# Properties

 Label 7728.2.a.bv Level $7728$ Weight $2$ Character orbit 7728.a Self dual yes Analytic conductor $61.708$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.1509.1 Defining polynomial: $$x^{3} - x^{2} - 7x + 4$$ x^3 - x^2 - 7*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1932) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} - \beta_1 q^{5} - q^{7} + q^{9}+O(q^{10})$$ q + q^3 - b1 * q^5 - q^7 + q^9 $$q + q^{3} - \beta_1 q^{5} - q^{7} + q^{9} + (2 \beta_1 - 1) q^{11} + ( - \beta_{2} + 1) q^{13} - \beta_1 q^{15} + (\beta_{2} - \beta_1 + 1) q^{17} - q^{19} - q^{21} - q^{23} + \beta_{2} q^{25} + q^{27} + ( - \beta_{2} - \beta_1 - 3) q^{29} + (\beta_{2} - \beta_1 - 1) q^{31} + (2 \beta_1 - 1) q^{33} + \beta_1 q^{35} + ( - \beta_{2} + 3 \beta_1 - 3) q^{37} + ( - \beta_{2} + 1) q^{39} + (2 \beta_{2} + 2 \beta_1 - 1) q^{41} + ( - \beta_{2} + 2 \beta_1 - 5) q^{43} - \beta_1 q^{45} + (\beta_{2} + \beta_1 - 4) q^{47} + q^{49} + (\beta_{2} - \beta_1 + 1) q^{51} + ( - \beta_{2} - 4) q^{53} + ( - 2 \beta_{2} + \beta_1 - 10) q^{55} - q^{57} + ( - \beta_{2} - 4) q^{59} + (\beta_1 + 7) q^{61} - q^{63} + (\beta_{2} + \beta_1 + 1) q^{65} + \beta_1 q^{67} - q^{69} + (3 \beta_{2} - 2 \beta_1 + 3) q^{71} + (\beta_{2} + \beta_1 + 1) q^{73} + \beta_{2} q^{75} + ( - 2 \beta_1 + 1) q^{77} + (\beta_{2} - 5 \beta_1 - 1) q^{79} + q^{81} + ( - \beta_{2} - \beta_1 - 3) q^{83} + ( - 3 \beta_1 + 4) q^{85} + ( - \beta_{2} - \beta_1 - 3) q^{87} + ( - \beta_{2} + 5) q^{89} + (\beta_{2} - 1) q^{91} + (\beta_{2} - \beta_1 - 1) q^{93} + \beta_1 q^{95} + ( - 3 \beta_{2} + \beta_1 + 3) q^{97} + (2 \beta_1 - 1) q^{99}+O(q^{100})$$ q + q^3 - b1 * q^5 - q^7 + q^9 + (2*b1 - 1) * q^11 + (-b2 + 1) * q^13 - b1 * q^15 + (b2 - b1 + 1) * q^17 - q^19 - q^21 - q^23 + b2 * q^25 + q^27 + (-b2 - b1 - 3) * q^29 + (b2 - b1 - 1) * q^31 + (2*b1 - 1) * q^33 + b1 * q^35 + (-b2 + 3*b1 - 3) * q^37 + (-b2 + 1) * q^39 + (2*b2 + 2*b1 - 1) * q^41 + (-b2 + 2*b1 - 5) * q^43 - b1 * q^45 + (b2 + b1 - 4) * q^47 + q^49 + (b2 - b1 + 1) * q^51 + (-b2 - 4) * q^53 + (-2*b2 + b1 - 10) * q^55 - q^57 + (-b2 - 4) * q^59 + (b1 + 7) * q^61 - q^63 + (b2 + b1 + 1) * q^65 + b1 * q^67 - q^69 + (3*b2 - 2*b1 + 3) * q^71 + (b2 + b1 + 1) * q^73 + b2 * q^75 + (-2*b1 + 1) * q^77 + (b2 - 5*b1 - 1) * q^79 + q^81 + (-b2 - b1 - 3) * q^83 + (-3*b1 + 4) * q^85 + (-b2 - b1 - 3) * q^87 + (-b2 + 5) * q^89 + (b2 - 1) * q^91 + (b2 - b1 - 1) * q^93 + b1 * q^95 + (-3*b2 + b1 + 3) * q^97 + (2*b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} - q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 - q^5 - 3 * q^7 + 3 * q^9 $$3 q + 3 q^{3} - q^{5} - 3 q^{7} + 3 q^{9} - q^{11} + 3 q^{13} - q^{15} + 2 q^{17} - 3 q^{19} - 3 q^{21} - 3 q^{23} + 3 q^{27} - 10 q^{29} - 4 q^{31} - q^{33} + q^{35} - 6 q^{37} + 3 q^{39} - q^{41} - 13 q^{43} - q^{45} - 11 q^{47} + 3 q^{49} + 2 q^{51} - 12 q^{53} - 29 q^{55} - 3 q^{57} - 12 q^{59} + 22 q^{61} - 3 q^{63} + 4 q^{65} + q^{67} - 3 q^{69} + 7 q^{71} + 4 q^{73} + q^{77} - 8 q^{79} + 3 q^{81} - 10 q^{83} + 9 q^{85} - 10 q^{87} + 15 q^{89} - 3 q^{91} - 4 q^{93} + q^{95} + 10 q^{97} - q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 - q^5 - 3 * q^7 + 3 * q^9 - q^11 + 3 * q^13 - q^15 + 2 * q^17 - 3 * q^19 - 3 * q^21 - 3 * q^23 + 3 * q^27 - 10 * q^29 - 4 * q^31 - q^33 + q^35 - 6 * q^37 + 3 * q^39 - q^41 - 13 * q^43 - q^45 - 11 * q^47 + 3 * q^49 + 2 * q^51 - 12 * q^53 - 29 * q^55 - 3 * q^57 - 12 * q^59 + 22 * q^61 - 3 * q^63 + 4 * q^65 + q^67 - 3 * q^69 + 7 * q^71 + 4 * q^73 + q^77 - 8 * q^79 + 3 * q^81 - 10 * q^83 + 9 * q^85 - 10 * q^87 + 15 * q^89 - 3 * q^91 - 4 * q^93 + q^95 + 10 * q^97 - q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 7x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 5$$ v^2 - 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 5$$ b2 + 5

## 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
 2.92542 0.551929 −2.47735
0 1.00000 0 −2.92542 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 −0.551929 0 −1.00000 0 1.00000 0
1.3 0 1.00000 0 2.47735 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.bv 3
4.b odd 2 1 1932.2.a.i 3
12.b even 2 1 5796.2.a.p 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1932.2.a.i 3 4.b odd 2 1
5796.2.a.p 3 12.b even 2 1
7728.2.a.bv 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7728))$$:

 $$T_{5}^{3} + T_{5}^{2} - 7T_{5} - 4$$ T5^3 + T5^2 - 7*T5 - 4 $$T_{11}^{3} + T_{11}^{2} - 29T_{11} + 3$$ T11^3 + T11^2 - 29*T11 + 3 $$T_{13}^{3} - 3T_{13}^{2} - 15T_{13} - 2$$ T13^3 - 3*T13^2 - 15*T13 - 2 $$T_{17}^{3} - 2T_{17}^{2} - 19T_{17} + 32$$ T17^3 - 2*T17^2 - 19*T17 + 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T - 1)^{3}$$
$5$ $$T^{3} + T^{2} - 7T - 4$$
$7$ $$(T + 1)^{3}$$
$11$ $$T^{3} + T^{2} - 29T + 3$$
$13$ $$T^{3} - 3 T^{2} - 15 T - 2$$
$17$ $$T^{3} - 2 T^{2} - 19 T + 32$$
$19$ $$(T + 1)^{3}$$
$23$ $$(T + 1)^{3}$$
$29$ $$T^{3} + 10 T^{2} + 3 T - 18$$
$31$ $$T^{3} + 4 T^{2} - 15 T - 6$$
$37$ $$T^{3} + 6 T^{2} - 57 T + 86$$
$41$ $$T^{3} + T^{2} - 121 T - 409$$
$43$ $$T^{3} + 13 T^{2} + 19 T - 24$$
$47$ $$T^{3} + 11 T^{2} + 10 T - 108$$
$53$ $$T^{3} + 12 T^{2} + 30 T - 27$$
$59$ $$T^{3} + 12 T^{2} + 30 T - 27$$
$61$ $$T^{3} - 22 T^{2} + 154 T - 339$$
$67$ $$T^{3} - T^{2} - 7T + 4$$
$71$ $$T^{3} - 7 T^{2} - 145 T + 1084$$
$73$ $$T^{3} - 4 T^{2} - 25 T - 8$$
$79$ $$T^{3} + 8 T^{2} - 155 T - 1278$$
$83$ $$T^{3} + 10 T^{2} + 3 T - 18$$
$89$ $$T^{3} - 15 T^{2} + 57 T - 54$$
$97$ $$T^{3} - 10 T^{2} - 121 T - 242$$