# Properties

 Label 3630.2.a.bc Level $3630$ Weight $2$ Character orbit 3630.a Self dual yes Analytic conductor $28.986$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

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## 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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 330) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.61803 −0.618034
−1.00000 −1.00000 1.00000 −1.00000 1.00000 1.38197 −1.00000 1.00000 1.00000
1.2 −1.00000 −1.00000 1.00000 −1.00000 1.00000 3.61803 −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.bc 2
11.b odd 2 1 3630.2.a.bi 2
11.c even 5 2 330.2.m.d 4
33.h odd 10 2 990.2.n.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.m.d 4 11.c even 5 2
990.2.n.a 4 33.h odd 10 2
3630.2.a.bc 2 1.a even 1 1 trivial
3630.2.a.bi 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3630))$$:

 $$T_{7}^{2} - 5 T_{7} + 5$$ $$T_{13}^{2} + T_{13} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$5 - 5 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$-1 + T + T^{2}$$
$17$ $$-16 + 4 T + T^{2}$$
$19$ $$-1 - T + T^{2}$$
$23$ $$-9 - 3 T + T^{2}$$
$29$ $$-36 + 6 T + T^{2}$$
$31$ $$44 + 14 T + T^{2}$$
$37$ $$-1 - T + T^{2}$$
$41$ $$-59 + 3 T + T^{2}$$
$43$ $$4 - 6 T + T^{2}$$
$47$ $$1 + 3 T + T^{2}$$
$53$ $$-31 + T + T^{2}$$
$59$ $$181 + 27 T + T^{2}$$
$61$ $$76 - 18 T + T^{2}$$
$67$ $$-116 - 6 T + T^{2}$$
$71$ $$4 + 14 T + T^{2}$$
$73$ $$-76 - 4 T + T^{2}$$
$79$ $$-20 + T^{2}$$
$83$ $$-64 - 8 T + T^{2}$$
$89$ $$61 + 17 T + T^{2}$$
$97$ $$-44 - 12 T + T^{2}$$
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