Properties

Label 900.2.a.h
Level $900$
Weight $2$
Character orbit 900.a
Self dual yes
Analytic conductor $7.187$
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] = [900,2,Mod(1,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("900.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.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: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{7} + 4 q^{11} - 4 q^{17} - 4 q^{23} + 6 q^{29} + 4 q^{31} - 8 q^{37} + 10 q^{41} + 4 q^{43} + 4 q^{47} + 9 q^{49} + 12 q^{53} - 4 q^{59} + 2 q^{61} - 4 q^{67} - 8 q^{73} + 16 q^{77} - 12 q^{79} - 4 q^{83} + 10 q^{89} + 8 q^{97}+O(q^{100}) \) 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 4.00000 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 900.2.a.h 1
3.b odd 2 1 300.2.a.d 1
4.b odd 2 1 3600.2.a.d 1
5.b even 2 1 900.2.a.a 1
5.c odd 4 2 180.2.d.a 2
12.b even 2 1 1200.2.a.a 1
15.d odd 2 1 300.2.a.a 1
15.e even 4 2 60.2.d.a 2
20.d odd 2 1 3600.2.a.bm 1
20.e even 4 2 720.2.f.c 2
24.f even 2 1 4800.2.a.bk 1
24.h odd 2 1 4800.2.a.bj 1
40.i odd 4 2 2880.2.f.l 2
40.k even 4 2 2880.2.f.p 2
45.k odd 12 4 1620.2.r.d 4
45.l even 12 4 1620.2.r.c 4
60.h even 2 1 1200.2.a.s 1
60.l odd 4 2 240.2.f.b 2
105.k odd 4 2 2940.2.k.c 2
105.w odd 12 4 2940.2.bb.e 4
105.x even 12 4 2940.2.bb.d 4
120.i odd 2 1 4800.2.a.bn 1
120.m even 2 1 4800.2.a.bf 1
120.q odd 4 2 960.2.f.c 2
120.w even 4 2 960.2.f.f 2
240.z odd 4 2 3840.2.d.be 2
240.bb even 4 2 3840.2.d.o 2
240.bd odd 4 2 3840.2.d.b 2
240.bf even 4 2 3840.2.d.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.d.a 2 15.e even 4 2
180.2.d.a 2 5.c odd 4 2
240.2.f.b 2 60.l odd 4 2
300.2.a.a 1 15.d odd 2 1
300.2.a.d 1 3.b odd 2 1
720.2.f.c 2 20.e even 4 2
900.2.a.a 1 5.b even 2 1
900.2.a.h 1 1.a even 1 1 trivial
960.2.f.c 2 120.q odd 4 2
960.2.f.f 2 120.w even 4 2
1200.2.a.a 1 12.b even 2 1
1200.2.a.s 1 60.h even 2 1
1620.2.r.c 4 45.l even 12 4
1620.2.r.d 4 45.k odd 12 4
2880.2.f.l 2 40.i odd 4 2
2880.2.f.p 2 40.k even 4 2
2940.2.k.c 2 105.k odd 4 2
2940.2.bb.d 4 105.x even 12 4
2940.2.bb.e 4 105.w odd 12 4
3600.2.a.d 1 4.b odd 2 1
3600.2.a.bm 1 20.d odd 2 1
3840.2.d.b 2 240.bd odd 4 2
3840.2.d.o 2 240.bb even 4 2
3840.2.d.r 2 240.bf even 4 2
3840.2.d.be 2 240.z odd 4 2
4800.2.a.bf 1 120.m even 2 1
4800.2.a.bj 1 24.h odd 2 1
4800.2.a.bk 1 24.f even 2 1
4800.2.a.bn 1 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(900))\):

\( T_{7} - 4 \) Copy content Toggle raw display
\( T_{11} - 4 \) 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
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 4 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 4 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T + 8 \) Copy content Toggle raw display
$41$ \( T - 10 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T - 4 \) Copy content Toggle raw display
$53$ \( T - 12 \) 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
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 8 \) Copy content Toggle raw display
$79$ \( T + 12 \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T - 10 \) Copy content Toggle raw display
$97$ \( T - 8 \) Copy content Toggle raw display
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