Properties

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

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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: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 16)
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.

\(f(q)\) \(=\) \( q + \beta q^{3} -\beta q^{5} -2 q^{7} + q^{9} +O(q^{10})\) \( q + \beta q^{3} -\beta q^{5} -2 q^{7} + q^{9} + \beta q^{11} + \beta q^{13} + 2 q^{15} + 2 q^{17} + 3 \beta q^{19} -2 \beta q^{21} -6 q^{23} + 3 q^{25} + 4 \beta q^{27} + 3 \beta q^{29} + 8 q^{31} -2 q^{33} + 2 \beta q^{35} + 3 \beta q^{37} -2 q^{39} + 5 \beta q^{43} -\beta q^{45} + 8 q^{47} -3 q^{49} + 2 \beta q^{51} + 5 \beta q^{53} + 2 q^{55} -6 q^{57} + 3 \beta q^{59} -9 \beta q^{61} -2 q^{63} + 2 q^{65} + 5 \beta q^{67} -6 \beta q^{69} -10 q^{71} -4 q^{73} + 3 \beta q^{75} -2 \beta q^{77} -5 q^{81} -\beta q^{83} -2 \beta q^{85} -6 q^{87} -4 q^{89} -2 \beta q^{91} + 8 \beta q^{93} + 6 q^{95} -2 q^{97} + \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 4q^{7} + 2q^{9} + 4q^{15} + 4q^{17} - 12q^{23} + 6q^{25} + 16q^{31} - 4q^{33} - 4q^{39} + 16q^{47} - 6q^{49} + 4q^{55} - 12q^{57} - 4q^{63} + 4q^{65} - 20q^{71} - 8q^{73} - 10q^{81} - 12q^{87} - 8q^{89} + 12q^{95} - 4q^{97} + O(q^{100}) \)

Character values

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

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

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 1.41421i 0 1.41421i 0 −2.00000 0 1.00000 0
513.2 0 1.41421i 0 1.41421i 0 −2.00000 0 1.00000 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.b 2
4.b odd 2 1 1024.2.b.e 2
8.b even 2 1 inner 1024.2.b.b 2
8.d odd 2 1 1024.2.b.e 2
16.e even 4 2 1024.2.a.e 2
16.f odd 4 2 1024.2.a.b 2
32.g even 8 2 64.2.e.a 2
32.g even 8 2 128.2.e.a 2
32.h odd 8 2 16.2.e.a 2
32.h odd 8 2 128.2.e.b 2
48.i odd 4 2 9216.2.a.s 2
48.k even 4 2 9216.2.a.d 2
96.o even 8 2 144.2.k.a 2
96.o even 8 2 1152.2.k.b 2
96.p odd 8 2 576.2.k.a 2
96.p odd 8 2 1152.2.k.a 2
160.u even 8 2 400.2.q.b 2
160.v odd 8 2 1600.2.q.b 2
160.y odd 8 2 400.2.l.c 2
160.z even 8 2 1600.2.l.a 2
160.ba even 8 2 400.2.q.a 2
160.bb odd 8 2 1600.2.q.a 2
224.x even 8 2 784.2.m.b 2
224.be even 24 4 784.2.x.c 4
224.bf odd 24 4 784.2.x.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.2.e.a 2 32.h odd 8 2
64.2.e.a 2 32.g even 8 2
128.2.e.a 2 32.g even 8 2
128.2.e.b 2 32.h odd 8 2
144.2.k.a 2 96.o even 8 2
400.2.l.c 2 160.y odd 8 2
400.2.q.a 2 160.ba even 8 2
400.2.q.b 2 160.u even 8 2
576.2.k.a 2 96.p odd 8 2
784.2.m.b 2 224.x even 8 2
784.2.x.c 4 224.be even 24 4
784.2.x.f 4 224.bf odd 24 4
1024.2.a.b 2 16.f odd 4 2
1024.2.a.e 2 16.e even 4 2
1024.2.b.b 2 1.a even 1 1 trivial
1024.2.b.b 2 8.b even 2 1 inner
1024.2.b.e 2 4.b odd 2 1
1024.2.b.e 2 8.d odd 2 1
1152.2.k.a 2 96.p odd 8 2
1152.2.k.b 2 96.o even 8 2
1600.2.l.a 2 160.z even 8 2
1600.2.q.a 2 160.bb odd 8 2
1600.2.q.b 2 160.v odd 8 2
9216.2.a.d 2 48.k even 4 2
9216.2.a.s 2 48.i odd 4 2

Hecke kernels

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

\( T_{3}^{2} + 2 \)
\( T_{5}^{2} + 2 \)
\( T_{7} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 2 + T^{2} \)
$5$ \( 2 + T^{2} \)
$7$ \( ( 2 + T )^{2} \)
$11$ \( 2 + T^{2} \)
$13$ \( 2 + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( 18 + T^{2} \)
$23$ \( ( 6 + T )^{2} \)
$29$ \( 18 + T^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( 18 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 50 + T^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( 50 + T^{2} \)
$59$ \( 18 + T^{2} \)
$61$ \( 162 + T^{2} \)
$67$ \( 50 + T^{2} \)
$71$ \( ( 10 + T )^{2} \)
$73$ \( ( 4 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( 2 + T^{2} \)
$89$ \( ( 4 + T )^{2} \)
$97$ \( ( 2 + T )^{2} \)
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