Properties

Label 512.2.e.d
Level $512$
Weight $2$
Character orbit 512.e
Analytic conductor $4.088$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.08834058349\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( -1 - i ) q^{5} + 3 i q^{9} +O(q^{10})\) \( q + ( -1 - i ) q^{5} + 3 i q^{9} + ( 5 - 5 i ) q^{13} + 8 q^{17} -3 i q^{25} + ( 3 - 3 i ) q^{29} + ( -7 - 7 i ) q^{37} + 8 i q^{41} + ( 3 - 3 i ) q^{45} + 7 q^{49} + ( 9 + 9 i ) q^{53} + ( -11 + 11 i ) q^{61} -10 q^{65} -6 i q^{73} -9 q^{81} + ( -8 - 8 i ) q^{85} -10 i q^{89} -8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + O(q^{10}) \) \( 2 q - 2 q^{5} + 10 q^{13} + 16 q^{17} + 6 q^{29} - 14 q^{37} + 6 q^{45} + 14 q^{49} + 18 q^{53} - 22 q^{61} - 20 q^{65} - 18 q^{81} - 16 q^{85} - 16 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/512\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(511\)
\(\chi(n)\) \(i\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
1.00000i
1.00000i
0 0 0 −1.00000 + 1.00000i 0 0 0 3.00000i 0
385.1 0 0 0 −1.00000 1.00000i 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 \(\Q(\sqrt{-1}) \)
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 512.2.e.d 2
3.b odd 2 1 4608.2.k.r 2
4.b odd 2 1 CM 512.2.e.d 2
8.b even 2 1 512.2.e.e yes 2
8.d odd 2 1 512.2.e.e yes 2
12.b even 2 1 4608.2.k.r 2
16.e even 4 1 inner 512.2.e.d 2
16.e even 4 1 512.2.e.e yes 2
16.f odd 4 1 inner 512.2.e.d 2
16.f odd 4 1 512.2.e.e yes 2
24.f even 2 1 4608.2.k.g 2
24.h odd 2 1 4608.2.k.g 2
32.g even 8 2 1024.2.a.c 2
32.g even 8 2 1024.2.b.c 2
32.h odd 8 2 1024.2.a.c 2
32.h odd 8 2 1024.2.b.c 2
48.i odd 4 1 4608.2.k.g 2
48.i odd 4 1 4608.2.k.r 2
48.k even 4 1 4608.2.k.g 2
48.k even 4 1 4608.2.k.r 2
96.o even 8 2 9216.2.a.p 2
96.p odd 8 2 9216.2.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.e.d 2 1.a even 1 1 trivial
512.2.e.d 2 4.b odd 2 1 CM
512.2.e.d 2 16.e even 4 1 inner
512.2.e.d 2 16.f odd 4 1 inner
512.2.e.e yes 2 8.b even 2 1
512.2.e.e yes 2 8.d odd 2 1
512.2.e.e yes 2 16.e even 4 1
512.2.e.e yes 2 16.f odd 4 1
1024.2.a.c 2 32.g even 8 2
1024.2.a.c 2 32.h odd 8 2
1024.2.b.c 2 32.g even 8 2
1024.2.b.c 2 32.h odd 8 2
4608.2.k.g 2 24.f even 2 1
4608.2.k.g 2 24.h odd 2 1
4608.2.k.g 2 48.i odd 4 1
4608.2.k.g 2 48.k even 4 1
4608.2.k.r 2 3.b odd 2 1
4608.2.k.r 2 12.b even 2 1
4608.2.k.r 2 48.i odd 4 1
4608.2.k.r 2 48.k even 4 1
9216.2.a.p 2 96.o even 8 2
9216.2.a.p 2 96.p odd 8 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(512, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 2 + 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 50 - 10 T + T^{2} \)
$17$ \( ( -8 + T )^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( 18 - 6 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 98 + 14 T + T^{2} \)
$41$ \( 64 + T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 162 - 18 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( 242 + 22 T + T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( 100 + T^{2} \)
$97$ \( ( 8 + T )^{2} \)
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