# Properties

 Label 3525.2.a.o 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

 $$f(q)$$ $$=$$ $$q + 2q^{2} + q^{3} + 2q^{4} + 2q^{6} + 4q^{7} + q^{9} + O(q^{10})$$ $$q + 2q^{2} + q^{3} + 2q^{4} + 2q^{6} + 4q^{7} + q^{9} + 2q^{12} + 5q^{13} + 8q^{14} - 4q^{16} + 6q^{17} + 2q^{18} - 2q^{19} + 4q^{21} + q^{23} + 10q^{26} + q^{27} + 8q^{28} - 6q^{29} - 8q^{31} - 8q^{32} + 12q^{34} + 2q^{36} + 2q^{37} - 4q^{38} + 5q^{39} - 2q^{41} + 8q^{42} - 5q^{43} + 2q^{46} + q^{47} - 4q^{48} + 9q^{49} + 6q^{51} + 10q^{52} + 12q^{53} + 2q^{54} - 2q^{57} - 12q^{58} - 3q^{59} - q^{61} - 16q^{62} + 4q^{63} - 8q^{64} + 8q^{67} + 12q^{68} + q^{69} + q^{71} - 13q^{73} + 4q^{74} - 4q^{76} + 10q^{78} - q^{79} + q^{81} - 4q^{82} + 12q^{83} + 8q^{84} - 10q^{86} - 6q^{87} - 15q^{89} + 20q^{91} + 2q^{92} - 8q^{93} + 2q^{94} - 8q^{96} - 8q^{97} + 18q^{98} + O(q^{100})$$

## 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
2.00000 1.00000 2.00000 0 2.00000 4.00000 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.o 1
5.b even 2 1 3525.2.a.a 1
5.c odd 4 2 705.2.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.c.a 2 5.c odd 4 2
3525.2.a.a 1 5.b even 2 1
3525.2.a.o 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3525))$$:

 $$T_{2} - 2$$ $$T_{7} - 4$$ $$T_{11}$$

## Hecke characteristic polynomials

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