Properties

Label 3969.1.q.a
Level $3969$
Weight $1$
Character orbit 3969.q
Analytic conductor $1.981$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -7
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3969.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.98078903514\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of 12.2.136738899331083.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( - \zeta_{24}^{9} + \zeta_{24}^{7}) q^{2} + ( - \zeta_{24}^{6} + \zeta_{24}^{4} - \zeta_{24}^{2}) q^{4} + (\zeta_{24}^{11} - \zeta_{24}^{9} - \zeta_{24}^{3} - \zeta_{24}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{24}^{9} + \zeta_{24}^{7}) q^{2} + ( - \zeta_{24}^{6} + \zeta_{24}^{4} - \zeta_{24}^{2}) q^{4} + (\zeta_{24}^{11} - \zeta_{24}^{9} - \zeta_{24}^{3} - \zeta_{24}) q^{8} + ( - \zeta_{24}^{11} - \zeta_{24}^{9}) q^{11} + ( - \zeta_{24}^{10} + \zeta_{24}^{8} - \zeta_{24}^{6} + \zeta_{24}^{4} - 1) q^{16} + ( - \zeta_{24}^{8} + \zeta_{24}^{4}) q^{22} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{23} - \zeta_{24}^{4} q^{25} + (\zeta_{24}^{9} + \zeta_{24}^{3}) q^{29} + (\zeta_{24}^{11} + \zeta_{24}^{9} - \zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24}) q^{32} + ( - \zeta_{24}^{10} - \zeta_{24}^{6}) q^{37} + ( - \zeta_{24}^{10} + \zeta_{24}^{2}) q^{43} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{44} + (\zeta_{24}^{8} - \zeta_{24}^{6} - \zeta_{24}^{2} + 1) q^{46} + ( - \zeta_{24}^{11} - \zeta_{24}) q^{50} + (\zeta_{24}^{11} + \zeta_{24}^{9}) q^{53} + (\zeta_{24}^{10} + \zeta_{24}^{6} - \zeta_{24}^{4} + 1) q^{58} + ( - \zeta_{24}^{10} + \zeta_{24}^{8} - \zeta_{24}^{4} - \zeta_{24}^{2} + 1) q^{64} - \zeta_{24}^{4} q^{67} + ( - \zeta_{24}^{11} - \zeta_{24}) q^{71} + ( - \zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24}) q^{74} - \zeta_{24}^{8} q^{79} + ( - \zeta_{24}^{11} + \zeta_{24}^{9} - \zeta_{24}^{7} + \zeta_{24}^{5}) q^{86} + (\zeta_{24}^{10} - \zeta_{24}^{6} - \zeta_{24}^{2}) q^{88} + (\zeta_{24}^{11} - \zeta_{24}^{9} + \zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24}) q^{92} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} - 8 q^{16} + 8 q^{22} - 4 q^{25} + 4 q^{46} + 4 q^{58} - 16 q^{64} - 4 q^{67} + 4 q^{79}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3969\mathbb{Z}\right)^\times\).

\(n\) \(2108\) \(3727\)
\(\chi(n)\) \(-1\) \(\zeta_{24}^{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2186.1
0.258819 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 + 0.965926i
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 0.965926i
−1.67303 0.965926i 0 1.36603 + 2.36603i 0 0 0 3.34607i 0 0
2186.2 −0.448288 0.258819i 0 −0.366025 0.633975i 0 0 0 0.896575i 0 0
2186.3 0.448288 + 0.258819i 0 −0.366025 0.633975i 0 0 0 0.896575i 0 0
2186.4 1.67303 + 0.965926i 0 1.36603 + 2.36603i 0 0 0 3.34607i 0 0
3644.1 −1.67303 + 0.965926i 0 1.36603 2.36603i 0 0 0 3.34607i 0 0
3644.2 −0.448288 + 0.258819i 0 −0.366025 + 0.633975i 0 0 0 0.896575i 0 0
3644.3 0.448288 0.258819i 0 −0.366025 + 0.633975i 0 0 0 0.896575i 0 0
3644.4 1.67303 0.965926i 0 1.36603 2.36603i 0 0 0 3.34607i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3644.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3969.1.q.a 8
3.b odd 2 1 inner 3969.1.q.a 8
7.b odd 2 1 CM 3969.1.q.a 8
7.c even 3 1 3969.1.b.a 4
7.c even 3 1 inner 3969.1.q.a 8
7.d odd 6 1 3969.1.b.a 4
7.d odd 6 1 inner 3969.1.q.a 8
9.c even 3 1 3969.1.j.c 8
9.c even 3 1 3969.1.n.c 8
9.d odd 6 1 3969.1.j.c 8
9.d odd 6 1 3969.1.n.c 8
21.c even 2 1 inner 3969.1.q.a 8
21.g even 6 1 3969.1.b.a 4
21.g even 6 1 inner 3969.1.q.a 8
21.h odd 6 1 3969.1.b.a 4
21.h odd 6 1 inner 3969.1.q.a 8
63.g even 3 1 3969.1.j.c 8
63.g even 3 1 3969.1.r.d 8
63.h even 3 1 3969.1.n.c 8
63.h even 3 1 3969.1.r.d 8
63.i even 6 1 3969.1.n.c 8
63.i even 6 1 3969.1.r.d 8
63.j odd 6 1 3969.1.n.c 8
63.j odd 6 1 3969.1.r.d 8
63.k odd 6 1 3969.1.j.c 8
63.k odd 6 1 3969.1.r.d 8
63.l odd 6 1 3969.1.j.c 8
63.l odd 6 1 3969.1.n.c 8
63.n odd 6 1 3969.1.j.c 8
63.n odd 6 1 3969.1.r.d 8
63.o even 6 1 3969.1.j.c 8
63.o even 6 1 3969.1.n.c 8
63.s even 6 1 3969.1.j.c 8
63.s even 6 1 3969.1.r.d 8
63.t odd 6 1 3969.1.n.c 8
63.t odd 6 1 3969.1.r.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3969.1.b.a 4 7.c even 3 1
3969.1.b.a 4 7.d odd 6 1
3969.1.b.a 4 21.g even 6 1
3969.1.b.a 4 21.h odd 6 1
3969.1.j.c 8 9.c even 3 1
3969.1.j.c 8 9.d odd 6 1
3969.1.j.c 8 63.g even 3 1
3969.1.j.c 8 63.k odd 6 1
3969.1.j.c 8 63.l odd 6 1
3969.1.j.c 8 63.n odd 6 1
3969.1.j.c 8 63.o even 6 1
3969.1.j.c 8 63.s even 6 1
3969.1.n.c 8 9.c even 3 1
3969.1.n.c 8 9.d odd 6 1
3969.1.n.c 8 63.h even 3 1
3969.1.n.c 8 63.i even 6 1
3969.1.n.c 8 63.j odd 6 1
3969.1.n.c 8 63.l odd 6 1
3969.1.n.c 8 63.o even 6 1
3969.1.n.c 8 63.t odd 6 1
3969.1.q.a 8 1.a even 1 1 trivial
3969.1.q.a 8 3.b odd 2 1 inner
3969.1.q.a 8 7.b odd 2 1 CM
3969.1.q.a 8 7.c even 3 1 inner
3969.1.q.a 8 7.d odd 6 1 inner
3969.1.q.a 8 21.c even 2 1 inner
3969.1.q.a 8 21.g even 6 1 inner
3969.1.q.a 8 21.h odd 6 1 inner
3969.1.r.d 8 63.g even 3 1
3969.1.r.d 8 63.h even 3 1
3969.1.r.d 8 63.i even 6 1
3969.1.r.d 8 63.j odd 6 1
3969.1.r.d 8 63.k odd 6 1
3969.1.r.d 8 63.n odd 6 1
3969.1.r.d 8 63.s even 6 1
3969.1.r.d 8 63.t odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3969, [\chi])\).

Hecke characteristic polynomials

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