Properties

Label 768.3.h.f
Level $768$
Weight $3$
Character orbit 768.h
Analytic conductor $20.926$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 768.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(20.9264843029\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 96)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 + \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{3} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{5} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{7} + ( 3 + 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{9} +O(q^{10})\) \( q + ( 1 + \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{3} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{5} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{7} + ( 3 + 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{9} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{11} -6 \zeta_{24}^{6} q^{13} + ( 2 \zeta_{24} - 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{15} + ( 16 \zeta_{24} - 16 \zeta_{24}^{3} - 16 \zeta_{24}^{5} ) q^{17} + ( -6 + 12 \zeta_{24}^{4} ) q^{19} + ( 18 \zeta_{24} + 18 \zeta_{24}^{3} - 18 \zeta_{24}^{5} - 18 \zeta_{24}^{6} ) q^{21} + ( -12 \zeta_{24} + 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} - 24 \zeta_{24}^{7} ) q^{23} -17 q^{25} + ( 15 - 3 \zeta_{24} - 3 \zeta_{24}^{3} - 30 \zeta_{24}^{4} - 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{27} + ( 22 \zeta_{24} + 22 \zeta_{24}^{3} - 22 \zeta_{24}^{5} ) q^{29} + ( 36 \zeta_{24}^{2} - 18 \zeta_{24}^{6} ) q^{31} + ( -36 - 18 \zeta_{24} + 18 \zeta_{24}^{3} + 18 \zeta_{24}^{5} ) q^{33} + ( -12 \zeta_{24} - 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} + 24 \zeta_{24}^{7} ) q^{35} + 38 \zeta_{24}^{6} q^{37} + ( -6 \zeta_{24} - 12 \zeta_{24}^{2} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{6} - 12 \zeta_{24}^{7} ) q^{39} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} ) q^{41} + ( 6 - 12 \zeta_{24}^{4} ) q^{43} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 24 \zeta_{24}^{6} ) q^{45} + ( -24 \zeta_{24} + 24 \zeta_{24}^{3} - 24 \zeta_{24}^{5} - 48 \zeta_{24}^{7} ) q^{47} + 59 q^{49} + ( 32 - 16 \zeta_{24} - 16 \zeta_{24}^{3} - 64 \zeta_{24}^{4} - 16 \zeta_{24}^{5} + 32 \zeta_{24}^{7} ) q^{51} + ( -10 \zeta_{24} - 10 \zeta_{24}^{3} + 10 \zeta_{24}^{5} ) q^{53} + ( 48 \zeta_{24}^{2} - 24 \zeta_{24}^{6} ) q^{55} + ( 18 - 18 \zeta_{24} + 18 \zeta_{24}^{3} + 18 \zeta_{24}^{5} ) q^{57} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{59} -22 \zeta_{24}^{6} q^{61} + ( -36 \zeta_{24} + 36 \zeta_{24}^{2} + 36 \zeta_{24}^{3} - 36 \zeta_{24}^{5} - 18 \zeta_{24}^{6} - 72 \zeta_{24}^{7} ) q^{63} + ( -12 \zeta_{24} + 12 \zeta_{24}^{3} + 12 \zeta_{24}^{5} ) q^{65} + ( 66 - 132 \zeta_{24}^{4} ) q^{67} + ( -36 \zeta_{24} - 36 \zeta_{24}^{3} + 36 \zeta_{24}^{5} - 72 \zeta_{24}^{6} ) q^{69} + ( 12 \zeta_{24} - 12 \zeta_{24}^{3} + 12 \zeta_{24}^{5} + 24 \zeta_{24}^{7} ) q^{71} + 30 q^{73} + ( -17 - 17 \zeta_{24} - 17 \zeta_{24}^{3} + 34 \zeta_{24}^{4} - 17 \zeta_{24}^{5} + 34 \zeta_{24}^{7} ) q^{75} + ( -108 \zeta_{24} - 108 \zeta_{24}^{3} + 108 \zeta_{24}^{5} ) q^{77} + ( 36 \zeta_{24}^{2} - 18 \zeta_{24}^{6} ) q^{79} + ( -63 + 36 \zeta_{24} - 36 \zeta_{24}^{3} - 36 \zeta_{24}^{5} ) q^{81} + ( -30 \zeta_{24} - 30 \zeta_{24}^{3} - 30 \zeta_{24}^{5} + 60 \zeta_{24}^{7} ) q^{83} + 64 \zeta_{24}^{6} q^{85} + ( -22 \zeta_{24} + 88 \zeta_{24}^{2} + 22 \zeta_{24}^{3} - 22 \zeta_{24}^{5} - 44 \zeta_{24}^{6} - 44 \zeta_{24}^{7} ) q^{87} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} ) q^{89} + ( 36 - 72 \zeta_{24}^{4} ) q^{91} + ( 54 \zeta_{24} + 54 \zeta_{24}^{3} - 54 \zeta_{24}^{5} - 54 \zeta_{24}^{6} ) q^{93} + ( -12 \zeta_{24} + 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} - 24 \zeta_{24}^{7} ) q^{95} + 90 q^{97} + ( -72 - 18 \zeta_{24} - 18 \zeta_{24}^{3} + 144 \zeta_{24}^{4} - 18 \zeta_{24}^{5} + 36 \zeta_{24}^{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 24q^{9} + O(q^{10}) \) \( 8q + 24q^{9} - 136q^{25} - 288q^{33} + 472q^{49} + 144q^{57} + 240q^{73} - 504q^{81} + 720q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
641.1
−0.258819 + 0.965926i
−0.965926 0.258819i
−0.258819 0.965926i
−0.965926 + 0.258819i
0.965926 + 0.258819i
0.258819 0.965926i
0.965926 0.258819i
0.258819 + 0.965926i
0 −2.44949 1.73205i 0 −2.82843 0 −10.3923 0 3.00000 + 8.48528i 0
641.2 0 −2.44949 1.73205i 0 2.82843 0 10.3923 0 3.00000 + 8.48528i 0
641.3 0 −2.44949 + 1.73205i 0 −2.82843 0 −10.3923 0 3.00000 8.48528i 0
641.4 0 −2.44949 + 1.73205i 0 2.82843 0 10.3923 0 3.00000 8.48528i 0
641.5 0 2.44949 1.73205i 0 −2.82843 0 10.3923 0 3.00000 8.48528i 0
641.6 0 2.44949 1.73205i 0 2.82843 0 −10.3923 0 3.00000 8.48528i 0
641.7 0 2.44949 + 1.73205i 0 −2.82843 0 10.3923 0 3.00000 + 8.48528i 0
641.8 0 2.44949 + 1.73205i 0 2.82843 0 −10.3923 0 3.00000 + 8.48528i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 641.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.3.h.f 8
3.b odd 2 1 inner 768.3.h.f 8
4.b odd 2 1 inner 768.3.h.f 8
8.b even 2 1 inner 768.3.h.f 8
8.d odd 2 1 inner 768.3.h.f 8
12.b even 2 1 inner 768.3.h.f 8
16.e even 4 1 96.3.e.a 4
16.e even 4 1 192.3.e.e 4
16.f odd 4 1 96.3.e.a 4
16.f odd 4 1 192.3.e.e 4
24.f even 2 1 inner 768.3.h.f 8
24.h odd 2 1 inner 768.3.h.f 8
48.i odd 4 1 96.3.e.a 4
48.i odd 4 1 192.3.e.e 4
48.k even 4 1 96.3.e.a 4
48.k even 4 1 192.3.e.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.3.e.a 4 16.e even 4 1
96.3.e.a 4 16.f odd 4 1
96.3.e.a 4 48.i odd 4 1
96.3.e.a 4 48.k even 4 1
192.3.e.e 4 16.e even 4 1
192.3.e.e 4 16.f odd 4 1
192.3.e.e 4 48.i odd 4 1
192.3.e.e 4 48.k even 4 1
768.3.h.f 8 1.a even 1 1 trivial
768.3.h.f 8 3.b odd 2 1 inner
768.3.h.f 8 4.b odd 2 1 inner
768.3.h.f 8 8.b even 2 1 inner
768.3.h.f 8 8.d odd 2 1 inner
768.3.h.f 8 12.b even 2 1 inner
768.3.h.f 8 24.f even 2 1 inner
768.3.h.f 8 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{2} - 8 \)
\( T_{7}^{2} - 108 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 81 - 6 T^{2} + T^{4} )^{2} \)
$5$ \( ( -8 + T^{2} )^{4} \)
$7$ \( ( -108 + T^{2} )^{4} \)
$11$ \( ( -216 + T^{2} )^{4} \)
$13$ \( ( 36 + T^{2} )^{4} \)
$17$ \( ( 512 + T^{2} )^{4} \)
$19$ \( ( 108 + T^{2} )^{4} \)
$23$ \( ( 864 + T^{2} )^{4} \)
$29$ \( ( -968 + T^{2} )^{4} \)
$31$ \( ( -972 + T^{2} )^{4} \)
$37$ \( ( 1444 + T^{2} )^{4} \)
$41$ \( ( 32 + T^{2} )^{4} \)
$43$ \( ( 108 + T^{2} )^{4} \)
$47$ \( ( 3456 + T^{2} )^{4} \)
$53$ \( ( -200 + T^{2} )^{4} \)
$59$ \( ( -216 + T^{2} )^{4} \)
$61$ \( ( 484 + T^{2} )^{4} \)
$67$ \( ( 13068 + T^{2} )^{4} \)
$71$ \( ( 864 + T^{2} )^{4} \)
$73$ \( ( -30 + T )^{8} \)
$79$ \( ( -972 + T^{2} )^{4} \)
$83$ \( ( -5400 + T^{2} )^{4} \)
$89$ \( ( 32 + T^{2} )^{4} \)
$97$ \( ( -90 + T )^{8} \)
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