# Properties

 Label 300.1.g.b Level $300$ Weight $1$ Character orbit 300.g Self dual yes Analytic conductor $0.150$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -3 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,1,Mod(101,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("300.101");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 300.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: $$0.149719503790$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.300.1 Artin image: $S_3$ Artin field: Galois closure of 3.1.300.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{7} + q^{9}+O(q^{10})$$ q + q^3 - q^7 + q^9 $$q + q^{3} - q^{7} + q^{9} - q^{13} - q^{19} - q^{21} + q^{27} - q^{31} + 2 q^{37} - q^{39} - q^{43} - q^{57} - q^{61} - q^{63} - q^{67} + 2 q^{73} + 2 q^{79} + q^{81} + q^{91} - q^{93} - q^{97}+O(q^{100})$$ q + q^3 - q^7 + q^9 - q^13 - q^19 - q^21 + q^27 - q^31 + 2 * q^37 - q^39 - q^43 - q^57 - q^61 - q^63 - q^67 + 2 * q^73 + 2 * q^79 + q^81 + q^91 - q^93 - q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 0
0 1.00000 0 0 0 −1.00000 0 1.00000 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.1.g.b yes 1
3.b odd 2 1 CM 300.1.g.b yes 1
4.b odd 2 1 1200.1.l.a 1
5.b even 2 1 300.1.g.a 1
5.c odd 4 2 300.1.b.a 2
12.b even 2 1 1200.1.l.a 1
15.d odd 2 1 300.1.g.a 1
15.e even 4 2 300.1.b.a 2
20.d odd 2 1 1200.1.l.b 1
20.e even 4 2 1200.1.c.a 2
60.h even 2 1 1200.1.l.b 1
60.l odd 4 2 1200.1.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.1.b.a 2 5.c odd 4 2
300.1.b.a 2 15.e even 4 2
300.1.g.a 1 5.b even 2 1
300.1.g.a 1 15.d odd 2 1
300.1.g.b yes 1 1.a even 1 1 trivial
300.1.g.b yes 1 3.b odd 2 1 CM
1200.1.c.a 2 20.e even 4 2
1200.1.c.a 2 60.l odd 4 2
1200.1.l.a 1 4.b odd 2 1
1200.1.l.a 1 12.b even 2 1
1200.1.l.b 1 20.d odd 2 1
1200.1.l.b 1 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} + 1$$ acting on $$S_{1}^{\mathrm{new}}(300, [\chi])$$.

## Hecke characteristic polynomials

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