# Properties

 Label 740.1.p.a Level $740$ Weight $1$ Character orbit 740.p Analytic conductor $0.369$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ 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(43,740)]

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([2, 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("740.43");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 740.p (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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.101306000.4

## $q$-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.

 $$f(q)$$ $$=$$ $$q - i q^{2} - q^{4} + i q^{5} + i q^{8} + i q^{9} +O(q^{10})$$ q - z * q^2 - q^4 + z * q^5 + z * q^8 + z * q^9 $$q - i q^{2} - q^{4} + i q^{5} + i q^{8} + i q^{9} + q^{10} + q^{16} + 2 q^{17} + q^{18} - i q^{20} - q^{25} + (i - 1) q^{29} - i q^{32} - 2 i q^{34} - i q^{36} - i q^{37} - q^{40} - q^{45} + i q^{49} + i q^{50} + ( - i + 1) q^{53} + (i + 1) q^{58} + ( - i - 1) q^{61} - q^{64} - 2 q^{68} - q^{72} + ( - i - 1) q^{73} - q^{74} + i q^{80} - q^{81} + 2 i q^{85} + ( - i + 1) q^{89} + i q^{90} + q^{98} +O(q^{100})$$ q - z * q^2 - q^4 + z * q^5 + z * q^8 + z * q^9 + q^10 + q^16 + 2 * q^17 + q^18 - z * q^20 - q^25 + (z - 1) * q^29 - z * q^32 - 2*z * q^34 - z * q^36 - z * q^37 - q^40 - q^45 + z * q^49 + z * q^50 + (-z + 1) * q^53 + (z + 1) * q^58 + (-z - 1) * q^61 - q^64 - 2 * q^68 - q^72 + (-z - 1) * q^73 - q^74 + z * q^80 - q^81 + 2*z * q^85 + (-z + 1) * q^89 + z * q^90 + q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4}+O(q^{10})$$ 2 * q - 2 * q^4 $$2 q - 2 q^{4} + 2 q^{10} + 2 q^{16} + 4 q^{17} + 2 q^{18} - 2 q^{25} - 2 q^{29} - 2 q^{40} - 2 q^{45} + 2 q^{53} + 2 q^{58} - 2 q^{61} - 2 q^{64} - 4 q^{68} - 2 q^{72} - 2 q^{73} - 2 q^{74} - 2 q^{81} + 2 q^{89} + 2 q^{98}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^10 + 2 * q^16 + 4 * q^17 + 2 * q^18 - 2 * q^25 - 2 * q^29 - 2 * q^40 - 2 * q^45 + 2 * q^53 + 2 * q^58 - 2 * q^61 - 2 * q^64 - 4 * q^68 - 2 * q^72 - 2 * q^73 - 2 * q^74 - 2 * q^81 + 2 * q^89 + 2 * q^98

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 − 1.00000i 1.00000i
1.00000i 0 −1.00000 1.00000i 0 0 1.00000i 1.00000i 1.00000
327.1 1.00000i 0 −1.00000 1.00000i 0 0 1.00000i 1.00000i 1.00000
 $$n$$: 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.f even 4 1 inner
740.p 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.p.a 2
4.b odd 2 1 CM 740.1.p.a 2
5.b even 2 1 3700.1.p.a 2
5.c odd 4 1 740.1.s.a yes 2
5.c odd 4 1 3700.1.s.a 2
20.d odd 2 1 3700.1.p.a 2
20.e even 4 1 740.1.s.a yes 2
20.e even 4 1 3700.1.s.a 2
37.d odd 4 1 740.1.s.a yes 2
148.g even 4 1 740.1.s.a yes 2
185.f even 4 1 inner 740.1.p.a 2
185.j odd 4 1 3700.1.s.a 2
185.k even 4 1 3700.1.p.a 2
740.k even 4 1 3700.1.s.a 2
740.p odd 4 1 inner 740.1.p.a 2
740.s odd 4 1 3700.1.p.a 2

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(740, [\chi])$$.

## Hecke characteristic polynomials

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