# Properties

 Label 4598.2.a.w Level $4598$ Weight $2$ Character orbit 4598.a Self dual yes Analytic conductor $36.715$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.73205 1.73205
−1.00000 −1.00000 1.00000 −0.732051 1.00000 0.267949 −1.00000 −2.00000 0.732051
1.2 −1.00000 −1.00000 1.00000 2.73205 1.00000 3.73205 −1.00000 −2.00000 −2.73205
 $$n$$: 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.w 2
11.b odd 2 1 4598.2.a.bf yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4598.2.a.w 2 1.a even 1 1 trivial
4598.2.a.bf yes 2 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4598))$$:

 $$T_{3} + 1$$ $$T_{5}^{2} - 2 T_{5} - 2$$ $$T_{7}^{2} - 4 T_{7} + 1$$ $$T_{13}^{2} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$-2 - 2 T + T^{2}$$
$7$ $$1 - 4 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$-12 + T^{2}$$
$17$ $$4 + 8 T + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$-3 + T^{2}$$
$31$ $$-8 - 4 T + T^{2}$$
$37$ $$-3 + 6 T + T^{2}$$
$41$ $$78 + 18 T + T^{2}$$
$43$ $$-44 + 4 T + T^{2}$$
$47$ $$69 - 18 T + T^{2}$$
$53$ $$-39 + 6 T + T^{2}$$
$59$ $$-107 - 2 T + T^{2}$$
$61$ $$6 - 6 T + T^{2}$$
$67$ $$( -14 + T )^{2}$$
$71$ $$54 - 18 T + T^{2}$$
$73$ $$( -2 + T )^{2}$$
$79$ $$-18 - 6 T + T^{2}$$
$83$ $$54 + 18 T + T^{2}$$
$89$ $$22 - 14 T + T^{2}$$
$97$ $$88 - 20 T + T^{2}$$