# Properties

 Label 5824.2.a.bl Level $5824$ Weight $2$ Character orbit 5824.a Self dual yes Analytic conductor $46.505$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5824 = 2^{6} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5824.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$46.5048741372$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + ( -3 - \beta ) q^{5} + q^{7} - q^{9} +O(q^{10})$$ $$q + \beta q^{3} + ( -3 - \beta ) q^{5} + q^{7} - q^{9} + 3 \beta q^{11} + q^{13} + ( -2 - 3 \beta ) q^{15} -\beta q^{17} + ( 3 - 3 \beta ) q^{19} + \beta q^{21} + ( -3 + 2 \beta ) q^{23} + ( 6 + 6 \beta ) q^{25} -4 \beta q^{27} + ( -3 - 2 \beta ) q^{29} + ( -1 - 3 \beta ) q^{31} + 6 q^{33} + ( -3 - \beta ) q^{35} + ( 2 + 3 \beta ) q^{37} + \beta q^{39} + ( 6 - 2 \beta ) q^{41} + 5 q^{43} + ( 3 + \beta ) q^{45} + ( 3 + \beta ) q^{47} + q^{49} -2 q^{51} + ( 3 + 2 \beta ) q^{53} + ( -6 - 9 \beta ) q^{55} + ( -6 + 3 \beta ) q^{57} + ( -6 - 4 \beta ) q^{59} -6 q^{61} - q^{63} + ( -3 - \beta ) q^{65} + ( 6 - 6 \beta ) q^{67} + ( 4 - 3 \beta ) q^{69} + ( -6 + 5 \beta ) q^{71} + ( -5 + 3 \beta ) q^{73} + ( 12 + 6 \beta ) q^{75} + 3 \beta q^{77} + ( 7 - 6 \beta ) q^{79} -5 q^{81} + ( -9 + 3 \beta ) q^{83} + ( 2 + 3 \beta ) q^{85} + ( -4 - 3 \beta ) q^{87} + ( 3 + \beta ) q^{89} + q^{91} + ( -6 - \beta ) q^{93} + ( -3 + 6 \beta ) q^{95} + ( -1 - 9 \beta ) q^{97} -3 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{5} + 2q^{7} - 2q^{9} + O(q^{10})$$ $$2q - 6q^{5} + 2q^{7} - 2q^{9} + 2q^{13} - 4q^{15} + 6q^{19} - 6q^{23} + 12q^{25} - 6q^{29} - 2q^{31} + 12q^{33} - 6q^{35} + 4q^{37} + 12q^{41} + 10q^{43} + 6q^{45} + 6q^{47} + 2q^{49} - 4q^{51} + 6q^{53} - 12q^{55} - 12q^{57} - 12q^{59} - 12q^{61} - 2q^{63} - 6q^{65} + 12q^{67} + 8q^{69} - 12q^{71} - 10q^{73} + 24q^{75} + 14q^{79} - 10q^{81} - 18q^{83} + 4q^{85} - 8q^{87} + 6q^{89} + 2q^{91} - 12q^{93} - 6q^{95} - 2q^{97} + O(q^{100})$$

## 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
 −1.41421 1.41421
0 −1.41421 0 −1.58579 0 1.00000 0 −1.00000 0
1.2 0 1.41421 0 −4.41421 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
$$2$$ $$1$$
$$7$$ $$-1$$
$$13$$ $$-1$$

## 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 5824.2.a.bl 2
4.b odd 2 1 5824.2.a.bk 2
8.b even 2 1 91.2.a.c 2
8.d odd 2 1 1456.2.a.q 2
24.h odd 2 1 819.2.a.h 2
40.f even 2 1 2275.2.a.j 2
56.h odd 2 1 637.2.a.g 2
56.j odd 6 2 637.2.e.g 4
56.p even 6 2 637.2.e.f 4
104.e even 2 1 1183.2.a.d 2
104.j odd 4 2 1183.2.c.d 4
168.i even 2 1 5733.2.a.s 2
728.l odd 2 1 8281.2.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.c 2 8.b even 2 1
637.2.a.g 2 56.h odd 2 1
637.2.e.f 4 56.p even 6 2
637.2.e.g 4 56.j odd 6 2
819.2.a.h 2 24.h odd 2 1
1183.2.a.d 2 104.e even 2 1
1183.2.c.d 4 104.j odd 4 2
1456.2.a.q 2 8.d odd 2 1
2275.2.a.j 2 40.f even 2 1
5733.2.a.s 2 168.i even 2 1
5824.2.a.bk 2 4.b odd 2 1
5824.2.a.bl 2 1.a even 1 1 trivial
8281.2.a.v 2 728.l odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5824))$$:

 $$T_{3}^{2} - 2$$ $$T_{5}^{2} + 6 T_{5} + 7$$ $$T_{11}^{2} - 18$$ $$T_{17}^{2} - 2$$ $$T_{19}^{2} - 6 T_{19} - 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-2 + T^{2}$$
$5$ $$7 + 6 T + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$-18 + T^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$-2 + T^{2}$$
$19$ $$-9 - 6 T + T^{2}$$
$23$ $$1 + 6 T + T^{2}$$
$29$ $$1 + 6 T + T^{2}$$
$31$ $$-17 + 2 T + T^{2}$$
$37$ $$-14 - 4 T + T^{2}$$
$41$ $$28 - 12 T + T^{2}$$
$43$ $$( -5 + T )^{2}$$
$47$ $$7 - 6 T + T^{2}$$
$53$ $$1 - 6 T + T^{2}$$
$59$ $$4 + 12 T + T^{2}$$
$61$ $$( 6 + T )^{2}$$
$67$ $$-36 - 12 T + T^{2}$$
$71$ $$-14 + 12 T + T^{2}$$
$73$ $$7 + 10 T + T^{2}$$
$79$ $$-23 - 14 T + T^{2}$$
$83$ $$63 + 18 T + T^{2}$$
$89$ $$7 - 6 T + T^{2}$$
$97$ $$-161 + 2 T + T^{2}$$