Properties

Label 512.2.e.j
Level $512$
Weight $2$
Character orbit 512.e
Analytic conductor $4.088$
Analytic rank $0$
Dimension $8$
CM no
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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{3} + ( 1 - \zeta_{16}^{4} + 2 \zeta_{16}^{6} ) q^{5} + ( 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} ) q^{7} + ( -2 \zeta_{16}^{2} + \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{3} + ( 1 - \zeta_{16}^{4} + 2 \zeta_{16}^{6} ) q^{5} + ( 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} ) q^{7} + ( -2 \zeta_{16}^{2} + \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{9} + ( -3 \zeta_{16} + 3 \zeta_{16}^{3} + \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{11} + ( -1 - 2 \zeta_{16}^{2} - \zeta_{16}^{4} ) q^{13} + ( 2 \zeta_{16} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{15} + ( -2 \zeta_{16}^{2} + 2 \zeta_{16}^{6} ) q^{17} + ( -3 \zeta_{16} - 3 \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{19} -4 \zeta_{16}^{6} q^{21} + ( 4 \zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{23} + ( 4 \zeta_{16}^{2} - \zeta_{16}^{4} + 4 \zeta_{16}^{6} ) q^{25} + ( -2 \zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{27} + ( 1 + 6 \zeta_{16}^{2} + \zeta_{16}^{4} ) q^{29} + ( -4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} ) q^{31} + ( -4 + 6 \zeta_{16}^{2} - 6 \zeta_{16}^{6} ) q^{33} + ( -2 \zeta_{16} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{35} + ( -5 + 5 \zeta_{16}^{4} + 2 \zeta_{16}^{6} ) q^{37} + ( 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} ) q^{39} -4 \zeta_{16}^{4} q^{41} + ( -\zeta_{16} + \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{43} + ( 5 - 6 \zeta_{16}^{2} + 5 \zeta_{16}^{4} ) q^{45} + ( 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} ) q^{47} + ( -1 - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{49} + ( 4 \zeta_{16}^{5} - 4 \zeta_{16}^{7} ) q^{51} + ( -1 + \zeta_{16}^{4} - 6 \zeta_{16}^{6} ) q^{53} + ( -8 \zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 8 \zeta_{16}^{7} ) q^{55} + ( 2 \zeta_{16}^{2} + 4 \zeta_{16}^{4} + 2 \zeta_{16}^{6} ) q^{57} + ( -\zeta_{16} + \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{59} + ( -5 + 6 \zeta_{16}^{2} - 5 \zeta_{16}^{4} ) q^{61} + ( 2 \zeta_{16} + 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{63} + 2 q^{65} + ( \zeta_{16} + \zeta_{16}^{3} - 5 \zeta_{16}^{5} + 5 \zeta_{16}^{7} ) q^{67} + ( 8 - 8 \zeta_{16}^{4} + 4 \zeta_{16}^{6} ) q^{69} + ( 4 \zeta_{16} - 6 \zeta_{16}^{3} - 6 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{71} + ( -6 \zeta_{16}^{2} + 2 \zeta_{16}^{4} - 6 \zeta_{16}^{6} ) q^{73} + ( 9 \zeta_{16} - 9 \zeta_{16}^{3} + \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{75} + ( -8 - 4 \zeta_{16}^{2} - 8 \zeta_{16}^{4} ) q^{77} + ( -8 \zeta_{16} + 8 \zeta_{16}^{7} ) q^{79} + ( 3 - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{6} ) q^{81} + ( 7 \zeta_{16} + 7 \zeta_{16}^{3} + \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{83} + ( 4 - 4 \zeta_{16}^{4} + 4 \zeta_{16}^{6} ) q^{85} + ( 4 \zeta_{16} - 6 \zeta_{16}^{3} - 6 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{87} + ( -2 \zeta_{16}^{2} - 2 \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{89} + ( 2 \zeta_{16} - 2 \zeta_{16}^{3} - 6 \zeta_{16}^{5} - 6 \zeta_{16}^{7} ) q^{91} + ( 8 - 8 \zeta_{16}^{2} + 8 \zeta_{16}^{4} ) q^{93} + ( 2 \zeta_{16} - 2 \zeta_{16}^{7} ) q^{95} + ( 8 - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{6} ) q^{97} + ( 7 \zeta_{16} + 7 \zeta_{16}^{3} - 7 \zeta_{16}^{5} + 7 \zeta_{16}^{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{5} + O(q^{10}) \) \( 8q + 8q^{5} - 8q^{13} + 8q^{29} - 32q^{33} - 40q^{37} + 40q^{45} - 8q^{49} - 8q^{53} - 40q^{61} + 16q^{65} + 64q^{69} - 64q^{77} + 24q^{81} + 32q^{85} + 64q^{93} + 64q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(5\) \(511\)
\(\chi(n)\) \(-\zeta_{16}^{2}\) \(1\)

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
−0.382683 + 0.923880i
0.923880 + 0.382683i
−0.923880 0.382683i
0.382683 0.923880i
−0.382683 0.923880i
0.923880 0.382683i
−0.923880 + 0.382683i
0.382683 + 0.923880i
0 −1.84776 1.84776i 0 2.41421 2.41421i 0 1.53073i 0 3.82843i 0
129.2 0 −0.765367 0.765367i 0 −0.414214 + 0.414214i 0 3.69552i 0 1.82843i 0
129.3 0 0.765367 + 0.765367i 0 −0.414214 + 0.414214i 0 3.69552i 0 1.82843i 0
129.4 0 1.84776 + 1.84776i 0 2.41421 2.41421i 0 1.53073i 0 3.82843i 0
385.1 0 −1.84776 + 1.84776i 0 2.41421 + 2.41421i 0 1.53073i 0 3.82843i 0
385.2 0 −0.765367 + 0.765367i 0 −0.414214 0.414214i 0 3.69552i 0 1.82843i 0
385.3 0 0.765367 0.765367i 0 −0.414214 0.414214i 0 3.69552i 0 1.82843i 0
385.4 0 1.84776 1.84776i 0 2.41421 + 2.41421i 0 1.53073i 0 3.82843i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 385.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
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.j yes 8
3.b odd 2 1 4608.2.k.bd 8
4.b odd 2 1 inner 512.2.e.j yes 8
8.b even 2 1 512.2.e.i 8
8.d odd 2 1 512.2.e.i 8
12.b even 2 1 4608.2.k.bd 8
16.e even 4 1 512.2.e.i 8
16.e even 4 1 inner 512.2.e.j yes 8
16.f odd 4 1 512.2.e.i 8
16.f odd 4 1 inner 512.2.e.j yes 8
24.f even 2 1 4608.2.k.bi 8
24.h odd 2 1 4608.2.k.bi 8
32.g even 8 1 1024.2.a.h 4
32.g even 8 1 1024.2.a.i 4
32.g even 8 2 1024.2.b.g 8
32.h odd 8 1 1024.2.a.h 4
32.h odd 8 1 1024.2.a.i 4
32.h odd 8 2 1024.2.b.g 8
48.i odd 4 1 4608.2.k.bd 8
48.i odd 4 1 4608.2.k.bi 8
48.k even 4 1 4608.2.k.bd 8
48.k even 4 1 4608.2.k.bi 8
96.o even 8 1 9216.2.a.w 4
96.o even 8 1 9216.2.a.bp 4
96.p odd 8 1 9216.2.a.w 4
96.p odd 8 1 9216.2.a.bp 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.e.i 8 8.b even 2 1
512.2.e.i 8 8.d odd 2 1
512.2.e.i 8 16.e even 4 1
512.2.e.i 8 16.f odd 4 1
512.2.e.j yes 8 1.a even 1 1 trivial
512.2.e.j yes 8 4.b odd 2 1 inner
512.2.e.j yes 8 16.e even 4 1 inner
512.2.e.j yes 8 16.f odd 4 1 inner
1024.2.a.h 4 32.g even 8 1
1024.2.a.h 4 32.h odd 8 1
1024.2.a.i 4 32.g even 8 1
1024.2.a.i 4 32.h odd 8 1
1024.2.b.g 8 32.g even 8 2
1024.2.b.g 8 32.h odd 8 2
4608.2.k.bd 8 3.b odd 2 1
4608.2.k.bd 8 12.b even 2 1
4608.2.k.bd 8 48.i odd 4 1
4608.2.k.bd 8 48.k even 4 1
4608.2.k.bi 8 24.f even 2 1
4608.2.k.bi 8 24.h odd 2 1
4608.2.k.bi 8 48.i odd 4 1
4608.2.k.bi 8 48.k even 4 1
9216.2.a.w 4 96.o even 8 1
9216.2.a.w 4 96.p odd 8 1
9216.2.a.bp 4 96.o even 8 1
9216.2.a.bp 4 96.p odd 8 1

Hecke kernels

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

\( T_{3}^{8} + 48 T_{3}^{4} + 64 \)
\( T_{5}^{4} - 4 T_{5}^{3} + 8 T_{5}^{2} + 8 T_{5} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 64 + 48 T^{4} + T^{8} \)
$5$ \( ( 4 + 8 T + 8 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$7$ \( ( 32 + 16 T^{2} + T^{4} )^{2} \)
$11$ \( 153664 + 816 T^{4} + T^{8} \)
$13$ \( ( 4 - 8 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$17$ \( ( -8 + T^{2} )^{4} \)
$19$ \( 64 + 1584 T^{4} + T^{8} \)
$23$ \( ( 1568 + 80 T^{2} + T^{4} )^{2} \)
$29$ \( ( 1156 + 136 T + 8 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$31$ \( ( 512 - 64 T^{2} + T^{4} )^{2} \)
$37$ \( ( 2116 + 920 T + 200 T^{2} + 20 T^{3} + T^{4} )^{2} \)
$41$ \( ( 16 + T^{2} )^{4} \)
$43$ \( 64 + 48 T^{4} + T^{8} \)
$47$ \( ( 512 - 64 T^{2} + T^{4} )^{2} \)
$53$ \( ( 1156 - 136 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$59$ \( 64 + 48 T^{4} + T^{8} \)
$61$ \( ( 196 + 280 T + 200 T^{2} + 20 T^{3} + T^{4} )^{2} \)
$67$ \( 153664 + 10032 T^{4} + T^{8} \)
$71$ \( ( 9248 + 208 T^{2} + T^{4} )^{2} \)
$73$ \( ( 4624 + 152 T^{2} + T^{4} )^{2} \)
$79$ \( ( 8192 - 256 T^{2} + T^{4} )^{2} \)
$83$ \( 5345344 + 35376 T^{4} + T^{8} \)
$89$ \( ( 16 + 24 T^{2} + T^{4} )^{2} \)
$97$ \( ( 56 - 16 T + T^{2} )^{4} \)
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