Properties

Label 1024.2.b.f
Level $1024$
Weight $2$
Character orbit 1024.b
Analytic conductor $8.177$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.17668116698\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta q^{3} - \beta q^{5} + 4 q^{7} - 5 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta q^{3} - \beta q^{5} + 4 q^{7} - 5 q^{9} - 2 \beta q^{11} + 3 \beta q^{13} + 4 q^{15} + 2 \beta q^{19} + 8 \beta q^{21} + 4 q^{23} + 3 q^{25} - 4 \beta q^{27} + 3 \beta q^{29} + 8 q^{31} + 8 q^{33} - 4 \beta q^{35} + \beta q^{37} - 12 q^{39} - 8 q^{41} + 2 \beta q^{43} + 5 \beta q^{45} - 8 q^{47} + 9 q^{49} - \beta q^{53} - 4 q^{55} - 8 q^{57} + 6 \beta q^{59} - 3 \beta q^{61} - 20 q^{63} + 6 q^{65} - 2 \beta q^{67} + 8 \beta q^{69} + 12 q^{71} + 2 q^{73} + 6 \beta q^{75} - 8 \beta q^{77} + q^{81} - 10 \beta q^{83} - 12 q^{87} - 14 q^{89} + 12 \beta q^{91} + 16 \beta q^{93} + 4 q^{95} - 16 q^{97} + 10 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{7} - 10 q^{9} + 8 q^{15} + 8 q^{23} + 6 q^{25} + 16 q^{31} + 16 q^{33} - 24 q^{39} - 16 q^{41} - 16 q^{47} + 18 q^{49} - 8 q^{55} - 16 q^{57} - 40 q^{63} + 12 q^{65} + 24 q^{71} + 4 q^{73} + 2 q^{81} - 24 q^{87} - 28 q^{89} + 8 q^{95} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
513.1
1.41421i
1.41421i
0 2.82843i 0 1.41421i 0 4.00000 0 −5.00000 0
513.2 0 2.82843i 0 1.41421i 0 4.00000 0 −5.00000 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 1024.2.b.f 2
4.b odd 2 1 1024.2.b.a 2
8.b even 2 1 inner 1024.2.b.f 2
8.d odd 2 1 1024.2.b.a 2
16.e even 4 2 1024.2.a.a 2
16.f odd 4 2 1024.2.a.f 2
32.g even 8 2 512.2.e.b yes 2
32.g even 8 2 512.2.e.g yes 2
32.h odd 8 2 512.2.e.a 2
32.h odd 8 2 512.2.e.h yes 2
48.i odd 4 2 9216.2.a.b 2
48.k even 4 2 9216.2.a.u 2
96.o even 8 2 4608.2.k.f 2
96.o even 8 2 4608.2.k.s 2
96.p odd 8 2 4608.2.k.j 2
96.p odd 8 2 4608.2.k.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.e.a 2 32.h odd 8 2
512.2.e.b yes 2 32.g even 8 2
512.2.e.g yes 2 32.g even 8 2
512.2.e.h yes 2 32.h odd 8 2
1024.2.a.a 2 16.e even 4 2
1024.2.a.f 2 16.f odd 4 2
1024.2.b.a 2 4.b odd 2 1
1024.2.b.a 2 8.d odd 2 1
1024.2.b.f 2 1.a even 1 1 trivial
1024.2.b.f 2 8.b even 2 1 inner
4608.2.k.f 2 96.o even 8 2
4608.2.k.j 2 96.p odd 8 2
4608.2.k.o 2 96.p odd 8 2
4608.2.k.s 2 96.o even 8 2
9216.2.a.b 2 48.i odd 4 2
9216.2.a.u 2 48.k even 4 2

Hecke kernels

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

\( T_{3}^{2} + 8 \) Copy content Toggle raw display
\( T_{5}^{2} + 2 \) Copy content Toggle raw display
\( T_{7} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 8 \) Copy content Toggle raw display
$5$ \( T^{2} + 2 \) Copy content Toggle raw display
$7$ \( (T - 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 8 \) Copy content Toggle raw display
$13$ \( T^{2} + 18 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 8 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 18 \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2 \) Copy content Toggle raw display
$41$ \( (T + 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 8 \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 2 \) Copy content Toggle raw display
$59$ \( T^{2} + 72 \) Copy content Toggle raw display
$61$ \( T^{2} + 18 \) Copy content Toggle raw display
$67$ \( T^{2} + 8 \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 200 \) Copy content Toggle raw display
$89$ \( (T + 14)^{2} \) Copy content Toggle raw display
$97$ \( (T + 16)^{2} \) Copy content Toggle raw display
show more
show less