# Properties

 Label 1519.1.c.a Level $1519$ Weight $1$ Character orbit 1519.c Self dual yes Analytic conductor $0.758$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -31 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1519,1,Mod(1177,1519)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1519, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1519.1177");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1519 = 7^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1519.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

sage: f = N # Warning: the index may be different

gp: f = lf \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$0.758079754190$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 31) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.31.1 Artin image: $D_6$ Artin field: Galois closure of 6.0.329623.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{5} + q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^5 + q^8 + q^9 $$q - q^{2} + q^{5} + q^{8} + q^{9} - q^{10} - q^{16} - q^{18} + q^{19} - q^{31} - q^{38} + q^{40} + q^{41} + q^{45} - 2 q^{47} + q^{59} + q^{62} + q^{64} + 2 q^{67} - q^{71} + q^{72} - q^{80} + q^{81} - q^{82} - q^{90} + 2 q^{94} + q^{95} + q^{97}+O(q^{100})$$ q - q^2 + q^5 + q^8 + q^9 - q^10 - q^16 - q^18 + q^19 - q^31 - q^38 + q^40 + q^41 + q^45 - 2 * q^47 + q^59 + q^62 + q^64 + 2 * q^67 - q^71 + q^72 - q^80 + q^81 - q^82 - q^90 + 2 * q^94 + q^95 + q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1519\mathbb{Z}\right)^\times$$.

 $$n$$ $$344$$ $$1179$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1177.1
 0
−1.00000 0 0 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
31.b odd 2 1 CM by $$\Q(\sqrt{-31})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1519.1.c.a 1
7.b odd 2 1 31.1.b.a 1
7.c even 3 2 1519.1.n.a 2
7.d odd 6 2 1519.1.n.b 2
21.c even 2 1 279.1.d.b 1
28.d even 2 1 496.1.e.a 1
31.b odd 2 1 CM 1519.1.c.a 1
35.c odd 2 1 775.1.d.b 1
35.f even 4 2 775.1.c.a 2
56.e even 2 1 1984.1.e.b 1
56.h odd 2 1 1984.1.e.a 1
63.l odd 6 2 2511.1.m.e 2
63.o even 6 2 2511.1.m.a 2
77.b even 2 1 3751.1.d.b 1
77.j odd 10 4 3751.1.t.c 4
77.l even 10 4 3751.1.t.a 4
217.d even 2 1 31.1.b.a 1
217.m even 6 2 1519.1.n.b 2
217.n odd 6 2 1519.1.n.a 2
217.s even 6 2 961.1.e.a 2
217.u odd 6 2 961.1.e.a 2
217.v even 10 4 961.1.f.a 4
217.w odd 10 4 961.1.f.a 4
217.bd odd 30 8 961.1.h.a 8
217.be even 30 8 961.1.h.a 8
651.e odd 2 1 279.1.d.b 1
868.c odd 2 1 496.1.e.a 1
1085.b even 2 1 775.1.d.b 1
1085.o odd 4 2 775.1.c.a 2
1736.h even 2 1 1984.1.e.a 1
1736.n odd 2 1 1984.1.e.b 1
1953.cb even 6 2 2511.1.m.e 2
1953.cu odd 6 2 2511.1.m.a 2
2387.d odd 2 1 3751.1.d.b 1
2387.cb even 10 4 3751.1.t.c 4
2387.cv odd 10 4 3751.1.t.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.1.b.a 1 7.b odd 2 1
31.1.b.a 1 217.d even 2 1
279.1.d.b 1 21.c even 2 1
279.1.d.b 1 651.e odd 2 1
496.1.e.a 1 28.d even 2 1
496.1.e.a 1 868.c odd 2 1
775.1.c.a 2 35.f even 4 2
775.1.c.a 2 1085.o odd 4 2
775.1.d.b 1 35.c odd 2 1
775.1.d.b 1 1085.b even 2 1
961.1.e.a 2 217.s even 6 2
961.1.e.a 2 217.u odd 6 2
961.1.f.a 4 217.v even 10 4
961.1.f.a 4 217.w odd 10 4
961.1.h.a 8 217.bd odd 30 8
961.1.h.a 8 217.be even 30 8
1519.1.c.a 1 1.a even 1 1 trivial
1519.1.c.a 1 31.b odd 2 1 CM
1519.1.n.a 2 7.c even 3 2
1519.1.n.a 2 217.n odd 6 2
1519.1.n.b 2 7.d odd 6 2
1519.1.n.b 2 217.m even 6 2
1984.1.e.a 1 56.h odd 2 1
1984.1.e.a 1 1736.h even 2 1
1984.1.e.b 1 56.e even 2 1
1984.1.e.b 1 1736.n odd 2 1
2511.1.m.a 2 63.o even 6 2
2511.1.m.a 2 1953.cu odd 6 2
2511.1.m.e 2 63.l odd 6 2
2511.1.m.e 2 1953.cb even 6 2
3751.1.d.b 1 77.b even 2 1
3751.1.d.b 1 2387.d odd 2 1
3751.1.t.a 4 77.l even 10 4
3751.1.t.a 4 2387.cv odd 10 4
3751.1.t.c 4 77.j odd 10 4
3751.1.t.c 4 2387.cb even 10 4

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1519, [\chi])$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{5} - 1$$ T5 - 1

## Hecke characteristic polynomials

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