Properties

Label 1280.2.f.g
Level $1280$
Weight $2$
Character orbit 1280.f
Analytic conductor $10.221$
Analytic rank $0$
Dimension $4$
CM discriminant -20
Inner twists $4$

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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: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{3} ) q^{3} -\beta_{2} q^{5} + ( 3 \beta_{1} - \beta_{2} ) q^{7} + ( 3 + 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{3} ) q^{3} -\beta_{2} q^{5} + ( 3 \beta_{1} - \beta_{2} ) q^{7} + ( 3 + 2 \beta_{3} ) q^{9} + ( 5 \beta_{1} + \beta_{2} ) q^{15} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{21} + ( -\beta_{1} + 3 \beta_{2} ) q^{23} -5 q^{25} + ( -10 - 2 \beta_{3} ) q^{27} + 6 \beta_{1} q^{29} + ( -5 + 3 \beta_{3} ) q^{35} + 2 \beta_{3} q^{41} + ( 9 + \beta_{3} ) q^{43} + ( -10 \beta_{1} - 3 \beta_{2} ) q^{45} + ( -7 \beta_{1} - 3 \beta_{2} ) q^{47} + ( -7 + 6 \beta_{3} ) q^{49} -6 \beta_{2} q^{61} + ( -\beta_{1} + 3 \beta_{2} ) q^{63} + ( 3 - 5 \beta_{3} ) q^{67} + ( -14 \beta_{1} - 2 \beta_{2} ) q^{69} + ( 5 + 5 \beta_{3} ) q^{75} + ( 11 + 6 \beta_{3} ) q^{81} + ( 11 + 3 \beta_{3} ) q^{83} + ( -6 \beta_{1} - 6 \beta_{2} ) q^{87} + 6 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 12 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{3} + 12 q^{9} - 20 q^{25} - 40 q^{27} - 20 q^{35} + 36 q^{43} - 28 q^{49} + 12 q^{67} + 20 q^{75} + 44 q^{81} + 44 q^{83} + 24 q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} + 4 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{2} + 2 \beta_{1}\)

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
0.618034i
0.618034i
1.61803i
1.61803i
0 −3.23607 0 2.23607i 0 0.763932i 0 7.47214 0
129.2 0 −3.23607 0 2.23607i 0 0.763932i 0 7.47214 0
129.3 0 1.23607 0 2.23607i 0 5.23607i 0 −1.47214 0
129.4 0 1.23607 0 2.23607i 0 5.23607i 0 −1.47214 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
8.d odd 2 1 inner
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.g 4
4.b odd 2 1 1280.2.f.h 4
5.b even 2 1 1280.2.f.h 4
8.b even 2 1 1280.2.f.h 4
8.d odd 2 1 inner 1280.2.f.g 4
16.e even 4 1 160.2.c.b 4
16.e even 4 1 320.2.c.d 4
16.f odd 4 1 160.2.c.b 4
16.f odd 4 1 320.2.c.d 4
20.d odd 2 1 CM 1280.2.f.g 4
40.e odd 2 1 1280.2.f.h 4
40.f even 2 1 inner 1280.2.f.g 4
48.i odd 4 1 1440.2.f.i 4
48.i odd 4 1 2880.2.f.w 4
48.k even 4 1 1440.2.f.i 4
48.k even 4 1 2880.2.f.w 4
80.i odd 4 1 800.2.a.n 2
80.i odd 4 1 1600.2.a.bd 2
80.j even 4 1 800.2.a.n 2
80.j even 4 1 1600.2.a.bd 2
80.k odd 4 1 160.2.c.b 4
80.k odd 4 1 320.2.c.d 4
80.q even 4 1 160.2.c.b 4
80.q even 4 1 320.2.c.d 4
80.s even 4 1 800.2.a.j 2
80.s even 4 1 1600.2.a.z 2
80.t odd 4 1 800.2.a.j 2
80.t odd 4 1 1600.2.a.z 2
240.t even 4 1 1440.2.f.i 4
240.t even 4 1 2880.2.f.w 4
240.z odd 4 1 7200.2.a.cb 2
240.bb even 4 1 7200.2.a.cr 2
240.bd odd 4 1 7200.2.a.cr 2
240.bf even 4 1 7200.2.a.cb 2
240.bm odd 4 1 1440.2.f.i 4
240.bm odd 4 1 2880.2.f.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.c.b 4 16.e even 4 1
160.2.c.b 4 16.f odd 4 1
160.2.c.b 4 80.k odd 4 1
160.2.c.b 4 80.q even 4 1
320.2.c.d 4 16.e even 4 1
320.2.c.d 4 16.f odd 4 1
320.2.c.d 4 80.k odd 4 1
320.2.c.d 4 80.q even 4 1
800.2.a.j 2 80.s even 4 1
800.2.a.j 2 80.t odd 4 1
800.2.a.n 2 80.i odd 4 1
800.2.a.n 2 80.j even 4 1
1280.2.f.g 4 1.a even 1 1 trivial
1280.2.f.g 4 8.d odd 2 1 inner
1280.2.f.g 4 20.d odd 2 1 CM
1280.2.f.g 4 40.f even 2 1 inner
1280.2.f.h 4 4.b odd 2 1
1280.2.f.h 4 5.b even 2 1
1280.2.f.h 4 8.b even 2 1
1280.2.f.h 4 40.e odd 2 1
1440.2.f.i 4 48.i odd 4 1
1440.2.f.i 4 48.k even 4 1
1440.2.f.i 4 240.t even 4 1
1440.2.f.i 4 240.bm odd 4 1
1600.2.a.z 2 80.s even 4 1
1600.2.a.z 2 80.t odd 4 1
1600.2.a.bd 2 80.i odd 4 1
1600.2.a.bd 2 80.j even 4 1
2880.2.f.w 4 48.i odd 4 1
2880.2.f.w 4 48.k even 4 1
2880.2.f.w 4 240.t even 4 1
2880.2.f.w 4 240.bm odd 4 1
7200.2.a.cb 2 240.z odd 4 1
7200.2.a.cb 2 240.bf even 4 1
7200.2.a.cr 2 240.bb even 4 1
7200.2.a.cr 2 240.bd odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 2 T_{3} - 4 \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -4 + 2 T + T^{2} )^{2} \)
$5$ \( ( 5 + T^{2} )^{2} \)
$7$ \( 16 + 28 T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( 1936 + 92 T^{2} + T^{4} \)
$29$ \( ( 36 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( -20 + T^{2} )^{2} \)
$43$ \( ( 76 - 18 T + T^{2} )^{2} \)
$47$ \( 16 + 188 T^{2} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 180 + T^{2} )^{2} \)
$67$ \( ( -116 - 6 T + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( ( 76 - 22 T + T^{2} )^{2} \)
$89$ \( ( -6 + T )^{4} \)
$97$ \( T^{4} \)
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