Properties

Label 3969.1.bz.a
Level $3969$
Weight $1$
Character orbit 3969.bz
Analytic conductor $1.981$
Analytic rank $0$
Dimension $12$
Projective image $D_{14}$
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.bz (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: \( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 441)
Projective image: \(D_{14}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{14} - \cdots)\)

$q$-expansion

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

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

Character values

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

\(n\) \(2108\) \(3727\)
\(\chi(n)\) \(-\zeta_{42}^{7}\) \(-\zeta_{42}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
−0.988831 0.149042i
−0.988831 + 0.149042i
0.955573 0.294755i
0.826239 0.563320i
−0.733052 + 0.680173i
0.365341 0.930874i
0.0747301 0.997204i
0.0747301 + 0.997204i
0.365341 + 0.930874i
−0.733052 0.680173i
0.826239 + 0.563320i
0.955573 + 0.294755i
0 0 0.733052 0.680173i 0 0 −0.365341 0.930874i 0 0 0
433.1 0 0 0.733052 + 0.680173i 0 0 −0.365341 + 0.930874i 0 0 0
622.1 0 0 −0.0747301 0.997204i 0 0 0.733052 + 0.680173i 0 0 0
1000.1 0 0 0.988831 0.149042i 0 0 −0.0747301 0.997204i 0 0 0
1189.1 0 0 −0.826239 0.563320i 0 0 −0.955573 0.294755i 0 0 0
1756.1 0 0 −0.955573 + 0.294755i 0 0 0.988831 0.149042i 0 0 0
2134.1 0 0 −0.365341 0.930874i 0 0 −0.826239 0.563320i 0 0 0
2323.1 0 0 −0.365341 + 0.930874i 0 0 −0.826239 + 0.563320i 0 0 0
2701.1 0 0 −0.955573 0.294755i 0 0 0.988831 + 0.149042i 0 0 0
3268.1 0 0 −0.826239 + 0.563320i 0 0 −0.955573 + 0.294755i 0 0 0
3457.1 0 0 0.988831 + 0.149042i 0 0 −0.0747301 + 0.997204i 0 0 0
3835.1 0 0 −0.0747301 + 0.997204i 0 0 0.733052 0.680173i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3835.1
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.f odd 14 1 inner
147.k even 14 1 inner
441.bh even 42 1 inner
441.bk 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.bz.a 12
3.b odd 2 1 CM 3969.1.bz.a 12
9.c even 3 1 441.1.v.a 6
9.c even 3 1 inner 3969.1.bz.a 12
9.d odd 6 1 441.1.v.a 6
9.d odd 6 1 inner 3969.1.bz.a 12
49.f odd 14 1 inner 3969.1.bz.a 12
63.g even 3 1 3087.1.bj.b 12
63.h even 3 1 3087.1.bj.b 12
63.i even 6 1 3087.1.bj.a 12
63.j odd 6 1 3087.1.bj.b 12
63.k odd 6 1 3087.1.bj.a 12
63.l odd 6 1 3087.1.v.a 6
63.n odd 6 1 3087.1.bj.b 12
63.o even 6 1 3087.1.v.a 6
63.s even 6 1 3087.1.bj.a 12
63.t odd 6 1 3087.1.bj.a 12
147.k even 14 1 inner 3969.1.bz.a 12
441.y even 21 1 3087.1.bj.a 12
441.z even 21 1 3087.1.bj.a 12
441.ba even 21 1 3087.1.v.a 6
441.bc odd 42 1 3087.1.bj.b 12
441.bd even 42 1 3087.1.bj.b 12
441.be odd 42 1 3087.1.v.a 6
441.bh even 42 1 441.1.v.a 6
441.bh even 42 1 inner 3969.1.bz.a 12
441.bi odd 42 1 3087.1.bj.a 12
441.bk odd 42 1 441.1.v.a 6
441.bk odd 42 1 inner 3969.1.bz.a 12
441.bl odd 42 1 3087.1.bj.b 12
441.bm odd 42 1 3087.1.bj.a 12
441.bn even 42 1 3087.1.bj.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.1.v.a 6 9.c even 3 1
441.1.v.a 6 9.d odd 6 1
441.1.v.a 6 441.bh even 42 1
441.1.v.a 6 441.bk odd 42 1
3087.1.v.a 6 63.l odd 6 1
3087.1.v.a 6 63.o even 6 1
3087.1.v.a 6 441.ba even 21 1
3087.1.v.a 6 441.be odd 42 1
3087.1.bj.a 12 63.i even 6 1
3087.1.bj.a 12 63.k odd 6 1
3087.1.bj.a 12 63.s even 6 1
3087.1.bj.a 12 63.t odd 6 1
3087.1.bj.a 12 441.y even 21 1
3087.1.bj.a 12 441.z even 21 1
3087.1.bj.a 12 441.bi odd 42 1
3087.1.bj.a 12 441.bm odd 42 1
3087.1.bj.b 12 63.g even 3 1
3087.1.bj.b 12 63.h even 3 1
3087.1.bj.b 12 63.j odd 6 1
3087.1.bj.b 12 63.n odd 6 1
3087.1.bj.b 12 441.bc odd 42 1
3087.1.bj.b 12 441.bd even 42 1
3087.1.bj.b 12 441.bl odd 42 1
3087.1.bj.b 12 441.bn even 42 1
3969.1.bz.a 12 1.a even 1 1 trivial
3969.1.bz.a 12 3.b odd 2 1 CM
3969.1.bz.a 12 9.c even 3 1 inner
3969.1.bz.a 12 9.d odd 6 1 inner
3969.1.bz.a 12 49.f odd 14 1 inner
3969.1.bz.a 12 147.k even 14 1 inner
3969.1.bz.a 12 441.bh even 42 1 inner
3969.1.bz.a 12 441.bk odd 42 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + T^{11} - T^{9} - T^{8} + T^{6} - T^{4} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + 14 T^{9} + 7 T^{7} + 63 T^{6} + \cdots + 49 \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( (T^{6} + 7 T^{4} + 14 T^{2} + 7)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} - 7 T^{10} + 35 T^{8} - 84 T^{6} + \cdots + 49 \) Copy content Toggle raw display
$37$ \( (T^{6} + 5 T^{5} + 11 T^{4} + 13 T^{3} + \cdots + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} + 2 T^{11} - 8 T^{9} - 9 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} - 7 T^{11} + 28 T^{10} - 77 T^{9} + \cdots + 49 \) Copy content Toggle raw display
$67$ \( (T^{6} - T^{5} + 3 T^{4} + 5 T^{2} - 2 T + 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( (T^{6} + 7 T^{2} + 14 T + 7)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - T^{5} + 3 T^{4} + 5 T^{2} - 2 T + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} - 7 T^{10} + 35 T^{8} - 84 T^{6} + \cdots + 49 \) Copy content Toggle raw display
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