Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [450,6,Mod(1,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(72.1727189158\)
Analytic rank: \(1\)
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 4 q^{2} + 16 q^{4} + 79 q^{7} + 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 16 q^{4} + 79 q^{7} + 64 q^{8} - 150 q^{11} - 137 q^{13} + 316 q^{14} + 256 q^{16} - 2034 q^{17} - 1969 q^{19} - 600 q^{22} - 1350 q^{23} - 548 q^{26} + 1264 q^{28} + 2946 q^{29} + 713 q^{31} + 1024 q^{32} - 8136 q^{34} + 3238 q^{37} - 7876 q^{38} - 6564 q^{41} + 19579 q^{43} - 2400 q^{44} - 5400 q^{46} - 21150 q^{47} - 10566 q^{49} - 2192 q^{52} - 25896 q^{53} + 5056 q^{56} + 11784 q^{58} - 25350 q^{59} + 50615 q^{61} + 2852 q^{62} + 4096 q^{64} + 22519 q^{67} - 32544 q^{68} - 33900 q^{71} - 82442 q^{73} + 12952 q^{74} - 31504 q^{76} - 11850 q^{77} - 81472 q^{79} - 26256 q^{82} + 25782 q^{83} + 78316 q^{86} - 9600 q^{88} - 103728 q^{89} - 10823 q^{91} - 21600 q^{92} - 84600 q^{94} + 57343 q^{97} - 42264 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 79.0000 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.t 1
3.b odd 2 1 150.6.a.g 1
5.b even 2 1 450.6.a.e 1
5.c odd 4 2 450.6.c.g 2
15.d odd 2 1 150.6.a.i yes 1
15.e even 4 2 150.6.c.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.6.a.g 1 3.b odd 2 1
150.6.a.i yes 1 15.d odd 2 1
150.6.c.c 2 15.e even 4 2
450.6.a.e 1 5.b even 2 1
450.6.a.t 1 1.a even 1 1 trivial
450.6.c.g 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(450))\):

\( T_{7} - 79 \) Copy content Toggle raw display
\( T_{11} + 150 \) Copy content Toggle raw display
\( T_{17} + 2034 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 79 \) Copy content Toggle raw display
$11$ \( T + 150 \) Copy content Toggle raw display
$13$ \( T + 137 \) Copy content Toggle raw display
$17$ \( T + 2034 \) Copy content Toggle raw display
$19$ \( T + 1969 \) Copy content Toggle raw display
$23$ \( T + 1350 \) Copy content Toggle raw display
$29$ \( T - 2946 \) Copy content Toggle raw display
$31$ \( T - 713 \) Copy content Toggle raw display
$37$ \( T - 3238 \) Copy content Toggle raw display
$41$ \( T + 6564 \) Copy content Toggle raw display
$43$ \( T - 19579 \) Copy content Toggle raw display
$47$ \( T + 21150 \) Copy content Toggle raw display
$53$ \( T + 25896 \) Copy content Toggle raw display
$59$ \( T + 25350 \) Copy content Toggle raw display
$61$ \( T - 50615 \) Copy content Toggle raw display
$67$ \( T - 22519 \) Copy content Toggle raw display
$71$ \( T + 33900 \) Copy content Toggle raw display
$73$ \( T + 82442 \) Copy content Toggle raw display
$79$ \( T + 81472 \) Copy content Toggle raw display
$83$ \( T - 25782 \) Copy content Toggle raw display
$89$ \( T + 103728 \) Copy content Toggle raw display
$97$ \( T - 57343 \) Copy content Toggle raw display
show more
show less