# Properties

 Label 4830.2.a.cb Level $4830$ Weight $2$ Character orbit 4830.a Self dual yes Analytic conductor $38.568$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

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## 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: 3.3.229.1 Defining polynomial: $$x^{3} - 4 x - 1$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 4 x - 1$$:

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

## 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
 2.11491 −1.86081 −0.254102
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.cb 3

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

 $$T_{11}^{3} - 2 T_{11}^{2} - 20 T_{11} - 16$$ $$T_{13}^{3} - 2 T_{13}^{2} - 24 T_{13} + 32$$ $$T_{17}^{3} - 16 T_{17} - 8$$ $$T_{19}^{3} - 6 T_{19}^{2} - 4 T_{19} + 16$$ $$T_{29}^{3} - 4 T_{29}^{2} - 16 T_{29} + 56$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$( -1 + T )^{3}$$
$11$ $$-16 - 20 T - 2 T^{2} + T^{3}$$
$13$ $$32 - 24 T - 2 T^{2} + T^{3}$$
$17$ $$-8 - 16 T + T^{3}$$
$19$ $$16 - 4 T - 6 T^{2} + T^{3}$$
$23$ $$( 1 + T )^{3}$$
$29$ $$56 - 16 T - 4 T^{2} + T^{3}$$
$31$ $$32 - 4 T - 6 T^{2} + T^{3}$$
$37$ $$184 - 52 T - 6 T^{2} + T^{3}$$
$41$ $$-56 - 52 T + 6 T^{2} + T^{3}$$
$43$ $$-208 + 140 T - 22 T^{2} + T^{3}$$
$47$ $$32 - 4 T - 6 T^{2} + T^{3}$$
$53$ $$184 - 52 T - 6 T^{2} + T^{3}$$
$59$ $$-64 - 64 T + T^{3}$$
$61$ $$( 2 + T )^{3}$$
$67$ $$16 - 20 T + 2 T^{2} + T^{3}$$
$71$ $$1184 - 180 T - 6 T^{2} + T^{3}$$
$73$ $$248 - 36 T - 14 T^{2} + T^{3}$$
$79$ $$T^{3}$$
$83$ $$-592 - 156 T - 2 T^{2} + T^{3}$$
$89$ $$-224 + 16 T + 14 T^{2} + T^{3}$$
$97$ $$-1184 - 200 T + 2 T^{2} + T^{3}$$
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