# Properties

 Label 3630.2.a.o 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

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{7} + q^{8} + q^{9} - q^{10} - q^{12} - 7q^{13} + q^{14} + q^{15} + q^{16} - q^{17} + q^{18} + 7q^{19} - q^{20} - q^{21} + 2q^{23} - q^{24} + q^{25} - 7q^{26} - q^{27} + q^{28} - 9q^{29} + q^{30} + q^{32} - q^{34} - q^{35} + q^{36} - 7q^{37} + 7q^{38} + 7q^{39} - q^{40} - 12q^{41} - q^{42} + 12q^{43} - q^{45} + 2q^{46} + 2q^{47} - q^{48} - 6q^{49} + q^{50} + q^{51} - 7q^{52} + 6q^{53} - q^{54} + q^{56} - 7q^{57} - 9q^{58} - 10q^{59} + q^{60} - 2q^{61} + q^{63} + q^{64} + 7q^{65} - 6q^{67} - q^{68} - 2q^{69} - q^{70} + 5q^{71} + q^{72} + 4q^{73} - 7q^{74} - q^{75} + 7q^{76} + 7q^{78} - 14q^{79} - q^{80} + q^{81} - 12q^{82} + q^{83} - q^{84} + q^{85} + 12q^{86} + 9q^{87} - 8q^{89} - q^{90} - 7q^{91} + 2q^{92} + 2q^{94} - 7q^{95} - q^{96} - 6q^{98} + O(q^{100})$$

## 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 1.00000 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.o yes 1
11.b odd 2 1 3630.2.a.c 1

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

 $$T_{7} - 1$$ $$T_{13} + 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$1 + T$$
$5$ $$1 + T$$
$7$ $$-1 + T$$
$11$ $$T$$
$13$ $$7 + T$$
$17$ $$1 + T$$
$19$ $$-7 + T$$
$23$ $$-2 + T$$
$29$ $$9 + T$$
$31$ $$T$$
$37$ $$7 + T$$
$41$ $$12 + T$$
$43$ $$-12 + T$$
$47$ $$-2 + T$$
$53$ $$-6 + T$$
$59$ $$10 + T$$
$61$ $$2 + T$$
$67$ $$6 + T$$
$71$ $$-5 + T$$
$73$ $$-4 + T$$
$79$ $$14 + T$$
$83$ $$-1 + T$$
$89$ $$8 + T$$
$97$ $$T$$