# Properties

 Label 1260.1.bj.d Level $1260$ Weight $1$ Character orbit 1260.bj 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(79,1260)]

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, 4, 3, 2]))

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

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

chi := DirichletCharacter("1260.79");

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.bj (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: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.79380.1 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + \zeta_{6} q^{3} + \zeta_{6}^{2} q^{4} + q^{5} + \zeta_{6}^{2} q^{6} - \zeta_{6}^{2} q^{7} - q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10})$$ q + z * q^2 + z * q^3 + z^2 * q^4 + q^5 + z^2 * q^6 - z^2 * q^7 - q^8 + z^2 * q^9 $$q + \zeta_{6} q^{2} + \zeta_{6} q^{3} + \zeta_{6}^{2} q^{4} + q^{5} + \zeta_{6}^{2} q^{6} - \zeta_{6}^{2} q^{7} - q^{8} + \zeta_{6}^{2} q^{9} + \zeta_{6} q^{10} - q^{12} + q^{14} + \zeta_{6} q^{15} - \zeta_{6} q^{16} - q^{18} + \zeta_{6}^{2} q^{20} + q^{21} - q^{23} - \zeta_{6} q^{24} + q^{25} - q^{27} + \zeta_{6} q^{28} - \zeta_{6}^{2} q^{29} + \zeta_{6}^{2} q^{30} - \zeta_{6}^{2} q^{32} - \zeta_{6}^{2} q^{35} - \zeta_{6} q^{36} - q^{40} + \zeta_{6} q^{41} + \zeta_{6} q^{42} + \zeta_{6}^{2} q^{43} + \zeta_{6}^{2} q^{45} - 2 \zeta_{6} q^{46} - \zeta_{6} q^{47} - \zeta_{6}^{2} q^{48} - \zeta_{6} q^{49} + \zeta_{6} q^{50} - \zeta_{6} q^{54} + \zeta_{6}^{2} q^{56} + q^{58} - q^{60} - \zeta_{6} q^{61} + \zeta_{6} q^{63} + q^{64} - \zeta_{6}^{2} q^{67} - 2 \zeta_{6} q^{69} + q^{70} - \zeta_{6}^{2} q^{72} + \zeta_{6} q^{75} - \zeta_{6} q^{80} - \zeta_{6} q^{81} + \zeta_{6}^{2} q^{82} + \zeta_{6}^{2} q^{83} + \zeta_{6}^{2} q^{84} - q^{86} + q^{87} + \zeta_{6}^{2} q^{89} - q^{90} - 2 \zeta_{6}^{2} q^{92} - \zeta_{6}^{2} q^{94} + q^{96} - \zeta_{6}^{2} q^{98} +O(q^{100})$$ q + z * q^2 + z * q^3 + z^2 * q^4 + q^5 + z^2 * q^6 - z^2 * q^7 - q^8 + z^2 * q^9 + z * q^10 - q^12 + q^14 + z * q^15 - z * q^16 - q^18 + z^2 * q^20 + q^21 - q^23 - z * q^24 + q^25 - q^27 + z * q^28 - z^2 * q^29 + z^2 * q^30 - z^2 * q^32 - z^2 * q^35 - z * q^36 - q^40 + z * q^41 + z * q^42 + z^2 * q^43 + z^2 * q^45 - 2*z * q^46 - z * q^47 - z^2 * q^48 - z * q^49 + z * q^50 - z * q^54 + z^2 * q^56 + q^58 - q^60 - z * q^61 + z * q^63 + q^64 - z^2 * q^67 - 2*z * q^69 + q^70 - z^2 * q^72 + z * q^75 - z * q^80 - z * q^81 + z^2 * q^82 + z^2 * q^83 + z^2 * q^84 - q^86 + q^87 + z^2 * q^89 - q^90 - 2*z^2 * q^92 - z^2 * q^94 + q^96 - z^2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{3} - q^{4} + 2 q^{5} - q^{6} + q^{7} - 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q + q^2 + q^3 - q^4 + 2 * q^5 - q^6 + q^7 - 2 * q^8 - q^9 $$2 q + q^{2} + q^{3} - q^{4} + 2 q^{5} - q^{6} + q^{7} - 2 q^{8} - q^{9} + q^{10} - 2 q^{12} + 2 q^{14} + q^{15} - q^{16} - 2 q^{18} - q^{20} + 2 q^{21} - 4 q^{23} - q^{24} + 2 q^{25} - 2 q^{27} + q^{28} + q^{29} - q^{30} + q^{32} + q^{35} - q^{36} - 2 q^{40} + q^{41} + q^{42} - q^{43} - q^{45} - 2 q^{46} - q^{47} + q^{48} - q^{49} + q^{50} - q^{54} - q^{56} + 2 q^{58} - 2 q^{60} - 2 q^{61} + q^{63} + 2 q^{64} + 2 q^{67} - 2 q^{69} + 2 q^{70} + q^{72} + q^{75} - q^{80} - q^{81} - q^{82} - q^{83} - q^{84} - 2 q^{86} + 2 q^{87} - 2 q^{89} - 2 q^{90} + 2 q^{92} + q^{94} + 2 q^{96} + q^{98}+O(q^{100})$$ 2 * q + q^2 + q^3 - q^4 + 2 * q^5 - q^6 + q^7 - 2 * q^8 - q^9 + q^10 - 2 * q^12 + 2 * q^14 + q^15 - q^16 - 2 * q^18 - q^20 + 2 * q^21 - 4 * q^23 - q^24 + 2 * q^25 - 2 * q^27 + q^28 + q^29 - q^30 + q^32 + q^35 - q^36 - 2 * q^40 + q^41 + q^42 - q^43 - q^45 - 2 * q^46 - q^47 + q^48 - q^49 + q^50 - q^54 - q^56 + 2 * q^58 - 2 * q^60 - 2 * q^61 + q^63 + 2 * q^64 + 2 * q^67 - 2 * q^69 + 2 * q^70 + q^72 + q^75 - q^80 - q^81 - q^82 - q^83 - q^84 - 2 * q^86 + 2 * q^87 - 2 * q^89 - 2 * q^90 + 2 * q^92 + q^94 + 2 * q^96 + q^98

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i 0.500000 0.866025i −1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
319.1 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i 1.00000 −0.500000 0.866025i 0.500000 + 0.866025i −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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
63.g even 3 1 inner
1260.bj 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.bj.d yes 2
3.b odd 2 1 3780.1.bj.a 2
4.b odd 2 1 1260.1.bj.a 2
5.b even 2 1 1260.1.bj.a 2
7.c even 3 1 1260.1.cw.b yes 2
9.c even 3 1 1260.1.cw.b yes 2
9.d odd 6 1 3780.1.cw.d 2
12.b even 2 1 3780.1.bj.c 2
15.d odd 2 1 3780.1.bj.c 2
20.d odd 2 1 CM 1260.1.bj.d yes 2
21.h odd 6 1 3780.1.cw.d 2
28.g odd 6 1 1260.1.cw.c yes 2
35.j even 6 1 1260.1.cw.c yes 2
36.f odd 6 1 1260.1.cw.c yes 2
36.h even 6 1 3780.1.cw.a 2
45.h odd 6 1 3780.1.cw.a 2
45.j even 6 1 1260.1.cw.c yes 2
60.h even 2 1 3780.1.bj.a 2
63.g even 3 1 inner 1260.1.bj.d yes 2
63.n odd 6 1 3780.1.bj.a 2
84.n even 6 1 3780.1.cw.a 2
105.o odd 6 1 3780.1.cw.a 2
140.p odd 6 1 1260.1.cw.b yes 2
180.n even 6 1 3780.1.cw.d 2
180.p odd 6 1 1260.1.cw.b yes 2
252.o even 6 1 3780.1.bj.c 2
252.bl odd 6 1 1260.1.bj.a 2
315.v odd 6 1 3780.1.bj.c 2
315.bo even 6 1 1260.1.bj.a 2
420.ba even 6 1 3780.1.cw.d 2
1260.bj odd 6 1 inner 1260.1.bj.d yes 2
1260.dh even 6 1 3780.1.bj.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.1.bj.a 2 4.b odd 2 1
1260.1.bj.a 2 5.b even 2 1
1260.1.bj.a 2 252.bl odd 6 1
1260.1.bj.a 2 315.bo even 6 1
1260.1.bj.d yes 2 1.a even 1 1 trivial
1260.1.bj.d yes 2 20.d odd 2 1 CM
1260.1.bj.d yes 2 63.g even 3 1 inner
1260.1.bj.d yes 2 1260.bj odd 6 1 inner
1260.1.cw.b yes 2 7.c even 3 1
1260.1.cw.b yes 2 9.c even 3 1
1260.1.cw.b yes 2 140.p odd 6 1
1260.1.cw.b yes 2 180.p odd 6 1
1260.1.cw.c yes 2 28.g odd 6 1
1260.1.cw.c yes 2 35.j even 6 1
1260.1.cw.c yes 2 36.f odd 6 1
1260.1.cw.c yes 2 45.j even 6 1
3780.1.bj.a 2 3.b odd 2 1
3780.1.bj.a 2 60.h even 2 1
3780.1.bj.a 2 63.n odd 6 1
3780.1.bj.a 2 1260.dh even 6 1
3780.1.bj.c 2 12.b even 2 1
3780.1.bj.c 2 15.d odd 2 1
3780.1.bj.c 2 252.o even 6 1
3780.1.bj.c 2 315.v odd 6 1
3780.1.cw.a 2 36.h even 6 1
3780.1.cw.a 2 45.h odd 6 1
3780.1.cw.a 2 84.n even 6 1
3780.1.cw.a 2 105.o odd 6 1
3780.1.cw.d 2 9.d odd 6 1
3780.1.cw.d 2 21.h odd 6 1
3780.1.cw.d 2 180.n even 6 1
3780.1.cw.d 2 420.ba even 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

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