# Properties

 Label 4830.2.a.bo 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: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ 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 $$\beta = \sqrt{13}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 2.30278 −1.30278
−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.bo 2

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

 $$T_{11}^{2} + 2 T_{11} - 12$$ $$T_{13}^{2} + 6 T_{13} - 4$$ $$T_{17}$$ $$T_{19} - 2$$ $$T_{29}^{2} + 10 T_{29} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$-12 + 2 T + T^{2}$$
$13$ $$-4 + 6 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( -2 + T )^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$12 + 10 T + T^{2}$$
$31$ $$( -2 + T )^{2}$$
$37$ $$-52 + T^{2}$$
$41$ $$-36 + 8 T + T^{2}$$
$43$ $$-4 + 6 T + T^{2}$$
$47$ $$-36 - 8 T + T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$( 12 + T )^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$-4 + 6 T + T^{2}$$
$71$ $$108 + 22 T + T^{2}$$
$73$ $$( -2 + T )^{2}$$
$79$ $$-208 + T^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$156 + 26 T + T^{2}$$
$97$ $$-68 + 14 T + T^{2}$$