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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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: \(0\)
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 4 q^{2} + 16 q^{4} - 98 q^{7} + 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 16 q^{4} - 98 q^{7} + 64 q^{8} - 354 q^{11} - 404 q^{13} - 392 q^{14} + 256 q^{16} + 654 q^{17} + 1796 q^{19} - 1416 q^{22} - 1080 q^{23} - 1616 q^{26} - 1568 q^{28} + 5754 q^{29} + 10196 q^{31} + 1024 q^{32} + 2616 q^{34} - 5552 q^{37} + 7184 q^{38} + 12960 q^{41} + 8968 q^{43} - 5664 q^{44} - 4320 q^{46} - 5400 q^{47} - 7203 q^{49} - 6464 q^{52} + 8214 q^{53} - 6272 q^{56} + 23016 q^{58} - 3954 q^{59} + 962 q^{61} + 40784 q^{62} + 4096 q^{64} + 17956 q^{67} + 10464 q^{68} + 56148 q^{71} + 85690 q^{73} - 22208 q^{74} + 28736 q^{76} + 34692 q^{77} - 26044 q^{79} + 51840 q^{82} - 93468 q^{83} + 35872 q^{86} - 22656 q^{88} - 73428 q^{89} + 39592 q^{91} - 17280 q^{92} - 21600 q^{94} - 128978 q^{97} - 28812 q^{98}+O(q^{100}) \) 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 −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 \(S_{6}^{\mathrm{new}}(\Gamma_0(450))\):

\( T_{7} + 98 \) Copy content Toggle raw display
\( T_{11} + 354 \) Copy content Toggle raw display
\( T_{17} - 654 \) 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
$7$ \( T + 98 \) Copy content Toggle raw display
$11$ \( T + 354 \) Copy content Toggle raw display
$13$ \( T + 404 \) Copy content Toggle raw display
$17$ \( T - 654 \) Copy content Toggle raw display
$19$ \( T - 1796 \) Copy content Toggle raw display
$23$ \( T + 1080 \) Copy content Toggle raw display
$29$ \( T - 5754 \) Copy content Toggle raw display
$31$ \( T - 10196 \) Copy content Toggle raw display
$37$ \( T + 5552 \) Copy content Toggle raw display
$41$ \( T - 12960 \) Copy content Toggle raw display
$43$ \( T - 8968 \) Copy content Toggle raw display
$47$ \( T + 5400 \) Copy content Toggle raw display
$53$ \( T - 8214 \) Copy content Toggle raw display
$59$ \( T + 3954 \) Copy content Toggle raw display
$61$ \( T - 962 \) Copy content Toggle raw display
$67$ \( T - 17956 \) Copy content Toggle raw display
$71$ \( T - 56148 \) Copy content Toggle raw display
$73$ \( T - 85690 \) Copy content Toggle raw display
$79$ \( T + 26044 \) Copy content Toggle raw display
$83$ \( T + 93468 \) Copy content Toggle raw display
$89$ \( T + 73428 \) Copy content Toggle raw display
$97$ \( T + 128978 \) Copy content Toggle raw display
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