Properties

 Label 4096.2.a.s Level $4096$ Weight $2$ Character orbit 4096.a Self dual yes Analytic conductor $32.707$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$4096 = 2^{12}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4096.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$32.7067246679$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{48})^+$$ Defining polynomial: $$x^{8} - 8x^{6} + 20x^{4} - 16x^{2} + 1$$ x^8 - 8*x^6 + 20*x^4 - 16*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1024) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 1) q^{3} + ( - \beta_{4} - \beta_{2}) q^{5} + (\beta_{7} - \beta_{5} - \beta_{2}) q^{7} + (\beta_{3} + 2 \beta_1) q^{9}+O(q^{10})$$ q + (b1 + 1) * q^3 + (-b4 - b2) * q^5 + (b7 - b5 - b2) * q^7 + (b3 + 2*b1) * q^9 $$q + (\beta_1 + 1) q^{3} + ( - \beta_{4} - \beta_{2}) q^{5} + (\beta_{7} - \beta_{5} - \beta_{2}) q^{7} + (\beta_{3} + 2 \beta_1) q^{9} + ( - 2 \beta_{6} + \beta_1 + 1) q^{11} + (2 \beta_{7} + \beta_{5} + \beta_{2}) q^{13} + ( - \beta_{7} - \beta_{4} - \beta_{2}) q^{15} + (\beta_{6} - \beta_{3} + 3 \beta_1 + 1) q^{17} + (3 \beta_{6} - \beta_{3} + 2 \beta_1) q^{19} + ( - \beta_{7} - 2 \beta_{5} + 2 \beta_{4} - 3 \beta_{2}) q^{21} + (\beta_{7} + 3 \beta_{5} + \beta_{2}) q^{23} + ( - 3 \beta_{6} - 2 \beta_{3} - \beta_1 - 1) q^{25} + (\beta_{6} + 3 \beta_{3} + \beta_1 + 1) q^{27} + (\beta_{7} - 3 \beta_{5} - 2 \beta_{4} - 2 \beta_{2}) q^{29} + (2 \beta_{7} - 4 \beta_{4} - 4 \beta_{2}) q^{31} + ( - 2 \beta_{6} + \beta_{3} + 2 \beta_1 + 5) q^{33} + (\beta_{6} + \beta_{3} - \beta_1 - 1) q^{35} + ( - 2 \beta_{7} - \beta_{5} + 2 \beta_{4} + 3 \beta_{2}) q^{37} + (\beta_{7} + 2 \beta_{5} + \beta_{4} + 3 \beta_{2}) q^{39} + ( - \beta_{6} - 2 \beta_{3} - \beta_1 + 4) q^{41} + ( - 2 \beta_{6} - 3 \beta_1 + 7) q^{43} + ( - \beta_{7} + \beta_{4} + 2 \beta_{2}) q^{45} + ( - 2 \beta_{7} - \beta_{5} - \beta_{4} + 4 \beta_{2}) q^{47} + (4 \beta_{3} + 2 \beta_1 + 1) q^{49} + (2 \beta_{3} + 2 \beta_1 + 6) q^{51} + (3 \beta_{7} - 2 \beta_{5} - \beta_{4} - 4 \beta_{2}) q^{53} + ( - 5 \beta_{7} - 2 \beta_{5} - 3 \beta_{4} - 3 \beta_{2}) q^{55} + (2 \beta_{6} + \beta_{3} + 1) q^{57} + ( - 3 \beta_{6} - \beta_{3} - 6 \beta_1 + 4) q^{59} + (3 \beta_{7} + 4 \beta_{5} - \beta_{4} + 2 \beta_{2}) q^{61} + ( - 3 \beta_{7} - 4 \beta_{5} + 3 \beta_{4} - 7 \beta_{2}) q^{63} + (5 \beta_{6} + 2 \beta_{3} + \beta_1 - 2) q^{65} + (2 \beta_{6} - 2 \beta_{3} + 5 \beta_1 + 3) q^{67} + (3 \beta_{7} + 4 \beta_{5} - 2 \beta_{4} + 5 \beta_{2}) q^{69} + (5 \beta_{7} + \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{71} + ( - 4 \beta_{6} - 3 \beta_{3} - 1) q^{73} + ( - 5 \beta_{6} - 3 \beta_{3} - 6 \beta_1) q^{75} + ( - \beta_{7} - 2 \beta_{5} + 4 \beta_{4} - 3 \beta_{2}) q^{77} + ( - 2 \beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_{2}) q^{79} + (4 \beta_{6} + \beta_{3} + 2 \beta_1 + 2) q^{81} + (\beta_{6} - 3 \beta_{3} - 2 \beta_1 + 6) q^{83} + ( - 2 \beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_{2}) q^{85} + ( - 5 \beta_{7} - 3 \beta_{5} + 2 \beta_{4} - 5 \beta_{2}) q^{87} + (2 \beta_{6} - 3 \beta_{3} + 2 \beta_1 - 1) q^{89} + ( - 3 \beta_{6} - \beta_{3} - 5 \beta_1 + 1) q^{91} + ( - 4 \beta_{7} - 2 \beta_{4} - 4 \beta_{2}) q^{93} + (3 \beta_{7} + 3 \beta_{5} + 2 \beta_{4} + \beta_{2}) q^{95} + (5 \beta_{6} + 3 \beta_{3} + 5 \beta_1 + 1) q^{97} + (5 \beta_{6} + 3 \beta_{3} + 6 \beta_1 + 8) q^{99}+O(q^{100})$$ q + (b1 + 1) * q^3 + (-b4 - b2) * q^5 + (b7 - b5 - b2) * q^7 + (b3 + 2*b1) * q^9 + (-2*b6 + b1 + 1) * q^11 + (2*b7 + b5 + b2) * q^13 + (-b7 - b4 - b2) * q^15 + (b6 - b3 + 3*b1 + 1) * q^17 + (3*b6 - b3 + 2*b1) * q^19 + (-b7 - 2*b5 + 2*b4 - 3*b2) * q^21 + (b7 + 3*b5 + b2) * q^23 + (-3*b6 - 2*b3 - b1 - 1) * q^25 + (b6 + 3*b3 + b1 + 1) * q^27 + (b7 - 3*b5 - 2*b4 - 2*b2) * q^29 + (2*b7 - 4*b4 - 4*b2) * q^31 + (-2*b6 + b3 + 2*b1 + 5) * q^33 + (b6 + b3 - b1 - 1) * q^35 + (-2*b7 - b5 + 2*b4 + 3*b2) * q^37 + (b7 + 2*b5 + b4 + 3*b2) * q^39 + (-b6 - 2*b3 - b1 + 4) * q^41 + (-2*b6 - 3*b1 + 7) * q^43 + (-b7 + b4 + 2*b2) * q^45 + (-2*b7 - b5 - b4 + 4*b2) * q^47 + (4*b3 + 2*b1 + 1) * q^49 + (2*b3 + 2*b1 + 6) * q^51 + (3*b7 - 2*b5 - b4 - 4*b2) * q^53 + (-5*b7 - 2*b5 - 3*b4 - 3*b2) * q^55 + (2*b6 + b3 + 1) * q^57 + (-3*b6 - b3 - 6*b1 + 4) * q^59 + (3*b7 + 4*b5 - b4 + 2*b2) * q^61 + (-3*b7 - 4*b5 + 3*b4 - 7*b2) * q^63 + (5*b6 + 2*b3 + b1 - 2) * q^65 + (2*b6 - 2*b3 + 5*b1 + 3) * q^67 + (3*b7 + 4*b5 - 2*b4 + 5*b2) * q^69 + (5*b7 + b5 - 2*b4 + b2) * q^71 + (-4*b6 - 3*b3 - 1) * q^73 + (-5*b6 - 3*b3 - 6*b1) * q^75 + (-b7 - 2*b5 + 4*b4 - 3*b2) * q^77 + (-2*b7 - b5 - b4 + 2*b2) * q^79 + (4*b6 + b3 + 2*b1 + 2) * q^81 + (b6 - 3*b3 - 2*b1 + 6) * q^83 + (-2*b7 + b5 - b4 - 2*b2) * q^85 + (-5*b7 - 3*b5 + 2*b4 - 5*b2) * q^87 + (2*b6 - 3*b3 + 2*b1 - 1) * q^89 + (-3*b6 - b3 - 5*b1 + 1) * q^91 + (-4*b7 - 2*b4 - 4*b2) * q^93 + (3*b7 + 3*b5 + 2*b4 + b2) * q^95 + (5*b6 + 3*b3 + 5*b1 + 1) * q^97 + (5*b6 + 3*b3 + 6*b1 + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{3}+O(q^{10})$$ 8 * q + 8 * q^3 $$8 q + 8 q^{3} + 8 q^{11} + 8 q^{17} - 8 q^{25} + 8 q^{27} + 40 q^{33} - 8 q^{35} + 32 q^{41} + 56 q^{43} + 8 q^{49} + 48 q^{51} + 8 q^{57} + 32 q^{59} - 16 q^{65} + 24 q^{67} - 8 q^{73} + 16 q^{81} + 48 q^{83} - 8 q^{89} + 8 q^{91} + 8 q^{97} + 64 q^{99}+O(q^{100})$$ 8 * q + 8 * q^3 + 8 * q^11 + 8 * q^17 - 8 * q^25 + 8 * q^27 + 40 * q^33 - 8 * q^35 + 32 * q^41 + 56 * q^43 + 8 * q^49 + 48 * q^51 + 8 * q^57 + 32 * q^59 - 16 * q^65 + 24 * q^67 - 8 * q^73 + 16 * q^81 + 48 * q^83 - 8 * q^89 + 8 * q^91 + 8 * q^97 + 64 * q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{48} + \zeta_{48}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{2}$$ $$=$$ $$\nu^{3} - 3\nu$$ v^3 - 3*v $$\beta_{3}$$ $$=$$ $$\nu^{4} - 4\nu^{2} + 2$$ v^4 - 4*v^2 + 2 $$\beta_{4}$$ $$=$$ $$\nu^{5} - 6\nu^{3} + 7\nu$$ v^5 - 6*v^3 + 7*v $$\beta_{5}$$ $$=$$ $$\nu^{5} - 6\nu^{3} + 9\nu$$ v^5 - 6*v^3 + 9*v $$\beta_{6}$$ $$=$$ $$\nu^{6} - 6\nu^{4} + 8\nu^{2}$$ v^6 - 6*v^4 + 8*v^2 $$\beta_{7}$$ $$=$$ $$\nu^{7} - 7\nu^{5} + 14\nu^{3} - 8\nu$$ v^7 - 7*v^5 + 14*v^3 - 8*v
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} ) / 2$$ (b5 - b4) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 + 2$$ b1 + 2 $$\nu^{3}$$ $$=$$ $$( 3\beta_{5} - 3\beta_{4} + 2\beta_{2} ) / 2$$ (3*b5 - 3*b4 + 2*b2) / 2 $$\nu^{4}$$ $$=$$ $$\beta_{3} + 4\beta _1 + 6$$ b3 + 4*b1 + 6 $$\nu^{5}$$ $$=$$ $$( 11\beta_{5} - 9\beta_{4} + 12\beta_{2} ) / 2$$ (11*b5 - 9*b4 + 12*b2) / 2 $$\nu^{6}$$ $$=$$ $$\beta_{6} + 6\beta_{3} + 16\beta _1 + 20$$ b6 + 6*b3 + 16*b1 + 20 $$\nu^{7}$$ $$=$$ $$( 2\beta_{7} + 43\beta_{5} - 29\beta_{4} + 56\beta_{2} ) / 2$$ (2*b7 + 43*b5 - 29*b4 + 56*b2) / 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
 0.261052 −0.261052 −1.21752 1.21752 −1.58671 1.58671 −1.98289 1.98289
0 −0.931852 0 −0.956470 0 −3.32633 0 −2.13165 0
1.2 0 −0.931852 0 0.956470 0 3.32633 0 −2.13165 0
1.3 0 0.482362 0 −1.47858 0 0.191104 0 −2.76733 0
1.4 0 0.482362 0 1.47858 0 −0.191104 0 −2.76733 0
1.5 0 1.51764 0 −3.56960 0 1.45158 0 −0.696775 0
1.6 0 1.51764 0 3.56960 0 −1.45158 0 −0.696775 0
1.7 0 2.93185 0 −0.396183 0 4.33496 0 5.59575 0
1.8 0 2.93185 0 0.396183 0 −4.33496 0 5.59575 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

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

Inner twists

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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4096.2.a.s 8
4.b odd 2 1 4096.2.a.i 8
8.b even 2 1 4096.2.a.i 8
8.d odd 2 1 inner 4096.2.a.s 8
64.i even 16 2 1024.2.g.a 16
64.i even 16 2 1024.2.g.d yes 16
64.i even 16 2 1024.2.g.f yes 16
64.i even 16 2 1024.2.g.g yes 16
64.j odd 16 2 1024.2.g.a 16
64.j odd 16 2 1024.2.g.d yes 16
64.j odd 16 2 1024.2.g.f yes 16
64.j odd 16 2 1024.2.g.g yes 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1024.2.g.a 16 64.i even 16 2
1024.2.g.a 16 64.j odd 16 2
1024.2.g.d yes 16 64.i even 16 2
1024.2.g.d yes 16 64.j odd 16 2
1024.2.g.f yes 16 64.i even 16 2
1024.2.g.f yes 16 64.j odd 16 2
1024.2.g.g yes 16 64.i even 16 2
1024.2.g.g yes 16 64.j odd 16 2
4096.2.a.i 8 4.b odd 2 1
4096.2.a.i 8 8.b even 2 1
4096.2.a.s 8 1.a even 1 1 trivial
4096.2.a.s 8 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4096))$$:

 $$T_{3}^{4} - 4T_{3}^{3} + 2T_{3}^{2} + 4T_{3} - 2$$ T3^4 - 4*T3^3 + 2*T3^2 + 4*T3 - 2 $$T_{5}^{8} - 16T_{5}^{6} + 44T_{5}^{4} - 32T_{5}^{2} + 4$$ T5^8 - 16*T5^6 + 44*T5^4 - 32*T5^2 + 4 $$T_{7}^{8} - 32T_{7}^{6} + 272T_{7}^{4} - 448T_{7}^{2} + 16$$ T7^8 - 32*T7^6 + 272*T7^4 - 448*T7^2 + 16 $$T_{23}^{8} - 112T_{23}^{6} + 3920T_{23}^{4} - 43904T_{23}^{2} + 80656$$ T23^8 - 112*T23^6 + 3920*T23^4 - 43904*T23^2 + 80656

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} - 4 T^{3} + 2 T^{2} + 4 T - 2)^{2}$$
$5$ $$T^{8} - 16 T^{6} + 44 T^{4} - 32 T^{2} + \cdots + 4$$
$7$ $$T^{8} - 32 T^{6} + 272 T^{4} + \cdots + 16$$
$11$ $$(T^{4} - 4 T^{3} - 22 T^{2} + 52 T + 142)^{2}$$
$13$ $$T^{8} - 32 T^{6} + 332 T^{4} + \cdots + 2116$$
$17$ $$(T^{4} - 4 T^{3} - 28 T^{2} + 160 T - 200)^{2}$$
$19$ $$(T^{4} - 34 T^{2} - 60 T + 46)^{2}$$
$23$ $$T^{8} - 112 T^{6} + 3920 T^{4} + \cdots + 80656$$
$29$ $$T^{8} - 176 T^{6} + 8684 T^{4} + \cdots + 1119364$$
$31$ $$T^{8} - 224 T^{6} + 15680 T^{4} + \cdots + 1024$$
$37$ $$T^{8} - 112 T^{6} + 2540 T^{4} + \cdots + 8836$$
$41$ $$(T^{4} - 16 T^{3} + 68 T^{2} - 32 T - 92)^{2}$$
$43$ $$(T^{4} - 28 T^{3} + 266 T^{2} - 980 T + 1150)^{2}$$
$47$ $$T^{8} - 256 T^{6} + 17984 T^{4} + \cdots + 1364224$$
$53$ $$T^{8} - 208 T^{6} + 11660 T^{4} + \cdots + 21316$$
$59$ $$(T^{4} - 16 T^{3} - 18 T^{2} + 980 T - 2450)^{2}$$
$61$ $$T^{8} - 208 T^{6} + 11468 T^{4} + \cdots + 2500$$
$67$ $$(T^{4} - 12 T^{3} - 46 T^{2} + 996 T - 2978)^{2}$$
$71$ $$T^{8} - 176 T^{6} + 7184 T^{4} + \cdots + 595984$$
$73$ $$(T^{4} + 4 T^{3} - 112 T^{2} - 808 T - 1436)^{2}$$
$79$ $$T^{8} - 128 T^{6} + 4928 T^{4} + \cdots + 160000$$
$83$ $$(T^{4} - 24 T^{3} + 134 T^{2} + 228 T - 2162)^{2}$$
$89$ $$(T^{4} + 4 T^{3} - 64 T^{2} - 136 T + 292)^{2}$$
$97$ $$(T^{4} - 4 T^{3} - 148 T^{2} + 304 T + 376)^{2}$$