# Properties

 Label 3920.1.j.d Level $3920$ Weight $1$ Character orbit 3920.j Self dual yes Analytic conductor $1.956$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -20 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3920 = 2^{4} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3920.j (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.95633484952$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 560) Projective image $$D_{6}$$ Projective field Galois closure of 6.0.3841600.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{3} + q^{5} + 2 q^{9} +O(q^{10})$$ $$q -\beta q^{3} + q^{5} + 2 q^{9} -\beta q^{15} + \beta q^{23} + q^{25} -\beta q^{27} - q^{29} - q^{41} + \beta q^{43} + 2 q^{45} - q^{61} -\beta q^{67} -3 q^{69} -\beta q^{75} + q^{81} + \beta q^{83} + \beta q^{87} + q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} + 4q^{9} + O(q^{10})$$ $$2q + 2q^{5} + 4q^{9} + 2q^{25} - 2q^{29} - 2q^{41} + 4q^{45} - 2q^{61} - 6q^{69} + 2q^{81} + 2q^{89} + O(q^{100})$$

## Character values

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

 $$n$$ $$981$$ $$1471$$ $$3041$$ $$3137$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3039.1
 1.73205 −1.73205
0 −1.73205 0 1.00000 0 0 0 2.00000 0
3039.2 0 1.73205 0 1.00000 0 0 0 2.00000 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3920.1.j.d 2
4.b odd 2 1 inner 3920.1.j.d 2
5.b even 2 1 inner 3920.1.j.d 2
7.b odd 2 1 3920.1.j.b 2
7.c even 3 2 3920.1.bt.c 4
7.d odd 6 2 560.1.bt.a 4
20.d odd 2 1 CM 3920.1.j.d 2
28.d even 2 1 3920.1.j.b 2
28.f even 6 2 560.1.bt.a 4
28.g odd 6 2 3920.1.bt.c 4
35.c odd 2 1 3920.1.j.b 2
35.i odd 6 2 560.1.bt.a 4
35.j even 6 2 3920.1.bt.c 4
35.k even 12 2 2800.1.ce.a 2
35.k even 12 2 2800.1.ce.b 2
56.j odd 6 2 2240.1.bt.c 4
56.m even 6 2 2240.1.bt.c 4
140.c even 2 1 3920.1.j.b 2
140.p odd 6 2 3920.1.bt.c 4
140.s even 6 2 560.1.bt.a 4
140.x odd 12 2 2800.1.ce.a 2
140.x odd 12 2 2800.1.ce.b 2
280.ba even 6 2 2240.1.bt.c 4
280.bk odd 6 2 2240.1.bt.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.1.bt.a 4 7.d odd 6 2
560.1.bt.a 4 28.f even 6 2
560.1.bt.a 4 35.i odd 6 2
560.1.bt.a 4 140.s even 6 2
2240.1.bt.c 4 56.j odd 6 2
2240.1.bt.c 4 56.m even 6 2
2240.1.bt.c 4 280.ba even 6 2
2240.1.bt.c 4 280.bk odd 6 2
2800.1.ce.a 2 35.k even 12 2
2800.1.ce.a 2 140.x odd 12 2
2800.1.ce.b 2 35.k even 12 2
2800.1.ce.b 2 140.x odd 12 2
3920.1.j.b 2 7.b odd 2 1
3920.1.j.b 2 28.d even 2 1
3920.1.j.b 2 35.c odd 2 1
3920.1.j.b 2 140.c even 2 1
3920.1.j.d 2 1.a even 1 1 trivial
3920.1.j.d 2 4.b odd 2 1 inner
3920.1.j.d 2 5.b even 2 1 inner
3920.1.j.d 2 20.d odd 2 1 CM
3920.1.bt.c 4 7.c even 3 2
3920.1.bt.c 4 28.g odd 6 2
3920.1.bt.c 4 35.j even 6 2
3920.1.bt.c 4 140.p odd 6 2

## Hecke kernels

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

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

## Hecke characteristic polynomials

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