# Properties

 Label 3040.1.dv.b Level $3040$ Weight $1$ Character orbit 3040.dv Analytic conductor $1.517$ Analytic rank $0$ Dimension $6$ Projective image $D_{9}$ CM discriminant -40 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3040,1,Mod(719,3040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3040, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 9, 9, 4]))

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

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

chi := DirichletCharacter("3040.719");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3040 = 2^{5} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3040.dv (of order $$18$$, degree $$6$$, 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: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 760) Projective image: $$D_{9}$$ Projective field: Galois closure of 9.1.43477921384960000.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{18}^{7} q^{5} + ( - \zeta_{18}^{8} - \zeta_{18}^{4}) q^{7} + \zeta_{18}^{8} q^{9} +O(q^{10})$$ q - z^7 * q^5 + (-z^8 - z^4) * q^7 + z^8 * q^9 $$q - \zeta_{18}^{7} q^{5} + ( - \zeta_{18}^{8} - \zeta_{18}^{4}) q^{7} + \zeta_{18}^{8} q^{9} + ( - \zeta_{18}^{4} - \zeta_{18}^{2}) q^{11} + \zeta_{18}^{5} q^{13} + \zeta_{18} q^{19} + (\zeta_{18}^{7} - \zeta_{18}^{6}) q^{23} - \zeta_{18}^{5} q^{25} + ( - \zeta_{18}^{6} - \zeta_{18}^{2}) q^{35} + ( - \zeta_{18}^{5} + \zeta_{18}^{4}) q^{37} + ( - \zeta_{18}^{5} - \zeta_{18}^{3}) q^{41} + \zeta_{18}^{6} q^{45} + \zeta_{18}^{8} q^{47} + (\zeta_{18}^{8} + \cdots - \zeta_{18}^{3}) q^{49} + \cdots + (\zeta_{18}^{3} + \zeta_{18}) q^{99} +O(q^{100})$$ q - z^7 * q^5 + (-z^8 - z^4) * q^7 + z^8 * q^9 + (-z^4 - z^2) * q^11 + z^5 * q^13 + z * q^19 + (z^7 - z^6) * q^23 - z^5 * q^25 + (-z^6 - z^2) * q^35 + (-z^5 + z^4) * q^37 + (-z^5 - z^3) * q^41 + z^6 * q^45 + z^8 * q^47 + (z^8 - z^7 - z^3) * q^49 + (z^4 + 1) * q^53 + (-z^2 - 1) * q^55 - z * q^59 + (z^7 + z^3) * q^63 + z^3 * q^65 + (z^8 + z^6 - z^3 - z) * q^77 - z^7 * q^81 + (-z + 1) * q^89 + (z^4 + 1) * q^91 - z^8 * q^95 + (z^3 + z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q+O(q^{10})$$ 6 * q $$6 q + 3 q^{23} + 3 q^{35} - 3 q^{41} - 3 q^{45} - 3 q^{49} + 6 q^{53} - 6 q^{55} + 3 q^{63} + 3 q^{65} - 6 q^{77} + 6 q^{89} + 6 q^{91} + 3 q^{99}+O(q^{100})$$ 6 * q + 3 * q^23 + 3 * q^35 - 3 * q^41 - 3 * q^45 - 3 * q^49 + 6 * q^53 - 6 * q^55 + 3 * q^63 + 3 * q^65 - 6 * q^77 + 6 * q^89 + 6 * q^91 + 3 * q^99

## Character values

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

 $$n$$ $$191$$ $$1217$$ $$1921$$ $$2661$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{18}^{2}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
719.1
 −0.766044 − 0.642788i 0.939693 − 0.342020i 0.939693 + 0.342020i −0.173648 − 0.984808i −0.173648 + 0.984808i −0.766044 + 0.642788i
0 0 0 0.173648 0.984808i 0 0.173648 + 0.300767i 0 0.766044 0.642788i 0
879.1 0 0 0 0.766044 + 0.642788i 0 0.766044 + 1.32683i 0 −0.939693 0.342020i 0
1999.1 0 0 0 0.766044 0.642788i 0 0.766044 1.32683i 0 −0.939693 + 0.342020i 0
2479.1 0 0 0 −0.939693 0.342020i 0 −0.939693 + 1.62760i 0 0.173648 0.984808i 0
2639.1 0 0 0 −0.939693 + 0.342020i 0 −0.939693 1.62760i 0 0.173648 + 0.984808i 0
2799.1 0 0 0 0.173648 + 0.984808i 0 0.173648 0.300767i 0 0.766044 + 0.642788i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 719.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by $$\Q(\sqrt{-10})$$
19.e even 9 1 inner
760.bz odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3040.1.dv.b 6
4.b odd 2 1 760.1.bz.a 6
5.b even 2 1 3040.1.dv.a 6
8.b even 2 1 760.1.bz.b yes 6
8.d odd 2 1 3040.1.dv.a 6
19.e even 9 1 inner 3040.1.dv.b 6
20.d odd 2 1 760.1.bz.b yes 6
20.e even 4 2 3800.1.cv.f 12
40.e odd 2 1 CM 3040.1.dv.b 6
40.f even 2 1 760.1.bz.a 6
40.i odd 4 2 3800.1.cv.f 12
76.l odd 18 1 760.1.bz.a 6
95.p even 18 1 3040.1.dv.a 6
152.t even 18 1 760.1.bz.b yes 6
152.u odd 18 1 3040.1.dv.a 6
380.ba odd 18 1 760.1.bz.b yes 6
380.bj even 36 2 3800.1.cv.f 12
760.bz odd 18 1 inner 3040.1.dv.b 6
760.cj even 18 1 760.1.bz.a 6
760.cq odd 36 2 3800.1.cv.f 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.1.bz.a 6 4.b odd 2 1
760.1.bz.a 6 40.f even 2 1
760.1.bz.a 6 76.l odd 18 1
760.1.bz.a 6 760.cj even 18 1
760.1.bz.b yes 6 8.b even 2 1
760.1.bz.b yes 6 20.d odd 2 1
760.1.bz.b yes 6 152.t even 18 1
760.1.bz.b yes 6 380.ba odd 18 1
3040.1.dv.a 6 5.b even 2 1
3040.1.dv.a 6 8.d odd 2 1
3040.1.dv.a 6 95.p even 18 1
3040.1.dv.a 6 152.u odd 18 1
3040.1.dv.b 6 1.a even 1 1 trivial
3040.1.dv.b 6 19.e even 9 1 inner
3040.1.dv.b 6 40.e odd 2 1 CM
3040.1.dv.b 6 760.bz odd 18 1 inner
3800.1.cv.f 12 20.e even 4 2
3800.1.cv.f 12 40.i odd 4 2
3800.1.cv.f 12 380.bj even 36 2
3800.1.cv.f 12 760.cq odd 36 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{6} + 3T_{7}^{4} - 2T_{7}^{3} + 9T_{7}^{2} - 3T_{7} + 1$$ acting on $$S_{1}^{\mathrm{new}}(3040, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$T^{6} + T^{3} + 1$$
$7$ $$T^{6} + 3 T^{4} + \cdots + 1$$
$11$ $$T^{6} + 3 T^{4} + \cdots + 1$$
$13$ $$T^{6} - T^{3} + 1$$
$17$ $$T^{6}$$
$19$ $$T^{6} - T^{3} + 1$$
$23$ $$T^{6} - 3 T^{5} + \cdots + 1$$
$29$ $$T^{6}$$
$31$ $$T^{6}$$
$37$ $$(T^{3} - 3 T + 1)^{2}$$
$41$ $$T^{6} + 3 T^{5} + \cdots + 1$$
$43$ $$T^{6}$$
$47$ $$T^{6} + T^{3} + 1$$
$53$ $$T^{6} - 6 T^{5} + \cdots + 1$$
$59$ $$T^{6} + T^{3} + 1$$
$61$ $$T^{6}$$
$67$ $$T^{6}$$
$71$ $$T^{6}$$
$73$ $$T^{6}$$
$79$ $$T^{6}$$
$83$ $$T^{6}$$
$89$ $$T^{6} - 6 T^{5} + \cdots + 1$$
$97$ $$T^{6}$$