# Properties

 Label 225.4.a.d Level $225$ Weight $4$ Character orbit 225.a Self dual yes Analytic conductor $13.275$ Analytic rank $0$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 9) Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 8 q^{4} - 20 q^{7}+O(q^{10})$$ q - 8 * q^4 - 20 * q^7 $$q - 8 q^{4} - 20 q^{7} + 70 q^{13} + 64 q^{16} + 56 q^{19} + 160 q^{28} + 308 q^{31} - 110 q^{37} + 520 q^{43} + 57 q^{49} - 560 q^{52} + 182 q^{61} - 512 q^{64} + 880 q^{67} - 1190 q^{73} - 448 q^{76} + 884 q^{79} - 1400 q^{91} + 1330 q^{97}+O(q^{100})$$ q - 8 * q^4 - 20 * q^7 + 70 * q^13 + 64 * q^16 + 56 * q^19 + 160 * q^28 + 308 * q^31 - 110 * q^37 + 520 * q^43 + 57 * q^49 - 560 * q^52 + 182 * q^61 - 512 * q^64 + 880 * q^67 - 1190 * q^73 - 448 * q^76 + 884 * q^79 - 1400 * q^91 + 1330 * 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 −8.00000 0 0 −20.0000 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 225.4.a.d 1
3.b odd 2 1 CM 225.4.a.d 1
5.b even 2 1 9.4.a.a 1
5.c odd 4 2 225.4.b.g 2
15.d odd 2 1 9.4.a.a 1
15.e even 4 2 225.4.b.g 2
20.d odd 2 1 144.4.a.d 1
35.c odd 2 1 441.4.a.f 1
35.i odd 6 2 441.4.e.j 2
35.j even 6 2 441.4.e.i 2
40.e odd 2 1 576.4.a.l 1
40.f even 2 1 576.4.a.m 1
45.h odd 6 2 81.4.c.b 2
45.j even 6 2 81.4.c.b 2
55.d odd 2 1 1089.4.a.g 1
60.h even 2 1 144.4.a.d 1
65.d even 2 1 1521.4.a.g 1
105.g even 2 1 441.4.a.f 1
105.o odd 6 2 441.4.e.i 2
105.p even 6 2 441.4.e.j 2
120.i odd 2 1 576.4.a.m 1
120.m even 2 1 576.4.a.l 1
165.d even 2 1 1089.4.a.g 1
195.e odd 2 1 1521.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.a.a 1 5.b even 2 1
9.4.a.a 1 15.d odd 2 1
81.4.c.b 2 45.h odd 6 2
81.4.c.b 2 45.j even 6 2
144.4.a.d 1 20.d odd 2 1
144.4.a.d 1 60.h even 2 1
225.4.a.d 1 1.a even 1 1 trivial
225.4.a.d 1 3.b odd 2 1 CM
225.4.b.g 2 5.c odd 4 2
225.4.b.g 2 15.e even 4 2
441.4.a.f 1 35.c odd 2 1
441.4.a.f 1 105.g even 2 1
441.4.e.i 2 35.j even 6 2
441.4.e.i 2 105.o odd 6 2
441.4.e.j 2 35.i odd 6 2
441.4.e.j 2 105.p even 6 2
576.4.a.l 1 40.e odd 2 1
576.4.a.l 1 120.m even 2 1
576.4.a.m 1 40.f even 2 1
576.4.a.m 1 120.i odd 2 1
1089.4.a.g 1 55.d odd 2 1
1089.4.a.g 1 165.d even 2 1
1521.4.a.g 1 65.d even 2 1
1521.4.a.g 1 195.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(225))$$:

 $$T_{2}$$ T2 $$T_{7} + 20$$ T7 + 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 20$$
$11$ $$T$$
$13$ $$T - 70$$
$17$ $$T$$
$19$ $$T - 56$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T - 308$$
$37$ $$T + 110$$
$41$ $$T$$
$43$ $$T - 520$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T - 182$$
$67$ $$T - 880$$
$71$ $$T$$
$73$ $$T + 1190$$
$79$ $$T - 884$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T - 1330$$