Properties

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

Related objects

Downloads

Learn more about

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 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.

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

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.l 2
4.b odd 2 1 1280.4.d.e 2
8.b even 2 1 inner 1280.4.d.l 2
8.d odd 2 1 1280.4.d.e 2
16.e even 4 1 80.4.a.d 1
16.e even 4 1 320.4.a.h 1
16.f odd 4 1 5.4.a.a 1
16.f odd 4 1 320.4.a.g 1
48.i odd 4 1 720.4.a.u 1
48.k even 4 1 45.4.a.d 1
80.i odd 4 1 400.4.c.k 2
80.j even 4 1 25.4.b.a 2
80.k odd 4 1 25.4.a.c 1
80.k odd 4 1 1600.4.a.bi 1
80.q even 4 1 400.4.a.m 1
80.q even 4 1 1600.4.a.s 1
80.s even 4 1 25.4.b.a 2
80.t odd 4 1 400.4.c.k 2
112.j even 4 1 245.4.a.a 1
112.u odd 12 2 245.4.e.f 2
112.v even 12 2 245.4.e.g 2
144.u even 12 2 405.4.e.c 2
144.v odd 12 2 405.4.e.l 2
176.i even 4 1 605.4.a.d 1
208.o odd 4 1 845.4.a.b 1
240.t even 4 1 225.4.a.b 1
240.z odd 4 1 225.4.b.c 2
240.bd odd 4 1 225.4.b.c 2
272.k odd 4 1 1445.4.a.a 1
304.m even 4 1 1805.4.a.h 1
336.v odd 4 1 2205.4.a.q 1
560.be even 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.f odd 4 1
25.4.a.c 1 80.k odd 4 1
25.4.b.a 2 80.j even 4 1
25.4.b.a 2 80.s even 4 1
45.4.a.d 1 48.k even 4 1
80.4.a.d 1 16.e even 4 1
225.4.a.b 1 240.t even 4 1
225.4.b.c 2 240.z odd 4 1
225.4.b.c 2 240.bd odd 4 1
245.4.a.a 1 112.j even 4 1
245.4.e.f 2 112.u odd 12 2
245.4.e.g 2 112.v even 12 2
320.4.a.g 1 16.f odd 4 1
320.4.a.h 1 16.e even 4 1
400.4.a.m 1 80.q even 4 1
400.4.c.k 2 80.i odd 4 1
400.4.c.k 2 80.t odd 4 1
405.4.e.c 2 144.u even 12 2
405.4.e.l 2 144.v odd 12 2
605.4.a.d 1 176.i even 4 1
720.4.a.u 1 48.i odd 4 1
845.4.a.b 1 208.o odd 4 1
1225.4.a.k 1 560.be even 4 1
1280.4.d.e 2 4.b odd 2 1
1280.4.d.e 2 8.d odd 2 1
1280.4.d.l 2 1.a even 1 1 trivial
1280.4.d.l 2 8.b even 2 1 inner
1445.4.a.a 1 272.k odd 4 1
1600.4.a.s 1 80.q even 4 1
1600.4.a.bi 1 80.k odd 4 1
1805.4.a.h 1 304.m even 4 1
2205.4.a.q 1 336.v odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 4 \)
\( T_{7} - 6 \)
\( T_{11}^{2} + 1024 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 50 T^{2} + 729 T^{4} \)
$5$ \( 1 + 25 T^{2} \)
$7$ \( ( 1 - 6 T + 343 T^{2} )^{2} \)
$11$ \( 1 - 1638 T^{2} + 1771561 T^{4} \)
$13$ \( 1 - 2950 T^{2} + 4826809 T^{4} \)
$17$ \( ( 1 - 26 T + 4913 T^{2} )^{2} \)
$19$ \( 1 - 3718 T^{2} + 47045881 T^{4} \)
$23$ \( ( 1 + 78 T + 12167 T^{2} )^{2} \)
$29$ \( 1 - 46278 T^{2} + 594823321 T^{4} \)
$31$ \( ( 1 - 108 T + 29791 T^{2} )^{2} \)
$37$ \( 1 - 30550 T^{2} + 2565726409 T^{4} \)
$41$ \( ( 1 + 22 T + 68921 T^{2} )^{2} \)
$43$ \( 1 + 36350 T^{2} + 6321363049 T^{4} \)
$47$ \( ( 1 - 514 T + 103823 T^{2} )^{2} \)
$53$ \( 1 - 297750 T^{2} + 22164361129 T^{4} \)
$59$ \( 1 - 160758 T^{2} + 42180533641 T^{4} \)
$61$ \( 1 - 185638 T^{2} + 51520374361 T^{4} \)
$67$ \( 1 - 585650 T^{2} + 90458382169 T^{4} \)
$71$ \( ( 1 - 412 T + 357911 T^{2} )^{2} \)
$73$ \( ( 1 - 878 T + 389017 T^{2} )^{2} \)
$79$ \( ( 1 + 600 T + 493039 T^{2} )^{2} \)
$83$ \( 1 - 1064050 T^{2} + 326940373369 T^{4} \)
$89$ \( ( 1 - 150 T + 704969 T^{2} )^{2} \)
$97$ \( ( 1 - 386 T + 912673 T^{2} )^{2} \)
show more
show less