# Properties

 Label 3920.2.a.br Level $3920$ Weight $2$ Character orbit 3920.a Self dual yes Analytic conductor $31.301$ Analytic rank $1$ Dimension $2$ 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: $$2$$ 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 245) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{3} + q^{5} -2 \beta q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{3} + q^{5} -2 \beta q^{9} + ( 3 - 2 \beta ) q^{11} + ( 3 + \beta ) q^{13} + ( -1 + \beta ) q^{15} + ( -1 + 3 \beta ) q^{17} -6 q^{19} + ( -6 - \beta ) q^{23} + q^{25} + ( -1 - \beta ) q^{27} + ( -3 - 4 \beta ) q^{29} + ( -6 - 3 \beta ) q^{31} + ( -7 + 5 \beta ) q^{33} + ( -2 + 3 \beta ) q^{37} + ( -1 + 2 \beta ) q^{39} + ( -2 - 3 \beta ) q^{41} -2 q^{43} -2 \beta q^{45} + ( 3 + 3 \beta ) q^{47} + ( 7 - 4 \beta ) q^{51} + 3 \beta q^{53} + ( 3 - 2 \beta ) q^{55} + ( 6 - 6 \beta ) q^{57} + ( -2 + 3 \beta ) q^{59} -2 \beta q^{61} + ( 3 + \beta ) q^{65} + ( 4 + 3 \beta ) q^{67} + ( 4 - 5 \beta ) q^{69} + ( 6 - 2 \beta ) q^{71} -6 \beta q^{73} + ( -1 + \beta ) q^{75} + ( 7 - 6 \beta ) q^{79} + ( -1 + 6 \beta ) q^{81} + ( -1 + 3 \beta ) q^{85} + ( -5 + \beta ) q^{87} -8 q^{89} -3 \beta q^{93} -6 q^{95} + ( 9 + 3 \beta ) q^{97} + ( 8 - 6 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{5} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{5} + 6q^{11} + 6q^{13} - 2q^{15} - 2q^{17} - 12q^{19} - 12q^{23} + 2q^{25} - 2q^{27} - 6q^{29} - 12q^{31} - 14q^{33} - 4q^{37} - 2q^{39} - 4q^{41} - 4q^{43} + 6q^{47} + 14q^{51} + 6q^{55} + 12q^{57} - 4q^{59} + 6q^{65} + 8q^{67} + 8q^{69} + 12q^{71} - 2q^{75} + 14q^{79} - 2q^{81} - 2q^{85} - 10q^{87} - 16q^{89} - 12q^{95} + 18q^{97} + 16q^{99} + O(q^{100})$$

## 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.br 2
4.b odd 2 1 245.2.a.f yes 2
7.b odd 2 1 3920.2.a.bw 2
12.b even 2 1 2205.2.a.t 2
20.d odd 2 1 1225.2.a.p 2
20.e even 4 2 1225.2.b.j 4
28.d even 2 1 245.2.a.e 2
28.f even 6 2 245.2.e.g 4
28.g odd 6 2 245.2.e.f 4
84.h odd 2 1 2205.2.a.v 2
140.c even 2 1 1225.2.a.r 2
140.j odd 4 2 1225.2.b.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.a.e 2 28.d even 2 1
245.2.a.f yes 2 4.b odd 2 1
245.2.e.f 4 28.g odd 6 2
245.2.e.g 4 28.f even 6 2
1225.2.a.p 2 20.d odd 2 1
1225.2.a.r 2 140.c even 2 1
1225.2.b.i 4 140.j odd 4 2
1225.2.b.j 4 20.e even 4 2
2205.2.a.t 2 12.b even 2 1
2205.2.a.v 2 84.h odd 2 1
3920.2.a.br 2 1.a even 1 1 trivial
3920.2.a.bw 2 7.b odd 2 1

## Hecke kernels

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

 $$T_{3}^{2} + 2 T_{3} - 1$$ $$T_{11}^{2} - 6 T_{11} + 1$$ $$T_{13}^{2} - 6 T_{13} + 7$$ $$T_{17}^{2} + 2 T_{17} - 17$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-1 + 2 T + T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$1 - 6 T + T^{2}$$
$13$ $$7 - 6 T + T^{2}$$
$17$ $$-17 + 2 T + T^{2}$$
$19$ $$( 6 + T )^{2}$$
$23$ $$34 + 12 T + T^{2}$$
$29$ $$-23 + 6 T + T^{2}$$
$31$ $$18 + 12 T + T^{2}$$
$37$ $$-14 + 4 T + T^{2}$$
$41$ $$-14 + 4 T + T^{2}$$
$43$ $$( 2 + T )^{2}$$
$47$ $$-9 - 6 T + T^{2}$$
$53$ $$-18 + T^{2}$$
$59$ $$-14 + 4 T + T^{2}$$
$61$ $$-8 + T^{2}$$
$67$ $$-2 - 8 T + T^{2}$$
$71$ $$28 - 12 T + T^{2}$$
$73$ $$-72 + T^{2}$$
$79$ $$-23 - 14 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$( 8 + T )^{2}$$
$97$ $$63 - 18 T + T^{2}$$
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