# Properties

 Label 3920.2.a.p 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

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{5} - 2q^{9} + O(q^{10})$$ $$q - q^{3} - q^{5} - 2q^{9} + 6q^{11} - 4q^{13} + q^{15} - 2q^{19} + 3q^{23} + q^{25} + 5q^{27} - 3q^{29} - 8q^{31} - 6q^{33} - 4q^{37} + 4q^{39} + 9q^{41} + 7q^{43} + 2q^{45} - 6q^{53} - 6q^{55} + 2q^{57} + 6q^{59} + 5q^{61} + 4q^{65} - 5q^{67} - 3q^{69} + 6q^{71} - 16q^{73} - q^{75} - 2q^{79} + q^{81} - 3q^{83} + 3q^{87} - 15q^{89} + 8q^{93} + 2q^{95} + 14q^{97} - 12q^{99} + O(q^{100})$$

## 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 −1.00000 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

## 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.p 1
4.b odd 2 1 490.2.a.c 1
7.b odd 2 1 3920.2.a.bc 1
7.c even 3 2 560.2.q.g 2
12.b even 2 1 4410.2.a.bm 1
20.d odd 2 1 2450.2.a.w 1
20.e even 4 2 2450.2.c.g 2
28.d even 2 1 490.2.a.b 1
28.f even 6 2 490.2.e.h 2
28.g odd 6 2 70.2.e.c 2
84.h odd 2 1 4410.2.a.bd 1
84.n even 6 2 630.2.k.b 2
140.c even 2 1 2450.2.a.bc 1
140.j odd 4 2 2450.2.c.l 2
140.p odd 6 2 350.2.e.e 2
140.w even 12 4 350.2.j.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.c 2 28.g odd 6 2
350.2.e.e 2 140.p odd 6 2
350.2.j.b 4 140.w even 12 4
490.2.a.b 1 28.d even 2 1
490.2.a.c 1 4.b odd 2 1
490.2.e.h 2 28.f even 6 2
560.2.q.g 2 7.c even 3 2
630.2.k.b 2 84.n even 6 2
2450.2.a.w 1 20.d odd 2 1
2450.2.a.bc 1 140.c even 2 1
2450.2.c.g 2 20.e even 4 2
2450.2.c.l 2 140.j odd 4 2
3920.2.a.p 1 1.a even 1 1 trivial
3920.2.a.bc 1 7.b odd 2 1
4410.2.a.bd 1 84.h odd 2 1
4410.2.a.bm 1 12.b even 2 1

## Hecke kernels

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

 $$T_{3} + 1$$ $$T_{11} - 6$$ $$T_{13} + 4$$ $$T_{17}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$1 + T$$
$5$ $$1 + T$$
$7$ $$T$$
$11$ $$-6 + T$$
$13$ $$4 + T$$
$17$ $$T$$
$19$ $$2 + T$$
$23$ $$-3 + T$$
$29$ $$3 + T$$
$31$ $$8 + T$$
$37$ $$4 + T$$
$41$ $$-9 + T$$
$43$ $$-7 + T$$
$47$ $$T$$
$53$ $$6 + T$$
$59$ $$-6 + T$$
$61$ $$-5 + T$$
$67$ $$5 + T$$
$71$ $$-6 + T$$
$73$ $$16 + T$$
$79$ $$2 + T$$
$83$ $$3 + T$$
$89$ $$15 + T$$
$97$ $$-14 + T$$