# Properties

 Label 7728.2.a.bn 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: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 −0.618034 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 1.61803 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.bn 2
4.b odd 2 1 483.2.a.d 2
12.b even 2 1 1449.2.a.h 2
28.d even 2 1 3381.2.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.d 2 4.b odd 2 1
1449.2.a.h 2 12.b even 2 1
3381.2.a.r 2 28.d even 2 1
7728.2.a.bn 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7728))$$:

 $$T_{5}^{2} - T_{5} - 1$$ T5^2 - T5 - 1 $$T_{11}^{2} - 5$$ T11^2 - 5 $$T_{13}^{2} + 7T_{13} + 11$$ T13^2 + 7*T13 + 11 $$T_{17}^{2} - 45$$ T17^2 - 45

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} - T - 1$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} - 5$$
$13$ $$T^{2} + 7T + 11$$
$17$ $$T^{2} - 45$$
$19$ $$T^{2} + 2T - 19$$
$23$ $$(T - 1)^{2}$$
$29$ $$T^{2} + 12T + 31$$
$31$ $$T^{2} - 45$$
$37$ $$(T + 11)^{2}$$
$41$ $$T^{2} - 6T - 11$$
$43$ $$T^{2} - T - 1$$
$47$ $$T^{2} - 10T + 20$$
$53$ $$T^{2} + 15T + 25$$
$59$ $$T^{2} + 21T + 109$$
$61$ $$T^{2} + 3T - 9$$
$67$ $$T^{2} + T - 31$$
$71$ $$T^{2} + 11T + 29$$
$73$ $$T^{2} + 12T - 9$$
$79$ $$T^{2} - 10T + 5$$
$83$ $$T^{2} + 4T - 121$$
$89$ $$T^{2} - 21T + 109$$
$97$ $$T^{2} + 6T - 171$$