# Properties

 Label 4830.2.a.bt 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: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ 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 $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.73205 1.73205
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.bt 2

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

 $$T_{11}^{2} + 4 T_{11} - 8$$ $$T_{13}^{2} - 12$$ $$T_{17} + 2$$ $$T_{19} - 4$$ $$T_{29}^{2} + 8 T_{29} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-8 + 4 T + T^{2}$$
$13$ $$-12 + T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$4 + 8 T + T^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$-44 + 4 T + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$-8 - 4 T + T^{2}$$
$47$ $$-48 + T^{2}$$
$53$ $$( -2 + T )^{2}$$
$59$ $$-48 + T^{2}$$
$61$ $$( -10 + T )^{2}$$
$67$ $$-8 - 4 T + T^{2}$$
$71$ $$24 + 12 T + T^{2}$$
$73$ $$( -2 + T )^{2}$$
$79$ $$-32 - 8 T + T^{2}$$
$83$ $$( -4 + T )^{2}$$
$89$ $$52 - 16 T + T^{2}$$
$97$ $$4 - 8 T + T^{2}$$