# Properties

 Label 4830.2.a.ca Level $4830$ Weight $2$ Character orbit 4830.a Self dual yes Analytic conductor $38.568$ Analytic rank $1$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} - 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + 6$$$$)/2$$

## 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
 2.34292 −1.81361 0.470683
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.ca 3

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

 $$T_{11}^{3} - 28 T_{11} - 16$$ $$T_{13}^{3} + 2 T_{13}^{2} - 28 T_{13} + 8$$ $$T_{17}^{3} + 10 T_{17}^{2} + 16 T_{17} - 16$$ $$T_{19}^{3} + 4 T_{19}^{2} - 12 T_{19} - 16$$ $$T_{29}^{3} + 10 T_{29}^{2} - 32 T_{29} - 352$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$-16 - 28 T + T^{3}$$
$13$ $$8 - 28 T + 2 T^{2} + T^{3}$$
$17$ $$-16 + 16 T + 10 T^{2} + T^{3}$$
$19$ $$-16 - 12 T + 4 T^{2} + T^{3}$$
$23$ $$( 1 + T )^{3}$$
$29$ $$-352 - 32 T + 10 T^{2} + T^{3}$$
$31$ $$64 + 68 T + 16 T^{2} + T^{3}$$
$37$ $$( 2 + T )^{3}$$
$41$ $$344 - 100 T - 6 T^{2} + T^{3}$$
$43$ $$-16 - 28 T + T^{3}$$
$47$ $$64 + 68 T + 16 T^{2} + T^{3}$$
$53$ $$( 2 + T )^{3}$$
$59$ $$-64 - 112 T - 4 T^{2} + T^{3}$$
$61$ $$-152 + 92 T + 22 T^{2} + T^{3}$$
$67$ $$272 + 164 T + 24 T^{2} + T^{3}$$
$71$ $$-352 - 124 T - 4 T^{2} + T^{3}$$
$73$ $$88 - 100 T - 6 T^{2} + T^{3}$$
$79$ $$2048 - 224 T - 8 T^{2} + T^{3}$$
$83$ $$-976 - 124 T + 12 T^{2} + T^{3}$$
$89$ $$-8 + 4 T + 14 T^{2} + T^{3}$$
$97$ $$-136 - 60 T - 2 T^{2} + T^{3}$$