Properties

Label 441.6.a.h
Level $441$
Weight $6$
Character orbit 441.a
Self dual yes
Analytic conductor $70.729$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.7292645375\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q + 2 q^{2} - 28 q^{4} + 11 q^{5} - 120 q^{8} + O(q^{10}) \) \( q + 2 q^{2} - 28 q^{4} + 11 q^{5} - 120 q^{8} + 22 q^{10} - 269 q^{11} + 308 q^{13} + 656 q^{16} + 1896 q^{17} + 164 q^{19} - 308 q^{20} - 538 q^{22} + 3264 q^{23} - 3004 q^{25} + 616 q^{26} - 2417 q^{29} - 2841 q^{31} + 5152 q^{32} + 3792 q^{34} - 11328 q^{37} + 328 q^{38} - 1320 q^{40} - 16856 q^{41} - 7894 q^{43} + 7532 q^{44} + 6528 q^{46} + 21102 q^{47} - 6008 q^{50} - 8624 q^{52} + 29691 q^{53} - 2959 q^{55} - 4834 q^{58} - 8163 q^{59} - 15166 q^{61} - 5682 q^{62} - 10688 q^{64} + 3388 q^{65} - 32078 q^{67} - 53088 q^{68} + 38274 q^{71} - 34866 q^{73} - 22656 q^{74} - 4592 q^{76} + 13529 q^{79} + 7216 q^{80} - 33712 q^{82} - 68103 q^{83} + 20856 q^{85} - 15788 q^{86} + 32280 q^{88} - 114922 q^{89} - 91392 q^{92} + 42204 q^{94} + 1804 q^{95} - 154959 q^{97} + O(q^{100}) \)

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
2.00000 0 −28.0000 11.0000 0 0 −120.000 0 22.0000
\(n\): 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 441.6.a.h 1
3.b odd 2 1 147.6.a.d 1
7.b odd 2 1 441.6.a.g 1
7.d odd 6 2 63.6.e.a 2
21.c even 2 1 147.6.a.c 1
21.g even 6 2 21.6.e.a 2
21.h odd 6 2 147.6.e.g 2
84.j odd 6 2 336.6.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.a 2 21.g even 6 2
63.6.e.a 2 7.d odd 6 2
147.6.a.c 1 21.c even 2 1
147.6.a.d 1 3.b odd 2 1
147.6.e.g 2 21.h odd 6 2
336.6.q.b 2 84.j odd 6 2
441.6.a.g 1 7.b odd 2 1
441.6.a.h 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2} - 2 \)
\( T_{5} - 11 \)
\( T_{13} - 308 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( T \)
$5$ \( -11 + T \)
$7$ \( T \)
$11$ \( 269 + T \)
$13$ \( -308 + T \)
$17$ \( -1896 + T \)
$19$ \( -164 + T \)
$23$ \( -3264 + T \)
$29$ \( 2417 + T \)
$31$ \( 2841 + T \)
$37$ \( 11328 + T \)
$41$ \( 16856 + T \)
$43$ \( 7894 + T \)
$47$ \( -21102 + T \)
$53$ \( -29691 + T \)
$59$ \( 8163 + T \)
$61$ \( 15166 + T \)
$67$ \( 32078 + T \)
$71$ \( -38274 + T \)
$73$ \( 34866 + T \)
$79$ \( -13529 + T \)
$83$ \( 68103 + T \)
$89$ \( 114922 + T \)
$97$ \( 154959 + T \)
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