# Properties

 Label 4830.2.a.bp 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: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 $$\beta = \sqrt{17}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 2.56155 −1.56155
−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.bp 2

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

 $$T_{11} - 4$$ $$T_{13}^{2} - 6 T_{13} - 8$$ $$T_{17}^{2} - 6 T_{17} - 8$$ $$T_{19}^{2} + 6 T_{19} - 8$$ $$T_{29} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$( -4 + T )^{2}$$
$13$ $$-8 - 6 T + T^{2}$$
$17$ $$-8 - 6 T + T^{2}$$
$19$ $$-8 + 6 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$-16 + 2 T + T^{2}$$
$37$ $$( 2 + T )^{2}$$
$41$ $$-52 - 8 T + T^{2}$$
$43$ $$-64 - 4 T + T^{2}$$
$47$ $$-16 - 2 T + T^{2}$$
$53$ $$( -10 + T )^{2}$$
$59$ $$-64 + 4 T + T^{2}$$
$61$ $$-68 + T^{2}$$
$67$ $$-64 - 4 T + T^{2}$$
$71$ $$-32 - 12 T + T^{2}$$
$73$ $$-52 + 8 T + T^{2}$$
$79$ $$-32 + 12 T + T^{2}$$
$83$ $$-144 + 6 T + T^{2}$$
$89$ $$-16 + 2 T + T^{2}$$
$97$ $$32 + 14 T + T^{2}$$