Properties

 Label 7728.2.a.x Level $7728$ Weight $2$ Character orbit 7728.a Self dual yes Analytic conductor $61.708$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7728.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$61.7083906820$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 483) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( - \beta - 2) q^{5} + q^{7} + q^{9}+O(q^{10})$$ q - q^3 + (-b - 2) * q^5 + q^7 + q^9 $$q - q^{3} + ( - \beta - 2) q^{5} + q^{7} + q^{9} + 5 q^{11} + ( - \beta + 1) q^{13} + (\beta + 2) q^{15} + (2 \beta - 3) q^{17} + ( - 2 \beta - 1) q^{19} - q^{21} - q^{23} + (5 \beta + 2) q^{25} - q^{27} + ( - 4 \beta + 1) q^{29} - 3 q^{31} - 5 q^{33} + ( - \beta - 2) q^{35} - 9 q^{37} + (\beta - 1) q^{39} + (4 \beta - 7) q^{41} + (5 \beta + 1) q^{43} + ( - \beta - 2) q^{45} + ( - 2 \beta + 6) q^{47} + q^{49} + ( - 2 \beta + 3) q^{51} + (5 \beta - 6) q^{53} + ( - 5 \beta - 10) q^{55} + (2 \beta + 1) q^{57} + 3 \beta q^{59} + ( - 3 \beta - 5) q^{61} + q^{63} + (2 \beta + 1) q^{65} + ( - 3 \beta + 8) q^{67} + q^{69} + (3 \beta + 3) q^{71} + (4 \beta + 3) q^{73} + ( - 5 \beta - 2) q^{75} + 5 q^{77} + q^{79} + q^{81} + (2 \beta - 3) q^{83} - 3 \beta q^{85} + (4 \beta - 1) q^{87} + ( - 3 \beta + 7) q^{89} + ( - \beta + 1) q^{91} + 3 q^{93} + (7 \beta + 8) q^{95} + (2 \beta - 15) q^{97} + 5 q^{99} +O(q^{100})$$ q - q^3 + (-b - 2) * q^5 + q^7 + q^9 + 5 * q^11 + (-b + 1) * q^13 + (b + 2) * q^15 + (2*b - 3) * q^17 + (-2*b - 1) * q^19 - q^21 - q^23 + (5*b + 2) * q^25 - q^27 + (-4*b + 1) * q^29 - 3 * q^31 - 5 * q^33 + (-b - 2) * q^35 - 9 * q^37 + (b - 1) * q^39 + (4*b - 7) * q^41 + (5*b + 1) * q^43 + (-b - 2) * q^45 + (-2*b + 6) * q^47 + q^49 + (-2*b + 3) * q^51 + (5*b - 6) * q^53 + (-5*b - 10) * q^55 + (2*b + 1) * q^57 + 3*b * q^59 + (-3*b - 5) * q^61 + q^63 + (2*b + 1) * q^65 + (-3*b + 8) * q^67 + q^69 + (3*b + 3) * q^71 + (4*b + 3) * q^73 + (-5*b - 2) * q^75 + 5 * q^77 + q^79 + q^81 + (2*b - 3) * q^83 - 3*b * q^85 + (4*b - 1) * q^87 + (-3*b + 7) * q^89 + (-b + 1) * q^91 + 3 * q^93 + (7*b + 8) * q^95 + (2*b - 15) * q^97 + 5 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 5 * q^5 + 2 * q^7 + 2 * q^9 $$2 q - 2 q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9} + 10 q^{11} + q^{13} + 5 q^{15} - 4 q^{17} - 4 q^{19} - 2 q^{21} - 2 q^{23} + 9 q^{25} - 2 q^{27} - 2 q^{29} - 6 q^{31} - 10 q^{33} - 5 q^{35} - 18 q^{37} - q^{39} - 10 q^{41} + 7 q^{43} - 5 q^{45} + 10 q^{47} + 2 q^{49} + 4 q^{51} - 7 q^{53} - 25 q^{55} + 4 q^{57} + 3 q^{59} - 13 q^{61} + 2 q^{63} + 4 q^{65} + 13 q^{67} + 2 q^{69} + 9 q^{71} + 10 q^{73} - 9 q^{75} + 10 q^{77} + 2 q^{79} + 2 q^{81} - 4 q^{83} - 3 q^{85} + 2 q^{87} + 11 q^{89} + q^{91} + 6 q^{93} + 23 q^{95} - 28 q^{97} + 10 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 5 * q^5 + 2 * q^7 + 2 * q^9 + 10 * q^11 + q^13 + 5 * q^15 - 4 * q^17 - 4 * q^19 - 2 * q^21 - 2 * q^23 + 9 * q^25 - 2 * q^27 - 2 * q^29 - 6 * q^31 - 10 * q^33 - 5 * q^35 - 18 * q^37 - q^39 - 10 * q^41 + 7 * q^43 - 5 * q^45 + 10 * q^47 + 2 * q^49 + 4 * q^51 - 7 * q^53 - 25 * q^55 + 4 * q^57 + 3 * q^59 - 13 * q^61 + 2 * q^63 + 4 * q^65 + 13 * q^67 + 2 * q^69 + 9 * q^71 + 10 * q^73 - 9 * q^75 + 10 * q^77 + 2 * q^79 + 2 * q^81 - 4 * q^83 - 3 * q^85 + 2 * q^87 + 11 * q^89 + q^91 + 6 * q^93 + 23 * q^95 - 28 * q^97 + 10 * 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
 2.30278 −1.30278
0 −1.00000 0 −4.30278 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 −0.697224 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$
$$23$$ $$1$$

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 7728.2.a.x 2
4.b odd 2 1 483.2.a.f 2
12.b even 2 1 1449.2.a.j 2
28.d even 2 1 3381.2.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.f 2 4.b odd 2 1
1449.2.a.j 2 12.b even 2 1
3381.2.a.p 2 28.d even 2 1
7728.2.a.x 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7728))$$:

 $$T_{5}^{2} + 5T_{5} + 3$$ T5^2 + 5*T5 + 3 $$T_{11} - 5$$ T11 - 5 $$T_{13}^{2} - T_{13} - 3$$ T13^2 - T13 - 3 $$T_{17}^{2} + 4T_{17} - 9$$ T17^2 + 4*T17 - 9

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 5T + 3$$
$7$ $$(T - 1)^{2}$$
$11$ $$(T - 5)^{2}$$
$13$ $$T^{2} - T - 3$$
$17$ $$T^{2} + 4T - 9$$
$19$ $$T^{2} + 4T - 9$$
$23$ $$(T + 1)^{2}$$
$29$ $$T^{2} + 2T - 51$$
$31$ $$(T + 3)^{2}$$
$37$ $$(T + 9)^{2}$$
$41$ $$T^{2} + 10T - 27$$
$43$ $$T^{2} - 7T - 69$$
$47$ $$T^{2} - 10T + 12$$
$53$ $$T^{2} + 7T - 69$$
$59$ $$T^{2} - 3T - 27$$
$61$ $$T^{2} + 13T + 13$$
$67$ $$T^{2} - 13T + 13$$
$71$ $$T^{2} - 9T - 9$$
$73$ $$T^{2} - 10T - 27$$
$79$ $$(T - 1)^{2}$$
$83$ $$T^{2} + 4T - 9$$
$89$ $$T^{2} - 11T + 1$$
$97$ $$T^{2} + 28T + 183$$