Properties

Label 3072.1.p.d
Level $3072$
Weight $1$
Character orbit 3072.p
Analytic conductor $1.533$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -3
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3072.p (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.53312771881\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.57982058496.8

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{16} q^{3} + ( -\zeta_{16}^{5} - \zeta_{16}^{7} ) q^{7} + \zeta_{16}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{16} q^{3} + ( -\zeta_{16}^{5} - \zeta_{16}^{7} ) q^{7} + \zeta_{16}^{2} q^{9} + ( 1 - \zeta_{16}^{2} ) q^{13} + ( -\zeta_{16}^{3} - \zeta_{16}^{7} ) q^{19} + ( -1 + \zeta_{16}^{6} ) q^{21} + \zeta_{16}^{6} q^{25} -\zeta_{16}^{3} q^{27} + ( \zeta_{16}^{3} - \zeta_{16}^{5} ) q^{31} + ( \zeta_{16}^{2} + \zeta_{16}^{4} ) q^{37} + ( -\zeta_{16} + \zeta_{16}^{3} ) q^{39} + ( -\zeta_{16} + \zeta_{16}^{5} ) q^{43} + ( -\zeta_{16}^{2} - \zeta_{16}^{4} - \zeta_{16}^{6} ) q^{49} + ( -1 + \zeta_{16}^{4} ) q^{57} + ( \zeta_{16}^{4} - \zeta_{16}^{6} ) q^{61} + ( \zeta_{16} - \zeta_{16}^{7} ) q^{63} + ( -1 - \zeta_{16}^{4} ) q^{73} -\zeta_{16}^{7} q^{75} + ( -\zeta_{16} - \zeta_{16}^{7} ) q^{79} + \zeta_{16}^{4} q^{81} + ( -\zeta_{16} - \zeta_{16}^{5} ) q^{91} + ( -\zeta_{16}^{4} + \zeta_{16}^{6} ) q^{93} + ( -\zeta_{16}^{2} + \zeta_{16}^{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{13} - 8q^{21} - 8q^{57} - 8q^{73} + O(q^{100}) \)

Character values

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

\(n\) \(1025\) \(2047\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(\zeta_{16}^{6}\)

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
0.923880 + 0.382683i
−0.923880 0.382683i
0.923880 0.382683i
−0.923880 + 0.382683i
0.382683 0.923880i
−0.382683 + 0.923880i
0.382683 + 0.923880i
−0.382683 0.923880i
0 −0.923880 0.382683i 0 0 0 1.30656 1.30656i 0 0.707107 + 0.707107i 0
641.2 0 0.923880 + 0.382683i 0 0 0 −1.30656 + 1.30656i 0 0.707107 + 0.707107i 0
1409.1 0 −0.923880 + 0.382683i 0 0 0 1.30656 + 1.30656i 0 0.707107 0.707107i 0
1409.2 0 0.923880 0.382683i 0 0 0 −1.30656 1.30656i 0 0.707107 0.707107i 0
2177.1 0 −0.382683 + 0.923880i 0 0 0 −0.541196 + 0.541196i 0 −0.707107 0.707107i 0
2177.2 0 0.382683 0.923880i 0 0 0 0.541196 0.541196i 0 −0.707107 0.707107i 0
2945.1 0 −0.382683 0.923880i 0 0 0 −0.541196 0.541196i 0 −0.707107 + 0.707107i 0
2945.2 0 0.382683 + 0.923880i 0 0 0 0.541196 + 0.541196i 0 −0.707107 + 0.707107i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2945.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
4.b odd 2 1 inner
12.b even 2 1 inner
32.g even 8 1 inner
32.h odd 8 1 inner
96.o even 8 1 inner
96.p odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3072.1.p.d yes 8
3.b odd 2 1 CM 3072.1.p.d yes 8
4.b odd 2 1 inner 3072.1.p.d yes 8
8.b even 2 1 3072.1.p.a 8
8.d odd 2 1 3072.1.p.a 8
12.b even 2 1 inner 3072.1.p.d yes 8
16.e even 4 1 3072.1.p.b yes 8
16.e even 4 1 3072.1.p.c yes 8
16.f odd 4 1 3072.1.p.b yes 8
16.f odd 4 1 3072.1.p.c yes 8
24.f even 2 1 3072.1.p.a 8
24.h odd 2 1 3072.1.p.a 8
32.g even 8 1 3072.1.p.a 8
32.g even 8 1 3072.1.p.b yes 8
32.g even 8 1 3072.1.p.c yes 8
32.g even 8 1 inner 3072.1.p.d yes 8
32.h odd 8 1 3072.1.p.a 8
32.h odd 8 1 3072.1.p.b yes 8
32.h odd 8 1 3072.1.p.c yes 8
32.h odd 8 1 inner 3072.1.p.d yes 8
48.i odd 4 1 3072.1.p.b yes 8
48.i odd 4 1 3072.1.p.c yes 8
48.k even 4 1 3072.1.p.b yes 8
48.k even 4 1 3072.1.p.c yes 8
96.o even 8 1 3072.1.p.a 8
96.o even 8 1 3072.1.p.b yes 8
96.o even 8 1 3072.1.p.c yes 8
96.o even 8 1 inner 3072.1.p.d yes 8
96.p odd 8 1 3072.1.p.a 8
96.p odd 8 1 3072.1.p.b yes 8
96.p odd 8 1 3072.1.p.c yes 8
96.p odd 8 1 inner 3072.1.p.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3072.1.p.a 8 8.b even 2 1
3072.1.p.a 8 8.d odd 2 1
3072.1.p.a 8 24.f even 2 1
3072.1.p.a 8 24.h odd 2 1
3072.1.p.a 8 32.g even 8 1
3072.1.p.a 8 32.h odd 8 1
3072.1.p.a 8 96.o even 8 1
3072.1.p.a 8 96.p odd 8 1
3072.1.p.b yes 8 16.e even 4 1
3072.1.p.b yes 8 16.f odd 4 1
3072.1.p.b yes 8 32.g even 8 1
3072.1.p.b yes 8 32.h odd 8 1
3072.1.p.b yes 8 48.i odd 4 1
3072.1.p.b yes 8 48.k even 4 1
3072.1.p.b yes 8 96.o even 8 1
3072.1.p.b yes 8 96.p odd 8 1
3072.1.p.c yes 8 16.e even 4 1
3072.1.p.c yes 8 16.f odd 4 1
3072.1.p.c yes 8 32.g even 8 1
3072.1.p.c yes 8 32.h odd 8 1
3072.1.p.c yes 8 48.i odd 4 1
3072.1.p.c yes 8 48.k even 4 1
3072.1.p.c yes 8 96.o even 8 1
3072.1.p.c yes 8 96.p odd 8 1
3072.1.p.d yes 8 1.a even 1 1 trivial
3072.1.p.d yes 8 3.b odd 2 1 CM
3072.1.p.d yes 8 4.b odd 2 1 inner
3072.1.p.d yes 8 12.b even 2 1 inner
3072.1.p.d yes 8 32.g even 8 1 inner
3072.1.p.d yes 8 32.h odd 8 1 inner
3072.1.p.d yes 8 96.o even 8 1 inner
3072.1.p.d yes 8 96.p odd 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{4} - 4 T_{13}^{3} + 6 T_{13}^{2} - 4 T_{13} + 2 \) acting on \(S_{1}^{\mathrm{new}}(3072, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 1 + T^{8} \)
$5$ \( T^{8} \)
$7$ \( 4 + 12 T^{4} + T^{8} \)
$11$ \( T^{8} \)
$13$ \( ( 2 - 4 T + 6 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$17$ \( T^{8} \)
$19$ \( 16 + T^{8} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( ( 2 - 4 T^{2} + T^{4} )^{2} \)
$37$ \( ( 2 + 4 T + 2 T^{2} + T^{4} )^{2} \)
$41$ \( T^{8} \)
$43$ \( 16 + T^{8} \)
$47$ \( T^{8} \)
$53$ \( T^{8} \)
$59$ \( T^{8} \)
$61$ \( ( 2 - 4 T + 2 T^{2} + T^{4} )^{2} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( ( 2 + 2 T + T^{2} )^{4} \)
$79$ \( ( 2 + 4 T^{2} + T^{4} )^{2} \)
$83$ \( T^{8} \)
$89$ \( T^{8} \)
$97$ \( ( -2 + T^{2} )^{4} \)
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