Properties

Label 648.1.j.b
Level $648$
Weight $1$
Character orbit 648.j
Analytic conductor $0.323$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -24
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 648.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.323394128186\)
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: no (minimal twist has level 216)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.216.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of 12.0.101559956668416.7

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} -\zeta_{6} q^{5} -\zeta_{6}^{2} q^{7} - q^{8} +O(q^{10})\) \( q -\zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} -\zeta_{6} q^{5} -\zeta_{6}^{2} q^{7} - q^{8} - q^{10} + \zeta_{6}^{2} q^{11} -\zeta_{6} q^{14} + \zeta_{6}^{2} q^{16} + \zeta_{6}^{2} q^{20} + \zeta_{6} q^{22} - q^{28} -2 \zeta_{6}^{2} q^{29} + \zeta_{6} q^{31} + \zeta_{6} q^{32} - q^{35} + \zeta_{6} q^{40} + q^{44} + q^{53} + q^{55} + \zeta_{6}^{2} q^{56} -2 \zeta_{6} q^{58} + 2 \zeta_{6} q^{59} + q^{62} + q^{64} + \zeta_{6}^{2} q^{70} - q^{73} + \zeta_{6} q^{77} + 2 \zeta_{6}^{2} q^{79} + q^{80} + \zeta_{6}^{2} q^{83} -\zeta_{6}^{2} q^{88} -\zeta_{6}^{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} - q^{5} + q^{7} - 2q^{8} + O(q^{10}) \) \( 2q + q^{2} - q^{4} - q^{5} + q^{7} - 2q^{8} - 2q^{10} - q^{11} - q^{14} - q^{16} - q^{20} + q^{22} - 2q^{28} + 2q^{29} + q^{31} + q^{32} - 2q^{35} + q^{40} + 2q^{44} + 2q^{53} + 2q^{55} - q^{56} - 2q^{58} + 2q^{59} + 2q^{62} + 2q^{64} - q^{70} - 2q^{73} + q^{77} - 2q^{79} + 2q^{80} - q^{83} + q^{88} + q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(-1\) \(1\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.500000 + 0.866025i −1.00000 0 −1.00000
269.1 0.500000 0.866025i 0 −0.500000 0.866025i −0.500000 0.866025i 0 0.500000 0.866025i −1.00000 0 −1.00000
\(n\): 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 \(\Q(\sqrt{-6}) \)
9.c even 3 1 inner
72.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.1.j.b 2
3.b odd 2 1 648.1.j.a 2
4.b odd 2 1 2592.1.n.a 2
8.b even 2 1 648.1.j.a 2
8.d odd 2 1 2592.1.n.b 2
9.c even 3 1 216.1.h.a 1
9.c even 3 1 inner 648.1.j.b 2
9.d odd 6 1 216.1.h.b yes 1
9.d odd 6 1 648.1.j.a 2
12.b even 2 1 2592.1.n.b 2
24.f even 2 1 2592.1.n.a 2
24.h odd 2 1 CM 648.1.j.b 2
36.f odd 6 1 864.1.h.b 1
36.f odd 6 1 2592.1.n.a 2
36.h even 6 1 864.1.h.a 1
36.h even 6 1 2592.1.n.b 2
72.j odd 6 1 216.1.h.a 1
72.j odd 6 1 inner 648.1.j.b 2
72.l even 6 1 864.1.h.b 1
72.l even 6 1 2592.1.n.a 2
72.n even 6 1 216.1.h.b yes 1
72.n even 6 1 648.1.j.a 2
72.p odd 6 1 864.1.h.a 1
72.p odd 6 1 2592.1.n.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.1.h.a 1 9.c even 3 1
216.1.h.a 1 72.j odd 6 1
216.1.h.b yes 1 9.d odd 6 1
216.1.h.b yes 1 72.n even 6 1
648.1.j.a 2 3.b odd 2 1
648.1.j.a 2 8.b even 2 1
648.1.j.a 2 9.d odd 6 1
648.1.j.a 2 72.n even 6 1
648.1.j.b 2 1.a even 1 1 trivial
648.1.j.b 2 9.c even 3 1 inner
648.1.j.b 2 24.h odd 2 1 CM
648.1.j.b 2 72.j odd 6 1 inner
864.1.h.a 1 36.h even 6 1
864.1.h.a 1 72.p odd 6 1
864.1.h.b 1 36.f odd 6 1
864.1.h.b 1 72.l even 6 1
2592.1.n.a 2 4.b odd 2 1
2592.1.n.a 2 24.f even 2 1
2592.1.n.a 2 36.f odd 6 1
2592.1.n.a 2 72.l even 6 1
2592.1.n.b 2 8.d odd 2 1
2592.1.n.b 2 12.b even 2 1
2592.1.n.b 2 36.h even 6 1
2592.1.n.b 2 72.p 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 \(S_{1}^{\mathrm{new}}(648, [\chi])\).

Hecke characteristic polynomials

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