# Properties

 Label 225.6.a.e Level $225$ Weight $6$ Character orbit 225.a Self dual yes Analytic conductor $36.086$ Analytic rank $1$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.0863594579$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 32 q^{4} + 25 q^{7}+O(q^{10})$$ q - 32 * q^4 + 25 * q^7 $$q - 32 q^{4} + 25 q^{7} + 775 q^{13} + 1024 q^{16} - 1711 q^{19} - 800 q^{28} + 2723 q^{31} - 16550 q^{37} - 22475 q^{43} - 16182 q^{49} - 24800 q^{52} + 56927 q^{61} - 32768 q^{64} - 73475 q^{67} + 1450 q^{73} + 54752 q^{76} - 100564 q^{79} + 19375 q^{91} - 177725 q^{97}+O(q^{100})$$ q - 32 * q^4 + 25 * q^7 + 775 * q^13 + 1024 * q^16 - 1711 * q^19 - 800 * q^28 + 2723 * q^31 - 16550 * q^37 - 22475 * q^43 - 16182 * q^49 - 24800 * q^52 + 56927 * q^61 - 32768 * q^64 - 73475 * q^67 + 1450 * q^73 + 54752 * q^76 - 100564 * q^79 + 19375 * q^91 - 177725 * 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
 0
0 0 −32.0000 0 0 25.0000 0 0 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
$$3$$ $$1$$
$$5$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.6.a.e yes 1
3.b odd 2 1 CM 225.6.a.e yes 1
5.b even 2 1 225.6.a.d 1
5.c odd 4 2 225.6.b.f 2
15.d odd 2 1 225.6.a.d 1
15.e even 4 2 225.6.b.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.6.a.d 1 5.b even 2 1
225.6.a.d 1 15.d odd 2 1
225.6.a.e yes 1 1.a even 1 1 trivial
225.6.a.e yes 1 3.b odd 2 1 CM
225.6.b.f 2 5.c odd 4 2
225.6.b.f 2 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(225))$$:

 $$T_{2}$$ T2 $$T_{7} - 25$$ T7 - 25

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 25$$
$11$ $$T$$
$13$ $$T - 775$$
$17$ $$T$$
$19$ $$T + 1711$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T - 2723$$
$37$ $$T + 16550$$
$41$ $$T$$
$43$ $$T + 22475$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T - 56927$$
$67$ $$T + 73475$$
$71$ $$T$$
$73$ $$T - 1450$$
$79$ $$T + 100564$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T + 177725$$