Properties

Label 648.1.b.a
Level $648$
Weight $1$
Character orbit 648.b
Self dual yes
Analytic conductor $0.323$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -8
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 648.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.323394128186\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.648.1
Artin image $D_6$
Artin field Galois closure of 6.0.1259712.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{8} + O(q^{10}) \) \( q - q^{2} + q^{4} - q^{8} + q^{11} + q^{16} + q^{17} - q^{19} - q^{22} + q^{25} - q^{32} - q^{34} + q^{38} + q^{41} - q^{43} + q^{44} + q^{49} - q^{50} + q^{59} + q^{64} - q^{67} + q^{68} - q^{73} - q^{76} - q^{82} - 2q^{83} + q^{86} - q^{88} - 2q^{89} - q^{97} - q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
−1.00000 0 1.00000 0 0 0 −1.00000 0 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 \(\Q(\sqrt{-2}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.1.b.a 1
3.b odd 2 1 648.1.b.b 1
4.b odd 2 1 2592.1.b.a 1
8.b even 2 1 2592.1.b.a 1
8.d odd 2 1 CM 648.1.b.a 1
9.c even 3 2 216.1.p.a 2
9.d odd 6 2 72.1.p.a 2
12.b even 2 1 2592.1.b.b 1
24.f even 2 1 648.1.b.b 1
24.h odd 2 1 2592.1.b.b 1
36.f odd 6 2 864.1.t.a 2
36.h even 6 2 288.1.t.a 2
45.h odd 6 2 1800.1.bk.d 2
45.l even 12 4 1800.1.ba.b 4
63.i even 6 2 3528.1.ce.b 2
63.j odd 6 2 3528.1.ce.a 2
63.n odd 6 2 3528.1.ba.b 2
63.o even 6 2 3528.1.cg.a 2
63.s even 6 2 3528.1.ba.a 2
72.j odd 6 2 288.1.t.a 2
72.l even 6 2 72.1.p.a 2
72.n even 6 2 864.1.t.a 2
72.p odd 6 2 216.1.p.a 2
144.u even 12 4 2304.1.o.c 4
144.w odd 12 4 2304.1.o.c 4
360.bd even 6 2 1800.1.bk.d 2
360.bt odd 12 4 1800.1.ba.b 4
504.u odd 6 2 3528.1.ba.a 2
504.bt even 6 2 3528.1.ce.a 2
504.cm odd 6 2 3528.1.ce.b 2
504.co odd 6 2 3528.1.cg.a 2
504.cy even 6 2 3528.1.ba.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.1.p.a 2 9.d odd 6 2
72.1.p.a 2 72.l even 6 2
216.1.p.a 2 9.c even 3 2
216.1.p.a 2 72.p odd 6 2
288.1.t.a 2 36.h even 6 2
288.1.t.a 2 72.j odd 6 2
648.1.b.a 1 1.a even 1 1 trivial
648.1.b.a 1 8.d odd 2 1 CM
648.1.b.b 1 3.b odd 2 1
648.1.b.b 1 24.f even 2 1
864.1.t.a 2 36.f odd 6 2
864.1.t.a 2 72.n even 6 2
1800.1.ba.b 4 45.l even 12 4
1800.1.ba.b 4 360.bt odd 12 4
1800.1.bk.d 2 45.h odd 6 2
1800.1.bk.d 2 360.bd even 6 2
2304.1.o.c 4 144.u even 12 4
2304.1.o.c 4 144.w odd 12 4
2592.1.b.a 1 4.b odd 2 1
2592.1.b.a 1 8.b even 2 1
2592.1.b.b 1 12.b even 2 1
2592.1.b.b 1 24.h odd 2 1
3528.1.ba.a 2 63.s even 6 2
3528.1.ba.a 2 504.u odd 6 2
3528.1.ba.b 2 63.n odd 6 2
3528.1.ba.b 2 504.cy even 6 2
3528.1.ce.a 2 63.j odd 6 2
3528.1.ce.a 2 504.bt even 6 2
3528.1.ce.b 2 63.i even 6 2
3528.1.ce.b 2 504.cm odd 6 2
3528.1.cg.a 2 63.o even 6 2
3528.1.cg.a 2 504.co odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} - 1 \) acting on \(S_{1}^{\mathrm{new}}(648, [\chi])\).

Hecke characteristic polynomials

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