Properties

Label 1280.4.d.n
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

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 20)
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.

\(f(q)\) \(=\) \( q + 4 i q^{3} - 5 i q^{5} + 16 q^{7} + 11 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 4 i q^{3} - 5 i q^{5} + 16 q^{7} + 11 q^{9} + 60 i q^{11} + 86 i q^{13} + 20 q^{15} + 18 q^{17} + 44 i q^{19} + 64 i q^{21} - 48 q^{23} - 25 q^{25} + 152 i q^{27} - 186 i q^{29} + 176 q^{31} - 240 q^{33} - 80 i q^{35} - 254 i q^{37} - 344 q^{39} - 186 q^{41} + 100 i q^{43} - 55 i q^{45} + 168 q^{47} - 87 q^{49} + 72 i q^{51} + 498 i q^{53} + 300 q^{55} - 176 q^{57} + 252 i q^{59} - 58 i q^{61} + 176 q^{63} + 430 q^{65} - 1036 i q^{67} - 192 i q^{69} - 168 q^{71} - 506 q^{73} - 100 i q^{75} + 960 i q^{77} + 272 q^{79} - 311 q^{81} + 948 i q^{83} - 90 i q^{85} + 744 q^{87} + 1014 q^{89} + 1376 i q^{91} + 704 i q^{93} + 220 q^{95} - 766 q^{97} + 660 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 32 q^{7} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 32 q^{7} + 22 q^{9} + 40 q^{15} + 36 q^{17} - 96 q^{23} - 50 q^{25} + 352 q^{31} - 480 q^{33} - 688 q^{39} - 372 q^{41} + 336 q^{47} - 174 q^{49} + 600 q^{55} - 352 q^{57} + 352 q^{63} + 860 q^{65} - 336 q^{71} - 1012 q^{73} + 544 q^{79} - 622 q^{81} + 1488 q^{87} + 2028 q^{89} + 440 q^{95} - 1532 q^{97}+O(q^{100}) \) 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 4.00000i 0 5.00000i 0 16.0000 0 11.0000 0
641.2 0 4.00000i 0 5.00000i 0 16.0000 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
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.n 2
4.b odd 2 1 1280.4.d.c 2
8.b even 2 1 inner 1280.4.d.n 2
8.d odd 2 1 1280.4.d.c 2
16.e even 4 1 20.4.a.a 1
16.e even 4 1 320.4.a.d 1
16.f odd 4 1 80.4.a.c 1
16.f odd 4 1 320.4.a.k 1
48.i odd 4 1 180.4.a.a 1
48.k even 4 1 720.4.a.k 1
80.i odd 4 1 100.4.c.a 2
80.j even 4 1 400.4.c.j 2
80.k odd 4 1 400.4.a.o 1
80.k odd 4 1 1600.4.a.p 1
80.q even 4 1 100.4.a.a 1
80.q even 4 1 1600.4.a.bl 1
80.s even 4 1 400.4.c.j 2
80.t odd 4 1 100.4.c.a 2
112.l odd 4 1 980.4.a.c 1
112.w even 12 2 980.4.i.e 2
112.x odd 12 2 980.4.i.n 2
144.w odd 12 2 1620.4.i.j 2
144.x even 12 2 1620.4.i.d 2
176.l odd 4 1 2420.4.a.d 1
240.bb even 4 1 900.4.d.k 2
240.bf even 4 1 900.4.d.k 2
240.bm odd 4 1 900.4.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.a.a 1 16.e even 4 1
80.4.a.c 1 16.f odd 4 1
100.4.a.a 1 80.q even 4 1
100.4.c.a 2 80.i odd 4 1
100.4.c.a 2 80.t odd 4 1
180.4.a.a 1 48.i odd 4 1
320.4.a.d 1 16.e even 4 1
320.4.a.k 1 16.f odd 4 1
400.4.a.o 1 80.k odd 4 1
400.4.c.j 2 80.j even 4 1
400.4.c.j 2 80.s even 4 1
720.4.a.k 1 48.k even 4 1
900.4.a.m 1 240.bm odd 4 1
900.4.d.k 2 240.bb even 4 1
900.4.d.k 2 240.bf even 4 1
980.4.a.c 1 112.l odd 4 1
980.4.i.e 2 112.w even 12 2
980.4.i.n 2 112.x odd 12 2
1280.4.d.c 2 4.b odd 2 1
1280.4.d.c 2 8.d odd 2 1
1280.4.d.n 2 1.a even 1 1 trivial
1280.4.d.n 2 8.b even 2 1 inner
1600.4.a.p 1 80.k odd 4 1
1600.4.a.bl 1 80.q even 4 1
1620.4.i.d 2 144.x even 12 2
1620.4.i.j 2 144.w odd 12 2
2420.4.a.d 1 176.l odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{7} - 16 \) Copy content Toggle raw display
\( T_{11}^{2} + 3600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{2} + 25 \) Copy content Toggle raw display
$7$ \( (T - 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 3600 \) Copy content Toggle raw display
$13$ \( T^{2} + 7396 \) Copy content Toggle raw display
$17$ \( (T - 18)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1936 \) Copy content Toggle raw display
$23$ \( (T + 48)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 34596 \) Copy content Toggle raw display
$31$ \( (T - 176)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 64516 \) Copy content Toggle raw display
$41$ \( (T + 186)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 10000 \) Copy content Toggle raw display
$47$ \( (T - 168)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 248004 \) Copy content Toggle raw display
$59$ \( T^{2} + 63504 \) Copy content Toggle raw display
$61$ \( T^{2} + 3364 \) Copy content Toggle raw display
$67$ \( T^{2} + 1073296 \) Copy content Toggle raw display
$71$ \( (T + 168)^{2} \) Copy content Toggle raw display
$73$ \( (T + 506)^{2} \) Copy content Toggle raw display
$79$ \( (T - 272)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 898704 \) Copy content Toggle raw display
$89$ \( (T - 1014)^{2} \) Copy content Toggle raw display
$97$ \( (T + 766)^{2} \) Copy content Toggle raw display
show more
show less