Properties

 Label 260.1.g.b Level $260$ Weight $1$ Character orbit 260.g Self dual yes Analytic conductor $0.130$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -4, -260, 65 Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [260,1,Mod(259,260)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(260, 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("260.259");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$260 = 2^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 260.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: $$0.129756903285$$ 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 $$\Q(i, \sqrt{65})$$ Artin image: $D_4$ Artin field: Galois closure of 4.0.1040.2

$q$-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 $$q + q^{2} + q^{4} - q^{5} + q^{8} - q^{9} - q^{10} - q^{13} + q^{16} - q^{18} - q^{20} + q^{25} - q^{26} - 2 q^{29} + q^{32} - q^{36} + 2 q^{37} - q^{40} + q^{45} + q^{49} + q^{50} - q^{52} - 2 q^{58} - 2 q^{61} + q^{64} + q^{65} - q^{72} + 2 q^{73} + 2 q^{74} - q^{80} + q^{81} + q^{90} - 2 q^{97} + q^{98}+O(q^{100})$$ q + q^2 + q^4 - q^5 + q^8 - q^9 - q^10 - q^13 + q^16 - q^18 - q^20 + q^25 - q^26 - 2 * q^29 + q^32 - q^36 + 2 * q^37 - q^40 + q^45 + q^49 + q^50 - q^52 - 2 * q^58 - 2 * q^61 + q^64 + q^65 - q^72 + 2 * q^73 + 2 * q^74 - q^80 + q^81 + q^90 - 2 * q^97 + q^98

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/260\mathbb{Z}\right)^\times$$.

 $$n$$ $$41$$ $$131$$ $$157$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
259.1
 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 $$\Q(\sqrt{-1})$$
65.d even 2 1 RM by $$\Q(\sqrt{65})$$
260.g odd 2 1 CM by $$\Q(\sqrt{-65})$$

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 260.1.g.b yes 1
3.b odd 2 1 2340.1.i.a 1
4.b odd 2 1 CM 260.1.g.b yes 1
5.b even 2 1 260.1.g.a 1
5.c odd 4 2 1300.1.e.e 2
12.b even 2 1 2340.1.i.a 1
13.b even 2 1 260.1.g.a 1
13.c even 3 2 3380.1.w.a 2
13.d odd 4 2 3380.1.h.a 2
13.e even 6 2 3380.1.w.d 2
13.f odd 12 4 3380.1.v.a 4
15.d odd 2 1 2340.1.i.b 1
20.d odd 2 1 260.1.g.a 1
20.e even 4 2 1300.1.e.e 2
39.d odd 2 1 2340.1.i.b 1
52.b odd 2 1 260.1.g.a 1
52.f even 4 2 3380.1.h.a 2
52.i odd 6 2 3380.1.w.d 2
52.j odd 6 2 3380.1.w.a 2
52.l even 12 4 3380.1.v.a 4
60.h even 2 1 2340.1.i.b 1
65.d even 2 1 RM 260.1.g.b yes 1
65.g odd 4 2 3380.1.h.a 2
65.h odd 4 2 1300.1.e.e 2
65.l even 6 2 3380.1.w.a 2
65.n even 6 2 3380.1.w.d 2
65.s odd 12 4 3380.1.v.a 4
156.h even 2 1 2340.1.i.b 1
195.e odd 2 1 2340.1.i.a 1
260.g odd 2 1 CM 260.1.g.b yes 1
260.p even 4 2 1300.1.e.e 2
260.u even 4 2 3380.1.h.a 2
260.v odd 6 2 3380.1.w.d 2
260.w odd 6 2 3380.1.w.a 2
260.bc even 12 4 3380.1.v.a 4
780.d even 2 1 2340.1.i.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.1.g.a 1 5.b even 2 1
260.1.g.a 1 13.b even 2 1
260.1.g.a 1 20.d odd 2 1
260.1.g.a 1 52.b odd 2 1
260.1.g.b yes 1 1.a even 1 1 trivial
260.1.g.b yes 1 4.b odd 2 1 CM
260.1.g.b yes 1 65.d even 2 1 RM
260.1.g.b yes 1 260.g odd 2 1 CM
1300.1.e.e 2 5.c odd 4 2
1300.1.e.e 2 20.e even 4 2
1300.1.e.e 2 65.h odd 4 2
1300.1.e.e 2 260.p even 4 2
2340.1.i.a 1 3.b odd 2 1
2340.1.i.a 1 12.b even 2 1
2340.1.i.a 1 195.e odd 2 1
2340.1.i.a 1 780.d even 2 1
2340.1.i.b 1 15.d odd 2 1
2340.1.i.b 1 39.d odd 2 1
2340.1.i.b 1 60.h even 2 1
2340.1.i.b 1 156.h even 2 1
3380.1.h.a 2 13.d odd 4 2
3380.1.h.a 2 52.f even 4 2
3380.1.h.a 2 65.g odd 4 2
3380.1.h.a 2 260.u even 4 2
3380.1.v.a 4 13.f odd 12 4
3380.1.v.a 4 52.l even 12 4
3380.1.v.a 4 65.s odd 12 4
3380.1.v.a 4 260.bc even 12 4
3380.1.w.a 2 13.c even 3 2
3380.1.w.a 2 52.j odd 6 2
3380.1.w.a 2 65.l even 6 2
3380.1.w.a 2 260.w odd 6 2
3380.1.w.d 2 13.e even 6 2
3380.1.w.d 2 52.i odd 6 2
3380.1.w.d 2 65.n even 6 2
3380.1.w.d 2 260.v odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{37} - 2$$ acting on $$S_{1}^{\mathrm{new}}(260, [\chi])$$.

Hecke characteristic polynomials

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