# Properties

 Label 585.2.c.c Level $585$ Weight $2$ Character orbit 585.c Analytic conductor $4.671$ Analytic rank $0$ Dimension $10$ 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: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 16x^{8} + 90x^{6} + 208x^{4} + 169x^{2} + 16$$ x^10 + 16*x^8 + 90*x^6 + 208*x^4 + 169*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 195) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{7} + \beta_{6} - \beta_{4} - 1) q^{4} + ( - \beta_{7} - \beta_{6}) q^{5} + ( - \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} - \beta_{4}) q^{7} + (\beta_{6} + \beta_{4} - \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b7 + b6 - b4 - 1) * q^4 + (-b7 - b6) * q^5 + (-b9 + b8 - b6 - b5 - b4) * q^7 + (b6 + b4 - b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{7} + \beta_{6} - \beta_{4} - 1) q^{4} + ( - \beta_{7} - \beta_{6}) q^{5} + ( - \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} - \beta_{4}) q^{7} + (\beta_{6} + \beta_{4} - \beta_1) q^{8} + ( - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{10} + (\beta_{9} + \beta_{8} - \beta_{7} - 2) q^{11} - \beta_{2} q^{13} + ( - 2 \beta_{7} + 2 \beta_{3} + 2) q^{14} + (\beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{4} - 2 \beta_{3} - 1) q^{16} + ( - \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_1) q^{17} + ( - 2 \beta_{7} + 2 \beta_{3} - 2) q^{19} + ( - \beta_{9} - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{3} - 2 \beta_{2} + \cdots - 2) q^{20}+ \cdots + ( - 4 \beta_{9} + 4 \beta_{8} - 8 \beta_{6} - 12 \beta_{5} - 8 \beta_{4} - \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b7 + b6 - b4 - 1) * q^4 + (-b7 - b6) * q^5 + (-b9 + b8 - b6 - b5 - b4) * q^7 + (b6 + b4 - b1) * q^8 + (-b8 + b7 + b6 + b5 - b4 + b3 + b2 + b1 + 1) * q^10 + (b9 + b8 - b7 - 2) * q^11 - b2 * q^13 + (-2*b7 + 2*b3 + 2) * q^14 + (b9 + b8 - b7 - b6 + b4 - 2*b3 - 1) * q^16 + (-b9 + b8 - b6 - b5 - b4 + 2*b1) * q^17 + (-2*b7 + 2*b3 - 2) * q^19 + (-b9 - b8 + 2*b7 + 2*b6 - b4 + b3 - 2*b2 - b1 - 2) * q^20 + (b9 - b8 + 2*b6 + 4*b5 + 2*b4 - 2*b1) * q^22 + (-b9 + b8 - b6 + b5 - b4 - 2*b2 + 2*b1) * q^23 + (2*b8 - b7 - 2*b6 - b5 - b2 + 2*b1) * q^25 - b3 * q^26 + (2*b6 + 4*b5 + 2*b4 - 2*b2 + 2*b1) * q^28 + (2*b7 + 2*b6 - 2*b4 + 2) * q^29 + (-2*b7 - 2*b3 + 2) * q^31 + (4*b2 - b1) * q^32 + (2*b6 - 2*b4 + 2*b3 - 4) * q^34 + (b9 + b8 - 3*b6 - 3*b5 - b4 - 2*b1 - 2) * q^35 + (b9 - b8 + b6 - b5 + b4 + 4*b2 - 2*b1) * q^37 + (2*b9 - 2*b8 + 4*b6 + 6*b5 + 4*b4 - 2*b2 - 2*b1) * q^38 + (-b8 + b6 + b4 - b3 - 3*b1 + 4) * q^40 + (b9 + b8 - b7 - 2*b6 + 2*b4 - 2*b3 - 2) * q^41 + (2*b6 + 2*b5 + 2*b4 - 2*b2) * q^43 + (2*b7 - b6 + b4 - 4*b3 + 4) * q^44 + (-2*b9 - 2*b8 + 4*b7 + 4*b6 - 4*b4) * q^46 + (2*b9 - 2*b8 + b6 + 2*b5 + b4 + 4*b2 - 2*b1) * q^47 + (-b9 - b8 + 3*b7 - b6 + b4 - 2*b3 - 3) * q^49 + (2*b6 + 2*b5 - 2*b4 + b3 - 2*b2 + 2*b1 - 4) * q^50 + (-b6 - b5 - b4 + b2) * q^52 + (-b9 + b8 - b6 - 3*b5 - b4 - 4*b2 + 2*b1) * q^53 + (-b9 - b8 + b6 - b5 + b4 + 4*b3 - 3*b2 + 1) * q^55 + (-2*b9 - 2*b8 + 2*b7 + 2*b6 - 2*b4 - 2*b3 + 2) * q^56 + (2*b6 + 2*b4 - 2*b1) * q^58 + (-2*b9 - 2*b8 + 4*b7 + 3*b6 - 3*b4 + 2*b3 + 4) * q^59 + (b9 + b8 + b7 + b6 - b4 - 2*b3 + 2) * q^61 + (2*b9 - 2*b8 + 2*b5 + 10*b2 + 2*b1) * q^62 + (2*b9 + 2*b8 - 3*b7 - 3*b6 + 3*b4 + 1) * q^64 + (b5 + b4) * q^65 + (2*b6 + 2*b5 + 2*b4 + 6*b2 + 2*b1) * q^67 + (4*b6 + 4*b5 + 4*b4 - 2*b2 - 4*b1) * q^68 + (2*b8 - 4*b7 - 2*b6 + 4*b3 - 4*b2 + 4) * q^70 + (-3*b9 - 3*b8 + 3*b7 + 2*b6 - 2*b4 + 4*b3 + 2) * q^71 + (-4*b6 - 4*b5 - 4*b4 + 2*b2) * q^73 + (2*b9 + 2*b8 - 4*b7 - 4*b6 + 4*b4 + 2*b3) * q^74 + (-2*b9 - 2*b8 + 2*b7 - 2*b6 + 2*b4 - 6*b3 + 2) * q^76 + (-b9 + b8 + b6 - b5 + b4 + 6*b2 - 2*b1) * q^77 + (-b9 - b8 - 3*b7 - 3*b6 + 3*b4 - 2*b3) * q^79 + (-b9 - b8 - 2*b6 - 2*b5 + b4 + 2*b1 + 2) * q^80 + (-b9 + b8 - 4*b6 - 2*b5 - 4*b4 + 2*b2 + 2*b1) * q^82 + (2*b9 - 2*b8 - b6 - 2*b5 - b4 + 2*b1) * q^83 + (b9 - b8 + 2*b7 - b6 - b5 - 3*b4 + 2*b3 + 2*b2) * q^85 + (-2*b6 + 2*b4 - 6*b3) * q^86 + (-b9 + b8 - 4*b6 - 2*b5 - 4*b4 + 6*b2 + 2*b1) * q^88 + (b9 + b8 - 5*b7 - 4*b6 + 4*b4 - 2*b3 + 2) * q^89 + (b9 + b8 - b7 - b6 + b4) * q^91 + (-2*b9 + 2*b8 - 2*b5 - 4*b1) * q^92 + (-b9 - b8 + 4*b7 + 2*b3 + 4) * q^94 + (2*b8 + 2*b6 + 4*b3 - 4*b2 + 4) * q^95 + (-b9 + b8 - 3*b6 - 3*b5 - 3*b4 - 4*b2 + 6*b1) * q^97 + (-4*b9 + 4*b8 - 8*b6 - 12*b5 - 8*b4 - b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 12 q^{4} + 2 q^{5}+O(q^{10})$$ 10 * q - 12 * q^4 + 2 * q^5 $$10 q - 12 q^{4} + 2 q^{5} + 4 q^{10} - 10 q^{11} + 24 q^{14} - 16 q^{19} - 32 q^{20} + 10 q^{25} + 16 q^{29} + 24 q^{31} - 40 q^{34} - 12 q^{35} + 36 q^{40} - 10 q^{41} + 36 q^{44} - 24 q^{46} - 44 q^{49} - 40 q^{50} + 2 q^{55} + 16 q^{59} + 26 q^{61} + 32 q^{64} + 56 q^{70} - 10 q^{71} + 24 q^{74} - 2 q^{79} + 12 q^{80} - 4 q^{85} + 38 q^{89} + 10 q^{91} + 24 q^{94} + 48 q^{95}+O(q^{100})$$ 10 * q - 12 * q^4 + 2 * q^5 + 4 * q^10 - 10 * q^11 + 24 * q^14 - 16 * q^19 - 32 * q^20 + 10 * q^25 + 16 * q^29 + 24 * q^31 - 40 * q^34 - 12 * q^35 + 36 * q^40 - 10 * q^41 + 36 * q^44 - 24 * q^46 - 44 * q^49 - 40 * q^50 + 2 * q^55 + 16 * q^59 + 26 * q^61 + 32 * q^64 + 56 * q^70 - 10 * q^71 + 24 * q^74 - 2 * q^79 + 12 * q^80 - 4 * q^85 + 38 * q^89 + 10 * q^91 + 24 * q^94 + 48 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 16x^{8} + 90x^{6} + 208x^{4} + 169x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} + 8\nu^{3} + 13\nu ) / 4$$ (v^5 + 8*v^3 + 13*v) / 4 $$\beta_{3}$$ $$=$$ $$( \nu^{6} + 8\nu^{4} + 13\nu^{2} ) / 4$$ (v^6 + 8*v^4 + 13*v^2) / 4 $$\beta_{4}$$ $$=$$ $$( \nu^{8} + 13\nu^{6} + 53\nu^{4} + 4\nu^{3} + 65\nu^{2} + 20\nu ) / 8$$ (v^8 + 13*v^6 + 53*v^4 + 4*v^3 + 65*v^2 + 20*v) / 8 $$\beta_{5}$$ $$=$$ $$( \nu^{7} + 11\nu^{5} + 33\nu^{3} + 19\nu ) / 4$$ (v^7 + 11*v^5 + 33*v^3 + 19*v) / 4 $$\beta_{6}$$ $$=$$ $$( -\nu^{8} - 13\nu^{6} - 53\nu^{4} + 4\nu^{3} - 65\nu^{2} + 20\nu ) / 8$$ (-v^8 - 13*v^6 - 53*v^4 + 4*v^3 - 65*v^2 + 20*v) / 8 $$\beta_{7}$$ $$=$$ $$( \nu^{8} + 13\nu^{6} + 53\nu^{4} + 69\nu^{2} + 12 ) / 4$$ (v^8 + 13*v^6 + 53*v^4 + 69*v^2 + 12) / 4 $$\beta_{8}$$ $$=$$ $$( \nu^{9} + 15\nu^{7} + 2\nu^{6} + 77\nu^{5} + 20\nu^{4} + 155\nu^{3} + 54\nu^{2} + 96\nu + 32 ) / 8$$ (v^9 + 15*v^7 + 2*v^6 + 77*v^5 + 20*v^4 + 155*v^3 + 54*v^2 + 96*v + 32) / 8 $$\beta_{9}$$ $$=$$ $$( -\nu^{9} - 15\nu^{7} + 2\nu^{6} - 77\nu^{5} + 20\nu^{4} - 155\nu^{3} + 54\nu^{2} - 96\nu + 32 ) / 8$$ (-v^9 - 15*v^7 + 2*v^6 - 77*v^5 + 20*v^4 - 155*v^3 + 54*v^2 - 96*v + 32) / 8
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{7} + \beta_{6} - \beta_{4} - 3$$ b7 + b6 - b4 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{4} - 5\beta_1$$ b6 + b4 - 5*b1 $$\nu^{4}$$ $$=$$ $$\beta_{9} + \beta_{8} - 7\beta_{7} - 7\beta_{6} + 7\beta_{4} - 2\beta_{3} + 13$$ b9 + b8 - 7*b7 - 7*b6 + 7*b4 - 2*b3 + 13 $$\nu^{5}$$ $$=$$ $$-8\beta_{6} - 8\beta_{4} + 4\beta_{2} + 27\beta_1$$ -8*b6 - 8*b4 + 4*b2 + 27*b1 $$\nu^{6}$$ $$=$$ $$-8\beta_{9} - 8\beta_{8} + 43\beta_{7} + 43\beta_{6} - 43\beta_{4} + 20\beta_{3} - 65$$ -8*b9 - 8*b8 + 43*b7 + 43*b6 - 43*b4 + 20*b3 - 65 $$\nu^{7}$$ $$=$$ $$55\beta_{6} + 4\beta_{5} + 55\beta_{4} - 44\beta_{2} - 151\beta_1$$ 55*b6 + 4*b5 + 55*b4 - 44*b2 - 151*b1 $$\nu^{8}$$ $$=$$ $$51\beta_{9} + 51\beta_{8} - 253\beta_{7} - 257\beta_{6} + 257\beta_{4} - 154\beta_{3} + 351$$ 51*b9 + 51*b8 - 253*b7 - 257*b6 + 257*b4 - 154*b3 + 351 $$\nu^{9}$$ $$=$$ $$-4\beta_{9} + 4\beta_{8} - 364\beta_{6} - 60\beta_{5} - 364\beta_{4} + 352\beta_{2} + 865\beta_1$$ -4*b9 + 4*b8 - 364*b6 - 60*b5 - 364*b4 + 352*b2 + 865*b1

## 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
 − 2.47948i − 2.26036i − 1.77159i − 1.22308i − 0.329386i 0.329386i 1.22308i 1.77159i 2.26036i 2.47948i
2.47948i 0 −4.14785 1.72481 1.42303i 0 0.949959i 5.32555i 0 −3.52839 4.27665i
469.2 2.26036i 0 −3.10922 2.23266 0.123438i 0 4.96953i 2.50723i 0 −0.279015 5.04661i
469.3 1.77159i 0 −1.13853 −1.51036 + 1.64888i 0 0.437634i 1.52618i 0 2.92114 + 2.67573i
469.4 1.22308i 0 0.504085 0.638796 + 2.14288i 0 4.18388i 3.06269i 0 2.62091 0.781296i
469.5 0.329386i 0 1.89150 −2.08591 + 0.805596i 0 3.70203i 1.28181i 0 0.265352 + 0.687069i
469.6 0.329386i 0 1.89150 −2.08591 0.805596i 0 3.70203i 1.28181i 0 0.265352 0.687069i
469.7 1.22308i 0 0.504085 0.638796 2.14288i 0 4.18388i 3.06269i 0 2.62091 + 0.781296i
469.8 1.77159i 0 −1.13853 −1.51036 1.64888i 0 0.437634i 1.52618i 0 2.92114 2.67573i
469.9 2.26036i 0 −3.10922 2.23266 + 0.123438i 0 4.96953i 2.50723i 0 −0.279015 + 5.04661i
469.10 2.47948i 0 −4.14785 1.72481 + 1.42303i 0 0.949959i 5.32555i 0 −3.52839 + 4.27665i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 469.10 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.c 10
3.b odd 2 1 195.2.c.b 10
5.b even 2 1 inner 585.2.c.c 10
5.c odd 4 1 2925.2.a.bl 5
5.c odd 4 1 2925.2.a.bm 5
12.b even 2 1 3120.2.l.p 10
15.d odd 2 1 195.2.c.b 10
15.e even 4 1 975.2.a.r 5
15.e even 4 1 975.2.a.s 5
60.h even 2 1 3120.2.l.p 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.c.b 10 3.b odd 2 1
195.2.c.b 10 15.d odd 2 1
585.2.c.c 10 1.a even 1 1 trivial
585.2.c.c 10 5.b even 2 1 inner
975.2.a.r 5 15.e even 4 1
975.2.a.s 5 15.e even 4 1
2925.2.a.bl 5 5.c odd 4 1
2925.2.a.bm 5 5.c odd 4 1
3120.2.l.p 10 12.b even 2 1
3120.2.l.p 10 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} + 16T_{2}^{8} + 90T_{2}^{6} + 208T_{2}^{4} + 169T_{2}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(585, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + 16 T^{8} + 90 T^{6} + 208 T^{4} + \cdots + 16$$
$3$ $$T^{10}$$
$5$ $$T^{10} - 2 T^{9} - 3 T^{8} + 8 T^{7} + \cdots + 3125$$
$7$ $$T^{10} + 57 T^{8} + 1072 T^{6} + \cdots + 1024$$
$11$ $$(T^{5} + 5 T^{4} - 24 T^{3} - 100 T^{2} + \cdots + 452)^{2}$$
$13$ $$(T^{2} + 1)^{5}$$
$17$ $$T^{10} + 73 T^{8} + 1744 T^{6} + \cdots + 6400$$
$19$ $$(T^{5} + 8 T^{4} - 40 T^{3} - 448 T^{2} + \cdots + 1280)^{2}$$
$23$ $$T^{10} + 161 T^{8} + 8096 T^{6} + \cdots + 102400$$
$29$ $$(T^{5} - 8 T^{4} - 24 T^{3} + 192 T^{2} + \cdots - 640)^{2}$$
$31$ $$(T^{5} - 12 T^{4} - 40 T^{3} + 736 T^{2} + \cdots + 128)^{2}$$
$37$ $$T^{10} + 209 T^{8} + 11312 T^{6} + \cdots + 350464$$
$41$ $$(T^{5} + 5 T^{4} - 48 T^{3} - 204 T^{2} + \cdots + 196)^{2}$$
$43$ $$T^{10} + 128 T^{8} + 4960 T^{6} + \cdots + 200704$$
$47$ $$T^{10} + 204 T^{8} + 10360 T^{6} + \cdots + 64$$
$53$ $$T^{10} + 217 T^{8} + 2608 T^{6} + \cdots + 256$$
$59$ $$(T^{5} - 8 T^{4} - 108 T^{3} + 472 T^{2} + \cdots - 400)^{2}$$
$61$ $$(T^{5} - 13 T^{4} + 128 T^{2} - 256)^{2}$$
$67$ $$T^{10} + 320 T^{8} + \cdots + 15745024$$
$71$ $$(T^{5} + 5 T^{4} - 208 T^{3} - 1220 T^{2} + \cdots + 28868)^{2}$$
$73$ $$T^{10} + 436 T^{8} + \cdots + 223323136$$
$79$ $$(T^{5} + T^{4} - 248 T^{3} - 728 T^{2} + \cdots + 400)^{2}$$
$83$ $$T^{10} + 588 T^{8} + \cdots + 3685946944$$
$89$ $$(T^{5} - 19 T^{4} - 56 T^{3} + 1108 T^{2} + \cdots + 4900)^{2}$$
$97$ $$T^{10} + 649 T^{8} + \cdots + 1260534016$$