Properties

Label 1280.3.e.a
Level $1280$
Weight $3$
Character orbit 1280.e
Analytic conductor $34.877$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1280.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(34.8774738381\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -4 + 3 i ) q^{5} + 9 q^{9} +O(q^{10})\) \( q + ( -4 + 3 i ) q^{5} + 9 q^{9} -24 q^{13} + 16 i q^{17} + ( 7 - 24 i ) q^{25} -42 i q^{29} + 24 q^{37} + 18 q^{41} + ( -36 + 27 i ) q^{45} -49 q^{49} -56 q^{53} -22 i q^{61} + ( 96 - 72 i ) q^{65} -96 i q^{73} + 81 q^{81} + ( -48 - 64 i ) q^{85} + 78 q^{89} -144 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{5} + 18 q^{9} + O(q^{10}) \) \( 2 q - 8 q^{5} + 18 q^{9} - 48 q^{13} + 14 q^{25} + 48 q^{37} + 36 q^{41} - 72 q^{45} - 98 q^{49} - 112 q^{53} + 192 q^{65} + 162 q^{81} - 96 q^{85} + 156 q^{89} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
639.1
1.00000i
1.00000i
0 0 0 −4.00000 3.00000i 0 0 0 9.00000 0
639.2 0 0 0 −4.00000 + 3.00000i 0 0 0 9.00000 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
40.e odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.3.e.a 2
4.b odd 2 1 CM 1280.3.e.a 2
5.b even 2 1 1280.3.e.d 2
8.b even 2 1 1280.3.e.d 2
8.d odd 2 1 1280.3.e.d 2
16.e even 4 1 20.3.d.c 2
16.e even 4 1 320.3.h.d 2
16.f odd 4 1 20.3.d.c 2
16.f odd 4 1 320.3.h.d 2
20.d odd 2 1 1280.3.e.d 2
40.e odd 2 1 inner 1280.3.e.a 2
40.f even 2 1 inner 1280.3.e.a 2
48.i odd 4 1 180.3.f.c 2
48.k even 4 1 180.3.f.c 2
80.i odd 4 1 100.3.b.a 1
80.i odd 4 1 1600.3.b.c 1
80.j even 4 1 100.3.b.b 1
80.j even 4 1 1600.3.b.a 1
80.k odd 4 1 20.3.d.c 2
80.k odd 4 1 320.3.h.d 2
80.q even 4 1 20.3.d.c 2
80.q even 4 1 320.3.h.d 2
80.s even 4 1 100.3.b.a 1
80.s even 4 1 1600.3.b.c 1
80.t odd 4 1 100.3.b.b 1
80.t odd 4 1 1600.3.b.a 1
240.t even 4 1 180.3.f.c 2
240.z odd 4 1 900.3.c.d 1
240.bb even 4 1 900.3.c.d 1
240.bd odd 4 1 900.3.c.a 1
240.bf even 4 1 900.3.c.a 1
240.bm odd 4 1 180.3.f.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.d.c 2 16.e even 4 1
20.3.d.c 2 16.f odd 4 1
20.3.d.c 2 80.k odd 4 1
20.3.d.c 2 80.q even 4 1
100.3.b.a 1 80.i odd 4 1
100.3.b.a 1 80.s even 4 1
100.3.b.b 1 80.j even 4 1
100.3.b.b 1 80.t odd 4 1
180.3.f.c 2 48.i odd 4 1
180.3.f.c 2 48.k even 4 1
180.3.f.c 2 240.t even 4 1
180.3.f.c 2 240.bm odd 4 1
320.3.h.d 2 16.e even 4 1
320.3.h.d 2 16.f odd 4 1
320.3.h.d 2 80.k odd 4 1
320.3.h.d 2 80.q even 4 1
900.3.c.a 1 240.bd odd 4 1
900.3.c.a 1 240.bf even 4 1
900.3.c.d 1 240.z odd 4 1
900.3.c.d 1 240.bb even 4 1
1280.3.e.a 2 1.a even 1 1 trivial
1280.3.e.a 2 4.b odd 2 1 CM
1280.3.e.a 2 40.e odd 2 1 inner
1280.3.e.a 2 40.f even 2 1 inner
1280.3.e.d 2 5.b even 2 1
1280.3.e.d 2 8.b even 2 1
1280.3.e.d 2 8.d odd 2 1
1280.3.e.d 2 20.d odd 2 1
1600.3.b.a 1 80.j even 4 1
1600.3.b.a 1 80.t odd 4 1
1600.3.b.c 1 80.i odd 4 1
1600.3.b.c 1 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 \(S_{3}^{\mathrm{new}}(1280, [\chi])\):

\( T_{3} \)
\( T_{7} \)
\( T_{11} \)
\( T_{13} + 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 25 + 8 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 24 + T )^{2} \)
$17$ \( 256 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( 1764 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( -24 + T )^{2} \)
$41$ \( ( -18 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( ( 56 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( 484 + T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 9216 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( ( -78 + T )^{2} \)
$97$ \( 20736 + T^{2} \)
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