Properties

Label 1200.4.a.bb
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 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)
 
\(f(q)\) \(=\) \( q + 3 q^{3} + q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + q^{7} + 9 q^{9} - 42 q^{11} - 67 q^{13} + 54 q^{17} + 115 q^{19} + 3 q^{21} + 162 q^{23} + 27 q^{27} - 210 q^{29} + 193 q^{31} - 126 q^{33} - 286 q^{37} - 201 q^{39} + 12 q^{41} - 263 q^{43} - 414 q^{47} - 342 q^{49} + 162 q^{51} - 192 q^{53} + 345 q^{57} - 690 q^{59} - 733 q^{61} + 9 q^{63} - 299 q^{67} + 486 q^{69} + 228 q^{71} + 938 q^{73} - 42 q^{77} + 160 q^{79} + 81 q^{81} + 462 q^{83} - 630 q^{87} - 240 q^{89} - 67 q^{91} + 579 q^{93} - 511 q^{97} - 378 q^{99}+O(q^{100}) \) 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 1.00000 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.bb 1
4.b odd 2 1 150.4.a.a 1
5.b even 2 1 1200.4.a.i 1
5.c odd 4 2 1200.4.f.c 2
12.b even 2 1 450.4.a.o 1
20.d odd 2 1 150.4.a.h yes 1
20.e even 4 2 150.4.c.e 2
60.h even 2 1 450.4.a.f 1
60.l odd 4 2 450.4.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.4.a.a 1 4.b odd 2 1
150.4.a.h yes 1 20.d odd 2 1
150.4.c.e 2 20.e even 4 2
450.4.a.f 1 60.h even 2 1
450.4.a.o 1 12.b even 2 1
450.4.c.a 2 60.l odd 4 2
1200.4.a.i 1 5.b even 2 1
1200.4.a.bb 1 1.a even 1 1 trivial
1200.4.f.c 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1200))\):

\( T_{7} - 1 \) Copy content Toggle raw display
\( T_{11} + 42 \) 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
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T + 42 \) Copy content Toggle raw display
$13$ \( T + 67 \) Copy content Toggle raw display
$17$ \( T - 54 \) Copy content Toggle raw display
$19$ \( T - 115 \) Copy content Toggle raw display
$23$ \( T - 162 \) Copy content Toggle raw display
$29$ \( T + 210 \) Copy content Toggle raw display
$31$ \( T - 193 \) Copy content Toggle raw display
$37$ \( T + 286 \) Copy content Toggle raw display
$41$ \( T - 12 \) Copy content Toggle raw display
$43$ \( T + 263 \) Copy content Toggle raw display
$47$ \( T + 414 \) Copy content Toggle raw display
$53$ \( T + 192 \) Copy content Toggle raw display
$59$ \( T + 690 \) Copy content Toggle raw display
$61$ \( T + 733 \) Copy content Toggle raw display
$67$ \( T + 299 \) Copy content Toggle raw display
$71$ \( T - 228 \) Copy content Toggle raw display
$73$ \( T - 938 \) Copy content Toggle raw display
$79$ \( T - 160 \) Copy content Toggle raw display
$83$ \( T - 462 \) Copy content Toggle raw display
$89$ \( T + 240 \) Copy content Toggle raw display
$97$ \( T + 511 \) Copy content Toggle raw display
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