Properties

 Label 585.2.c.b Level $585$ Weight $2$ Character orbit 585.c Analytic conductor $4.671$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$585 = 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 585.c (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$4.67124851824$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 4*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} + \beta_{3}) q^{2} + ( - \beta_{2} - \beta_1 - 2) q^{4} + (\beta_{4} - \beta_1) q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{7} + (3 \beta_{4} - 4 \beta_{3}) q^{8}+O(q^{10})$$ q + (-b5 + b3) * q^2 + (-b2 - b1 - 2) * q^4 + (b4 - b1) * q^5 + (-b5 + b4 - b3) * q^7 + (3*b4 - 4*b3) * q^8 $$q + ( - \beta_{5} + \beta_{3}) q^{2} + ( - \beta_{2} - \beta_1 - 2) q^{4} + (\beta_{4} - \beta_1) q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{7} + (3 \beta_{4} - 4 \beta_{3}) q^{8} + (\beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 1) q^{10} + (\beta_1 + 2) q^{11} - \beta_{3} q^{13} + (\beta_{2} + \beta_1 - 1) q^{14} + (4 \beta_{2} + 2 \beta_1 + 3) q^{16} + (2 \beta_{5} + 2 \beta_{3}) q^{17} + \beta_1 q^{19} + ( - \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{20} + ( - 2 \beta_{5} - \beta_{4} + 5 \beta_{3}) q^{22} + (\beta_{4} - 5 \beta_{3}) q^{23} + ( - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{25} + (\beta_1 + 1) q^{26} + ( - \beta_{5} - \beta_{4} + \beta_{3}) q^{28} + (3 \beta_{2} - 3 \beta_1 + 3) q^{29} + ( - 2 \beta_{2} + \beta_1 - 4) q^{31} + ( - 3 \beta_{5} - 4 \beta_{4} + 5 \beta_{3}) q^{32} + (2 \beta_{2} - 2 \beta_1 + 4) q^{34} + ( - 2 \beta_{5} - 2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{35} + ( - \beta_{5} + 3 \beta_{4} - \beta_{3}) q^{37} + ( - \beta_{4} + 3 \beta_{3}) q^{38} + ( - 4 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} - 3 \beta_1 - 6) q^{40} + (2 \beta_{2} + 2 \beta_1 + 2) q^{41} + (2 \beta_{5} + 3 \beta_{4} + \beta_{3}) q^{43} + ( - 4 \beta_{2} - 3 \beta_1 - 8) q^{44} + (2 \beta_{2} + 5 \beta_1 + 6) q^{46} + ( - \beta_{5} + \beta_{4} + 3 \beta_{3}) q^{47} + (2 \beta_1 + 3) q^{49} + ( - \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 4) q^{50} + ( - \beta_{5} - \beta_{4} + 2 \beta_{3}) q^{52} + (4 \beta_{5} - 2 \beta_{4} - 2 \beta_{3}) q^{53} + (3 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 3) q^{55} + ( - \beta_{2} + \beta_1 - 7) q^{56} + ( - 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3}) q^{58} + (2 \beta_{2} + 3 \beta_1 - 2) q^{59} + ( - 3 \beta_{2} + \beta_1 + 1) q^{61} + (4 \beta_{5} + 3 \beta_{4} - 3 \beta_{3}) q^{62} + ( - 3 \beta_{2} - \beta_1 - 12) q^{64} + ( - \beta_{5} - \beta_{2}) q^{65} + (5 \beta_{5} - \beta_{4} - 3 \beta_{3}) q^{67} + ( - 2 \beta_{4} + 4 \beta_{3}) q^{68} + (\beta_{5} - \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{70} + ( - 6 \beta_{2} - \beta_1 + 2) q^{71} + ( - \beta_{5} + 3 \beta_{4} - 9 \beta_{3}) q^{73} + (5 \beta_{2} + \beta_1 + 1) q^{74} + ( - 2 \beta_{2} - \beta_1 - 4) q^{76} + 2 \beta_{4} q^{77} + (2 \beta_{2} - 4 \beta_1 + 6) q^{79} + (4 \beta_{5} + \beta_{4} - 10 \beta_{3} - 6 \beta_{2} - \beta_1 - 10) q^{80} + ( - 2 \beta_{5} - 6 \beta_{4} + 10 \beta_{3}) q^{82} + ( - \beta_{5} - \beta_{4} - 7 \beta_{3}) q^{83} + (4 \beta_{5} - 2 \beta_{4} + 6 \beta_{3} - 2) q^{85} + (8 \beta_{2} - \beta_1 + 8) q^{86} + (4 \beta_{5} + 9 \beta_{4} - 11 \beta_{3}) q^{88} + ( - 4 \beta_{2} + 4 \beta_1 + 2) q^{89} + ( - \beta_{2} + \beta_1 - 1) q^{91} + ( - 6 \beta_{5} - 7 \beta_{4} + 13 \beta_{3}) q^{92} + (\beta_{2} - 3 \beta_1 - 5) q^{94} + (\beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 3) q^{95} + (6 \beta_{5} - 4 \beta_{4} + 6 \beta_{3}) q^{97} + ( - 3 \beta_{5} - 2 \beta_{4} + 9 \beta_{3}) q^{98}+O(q^{100})$$ q + (-b5 + b3) * q^2 + (-b2 - b1 - 2) * q^4 + (b4 - b1) * q^5 + (-b5 + b4 - b3) * q^7 + (3*b4 - 4*b3) * q^8 + (b4 - 3*b3 + 2*b2 + 1) * q^10 + (b1 + 2) * q^11 - b3 * q^13 + (b2 + b1 - 1) * q^14 + (4*b2 + 2*b1 + 3) * q^16 + (2*b5 + 2*b3) * q^17 + b1 * q^19 + (-b5 - 2*b4 + 3*b3 + 2*b2 + b1 + 4) * q^20 + (-2*b5 - b4 + 5*b3) * q^22 + (b4 - 5*b3) * q^23 + (-2*b4 + 2*b3 + 2*b2 - 2*b1 + 1) * q^25 + (b1 + 1) * q^26 + (-b5 - b4 + b3) * q^28 + (3*b2 - 3*b1 + 3) * q^29 + (-2*b2 + b1 - 4) * q^31 + (-3*b5 - 4*b4 + 5*b3) * q^32 + (2*b2 - 2*b1 + 4) * q^34 + (-2*b5 - 2*b3 + b2 - b1 - 1) * q^35 + (-b5 + 3*b4 - b3) * q^37 + (-b4 + 3*b3) * q^38 + (-4*b5 - 3*b4 + 3*b3 - b2 - 3*b1 - 6) * q^40 + (2*b2 + 2*b1 + 2) * q^41 + (2*b5 + 3*b4 + b3) * q^43 + (-4*b2 - 3*b1 - 8) * q^44 + (2*b2 + 5*b1 + 6) * q^46 + (-b5 + b4 + 3*b3) * q^47 + (2*b1 + 3) * q^49 + (-b5 - 2*b4 - 3*b3 - 4*b2 - 2*b1 - 4) * q^50 + (-b5 - b4 + 2*b3) * q^52 + (4*b5 - 2*b4 - 2*b3) * q^53 + (3*b4 - b3 - b2 - b1 - 3) * q^55 + (-b2 + b1 - 7) * q^56 + (-3*b5 - 3*b4 - 3*b3) * q^58 + (2*b2 + 3*b1 - 2) * q^59 + (-3*b2 + b1 + 1) * q^61 + (4*b5 + 3*b4 - 3*b3) * q^62 + (-3*b2 - b1 - 12) * q^64 + (-b5 - b2) * q^65 + (5*b5 - b4 - 3*b3) * q^67 + (-2*b4 + 4*b3) * q^68 + (b5 - b4 - 3*b3 - 2*b2 + 2*b1 - 4) * q^70 + (-6*b2 - b1 + 2) * q^71 + (-b5 + 3*b4 - 9*b3) * q^73 + (5*b2 + b1 + 1) * q^74 + (-2*b2 - b1 - 4) * q^76 + 2*b4 * q^77 + (2*b2 - 4*b1 + 6) * q^79 + (4*b5 + b4 - 10*b3 - 6*b2 - b1 - 10) * q^80 + (-2*b5 - 6*b4 + 10*b3) * q^82 + (-b5 - b4 - 7*b3) * q^83 + (4*b5 - 2*b4 + 6*b3 - 2) * q^85 + (8*b2 - b1 + 8) * q^86 + (4*b5 + 9*b4 - 11*b3) * q^88 + (-4*b2 + 4*b1 + 2) * q^89 + (-b2 + b1 - 1) * q^91 + (-6*b5 - 7*b4 + 13*b3) * q^92 + (b2 - 3*b1 - 5) * q^94 + (b4 - b3 - b2 + b1 - 3) * q^95 + (6*b5 - 4*b4 + 6*b3) * q^97 + (-3*b5 - 2*b4 + 9*b3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 10 q^{4}+O(q^{10})$$ 6 * q - 10 * q^4 $$6 q - 10 q^{4} + 2 q^{10} + 12 q^{11} - 8 q^{14} + 10 q^{16} + 20 q^{20} + 2 q^{25} + 6 q^{26} + 12 q^{29} - 20 q^{31} + 20 q^{34} - 8 q^{35} - 34 q^{40} + 8 q^{41} - 40 q^{44} + 32 q^{46} + 18 q^{49} - 16 q^{50} - 16 q^{55} - 40 q^{56} - 16 q^{59} + 12 q^{61} - 66 q^{64} + 2 q^{65} - 20 q^{70} + 24 q^{71} - 4 q^{74} - 20 q^{76} + 32 q^{79} - 48 q^{80} - 12 q^{85} + 32 q^{86} + 20 q^{89} - 4 q^{91} - 32 q^{94} - 16 q^{95}+O(q^{100})$$ 6 * q - 10 * q^4 + 2 * q^10 + 12 * q^11 - 8 * q^14 + 10 * q^16 + 20 * q^20 + 2 * q^25 + 6 * q^26 + 12 * q^29 - 20 * q^31 + 20 * q^34 - 8 * q^35 - 34 * q^40 + 8 * q^41 - 40 * q^44 + 32 * q^46 + 18 * q^49 - 16 * q^50 - 16 * q^55 - 40 * q^56 - 16 * q^59 + 12 * q^61 - 66 * q^64 + 2 * q^65 - 20 * q^70 + 24 * q^71 - 4 * q^74 - 20 * q^76 + 32 * q^79 - 48 * q^80 - 12 * q^85 + 32 * q^86 + 20 * q^89 - 4 * q^91 - 32 * q^94 - 16 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23$$ (-v^5 + 8*v^4 - 4*v^3 - v^2 + 2*v + 38) / 23 $$\beta_{2}$$ $$=$$ $$( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23$$ (-5*v^5 + 17*v^4 - 20*v^3 - 5*v^2 + 10*v + 29) / 23 $$\beta_{3}$$ $$=$$ $$( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23$$ (7*v^5 - 10*v^4 + 5*v^3 + 30*v^2 + 32*v - 13) / 23 $$\beta_{4}$$ $$=$$ $$( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23$$ (-11*v^5 + 19*v^4 - 21*v^3 - 11*v^2 - 70*v + 27) / 23 $$\beta_{5}$$ $$=$$ $$( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23$$ (-14*v^5 + 20*v^4 - 10*v^3 - 37*v^2 - 64*v + 26) / 23
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2$$ (b5 - b4 + b3 + b2 - b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2\beta_{3}$$ b5 + 2*b3 $$\nu^{3}$$ $$=$$ $$2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2$$ 2*b5 - b4 + 2*b3 - b2 + 2*b1 - 2 $$\nu^{4}$$ $$=$$ $$-\beta_{2} + 5\beta _1 - 7$$ -b2 + 5*b1 - 7 $$\nu^{5}$$ $$=$$ $$-8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9$$ -8*b5 + 3*b4 - 9*b3 - 3*b2 + 8*b1 - 9

Character values

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

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
469.1
 0.403032 − 0.403032i −0.854638 + 0.854638i 1.45161 + 1.45161i 1.45161 − 1.45161i −0.854638 − 0.854638i 0.403032 + 0.403032i
2.67513i 0 −5.15633 −1.67513 + 1.48119i 0 0.806063i 8.44358i 0 3.96239 + 4.48119i
469.2 1.53919i 0 −0.369102 −0.539189 2.17009i 0 1.70928i 2.51026i 0 −3.34017 + 0.829914i
469.3 1.21432i 0 0.525428 2.21432 + 0.311108i 0 2.90321i 3.06668i 0 0.377784 2.68889i
469.4 1.21432i 0 0.525428 2.21432 0.311108i 0 2.90321i 3.06668i 0 0.377784 + 2.68889i
469.5 1.53919i 0 −0.369102 −0.539189 + 2.17009i 0 1.70928i 2.51026i 0 −3.34017 0.829914i
469.6 2.67513i 0 −5.15633 −1.67513 1.48119i 0 0.806063i 8.44358i 0 3.96239 4.48119i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 469.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.c.b 6
3.b odd 2 1 65.2.b.a 6
5.b even 2 1 inner 585.2.c.b 6
5.c odd 4 1 2925.2.a.bf 3
5.c odd 4 1 2925.2.a.bj 3
12.b even 2 1 1040.2.d.c 6
15.d odd 2 1 65.2.b.a 6
15.e even 4 1 325.2.a.j 3
15.e even 4 1 325.2.a.k 3
39.d odd 2 1 845.2.b.c 6
39.f even 4 1 845.2.d.a 6
39.f even 4 1 845.2.d.b 6
39.h odd 6 2 845.2.n.g 12
39.i odd 6 2 845.2.n.f 12
39.k even 12 2 845.2.l.d 12
39.k even 12 2 845.2.l.e 12
60.h even 2 1 1040.2.d.c 6
60.l odd 4 1 5200.2.a.cb 3
60.l odd 4 1 5200.2.a.cj 3
195.e odd 2 1 845.2.b.c 6
195.n even 4 1 845.2.d.a 6
195.n even 4 1 845.2.d.b 6
195.s even 4 1 4225.2.a.ba 3
195.s even 4 1 4225.2.a.bh 3
195.x odd 6 2 845.2.n.f 12
195.y odd 6 2 845.2.n.g 12
195.bh even 12 2 845.2.l.d 12
195.bh even 12 2 845.2.l.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.b.a 6 3.b odd 2 1
65.2.b.a 6 15.d odd 2 1
325.2.a.j 3 15.e even 4 1
325.2.a.k 3 15.e even 4 1
585.2.c.b 6 1.a even 1 1 trivial
585.2.c.b 6 5.b even 2 1 inner
845.2.b.c 6 39.d odd 2 1
845.2.b.c 6 195.e odd 2 1
845.2.d.a 6 39.f even 4 1
845.2.d.a 6 195.n even 4 1
845.2.d.b 6 39.f even 4 1
845.2.d.b 6 195.n even 4 1
845.2.l.d 12 39.k even 12 2
845.2.l.d 12 195.bh even 12 2
845.2.l.e 12 39.k even 12 2
845.2.l.e 12 195.bh even 12 2
845.2.n.f 12 39.i odd 6 2
845.2.n.f 12 195.x odd 6 2
845.2.n.g 12 39.h odd 6 2
845.2.n.g 12 195.y odd 6 2
1040.2.d.c 6 12.b even 2 1
1040.2.d.c 6 60.h even 2 1
2925.2.a.bf 3 5.c odd 4 1
2925.2.a.bj 3 5.c odd 4 1
4225.2.a.ba 3 195.s even 4 1
4225.2.a.bh 3 195.s even 4 1
5200.2.a.cb 3 60.l odd 4 1
5200.2.a.cj 3 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 11T_{2}^{4} + 31T_{2}^{2} + 25$$ acting on $$S_{2}^{\mathrm{new}}(585, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 11 T^{4} + 31 T^{2} + 25$$
$3$ $$T^{6}$$
$5$ $$T^{6} - T^{4} - 16 T^{3} - 5 T^{2} + \cdots + 125$$
$7$ $$T^{6} + 12 T^{4} + 32 T^{2} + 16$$
$11$ $$(T^{3} - 6 T^{2} + 8 T + 2)^{2}$$
$13$ $$(T^{2} + 1)^{3}$$
$17$ $$T^{6} + 44 T^{4} + 112 T^{2} + \cdots + 64$$
$19$ $$(T^{3} - 4 T + 2)^{2}$$
$23$ $$T^{6} + 72 T^{4} + 1436 T^{2} + \cdots + 7396$$
$29$ $$(T^{3} - 6 T^{2} - 36 T + 108)^{2}$$
$31$ $$(T^{3} + 10 T^{2} + 20 T - 26)^{2}$$
$37$ $$T^{6} + 56 T^{4} + 784 T^{2} + \cdots + 2704$$
$41$ $$(T^{3} - 4 T^{2} - 32 T - 32)^{2}$$
$43$ $$T^{6} + 128 T^{4} + 5452 T^{2} + \cdots + 77284$$
$47$ $$T^{6} + 44 T^{4} + 384 T^{2} + \cdots + 400$$
$53$ $$T^{6} + 144 T^{4} + 6464 T^{2} + \cdots + 92416$$
$59$ $$(T^{3} + 8 T^{2} - 40 T - 262)^{2}$$
$61$ $$(T^{3} - 6 T^{2} - 16 T - 4)^{2}$$
$67$ $$T^{6} + 220 T^{4} + 15680 T^{2} + \cdots + 364816$$
$71$ $$(T^{3} - 12 T^{2} - 88 T + 754)^{2}$$
$73$ $$T^{6} + 248 T^{4} + 15568 T^{2} + \cdots + 55696$$
$79$ $$(T^{3} - 16 T^{2} + 24 T + 16)^{2}$$
$83$ $$T^{6} + 180 T^{4} + 9200 T^{2} + \cdots + 99856$$
$89$ $$(T^{3} - 10 T^{2} - 52 T + 200)^{2}$$
$97$ $$T^{6} + 364 T^{4} + 12656 T^{2} + \cdots + 40000$$