# Properties

 Label 1280.2.f.i Level $1280$ Weight $2$ Character orbit 1280.f Analytic conductor $10.221$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$10.2208514587$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 4*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 640) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 1) q^{3} + (\beta_{3} + \beta_1) q^{5} + ( - \beta_{5} - \beta_{3} + \beta_{2}) q^{7} + ( - \beta_{5} + \beta_{3} + 2 \beta_1 + 1) q^{9}+O(q^{10})$$ q + (-b1 - 1) * q^3 + (b3 + b1) * q^5 + (-b5 - b3 + b2) * q^7 + (-b5 + b3 + 2*b1 + 1) * q^9 $$q + ( - \beta_1 - 1) q^{3} + (\beta_{3} + \beta_1) q^{5} + ( - \beta_{5} - \beta_{3} + \beta_{2}) q^{7} + ( - \beta_{5} + \beta_{3} + 2 \beta_1 + 1) q^{9} + ( - \beta_{4} + 2 \beta_{2}) q^{11} + (\beta_{5} - \beta_{3} - 2) q^{13} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{15} + 2 \beta_{2} q^{17} + (2 \beta_{5} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2}) q^{19} + (\beta_{5} + \beta_{4} + \beta_{3} - 4 \beta_{2}) q^{21} + ( - \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2}) q^{23} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_1 + 1) q^{25} + (2 \beta_{5} - 2 \beta_{3} - 2 \beta_1 - 2) q^{27} - \beta_{4} q^{29} + (2 \beta_{5} - 2 \beta_{3} + 4) q^{31} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2}) q^{33} + (\beta_{4} + 2 \beta_{2} - \beta_1 + 3) q^{35} + ( - \beta_{5} + \beta_{3} - 2 \beta_1 - 4) q^{37} + 4 \beta_1 q^{39} + ( - \beta_{5} + \beta_{3} - 2 \beta_1 - 2) q^{41} + (2 \beta_{5} - 2 \beta_{3} - 3 \beta_1 + 5) q^{43} + ( - \beta_{5} - \beta_{4} + 3 \beta_1 + 6) q^{45} + ( - \beta_{5} - 2 \beta_{4} - \beta_{3} + 3 \beta_{2}) q^{47} + (\beta_{5} - \beta_{3} - 2 \beta_1 - 1) q^{49} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2}) q^{51} + (\beta_{5} - \beta_{3} + 6) q^{53} + (\beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{55} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 6 \beta_{2}) q^{57} + (2 \beta_{5} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2}) q^{59} + ( - \beta_{5} - \beta_{3} + 4 \beta_{2}) q^{61} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + 5 \beta_{2}) q^{63} + ( - \beta_{5} + 3 \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{65} + (2 \beta_{5} - 2 \beta_{3} - 5 \beta_1 + 3) q^{67} + (\beta_{5} + \beta_{4} + \beta_{3}) q^{69} + (2 \beta_{5} - 2 \beta_{3} - 4 \beta_1 + 4) q^{71} + (2 \beta_{5} + 6 \beta_{4} + 2 \beta_{3} - 6 \beta_{2}) q^{73} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 3 \beta_1 - 7) q^{75} + ( - 6 \beta_1 - 2) q^{77} + (2 \beta_{5} - 2 \beta_{3} - 4) q^{79} + (\beta_{5} - \beta_{3} + 2 \beta_1 + 1) q^{81} + (2 \beta_{5} - 2 \beta_{3} - \beta_1 - 9) q^{83} + ( - 2 \beta_{5} - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 2) q^{85} + 2 \beta_{2} q^{87} + (4 \beta_{5} - 4 \beta_{3} - 4 \beta_1 - 2) q^{89} + (2 \beta_{5} - 4 \beta_{4} + 2 \beta_{3}) q^{91} - 8 q^{93} + ( - 2 \beta_{5} - 3 \beta_{4} - 2 \beta_{2} + 2 \beta_1 - 6) q^{95} + (4 \beta_{4} + 2 \beta_{2}) q^{97} + ( - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + 6 \beta_{2}) q^{99}+O(q^{100})$$ q + (-b1 - 1) * q^3 + (b3 + b1) * q^5 + (-b5 - b3 + b2) * q^7 + (-b5 + b3 + 2*b1 + 1) * q^9 + (-b4 + 2*b2) * q^11 + (b5 - b3 - 2) * q^13 + (b5 - b4 - b3 + b2 - 2*b1 - 2) * q^15 + 2*b2 * q^17 + (2*b5 + b4 + 2*b3 - 2*b2) * q^19 + (b5 + b4 + b3 - 4*b2) * q^21 + (-b5 - 2*b4 - b3 + b2) * q^23 + (-b5 - b4 - b3 + 2*b1 + 1) * q^25 + (2*b5 - 2*b3 - 2*b1 - 2) * q^27 - b4 * q^29 + (2*b5 - 2*b3 + 4) * q^31 + (2*b5 - 2*b4 + 2*b3 - 2*b2) * q^33 + (b4 + 2*b2 - b1 + 3) * q^35 + (-b5 + b3 - 2*b1 - 4) * q^37 + 4*b1 * q^39 + (-b5 + b3 - 2*b1 - 2) * q^41 + (2*b5 - 2*b3 - 3*b1 + 5) * q^43 + (-b5 - b4 + 3*b1 + 6) * q^45 + (-b5 - 2*b4 - b3 + 3*b2) * q^47 + (b5 - b3 - 2*b1 - 1) * q^49 + (2*b5 - 2*b4 + 2*b3 - 4*b2) * q^51 + (b5 - b3 + 6) * q^53 + (b4 - 2*b3 + 2*b2 + 2*b1 - 2) * q^55 + (-2*b5 - 2*b4 - 2*b3 + 6*b2) * q^57 + (2*b5 + b4 + 2*b3 - 2*b2) * q^59 + (-b5 - b3 + 4*b2) * q^61 + (-b5 + 2*b4 - b3 + 5*b2) * q^63 + (-b5 + 3*b4 - b3 - 2*b2 - 2*b1 - 2) * q^65 + (2*b5 - 2*b3 - 5*b1 + 3) * q^67 + (b5 + b4 + b3) * q^69 + (2*b5 - 2*b3 - 4*b1 + 4) * q^71 + (2*b5 + 6*b4 + 2*b3 - 6*b2) * q^73 + (2*b5 + 2*b4 - 2*b3 - 3*b1 - 7) * q^75 + (-6*b1 - 2) * q^77 + (2*b5 - 2*b3 - 4) * q^79 + (b5 - b3 + 2*b1 + 1) * q^81 + (2*b5 - 2*b3 - b1 - 9) * q^83 + (-2*b5 - 2*b3 + 4*b2 + 2*b1 - 2) * q^85 + 2*b2 * q^87 + (4*b5 - 4*b3 - 4*b1 - 2) * q^89 + (2*b5 - 4*b4 + 2*b3) * q^91 - 8 * q^93 + (-2*b5 - 3*b4 - 2*b2 + 2*b1 - 6) * q^95 + (4*b4 + 2*b2) * q^97 + (-2*b5 + b4 - 2*b3 + 6*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 4 q^{3} + 6 q^{9}+O(q^{10})$$ 6 * q - 4 * q^3 + 6 * q^9 $$6 q - 4 q^{3} + 6 q^{9} - 16 q^{13} - 12 q^{15} + 2 q^{25} - 16 q^{27} + 16 q^{31} + 20 q^{35} - 16 q^{37} - 8 q^{39} - 4 q^{41} + 28 q^{43} + 32 q^{45} - 6 q^{49} + 32 q^{53} - 20 q^{55} - 8 q^{65} + 20 q^{67} + 24 q^{71} - 44 q^{75} - 32 q^{79} - 2 q^{81} - 60 q^{83} - 16 q^{85} - 20 q^{89} - 48 q^{93} - 36 q^{95}+O(q^{100})$$ 6 * q - 4 * q^3 + 6 * q^9 - 16 * q^13 - 12 * q^15 + 2 * q^25 - 16 * q^27 + 16 * q^31 + 20 * q^35 - 16 * q^37 - 8 * q^39 - 4 * q^41 + 28 * q^43 + 32 * q^45 - 6 * q^49 + 32 * q^53 - 20 * q^55 - 8 * q^65 + 20 * q^67 + 24 * q^71 - 44 * q^75 - 32 * q^79 - 2 * q^81 - 60 * q^83 - 16 * q^85 - 20 * q^89 - 48 * q^93 - 36 * q^95

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

 $$\beta_{1}$$ $$=$$ $$( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23$$ (-4*v^5 + 9*v^4 - 16*v^3 - 4*v^2 + 8*v - 9) / 23 $$\beta_{2}$$ $$=$$ $$( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} - 38\nu + 14 ) / 23$$ (-4*v^5 + 9*v^4 - 16*v^3 - 4*v^2 - 38*v + 14) / 23 $$\beta_{3}$$ $$=$$ $$( -6\nu^{5} + 2\nu^{4} - \nu^{3} - 6\nu^{2} - 80\nu - 2 ) / 23$$ (-6*v^5 + 2*v^4 - v^3 - 6*v^2 - 80*v - 2) / 23 $$\beta_{4}$$ $$=$$ $$( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 60\nu^{2} - 64\nu + 26 ) / 23$$ (-14*v^5 + 20*v^4 - 10*v^3 - 60*v^2 - 64*v + 26) / 23 $$\beta_{5}$$ $$=$$ $$( -16\nu^{5} + 36\nu^{4} - 41\nu^{3} - 16\nu^{2} - 60\nu + 56 ) / 23$$ (-16*v^5 + 36*v^4 - 41*v^3 - 16*v^2 - 60*v + 56) / 23
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta _1 + 1 ) / 2$$ (-b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} - \beta_{4} + \beta_{3} - 2\beta_{2} ) / 2$$ (b5 - b4 + b3 - 2*b2) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{5} - 2\beta_{2} - 2\beta _1 - 2$$ b5 - 2*b2 - 2*b1 - 2 $$\nu^{4}$$ $$=$$ $$2\beta_{5} - 2\beta_{3} - 5\beta _1 - 7$$ 2*b5 - 2*b3 - 5*b1 - 7 $$\nu^{5}$$ $$=$$ $$( \beta_{4} - 10\beta_{3} + 16\beta_{2} - 16\beta _1 - 18 ) / 2$$ (b4 - 10*b3 + 16*b2 - 16*b1 - 18) / 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 1.45161 − 1.45161i 1.45161 + 1.45161i 0.403032 + 0.403032i 0.403032 − 0.403032i −0.854638 + 0.854638i −0.854638 − 0.854638i
0 −2.90321 0 2.21432 0.311108i 0 3.52543i 0 5.42864 0
129.2 0 −2.90321 0 2.21432 + 0.311108i 0 3.52543i 0 5.42864 0
129.3 0 −0.806063 0 −1.67513 1.48119i 0 2.15633i 0 −2.35026 0
129.4 0 −0.806063 0 −1.67513 + 1.48119i 0 2.15633i 0 −2.35026 0
129.5 0 1.70928 0 −0.539189 2.17009i 0 2.63090i 0 −0.0783777 0
129.6 0 1.70928 0 −0.539189 + 2.17009i 0 2.63090i 0 −0.0783777 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 129.6 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.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.2.f.i 6
4.b odd 2 1 1280.2.f.k 6
5.b even 2 1 1280.2.f.l 6
8.b even 2 1 1280.2.f.l 6
8.d odd 2 1 1280.2.f.j 6
16.e even 4 1 640.2.c.a 6
16.e even 4 1 640.2.c.d yes 6
16.f odd 4 1 640.2.c.b yes 6
16.f odd 4 1 640.2.c.c yes 6
20.d odd 2 1 1280.2.f.j 6
40.e odd 2 1 1280.2.f.k 6
40.f even 2 1 inner 1280.2.f.i 6
80.i odd 4 1 3200.2.a.bs 3
80.i odd 4 1 3200.2.a.bv 3
80.j even 4 1 3200.2.a.bt 3
80.j even 4 1 3200.2.a.bu 3
80.k odd 4 1 640.2.c.b yes 6
80.k odd 4 1 640.2.c.c yes 6
80.q even 4 1 640.2.c.a 6
80.q even 4 1 640.2.c.d yes 6
80.s even 4 1 3200.2.a.bo 3
80.s even 4 1 3200.2.a.br 3
80.t odd 4 1 3200.2.a.bp 3
80.t odd 4 1 3200.2.a.bq 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.c.a 6 16.e even 4 1
640.2.c.a 6 80.q even 4 1
640.2.c.b yes 6 16.f odd 4 1
640.2.c.b yes 6 80.k odd 4 1
640.2.c.c yes 6 16.f odd 4 1
640.2.c.c yes 6 80.k odd 4 1
640.2.c.d yes 6 16.e even 4 1
640.2.c.d yes 6 80.q even 4 1
1280.2.f.i 6 1.a even 1 1 trivial
1280.2.f.i 6 40.f even 2 1 inner
1280.2.f.j 6 8.d odd 2 1
1280.2.f.j 6 20.d odd 2 1
1280.2.f.k 6 4.b odd 2 1
1280.2.f.k 6 40.e odd 2 1
1280.2.f.l 6 5.b even 2 1
1280.2.f.l 6 8.b even 2 1
3200.2.a.bo 3 80.s even 4 1
3200.2.a.bp 3 80.t odd 4 1
3200.2.a.bq 3 80.t odd 4 1
3200.2.a.br 3 80.s even 4 1
3200.2.a.bs 3 80.i odd 4 1
3200.2.a.bt 3 80.j even 4 1
3200.2.a.bu 3 80.j even 4 1
3200.2.a.bv 3 80.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1280, [\chi])$$:

 $$T_{3}^{3} + 2T_{3}^{2} - 4T_{3} - 4$$ T3^3 + 2*T3^2 - 4*T3 - 4 $$T_{13}^{3} + 8T_{13}^{2} + 8T_{13} - 16$$ T13^3 + 8*T13^2 + 8*T13 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$(T^{3} + 2 T^{2} - 4 T - 4)^{2}$$
$5$ $$T^{6} - T^{4} - 16 T^{3} - 5 T^{2} + \cdots + 125$$
$7$ $$T^{6} + 24 T^{4} + 176 T^{2} + \cdots + 400$$
$11$ $$T^{6} + 44 T^{4} + 432 T^{2} + \cdots + 64$$
$13$ $$(T^{3} + 8 T^{2} + 8 T - 16)^{2}$$
$17$ $$T^{6} + 48 T^{4} + 512 T^{2} + \cdots + 1024$$
$19$ $$T^{6} + 76 T^{4} + 1712 T^{2} + \cdots + 10816$$
$23$ $$T^{6} + 40 T^{4} + 80 T^{2} + 16$$
$29$ $$(T^{2} + 4)^{3}$$
$31$ $$(T^{3} - 8 T^{2} - 32 T + 128)^{2}$$
$37$ $$(T^{3} + 8 T^{2} - 32 T - 272)^{2}$$
$41$ $$(T^{3} + 2 T^{2} - 52 T - 184)^{2}$$
$43$ $$(T^{3} - 14 T^{2} + 20 T + 100)^{2}$$
$47$ $$T^{6} + 72 T^{4} + 1712 T^{2} + \cdots + 13456$$
$53$ $$(T^{3} - 16 T^{2} + 72 T - 80)^{2}$$
$59$ $$T^{6} + 76 T^{4} + 1712 T^{2} + \cdots + 10816$$
$61$ $$T^{6} + 156 T^{4} + 944 T^{2} + \cdots + 64$$
$67$ $$(T^{3} - 10 T^{2} - 60 T + 604)^{2}$$
$71$ $$(T^{3} - 12 T^{2} - 16 T + 320)^{2}$$
$73$ $$T^{6} + 400 T^{4} + 47360 T^{2} + \cdots + 1401856$$
$79$ $$(T^{3} + 16 T^{2} + 32 T - 128)^{2}$$
$83$ $$(T^{3} + 30 T^{2} + 260 T + 524)^{2}$$
$89$ $$(T^{3} + 10 T^{2} - 116 T - 1096)^{2}$$
$97$ $$T^{6} + 304 T^{4} + 23552 T^{2} + \cdots + 369664$$