# Properties

 Label 3040.1.b.a Level $3040$ Weight $1$ Character orbit 3040.b Analytic conductor $1.517$ Analytic rank $0$ Dimension $8$ Projective image $D_{8}$ CM discriminant -95 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3040,1,Mod(1329,3040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))

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

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

chi := DirichletCharacter("3040.1329");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3040 = 2^{5} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3040.b (of order $$2$$, degree $$1$$, 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: $$1.51715763840$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{16})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 1$$ x^8 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 760) Projective image: $$D_{8}$$ Projective field: Galois closure of 8.0.66724352000.2

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{16}^{5} - \zeta_{16}^{3}) q^{3} + \zeta_{16}^{4} q^{5} + (\zeta_{16}^{6} - \zeta_{16}^{2} - 1) q^{9}+O(q^{10})$$ q + (-z^5 - z^3) * q^3 + z^4 * q^5 + (z^6 - z^2 - 1) * q^9 $$q + ( - \zeta_{16}^{5} - \zeta_{16}^{3}) q^{3} + \zeta_{16}^{4} q^{5} + (\zeta_{16}^{6} - \zeta_{16}^{2} - 1) q^{9} + (\zeta_{16}^{6} + \zeta_{16}^{2}) q^{11} + (\zeta_{16}^{7} + \zeta_{16}) q^{13} + ( - \zeta_{16}^{7} + \zeta_{16}) q^{15} + \zeta_{16}^{4} q^{19} - q^{25} + (\zeta_{16}^{7} + \zeta_{16}^{5} + \cdots + \zeta_{16}) q^{27}+ \cdots + ( - \zeta_{16}^{6} + \cdots - \zeta_{16}^{2}) q^{99}+O(q^{100})$$ q + (-z^5 - z^3) * q^3 + z^4 * q^5 + (z^6 - z^2 - 1) * q^9 + (z^6 + z^2) * q^11 + (z^7 + z) * q^13 + (-z^7 + z) * q^15 + z^4 * q^19 - q^25 + (z^7 + z^5 + z^3 + z) * q^27 + (-z^7 - z^5 + z^3 + z) * q^33 + (z^5 + z^3) * q^37 + (-z^6 + z^2) * q^39 + (-z^6 - z^4 - z^2) * q^45 + q^49 + (-z^5 - z^3) * q^53 + (z^6 - z^2) * q^55 + (-z^7 + z) * q^57 + (z^6 + z^2) * q^61 + (z^5 - z^3) * q^65 + (z^7 + z) * q^67 + (z^5 + z^3) * q^75 + (-z^6 + z^2 + 1) * q^81 - q^95 + (z^5 - z^3) * q^97 + (-z^6 - 2*z^4 - z^2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{9}+O(q^{10})$$ 8 * q - 8 * q^9 $$8 q - 8 q^{9} - 8 q^{25} + 8 q^{49} + 8 q^{81} - 8 q^{95}+O(q^{100})$$ 8 * q - 8 * q^9 - 8 * q^25 + 8 * q^49 + 8 * q^81 - 8 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3040\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$1217$$ $$1921$$ $$2661$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1329.1
 −0.923880 + 0.382683i 0.923880 + 0.382683i −0.382683 − 0.923880i 0.382683 − 0.923880i 0.382683 + 0.923880i −0.382683 + 0.923880i 0.923880 − 0.382683i −0.923880 − 0.382683i
0 1.84776i 0 1.00000i 0 0 0 −2.41421 0
1329.2 0 1.84776i 0 1.00000i 0 0 0 −2.41421 0
1329.3 0 0.765367i 0 1.00000i 0 0 0 0.414214 0
1329.4 0 0.765367i 0 1.00000i 0 0 0 0.414214 0
1329.5 0 0.765367i 0 1.00000i 0 0 0 0.414214 0
1329.6 0 0.765367i 0 1.00000i 0 0 0 0.414214 0
1329.7 0 1.84776i 0 1.00000i 0 0 0 −2.41421 0
1329.8 0 1.84776i 0 1.00000i 0 0 0 −2.41421 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1329.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by $$\Q(\sqrt{-95})$$
5.b even 2 1 inner
8.b even 2 1 inner
19.b odd 2 1 inner
40.f even 2 1 inner
152.g odd 2 1 inner
760.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3040.1.b.a 8
4.b odd 2 1 760.1.b.a 8
5.b even 2 1 inner 3040.1.b.a 8
8.b even 2 1 inner 3040.1.b.a 8
8.d odd 2 1 760.1.b.a 8
19.b odd 2 1 inner 3040.1.b.a 8
20.d odd 2 1 760.1.b.a 8
20.e even 4 2 3800.1.o.g 8
40.e odd 2 1 760.1.b.a 8
40.f even 2 1 inner 3040.1.b.a 8
40.k even 4 2 3800.1.o.g 8
76.d even 2 1 760.1.b.a 8
95.d odd 2 1 CM 3040.1.b.a 8
152.b even 2 1 760.1.b.a 8
152.g odd 2 1 inner 3040.1.b.a 8
380.d even 2 1 760.1.b.a 8
380.j odd 4 2 3800.1.o.g 8
760.b odd 2 1 inner 3040.1.b.a 8
760.p even 2 1 760.1.b.a 8
760.y odd 4 2 3800.1.o.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.1.b.a 8 4.b odd 2 1
760.1.b.a 8 8.d odd 2 1
760.1.b.a 8 20.d odd 2 1
760.1.b.a 8 40.e odd 2 1
760.1.b.a 8 76.d even 2 1
760.1.b.a 8 152.b even 2 1
760.1.b.a 8 380.d even 2 1
760.1.b.a 8 760.p even 2 1
3040.1.b.a 8 1.a even 1 1 trivial
3040.1.b.a 8 5.b even 2 1 inner
3040.1.b.a 8 8.b even 2 1 inner
3040.1.b.a 8 19.b odd 2 1 inner
3040.1.b.a 8 40.f even 2 1 inner
3040.1.b.a 8 95.d odd 2 1 CM
3040.1.b.a 8 152.g odd 2 1 inner
3040.1.b.a 8 760.b odd 2 1 inner
3800.1.o.g 8 20.e even 4 2
3800.1.o.g 8 40.k even 4 2
3800.1.o.g 8 380.j odd 4 2
3800.1.o.g 8 760.y odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3040, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} + 4 T^{2} + 2)^{2}$$
$5$ $$(T^{2} + 1)^{4}$$
$7$ $$T^{8}$$
$11$ $$(T^{2} + 2)^{4}$$
$13$ $$(T^{4} + 4 T^{2} + 2)^{2}$$
$17$ $$T^{8}$$
$19$ $$(T^{2} + 1)^{4}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$(T^{4} + 4 T^{2} + 2)^{2}$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$(T^{4} + 4 T^{2} + 2)^{2}$$
$59$ $$T^{8}$$
$61$ $$(T^{2} + 2)^{4}$$
$67$ $$(T^{4} + 4 T^{2} + 2)^{2}$$
$71$ $$T^{8}$$
$73$ $$T^{8}$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$T^{8}$$
$97$ $$(T^{4} - 4 T^{2} + 2)^{2}$$