Properties

Label 3267.1.q.a
Level $3267$
Weight $1$
Character orbit 3267.q
Analytic conductor $1.630$
Analytic rank $0$
Dimension $6$
Projective image $D_{18}$
CM discriminant -11
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3267 = 3^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3267.q (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.63044539627\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{18}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{18}^{7} q^{3} + \zeta_{18}^{4} q^{4} + (\zeta_{18}^{6} + \zeta_{18}^{5}) q^{5} - \zeta_{18}^{5} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{18}^{7} q^{3} + \zeta_{18}^{4} q^{4} + (\zeta_{18}^{6} + \zeta_{18}^{5}) q^{5} - \zeta_{18}^{5} q^{9} - \zeta_{18}^{2} q^{12} + ( - \zeta_{18}^{4} - \zeta_{18}^{3}) q^{15} + \zeta_{18}^{8} q^{16} + ( - \zeta_{18} - 1) q^{20} + (\zeta_{18}^{7} - \zeta_{18}) q^{23} + ( - \zeta_{18}^{3} - \zeta_{18}^{2} - \zeta_{18}) q^{25} + \zeta_{18}^{3} q^{27} + ( - \zeta_{18}^{6} - \zeta_{18}^{2}) q^{31} + q^{36} + ( - \zeta_{18}^{8} + \zeta_{18}^{7}) q^{37} + (\zeta_{18}^{2} + \zeta_{18}) q^{45} + ( - \zeta_{18}^{6} + \zeta_{18}^{4}) q^{47} - \zeta_{18}^{6} q^{48} + \zeta_{18} q^{49} + (\zeta_{18}^{7} + \zeta_{18}^{2}) q^{53} + (\zeta_{18}^{8} + \zeta_{18}^{3}) q^{59} + ( - \zeta_{18}^{8} - \zeta_{18}^{7}) q^{60} - \zeta_{18}^{3} q^{64} + (\zeta_{18}^{5} - 1) q^{67} + ( - \zeta_{18}^{8} - \zeta_{18}^{5}) q^{69} + ( - \zeta_{18}^{4} + \zeta_{18}^{2}) q^{71} + ( - \zeta_{18}^{8} + \zeta_{18} + 1) q^{75} + ( - \zeta_{18}^{5} - \zeta_{18}^{4}) q^{80} - \zeta_{18} q^{81} + ( - \zeta_{18}^{3} - 1) q^{89} + ( - \zeta_{18}^{5} - \zeta_{18}^{2}) q^{92} + (\zeta_{18}^{4} + 1) q^{93} + ( - \zeta_{18}^{7} + 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{5} - 3 q^{15} - 6 q^{20} - 3 q^{25} + 3 q^{27} + 3 q^{31} + 6 q^{36} + 3 q^{47} + 3 q^{48} + 3 q^{59} - 3 q^{64} - 6 q^{67} + 6 q^{75} - 9 q^{89} + 6 q^{93} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3267\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(3026\)
\(\chi(n)\) \(1\) \(-\zeta_{18}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
122.1
−0.766044 0.642788i
0.939693 + 0.342020i
−0.173648 + 0.984808i
−0.173648 0.984808i
0.939693 0.342020i
−0.766044 + 0.642788i
0 −0.173648 + 0.984808i −0.939693 + 0.342020i 0.439693 0.524005i 0 0 0 −0.939693 0.342020i 0
848.1 0 −0.766044 + 0.642788i 0.173648 + 0.984808i −0.673648 + 1.85083i 0 0 0 0.173648 0.984808i 0
1211.1 0 0.939693 0.342020i 0.766044 + 0.642788i −1.26604 0.223238i 0 0 0 0.766044 0.642788i 0
1937.1 0 0.939693 + 0.342020i 0.766044 0.642788i −1.26604 + 0.223238i 0 0 0 0.766044 + 0.642788i 0
2300.1 0 −0.766044 0.642788i 0.173648 0.984808i −0.673648 1.85083i 0 0 0 0.173648 + 0.984808i 0
3026.1 0 −0.173648 0.984808i −0.939693 0.342020i 0.439693 + 0.524005i 0 0 0 −0.939693 + 0.342020i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3026.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
27.f odd 18 1 inner
297.o even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3267.1.q.a 6
11.b odd 2 1 CM 3267.1.q.a 6
11.c even 5 4 3267.1.be.a 24
11.d odd 10 4 3267.1.be.a 24
27.f odd 18 1 inner 3267.1.q.a 6
297.o even 18 1 inner 3267.1.q.a 6
297.v odd 90 4 3267.1.be.a 24
297.x even 90 4 3267.1.be.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3267.1.q.a 6 1.a even 1 1 trivial
3267.1.q.a 6 11.b odd 2 1 CM
3267.1.q.a 6 27.f odd 18 1 inner
3267.1.q.a 6 297.o even 18 1 inner
3267.1.be.a 24 11.c even 5 4
3267.1.be.a 24 11.d odd 10 4
3267.1.be.a 24 297.v odd 90 4
3267.1.be.a 24 297.x even 90 4

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3267, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{5} + 6 T^{4} + 6 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 9T^{3} + 27 \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{6} + 3 T^{4} - 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} \) Copy content Toggle raw display
$47$ \( T^{6} - 3 T^{5} + 6 T^{4} - 6 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$53$ \( T^{6} + 6 T^{4} + 9 T^{2} + 3 \) Copy content Toggle raw display
$59$ \( T^{6} - 3 T^{5} + 6 T^{4} - 6 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$61$ \( T^{6} \) Copy content Toggle raw display
$67$ \( T^{6} + 6 T^{5} + 15 T^{4} + 19 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{6} - 3 T^{4} + 9 T^{2} - 9 T + 3 \) Copy content Toggle raw display
$73$ \( T^{6} \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( (T^{2} + 3 T + 3)^{3} \) Copy content Toggle raw display
$97$ \( T^{6} - 6 T^{5} + 15 T^{4} - 19 T^{3} + \cdots + 1 \) Copy content Toggle raw display
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