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Properties

Label 900.3.u.c
Level $900$
Weight $3$
Character orbit 900.u
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(149,900)]
 
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, 5, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.149");
 
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.u (of order \(6\), degree \(2\), not 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 52 q^{9} + 96 q^{11} - 144 q^{19} - 256 q^{21} - 300 q^{29} - 24 q^{31} - 80 q^{39} + 180 q^{41} - 96 q^{49} - 288 q^{51} - 96 q^{59} - 156 q^{61} + 300 q^{69} - 240 q^{79} + 868 q^{81} + 240 q^{91} + 240 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1 0 −2.96464 + 0.459278i 0 0 0 7.16186 − 4.13490i 0 8.57813 − 2.72318i 0
149.2 0 −2.87949 − 0.841761i 0 0 0 10.3705 − 5.98742i 0 7.58288 + 4.84768i 0
149.3 0 −1.39478 + 2.65605i 0 0 0 2.45229 − 1.41583i 0 −5.10917 − 7.40921i 0
149.4 0 −0.920635 − 2.85525i 0 0 0 1.02985 − 0.594587i 0 −7.30486 + 5.25728i 0
149.5 0 −0.783177 − 2.89597i 0 0 0 4.08158 − 2.35650i 0 −7.77327 + 4.53611i 0
149.6 0 −0.114662 − 2.99781i 0 0 0 −1.38806 + 0.801399i 0 −8.97371 + 0.687471i 0
149.7 0 0.114662 + 2.99781i 0 0 0 1.38806 − 0.801399i 0 −8.97371 + 0.687471i 0
149.8 0 0.783177 + 2.89597i 0 0 0 −4.08158 + 2.35650i 0 −7.77327 + 4.53611i 0
149.9 0 0.920635 + 2.85525i 0 0 0 −1.02985 + 0.594587i 0 −7.30486 + 5.25728i 0
149.10 0 1.39478 − 2.65605i 0 0 0 −2.45229 + 1.41583i 0 −5.10917 − 7.40921i 0
149.11 0 2.87949 + 0.841761i 0 0 0 −10.3705 + 5.98742i 0 7.58288 + 4.84768i 0
149.12 0 2.96464 − 0.459278i 0 0 0 −7.16186 + 4.13490i 0 8.57813 − 2.72318i 0
749.1 0 −2.96464 − 0.459278i 0 0 0 7.16186 + 4.13490i 0 8.57813 + 2.72318i 0
749.2 0 −2.87949 + 0.841761i 0 0 0 10.3705 + 5.98742i 0 7.58288 − 4.84768i 0
749.3 0 −1.39478 − 2.65605i 0 0 0 2.45229 + 1.41583i 0 −5.10917 + 7.40921i 0
749.4 0 −0.920635 + 2.85525i 0 0 0 1.02985 + 0.594587i 0 −7.30486 − 5.25728i 0
749.5 0 −0.783177 + 2.89597i 0 0 0 4.08158 + 2.35650i 0 −7.77327 − 4.53611i 0
749.6 0 −0.114662 + 2.99781i 0 0 0 −1.38806 − 0.801399i 0 −8.97371 − 0.687471i 0
749.7 0 0.114662 − 2.99781i 0 0 0 1.38806 + 0.801399i 0 −8.97371 − 0.687471i 0
749.8 0 0.783177 − 2.89597i 0 0 0 −4.08158 − 2.35650i 0 −7.77327 − 4.53611i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.12
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.u.c 24
3.b odd 2 1 2700.3.u.c 24
5.b even 2 1 inner 900.3.u.c 24
5.c odd 4 1 180.3.o.b ✓ 12
5.c odd 4 1 900.3.p.c 12
9.c even 3 1 2700.3.u.c 24
9.d odd 6 1 inner 900.3.u.c 24
15.d odd 2 1 2700.3.u.c 24
15.e even 4 1 540.3.o.b 12
15.e even 4 1 2700.3.p.c 12
20.e even 4 1 720.3.bs.b 12
45.h odd 6 1 inner 900.3.u.c 24
45.j even 6 1 2700.3.u.c 24
45.k odd 12 1 540.3.o.b 12
45.k odd 12 1 1620.3.g.b 12
45.k odd 12 1 2700.3.p.c 12
45.l even 12 1 180.3.o.b ✓ 12
45.l even 12 1 900.3.p.c 12
45.l even 12 1 1620.3.g.b 12
60.l odd 4 1 2160.3.bs.b 12
180.v odd 12 1 720.3.bs.b 12
180.x even 12 1 2160.3.bs.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.o.b ✓ 12 5.c odd 4 1
180.3.o.b ✓ 12 45.l even 12 1
540.3.o.b 12 15.e even 4 1
540.3.o.b 12 45.k odd 12 1
720.3.bs.b 12 20.e even 4 1
720.3.bs.b 12 180.v odd 12 1
900.3.p.c 12 5.c odd 4 1
900.3.p.c 12 45.l even 12 1
900.3.u.c 24 1.a even 1 1 trivial
900.3.u.c 24 5.b even 2 1 inner
900.3.u.c 24 9.d odd 6 1 inner
900.3.u.c 24 45.h odd 6 1 inner
1620.3.g.b 12 45.k odd 12 1
1620.3.g.b 12 45.l even 12 1
2160.3.bs.b 12 60.l odd 4 1
2160.3.bs.b 12 180.x even 12 1
2700.3.p.c 12 15.e even 4 1
2700.3.p.c 12 45.k odd 12 1
2700.3.u.c 24 3.b odd 2 1
2700.3.u.c 24 9.c even 3 1
2700.3.u.c 24 15.d odd 2 1
2700.3.u.c 24 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} - 246 T_{7}^{22} + 43161 T_{7}^{20} - 3468650 T_{7}^{18} + 199575090 T_{7}^{16} + \cdots + 40263606220321 \) acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\). Copy content Toggle raw display