Properties

Label 512.2.b.b
Level $512$
Weight $2$
Character orbit 512.b
Analytic conductor $4.088$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.08834058349\)
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: yes
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.

\(f(q)\) \(=\) \( q + \beta q^{3} + 2 \beta q^{5} + 4 q^{7} + q^{9} +O(q^{10})\) \( q + \beta q^{3} + 2 \beta q^{5} + 4 q^{7} + q^{9} -\beta q^{11} + 2 \beta q^{13} -4 q^{15} -4 q^{17} -5 \beta q^{19} + 4 \beta q^{21} + 4 q^{23} -3 q^{25} + 4 \beta q^{27} + 6 \beta q^{29} -8 q^{31} + 2 q^{33} + 8 \beta q^{35} -2 \beta q^{37} -4 q^{39} -2 q^{41} -3 \beta q^{43} + 2 \beta q^{45} + 9 q^{49} -4 \beta q^{51} -2 \beta q^{53} + 4 q^{55} + 10 q^{57} -3 \beta q^{59} -6 \beta q^{61} + 4 q^{63} -8 q^{65} + 3 \beta q^{67} + 4 \beta q^{69} -4 q^{71} + 4 q^{73} -3 \beta q^{75} -4 \beta q^{77} + 8 q^{79} -5 q^{81} -7 \beta q^{83} -8 \beta q^{85} -12 q^{87} -12 q^{89} + 8 \beta q^{91} -8 \beta q^{93} + 20 q^{95} -4 q^{97} -\beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 8q^{7} + 2q^{9} - 8q^{15} - 8q^{17} + 8q^{23} - 6q^{25} - 16q^{31} + 4q^{33} - 8q^{39} - 4q^{41} + 18q^{49} + 8q^{55} + 20q^{57} + 8q^{63} - 16q^{65} - 8q^{71} + 8q^{73} + 16q^{79} - 10q^{81} - 24q^{87} - 24q^{89} + 40q^{95} - 8q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
1.41421i
1.41421i
0 1.41421i 0 2.82843i 0 4.00000 0 1.00000 0
257.2 0 1.41421i 0 2.82843i 0 4.00000 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 512.2.b.b 2
3.b odd 2 1 4608.2.d.b 2
4.b odd 2 1 512.2.b.a 2
8.b even 2 1 inner 512.2.b.b 2
8.d odd 2 1 512.2.b.a 2
12.b even 2 1 4608.2.d.a 2
16.e even 4 2 512.2.a.c 2
16.f odd 4 2 512.2.a.d yes 2
24.f even 2 1 4608.2.d.a 2
24.h odd 2 1 4608.2.d.b 2
32.g even 8 2 1024.2.e.b 2
32.g even 8 2 1024.2.e.e 2
32.h odd 8 2 1024.2.e.c 2
32.h odd 8 2 1024.2.e.d 2
48.i odd 4 2 4608.2.a.f 2
48.k even 4 2 4608.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.a.c 2 16.e even 4 2
512.2.a.d yes 2 16.f odd 4 2
512.2.b.a 2 4.b odd 2 1
512.2.b.a 2 8.d odd 2 1
512.2.b.b 2 1.a even 1 1 trivial
512.2.b.b 2 8.b even 2 1 inner
1024.2.e.b 2 32.g even 8 2
1024.2.e.c 2 32.h odd 8 2
1024.2.e.d 2 32.h odd 8 2
1024.2.e.e 2 32.g even 8 2
4608.2.a.f 2 48.i odd 4 2
4608.2.a.m 2 48.k even 4 2
4608.2.d.a 2 12.b even 2 1
4608.2.d.a 2 24.f even 2 1
4608.2.d.b 2 3.b odd 2 1
4608.2.d.b 2 24.h odd 2 1

Hecke kernels

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

\( T_{3}^{2} + 2 \)
\( T_{5}^{2} + 8 \)
\( T_{7} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 2 + T^{2} \)
$5$ \( 8 + T^{2} \)
$7$ \( ( -4 + T )^{2} \)
$11$ \( 2 + T^{2} \)
$13$ \( 8 + T^{2} \)
$17$ \( ( 4 + T )^{2} \)
$19$ \( 50 + T^{2} \)
$23$ \( ( -4 + T )^{2} \)
$29$ \( 72 + T^{2} \)
$31$ \( ( 8 + T )^{2} \)
$37$ \( 8 + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( 18 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 8 + T^{2} \)
$59$ \( 18 + T^{2} \)
$61$ \( 72 + T^{2} \)
$67$ \( 18 + T^{2} \)
$71$ \( ( 4 + T )^{2} \)
$73$ \( ( -4 + T )^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 98 + T^{2} \)
$89$ \( ( 12 + T )^{2} \)
$97$ \( ( 4 + T )^{2} \)
show more
show less