# Properties

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

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3120,1,Mod(2027,3120)]

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, 1, 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("3120.2027");

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.dq (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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{3} - q^{4} - q^{5} + i q^{6} - i q^{8} + q^{9} +O(q^{10})$$ q + z * q^2 + q^3 - q^4 - q^5 + z * q^6 - z * q^8 + q^9 $$q + i q^{2} + q^{3} - q^{4} - q^{5} + i q^{6} - i q^{8} + q^{9} - i q^{10} + (i + 1) q^{11} - q^{12} - i q^{13} - q^{15} + q^{16} + i q^{18} + q^{20} + (i - 1) q^{22} - i q^{24} + q^{25} + q^{26} + q^{27} - i q^{30} + i q^{32} + (i + 1) q^{33} - q^{36} - i q^{39} + i q^{40} + i q^{43} + ( - i - 1) q^{44} - q^{45} + (i + 1) q^{47} + q^{48} - i q^{49} + i q^{50} + i q^{52} + i q^{54} + ( - i - 1) q^{55} + ( - i + 1) q^{59} + q^{60} + (i + 1) q^{61} - q^{64} + i q^{65} + (i - 1) q^{66} - i q^{72} + q^{75} + q^{78} - q^{80} + q^{81} - q^{83} - 2 q^{86} + ( - i + 1) q^{88} - i q^{90} + (i - 1) q^{94} + i q^{96} + q^{98} + (i + 1) q^{99} +O(q^{100})$$ q + z * q^2 + q^3 - q^4 - q^5 + z * q^6 - z * q^8 + q^9 - z * q^10 + (z + 1) * q^11 - q^12 - z * q^13 - q^15 + q^16 + z * q^18 + q^20 + (z - 1) * q^22 - z * q^24 + q^25 + q^26 + q^27 - z * q^30 + z * q^32 + (z + 1) * q^33 - q^36 - z * q^39 + z * q^40 + z * q^43 + (-z - 1) * q^44 - q^45 + (z + 1) * q^47 + q^48 - z * q^49 + z * q^50 + z * 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 - z * q^72 + q^75 + q^78 - q^80 + q^81 - q^83 - 2 * q^86 + (-z + 1) * q^88 - z * q^90 + (z - 1) * q^94 + z * q^96 + q^98 + (z + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 2 * q^4 - 2 * q^5 + 2 * q^9 $$2 q + 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{9} + 2 q^{11} - 2 q^{12} - 2 q^{15} + 2 q^{16} + 2 q^{20} - 2 q^{22} + 2 q^{25} + 2 q^{26} + 2 q^{27} + 2 q^{33} - 2 q^{36} - 2 q^{44} - 2 q^{45} + 2 q^{47} + 2 q^{48} - 2 q^{55} + 2 q^{59} + 2 q^{60} + 2 q^{61} - 2 q^{64} - 2 q^{66} + 2 q^{75} + 2 q^{78} - 2 q^{80} + 2 q^{81} - 4 q^{83} - 4 q^{86} + 2 q^{88} - 2 q^{94} + 2 q^{98} + 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^4 - 2 * q^5 + 2 * q^9 + 2 * q^11 - 2 * q^12 - 2 * q^15 + 2 * q^16 + 2 * q^20 - 2 * q^22 + 2 * q^25 + 2 * q^26 + 2 * q^27 + 2 * q^33 - 2 * q^36 - 2 * q^44 - 2 * q^45 + 2 * q^47 + 2 * q^48 - 2 * q^55 + 2 * q^59 + 2 * q^60 + 2 * q^61 - 2 * q^64 - 2 * q^66 + 2 * q^75 + 2 * q^78 - 2 * q^80 + 2 * q^81 - 4 * q^83 - 4 * q^86 + 2 * q^88 - 2 * q^94 + 2 * q^98 + 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2027.1
 − 1.00000i 1.00000i
1.00000i 1.00000 −1.00000 −1.00000 1.00000i 0 1.00000i 1.00000 1.00000i
2963.1 1.00000i 1.00000 −1.00000 −1.00000 1.00000i 0 1.00000i 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.s even 4 1 inner
3120.dq 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.dq.b yes 2
3.b odd 2 1 3120.1.dq.c yes 2
5.c odd 4 1 3120.1.dl.a 2
13.b even 2 1 3120.1.dq.c yes 2
15.e even 4 1 3120.1.dl.d yes 2
16.f odd 4 1 3120.1.dl.a 2
39.d odd 2 1 CM 3120.1.dq.b yes 2
48.k even 4 1 3120.1.dl.d yes 2
65.h odd 4 1 3120.1.dl.d yes 2
80.s even 4 1 inner 3120.1.dq.b yes 2
195.s even 4 1 3120.1.dl.a 2
208.o odd 4 1 3120.1.dl.d yes 2
240.z odd 4 1 3120.1.dq.c yes 2
624.v even 4 1 3120.1.dl.a 2
1040.cq even 4 1 3120.1.dq.c yes 2
3120.dq odd 4 1 inner 3120.1.dq.b yes 2

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

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