# Properties

 Label 462.2.i.a Level $462$ Weight $2$ Character orbit 462.i Analytic conductor $3.689$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 + 0.866025i 0.5 − 0.866025i
0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.50000 2.59808i −1.00000 −2.50000 0.866025i −1.00000 −0.500000 0.866025i 1.50000 2.59808i
331.1 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i −1.50000 + 2.59808i −1.00000 −2.50000 + 0.866025i −1.00000 −0.500000 + 0.866025i 1.50000 + 2.59808i
 $$n$$: e.g. 2-40 or 990-1000 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.a 2
3.b odd 2 1 1386.2.k.j 2
7.c even 3 1 inner 462.2.i.a 2
7.c even 3 1 3234.2.a.o 1
7.d odd 6 1 3234.2.a.a 1
21.g even 6 1 9702.2.a.ce 1
21.h odd 6 1 1386.2.k.j 2
21.h odd 6 1 9702.2.a.be 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.a 2 1.a even 1 1 trivial
462.2.i.a 2 7.c even 3 1 inner
1386.2.k.j 2 3.b odd 2 1
1386.2.k.j 2 21.h odd 6 1
3234.2.a.a 1 7.d odd 6 1
3234.2.a.o 1 7.c even 3 1
9702.2.a.be 1 21.h odd 6 1
9702.2.a.ce 1 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 $$S_{2}^{\mathrm{new}}(462, [\chi])$$:

 $$T_{5}^{2} + 3T_{5} + 9$$ T5^2 + 3*T5 + 9 $$T_{13} - 6$$ T13 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$T^{2} + 5T + 7$$
$11$ $$T^{2} - T + 1$$
$13$ $$(T - 6)^{2}$$
$17$ $$T^{2} - 5T + 25$$
$19$ $$T^{2} + 6T + 36$$
$23$ $$T^{2} + 5T + 25$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} + 4T + 16$$
$37$ $$T^{2} - 2T + 4$$
$41$ $$(T - 5)^{2}$$
$43$ $$(T + 10)^{2}$$
$47$ $$T^{2} + 9T + 81$$
$53$ $$T^{2} + 2T + 4$$
$59$ $$T^{2} - 12T + 144$$
$61$ $$T^{2} - 5T + 25$$
$67$ $$T^{2} + 5T + 25$$
$71$ $$(T - 4)^{2}$$
$73$ $$T^{2} + 12T + 144$$
$79$ $$T^{2} - T + 1$$
$83$ $$(T - 1)^{2}$$
$89$ $$T^{2} + 6T + 36$$
$97$ $$(T - 9)^{2}$$