# Properties

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

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## 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 1932) 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 $$q + q^{3} + ( - \beta - 2) q^{5} + q^{7} + q^{9} + (2 \beta - 3) q^{11} + ( - \beta + 1) q^{13} + ( - \beta - 2) q^{15} + ( - 2 \beta - 3) q^{17} + (2 \beta + 3) q^{19} + q^{21} + q^{23} + (5 \beta + 2) q^{25} + q^{27} + (2 \beta - 3) q^{29} + ( - 2 \beta + 1) q^{31} + (2 \beta - 3) q^{33} + ( - \beta - 2) q^{35} + ( - 2 \beta + 3) q^{37} + ( - \beta + 1) q^{39} + ( - 2 \beta + 1) q^{41} + (\beta - 3) q^{43} + ( - \beta - 2) q^{45} + ( - 2 \beta + 2) q^{47} + q^{49} + ( - 2 \beta - 3) q^{51} + ( - 3 \beta + 2) q^{53} - 3 \beta q^{55} + (2 \beta + 3) q^{57} + 3 \beta q^{59} + ( - 3 \beta + 1) q^{61} + q^{63} + (2 \beta + 1) q^{65} + (\beta - 12) q^{67} + q^{69} + (3 \beta - 9) q^{71} + 11 q^{73} + (5 \beta + 2) q^{75} + (2 \beta - 3) q^{77} + (8 \beta - 3) q^{79} + q^{81} + (4 \beta - 11) q^{83} + (9 \beta + 12) q^{85} + (2 \beta - 3) q^{87} + (5 \beta - 9) q^{89} + ( - \beta + 1) q^{91} + ( - 2 \beta + 1) q^{93} + ( - 9 \beta - 12) q^{95} + (4 \beta + 9) q^{97} + (2 \beta - 3) q^{99} +O(q^{100})$$ q + q^3 + (-b - 2) * q^5 + q^7 + q^9 + (2*b - 3) * q^11 + (-b + 1) * q^13 + (-b - 2) * q^15 + (-2*b - 3) * q^17 + (2*b + 3) * q^19 + q^21 + q^23 + (5*b + 2) * q^25 + q^27 + (2*b - 3) * q^29 + (-2*b + 1) * q^31 + (2*b - 3) * q^33 + (-b - 2) * q^35 + (-2*b + 3) * q^37 + (-b + 1) * q^39 + (-2*b + 1) * q^41 + (b - 3) * q^43 + (-b - 2) * q^45 + (-2*b + 2) * q^47 + q^49 + (-2*b - 3) * q^51 + (-3*b + 2) * q^53 - 3*b * q^55 + (2*b + 3) * q^57 + 3*b * q^59 + (-3*b + 1) * q^61 + q^63 + (2*b + 1) * q^65 + (b - 12) * q^67 + q^69 + (3*b - 9) * q^71 + 11 * q^73 + (5*b + 2) * q^75 + (2*b - 3) * q^77 + (8*b - 3) * q^79 + q^81 + (4*b - 11) * q^83 + (9*b + 12) * q^85 + (2*b - 3) * q^87 + (5*b - 9) * q^89 + (-b + 1) * q^91 + (-2*b + 1) * q^93 + (-9*b - 12) * q^95 + (4*b + 9) * q^97 + (2*b - 3) * 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 $$2 q + 2 q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{11} + q^{13} - 5 q^{15} - 8 q^{17} + 8 q^{19} + 2 q^{21} + 2 q^{23} + 9 q^{25} + 2 q^{27} - 4 q^{29} - 4 q^{33} - 5 q^{35} + 4 q^{37} + q^{39} - 5 q^{43} - 5 q^{45} + 2 q^{47} + 2 q^{49} - 8 q^{51} + q^{53} - 3 q^{55} + 8 q^{57} + 3 q^{59} - q^{61} + 2 q^{63} + 4 q^{65} - 23 q^{67} + 2 q^{69} - 15 q^{71} + 22 q^{73} + 9 q^{75} - 4 q^{77} + 2 q^{79} + 2 q^{81} - 18 q^{83} + 33 q^{85} - 4 q^{87} - 13 q^{89} + q^{91} - 33 q^{95} + 22 q^{97} - 4 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 5 * q^5 + 2 * q^7 + 2 * q^9 - 4 * q^11 + q^13 - 5 * q^15 - 8 * q^17 + 8 * q^19 + 2 * q^21 + 2 * q^23 + 9 * q^25 + 2 * q^27 - 4 * q^29 - 4 * q^33 - 5 * q^35 + 4 * q^37 + q^39 - 5 * q^43 - 5 * q^45 + 2 * q^47 + 2 * q^49 - 8 * q^51 + q^53 - 3 * q^55 + 8 * q^57 + 3 * q^59 - q^61 + 2 * q^63 + 4 * q^65 - 23 * q^67 + 2 * q^69 - 15 * q^71 + 22 * q^73 + 9 * q^75 - 4 * q^77 + 2 * q^79 + 2 * q^81 - 18 * q^83 + 33 * q^85 - 4 * q^87 - 13 * q^89 + q^91 - 33 * q^95 + 22 * q^97 - 4 * 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.bf 2
4.b odd 2 1 1932.2.a.c 2
12.b even 2 1 5796.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1932.2.a.c 2 4.b odd 2 1
5796.2.a.m 2 12.b even 2 1
7728.2.a.bf 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 $$T_{11}^{2} + 4T_{11} - 9$$ T11^2 + 4*T11 - 9 $$T_{13}^{2} - T_{13} - 3$$ T13^2 - T13 - 3 $$T_{17}^{2} + 8T_{17} + 3$$ T17^2 + 8*T17 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} + 5T + 3$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} + 4T - 9$$
$13$ $$T^{2} - T - 3$$
$17$ $$T^{2} + 8T + 3$$
$19$ $$T^{2} - 8T + 3$$
$23$ $$(T - 1)^{2}$$
$29$ $$T^{2} + 4T - 9$$
$31$ $$T^{2} - 13$$
$37$ $$T^{2} - 4T - 9$$
$41$ $$T^{2} - 13$$
$43$ $$T^{2} + 5T + 3$$
$47$ $$T^{2} - 2T - 12$$
$53$ $$T^{2} - T - 29$$
$59$ $$T^{2} - 3T - 27$$
$61$ $$T^{2} + T - 29$$
$67$ $$T^{2} + 23T + 129$$
$71$ $$T^{2} + 15T + 27$$
$73$ $$(T - 11)^{2}$$
$79$ $$T^{2} - 2T - 207$$
$83$ $$T^{2} + 18T + 29$$
$89$ $$T^{2} + 13T - 39$$
$97$ $$T^{2} - 22T + 69$$
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