Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 - 2 i ) q^{3} -5 i q^{9} +O(q^{10})\) \( q + ( 2 - 2 i ) q^{3} -5 i q^{9} + ( -2 - 2 i ) q^{11} -6 q^{17} + ( 6 - 6 i ) q^{19} -5 i q^{25} + ( -4 - 4 i ) q^{27} -8 q^{33} -6 i q^{41} + ( 6 + 6 i ) q^{43} + 7 q^{49} + ( -12 + 12 i ) q^{51} -24 i q^{57} + ( 10 + 10 i ) q^{59} + ( -6 + 6 i ) q^{67} -2 i q^{73} + ( -10 - 10 i ) q^{75} - q^{81} + ( -2 + 2 i ) q^{83} + 18 i q^{89} -10 q^{97} + ( -10 + 10 i ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{3} + O(q^{10}) \) \( 2q + 4q^{3} - 4q^{11} - 12q^{17} + 12q^{19} - 8q^{27} - 16q^{33} + 12q^{43} + 14q^{49} - 24q^{51} + 20q^{59} - 12q^{67} - 20q^{75} - 2q^{81} - 4q^{83} - 20q^{97} - 20q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(i\) \(1\)

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
1.00000i
1.00000i
0 2.00000 + 2.00000i 0 0 0 0 0 5.00000i 0
769.1 0 2.00000 2.00000i 0 0 0 0 0 5.00000i 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 \(\Q(\sqrt{-2}) \)
16.e even 4 1 inner
16.f odd 4 1 inner

Twists

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

Hecke kernels

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

\( T_{3}^{2} - 4 T_{3} + 8 \)
\( T_{5} \)
\( T_{47} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 4 T + 8 T^{2} - 12 T^{3} + 9 T^{4} \)
$5$ \( 1 + 25 T^{4} \)
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( 1 + 4 T + 8 T^{2} + 44 T^{3} + 121 T^{4} \)
$13$ \( 1 + 169 T^{4} \)
$17$ \( ( 1 + 6 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 12 T + 72 T^{2} - 228 T^{3} + 361 T^{4} \)
$23$ \( ( 1 - 23 T^{2} )^{2} \)
$29$ \( 1 + 841 T^{4} \)
$31$ \( ( 1 + 31 T^{2} )^{2} \)
$37$ \( 1 + 1369 T^{4} \)
$41$ \( 1 - 46 T^{2} + 1681 T^{4} \)
$43$ \( 1 - 12 T + 72 T^{2} - 516 T^{3} + 1849 T^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( 1 + 2809 T^{4} \)
$59$ \( 1 - 20 T + 200 T^{2} - 1180 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 3721 T^{4} \)
$67$ \( 1 + 12 T + 72 T^{2} + 804 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 71 T^{2} )^{2} \)
$73$ \( 1 - 142 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 79 T^{2} )^{2} \)
$83$ \( 1 + 4 T + 8 T^{2} + 332 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 146 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 + 10 T + 97 T^{2} )^{2} \)
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