Properties

Label 3969.1.bt.a
Level $3969$
Weight $1$
Character orbit 3969.bt
Analytic conductor $1.981$
Analytic rank $0$
Dimension $12$
Projective image $D_{7}$
CM discriminant -3
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3969.bt (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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.373714754427.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{42}^{11} q^{4} + \zeta_{42}^{16} q^{7} +O(q^{10})\) \( q -\zeta_{42}^{11} q^{4} + \zeta_{42}^{16} q^{7} + ( \zeta_{42}^{8} - \zeta_{42}^{17} ) q^{13} -\zeta_{42} q^{16} + ( -\zeta_{42}^{9} + \zeta_{42}^{12} ) q^{19} -\zeta_{42}^{19} q^{25} + \zeta_{42}^{6} q^{28} + ( -\zeta_{42}^{11} - \zeta_{42}^{17} ) q^{31} + ( 1 + \zeta_{42}^{6} ) q^{37} + ( \zeta_{42}^{10} - \zeta_{42}^{13} ) q^{43} -\zeta_{42}^{11} q^{49} + ( -\zeta_{42}^{7} - \zeta_{42}^{19} ) q^{52} + ( -\zeta_{42}^{7} - \zeta_{42}^{13} ) q^{61} + \zeta_{42}^{12} q^{64} + ( \zeta_{42}^{8} + \zeta_{42}^{20} ) q^{67} + ( -\zeta_{42}^{9} - \zeta_{42}^{15} ) q^{73} + ( \zeta_{42}^{2} + \zeta_{42}^{20} ) q^{76} + ( \zeta_{42}^{4} + \zeta_{42}^{10} ) q^{79} + ( -\zeta_{42}^{3} + \zeta_{42}^{12} ) q^{91} + ( \zeta_{42}^{16} - \zeta_{42}^{19} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{4} + q^{7} + O(q^{10}) \) \( 12 q + q^{4} + q^{7} + 2 q^{13} + q^{16} - 4 q^{19} + q^{25} - 2 q^{28} + 2 q^{31} + 10 q^{37} + 2 q^{43} + q^{49} - 5 q^{52} - 5 q^{61} - 2 q^{64} + 2 q^{67} - 4 q^{73} + 2 q^{76} + 2 q^{79} - 4 q^{91} + 2 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(2108\) \(3727\)
\(\chi(n)\) \(-\zeta_{42}^{14}\) \(\zeta_{42}^{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
134.1
−0.988831 0.149042i
0.955573 + 0.294755i
0.365341 0.930874i
0.826239 + 0.563320i
−0.733052 0.680173i
−0.733052 + 0.680173i
0.826239 0.563320i
0.365341 + 0.930874i
0.955573 0.294755i
−0.988831 + 0.149042i
0.0747301 + 0.997204i
0.0747301 0.997204i
0 0 0.0747301 0.997204i 0 0 −0.733052 + 0.680173i 0 0 0
512.1 0 0 −0.988831 0.149042i 0 0 0.0747301 0.997204i 0 0 0
701.1 0 0 0.826239 0.563320i 0 0 0.955573 0.294755i 0 0 0
1268.1 0 0 0.955573 + 0.294755i 0 0 −0.988831 0.149042i 0 0 0
1646.1 0 0 0.365341 0.930874i 0 0 0.826239 0.563320i 0 0 0
1835.1 0 0 0.365341 + 0.930874i 0 0 0.826239 + 0.563320i 0 0 0
2213.1 0 0 0.955573 0.294755i 0 0 −0.988831 + 0.149042i 0 0 0
2780.1 0 0 0.826239 + 0.563320i 0 0 0.955573 + 0.294755i 0 0 0
2969.1 0 0 −0.988831 + 0.149042i 0 0 0.0747301 + 0.997204i 0 0 0
3347.1 0 0 0.0747301 + 0.997204i 0 0 −0.733052 0.680173i 0 0 0
3536.1 0 0 −0.733052 0.680173i 0 0 0.365341 0.930874i 0 0 0
3914.1 0 0 −0.733052 + 0.680173i 0 0 0.365341 + 0.930874i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3914.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
9.c even 3 1 inner
9.d odd 6 1 inner
49.e even 7 1 inner
147.l odd 14 1 inner
441.ba even 21 1 inner
441.be 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.bt.a 12
3.b odd 2 1 CM 3969.1.bt.a 12
9.c even 3 1 147.1.l.a 6
9.c even 3 1 inner 3969.1.bt.a 12
9.d odd 6 1 147.1.l.a 6
9.d odd 6 1 inner 3969.1.bt.a 12
36.f odd 6 1 2352.1.cj.a 6
36.h even 6 1 2352.1.cj.a 6
45.h odd 6 1 3675.1.bm.a 6
45.j even 6 1 3675.1.bm.a 6
45.k odd 12 2 3675.1.bj.a 12
45.l even 12 2 3675.1.bj.a 12
49.e even 7 1 inner 3969.1.bt.a 12
63.g even 3 1 1029.1.n.b 12
63.h even 3 1 1029.1.n.b 12
63.i even 6 1 1029.1.n.a 12
63.j odd 6 1 1029.1.n.b 12
63.k odd 6 1 1029.1.n.a 12
63.l odd 6 1 1029.1.l.a 6
63.n odd 6 1 1029.1.n.b 12
63.o even 6 1 1029.1.l.a 6
63.s even 6 1 1029.1.n.a 12
63.t odd 6 1 1029.1.n.a 12
147.l odd 14 1 inner 3969.1.bt.a 12
441.y even 21 1 1029.1.n.b 12
441.z even 21 1 1029.1.n.b 12
441.ba even 21 1 147.1.l.a 6
441.ba even 21 1 inner 3969.1.bt.a 12
441.bc odd 42 1 1029.1.n.a 12
441.bd even 42 1 1029.1.n.a 12
441.be odd 42 1 147.1.l.a 6
441.be odd 42 1 inner 3969.1.bt.a 12
441.bh even 42 1 1029.1.l.a 6
441.bi odd 42 1 1029.1.n.b 12
441.bk odd 42 1 1029.1.l.a 6
441.bl odd 42 1 1029.1.n.a 12
441.bm odd 42 1 1029.1.n.b 12
441.bn even 42 1 1029.1.n.a 12
1764.cn even 42 1 2352.1.cj.a 6
1764.cs odd 42 1 2352.1.cj.a 6
2205.di even 42 1 3675.1.bm.a 6
2205.dp odd 42 1 3675.1.bm.a 6
2205.ec even 84 2 3675.1.bj.a 12
2205.eg odd 84 2 3675.1.bj.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.1.l.a 6 9.c even 3 1
147.1.l.a 6 9.d odd 6 1
147.1.l.a 6 441.ba even 21 1
147.1.l.a 6 441.be odd 42 1
1029.1.l.a 6 63.l odd 6 1
1029.1.l.a 6 63.o even 6 1
1029.1.l.a 6 441.bh even 42 1
1029.1.l.a 6 441.bk odd 42 1
1029.1.n.a 12 63.i even 6 1
1029.1.n.a 12 63.k odd 6 1
1029.1.n.a 12 63.s even 6 1
1029.1.n.a 12 63.t odd 6 1
1029.1.n.a 12 441.bc odd 42 1
1029.1.n.a 12 441.bd even 42 1
1029.1.n.a 12 441.bl odd 42 1
1029.1.n.a 12 441.bn even 42 1
1029.1.n.b 12 63.g even 3 1
1029.1.n.b 12 63.h even 3 1
1029.1.n.b 12 63.j odd 6 1
1029.1.n.b 12 63.n odd 6 1
1029.1.n.b 12 441.y even 21 1
1029.1.n.b 12 441.z even 21 1
1029.1.n.b 12 441.bi odd 42 1
1029.1.n.b 12 441.bm odd 42 1
2352.1.cj.a 6 36.f odd 6 1
2352.1.cj.a 6 36.h even 6 1
2352.1.cj.a 6 1764.cn even 42 1
2352.1.cj.a 6 1764.cs odd 42 1
3675.1.bj.a 12 45.k odd 12 2
3675.1.bj.a 12 45.l even 12 2
3675.1.bj.a 12 2205.ec even 84 2
3675.1.bj.a 12 2205.eg odd 84 2
3675.1.bm.a 6 45.h odd 6 1
3675.1.bm.a 6 45.j even 6 1
3675.1.bm.a 6 2205.di even 42 1
3675.1.bm.a 6 2205.dp odd 42 1
3969.1.bt.a 12 1.a even 1 1 trivial
3969.1.bt.a 12 3.b odd 2 1 CM
3969.1.bt.a 12 9.c even 3 1 inner
3969.1.bt.a 12 9.d odd 6 1 inner
3969.1.bt.a 12 49.e even 7 1 inner
3969.1.bt.a 12 147.l odd 14 1 inner
3969.1.bt.a 12 441.ba even 21 1 inner
3969.1.bt.a 12 441.be odd 42 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( T^{12} \)
$5$ \( T^{12} \)
$7$ \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
$11$ \( T^{12} \)
$13$ \( 1 + 3 T + 7 T^{2} + 8 T^{3} + 3 T^{4} - 28 T^{5} + T^{6} + 7 T^{7} + 12 T^{8} - 6 T^{9} - 2 T^{11} + T^{12} \)
$17$ \( T^{12} \)
$19$ \( ( -1 - 2 T + T^{2} + T^{3} )^{4} \)
$23$ \( T^{12} \)
$29$ \( T^{12} \)
$31$ \( ( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} )^{2} \)
$37$ \( ( 1 - 3 T + 9 T^{2} - 13 T^{3} + 11 T^{4} - 5 T^{5} + T^{6} )^{2} \)
$41$ \( T^{12} \)
$43$ \( 1 - 4 T + 7 T^{2} - 20 T^{3} + 45 T^{4} - 42 T^{5} + 22 T^{6} - 9 T^{8} + 8 T^{9} - 2 T^{11} + T^{12} \)
$47$ \( T^{12} \)
$53$ \( T^{12} \)
$59$ \( T^{12} \)
$61$ \( 1 + 3 T + T^{3} + 31 T^{4} + 56 T^{5} + 57 T^{6} + 56 T^{7} + 47 T^{8} + 29 T^{9} + 14 T^{10} + 5 T^{11} + T^{12} \)
$67$ \( ( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} )^{2} \)
$71$ \( T^{12} \)
$73$ \( ( 1 + 4 T + 9 T^{2} + 8 T^{3} + 4 T^{4} + 2 T^{5} + T^{6} )^{2} \)
$79$ \( ( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} )^{2} \)
$83$ \( T^{12} \)
$89$ \( T^{12} \)
$97$ \( ( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} )^{2} \)
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