Properties

 Label 3630.2.a.n 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

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9} - q^{10} - q^{12} - 2q^{13} + q^{15} + q^{16} + 2q^{17} + q^{18} - 8q^{19} - q^{20} + 4q^{23} - q^{24} + q^{25} - 2q^{26} - q^{27} - 2q^{29} + q^{30} + 8q^{31} + q^{32} + 2q^{34} + q^{36} - 2q^{37} - 8q^{38} + 2q^{39} - q^{40} - 6q^{41} - 8q^{43} - q^{45} + 4q^{46} - 4q^{47} - q^{48} - 7q^{49} + q^{50} - 2q^{51} - 2q^{52} + 2q^{53} - q^{54} + 8q^{57} - 2q^{58} + 4q^{59} + q^{60} + 6q^{61} + 8q^{62} + q^{64} + 2q^{65} - 12q^{67} + 2q^{68} - 4q^{69} - 12q^{71} + q^{72} - 2q^{73} - 2q^{74} - q^{75} - 8q^{76} + 2q^{78} - q^{80} + q^{81} - 6q^{82} - 4q^{83} - 2q^{85} - 8q^{86} + 2q^{87} - 6q^{89} - q^{90} + 4q^{92} - 8q^{93} - 4q^{94} + 8q^{95} - q^{96} - 14q^{97} - 7q^{98} + O(q^{100})$$

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.n 1
11.b odd 2 1 330.2.a.a 1
33.d even 2 1 990.2.a.j 1
44.c even 2 1 2640.2.a.n 1
55.d odd 2 1 1650.2.a.r 1
55.e even 4 2 1650.2.c.l 2
132.d odd 2 1 7920.2.a.bb 1
165.d even 2 1 4950.2.a.k 1
165.l odd 4 2 4950.2.c.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.a.a 1 11.b odd 2 1
990.2.a.j 1 33.d even 2 1
1650.2.a.r 1 55.d odd 2 1
1650.2.c.l 2 55.e even 4 2
2640.2.a.n 1 44.c even 2 1
3630.2.a.n 1 1.a even 1 1 trivial
4950.2.a.k 1 165.d even 2 1
4950.2.c.g 2 165.l odd 4 2
7920.2.a.bb 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

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