Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{5} -4 q^{7} + 3 q^{9} +O(q^{10})\) \( q + i q^{5} -4 q^{7} + 3 q^{9} -4 i q^{11} + 2 i q^{13} + 2 q^{17} + 4 i q^{19} + 4 q^{23} - q^{25} + 2 i q^{29} + 8 q^{31} -4 i q^{35} + 6 i q^{37} + 6 q^{41} + 8 i q^{43} + 3 i q^{45} -4 q^{47} + 9 q^{49} + 6 i q^{53} + 4 q^{55} + 4 i q^{59} + 2 i q^{61} -12 q^{63} -2 q^{65} + 8 i q^{67} + 6 q^{73} + 16 i q^{77} + 9 q^{81} -16 i q^{83} + 2 i q^{85} + 6 q^{89} -8 i q^{91} -4 q^{95} -14 q^{97} -12 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{7} + 6q^{9} + O(q^{10}) \) \( 2q - 8q^{7} + 6q^{9} + 4q^{17} + 8q^{23} - 2q^{25} + 16q^{31} + 12q^{41} - 8q^{47} + 18q^{49} + 8q^{55} - 24q^{63} - 4q^{65} + 12q^{73} + 18q^{81} + 12q^{89} - 8q^{95} - 28q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
641.1
1.00000i
1.00000i
0 0 0 1.00000i 0 −4.00000 0 3.00000 0
641.2 0 0 0 1.00000i 0 −4.00000 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
8.b 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.d.a 2
4.b odd 2 1 1280.2.d.j 2
8.b even 2 1 inner 1280.2.d.a 2
8.d odd 2 1 1280.2.d.j 2
16.e even 4 1 80.2.a.a 1
16.e even 4 1 320.2.a.d 1
16.f odd 4 1 40.2.a.a 1
16.f odd 4 1 320.2.a.c 1
48.i odd 4 1 720.2.a.e 1
48.i odd 4 1 2880.2.a.bg 1
48.k even 4 1 360.2.a.a 1
48.k even 4 1 2880.2.a.t 1
80.i odd 4 1 400.2.c.d 2
80.i odd 4 1 1600.2.c.m 2
80.j even 4 1 200.2.c.b 2
80.j even 4 1 1600.2.c.k 2
80.k odd 4 1 200.2.a.c 1
80.k odd 4 1 1600.2.a.o 1
80.q even 4 1 400.2.a.e 1
80.q even 4 1 1600.2.a.k 1
80.s even 4 1 200.2.c.b 2
80.s even 4 1 1600.2.c.k 2
80.t odd 4 1 400.2.c.d 2
80.t odd 4 1 1600.2.c.m 2
112.j even 4 1 1960.2.a.g 1
112.l odd 4 1 3920.2.a.s 1
112.u odd 12 2 1960.2.q.h 2
112.v even 12 2 1960.2.q.i 2
144.u even 12 2 3240.2.q.x 2
144.v odd 12 2 3240.2.q.k 2
176.i even 4 1 4840.2.a.f 1
176.l odd 4 1 9680.2.a.q 1
208.o odd 4 1 6760.2.a.i 1
240.t even 4 1 1800.2.a.v 1
240.z odd 4 1 1800.2.f.a 2
240.bb even 4 1 3600.2.f.t 2
240.bd odd 4 1 1800.2.f.a 2
240.bf even 4 1 3600.2.f.t 2
240.bm odd 4 1 3600.2.a.h 1
560.be even 4 1 9800.2.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.a.a 1 16.f odd 4 1
80.2.a.a 1 16.e even 4 1
200.2.a.c 1 80.k odd 4 1
200.2.c.b 2 80.j even 4 1
200.2.c.b 2 80.s even 4 1
320.2.a.c 1 16.f odd 4 1
320.2.a.d 1 16.e even 4 1
360.2.a.a 1 48.k even 4 1
400.2.a.e 1 80.q even 4 1
400.2.c.d 2 80.i odd 4 1
400.2.c.d 2 80.t odd 4 1
720.2.a.e 1 48.i odd 4 1
1280.2.d.a 2 1.a even 1 1 trivial
1280.2.d.a 2 8.b even 2 1 inner
1280.2.d.j 2 4.b odd 2 1
1280.2.d.j 2 8.d odd 2 1
1600.2.a.k 1 80.q even 4 1
1600.2.a.o 1 80.k odd 4 1
1600.2.c.k 2 80.j even 4 1
1600.2.c.k 2 80.s even 4 1
1600.2.c.m 2 80.i odd 4 1
1600.2.c.m 2 80.t odd 4 1
1800.2.a.v 1 240.t even 4 1
1800.2.f.a 2 240.z odd 4 1
1800.2.f.a 2 240.bd odd 4 1
1960.2.a.g 1 112.j even 4 1
1960.2.q.h 2 112.u odd 12 2
1960.2.q.i 2 112.v even 12 2
2880.2.a.t 1 48.k even 4 1
2880.2.a.bg 1 48.i odd 4 1
3240.2.q.k 2 144.v odd 12 2
3240.2.q.x 2 144.u even 12 2
3600.2.a.h 1 240.bm odd 4 1
3600.2.f.t 2 240.bb even 4 1
3600.2.f.t 2 240.bf even 4 1
3920.2.a.s 1 112.l odd 4 1
4840.2.a.f 1 176.i even 4 1
6760.2.a.i 1 208.o odd 4 1
9680.2.a.q 1 176.l odd 4 1
9800.2.a.x 1 560.be even 4 1

Hecke kernels

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

\( T_{3} \)
\( T_{7} + 4 \)
\( T_{11}^{2} + 16 \)
\( T_{31} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 + T^{2} \)
$7$ \( ( 4 + T )^{2} \)
$11$ \( 16 + T^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( 16 + T^{2} \)
$23$ \( ( -4 + T )^{2} \)
$29$ \( 4 + T^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( ( 4 + T )^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( 16 + T^{2} \)
$61$ \( 4 + T^{2} \)
$67$ \( 64 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -6 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( 256 + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( ( 14 + T )^{2} \)
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