Properties

Label 450.4.a.l
Level $450$
Weight $4$
Character orbit 450.a
Self dual yes
Analytic conductor $26.551$
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] = [450,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(26.5508595026\)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} - 23 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} - 23 q^{7} + 8 q^{8} + 30 q^{11} - 29 q^{13} - 46 q^{14} + 16 q^{16} + 78 q^{17} + 149 q^{19} + 60 q^{22} + 150 q^{23} - 58 q^{26} - 92 q^{28} + 234 q^{29} - 217 q^{31} + 32 q^{32} + 156 q^{34} - 146 q^{37} + 298 q^{38} + 156 q^{41} + 433 q^{43} + 120 q^{44} + 300 q^{46} + 30 q^{47} + 186 q^{49} - 116 q^{52} - 552 q^{53} - 184 q^{56} + 468 q^{58} + 270 q^{59} + 275 q^{61} - 434 q^{62} + 64 q^{64} - 803 q^{67} + 312 q^{68} - 660 q^{71} + 646 q^{73} - 292 q^{74} + 596 q^{76} - 690 q^{77} + 992 q^{79} + 312 q^{82} - 846 q^{83} + 866 q^{86} + 240 q^{88} + 1488 q^{89} + 667 q^{91} + 600 q^{92} + 60 q^{94} + 319 q^{97} + 372 q^{98}+O(q^{100}) \) 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
2.00000 0 4.00000 0 0 −23.0000 8.00000 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 450.4.a.l 1
3.b odd 2 1 150.4.a.c 1
5.b even 2 1 450.4.a.i 1
5.c odd 4 2 450.4.c.h 2
12.b even 2 1 1200.4.a.r 1
15.d odd 2 1 150.4.a.g yes 1
15.e even 4 2 150.4.c.b 2
60.h even 2 1 1200.4.a.v 1
60.l odd 4 2 1200.4.f.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.4.a.c 1 3.b odd 2 1
150.4.a.g yes 1 15.d odd 2 1
150.4.c.b 2 15.e even 4 2
450.4.a.i 1 5.b even 2 1
450.4.a.l 1 1.a even 1 1 trivial
450.4.c.h 2 5.c odd 4 2
1200.4.a.r 1 12.b even 2 1
1200.4.a.v 1 60.h even 2 1
1200.4.f.q 2 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(450))\):

\( T_{7} + 23 \) Copy content Toggle raw display
\( T_{11} - 30 \) Copy content Toggle raw display
\( T_{17} - 78 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 23 \) Copy content Toggle raw display
$11$ \( T - 30 \) Copy content Toggle raw display
$13$ \( T + 29 \) Copy content Toggle raw display
$17$ \( T - 78 \) Copy content Toggle raw display
$19$ \( T - 149 \) Copy content Toggle raw display
$23$ \( T - 150 \) Copy content Toggle raw display
$29$ \( T - 234 \) Copy content Toggle raw display
$31$ \( T + 217 \) Copy content Toggle raw display
$37$ \( T + 146 \) Copy content Toggle raw display
$41$ \( T - 156 \) Copy content Toggle raw display
$43$ \( T - 433 \) Copy content Toggle raw display
$47$ \( T - 30 \) Copy content Toggle raw display
$53$ \( T + 552 \) Copy content Toggle raw display
$59$ \( T - 270 \) Copy content Toggle raw display
$61$ \( T - 275 \) Copy content Toggle raw display
$67$ \( T + 803 \) Copy content Toggle raw display
$71$ \( T + 660 \) Copy content Toggle raw display
$73$ \( T - 646 \) Copy content Toggle raw display
$79$ \( T - 992 \) Copy content Toggle raw display
$83$ \( T + 846 \) Copy content Toggle raw display
$89$ \( T - 1488 \) Copy content Toggle raw display
$97$ \( T - 319 \) Copy content Toggle raw display
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