# Properties

 Label 7728.2.a.cb Level $7728$ Weight $2$ Character orbit 7728.a Self dual yes Analytic conductor $61.708$ Analytic rank $1$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.2225.1 Defining polynomial: $$x^{4} - x^{3} - 5x^{2} + 2x + 4$$ x^4 - x^3 - 5*x^2 + 2*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 3864) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( - \beta_{3} - 1) q^{5} - q^{7} + q^{9}+O(q^{10})$$ q + q^3 + (-b3 - 1) * q^5 - q^7 + q^9 $$q + q^{3} + ( - \beta_{3} - 1) q^{5} - q^{7} + q^{9} + (\beta_{2} - \beta_1 - 1) q^{11} + (\beta_{3} + \beta_{2} - \beta_1) q^{13} + ( - \beta_{3} - 1) q^{15} + ( - \beta_{3} + \beta_1 - 2) q^{17} + ( - \beta_{2} + \beta_1 + 1) q^{19} - q^{21} + q^{23} + (3 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{25} + q^{27} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{29} + (\beta_{3} - 2 \beta_{2} + 3 \beta_1 + 4) q^{31} + (\beta_{2} - \beta_1 - 1) q^{33} + (\beta_{3} + 1) q^{35} + (2 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{37} + (\beta_{3} + \beta_{2} - \beta_1) q^{39} + (\beta_{3} - \beta_1 - 2) q^{41} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{43} + ( - \beta_{3} - 1) q^{45} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{47} + q^{49} + ( - \beta_{3} + \beta_1 - 2) q^{51} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{53} + (2 \beta_{3} - \beta_{2}) q^{55} + ( - \beta_{2} + \beta_1 + 1) q^{57} + ( - \beta_{2} - 2 \beta_1) q^{59} + (4 \beta_{3} + 3 \beta_1 + 3) q^{61} - q^{63} + ( - \beta_{3} - \beta_1 - 6) q^{65} + (\beta_{3} + 2 \beta_{2} - 1) q^{67} + q^{69} + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{71} + ( - \beta_{3} - 3 \beta_1 - 6) q^{73} + (3 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{75} + ( - \beta_{2} + \beta_1 + 1) q^{77} + ( - \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 4) q^{79} + q^{81} + (2 \beta_{3} - \beta_{2} + 3 \beta_1 - 3) q^{83} + (4 \beta_{3} - \beta_{2} + 2 \beta_1 + 8) q^{85} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{87} + (5 \beta_{3} - \beta_{2} + \beta_1 - 6) q^{89} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{91} + (\beta_{3} - 2 \beta_{2} + 3 \beta_1 + 4) q^{93} + ( - 2 \beta_{3} + \beta_{2}) q^{95} + ( - \beta_{3} - 3 \beta_1 - 6) q^{97} + (\beta_{2} - \beta_1 - 1) q^{99}+O(q^{100})$$ q + q^3 + (-b3 - 1) * q^5 - q^7 + q^9 + (b2 - b1 - 1) * q^11 + (b3 + b2 - b1) * q^13 + (-b3 - 1) * q^15 + (-b3 + b1 - 2) * q^17 + (-b2 + b1 + 1) * q^19 - q^21 + q^23 + (3*b3 - b2 + b1 + 1) * q^25 + q^27 + (b3 - 2*b2 - b1 - 2) * q^29 + (b3 - 2*b2 + 3*b1 + 4) * q^31 + (b2 - b1 - 1) * q^33 + (b3 + 1) * q^35 + (2*b3 - b2 + b1 + 1) * q^37 + (b3 + b2 - b1) * q^39 + (b3 - b1 - 2) * q^41 + (-2*b3 - 2*b2 + b1 - 3) * q^43 + (-b3 - 1) * q^45 + (-2*b3 + 2*b2 - 2*b1) * q^47 + q^49 + (-b3 + b1 - 2) * q^51 + (-b3 + 2*b2 + 2*b1 + 1) * q^53 + (2*b3 - b2) * q^55 + (-b2 + b1 + 1) * q^57 + (-b2 - 2*b1) * q^59 + (4*b3 + 3*b1 + 3) * q^61 - q^63 + (-b3 - b1 - 6) * q^65 + (b3 + 2*b2 - 1) * q^67 + q^69 + (b3 - 3*b2 + b1) * q^71 + (-b3 - 3*b1 - 6) * q^73 + (3*b3 - b2 + b1 + 1) * q^75 + (-b2 + b1 + 1) * q^77 + (-b3 + 2*b2 - 3*b1 - 4) * q^79 + q^81 + (2*b3 - b2 + 3*b1 - 3) * q^83 + (4*b3 - b2 + 2*b1 + 8) * q^85 + (b3 - 2*b2 - b1 - 2) * q^87 + (5*b3 - b2 + b1 - 6) * q^89 + (-b3 - b2 + b1) * q^91 + (b3 - 2*b2 + 3*b1 + 4) * q^93 + (-2*b3 + b2) * q^95 + (-b3 - 3*b1 - 6) * q^97 + (b2 - b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9 $$4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9} - 2 q^{11} + q^{13} - 3 q^{15} - 8 q^{17} + 2 q^{19} - 4 q^{21} + 4 q^{23} - q^{25} + 4 q^{27} - 10 q^{29} + 10 q^{31} - 2 q^{33} + 3 q^{35} + q^{39} - 8 q^{41} - 13 q^{43} - 3 q^{45} + 6 q^{47} + 4 q^{49} - 8 q^{51} + 5 q^{53} - 3 q^{55} + 2 q^{57} + q^{59} + 5 q^{61} - 4 q^{63} - 22 q^{65} - 3 q^{67} + 4 q^{69} - 5 q^{71} - 20 q^{73} - q^{75} + 2 q^{77} - 10 q^{79} + 4 q^{81} - 18 q^{83} + 25 q^{85} - 10 q^{87} - 31 q^{89} - q^{91} + 10 q^{93} + 3 q^{95} - 20 q^{97} - 2 q^{99}+O(q^{100})$$ 4 * q + 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9 - 2 * q^11 + q^13 - 3 * q^15 - 8 * q^17 + 2 * q^19 - 4 * q^21 + 4 * q^23 - q^25 + 4 * q^27 - 10 * q^29 + 10 * q^31 - 2 * q^33 + 3 * q^35 + q^39 - 8 * q^41 - 13 * q^43 - 3 * q^45 + 6 * q^47 + 4 * q^49 - 8 * q^51 + 5 * q^53 - 3 * q^55 + 2 * q^57 + q^59 + 5 * q^61 - 4 * q^63 - 22 * q^65 - 3 * q^67 + 4 * q^69 - 5 * q^71 - 20 * q^73 - q^75 + 2 * q^77 - 10 * q^79 + 4 * q^81 - 18 * q^83 + 25 * q^85 - 10 * q^87 - 31 * q^89 - q^91 + 10 * q^93 + 3 * q^95 - 20 * q^97 - 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 5x^{2} + 2x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} - \nu^{2} - \nu ) / 2$$ (v^3 - v^2 - v) / 2 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} - \nu^{2} + 7\nu + 4 ) / 2$$ (-v^3 - v^2 + 7*v + 4) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} - 3\nu^{2} - 3\nu + 6 ) / 2$$ (v^3 - 3*v^2 - 3*v + 6) / 2
 $$\nu$$ $$=$$ $$( -\beta_{3} + \beta_{2} + 2\beta _1 + 1 ) / 4$$ (-b3 + b2 + 2*b1 + 1) / 4 $$\nu^{2}$$ $$=$$ $$( -3\beta_{3} - \beta_{2} + 2\beta _1 + 11 ) / 4$$ (-3*b3 - b2 + 2*b1 + 11) / 4 $$\nu^{3}$$ $$=$$ $$-\beta_{3} + 3\beta _1 + 3$$ -b3 + 3*b1 + 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
 −0.820249 1.13856 −1.75660 2.43828
0 1.00000 0 −3.94523 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 −1.08564 0 −1.00000 0 1.00000 0
1.3 0 1.00000 0 0.703671 0 −1.00000 0 1.00000 0
1.4 0 1.00000 0 1.32719 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.cb 4
4.b odd 2 1 3864.2.a.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.2.a.r 4 4.b odd 2 1
7728.2.a.cb 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7728))$$:

 $$T_{5}^{4} + 3T_{5}^{3} - 5T_{5}^{2} - 4T_{5} + 4$$ T5^4 + 3*T5^3 - 5*T5^2 - 4*T5 + 4 $$T_{11}^{4} + 2T_{11}^{3} - 15T_{11}^{2} - 36T_{11} - 16$$ T11^4 + 2*T11^3 - 15*T11^2 - 36*T11 - 16 $$T_{13}^{4} - T_{13}^{3} - 29T_{13}^{2} + 24T_{13} + 76$$ T13^4 - T13^3 - 29*T13^2 + 24*T13 + 76 $$T_{17}^{4} + 8T_{17}^{3} + 3T_{17}^{2} - 92T_{17} - 164$$ T17^4 + 8*T17^3 + 3*T17^2 - 92*T17 - 164

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T - 1)^{4}$$
$5$ $$T^{4} + 3 T^{3} - 5 T^{2} - 4 T + 4$$
$7$ $$(T + 1)^{4}$$
$11$ $$T^{4} + 2 T^{3} - 15 T^{2} - 36 T - 16$$
$13$ $$T^{4} - T^{3} - 29 T^{2} + 24 T + 76$$
$17$ $$T^{4} + 8 T^{3} + 3 T^{2} - 92 T - 164$$
$19$ $$T^{4} - 2 T^{3} - 15 T^{2} + 36 T - 16$$
$23$ $$(T - 1)^{4}$$
$29$ $$T^{4} + 10 T^{3} - 59 T^{2} + \cdots + 1444$$
$31$ $$T^{4} - 10 T^{3} - 59 T^{2} + \cdots + 1024$$
$37$ $$T^{4} - 41 T^{2} - 120 T - 76$$
$41$ $$T^{4} + 8 T^{3} + 3 T^{2} - 12 T - 4$$
$43$ $$T^{4} + 13 T^{3} - 37 T^{2} + \cdots + 1136$$
$47$ $$T^{4} - 6 T^{3} - 68 T^{2} + 256 T + 256$$
$53$ $$T^{4} - 5 T^{3} - 139 T^{2} + \cdots + 3884$$
$59$ $$T^{4} - T^{3} - 75 T^{2} - 128 T + 304$$
$61$ $$T^{4} - 5 T^{3} - 191 T^{2} + \cdots - 1076$$
$67$ $$T^{4} + 3 T^{3} - 71 T^{2} - 188 T + 16$$
$71$ $$T^{4} + 5 T^{3} - 111 T^{2} + 280 T + 64$$
$73$ $$T^{4} + 20 T^{3} + 55 T^{2} + \cdots - 2900$$
$79$ $$T^{4} + 10 T^{3} - 59 T^{2} + \cdots + 1024$$
$83$ $$T^{4} + 18 T^{3} + 25 T^{2} + \cdots + 464$$
$89$ $$T^{4} + 31 T^{3} + 157 T^{2} + \cdots - 22364$$
$97$ $$T^{4} + 20 T^{3} + 55 T^{2} + \cdots - 2900$$