Properties

 Label 6720.2.a.cp Level $6720$ Weight $2$ Character orbit 6720.a Self dual yes Analytic conductor $53.659$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6720.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$53.6594701583$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 3360) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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

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 −1.00000 0 −1.00000 0 1.00000 0 1.00000 0
1.2 0 −1.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$7$$ $$-1$$

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 6720.2.a.cp 2
4.b odd 2 1 6720.2.a.cv 2
8.b even 2 1 3360.2.a.bh yes 2
8.d odd 2 1 3360.2.a.bd 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.2.a.bd 2 8.d odd 2 1
3360.2.a.bh yes 2 8.b even 2 1
6720.2.a.cp 2 1.a even 1 1 trivial
6720.2.a.cv 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6720))$$:

 $$T_{11}$$ T11 $$T_{13}^{2} - 20$$ T13^2 - 20 $$T_{17}^{2} - 20$$ T17^2 - 20 $$T_{19}$$ T19 $$T_{23}^{2} - 4T_{23} - 16$$ T23^2 - 4*T23 - 16 $$T_{29}^{2} + 8T_{29} - 4$$ T29^2 + 8*T29 - 4 $$T_{31}^{2} + 4T_{31} - 16$$ T31^2 + 4*T31 - 16

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 20$$
$17$ $$T^{2} - 20$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 4T - 16$$
$29$ $$T^{2} + 8T - 4$$
$31$ $$T^{2} + 4T - 16$$
$37$ $$T^{2} + 8T - 4$$
$41$ $$T^{2} - 4T - 76$$
$43$ $$T^{2} + 12T + 16$$
$47$ $$T^{2} - 4T - 16$$
$53$ $$(T + 2)^{2}$$
$59$ $$(T - 4)^{2}$$
$61$ $$T^{2} + 16T + 44$$
$67$ $$T^{2} - 4T - 16$$
$71$ $$T^{2} - 4T - 176$$
$73$ $$T^{2} - 8T - 4$$
$79$ $$T^{2} - 8T - 64$$
$83$ $$T^{2} - 16T - 16$$
$89$ $$T^{2} - 4T - 76$$
$97$ $$T^{2} - 20$$