Properties

Label 3969.1.b.a
Level $3969$
Weight $1$
Character orbit 3969.b
Analytic conductor $1.981$
Analytic rank $0$
Dimension $4$
Projective image $D_{12}$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3969.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.98078903514\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of 12.2.136738899331083.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( 2 \beta_{1} - \beta_{3} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( 2 \beta_{1} - \beta_{3} ) q^{8} + \beta_{3} q^{11} + ( 2 - \beta_{2} ) q^{16} + q^{22} + ( -\beta_{1} + \beta_{3} ) q^{23} + q^{25} + ( -\beta_{1} + \beta_{3} ) q^{29} -2 \beta_{1} q^{32} -\beta_{2} q^{37} + \beta_{2} q^{43} + ( -\beta_{1} + \beta_{3} ) q^{44} + ( -1 + \beta_{2} ) q^{46} -\beta_{1} q^{50} -\beta_{3} q^{53} + ( -1 + \beta_{2} ) q^{58} + ( -2 + \beta_{2} ) q^{64} + q^{67} -\beta_{1} q^{71} + ( -2 \beta_{1} + \beta_{3} ) q^{74} - q^{79} + ( 2 \beta_{1} - \beta_{3} ) q^{86} + \beta_{2} q^{88} + 2 \beta_{1} q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + O(q^{10}) \) \( 4 q - 4 q^{4} + 8 q^{16} + 4 q^{22} + 4 q^{25} - 4 q^{46} - 4 q^{58} - 8 q^{64} + 4 q^{67} - 4 q^{79} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 4 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 4 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 4 \beta_{1}\)

Character values

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

\(n\) \(2108\) \(3727\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3725.1
1.93185i
0.517638i
0.517638i
1.93185i
1.93185i 0 −2.73205 0 0 0 3.34607i 0 0
3725.2 0.517638i 0 0.732051 0 0 0 0.896575i 0 0
3725.3 0.517638i 0 0.732051 0 0 0 0.896575i 0 0
3725.4 1.93185i 0 −2.73205 0 0 0 3.34607i 0 0
\(n\): e.g. 2-40 or 990-1000
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
21.c even 2 1 inner

Twists

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( 1 + 4 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( 2 + T^{2} )^{2} \)
$29$ \( ( 2 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( ( -3 + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( -3 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( 1 + 4 T^{2} + T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( ( -1 + T )^{4} \)
$71$ \( 1 + 4 T^{2} + T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( 1 + T )^{4} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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