Properties

Label 1200.3.l.e
Level $1200$
Weight $3$
Character orbit 1200.l
Self dual yes
Analytic conductor $32.698$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,3,Mod(401,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.401");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.l (of order \(2\), degree \(1\), not minimal)

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: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 3 q^{3} + 13 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + 13 q^{7} + 9 q^{9} + 23 q^{13} - 11 q^{19} + 39 q^{21} + 27 q^{27} - 59 q^{31} + 26 q^{37} + 69 q^{39} - 83 q^{43} + 120 q^{49} - 33 q^{57} - 121 q^{61} + 117 q^{63} + 13 q^{67} - 46 q^{73} + 142 q^{79} + 81 q^{81} + 299 q^{91} - 177 q^{93} + 167 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
0
0 3.00000 0 0 0 13.0000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.l.e 1
3.b odd 2 1 CM 1200.3.l.e 1
4.b odd 2 1 300.3.g.a 1
5.b even 2 1 1200.3.l.a 1
5.c odd 4 2 1200.3.c.b 2
12.b even 2 1 300.3.g.a 1
15.d odd 2 1 1200.3.l.a 1
15.e even 4 2 1200.3.c.b 2
20.d odd 2 1 300.3.g.c yes 1
20.e even 4 2 300.3.b.b 2
60.h even 2 1 300.3.g.c yes 1
60.l odd 4 2 300.3.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.b.b 2 20.e even 4 2
300.3.b.b 2 60.l odd 4 2
300.3.g.a 1 4.b odd 2 1
300.3.g.a 1 12.b even 2 1
300.3.g.c yes 1 20.d odd 2 1
300.3.g.c yes 1 60.h even 2 1
1200.3.c.b 2 5.c odd 4 2
1200.3.c.b 2 15.e even 4 2
1200.3.l.a 1 5.b even 2 1
1200.3.l.a 1 15.d odd 2 1
1200.3.l.e 1 1.a even 1 1 trivial
1200.3.l.e 1 3.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7} - 13 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} - 23 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 13 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 23 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T + 11 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 59 \) Copy content Toggle raw display
$37$ \( T - 26 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 83 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 121 \) Copy content Toggle raw display
$67$ \( T - 13 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 46 \) Copy content Toggle raw display
$79$ \( T - 142 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 167 \) Copy content Toggle raw display
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