Properties

Label 441.4.a.f
Level $441$
Weight $4$
Character orbit 441.a
Self dual yes
Analytic conductor $26.020$
Analytic rank $1$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.0198423125\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 9)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q - 8q^{4} + O(q^{10}) \) \( q - 8q^{4} + 70q^{13} + 64q^{16} - 56q^{19} - 125q^{25} - 308q^{31} + 110q^{37} - 520q^{43} - 560q^{52} - 182q^{61} - 512q^{64} - 880q^{67} - 1190q^{73} + 448q^{76} + 884q^{79} + 1330q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 −8.00000 0 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.a.f 1
3.b odd 2 1 CM 441.4.a.f 1
7.b odd 2 1 9.4.a.a 1
7.c even 3 2 441.4.e.j 2
7.d odd 6 2 441.4.e.i 2
21.c even 2 1 9.4.a.a 1
21.g even 6 2 441.4.e.i 2
21.h odd 6 2 441.4.e.j 2
28.d even 2 1 144.4.a.d 1
35.c odd 2 1 225.4.a.d 1
35.f even 4 2 225.4.b.g 2
56.e even 2 1 576.4.a.l 1
56.h odd 2 1 576.4.a.m 1
63.l odd 6 2 81.4.c.b 2
63.o even 6 2 81.4.c.b 2
77.b even 2 1 1089.4.a.g 1
84.h odd 2 1 144.4.a.d 1
91.b odd 2 1 1521.4.a.g 1
105.g even 2 1 225.4.a.d 1
105.k odd 4 2 225.4.b.g 2
168.e odd 2 1 576.4.a.l 1
168.i even 2 1 576.4.a.m 1
231.h odd 2 1 1089.4.a.g 1
273.g even 2 1 1521.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.a.a 1 7.b odd 2 1
9.4.a.a 1 21.c even 2 1
81.4.c.b 2 63.l odd 6 2
81.4.c.b 2 63.o even 6 2
144.4.a.d 1 28.d even 2 1
144.4.a.d 1 84.h odd 2 1
225.4.a.d 1 35.c odd 2 1
225.4.a.d 1 105.g even 2 1
225.4.b.g 2 35.f even 4 2
225.4.b.g 2 105.k odd 4 2
441.4.a.f 1 1.a even 1 1 trivial
441.4.a.f 1 3.b odd 2 1 CM
441.4.e.i 2 7.d odd 6 2
441.4.e.i 2 21.g even 6 2
441.4.e.j 2 7.c even 3 2
441.4.e.j 2 21.h odd 6 2
576.4.a.l 1 56.e even 2 1
576.4.a.l 1 168.e odd 2 1
576.4.a.m 1 56.h odd 2 1
576.4.a.m 1 168.i even 2 1
1089.4.a.g 1 77.b even 2 1
1089.4.a.g 1 231.h odd 2 1
1521.4.a.g 1 91.b odd 2 1
1521.4.a.g 1 273.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2} \)
\( T_{5} \)
\( T_{13} - 70 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( -70 + T \)
$17$ \( T \)
$19$ \( 56 + T \)
$23$ \( T \)
$29$ \( T \)
$31$ \( 308 + T \)
$37$ \( -110 + T \)
$41$ \( T \)
$43$ \( 520 + T \)
$47$ \( T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( 182 + T \)
$67$ \( 880 + T \)
$71$ \( T \)
$73$ \( 1190 + T \)
$79$ \( -884 + T \)
$83$ \( T \)
$89$ \( T \)
$97$ \( -1330 + T \)
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