Properties

Label 1280.2.d.c
Level 1280
Weight 2
Character orbit 1280.d
Analytic conductor 10.221
Analytic rank 1
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
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.

\(f(q)\) \(=\) \( q + 2 i q^{3} -i q^{5} -2 q^{7} - q^{9} +O(q^{10})\) \( q + 2 i q^{3} -i q^{5} -2 q^{7} - q^{9} -2 i q^{13} + 2 q^{15} -6 q^{17} + 4 i q^{19} -4 i q^{21} -6 q^{23} - q^{25} + 4 i q^{27} -6 i q^{29} -4 q^{31} + 2 i q^{35} + 2 i q^{37} + 4 q^{39} -6 q^{41} -10 i q^{43} + i q^{45} -6 q^{47} -3 q^{49} -12 i q^{51} -6 i q^{53} -8 q^{57} + 12 i q^{59} -2 i q^{61} + 2 q^{63} -2 q^{65} -2 i q^{67} -12 i q^{69} + 12 q^{71} -2 q^{73} -2 i q^{75} + 8 q^{79} -11 q^{81} -6 i q^{83} + 6 i q^{85} + 12 q^{87} + 6 q^{89} + 4 i q^{91} -8 i q^{93} + 4 q^{95} + 2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{7} - 2q^{9} + O(q^{10}) \) \( 2q - 4q^{7} - 2q^{9} + 4q^{15} - 12q^{17} - 12q^{23} - 2q^{25} - 8q^{31} + 8q^{39} - 12q^{41} - 12q^{47} - 6q^{49} - 16q^{57} + 4q^{63} - 4q^{65} + 24q^{71} - 4q^{73} + 16q^{79} - 22q^{81} + 24q^{87} + 12q^{89} + 8q^{95} + 4q^{97} + O(q^{100}) \)

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 2.00000i 0 1.00000i 0 −2.00000 0 −1.00000 0
641.2 0 2.00000i 0 1.00000i 0 −2.00000 0 −1.00000 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.c 2
4.b odd 2 1 1280.2.d.g 2
8.b even 2 1 inner 1280.2.d.c 2
8.d odd 2 1 1280.2.d.g 2
16.e even 4 1 20.2.a.a 1
16.e even 4 1 320.2.a.f 1
16.f odd 4 1 80.2.a.b 1
16.f odd 4 1 320.2.a.a 1
48.i odd 4 1 180.2.a.a 1
48.i odd 4 1 2880.2.a.m 1
48.k even 4 1 720.2.a.h 1
48.k even 4 1 2880.2.a.f 1
80.i odd 4 1 100.2.c.a 2
80.i odd 4 1 1600.2.c.d 2
80.j even 4 1 400.2.c.b 2
80.j even 4 1 1600.2.c.e 2
80.k odd 4 1 400.2.a.c 1
80.k odd 4 1 1600.2.a.w 1
80.q even 4 1 100.2.a.a 1
80.q even 4 1 1600.2.a.c 1
80.s even 4 1 400.2.c.b 2
80.s even 4 1 1600.2.c.e 2
80.t odd 4 1 100.2.c.a 2
80.t odd 4 1 1600.2.c.d 2
112.j even 4 1 3920.2.a.h 1
112.l odd 4 1 980.2.a.h 1
112.w even 12 2 980.2.i.i 2
112.x odd 12 2 980.2.i.c 2
144.w odd 12 2 1620.2.i.b 2
144.x even 12 2 1620.2.i.h 2
176.i even 4 1 9680.2.a.ba 1
176.l odd 4 1 2420.2.a.a 1
208.m odd 4 1 3380.2.f.b 2
208.p even 4 1 3380.2.a.c 1
208.r odd 4 1 3380.2.f.b 2
240.t even 4 1 3600.2.a.be 1
240.z odd 4 1 3600.2.f.j 2
240.bb even 4 1 900.2.d.c 2
240.bd odd 4 1 3600.2.f.j 2
240.bf even 4 1 900.2.d.c 2
240.bm odd 4 1 900.2.a.b 1
272.j even 4 1 5780.2.c.a 2
272.r even 4 1 5780.2.a.f 1
272.s even 4 1 5780.2.c.a 2
304.j odd 4 1 7220.2.a.f 1
336.y even 4 1 8820.2.a.g 1
560.r even 4 1 4900.2.e.f 2
560.bf odd 4 1 4900.2.a.e 1
560.bn even 4 1 4900.2.e.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 16.e even 4 1
80.2.a.b 1 16.f odd 4 1
100.2.a.a 1 80.q even 4 1
100.2.c.a 2 80.i odd 4 1
100.2.c.a 2 80.t odd 4 1
180.2.a.a 1 48.i odd 4 1
320.2.a.a 1 16.f odd 4 1
320.2.a.f 1 16.e even 4 1
400.2.a.c 1 80.k odd 4 1
400.2.c.b 2 80.j even 4 1
400.2.c.b 2 80.s even 4 1
720.2.a.h 1 48.k even 4 1
900.2.a.b 1 240.bm odd 4 1
900.2.d.c 2 240.bb even 4 1
900.2.d.c 2 240.bf even 4 1
980.2.a.h 1 112.l odd 4 1
980.2.i.c 2 112.x odd 12 2
980.2.i.i 2 112.w even 12 2
1280.2.d.c 2 1.a even 1 1 trivial
1280.2.d.c 2 8.b even 2 1 inner
1280.2.d.g 2 4.b odd 2 1
1280.2.d.g 2 8.d odd 2 1
1600.2.a.c 1 80.q even 4 1
1600.2.a.w 1 80.k odd 4 1
1600.2.c.d 2 80.i odd 4 1
1600.2.c.d 2 80.t odd 4 1
1600.2.c.e 2 80.j even 4 1
1600.2.c.e 2 80.s even 4 1
1620.2.i.b 2 144.w odd 12 2
1620.2.i.h 2 144.x even 12 2
2420.2.a.a 1 176.l odd 4 1
2880.2.a.f 1 48.k even 4 1
2880.2.a.m 1 48.i odd 4 1
3380.2.a.c 1 208.p even 4 1
3380.2.f.b 2 208.m odd 4 1
3380.2.f.b 2 208.r odd 4 1
3600.2.a.be 1 240.t even 4 1
3600.2.f.j 2 240.z odd 4 1
3600.2.f.j 2 240.bd odd 4 1
3920.2.a.h 1 112.j even 4 1
4900.2.a.e 1 560.bf odd 4 1
4900.2.e.f 2 560.r even 4 1
4900.2.e.f 2 560.bn even 4 1
5780.2.a.f 1 272.r even 4 1
5780.2.c.a 2 272.j even 4 1
5780.2.c.a 2 272.s even 4 1
7220.2.a.f 1 304.j odd 4 1
8820.2.a.g 1 336.y even 4 1
9680.2.a.ba 1 176.i even 4 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 2 T^{2} + 9 T^{4} \)
$5$ \( 1 + T^{2} \)
$7$ \( ( 1 + 2 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( 1 - 22 T^{2} + 169 T^{4} \)
$17$ \( ( 1 + 6 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 22 T^{2} + 361 T^{4} \)
$23$ \( ( 1 + 6 T + 23 T^{2} )^{2} \)
$29$ \( 1 - 22 T^{2} + 841 T^{4} \)
$31$ \( ( 1 + 4 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( 1 + 14 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 + 6 T + 47 T^{2} )^{2} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( 1 + 26 T^{2} + 3481 T^{4} \)
$61$ \( 1 - 118 T^{2} + 3721 T^{4} \)
$67$ \( 1 - 130 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 2 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 130 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 6 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 2 T + 97 T^{2} )^{2} \)
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