# Properties

 Label 740.1.t.b Level $740$ Weight $1$ Character orbit 740.t Analytic conductor $0.369$ Analytic rank $0$ Dimension $2$ Projective image $S_{4}$ CM/RM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [740,1,Mod(549,740)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(740, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 1]))

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

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

chi := DirichletCharacter("740.549");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 740.t (of order $$4$$, 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: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$S_{4}$$ Projective field: Galois closure of 4.0.5065300.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{3} + i q^{5} + i q^{7}+O(q^{10})$$ q + q^3 + z * q^5 + z * q^7 $$q + q^{3} + i q^{5} + i q^{7} - i q^{11} + i q^{15} + i q^{21} - q^{25} - q^{27} + ( - i + 1) q^{31} - i q^{33} - q^{35} + i q^{37} - i q^{41} + ( - i + 1) q^{43} - i q^{47} - i q^{53} + q^{55} + (i - 1) q^{61} - q^{71} - q^{73} - q^{75} + q^{77} + (i + 1) q^{79} - q^{81} - i q^{83} + (i - 1) q^{89} + ( - i + 1) q^{93} + (i - 1) q^{97} +O(q^{100})$$ q + q^3 + z * q^5 + z * q^7 - z * q^11 + z * q^15 + z * q^21 - q^25 - q^27 + (-z + 1) * q^31 - z * q^33 - q^35 + z * q^37 - z * q^41 + (-z + 1) * q^43 - z * q^47 - z * q^53 + q^55 + (z - 1) * q^61 - q^71 - q^73 - q^75 + q^77 + (z + 1) * q^79 - q^81 - z * q^83 + (z - 1) * q^89 + (-z + 1) * q^93 + (z - 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3}+O(q^{10})$$ 2 * q + 2 * q^3 $$2 q + 2 q^{3} - 2 q^{25} - 2 q^{27} + 2 q^{31} - 2 q^{35} + 2 q^{43} + 2 q^{55} - 2 q^{61} - 2 q^{71} - 2 q^{73} - 2 q^{75} + 2 q^{77} + 2 q^{79} - 2 q^{81} - 2 q^{89} + 2 q^{93} - 2 q^{97}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^25 - 2 * q^27 + 2 * q^31 - 2 * q^35 + 2 * q^43 + 2 * q^55 - 2 * q^61 - 2 * q^71 - 2 * q^73 - 2 * q^75 + 2 * q^77 + 2 * q^79 - 2 * q^81 - 2 * q^89 + 2 * q^93 - 2 * q^97

## Character values

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

 $$n$$ $$261$$ $$297$$ $$371$$ $$\chi(n)$$ $$-i$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
549.1
 − 1.00000i 1.00000i
0 1.00000 0 1.00000i 0 1.00000i 0 0 0
709.1 0 1.00000 0 1.00000i 0 1.00000i 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
185.j odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.1.t.b yes 2
4.b odd 2 1 2960.1.cj.a 2
5.b even 2 1 740.1.t.a 2
5.c odd 4 1 3700.1.j.a 2
5.c odd 4 1 3700.1.j.b 2
20.d odd 2 1 2960.1.cj.b 2
37.d odd 4 1 740.1.t.a 2
148.g even 4 1 2960.1.cj.b 2
185.f even 4 1 3700.1.j.a 2
185.j odd 4 1 inner 740.1.t.b yes 2
185.k even 4 1 3700.1.j.b 2
740.k even 4 1 2960.1.cj.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.1.t.a 2 5.b even 2 1
740.1.t.a 2 37.d odd 4 1
740.1.t.b yes 2 1.a even 1 1 trivial
740.1.t.b yes 2 185.j odd 4 1 inner
2960.1.cj.a 2 4.b odd 2 1
2960.1.cj.a 2 740.k even 4 1
2960.1.cj.b 2 20.d odd 2 1
2960.1.cj.b 2 148.g even 4 1
3700.1.j.a 2 5.c odd 4 1
3700.1.j.a 2 185.f even 4 1
3700.1.j.b 2 5.c odd 4 1
3700.1.j.b 2 185.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 1$$ acting on $$S_{1}^{\mathrm{new}}(740, [\chi])$$.

## Hecke characteristic polynomials

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