# Properties

 Label 4598.2.a.bg Level $4598$ Weight $2$ Character orbit 4598.a Self dual yes Analytic conductor $36.715$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4598 = 2 \cdot 11^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4598.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.7152148494$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.61803 −0.618034
1.00000 −1.61803 1.00000 −1.61803 −1.61803 3.85410 1.00000 −0.381966 −1.61803
1.2 1.00000 0.618034 1.00000 0.618034 0.618034 −2.85410 1.00000 −2.61803 0.618034
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$11$$ $$1$$
$$19$$ $$-1$$

## 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 4598.2.a.bg yes 2
11.b odd 2 1 4598.2.a.x 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4598.2.a.x 2 11.b odd 2 1
4598.2.a.bg yes 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4598))$$:

 $$T_{3}^{2} + T_{3} - 1$$ $$T_{5}^{2} + T_{5} - 1$$ $$T_{7}^{2} - T_{7} - 11$$ $$T_{13}^{2} - T_{13} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$-1 + T + T^{2}$$
$5$ $$-1 + T + T^{2}$$
$7$ $$-11 - T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$-1 - T + T^{2}$$
$17$ $$-4 + 2 T + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$( 2 + T )^{2}$$
$29$ $$-5 + 5 T + T^{2}$$
$31$ $$1 + 3 T + T^{2}$$
$37$ $$-4 + 2 T + T^{2}$$
$41$ $$-1 + T + T^{2}$$
$43$ $$11 + 13 T + T^{2}$$
$47$ $$4 + 6 T + T^{2}$$
$53$ $$16 + 12 T + T^{2}$$
$59$ $$-64 - 8 T + T^{2}$$
$61$ $$-44 - 2 T + T^{2}$$
$67$ $$-31 + T + T^{2}$$
$71$ $$79 - 19 T + T^{2}$$
$73$ $$16 + 12 T + T^{2}$$
$79$ $$-20 + T^{2}$$
$83$ $$11 - 7 T + T^{2}$$
$89$ $$-76 + 4 T + T^{2}$$
$97$ $$176 + 28 T + T^{2}$$