Properties

Label 450.6.a.l
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

\(f(q)\) \(=\) \( q - 4q^{2} + 16q^{4} + 172q^{7} - 64q^{8} + O(q^{10}) \) \( q - 4q^{2} + 16q^{4} + 172q^{7} - 64q^{8} - 132q^{11} + 946q^{13} - 688q^{14} + 256q^{16} - 222q^{17} + 500q^{19} + 528q^{22} + 3564q^{23} - 3784q^{26} + 2752q^{28} - 2190q^{29} + 2312q^{31} - 1024q^{32} + 888q^{34} + 11242q^{37} - 2000q^{38} - 1242q^{41} - 20624q^{43} - 2112q^{44} - 14256q^{46} + 6588q^{47} + 12777q^{49} + 15136q^{52} - 21066q^{53} - 11008q^{56} + 8760q^{58} - 7980q^{59} + 16622q^{61} - 9248q^{62} + 4096q^{64} - 1808q^{67} - 3552q^{68} + 24528q^{71} - 20474q^{73} - 44968q^{74} + 8000q^{76} - 22704q^{77} - 46240q^{79} + 4968q^{82} - 51576q^{83} + 82496q^{86} + 8448q^{88} + 110310q^{89} + 162712q^{91} + 57024q^{92} - 26352q^{94} + 78382q^{97} - 51108q^{98} + O(q^{100}) \)

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 0 172.000 −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.l 1
3.b odd 2 1 50.6.a.d 1
5.b even 2 1 90.6.a.d 1
5.c odd 4 2 450.6.c.h 2
12.b even 2 1 400.6.a.n 1
15.d odd 2 1 10.6.a.b 1
15.e even 4 2 50.6.b.a 2
20.d odd 2 1 720.6.a.j 1
60.h even 2 1 80.6.a.a 1
60.l odd 4 2 400.6.c.b 2
105.g even 2 1 490.6.a.a 1
120.i odd 2 1 320.6.a.b 1
120.m even 2 1 320.6.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.b 1 15.d odd 2 1
50.6.a.d 1 3.b odd 2 1
50.6.b.a 2 15.e even 4 2
80.6.a.a 1 60.h even 2 1
90.6.a.d 1 5.b even 2 1
320.6.a.b 1 120.i odd 2 1
320.6.a.o 1 120.m even 2 1
400.6.a.n 1 12.b even 2 1
400.6.c.b 2 60.l odd 4 2
450.6.a.l 1 1.a even 1 1 trivial
450.6.c.h 2 5.c odd 4 2
490.6.a.a 1 105.g even 2 1
720.6.a.j 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(450))\):

\( T_{7} - 172 \)
\( T_{11} + 132 \)
\( T_{17} + 222 \)