# Properties

 Label 3920.2.a.bo Level $3920$ Weight $2$ Character orbit 3920.a Self dual yes Analytic conductor $31.301$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 980) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{3} - q^{5} -2 \beta q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{3} - q^{5} -2 \beta q^{9} + ( 1 + 2 \beta ) q^{11} + ( 5 - \beta ) q^{13} + ( 1 - \beta ) q^{15} + ( 5 + \beta ) q^{17} + ( 2 - 4 \beta ) q^{19} + ( -2 + \beta ) q^{23} + q^{25} + ( -1 - \beta ) q^{27} + ( 1 + 4 \beta ) q^{29} + ( -6 + \beta ) q^{31} + ( 3 - \beta ) q^{33} + ( -2 - \beta ) q^{37} + ( -7 + 6 \beta ) q^{39} + ( 2 - \beta ) q^{41} + ( -6 - 4 \beta ) q^{43} + 2 \beta q^{45} + ( -1 + 7 \beta ) q^{47} + ( -3 + 4 \beta ) q^{51} + ( -8 + 3 \beta ) q^{53} + ( -1 - 2 \beta ) q^{55} + ( -10 + 6 \beta ) q^{57} + ( -2 - \beta ) q^{59} + ( 8 - 2 \beta ) q^{61} + ( -5 + \beta ) q^{65} + ( 4 + 5 \beta ) q^{67} + ( 4 - 3 \beta ) q^{69} + ( 2 - 6 \beta ) q^{71} + ( 8 - 2 \beta ) q^{73} + ( -1 + \beta ) q^{75} + ( 1 - 10 \beta ) q^{79} + ( -1 + 6 \beta ) q^{81} + 8 q^{83} + ( -5 - \beta ) q^{85} + ( 7 - 3 \beta ) q^{87} + 12 \beta q^{89} + ( 8 - 7 \beta ) q^{93} + ( -2 + 4 \beta ) q^{95} + ( 3 + 9 \beta ) q^{97} + ( -8 - 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 2q^{5} + O(q^{10})$$ $$2q - 2q^{3} - 2q^{5} + 2q^{11} + 10q^{13} + 2q^{15} + 10q^{17} + 4q^{19} - 4q^{23} + 2q^{25} - 2q^{27} + 2q^{29} - 12q^{31} + 6q^{33} - 4q^{37} - 14q^{39} + 4q^{41} - 12q^{43} - 2q^{47} - 6q^{51} - 16q^{53} - 2q^{55} - 20q^{57} - 4q^{59} + 16q^{61} - 10q^{65} + 8q^{67} + 8q^{69} + 4q^{71} + 16q^{73} - 2q^{75} + 2q^{79} - 2q^{81} + 16q^{83} - 10q^{85} + 14q^{87} + 16q^{93} - 4q^{95} + 6q^{97} - 16q^{99} + O(q^{100})$$

## 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
 −1.41421 1.41421
0 −2.41421 0 −1.00000 0 0 0 2.82843 0
1.2 0 0.414214 0 −1.00000 0 0 0 −2.82843 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.bo 2
4.b odd 2 1 980.2.a.k yes 2
7.b odd 2 1 3920.2.a.bx 2
12.b even 2 1 8820.2.a.bl 2
20.d odd 2 1 4900.2.a.x 2
20.e even 4 2 4900.2.e.r 4
28.d even 2 1 980.2.a.j 2
28.f even 6 2 980.2.i.l 4
28.g odd 6 2 980.2.i.k 4
84.h odd 2 1 8820.2.a.bg 2
140.c even 2 1 4900.2.a.z 2
140.j odd 4 2 4900.2.e.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.a.j 2 28.d even 2 1
980.2.a.k yes 2 4.b odd 2 1
980.2.i.k 4 28.g odd 6 2
980.2.i.l 4 28.f even 6 2
3920.2.a.bo 2 1.a even 1 1 trivial
3920.2.a.bx 2 7.b odd 2 1
4900.2.a.x 2 20.d odd 2 1
4900.2.a.z 2 140.c even 2 1
4900.2.e.q 4 140.j odd 4 2
4900.2.e.r 4 20.e even 4 2
8820.2.a.bg 2 84.h odd 2 1
8820.2.a.bl 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3920))$$:

 $$T_{3}^{2} + 2 T_{3} - 1$$ $$T_{11}^{2} - 2 T_{11} - 7$$ $$T_{13}^{2} - 10 T_{13} + 23$$ $$T_{17}^{2} - 10 T_{17} + 23$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-1 + 2 T + T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$-7 - 2 T + T^{2}$$
$13$ $$23 - 10 T + T^{2}$$
$17$ $$23 - 10 T + T^{2}$$
$19$ $$-28 - 4 T + T^{2}$$
$23$ $$2 + 4 T + T^{2}$$
$29$ $$-31 - 2 T + T^{2}$$
$31$ $$34 + 12 T + T^{2}$$
$37$ $$2 + 4 T + T^{2}$$
$41$ $$2 - 4 T + T^{2}$$
$43$ $$4 + 12 T + T^{2}$$
$47$ $$-97 + 2 T + T^{2}$$
$53$ $$46 + 16 T + T^{2}$$
$59$ $$2 + 4 T + T^{2}$$
$61$ $$56 - 16 T + T^{2}$$
$67$ $$-34 - 8 T + T^{2}$$
$71$ $$-68 - 4 T + T^{2}$$
$73$ $$56 - 16 T + T^{2}$$
$79$ $$-199 - 2 T + T^{2}$$
$83$ $$( -8 + T )^{2}$$
$89$ $$-288 + T^{2}$$
$97$ $$-153 - 6 T + T^{2}$$