Properties

Label 1280.4.d.f
Level $1280$
Weight $4$
Character orbit 1280.d
Analytic conductor $75.522$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \(2\)
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 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 i q^{3} -5 i q^{5} -6 q^{7} + 23 q^{9} +O(q^{10})\) \( q + 2 i q^{3} -5 i q^{5} -6 q^{7} + 23 q^{9} + 60 i q^{11} -50 i q^{13} + 10 q^{15} -30 q^{17} -40 i q^{19} -12 i q^{21} -178 q^{23} -25 q^{25} + 100 i q^{27} -166 i q^{29} + 20 q^{31} -120 q^{33} + 30 i q^{35} + 10 i q^{37} + 100 q^{39} + 250 q^{41} + 142 i q^{43} -115 i q^{45} + 214 q^{47} -307 q^{49} -60 i q^{51} + 490 i q^{53} + 300 q^{55} + 80 q^{57} -800 i q^{59} -250 i q^{61} -138 q^{63} -250 q^{65} + 774 i q^{67} -356 i q^{69} -100 q^{71} + 230 q^{73} -50 i q^{75} -360 i q^{77} -1320 q^{79} + 421 q^{81} -982 i q^{83} + 150 i q^{85} + 332 q^{87} -874 q^{89} + 300 i q^{91} + 40 i q^{93} -200 q^{95} -310 q^{97} + 1380 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 12q^{7} + 46q^{9} + O(q^{10}) \) \( 2q - 12q^{7} + 46q^{9} + 20q^{15} - 60q^{17} - 356q^{23} - 50q^{25} + 40q^{31} - 240q^{33} + 200q^{39} + 500q^{41} + 428q^{47} - 614q^{49} + 600q^{55} + 160q^{57} - 276q^{63} - 500q^{65} - 200q^{71} + 460q^{73} - 2640q^{79} + 842q^{81} + 664q^{87} - 1748q^{89} - 400q^{95} - 620q^{97} + O(q^{100}) \)

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
1.00000i
1.00000i
0 2.00000i 0 5.00000i 0 −6.00000 0 23.0000 0
641.2 0 2.00000i 0 5.00000i 0 −6.00000 0 23.0000 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
8.b even 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.f 2
4.b odd 2 1 1280.4.d.k 2
8.b even 2 1 inner 1280.4.d.f 2
8.d odd 2 1 1280.4.d.k 2
16.e even 4 1 160.4.a.a 1
16.e even 4 1 320.4.a.i 1
16.f odd 4 1 160.4.a.b yes 1
16.f odd 4 1 320.4.a.f 1
48.i odd 4 1 1440.4.a.o 1
48.k even 4 1 1440.4.a.n 1
80.i odd 4 1 800.4.c.f 2
80.j even 4 1 800.4.c.e 2
80.k odd 4 1 800.4.a.d 1
80.k odd 4 1 1600.4.a.bj 1
80.q even 4 1 800.4.a.h 1
80.q even 4 1 1600.4.a.r 1
80.s even 4 1 800.4.c.e 2
80.t odd 4 1 800.4.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.a 1 16.e even 4 1
160.4.a.b yes 1 16.f odd 4 1
320.4.a.f 1 16.f odd 4 1
320.4.a.i 1 16.e even 4 1
800.4.a.d 1 80.k odd 4 1
800.4.a.h 1 80.q even 4 1
800.4.c.e 2 80.j even 4 1
800.4.c.e 2 80.s even 4 1
800.4.c.f 2 80.i odd 4 1
800.4.c.f 2 80.t odd 4 1
1280.4.d.f 2 1.a even 1 1 trivial
1280.4.d.f 2 8.b even 2 1 inner
1280.4.d.k 2 4.b odd 2 1
1280.4.d.k 2 8.d odd 2 1
1440.4.a.n 1 48.k even 4 1
1440.4.a.o 1 48.i odd 4 1
1600.4.a.r 1 80.q even 4 1
1600.4.a.bj 1 80.k odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 4 \)
\( T_{7} + 6 \)
\( T_{11}^{2} + 3600 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( 25 + T^{2} \)
$7$ \( ( 6 + T )^{2} \)
$11$ \( 3600 + T^{2} \)
$13$ \( 2500 + T^{2} \)
$17$ \( ( 30 + T )^{2} \)
$19$ \( 1600 + T^{2} \)
$23$ \( ( 178 + T )^{2} \)
$29$ \( 27556 + T^{2} \)
$31$ \( ( -20 + T )^{2} \)
$37$ \( 100 + T^{2} \)
$41$ \( ( -250 + T )^{2} \)
$43$ \( 20164 + T^{2} \)
$47$ \( ( -214 + T )^{2} \)
$53$ \( 240100 + T^{2} \)
$59$ \( 640000 + T^{2} \)
$61$ \( 62500 + T^{2} \)
$67$ \( 599076 + T^{2} \)
$71$ \( ( 100 + T )^{2} \)
$73$ \( ( -230 + T )^{2} \)
$79$ \( ( 1320 + T )^{2} \)
$83$ \( 964324 + T^{2} \)
$89$ \( ( 874 + T )^{2} \)
$97$ \( ( 310 + T )^{2} \)
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