# Properties

 Label 4725.2.a.i Level $4725$ Weight $2$ Character orbit 4725.a Self dual yes Analytic conductor $37.729$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4725 = 3^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4725.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$37.7293149551$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 189) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{4} - q^{7} + O(q^{10})$$ $$q - 2q^{4} - q^{7} - 6q^{11} + 4q^{13} + 4q^{16} + 3q^{17} + 2q^{19} - 6q^{23} + 2q^{28} + 6q^{29} - 4q^{31} + 7q^{37} + 3q^{41} + q^{43} + 12q^{44} + 9q^{47} + q^{49} - 8q^{52} - 6q^{53} - 9q^{59} - 10q^{61} - 8q^{64} + 4q^{67} - 6q^{68} - 2q^{73} - 4q^{76} + 6q^{77} - q^{79} + 3q^{83} - 6q^{89} - 4q^{91} + 12q^{92} + 10q^{97} + O(q^{100})$$

## 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 −1.00000 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
$$3$$ $$1$$
$$5$$ $$1$$
$$7$$ $$1$$

## 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 4725.2.a.i 1
3.b odd 2 1 4725.2.a.k 1
5.b even 2 1 189.2.a.b 1
15.d odd 2 1 189.2.a.c yes 1
20.d odd 2 1 3024.2.a.f 1
35.c odd 2 1 1323.2.a.l 1
45.h odd 6 2 567.2.f.d 2
45.j even 6 2 567.2.f.e 2
60.h even 2 1 3024.2.a.y 1
105.g even 2 1 1323.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.a.b 1 5.b even 2 1
189.2.a.c yes 1 15.d odd 2 1
567.2.f.d 2 45.h odd 6 2
567.2.f.e 2 45.j even 6 2
1323.2.a.h 1 105.g even 2 1
1323.2.a.l 1 35.c odd 2 1
3024.2.a.f 1 20.d odd 2 1
3024.2.a.y 1 60.h even 2 1
4725.2.a.i 1 1.a even 1 1 trivial
4725.2.a.k 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4725))$$:

 $$T_{2}$$ $$T_{11} + 6$$ $$T_{13} - 4$$ $$T_{37} - 7$$

## Hecke characteristic polynomials

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