Properties

Label 441.6.a.j
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

\(f(q)\) \(=\) \( q + 6 q^{2} + 4 q^{4} + 78 q^{5} - 168 q^{8} + O(q^{10}) \) \( q + 6 q^{2} + 4 q^{4} + 78 q^{5} - 168 q^{8} + 468 q^{10} - 444 q^{11} + 442 q^{13} - 1136 q^{16} - 126 q^{17} - 2684 q^{19} + 312 q^{20} - 2664 q^{22} - 4200 q^{23} + 2959 q^{25} + 2652 q^{26} + 5442 q^{29} - 80 q^{31} - 1440 q^{32} - 756 q^{34} - 5434 q^{37} - 16104 q^{38} - 13104 q^{40} + 7962 q^{41} - 11524 q^{43} - 1776 q^{44} - 25200 q^{46} - 13920 q^{47} + 17754 q^{50} + 1768 q^{52} + 9594 q^{53} - 34632 q^{55} + 32652 q^{58} + 27492 q^{59} - 49478 q^{61} - 480 q^{62} + 27712 q^{64} + 34476 q^{65} - 59356 q^{67} - 504 q^{68} - 32040 q^{71} + 61846 q^{73} - 32604 q^{74} - 10736 q^{76} - 65776 q^{79} - 88608 q^{80} + 47772 q^{82} + 40188 q^{83} - 9828 q^{85} - 69144 q^{86} + 74592 q^{88} - 7974 q^{89} - 16800 q^{92} - 83520 q^{94} - 209352 q^{95} + 143662 q^{97} + O(q^{100}) \)

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
6.00000 0 4.00000 78.0000 0 0 −168.000 0 468.000
\(n\): 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.j 1
3.b odd 2 1 147.6.a.b 1
7.b odd 2 1 63.6.a.d 1
21.c even 2 1 21.6.a.a 1
21.g even 6 2 147.6.e.j 2
21.h odd 6 2 147.6.e.i 2
28.d even 2 1 1008.6.a.c 1
84.h odd 2 1 336.6.a.r 1
105.g even 2 1 525.6.a.d 1
105.k odd 4 2 525.6.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.a 1 21.c even 2 1
63.6.a.d 1 7.b odd 2 1
147.6.a.b 1 3.b odd 2 1
147.6.e.i 2 21.h odd 6 2
147.6.e.j 2 21.g even 6 2
336.6.a.r 1 84.h odd 2 1
441.6.a.j 1 1.a even 1 1 trivial
525.6.a.d 1 105.g even 2 1
525.6.d.b 2 105.k odd 4 2
1008.6.a.c 1 28.d even 2 1

Hecke kernels

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

\( T_{2} - 6 \)
\( T_{5} - 78 \)
\( T_{13} - 442 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -6 + T \)
$3$ \( T \)
$5$ \( -78 + T \)
$7$ \( T \)
$11$ \( 444 + T \)
$13$ \( -442 + T \)
$17$ \( 126 + T \)
$19$ \( 2684 + T \)
$23$ \( 4200 + T \)
$29$ \( -5442 + T \)
$31$ \( 80 + T \)
$37$ \( 5434 + T \)
$41$ \( -7962 + T \)
$43$ \( 11524 + T \)
$47$ \( 13920 + T \)
$53$ \( -9594 + T \)
$59$ \( -27492 + T \)
$61$ \( 49478 + T \)
$67$ \( 59356 + T \)
$71$ \( 32040 + T \)
$73$ \( -61846 + T \)
$79$ \( 65776 + T \)
$83$ \( -40188 + T \)
$89$ \( 7974 + T \)
$97$ \( -143662 + T \)
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