# Properties

 Label 2240.1.bt.a Level $2240$ Weight $1$ Character orbit 2240.bt Analytic conductor $1.118$ 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] = [2240,1,Mod(319,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, 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("2240.319");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2240.bt (of order $$6$$, degree $$2$$, not 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: $$1.11790562830$$ 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: 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6}^{2} q^{3} + \zeta_{6} q^{5} + \zeta_{6}^{2} q^{7}+O(q^{10})$$ q + z^2 * q^3 + z * q^5 + z^2 * q^7 $$q + \zeta_{6}^{2} q^{3} + \zeta_{6} q^{5} + \zeta_{6}^{2} q^{7} - q^{15} - \zeta_{6} q^{21} + \zeta_{6} q^{23} + \zeta_{6}^{2} q^{25} - q^{27} + q^{29} - q^{35} - q^{41} + q^{43} - 2 \zeta_{6} q^{47} - \zeta_{6} q^{49} - \zeta_{6} q^{61} + \zeta_{6}^{2} q^{67} - q^{69} - \zeta_{6} q^{75} - \zeta_{6}^{2} q^{81} + q^{83} + \zeta_{6}^{2} q^{87} + \zeta_{6} q^{89} +O(q^{100})$$ q + z^2 * q^3 + z * q^5 + z^2 * q^7 - q^15 - z * q^21 + z * q^23 + z^2 * q^25 - q^27 + q^29 - q^35 - q^41 + q^43 - 2*z * q^47 - z * q^49 - z * q^61 + z^2 * q^67 - q^69 - z * q^75 - z^2 * q^81 + q^83 + z^2 * q^87 + z * q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + q^{5} - q^{7}+O(q^{10})$$ 2 * q - q^3 + q^5 - q^7 $$2 q - q^{3} + q^{5} - q^{7} - 2 q^{15} - q^{21} + q^{23} - q^{25} - 2 q^{27} + 2 q^{29} - 2 q^{35} - 2 q^{41} + 2 q^{43} - 2 q^{47} - q^{49} - q^{61} - q^{67} - 2 q^{69} - q^{75} + q^{81} + 2 q^{83} - q^{87} + q^{89}+O(q^{100})$$ 2 * q - q^3 + q^5 - q^7 - 2 * q^15 - q^21 + q^23 - q^25 - 2 * q^27 + 2 * q^29 - 2 * q^35 - 2 * q^41 + 2 * q^43 - 2 * q^47 - q^49 - q^61 - q^67 - 2 * q^69 - q^75 + q^81 + 2 * q^83 - q^87 + q^89

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$$.

 $$n$$ $$897$$ $$1471$$ $$1541$$ $$1921$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
319.1
 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 0
639.1 0 −0.500000 0.866025i 0 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 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
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 2240.1.bt.a 2
4.b odd 2 1 2240.1.bt.b 2
5.b even 2 1 2240.1.bt.b 2
7.c even 3 1 inner 2240.1.bt.a 2
8.b even 2 1 140.1.p.a 2
8.d odd 2 1 140.1.p.b yes 2
20.d odd 2 1 CM 2240.1.bt.a 2
24.f even 2 1 1260.1.ci.a 2
24.h odd 2 1 1260.1.ci.b 2
28.g odd 6 1 2240.1.bt.b 2
35.j even 6 1 2240.1.bt.b 2
40.e odd 2 1 140.1.p.a 2
40.f even 2 1 140.1.p.b yes 2
40.i odd 4 2 700.1.u.a 4
40.k even 4 2 700.1.u.a 4
56.e even 2 1 980.1.p.b 2
56.h odd 2 1 980.1.p.a 2
56.j odd 6 1 980.1.f.d 1
56.j odd 6 1 980.1.p.a 2
56.k odd 6 1 140.1.p.b yes 2
56.k odd 6 1 980.1.f.b 1
56.m even 6 1 980.1.f.a 1
56.m even 6 1 980.1.p.b 2
56.p even 6 1 140.1.p.a 2
56.p even 6 1 980.1.f.c 1
120.i odd 2 1 1260.1.ci.a 2
120.m even 2 1 1260.1.ci.b 2
140.p odd 6 1 inner 2240.1.bt.a 2
168.s odd 6 1 1260.1.ci.b 2
168.v even 6 1 1260.1.ci.a 2
280.c odd 2 1 980.1.p.b 2
280.n even 2 1 980.1.p.a 2
280.ba even 6 1 980.1.f.d 1
280.ba even 6 1 980.1.p.a 2
280.bf even 6 1 140.1.p.b yes 2
280.bf even 6 1 980.1.f.b 1
280.bi odd 6 1 140.1.p.a 2
280.bi odd 6 1 980.1.f.c 1
280.bk odd 6 1 980.1.f.a 1
280.bk odd 6 1 980.1.p.b 2
280.br even 12 2 700.1.u.a 4
280.bt odd 12 2 700.1.u.a 4
840.cg odd 6 1 1260.1.ci.a 2
840.cv even 6 1 1260.1.ci.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.p.a 2 8.b even 2 1
140.1.p.a 2 40.e odd 2 1
140.1.p.a 2 56.p even 6 1
140.1.p.a 2 280.bi odd 6 1
140.1.p.b yes 2 8.d odd 2 1
140.1.p.b yes 2 40.f even 2 1
140.1.p.b yes 2 56.k odd 6 1
140.1.p.b yes 2 280.bf even 6 1
700.1.u.a 4 40.i odd 4 2
700.1.u.a 4 40.k even 4 2
700.1.u.a 4 280.br even 12 2
700.1.u.a 4 280.bt odd 12 2
980.1.f.a 1 56.m even 6 1
980.1.f.a 1 280.bk odd 6 1
980.1.f.b 1 56.k odd 6 1
980.1.f.b 1 280.bf even 6 1
980.1.f.c 1 56.p even 6 1
980.1.f.c 1 280.bi odd 6 1
980.1.f.d 1 56.j odd 6 1
980.1.f.d 1 280.ba even 6 1
980.1.p.a 2 56.h odd 2 1
980.1.p.a 2 56.j odd 6 1
980.1.p.a 2 280.n even 2 1
980.1.p.a 2 280.ba even 6 1
980.1.p.b 2 56.e even 2 1
980.1.p.b 2 56.m even 6 1
980.1.p.b 2 280.c odd 2 1
980.1.p.b 2 280.bk odd 6 1
1260.1.ci.a 2 24.f even 2 1
1260.1.ci.a 2 120.i odd 2 1
1260.1.ci.a 2 168.v even 6 1
1260.1.ci.a 2 840.cg odd 6 1
1260.1.ci.b 2 24.h odd 2 1
1260.1.ci.b 2 120.m even 2 1
1260.1.ci.b 2 168.s odd 6 1
1260.1.ci.b 2 840.cv even 6 1
2240.1.bt.a 2 1.a even 1 1 trivial
2240.1.bt.a 2 7.c even 3 1 inner
2240.1.bt.a 2 20.d odd 2 1 CM
2240.1.bt.a 2 140.p odd 6 1 inner
2240.1.bt.b 2 4.b odd 2 1
2240.1.bt.b 2 5.b even 2 1
2240.1.bt.b 2 28.g odd 6 1
2240.1.bt.b 2 35.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + T_{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(2240, [\chi])$$.

## Hecke characteristic polynomials

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