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Properties

Label 900.3.p.f
Level $900$
Weight $3$
Character orbit 900.p
Analytic conductor $24.523$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,3,Mod(101,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.101");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.p (of order \(6\), degree \(2\), minimal)

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: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{9} - 18 q^{11} - 26 q^{21} - 36 q^{29} + 30 q^{31} - 6 q^{39} - 36 q^{41} - 108 q^{49} + 124 q^{51} + 306 q^{59} + 48 q^{61} + 268 q^{69} - 114 q^{79} - 14 q^{81} - 84 q^{91} - 418 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 0 −2.88925 + 0.807601i 0 0 0 −6.19003 + 10.7214i 0 7.69556 − 4.66673i 0
101.2 0 −2.69243 + 1.32320i 0 0 0 1.36039 − 2.35627i 0 5.49831 − 7.12521i 0
101.3 0 −2.55773 − 1.56781i 0 0 0 5.75148 − 9.96186i 0 4.08395 + 8.02006i 0
101.4 0 −1.92469 − 2.30121i 0 0 0 0.382345 − 0.662241i 0 −1.59110 + 8.85824i 0
101.5 0 −0.947625 + 2.84640i 0 0 0 −0.333962 + 0.578439i 0 −7.20401 − 5.39464i 0
101.6 0 −0.0930019 + 2.99856i 0 0 0 3.67363 − 6.36292i 0 −8.98270 − 0.557743i 0
101.7 0 0.0930019 − 2.99856i 0 0 0 −3.67363 + 6.36292i 0 −8.98270 − 0.557743i 0
101.8 0 0.947625 − 2.84640i 0 0 0 0.333962 − 0.578439i 0 −7.20401 − 5.39464i 0
101.9 0 1.92469 + 2.30121i 0 0 0 −0.382345 + 0.662241i 0 −1.59110 + 8.85824i 0
101.10 0 2.55773 + 1.56781i 0 0 0 −5.75148 + 9.96186i 0 4.08395 + 8.02006i 0
101.11 0 2.69243 − 1.32320i 0 0 0 −1.36039 + 2.35627i 0 5.49831 − 7.12521i 0
101.12 0 2.88925 − 0.807601i 0 0 0 6.19003 − 10.7214i 0 7.69556 − 4.66673i 0
401.1 0 −2.88925 − 0.807601i 0 0 0 −6.19003 − 10.7214i 0 7.69556 + 4.66673i 0
401.2 0 −2.69243 − 1.32320i 0 0 0 1.36039 + 2.35627i 0 5.49831 + 7.12521i 0
401.3 0 −2.55773 + 1.56781i 0 0 0 5.75148 + 9.96186i 0 4.08395 − 8.02006i 0
401.4 0 −1.92469 + 2.30121i 0 0 0 0.382345 + 0.662241i 0 −1.59110 − 8.85824i 0
401.5 0 −0.947625 − 2.84640i 0 0 0 −0.333962 − 0.578439i 0 −7.20401 + 5.39464i 0
401.6 0 −0.0930019 − 2.99856i 0 0 0 3.67363 + 6.36292i 0 −8.98270 + 0.557743i 0
401.7 0 0.0930019 + 2.99856i 0 0 0 −3.67363 − 6.36292i 0 −8.98270 + 0.557743i 0
401.8 0 0.947625 + 2.84640i 0 0 0 0.333962 + 0.578439i 0 −7.20401 + 5.39464i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.d odd 6 1 inner
45.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.p.f 24
3.b odd 2 1 2700.3.p.f 24
5.b even 2 1 inner 900.3.p.f 24
5.c odd 4 2 180.3.t.a ✓ 24
9.c even 3 1 2700.3.p.f 24
9.d odd 6 1 inner 900.3.p.f 24
15.d odd 2 1 2700.3.p.f 24
15.e even 4 2 540.3.t.a 24
45.h odd 6 1 inner 900.3.p.f 24
45.j even 6 1 2700.3.p.f 24
45.k odd 12 2 540.3.t.a 24
45.k odd 12 2 1620.3.b.b 24
45.l even 12 2 180.3.t.a ✓ 24
45.l even 12 2 1620.3.b.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.t.a ✓ 24 5.c odd 4 2
180.3.t.a ✓ 24 45.l even 12 2
540.3.t.a 24 15.e even 4 2
540.3.t.a 24 45.k odd 12 2
900.3.p.f 24 1.a even 1 1 trivial
900.3.p.f 24 5.b even 2 1 inner
900.3.p.f 24 9.d odd 6 1 inner
900.3.p.f 24 45.h odd 6 1 inner
1620.3.b.b 24 45.k odd 12 2
1620.3.b.b 24 45.l even 12 2
2700.3.p.f 24 3.b odd 2 1
2700.3.p.f 24 9.c even 3 1
2700.3.p.f 24 15.d odd 2 1
2700.3.p.f 24 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 348 T_{7}^{22} + 82536 T_{7}^{20} + 10624700 T_{7}^{18} + 991303875 T_{7}^{16} + \cdots + 4469486461456 \) acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\). Copy content Toggle raw display