Properties

Label 600.4.a.o
Level $600$
Weight $4$
Character orbit 600.a
Self dual yes
Analytic conductor $35.401$
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] = [600,4,Mod(1,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, 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("600.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 600.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: \(35.4011460034\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
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)
 
\(f(q)\) \(=\) \( q + 3 q^{3} + 10 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + 10 q^{7} + 9 q^{9} - 46 q^{11} - 34 q^{13} + 66 q^{17} + 104 q^{19} + 30 q^{21} + 164 q^{23} + 27 q^{27} + 224 q^{29} - 72 q^{31} - 138 q^{33} - 22 q^{37} - 102 q^{39} + 194 q^{41} + 108 q^{43} - 480 q^{47} - 243 q^{49} + 198 q^{51} + 286 q^{53} + 312 q^{57} + 426 q^{59} + 698 q^{61} + 90 q^{63} + 328 q^{67} + 492 q^{69} + 188 q^{71} - 740 q^{73} - 460 q^{77} + 1168 q^{79} + 81 q^{81} + 412 q^{83} + 672 q^{87} + 1206 q^{89} - 340 q^{91} - 216 q^{93} - 1384 q^{97} - 414 q^{99}+O(q^{100}) \) 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 10.0000 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 600.4.a.o 1
3.b odd 2 1 1800.4.a.y 1
4.b odd 2 1 1200.4.a.f 1
5.b even 2 1 600.4.a.b 1
5.c odd 4 2 120.4.f.a 2
15.d odd 2 1 1800.4.a.j 1
15.e even 4 2 360.4.f.c 2
20.d odd 2 1 1200.4.a.bh 1
20.e even 4 2 240.4.f.b 2
40.i odd 4 2 960.4.f.l 2
40.k even 4 2 960.4.f.g 2
60.l odd 4 2 720.4.f.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.f.a 2 5.c odd 4 2
240.4.f.b 2 20.e even 4 2
360.4.f.c 2 15.e even 4 2
600.4.a.b 1 5.b even 2 1
600.4.a.o 1 1.a even 1 1 trivial
720.4.f.g 2 60.l odd 4 2
960.4.f.g 2 40.k even 4 2
960.4.f.l 2 40.i odd 4 2
1200.4.a.f 1 4.b odd 2 1
1200.4.a.bh 1 20.d odd 2 1
1800.4.a.j 1 15.d odd 2 1
1800.4.a.y 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(600))\):

\( T_{7} - 10 \) Copy content Toggle raw display
\( T_{11} + 46 \) 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
$7$ \( T - 10 \) Copy content Toggle raw display
$11$ \( T + 46 \) Copy content Toggle raw display
$13$ \( T + 34 \) Copy content Toggle raw display
$17$ \( T - 66 \) Copy content Toggle raw display
$19$ \( T - 104 \) Copy content Toggle raw display
$23$ \( T - 164 \) Copy content Toggle raw display
$29$ \( T - 224 \) Copy content Toggle raw display
$31$ \( T + 72 \) Copy content Toggle raw display
$37$ \( T + 22 \) Copy content Toggle raw display
$41$ \( T - 194 \) Copy content Toggle raw display
$43$ \( T - 108 \) Copy content Toggle raw display
$47$ \( T + 480 \) Copy content Toggle raw display
$53$ \( T - 286 \) Copy content Toggle raw display
$59$ \( T - 426 \) Copy content Toggle raw display
$61$ \( T - 698 \) Copy content Toggle raw display
$67$ \( T - 328 \) Copy content Toggle raw display
$71$ \( T - 188 \) Copy content Toggle raw display
$73$ \( T + 740 \) Copy content Toggle raw display
$79$ \( T - 1168 \) Copy content Toggle raw display
$83$ \( T - 412 \) Copy content Toggle raw display
$89$ \( T - 1206 \) Copy content Toggle raw display
$97$ \( T + 1384 \) Copy content Toggle raw display
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