# Properties

 Label 9200.2.a.i Level $9200$ Weight $2$ Character orbit 9200.a Self dual yes Analytic conductor $73.462$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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## 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 4600) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{3} + q^{7} + q^{9} + O(q^{10})$$ $$q - 2 q^{3} + q^{7} + q^{9} + 5 q^{11} - q^{13} + 4 q^{17} - 7 q^{19} - 2 q^{21} - q^{23} + 4 q^{27} + 5 q^{29} - 2 q^{31} - 10 q^{33} + 2 q^{37} + 2 q^{39} + 11 q^{41} + q^{43} - 8 q^{47} - 6 q^{49} - 8 q^{51} + 14 q^{57} + 14 q^{59} + 10 q^{61} + q^{63} - 8 q^{67} + 2 q^{69} + 10 q^{71} - 7 q^{73} + 5 q^{77} - 7 q^{79} - 11 q^{81} - 15 q^{83} - 10 q^{87} + 10 q^{89} - q^{91} + 4 q^{93} + 4 q^{97} + 5 q^{99} + O(q^{100})$$

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

## 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.i 1
4.b odd 2 1 4600.2.a.n yes 1
5.b even 2 1 9200.2.a.bd 1
20.d odd 2 1 4600.2.a.c 1
20.e even 4 2 4600.2.e.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4600.2.a.c 1 20.d odd 2 1
4600.2.a.n yes 1 4.b odd 2 1
4600.2.e.c 2 20.e even 4 2
9200.2.a.i 1 1.a even 1 1 trivial
9200.2.a.bd 1 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9200))$$:

 $$T_{3} + 2$$ $$T_{7} - 1$$ $$T_{11} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$2 + T$$
$5$ $$T$$
$7$ $$-1 + T$$
$11$ $$-5 + T$$
$13$ $$1 + T$$
$17$ $$-4 + T$$
$19$ $$7 + T$$
$23$ $$1 + T$$
$29$ $$-5 + T$$
$31$ $$2 + T$$
$37$ $$-2 + T$$
$41$ $$-11 + T$$
$43$ $$-1 + T$$
$47$ $$8 + T$$
$53$ $$T$$
$59$ $$-14 + T$$
$61$ $$-10 + T$$
$67$ $$8 + T$$
$71$ $$-10 + T$$
$73$ $$7 + T$$
$79$ $$7 + T$$
$83$ $$15 + T$$
$89$ $$-10 + T$$
$97$ $$-4 + T$$
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