Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.o (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 2 i - 1) q^{5} + 3 i q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 i - 1) q^{5} + 3 i q^{9} + (i + 1) q^{13} + (3 i + 3) q^{17} + (4 i - 3) q^{25} + 4 q^{29} + ( - 7 i + 7) q^{37} + 8 q^{41} + ( - 3 i + 6) q^{45} + 7 i q^{49} + (9 i + 9) q^{53} - 12 i q^{61} + ( - 3 i + 1) q^{65} + ( - 11 i + 11) q^{73} - 9 q^{81} + ( - 9 i + 3) q^{85} + 16 i q^{89} + (13 i + 13) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 2 q^{13} + 6 q^{17} - 6 q^{25} + 8 q^{29} + 14 q^{37} + 16 q^{41} + 12 q^{45} + 18 q^{53} + 2 q^{65} + 22 q^{73} - 18 q^{81} + 6 q^{85} + 26 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(i\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
1.00000i
1.00000i
0 0 0 −1.00000 2.00000i 0 0 0 3.00000i 0
383.1 0 0 0 −1.00000 + 2.00000i 0 0 0 3.00000i 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
40.i odd 4 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.2.o.g 2
4.b odd 2 1 CM 1280.2.o.g 2
5.c odd 4 1 1280.2.o.j 2
8.b even 2 1 1280.2.o.j 2
8.d odd 2 1 1280.2.o.j 2
16.e even 4 1 20.2.e.a 2
16.e even 4 1 320.2.n.e 2
16.f odd 4 1 20.2.e.a 2
16.f odd 4 1 320.2.n.e 2
20.e even 4 1 1280.2.o.j 2
40.i odd 4 1 inner 1280.2.o.g 2
40.k even 4 1 inner 1280.2.o.g 2
48.i odd 4 1 180.2.k.c 2
48.k even 4 1 180.2.k.c 2
80.i odd 4 1 20.2.e.a 2
80.i odd 4 1 1600.2.n.h 2
80.j even 4 1 100.2.e.b 2
80.j even 4 1 320.2.n.e 2
80.k odd 4 1 100.2.e.b 2
80.k odd 4 1 1600.2.n.h 2
80.q even 4 1 100.2.e.b 2
80.q even 4 1 1600.2.n.h 2
80.s even 4 1 20.2.e.a 2
80.s even 4 1 1600.2.n.h 2
80.t odd 4 1 100.2.e.b 2
80.t odd 4 1 320.2.n.e 2
112.j even 4 1 980.2.k.a 2
112.l odd 4 1 980.2.k.a 2
112.u odd 12 2 980.2.x.d 4
112.v even 12 2 980.2.x.c 4
112.w even 12 2 980.2.x.d 4
112.x odd 12 2 980.2.x.c 4
240.t even 4 1 900.2.k.c 2
240.z odd 4 1 180.2.k.c 2
240.bb even 4 1 180.2.k.c 2
240.bd odd 4 1 900.2.k.c 2
240.bf even 4 1 900.2.k.c 2
240.bm odd 4 1 900.2.k.c 2
560.u odd 4 1 980.2.k.a 2
560.bn even 4 1 980.2.k.a 2
560.cf even 12 2 980.2.x.d 4
560.ch even 12 2 980.2.x.c 4
560.cy odd 12 2 980.2.x.d 4
560.da odd 12 2 980.2.x.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.e.a 2 16.e even 4 1
20.2.e.a 2 16.f odd 4 1
20.2.e.a 2 80.i odd 4 1
20.2.e.a 2 80.s even 4 1
100.2.e.b 2 80.j even 4 1
100.2.e.b 2 80.k odd 4 1
100.2.e.b 2 80.q even 4 1
100.2.e.b 2 80.t odd 4 1
180.2.k.c 2 48.i odd 4 1
180.2.k.c 2 48.k even 4 1
180.2.k.c 2 240.z odd 4 1
180.2.k.c 2 240.bb even 4 1
320.2.n.e 2 16.e even 4 1
320.2.n.e 2 16.f odd 4 1
320.2.n.e 2 80.j even 4 1
320.2.n.e 2 80.t odd 4 1
900.2.k.c 2 240.t even 4 1
900.2.k.c 2 240.bd odd 4 1
900.2.k.c 2 240.bf even 4 1
900.2.k.c 2 240.bm odd 4 1
980.2.k.a 2 112.j even 4 1
980.2.k.a 2 112.l odd 4 1
980.2.k.a 2 560.u odd 4 1
980.2.k.a 2 560.bn even 4 1
980.2.x.c 4 112.v even 12 2
980.2.x.c 4 112.x odd 12 2
980.2.x.c 4 560.ch even 12 2
980.2.x.c 4 560.da odd 12 2
980.2.x.d 4 112.u odd 12 2
980.2.x.d 4 112.w even 12 2
980.2.x.d 4 560.cf even 12 2
980.2.x.d 4 560.cy odd 12 2
1280.2.o.g 2 1.a even 1 1 trivial
1280.2.o.g 2 4.b odd 2 1 CM
1280.2.o.g 2 40.i odd 4 1 inner
1280.2.o.g 2 40.k even 4 1 inner
1280.2.o.j 2 5.c odd 4 1
1280.2.o.j 2 8.b even 2 1
1280.2.o.j 2 8.d odd 2 1
1280.2.o.j 2 20.e even 4 1
1600.2.n.h 2 80.i odd 4 1
1600.2.n.h 2 80.k odd 4 1
1600.2.n.h 2 80.q even 4 1
1600.2.n.h 2 80.s even 4 1

Hecke kernels

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

\( T_{3} \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$41$ \( (T - 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 18T + 162 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 144 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 22T + 242 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 256 \) Copy content Toggle raw display
$97$ \( T^{2} - 26T + 338 \) Copy content Toggle raw display
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