# Properties

 Label 740.1.v.b Level $740$ Weight $1$ Character orbit 740.v Analytic conductor $0.369$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [740,1,Mod(159,740)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(740, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 5]))

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

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

chi := DirichletCharacter("740.159");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 740.v (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: $$0.369308109348$$ 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: $$D_{6}$$ Projective field: Galois closure of 6.2.138687914000.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - q^{5} - q^{8} + \zeta_{6} q^{9}+O(q^{10})$$ q + z * q^2 + z^2 * q^4 - q^5 - q^8 + z * q^9 $$q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - q^{5} - q^{8} + \zeta_{6} q^{9} - \zeta_{6} q^{10} + 2 \zeta_{6}^{2} q^{13} - \zeta_{6} q^{16} + \zeta_{6} q^{17} + \zeta_{6}^{2} q^{18} - \zeta_{6}^{2} q^{20} + q^{25} - 2 q^{26} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{29} - \zeta_{6}^{2} q^{32} + \zeta_{6}^{2} q^{34} - q^{36} - \zeta_{6}^{2} q^{37} + q^{40} - \zeta_{6}^{2} q^{41} - \zeta_{6} q^{45} + \zeta_{6} q^{49} + \zeta_{6} q^{50} - 2 \zeta_{6} q^{52} + ( - \zeta_{6}^{2} + 1) q^{58} + (\zeta_{6} + 1) q^{61} + q^{64} - 2 \zeta_{6}^{2} q^{65} - q^{68} - \zeta_{6} q^{72} + q^{74} + \zeta_{6} q^{80} + \zeta_{6}^{2} q^{81} + q^{82} - \zeta_{6} q^{85} + (\zeta_{6}^{2} - 1) q^{89} - \zeta_{6}^{2} q^{90} + q^{97} + \zeta_{6}^{2} q^{98} +O(q^{100})$$ q + z * q^2 + z^2 * q^4 - q^5 - q^8 + z * q^9 - z * q^10 + 2*z^2 * q^13 - z * q^16 + z * q^17 + z^2 * q^18 - z^2 * q^20 + q^25 - 2 * q^26 + (-z^2 - z) * q^29 - z^2 * q^32 + z^2 * q^34 - q^36 - z^2 * q^37 + q^40 - z^2 * q^41 - z * q^45 + z * q^49 + z * q^50 - 2*z * q^52 + (-z^2 + 1) * q^58 + (z + 1) * q^61 + q^64 - 2*z^2 * q^65 - q^68 - z * q^72 + q^74 + z * q^80 + z^2 * q^81 + q^82 - z * q^85 + (z^2 - 1) * q^89 - z^2 * q^90 + q^97 + z^2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{8} + q^{9}+O(q^{10})$$ 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^8 + q^9 $$2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{8} + q^{9} - q^{10} - 2 q^{13} - q^{16} + q^{17} - q^{18} + q^{20} + 2 q^{25} - 4 q^{26} + q^{32} - q^{34} - 2 q^{36} + q^{37} + 2 q^{40} + q^{41} - q^{45} + q^{49} + q^{50} - 2 q^{52} + 3 q^{58} + 3 q^{61} + 2 q^{64} + 2 q^{65} - 2 q^{68} - q^{72} + 2 q^{74} + q^{80} - q^{81} + 2 q^{82} - q^{85} - 3 q^{89} + q^{90} + 2 q^{97} - q^{98}+O(q^{100})$$ 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^8 + q^9 - q^10 - 2 * q^13 - q^16 + q^17 - q^18 + q^20 + 2 * q^25 - 4 * q^26 + q^32 - q^34 - 2 * q^36 + q^37 + 2 * q^40 + q^41 - q^45 + q^49 + q^50 - 2 * q^52 + 3 * q^58 + 3 * q^61 + 2 * q^64 + 2 * q^65 - 2 * q^68 - q^72 + 2 * q^74 + q^80 - q^81 + 2 * q^82 - q^85 - 3 * q^89 + q^90 + 2 * q^97 - q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/740\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$297$$ $$371$$ $$\chi(n)$$ $$-\zeta_{6}^{2}$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
159.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.00000 0 0 −1.00000 0.500000 + 0.866025i −0.500000 0.866025i
619.1 0.500000 0.866025i 0 −0.500000 0.866025i −1.00000 0 0 −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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
185.l even 6 1 inner
740.v odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.1.v.b yes 2
4.b odd 2 1 CM 740.1.v.b yes 2
5.b even 2 1 740.1.v.a 2
5.c odd 4 2 3700.1.ba.a 4
20.d odd 2 1 740.1.v.a 2
20.e even 4 2 3700.1.ba.a 4
37.e even 6 1 740.1.v.a 2
148.j odd 6 1 740.1.v.a 2
185.l even 6 1 inner 740.1.v.b yes 2
185.r odd 12 2 3700.1.ba.a 4
740.v odd 6 1 inner 740.1.v.b yes 2
740.bh even 12 2 3700.1.ba.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.1.v.a 2 5.b even 2 1
740.1.v.a 2 20.d odd 2 1
740.1.v.a 2 37.e even 6 1
740.1.v.a 2 148.j odd 6 1
740.1.v.b yes 2 1.a even 1 1 trivial
740.1.v.b yes 2 4.b odd 2 1 CM
740.1.v.b yes 2 185.l even 6 1 inner
740.1.v.b yes 2 740.v odd 6 1 inner
3700.1.ba.a 4 5.c odd 4 2
3700.1.ba.a 4 20.e even 4 2
3700.1.ba.a 4 185.r odd 12 2
3700.1.ba.a 4 740.bh even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{2} + 2T_{13} + 4$$ acting on $$S_{1}^{\mathrm{new}}(740, [\chi])$$.

## Hecke characteristic polynomials

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