Properties

Label 256.2.e.a
Level $256$
Weight $2$
Character orbit 256.e
Analytic conductor $2.044$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 256.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.04417029174\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8} \)
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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{3} + ( -1 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{5} + ( -2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{7} + ( 2 - 4 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{3} + ( -1 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{5} + ( -2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{7} + ( 2 - 4 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{9} + ( 3 \zeta_{24} + \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{11} + ( -3 + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{13} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{15} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{17} + ( -\zeta_{24} - 3 \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{19} + ( -2 + 2 \zeta_{24}^{6} ) q^{21} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{23} -\zeta_{24}^{6} q^{25} + ( -4 \zeta_{24} + 4 \zeta_{24}^{5} ) q^{27} + ( 1 - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{29} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{31} + ( -4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{33} + ( 2 \zeta_{24} + 6 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{35} + ( 3 + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{37} + ( 8 \zeta_{24} - 8 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{39} + ( -4 + 8 \zeta_{24}^{4} ) q^{41} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{43} + ( 7 - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} + 7 \zeta_{24}^{6} ) q^{45} + ( 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{47} + ( -1 - 8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{49} + ( 2 \zeta_{24} - 6 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{51} + ( 7 + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - 7 \zeta_{24}^{6} ) q^{53} + ( 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{55} + ( -2 + 4 \zeta_{24}^{4} ) q^{57} + ( -9 \zeta_{24} + \zeta_{24}^{3} + 9 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{59} + ( 1 + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{61} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{63} + ( -6 + 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{65} + ( 3 \zeta_{24} + \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{67} + ( -2 + 8 \zeta_{24}^{2} - 8 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{69} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{71} + ( -2 + 4 \zeta_{24}^{4} - 6 \zeta_{24}^{6} ) q^{73} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{75} + ( -2 - 8 \zeta_{24}^{2} - 8 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{77} + ( -8 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{79} + ( -1 - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{81} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{83} + ( -6 + 6 \zeta_{24}^{6} ) q^{85} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{87} + ( 2 - 4 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{89} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{91} + ( -8 + 8 \zeta_{24}^{2} + 8 \zeta_{24}^{4} - 8 \zeta_{24}^{6} ) q^{93} + ( 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{7} ) q^{95} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{97} + ( -5 \zeta_{24} - 3 \zeta_{24}^{3} - 5 \zeta_{24}^{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 16q^{13} - 16q^{21} + 16q^{37} + 48q^{45} - 8q^{49} + 48q^{53} + 16q^{61} - 48q^{65} - 48q^{69} - 48q^{77} - 8q^{81} - 48q^{85} - 32q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(-\zeta_{24}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
−0.258819 0.965926i
0.258819 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
0 −1.93185 1.93185i 0 1.73205 1.73205i 0 1.03528i 0 4.46410i 0
65.2 0 −0.517638 0.517638i 0 −1.73205 + 1.73205i 0 3.86370i 0 2.46410i 0
65.3 0 0.517638 + 0.517638i 0 −1.73205 + 1.73205i 0 3.86370i 0 2.46410i 0
65.4 0 1.93185 + 1.93185i 0 1.73205 1.73205i 0 1.03528i 0 4.46410i 0
193.1 0 −1.93185 + 1.93185i 0 1.73205 + 1.73205i 0 1.03528i 0 4.46410i 0
193.2 0 −0.517638 + 0.517638i 0 −1.73205 1.73205i 0 3.86370i 0 2.46410i 0
193.3 0 0.517638 0.517638i 0 −1.73205 1.73205i 0 3.86370i 0 2.46410i 0
193.4 0 1.93185 1.93185i 0 1.73205 + 1.73205i 0 1.03528i 0 4.46410i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.4
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 256.2.e.a 8
3.b odd 2 1 2304.2.k.f 8
4.b odd 2 1 inner 256.2.e.a 8
8.b even 2 1 256.2.e.b yes 8
8.d odd 2 1 256.2.e.b yes 8
12.b even 2 1 2304.2.k.f 8
16.e even 4 1 inner 256.2.e.a 8
16.e even 4 1 256.2.e.b yes 8
16.f odd 4 1 inner 256.2.e.a 8
16.f odd 4 1 256.2.e.b yes 8
24.f even 2 1 2304.2.k.k 8
24.h odd 2 1 2304.2.k.k 8
32.g even 8 1 1024.2.a.g 4
32.g even 8 1 1024.2.a.j 4
32.g even 8 2 1024.2.b.h 8
32.h odd 8 1 1024.2.a.g 4
32.h odd 8 1 1024.2.a.j 4
32.h odd 8 2 1024.2.b.h 8
48.i odd 4 1 2304.2.k.f 8
48.i odd 4 1 2304.2.k.k 8
48.k even 4 1 2304.2.k.f 8
48.k even 4 1 2304.2.k.k 8
96.o even 8 1 9216.2.a.bb 4
96.o even 8 1 9216.2.a.bk 4
96.p odd 8 1 9216.2.a.bb 4
96.p odd 8 1 9216.2.a.bk 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
256.2.e.a 8 1.a even 1 1 trivial
256.2.e.a 8 4.b odd 2 1 inner
256.2.e.a 8 16.e even 4 1 inner
256.2.e.a 8 16.f odd 4 1 inner
256.2.e.b yes 8 8.b even 2 1
256.2.e.b yes 8 8.d odd 2 1
256.2.e.b yes 8 16.e even 4 1
256.2.e.b yes 8 16.f odd 4 1
1024.2.a.g 4 32.g even 8 1
1024.2.a.g 4 32.h odd 8 1
1024.2.a.j 4 32.g even 8 1
1024.2.a.j 4 32.h odd 8 1
1024.2.b.h 8 32.g even 8 2
1024.2.b.h 8 32.h odd 8 2
2304.2.k.f 8 3.b odd 2 1
2304.2.k.f 8 12.b even 2 1
2304.2.k.f 8 48.i odd 4 1
2304.2.k.f 8 48.k even 4 1
2304.2.k.k 8 24.f even 2 1
2304.2.k.k 8 24.h odd 2 1
2304.2.k.k 8 48.i odd 4 1
2304.2.k.k 8 48.k even 4 1
9216.2.a.bb 4 96.o even 8 1
9216.2.a.bb 4 96.p odd 8 1
9216.2.a.bk 4 96.o even 8 1
9216.2.a.bk 4 96.p odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{4} + 8 T_{13}^{3} + 32 T_{13}^{2} + 16 T_{13} + 4 \) acting on \(S_{2}^{\mathrm{new}}(256, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 16 + 56 T^{4} + T^{8} \)
$5$ \( ( 36 + T^{4} )^{2} \)
$7$ \( ( 16 + 16 T^{2} + T^{4} )^{2} \)
$11$ \( 1296 + 504 T^{4} + T^{8} \)
$13$ \( ( 4 + 16 T + 32 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$17$ \( ( -12 + T^{2} )^{4} \)
$19$ \( 1296 + 504 T^{4} + T^{8} \)
$23$ \( ( 144 + 48 T^{2} + T^{4} )^{2} \)
$29$ \( ( 36 + T^{4} )^{2} \)
$31$ \( ( -32 + T^{2} )^{4} \)
$37$ \( ( 4 - 16 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$41$ \( ( 48 + T^{2} )^{4} \)
$43$ \( 104976 + 4536 T^{4} + T^{8} \)
$47$ \( ( -96 + T^{2} )^{4} \)
$53$ \( ( 4356 - 1584 T + 288 T^{2} - 24 T^{3} + T^{4} )^{2} \)
$59$ \( 37015056 + 16056 T^{4} + T^{8} \)
$61$ \( ( 4 - 16 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$67$ \( 456976 + 1784 T^{4} + T^{8} \)
$71$ \( ( 144 + 48 T^{2} + T^{4} )^{2} \)
$73$ \( ( 576 + 96 T^{2} + T^{4} )^{2} \)
$79$ \( ( 4096 - 256 T^{2} + T^{4} )^{2} \)
$83$ \( 104976 + 4536 T^{4} + T^{8} \)
$89$ \( ( 576 + 96 T^{2} + T^{4} )^{2} \)
$97$ \( ( -12 + T^{2} )^{4} \)
show more
show less