# Properties

 Label 2640.2.a.x Level $2640$ Weight $2$ Character orbit 2640.a Self dual yes Analytic conductor $21.081$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2640.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$21.0805061336$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{5} -2 q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} - q^{5} -2 q^{7} + q^{9} + q^{11} + ( 2 + \beta ) q^{13} + q^{15} + ( -2 - \beta ) q^{19} + 2 q^{21} -2 \beta q^{23} + q^{25} - q^{27} -\beta q^{29} + ( 4 + 2 \beta ) q^{31} - q^{33} + 2 q^{35} + ( 2 - 2 \beta ) q^{37} + ( -2 - \beta ) q^{39} + \beta q^{41} + ( -2 + 2 \beta ) q^{43} - q^{45} + 2 \beta q^{47} -3 q^{49} + ( -6 + 2 \beta ) q^{53} - q^{55} + ( 2 + \beta ) q^{57} + 2 \beta q^{59} + 2 q^{61} -2 q^{63} + ( -2 - \beta ) q^{65} -8 q^{67} + 2 \beta q^{69} -4 \beta q^{71} + ( 2 - 3 \beta ) q^{73} - q^{75} -2 q^{77} + ( 10 - \beta ) q^{79} + q^{81} + ( -12 + \beta ) q^{83} + \beta q^{87} + ( -6 + 2 \beta ) q^{89} + ( -4 - 2 \beta ) q^{91} + ( -4 - 2 \beta ) q^{93} + ( 2 + \beta ) q^{95} -10 q^{97} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 2q^{5} - 4q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 2q^{5} - 4q^{7} + 2q^{9} + 2q^{11} + 4q^{13} + 2q^{15} - 4q^{19} + 4q^{21} + 2q^{25} - 2q^{27} + 8q^{31} - 2q^{33} + 4q^{35} + 4q^{37} - 4q^{39} - 4q^{43} - 2q^{45} - 6q^{49} - 12q^{53} - 2q^{55} + 4q^{57} + 4q^{61} - 4q^{63} - 4q^{65} - 16q^{67} + 4q^{73} - 2q^{75} - 4q^{77} + 20q^{79} + 2q^{81} - 24q^{83} - 12q^{89} - 8q^{91} - 8q^{93} + 4q^{95} - 20q^{97} + 2q^{99} + O(q^{100})$$

## 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
 −1.73205 1.73205
0 −1.00000 0 −1.00000 0 −2.00000 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 −2.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$11$$ $$-1$$

## 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 2640.2.a.x 2
3.b odd 2 1 7920.2.a.bz 2
4.b odd 2 1 165.2.a.b 2
12.b even 2 1 495.2.a.c 2
20.d odd 2 1 825.2.a.e 2
20.e even 4 2 825.2.c.c 4
28.d even 2 1 8085.2.a.bd 2
44.c even 2 1 1815.2.a.i 2
60.h even 2 1 2475.2.a.r 2
60.l odd 4 2 2475.2.c.n 4
132.d odd 2 1 5445.2.a.s 2
220.g even 2 1 9075.2.a.bh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.b 2 4.b odd 2 1
495.2.a.c 2 12.b even 2 1
825.2.a.e 2 20.d odd 2 1
825.2.c.c 4 20.e even 4 2
1815.2.a.i 2 44.c even 2 1
2475.2.a.r 2 60.h even 2 1
2475.2.c.n 4 60.l odd 4 2
2640.2.a.x 2 1.a even 1 1 trivial
5445.2.a.s 2 132.d odd 2 1
7920.2.a.bz 2 3.b odd 2 1
8085.2.a.bd 2 28.d even 2 1
9075.2.a.bh 2 220.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2640))$$:

 $$T_{7} + 2$$ $$T_{13}^{2} - 4 T_{13} - 8$$ $$T_{17}$$ $$T_{19}^{2} + 4 T_{19} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( 2 + T )^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$-8 - 4 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$-8 + 4 T + T^{2}$$
$23$ $$-48 + T^{2}$$
$29$ $$-12 + T^{2}$$
$31$ $$-32 - 8 T + T^{2}$$
$37$ $$-44 - 4 T + T^{2}$$
$41$ $$-12 + T^{2}$$
$43$ $$-44 + 4 T + T^{2}$$
$47$ $$-48 + T^{2}$$
$53$ $$-12 + 12 T + T^{2}$$
$59$ $$-48 + T^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$( 8 + T )^{2}$$
$71$ $$-192 + T^{2}$$
$73$ $$-104 - 4 T + T^{2}$$
$79$ $$88 - 20 T + T^{2}$$
$83$ $$132 + 24 T + T^{2}$$
$89$ $$-12 + 12 T + T^{2}$$
$97$ $$( 10 + T )^{2}$$