# Properties

 Label 4864.2.a.bm Level $4864$ Weight $2$ Character orbit 4864.a Self dual yes Analytic conductor $38.839$ Analytic rank $1$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4864 = 2^{8} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4864.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$38.8392355432$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: 8.8.34309996544.1 Defining polynomial: $$x^{8} - 4 x^{7} - 8 x^{6} + 28 x^{5} + 31 x^{4} - 36 x^{3} - 22 x^{2} + 12 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 2432) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{6} q^{5} + \beta_{4} q^{7} + ( -\beta_{1} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{6} q^{5} + \beta_{4} q^{7} + ( -\beta_{1} + \beta_{7} ) q^{9} + ( -2 + \beta_{1} - \beta_{7} ) q^{11} + ( -\beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{13} + ( -\beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{15} + ( -2 - \beta_{1} - \beta_{3} ) q^{17} + q^{19} + ( \beta_{2} - 2 \beta_{4} - \beta_{6} ) q^{21} + ( \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{23} + ( 3 - \beta_{1} + 2 \beta_{3} - \beta_{7} ) q^{25} + ( \beta_{3} - \beta_{7} ) q^{27} + ( -2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{29} + ( \beta_{4} + \beta_{5} ) q^{31} + ( -5 \beta_{1} - \beta_{3} + \beta_{7} ) q^{33} + ( -4 - 3 \beta_{1} - 2 \beta_{3} + 3 \beta_{7} ) q^{35} + ( \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{37} + ( -\beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{39} + ( 2 + \beta_{1} + \beta_{3} + \beta_{7} ) q^{41} + ( -2 + \beta_{1} + 2 \beta_{3} - 3 \beta_{7} ) q^{43} + ( 2 \beta_{4} + \beta_{6} ) q^{45} + ( \beta_{2} - 2 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{47} + ( -1 - 3 \beta_{1} - \beta_{3} ) q^{49} + ( -1 + \beta_{3} - 2 \beta_{7} ) q^{51} + ( -\beta_{2} - 2 \beta_{5} - \beta_{6} ) q^{53} + ( -2 \beta_{4} - 3 \beta_{6} ) q^{55} + \beta_{1} q^{57} + ( -3 + 3 \beta_{1} - \beta_{7} ) q^{59} + ( -2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{61} + ( -2 \beta_{2} + 2 \beta_{4} + 3 \beta_{6} ) q^{63} + ( -2 - 4 \beta_{1} - 2 \beta_{3} ) q^{65} + ( -5 - 2 \beta_{1} - \beta_{3} + 2 \beta_{7} ) q^{67} + ( \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{69} + ( -2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{71} + ( 1 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{7} ) q^{73} + ( -10 - 3 \beta_{3} + \beta_{7} ) q^{75} + ( 2 \beta_{2} - 4 \beta_{4} - 3 \beta_{6} ) q^{77} + ( 2 \beta_{2} + \beta_{4} + 3 \beta_{5} ) q^{79} + ( -5 - 2 \beta_{3} - 2 \beta_{7} ) q^{81} + ( -6 - 2 \beta_{3} + 2 \beta_{7} ) q^{83} + ( -2 \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{85} + ( -\beta_{2} + 3 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{87} + ( -\beta_{1} + 3 \beta_{3} - \beta_{7} ) q^{89} + ( -1 - 5 \beta_{1} + 3 \beta_{7} ) q^{91} + 2 \beta_{5} q^{93} + \beta_{6} q^{95} + ( 4 - 2 \beta_{1} ) q^{97} + ( -4 + 5 \beta_{1} + 2 \beta_{3} - 3 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{3} + 4q^{9} + O(q^{10})$$ $$8q - 4q^{3} + 4q^{9} - 20q^{11} - 8q^{17} + 8q^{19} + 20q^{25} - 4q^{27} + 24q^{33} - 12q^{35} + 8q^{41} - 28q^{43} + 8q^{49} - 12q^{51} - 4q^{57} - 36q^{59} + 8q^{65} - 28q^{67} - 8q^{73} - 68q^{75} - 32q^{81} - 40q^{83} - 8q^{89} + 12q^{91} + 40q^{97} - 60q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} - 8 x^{6} + 28 x^{5} + 31 x^{4} - 36 x^{3} - 22 x^{2} + 12 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-5 \nu^{7} + 37 \nu^{6} - 39 \nu^{5} - 205 \nu^{4} + 256 \nu^{3} + 412 \nu^{2} - 126 \nu - 162$$$$)/52$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{7} + 37 \nu^{6} - 39 \nu^{5} - 205 \nu^{4} + 308 \nu^{3} + 308 \nu^{2} - 438 \nu - 58$$$$)/52$$ $$\beta_{3}$$ $$=$$ $$($$$$-9 \nu^{7} + 25 \nu^{6} + 117 \nu^{5} - 161 \nu^{4} - 600 \nu^{3} - 28 \nu^{2} + 522 \nu + 10$$$$)/52$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{7} + 24 \nu^{6} + 13 \nu^{5} - 114 \nu^{4} - 30 \nu^{3} - 4 \nu^{2} - 22 \nu + 98$$$$)/26$$ $$\beta_{5}$$ $$=$$ $$($$$$15 \nu^{7} - 59 \nu^{6} - 91 \nu^{5} + 251 \nu^{4} + 428 \nu^{3} + 220 \nu^{2} - 142 \nu - 138$$$$)/52$$ $$\beta_{6}$$ $$=$$ $$($$$$10 \nu^{7} - 35 \nu^{6} - 104 \nu^{5} + 267 \nu^{4} + 424 \nu^{3} - 304 \nu^{2} - 268 \nu + 38$$$$)/26$$ $$\beta_{7}$$ $$=$$ $$($$$$23 \nu^{7} - 87 \nu^{6} - 195 \nu^{5} + 579 \nu^{4} + 736 \nu^{3} - 460 \nu^{2} - 294 \nu + 38$$$$)/52$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + \beta_{1} + 3$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + \beta_{3} + \beta_{1} + 9$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-10 \beta_{7} + 10 \beta_{6} + 3 \beta_{5} - 4 \beta_{4} + 5 \beta_{3} - \beta_{2} + 3 \beta_{1} + 23$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-15 \beta_{7} + 16 \beta_{6} + 3 \beta_{5} - 10 \beta_{4} + 10 \beta_{3} + \beta_{2} + 5 \beta_{1} + 48$$ $$\nu^{5}$$ $$=$$ $$($$$$-116 \beta_{7} + 124 \beta_{6} + 35 \beta_{5} - 62 \beta_{4} + 83 \beta_{3} + 11 \beta_{2} + 31 \beta_{1} + 323$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-199 \beta_{7} + 222 \beta_{6} + 60 \beta_{5} - 120 \beta_{4} + 167 \beta_{3} + 38 \beta_{2} + 54 \beta_{1} + 611$$ $$\nu^{7}$$ $$=$$ $$-731 \beta_{7} + 829 \beta_{6} + 255 \beta_{5} - 421 \beta_{4} + 665 \beta_{3} + 178 \beta_{2} + 175 \beta_{1} + 2202$$

## 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
 −1.58857 −0.139869 0.545005 3.79159 −0.836706 3.15093 0.904430 −1.82681
0 −2.63640 0 −2.63403 0 −3.88603 0 3.95063 0
1.2 0 −2.63640 0 2.63403 0 3.88603 0 3.95063 0
1.3 0 −1.65222 0 −4.30533 0 2.73423 0 −0.270160 0
1.4 0 −1.65222 0 4.30533 0 −2.73423 0 −0.270160 0
1.5 0 0.222191 0 −1.55081 0 3.06334 0 −2.95063 0
1.6 0 0.222191 0 1.55081 0 −3.06334 0 −2.95063 0
1.7 0 2.06644 0 −1.45635 0 −0.196723 0 1.27016 0
1.8 0 2.06644 0 1.45635 0 0.196723 0 1.27016 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
$$2$$ $$-1$$
$$19$$ $$-1$$

## 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 4864.2.a.bm 8
4.b odd 2 1 4864.2.a.br 8
8.b even 2 1 4864.2.a.br 8
8.d odd 2 1 inner 4864.2.a.bm 8
16.e even 4 2 2432.2.c.i 16
16.f odd 4 2 2432.2.c.i 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2432.2.c.i 16 16.e even 4 2
2432.2.c.i 16 16.f odd 4 2
4864.2.a.bm 8 1.a even 1 1 trivial
4864.2.a.bm 8 8.d odd 2 1 inner
4864.2.a.br 8 4.b odd 2 1
4864.2.a.br 8 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4864))$$:

 $$T_{3}^{4} + 2 T_{3}^{3} - 5 T_{3}^{2} - 8 T_{3} + 2$$ $$T_{5}^{8} - 30 T_{5}^{6} + 249 T_{5}^{4} - 712 T_{5}^{2} + 656$$ $$T_{7}^{8} - 32 T_{7}^{6} + 326 T_{7}^{4} - 1072 T_{7}^{2} + 41$$ $$T_{11}^{4} + 10 T_{11}^{3} + 25 T_{11}^{2} - 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 2 - 8 T - 5 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$5$ $$656 - 712 T^{2} + 249 T^{4} - 30 T^{6} + T^{8}$$
$7$ $$41 - 1072 T^{2} + 326 T^{4} - 32 T^{6} + T^{8}$$
$11$ $$( -32 + 25 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$13$ $$2624 - 9232 T^{2} + 1505 T^{4} - 74 T^{6} + T^{8}$$
$17$ $$( -7 + 2 T + T^{2} )^{4}$$
$19$ $$( -1 + T )^{8}$$
$23$ $$2624 - 9232 T^{2} + 1505 T^{4} - 74 T^{6} + T^{8}$$
$29$ $$164 - 30124 T^{2} + 3325 T^{4} - 110 T^{6} + T^{8}$$
$31$ $$2624 - 2400 T^{2} + 660 T^{4} - 52 T^{6} + T^{8}$$
$37$ $$4410944 - 673056 T^{2} + 21460 T^{4} - 252 T^{6} + T^{8}$$
$41$ $$( 32 + 48 T - 30 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$43$ $$( -1756 - 1172 T - 59 T^{2} + 14 T^{3} + T^{4} )^{2}$$
$47$ $$758336 - 154704 T^{2} + 10233 T^{4} - 230 T^{6} + T^{8}$$
$53$ $$393764 - 178164 T^{2} + 11485 T^{4} - 198 T^{6} + T^{8}$$
$59$ $$( -4 - 20 T + 69 T^{2} + 18 T^{3} + T^{4} )^{2}$$
$61$ $$189584 - 124040 T^{2} + 8265 T^{4} - 174 T^{6} + T^{8}$$
$67$ $$( -562 - 260 T + 23 T^{2} + 14 T^{3} + T^{4} )^{2}$$
$71$ $$52910336 - 3418112 T^{2} + 64864 T^{4} - 448 T^{6} + T^{8}$$
$73$ $$( -191 - 252 T - 74 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$79$ $$13983296 - 2761568 T^{2} + 66900 T^{4} - 468 T^{6} + T^{8}$$
$83$ $$( -64 - 32 T + 52 T^{2} + 20 T^{3} + T^{4} )^{2}$$
$89$ $$( -1864 - 1192 T - 158 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$97$ $$( -32 - 224 T + 124 T^{2} - 20 T^{3} + T^{4} )^{2}$$