# Properties

 Label 1280.3.e.g Level $1280$ Weight $3$ Character orbit 1280.e Analytic conductor $34.877$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1280 = 2^{8} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1280.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$34.8774738381$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.1827904.1 Defining polynomial: $$x^{6} + 9 x^{4} + 14 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 160) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -1 + \beta_{4} ) q^{5} + ( 2 + \beta_{4} + \beta_{5} ) q^{7} + ( -4 - 3 \beta_{2} + \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -1 + \beta_{4} ) q^{5} + ( 2 + \beta_{4} + \beta_{5} ) q^{7} + ( -4 - 3 \beta_{2} + \beta_{4} + \beta_{5} ) q^{9} + ( 2 + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{11} + ( -1 - 3 \beta_{2} - \beta_{4} - \beta_{5} ) q^{13} + ( 6 - 4 \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{15} + ( 2 + 2 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{17} + ( -2 + 6 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} ) q^{19} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{21} + ( -12 - 2 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} ) q^{23} + ( 4 + 6 \beta_{1} + 5 \beta_{2} + 7 \beta_{3} + \beta_{4} - \beta_{5} ) q^{25} + ( -2 - 6 \beta_{1} - 2 \beta_{2} + 14 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{27} + ( -8 \beta_{1} + \beta_{3} ) q^{29} + ( 4 + 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{31} + ( 12 \beta_{1} - 8 \beta_{3} ) q^{33} + ( 16 + 5 \beta_{1} + 10 \beta_{3} + 4 \beta_{4} ) q^{35} + ( 33 - 5 \beta_{2} - 3 \beta_{4} - 3 \beta_{5} ) q^{37} + ( -4 - 8 \beta_{1} - 4 \beta_{2} + 16 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{39} + ( 13 + 5 \beta_{2} + \beta_{4} + \beta_{5} ) q^{41} + ( -6 - 5 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} ) q^{43} + ( 46 + 8 \beta_{1} + 15 \beta_{2} + \beta_{3} + 4 \beta_{4} - 3 \beta_{5} ) q^{45} + ( -42 + 8 \beta_{2} - \beta_{4} - \beta_{5} ) q^{47} + ( -12 - 5 \beta_{2} + 7 \beta_{4} + 7 \beta_{5} ) q^{49} + ( 36 + 12 \beta_{2} - 4 \beta_{4} - 4 \beta_{5} ) q^{51} + ( -5 + 17 \beta_{2} - 5 \beta_{4} - 5 \beta_{5} ) q^{53} + ( -54 - 12 \beta_{1} - 10 \beta_{2} - 14 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} ) q^{55} + ( 8 + 12 \beta_{1} + 8 \beta_{2} - 32 \beta_{3} + 8 \beta_{4} - 8 \beta_{5} ) q^{57} + ( 62 + 6 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} ) q^{59} + ( 7 + 8 \beta_{1} + 7 \beta_{2} - 8 \beta_{3} + 7 \beta_{4} - 7 \beta_{5} ) q^{61} + ( 28 - 2 \beta_{2} + 7 \beta_{4} + 7 \beta_{5} ) q^{63} + ( 7 - 2 \beta_{1} + 15 \beta_{2} - 19 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{65} + ( 2 - 11 \beta_{1} + 2 \beta_{2} - 30 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{67} + ( 1 - 24 \beta_{1} + \beta_{2} + 7 \beta_{3} + \beta_{4} - \beta_{5} ) q^{69} + ( 8 + 8 \beta_{1} + 8 \beta_{2} - 12 \beta_{3} + 8 \beta_{4} - 8 \beta_{5} ) q^{71} + ( 4 - 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{73} + ( -48 + 15 \beta_{1} - 20 \beta_{3} + 8 \beta_{4} ) q^{75} + ( -64 + 8 \beta_{2} - 12 \beta_{4} - 12 \beta_{5} ) q^{77} + ( -4 - 16 \beta_{1} - 4 \beta_{2} - 44 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{79} + ( 46 + 19 \beta_{2} + 7 \beta_{4} + 7 \beta_{5} ) q^{81} + ( -14 - 11 \beta_{1} - 14 \beta_{2} + 18 \beta_{3} - 14 \beta_{4} + 14 \beta_{5} ) q^{83} + ( -42 + 24 \beta_{1} + 10 \beta_{2} + 8 \beta_{3} + 2 \beta_{4} - 14 \beta_{5} ) q^{85} + ( 106 + 26 \beta_{2} - 8 \beta_{4} - 8 \beta_{5} ) q^{87} + ( -10 + 8 \beta_{2} + 16 \beta_{4} + 16 \beta_{5} ) q^{89} + ( -32 + 8 \beta_{2} ) q^{91} + ( 56 + 8 \beta_{2} - 8 \beta_{4} - 8 \beta_{5} ) q^{93} + ( -10 + 4 \beta_{1} - 30 \beta_{2} + 38 \beta_{3} - 10 \beta_{4} + 6 \beta_{5} ) q^{95} + ( 6 + 8 \beta_{1} + 6 \beta_{2} + 18 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} ) q^{97} + ( -154 - 34 \beta_{2} - 6 \beta_{4} - 6 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 8 q^{5} + 12 q^{7} - 18 q^{9} + O(q^{10})$$ $$6 q - 8 q^{5} + 12 q^{7} - 18 q^{9} + 8 q^{11} + 36 q^{15} - 24 q^{19} - 68 q^{23} + 10 q^{25} + 88 q^{35} + 208 q^{37} + 68 q^{41} + 232 q^{45} - 268 q^{47} - 62 q^{49} + 192 q^{51} - 64 q^{53} - 288 q^{55} + 360 q^{59} + 172 q^{63} - 304 q^{75} - 400 q^{77} + 238 q^{81} - 304 q^{85} + 584 q^{87} - 76 q^{89} - 208 q^{91} + 320 q^{93} + 32 q^{95} - 856 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 9 x^{4} + 14 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{4} + 10 \nu^{2} - 7$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{5} + 20 \nu^{3} + 38 \nu$$$$)/5$$ $$\beta_{4}$$ $$=$$ $$($$$$7 \nu^{5} + 2 \nu^{4} + 60 \nu^{3} + 20 \nu^{2} + 73 \nu + 23$$$$)/5$$ $$\beta_{5}$$ $$=$$ $$($$$$-7 \nu^{5} + 4 \nu^{4} - 60 \nu^{3} + 30 \nu^{2} - 73 \nu + 21$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} - 3 \beta_{2} - 13$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} + 7 \beta_{3} - \beta_{2} - 12 \beta_{1} - 1$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-5 \beta_{5} - 5 \beta_{4} + 25 \beta_{2} + 79$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{5} + 5 \beta_{4} - 30 \beta_{3} + 5 \beta_{2} + 41 \beta_{1} + 5$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$261$$ $$511$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
639.1
 − 2.65109i − 1.37720i − 0.273891i 0.273891i 1.37720i 2.65109i
0 5.30219i 0 −4.75441 + 1.54778i 0 −0.206625 0 −19.1132 0
639.2 0 2.75441i 0 −2.54778 + 4.30219i 0 −3.84997 0 1.41325 0
639.3 0 0.547781i 0 3.30219 3.75441i 0 10.0566 0 8.69994 0
639.4 0 0.547781i 0 3.30219 + 3.75441i 0 10.0566 0 8.69994 0
639.5 0 2.75441i 0 −2.54778 4.30219i 0 −3.84997 0 1.41325 0
639.6 0 5.30219i 0 −4.75441 1.54778i 0 −0.206625 0 −19.1132 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 639.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.3.e.g 6
4.b odd 2 1 1280.3.e.f 6
5.b even 2 1 1280.3.e.h 6
8.b even 2 1 1280.3.e.i 6
8.d odd 2 1 1280.3.e.h 6
16.e even 4 1 160.3.h.a 6
16.e even 4 1 320.3.h.g 6
16.f odd 4 1 160.3.h.b yes 6
16.f odd 4 1 320.3.h.f 6
20.d odd 2 1 1280.3.e.i 6
40.e odd 2 1 inner 1280.3.e.g 6
40.f even 2 1 1280.3.e.f 6
48.i odd 4 1 1440.3.j.a 6
48.k even 4 1 1440.3.j.b 6
80.i odd 4 1 800.3.b.h 6
80.i odd 4 1 1600.3.b.w 6
80.j even 4 1 800.3.b.i 6
80.j even 4 1 1600.3.b.v 6
80.k odd 4 1 160.3.h.a 6
80.k odd 4 1 320.3.h.g 6
80.q even 4 1 160.3.h.b yes 6
80.q even 4 1 320.3.h.f 6
80.s even 4 1 800.3.b.h 6
80.s even 4 1 1600.3.b.w 6
80.t odd 4 1 800.3.b.i 6
80.t odd 4 1 1600.3.b.v 6
240.t even 4 1 1440.3.j.a 6
240.bm odd 4 1 1440.3.j.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.3.h.a 6 16.e even 4 1
160.3.h.a 6 80.k odd 4 1
160.3.h.b yes 6 16.f odd 4 1
160.3.h.b yes 6 80.q even 4 1
320.3.h.f 6 16.f odd 4 1
320.3.h.f 6 80.q even 4 1
320.3.h.g 6 16.e even 4 1
320.3.h.g 6 80.k odd 4 1
800.3.b.h 6 80.i odd 4 1
800.3.b.h 6 80.s even 4 1
800.3.b.i 6 80.j even 4 1
800.3.b.i 6 80.t odd 4 1
1280.3.e.f 6 4.b odd 2 1
1280.3.e.f 6 40.f even 2 1
1280.3.e.g 6 1.a even 1 1 trivial
1280.3.e.g 6 40.e odd 2 1 inner
1280.3.e.h 6 5.b even 2 1
1280.3.e.h 6 8.d odd 2 1
1280.3.e.i 6 8.b even 2 1
1280.3.e.i 6 20.d odd 2 1
1440.3.j.a 6 48.i odd 4 1
1440.3.j.a 6 240.t even 4 1
1440.3.j.b 6 48.k even 4 1
1440.3.j.b 6 240.bm odd 4 1
1600.3.b.v 6 80.j even 4 1
1600.3.b.v 6 80.t odd 4 1
1600.3.b.w 6 80.i odd 4 1
1600.3.b.w 6 80.s even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1280, [\chi])$$:

 $$T_{3}^{6} + 36 T_{3}^{4} + 224 T_{3}^{2} + 64$$ $$T_{7}^{3} - 6 T_{7}^{2} - 40 T_{7} - 8$$ $$T_{11}^{3} - 4 T_{11}^{2} - 272 T_{11} + 1600$$ $$T_{13}^{3} - 208 T_{13} - 832$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$64 + 224 T^{2} + 36 T^{4} + T^{6}$$
$5$ $$15625 + 5000 T + 675 T^{2} + 80 T^{3} + 27 T^{4} + 8 T^{5} + T^{6}$$
$7$ $$( -8 - 40 T - 6 T^{2} + T^{3} )^{2}$$
$11$ $$( 1600 - 272 T - 4 T^{2} + T^{3} )^{2}$$
$13$ $$( -832 - 208 T + T^{3} )^{2}$$
$17$ $$6553600 + 270336 T^{2} + 1088 T^{4} + T^{6}$$
$19$ $$( 3392 - 784 T + 12 T^{2} + T^{3} )^{2}$$
$23$ $$( -9160 - 152 T + 34 T^{2} + T^{3} )^{2}$$
$29$ $$4494400 + 805424 T^{2} + 2380 T^{4} + T^{6}$$
$31$ $$419430400 + 1900544 T^{2} + 2560 T^{4} + T^{6}$$
$37$ $$( -20800 + 2704 T - 104 T^{2} + T^{3} )^{2}$$
$41$ $$( 5000 - 100 T - 34 T^{2} + T^{3} )^{2}$$
$43$ $$39438400 + 2459744 T^{2} + 4260 T^{4} + T^{6}$$
$47$ $$( 22216 + 4824 T + 134 T^{2} + T^{3} )^{2}$$
$53$ $$( -224320 - 5968 T + 32 T^{2} + T^{3} )^{2}$$
$59$ $$( -159424 + 9968 T - 180 T^{2} + T^{3} )^{2}$$
$61$ $$6649423936 + 11154224 T^{2} + 5964 T^{4} + T^{6}$$
$67$ $$613057600 + 30359904 T^{2} + 14180 T^{4} + T^{6}$$
$71$ $$14231535616 + 19738624 T^{2} + 8384 T^{4} + T^{6}$$
$73$ $$3114532864 + 6447104 T^{2} + 4416 T^{4} + T^{6}$$
$79$ $$37060870144 + 159514624 T^{2} + 25856 T^{4} + T^{6}$$
$83$ $$233675560000 + 145948000 T^{2} + 24164 T^{4} + T^{6}$$
$89$ $$( -155000 - 13940 T + 38 T^{2} + T^{3} )^{2}$$
$97$ $$44302336 + 1384448 T^{2} + 7488 T^{4} + T^{6}$$