# Properties

 Label 4830.2.a.bu Level $4830$ Weight $2$ Character orbit 4830.a Self dual yes Analytic conductor $38.568$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4830.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$38.5677441763$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - q^{7} + q^{8} + q^{9} +O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - q^{7} + q^{8} + q^{9} - q^{10} + ( 3 + \beta ) q^{11} - q^{12} - q^{14} + q^{15} + q^{16} + ( -3 - \beta ) q^{17} + q^{18} + ( -5 - \beta ) q^{19} - q^{20} + q^{21} + ( 3 + \beta ) q^{22} + q^{23} - q^{24} + q^{25} - q^{27} - q^{28} + ( -3 - 3 \beta ) q^{29} + q^{30} + ( -1 + 3 \beta ) q^{31} + q^{32} + ( -3 - \beta ) q^{33} + ( -3 - \beta ) q^{34} + q^{35} + q^{36} + ( 4 + 2 \beta ) q^{37} + ( -5 - \beta ) q^{38} - q^{40} + ( -4 + 2 \beta ) q^{41} + q^{42} + ( 1 - 5 \beta ) q^{43} + ( 3 + \beta ) q^{44} - q^{45} + q^{46} + ( 3 - \beta ) q^{47} - q^{48} + q^{49} + q^{50} + ( 3 + \beta ) q^{51} + 6 \beta q^{53} - q^{54} + ( -3 - \beta ) q^{55} - q^{56} + ( 5 + \beta ) q^{57} + ( -3 - 3 \beta ) q^{58} + ( -6 - 2 \beta ) q^{59} + q^{60} + ( -2 - 4 \beta ) q^{61} + ( -1 + 3 \beta ) q^{62} - q^{63} + q^{64} + ( -3 - \beta ) q^{66} + ( -7 + 3 \beta ) q^{67} + ( -3 - \beta ) q^{68} - q^{69} + q^{70} + ( -3 - \beta ) q^{71} + q^{72} -10 q^{73} + ( 4 + 2 \beta ) q^{74} - q^{75} + ( -5 - \beta ) q^{76} + ( -3 - \beta ) q^{77} - q^{80} + q^{81} + ( -4 + 2 \beta ) q^{82} + ( 1 + \beta ) q^{83} + q^{84} + ( 3 + \beta ) q^{85} + ( 1 - 5 \beta ) q^{86} + ( 3 + 3 \beta ) q^{87} + ( 3 + \beta ) q^{88} + ( -10 + 2 \beta ) q^{89} - q^{90} + q^{92} + ( 1 - 3 \beta ) q^{93} + ( 3 - \beta ) q^{94} + ( 5 + \beta ) q^{95} - q^{96} + 12 q^{97} + q^{98} + ( 3 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} - 2q^{10} + 6q^{11} - 2q^{12} - 2q^{14} + 2q^{15} + 2q^{16} - 6q^{17} + 2q^{18} - 10q^{19} - 2q^{20} + 2q^{21} + 6q^{22} + 2q^{23} - 2q^{24} + 2q^{25} - 2q^{27} - 2q^{28} - 6q^{29} + 2q^{30} - 2q^{31} + 2q^{32} - 6q^{33} - 6q^{34} + 2q^{35} + 2q^{36} + 8q^{37} - 10q^{38} - 2q^{40} - 8q^{41} + 2q^{42} + 2q^{43} + 6q^{44} - 2q^{45} + 2q^{46} + 6q^{47} - 2q^{48} + 2q^{49} + 2q^{50} + 6q^{51} - 2q^{54} - 6q^{55} - 2q^{56} + 10q^{57} - 6q^{58} - 12q^{59} + 2q^{60} - 4q^{61} - 2q^{62} - 2q^{63} + 2q^{64} - 6q^{66} - 14q^{67} - 6q^{68} - 2q^{69} + 2q^{70} - 6q^{71} + 2q^{72} - 20q^{73} + 8q^{74} - 2q^{75} - 10q^{76} - 6q^{77} - 2q^{80} + 2q^{81} - 8q^{82} + 2q^{83} + 2q^{84} + 6q^{85} + 2q^{86} + 6q^{87} + 6q^{88} - 20q^{89} - 2q^{90} + 2q^{92} + 2q^{93} + 6q^{94} + 10q^{95} - 2q^{96} + 24q^{97} + 2q^{98} + 6q^{99} + O(q^{100})$$

## 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
 −0.618034 1.61803
1.00000 −1.00000 1.00000 −1.00000 −1.00000 −1.00000 1.00000 1.00000 −1.00000
1.2 1.00000 −1.00000 1.00000 −1.00000 −1.00000 −1.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$7$$ $$1$$
$$23$$ $$-1$$

## 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 4830.2.a.bu 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4830.2.a.bu 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4830))$$:

 $$T_{11}^{2} - 6 T_{11} + 4$$ $$T_{13}$$ $$T_{17}^{2} + 6 T_{17} + 4$$ $$T_{19}^{2} + 10 T_{19} + 20$$ $$T_{29}^{2} + 6 T_{29} - 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$4 - 6 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$4 + 6 T + T^{2}$$
$19$ $$20 + 10 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$-36 + 6 T + T^{2}$$
$31$ $$-44 + 2 T + T^{2}$$
$37$ $$-4 - 8 T + T^{2}$$
$41$ $$-4 + 8 T + T^{2}$$
$43$ $$-124 - 2 T + T^{2}$$
$47$ $$4 - 6 T + T^{2}$$
$53$ $$-180 + T^{2}$$
$59$ $$16 + 12 T + T^{2}$$
$61$ $$-76 + 4 T + T^{2}$$
$67$ $$4 + 14 T + T^{2}$$
$71$ $$4 + 6 T + T^{2}$$
$73$ $$( 10 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$-4 - 2 T + T^{2}$$
$89$ $$80 + 20 T + T^{2}$$
$97$ $$( -12 + T )^{2}$$