# Properties

 Label 1470.2.i.i Level $1470$ Weight $2$ Character orbit 1470.i Analytic conductor $11.738$ Analytic rank $1$ 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: $$1$$ 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 $$q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} + \zeta_{6} q^{5} - q^{6} + q^{8} - \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{10} + ( - \zeta_{6} + 1) q^{11} + \zeta_{6} q^{12} - 7 q^{13} + q^{15} - \zeta_{6} q^{16} + (4 \zeta_{6} - 4) q^{17} + (\zeta_{6} - 1) q^{18} + \zeta_{6} q^{19} - q^{20} - q^{22} - \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{24} + (\zeta_{6} - 1) q^{25} + 7 \zeta_{6} q^{26} - q^{27} - 8 q^{29} - \zeta_{6} q^{30} + ( - 6 \zeta_{6} + 6) q^{31} + (\zeta_{6} - 1) q^{32} - \zeta_{6} q^{33} + 4 q^{34} + q^{36} + 3 \zeta_{6} q^{37} + ( - \zeta_{6} + 1) q^{38} + (7 \zeta_{6} - 7) q^{39} + \zeta_{6} q^{40} - 9 q^{41} - 4 q^{43} + \zeta_{6} q^{44} + ( - \zeta_{6} + 1) q^{45} + (\zeta_{6} - 1) q^{46} - 3 \zeta_{6} q^{47} - q^{48} + q^{50} + 4 \zeta_{6} q^{51} + ( - 7 \zeta_{6} + 7) q^{52} + ( - \zeta_{6} + 1) q^{53} + \zeta_{6} q^{54} + q^{55} + q^{57} + 8 \zeta_{6} q^{58} + ( - 12 \zeta_{6} + 12) q^{59} + (\zeta_{6} - 1) q^{60} - 4 \zeta_{6} q^{61} - 6 q^{62} + q^{64} - 7 \zeta_{6} q^{65} + (\zeta_{6} - 1) q^{66} + (12 \zeta_{6} - 12) q^{67} - 4 \zeta_{6} q^{68} - q^{69} - 14 q^{71} - \zeta_{6} q^{72} + (14 \zeta_{6} - 14) q^{73} + ( - 3 \zeta_{6} + 3) q^{74} + \zeta_{6} q^{75} - q^{76} + 7 q^{78} - 4 \zeta_{6} q^{79} + ( - \zeta_{6} + 1) q^{80} + (\zeta_{6} - 1) q^{81} + 9 \zeta_{6} q^{82} - 12 q^{83} - 4 q^{85} + 4 \zeta_{6} q^{86} + (8 \zeta_{6} - 8) q^{87} + ( - \zeta_{6} + 1) q^{88} - 2 \zeta_{6} q^{89} - q^{90} + q^{92} - 6 \zeta_{6} q^{93} + (3 \zeta_{6} - 3) q^{94} + (\zeta_{6} - 1) q^{95} + \zeta_{6} q^{96} + 16 q^{97} - q^{99} +O(q^{100})$$ 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 - 7 * q^13 + q^15 - z * q^16 + (4*z - 4) * q^17 + (z - 1) * q^18 + z * q^19 - q^20 - q^22 - z * q^23 + (-z + 1) * q^24 + (z - 1) * q^25 + 7*z * q^26 - q^27 - 8 * q^29 - z * q^30 + (-6*z + 6) * q^31 + (z - 1) * q^32 - z * q^33 + 4 * q^34 + q^36 + 3*z * q^37 + (-z + 1) * q^38 + (7*z - 7) * q^39 + z * q^40 - 9 * q^41 - 4 * q^43 + z * q^44 + (-z + 1) * q^45 + (z - 1) * q^46 - 3*z * q^47 - q^48 + q^50 + 4*z * q^51 + (-7*z + 7) * q^52 + (-z + 1) * q^53 + z * q^54 + q^55 + q^57 + 8*z * q^58 + (-12*z + 12) * q^59 + (z - 1) * q^60 - 4*z * q^61 - 6 * q^62 + q^64 - 7*z * q^65 + (z - 1) * q^66 + (12*z - 12) * q^67 - 4*z * q^68 - q^69 - 14 * q^71 - z * q^72 + (14*z - 14) * q^73 + (-3*z + 3) * q^74 + z * q^75 - q^76 + 7 * q^78 - 4*z * q^79 + (-z + 1) * q^80 + (z - 1) * q^81 + 9*z * q^82 - 12 * q^83 - 4 * q^85 + 4*z * q^86 + (8*z - 8) * q^87 + (-z + 1) * q^88 - 2*z * q^89 - q^90 + q^92 - 6*z * q^93 + (3*z - 3) * q^94 + (z - 1) * 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 $$2 q - q^{2} + q^{3} - q^{4} + q^{5} - 2 q^{6} + 2 q^{8} - q^{9} + q^{10} + q^{11} + q^{12} - 14 q^{13} + 2 q^{15} - q^{16} - 4 q^{17} - q^{18} + q^{19} - 2 q^{20} - 2 q^{22} - q^{23} + q^{24} - q^{25} + 7 q^{26} - 2 q^{27} - 16 q^{29} - q^{30} + 6 q^{31} - q^{32} - q^{33} + 8 q^{34} + 2 q^{36} + 3 q^{37} + q^{38} - 7 q^{39} + q^{40} - 18 q^{41} - 8 q^{43} + q^{44} + q^{45} - q^{46} - 3 q^{47} - 2 q^{48} + 2 q^{50} + 4 q^{51} + 7 q^{52} + q^{53} + q^{54} + 2 q^{55} + 2 q^{57} + 8 q^{58} + 12 q^{59} - q^{60} - 4 q^{61} - 12 q^{62} + 2 q^{64} - 7 q^{65} - q^{66} - 12 q^{67} - 4 q^{68} - 2 q^{69} - 28 q^{71} - q^{72} - 14 q^{73} + 3 q^{74} + q^{75} - 2 q^{76} + 14 q^{78} - 4 q^{79} + q^{80} - q^{81} + 9 q^{82} - 24 q^{83} - 8 q^{85} + 4 q^{86} - 8 q^{87} + q^{88} - 2 q^{89} - 2 q^{90} + 2 q^{92} - 6 q^{93} - 3 q^{94} - q^{95} + q^{96} + 32 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q - q^2 + q^3 - q^4 + q^5 - 2 * q^6 + 2 * q^8 - q^9 + q^10 + q^11 + q^12 - 14 * q^13 + 2 * q^15 - q^16 - 4 * q^17 - q^18 + q^19 - 2 * q^20 - 2 * q^22 - q^23 + q^24 - q^25 + 7 * q^26 - 2 * q^27 - 16 * q^29 - q^30 + 6 * q^31 - q^32 - q^33 + 8 * q^34 + 2 * q^36 + 3 * q^37 + q^38 - 7 * q^39 + q^40 - 18 * q^41 - 8 * q^43 + q^44 + q^45 - q^46 - 3 * q^47 - 2 * q^48 + 2 * q^50 + 4 * q^51 + 7 * q^52 + q^53 + q^54 + 2 * q^55 + 2 * q^57 + 8 * q^58 + 12 * q^59 - q^60 - 4 * q^61 - 12 * q^62 + 2 * q^64 - 7 * q^65 - q^66 - 12 * q^67 - 4 * q^68 - 2 * q^69 - 28 * q^71 - q^72 - 14 * q^73 + 3 * q^74 + q^75 - 2 * q^76 + 14 * q^78 - 4 * q^79 + q^80 - q^81 + 9 * q^82 - 24 * q^83 - 8 * q^85 + 4 * q^86 - 8 * q^87 + q^88 - 2 * q^89 - 2 * q^90 + 2 * q^92 - 6 * q^93 - 3 * q^94 - 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.i 2
7.b odd 2 1 210.2.i.a 2
7.c even 3 1 1470.2.a.k 1
7.c even 3 1 inner 1470.2.i.i 2
7.d odd 6 1 210.2.i.a 2
7.d odd 6 1 1470.2.a.r 1
21.c even 2 1 630.2.k.h 2
21.g even 6 1 630.2.k.h 2
21.g even 6 1 4410.2.a.g 1
21.h odd 6 1 4410.2.a.q 1
28.d even 2 1 1680.2.bg.k 2
28.f even 6 1 1680.2.bg.k 2
35.c odd 2 1 1050.2.i.s 2
35.f even 4 2 1050.2.o.j 4
35.i odd 6 1 1050.2.i.s 2
35.i odd 6 1 7350.2.a.j 1
35.j even 6 1 7350.2.a.ba 1
35.k even 12 2 1050.2.o.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.a 2 7.b odd 2 1
210.2.i.a 2 7.d odd 6 1
630.2.k.h 2 21.c even 2 1
630.2.k.h 2 21.g even 6 1
1050.2.i.s 2 35.c odd 2 1
1050.2.i.s 2 35.i odd 6 1
1050.2.o.j 4 35.f even 4 2
1050.2.o.j 4 35.k even 12 2
1470.2.a.k 1 7.c even 3 1
1470.2.a.r 1 7.d odd 6 1
1470.2.i.i 2 1.a even 1 1 trivial
1470.2.i.i 2 7.c even 3 1 inner
1680.2.bg.k 2 28.d even 2 1
1680.2.bg.k 2 28.f even 6 1
4410.2.a.g 1 21.g even 6 1
4410.2.a.q 1 21.h odd 6 1
7350.2.a.j 1 35.i odd 6 1
7350.2.a.ba 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 $$S_{2}^{\mathrm{new}}(1470, [\chi])$$:

 $$T_{11}^{2} - T_{11} + 1$$ T11^2 - T11 + 1 $$T_{13} + 7$$ T13 + 7 $$T_{17}^{2} + 4T_{17} + 16$$ T17^2 + 4*T17 + 16 $$T_{19}^{2} - T_{19} + 1$$ T19^2 - T19 + 1 $$T_{31}^{2} - 6T_{31} + 36$$ T31^2 - 6*T31 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} - T + 1$$
$13$ $$(T + 7)^{2}$$
$17$ $$T^{2} + 4T + 16$$
$19$ $$T^{2} - T + 1$$
$23$ $$T^{2} + T + 1$$
$29$ $$(T + 8)^{2}$$
$31$ $$T^{2} - 6T + 36$$
$37$ $$T^{2} - 3T + 9$$
$41$ $$(T + 9)^{2}$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} + 3T + 9$$
$53$ $$T^{2} - T + 1$$
$59$ $$T^{2} - 12T + 144$$
$61$ $$T^{2} + 4T + 16$$
$67$ $$T^{2} + 12T + 144$$
$71$ $$(T + 14)^{2}$$
$73$ $$T^{2} + 14T + 196$$
$79$ $$T^{2} + 4T + 16$$
$83$ $$(T + 12)^{2}$$
$89$ $$T^{2} + 2T + 4$$
$97$ $$(T - 16)^{2}$$