# Properties

 Label 3630.2.a.y Level $3630$ Weight $2$ Character orbit 3630.a Self dual yes Analytic conductor $28.986$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3630.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.9856959337$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - 3q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - 3q^{7} + q^{8} + q^{9} + q^{10} + q^{12} - 5q^{13} - 3q^{14} + q^{15} + q^{16} - 7q^{17} + q^{18} - 7q^{19} + q^{20} - 3q^{21} + q^{24} + q^{25} - 5q^{26} + q^{27} - 3q^{28} + 7q^{29} + q^{30} + 6q^{31} + q^{32} - 7q^{34} - 3q^{35} + q^{36} - 5q^{37} - 7q^{38} - 5q^{39} + q^{40} - 10q^{41} - 3q^{42} + 6q^{43} + q^{45} - 10q^{47} + q^{48} + 2q^{49} + q^{50} - 7q^{51} - 5q^{52} - 12q^{53} + q^{54} - 3q^{56} - 7q^{57} + 7q^{58} + q^{60} + 12q^{61} + 6q^{62} - 3q^{63} + q^{64} - 5q^{65} - 2q^{67} - 7q^{68} - 3q^{70} - 9q^{71} + q^{72} - 6q^{73} - 5q^{74} + q^{75} - 7q^{76} - 5q^{78} - 10q^{79} + q^{80} + q^{81} - 10q^{82} + 13q^{83} - 3q^{84} - 7q^{85} + 6q^{86} + 7q^{87} + 4q^{89} + q^{90} + 15q^{91} + 6q^{93} - 10q^{94} - 7q^{95} + q^{96} + 2q^{97} + 2q^{98} + O(q^{100})$$

## 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
1.00000 1.00000 1.00000 1.00000 1.00000 −3.00000 1.00000 1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$11$$ $$-1$$

## 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 3630.2.a.y yes 1
11.b odd 2 1 3630.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3630.2.a.l 1 11.b odd 2 1
3630.2.a.y yes 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3630))$$:

 $$T_{7} + 3$$ $$T_{13} + 5$$

## Hecke characteristic polynomials

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