Properties

Label 1280.2.a.n
Level $1280$
Weight $2$
Character orbit 1280.a
Self dual yes
Analytic conductor $10.221$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{3} - q^{5} + ( -1 + \beta ) q^{7} + ( 1 + 2 \beta ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{3} - q^{5} + ( -1 + \beta ) q^{7} + ( 1 + 2 \beta ) q^{9} + 2 q^{11} + 2 \beta q^{13} + ( -1 - \beta ) q^{15} + 2 \beta q^{17} + ( 4 - 2 \beta ) q^{19} + 2 q^{21} + ( -1 - 3 \beta ) q^{23} + q^{25} + 4 q^{27} -4 \beta q^{29} + ( 2 + 2 \beta ) q^{31} + ( 2 + 2 \beta ) q^{33} + ( 1 - \beta ) q^{35} -2 q^{37} + ( 6 + 2 \beta ) q^{39} + ( 2 - 2 \beta ) q^{41} + ( 7 - \beta ) q^{43} + ( -1 - 2 \beta ) q^{45} + ( -5 + \beta ) q^{47} + ( -3 - 2 \beta ) q^{49} + ( 6 + 2 \beta ) q^{51} + ( 8 + 2 \beta ) q^{53} -2 q^{55} + ( -2 + 2 \beta ) q^{57} + ( 4 + 2 \beta ) q^{59} + ( 2 + 4 \beta ) q^{61} + ( 5 - \beta ) q^{63} -2 \beta q^{65} + ( 9 + \beta ) q^{67} + ( -10 - 4 \beta ) q^{69} + ( 2 + 2 \beta ) q^{71} + ( -4 - 2 \beta ) q^{73} + ( 1 + \beta ) q^{75} + ( -2 + 2 \beta ) q^{77} + ( 8 - 4 \beta ) q^{79} + ( 1 - 2 \beta ) q^{81} + ( 3 - \beta ) q^{83} -2 \beta q^{85} + ( -12 - 4 \beta ) q^{87} + ( -2 - 4 \beta ) q^{89} + ( 6 - 2 \beta ) q^{91} + ( 8 + 4 \beta ) q^{93} + ( -4 + 2 \beta ) q^{95} + ( -4 - 6 \beta ) q^{97} + ( 2 + 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{11} - 2 q^{15} + 8 q^{19} + 4 q^{21} - 2 q^{23} + 2 q^{25} + 8 q^{27} + 4 q^{31} + 4 q^{33} + 2 q^{35} - 4 q^{37} + 12 q^{39} + 4 q^{41} + 14 q^{43} - 2 q^{45} - 10 q^{47} - 6 q^{49} + 12 q^{51} + 16 q^{53} - 4 q^{55} - 4 q^{57} + 8 q^{59} + 4 q^{61} + 10 q^{63} + 18 q^{67} - 20 q^{69} + 4 q^{71} - 8 q^{73} + 2 q^{75} - 4 q^{77} + 16 q^{79} + 2 q^{81} + 6 q^{83} - 24 q^{87} - 4 q^{89} + 12 q^{91} + 16 q^{93} - 8 q^{95} - 8 q^{97} + 4 q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.73205
1.73205
0 −0.732051 0 −1.00000 0 −2.73205 0 −2.46410 0
1.2 0 2.73205 0 −1.00000 0 0.732051 0 4.46410 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.2.a.n 2
4.b odd 2 1 1280.2.a.a 2
5.b even 2 1 6400.2.a.be 2
8.b even 2 1 1280.2.a.d 2
8.d odd 2 1 1280.2.a.o 2
16.e even 4 2 160.2.d.a 4
16.f odd 4 2 40.2.d.a 4
20.d odd 2 1 6400.2.a.ce 2
40.e odd 2 1 6400.2.a.z 2
40.f even 2 1 6400.2.a.cj 2
48.i odd 4 2 1440.2.k.e 4
48.k even 4 2 360.2.k.e 4
80.i odd 4 2 800.2.f.c 4
80.j even 4 2 200.2.f.e 4
80.k odd 4 2 200.2.d.f 4
80.q even 4 2 800.2.d.e 4
80.s even 4 2 200.2.f.c 4
80.t odd 4 2 800.2.f.e 4
240.t even 4 2 1800.2.k.j 4
240.z odd 4 2 1800.2.d.p 4
240.bb even 4 2 7200.2.d.o 4
240.bd odd 4 2 1800.2.d.l 4
240.bf even 4 2 7200.2.d.n 4
240.bm odd 4 2 7200.2.k.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.d.a 4 16.f odd 4 2
160.2.d.a 4 16.e even 4 2
200.2.d.f 4 80.k odd 4 2
200.2.f.c 4 80.s even 4 2
200.2.f.e 4 80.j even 4 2
360.2.k.e 4 48.k even 4 2
800.2.d.e 4 80.q even 4 2
800.2.f.c 4 80.i odd 4 2
800.2.f.e 4 80.t odd 4 2
1280.2.a.a 2 4.b odd 2 1
1280.2.a.d 2 8.b even 2 1
1280.2.a.n 2 1.a even 1 1 trivial
1280.2.a.o 2 8.d odd 2 1
1440.2.k.e 4 48.i odd 4 2
1800.2.d.l 4 240.bd odd 4 2
1800.2.d.p 4 240.z odd 4 2
1800.2.k.j 4 240.t even 4 2
6400.2.a.z 2 40.e odd 2 1
6400.2.a.be 2 5.b even 2 1
6400.2.a.ce 2 20.d odd 2 1
6400.2.a.cj 2 40.f even 2 1
7200.2.d.n 4 240.bf even 4 2
7200.2.d.o 4 240.bb even 4 2
7200.2.k.j 4 240.bm odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1280))\):

\( T_{3}^{2} - 2 T_{3} - 2 \)
\( T_{7}^{2} + 2 T_{7} - 2 \)
\( T_{11} - 2 \)
\( T_{13}^{2} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -2 - 2 T + T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -2 + 2 T + T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( -12 + T^{2} \)
$17$ \( -12 + T^{2} \)
$19$ \( 4 - 8 T + T^{2} \)
$23$ \( -26 + 2 T + T^{2} \)
$29$ \( -48 + T^{2} \)
$31$ \( -8 - 4 T + T^{2} \)
$37$ \( ( 2 + T )^{2} \)
$41$ \( -8 - 4 T + T^{2} \)
$43$ \( 46 - 14 T + T^{2} \)
$47$ \( 22 + 10 T + T^{2} \)
$53$ \( 52 - 16 T + T^{2} \)
$59$ \( 4 - 8 T + T^{2} \)
$61$ \( -44 - 4 T + T^{2} \)
$67$ \( 78 - 18 T + T^{2} \)
$71$ \( -8 - 4 T + T^{2} \)
$73$ \( 4 + 8 T + T^{2} \)
$79$ \( 16 - 16 T + T^{2} \)
$83$ \( 6 - 6 T + T^{2} \)
$89$ \( -44 + 4 T + T^{2} \)
$97$ \( -92 + 8 T + T^{2} \)
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