# Properties

 Label 1519.1.n.a 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: $C_6\times S_3$ 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6}^{2} q^{2} + \zeta_{6}^{2} q^{5} + q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10})$$ 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.a 2
7.b odd 2 1 1519.1.n.b 2
7.c even 3 1 1519.1.c.a 1
7.c even 3 1 inner 1519.1.n.a 2
7.d odd 6 1 31.1.b.a 1
7.d odd 6 1 1519.1.n.b 2
21.g even 6 1 279.1.d.b 1
28.f even 6 1 496.1.e.a 1
31.b odd 2 1 CM 1519.1.n.a 2
35.i odd 6 1 775.1.d.b 1
35.k even 12 2 775.1.c.a 2
56.j odd 6 1 1984.1.e.a 1
56.m even 6 1 1984.1.e.b 1
63.i even 6 1 2511.1.m.a 2
63.k odd 6 1 2511.1.m.e 2
63.s even 6 1 2511.1.m.a 2
63.t odd 6 1 2511.1.m.e 2
77.i even 6 1 3751.1.d.b 1
77.n even 30 4 3751.1.t.a 4
77.p odd 30 4 3751.1.t.c 4
217.d even 2 1 1519.1.n.b 2
217.j odd 6 1 961.1.e.a 2
217.l even 6 1 961.1.e.a 2
217.m even 6 1 31.1.b.a 1
217.m even 6 1 1519.1.n.b 2
217.n odd 6 1 1519.1.c.a 1
217.n odd 6 1 inner 1519.1.n.a 2
217.q odd 6 1 961.1.e.a 2
217.r even 6 1 961.1.e.a 2
217.bf even 30 4 961.1.h.a 8
217.bi odd 30 4 961.1.f.a 4
217.bj odd 30 4 961.1.h.a 8
217.bk even 30 4 961.1.h.a 8
217.bl even 30 4 961.1.f.a 4
217.bm odd 30 4 961.1.h.a 8
651.be odd 6 1 279.1.d.b 1
868.bj odd 6 1 496.1.e.a 1
1085.bn even 6 1 775.1.d.b 1
1085.cp odd 12 2 775.1.c.a 2
1736.bh odd 6 1 1984.1.e.b 1
1736.ca even 6 1 1984.1.e.a 1
1953.z odd 6 1 2511.1.m.a 2
1953.bm even 6 1 2511.1.m.e 2
1953.dw odd 6 1 2511.1.m.a 2
1953.eg even 6 1 2511.1.m.e 2
2387.bj odd 6 1 3751.1.d.b 1
2387.gt odd 30 4 3751.1.t.a 4
2387.is even 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.d odd 6 1
31.1.b.a 1 217.m even 6 1
279.1.d.b 1 21.g even 6 1
279.1.d.b 1 651.be odd 6 1
496.1.e.a 1 28.f even 6 1
496.1.e.a 1 868.bj odd 6 1
775.1.c.a 2 35.k even 12 2
775.1.c.a 2 1085.cp odd 12 2
775.1.d.b 1 35.i odd 6 1
775.1.d.b 1 1085.bn even 6 1
961.1.e.a 2 217.j odd 6 1
961.1.e.a 2 217.l even 6 1
961.1.e.a 2 217.q odd 6 1
961.1.e.a 2 217.r even 6 1
961.1.f.a 4 217.bi odd 30 4
961.1.f.a 4 217.bl even 30 4
961.1.h.a 8 217.bf even 30 4
961.1.h.a 8 217.bj odd 30 4
961.1.h.a 8 217.bk even 30 4
961.1.h.a 8 217.bm odd 30 4
1519.1.c.a 1 7.c even 3 1
1519.1.c.a 1 217.n odd 6 1
1519.1.n.a 2 1.a even 1 1 trivial
1519.1.n.a 2 7.c even 3 1 inner
1519.1.n.a 2 31.b odd 2 1 CM
1519.1.n.a 2 217.n odd 6 1 inner
1519.1.n.b 2 7.b odd 2 1
1519.1.n.b 2 7.d odd 6 1
1519.1.n.b 2 217.d even 2 1
1519.1.n.b 2 217.m even 6 1
1984.1.e.a 1 56.j odd 6 1
1984.1.e.a 1 1736.ca even 6 1
1984.1.e.b 1 56.m even 6 1
1984.1.e.b 1 1736.bh odd 6 1
2511.1.m.a 2 63.i even 6 1
2511.1.m.a 2 63.s even 6 1
2511.1.m.a 2 1953.z odd 6 1
2511.1.m.a 2 1953.dw odd 6 1
2511.1.m.e 2 63.k odd 6 1
2511.1.m.e 2 63.t odd 6 1
2511.1.m.e 2 1953.bm even 6 1
2511.1.m.e 2 1953.eg even 6 1
3751.1.d.b 1 77.i even 6 1
3751.1.d.b 1 2387.bj odd 6 1
3751.1.t.a 4 77.n even 30 4
3751.1.t.a 4 2387.gt odd 30 4
3751.1.t.c 4 77.p odd 30 4
3751.1.t.c 4 2387.is even 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

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