# Properties

 Label 260.1.s.a Level $260$ Weight $1$ Character orbit 260.s Analytic conductor $0.130$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [260,1,Mod(187,260)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(260, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 3]))

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

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

chi := DirichletCharacter("260.187");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$260 = 2^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 260.s (of order $$4$$, degree $$2$$, 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: $$0.129756903285$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.1098500.1 Artin image: $C_4\wr C_2$ Artin field: Galois closure of 8.0.70304000.3

## $q$-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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + i q^{5} - q^{8} + i q^{9}+O(q^{10})$$ q - q^2 + q^4 + z * q^5 - q^8 + z * q^9 $$q - q^{2} + q^{4} + i q^{5} - q^{8} + i q^{9} - i q^{10} - i q^{13} + q^{16} + (i + 1) q^{17} - i q^{18} + i q^{20} - q^{25} + i q^{26} - i q^{29} - q^{32} + ( - i - 1) q^{34} + i q^{36} - i q^{40} + ( - i - 1) q^{41} - q^{45} - q^{49} + q^{50} - i q^{52} + ( - i - 1) q^{53} + 2 i q^{58} + q^{64} + q^{65} + (i + 1) q^{68} - i q^{72} + q^{73} + i q^{80} - q^{81} + (i + 1) q^{82} + (i - 1) q^{85} + (i + 1) q^{89} + q^{90} + q^{98} +O(q^{100})$$ q - q^2 + q^4 + z * q^5 - q^8 + z * q^9 - z * q^10 - z * q^13 + q^16 + (z + 1) * q^17 - z * q^18 + z * q^20 - q^25 + z * q^26 - z * q^29 - q^32 + (-z - 1) * q^34 + z * q^36 - z * q^40 + (-z - 1) * q^41 - q^45 - q^49 + q^50 - z * q^52 + (-z - 1) * q^53 + 2*z * q^58 + q^64 + q^65 + (z + 1) * q^68 - z * q^72 + q^73 + z * q^80 - q^81 + (z + 1) * q^82 + (z - 1) * q^85 + (z + 1) * q^89 + q^90 + q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 2 q^{16} + 2 q^{17} - 2 q^{25} - 2 q^{32} - 2 q^{34} - 2 q^{41} - 2 q^{45} - 2 q^{49} + 2 q^{50} - 2 q^{53} + 2 q^{64} + 2 q^{65} + 2 q^{68} + 4 q^{73} - 2 q^{81} + 2 q^{82} - 2 q^{85} + 2 q^{89} + 2 q^{90} + 2 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 2 * q^16 + 2 * q^17 - 2 * q^25 - 2 * q^32 - 2 * q^34 - 2 * q^41 - 2 * q^45 - 2 * q^49 + 2 * q^50 - 2 * q^53 + 2 * q^64 + 2 * q^65 + 2 * q^68 + 4 * q^73 - 2 * q^81 + 2 * q^82 - 2 * q^85 + 2 * q^89 + 2 * q^90 + 2 * q^98

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
187.1
 − 1.00000i 1.00000i
−1.00000 0 1.00000 1.00000i 0 0 −1.00000 1.00000i 1.00000i
203.1 −1.00000 0 1.00000 1.00000i 0 0 −1.00000 1.00000i 1.00000i
 $$n$$: 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.k even 4 1 inner
260.s odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 260.1.s.a yes 2
3.b odd 2 1 2340.1.bo.c 2
4.b odd 2 1 CM 260.1.s.a yes 2
5.b even 2 1 1300.1.s.a 2
5.c odd 4 1 260.1.l.a 2
5.c odd 4 1 1300.1.l.a 2
12.b even 2 1 2340.1.bo.c 2
13.b even 2 1 3380.1.s.a 2
13.c even 3 2 3380.1.be.d 4
13.d odd 4 1 260.1.l.a 2
13.d odd 4 1 3380.1.l.a 2
13.e even 6 2 3380.1.be.a 4
13.f odd 12 2 3380.1.bl.a 4
13.f odd 12 2 3380.1.bl.e 4
15.e even 4 1 2340.1.v.a 2
20.d odd 2 1 1300.1.s.a 2
20.e even 4 1 260.1.l.a 2
20.e even 4 1 1300.1.l.a 2
39.f even 4 1 2340.1.v.a 2
52.b odd 2 1 3380.1.s.a 2
52.f even 4 1 260.1.l.a 2
52.f even 4 1 3380.1.l.a 2
52.i odd 6 2 3380.1.be.a 4
52.j odd 6 2 3380.1.be.d 4
52.l even 12 2 3380.1.bl.a 4
52.l even 12 2 3380.1.bl.e 4
60.l odd 4 1 2340.1.v.a 2
65.f even 4 1 1300.1.s.a 2
65.f even 4 1 3380.1.s.a 2
65.g odd 4 1 1300.1.l.a 2
65.h odd 4 1 3380.1.l.a 2
65.k even 4 1 inner 260.1.s.a yes 2
65.o even 12 2 3380.1.be.d 4
65.q odd 12 2 3380.1.bl.e 4
65.r odd 12 2 3380.1.bl.a 4
65.t even 12 2 3380.1.be.a 4
156.l odd 4 1 2340.1.v.a 2
195.j odd 4 1 2340.1.bo.c 2
260.l odd 4 1 1300.1.s.a 2
260.l odd 4 1 3380.1.s.a 2
260.p even 4 1 3380.1.l.a 2
260.s odd 4 1 inner 260.1.s.a yes 2
260.u even 4 1 1300.1.l.a 2
260.be odd 12 2 3380.1.be.d 4
260.bg even 12 2 3380.1.bl.a 4
260.bj even 12 2 3380.1.bl.e 4
260.bl odd 12 2 3380.1.be.a 4
780.bn even 4 1 2340.1.bo.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.1.l.a 2 5.c odd 4 1
260.1.l.a 2 13.d odd 4 1
260.1.l.a 2 20.e even 4 1
260.1.l.a 2 52.f even 4 1
260.1.s.a yes 2 1.a even 1 1 trivial
260.1.s.a yes 2 4.b odd 2 1 CM
260.1.s.a yes 2 65.k even 4 1 inner
260.1.s.a yes 2 260.s odd 4 1 inner
1300.1.l.a 2 5.c odd 4 1
1300.1.l.a 2 20.e even 4 1
1300.1.l.a 2 65.g odd 4 1
1300.1.l.a 2 260.u even 4 1
1300.1.s.a 2 5.b even 2 1
1300.1.s.a 2 20.d odd 2 1
1300.1.s.a 2 65.f even 4 1
1300.1.s.a 2 260.l odd 4 1
2340.1.v.a 2 15.e even 4 1
2340.1.v.a 2 39.f even 4 1
2340.1.v.a 2 60.l odd 4 1
2340.1.v.a 2 156.l odd 4 1
2340.1.bo.c 2 3.b odd 2 1
2340.1.bo.c 2 12.b even 2 1
2340.1.bo.c 2 195.j odd 4 1
2340.1.bo.c 2 780.bn even 4 1
3380.1.l.a 2 13.d odd 4 1
3380.1.l.a 2 52.f even 4 1
3380.1.l.a 2 65.h odd 4 1
3380.1.l.a 2 260.p even 4 1
3380.1.s.a 2 13.b even 2 1
3380.1.s.a 2 52.b odd 2 1
3380.1.s.a 2 65.f even 4 1
3380.1.s.a 2 260.l odd 4 1
3380.1.be.a 4 13.e even 6 2
3380.1.be.a 4 52.i odd 6 2
3380.1.be.a 4 65.t even 12 2
3380.1.be.a 4 260.bl odd 12 2
3380.1.be.d 4 13.c even 3 2
3380.1.be.d 4 52.j odd 6 2
3380.1.be.d 4 65.o even 12 2
3380.1.be.d 4 260.be odd 12 2
3380.1.bl.a 4 13.f odd 12 2
3380.1.bl.a 4 52.l even 12 2
3380.1.bl.a 4 65.r odd 12 2
3380.1.bl.a 4 260.bg even 12 2
3380.1.bl.e 4 13.f odd 12 2
3380.1.bl.e 4 52.l even 12 2
3380.1.bl.e 4 65.q odd 12 2
3380.1.bl.e 4 260.bj even 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(260, [\chi])$$.

## Hecke characteristic polynomials

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