# Properties

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

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## 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} - 2q^{9} + O(q^{10})$$ $$q - q^{3} + q^{5} - 2q^{9} + 5q^{11} - q^{13} - q^{15} - 3q^{17} - 6q^{19} + 6q^{23} + q^{25} + 5q^{27} - 9q^{29} - 5q^{33} + 6q^{37} + q^{39} - 8q^{41} - 6q^{43} - 2q^{45} + 3q^{47} + 3q^{51} - 12q^{53} + 5q^{55} + 6q^{57} + 8q^{59} + 4q^{61} - q^{65} + 4q^{67} - 6q^{69} - 8q^{71} - 10q^{73} - q^{75} + 3q^{79} + q^{81} - 12q^{83} - 3q^{85} + 9q^{87} + 16q^{89} - 6q^{95} - 7q^{97} - 10q^{99} + O(q^{100})$$

## 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.r 1
4.b odd 2 1 1960.2.a.k 1
7.b odd 2 1 560.2.a.e 1
20.d odd 2 1 9800.2.a.n 1
21.c even 2 1 5040.2.a.be 1
28.d even 2 1 280.2.a.b 1
28.f even 6 2 1960.2.q.m 2
28.g odd 6 2 1960.2.q.e 2
35.c odd 2 1 2800.2.a.i 1
35.f even 4 2 2800.2.g.m 2
56.e even 2 1 2240.2.a.v 1
56.h odd 2 1 2240.2.a.j 1
84.h odd 2 1 2520.2.a.p 1
140.c even 2 1 1400.2.a.k 1
140.j odd 4 2 1400.2.g.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.b 1 28.d even 2 1
560.2.a.e 1 7.b odd 2 1
1400.2.a.k 1 140.c even 2 1
1400.2.g.e 2 140.j odd 4 2
1960.2.a.k 1 4.b odd 2 1
1960.2.q.e 2 28.g odd 6 2
1960.2.q.m 2 28.f even 6 2
2240.2.a.j 1 56.h odd 2 1
2240.2.a.v 1 56.e even 2 1
2520.2.a.p 1 84.h odd 2 1
2800.2.a.i 1 35.c odd 2 1
2800.2.g.m 2 35.f even 4 2
3920.2.a.r 1 1.a even 1 1 trivial
5040.2.a.be 1 21.c even 2 1
9800.2.a.n 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3920))$$:

 $$T_{3} + 1$$ $$T_{11} - 5$$ $$T_{13} + 1$$ $$T_{17} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$1 + T$$
$5$ $$-1 + T$$
$7$ $$T$$
$11$ $$-5 + T$$
$13$ $$1 + T$$
$17$ $$3 + T$$
$19$ $$6 + T$$
$23$ $$-6 + T$$
$29$ $$9 + T$$
$31$ $$T$$
$37$ $$-6 + T$$
$41$ $$8 + T$$
$43$ $$6 + T$$
$47$ $$-3 + T$$
$53$ $$12 + T$$
$59$ $$-8 + T$$
$61$ $$-4 + T$$
$67$ $$-4 + T$$
$71$ $$8 + T$$
$73$ $$10 + T$$
$79$ $$-3 + T$$
$83$ $$12 + T$$
$89$ $$-16 + T$$
$97$ $$7 + T$$
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