Properties

Label 3072.1.i.b
Level $3072$
Weight $1$
Character orbit 3072.i
Analytic conductor $1.533$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -8
Inner twists $4$

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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: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1536)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.9216.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -i q^{3} - q^{9} +O(q^{10})\) \( q -i q^{3} - q^{9} + ( -1 + i ) q^{11} + 2 i q^{17} + ( -1 + i ) q^{19} + i q^{25} + i q^{27} + ( 1 + i ) q^{33} + ( -1 - i ) q^{43} + q^{49} + 2 q^{51} + ( 1 + i ) q^{57} + ( 1 - i ) q^{59} + ( -1 + i ) q^{67} + q^{75} + q^{81} + ( -1 - i ) q^{83} -2 q^{89} + ( 1 - i ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 2q^{11} - 2q^{19} + 2q^{33} - 2q^{43} + 2q^{49} + 4q^{51} + 2q^{57} + 2q^{59} - 2q^{67} + 2q^{75} + 2q^{81} - 2q^{83} - 4q^{89} + 2q^{99} + O(q^{100}) \)

Character values

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

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

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
1.00000i
1.00000i
0 1.00000i 0 0 0 0 0 −1.00000 0
1793.1 0 1.00000i 0 0 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
48.i odd 4 1 inner
48.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3072.1.i.b 2
3.b odd 2 1 3072.1.i.d 2
4.b odd 2 1 3072.1.i.c 2
8.b even 2 1 3072.1.i.c 2
8.d odd 2 1 CM 3072.1.i.b 2
12.b even 2 1 3072.1.i.a 2
16.e even 4 1 3072.1.i.a 2
16.e even 4 1 3072.1.i.d 2
16.f odd 4 1 3072.1.i.a 2
16.f odd 4 1 3072.1.i.d 2
24.f even 2 1 3072.1.i.d 2
24.h odd 2 1 3072.1.i.a 2
32.g even 8 2 1536.1.e.b 4
32.g even 8 2 1536.1.h.c 4
32.h odd 8 2 1536.1.e.b 4
32.h odd 8 2 1536.1.h.c 4
48.i odd 4 1 inner 3072.1.i.b 2
48.i odd 4 1 3072.1.i.c 2
48.k even 4 1 inner 3072.1.i.b 2
48.k even 4 1 3072.1.i.c 2
96.o even 8 2 1536.1.e.b 4
96.o even 8 2 1536.1.h.c 4
96.p odd 8 2 1536.1.e.b 4
96.p odd 8 2 1536.1.h.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.1.e.b 4 32.g even 8 2
1536.1.e.b 4 32.h odd 8 2
1536.1.e.b 4 96.o even 8 2
1536.1.e.b 4 96.p odd 8 2
1536.1.h.c 4 32.g even 8 2
1536.1.h.c 4 32.h odd 8 2
1536.1.h.c 4 96.o even 8 2
1536.1.h.c 4 96.p odd 8 2
3072.1.i.a 2 12.b even 2 1
3072.1.i.a 2 16.e even 4 1
3072.1.i.a 2 16.f odd 4 1
3072.1.i.a 2 24.h odd 2 1
3072.1.i.b 2 1.a even 1 1 trivial
3072.1.i.b 2 8.d odd 2 1 CM
3072.1.i.b 2 48.i odd 4 1 inner
3072.1.i.b 2 48.k even 4 1 inner
3072.1.i.c 2 4.b odd 2 1
3072.1.i.c 2 8.b even 2 1
3072.1.i.c 2 48.i odd 4 1
3072.1.i.c 2 48.k even 4 1
3072.1.i.d 2 3.b odd 2 1
3072.1.i.d 2 16.e even 4 1
3072.1.i.d 2 16.f odd 4 1
3072.1.i.d 2 24.f even 2 1

Hecke kernels

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

\( T_{5} \)
\( T_{11}^{2} + 2 T_{11} + 2 \)
\( T_{19}^{2} + 2 T_{19} + 2 \)

Hecke characteristic polynomials

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