# Properties

 Label 5520.2.a.bs 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: $$2$$ Twist minimal: no (minimal twist has level 690) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} + (\beta + 1) q^{7} + q^{9}+O(q^{10})$$ q + q^3 + q^5 + (b + 1) * q^7 + q^9 $$q + q^{3} + q^{5} + (\beta + 1) q^{7} + q^{9} + ( - \beta - 1) q^{11} + 2 q^{13} + q^{15} + (\beta + 3) q^{17} - 4 q^{19} + (\beta + 1) q^{21} - q^{23} + q^{25} + q^{27} + 2 q^{29} + ( - \beta - 1) q^{33} + (\beta + 1) q^{35} + ( - \beta - 3) q^{37} + 2 q^{39} + 2 q^{41} + q^{45} + 8 q^{47} + (2 \beta + 11) q^{49} + (\beta + 3) q^{51} + ( - 2 \beta + 4) q^{53} + ( - \beta - 1) q^{55} - 4 q^{57} + (2 \beta + 6) q^{59} + ( - \beta + 5) q^{61} + (\beta + 1) q^{63} + 2 q^{65} + 8 q^{67} - q^{69} + ( - 2 \beta + 2) q^{71} + (2 \beta + 4) q^{73} + q^{75} + ( - 2 \beta - 18) q^{77} + (\beta + 1) q^{79} + q^{81} + ( - 3 \beta + 1) q^{83} + (\beta + 3) q^{85} + 2 q^{87} + (\beta - 1) q^{89} + (2 \beta + 2) q^{91} - 4 q^{95} + (2 \beta - 8) q^{97} + ( - \beta - 1) q^{99}+O(q^{100})$$ q + q^3 + q^5 + (b + 1) * q^7 + q^9 + (-b - 1) * q^11 + 2 * q^13 + q^15 + (b + 3) * q^17 - 4 * q^19 + (b + 1) * q^21 - q^23 + q^25 + q^27 + 2 * q^29 + (-b - 1) * q^33 + (b + 1) * q^35 + (-b - 3) * q^37 + 2 * q^39 + 2 * q^41 + q^45 + 8 * q^47 + (2*b + 11) * q^49 + (b + 3) * q^51 + (-2*b + 4) * q^53 + (-b - 1) * q^55 - 4 * q^57 + (2*b + 6) * q^59 + (-b + 5) * q^61 + (b + 1) * q^63 + 2 * q^65 + 8 * q^67 - q^69 + (-2*b + 2) * q^71 + (2*b + 4) * q^73 + q^75 + (-2*b - 18) * q^77 + (b + 1) * q^79 + q^81 + (-3*b + 1) * q^83 + (b + 3) * q^85 + 2 * q^87 + (b - 1) * q^89 + (2*b + 2) * q^91 - 4 * q^95 + (2*b - 8) * q^97 + (-b - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{11} + 4 q^{13} + 2 q^{15} + 6 q^{17} - 8 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{29} - 2 q^{33} + 2 q^{35} - 6 q^{37} + 4 q^{39} + 4 q^{41} + 2 q^{45} + 16 q^{47} + 22 q^{49} + 6 q^{51} + 8 q^{53} - 2 q^{55} - 8 q^{57} + 12 q^{59} + 10 q^{61} + 2 q^{63} + 4 q^{65} + 16 q^{67} - 2 q^{69} + 4 q^{71} + 8 q^{73} + 2 q^{75} - 36 q^{77} + 2 q^{79} + 2 q^{81} + 2 q^{83} + 6 q^{85} + 4 q^{87} - 2 q^{89} + 4 q^{91} - 8 q^{95} - 16 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 2 * q^11 + 4 * q^13 + 2 * q^15 + 6 * q^17 - 8 * q^19 + 2 * q^21 - 2 * q^23 + 2 * q^25 + 2 * q^27 + 4 * q^29 - 2 * q^33 + 2 * q^35 - 6 * q^37 + 4 * q^39 + 4 * q^41 + 2 * q^45 + 16 * q^47 + 22 * q^49 + 6 * q^51 + 8 * q^53 - 2 * q^55 - 8 * q^57 + 12 * q^59 + 10 * q^61 + 2 * q^63 + 4 * q^65 + 16 * q^67 - 2 * q^69 + 4 * q^71 + 8 * q^73 + 2 * q^75 - 36 * q^77 + 2 * q^79 + 2 * q^81 + 2 * q^83 + 6 * q^85 + 4 * q^87 - 2 * q^89 + 4 * q^91 - 8 * 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
 −1.56155 2.56155
0 1.00000 0 1.00000 0 −3.12311 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 5.12311 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.bs 2
4.b odd 2 1 690.2.a.l 2
12.b even 2 1 2070.2.a.t 2
20.d odd 2 1 3450.2.a.bi 2
20.e even 4 2 3450.2.d.v 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.a.l 2 4.b odd 2 1
2070.2.a.t 2 12.b even 2 1
3450.2.a.bi 2 20.d odd 2 1
3450.2.d.v 4 20.e even 4 2
5520.2.a.bs 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5520))$$:

 $$T_{7}^{2} - 2T_{7} - 16$$ T7^2 - 2*T7 - 16 $$T_{11}^{2} + 2T_{11} - 16$$ T11^2 + 2*T11 - 16 $$T_{13} - 2$$ T13 - 2 $$T_{17}^{2} - 6T_{17} - 8$$ T17^2 - 6*T17 - 8 $$T_{19} + 4$$ T19 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 2T - 16$$
$11$ $$T^{2} + 2T - 16$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} - 6T - 8$$
$19$ $$(T + 4)^{2}$$
$23$ $$(T + 1)^{2}$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 6T - 8$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2}$$
$47$ $$(T - 8)^{2}$$
$53$ $$T^{2} - 8T - 52$$
$59$ $$T^{2} - 12T - 32$$
$61$ $$T^{2} - 10T + 8$$
$67$ $$(T - 8)^{2}$$
$71$ $$T^{2} - 4T - 64$$
$73$ $$T^{2} - 8T - 52$$
$79$ $$T^{2} - 2T - 16$$
$83$ $$T^{2} - 2T - 152$$
$89$ $$T^{2} + 2T - 16$$
$97$ $$T^{2} + 16T - 4$$