# Properties

 Label 3120.1.dl.d Level $3120$ Weight $1$ Character orbit 3120.dl Analytic conductor $1.557$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -39 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3120,1,Mod(467,3120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3120, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("3120.467");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3120.dl (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: $$1.55708283941$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.9984000.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{2} + i q^{3} + q^{4} + i q^{5} + i q^{6} + q^{8} - q^{9}+O(q^{10})$$ q + q^2 + z * q^3 + q^4 + z * q^5 + z * q^6 + q^8 - q^9 $$q + q^{2} + i q^{3} + q^{4} + i q^{5} + i q^{6} + q^{8} - q^{9} + i q^{10} + (i - 1) q^{11} + i q^{12} - q^{13} - q^{15} + q^{16} - q^{18} + i q^{20} + (i - 1) q^{22} + i q^{24} - q^{25} - q^{26} - i q^{27} - q^{30} + q^{32} + ( - i - 1) q^{33} - q^{36} - i q^{39} + i q^{40} + q^{43} + (i - 1) q^{44} - i q^{45} + (i + 1) q^{47} + i q^{48} - i q^{49} - q^{50} - q^{52} - i q^{54} + ( - i - 1) q^{55} + (i + 1) q^{59} - q^{60} + ( - i + 1) q^{61} + q^{64} - i q^{65} + ( - i - 1) q^{66} - q^{72} - i q^{75} - i q^{78} + i q^{80} + q^{81} + i q^{83} + 2 q^{86} + (i - 1) q^{88} - i q^{90} + (i + 1) q^{94} + i q^{96} - i q^{98} + ( - i + 1) q^{99} +O(q^{100})$$ q + q^2 + z * q^3 + q^4 + z * q^5 + z * q^6 + q^8 - q^9 + z * q^10 + (z - 1) * q^11 + z * q^12 - q^13 - q^15 + q^16 - q^18 + z * q^20 + (z - 1) * q^22 + z * q^24 - q^25 - q^26 - z * q^27 - q^30 + q^32 + (-z - 1) * q^33 - q^36 - z * q^39 + z * q^40 + q^43 + (z - 1) * q^44 - z * q^45 + (z + 1) * q^47 + z * q^48 - z * q^49 - q^50 - q^52 - z * q^54 + (-z - 1) * q^55 + (z + 1) * q^59 - q^60 + (-z + 1) * q^61 + q^64 - z * q^65 + (-z - 1) * q^66 - q^72 - z * q^75 - z * q^78 + z * q^80 + q^81 + z * q^83 + 2 * q^86 + (z - 1) * q^88 - z * q^90 + (z + 1) * q^94 + z * q^96 - z * q^98 + (-z + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} - 2 q^{11} - 2 q^{13} - 2 q^{15} + 2 q^{16} - 2 q^{18} - 2 q^{22} - 2 q^{25} - 2 q^{26} - 2 q^{30} + 2 q^{32} - 2 q^{33} - 2 q^{36} + 4 q^{43} - 2 q^{44} + 2 q^{47} - 2 q^{50} - 2 q^{52} - 2 q^{55} + 2 q^{59} - 2 q^{60} + 2 q^{61} + 2 q^{64} - 2 q^{66} - 2 q^{72} + 2 q^{81} + 4 q^{86} - 2 q^{88} + 2 q^{94} + 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 - 2 * q^11 - 2 * q^13 - 2 * q^15 + 2 * q^16 - 2 * q^18 - 2 * q^22 - 2 * q^25 - 2 * q^26 - 2 * q^30 + 2 * q^32 - 2 * q^33 - 2 * q^36 + 4 * q^43 - 2 * q^44 + 2 * q^47 - 2 * q^50 - 2 * q^52 - 2 * q^55 + 2 * q^59 - 2 * q^60 + 2 * q^61 + 2 * q^64 - 2 * q^66 - 2 * q^72 + 2 * q^81 + 4 * q^86 - 2 * q^88 + 2 * q^94 + 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times$$.

 $$n$$ $$1951$$ $$2081$$ $$2341$$ $$2497$$ $$2641$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$i$$ $$-i$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
467.1
 − 1.00000i 1.00000i
1.00000 1.00000i 1.00000 1.00000i 1.00000i 0 1.00000 −1.00000 1.00000i
1403.1 1.00000 1.00000i 1.00000 1.00000i 1.00000i 0 1.00000 −1.00000 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
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$
80.j even 4 1 inner
3120.dl odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3120.1.dl.d yes 2
3.b odd 2 1 3120.1.dl.a 2
5.c odd 4 1 3120.1.dq.c yes 2
13.b even 2 1 3120.1.dl.a 2
15.e even 4 1 3120.1.dq.b yes 2
16.f odd 4 1 3120.1.dq.c yes 2
39.d odd 2 1 CM 3120.1.dl.d yes 2
48.k even 4 1 3120.1.dq.b yes 2
65.h odd 4 1 3120.1.dq.b yes 2
80.j even 4 1 inner 3120.1.dl.d yes 2
195.s even 4 1 3120.1.dq.c yes 2
208.o odd 4 1 3120.1.dq.b yes 2
240.bd odd 4 1 3120.1.dl.a 2
624.v even 4 1 3120.1.dq.c yes 2
1040.y even 4 1 3120.1.dl.a 2
3120.dl odd 4 1 inner 3120.1.dl.d yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3120.1.dl.a 2 3.b odd 2 1
3120.1.dl.a 2 13.b even 2 1
3120.1.dl.a 2 240.bd odd 4 1
3120.1.dl.a 2 1040.y even 4 1
3120.1.dl.d yes 2 1.a even 1 1 trivial
3120.1.dl.d yes 2 39.d odd 2 1 CM
3120.1.dl.d yes 2 80.j even 4 1 inner
3120.1.dl.d yes 2 3120.dl odd 4 1 inner
3120.1.dq.b yes 2 15.e even 4 1
3120.1.dq.b yes 2 48.k even 4 1
3120.1.dq.b yes 2 65.h odd 4 1
3120.1.dq.b yes 2 208.o odd 4 1
3120.1.dq.c yes 2 5.c odd 4 1
3120.1.dq.c yes 2 16.f odd 4 1
3120.1.dq.c yes 2 195.s even 4 1
3120.1.dq.c yes 2 624.v even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3120, [\chi])$$:

 $$T_{11}^{2} + 2T_{11} + 2$$ T11^2 + 2*T11 + 2 $$T_{41}$$ T41

## Hecke characteristic polynomials

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