Properties

Label 450.6.a.b
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 30)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4q^{2} + 16q^{4} - 164q^{7} - 64q^{8} + O(q^{10}) \) \( q - 4q^{2} + 16q^{4} - 164q^{7} - 64q^{8} - 720q^{11} - 698q^{13} + 656q^{14} + 256q^{16} - 2226q^{17} + 356q^{19} + 2880q^{22} - 1800q^{23} + 2792q^{26} - 2624q^{28} - 714q^{29} + 848q^{31} - 1024q^{32} + 8904q^{34} + 11302q^{37} - 1424q^{38} - 9354q^{41} + 5956q^{43} - 11520q^{44} + 7200q^{46} - 11160q^{47} + 10089q^{49} - 11168q^{52} + 14106q^{53} + 10496q^{56} + 2856q^{58} - 7920q^{59} - 13450q^{61} - 3392q^{62} + 4096q^{64} + 65476q^{67} - 35616q^{68} - 34560q^{71} - 86258q^{73} - 45208q^{74} + 5696q^{76} + 118080q^{77} - 108832q^{79} + 37416q^{82} + 10668q^{83} - 23824q^{86} + 46080q^{88} - 10818q^{89} + 114472q^{91} - 28800q^{92} + 44640q^{94} - 4418q^{97} - 40356q^{98} + O(q^{100}) \)

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 −164.000 −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.b 1
3.b odd 2 1 150.6.a.h 1
5.b even 2 1 90.6.a.g 1
5.c odd 4 2 450.6.c.b 2
15.d odd 2 1 30.6.a.a 1
15.e even 4 2 150.6.c.d 2
20.d odd 2 1 720.6.a.m 1
60.h even 2 1 240.6.a.a 1
120.i odd 2 1 960.6.a.n 1
120.m even 2 1 960.6.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.a.a 1 15.d odd 2 1
90.6.a.g 1 5.b even 2 1
150.6.a.h 1 3.b odd 2 1
150.6.c.d 2 15.e even 4 2
240.6.a.a 1 60.h even 2 1
450.6.a.b 1 1.a even 1 1 trivial
450.6.c.b 2 5.c odd 4 2
720.6.a.m 1 20.d odd 2 1
960.6.a.n 1 120.i odd 2 1
960.6.a.u 1 120.m even 2 1

Hecke kernels

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

\( T_{7} + 164 \)
\( T_{11} + 720 \)
\( T_{17} + 2226 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( 164 + T \)
$11$ \( 720 + T \)
$13$ \( 698 + T \)
$17$ \( 2226 + T \)
$19$ \( -356 + T \)
$23$ \( 1800 + T \)
$29$ \( 714 + T \)
$31$ \( -848 + T \)
$37$ \( -11302 + T \)
$41$ \( 9354 + T \)
$43$ \( -5956 + T \)
$47$ \( 11160 + T \)
$53$ \( -14106 + T \)
$59$ \( 7920 + T \)
$61$ \( 13450 + T \)
$67$ \( -65476 + T \)
$71$ \( 34560 + T \)
$73$ \( 86258 + T \)
$79$ \( 108832 + T \)
$83$ \( -10668 + T \)
$89$ \( 10818 + T \)
$97$ \( 4418 + T \)
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