# Properties

 Label 3920.1.bh.b Level $3920$ Weight $1$ Character orbit 3920.bh Analytic conductor $1.956$ Analytic rank $0$ Dimension $4$ Projective image $S_{4}$ CM/RM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.95633484952$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 980) Projective image $$S_{4}$$ Projective field Galois closure of 4.2.171500.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{8} q^{3} -\zeta_{8}^{3} q^{5} +O(q^{10})$$ $$q -\zeta_{8} q^{3} -\zeta_{8}^{3} q^{5} + q^{11} + \zeta_{8} q^{13} - q^{15} -\zeta_{8}^{3} q^{17} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{19} + ( 1 + \zeta_{8}^{2} ) q^{23} -\zeta_{8}^{2} q^{25} + \zeta_{8}^{3} q^{27} + \zeta_{8}^{2} q^{29} -\zeta_{8} q^{33} + ( -1 + \zeta_{8}^{2} ) q^{37} -\zeta_{8}^{2} q^{39} + ( 1 + \zeta_{8}^{2} ) q^{43} -\zeta_{8}^{3} q^{47} - q^{51} -\zeta_{8}^{3} q^{55} + ( -1 + \zeta_{8}^{2} ) q^{57} + q^{65} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{69} + \zeta_{8}^{3} q^{75} -\zeta_{8}^{2} q^{79} + q^{81} -\zeta_{8}^{2} q^{85} -\zeta_{8}^{3} q^{87} + ( -1 - \zeta_{8}^{2} ) q^{95} + \zeta_{8}^{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 4q^{11} - 4q^{15} + 4q^{23} - 4q^{37} + 4q^{43} - 4q^{51} - 4q^{57} + 4q^{65} + 4q^{81} - 4q^{95} + O(q^{100})$$

## Character values

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

 $$n$$ $$981$$ $$1471$$ $$3041$$ $$3137$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\zeta_{8}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2353.1
 0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i
0 −0.707107 0.707107i 0 0.707107 0.707107i 0 0 0 0 0
2353.2 0 0.707107 + 0.707107i 0 −0.707107 + 0.707107i 0 0 0 0 0
3137.1 0 −0.707107 + 0.707107i 0 0.707107 + 0.707107i 0 0 0 0 0
3137.2 0 0.707107 0.707107i 0 −0.707107 0.707107i 0 0 0 0 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
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3920.1.bh.b 4
4.b odd 2 1 980.1.l.a 4
5.c odd 4 1 inner 3920.1.bh.b 4
7.b odd 2 1 inner 3920.1.bh.b 4
7.c even 3 2 3920.1.cl.a 8
7.d odd 6 2 3920.1.cl.a 8
20.e even 4 1 980.1.l.a 4
28.d even 2 1 980.1.l.a 4
28.f even 6 2 980.1.w.a 8
28.g odd 6 2 980.1.w.a 8
35.f even 4 1 inner 3920.1.bh.b 4
35.k even 12 2 3920.1.cl.a 8
35.l odd 12 2 3920.1.cl.a 8
140.j odd 4 1 980.1.l.a 4
140.w even 12 2 980.1.w.a 8
140.x odd 12 2 980.1.w.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.1.l.a 4 4.b odd 2 1
980.1.l.a 4 20.e even 4 1
980.1.l.a 4 28.d even 2 1
980.1.l.a 4 140.j odd 4 1
980.1.w.a 8 28.f even 6 2
980.1.w.a 8 28.g odd 6 2
980.1.w.a 8 140.w even 12 2
980.1.w.a 8 140.x odd 12 2
3920.1.bh.b 4 1.a even 1 1 trivial
3920.1.bh.b 4 5.c odd 4 1 inner
3920.1.bh.b 4 7.b odd 2 1 inner
3920.1.bh.b 4 35.f even 4 1 inner
3920.1.cl.a 8 7.c even 3 2
3920.1.cl.a 8 7.d odd 6 2
3920.1.cl.a 8 35.k even 12 2
3920.1.cl.a 8 35.l odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11} - 1$$ acting on $$S_{1}^{\mathrm{new}}(3920, [\chi])$$.

## Hecke characteristic polynomials

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