# Properties

 Label 3920.2.a.bk Level $3920$ Weight $2$ Character orbit 3920.a Self dual yes Analytic conductor $31.301$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3920 = 2^{4} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3920.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$31.3013575923$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3q^{3} + q^{5} + 6q^{9} + O(q^{10})$$ $$q + 3q^{3} + q^{5} + 6q^{9} + 2q^{11} + 3q^{15} + 4q^{17} - 6q^{19} - 3q^{23} + q^{25} + 9q^{27} + 9q^{29} - 4q^{31} + 6q^{33} - 4q^{37} + 7q^{41} + 5q^{43} + 6q^{45} + 8q^{47} + 12q^{51} - 2q^{53} + 2q^{55} - 18q^{57} + 10q^{59} - q^{61} + 9q^{67} - 9q^{69} - 2q^{71} + 4q^{73} + 3q^{75} - 10q^{79} + 9q^{81} - 7q^{83} + 4q^{85} + 27q^{87} - q^{89} - 12q^{93} - 6q^{95} - 14q^{97} + 12q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 3.00000 0 1.00000 0 0 0 6.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3920.2.a.bk 1
4.b odd 2 1 490.2.a.e 1
7.b odd 2 1 3920.2.a.b 1
7.d odd 6 2 560.2.q.i 2
12.b even 2 1 4410.2.a.h 1
20.d odd 2 1 2450.2.a.q 1
20.e even 4 2 2450.2.c.a 2
28.d even 2 1 490.2.a.k 1
28.f even 6 2 70.2.e.a 2
28.g odd 6 2 490.2.e.f 2
84.h odd 2 1 4410.2.a.r 1
84.j odd 6 2 630.2.k.f 2
140.c even 2 1 2450.2.a.b 1
140.j odd 4 2 2450.2.c.s 2
140.s even 6 2 350.2.e.l 2
140.x odd 12 4 350.2.j.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.a 2 28.f even 6 2
350.2.e.l 2 140.s even 6 2
350.2.j.f 4 140.x odd 12 4
490.2.a.e 1 4.b odd 2 1
490.2.a.k 1 28.d even 2 1
490.2.e.f 2 28.g odd 6 2
560.2.q.i 2 7.d odd 6 2
630.2.k.f 2 84.j odd 6 2
2450.2.a.b 1 140.c even 2 1
2450.2.a.q 1 20.d odd 2 1
2450.2.c.a 2 20.e even 4 2
2450.2.c.s 2 140.j odd 4 2
3920.2.a.b 1 7.b odd 2 1
3920.2.a.bk 1 1.a even 1 1 trivial
4410.2.a.h 1 12.b even 2 1
4410.2.a.r 1 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3920))$$:

 $$T_{3} - 3$$ $$T_{11} - 2$$ $$T_{13}$$ $$T_{17} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-3 + T$$
$5$ $$-1 + T$$
$7$ $$T$$
$11$ $$-2 + T$$
$13$ $$T$$
$17$ $$-4 + T$$
$19$ $$6 + T$$
$23$ $$3 + T$$
$29$ $$-9 + T$$
$31$ $$4 + T$$
$37$ $$4 + T$$
$41$ $$-7 + T$$
$43$ $$-5 + T$$
$47$ $$-8 + T$$
$53$ $$2 + T$$
$59$ $$-10 + T$$
$61$ $$1 + T$$
$67$ $$-9 + T$$
$71$ $$2 + T$$
$73$ $$-4 + T$$
$79$ $$10 + T$$
$83$ $$7 + T$$
$89$ $$1 + T$$
$97$ $$14 + T$$