# Properties

 Label 300.9.g.a Level $300$ Weight $9$ Character orbit 300.g Self dual yes Analytic conductor $122.214$ 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] = [300,9,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, 9, names="a")

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

chi := DirichletCharacter("300.101");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ 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: $$122.213583018$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 12) 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)

 $$f(q)$$ $$=$$ $$q - 81 q^{3} - 4034 q^{7} + 6561 q^{9}+O(q^{10})$$ q - 81 * q^3 - 4034 * q^7 + 6561 * q^9 $$q - 81 q^{3} - 4034 q^{7} + 6561 q^{9} + 35806 q^{13} - 258526 q^{19} + 326754 q^{21} - 531441 q^{27} - 1809406 q^{31} - 503522 q^{37} - 2900286 q^{39} - 3492194 q^{43} + 10508355 q^{49} + 20940606 q^{57} - 23826526 q^{61} - 26467074 q^{63} + 5421406 q^{67} - 16169282 q^{73} - 18887038 q^{79} + 43046721 q^{81} - 144441404 q^{91} + 146561886 q^{93} - 176908034 q^{97}+O(q^{100})$$ q - 81 * q^3 - 4034 * q^7 + 6561 * q^9 + 35806 * q^13 - 258526 * q^19 + 326754 * q^21 - 531441 * q^27 - 1809406 * q^31 - 503522 * q^37 - 2900286 * q^39 - 3492194 * q^43 + 10508355 * q^49 + 20940606 * q^57 - 23826526 * q^61 - 26467074 * q^63 + 5421406 * q^67 - 16169282 * q^73 - 18887038 * q^79 + 43046721 * q^81 - 144441404 * q^91 + 146561886 * q^93 - 176908034 * q^97

## Character values

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

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$1$$ $$0$$ $$0$$

## 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 −81.0000 0 0 0 −4034.00 0 6561.00 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 300.9.g.a 1
3.b odd 2 1 CM 300.9.g.a 1
5.b even 2 1 12.9.c.a 1
5.c odd 4 2 300.9.b.b 2
15.d odd 2 1 12.9.c.a 1
15.e even 4 2 300.9.b.b 2
20.d odd 2 1 48.9.e.a 1
40.e odd 2 1 192.9.e.b 1
40.f even 2 1 192.9.e.a 1
45.h odd 6 2 324.9.g.a 2
45.j even 6 2 324.9.g.a 2
60.h even 2 1 48.9.e.a 1
120.i odd 2 1 192.9.e.a 1
120.m even 2 1 192.9.e.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.9.c.a 1 5.b even 2 1
12.9.c.a 1 15.d odd 2 1
48.9.e.a 1 20.d odd 2 1
48.9.e.a 1 60.h even 2 1
192.9.e.a 1 40.f even 2 1
192.9.e.a 1 120.i odd 2 1
192.9.e.b 1 40.e odd 2 1
192.9.e.b 1 120.m even 2 1
300.9.b.b 2 5.c odd 4 2
300.9.b.b 2 15.e even 4 2
300.9.g.a 1 1.a even 1 1 trivial
300.9.g.a 1 3.b odd 2 1 CM
324.9.g.a 2 45.h odd 6 2
324.9.g.a 2 45.j even 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} + 4034$$ acting on $$S_{9}^{\mathrm{new}}(300, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 81$$
$5$ $$T$$
$7$ $$T + 4034$$
$11$ $$T$$
$13$ $$T - 35806$$
$17$ $$T$$
$19$ $$T + 258526$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T + 1809406$$
$37$ $$T + 503522$$
$41$ $$T$$
$43$ $$T + 3492194$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T + 23826526$$
$67$ $$T - 5421406$$
$71$ $$T$$
$73$ $$T + 16169282$$
$79$ $$T + 18887038$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T + 176908034$$