Properties

Label 450.6.a.o
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 90)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{2} + 16q^{4} - 98q^{7} + 64q^{8} + O(q^{10}) \) \( q + 4q^{2} + 16q^{4} - 98q^{7} + 64q^{8} - 354q^{11} - 404q^{13} - 392q^{14} + 256q^{16} + 654q^{17} + 1796q^{19} - 1416q^{22} - 1080q^{23} - 1616q^{26} - 1568q^{28} + 5754q^{29} + 10196q^{31} + 1024q^{32} + 2616q^{34} - 5552q^{37} + 7184q^{38} + 12960q^{41} + 8968q^{43} - 5664q^{44} - 4320q^{46} - 5400q^{47} - 7203q^{49} - 6464q^{52} + 8214q^{53} - 6272q^{56} + 23016q^{58} - 3954q^{59} + 962q^{61} + 40784q^{62} + 4096q^{64} + 17956q^{67} + 10464q^{68} + 56148q^{71} + 85690q^{73} - 22208q^{74} + 28736q^{76} + 34692q^{77} - 26044q^{79} + 51840q^{82} - 93468q^{83} + 35872q^{86} - 22656q^{88} - 73428q^{89} + 39592q^{91} - 17280q^{92} - 21600q^{94} - 128978q^{97} - 28812q^{98} + O(q^{100}) \)

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 −98.0000 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.o 1
3.b odd 2 1 450.6.a.d 1
5.b even 2 1 90.6.a.c 1
5.c odd 4 2 450.6.c.d 2
15.d odd 2 1 90.6.a.e yes 1
15.e even 4 2 450.6.c.l 2
20.d odd 2 1 720.6.a.o 1
60.h even 2 1 720.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.6.a.c 1 5.b even 2 1
90.6.a.e yes 1 15.d odd 2 1
450.6.a.d 1 3.b odd 2 1
450.6.a.o 1 1.a even 1 1 trivial
450.6.c.d 2 5.c odd 4 2
450.6.c.l 2 15.e even 4 2
720.6.a.c 1 60.h even 2 1
720.6.a.o 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(450))\):

\( T_{7} + 98 \)
\( T_{11} + 354 \)
\( T_{17} - 654 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( 98 + T \)
$11$ \( 354 + T \)
$13$ \( 404 + T \)
$17$ \( -654 + T \)
$19$ \( -1796 + T \)
$23$ \( 1080 + T \)
$29$ \( -5754 + T \)
$31$ \( -10196 + T \)
$37$ \( 5552 + T \)
$41$ \( -12960 + T \)
$43$ \( -8968 + T \)
$47$ \( 5400 + T \)
$53$ \( -8214 + T \)
$59$ \( 3954 + T \)
$61$ \( -962 + T \)
$67$ \( -17956 + T \)
$71$ \( -56148 + T \)
$73$ \( -85690 + T \)
$79$ \( 26044 + T \)
$83$ \( 93468 + T \)
$89$ \( 73428 + T \)
$97$ \( 128978 + T \)
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