# Properties

 Label 3640.2.a.g Level $3640$ Weight $2$ Character orbit 3640.a Self dual yes Analytic conductor $29.066$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3640.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.0655463357$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2 q^{3} - q^{5} - q^{7} + q^{9} + O(q^{10})$$ $$q + 2 q^{3} - q^{5} - q^{7} + q^{9} - 2 q^{11} - q^{13} - 2 q^{15} + 6 q^{17} - 6 q^{19} - 2 q^{21} + 6 q^{23} + q^{25} - 4 q^{27} - 6 q^{29} - 6 q^{31} - 4 q^{33} + q^{35} + 6 q^{37} - 2 q^{39} - 10 q^{41} - 6 q^{43} - q^{45} + q^{49} + 12 q^{51} + 6 q^{53} + 2 q^{55} - 12 q^{57} - 10 q^{59} + 2 q^{61} - q^{63} + q^{65} + 8 q^{67} + 12 q^{69} - 10 q^{71} - 14 q^{73} + 2 q^{75} + 2 q^{77} - 12 q^{79} - 11 q^{81} + 12 q^{83} - 6 q^{85} - 12 q^{87} - 10 q^{89} + q^{91} - 12 q^{93} + 6 q^{95} - 2 q^{97} - 2 q^{99} + O(q^{100})$$

## 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
 0
0 2.00000 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
$$2$$ $$1$$
$$5$$ $$1$$
$$7$$ $$1$$
$$13$$ $$1$$

## 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 3640.2.a.g 1
4.b odd 2 1 7280.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.2.a.g 1 1.a even 1 1 trivial
7280.2.a.c 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3640))$$:

 $$T_{3} - 2$$ $$T_{11} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-2 + T$$
$5$ $$1 + T$$
$7$ $$1 + T$$
$11$ $$2 + T$$
$13$ $$1 + T$$
$17$ $$-6 + T$$
$19$ $$6 + T$$
$23$ $$-6 + T$$
$29$ $$6 + T$$
$31$ $$6 + T$$
$37$ $$-6 + T$$
$41$ $$10 + T$$
$43$ $$6 + T$$
$47$ $$T$$
$53$ $$-6 + T$$
$59$ $$10 + T$$
$61$ $$-2 + T$$
$67$ $$-8 + T$$
$71$ $$10 + T$$
$73$ $$14 + T$$
$79$ $$12 + T$$
$83$ $$-12 + T$$
$89$ $$10 + T$$
$97$ $$2 + T$$