# Properties

 Label 3630.2.a.l Level $3630$ Weight $2$ Character orbit 3630.a Self dual yes Analytic conductor $28.986$ Analytic rank $0$ 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: $$0$$ 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.l 1
11.b odd 2 1 3630.2.a.y yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3630.2.a.l 1 1.a even 1 1 trivial
3630.2.a.y yes 1 11.b odd 2 1

## 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

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