Properties

Label 441.6.a.k
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 7)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 10 q^{2} + 68 q^{4} - 56 q^{5} + 360 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 10 q^{2} + 68 q^{4} - 56 q^{5} + 360 q^{8} - 560 q^{10} - 232 q^{11} + 140 q^{13} + 1424 q^{16} - 1722 q^{17} + 98 q^{19} - 3808 q^{20} - 2320 q^{22} - 1824 q^{23} + 11 q^{25} + 1400 q^{26} - 3418 q^{29} + 7644 q^{31} + 2720 q^{32} - 17220 q^{34} - 10398 q^{37} + 980 q^{38} - 20160 q^{40} - 17962 q^{41} + 10880 q^{43} - 15776 q^{44} - 18240 q^{46} + 9324 q^{47} + 110 q^{50} + 9520 q^{52} - 2262 q^{53} + 12992 q^{55} - 34180 q^{58} - 2730 q^{59} - 25648 q^{61} + 76440 q^{62} - 18368 q^{64} - 7840 q^{65} - 48404 q^{67} - 117096 q^{68} + 58560 q^{71} - 68082 q^{73} - 103980 q^{74} + 6664 q^{76} + 31784 q^{79} - 79744 q^{80} - 179620 q^{82} - 20538 q^{83} + 96432 q^{85} + 108800 q^{86} - 83520 q^{88} - 50582 q^{89} - 124032 q^{92} + 93240 q^{94} - 5488 q^{95} + 58506 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 −56.0000 0 0 360.000 0 −560.000
\(n\): 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.k 1
3.b odd 2 1 49.6.a.a 1
7.b odd 2 1 63.6.a.e 1
12.b even 2 1 784.6.a.c 1
21.c even 2 1 7.6.a.a 1
21.g even 6 2 49.6.c.c 2
21.h odd 6 2 49.6.c.b 2
28.d even 2 1 1008.6.a.y 1
84.h odd 2 1 112.6.a.g 1
105.g even 2 1 175.6.a.b 1
105.k odd 4 2 175.6.b.a 2
168.e odd 2 1 448.6.a.c 1
168.i even 2 1 448.6.a.m 1
231.h odd 2 1 847.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.a 1 21.c even 2 1
49.6.a.a 1 3.b odd 2 1
49.6.c.b 2 21.h odd 6 2
49.6.c.c 2 21.g even 6 2
63.6.a.e 1 7.b odd 2 1
112.6.a.g 1 84.h odd 2 1
175.6.a.b 1 105.g even 2 1
175.6.b.a 2 105.k odd 4 2
441.6.a.k 1 1.a even 1 1 trivial
448.6.a.c 1 168.e odd 2 1
448.6.a.m 1 168.i even 2 1
784.6.a.c 1 12.b even 2 1
847.6.a.b 1 231.h odd 2 1
1008.6.a.y 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2} - 10 \) Copy content Toggle raw display
\( T_{5} + 56 \) Copy content Toggle raw display
\( T_{13} - 140 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 10 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 56 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 232 \) Copy content Toggle raw display
$13$ \( T - 140 \) Copy content Toggle raw display
$17$ \( T + 1722 \) Copy content Toggle raw display
$19$ \( T - 98 \) Copy content Toggle raw display
$23$ \( T + 1824 \) Copy content Toggle raw display
$29$ \( T + 3418 \) Copy content Toggle raw display
$31$ \( T - 7644 \) Copy content Toggle raw display
$37$ \( T + 10398 \) Copy content Toggle raw display
$41$ \( T + 17962 \) Copy content Toggle raw display
$43$ \( T - 10880 \) Copy content Toggle raw display
$47$ \( T - 9324 \) Copy content Toggle raw display
$53$ \( T + 2262 \) Copy content Toggle raw display
$59$ \( T + 2730 \) Copy content Toggle raw display
$61$ \( T + 25648 \) Copy content Toggle raw display
$67$ \( T + 48404 \) Copy content Toggle raw display
$71$ \( T - 58560 \) Copy content Toggle raw display
$73$ \( T + 68082 \) Copy content Toggle raw display
$79$ \( T - 31784 \) Copy content Toggle raw display
$83$ \( T + 20538 \) Copy content Toggle raw display
$89$ \( T + 50582 \) Copy content Toggle raw display
$97$ \( T - 58506 \) Copy content Toggle raw display
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