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

Downloads

Learn more

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: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{3} + (\beta_{3} + \beta_1) q^{5} + ( - \beta_{5} - \beta_{3} + \beta_{2}) q^{7} + ( - \beta_{5} + \beta_{3} + 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 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}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{3} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{3} + 6 q^{9} - 16 q^{13} - 12 q^{15} + 2 q^{25} - 16 q^{27} + 16 q^{31} + 20 q^{35} - 16 q^{37} - 8 q^{39} - 4 q^{41} + 28 q^{43} + 32 q^{45} - 6 q^{49} + 32 q^{53} - 20 q^{55} - 8 q^{65} + 20 q^{67} + 24 q^{71} - 44 q^{75} - 32 q^{79} - 2 q^{81} - 60 q^{83} - 16 q^{85} - 20 q^{89} - 48 q^{93} - 36 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} - 38\nu + 14 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -6\nu^{5} + 2\nu^{4} - \nu^{3} - 6\nu^{2} - 80\nu - 2 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 60\nu^{2} - 64\nu + 26 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -16\nu^{5} + 36\nu^{4} - 41\nu^{3} - 16\nu^{2} - 60\nu + 56 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} - 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - 2\beta_{2} - 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 2\beta_{3} - 5\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{4} - 10\beta_{3} + 16\beta_{2} - 16\beta _1 - 18 ) / 2 \) Copy content Toggle raw display

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:

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 \(S_{2}^{\mathrm{new}}(1280, [\chi])\):

\( T_{3}^{3} + 2T_{3}^{2} - 4T_{3} - 4 \) Copy content Toggle raw display
\( T_{13}^{3} + 8T_{13}^{2} + 8T_{13} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

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