Properties

Label 1152.1.t.a
Level $1152$
Weight $1$
Character orbit 1152.t
Analytic conductor $0.575$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -8
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1152.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.574922894553\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.13436928.2

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{12} q^{3} + \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{12} q^{3} + \zeta_{12}^{2} q^{9} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{11} + q^{17} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{19} + \zeta_{12}^{4} q^{25} -\zeta_{12}^{3} q^{27} + ( 1 - \zeta_{12}^{4} ) q^{33} + \zeta_{12}^{2} q^{41} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{43} -\zeta_{12}^{2} q^{49} -\zeta_{12} q^{51} + ( -1 - \zeta_{12}^{2} ) q^{57} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{59} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{67} + q^{73} -\zeta_{12}^{5} q^{75} + \zeta_{12}^{4} q^{81} -2 q^{89} + \zeta_{12}^{4} q^{97} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{9} + 4q^{17} - 2q^{25} + 6q^{33} + 2q^{41} - 2q^{49} - 6q^{57} + 4q^{73} - 2q^{81} - 8q^{89} - 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 −0.866025 + 0.500000i 0 0 0 0 0 0.500000 0.866025i 0
319.2 0 0.866025 0.500000i 0 0 0 0 0 0.500000 0.866025i 0
1087.1 0 −0.866025 0.500000i 0 0 0 0 0 0.500000 + 0.866025i 0
1087.2 0 0.866025 + 0.500000i 0 0 0 0 0 0.500000 + 0.866025i 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 \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.b even 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner
72.n even 6 1 inner
72.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.1.t.a 4
3.b odd 2 1 3456.1.t.a 4
4.b odd 2 1 inner 1152.1.t.a 4
8.b even 2 1 inner 1152.1.t.a 4
8.d odd 2 1 CM 1152.1.t.a 4
9.c even 3 1 inner 1152.1.t.a 4
9.d odd 6 1 3456.1.t.a 4
12.b even 2 1 3456.1.t.a 4
16.e even 4 1 2304.1.o.a 2
16.e even 4 1 2304.1.o.b 2
16.f odd 4 1 2304.1.o.a 2
16.f odd 4 1 2304.1.o.b 2
24.f even 2 1 3456.1.t.a 4
24.h odd 2 1 3456.1.t.a 4
36.f odd 6 1 inner 1152.1.t.a 4
36.h even 6 1 3456.1.t.a 4
72.j odd 6 1 3456.1.t.a 4
72.l even 6 1 3456.1.t.a 4
72.n even 6 1 inner 1152.1.t.a 4
72.p odd 6 1 inner 1152.1.t.a 4
144.v odd 12 1 2304.1.o.a 2
144.v odd 12 1 2304.1.o.b 2
144.x even 12 1 2304.1.o.a 2
144.x even 12 1 2304.1.o.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.1.t.a 4 1.a even 1 1 trivial
1152.1.t.a 4 4.b odd 2 1 inner
1152.1.t.a 4 8.b even 2 1 inner
1152.1.t.a 4 8.d odd 2 1 CM
1152.1.t.a 4 9.c even 3 1 inner
1152.1.t.a 4 36.f odd 6 1 inner
1152.1.t.a 4 72.n even 6 1 inner
1152.1.t.a 4 72.p odd 6 1 inner
2304.1.o.a 2 16.e even 4 1
2304.1.o.a 2 16.f odd 4 1
2304.1.o.a 2 144.v odd 12 1
2304.1.o.a 2 144.x even 12 1
2304.1.o.b 2 16.e even 4 1
2304.1.o.b 2 16.f odd 4 1
2304.1.o.b 2 144.v odd 12 1
2304.1.o.b 2 144.x even 12 1
3456.1.t.a 4 3.b odd 2 1
3456.1.t.a 4 9.d odd 6 1
3456.1.t.a 4 12.b even 2 1
3456.1.t.a 4 24.f even 2 1
3456.1.t.a 4 24.h odd 2 1
3456.1.t.a 4 36.h even 6 1
3456.1.t.a 4 72.j odd 6 1
3456.1.t.a 4 72.l even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1152, [\chi])\).

Hecke characteristic polynomials

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