Properties

 Label 3640.2.a.o Level $3640$ Weight $2$ Character orbit 3640.a Self dual yes Analytic conductor $29.066$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4 x - 1$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{2} ) q^{3} - q^{5} - q^{7} + ( 1 + \beta_{1} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{2} ) q^{3} - q^{5} - q^{7} + ( 1 + \beta_{1} ) q^{9} + ( \beta_{1} + \beta_{2} ) q^{11} + q^{13} + ( 1 + \beta_{2} ) q^{15} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{17} -\beta_{2} q^{19} + ( 1 + \beta_{2} ) q^{21} + ( 2 - 3 \beta_{1} + \beta_{2} ) q^{23} + q^{25} + ( 1 - 2 \beta_{1} + 2 \beta_{2} ) q^{27} + ( -1 + 3 \beta_{1} + 2 \beta_{2} ) q^{29} + ( 5 + \beta_{2} ) q^{31} + ( -4 - 3 \beta_{1} + \beta_{2} ) q^{33} + q^{35} + ( -5 - 2 \beta_{1} + \beta_{2} ) q^{37} + ( -1 - \beta_{2} ) q^{39} + ( 2 - 3 \beta_{1} ) q^{41} + ( -2 + 2 \beta_{2} ) q^{43} + ( -1 - \beta_{1} ) q^{45} + ( 2 + 5 \beta_{1} - 3 \beta_{2} ) q^{47} + q^{49} + ( 7 + 5 \beta_{1} + \beta_{2} ) q^{51} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{53} + ( -\beta_{1} - \beta_{2} ) q^{55} + ( 3 + \beta_{1} - \beta_{2} ) q^{57} + ( -1 - 3 \beta_{1} + 4 \beta_{2} ) q^{59} + ( -6 - \beta_{1} - \beta_{2} ) q^{61} + ( -1 - \beta_{1} ) q^{63} - q^{65} + ( 4 + \beta_{1} ) q^{67} + ( -2 + 5 \beta_{1} - \beta_{2} ) q^{69} + ( 2 + 2 \beta_{1} + 4 \beta_{2} ) q^{71} + ( 7 \beta_{1} - \beta_{2} ) q^{73} + ( -1 - \beta_{2} ) q^{75} + ( -\beta_{1} - \beta_{2} ) q^{77} + ( 4 + \beta_{1} + 2 \beta_{2} ) q^{79} + ( -8 - \beta_{1} + \beta_{2} ) q^{81} + ( -6 - 4 \beta_{1} ) q^{83} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{85} + ( -8 - 8 \beta_{1} + 3 \beta_{2} ) q^{87} + ( -5 + 3 \beta_{1} - 2 \beta_{2} ) q^{89} - q^{91} + ( -8 - \beta_{1} - 4 \beta_{2} ) q^{93} + \beta_{2} q^{95} + ( -4 + \beta_{1} - 5 \beta_{2} ) q^{97} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} + O(q^{10})$$ $$3 q - 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} - q^{11} + 3 q^{13} + 2 q^{15} - 5 q^{17} + q^{19} + 2 q^{21} + 5 q^{23} + 3 q^{25} + q^{27} - 5 q^{29} + 14 q^{31} - 13 q^{33} + 3 q^{35} - 16 q^{37} - 2 q^{39} + 6 q^{41} - 8 q^{43} - 3 q^{45} + 9 q^{47} + 3 q^{49} + 20 q^{51} - 8 q^{53} + q^{55} + 10 q^{57} - 7 q^{59} - 17 q^{61} - 3 q^{63} - 3 q^{65} + 12 q^{67} - 5 q^{69} + 2 q^{71} + q^{73} - 2 q^{75} + q^{77} + 10 q^{79} - 25 q^{81} - 18 q^{83} + 5 q^{85} - 27 q^{87} - 13 q^{89} - 3 q^{91} - 20 q^{93} - q^{95} - 7 q^{97} + 10 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 4 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$

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
 2.11491 −1.86081 −0.254102
0 −2.47283 0 −1.00000 0 −1.00000 0 3.11491 0
1.2 0 −1.46260 0 −1.00000 0 −1.00000 0 −0.860806 0
1.3 0 1.93543 0 −1.00000 0 −1.00000 0 0.745898 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.o 3
4.b odd 2 1 7280.2.a.bp 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.2.a.o 3 1.a even 1 1 trivial
7280.2.a.bp 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3640))$$:

 $$T_{3}^{3} + 2 T_{3}^{2} - 4 T_{3} - 7$$ $$T_{11}^{3} + T_{11}^{2} - 12 T_{11} - 16$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$-7 - 4 T + 2 T^{2} + T^{3}$$
$5$ $$( 1 + T )^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$-16 - 12 T + T^{2} + T^{3}$$
$13$ $$( -1 + T )^{3}$$
$17$ $$14 - 19 T + 5 T^{2} + T^{3}$$
$19$ $$-2 - 5 T - T^{2} + T^{3}$$
$23$ $$-4 - 24 T - 5 T^{2} + T^{3}$$
$29$ $$-358 - 67 T + 5 T^{2} + T^{3}$$
$31$ $$-73 + 60 T - 14 T^{2} + T^{3}$$
$37$ $$47 + 70 T + 16 T^{2} + T^{3}$$
$41$ $$91 - 24 T - 6 T^{2} + T^{3}$$
$43$ $$-8 + 8 T^{2} + T^{3}$$
$47$ $$676 - 76 T - 9 T^{2} + T^{3}$$
$53$ $$-208 - 28 T + 8 T^{2} + T^{3}$$
$59$ $$-112 - 69 T + 7 T^{2} + T^{3}$$
$61$ $$124 + 84 T + 17 T^{2} + T^{3}$$
$67$ $$-49 + 44 T - 12 T^{2} + T^{3}$$
$71$ $$16 - 124 T - 2 T^{2} + T^{3}$$
$73$ $$208 - 180 T - T^{2} + T^{3}$$
$79$ $$59 + 2 T - 10 T^{2} + T^{3}$$
$83$ $$-104 + 44 T + 18 T^{2} + T^{3}$$
$89$ $$-2 + 17 T + 13 T^{2} + T^{3}$$
$97$ $$-788 - 106 T + 7 T^{2} + T^{3}$$