Properties

 Label 3640.2.a.p Level $3640$ Weight $2$ Character orbit 3640.a Self dual yes Analytic conductor $29.066$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1101.1 Defining polynomial: $$x^{3} - x^{2} - 9 x + 12$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} - q^{5} + q^{7} + ( 4 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} - q^{5} + q^{7} + ( 4 - \beta_{1} + \beta_{2} ) q^{9} - q^{13} + \beta_{1} q^{15} + ( 1 - \beta_{1} + \beta_{2} ) q^{17} + ( -1 - \beta_{1} + \beta_{2} ) q^{19} -\beta_{1} q^{21} + ( 2 + 2 \beta_{2} ) q^{23} + q^{25} + ( 5 - 2 \beta_{1} - \beta_{2} ) q^{27} + ( -4 + \beta_{1} - 2 \beta_{2} ) q^{29} + ( 4 - \beta_{1} ) q^{31} - q^{35} + ( -\beta_{1} - 2 \beta_{2} ) q^{37} + \beta_{1} q^{39} + ( 1 - \beta_{1} + \beta_{2} ) q^{41} + ( -4 + 2 \beta_{1} ) q^{43} + ( -4 + \beta_{1} - \beta_{2} ) q^{45} + ( 2 - 2 \beta_{2} ) q^{47} + q^{49} + ( 5 - 2 \beta_{1} - \beta_{2} ) q^{51} + 2 q^{53} + ( 5 - \beta_{2} ) q^{57} + ( 4 - 3 \beta_{1} ) q^{59} -10 q^{61} + ( 4 - \beta_{1} + \beta_{2} ) q^{63} + q^{65} + ( 1 + \beta_{1} + 3 \beta_{2} ) q^{67} + ( -4 - 2 \beta_{1} - 4 \beta_{2} ) q^{69} + 2 \beta_{1} q^{71} + ( 4 - 4 \beta_{1} + 2 \beta_{2} ) q^{73} -\beta_{1} q^{75} + ( 1 + \beta_{1} - \beta_{2} ) q^{79} + ( 4 - 4 \beta_{1} + \beta_{2} ) q^{81} + ( -2 - 2 \beta_{2} ) q^{83} + ( -1 + \beta_{1} - \beta_{2} ) q^{85} + ( -3 + 5 \beta_{1} + 3 \beta_{2} ) q^{87} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{89} - q^{91} + ( 7 - 5 \beta_{1} + \beta_{2} ) q^{93} + ( 1 + \beta_{1} - \beta_{2} ) q^{95} + ( 2 + 2 \beta_{1} + 4 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} - 3 q^{5} + 3 q^{7} + 10 q^{9} + O(q^{10})$$ $$3 q - q^{3} - 3 q^{5} + 3 q^{7} + 10 q^{9} - 3 q^{13} + q^{15} + q^{17} - 5 q^{19} - q^{21} + 4 q^{23} + 3 q^{25} + 14 q^{27} - 9 q^{29} + 11 q^{31} - 3 q^{35} + q^{37} + q^{39} + q^{41} - 10 q^{43} - 10 q^{45} + 8 q^{47} + 3 q^{49} + 14 q^{51} + 6 q^{53} + 16 q^{57} + 9 q^{59} - 30 q^{61} + 10 q^{63} + 3 q^{65} + q^{67} - 10 q^{69} + 2 q^{71} + 6 q^{73} - q^{75} + 5 q^{79} + 7 q^{81} - 4 q^{83} - q^{85} - 7 q^{87} - q^{89} - 3 q^{91} + 15 q^{93} + 5 q^{95} + 4 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 9 x + 12$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta_{1} + 7$$

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
 2.68740 1.43163 −3.11903
0 −2.68740 0 −1.00000 0 1.00000 0 4.22212 0
1.2 0 −1.43163 0 −1.00000 0 1.00000 0 −0.950444 0
1.3 0 3.11903 0 −1.00000 0 1.00000 0 6.72833 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.p 3
4.b odd 2 1 7280.2.a.bn 3

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

 $$T_{3}^{3} + T_{3}^{2} - 9 T_{3} - 12$$ $$T_{11}$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$-12 - 9 T + T^{2} + T^{3}$$
$5$ $$( 1 + T )^{3}$$
$7$ $$( -1 + T )^{3}$$
$11$ $$T^{3}$$
$13$ $$( 1 + T )^{3}$$
$17$ $$18 - 15 T - T^{2} + T^{3}$$
$19$ $$-8 - 7 T + 5 T^{2} + T^{3}$$
$23$ $$48 - 36 T - 4 T^{2} + T^{3}$$
$29$ $$-202 - 15 T + 9 T^{2} + T^{3}$$
$31$ $$-24 + 31 T - 11 T^{2} + T^{3}$$
$37$ $$186 - 59 T - T^{2} + T^{3}$$
$41$ $$18 - 15 T - T^{2} + T^{3}$$
$43$ $$-16 - 4 T + 10 T^{2} + T^{3}$$
$47$ $$96 - 20 T - 8 T^{2} + T^{3}$$
$53$ $$( -2 + T )^{3}$$
$59$ $$-16 - 57 T - 9 T^{2} + T^{3}$$
$61$ $$( 10 + T )^{3}$$
$67$ $$-332 - 115 T - T^{2} + T^{3}$$
$71$ $$96 - 36 T - 2 T^{2} + T^{3}$$
$73$ $$-128 - 144 T - 6 T^{2} + T^{3}$$
$79$ $$8 - 7 T - 5 T^{2} + T^{3}$$
$83$ $$-48 - 36 T + 4 T^{2} + T^{3}$$
$89$ $$386 - 151 T + T^{2} + T^{3}$$
$97$ $$-1016 - 232 T - 4 T^{2} + T^{3}$$