Properties

Label 441.6.a.b
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

\(f(q)\) \(=\) \( q - 10 q^{2} + 68 q^{4} - 106 q^{5} - 360 q^{8} + O(q^{10}) \) \( q - 10 q^{2} + 68 q^{4} - 106 q^{5} - 360 q^{8} + 1060 q^{10} - 92 q^{11} - 670 q^{13} + 1424 q^{16} - 222 q^{17} + 908 q^{19} - 7208 q^{20} + 920 q^{22} + 1176 q^{23} + 8111 q^{25} + 6700 q^{26} - 1118 q^{29} - 3696 q^{31} - 2720 q^{32} + 2220 q^{34} + 4182 q^{37} - 9080 q^{38} + 38160 q^{40} - 6662 q^{41} - 3700 q^{43} - 6256 q^{44} - 11760 q^{46} - 7056 q^{47} - 81110 q^{50} - 45560 q^{52} + 37578 q^{53} + 9752 q^{55} + 11180 q^{58} + 32700 q^{59} + 10802 q^{61} + 36960 q^{62} - 18368 q^{64} + 71020 q^{65} + 64996 q^{67} - 15096 q^{68} + 61320 q^{71} - 38922 q^{73} - 41820 q^{74} + 61744 q^{76} - 88096 q^{79} - 150944 q^{80} + 66620 q^{82} + 71892 q^{83} + 23532 q^{85} + 37000 q^{86} + 33120 q^{88} + 111818 q^{89} + 79968 q^{92} + 70560 q^{94} - 96248 q^{95} + 150846 q^{97} + O(q^{100}) \)

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
−10.0000 0 68.0000 −106.000 0 0 −360.000 0 1060.00
\(n\): 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.b 1
3.b odd 2 1 147.6.a.g 1
7.b odd 2 1 63.6.a.a 1
21.c even 2 1 21.6.a.d 1
21.g even 6 2 147.6.e.a 2
21.h odd 6 2 147.6.e.b 2
28.d even 2 1 1008.6.a.bc 1
84.h odd 2 1 336.6.a.a 1
105.g even 2 1 525.6.a.a 1
105.k odd 4 2 525.6.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.d 1 21.c even 2 1
63.6.a.a 1 7.b odd 2 1
147.6.a.g 1 3.b odd 2 1
147.6.e.a 2 21.g even 6 2
147.6.e.b 2 21.h odd 6 2
336.6.a.a 1 84.h odd 2 1
441.6.a.b 1 1.a even 1 1 trivial
525.6.a.a 1 105.g even 2 1
525.6.d.a 2 105.k odd 4 2
1008.6.a.bc 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2} + 10 \)
\( T_{5} + 106 \)
\( T_{13} + 670 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 10 + T \)
$3$ \( T \)
$5$ \( 106 + T \)
$7$ \( T \)
$11$ \( 92 + T \)
$13$ \( 670 + T \)
$17$ \( 222 + T \)
$19$ \( -908 + T \)
$23$ \( -1176 + T \)
$29$ \( 1118 + T \)
$31$ \( 3696 + T \)
$37$ \( -4182 + T \)
$41$ \( 6662 + T \)
$43$ \( 3700 + T \)
$47$ \( 7056 + T \)
$53$ \( -37578 + T \)
$59$ \( -32700 + T \)
$61$ \( -10802 + T \)
$67$ \( -64996 + T \)
$71$ \( -61320 + T \)
$73$ \( 38922 + T \)
$79$ \( 88096 + T \)
$83$ \( -71892 + T \)
$89$ \( -111818 + T \)
$97$ \( -150846 + T \)
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