Properties

Label 3072.2.d.b
Level $3072$
Weight $2$
Character orbit 3072.d
Analytic conductor $24.530$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1536)
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 \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{8}^{2} q^{3} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{5} - q^{9} +O(q^{10})\) \( q + \zeta_{8}^{2} q^{3} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{5} - q^{9} + 4 \zeta_{8}^{2} q^{11} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{13} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{15} -4 \zeta_{8}^{2} q^{19} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{23} + 3 q^{25} -\zeta_{8}^{2} q^{27} + ( -5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{29} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{31} -4 q^{33} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{37} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{39} + 12 \zeta_{8}^{2} q^{43} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{45} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{47} -7 q^{49} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{53} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{55} + 4 q^{57} -4 \zeta_{8}^{2} q^{59} + ( 9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{61} -2 q^{65} -4 \zeta_{8}^{2} q^{67} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{69} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{71} + 10 q^{73} + 3 \zeta_{8}^{2} q^{75} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{79} + q^{81} -12 \zeta_{8}^{2} q^{83} + ( 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{87} -6 q^{89} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{93} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{95} + 8 q^{97} -4 \zeta_{8}^{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} + 12q^{25} - 16q^{33} - 28q^{49} + 16q^{57} - 8q^{65} + 40q^{73} + 4q^{81} - 24q^{89} + 32q^{97} + O(q^{100}) \)

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
−0.707107 + 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
0 1.00000i 0 1.41421i 0 0 0 −1.00000 0
1537.2 0 1.00000i 0 1.41421i 0 0 0 −1.00000 0
1537.3 0 1.00000i 0 1.41421i 0 0 0 −1.00000 0
1537.4 0 1.00000i 0 1.41421i 0 0 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
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.b 4
4.b odd 2 1 inner 3072.2.d.b 4
8.b even 2 1 inner 3072.2.d.b 4
8.d odd 2 1 inner 3072.2.d.b 4
16.e even 4 1 3072.2.a.c 2
16.e even 4 1 3072.2.a.e 2
16.f odd 4 1 3072.2.a.c 2
16.f odd 4 1 3072.2.a.e 2
32.g even 8 2 1536.2.j.a 4
32.g even 8 2 1536.2.j.d yes 4
32.h odd 8 2 1536.2.j.a 4
32.h odd 8 2 1536.2.j.d yes 4
48.i odd 4 1 9216.2.a.f 2
48.i odd 4 1 9216.2.a.r 2
48.k even 4 1 9216.2.a.f 2
48.k even 4 1 9216.2.a.r 2
96.o even 8 2 4608.2.k.z 4
96.o even 8 2 4608.2.k.ba 4
96.p odd 8 2 4608.2.k.z 4
96.p odd 8 2 4608.2.k.ba 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.j.a 4 32.g even 8 2
1536.2.j.a 4 32.h odd 8 2
1536.2.j.d yes 4 32.g even 8 2
1536.2.j.d yes 4 32.h odd 8 2
3072.2.a.c 2 16.e even 4 1
3072.2.a.c 2 16.f odd 4 1
3072.2.a.e 2 16.e even 4 1
3072.2.a.e 2 16.f odd 4 1
3072.2.d.b 4 1.a even 1 1 trivial
3072.2.d.b 4 4.b odd 2 1 inner
3072.2.d.b 4 8.b even 2 1 inner
3072.2.d.b 4 8.d odd 2 1 inner
4608.2.k.z 4 96.o even 8 2
4608.2.k.z 4 96.p odd 8 2
4608.2.k.ba 4 96.o even 8 2
4608.2.k.ba 4 96.p odd 8 2
9216.2.a.f 2 48.i odd 4 1
9216.2.a.f 2 48.k even 4 1
9216.2.a.r 2 48.i odd 4 1
9216.2.a.r 2 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 \(S_{2}^{\mathrm{new}}(3072, [\chi])\):

\( T_{5}^{2} + 2 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( ( 2 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( ( 16 + T^{2} )^{2} \)
$13$ \( ( 2 + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( ( 16 + T^{2} )^{2} \)
$23$ \( ( -32 + T^{2} )^{2} \)
$29$ \( ( 50 + T^{2} )^{2} \)
$31$ \( ( -32 + T^{2} )^{2} \)
$37$ \( ( 18 + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( 144 + T^{2} )^{2} \)
$47$ \( ( -128 + T^{2} )^{2} \)
$53$ \( ( 2 + T^{2} )^{2} \)
$59$ \( ( 16 + T^{2} )^{2} \)
$61$ \( ( 162 + T^{2} )^{2} \)
$67$ \( ( 16 + T^{2} )^{2} \)
$71$ \( ( -32 + T^{2} )^{2} \)
$73$ \( ( -10 + T )^{4} \)
$79$ \( ( -288 + T^{2} )^{2} \)
$83$ \( ( 144 + T^{2} )^{2} \)
$89$ \( ( 6 + T )^{4} \)
$97$ \( ( -8 + T )^{4} \)
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