# Properties

 Label 300.3.g.a Level $300$ Weight $3$ Character orbit 300.g Self dual yes Analytic conductor $8.174$ Analytic rank $0$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,3,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, 3, names="a")

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

chi := DirichletCharacter("300.101");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ 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: $$8.17440793081$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} - 13 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 - 13 * q^7 + 9 * q^9 $$q - 3 q^{3} - 13 q^{7} + 9 q^{9} + 23 q^{13} + 11 q^{19} + 39 q^{21} - 27 q^{27} + 59 q^{31} + 26 q^{37} - 69 q^{39} + 83 q^{43} + 120 q^{49} - 33 q^{57} - 121 q^{61} - 117 q^{63} - 13 q^{67} - 46 q^{73} - 142 q^{79} + 81 q^{81} - 299 q^{91} - 177 q^{93} + 167 q^{97}+O(q^{100})$$ q - 3 * q^3 - 13 * q^7 + 9 * q^9 + 23 * q^13 + 11 * q^19 + 39 * q^21 - 27 * q^27 + 59 * q^31 + 26 * q^37 - 69 * q^39 + 83 * q^43 + 120 * q^49 - 33 * q^57 - 121 * q^61 - 117 * q^63 - 13 * q^67 - 46 * q^73 - 142 * q^79 + 81 * q^81 - 299 * q^91 - 177 * q^93 + 167 * 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 −3.00000 0 0 0 −13.0000 0 9.00000 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 300.3.g.a 1
3.b odd 2 1 CM 300.3.g.a 1
4.b odd 2 1 1200.3.l.e 1
5.b even 2 1 300.3.g.c yes 1
5.c odd 4 2 300.3.b.b 2
12.b even 2 1 1200.3.l.e 1
15.d odd 2 1 300.3.g.c yes 1
15.e even 4 2 300.3.b.b 2
20.d odd 2 1 1200.3.l.a 1
20.e even 4 2 1200.3.c.b 2
60.h even 2 1 1200.3.l.a 1
60.l odd 4 2 1200.3.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.b.b 2 5.c odd 4 2
300.3.b.b 2 15.e even 4 2
300.3.g.a 1 1.a even 1 1 trivial
300.3.g.a 1 3.b odd 2 1 CM
300.3.g.c yes 1 5.b even 2 1
300.3.g.c yes 1 15.d odd 2 1
1200.3.c.b 2 20.e even 4 2
1200.3.c.b 2 60.l odd 4 2
1200.3.l.a 1 20.d odd 2 1
1200.3.l.a 1 60.h even 2 1
1200.3.l.e 1 4.b odd 2 1
1200.3.l.e 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(300, [\chi])$$:

 $$T_{7} + 13$$ T7 + 13 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T$$
$7$ $$T + 13$$
$11$ $$T$$
$13$ $$T - 23$$
$17$ $$T$$
$19$ $$T - 11$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T - 59$$
$37$ $$T - 26$$
$41$ $$T$$
$43$ $$T - 83$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T + 121$$
$67$ $$T + 13$$
$71$ $$T$$
$73$ $$T + 46$$
$79$ $$T + 142$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T - 167$$