Properties

Label 441.4.a.g
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 147)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - 7q^{4} - 12q^{5} - 15q^{8} + O(q^{10}) \) \( q + q^{2} - 7q^{4} - 12q^{5} - 15q^{8} - 12q^{10} - 20q^{11} - 84q^{13} + 41q^{16} + 96q^{17} + 12q^{19} + 84q^{20} - 20q^{22} + 176q^{23} + 19q^{25} - 84q^{26} - 58q^{29} - 264q^{31} + 161q^{32} + 96q^{34} + 258q^{37} + 12q^{38} + 180q^{40} + 156q^{43} + 140q^{44} + 176q^{46} + 408q^{47} + 19q^{50} + 588q^{52} + 722q^{53} + 240q^{55} - 58q^{58} - 492q^{59} - 492q^{61} - 264q^{62} - 167q^{64} + 1008q^{65} + 412q^{67} - 672q^{68} - 296q^{71} + 240q^{73} + 258q^{74} - 84q^{76} + 776q^{79} - 492q^{80} - 924q^{83} - 1152q^{85} + 156q^{86} + 300q^{88} + 744q^{89} - 1232q^{92} + 408q^{94} - 144q^{95} - 168q^{97} + O(q^{100}) \)

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
1.00000 0 −7.00000 −12.0000 0 0 −15.0000 0 −12.0000
\(n\): 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.g 1
3.b odd 2 1 147.4.a.d 1
7.b odd 2 1 441.4.a.h 1
7.c even 3 2 441.4.e.g 2
7.d odd 6 2 441.4.e.f 2
12.b even 2 1 2352.4.a.bi 1
21.c even 2 1 147.4.a.e yes 1
21.g even 6 2 147.4.e.e 2
21.h odd 6 2 147.4.e.f 2
84.h odd 2 1 2352.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.d 1 3.b odd 2 1
147.4.a.e yes 1 21.c even 2 1
147.4.e.e 2 21.g even 6 2
147.4.e.f 2 21.h odd 6 2
441.4.a.g 1 1.a even 1 1 trivial
441.4.a.h 1 7.b odd 2 1
441.4.e.f 2 7.d odd 6 2
441.4.e.g 2 7.c even 3 2
2352.4.a.b 1 84.h odd 2 1
2352.4.a.bi 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2} - 1 \)
\( T_{5} + 12 \)
\( T_{13} + 84 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( 12 + T \)
$7$ \( T \)
$11$ \( 20 + T \)
$13$ \( 84 + T \)
$17$ \( -96 + T \)
$19$ \( -12 + T \)
$23$ \( -176 + T \)
$29$ \( 58 + T \)
$31$ \( 264 + T \)
$37$ \( -258 + T \)
$41$ \( T \)
$43$ \( -156 + T \)
$47$ \( -408 + T \)
$53$ \( -722 + T \)
$59$ \( 492 + T \)
$61$ \( 492 + T \)
$67$ \( -412 + T \)
$71$ \( 296 + T \)
$73$ \( -240 + T \)
$79$ \( -776 + T \)
$83$ \( 924 + T \)
$89$ \( -744 + T \)
$97$ \( 168 + T \)
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