# Properties

 Label 462.2.i.g Level $462$ Weight $2$ Character orbit 462.i Analytic conductor $3.689$ 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] = [462,2,Mod(67,462)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(462, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

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

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

chi := DirichletCharacter("462.67");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.i (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: $$3.68908857338$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.21870000.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} + 24x^{4} - 43x^{3} + 138x^{2} - 117x + 73$$ x^6 - 3*x^5 + 24*x^4 - 43*x^3 + 138*x^2 - 117*x + 73 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} + 24x^{4} - 43x^{3} + 138x^{2} - 117x + 73$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -2\nu^{5} + 5\nu^{4} - 30\nu^{3} + 40\nu^{2} - 70\nu + 13 ) / 31$$ (-2*v^5 + 5*v^4 - 30*v^3 + 40*v^2 - 70*v + 13) / 31 $$\beta_{3}$$ $$=$$ $$( -\nu^{5} + 18\nu^{4} - 15\nu^{3} + 144\nu^{2} + 151\nu - 164 ) / 62$$ (-v^5 + 18*v^4 - 15*v^3 + 144*v^2 + 151*v - 164) / 62 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} - 13\nu^{4} + 47\nu^{3} - 197\nu^{2} + 461\nu - 133 ) / 62$$ (-v^5 - 13*v^4 + 47*v^3 - 197*v^2 + 461*v - 133) / 62 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} - 13\nu^{4} + 16\nu^{3} - 166\nu^{2} + 151\nu - 102 ) / 31$$ (-v^5 - 13*v^4 + 16*v^3 - 166*v^2 + 151*v - 102) / 31
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-2\beta_{5} + 2\beta_{4} - 2\beta_{3} + \beta_{2} + 2\beta _1 - 8$$ -2*b5 + 2*b4 - 2*b3 + b2 + 2*b1 - 8 $$\nu^{3}$$ $$=$$ $$-3\beta_{5} + 4\beta_{4} - 2\beta_{3} + \beta_{2} - 8\beta _1 - 7$$ -3*b5 + 4*b4 - 2*b3 + b2 - 8*b1 - 7 $$\nu^{4}$$ $$=$$ $$16\beta_{5} - 16\beta_{4} + 20\beta_{3} - 9\beta_{2} - 28\beta _1 + 75$$ 16*b5 - 16*b4 + 20*b3 - 9*b2 - 28*b1 + 75 $$\nu^{5}$$ $$=$$ $$45\beta_{5} - 60\beta_{4} + 40\beta_{3} - 33\beta_{2} + 55\beta _1 + 139$$ 45*b5 - 60*b4 + 40*b3 - 33*b2 + 55*b1 + 139

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/462\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 + 3.23735i 0.5 − 3.05087i 0.5 + 0.679547i 0.5 − 3.23735i 0.5 + 3.05087i 0.5 − 0.679547i
−0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −2.20942 3.82682i −1.00000 2.20942 1.45550i 1.00000 −0.500000 0.866025i −2.20942 + 3.82682i
67.2 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0.806615 + 1.39710i −1.00000 −0.806615 2.51980i 1.00000 −0.500000 0.866025i 0.806615 1.39710i
67.3 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 1.40280 + 2.42972i −1.00000 −1.40280 + 2.24325i 1.00000 −0.500000 0.866025i 1.40280 2.42972i
331.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −2.20942 + 3.82682i −1.00000 2.20942 + 1.45550i 1.00000 −0.500000 + 0.866025i −2.20942 3.82682i
331.2 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0.806615 1.39710i −1.00000 −0.806615 + 2.51980i 1.00000 −0.500000 + 0.866025i 0.806615 + 1.39710i
331.3 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 1.40280 2.42972i −1.00000 −1.40280 2.24325i 1.00000 −0.500000 + 0.866025i 1.40280 + 2.42972i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 331.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.i.g 6
3.b odd 2 1 1386.2.k.v 6
7.c even 3 1 inner 462.2.i.g 6
7.c even 3 1 3234.2.a.bf 3
7.d odd 6 1 3234.2.a.bh 3
21.g even 6 1 9702.2.a.dw 3
21.h odd 6 1 1386.2.k.v 6
21.h odd 6 1 9702.2.a.dv 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.g 6 1.a even 1 1 trivial
462.2.i.g 6 7.c even 3 1 inner
1386.2.k.v 6 3.b odd 2 1
1386.2.k.v 6 21.h odd 6 1
3234.2.a.bf 3 7.c even 3 1
3234.2.a.bh 3 7.d odd 6 1
9702.2.a.dv 3 21.h odd 6 1
9702.2.a.dw 3 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(462, [\chi])$$:

 $$T_{5}^{6} + 15T_{5}^{4} - 40T_{5}^{3} + 225T_{5}^{2} - 300T_{5} + 400$$ T5^6 + 15*T5^4 - 40*T5^3 + 225*T5^2 - 300*T5 + 400 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{3}$$
$3$ $$(T^{2} - T + 1)^{3}$$
$5$ $$T^{6} + 15 T^{4} - 40 T^{3} + \cdots + 400$$
$7$ $$T^{6} + 6 T^{4} - 20 T^{3} + 42 T^{2} + \cdots + 343$$
$11$ $$(T^{2} - T + 1)^{3}$$
$13$ $$T^{6}$$
$17$ $$T^{6} + 3 T^{5} + 66 T^{4} + \cdots + 19321$$
$19$ $$T^{6} - 9 T^{5} + 69 T^{4} - 164 T^{3} + \cdots + 784$$
$23$ $$T^{6} + 3 T^{5} + 36 T^{4} + \cdots + 7921$$
$29$ $$(T^{3} - 9 T^{2} - 48 T + 348)^{2}$$
$31$ $$T^{6} - 6 T^{5} + 84 T^{4} + \cdots + 36864$$
$37$ $$T^{6} + 21 T^{5} + 309 T^{4} + \cdots + 51984$$
$41$ $$(T^{3} - 12 T^{2} - 27 T + 306)^{2}$$
$43$ $$(T^{3} - 3 T^{2} - 132 T + 404)^{2}$$
$47$ $$T^{6} + 3 T^{5} + 36 T^{4} - 63 T^{3} + \cdots + 81$$
$53$ $$(T^{2} + 8 T + 64)^{3}$$
$59$ $$T^{6} - 9 T^{5} + 129 T^{4} + \cdots + 2304$$
$61$ $$T^{6} - 6 T^{5} + 99 T^{4} + \cdots + 9604$$
$67$ $$T^{6} + 6 T^{5} + 99 T^{4} + \cdots + 44944$$
$71$ $$(T^{3} - 9 T^{2} - 108 T + 108)^{2}$$
$73$ $$T^{6} + 120 T^{4} - 960 T^{3} + \cdots + 230400$$
$79$ $$T^{6} + 12 T^{5} + 111 T^{4} + \cdots + 256$$
$83$ $$(T^{3} - 18 T^{2} + 33 T + 164)^{2}$$
$89$ $$T^{6} - 12 T^{5} + 156 T^{4} + \cdots + 256$$
$97$ $$(T - 7)^{6}$$