# Properties

 Label 1260.1.de.a Level $1260$ Weight $1$ Character orbit 1260.de Analytic conductor $0.629$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -20 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1260,1,Mod(59,1260)]

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, 5, 3, 1]))

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

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

chi := DirichletCharacter("1260.59");

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.de (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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.661624362000.11

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{5} q^{2} + \zeta_{12}^{5} q^{3} - \zeta_{12}^{4} q^{4} - q^{5} - \zeta_{12}^{4} q^{6} + \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} - \zeta_{12}^{4} q^{9} +O(q^{10})$$ q + z^5 * q^2 + z^5 * q^3 - z^4 * q^4 - q^5 - z^4 * q^6 + z * q^7 + z^3 * q^8 - z^4 * q^9 $$q + \zeta_{12}^{5} q^{2} + \zeta_{12}^{5} q^{3} - \zeta_{12}^{4} q^{4} - q^{5} - \zeta_{12}^{4} q^{6} + \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} - \zeta_{12}^{4} q^{9} - \zeta_{12}^{5} q^{10} + \zeta_{12}^{3} q^{12} - q^{14} - \zeta_{12}^{5} q^{15} - \zeta_{12}^{2} q^{16} + \zeta_{12}^{3} q^{18} + \zeta_{12}^{4} q^{20} - q^{21} - \zeta_{12}^{3} q^{23} - \zeta_{12}^{2} q^{24} + q^{25} + \zeta_{12}^{3} q^{27} - \zeta_{12}^{5} q^{28} + (\zeta_{12}^{2} + 1) q^{29} + \zeta_{12}^{4} q^{30} + \zeta_{12} q^{32} - \zeta_{12} q^{35} - \zeta_{12}^{2} q^{36} - \zeta_{12}^{3} q^{40} + \zeta_{12}^{2} q^{41} - \zeta_{12}^{5} q^{42} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{43} + \zeta_{12}^{4} q^{45} + 2 \zeta_{12}^{2} q^{46} + (\zeta_{12}^{3} + \zeta_{12}) q^{47} + \zeta_{12} q^{48} + \zeta_{12}^{2} q^{49} + \zeta_{12}^{5} q^{50} - \zeta_{12}^{2} q^{54} + \zeta_{12}^{4} q^{56} + (\zeta_{12}^{5} - \zeta_{12}) q^{58} - \zeta_{12}^{3} q^{60} - \zeta_{12}^{5} q^{63} - q^{64} + 2 \zeta_{12}^{2} q^{69} + q^{70} + \zeta_{12} q^{72} + \zeta_{12}^{5} q^{75} + \zeta_{12}^{2} q^{80} - \zeta_{12}^{2} q^{81} - \zeta_{12} q^{82} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{83} + \zeta_{12}^{4} q^{84} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{86} + (\zeta_{12}^{5} - \zeta_{12}) q^{87} + \zeta_{12}^{4} q^{89} - \zeta_{12}^{3} q^{90} - 2 \zeta_{12} q^{92} + ( - \zeta_{12}^{2} - 1) q^{94} - q^{96} - \zeta_{12} q^{98} +O(q^{100})$$ q + z^5 * q^2 + z^5 * q^3 - z^4 * q^4 - q^5 - z^4 * q^6 + z * q^7 + z^3 * q^8 - z^4 * q^9 - z^5 * q^10 + z^3 * q^12 - q^14 - z^5 * q^15 - z^2 * q^16 + z^3 * q^18 + z^4 * q^20 - q^21 - z^3 * q^23 - z^2 * q^24 + q^25 + z^3 * q^27 - z^5 * q^28 + (z^2 + 1) * q^29 + z^4 * q^30 + z * q^32 - z * q^35 - z^2 * q^36 - z^3 * q^40 + z^2 * q^41 - z^5 * q^42 + (z^5 + z^3) * q^43 + z^4 * q^45 + 2*z^2 * q^46 + (z^3 + z) * q^47 + z * q^48 + z^2 * q^49 + z^5 * q^50 - z^2 * q^54 + z^4 * q^56 + (z^5 - z) * q^58 - z^3 * q^60 - z^5 * q^63 - q^64 + 2*z^2 * q^69 + q^70 + z * q^72 + z^5 * q^75 + z^2 * q^80 - z^2 * q^81 - z * q^82 + (-z^5 - z^3) * q^83 + z^4 * q^84 + (-z^4 - z^2) * q^86 + (z^5 - z) * q^87 + z^4 * q^89 - z^3 * q^90 - 2*z * q^92 + (-z^2 - 1) * q^94 - q^96 - z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} - 4 q^{5} + 2 q^{6} + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^4 - 4 * q^5 + 2 * q^6 + 2 * q^9 $$4 q + 2 q^{4} - 4 q^{5} + 2 q^{6} + 2 q^{9} - 4 q^{14} - 2 q^{16} - 2 q^{20} - 4 q^{21} - 2 q^{24} + 4 q^{25} + 6 q^{29} - 2 q^{30} - 2 q^{36} + 2 q^{41} - 2 q^{45} + 4 q^{46} + 2 q^{49} - 2 q^{54} - 2 q^{56} - 4 q^{64} + 4 q^{69} + 4 q^{70} + 2 q^{80} - 2 q^{81} - 2 q^{84} - 4 q^{89} - 6 q^{94} - 4 q^{96}+O(q^{100})$$ 4 * q + 2 * q^4 - 4 * q^5 + 2 * q^6 + 2 * q^9 - 4 * q^14 - 2 * q^16 - 2 * q^20 - 4 * q^21 - 2 * q^24 + 4 * q^25 + 6 * q^29 - 2 * q^30 - 2 * q^36 + 2 * q^41 - 2 * q^45 + 4 * q^46 + 2 * q^49 - 2 * q^54 - 2 * q^56 - 4 * q^64 + 4 * q^69 + 4 * q^70 + 2 * q^80 - 2 * q^81 - 2 * q^84 - 4 * q^89 - 6 * q^94 - 4 * q^96

## Character values

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

 $$n$$ $$281$$ $$631$$ $$757$$ $$1081$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$ $$-1$$ $$-1$$ $$-\zeta_{12}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
59.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i −1.00000 0.500000 + 0.866025i 0.866025 0.500000i 1.00000i 0.500000 + 0.866025i 0.866025 + 0.500000i
59.2 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i −1.00000 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i 0.500000 + 0.866025i −0.866025 0.500000i
299.1 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i −1.00000 0.500000 0.866025i 0.866025 + 0.500000i 1.00000i 0.500000 0.866025i 0.866025 0.500000i
299.2 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i −1.00000 0.500000 0.866025i −0.866025 0.500000i 1.00000i 0.500000 0.866025i −0.866025 + 0.500000i
 $$n$$: 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 $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
63.s even 6 1 inner
252.bn odd 6 1 inner
315.u even 6 1 inner
1260.de 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.de.a yes 4
3.b odd 2 1 3780.1.de.b 4
4.b odd 2 1 inner 1260.1.de.a yes 4
5.b even 2 1 inner 1260.1.de.a yes 4
7.d odd 6 1 1260.1.br.a 4
9.c even 3 1 3780.1.br.b 4
9.d odd 6 1 1260.1.br.a 4
12.b even 2 1 3780.1.de.b 4
15.d odd 2 1 3780.1.de.b 4
20.d odd 2 1 CM 1260.1.de.a yes 4
21.g even 6 1 3780.1.br.b 4
28.f even 6 1 1260.1.br.a 4
35.i odd 6 1 1260.1.br.a 4
36.f odd 6 1 3780.1.br.b 4
36.h even 6 1 1260.1.br.a 4
45.h odd 6 1 1260.1.br.a 4
45.j even 6 1 3780.1.br.b 4
60.h even 2 1 3780.1.de.b 4
63.k odd 6 1 3780.1.de.b 4
63.s even 6 1 inner 1260.1.de.a yes 4
84.j odd 6 1 3780.1.br.b 4
105.p even 6 1 3780.1.br.b 4
140.s even 6 1 1260.1.br.a 4
180.n even 6 1 1260.1.br.a 4
180.p odd 6 1 3780.1.br.b 4
252.n even 6 1 3780.1.de.b 4
252.bn odd 6 1 inner 1260.1.de.a yes 4
315.u even 6 1 inner 1260.1.de.a yes 4
315.bn odd 6 1 3780.1.de.b 4
420.be odd 6 1 3780.1.br.b 4
1260.bc even 6 1 3780.1.de.b 4
1260.de odd 6 1 inner 1260.1.de.a yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.1.br.a 4 7.d odd 6 1
1260.1.br.a 4 9.d odd 6 1
1260.1.br.a 4 28.f even 6 1
1260.1.br.a 4 35.i odd 6 1
1260.1.br.a 4 36.h even 6 1
1260.1.br.a 4 45.h odd 6 1
1260.1.br.a 4 140.s even 6 1
1260.1.br.a 4 180.n even 6 1
1260.1.de.a yes 4 1.a even 1 1 trivial
1260.1.de.a yes 4 4.b odd 2 1 inner
1260.1.de.a yes 4 5.b even 2 1 inner
1260.1.de.a yes 4 20.d odd 2 1 CM
1260.1.de.a yes 4 63.s even 6 1 inner
1260.1.de.a yes 4 252.bn odd 6 1 inner
1260.1.de.a yes 4 315.u even 6 1 inner
1260.1.de.a yes 4 1260.de odd 6 1 inner
3780.1.br.b 4 9.c even 3 1
3780.1.br.b 4 21.g even 6 1
3780.1.br.b 4 36.f odd 6 1
3780.1.br.b 4 45.j even 6 1
3780.1.br.b 4 84.j odd 6 1
3780.1.br.b 4 105.p even 6 1
3780.1.br.b 4 180.p odd 6 1
3780.1.br.b 4 420.be odd 6 1
3780.1.de.b 4 3.b odd 2 1
3780.1.de.b 4 12.b even 2 1
3780.1.de.b 4 15.d odd 2 1
3780.1.de.b 4 60.h even 2 1
3780.1.de.b 4 63.k odd 6 1
3780.1.de.b 4 252.n even 6 1
3780.1.de.b 4 315.bn odd 6 1
3780.1.de.b 4 1260.bc even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{23}^{2} + 4$$ acting on $$S_{1}^{\mathrm{new}}(1260, [\chi])$$.

## Hecke characteristic polynomials

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