# Properties

 Label 585.2.n.c Level $585$ Weight $2$ Character orbit 585.n Analytic conductor $4.671$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [585,2,Mod(307,585)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(585, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

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

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

chi := DirichletCharacter("585.307");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$585 = 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 585.n (of order $$4$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$4.67124851824$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - i q^{2} + q^{4} + (i + 2) q^{5} - 2 q^{7} - 3 i q^{8} +O(q^{10})$$ q - i * q^2 + q^4 + (i + 2) * q^5 - 2 * q^7 - 3*i * q^8 $$q - i q^{2} + q^{4} + (i + 2) q^{5} - 2 q^{7} - 3 i q^{8} + ( - 2 i + 1) q^{10} + ( - i + 1) q^{11} + ( - 3 i - 2) q^{13} + 2 i q^{14} - q^{16} + ( - i + 1) q^{17} + ( - 5 i + 5) q^{19} + (i + 2) q^{20} + ( - i - 1) q^{22} + (3 i + 3) q^{23} + (4 i + 3) q^{25} + (2 i - 3) q^{26} - 2 q^{28} + (5 i + 5) q^{31} - 5 i q^{32} + ( - i - 1) q^{34} + ( - 2 i - 4) q^{35} + ( - 5 i - 5) q^{38} + ( - 6 i + 3) q^{40} + (7 i + 7) q^{41} + (i + 1) q^{43} + ( - i + 1) q^{44} + ( - 3 i + 3) q^{46} - 6 q^{47} - 3 q^{49} + ( - 3 i + 4) q^{50} + ( - 3 i - 2) q^{52} + (5 i - 5) q^{53} + ( - i + 3) q^{55} + 6 i q^{56} + ( - 7 i - 7) q^{59} - 14 q^{61} + ( - 5 i + 5) q^{62} - 7 q^{64} + ( - 8 i - 1) q^{65} - 4 i q^{67} + ( - i + 1) q^{68} + (4 i - 2) q^{70} + ( - i - 1) q^{71} + 10 i q^{73} + ( - 5 i + 5) q^{76} + (2 i - 2) q^{77} - 2 i q^{79} + ( - i - 2) q^{80} + ( - 7 i + 7) q^{82} - 6 q^{83} + ( - i + 3) q^{85} + ( - i + 1) q^{86} + ( - 3 i - 3) q^{88} + (5 i + 5) q^{89} + (6 i + 4) q^{91} + (3 i + 3) q^{92} + 6 i q^{94} + ( - 5 i + 15) q^{95} + 2 i q^{97} + 3 i q^{98} +O(q^{100})$$ q - i * q^2 + q^4 + (i + 2) * q^5 - 2 * q^7 - 3*i * q^8 + (-2*i + 1) * q^10 + (-i + 1) * q^11 + (-3*i - 2) * q^13 + 2*i * q^14 - q^16 + (-i + 1) * q^17 + (-5*i + 5) * q^19 + (i + 2) * q^20 + (-i - 1) * q^22 + (3*i + 3) * q^23 + (4*i + 3) * q^25 + (2*i - 3) * q^26 - 2 * q^28 + (5*i + 5) * q^31 - 5*i * q^32 + (-i - 1) * q^34 + (-2*i - 4) * q^35 + (-5*i - 5) * q^38 + (-6*i + 3) * q^40 + (7*i + 7) * q^41 + (i + 1) * q^43 + (-i + 1) * q^44 + (-3*i + 3) * q^46 - 6 * q^47 - 3 * q^49 + (-3*i + 4) * q^50 + (-3*i - 2) * q^52 + (5*i - 5) * q^53 + (-i + 3) * q^55 + 6*i * q^56 + (-7*i - 7) * q^59 - 14 * q^61 + (-5*i + 5) * q^62 - 7 * q^64 + (-8*i - 1) * q^65 - 4*i * q^67 + (-i + 1) * q^68 + (4*i - 2) * q^70 + (-i - 1) * q^71 + 10*i * q^73 + (-5*i + 5) * q^76 + (2*i - 2) * q^77 - 2*i * q^79 + (-i - 2) * q^80 + (-7*i + 7) * q^82 - 6 * q^83 + (-i + 3) * q^85 + (-i + 1) * q^86 + (-3*i - 3) * q^88 + (5*i + 5) * q^89 + (6*i + 4) * q^91 + (3*i + 3) * q^92 + 6*i * q^94 + (-5*i + 15) * q^95 + 2*i * q^97 + 3*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 4 q^{5} - 4 q^{7}+O(q^{10})$$ 2 * q + 2 * q^4 + 4 * q^5 - 4 * q^7 $$2 q + 2 q^{4} + 4 q^{5} - 4 q^{7} + 2 q^{10} + 2 q^{11} - 4 q^{13} - 2 q^{16} + 2 q^{17} + 10 q^{19} + 4 q^{20} - 2 q^{22} + 6 q^{23} + 6 q^{25} - 6 q^{26} - 4 q^{28} + 10 q^{31} - 2 q^{34} - 8 q^{35} - 10 q^{38} + 6 q^{40} + 14 q^{41} + 2 q^{43} + 2 q^{44} + 6 q^{46} - 12 q^{47} - 6 q^{49} + 8 q^{50} - 4 q^{52} - 10 q^{53} + 6 q^{55} - 14 q^{59} - 28 q^{61} + 10 q^{62} - 14 q^{64} - 2 q^{65} + 2 q^{68} - 4 q^{70} - 2 q^{71} + 10 q^{76} - 4 q^{77} - 4 q^{80} + 14 q^{82} - 12 q^{83} + 6 q^{85} + 2 q^{86} - 6 q^{88} + 10 q^{89} + 8 q^{91} + 6 q^{92} + 30 q^{95}+O(q^{100})$$ 2 * q + 2 * q^4 + 4 * q^5 - 4 * q^7 + 2 * q^10 + 2 * q^11 - 4 * q^13 - 2 * q^16 + 2 * q^17 + 10 * q^19 + 4 * q^20 - 2 * q^22 + 6 * q^23 + 6 * q^25 - 6 * q^26 - 4 * q^28 + 10 * q^31 - 2 * q^34 - 8 * q^35 - 10 * q^38 + 6 * q^40 + 14 * q^41 + 2 * q^43 + 2 * q^44 + 6 * q^46 - 12 * q^47 - 6 * q^49 + 8 * q^50 - 4 * q^52 - 10 * q^53 + 6 * q^55 - 14 * q^59 - 28 * q^61 + 10 * q^62 - 14 * q^64 - 2 * q^65 + 2 * q^68 - 4 * q^70 - 2 * q^71 + 10 * q^76 - 4 * q^77 - 4 * q^80 + 14 * q^82 - 12 * q^83 + 6 * q^85 + 2 * q^86 - 6 * q^88 + 10 * q^89 + 8 * q^91 + 6 * q^92 + 30 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/585\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$496$$ $$\chi(n)$$ $$1$$ $$i$$ $$i$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
307.1
 1.00000i − 1.00000i
1.00000i 0 1.00000 2.00000 + 1.00000i 0 −2.00000 3.00000i 0 1.00000 2.00000i
343.1 1.00000i 0 1.00000 2.00000 1.00000i 0 −2.00000 3.00000i 0 1.00000 + 2.00000i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.n.c 2
3.b odd 2 1 65.2.f.a 2
5.c odd 4 1 585.2.w.b 2
12.b even 2 1 1040.2.cd.b 2
13.d odd 4 1 585.2.w.b 2
15.d odd 2 1 325.2.f.a 2
15.e even 4 1 65.2.k.a yes 2
15.e even 4 1 325.2.k.a 2
39.d odd 2 1 845.2.f.a 2
39.f even 4 1 65.2.k.a yes 2
39.f even 4 1 845.2.k.a 2
39.h odd 6 2 845.2.t.b 4
39.i odd 6 2 845.2.t.a 4
39.k even 12 2 845.2.o.a 4
39.k even 12 2 845.2.o.b 4
60.l odd 4 1 1040.2.bg.a 2
65.f even 4 1 inner 585.2.n.c 2
156.l odd 4 1 1040.2.bg.a 2
195.j odd 4 1 325.2.f.a 2
195.j odd 4 1 845.2.f.a 2
195.n even 4 1 325.2.k.a 2
195.s even 4 1 845.2.k.a 2
195.u odd 4 1 65.2.f.a 2
195.bc odd 12 2 845.2.t.a 4
195.bf even 12 2 845.2.o.b 4
195.bl even 12 2 845.2.o.a 4
195.bn odd 12 2 845.2.t.b 4
780.u even 4 1 1040.2.cd.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.f.a 2 3.b odd 2 1
65.2.f.a 2 195.u odd 4 1
65.2.k.a yes 2 15.e even 4 1
65.2.k.a yes 2 39.f even 4 1
325.2.f.a 2 15.d odd 2 1
325.2.f.a 2 195.j odd 4 1
325.2.k.a 2 15.e even 4 1
325.2.k.a 2 195.n even 4 1
585.2.n.c 2 1.a even 1 1 trivial
585.2.n.c 2 65.f even 4 1 inner
585.2.w.b 2 5.c odd 4 1
585.2.w.b 2 13.d odd 4 1
845.2.f.a 2 39.d odd 2 1
845.2.f.a 2 195.j odd 4 1
845.2.k.a 2 39.f even 4 1
845.2.k.a 2 195.s even 4 1
845.2.o.a 4 39.k even 12 2
845.2.o.a 4 195.bl even 12 2
845.2.o.b 4 39.k even 12 2
845.2.o.b 4 195.bf even 12 2
845.2.t.a 4 39.i odd 6 2
845.2.t.a 4 195.bc odd 12 2
845.2.t.b 4 39.h odd 6 2
845.2.t.b 4 195.bn odd 12 2
1040.2.bg.a 2 60.l odd 4 1
1040.2.bg.a 2 156.l odd 4 1
1040.2.cd.b 2 12.b even 2 1
1040.2.cd.b 2 780.u even 4 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 4T + 5$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2} - 2T + 2$$
$13$ $$T^{2} + 4T + 13$$
$17$ $$T^{2} - 2T + 2$$
$19$ $$T^{2} - 10T + 50$$
$23$ $$T^{2} - 6T + 18$$
$29$ $$T^{2}$$
$31$ $$T^{2} - 10T + 50$$
$37$ $$T^{2}$$
$41$ $$T^{2} - 14T + 98$$
$43$ $$T^{2} - 2T + 2$$
$47$ $$(T + 6)^{2}$$
$53$ $$T^{2} + 10T + 50$$
$59$ $$T^{2} + 14T + 98$$
$61$ $$(T + 14)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$T^{2} + 2T + 2$$
$73$ $$T^{2} + 100$$
$79$ $$T^{2} + 4$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} - 10T + 50$$
$97$ $$T^{2} + 4$$