# Properties

 Label 5520.2.a.bj 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: $$\Q(\sqrt{15})$$ Defining polynomial: $$x^{2} - 15$$ x^2 - 15 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 $$\beta = \sqrt{15}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} - 3 q^{7} + q^{9}+O(q^{10})$$ q - q^3 + q^5 - 3 * q^7 + q^9 $$q - q^{3} + q^{5} - 3 q^{7} + q^{9} + ( - \beta - 1) q^{11} + ( - \beta + 1) q^{13} - q^{15} + \beta q^{17} + (\beta - 1) q^{19} + 3 q^{21} + q^{23} + q^{25} - q^{27} + ( - \beta - 2) q^{29} - 3 q^{31} + (\beta + 1) q^{33} - 3 q^{35} + q^{37} + (\beta - 1) q^{39} + ( - \beta - 2) q^{41} + ( - 2 \beta + 4) q^{43} + q^{45} + (\beta + 3) q^{47} + 2 q^{49} - \beta q^{51} - \beta q^{53} + ( - \beta - 1) q^{55} + ( - \beta + 1) q^{57} + ( - \beta - 2) q^{59} + (\beta + 5) q^{61} - 3 q^{63} + ( - \beta + 1) q^{65} + ( - 2 \beta - 3) q^{67} - q^{69} + (3 \beta + 2) q^{71} + ( - \beta + 1) q^{73} - q^{75} + (3 \beta + 3) q^{77} - 4 q^{79} + q^{81} + ( - \beta + 4) q^{83} + \beta q^{85} + (\beta + 2) q^{87} + (2 \beta - 6) q^{89} + (3 \beta - 3) q^{91} + 3 q^{93} + (\beta - 1) q^{95} + 8 q^{97} + ( - \beta - 1) q^{99} +O(q^{100})$$ q - q^3 + q^5 - 3 * q^7 + q^9 + (-b - 1) * q^11 + (-b + 1) * q^13 - q^15 + b * q^17 + (b - 1) * q^19 + 3 * q^21 + q^23 + q^25 - q^27 + (-b - 2) * q^29 - 3 * q^31 + (b + 1) * q^33 - 3 * q^35 + q^37 + (b - 1) * q^39 + (-b - 2) * q^41 + (-2*b + 4) * q^43 + q^45 + (b + 3) * q^47 + 2 * q^49 - b * q^51 - b * q^53 + (-b - 1) * q^55 + (-b + 1) * q^57 + (-b - 2) * q^59 + (b + 5) * q^61 - 3 * q^63 + (-b + 1) * q^65 + (-2*b - 3) * q^67 - q^69 + (3*b + 2) * q^71 + (-b + 1) * q^73 - q^75 + (3*b + 3) * q^77 - 4 * q^79 + q^81 + (-b + 4) * q^83 + b * q^85 + (b + 2) * q^87 + (2*b - 6) * q^89 + (3*b - 3) * q^91 + 3 * q^93 + (b - 1) * q^95 + 8 * q^97 + (-b - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{15} - 2 q^{19} + 6 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} - 6 q^{31} + 2 q^{33} - 6 q^{35} + 2 q^{37} - 2 q^{39} - 4 q^{41} + 8 q^{43} + 2 q^{45} + 6 q^{47} + 4 q^{49} - 2 q^{55} + 2 q^{57} - 4 q^{59} + 10 q^{61} - 6 q^{63} + 2 q^{65} - 6 q^{67} - 2 q^{69} + 4 q^{71} + 2 q^{73} - 2 q^{75} + 6 q^{77} - 8 q^{79} + 2 q^{81} + 8 q^{83} + 4 q^{87} - 12 q^{89} - 6 q^{91} + 6 q^{93} - 2 q^{95} + 16 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9 - 2 * q^11 + 2 * q^13 - 2 * q^15 - 2 * q^19 + 6 * q^21 + 2 * q^23 + 2 * q^25 - 2 * q^27 - 4 * q^29 - 6 * q^31 + 2 * q^33 - 6 * q^35 + 2 * q^37 - 2 * q^39 - 4 * q^41 + 8 * q^43 + 2 * q^45 + 6 * q^47 + 4 * q^49 - 2 * q^55 + 2 * q^57 - 4 * q^59 + 10 * q^61 - 6 * q^63 + 2 * q^65 - 6 * q^67 - 2 * q^69 + 4 * q^71 + 2 * q^73 - 2 * q^75 + 6 * q^77 - 8 * q^79 + 2 * q^81 + 8 * q^83 + 4 * q^87 - 12 * q^89 - 6 * q^91 + 6 * q^93 - 2 * q^95 + 16 * 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
 3.87298 −3.87298
0 −1.00000 0 1.00000 0 −3.00000 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −3.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$23$$ $$-1$$

## 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.bj 2
4.b odd 2 1 1380.2.a.i 2
12.b even 2 1 4140.2.a.p 2
20.d odd 2 1 6900.2.a.j 2
20.e even 4 2 6900.2.f.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.a.i 2 4.b odd 2 1
4140.2.a.p 2 12.b even 2 1
5520.2.a.bj 2 1.a even 1 1 trivial
6900.2.a.j 2 20.d odd 2 1
6900.2.f.o 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5520))$$:

 $$T_{7} + 3$$ T7 + 3 $$T_{11}^{2} + 2T_{11} - 14$$ T11^2 + 2*T11 - 14 $$T_{13}^{2} - 2T_{13} - 14$$ T13^2 - 2*T13 - 14 $$T_{17}^{2} - 15$$ T17^2 - 15 $$T_{19}^{2} + 2T_{19} - 14$$ T19^2 + 2*T19 - 14

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T + 3)^{2}$$
$11$ $$T^{2} + 2T - 14$$
$13$ $$T^{2} - 2T - 14$$
$17$ $$T^{2} - 15$$
$19$ $$T^{2} + 2T - 14$$
$23$ $$(T - 1)^{2}$$
$29$ $$T^{2} + 4T - 11$$
$31$ $$(T + 3)^{2}$$
$37$ $$(T - 1)^{2}$$
$41$ $$T^{2} + 4T - 11$$
$43$ $$T^{2} - 8T - 44$$
$47$ $$T^{2} - 6T - 6$$
$53$ $$T^{2} - 15$$
$59$ $$T^{2} + 4T - 11$$
$61$ $$T^{2} - 10T + 10$$
$67$ $$T^{2} + 6T - 51$$
$71$ $$T^{2} - 4T - 131$$
$73$ $$T^{2} - 2T - 14$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} - 8T + 1$$
$89$ $$T^{2} + 12T - 24$$
$97$ $$(T - 8)^{2}$$