# Properties

 Label 2511.1.m.a Level $2511$ Weight $1$ Character orbit 2511.m Analytic conductor $1.253$ 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] = [2511,1,Mod(433,2511)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2511, 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("2511.433");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2511 = 3^{4} \cdot 31$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2511.m (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: $$1.25315224672$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{5} - \zeta_{6}^{2} q^{7} - q^{8} +O(q^{10})$$ q + z^2 * q^2 - z * q^5 - z^2 * q^7 - q^8 $$q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{5} - \zeta_{6}^{2} q^{7} - q^{8} + q^{10} + \zeta_{6} q^{14} - \zeta_{6}^{2} q^{16} - q^{19} - \zeta_{6} q^{31} - q^{35} - \zeta_{6}^{2} q^{38} + \zeta_{6} q^{40} - \zeta_{6} q^{41} - \zeta_{6}^{2} q^{47} + \zeta_{6}^{2} q^{56} - \zeta_{6} q^{59} + q^{62} + q^{64} - \zeta_{6} q^{67} - \zeta_{6}^{2} q^{70} + q^{71} - q^{80} + q^{82} + 2 \zeta_{6} q^{94} + \zeta_{6} q^{95} - \zeta_{6}^{2} q^{97} +O(q^{100})$$ q + z^2 * q^2 - z * q^5 - z^2 * q^7 - q^8 + q^10 + z * q^14 - z^2 * q^16 - q^19 - z * q^31 - q^35 - z^2 * q^38 + z * q^40 - z * q^41 - z^2 * q^47 + z^2 * q^56 - z * q^59 + q^62 + q^64 - z * q^67 - z^2 * q^70 + q^71 - q^80 + q^82 + 2*z * q^94 + z * q^95 - z^2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{5} + q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q - q^2 - q^5 + q^7 - 2 * q^8 $$2 q - q^{2} - q^{5} + q^{7} - 2 q^{8} + 2 q^{10} + q^{14} + q^{16} - 2 q^{19} - q^{31} - 2 q^{35} + q^{38} + q^{40} - q^{41} + 2 q^{47} - q^{56} - q^{59} + 2 q^{62} + 2 q^{64} - 2 q^{67} + q^{70} + 2 q^{71} - 2 q^{80} + 2 q^{82} + 2 q^{94} + q^{95} + q^{97}+O(q^{100})$$ 2 * q - q^2 - q^5 + q^7 - 2 * q^8 + 2 * q^10 + q^14 + q^16 - 2 * q^19 - q^31 - 2 * q^35 + q^38 + q^40 - q^41 + 2 * q^47 - q^56 - q^59 + 2 * q^62 + 2 * q^64 - 2 * q^67 + q^70 + 2 * q^71 - 2 * q^80 + 2 * q^82 + 2 * q^94 + q^95 + q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2511\mathbb{Z}\right)^\times$$.

 $$n$$ $$406$$ $$1055$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
433.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 0 0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i −1.00000 0 1.00000
1270.1 −0.500000 + 0.866025i 0 0 −0.500000 0.866025i 0 0.500000 0.866025i −1.00000 0 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 $$\Q(\sqrt{-31})$$
9.c even 3 1 inner
279.m odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2511.1.m.a 2
3.b odd 2 1 2511.1.m.e 2
9.c even 3 1 279.1.d.b 1
9.c even 3 1 inner 2511.1.m.a 2
9.d odd 6 1 31.1.b.a 1
9.d odd 6 1 2511.1.m.e 2
31.b odd 2 1 CM 2511.1.m.a 2
36.h even 6 1 496.1.e.a 1
45.h odd 6 1 775.1.d.b 1
45.l even 12 2 775.1.c.a 2
63.i even 6 1 1519.1.n.a 2
63.j odd 6 1 1519.1.n.b 2
63.n odd 6 1 1519.1.n.b 2
63.o even 6 1 1519.1.c.a 1
63.s even 6 1 1519.1.n.a 2
72.j odd 6 1 1984.1.e.a 1
72.l even 6 1 1984.1.e.b 1
93.c even 2 1 2511.1.m.e 2
99.g even 6 1 3751.1.d.b 1
99.n odd 30 4 3751.1.t.c 4
99.p even 30 4 3751.1.t.a 4
279.m odd 6 1 279.1.d.b 1
279.m odd 6 1 inner 2511.1.m.a 2
279.o even 6 1 961.1.e.a 2
279.p odd 6 1 961.1.e.a 2
279.q odd 6 1 961.1.e.a 2
279.r even 6 1 961.1.e.a 2
279.s even 6 1 31.1.b.a 1
279.s even 6 1 2511.1.m.e 2
279.bd odd 30 4 961.1.h.a 8
279.be even 30 4 961.1.h.a 8
279.bf odd 30 4 961.1.f.a 4
279.bg even 30 4 961.1.f.a 4
279.bh even 30 4 961.1.h.a 8
279.bi odd 30 4 961.1.h.a 8
1116.bk odd 6 1 496.1.e.a 1
1395.ba even 6 1 775.1.d.b 1
1395.cg odd 12 2 775.1.c.a 2
1953.z odd 6 1 1519.1.n.a 2
1953.bf even 6 1 1519.1.n.b 2
1953.cs even 6 1 1519.1.n.b 2
1953.cu odd 6 1 1519.1.c.a 1
1953.dw odd 6 1 1519.1.n.a 2
2232.bi even 6 1 1984.1.e.a 1
2232.bp odd 6 1 1984.1.e.b 1
3069.ba odd 6 1 3751.1.d.b 1
3069.gm even 30 4 3751.1.t.c 4
3069.ib odd 30 4 3751.1.t.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.1.b.a 1 9.d odd 6 1
31.1.b.a 1 279.s even 6 1
279.1.d.b 1 9.c even 3 1
279.1.d.b 1 279.m odd 6 1
496.1.e.a 1 36.h even 6 1
496.1.e.a 1 1116.bk odd 6 1
775.1.c.a 2 45.l even 12 2
775.1.c.a 2 1395.cg odd 12 2
775.1.d.b 1 45.h odd 6 1
775.1.d.b 1 1395.ba even 6 1
961.1.e.a 2 279.o even 6 1
961.1.e.a 2 279.p odd 6 1
961.1.e.a 2 279.q odd 6 1
961.1.e.a 2 279.r even 6 1
961.1.f.a 4 279.bf odd 30 4
961.1.f.a 4 279.bg even 30 4
961.1.h.a 8 279.bd odd 30 4
961.1.h.a 8 279.be even 30 4
961.1.h.a 8 279.bh even 30 4
961.1.h.a 8 279.bi odd 30 4
1519.1.c.a 1 63.o even 6 1
1519.1.c.a 1 1953.cu odd 6 1
1519.1.n.a 2 63.i even 6 1
1519.1.n.a 2 63.s even 6 1
1519.1.n.a 2 1953.z odd 6 1
1519.1.n.a 2 1953.dw odd 6 1
1519.1.n.b 2 63.j odd 6 1
1519.1.n.b 2 63.n odd 6 1
1519.1.n.b 2 1953.bf even 6 1
1519.1.n.b 2 1953.cs even 6 1
1984.1.e.a 1 72.j odd 6 1
1984.1.e.a 1 2232.bi even 6 1
1984.1.e.b 1 72.l even 6 1
1984.1.e.b 1 2232.bp odd 6 1
2511.1.m.a 2 1.a even 1 1 trivial
2511.1.m.a 2 9.c even 3 1 inner
2511.1.m.a 2 31.b odd 2 1 CM
2511.1.m.a 2 279.m odd 6 1 inner
2511.1.m.e 2 3.b odd 2 1
2511.1.m.e 2 9.d odd 6 1
2511.1.m.e 2 93.c even 2 1
2511.1.m.e 2 279.s even 6 1
3751.1.d.b 1 99.g even 6 1
3751.1.d.b 1 3069.ba odd 6 1
3751.1.t.a 4 99.p even 30 4
3751.1.t.a 4 3069.ib odd 30 4
3751.1.t.c 4 99.n odd 30 4
3751.1.t.c 4 3069.gm even 30 4

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2511, [\chi])$$:

 $$T_{2}^{2} + T_{2} + 1$$ T2^2 + T2 + 1 $$T_{7}^{2} - T_{7} + 1$$ T7^2 - T7 + 1 $$T_{11}$$ T11

## Hecke characteristic polynomials

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