# Properties

 Label 3360.2.a.bg Level $3360$ Weight $2$ Character orbit 3360.a Self dual yes Analytic conductor $26.830$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{17}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} + q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} - q^{5} + q^{7} + q^{9} + ( 1 + \beta ) q^{11} + ( 1 - \beta ) q^{13} - q^{15} + 2 q^{17} + ( 3 - \beta ) q^{19} + q^{21} + q^{25} + q^{27} -2 q^{29} + ( 1 + \beta ) q^{31} + ( 1 + \beta ) q^{33} - q^{35} -2 q^{37} + ( 1 - \beta ) q^{39} + 2 q^{41} + ( 2 + 2 \beta ) q^{43} - q^{45} + q^{49} + 2 q^{51} + ( 1 + 3 \beta ) q^{53} + ( -1 - \beta ) q^{55} + ( 3 - \beta ) q^{57} -4 q^{59} -2 \beta q^{61} + q^{63} + ( -1 + \beta ) q^{65} + ( 6 - 2 \beta ) q^{67} + ( -1 - \beta ) q^{71} + ( 11 + \beta ) q^{73} + q^{75} + ( 1 + \beta ) q^{77} + ( 6 - 2 \beta ) q^{79} + q^{81} -4 q^{83} -2 q^{85} -2 q^{87} + ( 8 - 2 \beta ) q^{89} + ( 1 - \beta ) q^{91} + ( 1 + \beta ) q^{93} + ( -3 + \beta ) q^{95} + ( 9 - \beta ) q^{97} + ( 1 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 2 q^{15} + 4 q^{17} + 6 q^{19} + 2 q^{21} + 2 q^{25} + 2 q^{27} - 4 q^{29} + 2 q^{31} + 2 q^{33} - 2 q^{35} - 4 q^{37} + 2 q^{39} + 4 q^{41} + 4 q^{43} - 2 q^{45} + 2 q^{49} + 4 q^{51} + 2 q^{53} - 2 q^{55} + 6 q^{57} - 8 q^{59} + 2 q^{63} - 2 q^{65} + 12 q^{67} - 2 q^{71} + 22 q^{73} + 2 q^{75} + 2 q^{77} + 12 q^{79} + 2 q^{81} - 8 q^{83} - 4 q^{85} - 4 q^{87} + 16 q^{89} + 2 q^{91} + 2 q^{93} - 6 q^{95} + 18 q^{97} + 2 q^{99} + O(q^{100})$$

## 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
 −1.56155 2.56155
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
 $$n$$: 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.bg yes 2
4.b odd 2 1 3360.2.a.ba 2
8.b even 2 1 6720.2.a.ct 2
8.d odd 2 1 6720.2.a.cw 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.2.a.ba 2 4.b odd 2 1
3360.2.a.bg yes 2 1.a even 1 1 trivial
6720.2.a.ct 2 8.b even 2 1
6720.2.a.cw 2 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3360))$$:

 $$T_{11}^{2} - 2 T_{11} - 16$$ $$T_{13}^{2} - 2 T_{13} - 16$$ $$T_{17} - 2$$ $$T_{19}^{2} - 6 T_{19} - 8$$ $$T_{23}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$-16 - 2 T + T^{2}$$
$13$ $$-16 - 2 T + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$-8 - 6 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$-16 - 2 T + T^{2}$$
$37$ $$( 2 + T )^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$-64 - 4 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$-152 - 2 T + T^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$-68 + T^{2}$$
$67$ $$-32 - 12 T + T^{2}$$
$71$ $$-16 + 2 T + T^{2}$$
$73$ $$104 - 22 T + T^{2}$$
$79$ $$-32 - 12 T + T^{2}$$
$83$ $$( 4 + T )^{2}$$
$89$ $$-4 - 16 T + T^{2}$$
$97$ $$64 - 18 T + T^{2}$$