Properties

Label 672.1.ba.a
Level 672
Weight 1
Character orbit 672.ba
Analytic conductor 0.335
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM discriminant -24
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 672.ba (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.335371688489\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.1176.1
Artin image $C_6\times S_3$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6} q^{3} + \zeta_{6}^{2} q^{5} + \zeta_{6} q^{7} + \zeta_{6}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{6} q^{3} + \zeta_{6}^{2} q^{5} + \zeta_{6} q^{7} + \zeta_{6}^{2} q^{9} + \zeta_{6} q^{11} + q^{15} -\zeta_{6}^{2} q^{21} + q^{27} + q^{29} -\zeta_{6} q^{31} -\zeta_{6}^{2} q^{33} - q^{35} -\zeta_{6} q^{45} + \zeta_{6}^{2} q^{49} -\zeta_{6} q^{53} - q^{55} + \zeta_{6} q^{59} - q^{63} -2 \zeta_{6} q^{73} + \zeta_{6}^{2} q^{77} + \zeta_{6}^{2} q^{79} -\zeta_{6} q^{81} - q^{83} -\zeta_{6} q^{87} + \zeta_{6}^{2} q^{93} - q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} - q^{5} + q^{7} - q^{9} + O(q^{10}) \) \( 2q - q^{3} - q^{5} + q^{7} - q^{9} + q^{11} + 2q^{15} + q^{21} + 2q^{27} + 2q^{29} - q^{31} + q^{33} - 2q^{35} - q^{45} - q^{49} - q^{53} - 2q^{55} + q^{59} - 2q^{63} - 2q^{73} - q^{77} - q^{79} - q^{81} - 2q^{83} - q^{87} - q^{93} - 2q^{97} - 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(\zeta_{6}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0.500000 0.866025i 0 −0.500000 0.866025i 0
401.1 0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
7.c even 3 1 inner
168.s odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.1.ba.a 2
3.b odd 2 1 672.1.ba.b 2
4.b odd 2 1 168.1.s.b yes 2
7.c even 3 1 inner 672.1.ba.a 2
8.b even 2 1 672.1.ba.b 2
8.d odd 2 1 168.1.s.a 2
12.b even 2 1 168.1.s.a 2
21.h odd 6 1 672.1.ba.b 2
24.f even 2 1 168.1.s.b yes 2
24.h odd 2 1 CM 672.1.ba.a 2
28.d even 2 1 1176.1.s.b 2
28.f even 6 1 1176.1.n.b 1
28.f even 6 1 1176.1.s.b 2
28.g odd 6 1 168.1.s.b yes 2
28.g odd 6 1 1176.1.n.a 1
56.e even 2 1 1176.1.s.a 2
56.k odd 6 1 168.1.s.a 2
56.k odd 6 1 1176.1.n.d 1
56.m even 6 1 1176.1.n.c 1
56.m even 6 1 1176.1.s.a 2
56.p even 6 1 672.1.ba.b 2
84.h odd 2 1 1176.1.s.a 2
84.j odd 6 1 1176.1.n.c 1
84.j odd 6 1 1176.1.s.a 2
84.n even 6 1 168.1.s.a 2
84.n even 6 1 1176.1.n.d 1
168.e odd 2 1 1176.1.s.b 2
168.s odd 6 1 inner 672.1.ba.a 2
168.v even 6 1 168.1.s.b yes 2
168.v even 6 1 1176.1.n.a 1
168.be odd 6 1 1176.1.n.b 1
168.be odd 6 1 1176.1.s.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.1.s.a 2 8.d odd 2 1
168.1.s.a 2 12.b even 2 1
168.1.s.a 2 56.k odd 6 1
168.1.s.a 2 84.n even 6 1
168.1.s.b yes 2 4.b odd 2 1
168.1.s.b yes 2 24.f even 2 1
168.1.s.b yes 2 28.g odd 6 1
168.1.s.b yes 2 168.v even 6 1
672.1.ba.a 2 1.a even 1 1 trivial
672.1.ba.a 2 7.c even 3 1 inner
672.1.ba.a 2 24.h odd 2 1 CM
672.1.ba.a 2 168.s odd 6 1 inner
672.1.ba.b 2 3.b odd 2 1
672.1.ba.b 2 8.b even 2 1
672.1.ba.b 2 21.h odd 6 1
672.1.ba.b 2 56.p even 6 1
1176.1.n.a 1 28.g odd 6 1
1176.1.n.a 1 168.v even 6 1
1176.1.n.b 1 28.f even 6 1
1176.1.n.b 1 168.be odd 6 1
1176.1.n.c 1 56.m even 6 1
1176.1.n.c 1 84.j odd 6 1
1176.1.n.d 1 56.k odd 6 1
1176.1.n.d 1 84.n even 6 1
1176.1.s.a 2 56.e even 2 1
1176.1.s.a 2 56.m even 6 1
1176.1.s.a 2 84.h odd 2 1
1176.1.s.a 2 84.j odd 6 1
1176.1.s.b 2 28.d even 2 1
1176.1.s.b 2 28.f even 6 1
1176.1.s.b 2 168.e odd 2 1
1176.1.s.b 2 168.be odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + T_{5} + 1 \) acting on \(S_{1}^{\mathrm{new}}(672, [\chi])\).

Hecke characteristic polynomials

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