# Properties

 Label 4830.2.a.by Level $4830$ Weight $2$ Character orbit 4830.a Self dual yes Analytic conductor $38.568$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-2 \nu^{2} + 4 \nu + 3$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} - 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 2$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + 5$$$$)/2$$

## 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
 0.311108 −1.48119 2.17009
−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
1.3 −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.by 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4830.2.a.by 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4830))$$:

 $$T_{11}^{3} + 4 T_{11}^{2} - 16 T_{11} - 32$$ $$T_{13} - 2$$ $$T_{17}^{3} + 2 T_{17}^{2} - 36 T_{17} - 104$$ $$T_{19}^{3} - 4 T_{19}^{2} - 16 T_{19} + 32$$ $$T_{29}^{3} - 2 T_{29}^{2} - 20 T_{29} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$( -1 + T )^{3}$$
$11$ $$-32 - 16 T + 4 T^{2} + T^{3}$$
$13$ $$( -2 + T )^{3}$$
$17$ $$-104 - 36 T + 2 T^{2} + T^{3}$$
$19$ $$32 - 16 T - 4 T^{2} + T^{3}$$
$23$ $$( -1 + T )^{3}$$
$29$ $$8 - 20 T - 2 T^{2} + T^{3}$$
$31$ $$-160 - 16 T + 8 T^{2} + T^{3}$$
$37$ $$136 - 20 T - 10 T^{2} + T^{3}$$
$41$ $$40 - 52 T - 2 T^{2} + T^{3}$$
$43$ $$160 - 16 T - 8 T^{2} + T^{3}$$
$47$ $$-32 - 16 T + 4 T^{2} + T^{3}$$
$53$ $$200 - 52 T - 10 T^{2} + T^{3}$$
$59$ $$-320 - 16 T + 12 T^{2} + T^{3}$$
$61$ $$( -6 + T )^{3}$$
$67$ $$928 - 112 T - 8 T^{2} + T^{3}$$
$71$ $$-928 - 112 T + 8 T^{2} + T^{3}$$
$73$ $$344 - 20 T - 14 T^{2} + T^{3}$$
$79$ $$128 - 32 T - 8 T^{2} + T^{3}$$
$83$ $$544 - 160 T - 8 T^{2} + T^{3}$$
$89$ $$-8 - 52 T - 6 T^{2} + T^{3}$$
$97$ $$-184 + 172 T - 26 T^{2} + T^{3}$$