Properties

Label 450.6.a.a
Level $450$
Weight $6$
Character orbit 450.a
Self dual yes
Analytic conductor $72.173$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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 150)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4q^{2} + 16q^{4} - 233q^{7} - 64q^{8} + O(q^{10}) \) \( q - 4q^{2} + 16q^{4} - 233q^{7} - 64q^{8} + 498q^{11} - 809q^{13} + 932q^{14} + 256q^{16} - 1002q^{17} - 1705q^{19} - 1992q^{22} + 1554q^{23} + 3236q^{26} - 3728q^{28} - 7830q^{29} + 977q^{31} - 1024q^{32} + 4008q^{34} + 4822q^{37} + 6820q^{38} + 8148q^{41} - 19469q^{43} + 7968q^{44} - 6216q^{46} + 8418q^{47} + 37482q^{49} - 12944q^{52} + 17664q^{53} + 14912q^{56} + 31320q^{58} - 35910q^{59} + 3527q^{61} - 3908q^{62} + 4096q^{64} - 57473q^{67} - 16032q^{68} + 7548q^{71} + 646q^{73} - 19288q^{74} - 27280q^{76} - 116034q^{77} - 22720q^{79} - 32592q^{82} + 11574q^{83} + 77876q^{86} - 31872q^{88} + 78960q^{89} + 188497q^{91} + 24864q^{92} - 33672q^{94} - 54593q^{97} - 149928q^{98} + O(q^{100}) \)

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 −233.000 −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.a 1
3.b odd 2 1 150.6.a.l yes 1
5.b even 2 1 450.6.a.x 1
5.c odd 4 2 450.6.c.n 2
15.d odd 2 1 150.6.a.c 1
15.e even 4 2 150.6.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.6.a.c 1 15.d odd 2 1
150.6.a.l yes 1 3.b odd 2 1
150.6.c.e 2 15.e even 4 2
450.6.a.a 1 1.a even 1 1 trivial
450.6.a.x 1 5.b even 2 1
450.6.c.n 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(450))\):

\( T_{7} + 233 \)
\( T_{11} - 498 \)
\( T_{17} + 1002 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( 233 + T \)
$11$ \( -498 + T \)
$13$ \( 809 + T \)
$17$ \( 1002 + T \)
$19$ \( 1705 + T \)
$23$ \( -1554 + T \)
$29$ \( 7830 + T \)
$31$ \( -977 + T \)
$37$ \( -4822 + T \)
$41$ \( -8148 + T \)
$43$ \( 19469 + T \)
$47$ \( -8418 + T \)
$53$ \( -17664 + T \)
$59$ \( 35910 + T \)
$61$ \( -3527 + T \)
$67$ \( 57473 + T \)
$71$ \( -7548 + T \)
$73$ \( -646 + T \)
$79$ \( 22720 + T \)
$83$ \( -11574 + T \)
$89$ \( -78960 + T \)
$97$ \( 54593 + T \)
show more
show less