Properties

Label 450.6.a.g
Level $450$
Weight $6$
Character orbit 450.a
Self dual yes
Analytic conductor $72.173$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.1727189158\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 150)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4q^{2} + 16q^{4} + q^{7} - 64q^{8} + O(q^{10}) \) \( q - 4q^{2} + 16q^{4} + q^{7} - 64q^{8} + 210q^{11} + 667q^{13} - 4q^{14} + 256q^{16} + 114q^{17} + 581q^{19} - 840q^{22} - 4350q^{23} - 2668q^{26} + 16q^{28} + 126q^{29} + 7583q^{31} - 1024q^{32} - 456q^{34} + 3742q^{37} - 2324q^{38} + 2856q^{41} + 18241q^{43} + 3360q^{44} + 17400q^{46} - 23370q^{47} - 16806q^{49} + 10672q^{52} - 21684q^{53} - 64q^{56} - 504q^{58} + 32310q^{59} - 7165q^{61} - 30332q^{62} + 4096q^{64} - 59579q^{67} + 1824q^{68} + 43080q^{71} + 28942q^{73} - 14968q^{74} + 9296q^{76} + 210q^{77} + 27608q^{79} - 11424q^{82} - 1782q^{83} - 72964q^{86} - 13440q^{88} - 50208q^{89} + 667q^{91} - 69600q^{92} + 93480q^{94} - 142793q^{97} + 67224q^{98} + O(q^{100}) \)

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
−4.00000 0 16.0000 0 0 1.00000 −64.0000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)

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 450.6.a.g 1
3.b odd 2 1 150.6.a.k yes 1
5.b even 2 1 450.6.a.r 1
5.c odd 4 2 450.6.c.k 2
15.d odd 2 1 150.6.a.e 1
15.e even 4 2 150.6.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.6.a.e 1 15.d odd 2 1
150.6.a.k yes 1 3.b odd 2 1
150.6.c.a 2 15.e even 4 2
450.6.a.g 1 1.a even 1 1 trivial
450.6.a.r 1 5.b even 2 1
450.6.c.k 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(450))\):

\( T_{7} - 1 \)
\( T_{11} - 210 \)
\( T_{17} - 114 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( -1 + T \)
$11$ \( -210 + T \)
$13$ \( -667 + T \)
$17$ \( -114 + T \)
$19$ \( -581 + T \)
$23$ \( 4350 + T \)
$29$ \( -126 + T \)
$31$ \( -7583 + T \)
$37$ \( -3742 + T \)
$41$ \( -2856 + T \)
$43$ \( -18241 + T \)
$47$ \( 23370 + T \)
$53$ \( 21684 + T \)
$59$ \( -32310 + T \)
$61$ \( 7165 + T \)
$67$ \( 59579 + T \)
$71$ \( -43080 + T \)
$73$ \( -28942 + T \)
$79$ \( -27608 + T \)
$83$ \( 1782 + T \)
$89$ \( 50208 + T \)
$97$ \( 142793 + T \)
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