# Properties

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

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1260,1,Mod(499,1260)]

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, 2, 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("1260.499");

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.cw (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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.79380.1 Artin image: $C_3\times C_6\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{36} - \cdots)$$

## $q$-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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
499.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 0.500000 + 0.866025i 1.00000 −0.500000 0.866025i −0.500000 0.866025i 0.500000 + 0.866025i −1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
1159.1 −1.00000 0.500000 0.866025i 1.00000 −0.500000 + 0.866025i −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.h even 3 1 inner
1260.cw 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.cw.b yes 2
3.b odd 2 1 3780.1.cw.d 2
4.b odd 2 1 1260.1.cw.c yes 2
5.b even 2 1 1260.1.cw.c yes 2
7.c even 3 1 1260.1.bj.d yes 2
9.c even 3 1 1260.1.bj.d yes 2
9.d odd 6 1 3780.1.bj.a 2
12.b even 2 1 3780.1.cw.a 2
15.d odd 2 1 3780.1.cw.a 2
20.d odd 2 1 CM 1260.1.cw.b yes 2
21.h odd 6 1 3780.1.bj.a 2
28.g odd 6 1 1260.1.bj.a 2
35.j even 6 1 1260.1.bj.a 2
36.f odd 6 1 1260.1.bj.a 2
36.h even 6 1 3780.1.bj.c 2
45.h odd 6 1 3780.1.bj.c 2
45.j even 6 1 1260.1.bj.a 2
60.h even 2 1 3780.1.cw.d 2
63.h even 3 1 inner 1260.1.cw.b yes 2
63.j odd 6 1 3780.1.cw.d 2
84.n even 6 1 3780.1.bj.c 2
105.o odd 6 1 3780.1.bj.c 2
140.p odd 6 1 1260.1.bj.d yes 2
180.n even 6 1 3780.1.bj.a 2
180.p odd 6 1 1260.1.bj.d yes 2
252.u odd 6 1 1260.1.cw.c yes 2
252.bb even 6 1 3780.1.cw.a 2
315.r even 6 1 1260.1.cw.c yes 2
315.br odd 6 1 3780.1.cw.a 2
420.ba even 6 1 3780.1.bj.a 2
1260.bx even 6 1 3780.1.cw.d 2
1260.cw odd 6 1 inner 1260.1.cw.b yes 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

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