Properties

Label 450.6.a.f
Level $450$
Weight $6$
Character orbit 450.a
Self dual yes
Analytic conductor $72.173$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4q^{2} + 16q^{4} - 4q^{7} - 64q^{8} + O(q^{10}) \) \( q - 4q^{2} + 16q^{4} - 4q^{7} - 64q^{8} + 500q^{11} - 288q^{13} + 16q^{14} + 256q^{16} - 1516q^{17} - 1344q^{19} - 2000q^{22} + 4100q^{23} + 1152q^{26} - 64q^{28} + 2646q^{29} - 5612q^{31} - 1024q^{32} + 6064q^{34} - 7288q^{37} + 5376q^{38} + 18986q^{41} - 2404q^{43} + 8000q^{44} - 16400q^{46} - 8900q^{47} - 16791q^{49} - 4608q^{52} - 39804q^{53} + 256q^{56} - 10584q^{58} + 28300q^{59} + 18290q^{61} + 22448q^{62} + 4096q^{64} + 65956q^{67} - 24256q^{68} + 28800q^{71} - 30808q^{73} + 29152q^{74} - 21504q^{76} - 2000q^{77} + 60228q^{79} - 75944q^{82} + 2468q^{83} + 9616q^{86} - 32000q^{88} - 22678q^{89} + 1152q^{91} + 65600q^{92} + 35600q^{94} - 36968q^{97} + 67164q^{98} + O(q^{100}) \)

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 −4.00000 −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.f 1
3.b odd 2 1 150.6.a.j 1
5.b even 2 1 450.6.a.s 1
5.c odd 4 2 90.6.c.b 2
15.d odd 2 1 150.6.a.f 1
15.e even 4 2 30.6.c.a 2
20.e even 4 2 720.6.f.g 2
60.l odd 4 2 240.6.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.c.a 2 15.e even 4 2
90.6.c.b 2 5.c odd 4 2
150.6.a.f 1 15.d odd 2 1
150.6.a.j 1 3.b odd 2 1
240.6.f.a 2 60.l odd 4 2
450.6.a.f 1 1.a even 1 1 trivial
450.6.a.s 1 5.b even 2 1
720.6.f.g 2 20.e even 4 2

Hecke kernels

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

\( T_{7} + 4 \)
\( T_{11} - 500 \)
\( T_{17} + 1516 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( 4 + T \)
$11$ \( -500 + T \)
$13$ \( 288 + T \)
$17$ \( 1516 + T \)
$19$ \( 1344 + T \)
$23$ \( -4100 + T \)
$29$ \( -2646 + T \)
$31$ \( 5612 + T \)
$37$ \( 7288 + T \)
$41$ \( -18986 + T \)
$43$ \( 2404 + T \)
$47$ \( 8900 + T \)
$53$ \( 39804 + T \)
$59$ \( -28300 + T \)
$61$ \( -18290 + T \)
$67$ \( -65956 + T \)
$71$ \( -28800 + T \)
$73$ \( 30808 + T \)
$79$ \( -60228 + T \)
$83$ \( -2468 + T \)
$89$ \( 22678 + T \)
$97$ \( 36968 + T \)
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