Properties

Label 1800.4.a.bf
Level $1800$
Weight $4$
Character orbit 1800.a
Self dual yes
Analytic conductor $106.203$
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] = [1800,4,Mod(1,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("1800.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1800.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: \(106.203438010\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 600)
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)
 
\(f(q)\) \(=\) \( q + 19 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 19 q^{7} - 22 q^{11} - q^{13} - 58 q^{17} - 53 q^{19} + 58 q^{23} - 22 q^{29} - 35 q^{31} + 270 q^{37} + 468 q^{41} + 431 q^{43} - 230 q^{47} + 18 q^{49} - 446 q^{59} + 127 q^{61} + 811 q^{67} - 36 q^{71} - 522 q^{73} - 418 q^{77} + 1368 q^{79} - 1138 q^{83} - 144 q^{89} - 19 q^{91} + 1079 q^{97}+O(q^{100}) \) 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
0 0 0 0 0 19.0000 0 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 1800.4.a.bf 1
3.b odd 2 1 600.4.a.g 1
5.b even 2 1 1800.4.a.g 1
5.c odd 4 2 1800.4.f.g 2
12.b even 2 1 1200.4.a.x 1
15.d odd 2 1 600.4.a.j yes 1
15.e even 4 2 600.4.f.h 2
60.h even 2 1 1200.4.a.n 1
60.l odd 4 2 1200.4.f.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.4.a.g 1 3.b odd 2 1
600.4.a.j yes 1 15.d odd 2 1
600.4.f.h 2 15.e even 4 2
1200.4.a.n 1 60.h even 2 1
1200.4.a.x 1 12.b even 2 1
1200.4.f.f 2 60.l odd 4 2
1800.4.a.g 1 5.b even 2 1
1800.4.a.bf 1 1.a even 1 1 trivial
1800.4.f.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1800))\):

\( T_{7} - 19 \) Copy content Toggle raw display
\( T_{11} + 22 \) Copy content Toggle raw display
\( T_{17} + 58 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 19 \) Copy content Toggle raw display
$11$ \( T + 22 \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T + 58 \) Copy content Toggle raw display
$19$ \( T + 53 \) Copy content Toggle raw display
$23$ \( T - 58 \) Copy content Toggle raw display
$29$ \( T + 22 \) Copy content Toggle raw display
$31$ \( T + 35 \) Copy content Toggle raw display
$37$ \( T - 270 \) Copy content Toggle raw display
$41$ \( T - 468 \) Copy content Toggle raw display
$43$ \( T - 431 \) Copy content Toggle raw display
$47$ \( T + 230 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T + 446 \) Copy content Toggle raw display
$61$ \( T - 127 \) Copy content Toggle raw display
$67$ \( T - 811 \) Copy content Toggle raw display
$71$ \( T + 36 \) Copy content Toggle raw display
$73$ \( T + 522 \) Copy content Toggle raw display
$79$ \( T - 1368 \) Copy content Toggle raw display
$83$ \( T + 1138 \) Copy content Toggle raw display
$89$ \( T + 144 \) Copy content Toggle raw display
$97$ \( T - 1079 \) Copy content Toggle raw display
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