# 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: $$x^{2} - x - 1$$ 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.

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

## 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4598))$$:

 $$T_{3}^{2} + T_{3} - 1$$ T3^2 + T3 - 1 $$T_{5}^{2} + T_{5} - 1$$ T5^2 + T5 - 1 $$T_{7}^{2} - T_{7} - 11$$ T7^2 - T7 - 11 $$T_{13}^{2} - T_{13} - 1$$ T13^2 - T13 - 1

## Hecke characteristic polynomials

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