# Properties

 Label 1470.2.i.x Level $1470$ Weight $2$ Character orbit 1470.i Analytic conductor $11.738$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + (\beta_{2} + 1) q^{3} + ( - \beta_{2} - 1) q^{4} + \beta_{2} q^{5} + q^{6} - q^{8} + \beta_{2} q^{9}+O(q^{10})$$ q - b2 * q^2 + (b2 + 1) * q^3 + (-b2 - 1) * q^4 + b2 * q^5 + q^6 - q^8 + b2 * q^9 $$q - \beta_{2} q^{2} + (\beta_{2} + 1) q^{3} + ( - \beta_{2} - 1) q^{4} + \beta_{2} q^{5} + q^{6} - q^{8} + \beta_{2} q^{9} + (\beta_{2} + 1) q^{10} + (2 \beta_{2} + 3 \beta_1 + 2) q^{11} - \beta_{2} q^{12} + 4 \beta_{3} q^{13} - q^{15} + \beta_{2} q^{16} + ( - 4 \beta_{2} - \beta_1 - 4) q^{17} + (\beta_{2} + 1) q^{18} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{19} + q^{20} + ( - 3 \beta_{3} + 2) q^{22} + (2 \beta_{3} + 6 \beta_{2} + 2 \beta_1) q^{23} + ( - \beta_{2} - 1) q^{24} + ( - \beta_{2} - 1) q^{25} + (4 \beta_{3} + 4 \beta_1) q^{26} - q^{27} + ( - \beta_{3} + 4) q^{29} + \beta_{2} q^{30} + (6 \beta_{2} - 3 \beta_1 + 6) q^{31} + (\beta_{2} + 1) q^{32} + (3 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{33} + (\beta_{3} - 4) q^{34} + q^{36} + (3 \beta_{3} + 4 \beta_{2} + 3 \beta_1) q^{37} + ( - 4 \beta_{2} + 2 \beta_1 - 4) q^{38} - 4 \beta_1 q^{39} - \beta_{2} q^{40} + (6 \beta_{3} + 2) q^{41} + (5 \beta_{3} + 2) q^{43} + ( - 3 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{44} + ( - \beta_{2} - 1) q^{45} + (6 \beta_{2} + 2 \beta_1 + 6) q^{46} + (3 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{47} - q^{48} - q^{50} + ( - \beta_{3} - 4 \beta_{2} - \beta_1) q^{51} + 4 \beta_1 q^{52} + ( - 6 \beta_{2} - 4 \beta_1 - 6) q^{53} + \beta_{2} q^{54} + (3 \beta_{3} - 2) q^{55} + (2 \beta_{3} + 4) q^{57} + ( - \beta_{3} - 4 \beta_{2} - \beta_1) q^{58} + (6 \beta_{2} + 2 \beta_1 + 6) q^{59} + (\beta_{2} + 1) q^{60} + (4 \beta_{3} - 6 \beta_{2} + 4 \beta_1) q^{61} + (3 \beta_{3} + 6) q^{62} + q^{64} + ( - 4 \beta_{3} - 4 \beta_1) q^{65} + (2 \beta_{2} + 3 \beta_1 + 2) q^{66} + ( - 2 \beta_{2} + 5 \beta_1 - 2) q^{67} + (\beta_{3} + 4 \beta_{2} + \beta_1) q^{68} + (2 \beta_{3} - 6) q^{69} + ( - 6 \beta_{3} + 4) q^{71} - \beta_{2} q^{72} + (2 \beta_{2} + 2) q^{73} + (4 \beta_{2} + 3 \beta_1 + 4) q^{74} - \beta_{2} q^{75} + ( - 2 \beta_{3} - 4) q^{76} + 4 \beta_{3} q^{78} + 8 \beta_{2} q^{79} + ( - \beta_{2} - 1) q^{80} + ( - \beta_{2} - 1) q^{81} + (6 \beta_{3} - 2 \beta_{2} + 6 \beta_1) q^{82} + ( - 2 \beta_{3} + 8) q^{83} + ( - \beta_{3} + 4) q^{85} + (5 \beta_{3} - 2 \beta_{2} + 5 \beta_1) q^{86} + (4 \beta_{2} + \beta_1 + 4) q^{87} + ( - 2 \beta_{2} - 3 \beta_1 - 2) q^{88} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{89} - q^{90} + ( - 2 \beta_{3} + 6) q^{92} + ( - 3 \beta_{3} + 6 \beta_{2} - 3 \beta_1) q^{93} + (2 \beta_{2} + 3 \beta_1 + 2) q^{94} + (4 \beta_{2} - 2 \beta_1 + 4) q^{95} + \beta_{2} q^{96} + ( - 6 \beta_{3} + 2) q^{97} + (3 \beta_{3} - 2) q^{99}+O(q^{100})$$ q - b2 * q^2 + (b2 + 1) * q^3 + (-b2 - 1) * q^4 + b2 * q^5 + q^6 - q^8 + b2 * q^9 + (b2 + 1) * q^10 + (2*b2 + 3*b1 + 2) * q^11 - b2 * q^12 + 4*b3 * q^13 - q^15 + b2 * q^16 + (-4*b2 - b1 - 4) * q^17 + (b2 + 1) * q^18 + (2*b3 - 4*b2 + 2*b1) * q^19 + q^20 + (-3*b3 + 2) * q^22 + (2*b3 + 6*b2 + 2*b1) * q^23 + (-b2 - 1) * q^24 + (-b2 - 1) * q^25 + (4*b3 + 4*b1) * q^26 - q^27 + (-b3 + 4) * q^29 + b2 * q^30 + (6*b2 - 3*b1 + 6) * q^31 + (b2 + 1) * q^32 + (3*b3 + 2*b2 + 3*b1) * q^33 + (b3 - 4) * q^34 + q^36 + (3*b3 + 4*b2 + 3*b1) * q^37 + (-4*b2 + 2*b1 - 4) * q^38 - 4*b1 * q^39 - b2 * q^40 + (6*b3 + 2) * q^41 + (5*b3 + 2) * q^43 + (-3*b3 - 2*b2 - 3*b1) * q^44 + (-b2 - 1) * q^45 + (6*b2 + 2*b1 + 6) * q^46 + (3*b3 + 2*b2 + 3*b1) * q^47 - q^48 - q^50 + (-b3 - 4*b2 - b1) * q^51 + 4*b1 * q^52 + (-6*b2 - 4*b1 - 6) * q^53 + b2 * q^54 + (3*b3 - 2) * q^55 + (2*b3 + 4) * q^57 + (-b3 - 4*b2 - b1) * q^58 + (6*b2 + 2*b1 + 6) * q^59 + (b2 + 1) * q^60 + (4*b3 - 6*b2 + 4*b1) * q^61 + (3*b3 + 6) * q^62 + q^64 + (-4*b3 - 4*b1) * q^65 + (2*b2 + 3*b1 + 2) * q^66 + (-2*b2 + 5*b1 - 2) * q^67 + (b3 + 4*b2 + b1) * q^68 + (2*b3 - 6) * q^69 + (-6*b3 + 4) * q^71 - b2 * q^72 + (2*b2 + 2) * q^73 + (4*b2 + 3*b1 + 4) * q^74 - b2 * q^75 + (-2*b3 - 4) * q^76 + 4*b3 * q^78 + 8*b2 * q^79 + (-b2 - 1) * q^80 + (-b2 - 1) * q^81 + (6*b3 - 2*b2 + 6*b1) * q^82 + (-2*b3 + 8) * q^83 + (-b3 + 4) * q^85 + (5*b3 - 2*b2 + 5*b1) * q^86 + (4*b2 + b1 + 4) * q^87 + (-2*b2 - 3*b1 - 2) * q^88 + (2*b3 + 2*b2 + 2*b1) * q^89 - q^90 + (-2*b3 + 6) * q^92 + (-3*b3 + 6*b2 - 3*b1) * q^93 + (2*b2 + 3*b1 + 2) * q^94 + (4*b2 - 2*b1 + 4) * q^95 + b2 * q^96 + (-6*b3 + 2) * q^97 + (3*b3 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} + 4 q^{6} - 4 q^{8} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 - 2 * q^5 + 4 * q^6 - 4 * q^8 - 2 * q^9 $$4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} + 4 q^{6} - 4 q^{8} - 2 q^{9} + 2 q^{10} + 4 q^{11} + 2 q^{12} - 4 q^{15} - 2 q^{16} - 8 q^{17} + 2 q^{18} + 8 q^{19} + 4 q^{20} + 8 q^{22} - 12 q^{23} - 2 q^{24} - 2 q^{25} - 4 q^{27} + 16 q^{29} - 2 q^{30} + 12 q^{31} + 2 q^{32} - 4 q^{33} - 16 q^{34} + 4 q^{36} - 8 q^{37} - 8 q^{38} + 2 q^{40} + 8 q^{41} + 8 q^{43} + 4 q^{44} - 2 q^{45} + 12 q^{46} - 4 q^{47} - 4 q^{48} - 4 q^{50} + 8 q^{51} - 12 q^{53} - 2 q^{54} - 8 q^{55} + 16 q^{57} + 8 q^{58} + 12 q^{59} + 2 q^{60} + 12 q^{61} + 24 q^{62} + 4 q^{64} + 4 q^{66} - 4 q^{67} - 8 q^{68} - 24 q^{69} + 16 q^{71} + 2 q^{72} + 4 q^{73} + 8 q^{74} + 2 q^{75} - 16 q^{76} - 16 q^{79} - 2 q^{80} - 2 q^{81} + 4 q^{82} + 32 q^{83} + 16 q^{85} + 4 q^{86} + 8 q^{87} - 4 q^{88} - 4 q^{89} - 4 q^{90} + 24 q^{92} - 12 q^{93} + 4 q^{94} + 8 q^{95} - 2 q^{96} + 8 q^{97} - 8 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 - 2 * q^5 + 4 * q^6 - 4 * q^8 - 2 * q^9 + 2 * q^10 + 4 * q^11 + 2 * q^12 - 4 * q^15 - 2 * q^16 - 8 * q^17 + 2 * q^18 + 8 * q^19 + 4 * q^20 + 8 * q^22 - 12 * q^23 - 2 * q^24 - 2 * q^25 - 4 * q^27 + 16 * q^29 - 2 * q^30 + 12 * q^31 + 2 * q^32 - 4 * q^33 - 16 * q^34 + 4 * q^36 - 8 * q^37 - 8 * q^38 + 2 * q^40 + 8 * q^41 + 8 * q^43 + 4 * q^44 - 2 * q^45 + 12 * q^46 - 4 * q^47 - 4 * q^48 - 4 * q^50 + 8 * q^51 - 12 * q^53 - 2 * q^54 - 8 * q^55 + 16 * q^57 + 8 * q^58 + 12 * q^59 + 2 * q^60 + 12 * q^61 + 24 * q^62 + 4 * q^64 + 4 * q^66 - 4 * q^67 - 8 * q^68 - 24 * q^69 + 16 * q^71 + 2 * q^72 + 4 * q^73 + 8 * q^74 + 2 * q^75 - 16 * q^76 - 16 * q^79 - 2 * q^80 - 2 * q^81 + 4 * q^82 + 32 * q^83 + 16 * q^85 + 4 * q^86 + 8 * q^87 - 4 * q^88 - 4 * q^89 - 4 * q^90 + 24 * q^92 - 12 * q^93 + 4 * q^94 + 8 * q^95 - 2 * q^96 + 8 * q^97 - 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$491$$ $$1081$$ $$1177$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
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
361.2 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
961.2 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.x 4
7.b odd 2 1 1470.2.i.w 4
7.c even 3 1 1470.2.a.s 2
7.c even 3 1 inner 1470.2.i.x 4
7.d odd 6 1 1470.2.a.t yes 2
7.d odd 6 1 1470.2.i.w 4
21.g even 6 1 4410.2.a.bz 2
21.h odd 6 1 4410.2.a.bw 2
35.i odd 6 1 7350.2.a.dh 2
35.j even 6 1 7350.2.a.dl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.a.s 2 7.c even 3 1
1470.2.a.t yes 2 7.d odd 6 1
1470.2.i.w 4 7.b odd 2 1
1470.2.i.w 4 7.d odd 6 1
1470.2.i.x 4 1.a even 1 1 trivial
1470.2.i.x 4 7.c even 3 1 inner
4410.2.a.bw 2 21.h odd 6 1
4410.2.a.bz 2 21.g even 6 1
7350.2.a.dh 2 35.i odd 6 1
7350.2.a.dl 2 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 $$S_{2}^{\mathrm{new}}(1470, [\chi])$$:

 $$T_{11}^{4} - 4T_{11}^{3} + 30T_{11}^{2} + 56T_{11} + 196$$ T11^4 - 4*T11^3 + 30*T11^2 + 56*T11 + 196 $$T_{13}^{2} - 32$$ T13^2 - 32 $$T_{17}^{4} + 8T_{17}^{3} + 50T_{17}^{2} + 112T_{17} + 196$$ T17^4 + 8*T17^3 + 50*T17^2 + 112*T17 + 196 $$T_{19}^{4} - 8T_{19}^{3} + 56T_{19}^{2} - 64T_{19} + 64$$ T19^4 - 8*T19^3 + 56*T19^2 - 64*T19 + 64 $$T_{31}^{4} - 12T_{31}^{3} + 126T_{31}^{2} - 216T_{31} + 324$$ T31^4 - 12*T31^3 + 126*T31^2 - 216*T31 + 324

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 4 T^{3} + 30 T^{2} + 56 T + 196$$
$13$ $$(T^{2} - 32)^{2}$$
$17$ $$T^{4} + 8 T^{3} + 50 T^{2} + 112 T + 196$$
$19$ $$T^{4} - 8 T^{3} + 56 T^{2} - 64 T + 64$$
$23$ $$T^{4} + 12 T^{3} + 116 T^{2} + \cdots + 784$$
$29$ $$(T^{2} - 8 T + 14)^{2}$$
$31$ $$T^{4} - 12 T^{3} + 126 T^{2} + \cdots + 324$$
$37$ $$T^{4} + 8 T^{3} + 66 T^{2} - 16 T + 4$$
$41$ $$(T^{2} - 4 T - 68)^{2}$$
$43$ $$(T^{2} - 4 T - 46)^{2}$$
$47$ $$T^{4} + 4 T^{3} + 30 T^{2} - 56 T + 196$$
$53$ $$T^{4} + 12 T^{3} + 140 T^{2} + \cdots + 16$$
$59$ $$T^{4} - 12 T^{3} + 116 T^{2} + \cdots + 784$$
$61$ $$T^{4} - 12 T^{3} + 140 T^{2} + \cdots + 16$$
$67$ $$T^{4} + 4 T^{3} + 62 T^{2} + \cdots + 2116$$
$71$ $$(T^{2} - 8 T - 56)^{2}$$
$73$ $$(T^{2} - 2 T + 4)^{2}$$
$79$ $$(T^{2} + 8 T + 64)^{2}$$
$83$ $$(T^{2} - 16 T + 56)^{2}$$
$89$ $$T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16$$
$97$ $$(T^{2} - 4 T - 68)^{2}$$