# Properties

 Label 3360.2.a.bi Level $3360$ Weight $2$ Character orbit 3360.a Self dual yes Analytic conductor $26.830$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3360.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.8297350792$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} - q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + q^{5} - q^{7} + q^{9} + ( -1 + \beta_{2} ) q^{11} + ( 1 + \beta_{1} ) q^{13} - q^{15} + ( -\beta_{1} - \beta_{2} ) q^{17} + ( -3 - \beta_{2} ) q^{19} + q^{21} + ( -\beta_{1} + \beta_{2} ) q^{23} + q^{25} - q^{27} + ( -\beta_{1} - \beta_{2} ) q^{29} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{31} + ( 1 - \beta_{2} ) q^{33} - q^{35} + ( \beta_{1} + \beta_{2} ) q^{37} + ( -1 - \beta_{1} ) q^{39} + 2 \beta_{2} q^{41} + ( -2 - \beta_{1} - \beta_{2} ) q^{43} + q^{45} + ( \beta_{1} - \beta_{2} ) q^{47} + q^{49} + ( \beta_{1} + \beta_{2} ) q^{51} + ( -3 + \beta_{2} ) q^{53} + ( -1 + \beta_{2} ) q^{55} + ( 3 + \beta_{2} ) q^{57} -8 q^{59} + ( -2 - \beta_{1} + \beta_{2} ) q^{61} - q^{63} + ( 1 + \beta_{1} ) q^{65} + ( \beta_{1} + 3 \beta_{2} ) q^{67} + ( \beta_{1} - \beta_{2} ) q^{69} + ( -1 + \beta_{1} ) q^{71} + ( 3 - \beta_{1} ) q^{73} - q^{75} + ( 1 - \beta_{2} ) q^{77} -4 q^{79} + q^{81} + ( -2 - 2 \beta_{2} ) q^{83} + ( -\beta_{1} - \beta_{2} ) q^{85} + ( \beta_{1} + \beta_{2} ) q^{87} + 2 \beta_{1} q^{89} + ( -1 - \beta_{1} ) q^{91} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{93} + ( -3 - \beta_{2} ) q^{95} + ( 1 + \beta_{1} ) q^{97} + ( -1 + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{3} + 3q^{5} - 3q^{7} + 3q^{9} + O(q^{10})$$ $$3q - 3q^{3} + 3q^{5} - 3q^{7} + 3q^{9} - 4q^{11} + 2q^{13} - 3q^{15} + 2q^{17} - 8q^{19} + 3q^{21} + 3q^{25} - 3q^{27} + 2q^{29} - 8q^{31} + 4q^{33} - 3q^{35} - 2q^{37} - 2q^{39} - 2q^{41} - 4q^{43} + 3q^{45} + 3q^{49} - 2q^{51} - 10q^{53} - 4q^{55} + 8q^{57} - 24q^{59} - 6q^{61} - 3q^{63} + 2q^{65} - 4q^{67} - 4q^{71} + 10q^{73} - 3q^{75} + 4q^{77} - 12q^{79} + 3q^{81} - 4q^{83} + 2q^{85} - 2q^{87} - 2q^{89} - 2q^{91} + 8q^{93} - 8q^{95} + 2q^{97} - 4q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-2 \nu^{2} + 4 \nu + 3$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} - 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 2$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + 5$$$$)/2$$

## 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
 0.311108 −1.48119 2.17009
0 −1.00000 0 1.00000 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −1.00000 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 −1.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## 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 3360.2.a.bi 3
4.b odd 2 1 3360.2.a.bj yes 3
8.b even 2 1 6720.2.a.db 3
8.d odd 2 1 6720.2.a.da 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.2.a.bi 3 1.a even 1 1 trivial
3360.2.a.bj yes 3 4.b odd 2 1
6720.2.a.da 3 8.d odd 2 1
6720.2.a.db 3 8.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3360))$$:

 $$T_{11}^{3} + 4 T_{11}^{2} - 16 T_{11} - 32$$ $$T_{13}^{3} - 2 T_{13}^{2} - 36 T_{13} + 104$$ $$T_{17}^{3} - 2 T_{17}^{2} - 52 T_{17} + 40$$ $$T_{19}^{3} + 8 T_{19}^{2} - 32$$ $$T_{23}^{3} - 64 T_{23} + 128$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$-32 - 16 T + 4 T^{2} + T^{3}$$
$13$ $$104 - 36 T - 2 T^{2} + T^{3}$$
$17$ $$40 - 52 T - 2 T^{2} + T^{3}$$
$19$ $$-32 + 8 T^{2} + T^{3}$$
$23$ $$128 - 64 T + T^{3}$$
$29$ $$40 - 52 T - 2 T^{2} + T^{3}$$
$31$ $$-928 - 112 T + 8 T^{2} + T^{3}$$
$37$ $$-40 - 52 T + 2 T^{2} + T^{3}$$
$41$ $$-104 - 84 T + 2 T^{2} + T^{3}$$
$43$ $$-64 - 48 T + 4 T^{2} + T^{3}$$
$47$ $$-128 - 64 T + T^{3}$$
$53$ $$-40 + 12 T + 10 T^{2} + T^{3}$$
$59$ $$( 8 + T )^{3}$$
$61$ $$8 - 52 T + 6 T^{2} + T^{3}$$
$67$ $$-1472 - 208 T + 4 T^{2} + T^{3}$$
$71$ $$32 - 32 T + 4 T^{2} + T^{3}$$
$73$ $$8 - 4 T - 10 T^{2} + T^{3}$$
$79$ $$( 4 + T )^{3}$$
$83$ $$-64 - 80 T + 4 T^{2} + T^{3}$$
$89$ $$536 - 148 T + 2 T^{2} + T^{3}$$
$97$ $$104 - 36 T - 2 T^{2} + T^{3}$$