# Properties

 Label 3584.2.a.p Level $3584$ Weight $2$ Character orbit 3584.a Self dual yes Analytic conductor $28.618$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# 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: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 2x^{9} - 19x^{8} + 44x^{7} + 86x^{6} - 236x^{5} - 58x^{4} + 368x^{3} - 194x^{2} - 12x + 18$$ x^10 - 2*x^9 - 19*x^8 + 44*x^7 + 86*x^6 - 236*x^5 - 58*x^4 + 368*x^3 - 194*x^2 - 12*x + 18 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{8}$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{3} + \beta_{3} q^{5} + q^{7} + (\beta_{6} + 2) q^{9}+O(q^{10})$$ q - b5 * q^3 + b3 * q^5 + q^7 + (b6 + 2) * q^9 $$q - \beta_{5} q^{3} + \beta_{3} q^{5} + q^{7} + (\beta_{6} + 2) q^{9} - \beta_{7} q^{11} + \beta_{8} q^{13} + (\beta_{9} + \beta_{6} + \beta_{2} + 1) q^{15} + (\beta_{4} + 2) q^{17} + (\beta_{8} - \beta_{5} - \beta_{3} + \beta_1) q^{19} - \beta_{5} q^{21} + ( - \beta_{2} + 1) q^{23} + (\beta_{9} + \beta_{6} - \beta_{4} + \beta_{2} + 2) q^{25} + ( - \beta_{8} - 2 \beta_{5} + \beta_{3} - \beta_1) q^{27} + ( - \beta_{8} - \beta_{7} + \beta_{5} + 2 \beta_{3}) q^{29} + ( - \beta_{9} - \beta_{6} + \beta_{4}) q^{31} + ( - \beta_{9} - \beta_{6} - 2 \beta_{2} + 2) q^{33} + \beta_{3} q^{35} + (\beta_{8} + \beta_{7} + \beta_{5} + \beta_1) q^{37} + ( - 2 \beta_{4} + \beta_{2} - 1) q^{39} + ( - \beta_{9} + \beta_{2} + 2) q^{41} + (\beta_{7} - 2 \beta_{5} - 2 \beta_{3} - \beta_1) q^{43} + ( - \beta_{8} + 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{45} + ( - \beta_{9} - \beta_{2} - 2) q^{47} + q^{49} + ( - 2 \beta_{8} - 2 \beta_{5} + 2 \beta_{3} + \beta_1) q^{51} + ( - \beta_{8} - \beta_{7} + 3 \beta_{5} + 2 \beta_{3} - \beta_1) q^{53} + ( - 3 \beta_{6} + \beta_{4} - \beta_{2} - 2) q^{55} + ( - \beta_{9} - \beta_{6} - \beta_{4} - \beta_{2} + 3) q^{57} + (\beta_{8} - \beta_{5} - \beta_{3} - \beta_1) q^{59} + ( - \beta_{3} + 2 \beta_1) q^{61} + (\beta_{6} + 2) q^{63} + (2 \beta_{9} - \beta_{6} + 3) q^{65} + (\beta_{7} - 2 \beta_{5} + \beta_1) q^{67} + ( - \beta_{8} - 2 \beta_{7} - 2 \beta_{5} - \beta_{3} + 2 \beta_1) q^{69} + ( - \beta_{9} + \beta_{6}) q^{71} + (\beta_{9} - \beta_{4} + \beta_{2} + 4) q^{73} + (\beta_{8} + 2 \beta_{7} - 3 \beta_{5} + 3 \beta_{3} - 3 \beta_1) q^{75} - \beta_{7} q^{77} + ( - \beta_{9} + \beta_{6} + 4) q^{79} + (\beta_{9} + \beta_{6} + \beta_{4} + \beta_{2} + 6) q^{81} + ( - \beta_{5} + 2 \beta_1) q^{83} + ( - 2 \beta_{5} - 3 \beta_1) q^{85} + (\beta_{9} + 2 \beta_{4} - \beta_{2}) q^{87} + ( - \beta_{9} - \beta_{6} + 2) q^{89} + \beta_{8} q^{91} + (2 \beta_{5} - 2 \beta_{3} + \beta_1) q^{93} + (2 \beta_{9} - 2 \beta_{6} - \beta_{2} - 3) q^{95} + ( - 3 \beta_{6} - \beta_{2} + 4) q^{97} + ( - \beta_{7} - 2 \beta_{5} - 6 \beta_{3} + 4 \beta_1) q^{99}+O(q^{100})$$ q - b5 * q^3 + b3 * q^5 + q^7 + (b6 + 2) * q^9 - b7 * q^11 + b8 * q^13 + (b9 + b6 + b2 + 1) * q^15 + (b4 + 2) * q^17 + (b8 - b5 - b3 + b1) * q^19 - b5 * q^21 + (-b2 + 1) * q^23 + (b9 + b6 - b4 + b2 + 2) * q^25 + (-b8 - 2*b5 + b3 - b1) * q^27 + (-b8 - b7 + b5 + 2*b3) * q^29 + (-b9 - b6 + b4) * q^31 + (-b9 - b6 - 2*b2 + 2) * q^33 + b3 * q^35 + (b8 + b7 + b5 + b1) * q^37 + (-2*b4 + b2 - 1) * q^39 + (-b9 + b2 + 2) * q^41 + (b7 - 2*b5 - 2*b3 - b1) * q^43 + (-b8 + 2*b7 - 2*b5 + 2*b3 - 2*b1) * q^45 + (-b9 - b2 - 2) * q^47 + q^49 + (-2*b8 - 2*b5 + 2*b3 + b1) * q^51 + (-b8 - b7 + 3*b5 + 2*b3 - b1) * q^53 + (-3*b6 + b4 - b2 - 2) * q^55 + (-b9 - b6 - b4 - b2 + 3) * q^57 + (b8 - b5 - b3 - b1) * q^59 + (-b3 + 2*b1) * q^61 + (b6 + 2) * q^63 + (2*b9 - b6 + 3) * q^65 + (b7 - 2*b5 + b1) * q^67 + (-b8 - 2*b7 - 2*b5 - b3 + 2*b1) * q^69 + (-b9 + b6) * q^71 + (b9 - b4 + b2 + 4) * q^73 + (b8 + 2*b7 - 3*b5 + 3*b3 - 3*b1) * q^75 - b7 * q^77 + (-b9 + b6 + 4) * q^79 + (b9 + b6 + b4 + b2 + 6) * q^81 + (-b5 + 2*b1) * q^83 + (-2*b5 - 3*b1) * q^85 + (b9 + 2*b4 - b2) * q^87 + (-b9 - b6 + 2) * q^89 + b8 * q^91 + (2*b5 - 2*b3 + b1) * q^93 + (2*b9 - 2*b6 - b2 - 3) * q^95 + (-3*b6 - b2 + 4) * q^97 + (-b7 - 2*b5 - 6*b3 + 4*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + 10 q^{7} + 22 q^{9}+O(q^{10})$$ 10 * q + 10 * q^7 + 22 * q^9 $$10 q + 10 q^{7} + 22 q^{9} + 16 q^{15} + 16 q^{17} + 8 q^{23} + 30 q^{25} - 8 q^{31} + 12 q^{33} + 20 q^{41} - 24 q^{47} + 10 q^{49} - 32 q^{55} + 28 q^{57} + 22 q^{63} + 32 q^{65} + 48 q^{73} + 40 q^{79} + 62 q^{81} - 8 q^{87} + 16 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100})$$ 10 * q + 10 * q^7 + 22 * q^9 + 16 * q^15 + 16 * q^17 + 8 * q^23 + 30 * q^25 - 8 * q^31 + 12 * q^33 + 20 * q^41 - 24 * q^47 + 10 * q^49 - 32 * q^55 + 28 * q^57 + 22 * q^63 + 32 * q^65 + 48 * q^73 + 40 * q^79 + 62 * q^81 - 8 * q^87 + 16 * q^89 - 32 * q^95 + 32 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2x^{9} - 19x^{8} + 44x^{7} + 86x^{6} - 236x^{5} - 58x^{4} + 368x^{3} - 194x^{2} - 12x + 18$$ :

 $$\beta_{1}$$ $$=$$ $$( - 365 \nu^{9} + 475 \nu^{8} + 7148 \nu^{7} - 11086 \nu^{6} - 37174 \nu^{5} + 59872 \nu^{4} + 55808 \nu^{3} - 95212 \nu^{2} + 9046 \nu + 12126 ) / 723$$ (-365*v^9 + 475*v^8 + 7148*v^7 - 11086*v^6 - 37174*v^5 + 59872*v^4 + 55808*v^3 - 95212*v^2 + 9046*v + 12126) / 723 $$\beta_{2}$$ $$=$$ $$( - 270 \nu^{9} + 424 \nu^{8} + 5380 \nu^{7} - 9399 \nu^{6} - 28522 \nu^{5} + 49165 \nu^{4} + 44102 \nu^{3} - 73818 \nu^{2} + 8712 \nu + 7963 ) / 241$$ (-270*v^9 + 424*v^8 + 5380*v^7 - 9399*v^6 - 28522*v^5 + 49165*v^4 + 44102*v^3 - 73818*v^2 + 8712*v + 7963) / 241 $$\beta_{3}$$ $$=$$ $$( - 821 \nu^{9} + 1009 \nu^{8} + 16502 \nu^{7} - 23377 \nu^{6} - 90850 \nu^{5} + 124135 \nu^{4} + 153497 \nu^{3} - 186769 \nu^{2} + 3130 \nu + 18516 ) / 723$$ (-821*v^9 + 1009*v^8 + 16502*v^7 - 23377*v^6 - 90850*v^5 + 124135*v^4 + 153497*v^3 - 186769*v^2 + 3130*v + 18516) / 723 $$\beta_{4}$$ $$=$$ $$( 910 \nu^{9} - 1072 \nu^{8} - 18204 \nu^{7} + 25064 \nu^{6} + 99468 \nu^{5} - 133080 \nu^{4} - 165278 \nu^{3} + 200594 \nu^{2} - 7512 \nu - 20724 ) / 241$$ (910*v^9 - 1072*v^8 - 18204*v^7 + 25064*v^6 + 99468*v^5 - 133080*v^4 - 165278*v^3 + 200594*v^2 - 7512*v - 20724) / 241 $$\beta_{5}$$ $$=$$ $$( 5947 \nu^{9} - 7145 \nu^{8} - 118678 \nu^{7} + 166772 \nu^{6} + 644474 \nu^{5} - 886628 \nu^{4} - 1055350 \nu^{3} + 1338686 \nu^{2} - 75278 \nu - 129066 ) / 1446$$ (5947*v^9 - 7145*v^8 - 118678*v^7 + 166772*v^6 + 644474*v^5 - 886628*v^4 - 1055350*v^3 + 1338686*v^2 - 75278*v - 129066) / 1446 $$\beta_{6}$$ $$=$$ $$( - 1938 \nu^{9} + 2390 \nu^{8} + 38670 \nu^{7} - 55671 \nu^{6} - 209566 \nu^{5} + 297519 \nu^{4} + 340408 \nu^{3} - 454368 \nu^{2} + 30528 \nu + 46317 ) / 241$$ (-1938*v^9 + 2390*v^8 + 38670*v^7 - 55671*v^6 - 209566*v^5 + 297519*v^4 + 340408*v^3 - 454368*v^2 + 30528*v + 46317) / 241 $$\beta_{7}$$ $$=$$ $$( 4723 \nu^{9} - 5737 \nu^{8} - 94128 \nu^{7} + 133996 \nu^{6} + 509326 \nu^{5} - 715642 \nu^{4} - 825312 \nu^{3} + 1091672 \nu^{2} - 74922 \nu - 109098 ) / 482$$ (4723*v^9 - 5737*v^8 - 94128*v^7 + 133996*v^6 + 509326*v^5 - 715642*v^4 - 825312*v^3 + 1091672*v^2 - 74922*v - 109098) / 482 $$\beta_{8}$$ $$=$$ $$( - 3738 \nu^{9} + 4574 \nu^{8} + 74617 \nu^{7} - 106522 \nu^{6} - 404854 \nu^{5} + 567767 \nu^{4} + 659887 \nu^{3} - 862379 \nu^{2} + 54386 \nu + 85506 ) / 241$$ (-3738*v^9 + 4574*v^8 + 74617*v^7 - 106522*v^6 - 404854*v^5 + 567767*v^4 + 659887*v^3 - 862379*v^2 + 54386*v + 85506) / 241 $$\beta_{9}$$ $$=$$ $$( 4952 \nu^{9} - 6002 \nu^{8} - 98816 \nu^{7} + 140021 \nu^{6} + 536250 \nu^{5} - 746099 \nu^{4} - 876072 \nu^{3} + 1132928 \nu^{2} - 65184 \nu - 114105 ) / 241$$ (4952*v^9 - 6002*v^8 - 98816*v^7 + 140021*v^6 + 536250*v^5 - 746099*v^4 - 876072*v^3 + 1132928*v^2 - 65184*v - 114105) / 241
 $$\nu$$ $$=$$ $$( \beta_{9} + \beta_{6} - 2\beta_{5} - \beta_{4} + \beta_1 ) / 4$$ (b9 + b6 - 2*b5 - b4 + b1) / 4 $$\nu^{2}$$ $$=$$ $$( -2\beta_{7} - 2\beta_{6} + 2\beta_{4} + 4\beta_{3} - \beta _1 + 18 ) / 4$$ (-2*b7 - 2*b6 + 2*b4 + 4*b3 - b1 + 18) / 4 $$\nu^{3}$$ $$=$$ $$( 8 \beta_{9} + 4 \beta_{8} + 6 \beta_{7} + 8 \beta_{6} - 16 \beta_{5} - 8 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 15 \beta _1 - 10 ) / 4$$ (8*b9 + 4*b8 + 6*b7 + 8*b6 - 16*b5 - 8*b4 - 4*b3 - 2*b2 + 15*b1 - 10) / 4 $$\nu^{4}$$ $$=$$ $$( - 2 \beta_{9} - 3 \beta_{8} - 17 \beta_{7} - 14 \beta_{6} + 9 \beta_{5} + 10 \beta_{4} + 26 \beta_{3} + 2 \beta_{2} - 11 \beta _1 + 76 ) / 2$$ (-2*b9 - 3*b8 - 17*b7 - 14*b6 + 9*b5 + 10*b4 + 26*b3 + 2*b2 - 11*b1 + 76) / 2 $$\nu^{5}$$ $$=$$ $$( 82 \beta_{9} + 66 \beta_{8} + 104 \beta_{7} + 91 \beta_{6} - 160 \beta_{5} - 91 \beta_{4} - 98 \beta_{3} - 31 \beta_{2} + 184 \beta _1 - 200 ) / 4$$ (82*b9 + 66*b8 + 104*b7 + 91*b6 - 160*b5 - 91*b4 - 98*b3 - 31*b2 + 184*b1 - 200) / 4 $$\nu^{6}$$ $$=$$ $$( - 48 \beta_{9} - 59 \beta_{8} - 235 \beta_{7} - 189 \beta_{6} + 171 \beta_{5} + 123 \beta_{4} + 320 \beta_{3} + 39 \beta_{2} - 185 \beta _1 + 798 ) / 2$$ (-48*b9 - 59*b8 - 235*b7 - 189*b6 + 171*b5 + 123*b4 + 320*b3 + 39*b2 - 185*b1 + 798) / 2 $$\nu^{7}$$ $$=$$ $$( 465 \beta_{9} + 441 \beta_{8} + 757 \beta_{7} + 593 \beta_{6} - 919 \beta_{5} - 569 \beta_{4} - 796 \beta_{3} - 208 \beta_{2} + 1134 \beta _1 - 1646 ) / 2$$ (465*b9 + 441*b8 + 757*b7 + 593*b6 - 919*b5 - 569*b4 - 796*b3 - 208*b2 + 1134*b1 - 1646) / 2 $$\nu^{8}$$ $$=$$ $$( - 841 \beta_{9} - 946 \beta_{8} - 3130 \beta_{7} - 2508 \beta_{6} + 2598 \beta_{5} + 1657 \beta_{4} + 4028 \beta_{3} + 601 \beta_{2} - 2804 \beta _1 + 9392 ) / 2$$ (-841*b9 - 946*b8 - 3130*b7 - 2508*b6 + 2598*b5 + 1657*b4 + 4028*b3 + 601*b2 - 2804*b1 + 9392) / 2 $$\nu^{9}$$ $$=$$ $$( 5590 \beta_{9} + 5650 \beta_{8} + 10524 \beta_{7} + 8044 \beta_{6} - 11434 \beta_{5} - 7333 \beta_{4} - 11638 \beta_{3} - 2722 \beta_{2} + 14289 \beta _1 - 24644 ) / 2$$ (5590*b9 + 5650*b8 + 10524*b7 + 8044*b6 - 11434*b5 - 7333*b4 - 11638*b3 - 2722*b2 + 14289*b1 - 24644) / 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
 0.508920 −2.00182 −0.269488 0.776589 −3.66546 −1.79086 2.97399 0.791624 2.40586 2.27064
0 −3.17593 0 −4.23755 0 1.00000 0 7.08655 0
1.2 0 −2.99347 0 0.369776 0 1.00000 0 5.96088 0
1.3 0 −2.47533 0 2.59708 0 1.00000 0 3.12724 0
1.4 0 −0.783186 0 −3.56843 0 1.00000 0 −2.38662 0
1.5 0 −0.460386 0 1.55819 0 1.00000 0 −2.78804 0
1.6 0 0.460386 0 −1.55819 0 1.00000 0 −2.78804 0
1.7 0 0.783186 0 3.56843 0 1.00000 0 −2.38662 0
1.8 0 2.47533 0 −2.59708 0 1.00000 0 3.12724 0
1.9 0 2.99347 0 −0.369776 0 1.00000 0 5.96088 0
1.10 0 3.17593 0 4.23755 0 1.00000 0 7.08655 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3584.2.a.p yes 10
4.b odd 2 1 3584.2.a.o 10
8.b even 2 1 inner 3584.2.a.p yes 10
8.d odd 2 1 3584.2.a.o 10
16.e even 4 2 3584.2.b.g 10
16.f odd 4 2 3584.2.b.h 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.a.o 10 4.b odd 2 1
3584.2.a.o 10 8.d odd 2 1
3584.2.a.p yes 10 1.a even 1 1 trivial
3584.2.a.p yes 10 8.b even 2 1 inner
3584.2.b.g 10 16.e even 4 2
3584.2.b.h 10 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3584))$$:

 $$T_{3}^{10} - 26T_{3}^{8} + 228T_{3}^{6} - 728T_{3}^{4} + 484T_{3}^{2} - 72$$ T3^10 - 26*T3^8 + 228*T3^6 - 728*T3^4 + 484*T3^2 - 72 $$T_{5}^{10} - 40T_{5}^{8} + 532T_{5}^{6} - 2672T_{5}^{4} + 4100T_{5}^{2} - 512$$ T5^10 - 40*T5^8 + 532*T5^6 - 2672*T5^4 + 4100*T5^2 - 512 $$T_{23}^{5} - 4T_{23}^{4} - 60T_{23}^{3} + 112T_{23}^{2} + 644T_{23} - 1296$$ T23^5 - 4*T23^4 - 60*T23^3 + 112*T23^2 + 644*T23 - 1296

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$T^{10} - 26 T^{8} + 228 T^{6} + \cdots - 72$$
$5$ $$T^{10} - 40 T^{8} + 532 T^{6} + \cdots - 512$$
$7$ $$(T - 1)^{10}$$
$11$ $$T^{10} - 98 T^{8} + 3648 T^{6} + \cdots - 1492992$$
$13$ $$T^{10} - 104 T^{8} + 3540 T^{6} + \cdots - 4608$$
$17$ $$(T^{5} - 8 T^{4} - 24 T^{3} + 256 T^{2} + \cdots - 1152)^{2}$$
$19$ $$T^{10} - 122 T^{8} + 5028 T^{6} + \cdots - 5832$$
$23$ $$(T^{5} - 4 T^{4} - 60 T^{3} + 112 T^{2} + \cdots - 1296)^{2}$$
$29$ $$T^{10} - 184 T^{8} + 11296 T^{6} + \cdots - 18432$$
$31$ $$(T^{5} + 4 T^{4} - 96 T^{3} - 640 T^{2} + \cdots - 384)^{2}$$
$37$ $$T^{10} - 216 T^{8} + 15648 T^{6} + \cdots - 1492992$$
$41$ $$(T^{5} - 10 T^{4} - 120 T^{3} + \cdots - 28128)^{2}$$
$43$ $$T^{10} - 274 T^{8} + 25984 T^{6} + \cdots - 18432$$
$47$ $$(T^{5} + 12 T^{4} - 48 T^{3} - 768 T^{2} + \cdots + 10368)^{2}$$
$53$ $$T^{10} - 280 T^{8} + 22240 T^{6} + \cdots - 5752832$$
$59$ $$T^{10} - 202 T^{8} + 6628 T^{6} + \cdots - 52488$$
$61$ $$T^{10} - 200 T^{8} + 13140 T^{6} + \cdots - 7558272$$
$67$ $$T^{10} - 194 T^{8} + 11904 T^{6} + \cdots - 2654208$$
$71$ $$(T^{5} - 120 T^{3} + 416 T^{2} + \cdots - 1152)^{2}$$
$73$ $$(T^{5} - 24 T^{4} + 104 T^{3} + \cdots + 10496)^{2}$$
$79$ $$(T^{5} - 20 T^{4} + 40 T^{3} + 1216 T^{2} + \cdots + 11584)^{2}$$
$83$ $$T^{10} - 170 T^{8} + 10020 T^{6} + \cdots - 7235208$$
$89$ $$(T^{5} - 8 T^{4} - 72 T^{3} + 288 T^{2} + \cdots + 2304)^{2}$$
$97$ $$(T^{5} - 16 T^{4} - 264 T^{3} + 4288 T^{2} + \cdots - 1536)^{2}$$