Properties

Label 1280.2.d.l
Level $1280$
Weight $2$
Character orbit 1280.d
Analytic conductor $10.221$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{3} + \zeta_{8}^{2} q^{5} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{7} -5 q^{9} +O(q^{10})\) \( q + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{3} + \zeta_{8}^{2} q^{5} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{7} -5 q^{9} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{11} + 2 \zeta_{8}^{2} q^{13} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{15} + 2 q^{17} -8 \zeta_{8}^{2} q^{21} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{23} - q^{25} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{27} -6 \zeta_{8}^{2} q^{29} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{31} -16 q^{33} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{35} -10 \zeta_{8}^{2} q^{37} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{39} -2 q^{41} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{43} -5 \zeta_{8}^{2} q^{45} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{47} + q^{49} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{51} + 6 \zeta_{8}^{2} q^{53} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{55} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{59} + 2 \zeta_{8}^{2} q^{61} + ( 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{63} -2 q^{65} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{67} + 8 \zeta_{8}^{2} q^{69} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{71} + 6 q^{73} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{75} -16 \zeta_{8}^{2} q^{77} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{79} + q^{81} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{83} + 2 \zeta_{8}^{2} q^{85} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{87} -10 q^{89} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{91} -16 \zeta_{8}^{2} q^{93} + 2 q^{97} + ( -20 \zeta_{8} - 20 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 20q^{9} + O(q^{10}) \) \( 4q - 20q^{9} + 8q^{17} - 4q^{25} - 64q^{33} - 8q^{41} + 4q^{49} - 8q^{65} + 24q^{73} + 4q^{81} - 40q^{89} + 8q^{97} + O(q^{100}) \)

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
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0 2.82843i 0 1.00000i 0 −2.82843 0 −5.00000 0
641.2 0 2.82843i 0 1.00000i 0 2.82843 0 −5.00000 0
641.3 0 2.82843i 0 1.00000i 0 2.82843 0 −5.00000 0
641.4 0 2.82843i 0 1.00000i 0 −2.82843 0 −5.00000 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 inner
8.b even 2 1 inner
8.d odd 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.l 4
4.b odd 2 1 inner 1280.2.d.l 4
8.b even 2 1 inner 1280.2.d.l 4
8.d odd 2 1 inner 1280.2.d.l 4
16.e even 4 1 160.2.a.c 2
16.e even 4 1 320.2.a.g 2
16.f odd 4 1 160.2.a.c 2
16.f odd 4 1 320.2.a.g 2
48.i odd 4 1 1440.2.a.o 2
48.i odd 4 1 2880.2.a.bk 2
48.k even 4 1 1440.2.a.o 2
48.k even 4 1 2880.2.a.bk 2
80.i odd 4 1 800.2.c.f 4
80.i odd 4 1 1600.2.c.n 4
80.j even 4 1 800.2.c.f 4
80.j even 4 1 1600.2.c.n 4
80.k odd 4 1 800.2.a.m 2
80.k odd 4 1 1600.2.a.bc 2
80.q even 4 1 800.2.a.m 2
80.q even 4 1 1600.2.a.bc 2
80.s even 4 1 800.2.c.f 4
80.s even 4 1 1600.2.c.n 4
80.t odd 4 1 800.2.c.f 4
80.t odd 4 1 1600.2.c.n 4
112.j even 4 1 7840.2.a.bf 2
112.l odd 4 1 7840.2.a.bf 2
240.t even 4 1 7200.2.a.cm 2
240.z odd 4 1 7200.2.f.bh 4
240.bb even 4 1 7200.2.f.bh 4
240.bd odd 4 1 7200.2.f.bh 4
240.bf even 4 1 7200.2.f.bh 4
240.bm odd 4 1 7200.2.a.cm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.a.c 2 16.e even 4 1
160.2.a.c 2 16.f odd 4 1
320.2.a.g 2 16.e even 4 1
320.2.a.g 2 16.f odd 4 1
800.2.a.m 2 80.k odd 4 1
800.2.a.m 2 80.q even 4 1
800.2.c.f 4 80.i odd 4 1
800.2.c.f 4 80.j even 4 1
800.2.c.f 4 80.s even 4 1
800.2.c.f 4 80.t odd 4 1
1280.2.d.l 4 1.a even 1 1 trivial
1280.2.d.l 4 4.b odd 2 1 inner
1280.2.d.l 4 8.b even 2 1 inner
1280.2.d.l 4 8.d odd 2 1 inner
1440.2.a.o 2 48.i odd 4 1
1440.2.a.o 2 48.k even 4 1
1600.2.a.bc 2 80.k odd 4 1
1600.2.a.bc 2 80.q even 4 1
1600.2.c.n 4 80.i odd 4 1
1600.2.c.n 4 80.j even 4 1
1600.2.c.n 4 80.s even 4 1
1600.2.c.n 4 80.t odd 4 1
2880.2.a.bk 2 48.i odd 4 1
2880.2.a.bk 2 48.k even 4 1
7200.2.a.cm 2 240.t even 4 1
7200.2.a.cm 2 240.bm odd 4 1
7200.2.f.bh 4 240.z odd 4 1
7200.2.f.bh 4 240.bb even 4 1
7200.2.f.bh 4 240.bd odd 4 1
7200.2.f.bh 4 240.bf even 4 1
7840.2.a.bf 2 112.j even 4 1
7840.2.a.bf 2 112.l odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 8 \)
\( T_{7}^{2} - 8 \)
\( T_{11}^{2} + 32 \)
\( T_{31}^{2} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 8 + T^{2} )^{2} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( ( -8 + T^{2} )^{2} \)
$11$ \( ( 32 + T^{2} )^{2} \)
$13$ \( ( 4 + T^{2} )^{2} \)
$17$ \( ( -2 + T )^{4} \)
$19$ \( T^{4} \)
$23$ \( ( -8 + T^{2} )^{2} \)
$29$ \( ( 36 + T^{2} )^{2} \)
$31$ \( ( -32 + T^{2} )^{2} \)
$37$ \( ( 100 + T^{2} )^{2} \)
$41$ \( ( 2 + T )^{4} \)
$43$ \( ( 72 + T^{2} )^{2} \)
$47$ \( ( -8 + T^{2} )^{2} \)
$53$ \( ( 36 + T^{2} )^{2} \)
$59$ \( ( 128 + T^{2} )^{2} \)
$61$ \( ( 4 + T^{2} )^{2} \)
$67$ \( ( 8 + T^{2} )^{2} \)
$71$ \( ( -32 + T^{2} )^{2} \)
$73$ \( ( -6 + T )^{4} \)
$79$ \( ( -128 + T^{2} )^{2} \)
$83$ \( ( 8 + T^{2} )^{2} \)
$89$ \( ( 10 + T )^{4} \)
$97$ \( ( -2 + T )^{4} \)
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