# Properties

 Label 3525.2.a.e Level $3525$ Weight $2$ Character orbit 3525.a Self dual yes Analytic conductor $28.147$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} - q^{4} + q^{6} + 3q^{7} + 3q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} - q^{3} - q^{4} + q^{6} + 3q^{7} + 3q^{8} + q^{9} - 2q^{11} + q^{12} + q^{13} - 3q^{14} - q^{16} - 3q^{17} - q^{18} - 3q^{19} - 3q^{21} + 2q^{22} + 9q^{23} - 3q^{24} - q^{26} - q^{27} - 3q^{28} - 5q^{29} - 8q^{31} - 5q^{32} + 2q^{33} + 3q^{34} - q^{36} - 4q^{37} + 3q^{38} - q^{39} + 9q^{41} + 3q^{42} - 2q^{43} + 2q^{44} - 9q^{46} - q^{47} + q^{48} + 2q^{49} + 3q^{51} - q^{52} - 3q^{53} + q^{54} + 9q^{56} + 3q^{57} + 5q^{58} - 9q^{59} + q^{61} + 8q^{62} + 3q^{63} + 7q^{64} - 2q^{66} - 8q^{67} + 3q^{68} - 9q^{69} + 13q^{71} + 3q^{72} + 2q^{73} + 4q^{74} + 3q^{76} - 6q^{77} + q^{78} - 6q^{79} + q^{81} - 9q^{82} + 4q^{83} + 3q^{84} + 2q^{86} + 5q^{87} - 6q^{88} + 10q^{89} + 3q^{91} - 9q^{92} + 8q^{93} + q^{94} + 5q^{96} + 8q^{97} - 2q^{98} - 2q^{99} + O(q^{100})$$

## 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 3.00000 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.e 1
5.b even 2 1 705.2.a.e 1
15.d odd 2 1 2115.2.a.d 1

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

 $$T_{2} + 1$$ $$T_{7} - 3$$ $$T_{11} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$-3 + T$$
$11$ $$2 + T$$
$13$ $$-1 + T$$
$17$ $$3 + T$$
$19$ $$3 + T$$
$23$ $$-9 + T$$
$29$ $$5 + T$$
$31$ $$8 + T$$
$37$ $$4 + T$$
$41$ $$-9 + T$$
$43$ $$2 + T$$
$47$ $$1 + T$$
$53$ $$3 + T$$
$59$ $$9 + T$$
$61$ $$-1 + T$$
$67$ $$8 + T$$
$71$ $$-13 + T$$
$73$ $$-2 + T$$
$79$ $$6 + T$$
$83$ $$-4 + T$$
$89$ $$-10 + T$$
$97$ $$-8 + T$$