Properties

Label 441.1.q.a
Level $441$
Weight $1$
Character orbit 441.q
Analytic conductor $0.220$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -7
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 441.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.220087670571\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.189.1
Artin image: $C_3\times SD_{16}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i 0 0.500000 0.866025i 0 0 0 0 0 0
116.2 1.22474 0.707107i 0 0.500000 0.866025i 0 0 0 0 0 0
422.1 −1.22474 0.707107i 0 0.500000 + 0.866025i 0 0 0 0 0 0
422.2 1.22474 + 0.707107i 0 0.500000 + 0.866025i 0 0 0 0 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
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 441.1.q.a 4
3.b odd 2 1 inner 441.1.q.a 4
7.b odd 2 1 CM 441.1.q.a 4
7.c even 3 1 441.1.b.a 2
7.c even 3 1 inner 441.1.q.a 4
7.d odd 6 1 441.1.b.a 2
7.d odd 6 1 inner 441.1.q.a 4
9.c even 3 1 3969.1.j.b 4
9.c even 3 1 3969.1.n.b 4
9.d odd 6 1 3969.1.j.b 4
9.d odd 6 1 3969.1.n.b 4
21.c even 2 1 inner 441.1.q.a 4
21.g even 6 1 441.1.b.a 2
21.g even 6 1 inner 441.1.q.a 4
21.h odd 6 1 441.1.b.a 2
21.h odd 6 1 inner 441.1.q.a 4
63.g even 3 1 3969.1.j.b 4
63.g even 3 1 3969.1.r.c 4
63.h even 3 1 3969.1.n.b 4
63.h even 3 1 3969.1.r.c 4
63.i even 6 1 3969.1.n.b 4
63.i even 6 1 3969.1.r.c 4
63.j odd 6 1 3969.1.n.b 4
63.j odd 6 1 3969.1.r.c 4
63.k odd 6 1 3969.1.j.b 4
63.k odd 6 1 3969.1.r.c 4
63.l odd 6 1 3969.1.j.b 4
63.l odd 6 1 3969.1.n.b 4
63.n odd 6 1 3969.1.j.b 4
63.n odd 6 1 3969.1.r.c 4
63.o even 6 1 3969.1.j.b 4
63.o even 6 1 3969.1.n.b 4
63.s even 6 1 3969.1.j.b 4
63.s even 6 1 3969.1.r.c 4
63.t odd 6 1 3969.1.n.b 4
63.t odd 6 1 3969.1.r.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.1.b.a 2 7.c even 3 1
441.1.b.a 2 7.d odd 6 1
441.1.b.a 2 21.g even 6 1
441.1.b.a 2 21.h odd 6 1
441.1.q.a 4 1.a even 1 1 trivial
441.1.q.a 4 3.b odd 2 1 inner
441.1.q.a 4 7.b odd 2 1 CM
441.1.q.a 4 7.c even 3 1 inner
441.1.q.a 4 7.d odd 6 1 inner
441.1.q.a 4 21.c even 2 1 inner
441.1.q.a 4 21.g even 6 1 inner
441.1.q.a 4 21.h odd 6 1 inner
3969.1.j.b 4 9.c even 3 1
3969.1.j.b 4 9.d odd 6 1
3969.1.j.b 4 63.g even 3 1
3969.1.j.b 4 63.k odd 6 1
3969.1.j.b 4 63.l odd 6 1
3969.1.j.b 4 63.n odd 6 1
3969.1.j.b 4 63.o even 6 1
3969.1.j.b 4 63.s even 6 1
3969.1.n.b 4 9.c even 3 1
3969.1.n.b 4 9.d odd 6 1
3969.1.n.b 4 63.h even 3 1
3969.1.n.b 4 63.i even 6 1
3969.1.n.b 4 63.j odd 6 1
3969.1.n.b 4 63.l odd 6 1
3969.1.n.b 4 63.o even 6 1
3969.1.n.b 4 63.t odd 6 1
3969.1.r.c 4 63.g even 3 1
3969.1.r.c 4 63.h even 3 1
3969.1.r.c 4 63.i even 6 1
3969.1.r.c 4 63.j odd 6 1
3969.1.r.c 4 63.k odd 6 1
3969.1.r.c 4 63.n odd 6 1
3969.1.r.c 4 63.s even 6 1
3969.1.r.c 4 63.t odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(441, [\chi])\).

Hecke characteristic polynomials

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