# Properties

 Label 3630.2.a.bg Level $3630$ Weight $2$ Character orbit 3630.a Self dual yes Analytic conductor $28.986$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + ( 2 + \beta ) q^{7} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + ( 2 + \beta ) q^{7} - q^{8} + q^{9} + q^{10} + q^{12} + ( 1 - 2 \beta ) q^{13} + ( -2 - \beta ) q^{14} - q^{15} + q^{16} -\beta q^{17} - q^{18} + 7 q^{19} - q^{20} + ( 2 + \beta ) q^{21} + ( -5 + \beta ) q^{23} - q^{24} + q^{25} + ( -1 + 2 \beta ) q^{26} + q^{27} + ( 2 + \beta ) q^{28} + 5 q^{29} + q^{30} + ( 1 - 3 \beta ) q^{31} - q^{32} + \beta q^{34} + ( -2 - \beta ) q^{35} + q^{36} + ( -2 + 3 \beta ) q^{37} -7 q^{38} + ( 1 - 2 \beta ) q^{39} + q^{40} + ( 1 + 3 \beta ) q^{41} + ( -2 - \beta ) q^{42} + ( 3 + 3 \beta ) q^{43} - q^{45} + ( 5 - \beta ) q^{46} + ( -2 - 4 \beta ) q^{47} + q^{48} + 4 \beta q^{49} - q^{50} -\beta q^{51} + ( 1 - 2 \beta ) q^{52} + ( -1 - \beta ) q^{53} - q^{54} + ( -2 - \beta ) q^{56} + 7 q^{57} -5 q^{58} + ( -3 + 5 \beta ) q^{59} - q^{60} + ( 3 - 7 \beta ) q^{61} + ( -1 + 3 \beta ) q^{62} + ( 2 + \beta ) q^{63} + q^{64} + ( -1 + 2 \beta ) q^{65} + ( 8 + 2 \beta ) q^{67} -\beta q^{68} + ( -5 + \beta ) q^{69} + ( 2 + \beta ) q^{70} + ( 1 + 2 \beta ) q^{71} - q^{72} + ( -1 + 5 \beta ) q^{73} + ( 2 - 3 \beta ) q^{74} + q^{75} + 7 q^{76} + ( -1 + 2 \beta ) q^{78} + 6 \beta q^{79} - q^{80} + q^{81} + ( -1 - 3 \beta ) q^{82} + ( 13 + 2 \beta ) q^{83} + ( 2 + \beta ) q^{84} + \beta q^{85} + ( -3 - 3 \beta ) q^{86} + 5 q^{87} + ( 6 - 2 \beta ) q^{89} + q^{90} + ( -4 - 3 \beta ) q^{91} + ( -5 + \beta ) q^{92} + ( 1 - 3 \beta ) q^{93} + ( 2 + 4 \beta ) q^{94} -7 q^{95} - q^{96} + ( 7 - \beta ) q^{97} -4 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} + 4q^{7} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} + 4q^{7} - 2q^{8} + 2q^{9} + 2q^{10} + 2q^{12} + 2q^{13} - 4q^{14} - 2q^{15} + 2q^{16} - 2q^{18} + 14q^{19} - 2q^{20} + 4q^{21} - 10q^{23} - 2q^{24} + 2q^{25} - 2q^{26} + 2q^{27} + 4q^{28} + 10q^{29} + 2q^{30} + 2q^{31} - 2q^{32} - 4q^{35} + 2q^{36} - 4q^{37} - 14q^{38} + 2q^{39} + 2q^{40} + 2q^{41} - 4q^{42} + 6q^{43} - 2q^{45} + 10q^{46} - 4q^{47} + 2q^{48} - 2q^{50} + 2q^{52} - 2q^{53} - 2q^{54} - 4q^{56} + 14q^{57} - 10q^{58} - 6q^{59} - 2q^{60} + 6q^{61} - 2q^{62} + 4q^{63} + 2q^{64} - 2q^{65} + 16q^{67} - 10q^{69} + 4q^{70} + 2q^{71} - 2q^{72} - 2q^{73} + 4q^{74} + 2q^{75} + 14q^{76} - 2q^{78} - 2q^{80} + 2q^{81} - 2q^{82} + 26q^{83} + 4q^{84} - 6q^{86} + 10q^{87} + 12q^{89} + 2q^{90} - 8q^{91} - 10q^{92} + 2q^{93} + 4q^{94} - 14q^{95} - 2q^{96} + 14q^{97} + O(q^{100})$$

## 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
 −1.73205 1.73205
−1.00000 1.00000 1.00000 −1.00000 −1.00000 0.267949 −1.00000 1.00000 1.00000
1.2 −1.00000 1.00000 1.00000 −1.00000 −1.00000 3.73205 −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.bg 2
11.b odd 2 1 3630.2.a.bo yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3630.2.a.bg 2 1.a even 1 1 trivial
3630.2.a.bo yes 2 11.b odd 2 1

## Hecke kernels

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

 $$T_{7}^{2} - 4 T_{7} + 1$$ $$T_{13}^{2} - 2 T_{13} - 11$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$1 - 4 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$-11 - 2 T + T^{2}$$
$17$ $$-3 + T^{2}$$
$19$ $$( -7 + T )^{2}$$
$23$ $$22 + 10 T + T^{2}$$
$29$ $$( -5 + T )^{2}$$
$31$ $$-26 - 2 T + T^{2}$$
$37$ $$-23 + 4 T + T^{2}$$
$41$ $$-26 - 2 T + T^{2}$$
$43$ $$-18 - 6 T + T^{2}$$
$47$ $$-44 + 4 T + T^{2}$$
$53$ $$-2 + 2 T + T^{2}$$
$59$ $$-66 + 6 T + T^{2}$$
$61$ $$-138 - 6 T + T^{2}$$
$67$ $$52 - 16 T + T^{2}$$
$71$ $$-11 - 2 T + T^{2}$$
$73$ $$-74 + 2 T + T^{2}$$
$79$ $$-108 + T^{2}$$
$83$ $$157 - 26 T + T^{2}$$
$89$ $$24 - 12 T + T^{2}$$
$97$ $$46 - 14 T + T^{2}$$