Properties

Label 327.1.d.a
Level $327$
Weight $1$
Character orbit 327.d
Self dual yes
Analytic conductor $0.163$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -327
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 327 = 3 \cdot 109 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 327.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.163194259131\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.327.1
Artin image: $S_3$
Artin field: Galois closure of 3.1.327.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{3} - q^{6} - q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} - q^{6} - q^{7} + q^{8} + q^{9} + 2q^{11} + q^{14} - q^{16} - q^{17} - q^{18} - q^{21} - 2q^{22} - q^{23} + q^{24} + q^{25} + q^{27} - q^{31} + 2q^{33} + q^{34} - q^{41} + q^{42} - q^{43} + q^{46} - q^{47} - q^{48} - q^{50} - q^{51} + 2q^{53} - q^{54} - q^{56} - q^{59} - q^{61} + q^{62} - q^{63} + q^{64} - 2q^{66} - q^{69} + q^{72} - q^{73} + q^{75} - 2q^{77} + q^{81} + q^{82} + q^{86} + 2q^{88} - q^{93} + q^{94} - q^{97} + 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/327\mathbb{Z}\right)^\times\).

\(n\) \(110\) \(115\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
326.1
0
−1.00000 1.00000 0 0 −1.00000 −1.00000 1.00000 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
327.d odd 2 1 CM by \(\Q(\sqrt{-327}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 327.1.d.a 1
3.b odd 2 1 327.1.d.c yes 1
109.b even 2 1 327.1.d.c yes 1
327.d odd 2 1 CM 327.1.d.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
327.1.d.a 1 1.a even 1 1 trivial
327.1.d.a 1 327.d odd 2 1 CM
327.1.d.c yes 1 3.b odd 2 1
327.1.d.c yes 1 109.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(327, [\chi])\).

Hecke characteristic polynomials

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