# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [585,2,Mod(1,585)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(585, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("585.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$4.67124851824$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 0 −1.00000 1.00000 0 −4.00000 −3.00000 0 1.00000
 $$n$$: 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.h 1
3.b odd 2 1 65.2.a.a 1
4.b odd 2 1 9360.2.a.ca 1
5.b even 2 1 2925.2.a.f 1
5.c odd 4 2 2925.2.c.h 2
12.b even 2 1 1040.2.a.f 1
13.b even 2 1 7605.2.a.f 1
15.d odd 2 1 325.2.a.d 1
15.e even 4 2 325.2.b.b 2
21.c even 2 1 3185.2.a.e 1
24.f even 2 1 4160.2.a.f 1
24.h odd 2 1 4160.2.a.q 1
33.d even 2 1 7865.2.a.c 1
39.d odd 2 1 845.2.a.a 1
39.f even 4 2 845.2.c.a 2
39.h odd 6 2 845.2.e.a 2
39.i odd 6 2 845.2.e.b 2
39.k even 12 4 845.2.m.b 4
60.h even 2 1 5200.2.a.d 1
195.e odd 2 1 4225.2.a.g 1

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T + 4$$
$11$ $$T + 2$$
$13$ $$T + 1$$
$17$ $$T + 2$$
$19$ $$T + 6$$
$23$ $$T - 6$$
$29$ $$T + 2$$
$31$ $$T + 10$$
$37$ $$T + 2$$
$41$ $$T - 6$$
$43$ $$T - 10$$
$47$ $$T + 4$$
$53$ $$T + 2$$
$59$ $$T + 6$$
$61$ $$T - 2$$
$67$ $$T + 4$$
$71$ $$T + 6$$
$73$ $$T + 6$$
$79$ $$T + 12$$
$83$ $$T - 16$$
$89$ $$T + 2$$
$97$ $$T + 2$$