Properties

Label 3072.2.d.h
Level $3072$
Weight $2$
Character orbit 3072.d
Analytic conductor $24.530$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3072.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5300435009\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 768)
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.

\(f(q)\) \(=\) \( q + \zeta_{24}^{6} q^{3} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{5} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{7} - q^{9} +O(q^{10})\) \( q + \zeta_{24}^{6} q^{3} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{5} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{7} - q^{9} + ( 2 - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{11} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{13} + ( 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{15} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{17} + ( 2 - 4 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{19} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{21} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{23} + ( -3 + 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{25} -\zeta_{24}^{6} q^{27} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{29} + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{31} + ( 2 + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{33} + ( -2 + 4 \zeta_{24}^{4} - 6 \zeta_{24}^{6} ) q^{35} + ( \zeta_{24} - \zeta_{24}^{3} - 5 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{37} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{39} + ( -8 - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{41} + ( 2 - 4 \zeta_{24}^{4} ) q^{43} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{45} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{47} - q^{49} + ( -2 + 4 \zeta_{24}^{4} ) q^{51} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{53} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{55} + ( -4 + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{57} + ( -8 + 16 \zeta_{24}^{4} ) q^{59} + ( \zeta_{24} - \zeta_{24}^{3} - 5 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{61} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{63} + ( 6 - 12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{65} + ( 4 - 8 \zeta_{24}^{4} + 8 \zeta_{24}^{6} ) q^{67} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{69} + ( -10 \zeta_{24} - 10 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{71} -4 q^{73} + ( -4 + 8 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{75} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{77} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{79} + q^{81} + ( 2 - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{83} + ( 8 \zeta_{24} - 8 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{85} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{87} + ( -2 - 8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{89} + ( 6 - 12 \zeta_{24}^{4} ) q^{91} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{93} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 16 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{95} + ( 8 + 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{97} + ( -2 + 4 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{9} + O(q^{10}) \) \( 8q - 8q^{9} - 24q^{25} + 16q^{33} - 64q^{41} - 8q^{49} - 32q^{57} + 48q^{65} - 32q^{73} + 8q^{81} - 16q^{89} + 64q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times\).

\(n\) \(1025\) \(2047\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1537.1
−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 1.00000i 0 3.86370i 0 −2.44949 0 −1.00000 0
1537.2 0 1.00000i 0 1.03528i 0 −2.44949 0 −1.00000 0
1537.3 0 1.00000i 0 1.03528i 0 2.44949 0 −1.00000 0
1537.4 0 1.00000i 0 3.86370i 0 2.44949 0 −1.00000 0
1537.5 0 1.00000i 0 3.86370i 0 2.44949 0 −1.00000 0
1537.6 0 1.00000i 0 1.03528i 0 2.44949 0 −1.00000 0
1537.7 0 1.00000i 0 1.03528i 0 −2.44949 0 −1.00000 0
1537.8 0 1.00000i 0 3.86370i 0 −2.44949 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1537.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3072.2.d.h 8
4.b odd 2 1 inner 3072.2.d.h 8
8.b even 2 1 inner 3072.2.d.h 8
8.d odd 2 1 inner 3072.2.d.h 8
16.e even 4 1 3072.2.a.l 4
16.e even 4 1 3072.2.a.r 4
16.f odd 4 1 3072.2.a.l 4
16.f odd 4 1 3072.2.a.r 4
32.g even 8 2 768.2.j.e 8
32.g even 8 2 768.2.j.f yes 8
32.h odd 8 2 768.2.j.e 8
32.h odd 8 2 768.2.j.f yes 8
48.i odd 4 1 9216.2.a.bc 4
48.i odd 4 1 9216.2.a.bi 4
48.k even 4 1 9216.2.a.bc 4
48.k even 4 1 9216.2.a.bi 4
96.o even 8 2 2304.2.k.e 8
96.o even 8 2 2304.2.k.l 8
96.p odd 8 2 2304.2.k.e 8
96.p odd 8 2 2304.2.k.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.2.j.e 8 32.g even 8 2
768.2.j.e 8 32.h odd 8 2
768.2.j.f yes 8 32.g even 8 2
768.2.j.f yes 8 32.h odd 8 2
2304.2.k.e 8 96.o even 8 2
2304.2.k.e 8 96.p odd 8 2
2304.2.k.l 8 96.o even 8 2
2304.2.k.l 8 96.p odd 8 2
3072.2.a.l 4 16.e even 4 1
3072.2.a.l 4 16.f odd 4 1
3072.2.a.r 4 16.e even 4 1
3072.2.a.r 4 16.f odd 4 1
3072.2.d.h 8 1.a even 1 1 trivial
3072.2.d.h 8 4.b odd 2 1 inner
3072.2.d.h 8 8.b even 2 1 inner
3072.2.d.h 8 8.d odd 2 1 inner
9216.2.a.bc 4 48.i odd 4 1
9216.2.a.bc 4 48.k even 4 1
9216.2.a.bi 4 48.i odd 4 1
9216.2.a.bi 4 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3072, [\chi])\):

\( T_{5}^{4} + 16 T_{5}^{2} + 16 \)
\( T_{7}^{2} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 1 + T^{2} )^{4} \)
$5$ \( ( 16 + 16 T^{2} + T^{4} )^{2} \)
$7$ \( ( -6 + T^{2} )^{4} \)
$11$ \( ( 64 + 32 T^{2} + T^{4} )^{2} \)
$13$ \( ( 18 + T^{2} )^{4} \)
$17$ \( ( -12 + T^{2} )^{4} \)
$19$ \( ( 16 + 56 T^{2} + T^{4} )^{2} \)
$23$ \( ( -8 + T^{2} )^{4} \)
$29$ \( ( 2704 + 112 T^{2} + T^{4} )^{2} \)
$31$ \( ( -54 + T^{2} )^{4} \)
$37$ \( ( 36 + 84 T^{2} + T^{4} )^{2} \)
$41$ \( ( 52 + 16 T + T^{2} )^{4} \)
$43$ \( ( 12 + T^{2} )^{4} \)
$47$ \( ( -8 + T^{2} )^{4} \)
$53$ \( ( 1936 + 112 T^{2} + T^{4} )^{2} \)
$59$ \( ( 192 + T^{2} )^{4} \)
$61$ \( ( 36 + 84 T^{2} + T^{4} )^{2} \)
$67$ \( ( 256 + 224 T^{2} + T^{4} )^{2} \)
$71$ \( ( 10816 - 304 T^{2} + T^{4} )^{2} \)
$73$ \( ( 4 + T )^{8} \)
$79$ \( ( -6 + T^{2} )^{4} \)
$83$ \( ( 64 + 32 T^{2} + T^{4} )^{2} \)
$89$ \( ( -44 + 4 T + T^{2} )^{4} \)
$97$ \( ( 16 - 16 T + T^{2} )^{4} \)
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