# Properties

 Label 585.2.a.m Level $585$ Weight $2$ Character orbit 585.a Self dual yes Analytic conductor $4.671$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} - q^{5} + (2 \beta + 2) q^{7} + (\beta + 3) q^{8}+O(q^{10})$$ q + (b + 1) * q^2 + (2*b + 1) * q^4 - q^5 + (2*b + 2) * q^7 + (b + 3) * q^8 $$q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} - q^{5} + (2 \beta + 2) q^{7} + (\beta + 3) q^{8} + ( - \beta - 1) q^{10} + ( - \beta - 2) q^{11} - q^{13} + (4 \beta + 6) q^{14} + 3 q^{16} + ( - 2 \beta + 2) q^{17} + ( - \beta + 2) q^{19} + ( - 2 \beta - 1) q^{20} + ( - 3 \beta - 4) q^{22} - \beta q^{23} + q^{25} + ( - \beta - 1) q^{26} + (6 \beta + 10) q^{28} + 4 \beta q^{29} + ( - 3 \beta + 6) q^{31} + (\beta - 3) q^{32} - 2 q^{34} + ( - 2 \beta - 2) q^{35} - 6 \beta q^{37} + \beta q^{38} + ( - \beta - 3) q^{40} + ( - 2 \beta + 6) q^{41} + ( - 5 \beta - 4) q^{43} + ( - 5 \beta - 6) q^{44} + ( - \beta - 2) q^{46} + (2 \beta + 2) q^{47} + (8 \beta + 5) q^{49} + (\beta + 1) q^{50} + ( - 2 \beta - 1) q^{52} + ( - 6 \beta + 6) q^{53} + (\beta + 2) q^{55} + (8 \beta + 10) q^{56} + (4 \beta + 8) q^{58} + (3 \beta - 6) q^{59} - 8 q^{61} + 3 \beta q^{62} + ( - 2 \beta - 7) q^{64} + q^{65} - 2 q^{67} + (2 \beta - 6) q^{68} + ( - 4 \beta - 6) q^{70} + ( - 7 \beta - 2) q^{71} + 6 \beta q^{73} + ( - 6 \beta - 12) q^{74} + (3 \beta - 2) q^{76} + ( - 6 \beta - 8) q^{77} - 6 \beta q^{79} - 3 q^{80} + (4 \beta + 2) q^{82} + ( - 2 \beta + 6) q^{83} + (2 \beta - 2) q^{85} + ( - 9 \beta - 14) q^{86} + ( - 5 \beta - 8) q^{88} - 6 q^{89} + ( - 2 \beta - 2) q^{91} + ( - \beta - 4) q^{92} + (4 \beta + 6) q^{94} + (\beta - 2) q^{95} + ( - 4 \beta - 2) q^{97} + (13 \beta + 21) q^{98} +O(q^{100})$$ q + (b + 1) * q^2 + (2*b + 1) * q^4 - q^5 + (2*b + 2) * q^7 + (b + 3) * q^8 + (-b - 1) * q^10 + (-b - 2) * q^11 - q^13 + (4*b + 6) * q^14 + 3 * q^16 + (-2*b + 2) * q^17 + (-b + 2) * q^19 + (-2*b - 1) * q^20 + (-3*b - 4) * q^22 - b * q^23 + q^25 + (-b - 1) * q^26 + (6*b + 10) * q^28 + 4*b * q^29 + (-3*b + 6) * q^31 + (b - 3) * q^32 - 2 * q^34 + (-2*b - 2) * q^35 - 6*b * q^37 + b * q^38 + (-b - 3) * q^40 + (-2*b + 6) * q^41 + (-5*b - 4) * q^43 + (-5*b - 6) * q^44 + (-b - 2) * q^46 + (2*b + 2) * q^47 + (8*b + 5) * q^49 + (b + 1) * q^50 + (-2*b - 1) * q^52 + (-6*b + 6) * q^53 + (b + 2) * q^55 + (8*b + 10) * q^56 + (4*b + 8) * q^58 + (3*b - 6) * q^59 - 8 * q^61 + 3*b * q^62 + (-2*b - 7) * q^64 + q^65 - 2 * q^67 + (2*b - 6) * q^68 + (-4*b - 6) * q^70 + (-7*b - 2) * q^71 + 6*b * q^73 + (-6*b - 12) * q^74 + (3*b - 2) * q^76 + (-6*b - 8) * q^77 - 6*b * q^79 - 3 * q^80 + (4*b + 2) * q^82 + (-2*b + 6) * q^83 + (2*b - 2) * q^85 + (-9*b - 14) * q^86 + (-5*b - 8) * q^88 - 6 * q^89 + (-2*b - 2) * q^91 + (-b - 4) * q^92 + (4*b + 6) * q^94 + (b - 2) * q^95 + (-4*b - 2) * q^97 + (13*b + 21) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 + 6 * q^8 $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 6 q^{8} - 2 q^{10} - 4 q^{11} - 2 q^{13} + 12 q^{14} + 6 q^{16} + 4 q^{17} + 4 q^{19} - 2 q^{20} - 8 q^{22} + 2 q^{25} - 2 q^{26} + 20 q^{28} + 12 q^{31} - 6 q^{32} - 4 q^{34} - 4 q^{35} - 6 q^{40} + 12 q^{41} - 8 q^{43} - 12 q^{44} - 4 q^{46} + 4 q^{47} + 10 q^{49} + 2 q^{50} - 2 q^{52} + 12 q^{53} + 4 q^{55} + 20 q^{56} + 16 q^{58} - 12 q^{59} - 16 q^{61} - 14 q^{64} + 2 q^{65} - 4 q^{67} - 12 q^{68} - 12 q^{70} - 4 q^{71} - 24 q^{74} - 4 q^{76} - 16 q^{77} - 6 q^{80} + 4 q^{82} + 12 q^{83} - 4 q^{85} - 28 q^{86} - 16 q^{88} - 12 q^{89} - 4 q^{91} - 8 q^{92} + 12 q^{94} - 4 q^{95} - 4 q^{97} + 42 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 + 6 * q^8 - 2 * q^10 - 4 * q^11 - 2 * q^13 + 12 * q^14 + 6 * q^16 + 4 * q^17 + 4 * q^19 - 2 * q^20 - 8 * q^22 + 2 * q^25 - 2 * q^26 + 20 * q^28 + 12 * q^31 - 6 * q^32 - 4 * q^34 - 4 * q^35 - 6 * q^40 + 12 * q^41 - 8 * q^43 - 12 * q^44 - 4 * q^46 + 4 * q^47 + 10 * q^49 + 2 * q^50 - 2 * q^52 + 12 * q^53 + 4 * q^55 + 20 * q^56 + 16 * q^58 - 12 * q^59 - 16 * q^61 - 14 * q^64 + 2 * q^65 - 4 * q^67 - 12 * q^68 - 12 * q^70 - 4 * q^71 - 24 * q^74 - 4 * q^76 - 16 * q^77 - 6 * q^80 + 4 * q^82 + 12 * q^83 - 4 * q^85 - 28 * q^86 - 16 * q^88 - 12 * q^89 - 4 * q^91 - 8 * q^92 + 12 * q^94 - 4 * q^95 - 4 * q^97 + 42 * 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
 −1.41421 1.41421
−0.414214 0 −1.82843 −1.00000 0 −0.828427 1.58579 0 0.414214
1.2 2.41421 0 3.82843 −1.00000 0 4.82843 4.41421 0 −2.41421
 $$n$$: 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.m 2
3.b odd 2 1 65.2.a.b 2
4.b odd 2 1 9360.2.a.cd 2
5.b even 2 1 2925.2.a.u 2
5.c odd 4 2 2925.2.c.r 4
12.b even 2 1 1040.2.a.j 2
13.b even 2 1 7605.2.a.x 2
15.d odd 2 1 325.2.a.i 2
15.e even 4 2 325.2.b.f 4
21.c even 2 1 3185.2.a.j 2
24.f even 2 1 4160.2.a.z 2
24.h odd 2 1 4160.2.a.bf 2
33.d even 2 1 7865.2.a.j 2
39.d odd 2 1 845.2.a.g 2
39.f even 4 2 845.2.c.b 4
39.h odd 6 2 845.2.e.c 4
39.i odd 6 2 845.2.e.h 4
39.k even 12 4 845.2.m.f 8
60.h even 2 1 5200.2.a.bu 2
195.e odd 2 1 4225.2.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.b 2 3.b odd 2 1
325.2.a.i 2 15.d odd 2 1
325.2.b.f 4 15.e even 4 2
585.2.a.m 2 1.a even 1 1 trivial
845.2.a.g 2 39.d odd 2 1
845.2.c.b 4 39.f even 4 2
845.2.e.c 4 39.h odd 6 2
845.2.e.h 4 39.i odd 6 2
845.2.m.f 8 39.k even 12 4
1040.2.a.j 2 12.b even 2 1
2925.2.a.u 2 5.b even 2 1
2925.2.c.r 4 5.c odd 4 2
3185.2.a.j 2 21.c even 2 1
4160.2.a.z 2 24.f even 2 1
4160.2.a.bf 2 24.h odd 2 1
4225.2.a.r 2 195.e odd 2 1
5200.2.a.bu 2 60.h even 2 1
7605.2.a.x 2 13.b even 2 1
7865.2.a.j 2 33.d even 2 1
9360.2.a.cd 2 4.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(585))$$:

 $$T_{2}^{2} - 2T_{2} - 1$$ T2^2 - 2*T2 - 1 $$T_{7}^{2} - 4T_{7} - 4$$ T7^2 - 4*T7 - 4 $$T_{11}^{2} + 4T_{11} + 2$$ T11^2 + 4*T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 1$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 4T - 4$$
$11$ $$T^{2} + 4T + 2$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 4T - 4$$
$19$ $$T^{2} - 4T + 2$$
$23$ $$T^{2} - 2$$
$29$ $$T^{2} - 32$$
$31$ $$T^{2} - 12T + 18$$
$37$ $$T^{2} - 72$$
$41$ $$T^{2} - 12T + 28$$
$43$ $$T^{2} + 8T - 34$$
$47$ $$T^{2} - 4T - 4$$
$53$ $$T^{2} - 12T - 36$$
$59$ $$T^{2} + 12T + 18$$
$61$ $$(T + 8)^{2}$$
$67$ $$(T + 2)^{2}$$
$71$ $$T^{2} + 4T - 94$$
$73$ $$T^{2} - 72$$
$79$ $$T^{2} - 72$$
$83$ $$T^{2} - 12T + 28$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 4T - 28$$