Properties

Label 1280.4.d.v
Level $1280$
Weight $4$
Character orbit 1280.d
Analytic conductor $75.522$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1280.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(75.5224448073\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
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 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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + 5 \beta_{1} q^{5} -7 \beta_{3} q^{7} + 7 q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + 5 \beta_{1} q^{5} -7 \beta_{3} q^{7} + 7 q^{9} -2 \beta_{2} q^{11} -62 \beta_{1} q^{13} + 5 \beta_{3} q^{15} -46 q^{17} -24 \beta_{2} q^{19} + 140 \beta_{1} q^{21} + 43 \beta_{3} q^{23} -25 q^{25} -34 \beta_{2} q^{27} -90 \beta_{1} q^{29} + 34 \beta_{3} q^{31} -40 q^{33} -35 \beta_{2} q^{35} + 214 \beta_{1} q^{37} -62 \beta_{3} q^{39} + 10 q^{41} -15 \beta_{2} q^{43} + 35 \beta_{1} q^{45} -89 \beta_{3} q^{47} + 637 q^{49} + 46 \beta_{2} q^{51} + 678 \beta_{1} q^{53} + 10 \beta_{3} q^{55} -480 q^{57} -92 \beta_{2} q^{59} + 250 \beta_{1} q^{61} -49 \beta_{3} q^{63} + 310 q^{65} -11 \beta_{2} q^{67} -860 \beta_{1} q^{69} -82 \beta_{3} q^{71} -522 q^{73} + 25 \beta_{2} q^{75} + 280 \beta_{1} q^{77} -196 \beta_{3} q^{79} -491 q^{81} -85 \beta_{2} q^{83} -230 \beta_{1} q^{85} -90 \beta_{3} q^{87} -970 q^{89} + 434 \beta_{2} q^{91} -680 \beta_{1} q^{93} + 120 \beta_{3} q^{95} -934 q^{97} -14 \beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 28 q^{9} + O(q^{10}) \) \( 4 q + 28 q^{9} - 184 q^{17} - 100 q^{25} - 160 q^{33} + 40 q^{41} + 2548 q^{49} - 1920 q^{57} + 1240 q^{65} - 2088 q^{73} - 1964 q^{81} - 3880 q^{89} - 3736 q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
641.1
1.61803i
0.618034i
0.618034i
1.61803i
0 4.47214i 0 5.00000i 0 31.3050 0 7.00000 0
641.2 0 4.47214i 0 5.00000i 0 −31.3050 0 7.00000 0
641.3 0 4.47214i 0 5.00000i 0 −31.3050 0 7.00000 0
641.4 0 4.47214i 0 5.00000i 0 31.3050 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
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 1280.4.d.v 4
4.b odd 2 1 inner 1280.4.d.v 4
8.b even 2 1 inner 1280.4.d.v 4
8.d odd 2 1 inner 1280.4.d.v 4
16.e even 4 1 160.4.a.d 2
16.e even 4 1 320.4.a.q 2
16.f odd 4 1 160.4.a.d 2
16.f odd 4 1 320.4.a.q 2
48.i odd 4 1 1440.4.a.bb 2
48.k even 4 1 1440.4.a.bb 2
80.i odd 4 1 800.4.c.l 4
80.j even 4 1 800.4.c.l 4
80.k odd 4 1 800.4.a.o 2
80.k odd 4 1 1600.4.a.cg 2
80.q even 4 1 800.4.a.o 2
80.q even 4 1 1600.4.a.cg 2
80.s even 4 1 800.4.c.l 4
80.t odd 4 1 800.4.c.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.d 2 16.e even 4 1
160.4.a.d 2 16.f odd 4 1
320.4.a.q 2 16.e even 4 1
320.4.a.q 2 16.f odd 4 1
800.4.a.o 2 80.k odd 4 1
800.4.a.o 2 80.q even 4 1
800.4.c.l 4 80.i odd 4 1
800.4.c.l 4 80.j even 4 1
800.4.c.l 4 80.s even 4 1
800.4.c.l 4 80.t odd 4 1
1280.4.d.v 4 1.a even 1 1 trivial
1280.4.d.v 4 4.b odd 2 1 inner
1280.4.d.v 4 8.b even 2 1 inner
1280.4.d.v 4 8.d odd 2 1 inner
1440.4.a.bb 2 48.i odd 4 1
1440.4.a.bb 2 48.k even 4 1
1600.4.a.cg 2 80.k odd 4 1
1600.4.a.cg 2 80.q even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1280, [\chi])\):

\( T_{3}^{2} + 20 \)
\( T_{7}^{2} - 980 \)
\( T_{11}^{2} + 80 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 20 + T^{2} )^{2} \)
$5$ \( ( 25 + T^{2} )^{2} \)
$7$ \( ( -980 + T^{2} )^{2} \)
$11$ \( ( 80 + T^{2} )^{2} \)
$13$ \( ( 3844 + T^{2} )^{2} \)
$17$ \( ( 46 + T )^{4} \)
$19$ \( ( 11520 + T^{2} )^{2} \)
$23$ \( ( -36980 + T^{2} )^{2} \)
$29$ \( ( 8100 + T^{2} )^{2} \)
$31$ \( ( -23120 + T^{2} )^{2} \)
$37$ \( ( 45796 + T^{2} )^{2} \)
$41$ \( ( -10 + T )^{4} \)
$43$ \( ( 4500 + T^{2} )^{2} \)
$47$ \( ( -158420 + T^{2} )^{2} \)
$53$ \( ( 459684 + T^{2} )^{2} \)
$59$ \( ( 169280 + T^{2} )^{2} \)
$61$ \( ( 62500 + T^{2} )^{2} \)
$67$ \( ( 2420 + T^{2} )^{2} \)
$71$ \( ( -134480 + T^{2} )^{2} \)
$73$ \( ( 522 + T )^{4} \)
$79$ \( ( -768320 + T^{2} )^{2} \)
$83$ \( ( 144500 + T^{2} )^{2} \)
$89$ \( ( 970 + T )^{4} \)
$97$ \( ( 934 + T )^{4} \)
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