Properties

 Label 3920.2.a.ba 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 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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.ba 1
4.b odd 2 1 245.2.a.c 1
7.b odd 2 1 560.2.a.b 1
12.b even 2 1 2205.2.a.e 1
20.d odd 2 1 1225.2.a.e 1
20.e even 4 2 1225.2.b.d 2
21.c even 2 1 5040.2.a.v 1
28.d even 2 1 35.2.a.a 1
28.f even 6 2 245.2.e.a 2
28.g odd 6 2 245.2.e.b 2
35.c odd 2 1 2800.2.a.z 1
35.f even 4 2 2800.2.g.l 2
56.e even 2 1 2240.2.a.k 1
56.h odd 2 1 2240.2.a.u 1
84.h odd 2 1 315.2.a.b 1
140.c even 2 1 175.2.a.b 1
140.j odd 4 2 175.2.b.a 2
308.g odd 2 1 4235.2.a.c 1
364.h even 2 1 5915.2.a.f 1
420.o odd 2 1 1575.2.a.f 1
420.w even 4 2 1575.2.d.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.a 1 28.d even 2 1
175.2.a.b 1 140.c even 2 1
175.2.b.a 2 140.j odd 4 2
245.2.a.c 1 4.b odd 2 1
245.2.e.a 2 28.f even 6 2
245.2.e.b 2 28.g odd 6 2
315.2.a.b 1 84.h odd 2 1
560.2.a.b 1 7.b odd 2 1
1225.2.a.e 1 20.d odd 2 1
1225.2.b.d 2 20.e even 4 2
1575.2.a.f 1 420.o odd 2 1
1575.2.d.c 2 420.w even 4 2
2205.2.a.e 1 12.b even 2 1
2240.2.a.k 1 56.e even 2 1
2240.2.a.u 1 56.h odd 2 1
2800.2.a.z 1 35.c odd 2 1
2800.2.g.l 2 35.f even 4 2
3920.2.a.ba 1 1.a even 1 1 trivial
4235.2.a.c 1 308.g odd 2 1
5040.2.a.v 1 21.c even 2 1
5915.2.a.f 1 364.h even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

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