# Properties

 Label 2300.1.f.a Level $2300$ Weight $1$ Character orbit 2300.f Analytic conductor $1.148$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -115 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,1,Mod(1701,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2300.1701");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2300 = 2^{2} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2300.f (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: no Analytic conductor: $$1.14784952906$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 460) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.460.1 Artin image: $C_4\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - i q^{7} - q^{9} +O(q^{10})$$ q - z * q^7 - q^9 $$q - i q^{7} - q^{9} - i q^{17} - i q^{23} + q^{29} - q^{31} - i q^{37} - q^{41} - 2 i q^{43} + i q^{53} + q^{59} + i q^{63} - i q^{67} - q^{71} + q^{81} + i q^{83} + 2 i q^{97} +O(q^{100})$$ q - z * q^7 - q^9 - z * q^17 - z * q^23 + q^29 - q^31 - z * q^37 - q^41 - 2*z * q^43 + z * q^53 + q^59 + z * q^63 - z * q^67 - q^71 + q^81 + z * q^83 + 2*z * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} + 2 q^{29} - 2 q^{31} - 2 q^{41} + 2 q^{59} - 2 q^{71} + 2 q^{81}+O(q^{100})$$ 2 * q - 2 * q^9 + 2 * q^29 - 2 * q^31 - 2 * q^41 + 2 * q^59 - 2 * q^71 + 2 * q^81

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$1151$$ $$1201$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1701.1
 1.00000i − 1.00000i
0 0 0 0 0 1.00000i 0 −1.00000 0
1701.2 0 0 0 0 0 1.00000i 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
115.c odd 2 1 CM by $$\Q(\sqrt{-115})$$
5.b even 2 1 inner
23.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2300.1.f.a 2
5.b even 2 1 inner 2300.1.f.a 2
5.c odd 4 1 460.1.d.a 1
5.c odd 4 1 460.1.d.b yes 1
20.e even 4 1 1840.1.g.a 1
20.e even 4 1 1840.1.g.b 1
23.b odd 2 1 inner 2300.1.f.a 2
115.c odd 2 1 CM 2300.1.f.a 2
115.e even 4 1 460.1.d.a 1
115.e even 4 1 460.1.d.b yes 1
460.k odd 4 1 1840.1.g.a 1
460.k odd 4 1 1840.1.g.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.1.d.a 1 5.c odd 4 1
460.1.d.a 1 115.e even 4 1
460.1.d.b yes 1 5.c odd 4 1
460.1.d.b yes 1 115.e even 4 1
1840.1.g.a 1 20.e even 4 1
1840.1.g.a 1 460.k odd 4 1
1840.1.g.b 1 20.e even 4 1
1840.1.g.b 1 460.k odd 4 1
2300.1.f.a 2 1.a even 1 1 trivial
2300.1.f.a 2 5.b even 2 1 inner
2300.1.f.a 2 23.b odd 2 1 inner
2300.1.f.a 2 115.c odd 2 1 CM

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(2300, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 1$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 1$$
$29$ $$(T - 1)^{2}$$
$31$ $$(T + 1)^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$(T + 1)^{2}$$
$43$ $$T^{2} + 4$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 1$$
$59$ $$(T - 1)^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2} + 1$$
$71$ $$(T + 1)^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 1$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 4$$