# Properties

 Label 740.1.g.a Level $740$ Weight $1$ Character orbit 740.g Self dual yes Analytic conductor $0.369$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -4, -740, 185 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [740,1,Mod(739,740)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(740, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("740.739");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 740.g (of order $$2$$, degree $$1$$, 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: yes Analytic conductor: $$0.369308109348$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(i, \sqrt{185})$$ Artin image: $D_4$ Artin field: Galois closure of 4.0.2960.2

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
739.1
 0
−1.00000 0 1.00000 −1.00000 0 0 −1.00000 −1.00000 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.d even 2 1 RM by $$\Q(\sqrt{185})$$
740.g odd 2 1 CM by $$\Q(\sqrt{-185})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.1.g.a 1
4.b odd 2 1 CM 740.1.g.a 1
5.b even 2 1 740.1.g.b yes 1
5.c odd 4 2 3700.1.b.e 2
20.d odd 2 1 740.1.g.b yes 1
20.e even 4 2 3700.1.b.e 2
37.b even 2 1 740.1.g.b yes 1
148.b odd 2 1 740.1.g.b yes 1
185.d even 2 1 RM 740.1.g.a 1
185.h odd 4 2 3700.1.b.e 2
740.g odd 2 1 CM 740.1.g.a 1
740.m even 4 2 3700.1.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.1.g.a 1 1.a even 1 1 trivial
740.1.g.a 1 4.b odd 2 1 CM
740.1.g.a 1 185.d even 2 1 RM
740.1.g.a 1 740.g odd 2 1 CM
740.1.g.b yes 1 5.b even 2 1
740.1.g.b yes 1 20.d odd 2 1
740.1.g.b yes 1 37.b even 2 1
740.1.g.b yes 1 148.b odd 2 1
3700.1.b.e 2 5.c odd 4 2
3700.1.b.e 2 20.e even 4 2
3700.1.b.e 2 185.h odd 4 2
3700.1.b.e 2 740.m even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(740, [\chi])$$:

 $$T_{3}$$ T3 $$T_{13} - 2$$ T13 - 2 $$T_{97} + 2$$ T97 + 2

## Hecke characteristic polynomials

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