Properties

Label 450.6.a.h
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 10)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4q^{2} + 16q^{4} + 22q^{7} - 64q^{8} + O(q^{10}) \) \( q - 4q^{2} + 16q^{4} + 22q^{7} - 64q^{8} + 768q^{11} + 46q^{13} - 88q^{14} + 256q^{16} + 378q^{17} + 1100q^{19} - 3072q^{22} - 1986q^{23} - 184q^{26} + 352q^{28} + 5610q^{29} - 3988q^{31} - 1024q^{32} - 1512q^{34} + 142q^{37} - 4400q^{38} - 1542q^{41} + 5026q^{43} + 12288q^{44} + 7944q^{46} + 24738q^{47} - 16323q^{49} + 736q^{52} - 14166q^{53} - 1408q^{56} - 22440q^{58} - 28380q^{59} + 5522q^{61} + 15952q^{62} + 4096q^{64} + 24742q^{67} + 6048q^{68} - 42372q^{71} + 52126q^{73} - 568q^{74} + 17600q^{76} + 16896q^{77} - 39640q^{79} + 6168q^{82} - 59826q^{83} - 20104q^{86} - 49152q^{88} - 57690q^{89} + 1012q^{91} - 31776q^{92} - 98952q^{94} + 144382q^{97} + 65292q^{98} + O(q^{100}) \)

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 22.0000 −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.h 1
3.b odd 2 1 50.6.a.g 1
5.b even 2 1 90.6.a.f 1
5.c odd 4 2 450.6.c.o 2
12.b even 2 1 400.6.a.a 1
15.d odd 2 1 10.6.a.a 1
15.e even 4 2 50.6.b.d 2
20.d odd 2 1 720.6.a.r 1
60.h even 2 1 80.6.a.h 1
60.l odd 4 2 400.6.c.a 2
105.g even 2 1 490.6.a.j 1
120.i odd 2 1 320.6.a.p 1
120.m even 2 1 320.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.a 1 15.d odd 2 1
50.6.a.g 1 3.b odd 2 1
50.6.b.d 2 15.e even 4 2
80.6.a.h 1 60.h even 2 1
90.6.a.f 1 5.b even 2 1
320.6.a.a 1 120.m even 2 1
320.6.a.p 1 120.i odd 2 1
400.6.a.a 1 12.b even 2 1
400.6.c.a 2 60.l odd 4 2
450.6.a.h 1 1.a even 1 1 trivial
450.6.c.o 2 5.c odd 4 2
490.6.a.j 1 105.g even 2 1
720.6.a.r 1 20.d odd 2 1

Hecke kernels

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

\( T_{7} - 22 \)
\( T_{11} - 768 \)
\( T_{17} - 378 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( -22 + T \)
$11$ \( -768 + T \)
$13$ \( -46 + T \)
$17$ \( -378 + T \)
$19$ \( -1100 + T \)
$23$ \( 1986 + T \)
$29$ \( -5610 + T \)
$31$ \( 3988 + T \)
$37$ \( -142 + T \)
$41$ \( 1542 + T \)
$43$ \( -5026 + T \)
$47$ \( -24738 + T \)
$53$ \( 14166 + T \)
$59$ \( 28380 + T \)
$61$ \( -5522 + T \)
$67$ \( -24742 + T \)
$71$ \( 42372 + T \)
$73$ \( -52126 + T \)
$79$ \( 39640 + T \)
$83$ \( 59826 + T \)
$89$ \( 57690 + T \)
$97$ \( -144382 + T \)
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