Properties

Label 450.6.a.x
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 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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} + 16 q^{4} + 233 q^{7} + 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 16 q^{4} + 233 q^{7} + 64 q^{8} + 498 q^{11} + 809 q^{13} + 932 q^{14} + 256 q^{16} + 1002 q^{17} - 1705 q^{19} + 1992 q^{22} - 1554 q^{23} + 3236 q^{26} + 3728 q^{28} - 7830 q^{29} + 977 q^{31} + 1024 q^{32} + 4008 q^{34} - 4822 q^{37} - 6820 q^{38} + 8148 q^{41} + 19469 q^{43} + 7968 q^{44} - 6216 q^{46} - 8418 q^{47} + 37482 q^{49} + 12944 q^{52} - 17664 q^{53} + 14912 q^{56} - 31320 q^{58} - 35910 q^{59} + 3527 q^{61} + 3908 q^{62} + 4096 q^{64} + 57473 q^{67} + 16032 q^{68} + 7548 q^{71} - 646 q^{73} - 19288 q^{74} - 27280 q^{76} + 116034 q^{77} - 22720 q^{79} + 32592 q^{82} - 11574 q^{83} + 77876 q^{86} + 31872 q^{88} + 78960 q^{89} + 188497 q^{91} - 24864 q^{92} - 33672 q^{94} + 54593 q^{97} + 149928 q^{98}+O(q^{100}) \) 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 0 233.000 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.x 1
3.b odd 2 1 150.6.a.c 1
5.b even 2 1 450.6.a.a 1
5.c odd 4 2 450.6.c.n 2
15.d odd 2 1 150.6.a.l yes 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 3.b odd 2 1
150.6.a.l yes 1 15.d odd 2 1
150.6.c.e 2 15.e even 4 2
450.6.a.a 1 5.b even 2 1
450.6.a.x 1 1.a even 1 1 trivial
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 \(S_{6}^{\mathrm{new}}(\Gamma_0(450))\):

\( T_{7} - 233 \) Copy content Toggle raw display
\( T_{11} - 498 \) Copy content Toggle raw display
\( T_{17} - 1002 \) 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
$7$ \( T - 233 \) Copy content Toggle raw display
$11$ \( T - 498 \) Copy content Toggle raw display
$13$ \( T - 809 \) Copy content Toggle raw display
$17$ \( T - 1002 \) Copy content Toggle raw display
$19$ \( T + 1705 \) Copy content Toggle raw display
$23$ \( T + 1554 \) Copy content Toggle raw display
$29$ \( T + 7830 \) Copy content Toggle raw display
$31$ \( T - 977 \) Copy content Toggle raw display
$37$ \( T + 4822 \) Copy content Toggle raw display
$41$ \( T - 8148 \) Copy content Toggle raw display
$43$ \( T - 19469 \) Copy content Toggle raw display
$47$ \( T + 8418 \) Copy content Toggle raw display
$53$ \( T + 17664 \) Copy content Toggle raw display
$59$ \( T + 35910 \) Copy content Toggle raw display
$61$ \( T - 3527 \) Copy content Toggle raw display
$67$ \( T - 57473 \) Copy content Toggle raw display
$71$ \( T - 7548 \) Copy content Toggle raw display
$73$ \( T + 646 \) Copy content Toggle raw display
$79$ \( T + 22720 \) Copy content Toggle raw display
$83$ \( T + 11574 \) Copy content Toggle raw display
$89$ \( T - 78960 \) Copy content Toggle raw display
$97$ \( T - 54593 \) Copy content Toggle raw display
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