# Properties

 Label 3630.2.a.f 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: no (minimal twist has level 330) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{8} + q^{9} - q^{10} - q^{12} - 6q^{13} - q^{15} + q^{16} - 2q^{17} - q^{18} + 4q^{19} + q^{20} + q^{24} + q^{25} + 6q^{26} - q^{27} + 10q^{29} + q^{30} - q^{32} + 2q^{34} + q^{36} + 6q^{37} - 4q^{38} + 6q^{39} - q^{40} - 2q^{41} - 4q^{43} + q^{45} - 8q^{47} - q^{48} - 7q^{49} - q^{50} + 2q^{51} - 6q^{52} - 10q^{53} + q^{54} - 4q^{57} - 10q^{58} - 4q^{59} - q^{60} + 2q^{61} + q^{64} - 6q^{65} - 4q^{67} - 2q^{68} - 8q^{71} - q^{72} - 2q^{73} - 6q^{74} - q^{75} + 4q^{76} - 6q^{78} + 8q^{79} + q^{80} + q^{81} + 2q^{82} + 12q^{83} - 2q^{85} + 4q^{86} - 10q^{87} - 6q^{89} - q^{90} + 8q^{94} + 4q^{95} + q^{96} + 18q^{97} + 7q^{98} + O(q^{100})$$

## 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 0 −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.f 1
11.b odd 2 1 330.2.a.d 1
33.d even 2 1 990.2.a.b 1
44.c even 2 1 2640.2.a.t 1
55.d odd 2 1 1650.2.a.h 1
55.e even 4 2 1650.2.c.g 2
132.d odd 2 1 7920.2.a.m 1
165.d even 2 1 4950.2.a.bg 1
165.l odd 4 2 4950.2.c.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.a.d 1 11.b odd 2 1
990.2.a.b 1 33.d even 2 1
1650.2.a.h 1 55.d odd 2 1
1650.2.c.g 2 55.e even 4 2
2640.2.a.t 1 44.c even 2 1
3630.2.a.f 1 1.a even 1 1 trivial
4950.2.a.bg 1 165.d even 2 1
4950.2.c.j 2 165.l odd 4 2
7920.2.a.m 1 132.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3630))$$:

 $$T_{7}$$ $$T_{13} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$1 + T$$
$5$ $$-1 + T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$6 + T$$
$17$ $$2 + T$$
$19$ $$-4 + T$$
$23$ $$T$$
$29$ $$-10 + T$$
$31$ $$T$$
$37$ $$-6 + T$$
$41$ $$2 + T$$
$43$ $$4 + T$$
$47$ $$8 + T$$
$53$ $$10 + T$$
$59$ $$4 + T$$
$61$ $$-2 + T$$
$67$ $$4 + T$$
$71$ $$8 + T$$
$73$ $$2 + T$$
$79$ $$-8 + T$$
$83$ $$-12 + T$$
$89$ $$6 + T$$
$97$ $$-18 + T$$