*x^4 - 15*x^3 + 19*x^2 - 12*x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} - \nu^{4} + 5\nu^{3} + \nu^{2} + 6 ) / 3$$ (v^5 - v^4 + 5*v^3 + v^2 + 6) / 3 $$\beta_{3}$$ $$=$$ $$( -\nu^{5} + \nu^{4} - 5\nu^{3} + 2\nu^{2} - 3\nu ) / 3$$ (-v^5 + v^4 - 5*v^3 + 2*v^2 - 3*v) / 3 $$\beta_{4}$$ $$=$$ $$( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 6 ) / 3$$ (-2*v^5 + 5*v^4 - 16*v^3 + 19*v^2 - 21*v + 6) / 3 $$\beta_{5}$$ $$=$$ $$( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 33\nu - 9 ) / 3$$ (2*v^5 - 5*v^4 + 19*v^3 - 22*v^2 + 33*v - 9) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta _1 - 2$$ b3 + b2 + b1 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3\beta _1 - 1$$ b5 + b4 + b3 + b2 - 3*b1 - 1 $$\nu^{4}$$ $$=$$ $$2\beta_{5} + 3\beta_{4} - 5\beta_{3} - 3\beta_{2} - 6\beta _1 + 6$$ 2*b5 + 3*b4 - 5*b3 - 3*b2 - 6*b1 + 6 $$\nu^{5}$$ $$=$$ $$-3\beta_{5} - 2\beta_{4} - 11\beta_{3} - 6\beta_{2} + 8\beta _1 + 7$$ -3*b5 - 2*b4 - 11*b3 - 6*b2 + 8*b1 + 7

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$1$$ $$\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
148.1
 0.5 − 0.224437i 0.5 + 1.41036i 0.5 − 2.05195i 0.5 + 0.224437i 0.5 − 1.41036i 0.5 + 2.05195i
−0.849814 1.47192i −0.349814 1.69636i −0.444368 + 0.769668i −1.79418 + 3.10761i −2.19963 + 1.95649i 0 −1.88874 −2.75526 + 1.18682i 6.09888
148.2 0.119562 + 0.207087i 0.619562 + 1.61745i 0.971410 1.68253i 0.590972 1.02359i −0.260877 + 0.321688i 0 0.942820 −2.23229 + 2.00422i 0.282630
148.3 1.23025 + 2.13086i 1.73025 + 0.0789082i −2.02704 + 3.51094i −1.29679 + 2.24611i 1.96050 + 3.78400i 0 −5.05408 2.98755 + 0.273062i −6.38151
295.1 −0.849814 + 1.47192i −0.349814 + 1.69636i −0.444368 0.769668i −1.79418 3.10761i −2.19963 1.95649i 0 −1.88874 −2.75526 1.18682i 6.09888
295.2 0.119562 0.207087i 0.619562 1.61745i 0.971410 + 1.68253i 0.590972 + 1.02359i −0.260877 0.321688i 0 0.942820 −2.23229 2.00422i 0.282630
295.3 1.23025 2.13086i 1.73025 0.0789082i −2.02704 3.51094i −1.29679 2.24611i 1.96050 3.78400i 0 −5.05408 2.98755 0.273062i −6.38151
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 148.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.f.d 6
3.b odd 2 1 1323.2.f.c 6
7.b odd 2 1 63.2.f.b 6
7.c even 3 1 441.2.g.d 6
7.c even 3 1 441.2.h.b 6
7.d odd 6 1 441.2.g.e 6
7.d odd 6 1 441.2.h.c 6
9.c even 3 1 inner 441.2.f.d 6
9.c even 3 1 3969.2.a.m 3
9.d odd 6 1 1323.2.f.c 6
9.d odd 6 1 3969.2.a.p 3
21.c even 2 1 189.2.f.a 6
21.g even 6 1 1323.2.g.c 6
21.g even 6 1 1323.2.h.d 6
21.h odd 6 1 1323.2.g.b 6
21.h odd 6 1 1323.2.h.e 6
28.d even 2 1 1008.2.r.k 6
63.g even 3 1 441.2.h.b 6
63.h even 3 1 441.2.g.d 6
63.i even 6 1 1323.2.g.c 6
63.j odd 6 1 1323.2.g.b 6
63.k odd 6 1 441.2.h.c 6
63.l odd 6 1 63.2.f.b 6
63.l odd 6 1 567.2.a.d 3
63.n odd 6 1 1323.2.h.e 6
63.o even 6 1 189.2.f.a 6
63.o even 6 1 567.2.a.g 3
63.s even 6 1 1323.2.h.d 6
63.t odd 6 1 441.2.g.e 6
84.h odd 2 1 3024.2.r.g 6
252.s odd 6 1 3024.2.r.g 6
252.s odd 6 1 9072.2.a.cd 3
252.bi even 6 1 1008.2.r.k 6
252.bi even 6 1 9072.2.a.bq 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.b 6 7.b odd 2 1
63.2.f.b 6 63.l odd 6 1
189.2.f.a 6 21.c even 2 1
189.2.f.a 6 63.o even 6 1
441.2.f.d 6 1.a even 1 1 trivial
441.2.f.d 6 9.c even 3 1 inner
441.2.g.d 6 7.c even 3 1
441.2.g.d 6 63.h even 3 1
441.2.g.e 6 7.d odd 6 1
441.2.g.e 6 63.t odd 6 1
441.2.h.b 6 7.c even 3 1
441.2.h.b 6 63.g even 3 1
441.2.h.c 6 7.d odd 6 1
441.2.h.c 6 63.k odd 6 1
567.2.a.d 3 63.l odd 6 1
567.2.a.g 3 63.o even 6 1
1008.2.r.k 6 28.d even 2 1
1008.2.r.k 6 252.bi even 6 1
1323.2.f.c 6 3.b odd 2 1
1323.2.f.c 6 9.d odd 6 1
1323.2.g.b 6 21.h odd 6 1
1323.2.g.b 6 63.j odd 6 1
1323.2.g.c 6 21.g even 6 1
1323.2.g.c 6 63.i even 6 1
1323.2.h.d 6 21.g even 6 1
1323.2.h.d 6 63.s even 6 1
1323.2.h.e 6 21.h odd 6 1
1323.2.h.e 6 63.n odd 6 1
3024.2.r.g 6 84.h odd 2 1
3024.2.r.g 6 252.s odd 6 1
3969.2.a.m 3 9.c even 3 1
3969.2.a.p 3 9.d odd 6 1
9072.2.a.bq 3 252.bi even 6 1
9072.2.a.cd 3 252.s odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{6} - T_{2}^{5} + 5T_{2}^{4} + 2T_{2}^{3} + 17T_{2}^{2} - 4T_{2} + 1$$ T2^6 - T2^5 + 5*T2^4 + 2*T2^3 + 17*T2^2 - 4*T2 + 1 $$T_{5}^{6} + 5T_{5}^{5} + 23T_{5}^{4} + 32T_{5}^{3} + 59T_{5}^{2} - 22T_{5} + 121$$ T5^6 + 5*T5^5 + 23*T5^4 + 32*T5^3 + 59*T5^2 - 22*T5 + 121

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - T^{5} + 5 T^{4} + 2 T^{3} + 17 T^{2} + \cdots + 1$$
$3$ $$T^{6} - 4 T^{5} + 10 T^{4} - 21 T^{3} + \cdots + 27$$
$5$ $$T^{6} + 5 T^{5} + 23 T^{4} + 32 T^{3} + \cdots + 121$$
$7$ $$T^{6}$$
$11$ $$T^{6} - 2 T^{5} + 23 T^{4} + \cdots + 2209$$
$13$ $$(T^{2} - T + 1)^{3}$$
$17$ $$(T^{3} - 12 T^{2} + 39 T - 27)^{2}$$
$19$ $$(T^{3} - 3 T^{2} - 6 T + 7)^{2}$$
$23$ $$T^{6} + 33 T^{4} - 18 T^{3} + 1089 T^{2} + \cdots + 81$$
$29$ $$T^{6} + T^{5} + 5 T^{4} - 2 T^{3} + 17 T^{2} + \cdots + 1$$
$31$ $$T^{6} + 3 T^{5} + 33 T^{4} - 126 T^{3} + \cdots + 729$$
$37$ $$(T^{3} + 3 T^{2} - 54 T + 81)^{2}$$
$41$ $$T^{6} + 22 T^{5} + 329 T^{4} + \cdots + 124609$$
$43$ $$T^{6} - 3 T^{5} + 75 T^{4} + \cdots + 14641$$
$47$ $$T^{6} + 9 T^{5} + 135 T^{4} + \cdots + 35721$$
$53$ $$(T^{3} + 18 T^{2} + 75 T + 9)^{2}$$
$59$ $$T^{6} + 9 T^{5} + 87 T^{4} + \cdots + 3969$$
$61$ $$T^{6} + 6 T^{5} + 57 T^{4} + \cdots + 4489$$
$67$ $$T^{6} + 207 T^{4} + 1366 T^{3} + \cdots + 466489$$
$71$ $$(T^{3} - 9 T^{2} - 6 T + 81)^{2}$$
$73$ $$(T^{3} + 3 T^{2} - 168 T + 243)^{2}$$
$79$ $$T^{6} + 15 T^{5} + 273 T^{4} + \cdots + 591361$$
$83$ $$T^{6} + 12 T^{5} + 105 T^{4} + \cdots + 729$$
$89$ $$(T^{3} - 2 T^{2} - 151 T - 379)^{2}$$
$97$ $$T^{6} - 3 T^{5} + 123 T^{4} + \cdots + 363609$$