# Properties

 Label 3630.2.a.bd 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.

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

## 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
 −1.73205 1.73205
−1.00000 −1.00000 1.00000 1.00000 1.00000 −1.73205 −1.00000 1.00000 −1.00000
1.2 −1.00000 −1.00000 1.00000 1.00000 1.00000 1.73205 −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.bd 2
11.b odd 2 1 3630.2.a.bn yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3630.2.a.bd 2 1.a even 1 1 trivial
3630.2.a.bn 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3630))$$:

 $$T_{7}^{2} - 3$$ $$T_{13}^{2} - 6 T_{13} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$-3 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$-3 - 6 T + T^{2}$$
$17$ $$33 - 12 T + T^{2}$$
$19$ $$-3 - 6 T + T^{2}$$
$23$ $$-18 - 6 T + T^{2}$$
$29$ $$-3 - 6 T + T^{2}$$
$31$ $$-2 + 10 T + T^{2}$$
$37$ $$-23 + 4 T + T^{2}$$
$41$ $$-66 - 6 T + T^{2}$$
$43$ $$6 - 6 T + T^{2}$$
$47$ $$( 6 + T )^{2}$$
$53$ $$-18 - 6 T + T^{2}$$
$59$ $$-18 + 6 T + T^{2}$$
$61$ $$6 - 6 T + T^{2}$$
$67$ $$-92 - 8 T + T^{2}$$
$71$ $$-99 - 6 T + T^{2}$$
$73$ $$6 - 6 T + T^{2}$$
$79$ $$132 + 24 T + T^{2}$$
$83$ $$69 - 18 T + T^{2}$$
$89$ $$-72 - 12 T + T^{2}$$
$97$ $$-2 - 10 T + T^{2}$$