Properties

Label 3072.1.i.g
Level $3072$
Weight $1$
Character orbit 3072.i
Analytic conductor $1.533$
Analytic rank $0$
Dimension $4$
Projective image $D_{2}$
CM/RM discs -3, -8, 24
Inner twists $16$

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Newspace parameters

Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3072.i (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.53312771881\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 192)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-2}, \sqrt{-3})\)
Artin image: $OD_{16}:C_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{8} q^{3} + \zeta_{8}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{8} q^{3} + \zeta_{8}^{2} q^{9} -2 \zeta_{8} q^{19} -\zeta_{8}^{2} q^{25} -\zeta_{8}^{3} q^{27} -2 \zeta_{8}^{3} q^{43} + q^{49} + 2 \zeta_{8}^{2} q^{57} -2 \zeta_{8} q^{67} -2 \zeta_{8}^{2} q^{73} + \zeta_{8}^{3} q^{75} - q^{81} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 4q^{49} - 4q^{81} - 8q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(1025\) \(2047\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(-\zeta_{8}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0 −0.707107 0.707107i 0 0 0 0 0 1.00000i 0
257.2 0 0.707107 + 0.707107i 0 0 0 0 0 1.00000i 0
1793.1 0 −0.707107 + 0.707107i 0 0 0 0 0 1.00000i 0
1793.2 0 0.707107 0.707107i 0 0 0 0 0 1.00000i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
24.f even 2 1 RM by \(\Q(\sqrt{6}) \)
4.b odd 2 1 inner
8.b even 2 1 inner
12.b even 2 1 inner
16.e even 4 2 inner
16.f odd 4 2 inner
24.h odd 2 1 inner
48.i odd 4 2 inner
48.k even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3072.1.i.g 4
3.b odd 2 1 CM 3072.1.i.g 4
4.b odd 2 1 inner 3072.1.i.g 4
8.b even 2 1 inner 3072.1.i.g 4
8.d odd 2 1 CM 3072.1.i.g 4
12.b even 2 1 inner 3072.1.i.g 4
16.e even 4 2 inner 3072.1.i.g 4
16.f odd 4 2 inner 3072.1.i.g 4
24.f even 2 1 RM 3072.1.i.g 4
24.h odd 2 1 inner 3072.1.i.g 4
32.g even 8 2 192.1.h.a 2
32.g even 8 1 768.1.e.a 1
32.g even 8 1 768.1.e.b 1
32.h odd 8 2 192.1.h.a 2
32.h odd 8 1 768.1.e.a 1
32.h odd 8 1 768.1.e.b 1
48.i odd 4 2 inner 3072.1.i.g 4
48.k even 4 2 inner 3072.1.i.g 4
96.o even 8 2 192.1.h.a 2
96.o even 8 1 768.1.e.a 1
96.o even 8 1 768.1.e.b 1
96.p odd 8 2 192.1.h.a 2
96.p odd 8 1 768.1.e.a 1
96.p odd 8 1 768.1.e.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.1.h.a 2 32.g even 8 2
192.1.h.a 2 32.h odd 8 2
192.1.h.a 2 96.o even 8 2
192.1.h.a 2 96.p odd 8 2
768.1.e.a 1 32.g even 8 1
768.1.e.a 1 32.h odd 8 1
768.1.e.a 1 96.o even 8 1
768.1.e.a 1 96.p odd 8 1
768.1.e.b 1 32.g even 8 1
768.1.e.b 1 32.h odd 8 1
768.1.e.b 1 96.o even 8 1
768.1.e.b 1 96.p odd 8 1
3072.1.i.g 4 1.a even 1 1 trivial
3072.1.i.g 4 3.b odd 2 1 CM
3072.1.i.g 4 4.b odd 2 1 inner
3072.1.i.g 4 8.b even 2 1 inner
3072.1.i.g 4 8.d odd 2 1 CM
3072.1.i.g 4 12.b even 2 1 inner
3072.1.i.g 4 16.e even 4 2 inner
3072.1.i.g 4 16.f odd 4 2 inner
3072.1.i.g 4 24.f even 2 1 RM
3072.1.i.g 4 24.h odd 2 1 inner
3072.1.i.g 4 48.i odd 4 2 inner
3072.1.i.g 4 48.k even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3072, [\chi])\):

\( T_{5} \)
\( T_{11} \)
\( T_{19}^{4} + 16 \)

Hecke characteristic polynomials

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