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

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{2} + ( 1 + 2 \beta ) q^{4} - q^{5} + ( 2 + 2 \beta ) q^{7} + ( 3 + \beta ) q^{8} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{2} + ( 1 + 2 \beta ) q^{4} - q^{5} + ( 2 + 2 \beta ) q^{7} + ( 3 + \beta ) q^{8} + ( -1 - \beta ) q^{10} + ( -2 - \beta ) q^{11} - q^{13} + ( 6 + 4 \beta ) q^{14} + 3 q^{16} + ( 2 - 2 \beta ) q^{17} + ( 2 - \beta ) q^{19} + ( -1 - 2 \beta ) q^{20} + ( -4 - 3 \beta ) q^{22} -\beta q^{23} + q^{25} + ( -1 - \beta ) q^{26} + ( 10 + 6 \beta ) q^{28} + 4 \beta q^{29} + ( 6 - 3 \beta ) q^{31} + ( -3 + \beta ) q^{32} -2 q^{34} + ( -2 - 2 \beta ) q^{35} -6 \beta q^{37} + \beta q^{38} + ( -3 - \beta ) q^{40} + ( 6 - 2 \beta ) q^{41} + ( -4 - 5 \beta ) q^{43} + ( -6 - 5 \beta ) q^{44} + ( -2 - \beta ) q^{46} + ( 2 + 2 \beta ) q^{47} + ( 5 + 8 \beta ) q^{49} + ( 1 + \beta ) q^{50} + ( -1 - 2 \beta ) q^{52} + ( 6 - 6 \beta ) q^{53} + ( 2 + \beta ) q^{55} + ( 10 + 8 \beta ) q^{56} + ( 8 + 4 \beta ) q^{58} + ( -6 + 3 \beta ) q^{59} -8 q^{61} + 3 \beta q^{62} + ( -7 - 2 \beta ) q^{64} + q^{65} -2 q^{67} + ( -6 + 2 \beta ) q^{68} + ( -6 - 4 \beta ) q^{70} + ( -2 - 7 \beta ) q^{71} + 6 \beta q^{73} + ( -12 - 6 \beta ) q^{74} + ( -2 + 3 \beta ) q^{76} + ( -8 - 6 \beta ) q^{77} -6 \beta q^{79} -3 q^{80} + ( 2 + 4 \beta ) q^{82} + ( 6 - 2 \beta ) q^{83} + ( -2 + 2 \beta ) q^{85} + ( -14 - 9 \beta ) q^{86} + ( -8 - 5 \beta ) q^{88} -6 q^{89} + ( -2 - 2 \beta ) q^{91} + ( -4 - \beta ) q^{92} + ( 6 + 4 \beta ) q^{94} + ( -2 + \beta ) q^{95} + ( -2 - 4 \beta ) q^{97} + ( 21 + 13 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 6 q^{8} + O(q^{10})$$ $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 6 q^{8} - 2 q^{10} - 4 q^{11} - 2 q^{13} + 12 q^{14} + 6 q^{16} + 4 q^{17} + 4 q^{19} - 2 q^{20} - 8 q^{22} + 2 q^{25} - 2 q^{26} + 20 q^{28} + 12 q^{31} - 6 q^{32} - 4 q^{34} - 4 q^{35} - 6 q^{40} + 12 q^{41} - 8 q^{43} - 12 q^{44} - 4 q^{46} + 4 q^{47} + 10 q^{49} + 2 q^{50} - 2 q^{52} + 12 q^{53} + 4 q^{55} + 20 q^{56} + 16 q^{58} - 12 q^{59} - 16 q^{61} - 14 q^{64} + 2 q^{65} - 4 q^{67} - 12 q^{68} - 12 q^{70} - 4 q^{71} - 24 q^{74} - 4 q^{76} - 16 q^{77} - 6 q^{80} + 4 q^{82} + 12 q^{83} - 4 q^{85} - 28 q^{86} - 16 q^{88} - 12 q^{89} - 4 q^{91} - 8 q^{92} + 12 q^{94} - 4 q^{95} - 4 q^{97} + 42 q^{98} + O(q^{100})$$

## 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(585))$$:

 $$T_{2}^{2} - 2 T_{2} - 1$$ $$T_{7}^{2} - 4 T_{7} - 4$$ $$T_{11}^{2} + 4 T_{11} + 2$$

## Hecke characteristic polynomials

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