Properties

Label 441.4.a.k
Level $441$
Weight $4$
Character orbit 441.a
Self dual yes
Analytic conductor $26.020$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{2} + q^{4} - 3q^{5} - 21q^{8} + O(q^{10}) \) \( q + 3q^{2} + q^{4} - 3q^{5} - 21q^{8} - 9q^{10} + 15q^{11} + 64q^{13} - 71q^{16} + 84q^{17} + 16q^{19} - 3q^{20} + 45q^{22} + 84q^{23} - 116q^{25} + 192q^{26} + 297q^{29} + 253q^{31} - 45q^{32} + 252q^{34} - 316q^{37} + 48q^{38} + 63q^{40} + 360q^{41} + 26q^{43} + 15q^{44} + 252q^{46} - 30q^{47} - 348q^{50} + 64q^{52} - 363q^{53} - 45q^{55} + 891q^{58} - 15q^{59} + 118q^{61} + 759q^{62} + 433q^{64} - 192q^{65} - 370q^{67} + 84q^{68} + 342q^{71} - 362q^{73} - 948q^{74} + 16q^{76} + 467q^{79} + 213q^{80} + 1080q^{82} + 477q^{83} - 252q^{85} + 78q^{86} - 315q^{88} + 906q^{89} + 84q^{92} - 90q^{94} - 48q^{95} - 503q^{97} + O(q^{100}) \)

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
3.00000 0 1.00000 −3.00000 0 0 −21.0000 0 −9.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.a.k 1
3.b odd 2 1 147.4.a.a 1
7.b odd 2 1 441.4.a.l 1
7.c even 3 2 441.4.e.c 2
7.d odd 6 2 63.4.e.a 2
12.b even 2 1 2352.4.a.bd 1
21.c even 2 1 147.4.a.b 1
21.g even 6 2 21.4.e.a 2
21.h odd 6 2 147.4.e.h 2
84.h odd 2 1 2352.4.a.i 1
84.j odd 6 2 336.4.q.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.a 2 21.g even 6 2
63.4.e.a 2 7.d odd 6 2
147.4.a.a 1 3.b odd 2 1
147.4.a.b 1 21.c even 2 1
147.4.e.h 2 21.h odd 6 2
336.4.q.e 2 84.j odd 6 2
441.4.a.k 1 1.a even 1 1 trivial
441.4.a.l 1 7.b odd 2 1
441.4.e.c 2 7.c even 3 2
2352.4.a.i 1 84.h odd 2 1
2352.4.a.bd 1 12.b even 2 1

Hecke kernels

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

\( T_{2} - 3 \)
\( T_{5} + 3 \)
\( T_{13} - 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 + T \)
$3$ \( T \)
$5$ \( 3 + T \)
$7$ \( T \)
$11$ \( -15 + T \)
$13$ \( -64 + T \)
$17$ \( -84 + T \)
$19$ \( -16 + T \)
$23$ \( -84 + T \)
$29$ \( -297 + T \)
$31$ \( -253 + T \)
$37$ \( 316 + T \)
$41$ \( -360 + T \)
$43$ \( -26 + T \)
$47$ \( 30 + T \)
$53$ \( 363 + T \)
$59$ \( 15 + T \)
$61$ \( -118 + T \)
$67$ \( 370 + T \)
$71$ \( -342 + T \)
$73$ \( 362 + T \)
$79$ \( -467 + T \)
$83$ \( -477 + T \)
$89$ \( -906 + T \)
$97$ \( 503 + T \)
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