Properties

Label 450.6.a.i
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 150)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4q^{2} + 16q^{4} + 47q^{7} - 64q^{8} + O(q^{10}) \) \( q - 4q^{2} + 16q^{4} + 47q^{7} - 64q^{8} - 222q^{11} + 101q^{13} - 188q^{14} + 256q^{16} - 162q^{17} + 1685q^{19} + 888q^{22} - 306q^{23} - 404q^{26} + 752q^{28} - 7890q^{29} - 8593q^{31} - 1024q^{32} + 648q^{34} + 8642q^{37} - 6740q^{38} + 18168q^{41} + 14351q^{43} - 3552q^{44} + 1224q^{46} + 1098q^{47} - 14598q^{49} + 1616q^{52} - 17916q^{53} - 3008q^{56} + 31560q^{58} - 17610q^{59} - 21853q^{61} + 34372q^{62} + 4096q^{64} + 107q^{67} - 2592q^{68} + 40728q^{71} + 34706q^{73} - 34568q^{74} + 26960q^{76} - 10434q^{77} - 69160q^{79} - 72672q^{82} + 108534q^{83} - 57404q^{86} + 14208q^{88} - 35040q^{89} + 4747q^{91} - 4896q^{92} - 4392q^{94} - 823q^{97} + 58392q^{98} + O(q^{100}) \)

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 47.0000 −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.i 1
3.b odd 2 1 150.6.a.m yes 1
5.b even 2 1 450.6.a.p 1
5.c odd 4 2 450.6.c.e 2
15.d odd 2 1 150.6.a.a 1
15.e even 4 2 150.6.c.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.6.a.a 1 15.d odd 2 1
150.6.a.m yes 1 3.b odd 2 1
150.6.c.g 2 15.e even 4 2
450.6.a.i 1 1.a even 1 1 trivial
450.6.a.p 1 5.b even 2 1
450.6.c.e 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(450))\):

\( T_{7} - 47 \)
\( T_{11} + 222 \)
\( T_{17} + 162 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( -47 + T \)
$11$ \( 222 + T \)
$13$ \( -101 + T \)
$17$ \( 162 + T \)
$19$ \( -1685 + T \)
$23$ \( 306 + T \)
$29$ \( 7890 + T \)
$31$ \( 8593 + T \)
$37$ \( -8642 + T \)
$41$ \( -18168 + T \)
$43$ \( -14351 + T \)
$47$ \( -1098 + T \)
$53$ \( 17916 + T \)
$59$ \( 17610 + T \)
$61$ \( 21853 + T \)
$67$ \( -107 + T \)
$71$ \( -40728 + T \)
$73$ \( -34706 + T \)
$79$ \( 69160 + T \)
$83$ \( -108534 + T \)
$89$ \( 35040 + T \)
$97$ \( 823 + T \)
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