Properties

Label 1280.3.e.c
Level $1280$
Weight $3$
Character orbit 1280.e
Analytic conductor $34.877$
Analytic rank $0$
Dimension $2$
CM discriminant -20
Inner twists $4$

Related objects

Downloads

Learn more

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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + 4 i q^{3} + 5 i q^{5} + 4 q^{7} -7 q^{9} +O(q^{10})\) \( q + 4 i q^{3} + 5 i q^{5} + 4 q^{7} -7 q^{9} -20 q^{15} + 16 i q^{21} -44 q^{23} -25 q^{25} + 8 i q^{27} -22 i q^{29} + 20 i q^{35} -62 q^{41} + 76 i q^{43} -35 i q^{45} -4 q^{47} -33 q^{49} -58 i q^{61} -28 q^{63} + 116 i q^{67} -176 i q^{69} -100 i q^{75} -95 q^{81} -76 i q^{83} + 88 q^{87} + 142 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{7} - 14 q^{9} + O(q^{10}) \) \( 2 q + 8 q^{7} - 14 q^{9} - 40 q^{15} - 88 q^{23} - 50 q^{25} - 124 q^{41} - 8 q^{47} - 66 q^{49} - 56 q^{63} - 190 q^{81} + 176 q^{87} + 284 q^{89} + O(q^{100}) \)

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 4.00000i 0 5.00000i 0 4.00000 0 −7.00000 0
639.2 0 4.00000i 0 5.00000i 0 4.00000 0 −7.00000 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
8.b even 2 1 inner
40.e odd 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.c 2
4.b odd 2 1 1280.3.e.b 2
5.b even 2 1 1280.3.e.b 2
8.b even 2 1 inner 1280.3.e.c 2
8.d odd 2 1 1280.3.e.b 2
16.e even 4 1 20.3.d.a 1
16.e even 4 1 320.3.h.a 1
16.f odd 4 1 20.3.d.b yes 1
16.f odd 4 1 320.3.h.b 1
20.d odd 2 1 CM 1280.3.e.c 2
40.e odd 2 1 inner 1280.3.e.c 2
40.f even 2 1 1280.3.e.b 2
48.i odd 4 1 180.3.f.b 1
48.k even 4 1 180.3.f.a 1
80.i odd 4 1 100.3.b.c 2
80.i odd 4 1 1600.3.b.f 2
80.j even 4 1 100.3.b.c 2
80.j even 4 1 1600.3.b.f 2
80.k odd 4 1 20.3.d.a 1
80.k odd 4 1 320.3.h.a 1
80.q even 4 1 20.3.d.b yes 1
80.q even 4 1 320.3.h.b 1
80.s even 4 1 100.3.b.c 2
80.s even 4 1 1600.3.b.f 2
80.t odd 4 1 100.3.b.c 2
80.t odd 4 1 1600.3.b.f 2
240.t even 4 1 180.3.f.b 1
240.z odd 4 1 900.3.c.h 2
240.bb even 4 1 900.3.c.h 2
240.bd odd 4 1 900.3.c.h 2
240.bf even 4 1 900.3.c.h 2
240.bm odd 4 1 180.3.f.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.d.a 1 16.e even 4 1
20.3.d.a 1 80.k odd 4 1
20.3.d.b yes 1 16.f odd 4 1
20.3.d.b yes 1 80.q even 4 1
100.3.b.c 2 80.i odd 4 1
100.3.b.c 2 80.j even 4 1
100.3.b.c 2 80.s even 4 1
100.3.b.c 2 80.t odd 4 1
180.3.f.a 1 48.k even 4 1
180.3.f.a 1 240.bm odd 4 1
180.3.f.b 1 48.i odd 4 1
180.3.f.b 1 240.t even 4 1
320.3.h.a 1 16.e even 4 1
320.3.h.a 1 80.k odd 4 1
320.3.h.b 1 16.f odd 4 1
320.3.h.b 1 80.q even 4 1
900.3.c.h 2 240.z odd 4 1
900.3.c.h 2 240.bb even 4 1
900.3.c.h 2 240.bd odd 4 1
900.3.c.h 2 240.bf even 4 1
1280.3.e.b 2 4.b odd 2 1
1280.3.e.b 2 5.b even 2 1
1280.3.e.b 2 8.d odd 2 1
1280.3.e.b 2 40.f even 2 1
1280.3.e.c 2 1.a even 1 1 trivial
1280.3.e.c 2 8.b even 2 1 inner
1280.3.e.c 2 20.d odd 2 1 CM
1280.3.e.c 2 40.e odd 2 1 inner
1600.3.b.f 2 80.i odd 4 1
1600.3.b.f 2 80.j even 4 1
1600.3.b.f 2 80.s even 4 1
1600.3.b.f 2 80.t odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1280, [\chi])\):

\( T_{3}^{2} + 16 \)
\( T_{7} - 4 \)
\( T_{11} \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 16 + T^{2} \)
$5$ \( 25 + T^{2} \)
$7$ \( ( -4 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( ( 44 + T )^{2} \)
$29$ \( 484 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( ( 62 + T )^{2} \)
$43$ \( 5776 + T^{2} \)
$47$ \( ( 4 + T )^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 3364 + T^{2} \)
$67$ \( 13456 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( T^{2} \)
$79$ \( T^{2} \)
$83$ \( 5776 + T^{2} \)
$89$ \( ( -142 + T )^{2} \)
$97$ \( T^{2} \)
show more
show less