# Properties

 Label 1260.1.bt.d Level $1260$ Weight $1$ Character orbit 1260.bt Analytic conductor $0.629$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -35 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1260,1,Mod(349,1260)]

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([0, 4, 3, 3]))

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

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

chi := DirichletCharacter("1260.349");

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.bt (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 3.1.11340.2 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6}^{2} q^{3} + \zeta_{6} q^{5} - \zeta_{6}^{2} q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q - z^2 * q^3 + z * q^5 - z^2 * q^7 - z * q^9 $$q - \zeta_{6}^{2} q^{3} + \zeta_{6} q^{5} - \zeta_{6}^{2} q^{7} - \zeta_{6} q^{9} - \zeta_{6}^{2} q^{11} + \zeta_{6} q^{13} + q^{15} - q^{17} - \zeta_{6} q^{21} + \zeta_{6}^{2} q^{25} - q^{27} - \zeta_{6}^{2} q^{29} - \zeta_{6} q^{33} + q^{35} + 2 q^{39} - \zeta_{6}^{2} q^{45} + \zeta_{6}^{2} q^{47} - \zeta_{6} q^{49} + 2 \zeta_{6}^{2} q^{51} + q^{55} - q^{63} + 2 \zeta_{6}^{2} q^{65} - q^{71} + q^{73} + \zeta_{6} q^{75} - \zeta_{6} q^{77} - \zeta_{6}^{2} q^{79} + \zeta_{6}^{2} q^{81} + \zeta_{6}^{2} q^{83} - 2 \zeta_{6} q^{85} - \zeta_{6} q^{87} + 2 q^{91} + \zeta_{6}^{2} q^{97} - q^{99} +O(q^{100})$$ q - z^2 * q^3 + z * q^5 - z^2 * q^7 - z * q^9 - z^2 * q^11 + z * q^13 + q^15 - q^17 - z * q^21 + z^2 * q^25 - q^27 - z^2 * q^29 - z * q^33 + q^35 + 2 * q^39 - z^2 * q^45 + z^2 * q^47 - z * q^49 + 2*z^2 * q^51 + q^55 - q^63 + 2*z^2 * q^65 - q^71 + q^73 + z * q^75 - z * q^77 - z^2 * q^79 + z^2 * q^81 + z^2 * q^83 - 2*z * q^85 - z * q^87 + 2 * q^91 + z^2 * q^97 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + q^{5} + q^{7} - q^{9}+O(q^{10})$$ 2 * q + q^3 + q^5 + q^7 - q^9 $$2 q + q^{3} + q^{5} + q^{7} - q^{9} + q^{11} + 2 q^{13} + 2 q^{15} - 4 q^{17} - q^{21} - q^{25} - 2 q^{27} + q^{29} - q^{33} + 2 q^{35} + 4 q^{39} + q^{45} - q^{47} - q^{49} - 2 q^{51} + 2 q^{55} - 2 q^{63} - 2 q^{65} - 2 q^{71} + 2 q^{73} + q^{75} - q^{77} + q^{79} - q^{81} - q^{83} - 2 q^{85} - q^{87} + 4 q^{91} - q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q + q^3 + q^5 + q^7 - q^9 + q^11 + 2 * q^13 + 2 * q^15 - 4 * q^17 - q^21 - q^25 - 2 * q^27 + q^29 - q^33 + 2 * q^35 + 4 * q^39 + q^45 - q^47 - q^49 - 2 * q^51 + 2 * q^55 - 2 * q^63 - 2 * q^65 - 2 * q^71 + 2 * q^73 + q^75 - q^77 + q^79 - q^81 - q^83 - 2 * q^85 - q^87 + 4 * q^91 - q^97 - 2 * q^99

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
349.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i 0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 −0.500000 0.866025i 0
769.1 0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0
 $$n$$: 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
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$
9.c even 3 1 inner
315.bg 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.bt.d yes 2
3.b odd 2 1 3780.1.bt.a 2
5.b even 2 1 1260.1.bt.b 2
7.b odd 2 1 1260.1.bt.b 2
9.c even 3 1 inner 1260.1.bt.d yes 2
9.d odd 6 1 3780.1.bt.a 2
15.d odd 2 1 3780.1.bt.c 2
21.c even 2 1 3780.1.bt.c 2
35.c odd 2 1 CM 1260.1.bt.d yes 2
45.h odd 6 1 3780.1.bt.c 2
45.j even 6 1 1260.1.bt.b 2
63.l odd 6 1 1260.1.bt.b 2
63.o even 6 1 3780.1.bt.c 2
105.g even 2 1 3780.1.bt.a 2
315.z even 6 1 3780.1.bt.a 2
315.bg odd 6 1 inner 1260.1.bt.d yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.1.bt.b 2 5.b even 2 1
1260.1.bt.b 2 7.b odd 2 1
1260.1.bt.b 2 45.j even 6 1
1260.1.bt.b 2 63.l odd 6 1
1260.1.bt.d yes 2 1.a even 1 1 trivial
1260.1.bt.d yes 2 9.c even 3 1 inner
1260.1.bt.d yes 2 35.c odd 2 1 CM
1260.1.bt.d yes 2 315.bg odd 6 1 inner
3780.1.bt.a 2 3.b odd 2 1
3780.1.bt.a 2 9.d odd 6 1
3780.1.bt.a 2 105.g even 2 1
3780.1.bt.a 2 315.z even 6 1
3780.1.bt.c 2 15.d odd 2 1
3780.1.bt.c 2 21.c even 2 1
3780.1.bt.c 2 45.h odd 6 1
3780.1.bt.c 2 63.o even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1260, [\chi])$$:

 $$T_{11}^{2} - T_{11} + 1$$ T11^2 - T11 + 1 $$T_{13}^{2} - 2T_{13} + 4$$ T13^2 - 2*T13 + 4

## Hecke characteristic polynomials

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