# Properties

 Label 3525.2.a.f Level $3525$ Weight $2$ Character orbit 3525.a Self dual yes Analytic conductor $28.147$ Analytic rank $2$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3525 = 3 \cdot 5^{2} \cdot 47$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3525.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.1472667125$$ Analytic rank: $$2$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 705) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} - 2q^{4} - 2q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} - 2q^{4} - 2q^{7} + q^{9} - 6q^{11} + 2q^{12} - 5q^{13} + 4q^{16} - 6q^{17} + 2q^{19} + 2q^{21} - 9q^{23} - q^{27} + 4q^{28} - 6q^{29} + 2q^{31} + 6q^{33} - 2q^{36} + 4q^{37} + 5q^{39} - 11q^{43} + 12q^{44} + q^{47} - 4q^{48} - 3q^{49} + 6q^{51} + 10q^{52} - 6q^{53} - 2q^{57} + 9q^{59} - q^{61} - 2q^{63} - 8q^{64} + 4q^{67} + 12q^{68} + 9q^{69} - 15q^{71} - 11q^{73} - 4q^{76} + 12q^{77} + 11q^{79} + q^{81} - 4q^{84} + 6q^{87} - 3q^{89} + 10q^{91} + 18q^{92} - 2q^{93} - 2q^{97} - 6q^{99} + O(q^{100})$$

## 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 −1.00000 −2.00000 0 0 −2.00000 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
$$3$$ $$1$$
$$5$$ $$1$$
$$47$$ $$-1$$

## 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 3525.2.a.f 1
5.b even 2 1 705.2.a.d 1
15.d odd 2 1 2115.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.a.d 1 5.b even 2 1
2115.2.a.f 1 15.d odd 2 1
3525.2.a.f 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3525))$$:

 $$T_{2}$$ $$T_{7} + 2$$ $$T_{11} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$2 + T$$
$11$ $$6 + T$$
$13$ $$5 + T$$
$17$ $$6 + T$$
$19$ $$-2 + T$$
$23$ $$9 + T$$
$29$ $$6 + T$$
$31$ $$-2 + T$$
$37$ $$-4 + T$$
$41$ $$T$$
$43$ $$11 + T$$
$47$ $$-1 + T$$
$53$ $$6 + T$$
$59$ $$-9 + T$$
$61$ $$1 + T$$
$67$ $$-4 + T$$
$71$ $$15 + T$$
$73$ $$11 + T$$
$79$ $$-11 + T$$
$83$ $$T$$
$89$ $$3 + T$$
$97$ $$2 + T$$