# Properties

 Label 3630.2.a.i 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + 2q^{7} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + 2q^{7} - q^{8} + q^{9} + q^{10} + q^{12} + 4q^{13} - 2q^{14} - q^{15} + q^{16} + 6q^{17} - q^{18} - 2q^{19} - q^{20} + 2q^{21} + 4q^{23} - q^{24} + q^{25} - 4q^{26} + q^{27} + 2q^{28} + 2q^{29} + q^{30} - 8q^{31} - q^{32} - 6q^{34} - 2q^{35} + q^{36} + 10q^{37} + 2q^{38} + 4q^{39} + q^{40} - 2q^{41} - 2q^{42} + 6q^{43} - q^{45} - 4q^{46} - 8q^{47} + q^{48} - 3q^{49} - q^{50} + 6q^{51} + 4q^{52} - 10q^{53} - q^{54} - 2q^{56} - 2q^{57} - 2q^{58} - q^{60} + 12q^{61} + 8q^{62} + 2q^{63} + q^{64} - 4q^{65} - 4q^{67} + 6q^{68} + 4q^{69} + 2q^{70} - 8q^{71} - q^{72} + 8q^{73} - 10q^{74} + q^{75} - 2q^{76} - 4q^{78} + 6q^{79} - q^{80} + q^{81} + 2q^{82} - 8q^{83} + 2q^{84} - 6q^{85} - 6q^{86} + 2q^{87} - 6q^{89} + q^{90} + 8q^{91} + 4q^{92} - 8q^{93} + 8q^{94} + 2q^{95} - q^{96} + 10q^{97} + 3q^{98} + O(q^{100})$$

## 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 2.00000 −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.i 1
11.b odd 2 1 3630.2.a.t yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3630.2.a.i 1 1.a even 1 1 trivial
3630.2.a.t 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3630))$$:

 $$T_{7} - 2$$ $$T_{13} - 4$$

## Hecke characteristic polynomials

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