Properties

Label 1148.1.o.b
Level $1148$
Weight $1$
Character orbit 1148.o
Analytic conductor $0.573$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -164
Inner twists $4$

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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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.8036.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.216136256.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} -\zeta_{6}^{2} q^{3} + \zeta_{6}^{2} q^{4} + \zeta_{6} q^{5} - q^{6} -\zeta_{6} q^{7} + q^{8} +O(q^{10})\) \( q -\zeta_{6} q^{2} -\zeta_{6}^{2} q^{3} + \zeta_{6}^{2} q^{4} + \zeta_{6} q^{5} - q^{6} -\zeta_{6} q^{7} + q^{8} -\zeta_{6}^{2} q^{10} -\zeta_{6}^{2} q^{11} + \zeta_{6} q^{12} + \zeta_{6}^{2} q^{14} + q^{15} -\zeta_{6} q^{16} + \zeta_{6} q^{19} - q^{20} - q^{21} - q^{22} -\zeta_{6}^{2} q^{24} + q^{27} + q^{28} -\zeta_{6} q^{30} + \zeta_{6}^{2} q^{32} -\zeta_{6} q^{33} -\zeta_{6}^{2} q^{35} -2 \zeta_{6} q^{37} -\zeta_{6}^{2} q^{38} + \zeta_{6} q^{40} + q^{41} + \zeta_{6} q^{42} + \zeta_{6} q^{44} -2 \zeta_{6} q^{47} - q^{48} + \zeta_{6}^{2} q^{49} -\zeta_{6} q^{54} + q^{55} -\zeta_{6} q^{56} + q^{57} + \zeta_{6}^{2} q^{60} + \zeta_{6} q^{61} + q^{64} + \zeta_{6}^{2} q^{66} + 2 \zeta_{6}^{2} q^{67} - q^{70} - q^{71} + 2 \zeta_{6}^{2} q^{73} + 2 \zeta_{6}^{2} q^{74} - q^{76} - q^{77} + \zeta_{6} q^{79} -\zeta_{6}^{2} q^{80} -\zeta_{6}^{2} q^{81} -\zeta_{6} q^{82} -\zeta_{6}^{2} q^{84} -\zeta_{6}^{2} q^{88} + 2 \zeta_{6}^{2} q^{94} + \zeta_{6}^{2} q^{95} + \zeta_{6} q^{96} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + q^{3} - q^{4} + q^{5} - 2q^{6} - q^{7} + 2q^{8} + O(q^{10}) \) \( 2q - q^{2} + q^{3} - q^{4} + q^{5} - 2q^{6} - q^{7} + 2q^{8} + q^{10} + q^{11} + q^{12} - q^{14} + 2q^{15} - q^{16} + q^{19} - 2q^{20} - 2q^{21} - 2q^{22} + q^{24} + 2q^{27} + 2q^{28} - q^{30} - q^{32} - q^{33} + q^{35} - 2q^{37} + q^{38} + q^{40} + 2q^{41} + q^{42} + q^{44} - 2q^{47} - 2q^{48} - q^{49} - q^{54} + 2q^{55} - q^{56} + 2q^{57} - q^{60} + q^{61} + 2q^{64} - q^{66} - 2q^{67} - 2q^{70} - 2q^{71} - 2q^{73} - 2q^{74} - 2q^{76} - 2q^{77} + q^{79} + q^{80} + q^{81} - q^{82} + q^{84} + q^{88} - 2q^{94} - q^{95} + q^{96} + 2q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(493\) \(575\) \(785\)
\(\chi(n)\) \(-\zeta_{6}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0.500000 0.866025i −1.00000 −0.500000 + 0.866025i 1.00000 0 0.500000 + 0.866025i
655.1 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i −1.00000 −0.500000 0.866025i 1.00000 0 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 \(\Q(\sqrt{-41}) \)
7.c even 3 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.b yes 2
4.b odd 2 1 1148.1.o.a 2
7.c even 3 1 inner 1148.1.o.b yes 2
28.g odd 6 1 1148.1.o.a 2
41.b even 2 1 1148.1.o.a 2
164.d odd 2 1 CM 1148.1.o.b yes 2
287.j even 6 1 1148.1.o.a 2
1148.o odd 6 1 inner 1148.1.o.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.1.o.a 2 4.b odd 2 1
1148.1.o.a 2 28.g odd 6 1
1148.1.o.a 2 41.b even 2 1
1148.1.o.a 2 287.j even 6 1
1148.1.o.b yes 2 1.a even 1 1 trivial
1148.1.o.b yes 2 7.c even 3 1 inner
1148.1.o.b yes 2 164.d odd 2 1 CM
1148.1.o.b yes 2 1148.o odd 6 1 inner

Hecke kernels

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

Hecke characteristic polynomials

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