Properties

Label 1280.2.c.g
Level $1280$
Weight $2$
Character orbit 1280.c
Analytic conductor $10.221$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 320)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 3 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 5 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{3} + 5 \beta_{1}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1
1.93185i
0.517638i
1.93185i
0.517638i
0 2.44949i 0 −1.73205 1.41421i 0 1.41421i 0 −3.00000 0
769.2 0 2.44949i 0 1.73205 + 1.41421i 0 1.41421i 0 −3.00000 0
769.3 0 2.44949i 0 −1.73205 + 1.41421i 0 1.41421i 0 −3.00000 0
769.4 0 2.44949i 0 1.73205 1.41421i 0 1.41421i 0 −3.00000 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
5.b even 2 1 inner
8.d odd 2 1 inner
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 1280.2.c.g 4
4.b odd 2 1 1280.2.c.h 4
5.b even 2 1 inner 1280.2.c.g 4
5.c odd 4 2 6400.2.a.ct 4
8.b even 2 1 1280.2.c.h 4
8.d odd 2 1 inner 1280.2.c.g 4
16.e even 4 2 320.2.f.b 8
16.f odd 4 2 320.2.f.b 8
20.d odd 2 1 1280.2.c.h 4
20.e even 4 2 6400.2.a.cu 4
40.e odd 2 1 inner 1280.2.c.g 4
40.f even 2 1 1280.2.c.h 4
40.i odd 4 2 6400.2.a.cu 4
40.k even 4 2 6400.2.a.ct 4
48.i odd 4 2 2880.2.d.g 8
48.k even 4 2 2880.2.d.g 8
80.i odd 4 2 1600.2.d.i 8
80.j even 4 2 1600.2.d.i 8
80.k odd 4 2 320.2.f.b 8
80.q even 4 2 320.2.f.b 8
80.s even 4 2 1600.2.d.i 8
80.t odd 4 2 1600.2.d.i 8
240.t even 4 2 2880.2.d.g 8
240.bm odd 4 2 2880.2.d.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.f.b 8 16.e even 4 2
320.2.f.b 8 16.f odd 4 2
320.2.f.b 8 80.k odd 4 2
320.2.f.b 8 80.q even 4 2
1280.2.c.g 4 1.a even 1 1 trivial
1280.2.c.g 4 5.b even 2 1 inner
1280.2.c.g 4 8.d odd 2 1 inner
1280.2.c.g 4 40.e odd 2 1 inner
1280.2.c.h 4 4.b odd 2 1
1280.2.c.h 4 8.b even 2 1
1280.2.c.h 4 20.d odd 2 1
1280.2.c.h 4 40.f even 2 1
1600.2.d.i 8 80.i odd 4 2
1600.2.d.i 8 80.j even 4 2
1600.2.d.i 8 80.s even 4 2
1600.2.d.i 8 80.t odd 4 2
2880.2.d.g 8 48.i odd 4 2
2880.2.d.g 8 48.k even 4 2
2880.2.d.g 8 240.t even 4 2
2880.2.d.g 8 240.bm odd 4 2
6400.2.a.ct 4 5.c odd 4 2
6400.2.a.ct 4 40.k even 4 2
6400.2.a.cu 4 20.e even 4 2
6400.2.a.cu 4 40.i odd 4 2

Hecke kernels

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

\( T_{3}^{2} + 6 \)
\( T_{7}^{2} + 2 \)
\( T_{11} + 2 \)
\( T_{29}^{2} - 48 \)
\( T_{31}^{2} - 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 6 + T^{2} )^{2} \)
$5$ \( 25 - 2 T^{2} + T^{4} \)
$7$ \( ( 2 + T^{2} )^{2} \)
$11$ \( ( 2 + T )^{4} \)
$13$ \( ( 32 + T^{2} )^{2} \)
$17$ \( ( 24 + T^{2} )^{2} \)
$19$ \( ( -6 + T )^{4} \)
$23$ \( ( 50 + T^{2} )^{2} \)
$29$ \( ( -48 + T^{2} )^{2} \)
$31$ \( ( -48 + T^{2} )^{2} \)
$37$ \( ( 8 + T^{2} )^{2} \)
$41$ \( ( -4 + T )^{4} \)
$43$ \( ( 6 + T^{2} )^{2} \)
$47$ \( ( 18 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( ( -2 + T )^{4} \)
$61$ \( ( -12 + T^{2} )^{2} \)
$67$ \( ( 6 + T^{2} )^{2} \)
$71$ \( ( -48 + T^{2} )^{2} \)
$73$ \( ( 24 + T^{2} )^{2} \)
$79$ \( ( -48 + T^{2} )^{2} \)
$83$ \( ( 150 + T^{2} )^{2} \)
$89$ \( ( -2 + T )^{4} \)
$97$ \( ( 216 + T^{2} )^{2} \)
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