Properties

Label 3969.1.cb.a
Level $3969$
Weight $1$
Character orbit 3969.cb
Analytic conductor $1.981$
Analytic rank $0$
Dimension $12$
Projective image $D_{21}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3969.cb (of order \(42\), degree \(12\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.98078903514\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{21})\)
Defining polynomial: \(x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1323)
Projective image: \(D_{21}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{21} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{42}^{18} q^{4} + \zeta_{42}^{16} q^{7} +O(q^{10})\) \( q + \zeta_{42}^{18} q^{4} + \zeta_{42}^{16} q^{7} + ( -\zeta_{42} - \zeta_{42}^{3} ) q^{13} -\zeta_{42}^{15} q^{16} + ( \zeta_{42}^{2} - \zeta_{42}^{5} ) q^{19} -\zeta_{42}^{5} q^{25} -\zeta_{42}^{13} q^{28} + ( \zeta_{42}^{4} - \zeta_{42}^{17} ) q^{31} + ( 1 - \zeta_{42}^{13} ) q^{37} + ( \zeta_{42}^{6} - \zeta_{42}^{17} ) q^{43} -\zeta_{42}^{11} q^{49} + ( 1 - \zeta_{42}^{19} ) q^{52} + ( -\zeta_{42}^{13} + \zeta_{42}^{14} ) q^{61} + \zeta_{42}^{12} q^{64} + ( \zeta_{42}^{8} - \zeta_{42}^{13} ) q^{67} + ( -\zeta_{42} + \zeta_{42}^{16} ) q^{73} + ( \zeta_{42}^{2} + \zeta_{42}^{20} ) q^{76} + ( \zeta_{42}^{10} - \zeta_{42}^{11} ) q^{79} + ( -\zeta_{42}^{17} - \zeta_{42}^{19} ) q^{91} + ( \zeta_{42}^{2} + \zeta_{42}^{12} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{4} + q^{7} + O(q^{10}) \) \( 12 q - 2 q^{4} + q^{7} - q^{13} - 2 q^{16} + 2 q^{19} + q^{25} + q^{28} + 2 q^{31} + 13 q^{37} - q^{43} + q^{49} + 13 q^{52} - 5 q^{61} - 2 q^{64} + 2 q^{67} + 2 q^{73} + 2 q^{76} + 2 q^{79} + 2 q^{91} - q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(2108\) \(3727\)
\(\chi(n)\) \(-\zeta_{42}^{14}\) \(-\zeta_{42}^{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
296.1
−0.988831 0.149042i
0.0747301 0.997204i
0.365341 0.930874i
0.826239 0.563320i
0.955573 0.294755i
−0.988831 + 0.149042i
0.826239 + 0.563320i
0.955573 + 0.294755i
0.0747301 + 0.997204i
−0.733052 0.680173i
−0.733052 + 0.680173i
0.365341 + 0.930874i
0 0 −0.900969 + 0.433884i 0 0 −0.733052 + 0.680173i 0 0 0
431.1 0 0 −0.222521 0.974928i 0 0 0.365341 + 0.930874i 0 0 0
1430.1 0 0 −0.900969 0.433884i 0 0 0.955573 0.294755i 0 0 0
1565.1 0 0 −0.222521 + 0.974928i 0 0 −0.988831 + 0.149042i 0 0 0
1997.1 0 0 0.623490 + 0.781831i 0 0 0.0747301 + 0.997204i 0 0 0
2132.1 0 0 −0.900969 0.433884i 0 0 −0.733052 0.680173i 0 0 0
2564.1 0 0 −0.222521 0.974928i 0 0 −0.988831 0.149042i 0 0 0
2699.1 0 0 0.623490 0.781831i 0 0 0.0747301 0.997204i 0 0 0
3131.1 0 0 −0.222521 + 0.974928i 0 0 0.365341 0.930874i 0 0 0
3266.1 0 0 0.623490 + 0.781831i 0 0 0.826239 0.563320i 0 0 0
3698.1 0 0 0.623490 0.781831i 0 0 0.826239 + 0.563320i 0 0 0
3833.1 0 0 −0.900969 + 0.433884i 0 0 0.955573 + 0.294755i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3833.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
441.y even 21 1 inner
441.bm odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3969.1.cb.a 12
3.b odd 2 1 CM 3969.1.cb.a 12
9.c even 3 1 1323.1.br.a 12
9.c even 3 1 3969.1.bx.a 12
9.d odd 6 1 1323.1.br.a 12
9.d odd 6 1 3969.1.bx.a 12
49.g even 21 1 3969.1.bx.a 12
147.n odd 42 1 3969.1.bx.a 12
441.y even 21 1 inner 3969.1.cb.a 12
441.z even 21 1 1323.1.br.a 12
441.bi odd 42 1 1323.1.br.a 12
441.bm odd 42 1 inner 3969.1.cb.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1323.1.br.a 12 9.c even 3 1
1323.1.br.a 12 9.d odd 6 1
1323.1.br.a 12 441.z even 21 1
1323.1.br.a 12 441.bi odd 42 1
3969.1.bx.a 12 9.c even 3 1
3969.1.bx.a 12 9.d odd 6 1
3969.1.bx.a 12 49.g even 21 1
3969.1.bx.a 12 147.n odd 42 1
3969.1.cb.a 12 1.a even 1 1 trivial
3969.1.cb.a 12 3.b odd 2 1 CM
3969.1.cb.a 12 441.y even 21 1 inner
3969.1.cb.a 12 441.bm odd 42 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( T^{12} \)
$5$ \( T^{12} \)
$7$ \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
$11$ \( T^{12} \)
$13$ \( 1 + 15 T + 70 T^{2} + 104 T^{3} + 90 T^{4} + 35 T^{5} + 43 T^{6} + 7 T^{7} + 6 T^{8} + 6 T^{9} + T^{11} + T^{12} \)
$17$ \( T^{12} \)
$19$ \( ( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} )^{2} \)
$23$ \( T^{12} \)
$29$ \( T^{12} \)
$31$ \( ( 1 - 8 T + 8 T^{2} + 6 T^{3} - 6 T^{4} - T^{5} + T^{6} )^{2} \)
$37$ \( 1 - 6 T + 63 T^{2} - 260 T^{3} + 643 T^{4} - 1078 T^{5} + 1275 T^{6} - 1078 T^{7} + 650 T^{8} - 274 T^{9} + 77 T^{10} - 13 T^{11} + T^{12} \)
$41$ \( T^{12} \)
$43$ \( 1 - 13 T + 49 T^{2} - 29 T^{3} + 69 T^{4} - 20 T^{6} - 21 T^{7} + 6 T^{8} - T^{9} + T^{11} + T^{12} \)
$47$ \( T^{12} \)
$53$ \( T^{12} \)
$59$ \( T^{12} \)
$61$ \( 1 + 12 T + 45 T^{2} + 10 T^{3} + 61 T^{4} + 92 T^{5} + 105 T^{6} + 92 T^{7} + 68 T^{8} + 38 T^{9} + 17 T^{10} + 5 T^{11} + T^{12} \)
$67$ \( ( 1 - 8 T + 8 T^{2} + 6 T^{3} - 6 T^{4} - T^{5} + T^{6} )^{2} \)
$71$ \( T^{12} \)
$73$ \( 1 - 4 T + 7 T^{2} - 20 T^{3} + 45 T^{4} - 42 T^{5} + 22 T^{6} - 9 T^{8} + 8 T^{9} - 2 T^{11} + T^{12} \)
$79$ \( ( 1 - 8 T + 8 T^{2} + 6 T^{3} - 6 T^{4} - T^{5} + T^{6} )^{2} \)
$83$ \( T^{12} \)
$89$ \( T^{12} \)
$97$ \( 1 + 8 T + 56 T^{2} + 76 T^{3} + 118 T^{4} + 49 T^{5} + 78 T^{6} + 28 T^{7} + 34 T^{8} + 6 T^{9} + 7 T^{10} + T^{11} + T^{12} \)
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