# Properties

 Label 585.2.b.g Level $585$ Weight $2$ Character orbit 585.b 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.67124851824$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.5089536.1 Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 16*x^2 - 24*x + 18 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.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{2} + (\beta_{3} - 2) q^{4} + \beta_{4} q^{5} + ( - \beta_{4} - \beta_{2}) q^{7} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{8}+O(q^{10})$$ q - b5 * q^2 + (b3 - 2) * q^4 + b4 * q^5 + (-b4 - b2) * q^7 + (b5 + b4 + b2) * q^8 $$q - \beta_{5} q^{2} + (\beta_{3} - 2) q^{4} + \beta_{4} q^{5} + ( - \beta_{4} - \beta_{2}) q^{7} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{8} - \beta_1 q^{10} + ( - \beta_{5} + 2 \beta_{4} - \beta_{2}) q^{11} + ( - \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 - 1) q^{13} + (\beta_{3} + 2 \beta_1 - 1) q^{14} + ( - 2 \beta_1 + 1) q^{16} + (2 \beta_{3} - 2) q^{17} + (\beta_{5} + 2 \beta_{4} - \beta_{2}) q^{19} + ( - 2 \beta_{4} - \beta_{2}) q^{20} + (2 \beta_{3} - \beta_1 - 5) q^{22} + ( - \beta_1 - 3) q^{23} - q^{25} + (\beta_{5} + 4 \beta_{4} + \beta_{2} - 2 \beta_1 - 3) q^{26} + (2 \beta_{5} + 7 \beta_{4} + \beta_{2}) q^{28} + ( - \beta_{3} + 1) q^{29} + ( - \beta_{5} + 4 \beta_{4} + \beta_{2}) q^{31} + (\beta_{5} - 6 \beta_{4}) q^{32} + (4 \beta_{5} + 2 \beta_{4} + 2 \beta_{2}) q^{34} + ( - \beta_{3} + 1) q^{35} + ( - 7 \beta_{4} - \beta_{2}) q^{37} + ( - \beta_1 + 3) q^{38} + (\beta_{3} + \beta_1 - 1) q^{40} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2}) q^{41} + (2 \beta_{3} - \beta_1 + 3) q^{43} + (5 \beta_{5} + 2 \beta_{4} - \beta_{2}) q^{44} + (3 \beta_{5} - 4 \beta_{4} - \beta_{2}) q^{46} + ( - \beta_{4} - \beta_{2}) q^{47} + (2 \beta_1 + 1) q^{49} + \beta_{5} q^{50} + (\beta_{5} - 6 \beta_{4} - 2 \beta_{3} - 3 \beta_1 + 3) q^{52} + (2 \beta_{3} + 2 \beta_1 - 2) q^{53} + ( - \beta_{3} - \beta_1 - 2) q^{55} + ( - \beta_{3} - 4 \beta_1 + 7) q^{56} + ( - 2 \beta_{5} - \beta_{4} - \beta_{2}) q^{58} + ( - \beta_{5} - 4 \beta_{4} - \beta_{2}) q^{59} + ( - \beta_{3} + 2 \beta_1 - 1) q^{61} + ( - 5 \beta_1 - 3) q^{62} + ( - \beta_{3} + 2 \beta_1 + 6) q^{64} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 - 1) q^{65} + ( - 2 \beta_{5} + 7 \beta_{4} + \beta_{2}) q^{67} + ( - 2 \beta_{3} - 4 \beta_1 + 14) q^{68} + ( - 2 \beta_{5} - \beta_{4} - \beta_{2}) q^{70} + ( - \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{71} + ( - 4 \beta_{5} - 5 \beta_{4} + \beta_{2}) q^{73} + (\beta_{3} + 8 \beta_1 - 1) q^{74} + ( - \beta_{5} - 3 \beta_{2}) q^{76} + ( - 2 \beta_{3} + 4 \beta_1 - 4) q^{77} + ( - 2 \beta_{3} + 4 \beta_1 + 6) q^{79} + (2 \beta_{5} + \beta_{4}) q^{80} + ( - 4 \beta_1 - 6) q^{82} + ( - 2 \beta_{5} - 5 \beta_{4} + \beta_{2}) q^{83} + ( - 2 \beta_{4} - 2 \beta_{2}) q^{85} + ( - \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{86} + ( - 3 \beta_1 + 9) q^{88} + (4 \beta_{5} - 2 \beta_{4} - 2 \beta_{2}) q^{89} + (2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 5) q^{91} + ( - 2 \beta_{3} + 3 \beta_1 + 5) q^{92} + (\beta_{3} + 2 \beta_1 - 1) q^{94} + ( - \beta_{3} + \beta_1 - 2) q^{95} + ( - 4 \beta_{4} + 2 \beta_{2}) q^{97} + ( - \beta_{5} + 8 \beta_{4} + 2 \beta_{2}) q^{98}+O(q^{100})$$ q - b5 * q^2 + (b3 - 2) * q^4 + b4 * q^5 + (-b4 - b2) * q^7 + (b5 + b4 + b2) * q^8 - b1 * q^10 + (-b5 + 2*b4 - b2) * q^11 + (-b5 + b4 + b2 + b1 - 1) * q^13 + (b3 + 2*b1 - 1) * q^14 + (-2*b1 + 1) * q^16 + (2*b3 - 2) * q^17 + (b5 + 2*b4 - b2) * q^19 + (-2*b4 - b2) * q^20 + (2*b3 - b1 - 5) * q^22 + (-b1 - 3) * q^23 - q^25 + (b5 + 4*b4 + b2 - 2*b1 - 3) * q^26 + (2*b5 + 7*b4 + b2) * q^28 + (-b3 + 1) * q^29 + (-b5 + 4*b4 + b2) * q^31 + (b5 - 6*b4) * q^32 + (4*b5 + 2*b4 + 2*b2) * q^34 + (-b3 + 1) * q^35 + (-7*b4 - b2) * q^37 + (-b1 + 3) * q^38 + (b3 + b1 - 1) * q^40 + (-2*b5 + 2*b4 + 2*b2) * q^41 + (2*b3 - b1 + 3) * q^43 + (5*b5 + 2*b4 - b2) * q^44 + (3*b5 - 4*b4 - b2) * q^46 + (-b4 - b2) * q^47 + (2*b1 + 1) * q^49 + b5 * q^50 + (b5 - 6*b4 - 2*b3 - 3*b1 + 3) * q^52 + (2*b3 + 2*b1 - 2) * q^53 + (-b3 - b1 - 2) * q^55 + (-b3 - 4*b1 + 7) * q^56 + (-2*b5 - b4 - b2) * q^58 + (-b5 - 4*b4 - b2) * q^59 + (-b3 + 2*b1 - 1) * q^61 + (-5*b1 - 3) * q^62 + (-b3 + 2*b1 + 6) * q^64 + (-b5 - b4 + b3 - b1 - 1) * q^65 + (-2*b5 + 7*b4 + b2) * q^67 + (-2*b3 - 4*b1 + 14) * q^68 + (-2*b5 - b4 - b2) * q^70 + (-b5 - 2*b4 + b2) * q^71 + (-4*b5 - 5*b4 + b2) * q^73 + (b3 + 8*b1 - 1) * q^74 + (-b5 - 3*b2) * q^76 + (-2*b3 + 4*b1 - 4) * q^77 + (-2*b3 + 4*b1 + 6) * q^79 + (2*b5 + b4) * q^80 + (-4*b1 - 6) * q^82 + (-2*b5 - 5*b4 + b2) * q^83 + (-2*b4 - 2*b2) * q^85 + (-b5 - 2*b4 + b2) * q^86 + (-3*b1 + 9) * q^88 + (4*b5 - 2*b4 - 2*b2) * q^89 + (2*b5 + 2*b4 + b3 + 2*b2 + 5) * q^91 + (-2*b3 + 3*b1 + 5) * q^92 + (b3 + 2*b1 - 1) * q^94 + (-b3 + b1 - 2) * q^95 + (-4*b4 + 2*b2) * q^97 + (-b5 + 8*b4 + 2*b2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 10 q^{4}+O(q^{10})$$ 6 * q - 10 * q^4 $$6 q - 10 q^{4} + 2 q^{10} - 8 q^{13} - 8 q^{14} + 10 q^{16} - 8 q^{17} - 24 q^{22} - 16 q^{23} - 6 q^{25} - 14 q^{26} + 4 q^{29} + 4 q^{35} + 20 q^{38} - 6 q^{40} + 24 q^{43} + 2 q^{49} + 20 q^{52} - 12 q^{53} - 12 q^{55} + 48 q^{56} - 12 q^{61} - 8 q^{62} + 30 q^{64} - 2 q^{65} + 88 q^{68} - 20 q^{74} - 36 q^{77} + 24 q^{79} - 28 q^{82} + 60 q^{88} + 32 q^{91} + 20 q^{92} - 8 q^{94} - 16 q^{95}+O(q^{100})$$ 6 * q - 10 * q^4 + 2 * q^10 - 8 * q^13 - 8 * q^14 + 10 * q^16 - 8 * q^17 - 24 * q^22 - 16 * q^23 - 6 * q^25 - 14 * q^26 + 4 * q^29 + 4 * q^35 + 20 * q^38 - 6 * q^40 + 24 * q^43 + 2 * q^49 + 20 * q^52 - 12 * q^53 - 12 * q^55 + 48 * q^56 - 12 * q^61 - 8 * q^62 + 30 * q^64 - 2 * q^65 + 88 * q^68 - 20 * q^74 - 36 * q^77 + 24 * q^79 - 28 * q^82 + 60 * q^88 + 32 * q^91 + 20 * q^92 - 8 * q^94 - 16 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{5} + 24\nu^{4} - 6\nu^{3} - \nu^{2} + 6\nu + 285 ) / 131$$ (-v^5 + 24*v^4 - 6*v^3 - v^2 + 6*v + 285) / 131 $$\beta_{2}$$ $$=$$ $$( -5\nu^{5} - 11\nu^{4} + 101\nu^{3} - 136\nu^{2} + 292\nu - 147 ) / 393$$ (-5*v^5 - 11*v^4 + 101*v^3 - 136*v^2 + 292*v - 147) / 393 $$\beta_{3}$$ $$=$$ $$( 6\nu^{5} - 13\nu^{4} + 36\nu^{3} + 6\nu^{2} - 36\nu - 7 ) / 131$$ (6*v^5 - 13*v^4 + 36*v^3 + 6*v^2 - 36*v - 7) / 131 $$\beta_{4}$$ $$=$$ $$( 23\nu^{5} - 28\nu^{4} + 7\nu^{3} + 154\nu^{2} + 386\nu - 267 ) / 393$$ (23*v^5 - 28*v^4 + 7*v^3 + 154*v^2 + 386*v - 267) / 393 $$\beta_{5}$$ $$=$$ $$( -23\nu^{5} + 28\nu^{4} - 7\nu^{3} - 23\nu^{2} - 386\nu + 267 ) / 131$$ (-23*v^5 + 28*v^4 - 7*v^3 - 23*v^2 - 386*v + 267) / 131
 $$\nu$$ $$=$$ $$( \beta_{4} - \beta_{3} + \beta_{2} + 1 ) / 2$$ (b4 - b3 + b2 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{5} + 3\beta_{4}$$ b5 + 3*b4 $$\nu^{3}$$ $$=$$ $$\beta_{5} + 2\beta_{4} + 2\beta_{3} + 2\beta_{2} + \beta _1 - 2$$ b5 + 2*b4 + 2*b3 + 2*b2 + b1 - 2 $$\nu^{4}$$ $$=$$ $$\beta_{3} + 6\beta _1 - 13$$ b3 + 6*b1 - 13 $$\nu^{5}$$ $$=$$ $$-7\beta_{5} - 12\beta_{4} + 9\beta_{3} - 9\beta_{2} + 7\beta _1 - 12$$ -7*b5 - 12*b4 + 9*b3 - 9*b2 + 7*b1 - 12

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 1.66044 + 1.66044i 0.675970 − 0.675970i −1.33641 − 1.33641i −1.33641 + 1.33641i 0.675970 + 0.675970i 1.66044 − 1.66044i
2.51414i 0 −4.32088 1.00000i 0 3.32088i 5.83502i 0 2.51414
181.2 2.08613i 0 −2.35194 1.00000i 0 1.35194i 0.734191i 0 −2.08613
181.3 0.571993i 0 1.67282 1.00000i 0 2.67282i 2.10083i 0 0.571993
181.4 0.571993i 0 1.67282 1.00000i 0 2.67282i 2.10083i 0 0.571993
181.5 2.08613i 0 −2.35194 1.00000i 0 1.35194i 0.734191i 0 −2.08613
181.6 2.51414i 0 −4.32088 1.00000i 0 3.32088i 5.83502i 0 2.51414
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 181.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.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.b.g 6
3.b odd 2 1 65.2.c.a 6
12.b even 2 1 1040.2.k.d 6
13.b even 2 1 inner 585.2.b.g 6
13.d odd 4 1 7605.2.a.bs 3
13.d odd 4 1 7605.2.a.cc 3
15.d odd 2 1 325.2.c.g 6
15.e even 4 1 325.2.d.e 6
15.e even 4 1 325.2.d.f 6
39.d odd 2 1 65.2.c.a 6
39.f even 4 1 845.2.a.i 3
39.f even 4 1 845.2.a.k 3
39.h odd 6 2 845.2.m.h 12
39.i odd 6 2 845.2.m.h 12
39.k even 12 2 845.2.e.i 6
39.k even 12 2 845.2.e.k 6
156.h even 2 1 1040.2.k.d 6
195.e odd 2 1 325.2.c.g 6
195.n even 4 1 4225.2.a.bc 3
195.n even 4 1 4225.2.a.be 3
195.s even 4 1 325.2.d.e 6
195.s even 4 1 325.2.d.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.c.a 6 3.b odd 2 1
65.2.c.a 6 39.d odd 2 1
325.2.c.g 6 15.d odd 2 1
325.2.c.g 6 195.e odd 2 1
325.2.d.e 6 15.e even 4 1
325.2.d.e 6 195.s even 4 1
325.2.d.f 6 15.e even 4 1
325.2.d.f 6 195.s even 4 1
585.2.b.g 6 1.a even 1 1 trivial
585.2.b.g 6 13.b even 2 1 inner
845.2.a.i 3 39.f even 4 1
845.2.a.k 3 39.f even 4 1
845.2.e.i 6 39.k even 12 2
845.2.e.k 6 39.k even 12 2
845.2.m.h 12 39.h odd 6 2
845.2.m.h 12 39.i odd 6 2
1040.2.k.d 6 12.b even 2 1
1040.2.k.d 6 156.h even 2 1
4225.2.a.bc 3 195.n even 4 1
4225.2.a.be 3 195.n even 4 1
7605.2.a.bs 3 13.d odd 4 1
7605.2.a.cc 3 13.d odd 4 1

## Hecke kernels

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

 $$T_{2}^{6} + 11T_{2}^{4} + 31T_{2}^{2} + 9$$ T2^6 + 11*T2^4 + 31*T2^2 + 9 $$T_{7}^{6} + 20T_{7}^{4} + 112T_{7}^{2} + 144$$ T7^6 + 20*T7^4 + 112*T7^2 + 144 $$T_{17}^{3} + 4T_{17}^{2} - 32T_{17} - 96$$ T17^3 + 4*T17^2 - 32*T17 - 96

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 11 T^{4} + 31 T^{2} + 9$$
$3$ $$T^{6}$$
$5$ $$(T^{2} + 1)^{3}$$
$7$ $$T^{6} + 20 T^{4} + 112 T^{2} + \cdots + 144$$
$11$ $$T^{6} + 48 T^{4} + 684 T^{2} + \cdots + 2916$$
$13$ $$T^{6} + 8 T^{5} + 39 T^{4} + \cdots + 2197$$
$17$ $$(T^{3} + 4 T^{2} - 32 T - 96)^{2}$$
$19$ $$T^{6} + 44 T^{4} + 196 T^{2} + \cdots + 36$$
$23$ $$(T^{3} + 8 T^{2} + 16 T + 6)^{2}$$
$29$ $$(T^{3} - 2 T^{2} - 8 T + 12)^{2}$$
$31$ $$T^{6} + 56 T^{4} + 604 T^{2} + \cdots + 36$$
$37$ $$T^{6} + 152 T^{4} + 6256 T^{2} + \cdots + 51984$$
$41$ $$T^{6} + 92 T^{4} + 2224 T^{2} + \cdots + 5184$$
$43$ $$(T^{3} - 12 T^{2} + 12 T - 2)^{2}$$
$47$ $$T^{6} + 20 T^{4} + 112 T^{2} + \cdots + 144$$
$53$ $$(T^{3} + 6 T^{2} - 60 T + 72)^{2}$$
$59$ $$T^{6} + 84 T^{4} + 468 T^{2} + \cdots + 324$$
$61$ $$(T^{3} + 6 T^{2} - 12 T - 76)^{2}$$
$67$ $$T^{6} + 156 T^{4} + 3168 T^{2} + \cdots + 11664$$
$71$ $$T^{6} + 44 T^{4} + 196 T^{2} + \cdots + 36$$
$73$ $$T^{6} + 296 T^{4} + 23344 T^{2} + \cdots + 266256$$
$79$ $$(T^{3} - 12 T^{2} - 48 T + 32)^{2}$$
$83$ $$T^{6} + 156 T^{4} + 5760 T^{2} + \cdots + 1296$$
$89$ $$T^{6} + 192 T^{4} + 9216 T^{2} + \cdots + 82944$$
$97$ $$T^{6} + 140 T^{4} + 1456 T^{2} + \cdots + 576$$