# Properties

 Label 3630.2.a.s 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

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + 4q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + 4q^{7} + q^{8} + q^{9} + q^{10} - q^{12} + 2q^{13} + 4q^{14} - q^{15} + q^{16} + 2q^{17} + q^{18} + 8q^{19} + q^{20} - 4q^{21} - q^{24} + q^{25} + 2q^{26} - q^{27} + 4q^{28} - 2q^{29} - q^{30} - 8q^{31} + q^{32} + 2q^{34} + 4q^{35} + q^{36} - 10q^{37} + 8q^{38} - 2q^{39} + q^{40} + 10q^{41} - 4q^{42} + q^{45} - q^{48} + 9q^{49} + q^{50} - 2q^{51} + 2q^{52} + 14q^{53} - q^{54} + 4q^{56} - 8q^{57} - 2q^{58} - 4q^{59} - q^{60} - 14q^{61} - 8q^{62} + 4q^{63} + q^{64} + 2q^{65} - 4q^{67} + 2q^{68} + 4q^{70} + 8q^{71} + q^{72} - 10q^{73} - 10q^{74} - q^{75} + 8q^{76} - 2q^{78} - 12q^{79} + q^{80} + q^{81} + 10q^{82} - 4q^{83} - 4q^{84} + 2q^{85} + 2q^{87} - 6q^{89} + q^{90} + 8q^{91} + 8q^{93} + 8q^{95} - q^{96} - 14q^{97} + 9q^{98} + O(q^{100})$$

## 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 4.00000 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.s 1
11.b odd 2 1 330.2.a.b 1
33.d even 2 1 990.2.a.h 1
44.c even 2 1 2640.2.a.v 1
55.d odd 2 1 1650.2.a.t 1
55.e even 4 2 1650.2.c.j 2
132.d odd 2 1 7920.2.a.r 1
165.d even 2 1 4950.2.a.t 1
165.l odd 4 2 4950.2.c.o 2

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

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

## Hecke characteristic polynomials

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