Properties

Label 441.6.a.i
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 3)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 6 q^{2} + 4 q^{4} + 6 q^{5} - 168 q^{8} + O(q^{10}) \) \( q + 6 q^{2} + 4 q^{4} + 6 q^{5} - 168 q^{8} + 36 q^{10} + 564 q^{11} - 638 q^{13} - 1136 q^{16} + 882 q^{17} + 556 q^{19} + 24 q^{20} + 3384 q^{22} + 840 q^{23} - 3089 q^{25} - 3828 q^{26} - 4638 q^{29} - 4400 q^{31} - 1440 q^{32} + 5292 q^{34} - 2410 q^{37} + 3336 q^{38} - 1008 q^{40} - 6870 q^{41} + 9644 q^{43} + 2256 q^{44} + 5040 q^{46} - 18672 q^{47} - 18534 q^{50} - 2552 q^{52} - 33750 q^{53} + 3384 q^{55} - 27828 q^{58} - 18084 q^{59} - 39758 q^{61} - 26400 q^{62} + 27712 q^{64} - 3828 q^{65} - 23068 q^{67} + 3528 q^{68} + 4248 q^{71} + 41110 q^{73} - 14460 q^{74} + 2224 q^{76} + 21920 q^{79} - 6816 q^{80} - 41220 q^{82} + 82452 q^{83} + 5292 q^{85} + 57864 q^{86} - 94752 q^{88} - 94086 q^{89} + 3360 q^{92} - 112032 q^{94} + 3336 q^{95} - 49442 q^{97} + O(q^{100}) \)

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 6.00000 0 0 −168.000 0 36.0000
\(n\): 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.i 1
3.b odd 2 1 147.6.a.a 1
7.b odd 2 1 9.6.a.a 1
21.c even 2 1 3.6.a.a 1
21.g even 6 2 147.6.e.h 2
21.h odd 6 2 147.6.e.k 2
28.d even 2 1 144.6.a.f 1
35.c odd 2 1 225.6.a.a 1
35.f even 4 2 225.6.b.b 2
56.e even 2 1 576.6.a.t 1
56.h odd 2 1 576.6.a.s 1
63.l odd 6 2 81.6.c.a 2
63.o even 6 2 81.6.c.c 2
77.b even 2 1 1089.6.a.b 1
84.h odd 2 1 48.6.a.a 1
105.g even 2 1 75.6.a.e 1
105.k odd 4 2 75.6.b.b 2
168.e odd 2 1 192.6.a.l 1
168.i even 2 1 192.6.a.d 1
231.h odd 2 1 363.6.a.d 1
273.g even 2 1 507.6.a.b 1
336.v odd 4 2 768.6.d.h 2
336.y even 4 2 768.6.d.k 2
357.c even 2 1 867.6.a.a 1
399.h odd 2 1 1083.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.6.a.a 1 21.c even 2 1
9.6.a.a 1 7.b odd 2 1
48.6.a.a 1 84.h odd 2 1
75.6.a.e 1 105.g even 2 1
75.6.b.b 2 105.k odd 4 2
81.6.c.a 2 63.l odd 6 2
81.6.c.c 2 63.o even 6 2
144.6.a.f 1 28.d even 2 1
147.6.a.a 1 3.b odd 2 1
147.6.e.h 2 21.g even 6 2
147.6.e.k 2 21.h odd 6 2
192.6.a.d 1 168.i even 2 1
192.6.a.l 1 168.e odd 2 1
225.6.a.a 1 35.c odd 2 1
225.6.b.b 2 35.f even 4 2
363.6.a.d 1 231.h odd 2 1
441.6.a.i 1 1.a even 1 1 trivial
507.6.a.b 1 273.g even 2 1
576.6.a.s 1 56.h odd 2 1
576.6.a.t 1 56.e even 2 1
768.6.d.h 2 336.v odd 4 2
768.6.d.k 2 336.y even 4 2
867.6.a.a 1 357.c even 2 1
1083.6.a.c 1 399.h odd 2 1
1089.6.a.b 1 77.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 \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2} - 6 \)
\( T_{5} - 6 \)
\( T_{13} + 638 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -6 + T \)
$3$ \( T \)
$5$ \( -6 + T \)
$7$ \( T \)
$11$ \( -564 + T \)
$13$ \( 638 + T \)
$17$ \( -882 + T \)
$19$ \( -556 + T \)
$23$ \( -840 + T \)
$29$ \( 4638 + T \)
$31$ \( 4400 + T \)
$37$ \( 2410 + T \)
$41$ \( 6870 + T \)
$43$ \( -9644 + T \)
$47$ \( 18672 + T \)
$53$ \( 33750 + T \)
$59$ \( 18084 + T \)
$61$ \( 39758 + T \)
$67$ \( 23068 + T \)
$71$ \( -4248 + T \)
$73$ \( -41110 + T \)
$79$ \( -21920 + T \)
$83$ \( -82452 + T \)
$89$ \( 94086 + T \)
$97$ \( 49442 + T \)
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