Properties

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

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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q + 2q^{3} - q^{5} + q^{9} + O(q^{10})$$ $$q + 2q^{3} - q^{5} + q^{9} + q^{11} - 3q^{13} - 2q^{15} - 2q^{17} + 5q^{19} - 7q^{23} + q^{25} - 4q^{27} - 6q^{29} - 4q^{31} + 2q^{33} - 5q^{37} - 6q^{39} - 5q^{41} - 6q^{43} - q^{45} + 9q^{47} - 4q^{51} + 11q^{53} - q^{55} + 10q^{57} - 8q^{59} - 12q^{61} + 3q^{65} + 4q^{67} - 14q^{69} + 4q^{71} + 12q^{73} + 2q^{75} - 14q^{79} - 11q^{81} + 4q^{83} + 2q^{85} - 12q^{87} + 6q^{89} - 8q^{93} - 5q^{95} + 6q^{97} + q^{99} + O(q^{100})$$

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.bf 1
4.b odd 2 1 1960.2.a.a 1
7.b odd 2 1 3920.2.a.i 1
7.c even 3 2 560.2.q.c 2
20.d odd 2 1 9800.2.a.bi 1
28.d even 2 1 1960.2.a.m 1
28.f even 6 2 1960.2.q.c 2
28.g odd 6 2 280.2.q.c 2
84.n even 6 2 2520.2.bi.e 2
140.c even 2 1 9800.2.a.g 1
140.p odd 6 2 1400.2.q.a 2
140.w even 12 4 1400.2.bh.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.c 2 28.g odd 6 2
560.2.q.c 2 7.c even 3 2
1400.2.q.a 2 140.p odd 6 2
1400.2.bh.e 4 140.w even 12 4
1960.2.a.a 1 4.b odd 2 1
1960.2.a.m 1 28.d even 2 1
1960.2.q.c 2 28.f even 6 2
2520.2.bi.e 2 84.n even 6 2
3920.2.a.i 1 7.b odd 2 1
3920.2.a.bf 1 1.a even 1 1 trivial
9800.2.a.g 1 140.c even 2 1
9800.2.a.bi 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3920))$$:

 $$T_{3} - 2$$ $$T_{11} - 1$$ $$T_{13} + 3$$ $$T_{17} + 2$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-2 + T$$
$5$ $$1 + T$$
$7$ $$T$$
$11$ $$-1 + T$$
$13$ $$3 + T$$
$17$ $$2 + T$$
$19$ $$-5 + T$$
$23$ $$7 + T$$
$29$ $$6 + T$$
$31$ $$4 + T$$
$37$ $$5 + T$$
$41$ $$5 + T$$
$43$ $$6 + T$$
$47$ $$-9 + T$$
$53$ $$-11 + T$$
$59$ $$8 + T$$
$61$ $$12 + T$$
$67$ $$-4 + T$$
$71$ $$-4 + T$$
$73$ $$-12 + T$$
$79$ $$14 + T$$
$83$ $$-4 + T$$
$89$ $$-6 + T$$
$97$ $$-6 + T$$