# Properties

 Label 5808.2.a.ca Level $5808$ Weight $2$ Character orbit 5808.a Self dual yes Analytic conductor $46.377$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5808 = 2^{4} \cdot 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5808.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$46.3771134940$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 363) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} - 3 q^{5} + 2 \beta q^{7} + q^{9}+O(q^{10})$$ q + q^3 - 3 * q^5 + 2*b * q^7 + q^9 $$q + q^{3} - 3 q^{5} + 2 \beta q^{7} + q^{9} - \beta q^{13} - 3 q^{15} + \beta q^{17} - 4 \beta q^{19} + 2 \beta q^{21} + 6 q^{23} + 4 q^{25} + q^{27} - \beta q^{29} - 4 q^{31} - 6 \beta q^{35} - 11 q^{37} - \beta q^{39} - \beta q^{41} - 2 \beta q^{43} - 3 q^{45} + 5 q^{49} + \beta q^{51} - 9 q^{53} - 4 \beta q^{57} + 6 q^{59} + 2 \beta q^{63} + 3 \beta q^{65} + 2 q^{67} + 6 q^{69} + 6 q^{71} + 4 \beta q^{73} + 4 q^{75} + q^{81} - 3 \beta q^{85} - \beta q^{87} + 9 q^{89} - 6 q^{91} - 4 q^{93} + 12 \beta q^{95} - 7 q^{97} +O(q^{100})$$ q + q^3 - 3 * q^5 + 2*b * q^7 + q^9 - b * q^13 - 3 * q^15 + b * q^17 - 4*b * q^19 + 2*b * q^21 + 6 * q^23 + 4 * q^25 + q^27 - b * q^29 - 4 * q^31 - 6*b * q^35 - 11 * q^37 - b * q^39 - b * q^41 - 2*b * q^43 - 3 * q^45 + 5 * q^49 + b * q^51 - 9 * q^53 - 4*b * q^57 + 6 * q^59 + 2*b * q^63 + 3*b * q^65 + 2 * q^67 + 6 * q^69 + 6 * q^71 + 4*b * q^73 + 4 * q^75 + q^81 - 3*b * q^85 - b * q^87 + 9 * q^89 - 6 * q^91 - 4 * q^93 + 12*b * q^95 - 7 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 6 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 6 * q^5 + 2 * q^9 $$2 q + 2 q^{3} - 6 q^{5} + 2 q^{9} - 6 q^{15} + 12 q^{23} + 8 q^{25} + 2 q^{27} - 8 q^{31} - 22 q^{37} - 6 q^{45} + 10 q^{49} - 18 q^{53} + 12 q^{59} + 4 q^{67} + 12 q^{69} + 12 q^{71} + 8 q^{75} + 2 q^{81} + 18 q^{89} - 12 q^{91} - 8 q^{93} - 14 q^{97}+O(q^{100})$$ 2 * q + 2 * q^3 - 6 * q^5 + 2 * q^9 - 6 * q^15 + 12 * q^23 + 8 * q^25 + 2 * q^27 - 8 * q^31 - 22 * q^37 - 6 * q^45 + 10 * q^49 - 18 * q^53 + 12 * q^59 + 4 * q^67 + 12 * q^69 + 12 * q^71 + 8 * q^75 + 2 * q^81 + 18 * q^89 - 12 * q^91 - 8 * q^93 - 14 * 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
 −1.73205 1.73205
0 1.00000 0 −3.00000 0 −3.46410 0 1.00000 0
1.2 0 1.00000 0 −3.00000 0 3.46410 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$11$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5808.2.a.ca 2
4.b odd 2 1 363.2.a.f 2
11.b odd 2 1 inner 5808.2.a.ca 2
12.b even 2 1 1089.2.a.o 2
20.d odd 2 1 9075.2.a.bo 2
44.c even 2 1 363.2.a.f 2
44.g even 10 4 363.2.e.m 8
44.h odd 10 4 363.2.e.m 8
132.d odd 2 1 1089.2.a.o 2
220.g even 2 1 9075.2.a.bo 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.a.f 2 4.b odd 2 1
363.2.a.f 2 44.c even 2 1
363.2.e.m 8 44.g even 10 4
363.2.e.m 8 44.h odd 10 4
1089.2.a.o 2 12.b even 2 1
1089.2.a.o 2 132.d odd 2 1
5808.2.a.ca 2 1.a even 1 1 trivial
5808.2.a.ca 2 11.b odd 2 1 inner
9075.2.a.bo 2 20.d odd 2 1
9075.2.a.bo 2 220.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5808))$$:

 $$T_{5} + 3$$ T5 + 3 $$T_{7}^{2} - 12$$ T7^2 - 12 $$T_{13}^{2} - 3$$ T13^2 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T + 3)^{2}$$
$7$ $$T^{2} - 12$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 3$$
$17$ $$T^{2} - 3$$
$19$ $$T^{2} - 48$$
$23$ $$(T - 6)^{2}$$
$29$ $$T^{2} - 3$$
$31$ $$(T + 4)^{2}$$
$37$ $$(T + 11)^{2}$$
$41$ $$T^{2} - 3$$
$43$ $$T^{2} - 12$$
$47$ $$T^{2}$$
$53$ $$(T + 9)^{2}$$
$59$ $$(T - 6)^{2}$$
$61$ $$T^{2}$$
$67$ $$(T - 2)^{2}$$
$71$ $$(T - 6)^{2}$$
$73$ $$T^{2} - 48$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$(T - 9)^{2}$$
$97$ $$(T + 7)^{2}$$