# Properties

 Label 4830.2.a.bq Level $4830$ Weight $2$ Character orbit 4830.a Self dual yes Analytic conductor $38.568$ Analytic rank $0$ 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: $$0$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{7} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{7} - q^{8} + q^{9} - q^{10} + ( -1 - \beta ) q^{11} + q^{12} + ( 1 + \beta ) q^{13} + q^{14} + q^{15} + q^{16} + ( 2 - 2 \beta ) q^{17} - q^{18} -2 \beta q^{19} + q^{20} - q^{21} + ( 1 + \beta ) q^{22} + q^{23} - q^{24} + q^{25} + ( -1 - \beta ) q^{26} + q^{27} - q^{28} + ( 1 + 3 \beta ) q^{29} - q^{30} -2 \beta q^{31} - q^{32} + ( -1 - \beta ) q^{33} + ( -2 + 2 \beta ) q^{34} - q^{35} + q^{36} + ( 4 - 2 \beta ) q^{37} + 2 \beta q^{38} + ( 1 + \beta ) q^{39} - q^{40} + ( 4 + 2 \beta ) q^{41} + q^{42} + ( -3 + \beta ) q^{43} + ( -1 - \beta ) q^{44} + q^{45} - q^{46} -2 q^{47} + q^{48} + q^{49} - q^{50} + ( 2 - 2 \beta ) q^{51} + ( 1 + \beta ) q^{52} + 10 q^{53} - q^{54} + ( -1 - \beta ) q^{55} + q^{56} -2 \beta q^{57} + ( -1 - 3 \beta ) q^{58} + 4 \beta q^{59} + q^{60} -2 q^{61} + 2 \beta q^{62} - q^{63} + q^{64} + ( 1 + \beta ) q^{65} + ( 1 + \beta ) q^{66} + ( -11 + \beta ) q^{67} + ( 2 - 2 \beta ) q^{68} + q^{69} + q^{70} + ( 3 + 3 \beta ) q^{71} - q^{72} + ( 6 + 4 \beta ) q^{73} + ( -4 + 2 \beta ) q^{74} + q^{75} -2 \beta q^{76} + ( 1 + \beta ) q^{77} + ( -1 - \beta ) q^{78} -4 q^{79} + q^{80} + q^{81} + ( -4 - 2 \beta ) q^{82} + ( 4 - 2 \beta ) q^{83} - q^{84} + ( 2 - 2 \beta ) q^{85} + ( 3 - \beta ) q^{86} + ( 1 + 3 \beta ) q^{87} + ( 1 + \beta ) q^{88} + ( 5 + \beta ) q^{89} - q^{90} + ( -1 - \beta ) q^{91} + q^{92} -2 \beta q^{93} + 2 q^{94} -2 \beta q^{95} - q^{96} + ( -1 + 3 \beta ) q^{97} - q^{98} + ( -1 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} - 2q^{10} - 2q^{11} + 2q^{12} + 2q^{13} + 2q^{14} + 2q^{15} + 2q^{16} + 4q^{17} - 2q^{18} + 2q^{20} - 2q^{21} + 2q^{22} + 2q^{23} - 2q^{24} + 2q^{25} - 2q^{26} + 2q^{27} - 2q^{28} + 2q^{29} - 2q^{30} - 2q^{32} - 2q^{33} - 4q^{34} - 2q^{35} + 2q^{36} + 8q^{37} + 2q^{39} - 2q^{40} + 8q^{41} + 2q^{42} - 6q^{43} - 2q^{44} + 2q^{45} - 2q^{46} - 4q^{47} + 2q^{48} + 2q^{49} - 2q^{50} + 4q^{51} + 2q^{52} + 20q^{53} - 2q^{54} - 2q^{55} + 2q^{56} - 2q^{58} + 2q^{60} - 4q^{61} - 2q^{63} + 2q^{64} + 2q^{65} + 2q^{66} - 22q^{67} + 4q^{68} + 2q^{69} + 2q^{70} + 6q^{71} - 2q^{72} + 12q^{73} - 8q^{74} + 2q^{75} + 2q^{77} - 2q^{78} - 8q^{79} + 2q^{80} + 2q^{81} - 8q^{82} + 8q^{83} - 2q^{84} + 4q^{85} + 6q^{86} + 2q^{87} + 2q^{88} + 10q^{89} - 2q^{90} - 2q^{91} + 2q^{92} + 4q^{94} - 2q^{96} - 2q^{97} - 2q^{98} - 2q^{99} + O(q^{100})$$

## 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
 1.61803 −0.618034
−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.bq 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4830.2.a.bq 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4830))$$:

 $$T_{11}^{2} + 2 T_{11} - 4$$ $$T_{13}^{2} - 2 T_{13} - 4$$ $$T_{17}^{2} - 4 T_{17} - 16$$ $$T_{19}^{2} - 20$$ $$T_{29}^{2} - 2 T_{29} - 44$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-4 + 2 T + T^{2}$$
$13$ $$-4 - 2 T + T^{2}$$
$17$ $$-16 - 4 T + T^{2}$$
$19$ $$-20 + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$-44 - 2 T + T^{2}$$
$31$ $$-20 + T^{2}$$
$37$ $$-4 - 8 T + T^{2}$$
$41$ $$-4 - 8 T + T^{2}$$
$43$ $$4 + 6 T + T^{2}$$
$47$ $$( 2 + T )^{2}$$
$53$ $$( -10 + T )^{2}$$
$59$ $$-80 + T^{2}$$
$61$ $$( 2 + T )^{2}$$
$67$ $$116 + 22 T + T^{2}$$
$71$ $$-36 - 6 T + T^{2}$$
$73$ $$-44 - 12 T + T^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$-4 - 8 T + T^{2}$$
$89$ $$20 - 10 T + T^{2}$$
$97$ $$-44 + 2 T + T^{2}$$