# Properties

 Label 592.2.a.f Level $592$ Weight $2$ Character orbit 592.a Self dual yes Analytic conductor $4.727$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$592 = 2^{4} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 592.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.72714379966$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{3} - \beta q^{5} + (2 \beta - 2) q^{7} + (3 \beta + 1) q^{9} +O(q^{10})$$ q + (-b - 1) * q^3 - b * q^5 + (2*b - 2) * q^7 + (3*b + 1) * q^9 $$q + ( - \beta - 1) q^{3} - \beta q^{5} + (2 \beta - 2) q^{7} + (3 \beta + 1) q^{9} + \beta q^{11} + (\beta - 1) q^{13} + (2 \beta + 3) q^{15} - 6 q^{17} - 2 q^{19} + ( - 2 \beta - 4) q^{21} + ( - 3 \beta + 3) q^{23} + (\beta - 2) q^{25} + ( - 4 \beta - 7) q^{27} + ( - 3 \beta + 3) q^{29} + (\beta - 2) q^{31} + ( - 2 \beta - 3) q^{33} - 6 q^{35} + q^{37} + ( - \beta - 2) q^{39} + (3 \beta + 3) q^{41} + ( - 2 \beta + 4) q^{43} + ( - 4 \beta - 9) q^{45} - 2 \beta q^{47} + ( - 4 \beta + 9) q^{49} + (6 \beta + 6) q^{51} - 6 q^{53} + ( - \beta - 3) q^{55} + (2 \beta + 2) q^{57} + ( - 2 \beta - 6) q^{59} + (5 \beta - 4) q^{61} + (2 \beta + 16) q^{63} - 3 q^{65} + (5 \beta - 8) q^{67} + (3 \beta + 6) q^{69} - 6 q^{71} + ( - \beta - 10) q^{73} - q^{75} + 6 q^{77} + ( - 7 \beta + 7) q^{79} + (6 \beta + 16) q^{81} + (4 \beta - 12) q^{83} + 6 \beta q^{85} + (3 \beta + 6) q^{87} - 4 \beta q^{89} + ( - 2 \beta + 8) q^{91} - q^{93} + 2 \beta q^{95} + ( - 8 \beta + 2) q^{97} + (4 \beta + 9) q^{99} +O(q^{100})$$ q + (-b - 1) * q^3 - b * q^5 + (2*b - 2) * q^7 + (3*b + 1) * q^9 + b * q^11 + (b - 1) * q^13 + (2*b + 3) * q^15 - 6 * q^17 - 2 * q^19 + (-2*b - 4) * q^21 + (-3*b + 3) * q^23 + (b - 2) * q^25 + (-4*b - 7) * q^27 + (-3*b + 3) * q^29 + (b - 2) * q^31 + (-2*b - 3) * q^33 - 6 * q^35 + q^37 + (-b - 2) * q^39 + (3*b + 3) * q^41 + (-2*b + 4) * q^43 + (-4*b - 9) * q^45 - 2*b * q^47 + (-4*b + 9) * q^49 + (6*b + 6) * q^51 - 6 * q^53 + (-b - 3) * q^55 + (2*b + 2) * q^57 + (-2*b - 6) * q^59 + (5*b - 4) * q^61 + (2*b + 16) * q^63 - 3 * q^65 + (5*b - 8) * q^67 + (3*b + 6) * q^69 - 6 * q^71 + (-b - 10) * q^73 - q^75 + 6 * q^77 + (-7*b + 7) * q^79 + (6*b + 16) * q^81 + (4*b - 12) * q^83 + 6*b * q^85 + (3*b + 6) * q^87 - 4*b * q^89 + (-2*b + 8) * q^91 - q^93 + 2*b * q^95 + (-8*b + 2) * q^97 + (4*b + 9) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} - q^{5} - 2 q^{7} + 5 q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 - q^5 - 2 * q^7 + 5 * q^9 $$2 q - 3 q^{3} - q^{5} - 2 q^{7} + 5 q^{9} + q^{11} - q^{13} + 8 q^{15} - 12 q^{17} - 4 q^{19} - 10 q^{21} + 3 q^{23} - 3 q^{25} - 18 q^{27} + 3 q^{29} - 3 q^{31} - 8 q^{33} - 12 q^{35} + 2 q^{37} - 5 q^{39} + 9 q^{41} + 6 q^{43} - 22 q^{45} - 2 q^{47} + 14 q^{49} + 18 q^{51} - 12 q^{53} - 7 q^{55} + 6 q^{57} - 14 q^{59} - 3 q^{61} + 34 q^{63} - 6 q^{65} - 11 q^{67} + 15 q^{69} - 12 q^{71} - 21 q^{73} - 2 q^{75} + 12 q^{77} + 7 q^{79} + 38 q^{81} - 20 q^{83} + 6 q^{85} + 15 q^{87} - 4 q^{89} + 14 q^{91} - 2 q^{93} + 2 q^{95} - 4 q^{97} + 22 q^{99}+O(q^{100})$$ 2 * q - 3 * q^3 - q^5 - 2 * q^7 + 5 * q^9 + q^11 - q^13 + 8 * q^15 - 12 * q^17 - 4 * q^19 - 10 * q^21 + 3 * q^23 - 3 * q^25 - 18 * q^27 + 3 * q^29 - 3 * q^31 - 8 * q^33 - 12 * q^35 + 2 * q^37 - 5 * q^39 + 9 * q^41 + 6 * q^43 - 22 * q^45 - 2 * q^47 + 14 * q^49 + 18 * q^51 - 12 * q^53 - 7 * q^55 + 6 * q^57 - 14 * q^59 - 3 * q^61 + 34 * q^63 - 6 * q^65 - 11 * q^67 + 15 * q^69 - 12 * q^71 - 21 * q^73 - 2 * q^75 + 12 * q^77 + 7 * q^79 + 38 * q^81 - 20 * q^83 + 6 * q^85 + 15 * q^87 - 4 * q^89 + 14 * q^91 - 2 * q^93 + 2 * q^95 - 4 * q^97 + 22 * q^99

## 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
 2.30278 −1.30278
0 −3.30278 0 −2.30278 0 2.60555 0 7.90833 0
1.2 0 0.302776 0 1.30278 0 −4.60555 0 −2.90833 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$37$$ $$-1$$

## 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 592.2.a.f 2
3.b odd 2 1 5328.2.a.bf 2
4.b odd 2 1 74.2.a.a 2
8.b even 2 1 2368.2.a.ba 2
8.d odd 2 1 2368.2.a.s 2
12.b even 2 1 666.2.a.j 2
20.d odd 2 1 1850.2.a.u 2
20.e even 4 2 1850.2.b.i 4
28.d even 2 1 3626.2.a.a 2
44.c even 2 1 8954.2.a.p 2
148.b odd 2 1 2738.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.a 2 4.b odd 2 1
592.2.a.f 2 1.a even 1 1 trivial
666.2.a.j 2 12.b even 2 1
1850.2.a.u 2 20.d odd 2 1
1850.2.b.i 4 20.e even 4 2
2368.2.a.s 2 8.d odd 2 1
2368.2.a.ba 2 8.b even 2 1
2738.2.a.l 2 148.b odd 2 1
3626.2.a.a 2 28.d even 2 1
5328.2.a.bf 2 3.b odd 2 1
8954.2.a.p 2 44.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(592))$$:

 $$T_{3}^{2} + 3T_{3} - 1$$ T3^2 + 3*T3 - 1 $$T_{5}^{2} + T_{5} - 3$$ T5^2 + T5 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3T - 1$$
$5$ $$T^{2} + T - 3$$
$7$ $$T^{2} + 2T - 12$$
$11$ $$T^{2} - T - 3$$
$13$ $$T^{2} + T - 3$$
$17$ $$(T + 6)^{2}$$
$19$ $$(T + 2)^{2}$$
$23$ $$T^{2} - 3T - 27$$
$29$ $$T^{2} - 3T - 27$$
$31$ $$T^{2} + 3T - 1$$
$37$ $$(T - 1)^{2}$$
$41$ $$T^{2} - 9T - 9$$
$43$ $$T^{2} - 6T - 4$$
$47$ $$T^{2} + 2T - 12$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 14T + 36$$
$61$ $$T^{2} + 3T - 79$$
$67$ $$T^{2} + 11T - 51$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} + 21T + 107$$
$79$ $$T^{2} - 7T - 147$$
$83$ $$T^{2} + 20T + 48$$
$89$ $$T^{2} + 4T - 48$$
$97$ $$T^{2} + 4T - 204$$