# Properties

 Label 3920.1.bt.d Level $3920$ Weight $1$ Character orbit 3920.bt Analytic conductor $1.956$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -35 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3920 = 2^{4} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3920.bt (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.95633484952$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.313600.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} + \zeta_{12} q^{5} + ( -1 + \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} + \zeta_{12} q^{5} + ( -1 + \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{9} + ( -1 + \zeta_{12}^{4} ) q^{11} -\zeta_{12}^{3} q^{13} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{15} + \zeta_{12}^{5} q^{17} + \zeta_{12}^{2} q^{25} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{27} + q^{29} + ( -2 \zeta_{12} - \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{33} + ( 1 - \zeta_{12}^{4} ) q^{39} + ( -\zeta_{12} + \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{45} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{47} + ( -1 - \zeta_{12}^{2} ) q^{51} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{55} -\zeta_{12}^{4} q^{65} -2 \zeta_{12}^{5} q^{73} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{75} + ( 1 + \zeta_{12}^{2} ) q^{79} -\zeta_{12}^{2} q^{81} - q^{85} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{87} -\zeta_{12}^{3} q^{97} + ( -2 \zeta_{12}^{2} - 2 \zeta_{12}^{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{9} + O(q^{10})$$ $$4q - 4q^{9} - 6q^{11} + 2q^{25} + 4q^{29} + 6q^{39} - 6q^{51} + 2q^{65} + 6q^{79} - 2q^{81} - 4q^{85} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3920\mathbb{Z}\right)^\times$$.

 $$n$$ $$981$$ $$1471$$ $$3041$$ $$3137$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{12}^{4}$$ $$-1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 −0.866025 1.50000i 0 −0.866025 0.500000i 0 0 0 −1.00000 + 1.73205i 0
79.2 0 0.866025 + 1.50000i 0 0.866025 + 0.500000i 0 0 0 −1.00000 + 1.73205i 0
1439.1 0 −0.866025 + 1.50000i 0 −0.866025 + 0.500000i 0 0 0 −1.00000 1.73205i 0
1439.2 0 0.866025 1.50000i 0 0.866025 0.500000i 0 0 0 −1.00000 1.73205i 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
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$
5.b even 2 1 inner
7.b odd 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
140.p odd 6 1 inner
140.s even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3920.1.bt.d 4
4.b odd 2 1 3920.1.bt.e 4
5.b even 2 1 inner 3920.1.bt.d 4
7.b odd 2 1 inner 3920.1.bt.d 4
7.c even 3 1 3920.1.j.e 4
7.c even 3 1 3920.1.bt.e 4
7.d odd 6 1 3920.1.j.e 4
7.d odd 6 1 3920.1.bt.e 4
20.d odd 2 1 3920.1.bt.e 4
28.d even 2 1 3920.1.bt.e 4
28.f even 6 1 3920.1.j.e 4
28.f even 6 1 inner 3920.1.bt.d 4
28.g odd 6 1 3920.1.j.e 4
28.g odd 6 1 inner 3920.1.bt.d 4
35.c odd 2 1 CM 3920.1.bt.d 4
35.i odd 6 1 3920.1.j.e 4
35.i odd 6 1 3920.1.bt.e 4
35.j even 6 1 3920.1.j.e 4
35.j even 6 1 3920.1.bt.e 4
140.c even 2 1 3920.1.bt.e 4
140.p odd 6 1 3920.1.j.e 4
140.p odd 6 1 inner 3920.1.bt.d 4
140.s even 6 1 3920.1.j.e 4
140.s even 6 1 inner 3920.1.bt.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3920.1.j.e 4 7.c even 3 1
3920.1.j.e 4 7.d odd 6 1
3920.1.j.e 4 28.f even 6 1
3920.1.j.e 4 28.g odd 6 1
3920.1.j.e 4 35.i odd 6 1
3920.1.j.e 4 35.j even 6 1
3920.1.j.e 4 140.p odd 6 1
3920.1.j.e 4 140.s even 6 1
3920.1.bt.d 4 1.a even 1 1 trivial
3920.1.bt.d 4 5.b even 2 1 inner
3920.1.bt.d 4 7.b odd 2 1 inner
3920.1.bt.d 4 28.f even 6 1 inner
3920.1.bt.d 4 28.g odd 6 1 inner
3920.1.bt.d 4 35.c odd 2 1 CM
3920.1.bt.d 4 140.p odd 6 1 inner
3920.1.bt.d 4 140.s even 6 1 inner
3920.1.bt.e 4 4.b odd 2 1
3920.1.bt.e 4 7.c even 3 1
3920.1.bt.e 4 7.d odd 6 1
3920.1.bt.e 4 20.d odd 2 1
3920.1.bt.e 4 28.d even 2 1
3920.1.bt.e 4 35.i odd 6 1
3920.1.bt.e 4 35.j even 6 1
3920.1.bt.e 4 140.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3920, [\chi])$$:

 $$T_{3}^{4} + 3 T_{3}^{2} + 9$$ $$T_{11}^{2} + 3 T_{11} + 3$$ $$T_{13}^{2} + 1$$ $$T_{41}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + 3 T^{2} + T^{4}$$
$5$ $$1 - T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 3 + 3 T + T^{2} )^{2}$$
$13$ $$( 1 + T^{2} )^{2}$$
$17$ $$1 - T^{2} + T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$( -1 + T )^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$9 + 3 T^{2} + T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$16 - 4 T^{2} + T^{4}$$
$79$ $$( 3 - 3 T + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$( 1 + T^{2} )^{2}$$