# Properties

 Label 225.2.a.d Level 225 Weight 2 Character orbit 225.a Self dual Yes Analytic conductor 1.797 Analytic rank 0 Dimension 1 CM disc. -3 Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: Yes Analytic conductor: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q$$ $$\mathstrut -\mathstrut 2q^{4}$$ $$\mathstrut +\mathstrut 5q^{7}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$q$$ $$\mathstrut -\mathstrut 2q^{4}$$ $$\mathstrut +\mathstrut 5q^{7}$$ $$\mathstrut +\mathstrut 5q^{13}$$ $$\mathstrut +\mathstrut 4q^{16}$$ $$\mathstrut -\mathstrut q^{19}$$ $$\mathstrut -\mathstrut 10q^{28}$$ $$\mathstrut -\mathstrut 7q^{31}$$ $$\mathstrut -\mathstrut 10q^{37}$$ $$\mathstrut +\mathstrut 5q^{43}$$ $$\mathstrut +\mathstrut 18q^{49}$$ $$\mathstrut -\mathstrut 10q^{52}$$ $$\mathstrut -\mathstrut 13q^{61}$$ $$\mathstrut -\mathstrut 8q^{64}$$ $$\mathstrut +\mathstrut 5q^{67}$$ $$\mathstrut -\mathstrut 10q^{73}$$ $$\mathstrut +\mathstrut 2q^{76}$$ $$\mathstrut -\mathstrut 4q^{79}$$ $$\mathstrut +\mathstrut 25q^{91}$$ $$\mathstrut +\mathstrut 5q^{97}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## 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 0 −2.00000 0 0 5.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 CM by $$\Q(\sqrt{-3})$$ yes

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$-1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(225))$$:

 $$T_{2}$$ $$T_{7}$$ $$\mathstrut -\mathstrut 5$$