Properties

Label 663.1.g.b
Level $663$
Weight $1$
Character orbit 663.g
Self dual yes
Analytic conductor $0.331$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -51, -663, 13
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 663.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.330880103376\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{13}, \sqrt{-51})\)
Artin image: $D_4$
Artin field: Galois closure of 4.2.8619.2

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - q^{4} + q^{9} + O(q^{10}) \) \( q + q^{3} - q^{4} + q^{9} - q^{12} + q^{13} + q^{16} + q^{17} - 2q^{23} + q^{25} + q^{27} - 2q^{29} - q^{36} + q^{39} - 2q^{43} + q^{48} - q^{49} + q^{51} - q^{52} - q^{64} - q^{68} - 2q^{69} + q^{75} + q^{81} - 2q^{87} + 2q^{92} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/663\mathbb{Z}\right)^\times\).

\(n\) \(443\) \(547\) \(613\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
662.1
0
0 1.00000 −1.00000 0 0 0 0 1.00000 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
13.b even 2 1 RM by \(\Q(\sqrt{13}) \)
51.c odd 2 1 CM by \(\Q(\sqrt{-51}) \)
663.g odd 2 1 CM by \(\Q(\sqrt{-663}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.1.g.b yes 1
3.b odd 2 1 663.1.g.a 1
13.b even 2 1 RM 663.1.g.b yes 1
17.b even 2 1 663.1.g.a 1
39.d odd 2 1 663.1.g.a 1
51.c odd 2 1 CM 663.1.g.b yes 1
221.b even 2 1 663.1.g.a 1
663.g odd 2 1 CM 663.1.g.b yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.1.g.a 1 3.b odd 2 1
663.1.g.a 1 17.b even 2 1
663.1.g.a 1 39.d odd 2 1
663.1.g.a 1 221.b even 2 1
663.1.g.b yes 1 1.a even 1 1 trivial
663.1.g.b yes 1 13.b even 2 1 RM
663.1.g.b yes 1 51.c odd 2 1 CM
663.1.g.b yes 1 663.g odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(663, [\chi])\):

\( T_{2} \)
\( T_{23} + 2 \)
\( T_{107} - 2 \)

Hecke characteristic polynomials

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