# Properties

 Label 1260.1.ci.a Level $1260$ Weight $1$ Character orbit 1260.ci Analytic conductor $0.629$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -20 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1260,1,Mod(739,1260)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1260, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 3, 4]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1260.739");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1260.ci (of order $$6$$, 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: $$0.628821915918$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ 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 140) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.980.1 Artin image: $C_6\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{4} - \zeta_{6}^{2} q^{5} + \zeta_{6} q^{7} + q^{8} +O(q^{10})$$ q + z^2 * q^2 - z * q^4 - z^2 * q^5 + z * q^7 + q^8 $$q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{4} - \zeta_{6}^{2} q^{5} + \zeta_{6} q^{7} + q^{8} + \zeta_{6} q^{10} - q^{14} + \zeta_{6}^{2} q^{16} - q^{20} - \zeta_{6}^{2} q^{23} - \zeta_{6} q^{25} - \zeta_{6}^{2} q^{28} + q^{29} - \zeta_{6} q^{32} + q^{35} - \zeta_{6}^{2} q^{40} + q^{41} + q^{43} + \zeta_{6} q^{46} + 2 \zeta_{6}^{2} q^{47} + \zeta_{6}^{2} q^{49} + q^{50} + \zeta_{6} q^{56} + \zeta_{6}^{2} q^{58} - \zeta_{6}^{2} q^{61} + q^{64} - \zeta_{6} q^{67} + \zeta_{6}^{2} q^{70} + \zeta_{6} q^{80} + \zeta_{6}^{2} q^{82} - q^{83} + \zeta_{6}^{2} q^{86} + \zeta_{6}^{2} q^{89} - q^{92} - 2 \zeta_{6} q^{94} - \zeta_{6} q^{98} +O(q^{100})$$ q + z^2 * q^2 - z * q^4 - z^2 * q^5 + z * q^7 + q^8 + z * q^10 - q^14 + z^2 * q^16 - q^20 - z^2 * q^23 - z * q^25 - z^2 * q^28 + q^29 - z * q^32 + q^35 - z^2 * q^40 + q^41 + q^43 + z * q^46 + 2*z^2 * q^47 + z^2 * q^49 + q^50 + z * q^56 + z^2 * q^58 - z^2 * q^61 + q^64 - z * q^67 + z^2 * q^70 + z * q^80 + z^2 * q^82 - q^83 + z^2 * q^86 + z^2 * q^89 - q^92 - 2*z * q^94 - z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} + q^{5} + q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q - q^2 - q^4 + q^5 + q^7 + 2 * q^8 $$2 q - q^{2} - q^{4} + q^{5} + q^{7} + 2 q^{8} + q^{10} - 2 q^{14} - q^{16} - 2 q^{20} + q^{23} - q^{25} + q^{28} + 2 q^{29} - q^{32} + 2 q^{35} + q^{40} + 2 q^{41} + 2 q^{43} + q^{46} - 2 q^{47} - q^{49} + 2 q^{50} + q^{56} - q^{58} + q^{61} + 2 q^{64} - q^{67} - q^{70} + q^{80} - q^{82} - 2 q^{83} - q^{86} - q^{89} - 2 q^{92} - 2 q^{94} - q^{98}+O(q^{100})$$ 2 * q - q^2 - q^4 + q^5 + q^7 + 2 * q^8 + q^10 - 2 * q^14 - q^16 - 2 * q^20 + q^23 - q^25 + q^28 + 2 * q^29 - q^32 + 2 * q^35 + q^40 + 2 * q^41 + 2 * q^43 + q^46 - 2 * q^47 - q^49 + 2 * q^50 + q^56 - q^58 + q^61 + 2 * q^64 - q^67 - q^70 + q^80 - q^82 - 2 * q^83 - q^86 - q^89 - 2 * q^92 - 2 * q^94 - q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times$$.

 $$n$$ $$281$$ $$631$$ $$757$$ $$1081$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$\zeta_{6}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
739.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 0.500000 0.866025i 1.00000 0 0.500000 0.866025i
919.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 0.500000 + 0.866025i 1.00000 0 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
7.c even 3 1 inner
140.p odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.1.ci.a 2
3.b odd 2 1 140.1.p.b yes 2
4.b odd 2 1 1260.1.ci.b 2
5.b even 2 1 1260.1.ci.b 2
7.c even 3 1 inner 1260.1.ci.a 2
12.b even 2 1 140.1.p.a 2
15.d odd 2 1 140.1.p.a 2
15.e even 4 2 700.1.u.a 4
20.d odd 2 1 CM 1260.1.ci.a 2
21.c even 2 1 980.1.p.b 2
21.g even 6 1 980.1.f.a 1
21.g even 6 1 980.1.p.b 2
21.h odd 6 1 140.1.p.b yes 2
21.h odd 6 1 980.1.f.b 1
24.f even 2 1 2240.1.bt.a 2
24.h odd 2 1 2240.1.bt.b 2
28.g odd 6 1 1260.1.ci.b 2
35.j even 6 1 1260.1.ci.b 2
60.h even 2 1 140.1.p.b yes 2
60.l odd 4 2 700.1.u.a 4
84.h odd 2 1 980.1.p.a 2
84.j odd 6 1 980.1.f.d 1
84.j odd 6 1 980.1.p.a 2
84.n even 6 1 140.1.p.a 2
84.n even 6 1 980.1.f.c 1
105.g even 2 1 980.1.p.a 2
105.o odd 6 1 140.1.p.a 2
105.o odd 6 1 980.1.f.c 1
105.p even 6 1 980.1.f.d 1
105.p even 6 1 980.1.p.a 2
105.x even 12 2 700.1.u.a 4
120.i odd 2 1 2240.1.bt.a 2
120.m even 2 1 2240.1.bt.b 2
140.p odd 6 1 inner 1260.1.ci.a 2
168.s odd 6 1 2240.1.bt.b 2
168.v even 6 1 2240.1.bt.a 2
420.o odd 2 1 980.1.p.b 2
420.ba even 6 1 140.1.p.b yes 2
420.ba even 6 1 980.1.f.b 1
420.be odd 6 1 980.1.f.a 1
420.be odd 6 1 980.1.p.b 2
420.bp odd 12 2 700.1.u.a 4
840.cg odd 6 1 2240.1.bt.a 2
840.cv even 6 1 2240.1.bt.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.p.a 2 12.b even 2 1
140.1.p.a 2 15.d odd 2 1
140.1.p.a 2 84.n even 6 1
140.1.p.a 2 105.o odd 6 1
140.1.p.b yes 2 3.b odd 2 1
140.1.p.b yes 2 21.h odd 6 1
140.1.p.b yes 2 60.h even 2 1
140.1.p.b yes 2 420.ba even 6 1
700.1.u.a 4 15.e even 4 2
700.1.u.a 4 60.l odd 4 2
700.1.u.a 4 105.x even 12 2
700.1.u.a 4 420.bp odd 12 2
980.1.f.a 1 21.g even 6 1
980.1.f.a 1 420.be odd 6 1
980.1.f.b 1 21.h odd 6 1
980.1.f.b 1 420.ba even 6 1
980.1.f.c 1 84.n even 6 1
980.1.f.c 1 105.o odd 6 1
980.1.f.d 1 84.j odd 6 1
980.1.f.d 1 105.p even 6 1
980.1.p.a 2 84.h odd 2 1
980.1.p.a 2 84.j odd 6 1
980.1.p.a 2 105.g even 2 1
980.1.p.a 2 105.p even 6 1
980.1.p.b 2 21.c even 2 1
980.1.p.b 2 21.g even 6 1
980.1.p.b 2 420.o odd 2 1
980.1.p.b 2 420.be odd 6 1
1260.1.ci.a 2 1.a even 1 1 trivial
1260.1.ci.a 2 7.c even 3 1 inner
1260.1.ci.a 2 20.d odd 2 1 CM
1260.1.ci.a 2 140.p odd 6 1 inner
1260.1.ci.b 2 4.b odd 2 1
1260.1.ci.b 2 5.b even 2 1
1260.1.ci.b 2 28.g odd 6 1
1260.1.ci.b 2 35.j even 6 1
2240.1.bt.a 2 24.f even 2 1
2240.1.bt.a 2 120.i odd 2 1
2240.1.bt.a 2 168.v even 6 1
2240.1.bt.a 2 840.cg odd 6 1
2240.1.bt.b 2 24.h odd 2 1
2240.1.bt.b 2 120.m even 2 1
2240.1.bt.b 2 168.s odd 6 1
2240.1.bt.b 2 840.cv even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{23}^{2} - T_{23} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1260, [\chi])$$.

## Hecke characteristic polynomials

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