# Properties

 Label 3584.2.a.l Level $3584$ Weight $2$ Character orbit 3584.a Self dual yes Analytic conductor $28.618$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3584 = 2^{9} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3584.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.6183840844$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.18857984.1 Defining polynomial: $$x^{6} - 2x^{5} - 9x^{4} + 16x^{3} + 6x^{2} - 12x + 2$$ x^6 - 2*x^5 - 9*x^4 + 16*x^3 + 6*x^2 - 12*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 1) q^{3} + \beta_1 q^{5} + q^{7} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{9}+O(q^{10})$$ q + (b2 + 1) * q^3 + b1 * q^5 + q^7 + (-b4 + b3 + b2 + 2) * q^9 $$q + (\beta_{2} + 1) q^{3} + \beta_1 q^{5} + q^{7} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{9} + (\beta_{5} + \beta_{3} + \beta_1 + 1) q^{11} + \beta_{4} q^{13} + (\beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{15} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{17} + ( - \beta_{4} - \beta_{2} - \beta_1 + 3) q^{19} + (\beta_{2} + 1) q^{21} + (\beta_{5} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{23} + (\beta_{5} - 2 \beta_{3} - \beta_1) q^{25} + ( - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 4) q^{27} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{29} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{31} + ( - \beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{33} + \beta_1 q^{35} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 4) q^{37} + (\beta_{5} - 2 \beta_{2} + \beta_1 - 1) q^{39} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{41} + (\beta_{5} - 3 \beta_{3} - 2 \beta_{2} - \beta_1 + 3) q^{43} + (\beta_{4} + 4 \beta_{3} + 2 \beta_1) q^{45} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2) q^{47} + q^{49} + (2 \beta_1 + 4) q^{51} + ( - \beta_{5} - \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{53} + (2 \beta_{5} + 2 \beta_1 + 2) q^{55} + ( - \beta_{5} - 2 \beta_{3} + 4 \beta_{2} - 3 \beta_1 + 1) q^{57} + ( - 2 \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{59} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{61} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{63} + ( - \beta_{4} - 3 \beta_{3} + \beta_{2} - 1) q^{65} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_1 + 3) q^{67} + (2 \beta_{5} + 3 \beta_{4} - 6 \beta_{3} - 2 \beta_{2} + \beta_1 - 8) q^{69} + (3 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{71} + ( - 3 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 5 \beta_1 + 4) q^{73} + (2 \beta_{5} - \beta_{4} - 6 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 3) q^{75} + (\beta_{5} + \beta_{3} + \beta_1 + 1) q^{77} + ( - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 3 \beta_1 - 4) q^{79} + ( - 3 \beta_{5} + 6 \beta_{3} + 4 \beta_{2} - \beta_1 + 4) q^{81} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{83} + (2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{85} + ( - 2 \beta_{5} - 4 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{87} + (\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 3 \beta_1 + 2) q^{89} + \beta_{4} q^{91} + ( - 2 \beta_{4} + 8 \beta_{3} + 2 \beta_1 + 4) q^{93} + ( - \beta_{5} + 4 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 3) q^{95} + (\beta_{5} - \beta_{4} - 5 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{97} + ( - \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + \beta_1 + 7) q^{99}+O(q^{100})$$ q + (b2 + 1) * q^3 + b1 * q^5 + q^7 + (-b4 + b3 + b2 + 2) * q^9 + (b5 + b3 + b1 + 1) * q^11 + b4 * q^13 + (b4 + b3 + b2 + 2*b1 - 1) * q^15 + (b5 + b4 + b3 + b2 + b1) * q^17 + (-b4 - b2 - b1 + 3) * q^19 + (b2 + 1) * q^21 + (b5 - 2*b3 - 2*b2 + b1 - 1) * q^23 + (b5 - 2*b3 - b1) * q^25 + (-2*b5 - b4 + 2*b3 + 2*b2 - b1 + 4) * q^27 + (b5 - b4 + b3 - b2 - b1) * q^29 + (-2*b5 + 2*b2) * q^31 + (-b5 + b4 - b3 + 3*b2 + b1 + 2) * q^33 + b1 * q^35 + (-b5 - b4 - b3 + b2 - b1 + 4) * q^37 + (b5 - 2*b2 + b1 - 1) * q^39 + (2*b3 + 2*b2 + 2*b1) * q^41 + (b5 - 3*b3 - 2*b2 - b1 + 3) * q^43 + (b4 + 4*b3 + 2*b1) * q^45 + (-2*b5 - 2*b4 + 2*b3 + 2) * q^47 + q^49 + (2*b1 + 4) * q^51 + (-b5 - b4 - 5*b3 - 3*b2 - 3*b1) * q^53 + (2*b5 + 2*b1 + 2) * q^55 + (-b5 - 2*b3 + 4*b2 - 3*b1 + 1) * q^57 + (-2*b5 + b4 + 2*b3 + b2 + b1 + 3) * q^59 + (-2*b5 - 2*b4 + 2*b3 + 2*b2 - 3*b1) * q^61 + (-b4 + b3 + b2 + 2) * q^63 + (-b4 - 3*b3 + b2 - 1) * q^65 + (-b5 + 2*b4 - b3 - b1 + 3) * q^67 + (2*b5 + 3*b4 - 6*b3 - 2*b2 + b1 - 8) * q^69 + (3*b5 - b4 - b3 + b2 + b1 - 2) * q^71 + (-3*b5 - b4 + b3 + b2 - 5*b1 + 4) * q^73 + (2*b5 - b4 - 6*b3 - 3*b2 - 3*b1 + 3) * q^75 + (b5 + b3 + b1 + 1) * q^77 + (-b5 + b4 - b3 - b2 - 3*b1 - 4) * q^79 + (-3*b5 + 6*b3 + 4*b2 - b1 + 4) * q^81 + (2*b5 + 2*b4 - 2*b3 - b2 - 2*b1 + 1) * q^83 + (2*b5 - 2*b3 + 2*b2 + 2*b1) * q^85 + (-2*b5 - 4*b3 + 2*b2 - 4*b1) * q^87 + (b5 + b4 - b3 - b2 + 3*b1 + 2) * q^89 + b4 * q^91 + (-2*b4 + 8*b3 + 2*b1 + 4) * q^93 + (-b5 + 4*b3 - 2*b2 + 3*b1 - 3) * q^95 + (b5 - b4 - 5*b3 - b2 + b1 + 2) * q^97 + (-b5 - 2*b4 + 3*b3 + b1 + 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 4 q^{3} + 6 q^{7} + 10 q^{9}+O(q^{10})$$ 6 * q + 4 * q^3 + 6 * q^7 + 10 * q^9 $$6 q + 4 q^{3} + 6 q^{7} + 10 q^{9} + 8 q^{11} - 8 q^{15} + 20 q^{19} + 4 q^{21} + 2 q^{25} + 16 q^{27} + 4 q^{29} - 8 q^{31} + 4 q^{33} + 20 q^{37} - 4 q^{41} + 24 q^{43} + 8 q^{47} + 6 q^{49} + 24 q^{51} + 4 q^{53} + 16 q^{55} - 4 q^{57} + 12 q^{59} - 8 q^{61} + 10 q^{63} - 8 q^{65} + 16 q^{67} - 40 q^{69} - 8 q^{71} + 16 q^{73} + 28 q^{75} + 8 q^{77} - 24 q^{79} + 10 q^{81} + 12 q^{83} - 8 q^{87} + 16 q^{89} + 24 q^{93} - 16 q^{95} + 16 q^{97} + 40 q^{99}+O(q^{100})$$ 6 * q + 4 * q^3 + 6 * q^7 + 10 * q^9 + 8 * q^11 - 8 * q^15 + 20 * q^19 + 4 * q^21 + 2 * q^25 + 16 * q^27 + 4 * q^29 - 8 * q^31 + 4 * q^33 + 20 * q^37 - 4 * q^41 + 24 * q^43 + 8 * q^47 + 6 * q^49 + 24 * q^51 + 4 * q^53 + 16 * q^55 - 4 * q^57 + 12 * q^59 - 8 * q^61 + 10 * q^63 - 8 * q^65 + 16 * q^67 - 40 * q^69 - 8 * q^71 + 16 * q^73 + 28 * q^75 + 8 * q^77 - 24 * q^79 + 10 * q^81 + 12 * q^83 - 8 * q^87 + 16 * q^89 + 24 * q^93 - 16 * q^95 + 16 * q^97 + 40 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} - 9x^{4} + 16x^{3} + 6x^{2} - 12x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{5} + 6\nu^{4} - 18\nu^{3} - 52\nu^{2} + 65\nu + 14 ) / 19$$ (v^5 + 6*v^4 - 18*v^3 - 52*v^2 + 65*v + 14) / 19 $$\beta_{2}$$ $$=$$ $$( -3\nu^{5} + \nu^{4} + 35\nu^{3} + 4\nu^{2} - 81\nu - 23 ) / 19$$ (-3*v^5 + v^4 + 35*v^3 + 4*v^2 - 81*v - 23) / 19 $$\beta_{3}$$ $$=$$ $$( 5\nu^{5} - 8\nu^{4} - 52\nu^{3} + 63\nu^{2} + 78\nu - 44 ) / 19$$ (5*v^5 - 8*v^4 - 52*v^3 + 63*v^2 + 78*v - 44) / 19 $$\beta_{4}$$ $$=$$ $$( -16\nu^{5} + 18\nu^{4} + 155\nu^{3} - 118\nu^{2} - 147\nu + 42 ) / 19$$ (-16*v^5 + 18*v^4 + 155*v^3 - 118*v^2 - 147*v + 42) / 19 $$\beta_{5}$$ $$=$$ $$( 17\nu^{5} - 31\nu^{4} - 154\nu^{3} + 237\nu^{2} + 117\nu - 123 ) / 19$$ (17*v^5 - 31*v^4 - 154*v^3 + 237*v^2 + 117*v - 123) / 19
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 ) / 2$$ (b5 + b4 - b3 - b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + \beta_{4} + \beta_{3} + 3\beta_{2} + 3\beta _1 + 8 ) / 2$$ (b5 + b4 + b3 + 3*b2 + 3*b1 + 8) / 2 $$\nu^{3}$$ $$=$$ $$( 9\beta_{5} + 7\beta_{4} - 13\beta_{3} - 5\beta_{2} + 9\beta_1 ) / 2$$ (9*b5 + 7*b4 - 13*b3 - 5*b2 + 9*b1) / 2 $$\nu^{4}$$ $$=$$ $$( 11\beta_{5} + 9\beta_{4} + \beta_{3} + 27\beta_{2} + 33\beta _1 + 62 ) / 2$$ (11*b5 + 9*b4 + b3 + 27*b2 + 33*b1 + 62) / 2 $$\nu^{5}$$ $$=$$ $$( 83\beta_{5} + 59\beta_{4} - 123\beta_{3} - 31\beta_{2} + 93\beta _1 + 16 ) / 2$$ (83*b5 + 59*b4 - 123*b3 - 31*b2 + 93*b1 + 16) / 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0.752719 0.194171 1.62476 3.17133 −0.951290 −2.79169
0 −2.53584 0 1.47134 0 1.00000 0 3.43049 0
1.2 0 −1.01685 0 1.29145 0 1.00000 0 −1.96601 0
1.3 0 −0.101362 0 −2.19640 0 1.00000 0 −2.98973 0
1.4 0 1.81529 0 2.66965 0 1.00000 0 0.295267 0
1.5 0 2.61578 0 −3.96111 0 1.00000 0 3.84231 0
1.6 0 3.22299 0 0.725063 0 1.00000 0 7.38766 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3584.2.a.l yes 6
4.b odd 2 1 3584.2.a.e 6
8.b even 2 1 3584.2.a.f yes 6
8.d odd 2 1 3584.2.a.k yes 6
16.e even 4 2 3584.2.b.i 12
16.f odd 4 2 3584.2.b.k 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.a.e 6 4.b odd 2 1
3584.2.a.f yes 6 8.b even 2 1
3584.2.a.k yes 6 8.d odd 2 1
3584.2.a.l yes 6 1.a even 1 1 trivial
3584.2.b.i 12 16.e even 4 2
3584.2.b.k 12 16.f odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3584))$$:

 $$T_{3}^{6} - 4T_{3}^{5} - 6T_{3}^{4} + 32T_{3}^{3} - 2T_{3}^{2} - 40T_{3} - 4$$ T3^6 - 4*T3^5 - 6*T3^4 + 32*T3^3 - 2*T3^2 - 40*T3 - 4 $$T_{5}^{6} - 16T_{5}^{4} + 16T_{5}^{3} + 46T_{5}^{2} - 80T_{5} + 32$$ T5^6 - 16*T5^4 + 16*T5^3 + 46*T5^2 - 80*T5 + 32 $$T_{23}^{6} - 76T_{23}^{4} + 96T_{23}^{3} + 1060T_{23}^{2} - 832T_{23} - 2848$$ T23^6 - 76*T23^4 + 96*T23^3 + 1060*T23^2 - 832*T23 - 2848

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} - 4 T^{5} - 6 T^{4} + 32 T^{3} + \cdots - 4$$
$5$ $$T^{6} - 16 T^{4} + 16 T^{3} + 46 T^{2} + \cdots + 32$$
$7$ $$(T - 1)^{6}$$
$11$ $$T^{6} - 8 T^{5} - 2 T^{4} + 128 T^{3} + \cdots + 64$$
$13$ $$T^{6} - 24 T^{4} - 32 T^{3} + 46 T^{2} + \cdots - 32$$
$17$ $$T^{6} - 48 T^{4} + 32 T^{3} + \cdots - 896$$
$19$ $$T^{6} - 20 T^{5} + 130 T^{4} + \cdots - 548$$
$23$ $$T^{6} - 76 T^{4} + 96 T^{3} + \cdots - 2848$$
$29$ $$T^{6} - 4 T^{5} - 76 T^{4} + 128 T^{3} + \cdots + 128$$
$31$ $$T^{6} + 8 T^{5} - 72 T^{4} + \cdots + 11776$$
$37$ $$T^{6} - 20 T^{5} + 116 T^{4} + \cdots + 128$$
$41$ $$T^{6} + 4 T^{5} - 84 T^{4} + \cdots + 5696$$
$43$ $$T^{6} - 24 T^{5} + 142 T^{4} + \cdots - 46016$$
$47$ $$T^{6} - 8 T^{5} - 152 T^{4} + \cdots - 179200$$
$53$ $$T^{6} - 4 T^{5} - 268 T^{4} + \cdots - 48896$$
$59$ $$T^{6} - 12 T^{5} - 110 T^{4} + \cdots + 15548$$
$61$ $$T^{6} + 8 T^{5} - 192 T^{4} + \cdots + 49664$$
$67$ $$T^{6} - 16 T^{5} - 50 T^{4} + \cdots + 4352$$
$71$ $$T^{6} + 8 T^{5} - 248 T^{4} + \cdots - 51200$$
$73$ $$T^{6} - 16 T^{5} - 240 T^{4} + \cdots + 899200$$
$79$ $$T^{6} + 24 T^{5} + 64 T^{4} + \cdots - 21632$$
$83$ $$T^{6} - 12 T^{5} - 246 T^{4} + \cdots - 269732$$
$89$ $$T^{6} - 16 T^{5} - 32 T^{4} + \cdots + 6272$$
$97$ $$T^{6} - 16 T^{5} - 96 T^{4} + \cdots + 232576$$