# Properties

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

## $q$-expansion

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

## 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.k 1
11.b odd 2 1 330.2.a.e 1
33.d even 2 1 990.2.a.c 1
44.c even 2 1 2640.2.a.h 1
55.d odd 2 1 1650.2.a.b 1
55.e even 4 2 1650.2.c.i 2
132.d odd 2 1 7920.2.a.g 1
165.d even 2 1 4950.2.a.bk 1
165.l odd 4 2 4950.2.c.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.a.e 1 11.b odd 2 1
990.2.a.c 1 33.d even 2 1
1650.2.a.b 1 55.d odd 2 1
1650.2.c.i 2 55.e even 4 2
2640.2.a.h 1 44.c even 2 1
3630.2.a.k 1 1.a even 1 1 trivial
4950.2.a.bk 1 165.d even 2 1
4950.2.c.v 2 165.l odd 4 2
7920.2.a.g 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3630))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$-1 + T$$
$5$ $$-1 + T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$-2 + T$$
$17$ $$2 + T$$
$19$ $$-4 + T$$
$23$ $$T$$
$29$ $$-2 + T$$
$31$ $$T$$
$37$ $$2 + T$$
$41$ $$2 + T$$
$43$ $$-12 + T$$
$47$ $$-8 + T$$
$53$ $$-6 + T$$
$59$ $$12 + T$$
$61$ $$6 + T$$
$67$ $$-4 + T$$
$71$ $$T$$
$73$ $$-6 + T$$
$79$ $$-16 + T$$
$83$ $$4 + T$$
$89$ $$-10 + T$$
$97$ $$-2 + T$$