Properties

 Label 585.2.n.e Level $585$ Weight $2$ Character orbit 585.n Analytic conductor $4.671$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.619810816.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1$$ x^8 - 2*x^5 + 14*x^4 - 8*x^3 + 2*x^2 + 2*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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 a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{6} q^{2} + (\beta_{7} + \beta_{5} - \beta_{4} - 1) q^{4} + ( - \beta_{4} - \beta_1) q^{5} + ( - \beta_{7} - \beta_{5} + \cdots - 2 \beta_{3}) q^{7}+ \cdots + (\beta_{6} - \beta_{5} + \beta_{4} + \cdots + \beta_1) q^{8}+O(q^{10})$$ q - b6 * q^2 + (b7 + b5 - b4 - 1) * q^4 + (-b4 - b1) * q^5 + (-b7 - b5 - b4 - 2*b3) * q^7 + (b6 - b5 + b4 - b3 + b1) * q^8 $$q - \beta_{6} q^{2} + (\beta_{7} + \beta_{5} - \beta_{4} - 1) q^{4} + ( - \beta_{4} - \beta_1) q^{5} + ( - \beta_{7} - \beta_{5} + \cdots - 2 \beta_{3}) q^{7}+ \cdots + ( - \beta_{6} - 2 \beta_{5} + \cdots - 2 \beta_1) q^{98}+O(q^{100})$$ q - b6 * q^2 + (b7 + b5 - b4 - 1) * q^4 + (-b4 - b1) * q^5 + (-b7 - b5 - b4 - 2*b3) * q^7 + (b6 - b5 + b4 - b3 + b1) * q^8 + (b7 + 2*b5 + 2*b3 - 2*b2 - b1 + 1) * q^10 + (b6 - b5 - b3 - b2 - 1) * q^11 + (-b7 + b6 - 2*b5 + 2*b4 - 2*b3 + 2) * q^13 + (-2*b6 + 3*b5 - 3*b4 + 3*b3 - 2*b2 - 3*b1) * q^14 + (-2*b4 - 2*b3 - 1) * q^16 + (-b7 + b6 - 2*b5 + b4 - 2*b3 - 2*b2 + b1 - 2) * q^17 + (b3 - 2*b2 - 2) * q^19 + (b7 + b6 + b5 + b4 - b3 + 3*b2 + b1 + 1) * q^20 + (-b7 + b6 + b5 - 2*b2 - b1 + 2) * q^22 + (-b7 + b6 - b5 + 2*b4 - 2*b2 - b1 + 2) * q^23 + (-b7 - 2*b6 - b3 + 2*b2 - b1 - 1) * q^25 + (-b7 - 4*b6 + 2*b5 - 2*b4 + b2 - b1) * q^26 + (3*b7 + 5*b5 - b4 + 2*b3) * q^28 + (2*b6 - b5 + b4 - b3 - 2*b2 + b1) * q^29 + (b6 + 2*b5 - b4 - b2 + 1) * q^31 + (3*b6 - 2*b2) * q^32 + (-2*b7 + 2*b5 - 2*b4 - b2 - 2*b1 + 1) * q^34 + (-b7 - b6 - 3*b5 + 2*b4 - 3*b3 - b2 + 3) * q^35 + (b7 + b3 - 6) * q^37 + (2*b6 + b5 + b4 - b2 + 1) * q^38 + (b6 - 2*b5 + b4 + b2 + b1 + 2) * q^40 + (2*b7 + b6 + b5 + 2*b2 + 2*b1 - 2) * q^41 + (b7 + 3*b6 + b5 + 2*b4 + b1) * q^43 + (-2*b6 + 2*b5 + b3 + 2*b2 + 2) * q^44 + (-4*b6 + 4*b5 + b3 + 2*b2 + 2) * q^46 + (3*b7 + 3*b5 - 3*b4 - 2) * q^47 + (-4*b5 + 2*b4 + 2*b3 + 1) * q^49 + (3*b7 - b6 + 4*b5 - 4*b4 + 3*b3 + 2*b2 - b1 - 6) * q^50 + (3*b7 + 2*b5 - b4 + b3 + 3*b2 - b1 - 5) * q^52 + (b7 - 2*b6 + 3*b5 - b4 + 4*b3 + 3*b2 - b1 + 3) * q^53 + (-2*b6 - b5 + b4 + 3*b2 + b1 + 1) * q^55 + (2*b6 + b5 - b4 + b3 + 4*b2 + b1) * q^56 + (-3*b7 - b5 + b4 - 2*b3 + 4) * q^58 + (-4*b6 + b5 - 5*b4 - 2*b2 + 2) * q^59 + (3*b7 - 2*b5 - 2*b4 + b3) * q^61 + (-b7 - b6 + b4 + b3 + 4*b2 + b1 + 4) * q^62 + (-3*b7 - b5 - b4 - 4*b3 + 7) * q^64 + (-4*b5 + 2*b4 + 2*b3 - b2 - 2*b1 - 2) * q^65 + (2*b6 + b5 - b4 + b3 - b1) * q^67 + (-3*b6 + 3*b5 + 4*b3) * q^68 + (b7 - 5*b6 + 6*b5 - 5*b4 - b2 - 2*b1 - 7) * q^70 + (-b7 - 2*b6 + 4*b5 - 6*b4 - 2*b2 - b1 + 2) * q^71 + (-b5 + b4 - b3 + 3*b1) * q^73 + (8*b6 - 2*b5 + 2*b4 - 2*b3 - 2*b2 + b1) * q^74 + (-2*b7 - b6 - b5 + 2*b4 + b3 + b2 + 2*b1 + 1) * q^76 + (-b6 + b5 + 2*b3 + 3*b2 + 3) * q^77 + (-6*b6 + 5*b5 - 5*b4 + 5*b3 + 2*b2 - 2*b1) * q^79 + (-2*b5 + 3*b4 - 4*b3 + 2*b2 + b1 + 4) * q^80 + (-3*b7 + 6*b6 - 9*b5 + 3*b4 - 6*b3 + b2 + 3*b1 + 1) * q^82 + (-3*b7 + b5 + 3*b4 - 4) * q^83 + (b7 - 2*b6 - 2*b5 + 2*b4 + b3 - b2 + 3*b1 + 3) * q^85 + (-4*b7 + 2*b6 - 6*b5 + 4*b4 - 5*b3 + 6*b2 + 4*b1 + 6) * q^86 + (b5 - b4 + b2 - 1) * q^88 + (-b7 - 2*b6 - 5*b5 + 3*b4 + 5*b2 - b1 - 5) * q^89 + (-2*b7 - b6 - 2*b5 - 3*b4 + b3 + 5*b2 - 3*b1 + 1) * q^91 + (2*b7 - b5 + b4 + 7*b2 + 2*b1 - 7) * q^92 + (8*b6 - 3*b5 + 3*b4 - 3*b3 + 3*b1) * q^94 + (b7 - 3*b6 + 3*b5 - 2*b4 + 2*b3 + 2*b1) * q^95 + (6*b6 - 5*b5 + 5*b4 - 5*b3 - 2*b2 + 2*b1) * q^97 + (-b6 - 2*b5 + 2*b4 - 2*b3 - 10*b2 - 2*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{4} + 2 q^{5}+O(q^{10})$$ 8 * q - 8 * q^4 + 2 * q^5 $$8 q - 8 q^{4} + 2 q^{5} + 6 q^{10} - 6 q^{11} + 14 q^{13} - 8 q^{16} - 16 q^{17} - 14 q^{19} + 2 q^{20} + 10 q^{22} + 14 q^{23} - 12 q^{25} - 6 q^{26} - 8 q^{28} + 2 q^{31} + 24 q^{35} - 44 q^{37} + 2 q^{38} + 22 q^{40} - 16 q^{41} - 6 q^{43} + 10 q^{44} + 2 q^{46} - 16 q^{47} + 24 q^{49} - 44 q^{50} - 38 q^{52} + 24 q^{53} + 10 q^{55} + 24 q^{58} + 22 q^{59} + 20 q^{61} + 30 q^{62} + 48 q^{64} - 4 q^{68} - 68 q^{70} + 10 q^{71} + 6 q^{76} + 24 q^{77} + 26 q^{80} + 20 q^{82} - 48 q^{83} + 32 q^{85} + 46 q^{86} - 10 q^{88} - 28 q^{89} + 20 q^{91} - 50 q^{92} - 2 q^{95}+O(q^{100})$$ 8 * q - 8 * q^4 + 2 * q^5 + 6 * q^10 - 6 * q^11 + 14 * q^13 - 8 * q^16 - 16 * q^17 - 14 * q^19 + 2 * q^20 + 10 * q^22 + 14 * q^23 - 12 * q^25 - 6 * q^26 - 8 * q^28 + 2 * q^31 + 24 * q^35 - 44 * q^37 + 2 * q^38 + 22 * q^40 - 16 * q^41 - 6 * q^43 + 10 * q^44 + 2 * q^46 - 16 * q^47 + 24 * q^49 - 44 * q^50 - 38 * q^52 + 24 * q^53 + 10 * q^55 + 24 * q^58 + 22 * q^59 + 20 * q^61 + 30 * q^62 + 48 * q^64 - 4 * q^68 - 68 * q^70 + 10 * q^71 + 6 * q^76 + 24 * q^77 + 26 * q^80 + 20 * q^82 - 48 * q^83 + 32 * q^85 + 46 * q^86 - 10 * q^88 - 28 * q^89 + 20 * q^91 - 50 * q^92 - 2 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( 64\nu^{7} + 16\nu^{6} + 4\nu^{5} - 127\nu^{4} + 944\nu^{3} - 276\nu^{2} + 378\nu + 63 ) / 319$$ (64*v^7 + 16*v^6 + 4*v^5 - 127*v^4 + 944*v^3 - 276*v^2 + 378*v + 63) / 319 $$\beta_{2}$$ $$=$$ $$( -63\nu^{7} + 64\nu^{6} + 16\nu^{5} + 130\nu^{4} - 1009\nu^{3} + 1448\nu^{2} - 402\nu - 67 ) / 319$$ (-63*v^7 + 64*v^6 + 16*v^5 + 130*v^4 - 1009*v^3 + 1448*v^2 - 402*v - 67) / 319 $$\beta_{3}$$ $$=$$ $$( -67\nu^{7} + 63\nu^{6} - 64\nu^{5} + 118\nu^{4} - 1068\nu^{3} + 1545\nu^{2} - 1263\nu + 268 ) / 319$$ (-67*v^7 + 63*v^6 - 64*v^5 + 118*v^4 - 1068*v^3 + 1545*v^2 - 1263*v + 268) / 319 $$\beta_{4}$$ $$=$$ $$( 83\nu^{7} - 59\nu^{6} + 65\nu^{5} - 70\nu^{4} + 1304\nu^{3} - 1614\nu^{2} + 1198\nu + 306 ) / 319$$ (83*v^7 - 59*v^6 + 65*v^5 - 70*v^4 + 1304*v^3 - 1614*v^2 + 1198*v + 306) / 319 $$\beta_{5}$$ $$=$$ $$( -172\nu^{7} - 43\nu^{6} + 69\nu^{5} + 441\nu^{4} - 2218\nu^{3} + 662\nu^{2} + 619\nu - 269 ) / 319$$ (-172*v^7 - 43*v^6 + 69*v^5 + 441*v^4 - 2218*v^3 + 662*v^2 + 619*v - 269) / 319 $$\beta_{6}$$ $$=$$ $$( -196\nu^{7} - 49\nu^{6} - 92\nu^{5} + 369\nu^{4} - 2572\nu^{3} + 1244\nu^{2} - 1038\nu - 173 ) / 319$$ (-196*v^7 - 49*v^6 - 92*v^5 + 369*v^4 - 2572*v^3 + 1244*v^2 - 1038*v - 173) / 319 $$\beta_{7}$$ $$=$$ $$\nu^{7} - 2\nu^{4} + 14\nu^{3} - 8\nu^{2} + \nu + 2$$ v^7 - 2*v^4 + 14*v^3 - 8*v^2 + v + 2
 $$\nu$$ $$=$$ $$( -\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 ) / 2$$ (-b7 - b5 + b4 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2}$$ b6 - b5 + b4 - b3 + 2*b2 $$\nu^{3}$$ $$=$$ $$( 3\beta_{7} + 5\beta_{5} - 5\beta_{4} - 2\beta_{2} + 3\beta _1 + 2 ) / 2$$ (3*b7 + 5*b5 - 5*b4 - 2*b2 + 3*b1 + 2) / 2 $$\nu^{4}$$ $$=$$ $$-\beta_{7} - \beta_{5} + 5\beta_{4} + 4\beta_{3} - 7$$ -b7 - b5 + 5*b4 + 4*b3 - 7 $$\nu^{5}$$ $$=$$ $$( 11\beta_{7} + 2\beta_{6} + 9\beta_{5} - 11\beta_{4} - 12\beta_{3} + 12\beta_{2} - 11\beta _1 + 12 ) / 2$$ (11*b7 + 2*b6 + 9*b5 - 11*b4 - 12*b3 + 12*b2 - 11*b1 + 12) / 2 $$\nu^{6}$$ $$=$$ $$-15\beta_{6} + 16\beta_{5} - 16\beta_{4} + 16\beta_{3} - 28\beta_{2} + 7\beta_1$$ -15*b6 + 16*b5 - 16*b4 + 16*b3 - 28*b2 + 7*b1 $$\nu^{7}$$ $$=$$ $$( -43\beta_{7} + 16\beta_{6} - 89\beta_{5} + 105\beta_{4} + 60\beta_{2} - 43\beta _1 - 60 ) / 2$$ (-43*b7 + 16*b6 - 89*b5 + 105*b4 + 60*b2 - 43*b1 - 60) / 2

Character values

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

 $$n$$ $$326$$ $$352$$ $$496$$ $$\chi(n)$$ $$1$$ $$-\beta_{2}$$ $$-\beta_{2}$$

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
 0.561103 − 0.561103i −1.49094 + 1.49094i −0.252709 + 0.252709i 1.18254 − 1.18254i 1.18254 + 1.18254i −0.252709 − 0.252709i −1.49094 − 1.49094i 0.561103 + 0.561103i
2.03032i 0 −2.12221 −1.70032 + 1.45220i 0 −1.61845 0.248119i 0 2.94844 + 3.45220i
307.2 0.134632i 0 1.98187 1.29021 1.82630i 0 1.90970 0.536087i 0 0.245878 + 0.173703i
307.3 1.57942i 0 −0.494582 −0.146426 2.23127i 0 −4.24997 2.37769i 0 3.52412 0.231269i
307.4 2.31627i 0 −3.36509 1.55654 + 1.60536i 0 3.95872 3.16190i 0 −3.71844 + 3.60536i
343.1 2.31627i 0 −3.36509 1.55654 1.60536i 0 3.95872 3.16190i 0 −3.71844 3.60536i
343.2 1.57942i 0 −0.494582 −0.146426 + 2.23127i 0 −4.24997 2.37769i 0 3.52412 + 0.231269i
343.3 0.134632i 0 1.98187 1.29021 + 1.82630i 0 1.90970 0.536087i 0 0.245878 0.173703i
343.4 2.03032i 0 −2.12221 −1.70032 1.45220i 0 −1.61845 0.248119i 0 2.94844 3.45220i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 307.4 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.e 8
3.b odd 2 1 65.2.f.b 8
5.c odd 4 1 585.2.w.e 8
12.b even 2 1 1040.2.cd.n 8
13.d odd 4 1 585.2.w.e 8
15.d odd 2 1 325.2.f.b 8
15.e even 4 1 65.2.k.b yes 8
15.e even 4 1 325.2.k.b 8
39.d odd 2 1 845.2.f.b 8
39.f even 4 1 65.2.k.b yes 8
39.f even 4 1 845.2.k.b 8
39.h odd 6 2 845.2.t.d 16
39.i odd 6 2 845.2.t.c 16
39.k even 12 2 845.2.o.c 16
39.k even 12 2 845.2.o.d 16
60.l odd 4 1 1040.2.bg.n 8
65.f even 4 1 inner 585.2.n.e 8
156.l odd 4 1 1040.2.bg.n 8
195.j odd 4 1 325.2.f.b 8
195.j odd 4 1 845.2.f.b 8
195.n even 4 1 325.2.k.b 8
195.s even 4 1 845.2.k.b 8
195.u odd 4 1 65.2.f.b 8
195.bc odd 12 2 845.2.t.c 16
195.bf even 12 2 845.2.o.c 16
195.bl even 12 2 845.2.o.d 16
195.bn odd 12 2 845.2.t.d 16
780.u even 4 1 1040.2.cd.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.f.b 8 3.b odd 2 1
65.2.f.b 8 195.u odd 4 1
65.2.k.b yes 8 15.e even 4 1
65.2.k.b yes 8 39.f even 4 1
325.2.f.b 8 15.d odd 2 1
325.2.f.b 8 195.j odd 4 1
325.2.k.b 8 15.e even 4 1
325.2.k.b 8 195.n even 4 1
585.2.n.e 8 1.a even 1 1 trivial
585.2.n.e 8 65.f even 4 1 inner
585.2.w.e 8 5.c odd 4 1
585.2.w.e 8 13.d odd 4 1
845.2.f.b 8 39.d odd 2 1
845.2.f.b 8 195.j odd 4 1
845.2.k.b 8 39.f even 4 1
845.2.k.b 8 195.s even 4 1
845.2.o.c 16 39.k even 12 2
845.2.o.c 16 195.bf even 12 2
845.2.o.d 16 39.k even 12 2
845.2.o.d 16 195.bl even 12 2
845.2.t.c 16 39.i odd 6 2
845.2.t.c 16 195.bc odd 12 2
845.2.t.d 16 39.h odd 6 2
845.2.t.d 16 195.bn odd 12 2
1040.2.bg.n 8 60.l odd 4 1
1040.2.bg.n 8 156.l odd 4 1
1040.2.cd.n 8 12.b even 2 1
1040.2.cd.n 8 780.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 $$S_{2}^{\mathrm{new}}(585, [\chi])$$:

 $$T_{2}^{8} + 12T_{2}^{6} + 46T_{2}^{4} + 56T_{2}^{2} + 1$$ T2^8 + 12*T2^6 + 46*T2^4 + 56*T2^2 + 1 $$T_{7}^{4} - 20T_{7}^{2} + 4T_{7} + 52$$ T7^4 - 20*T7^2 + 4*T7 + 52 $$T_{11}^{8} + 6T_{11}^{7} + 18T_{11}^{6} + 8T_{11}^{5} - 4T_{11}^{4} - 12T_{11}^{3} + 32T_{11}^{2} - 16T_{11} + 4$$ T11^8 + 6*T11^7 + 18*T11^6 + 8*T11^5 - 4*T11^4 - 12*T11^3 + 32*T11^2 - 16*T11 + 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 12 T^{6} + \cdots + 1$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 2 T^{7} + \cdots + 625$$
$7$ $$(T^{4} - 20 T^{2} + \cdots + 52)^{2}$$
$11$ $$T^{8} + 6 T^{7} + \cdots + 4$$
$13$ $$T^{8} - 14 T^{7} + \cdots + 28561$$
$17$ $$T^{8} + 16 T^{7} + \cdots + 13456$$
$19$ $$T^{8} + 14 T^{7} + \cdots + 100$$
$23$ $$T^{8} - 14 T^{7} + \cdots + 40804$$
$29$ $$T^{8} + 44 T^{6} + \cdots + 10000$$
$31$ $$T^{8} - 2 T^{7} + \cdots + 16900$$
$37$ $$(T^{4} + 22 T^{3} + \cdots + 580)^{2}$$
$41$ $$T^{8} + 16 T^{7} + \cdots + 13456$$
$43$ $$T^{8} + 6 T^{7} + \cdots + 8836$$
$47$ $$(T^{4} + 8 T^{3} + \cdots - 164)^{2}$$
$53$ $$T^{8} - 24 T^{7} + \cdots + 19600$$
$59$ $$T^{8} - 22 T^{7} + \cdots + 119716$$
$61$ $$(T^{4} - 10 T^{3} + \cdots - 3628)^{2}$$
$67$ $$T^{8} + 100 T^{6} + \cdots + 21904$$
$71$ $$T^{8} - 10 T^{7} + \cdots + 1223236$$
$73$ $$T^{8} + 116 T^{6} + \cdots + 547600$$
$79$ $$T^{8} + 292 T^{6} + \cdots + 13719616$$
$83$ $$(T^{4} + 24 T^{3} + \cdots - 7372)^{2}$$
$89$ $$T^{8} + 28 T^{7} + \cdots + 1795600$$
$97$ $$T^{8} + 292 T^{6} + \cdots + 13719616$$