# Properties

 Label 3525.2.a.i 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

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} - q^{4} - q^{6} - q^{7} - 3q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} - q^{4} - q^{6} - q^{7} - 3q^{8} + q^{9} - 2q^{11} + q^{12} + 7q^{13} - q^{14} - q^{16} - q^{17} + q^{18} - q^{19} + q^{21} - 2q^{22} + q^{23} + 3q^{24} + 7q^{26} - q^{27} + q^{28} + 7q^{29} - 8q^{31} + 5q^{32} + 2q^{33} - q^{34} - q^{36} - q^{38} - 7q^{39} - 11q^{41} + q^{42} + 10q^{43} + 2q^{44} + q^{46} - q^{47} + q^{48} - 6q^{49} + q^{51} - 7q^{52} + 7q^{53} - q^{54} + 3q^{56} + q^{57} + 7q^{58} + 9q^{59} - 15q^{61} - 8q^{62} - q^{63} + 7q^{64} + 2q^{66} - 4q^{67} + q^{68} - q^{69} - 5q^{71} - 3q^{72} - 2q^{73} + q^{76} + 2q^{77} - 7q^{78} - 10q^{79} + q^{81} - 11q^{82} - 12q^{83} - q^{84} + 10q^{86} - 7q^{87} + 6q^{88} - 6q^{89} - 7q^{91} - q^{92} + 8q^{93} - q^{94} - 5q^{96} - 12q^{97} - 6q^{98} - 2q^{99} + O(q^{100})$$

## 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 −1.00000 −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.i 1
5.b even 2 1 705.2.a.b 1
15.d odd 2 1 2115.2.a.j 1

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$1 + T$$
$11$ $$2 + T$$
$13$ $$-7 + T$$
$17$ $$1 + T$$
$19$ $$1 + T$$
$23$ $$-1 + T$$
$29$ $$-7 + T$$
$31$ $$8 + T$$
$37$ $$T$$
$41$ $$11 + T$$
$43$ $$-10 + T$$
$47$ $$1 + T$$
$53$ $$-7 + T$$
$59$ $$-9 + T$$
$61$ $$15 + T$$
$67$ $$4 + T$$
$71$ $$5 + T$$
$73$ $$2 + T$$
$79$ $$10 + T$$
$83$ $$12 + T$$
$89$ $$6 + T$$
$97$ $$12 + T$$