# Properties

 Label 2760.1.bv.a Level $2760$ Weight $1$ Character orbit 2760.bv Analytic conductor $1.377$ Analytic rank $0$ Dimension $4$ Projective image $S_{4}$ CM/RM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2760,1,Mod(137,2760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2760, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 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("2760.137");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2760.bv (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: $$1.37741943487$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$S_{4}$$ Projective field: Galois closure of 4.2.414000.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{8}^{2} q^{3} - \zeta_{8}^{3} q^{5} - \zeta_{8}^{3} q^{7} - q^{9} +O(q^{10})$$ q - z^2 * q^3 - z^3 * q^5 - z^3 * q^7 - q^9 $$q - \zeta_{8}^{2} q^{3} - \zeta_{8}^{3} q^{5} - \zeta_{8}^{3} q^{7} - q^{9} + ( - \zeta_{8}^{2} + 1) q^{13} - \zeta_{8} q^{15} + \zeta_{8} q^{17} - \zeta_{8} q^{21} + \zeta_{8}^{3} q^{23} - \zeta_{8}^{2} q^{25} + \zeta_{8}^{2} q^{27} + q^{29} - q^{31} - \zeta_{8}^{2} q^{35} + \zeta_{8}^{3} q^{37} + ( - \zeta_{8}^{2} - 1) q^{39} - \zeta_{8}^{2} q^{41} + \zeta_{8} q^{43} + \zeta_{8}^{3} q^{45} - \zeta_{8}^{3} q^{51} + \zeta_{8}^{3} q^{53} - q^{59} + \zeta_{8}^{3} q^{63} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{65} + \zeta_{8}^{3} q^{67} + \zeta_{8} q^{69} - \zeta_{8}^{2} q^{71} - q^{75} + (\zeta_{8}^{3} - \zeta_{8}) q^{79} + q^{81} - \zeta_{8}^{3} q^{83} + q^{85} - \zeta_{8}^{2} q^{87} + (\zeta_{8}^{3} + \zeta_{8}) q^{89} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{91} + \zeta_{8}^{2} q^{93} +O(q^{100})$$ q - z^2 * q^3 - z^3 * q^5 - z^3 * q^7 - q^9 + (-z^2 + 1) * q^13 - z * q^15 + z * q^17 - z * q^21 + z^3 * q^23 - z^2 * q^25 + z^2 * q^27 + q^29 - q^31 - z^2 * q^35 + z^3 * q^37 + (-z^2 - 1) * q^39 - z^2 * q^41 + z * q^43 + z^3 * q^45 - z^3 * q^51 + z^3 * q^53 - q^59 + z^3 * q^63 + (-z^3 - z) * q^65 + z^3 * q^67 + z * q^69 - z^2 * q^71 - q^75 + (z^3 - z) * q^79 + q^81 - z^3 * q^83 + q^85 - z^2 * q^87 + (z^3 + z) * q^89 + (-z^3 - z) * q^91 + z^2 * q^93 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^9 $$4 q - 4 q^{9} + 4 q^{13} + 4 q^{29} - 4 q^{31} - 4 q^{39} - 4 q^{59} - 4 q^{75} + 4 q^{81} + 4 q^{85}+O(q^{100})$$ 4 * q - 4 * q^9 + 4 * q^13 + 4 * q^29 - 4 * q^31 - 4 * q^39 - 4 * q^59 - 4 * q^75 + 4 * q^81 + 4 * q^85

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2760\mathbb{Z}\right)^\times$$.

 $$n$$ $$1201$$ $$1381$$ $$1657$$ $$1841$$ $$2071$$ $$\chi(n)$$ $$-1$$ $$1$$ $$\zeta_{8}^{2}$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
137.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
0 1.00000i 0 −0.707107 + 0.707107i 0 −0.707107 + 0.707107i 0 −1.00000 0
137.2 0 1.00000i 0 0.707107 0.707107i 0 0.707107 0.707107i 0 −1.00000 0
1793.1 0 1.00000i 0 −0.707107 0.707107i 0 −0.707107 0.707107i 0 −1.00000 0
1793.2 0 1.00000i 0 0.707107 + 0.707107i 0 0.707107 + 0.707107i 0 −1.00000 0
 $$n$$: 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
15.e even 4 1 inner
23.b odd 2 1 inner
345.l odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2760.1.bv.a 4
3.b odd 2 1 2760.1.bv.b yes 4
5.c odd 4 1 2760.1.bv.b yes 4
15.e even 4 1 inner 2760.1.bv.a 4
23.b odd 2 1 inner 2760.1.bv.a 4
69.c even 2 1 2760.1.bv.b yes 4
115.e even 4 1 2760.1.bv.b yes 4
345.l odd 4 1 inner 2760.1.bv.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.1.bv.a 4 1.a even 1 1 trivial
2760.1.bv.a 4 15.e even 4 1 inner
2760.1.bv.a 4 23.b odd 2 1 inner
2760.1.bv.a 4 345.l odd 4 1 inner
2760.1.bv.b yes 4 3.b odd 2 1
2760.1.bv.b yes 4 5.c odd 4 1
2760.1.bv.b yes 4 69.c even 2 1
2760.1.bv.b yes 4 115.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{29} - 1$$ acting on $$S_{1}^{\mathrm{new}}(2760, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$T^{4} + 1$$
$7$ $$T^{4} + 1$$
$11$ $$T^{4}$$
$13$ $$(T^{2} - 2 T + 2)^{2}$$
$17$ $$T^{4} + 1$$
$19$ $$T^{4}$$
$23$ $$T^{4} + 1$$
$29$ $$(T - 1)^{4}$$
$31$ $$(T + 1)^{4}$$
$37$ $$T^{4} + 1$$
$41$ $$(T^{2} + 1)^{2}$$
$43$ $$T^{4} + 16$$
$47$ $$T^{4}$$
$53$ $$T^{4} + 1$$
$59$ $$(T + 1)^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4} + 1$$
$71$ $$(T^{2} + 1)^{2}$$
$73$ $$T^{4}$$
$79$ $$(T^{2} - 2)^{2}$$
$83$ $$T^{4} + 1$$
$89$ $$(T^{2} + 2)^{2}$$
$97$ $$T^{4}$$