# Properties

 Label 3525.2.a.l Level $3525$ Weight $2$ Character orbit 3525.a Self dual yes Analytic conductor $28.147$ Analytic rank $0$ 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: $$0$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} - q^{4} + q^{6} + 5q^{7} - 3q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} - q^{4} + q^{6} + 5q^{7} - 3q^{8} + q^{9} + 6q^{11} - q^{12} - 3q^{13} + 5q^{14} - q^{16} + 3q^{17} + q^{18} - q^{19} + 5q^{21} + 6q^{22} + 5q^{23} - 3q^{24} - 3q^{26} + q^{27} - 5q^{28} - 7q^{29} + 5q^{32} + 6q^{33} + 3q^{34} - q^{36} - q^{38} - 3q^{39} - 5q^{41} + 5q^{42} + 6q^{43} - 6q^{44} + 5q^{46} - q^{47} - q^{48} + 18q^{49} + 3q^{51} + 3q^{52} - 5q^{53} + q^{54} - 15q^{56} - q^{57} - 7q^{58} - 9q^{59} - 7q^{61} + 5q^{63} + 7q^{64} + 6q^{66} + 8q^{67} - 3q^{68} + 5q^{69} - 3q^{71} - 3q^{72} + 10q^{73} + q^{76} + 30q^{77} - 3q^{78} - 10q^{79} + q^{81} - 5q^{82} + 8q^{83} - 5q^{84} + 6q^{86} - 7q^{87} - 18q^{88} + 14q^{89} - 15q^{91} - 5q^{92} - q^{94} + 5q^{96} - 12q^{97} + 18q^{98} + 6q^{99} + O(q^{100})$$

## 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
1.00000 1.00000 −1.00000 0 1.00000 5.00000 −3.00000 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.l 1
5.b even 2 1 705.2.a.a 1
15.d odd 2 1 2115.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.a.a 1 5.b even 2 1
2115.2.a.g 1 15.d odd 2 1
3525.2.a.l 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3525))$$:

 $$T_{2} - 1$$ $$T_{7} - 5$$ $$T_{11} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$-5 + T$$
$11$ $$-6 + T$$
$13$ $$3 + T$$
$17$ $$-3 + T$$
$19$ $$1 + T$$
$23$ $$-5 + T$$
$29$ $$7 + T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$5 + T$$
$43$ $$-6 + T$$
$47$ $$1 + T$$
$53$ $$5 + T$$
$59$ $$9 + T$$
$61$ $$7 + T$$
$67$ $$-8 + T$$
$71$ $$3 + T$$
$73$ $$-10 + T$$
$79$ $$10 + T$$
$83$ $$-8 + T$$
$89$ $$-14 + T$$
$97$ $$12 + T$$