Properties

 Label 585.2.a.j Level $585$ Weight $2$ Character orbit 585.a Self dual yes Analytic conductor $4.671$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$585 = 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 585.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$4.67124851824$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + (\beta + 2) q^{4} - q^{5} + (\beta - 3) q^{7} + ( - \beta - 4) q^{8} +O(q^{10})$$ q - b * q^2 + (b + 2) * q^4 - q^5 + (b - 3) * q^7 + (-b - 4) * q^8 $$q - \beta q^{2} + (\beta + 2) q^{4} - q^{5} + (\beta - 3) q^{7} + ( - \beta - 4) q^{8} + \beta q^{10} + (\beta - 1) q^{11} - q^{13} + (2 \beta - 4) q^{14} + 3 \beta q^{16} + ( - \beta + 1) q^{17} - 2 \beta q^{19} + ( - \beta - 2) q^{20} - 4 q^{22} + (\beta - 5) q^{23} + q^{25} + \beta q^{26} - 2 q^{28} + (2 \beta + 2) q^{29} + 6 q^{31} + ( - \beta - 4) q^{32} + 4 q^{34} + ( - \beta + 3) q^{35} + ( - 3 \beta - 3) q^{37} + (2 \beta + 8) q^{38} + (\beta + 4) q^{40} + ( - \beta - 1) q^{41} + (2 \beta - 2) q^{43} + (2 \beta + 2) q^{44} + (4 \beta - 4) q^{46} + ( - 2 \beta - 6) q^{47} + ( - 5 \beta + 6) q^{49} - \beta q^{50} + ( - \beta - 2) q^{52} + (3 \beta - 3) q^{53} + ( - \beta + 1) q^{55} + ( - 2 \beta + 8) q^{56} + ( - 4 \beta - 8) q^{58} - 12 q^{59} + ( - 3 \beta + 1) q^{61} - 6 \beta q^{62} + ( - \beta + 4) q^{64} + q^{65} + ( - 6 \beta + 4) q^{67} + ( - 2 \beta - 2) q^{68} + ( - 2 \beta + 4) q^{70} + (\beta - 13) q^{71} - 6 q^{73} + (6 \beta + 12) q^{74} + ( - 6 \beta - 8) q^{76} + ( - 3 \beta + 7) q^{77} + (3 \beta - 3) q^{79} - 3 \beta q^{80} + (2 \beta + 4) q^{82} + (8 \beta - 4) q^{83} + (\beta - 1) q^{85} - 8 q^{86} - 4 \beta q^{88} + ( - 3 \beta - 3) q^{89} + ( - \beta + 3) q^{91} + ( - 2 \beta - 6) q^{92} + (8 \beta + 8) q^{94} + 2 \beta q^{95} + (7 \beta - 1) q^{97} + ( - \beta + 20) q^{98} +O(q^{100})$$ q - b * q^2 + (b + 2) * q^4 - q^5 + (b - 3) * q^7 + (-b - 4) * q^8 + b * q^10 + (b - 1) * q^11 - q^13 + (2*b - 4) * q^14 + 3*b * q^16 + (-b + 1) * q^17 - 2*b * q^19 + (-b - 2) * q^20 - 4 * q^22 + (b - 5) * q^23 + q^25 + b * q^26 - 2 * q^28 + (2*b + 2) * q^29 + 6 * q^31 + (-b - 4) * q^32 + 4 * q^34 + (-b + 3) * q^35 + (-3*b - 3) * q^37 + (2*b + 8) * q^38 + (b + 4) * q^40 + (-b - 1) * q^41 + (2*b - 2) * q^43 + (2*b + 2) * q^44 + (4*b - 4) * q^46 + (-2*b - 6) * q^47 + (-5*b + 6) * q^49 - b * q^50 + (-b - 2) * q^52 + (3*b - 3) * q^53 + (-b + 1) * q^55 + (-2*b + 8) * q^56 + (-4*b - 8) * q^58 - 12 * q^59 + (-3*b + 1) * q^61 - 6*b * q^62 + (-b + 4) * q^64 + q^65 + (-6*b + 4) * q^67 + (-2*b - 2) * q^68 + (-2*b + 4) * q^70 + (b - 13) * q^71 - 6 * q^73 + (6*b + 12) * q^74 + (-6*b - 8) * q^76 + (-3*b + 7) * q^77 + (3*b - 3) * q^79 - 3*b * q^80 + (2*b + 4) * q^82 + (8*b - 4) * q^83 + (b - 1) * q^85 - 8 * q^86 - 4*b * q^88 + (-3*b - 3) * q^89 + (-b + 3) * q^91 + (-2*b - 6) * q^92 + (8*b + 8) * q^94 + 2*b * q^95 + (7*b - 1) * q^97 + (-b + 20) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 5 q^{4} - 2 q^{5} - 5 q^{7} - 9 q^{8}+O(q^{10})$$ 2 * q - q^2 + 5 * q^4 - 2 * q^5 - 5 * q^7 - 9 * q^8 $$2 q - q^{2} + 5 q^{4} - 2 q^{5} - 5 q^{7} - 9 q^{8} + q^{10} - q^{11} - 2 q^{13} - 6 q^{14} + 3 q^{16} + q^{17} - 2 q^{19} - 5 q^{20} - 8 q^{22} - 9 q^{23} + 2 q^{25} + q^{26} - 4 q^{28} + 6 q^{29} + 12 q^{31} - 9 q^{32} + 8 q^{34} + 5 q^{35} - 9 q^{37} + 18 q^{38} + 9 q^{40} - 3 q^{41} - 2 q^{43} + 6 q^{44} - 4 q^{46} - 14 q^{47} + 7 q^{49} - q^{50} - 5 q^{52} - 3 q^{53} + q^{55} + 14 q^{56} - 20 q^{58} - 24 q^{59} - q^{61} - 6 q^{62} + 7 q^{64} + 2 q^{65} + 2 q^{67} - 6 q^{68} + 6 q^{70} - 25 q^{71} - 12 q^{73} + 30 q^{74} - 22 q^{76} + 11 q^{77} - 3 q^{79} - 3 q^{80} + 10 q^{82} - q^{85} - 16 q^{86} - 4 q^{88} - 9 q^{89} + 5 q^{91} - 14 q^{92} + 24 q^{94} + 2 q^{95} + 5 q^{97} + 39 q^{98}+O(q^{100})$$ 2 * q - q^2 + 5 * q^4 - 2 * q^5 - 5 * q^7 - 9 * q^8 + q^10 - q^11 - 2 * q^13 - 6 * q^14 + 3 * q^16 + q^17 - 2 * q^19 - 5 * q^20 - 8 * q^22 - 9 * q^23 + 2 * q^25 + q^26 - 4 * q^28 + 6 * q^29 + 12 * q^31 - 9 * q^32 + 8 * q^34 + 5 * q^35 - 9 * q^37 + 18 * q^38 + 9 * q^40 - 3 * q^41 - 2 * q^43 + 6 * q^44 - 4 * q^46 - 14 * q^47 + 7 * q^49 - q^50 - 5 * q^52 - 3 * q^53 + q^55 + 14 * q^56 - 20 * q^58 - 24 * q^59 - q^61 - 6 * q^62 + 7 * q^64 + 2 * q^65 + 2 * q^67 - 6 * q^68 + 6 * q^70 - 25 * q^71 - 12 * q^73 + 30 * q^74 - 22 * q^76 + 11 * q^77 - 3 * q^79 - 3 * q^80 + 10 * q^82 - q^85 - 16 * q^86 - 4 * q^88 - 9 * q^89 + 5 * q^91 - 14 * q^92 + 24 * q^94 + 2 * q^95 + 5 * q^97 + 39 * 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
 2.56155 −1.56155
−2.56155 0 4.56155 −1.00000 0 −0.438447 −6.56155 0 2.56155
1.2 1.56155 0 0.438447 −1.00000 0 −4.56155 −2.43845 0 −1.56155
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$1$$
$$13$$ $$1$$

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 585.2.a.j 2
3.b odd 2 1 585.2.a.l yes 2
4.b odd 2 1 9360.2.a.cl 2
5.b even 2 1 2925.2.a.bc 2
5.c odd 4 2 2925.2.c.o 4
12.b even 2 1 9360.2.a.cw 2
13.b even 2 1 7605.2.a.bi 2
15.d odd 2 1 2925.2.a.x 2
15.e even 4 2 2925.2.c.p 4
39.d odd 2 1 7605.2.a.bd 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.a.j 2 1.a even 1 1 trivial
585.2.a.l yes 2 3.b odd 2 1
2925.2.a.x 2 15.d odd 2 1
2925.2.a.bc 2 5.b even 2 1
2925.2.c.o 4 5.c odd 4 2
2925.2.c.p 4 15.e even 4 2
7605.2.a.bd 2 39.d odd 2 1
7605.2.a.bi 2 13.b even 2 1
9360.2.a.cl 2 4.b odd 2 1
9360.2.a.cw 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(585))$$:

 $$T_{2}^{2} + T_{2} - 4$$ T2^2 + T2 - 4 $$T_{7}^{2} + 5T_{7} + 2$$ T7^2 + 5*T7 + 2 $$T_{11}^{2} + T_{11} - 4$$ T11^2 + T11 - 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 4$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + 5T + 2$$
$11$ $$T^{2} + T - 4$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - T - 4$$
$19$ $$T^{2} + 2T - 16$$
$23$ $$T^{2} + 9T + 16$$
$29$ $$T^{2} - 6T - 8$$
$31$ $$(T - 6)^{2}$$
$37$ $$T^{2} + 9T - 18$$
$41$ $$T^{2} + 3T - 2$$
$43$ $$T^{2} + 2T - 16$$
$47$ $$T^{2} + 14T + 32$$
$53$ $$T^{2} + 3T - 36$$
$59$ $$(T + 12)^{2}$$
$61$ $$T^{2} + T - 38$$
$67$ $$T^{2} - 2T - 152$$
$71$ $$T^{2} + 25T + 152$$
$73$ $$(T + 6)^{2}$$
$79$ $$T^{2} + 3T - 36$$
$83$ $$T^{2} - 272$$
$89$ $$T^{2} + 9T - 18$$
$97$ $$T^{2} - 5T - 202$$