# Properties

 Label 5200.2.a.i Level $5200$ Weight $2$ Character orbit 5200.a Self dual yes Analytic conductor $41.522$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5200 = 2^{4} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$41.5222090511$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 650) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{3} + 5 q^{7} + q^{9}+O(q^{10})$$ q - 2 * q^3 + 5 * q^7 + q^9 $$q - 2 q^{3} + 5 q^{7} + q^{9} + 3 q^{11} - q^{13} - 3 q^{17} + 4 q^{19} - 10 q^{21} + 6 q^{23} + 4 q^{27} + 9 q^{29} - 5 q^{31} - 6 q^{33} - 2 q^{37} + 2 q^{39} + 2 q^{43} - 9 q^{47} + 18 q^{49} + 6 q^{51} + 9 q^{53} - 8 q^{57} + 9 q^{59} - q^{61} + 5 q^{63} + 5 q^{67} - 12 q^{69} - 14 q^{73} + 15 q^{77} + 16 q^{79} - 11 q^{81} - 15 q^{83} - 18 q^{87} - 6 q^{89} - 5 q^{91} + 10 q^{93} - 8 q^{97} + 3 q^{99}+O(q^{100})$$ q - 2 * q^3 + 5 * q^7 + q^9 + 3 * q^11 - q^13 - 3 * q^17 + 4 * q^19 - 10 * q^21 + 6 * q^23 + 4 * q^27 + 9 * q^29 - 5 * q^31 - 6 * q^33 - 2 * q^37 + 2 * q^39 + 2 * q^43 - 9 * q^47 + 18 * q^49 + 6 * q^51 + 9 * q^53 - 8 * q^57 + 9 * q^59 - q^61 + 5 * q^63 + 5 * q^67 - 12 * q^69 - 14 * q^73 + 15 * q^77 + 16 * q^79 - 11 * q^81 - 15 * q^83 - 18 * q^87 - 6 * q^89 - 5 * q^91 + 10 * q^93 - 8 * q^97 + 3 * 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
 0
0 −2.00000 0 0 0 5.00000 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$1$$

## 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 5200.2.a.i 1
4.b odd 2 1 650.2.a.f 1
5.b even 2 1 5200.2.a.bc 1
12.b even 2 1 5850.2.a.bc 1
20.d odd 2 1 650.2.a.h yes 1
20.e even 4 2 650.2.b.b 2
52.b odd 2 1 8450.2.a.x 1
60.h even 2 1 5850.2.a.bb 1
60.l odd 4 2 5850.2.e.ba 2
260.g odd 2 1 8450.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.a.f 1 4.b odd 2 1
650.2.a.h yes 1 20.d odd 2 1
650.2.b.b 2 20.e even 4 2
5200.2.a.i 1 1.a even 1 1 trivial
5200.2.a.bc 1 5.b even 2 1
5850.2.a.bb 1 60.h even 2 1
5850.2.a.bc 1 12.b even 2 1
5850.2.e.ba 2 60.l odd 4 2
8450.2.a.a 1 260.g odd 2 1
8450.2.a.x 1 52.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5200))$$:

 $$T_{3} + 2$$ T3 + 2 $$T_{7} - 5$$ T7 - 5 $$T_{11} - 3$$ T11 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 2$$
$5$ $$T$$
$7$ $$T - 5$$
$11$ $$T - 3$$
$13$ $$T + 1$$
$17$ $$T + 3$$
$19$ $$T - 4$$
$23$ $$T - 6$$
$29$ $$T - 9$$
$31$ $$T + 5$$
$37$ $$T + 2$$
$41$ $$T$$
$43$ $$T - 2$$
$47$ $$T + 9$$
$53$ $$T - 9$$
$59$ $$T - 9$$
$61$ $$T + 1$$
$67$ $$T - 5$$
$71$ $$T$$
$73$ $$T + 14$$
$79$ $$T - 16$$
$83$ $$T + 15$$
$89$ $$T + 6$$
$97$ $$T + 8$$