Properties

Label 1200.4.a.g
Level $1200$
Weight $4$
Character orbit 1200.a
Self dual yes
Analytic conductor $70.802$
Analytic rank $1$
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] = [1200,4,Mod(1,1200)]
 
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, 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("1200.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.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: \(70.8022920069\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \( q - 3 q^{3} - 5 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} - 5 q^{7} + 9 q^{9} - 14 q^{11} + q^{13} + 46 q^{17} - 19 q^{19} + 15 q^{21} + 46 q^{23} - 27 q^{27} + 14 q^{29} - 133 q^{31} + 42 q^{33} + 258 q^{37} - 3 q^{39} + 84 q^{41} + 167 q^{43} - 410 q^{47} - 318 q^{49} - 138 q^{51} + 456 q^{53} + 57 q^{57} + 194 q^{59} - 17 q^{61} - 45 q^{63} - 653 q^{67} - 138 q^{69} - 828 q^{71} + 570 q^{73} + 70 q^{77} + 552 q^{79} + 81 q^{81} - 142 q^{83} - 42 q^{87} - 1104 q^{89} - 5 q^{91} + 399 q^{93} + 841 q^{97} - 126 q^{99}+O(q^{100}) \) 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 −3.00000 0 0 0 −5.00000 0 9.00000 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 1200.4.a.g 1
4.b odd 2 1 600.4.a.n yes 1
5.b even 2 1 1200.4.a.bd 1
5.c odd 4 2 1200.4.f.i 2
12.b even 2 1 1800.4.a.v 1
20.d odd 2 1 600.4.a.e 1
20.e even 4 2 600.4.f.e 2
60.h even 2 1 1800.4.a.m 1
60.l odd 4 2 1800.4.f.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.4.a.e 1 20.d odd 2 1
600.4.a.n yes 1 4.b odd 2 1
600.4.f.e 2 20.e even 4 2
1200.4.a.g 1 1.a even 1 1 trivial
1200.4.a.bd 1 5.b even 2 1
1200.4.f.i 2 5.c odd 4 2
1800.4.a.m 1 60.h even 2 1
1800.4.a.v 1 12.b even 2 1
1800.4.f.l 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1200))\):

\( T_{7} + 5 \) Copy content Toggle raw display
\( T_{11} + 14 \) 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
$7$ \( T + 5 \) Copy content Toggle raw display
$11$ \( T + 14 \) Copy content Toggle raw display
$13$ \( T - 1 \) Copy content Toggle raw display
$17$ \( T - 46 \) Copy content Toggle raw display
$19$ \( T + 19 \) Copy content Toggle raw display
$23$ \( T - 46 \) Copy content Toggle raw display
$29$ \( T - 14 \) Copy content Toggle raw display
$31$ \( T + 133 \) Copy content Toggle raw display
$37$ \( T - 258 \) Copy content Toggle raw display
$41$ \( T - 84 \) Copy content Toggle raw display
$43$ \( T - 167 \) Copy content Toggle raw display
$47$ \( T + 410 \) Copy content Toggle raw display
$53$ \( T - 456 \) Copy content Toggle raw display
$59$ \( T - 194 \) Copy content Toggle raw display
$61$ \( T + 17 \) Copy content Toggle raw display
$67$ \( T + 653 \) Copy content Toggle raw display
$71$ \( T + 828 \) Copy content Toggle raw display
$73$ \( T - 570 \) Copy content Toggle raw display
$79$ \( T - 552 \) Copy content Toggle raw display
$83$ \( T + 142 \) Copy content Toggle raw display
$89$ \( T + 1104 \) Copy content Toggle raw display
$97$ \( T - 841 \) Copy content Toggle raw display
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