Properties

Label 1148.1.o.c
Level $1148$
Weight $1$
Character orbit 1148.o
Analytic conductor $0.573$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -164
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1148.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.572926634503\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.258309184.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{12}^{4} q^{2} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} -\zeta_{12}^{2} q^{4} + \zeta_{12}^{4} q^{5} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{6} + \zeta_{12} q^{7} + q^{8} + ( -1 + \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{9} +O(q^{10})\) \( q + \zeta_{12}^{4} q^{2} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} -\zeta_{12}^{2} q^{4} + \zeta_{12}^{4} q^{5} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{6} + \zeta_{12} q^{7} + q^{8} + ( -1 + \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{9} -\zeta_{12}^{2} q^{10} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{11} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{12} + \zeta_{12}^{5} q^{14} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{15} + \zeta_{12}^{4} q^{16} + ( -1 - \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{18} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{19} + q^{20} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{21} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{22} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{24} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{27} -\zeta_{12}^{3} q^{28} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{30} -\zeta_{12}^{2} q^{32} + ( 1 - \zeta_{12}^{2} - 2 \zeta_{12}^{4} ) q^{33} + \zeta_{12}^{5} q^{35} + ( 1 + \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{36} -2 \zeta_{12}^{4} q^{37} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{38} + \zeta_{12}^{4} q^{40} + q^{41} + ( -1 - \zeta_{12}^{2} ) q^{42} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{44} + ( -1 - \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{45} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{48} + \zeta_{12}^{2} q^{49} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{54} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{55} + \zeta_{12} q^{56} + ( -2 - \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{57} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{60} -\zeta_{12}^{4} q^{61} + ( -\zeta_{12} + \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{63} + q^{64} + ( 1 + 2 \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{66} -\zeta_{12}^{3} q^{70} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{71} + ( -1 + \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{72} + 2 \zeta_{12}^{2} q^{73} + 2 \zeta_{12}^{2} q^{74} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{76} + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{77} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{79} -\zeta_{12}^{2} q^{80} -\zeta_{12}^{2} q^{81} + \zeta_{12}^{4} q^{82} + ( 1 - \zeta_{12}^{4} ) q^{84} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{88} + ( 1 + \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{90} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{95} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{96} - q^{98} + ( 2 \zeta_{12} - 2 \zeta_{12}^{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 2q^{4} - 2q^{5} + 4q^{8} - 4q^{9} + O(q^{10}) \) \( 4q - 2q^{2} - 2q^{4} - 2q^{5} + 4q^{8} - 4q^{9} - 2q^{10} - 2q^{16} - 4q^{18} + 4q^{20} - 2q^{32} + 6q^{33} + 8q^{36} + 4q^{37} - 2q^{40} + 4q^{41} - 6q^{42} - 4q^{45} + 2q^{49} - 12q^{57} + 2q^{61} + 4q^{64} + 6q^{66} - 4q^{72} + 4q^{73} + 4q^{74} - 2q^{80} - 2q^{81} - 2q^{82} + 6q^{84} + 8q^{90} - 4q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times\).

\(n\) \(493\) \(575\) \(785\)
\(\chi(n)\) \(\zeta_{12}^{4}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.500000 + 0.866025i −0.866025 1.50000i −0.500000 0.866025i −0.500000 + 0.866025i 1.73205 −0.866025 0.500000i 1.00000 −1.00000 + 1.73205i −0.500000 0.866025i
163.2 −0.500000 + 0.866025i 0.866025 + 1.50000i −0.500000 0.866025i −0.500000 + 0.866025i −1.73205 0.866025 + 0.500000i 1.00000 −1.00000 + 1.73205i −0.500000 0.866025i
655.1 −0.500000 0.866025i −0.866025 + 1.50000i −0.500000 + 0.866025i −0.500000 0.866025i 1.73205 −0.866025 + 0.500000i 1.00000 −1.00000 1.73205i −0.500000 + 0.866025i
655.2 −0.500000 0.866025i 0.866025 1.50000i −0.500000 + 0.866025i −0.500000 0.866025i −1.73205 0.866025 0.500000i 1.00000 −1.00000 1.73205i −0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
164.d odd 2 1 CM by \(\Q(\sqrt{-41}) \)
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner
41.b even 2 1 inner
287.j even 6 1 inner
1148.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.1.o.c 4
4.b odd 2 1 inner 1148.1.o.c 4
7.c even 3 1 inner 1148.1.o.c 4
28.g odd 6 1 inner 1148.1.o.c 4
41.b even 2 1 inner 1148.1.o.c 4
164.d odd 2 1 CM 1148.1.o.c 4
287.j even 6 1 inner 1148.1.o.c 4
1148.o odd 6 1 inner 1148.1.o.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.1.o.c 4 1.a even 1 1 trivial
1148.1.o.c 4 4.b odd 2 1 inner
1148.1.o.c 4 7.c even 3 1 inner
1148.1.o.c 4 28.g odd 6 1 inner
1148.1.o.c 4 41.b even 2 1 inner
1148.1.o.c 4 164.d odd 2 1 CM
1148.1.o.c 4 287.j even 6 1 inner
1148.1.o.c 4 1148.o odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 3 T_{3}^{2} + 9 \) acting on \(S_{1}^{\mathrm{new}}(1148, [\chi])\).

Hecke characteristic polynomials

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