Properties

Label 1200.1.bj.b
Level $1200$
Weight $1$
Character orbit 1200.bj
Analytic conductor $0.599$
Analytic rank $0$
Dimension $8$
Projective image $D_{6}$
CM discriminant -3
Inner twists $16$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1200.bj (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.598878015160\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.21600000.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{24}^{3} q^{3} + \zeta_{24}^{9} q^{7} + \zeta_{24}^{6} q^{9} +O(q^{10})\) \( q -\zeta_{24}^{3} q^{3} + \zeta_{24}^{9} q^{7} + \zeta_{24}^{6} q^{9} + ( \zeta_{24}^{7} + \zeta_{24}^{11} ) q^{13} + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{19} + q^{21} -\zeta_{24}^{9} q^{27} + ( \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{31} + ( \zeta_{24}^{2} - \zeta_{24}^{10} ) q^{39} + \zeta_{24}^{3} q^{43} + ( \zeta_{24} + \zeta_{24}^{5} ) q^{57} - q^{61} -\zeta_{24}^{3} q^{63} -\zeta_{24}^{9} q^{67} - q^{81} + ( -\zeta_{24}^{4} - \zeta_{24}^{8} ) q^{91} + ( -\zeta_{24}^{7} - \zeta_{24}^{11} ) q^{93} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{21} - 8q^{61} - 8q^{81} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(\zeta_{24}^{6}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 + 0.258819i
0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.965926 0.258819i
0 −0.707107 + 0.707107i 0 0 0 −0.707107 0.707107i 0 1.00000i 0
143.2 0 −0.707107 + 0.707107i 0 0 0 −0.707107 0.707107i 0 1.00000i 0
143.3 0 0.707107 0.707107i 0 0 0 0.707107 + 0.707107i 0 1.00000i 0
143.4 0 0.707107 0.707107i 0 0 0 0.707107 + 0.707107i 0 1.00000i 0
1007.1 0 −0.707107 0.707107i 0 0 0 −0.707107 + 0.707107i 0 1.00000i 0
1007.2 0 −0.707107 0.707107i 0 0 0 −0.707107 + 0.707107i 0 1.00000i 0
1007.3 0 0.707107 + 0.707107i 0 0 0 0.707107 0.707107i 0 1.00000i 0
1007.4 0 0.707107 + 0.707107i 0 0 0 0.707107 0.707107i 0 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1007.4
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
5.b even 2 1 inner
5.c odd 4 2 inner
12.b even 2 1 inner
15.d odd 2 1 inner
15.e even 4 2 inner
20.d odd 2 1 inner
20.e even 4 2 inner
60.h even 2 1 inner
60.l odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.1.bj.b 8
3.b odd 2 1 CM 1200.1.bj.b 8
4.b odd 2 1 inner 1200.1.bj.b 8
5.b even 2 1 inner 1200.1.bj.b 8
5.c odd 4 2 inner 1200.1.bj.b 8
12.b even 2 1 inner 1200.1.bj.b 8
15.d odd 2 1 inner 1200.1.bj.b 8
15.e even 4 2 inner 1200.1.bj.b 8
20.d odd 2 1 inner 1200.1.bj.b 8
20.e even 4 2 inner 1200.1.bj.b 8
60.h even 2 1 inner 1200.1.bj.b 8
60.l odd 4 2 inner 1200.1.bj.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1200.1.bj.b 8 1.a even 1 1 trivial
1200.1.bj.b 8 3.b odd 2 1 CM
1200.1.bj.b 8 4.b odd 2 1 inner
1200.1.bj.b 8 5.b even 2 1 inner
1200.1.bj.b 8 5.c odd 4 2 inner
1200.1.bj.b 8 12.b even 2 1 inner
1200.1.bj.b 8 15.d odd 2 1 inner
1200.1.bj.b 8 15.e even 4 2 inner
1200.1.bj.b 8 20.d odd 2 1 inner
1200.1.bj.b 8 20.e even 4 2 inner
1200.1.bj.b 8 60.h even 2 1 inner
1200.1.bj.b 8 60.l odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1200, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 1 + T^{4} )^{2} \)
$5$ \( T^{8} \)
$7$ \( ( 1 + T^{4} )^{2} \)
$11$ \( T^{8} \)
$13$ \( ( 9 + T^{4} )^{2} \)
$17$ \( T^{8} \)
$19$ \( ( -3 + T^{2} )^{4} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( ( 3 + T^{2} )^{4} \)
$37$ \( T^{8} \)
$41$ \( T^{8} \)
$43$ \( ( 1 + T^{4} )^{2} \)
$47$ \( T^{8} \)
$53$ \( T^{8} \)
$59$ \( T^{8} \)
$61$ \( ( 1 + T )^{8} \)
$67$ \( ( 1 + T^{4} )^{2} \)
$71$ \( T^{8} \)
$73$ \( T^{8} \)
$79$ \( T^{8} \)
$83$ \( T^{8} \)
$89$ \( T^{8} \)
$97$ \( ( 9 + T^{4} )^{2} \)
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