# Properties

 Label 4830.2.a.bm 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: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ 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 $$\beta = \sqrt{33}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −2.37228 3.37228
−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.bm 2

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

 $$T_{11} + 4$$ $$T_{13}^{2} - 2 T_{13} - 32$$ $$T_{17}$$ $$T_{19}^{2} + 2 T_{19} - 32$$ $$T_{29}^{2} + 2 T_{29} - 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$-32 - 2 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$-32 + 2 T + T^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$-32 + 2 T + T^{2}$$
$31$ $$-32 + 2 T + T^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$-24 + 6 T + T^{2}$$
$43$ $$T^{2}$$
$47$ $$16 + 14 T + T^{2}$$
$53$ $$16 - 14 T + T^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$-132 + T^{2}$$
$67$ $$( -8 + T )^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$-8 - 10 T + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$88 - 22 T + T^{2}$$
$89$ $$48 + 18 T + T^{2}$$
$97$ $$16 - 14 T + T^{2}$$