# Properties

 Label 3920.2.a.cc Level $3920$ Weight $2$ Character orbit 3920.a Self dual yes Analytic conductor $31.301$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.1944.1 Defining polynomial: $$x^{3} - 9 x - 6$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + q^{5} + ( 3 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + q^{5} + ( 3 + \beta_{1} + \beta_{2} ) q^{9} + ( -1 - \beta_{1} - \beta_{2} ) q^{11} + ( 1 + \beta_{1} - \beta_{2} ) q^{13} + \beta_{1} q^{15} + 2 q^{17} + ( -1 + \beta_{1} - \beta_{2} ) q^{19} + ( -1 + \beta_{2} ) q^{23} + q^{25} + ( 6 + 3 \beta_{1} ) q^{27} + ( 4 - \beta_{1} - \beta_{2} ) q^{29} + ( -4 + 2 \beta_{1} ) q^{31} + ( -6 - 4 \beta_{1} ) q^{33} + ( 3 + \beta_{1} - \beta_{2} ) q^{37} + ( 6 + 2 \beta_{2} ) q^{39} + ( 3 + 2 \beta_{2} ) q^{41} + ( -4 + \beta_{1} ) q^{43} + ( 3 + \beta_{1} + \beta_{2} ) q^{45} + ( 5 + \beta_{1} + 3 \beta_{2} ) q^{47} + 2 \beta_{1} q^{51} + ( 3 - 3 \beta_{1} - \beta_{2} ) q^{53} + ( -1 - \beta_{1} - \beta_{2} ) q^{55} + ( 6 - 2 \beta_{1} + 2 \beta_{2} ) q^{57} -8 q^{59} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{61} + ( 1 + \beta_{1} - \beta_{2} ) q^{65} + ( -2 - 3 \beta_{1} ) q^{67} + ( \beta_{1} - \beta_{2} ) q^{69} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{73} + \beta_{1} q^{75} + ( 6 + 2 \beta_{1} - 2 \beta_{2} ) q^{79} + ( 9 + 6 \beta_{1} ) q^{81} + ( 10 - \beta_{1} ) q^{83} + 2 q^{85} + ( -6 + \beta_{1} ) q^{87} + ( \beta_{1} - \beta_{2} ) q^{89} + ( 12 - 2 \beta_{1} + 2 \beta_{2} ) q^{93} + ( -1 + \beta_{1} - \beta_{2} ) q^{95} -2 q^{97} + ( -21 - 7 \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{5} + 9q^{9} + O(q^{10})$$ $$3q + 3q^{5} + 9q^{9} - 3q^{11} + 3q^{13} + 6q^{17} - 3q^{19} - 3q^{23} + 3q^{25} + 18q^{27} + 12q^{29} - 12q^{31} - 18q^{33} + 9q^{37} + 18q^{39} + 9q^{41} - 12q^{43} + 9q^{45} + 15q^{47} + 9q^{53} - 3q^{55} + 18q^{57} - 24q^{59} - 6q^{61} + 3q^{65} - 6q^{67} + 18q^{79} + 27q^{81} + 30q^{83} + 6q^{85} - 18q^{87} + 36q^{93} - 3q^{95} - 6q^{97} - 63q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 9 x - 6$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 6$$

## 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
 −2.58423 −0.705720 3.28995
0 −2.58423 0 1.00000 0 0 0 3.67822 0
1.2 0 −0.705720 0 1.00000 0 0 0 −2.50196 0
1.3 0 3.28995 0 1.00000 0 0 0 7.82374 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.cc 3
4.b odd 2 1 1960.2.a.w 3
7.b odd 2 1 3920.2.a.cb 3
7.c even 3 2 560.2.q.l 6
20.d odd 2 1 9800.2.a.ce 3
28.d even 2 1 1960.2.a.v 3
28.f even 6 2 1960.2.q.w 6
28.g odd 6 2 280.2.q.e 6
84.n even 6 2 2520.2.bi.q 6
140.c even 2 1 9800.2.a.cf 3
140.p odd 6 2 1400.2.q.j 6
140.w even 12 4 1400.2.bh.i 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.e 6 28.g odd 6 2
560.2.q.l 6 7.c even 3 2
1400.2.q.j 6 140.p odd 6 2
1400.2.bh.i 12 140.w even 12 4
1960.2.a.v 3 28.d even 2 1
1960.2.a.w 3 4.b odd 2 1
1960.2.q.w 6 28.f even 6 2
2520.2.bi.q 6 84.n even 6 2
3920.2.a.cb 3 7.b odd 2 1
3920.2.a.cc 3 1.a even 1 1 trivial
9800.2.a.ce 3 20.d odd 2 1
9800.2.a.cf 3 140.c even 2 1

## Hecke kernels

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

 $$T_{3}^{3} - 9 T_{3} - 6$$ $$T_{11}^{3} + 3 T_{11}^{2} - 24 T_{11} - 44$$ $$T_{13}^{3} - 3 T_{13}^{2} - 24 T_{13} + 68$$ $$T_{17} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$-6 - 9 T + T^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$T^{3}$$
$11$ $$-44 - 24 T + 3 T^{2} + T^{3}$$
$13$ $$68 - 24 T - 3 T^{2} + T^{3}$$
$17$ $$( -2 + T )^{3}$$
$19$ $$16 - 24 T + 3 T^{2} + T^{3}$$
$23$ $$7 - 15 T + 3 T^{2} + T^{3}$$
$29$ $$26 + 21 T - 12 T^{2} + T^{3}$$
$31$ $$-128 + 12 T + 12 T^{2} + T^{3}$$
$37$ $$96 - 9 T^{2} + T^{3}$$
$41$ $$381 - 45 T - 9 T^{2} + T^{3}$$
$43$ $$22 + 39 T + 12 T^{2} + T^{3}$$
$47$ $$1588 - 96 T - 15 T^{2} + T^{3}$$
$53$ $$624 - 72 T - 9 T^{2} + T^{3}$$
$59$ $$( 8 + T )^{3}$$
$61$ $$-544 - 87 T + 6 T^{2} + T^{3}$$
$67$ $$8 - 69 T + 6 T^{2} + T^{3}$$
$71$ $$T^{3}$$
$73$ $$336 - 108 T + T^{3}$$
$79$ $$768 - 18 T^{2} + T^{3}$$
$83$ $$-904 + 291 T - 30 T^{2} + T^{3}$$
$89$ $$42 - 27 T + T^{3}$$
$97$ $$( 2 + T )^{3}$$