# Properties

 Label 1470.2.i.r 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: 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} + \zeta_{6} q^{5} + q^{6} - q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + z * q^2 + (-z + 1) * q^3 + (z - 1) * q^4 + z * q^5 + q^6 - q^8 - z * q^9 $$q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} + \zeta_{6} q^{5} + q^{6} - q^{8} - \zeta_{6} q^{9} + (\zeta_{6} - 1) q^{10} + (6 \zeta_{6} - 6) q^{11} + \zeta_{6} q^{12} - 6 q^{13} + q^{15} - \zeta_{6} q^{16} + ( - \zeta_{6} + 1) q^{18} - 4 \zeta_{6} q^{19} - q^{20} - 6 q^{22} + (\zeta_{6} - 1) q^{24} + (\zeta_{6} - 1) q^{25} - 6 \zeta_{6} q^{26} - q^{27} - 8 q^{29} + \zeta_{6} q^{30} + ( - 2 \zeta_{6} + 2) q^{31} + ( - \zeta_{6} + 1) q^{32} + 6 \zeta_{6} q^{33} + q^{36} - 4 \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{38} + (6 \zeta_{6} - 6) q^{39} - \zeta_{6} q^{40} + 10 q^{41} - 6 q^{43} - 6 \zeta_{6} q^{44} + ( - \zeta_{6} + 1) q^{45} - 2 \zeta_{6} q^{47} - q^{48} - q^{50} + ( - 6 \zeta_{6} + 6) q^{52} + (10 \zeta_{6} - 10) q^{53} - \zeta_{6} q^{54} - 6 q^{55} - 4 q^{57} - 8 \zeta_{6} q^{58} + (4 \zeta_{6} - 4) q^{59} + (\zeta_{6} - 1) q^{60} + 14 \zeta_{6} q^{61} + 2 q^{62} + q^{64} - 6 \zeta_{6} q^{65} + (6 \zeta_{6} - 6) q^{66} + (14 \zeta_{6} - 14) q^{67} + 8 q^{71} + \zeta_{6} q^{72} + ( - 6 \zeta_{6} + 6) q^{73} + ( - 4 \zeta_{6} + 4) q^{74} + \zeta_{6} q^{75} + 4 q^{76} - 6 q^{78} + 8 \zeta_{6} q^{79} + ( - \zeta_{6} + 1) q^{80} + (\zeta_{6} - 1) q^{81} + 10 \zeta_{6} q^{82} + 8 q^{83} - 6 \zeta_{6} q^{86} + (8 \zeta_{6} - 8) q^{87} + ( - 6 \zeta_{6} + 6) q^{88} - 18 \zeta_{6} q^{89} + q^{90} - 2 \zeta_{6} q^{93} + ( - 2 \zeta_{6} + 2) q^{94} + ( - 4 \zeta_{6} + 4) q^{95} - \zeta_{6} q^{96} - 2 q^{97} + 6 q^{99} +O(q^{100})$$ 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 + (6*z - 6) * q^11 + z * q^12 - 6 * q^13 + q^15 - z * q^16 + (-z + 1) * q^18 - 4*z * q^19 - q^20 - 6 * q^22 + (z - 1) * q^24 + (z - 1) * q^25 - 6*z * q^26 - q^27 - 8 * q^29 + z * q^30 + (-2*z + 2) * q^31 + (-z + 1) * q^32 + 6*z * q^33 + q^36 - 4*z * q^37 + (-4*z + 4) * q^38 + (6*z - 6) * q^39 - z * q^40 + 10 * q^41 - 6 * q^43 - 6*z * q^44 + (-z + 1) * q^45 - 2*z * q^47 - q^48 - q^50 + (-6*z + 6) * q^52 + (10*z - 10) * q^53 - z * q^54 - 6 * q^55 - 4 * q^57 - 8*z * q^58 + (4*z - 4) * q^59 + (z - 1) * q^60 + 14*z * q^61 + 2 * q^62 + q^64 - 6*z * q^65 + (6*z - 6) * q^66 + (14*z - 14) * q^67 + 8 * q^71 + z * q^72 + (-6*z + 6) * q^73 + (-4*z + 4) * q^74 + z * q^75 + 4 * q^76 - 6 * q^78 + 8*z * q^79 + (-z + 1) * q^80 + (z - 1) * q^81 + 10*z * q^82 + 8 * q^83 - 6*z * q^86 + (8*z - 8) * q^87 + (-6*z + 6) * q^88 - 18*z * q^89 + q^90 - 2*z * q^93 + (-2*z + 2) * q^94 + (-4*z + 4) * q^95 - z * q^96 - 2 * q^97 + 6 * 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 $$2 q + q^{2} + q^{3} - q^{4} + q^{5} + 2 q^{6} - 2 q^{8} - q^{9} - q^{10} - 6 q^{11} + q^{12} - 12 q^{13} + 2 q^{15} - q^{16} + q^{18} - 4 q^{19} - 2 q^{20} - 12 q^{22} - q^{24} - q^{25} - 6 q^{26} - 2 q^{27} - 16 q^{29} + q^{30} + 2 q^{31} + q^{32} + 6 q^{33} + 2 q^{36} - 4 q^{37} + 4 q^{38} - 6 q^{39} - q^{40} + 20 q^{41} - 12 q^{43} - 6 q^{44} + q^{45} - 2 q^{47} - 2 q^{48} - 2 q^{50} + 6 q^{52} - 10 q^{53} - q^{54} - 12 q^{55} - 8 q^{57} - 8 q^{58} - 4 q^{59} - q^{60} + 14 q^{61} + 4 q^{62} + 2 q^{64} - 6 q^{65} - 6 q^{66} - 14 q^{67} + 16 q^{71} + q^{72} + 6 q^{73} + 4 q^{74} + q^{75} + 8 q^{76} - 12 q^{78} + 8 q^{79} + q^{80} - q^{81} + 10 q^{82} + 16 q^{83} - 6 q^{86} - 8 q^{87} + 6 q^{88} - 18 q^{89} + 2 q^{90} - 2 q^{93} + 2 q^{94} + 4 q^{95} - q^{96} - 4 q^{97} + 12 q^{99}+O(q^{100})$$ 2 * q + q^2 + q^3 - q^4 + q^5 + 2 * q^6 - 2 * q^8 - q^9 - q^10 - 6 * q^11 + q^12 - 12 * q^13 + 2 * q^15 - q^16 + q^18 - 4 * q^19 - 2 * q^20 - 12 * q^22 - q^24 - q^25 - 6 * q^26 - 2 * q^27 - 16 * q^29 + q^30 + 2 * q^31 + q^32 + 6 * q^33 + 2 * q^36 - 4 * q^37 + 4 * q^38 - 6 * q^39 - q^40 + 20 * q^41 - 12 * q^43 - 6 * q^44 + q^45 - 2 * q^47 - 2 * q^48 - 2 * q^50 + 6 * q^52 - 10 * q^53 - q^54 - 12 * q^55 - 8 * q^57 - 8 * q^58 - 4 * q^59 - q^60 + 14 * q^61 + 4 * q^62 + 2 * q^64 - 6 * q^65 - 6 * q^66 - 14 * q^67 + 16 * q^71 + q^72 + 6 * q^73 + 4 * q^74 + q^75 + 8 * q^76 - 12 * q^78 + 8 * q^79 + q^80 - q^81 + 10 * q^82 + 16 * q^83 - 6 * q^86 - 8 * q^87 + 6 * q^88 - 18 * q^89 + 2 * q^90 - 2 * q^93 + 2 * q^94 + 4 * q^95 - q^96 - 4 * q^97 + 12 * 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.r 2
7.b odd 2 1 1470.2.i.k 2
7.c even 3 1 1470.2.a.c 1
7.c even 3 1 inner 1470.2.i.r 2
7.d odd 6 1 1470.2.a.i yes 1
7.d odd 6 1 1470.2.i.k 2
21.g even 6 1 4410.2.a.v 1
21.h odd 6 1 4410.2.a.be 1
35.i odd 6 1 7350.2.a.cf 1
35.j even 6 1 7350.2.a.da 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.a.c 1 7.c even 3 1
1470.2.a.i yes 1 7.d odd 6 1
1470.2.i.k 2 7.b odd 2 1
1470.2.i.k 2 7.d odd 6 1
1470.2.i.r 2 1.a even 1 1 trivial
1470.2.i.r 2 7.c even 3 1 inner
4410.2.a.v 1 21.g even 6 1
4410.2.a.be 1 21.h odd 6 1
7350.2.a.cf 1 35.i odd 6 1
7350.2.a.da 1 35.j even 6 1

## Hecke kernels

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

 $$T_{11}^{2} + 6T_{11} + 36$$ T11^2 + 6*T11 + 36 $$T_{13} + 6$$ T13 + 6 $$T_{17}$$ T17 $$T_{19}^{2} + 4T_{19} + 16$$ T19^2 + 4*T19 + 16 $$T_{31}^{2} - 2T_{31} + 4$$ T31^2 - 2*T31 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 6T + 36$$
$13$ $$(T + 6)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 4T + 16$$
$23$ $$T^{2}$$
$29$ $$(T + 8)^{2}$$
$31$ $$T^{2} - 2T + 4$$
$37$ $$T^{2} + 4T + 16$$
$41$ $$(T - 10)^{2}$$
$43$ $$(T + 6)^{2}$$
$47$ $$T^{2} + 2T + 4$$
$53$ $$T^{2} + 10T + 100$$
$59$ $$T^{2} + 4T + 16$$
$61$ $$T^{2} - 14T + 196$$
$67$ $$T^{2} + 14T + 196$$
$71$ $$(T - 8)^{2}$$
$73$ $$T^{2} - 6T + 36$$
$79$ $$T^{2} - 8T + 64$$
$83$ $$(T - 8)^{2}$$
$89$ $$T^{2} + 18T + 324$$
$97$ $$(T + 2)^{2}$$