# Properties

 Label 1470.2.i.m Level $1470$ Weight $2$ Character orbit 1470.i Analytic conductor $11.738$ 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] = [1470,2,Mod(361,1470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1470, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1470.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1470.i (of order $$3$$, degree $$2$$, not 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: $$11.7380090971$$ 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: no (minimal twist has level 210) 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times$$.

 $$n$$ $$491$$ $$1081$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −1.00000 0 −1.00000 −0.500000 0.866025i 0.500000 0.866025i
961.1 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −1.00000 0 −1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
 $$n$$: 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 1470.2.i.m 2
7.b odd 2 1 210.2.i.d 2
7.c even 3 1 1470.2.a.h 1
7.c even 3 1 inner 1470.2.i.m 2
7.d odd 6 1 210.2.i.d 2
7.d odd 6 1 1470.2.a.a 1
21.c even 2 1 630.2.k.c 2
21.g even 6 1 630.2.k.c 2
21.g even 6 1 4410.2.a.bj 1
21.h odd 6 1 4410.2.a.ba 1
28.d even 2 1 1680.2.bg.g 2
28.f even 6 1 1680.2.bg.g 2
35.c odd 2 1 1050.2.i.b 2
35.f even 4 2 1050.2.o.i 4
35.i odd 6 1 1050.2.i.b 2
35.i odd 6 1 7350.2.a.cp 1
35.j even 6 1 7350.2.a.bu 1
35.k even 12 2 1050.2.o.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.d 2 7.b odd 2 1
210.2.i.d 2 7.d odd 6 1
630.2.k.c 2 21.c even 2 1
630.2.k.c 2 21.g even 6 1
1050.2.i.b 2 35.c odd 2 1
1050.2.i.b 2 35.i odd 6 1
1050.2.o.i 4 35.f even 4 2
1050.2.o.i 4 35.k even 12 2
1470.2.a.a 1 7.d odd 6 1
1470.2.a.h 1 7.c even 3 1
1470.2.i.m 2 1.a even 1 1 trivial
1470.2.i.m 2 7.c even 3 1 inner
1680.2.bg.g 2 28.d even 2 1
1680.2.bg.g 2 28.f even 6 1
4410.2.a.ba 1 21.h odd 6 1
4410.2.a.bj 1 21.g even 6 1
7350.2.a.bu 1 35.j even 6 1
7350.2.a.cp 1 35.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1470, [\chi])$$:

 $$T_{11}^{2} - T_{11} + 1$$ T11^2 - T11 + 1 $$T_{13} + 1$$ T13 + 1 $$T_{17}$$ T17 $$T_{19}^{2} + 3T_{19} + 9$$ T19^2 + 3*T19 + 9 $$T_{31}^{2} + 2T_{31} + 4$$ T31^2 + 2*T31 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} - T + 1$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 3T + 9$$
$23$ $$T^{2} + 7T + 49$$
$29$ $$(T + 8)^{2}$$
$31$ $$T^{2} + 2T + 4$$
$37$ $$T^{2} + 11T + 121$$
$41$ $$(T - 11)^{2}$$
$43$ $$(T - 8)^{2}$$
$47$ $$T^{2} + 5T + 25$$
$53$ $$T^{2} - 11T + 121$$
$59$ $$T^{2} - 4T + 16$$
$61$ $$T^{2}$$
$67$ $$T^{2}$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} + 6T + 36$$
$79$ $$T^{2} - 8T + 64$$
$83$ $$(T + 8)^{2}$$
$89$ $$T^{2} + 10T + 100$$
$97$ $$(T - 16)^{2}$$