# Properties

 Label 3640.2.a.n Level $3640$ Weight $2$ Character orbit 3640.a Self dual yes Analytic conductor $29.066$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + q^{5} + q^{7} + (\beta - 2) q^{9}+O(q^{10})$$ q + b * q^3 + q^5 + q^7 + (b - 2) * q^9 $$q + \beta q^{3} + q^{5} + q^{7} + (\beta - 2) q^{9} - 4 \beta q^{11} + q^{13} + \beta q^{15} + ( - \beta - 3) q^{17} + ( - \beta - 1) q^{19} + \beta q^{21} + (4 \beta - 6) q^{23} + q^{25} + ( - 4 \beta + 1) q^{27} + (\beta - 4) q^{29} + 5 \beta q^{31} + ( - 4 \beta - 4) q^{33} + q^{35} + ( - 5 \beta + 4) q^{37} + \beta q^{39} + ( - \beta - 7) q^{41} + (2 \beta - 4) q^{43} + (\beta - 2) q^{45} + (4 \beta - 2) q^{47} + q^{49} + ( - 4 \beta - 1) q^{51} + ( - 4 \beta + 2) q^{53} - 4 \beta q^{55} + ( - 2 \beta - 1) q^{57} + ( - 9 \beta + 8) q^{59} + (4 \beta - 10) q^{61} + (\beta - 2) q^{63} + q^{65} + ( - \beta - 1) q^{67} + ( - 2 \beta + 4) q^{69} + (6 \beta - 12) q^{71} + (4 \beta - 4) q^{73} + \beta q^{75} - 4 \beta q^{77} + ( - 9 \beta + 11) q^{79} + ( - 6 \beta + 2) q^{81} - 6 q^{83} + ( - \beta - 3) q^{85} + ( - 3 \beta + 1) q^{87} + (7 \beta - 4) q^{89} + q^{91} + (5 \beta + 5) q^{93} + ( - \beta - 1) q^{95} + ( - 6 \beta - 6) q^{97} + (4 \beta - 4) q^{99} +O(q^{100})$$ q + b * q^3 + q^5 + q^7 + (b - 2) * q^9 - 4*b * q^11 + q^13 + b * q^15 + (-b - 3) * q^17 + (-b - 1) * q^19 + b * q^21 + (4*b - 6) * q^23 + q^25 + (-4*b + 1) * q^27 + (b - 4) * q^29 + 5*b * q^31 + (-4*b - 4) * q^33 + q^35 + (-5*b + 4) * q^37 + b * q^39 + (-b - 7) * q^41 + (2*b - 4) * q^43 + (b - 2) * q^45 + (4*b - 2) * q^47 + q^49 + (-4*b - 1) * q^51 + (-4*b + 2) * q^53 - 4*b * q^55 + (-2*b - 1) * q^57 + (-9*b + 8) * q^59 + (4*b - 10) * q^61 + (b - 2) * q^63 + q^65 + (-b - 1) * q^67 + (-2*b + 4) * q^69 + (6*b - 12) * q^71 + (4*b - 4) * q^73 + b * q^75 - 4*b * q^77 + (-9*b + 11) * q^79 + (-6*b + 2) * q^81 - 6 * q^83 + (-b - 3) * q^85 + (-3*b + 1) * q^87 + (7*b - 4) * q^89 + q^91 + (5*b + 5) * q^93 + (-b - 1) * q^95 + (-6*b - 6) * q^97 + (4*b - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 2 q^{5} + 2 q^{7} - 3 q^{9}+O(q^{10})$$ 2 * q + q^3 + 2 * q^5 + 2 * q^7 - 3 * q^9 $$2 q + q^{3} + 2 q^{5} + 2 q^{7} - 3 q^{9} - 4 q^{11} + 2 q^{13} + q^{15} - 7 q^{17} - 3 q^{19} + q^{21} - 8 q^{23} + 2 q^{25} - 2 q^{27} - 7 q^{29} + 5 q^{31} - 12 q^{33} + 2 q^{35} + 3 q^{37} + q^{39} - 15 q^{41} - 6 q^{43} - 3 q^{45} + 2 q^{49} - 6 q^{51} - 4 q^{55} - 4 q^{57} + 7 q^{59} - 16 q^{61} - 3 q^{63} + 2 q^{65} - 3 q^{67} + 6 q^{69} - 18 q^{71} - 4 q^{73} + q^{75} - 4 q^{77} + 13 q^{79} - 2 q^{81} - 12 q^{83} - 7 q^{85} - q^{87} - q^{89} + 2 q^{91} + 15 q^{93} - 3 q^{95} - 18 q^{97} - 4 q^{99}+O(q^{100})$$ 2 * q + q^3 + 2 * q^5 + 2 * q^7 - 3 * q^9 - 4 * q^11 + 2 * q^13 + q^15 - 7 * q^17 - 3 * q^19 + q^21 - 8 * q^23 + 2 * q^25 - 2 * q^27 - 7 * q^29 + 5 * q^31 - 12 * q^33 + 2 * q^35 + 3 * q^37 + q^39 - 15 * q^41 - 6 * q^43 - 3 * q^45 + 2 * q^49 - 6 * q^51 - 4 * q^55 - 4 * q^57 + 7 * q^59 - 16 * q^61 - 3 * q^63 + 2 * q^65 - 3 * q^67 + 6 * q^69 - 18 * q^71 - 4 * q^73 + q^75 - 4 * q^77 + 13 * q^79 - 2 * q^81 - 12 * q^83 - 7 * q^85 - q^87 - q^89 + 2 * q^91 + 15 * q^93 - 3 * q^95 - 18 * q^97 - 4 * q^99

## 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 −0.618034 0 1.00000 0 1.00000 0 −2.61803 0
1.2 0 1.61803 0 1.00000 0 1.00000 0 −0.381966 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.n 2
4.b odd 2 1 7280.2.a.z 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.2.a.n 2 1.a even 1 1 trivial
7280.2.a.z 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3640))$$:

 $$T_{3}^{2} - T_{3} - 1$$ T3^2 - T3 - 1 $$T_{11}^{2} + 4T_{11} - 16$$ T11^2 + 4*T11 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T - 1$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} + 4T - 16$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} + 7T + 11$$
$19$ $$T^{2} + 3T + 1$$
$23$ $$T^{2} + 8T - 4$$
$29$ $$T^{2} + 7T + 11$$
$31$ $$T^{2} - 5T - 25$$
$37$ $$T^{2} - 3T - 29$$
$41$ $$T^{2} + 15T + 55$$
$43$ $$T^{2} + 6T + 4$$
$47$ $$T^{2} - 20$$
$53$ $$T^{2} - 20$$
$59$ $$T^{2} - 7T - 89$$
$61$ $$T^{2} + 16T + 44$$
$67$ $$T^{2} + 3T + 1$$
$71$ $$T^{2} + 18T + 36$$
$73$ $$T^{2} + 4T - 16$$
$79$ $$T^{2} - 13T - 59$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} + T - 61$$
$97$ $$T^{2} + 18T + 36$$