# Properties

 Label 5520.2.a.br Level $5520$ Weight $2$ Character orbit 5520.a Self dual yes Analytic conductor $44.077$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5520.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.0774219157$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1380) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} - \beta q^{7} + q^{9} +O(q^{10})$$ q + q^3 + q^5 - b * q^7 + q^9 $$q + q^{3} + q^{5} - \beta q^{7} + q^{9} - 2 \beta q^{11} + 2 q^{13} + q^{15} + ( - \beta + 2) q^{17} + 2 q^{19} - \beta q^{21} - q^{23} + q^{25} + q^{27} + ( - 3 \beta + 2) q^{29} + 3 \beta q^{31} - 2 \beta q^{33} - \beta q^{35} + (\beta + 4) q^{37} + 2 q^{39} + (3 \beta + 2) q^{41} + q^{45} - 4 q^{47} + (\beta - 3) q^{49} + ( - \beta + 2) q^{51} + ( - \beta + 6) q^{53} - 2 \beta q^{55} + 2 q^{57} + (\beta - 2) q^{59} + ( - 2 \beta + 6) q^{61} - \beta q^{63} + 2 q^{65} + (3 \beta - 4) q^{67} - q^{69} + ( - \beta - 2) q^{71} + (4 \beta + 2) q^{73} + q^{75} + (2 \beta + 8) q^{77} + (2 \beta + 6) q^{79} + q^{81} + (3 \beta - 8) q^{83} + ( - \beta + 2) q^{85} + ( - 3 \beta + 2) q^{87} + (2 \beta - 2) q^{89} - 2 \beta q^{91} + 3 \beta q^{93} + 2 q^{95} + ( - 2 \beta + 8) q^{97} - 2 \beta q^{99} +O(q^{100})$$ q + q^3 + q^5 - b * q^7 + q^9 - 2*b * q^11 + 2 * q^13 + q^15 + (-b + 2) * q^17 + 2 * q^19 - b * q^21 - q^23 + q^25 + q^27 + (-3*b + 2) * q^29 + 3*b * q^31 - 2*b * q^33 - b * q^35 + (b + 4) * q^37 + 2 * q^39 + (3*b + 2) * q^41 + q^45 - 4 * q^47 + (b - 3) * q^49 + (-b + 2) * q^51 + (-b + 6) * q^53 - 2*b * q^55 + 2 * q^57 + (b - 2) * q^59 + (-2*b + 6) * q^61 - b * q^63 + 2 * q^65 + (3*b - 4) * q^67 - q^69 + (-b - 2) * q^71 + (4*b + 2) * q^73 + q^75 + (2*b + 8) * q^77 + (2*b + 6) * q^79 + q^81 + (3*b - 8) * q^83 + (-b + 2) * q^85 + (-3*b + 2) * q^87 + (2*b - 2) * q^89 - 2*b * q^91 + 3*b * q^93 + 2 * q^95 + (-2*b + 8) * q^97 - 2*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9} - 2 q^{11} + 4 q^{13} + 2 q^{15} + 3 q^{17} + 4 q^{19} - q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} + q^{29} + 3 q^{31} - 2 q^{33} - q^{35} + 9 q^{37} + 4 q^{39} + 7 q^{41} + 2 q^{45} - 8 q^{47} - 5 q^{49} + 3 q^{51} + 11 q^{53} - 2 q^{55} + 4 q^{57} - 3 q^{59} + 10 q^{61} - q^{63} + 4 q^{65} - 5 q^{67} - 2 q^{69} - 5 q^{71} + 8 q^{73} + 2 q^{75} + 18 q^{77} + 14 q^{79} + 2 q^{81} - 13 q^{83} + 3 q^{85} + q^{87} - 2 q^{89} - 2 q^{91} + 3 q^{93} + 4 q^{95} + 14 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9 - 2 * q^11 + 4 * q^13 + 2 * q^15 + 3 * q^17 + 4 * q^19 - q^21 - 2 * q^23 + 2 * q^25 + 2 * q^27 + q^29 + 3 * q^31 - 2 * q^33 - q^35 + 9 * q^37 + 4 * q^39 + 7 * q^41 + 2 * q^45 - 8 * q^47 - 5 * q^49 + 3 * q^51 + 11 * q^53 - 2 * q^55 + 4 * q^57 - 3 * q^59 + 10 * q^61 - q^63 + 4 * q^65 - 5 * q^67 - 2 * q^69 - 5 * q^71 + 8 * q^73 + 2 * q^75 + 18 * q^77 + 14 * q^79 + 2 * q^81 - 13 * q^83 + 3 * q^85 + q^87 - 2 * q^89 - 2 * q^91 + 3 * q^93 + 4 * q^95 + 14 * q^97 - 2 * 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
 2.56155 −1.56155
0 1.00000 0 1.00000 0 −2.56155 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 1.56155 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$23$$ $$1$$

## 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 5520.2.a.br 2
4.b odd 2 1 1380.2.a.g 2
12.b even 2 1 4140.2.a.n 2
20.d odd 2 1 6900.2.a.u 2
20.e even 4 2 6900.2.f.p 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.a.g 2 4.b odd 2 1
4140.2.a.n 2 12.b even 2 1
5520.2.a.br 2 1.a even 1 1 trivial
6900.2.a.u 2 20.d odd 2 1
6900.2.f.p 4 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5520))$$:

 $$T_{7}^{2} + T_{7} - 4$$ T7^2 + T7 - 4 $$T_{11}^{2} + 2T_{11} - 16$$ T11^2 + 2*T11 - 16 $$T_{13} - 2$$ T13 - 2 $$T_{17}^{2} - 3T_{17} - 2$$ T17^2 - 3*T17 - 2 $$T_{19} - 2$$ T19 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} + T - 4$$
$11$ $$T^{2} + 2T - 16$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} - 3T - 2$$
$19$ $$(T - 2)^{2}$$
$23$ $$(T + 1)^{2}$$
$29$ $$T^{2} - T - 38$$
$31$ $$T^{2} - 3T - 36$$
$37$ $$T^{2} - 9T + 16$$
$41$ $$T^{2} - 7T - 26$$
$43$ $$T^{2}$$
$47$ $$(T + 4)^{2}$$
$53$ $$T^{2} - 11T + 26$$
$59$ $$T^{2} + 3T - 2$$
$61$ $$T^{2} - 10T + 8$$
$67$ $$T^{2} + 5T - 32$$
$71$ $$T^{2} + 5T + 2$$
$73$ $$T^{2} - 8T - 52$$
$79$ $$T^{2} - 14T + 32$$
$83$ $$T^{2} + 13T + 4$$
$89$ $$T^{2} + 2T - 16$$
$97$ $$T^{2} - 14T + 32$$