# Properties

 Label 3920.2.a.bd 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 1960) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{3} - q^{5} + q^{9} + O(q^{10})$$ $$q + 2q^{3} - q^{5} + q^{9} - 4q^{11} - 2q^{13} - 2q^{15} + 2q^{19} + 4q^{23} + q^{25} - 4q^{27} + 10q^{29} + 4q^{31} - 8q^{33} - 2q^{37} - 4q^{39} + 12q^{41} + 4q^{43} - q^{45} + 4q^{47} + 2q^{53} + 4q^{55} + 4q^{57} + 10q^{59} - 6q^{61} + 2q^{65} - 4q^{67} + 8q^{69} + 12q^{71} + 4q^{73} + 2q^{75} + 4q^{79} - 11q^{81} + 14q^{83} + 20q^{87} - 8q^{89} + 8q^{93} - 2q^{95} + 8q^{97} - 4q^{99} + O(q^{100})$$

## 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 2.00000 0 −1.00000 0 0 0 1.00000 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.bd 1
4.b odd 2 1 1960.2.a.b 1
7.b odd 2 1 3920.2.a.f 1
20.d odd 2 1 9800.2.a.bn 1
28.d even 2 1 1960.2.a.n yes 1
28.f even 6 2 1960.2.q.b 2
28.g odd 6 2 1960.2.q.n 2
140.c even 2 1 9800.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.a.b 1 4.b odd 2 1
1960.2.a.n yes 1 28.d even 2 1
1960.2.q.b 2 28.f even 6 2
1960.2.q.n 2 28.g odd 6 2
3920.2.a.f 1 7.b odd 2 1
3920.2.a.bd 1 1.a even 1 1 trivial
9800.2.a.i 1 140.c even 2 1
9800.2.a.bn 1 20.d odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-2 + T$$
$5$ $$1 + T$$
$7$ $$T$$
$11$ $$4 + T$$
$13$ $$2 + T$$
$17$ $$T$$
$19$ $$-2 + T$$
$23$ $$-4 + T$$
$29$ $$-10 + T$$
$31$ $$-4 + T$$
$37$ $$2 + T$$
$41$ $$-12 + T$$
$43$ $$-4 + T$$
$47$ $$-4 + T$$
$53$ $$-2 + T$$
$59$ $$-10 + T$$
$61$ $$6 + T$$
$67$ $$4 + T$$
$71$ $$-12 + T$$
$73$ $$-4 + T$$
$79$ $$-4 + T$$
$83$ $$-14 + T$$
$89$ $$8 + T$$
$97$ $$-8 + T$$