Properties

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

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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: \( 2^{2} \)
Twist minimal: no (minimal twist has level 64)
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{8} q^{3} + \zeta_{8}^{2} q^{9} +O(q^{10})\) \( q + 2 \zeta_{8} q^{3} + \zeta_{8}^{2} q^{9} -6 \zeta_{8}^{3} q^{11} + 6 q^{17} -2 \zeta_{8} q^{19} + 5 \zeta_{8}^{2} q^{25} -4 \zeta_{8}^{3} q^{27} + 12 q^{33} + 6 \zeta_{8}^{2} q^{41} + 10 \zeta_{8}^{3} q^{43} + 7 q^{49} + 12 \zeta_{8} q^{51} -4 \zeta_{8}^{2} q^{57} + 6 \zeta_{8}^{3} q^{59} -14 \zeta_{8} q^{67} -2 \zeta_{8}^{2} q^{73} + 10 \zeta_{8}^{3} q^{75} + 11 q^{81} -18 \zeta_{8} q^{83} + 18 \zeta_{8}^{2} q^{89} + 10 q^{97} + 6 \zeta_{8} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 24q^{17} + 48q^{33} + 28q^{49} + 44q^{81} + 40q^{97} + O(q^{100}) \)

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 −1.41421 1.41421i 0 0 0 0 0 1.00000i 0
257.2 0 1.41421 + 1.41421i 0 0 0 0 0 1.00000i 0
769.1 0 −1.41421 + 1.41421i 0 0 0 0 0 1.00000i 0
769.2 0 1.41421 1.41421i 0 0 0 0 0 1.00000i 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.b even 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.l 4
4.b odd 2 1 inner 1024.2.e.l 4
8.b even 2 1 inner 1024.2.e.l 4
8.d odd 2 1 CM 1024.2.e.l 4
16.e even 4 2 inner 1024.2.e.l 4
16.f odd 4 2 inner 1024.2.e.l 4
32.g even 8 2 64.2.b.a 2
32.g even 8 1 256.2.a.a 1
32.g even 8 1 256.2.a.d 1
32.h odd 8 2 64.2.b.a 2
32.h odd 8 1 256.2.a.a 1
32.h odd 8 1 256.2.a.d 1
96.o even 8 2 576.2.d.a 2
96.o even 8 1 2304.2.a.h 1
96.o even 8 1 2304.2.a.i 1
96.p odd 8 2 576.2.d.a 2
96.p odd 8 1 2304.2.a.h 1
96.p odd 8 1 2304.2.a.i 1
160.u even 8 1 1600.2.f.a 2
160.u even 8 1 1600.2.f.b 2
160.v odd 8 1 1600.2.f.a 2
160.v odd 8 1 1600.2.f.b 2
160.y odd 8 2 1600.2.d.a 2
160.y odd 8 1 6400.2.a.a 1
160.y odd 8 1 6400.2.a.x 1
160.z even 8 2 1600.2.d.a 2
160.z even 8 1 6400.2.a.a 1
160.z even 8 1 6400.2.a.x 1
160.ba even 8 1 1600.2.f.a 2
160.ba even 8 1 1600.2.f.b 2
160.bb odd 8 1 1600.2.f.a 2
160.bb odd 8 1 1600.2.f.b 2
224.v odd 8 2 3136.2.b.b 2
224.x even 8 2 3136.2.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.2.b.a 2 32.g even 8 2
64.2.b.a 2 32.h odd 8 2
256.2.a.a 1 32.g even 8 1
256.2.a.a 1 32.h odd 8 1
256.2.a.d 1 32.g even 8 1
256.2.a.d 1 32.h odd 8 1
576.2.d.a 2 96.o even 8 2
576.2.d.a 2 96.p odd 8 2
1024.2.e.l 4 1.a even 1 1 trivial
1024.2.e.l 4 4.b odd 2 1 inner
1024.2.e.l 4 8.b even 2 1 inner
1024.2.e.l 4 8.d odd 2 1 CM
1024.2.e.l 4 16.e even 4 2 inner
1024.2.e.l 4 16.f odd 4 2 inner
1600.2.d.a 2 160.y odd 8 2
1600.2.d.a 2 160.z even 8 2
1600.2.f.a 2 160.u even 8 1
1600.2.f.a 2 160.v odd 8 1
1600.2.f.a 2 160.ba even 8 1
1600.2.f.a 2 160.bb odd 8 1
1600.2.f.b 2 160.u even 8 1
1600.2.f.b 2 160.v odd 8 1
1600.2.f.b 2 160.ba even 8 1
1600.2.f.b 2 160.bb odd 8 1
2304.2.a.h 1 96.o even 8 1
2304.2.a.h 1 96.p odd 8 1
2304.2.a.i 1 96.o even 8 1
2304.2.a.i 1 96.p odd 8 1
3136.2.b.b 2 224.v odd 8 2
3136.2.b.b 2 224.x even 8 2
6400.2.a.a 1 160.y odd 8 1
6400.2.a.a 1 160.z even 8 1
6400.2.a.x 1 160.y odd 8 1
6400.2.a.x 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 \(S_{2}^{\mathrm{new}}(1024, [\chi])\):

\( T_{3}^{4} + 16 \)
\( T_{5} \)
\( T_{47} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 16 + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( 1296 + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( -6 + T )^{4} \)
$19$ \( 16 + T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 36 + T^{2} )^{2} \)
$43$ \( 10000 + T^{4} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( 1296 + T^{4} \)
$61$ \( T^{4} \)
$67$ \( 38416 + T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( 4 + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( 104976 + T^{4} \)
$89$ \( ( 324 + T^{2} )^{2} \)
$97$ \( ( -10 + T )^{4} \)
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