Properties

Label 3600.2.a.u
Level $3600$
Weight $2$
Character orbit 3600.a
Self dual yes
Analytic conductor $28.746$
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] = [3600,2,Mod(1,3600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3600, 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("3600.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3600.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: \(28.7461447277\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
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)
 
\(f(q)\) \(=\) \( q+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{11} + 2 q^{13} + 2 q^{17} - 4 q^{19} + 2 q^{29} + 10 q^{37} - 10 q^{41} + 4 q^{43} - 8 q^{47} - 7 q^{49} - 10 q^{53} - 4 q^{59} - 2 q^{61} + 12 q^{67} - 8 q^{71} - 10 q^{73} - 12 q^{83} + 6 q^{89} - 2 q^{97}+O(q^{100}) \) 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 0 0 0 0 0 0 0 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 3600.2.a.u 1
3.b odd 2 1 1200.2.a.e 1
4.b odd 2 1 225.2.a.b 1
5.b even 2 1 720.2.a.c 1
5.c odd 4 2 3600.2.f.e 2
12.b even 2 1 75.2.a.b 1
15.d odd 2 1 240.2.a.d 1
15.e even 4 2 1200.2.f.h 2
20.d odd 2 1 45.2.a.a 1
20.e even 4 2 225.2.b.b 2
24.f even 2 1 4800.2.a.t 1
24.h odd 2 1 4800.2.a.bz 1
40.e odd 2 1 2880.2.a.y 1
40.f even 2 1 2880.2.a.bc 1
60.h even 2 1 15.2.a.a 1
60.l odd 4 2 75.2.b.b 2
84.h odd 2 1 3675.2.a.j 1
120.i odd 2 1 960.2.a.a 1
120.m even 2 1 960.2.a.l 1
120.q odd 4 2 4800.2.f.bf 2
120.w even 4 2 4800.2.f.c 2
132.d odd 2 1 9075.2.a.g 1
140.c even 2 1 2205.2.a.i 1
180.n even 6 2 405.2.e.f 2
180.p odd 6 2 405.2.e.c 2
220.g even 2 1 5445.2.a.c 1
240.t even 4 2 3840.2.k.m 2
240.bm odd 4 2 3840.2.k.r 2
260.g odd 2 1 7605.2.a.g 1
420.o odd 2 1 735.2.a.c 1
420.ba even 6 2 735.2.i.e 2
420.be odd 6 2 735.2.i.d 2
660.g odd 2 1 1815.2.a.d 1
780.d even 2 1 2535.2.a.j 1
1020.b even 2 1 4335.2.a.c 1
1140.p odd 2 1 5415.2.a.j 1
1380.b odd 2 1 7935.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 60.h even 2 1
45.2.a.a 1 20.d odd 2 1
75.2.a.b 1 12.b even 2 1
75.2.b.b 2 60.l odd 4 2
225.2.a.b 1 4.b odd 2 1
225.2.b.b 2 20.e even 4 2
240.2.a.d 1 15.d odd 2 1
405.2.e.c 2 180.p odd 6 2
405.2.e.f 2 180.n even 6 2
720.2.a.c 1 5.b even 2 1
735.2.a.c 1 420.o odd 2 1
735.2.i.d 2 420.be odd 6 2
735.2.i.e 2 420.ba even 6 2
960.2.a.a 1 120.i odd 2 1
960.2.a.l 1 120.m even 2 1
1200.2.a.e 1 3.b odd 2 1
1200.2.f.h 2 15.e even 4 2
1815.2.a.d 1 660.g odd 2 1
2205.2.a.i 1 140.c even 2 1
2535.2.a.j 1 780.d even 2 1
2880.2.a.y 1 40.e odd 2 1
2880.2.a.bc 1 40.f even 2 1
3600.2.a.u 1 1.a even 1 1 trivial
3600.2.f.e 2 5.c odd 4 2
3675.2.a.j 1 84.h odd 2 1
3840.2.k.m 2 240.t even 4 2
3840.2.k.r 2 240.bm odd 4 2
4335.2.a.c 1 1020.b even 2 1
4800.2.a.t 1 24.f even 2 1
4800.2.a.bz 1 24.h odd 2 1
4800.2.f.c 2 120.w even 4 2
4800.2.f.bf 2 120.q odd 4 2
5415.2.a.j 1 1140.p odd 2 1
5445.2.a.c 1 220.g even 2 1
7605.2.a.g 1 260.g odd 2 1
7935.2.a.d 1 1380.b odd 2 1
9075.2.a.g 1 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3600))\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display
\( T_{17} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) 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
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 10 \) Copy content Toggle raw display
$41$ \( T + 10 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T + 8 \) Copy content Toggle raw display
$53$ \( T + 10 \) Copy content Toggle raw display
$59$ \( T + 4 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T - 12 \) Copy content Toggle raw display
$71$ \( T + 8 \) Copy content Toggle raw display
$73$ \( T + 10 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 12 \) 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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