# Properties

 Label 9200.2.a.y Level $9200$ Weight $2$ Character orbit 9200.a Self dual yes Analytic conductor $73.462$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9200 = 2^{4} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$73.4623698596$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 460) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - 4q^{7} - 2q^{9} + O(q^{10})$$ $$q + q^{3} - 4q^{7} - 2q^{9} + 6q^{11} + q^{13} - 2q^{19} - 4q^{21} + q^{23} - 5q^{27} + 9q^{29} - 5q^{31} + 6q^{33} - 2q^{37} + q^{39} - 9q^{41} - 4q^{43} - 3q^{47} + 9q^{49} + 6q^{53} - 2q^{57} + 2q^{61} + 8q^{63} - 10q^{67} + q^{69} + 3q^{71} + 7q^{73} - 24q^{77} + 10q^{79} + q^{81} - 12q^{83} + 9q^{87} - 4q^{91} - 5q^{93} - 8q^{97} - 12q^{99} + O(q^{100})$$

## 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
 0
0 1.00000 0 0 0 −4.00000 0 −2.00000 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$23$$ $$-1$$

## 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 9200.2.a.y 1
4.b odd 2 1 2300.2.a.d 1
5.b even 2 1 1840.2.a.c 1
20.d odd 2 1 460.2.a.c 1
20.e even 4 2 2300.2.c.d 2
40.e odd 2 1 7360.2.a.i 1
40.f even 2 1 7360.2.a.v 1
60.h even 2 1 4140.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.a.c 1 20.d odd 2 1
1840.2.a.c 1 5.b even 2 1
2300.2.a.d 1 4.b odd 2 1
2300.2.c.d 2 20.e even 4 2
4140.2.a.f 1 60.h even 2 1
7360.2.a.i 1 40.e odd 2 1
7360.2.a.v 1 40.f even 2 1
9200.2.a.y 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9200))$$:

 $$T_{3} - 1$$ $$T_{7} + 4$$ $$T_{11} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$4 + T$$
$11$ $$-6 + T$$
$13$ $$-1 + T$$
$17$ $$T$$
$19$ $$2 + T$$
$23$ $$-1 + T$$
$29$ $$-9 + T$$
$31$ $$5 + T$$
$37$ $$2 + T$$
$41$ $$9 + T$$
$43$ $$4 + T$$
$47$ $$3 + T$$
$53$ $$-6 + T$$
$59$ $$T$$
$61$ $$-2 + T$$
$67$ $$10 + T$$
$71$ $$-3 + T$$
$73$ $$-7 + T$$
$79$ $$-10 + T$$
$83$ $$12 + T$$
$89$ $$T$$
$97$ $$8 + T$$