# Properties

 Label 3360.2.a.bh 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: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} + q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + q^{5} + q^{7} + q^{9} + \beta q^{13} + q^{15} + \beta q^{17} + q^{21} + ( 2 - \beta ) q^{23} + q^{25} + q^{27} + ( 4 - \beta ) q^{29} + ( -2 + \beta ) q^{31} + q^{35} + ( 4 - \beta ) q^{37} + \beta q^{39} + ( 2 - 2 \beta ) q^{41} + ( 6 - \beta ) q^{43} + q^{45} + ( 2 + \beta ) q^{47} + q^{49} + \beta q^{51} + 2 q^{53} -4 q^{59} + ( 8 - \beta ) q^{61} + q^{63} + \beta q^{65} + ( -2 - \beta ) q^{67} + ( 2 - \beta ) q^{69} + ( 2 - 3 \beta ) q^{71} + ( 4 - \beta ) q^{73} + q^{75} + ( 4 - 2 \beta ) q^{79} + q^{81} + ( -8 + 2 \beta ) q^{83} + \beta q^{85} + ( 4 - \beta ) q^{87} + ( 2 + 2 \beta ) q^{89} + \beta q^{91} + ( -2 + \beta ) q^{93} + \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 2 q^{15} + 2 q^{21} + 4 q^{23} + 2 q^{25} + 2 q^{27} + 8 q^{29} - 4 q^{31} + 2 q^{35} + 8 q^{37} + 4 q^{41} + 12 q^{43} + 2 q^{45} + 4 q^{47} + 2 q^{49} + 4 q^{53} - 8 q^{59} + 16 q^{61} + 2 q^{63} - 4 q^{67} + 4 q^{69} + 4 q^{71} + 8 q^{73} + 2 q^{75} + 8 q^{79} + 2 q^{81} - 16 q^{83} + 8 q^{87} + 4 q^{89} - 4 q^{93} + O(q^{100})$$

## 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
 −0.618034 1.61803
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.bh yes 2
4.b odd 2 1 3360.2.a.bd 2
8.b even 2 1 6720.2.a.cp 2
8.d odd 2 1 6720.2.a.cv 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.2.a.bd 2 4.b odd 2 1
3360.2.a.bh yes 2 1.a even 1 1 trivial
6720.2.a.cp 2 8.b even 2 1
6720.2.a.cv 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3360))$$:

 $$T_{11}$$ $$T_{13}^{2} - 20$$ $$T_{17}^{2} - 20$$ $$T_{19}$$ $$T_{23}^{2} - 4 T_{23} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$-20 + T^{2}$$
$17$ $$-20 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$-16 - 4 T + T^{2}$$
$29$ $$-4 - 8 T + T^{2}$$
$31$ $$-16 + 4 T + T^{2}$$
$37$ $$-4 - 8 T + T^{2}$$
$41$ $$-76 - 4 T + T^{2}$$
$43$ $$16 - 12 T + T^{2}$$
$47$ $$-16 - 4 T + T^{2}$$
$53$ $$( -2 + T )^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$44 - 16 T + T^{2}$$
$67$ $$-16 + 4 T + T^{2}$$
$71$ $$-176 - 4 T + T^{2}$$
$73$ $$-4 - 8 T + T^{2}$$
$79$ $$-64 - 8 T + T^{2}$$
$83$ $$-16 + 16 T + T^{2}$$
$89$ $$-76 - 4 T + T^{2}$$
$97$ $$-20 + T^{2}$$