Properties

Label 1280.3.e.e
Level $1280$
Weight $3$
Character orbit 1280.e
Analytic conductor $34.877$
Analytic rank $0$
Dimension $4$
CM discriminant -20
Inner twists $8$

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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: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + 5 \beta_{1} q^{5} + 3 \beta_{3} q^{7} -11 q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + 5 \beta_{1} q^{5} + 3 \beta_{3} q^{7} -11 q^{9} + 5 \beta_{3} q^{15} -60 \beta_{1} q^{21} + 3 \beta_{3} q^{23} -25 q^{25} + 2 \beta_{2} q^{27} + 22 \beta_{1} q^{29} + 15 \beta_{2} q^{35} + 62 q^{41} + 9 \beta_{2} q^{43} -55 \beta_{1} q^{45} + 21 \beta_{3} q^{47} + 131 q^{49} -58 \beta_{1} q^{61} -33 \beta_{3} q^{63} + 15 \beta_{2} q^{67} -60 \beta_{1} q^{69} + 25 \beta_{2} q^{75} -59 q^{81} -33 \beta_{2} q^{83} + 22 \beta_{3} q^{87} + 142 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 44 q^{9} + O(q^{10}) \) \( 4 q - 44 q^{9} - 100 q^{25} + 248 q^{41} + 524 q^{49} - 236 q^{81} + 568 q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 2 \nu \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} + 8 \nu \)
\(\beta_{3}\)\(=\)\( 4 \nu^{2} + 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 2 \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 6\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{2} + 4 \beta_{1}\)\()/2\)

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
1.61803i
0.618034i
0.618034i
1.61803i
0 4.47214i 0 5.00000i 0 −13.4164 0 −11.0000 0
639.2 0 4.47214i 0 5.00000i 0 13.4164 0 −11.0000 0
639.3 0 4.47214i 0 5.00000i 0 13.4164 0 −11.0000 0
639.4 0 4.47214i 0 5.00000i 0 −13.4164 0 −11.0000 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 \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
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.e 4
4.b odd 2 1 inner 1280.3.e.e 4
5.b even 2 1 inner 1280.3.e.e 4
8.b even 2 1 inner 1280.3.e.e 4
8.d odd 2 1 inner 1280.3.e.e 4
16.e even 4 1 80.3.h.a 2
16.e even 4 1 320.3.h.c 2
16.f odd 4 1 80.3.h.a 2
16.f odd 4 1 320.3.h.c 2
20.d odd 2 1 CM 1280.3.e.e 4
40.e odd 2 1 inner 1280.3.e.e 4
40.f even 2 1 inner 1280.3.e.e 4
48.i odd 4 1 720.3.j.a 2
48.k even 4 1 720.3.j.a 2
80.i odd 4 1 400.3.b.b 2
80.i odd 4 1 1600.3.b.d 2
80.j even 4 1 400.3.b.b 2
80.j even 4 1 1600.3.b.d 2
80.k odd 4 1 80.3.h.a 2
80.k odd 4 1 320.3.h.c 2
80.q even 4 1 80.3.h.a 2
80.q even 4 1 320.3.h.c 2
80.s even 4 1 400.3.b.b 2
80.s even 4 1 1600.3.b.d 2
80.t odd 4 1 400.3.b.b 2
80.t odd 4 1 1600.3.b.d 2
240.t even 4 1 720.3.j.a 2
240.z odd 4 1 3600.3.e.o 2
240.bb even 4 1 3600.3.e.o 2
240.bd odd 4 1 3600.3.e.o 2
240.bf even 4 1 3600.3.e.o 2
240.bm odd 4 1 720.3.j.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.3.h.a 2 16.e even 4 1
80.3.h.a 2 16.f odd 4 1
80.3.h.a 2 80.k odd 4 1
80.3.h.a 2 80.q even 4 1
320.3.h.c 2 16.e even 4 1
320.3.h.c 2 16.f odd 4 1
320.3.h.c 2 80.k odd 4 1
320.3.h.c 2 80.q even 4 1
400.3.b.b 2 80.i odd 4 1
400.3.b.b 2 80.j even 4 1
400.3.b.b 2 80.s even 4 1
400.3.b.b 2 80.t odd 4 1
720.3.j.a 2 48.i odd 4 1
720.3.j.a 2 48.k even 4 1
720.3.j.a 2 240.t even 4 1
720.3.j.a 2 240.bm odd 4 1
1280.3.e.e 4 1.a even 1 1 trivial
1280.3.e.e 4 4.b odd 2 1 inner
1280.3.e.e 4 5.b even 2 1 inner
1280.3.e.e 4 8.b even 2 1 inner
1280.3.e.e 4 8.d odd 2 1 inner
1280.3.e.e 4 20.d odd 2 1 CM
1280.3.e.e 4 40.e odd 2 1 inner
1280.3.e.e 4 40.f even 2 1 inner
1600.3.b.d 2 80.i odd 4 1
1600.3.b.d 2 80.j even 4 1
1600.3.b.d 2 80.s even 4 1
1600.3.b.d 2 80.t odd 4 1
3600.3.e.o 2 240.z odd 4 1
3600.3.e.o 2 240.bb even 4 1
3600.3.e.o 2 240.bd odd 4 1
3600.3.e.o 2 240.bf even 4 1

Hecke kernels

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

\( T_{3}^{2} + 20 \)
\( T_{7}^{2} - 180 \)
\( T_{11} \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 20 + T^{2} )^{2} \)
$5$ \( ( 25 + T^{2} )^{2} \)
$7$ \( ( -180 + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( -180 + T^{2} )^{2} \)
$29$ \( ( 484 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( -62 + T )^{4} \)
$43$ \( ( 1620 + T^{2} )^{2} \)
$47$ \( ( -8820 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 3364 + T^{2} )^{2} \)
$67$ \( ( 4500 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( ( 21780 + T^{2} )^{2} \)
$89$ \( ( -142 + T )^{4} \)
$97$ \( T^{4} \)
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