Properties

Label 1024.2.e.k
Level $1024$
Weight $2$
Character orbit 1024.e
Analytic conductor $8.177$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.e (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.17668116698\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 \zeta_{8} q^{5} + 3 \zeta_{8}^{2} q^{9} +O(q^{10})\) \( q + 4 \zeta_{8} q^{5} + 3 \zeta_{8}^{2} q^{9} -4 \zeta_{8}^{3} q^{13} + 2 q^{17} + 11 \zeta_{8}^{2} q^{25} + 4 \zeta_{8}^{3} q^{29} -12 \zeta_{8} q^{37} + 10 \zeta_{8}^{2} q^{41} + 12 \zeta_{8}^{3} q^{45} + 7 q^{49} -4 \zeta_{8} q^{53} -12 \zeta_{8}^{3} q^{61} + 16 q^{65} + 6 \zeta_{8}^{2} q^{73} -9 q^{81} + 8 \zeta_{8} q^{85} + 10 \zeta_{8}^{2} q^{89} -18 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 8q^{17} + 28q^{49} + 64q^{65} - 36q^{81} - 72q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(\zeta_{8}^{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
0 0 0 −2.82843 + 2.82843i 0 0 0 3.00000i 0
257.2 0 0 0 2.82843 2.82843i 0 0 0 3.00000i 0
769.1 0 0 0 −2.82843 2.82843i 0 0 0 3.00000i 0
769.2 0 0 0 2.82843 + 2.82843i 0 0 0 3.00000i 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 CM by \(\Q(\sqrt{-1}) \)
8.b even 2 1 inner
8.d odd 2 1 inner
16.e even 4 2 inner
16.f odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1024.2.e.k 4
4.b odd 2 1 CM 1024.2.e.k 4
8.b even 2 1 inner 1024.2.e.k 4
8.d odd 2 1 inner 1024.2.e.k 4
16.e even 4 2 inner 1024.2.e.k 4
16.f odd 4 2 inner 1024.2.e.k 4
32.g even 8 2 128.2.b.b 2
32.g even 8 1 256.2.a.b 1
32.g even 8 1 256.2.a.c 1
32.h odd 8 2 128.2.b.b 2
32.h odd 8 1 256.2.a.b 1
32.h odd 8 1 256.2.a.c 1
96.o even 8 2 1152.2.d.d 2
96.o even 8 1 2304.2.a.a 1
96.o even 8 1 2304.2.a.p 1
96.p odd 8 2 1152.2.d.d 2
96.p odd 8 1 2304.2.a.a 1
96.p odd 8 1 2304.2.a.p 1
160.u even 8 1 3200.2.f.c 2
160.u even 8 1 3200.2.f.d 2
160.v odd 8 1 3200.2.f.c 2
160.v odd 8 1 3200.2.f.d 2
160.y odd 8 2 3200.2.d.e 2
160.y odd 8 1 6400.2.a.l 1
160.y odd 8 1 6400.2.a.m 1
160.z even 8 2 3200.2.d.e 2
160.z even 8 1 6400.2.a.l 1
160.z even 8 1 6400.2.a.m 1
160.ba even 8 1 3200.2.f.c 2
160.ba even 8 1 3200.2.f.d 2
160.bb odd 8 1 3200.2.f.c 2
160.bb odd 8 1 3200.2.f.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.b.b 2 32.g even 8 2
128.2.b.b 2 32.h odd 8 2
256.2.a.b 1 32.g even 8 1
256.2.a.b 1 32.h odd 8 1
256.2.a.c 1 32.g even 8 1
256.2.a.c 1 32.h odd 8 1
1024.2.e.k 4 1.a even 1 1 trivial
1024.2.e.k 4 4.b odd 2 1 CM
1024.2.e.k 4 8.b even 2 1 inner
1024.2.e.k 4 8.d odd 2 1 inner
1024.2.e.k 4 16.e even 4 2 inner
1024.2.e.k 4 16.f odd 4 2 inner
1152.2.d.d 2 96.o even 8 2
1152.2.d.d 2 96.p odd 8 2
2304.2.a.a 1 96.o even 8 1
2304.2.a.a 1 96.p odd 8 1
2304.2.a.p 1 96.o even 8 1
2304.2.a.p 1 96.p odd 8 1
3200.2.d.e 2 160.y odd 8 2
3200.2.d.e 2 160.z even 8 2
3200.2.f.c 2 160.u even 8 1
3200.2.f.c 2 160.v odd 8 1
3200.2.f.c 2 160.ba even 8 1
3200.2.f.c 2 160.bb odd 8 1
3200.2.f.d 2 160.u even 8 1
3200.2.f.d 2 160.v odd 8 1
3200.2.f.d 2 160.ba even 8 1
3200.2.f.d 2 160.bb odd 8 1
6400.2.a.l 1 160.y odd 8 1
6400.2.a.l 1 160.z even 8 1
6400.2.a.m 1 160.y odd 8 1
6400.2.a.m 1 160.z even 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1024, [\chi])\):

\( T_{3} \)
\( T_{5}^{4} + 256 \)
\( T_{47} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 256 + T^{4} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( 256 + T^{4} \)
$17$ \( ( -2 + T )^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( 256 + T^{4} \)
$31$ \( T^{4} \)
$37$ \( 20736 + T^{4} \)
$41$ \( ( 100 + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( 256 + T^{4} \)
$59$ \( T^{4} \)
$61$ \( 20736 + T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( 36 + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( ( 100 + T^{2} )^{2} \)
$97$ \( ( 18 + T )^{4} \)
show more
show less