Properties

Label 648.1.b.b
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 $2$

Related objects

Downloads

Learn more

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: $S_3$
Artin field: Galois closure of 3.1.648.1

$q$-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.b 1
3.b odd 2 1 648.1.b.a 1
4.b odd 2 1 2592.1.b.b 1
8.b even 2 1 2592.1.b.b 1
8.d odd 2 1 CM 648.1.b.b 1
9.c even 3 2 72.1.p.a 2
9.d odd 6 2 216.1.p.a 2
12.b even 2 1 2592.1.b.a 1
24.f even 2 1 648.1.b.a 1
24.h odd 2 1 2592.1.b.a 1
36.f odd 6 2 288.1.t.a 2
36.h even 6 2 864.1.t.a 2
45.j even 6 2 1800.1.bk.d 2
45.k odd 12 4 1800.1.ba.b 4
63.g even 3 2 3528.1.ba.b 2
63.h even 3 2 3528.1.ce.a 2
63.k odd 6 2 3528.1.ba.a 2
63.l odd 6 2 3528.1.cg.a 2
63.t odd 6 2 3528.1.ce.b 2
72.j odd 6 2 864.1.t.a 2
72.l even 6 2 216.1.p.a 2
72.n even 6 2 288.1.t.a 2
72.p odd 6 2 72.1.p.a 2
144.v odd 12 4 2304.1.o.c 4
144.x even 12 4 2304.1.o.c 4
360.z odd 6 2 1800.1.bk.d 2
360.bo even 12 4 1800.1.ba.b 4
504.ba odd 6 2 3528.1.ba.b 2
504.be even 6 2 3528.1.cg.a 2
504.bf even 6 2 3528.1.ce.b 2
504.ce odd 6 2 3528.1.ce.a 2
504.cz even 6 2 3528.1.ba.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.1.p.a 2 9.c even 3 2
72.1.p.a 2 72.p odd 6 2
216.1.p.a 2 9.d odd 6 2
216.1.p.a 2 72.l even 6 2
288.1.t.a 2 36.f odd 6 2
288.1.t.a 2 72.n even 6 2
648.1.b.a 1 3.b odd 2 1
648.1.b.a 1 24.f even 2 1
648.1.b.b 1 1.a even 1 1 trivial
648.1.b.b 1 8.d odd 2 1 CM
864.1.t.a 2 36.h even 6 2
864.1.t.a 2 72.j odd 6 2
1800.1.ba.b 4 45.k odd 12 4
1800.1.ba.b 4 360.bo even 12 4
1800.1.bk.d 2 45.j even 6 2
1800.1.bk.d 2 360.z odd 6 2
2304.1.o.c 4 144.v odd 12 4
2304.1.o.c 4 144.x even 12 4
2592.1.b.a 1 12.b even 2 1
2592.1.b.a 1 24.h odd 2 1
2592.1.b.b 1 4.b odd 2 1
2592.1.b.b 1 8.b even 2 1
3528.1.ba.a 2 63.k odd 6 2
3528.1.ba.a 2 504.cz even 6 2
3528.1.ba.b 2 63.g even 3 2
3528.1.ba.b 2 504.ba odd 6 2
3528.1.ce.a 2 63.h even 3 2
3528.1.ce.a 2 504.ce odd 6 2
3528.1.ce.b 2 63.t odd 6 2
3528.1.ce.b 2 504.bf even 6 2
3528.1.cg.a 2 63.l odd 6 2
3528.1.cg.a 2 504.be even 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

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