# Properties

 Label 1519.1.n.b Level $1519$ Weight $1$ Character orbit 1519.n Analytic conductor $0.758$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -31 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1519,1,Mod(30,1519)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1519, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1519.30");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1519 = 7^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1519.n (of order $$6$$, degree $$2$$, not 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: $$0.758079754190$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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: $C_3\times S_3$ Artin field: Galois closure of 6.0.71528191.2

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6}^{2} q^{2} - \zeta_{6}^{2} q^{5} + q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10})$$ q - z^2 * q^2 - z^2 * q^5 + q^8 + z^2 * q^9 $$q - \zeta_{6}^{2} q^{2} - \zeta_{6}^{2} q^{5} + q^{8} + \zeta_{6}^{2} q^{9} - \zeta_{6} q^{10} - \zeta_{6}^{2} q^{16} + \zeta_{6} q^{18} - \zeta_{6}^{2} q^{19} - \zeta_{6} q^{31} - \zeta_{6} q^{38} - \zeta_{6}^{2} q^{40} - q^{41} + \zeta_{6} q^{45} + \zeta_{6}^{2} q^{47} + \zeta_{6} q^{59} - q^{62} + q^{64} - \zeta_{6} q^{67} - q^{71} + \zeta_{6}^{2} q^{72} - \zeta_{6} q^{80} - \zeta_{6} q^{81} + \zeta_{6}^{2} q^{82} + q^{90} + 2 \zeta_{6} q^{94} - \zeta_{6} q^{95} - q^{97} +O(q^{100})$$ q - z^2 * q^2 - z^2 * q^5 + q^8 + z^2 * q^9 - z * q^10 - z^2 * q^16 + z * q^18 - z^2 * q^19 - z * q^31 - z * q^38 - z^2 * q^40 - q^41 + z * q^45 + z^2 * q^47 + z * q^59 - q^62 + q^64 - z * q^67 - q^71 + z^2 * q^72 - z * q^80 - z * q^81 + z^2 * q^82 + q^90 + 2*z * q^94 - z * q^95 - q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{5} + 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q + q^2 + q^5 + 2 * q^8 - q^9 $$2 q + q^{2} + q^{5} + 2 q^{8} - q^{9} - q^{10} + q^{16} + q^{18} + q^{19} - q^{31} - q^{38} + q^{40} - 2 q^{41} + q^{45} - 2 q^{47} + q^{59} - 2 q^{62} + 2 q^{64} - 2 q^{67} - 2 q^{71} - q^{72} - q^{80} - q^{81} - q^{82} + 2 q^{90} + 2 q^{94} - q^{95} - 2 q^{97}+O(q^{100})$$ 2 * q + q^2 + q^5 + 2 * q^8 - q^9 - q^10 + q^16 + q^18 + q^19 - q^31 - q^38 + q^40 - 2 * q^41 + q^45 - 2 * q^47 + q^59 - 2 * q^62 + 2 * q^64 - 2 * q^67 - 2 * q^71 - q^72 - q^80 - q^81 - q^82 + 2 * q^90 + 2 * q^94 - q^95 - 2 * q^97

## Character values

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

 $$n$$ $$344$$ $$1179$$ $$\chi(n)$$ $$-1$$ $$\zeta_{6}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
30.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 0 0 0.500000 0.866025i 0 0 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
557.1 0.500000 + 0.866025i 0 0 0.500000 + 0.866025i 0 0 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
 $$n$$: 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 $$\Q(\sqrt{-31})$$
7.c even 3 1 inner
217.n odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1519.1.n.b 2
7.b odd 2 1 1519.1.n.a 2
7.c even 3 1 31.1.b.a 1
7.c even 3 1 inner 1519.1.n.b 2
7.d odd 6 1 1519.1.c.a 1
7.d odd 6 1 1519.1.n.a 2
21.h odd 6 1 279.1.d.b 1
28.g odd 6 1 496.1.e.a 1
31.b odd 2 1 CM 1519.1.n.b 2
35.j even 6 1 775.1.d.b 1
35.l odd 12 2 775.1.c.a 2
56.k odd 6 1 1984.1.e.b 1
56.p even 6 1 1984.1.e.a 1
63.g even 3 1 2511.1.m.e 2
63.h even 3 1 2511.1.m.e 2
63.j odd 6 1 2511.1.m.a 2
63.n odd 6 1 2511.1.m.a 2
77.h odd 6 1 3751.1.d.b 1
77.m even 15 4 3751.1.t.c 4
77.o odd 30 4 3751.1.t.a 4
217.d even 2 1 1519.1.n.a 2
217.e even 3 1 961.1.e.a 2
217.g even 3 1 961.1.e.a 2
217.k odd 6 1 961.1.e.a 2
217.m even 6 1 1519.1.c.a 1
217.m even 6 1 1519.1.n.a 2
217.n odd 6 1 31.1.b.a 1
217.n odd 6 1 inner 1519.1.n.b 2
217.o odd 6 1 961.1.e.a 2
217.z even 15 4 961.1.h.a 8
217.ba even 15 4 961.1.h.a 8
217.bb even 15 4 961.1.f.a 4
217.bg odd 30 4 961.1.f.a 4
217.bh odd 30 4 961.1.h.a 8
217.bn odd 30 4 961.1.h.a 8
651.bg even 6 1 279.1.d.b 1
868.r even 6 1 496.1.e.a 1
1085.bh odd 6 1 775.1.d.b 1
1085.cf even 12 2 775.1.c.a 2
1736.br odd 6 1 1984.1.e.a 1
1736.ct even 6 1 1984.1.e.b 1
1953.bf even 6 1 2511.1.m.a 2
1953.ck odd 6 1 2511.1.m.e 2
1953.cs even 6 1 2511.1.m.a 2
1953.eb odd 6 1 2511.1.m.e 2
2387.bm even 6 1 3751.1.d.b 1
2387.gi even 30 4 3751.1.t.a 4
2387.il odd 30 4 3751.1.t.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.1.b.a 1 7.c even 3 1
31.1.b.a 1 217.n odd 6 1
279.1.d.b 1 21.h odd 6 1
279.1.d.b 1 651.bg even 6 1
496.1.e.a 1 28.g odd 6 1
496.1.e.a 1 868.r even 6 1
775.1.c.a 2 35.l odd 12 2
775.1.c.a 2 1085.cf even 12 2
775.1.d.b 1 35.j even 6 1
775.1.d.b 1 1085.bh odd 6 1
961.1.e.a 2 217.e even 3 1
961.1.e.a 2 217.g even 3 1
961.1.e.a 2 217.k odd 6 1
961.1.e.a 2 217.o odd 6 1
961.1.f.a 4 217.bb even 15 4
961.1.f.a 4 217.bg odd 30 4
961.1.h.a 8 217.z even 15 4
961.1.h.a 8 217.ba even 15 4
961.1.h.a 8 217.bh odd 30 4
961.1.h.a 8 217.bn odd 30 4
1519.1.c.a 1 7.d odd 6 1
1519.1.c.a 1 217.m even 6 1
1519.1.n.a 2 7.b odd 2 1
1519.1.n.a 2 7.d odd 6 1
1519.1.n.a 2 217.d even 2 1
1519.1.n.a 2 217.m even 6 1
1519.1.n.b 2 1.a even 1 1 trivial
1519.1.n.b 2 7.c even 3 1 inner
1519.1.n.b 2 31.b odd 2 1 CM
1519.1.n.b 2 217.n odd 6 1 inner
1984.1.e.a 1 56.p even 6 1
1984.1.e.a 1 1736.br odd 6 1
1984.1.e.b 1 56.k odd 6 1
1984.1.e.b 1 1736.ct even 6 1
2511.1.m.a 2 63.j odd 6 1
2511.1.m.a 2 63.n odd 6 1
2511.1.m.a 2 1953.bf even 6 1
2511.1.m.a 2 1953.cs even 6 1
2511.1.m.e 2 63.g even 3 1
2511.1.m.e 2 63.h even 3 1
2511.1.m.e 2 1953.ck odd 6 1
2511.1.m.e 2 1953.eb odd 6 1
3751.1.d.b 1 77.h odd 6 1
3751.1.d.b 1 2387.bm even 6 1
3751.1.t.a 4 77.o odd 30 4
3751.1.t.a 4 2387.gi even 30 4
3751.1.t.c 4 77.m even 15 4
3751.1.t.c 4 2387.il odd 30 4

## Hecke kernels

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

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

## Hecke characteristic polynomials

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