Properties

Label 1280.4.d.e
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: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 5)
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.

\(f(q)\) \(=\) \( q + 2 i q^{3} + 5 i q^{5} - 6 q^{7} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{3} + 5 i q^{5} - 6 q^{7} + 23 q^{9} - 32 i q^{11} - 38 i q^{13} - 10 q^{15} + 26 q^{17} + 100 i q^{19} - 12 i q^{21} + 78 q^{23} - 25 q^{25} + 100 i q^{27} - 50 i q^{29} - 108 q^{31} + 64 q^{33} - 30 i q^{35} - 266 i q^{37} + 76 q^{39} - 22 q^{41} - 442 i q^{43} + 115 i q^{45} - 514 q^{47} - 307 q^{49} + 52 i q^{51} - 2 i q^{53} + 160 q^{55} - 200 q^{57} - 500 i q^{59} - 518 i q^{61} - 138 q^{63} + 190 q^{65} + 126 i q^{67} + 156 i q^{69} - 412 q^{71} + 878 q^{73} - 50 i q^{75} + 192 i q^{77} + 600 q^{79} + 421 q^{81} + 282 i q^{83} + 130 i q^{85} + 100 q^{87} + 150 q^{89} + 228 i q^{91} - 216 i q^{93} - 500 q^{95} + 386 q^{97} - 736 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{7} + 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 12 q^{7} + 46 q^{9} - 20 q^{15} + 52 q^{17} + 156 q^{23} - 50 q^{25} - 216 q^{31} + 128 q^{33} + 152 q^{39} - 44 q^{41} - 1028 q^{47} - 614 q^{49} + 320 q^{55} - 400 q^{57} - 276 q^{63} + 380 q^{65} - 824 q^{71} + 1756 q^{73} + 1200 q^{79} + 842 q^{81} + 200 q^{87} + 300 q^{89} - 1000 q^{95} + 772 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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.e 2
4.b odd 2 1 1280.4.d.l 2
8.b even 2 1 inner 1280.4.d.e 2
8.d odd 2 1 1280.4.d.l 2
16.e even 4 1 5.4.a.a 1
16.e even 4 1 320.4.a.g 1
16.f odd 4 1 80.4.a.d 1
16.f odd 4 1 320.4.a.h 1
48.i odd 4 1 45.4.a.d 1
48.k even 4 1 720.4.a.u 1
80.i odd 4 1 25.4.b.a 2
80.j even 4 1 400.4.c.k 2
80.k odd 4 1 400.4.a.m 1
80.k odd 4 1 1600.4.a.s 1
80.q even 4 1 25.4.a.c 1
80.q even 4 1 1600.4.a.bi 1
80.s even 4 1 400.4.c.k 2
80.t odd 4 1 25.4.b.a 2
112.l odd 4 1 245.4.a.a 1
112.w even 12 2 245.4.e.f 2
112.x odd 12 2 245.4.e.g 2
144.w odd 12 2 405.4.e.c 2
144.x even 12 2 405.4.e.l 2
176.l odd 4 1 605.4.a.d 1
208.p even 4 1 845.4.a.b 1
240.bb even 4 1 225.4.b.c 2
240.bf even 4 1 225.4.b.c 2
240.bm odd 4 1 225.4.a.b 1
272.r even 4 1 1445.4.a.a 1
304.j odd 4 1 1805.4.a.h 1
336.y even 4 1 2205.4.a.q 1
560.bf odd 4 1 1225.4.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 16.e even 4 1
25.4.a.c 1 80.q even 4 1
25.4.b.a 2 80.i odd 4 1
25.4.b.a 2 80.t odd 4 1
45.4.a.d 1 48.i odd 4 1
80.4.a.d 1 16.f odd 4 1
225.4.a.b 1 240.bm odd 4 1
225.4.b.c 2 240.bb even 4 1
225.4.b.c 2 240.bf even 4 1
245.4.a.a 1 112.l odd 4 1
245.4.e.f 2 112.w even 12 2
245.4.e.g 2 112.x odd 12 2
320.4.a.g 1 16.e even 4 1
320.4.a.h 1 16.f odd 4 1
400.4.a.m 1 80.k odd 4 1
400.4.c.k 2 80.j even 4 1
400.4.c.k 2 80.s even 4 1
405.4.e.c 2 144.w odd 12 2
405.4.e.l 2 144.x even 12 2
605.4.a.d 1 176.l odd 4 1
720.4.a.u 1 48.k even 4 1
845.4.a.b 1 208.p even 4 1
1225.4.a.k 1 560.bf odd 4 1
1280.4.d.e 2 1.a even 1 1 trivial
1280.4.d.e 2 8.b even 2 1 inner
1280.4.d.l 2 4.b odd 2 1
1280.4.d.l 2 8.d odd 2 1
1445.4.a.a 1 272.r even 4 1
1600.4.a.s 1 80.k odd 4 1
1600.4.a.bi 1 80.q even 4 1
1805.4.a.h 1 304.j odd 4 1
2205.4.a.q 1 336.y even 4 1

Hecke kernels

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

\( T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7} + 6 \) Copy content Toggle raw display
\( T_{11}^{2} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 25 \) Copy content Toggle raw display
$7$ \( (T + 6)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1024 \) Copy content Toggle raw display
$13$ \( T^{2} + 1444 \) Copy content Toggle raw display
$17$ \( (T - 26)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 10000 \) Copy content Toggle raw display
$23$ \( (T - 78)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 2500 \) Copy content Toggle raw display
$31$ \( (T + 108)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 70756 \) Copy content Toggle raw display
$41$ \( (T + 22)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 195364 \) Copy content Toggle raw display
$47$ \( (T + 514)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 4 \) Copy content Toggle raw display
$59$ \( T^{2} + 250000 \) Copy content Toggle raw display
$61$ \( T^{2} + 268324 \) Copy content Toggle raw display
$67$ \( T^{2} + 15876 \) Copy content Toggle raw display
$71$ \( (T + 412)^{2} \) Copy content Toggle raw display
$73$ \( (T - 878)^{2} \) Copy content Toggle raw display
$79$ \( (T - 600)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 79524 \) Copy content Toggle raw display
$89$ \( (T - 150)^{2} \) Copy content Toggle raw display
$97$ \( (T - 386)^{2} \) Copy content Toggle raw display
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