Properties

Label 1200.2.a.d
Level $1200$
Weight $2$
Character orbit 1200.a
Self dual yes
Analytic conductor $9.582$
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] = [1200,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
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: \(9.58204824255\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
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)
 
\(f(q)\) \(=\) \( q - q^{3} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{9} - 4 q^{11} + 2 q^{13} - 2 q^{17} + 4 q^{19} - 8 q^{23} - q^{27} + 6 q^{29} - 8 q^{31} + 4 q^{33} - 6 q^{37} - 2 q^{39} - 6 q^{41} + 4 q^{43} - 7 q^{49} + 2 q^{51} + 2 q^{53} - 4 q^{57} - 4 q^{59} - 2 q^{61} - 4 q^{67} + 8 q^{69} - 8 q^{71} - 10 q^{73} + 8 q^{79} + q^{81} - 4 q^{83} - 6 q^{87} - 6 q^{89} + 8 q^{93} - 2 q^{97} - 4 q^{99}+O(q^{100}) \) 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 −1.00000 0 0 0 0 0 1.00000 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 1200.2.a.d 1
3.b odd 2 1 3600.2.a.v 1
4.b odd 2 1 600.2.a.h 1
5.b even 2 1 48.2.a.a 1
5.c odd 4 2 1200.2.f.b 2
8.b even 2 1 4800.2.a.cc 1
8.d odd 2 1 4800.2.a.q 1
12.b even 2 1 1800.2.a.m 1
15.d odd 2 1 144.2.a.b 1
15.e even 4 2 3600.2.f.r 2
20.d odd 2 1 24.2.a.a 1
20.e even 4 2 600.2.f.e 2
35.c odd 2 1 2352.2.a.i 1
35.i odd 6 2 2352.2.q.r 2
35.j even 6 2 2352.2.q.l 2
40.e odd 2 1 192.2.a.d 1
40.f even 2 1 192.2.a.b 1
40.i odd 4 2 4800.2.f.bg 2
40.k even 4 2 4800.2.f.d 2
45.h odd 6 2 1296.2.i.e 2
45.j even 6 2 1296.2.i.m 2
55.d odd 2 1 5808.2.a.s 1
60.h even 2 1 72.2.a.a 1
60.l odd 4 2 1800.2.f.c 2
65.d even 2 1 8112.2.a.be 1
80.k odd 4 2 768.2.d.e 2
80.q even 4 2 768.2.d.d 2
105.g even 2 1 7056.2.a.q 1
120.i odd 2 1 576.2.a.b 1
120.m even 2 1 576.2.a.d 1
140.c even 2 1 1176.2.a.i 1
140.p odd 6 2 1176.2.q.i 2
140.s even 6 2 1176.2.q.a 2
180.n even 6 2 648.2.i.b 2
180.p odd 6 2 648.2.i.g 2
220.g even 2 1 2904.2.a.c 1
240.t even 4 2 2304.2.d.i 2
240.bm odd 4 2 2304.2.d.k 2
260.g odd 2 1 4056.2.a.i 1
260.u even 4 2 4056.2.c.e 2
280.c odd 2 1 9408.2.a.cc 1
280.n even 2 1 9408.2.a.h 1
340.d odd 2 1 6936.2.a.p 1
380.d even 2 1 8664.2.a.j 1
420.o odd 2 1 3528.2.a.d 1
420.ba even 6 2 3528.2.s.j 2
420.be odd 6 2 3528.2.s.y 2
660.g odd 2 1 8712.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 20.d odd 2 1
48.2.a.a 1 5.b even 2 1
72.2.a.a 1 60.h even 2 1
144.2.a.b 1 15.d odd 2 1
192.2.a.b 1 40.f even 2 1
192.2.a.d 1 40.e odd 2 1
576.2.a.b 1 120.i odd 2 1
576.2.a.d 1 120.m even 2 1
600.2.a.h 1 4.b odd 2 1
600.2.f.e 2 20.e even 4 2
648.2.i.b 2 180.n even 6 2
648.2.i.g 2 180.p odd 6 2
768.2.d.d 2 80.q even 4 2
768.2.d.e 2 80.k odd 4 2
1176.2.a.i 1 140.c even 2 1
1176.2.q.a 2 140.s even 6 2
1176.2.q.i 2 140.p odd 6 2
1200.2.a.d 1 1.a even 1 1 trivial
1200.2.f.b 2 5.c odd 4 2
1296.2.i.e 2 45.h odd 6 2
1296.2.i.m 2 45.j even 6 2
1800.2.a.m 1 12.b even 2 1
1800.2.f.c 2 60.l odd 4 2
2304.2.d.i 2 240.t even 4 2
2304.2.d.k 2 240.bm odd 4 2
2352.2.a.i 1 35.c odd 2 1
2352.2.q.l 2 35.j even 6 2
2352.2.q.r 2 35.i odd 6 2
2904.2.a.c 1 220.g even 2 1
3528.2.a.d 1 420.o odd 2 1
3528.2.s.j 2 420.ba even 6 2
3528.2.s.y 2 420.be odd 6 2
3600.2.a.v 1 3.b odd 2 1
3600.2.f.r 2 15.e even 4 2
4056.2.a.i 1 260.g odd 2 1
4056.2.c.e 2 260.u even 4 2
4800.2.a.q 1 8.d odd 2 1
4800.2.a.cc 1 8.b even 2 1
4800.2.f.d 2 40.k even 4 2
4800.2.f.bg 2 40.i odd 4 2
5808.2.a.s 1 55.d odd 2 1
6936.2.a.p 1 340.d odd 2 1
7056.2.a.q 1 105.g even 2 1
8112.2.a.be 1 65.d even 2 1
8664.2.a.j 1 380.d even 2 1
8712.2.a.u 1 660.g odd 2 1
9408.2.a.h 1 280.n even 2 1
9408.2.a.cc 1 280.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1200))\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 4 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T + 8 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T + 6 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 2 \) Copy content Toggle raw display
$59$ \( T + 4 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T + 8 \) Copy content Toggle raw display
$73$ \( T + 10 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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