Properties

Label 450.6.a.n
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 50)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{2} + 16q^{4} - 142q^{7} + 64q^{8} + O(q^{10}) \) \( q + 4q^{2} + 16q^{4} - 142q^{7} + 64q^{8} - 777q^{11} + 884q^{13} - 568q^{14} + 256q^{16} + 27q^{17} + 1145q^{19} - 3108q^{22} - 1854q^{23} + 3536q^{26} - 2272q^{28} + 4920q^{29} + 1802q^{31} + 1024q^{32} + 108q^{34} + 13178q^{37} + 4580q^{38} + 15123q^{41} + 7844q^{43} - 12432q^{44} - 7416q^{46} + 6732q^{47} + 3357q^{49} + 14144q^{52} - 3414q^{53} - 9088q^{56} + 19680q^{58} - 33960q^{59} + 47402q^{61} + 7208q^{62} + 4096q^{64} - 13177q^{67} + 432q^{68} + 7548q^{71} - 59821q^{73} + 52712q^{74} + 18320q^{76} + 110334q^{77} + 75830q^{79} + 60492q^{82} - 46299q^{83} + 31376q^{86} - 49728q^{88} + 30585q^{89} - 125528q^{91} - 29664q^{92} + 26928q^{94} + 104018q^{97} + 13428q^{98} + O(q^{100}) \)

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 −142.000 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.n 1
3.b odd 2 1 50.6.a.a 1
5.b even 2 1 450.6.a.j 1
5.c odd 4 2 450.6.c.a 2
12.b even 2 1 400.6.a.j 1
15.d odd 2 1 50.6.a.f yes 1
15.e even 4 2 50.6.b.c 2
60.h even 2 1 400.6.a.e 1
60.l odd 4 2 400.6.c.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.6.a.a 1 3.b odd 2 1
50.6.a.f yes 1 15.d odd 2 1
50.6.b.c 2 15.e even 4 2
400.6.a.e 1 60.h even 2 1
400.6.a.j 1 12.b even 2 1
400.6.c.g 2 60.l odd 4 2
450.6.a.j 1 5.b even 2 1
450.6.a.n 1 1.a even 1 1 trivial
450.6.c.a 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(450))\):

\( T_{7} + 142 \)
\( T_{11} + 777 \)
\( T_{17} - 27 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( 142 + T \)
$11$ \( 777 + T \)
$13$ \( -884 + T \)
$17$ \( -27 + T \)
$19$ \( -1145 + T \)
$23$ \( 1854 + T \)
$29$ \( -4920 + T \)
$31$ \( -1802 + T \)
$37$ \( -13178 + T \)
$41$ \( -15123 + T \)
$43$ \( -7844 + T \)
$47$ \( -6732 + T \)
$53$ \( 3414 + T \)
$59$ \( 33960 + T \)
$61$ \( -47402 + T \)
$67$ \( 13177 + T \)
$71$ \( -7548 + T \)
$73$ \( 59821 + T \)
$79$ \( -75830 + T \)
$83$ \( 46299 + T \)
$89$ \( -30585 + T \)
$97$ \( -104018 + T \)
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