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Properties

Label 900.2.be.f
Level $900$
Weight $2$
Character orbit 900.be
Analytic conductor $7.187$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,2,Mod(257,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 10, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.257");
 
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.be (of order \(12\), degree \(4\), 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: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 12 q^{11} + 12 q^{21} + 8 q^{31} + 60 q^{41} + 36 q^{51} + 52 q^{61} - 36 q^{81} + 120 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1 0 −1.61701 + 0.620711i 0 0 0 −0.876983 + 3.27295i 0 2.22943 − 2.00739i 0
257.2 0 −1.09001 − 1.34606i 0 0 0 0.599926 − 2.23895i 0 −0.623735 + 2.93444i 0
257.3 0 −0.943342 + 1.45262i 0 0 0 0.0937177 − 0.349759i 0 −1.22021 − 2.74064i 0
257.4 0 −0.0906485 − 1.72968i 0 0 0 −0.819062 + 3.05678i 0 −2.98357 + 0.313585i 0
257.5 0 0.0906485 + 1.72968i 0 0 0 0.819062 − 3.05678i 0 −2.98357 + 0.313585i 0
257.6 0 0.943342 − 1.45262i 0 0 0 −0.0937177 + 0.349759i 0 −1.22021 − 2.74064i 0
257.7 0 1.09001 + 1.34606i 0 0 0 −0.599926 + 2.23895i 0 −0.623735 + 2.93444i 0
257.8 0 1.61701 − 0.620711i 0 0 0 0.876983 − 3.27295i 0 2.22943 − 2.00739i 0
293.1 0 −1.72968 + 0.0906485i 0 0 0 −3.05678 − 0.819062i 0 2.98357 − 0.313585i 0
293.2 0 −1.45262 − 0.943342i 0 0 0 −0.349759 − 0.0937177i 0 1.22021 + 2.74064i 0
293.3 0 −1.34606 + 1.09001i 0 0 0 2.23895 + 0.599926i 0 0.623735 − 2.93444i 0
293.4 0 −0.620711 − 1.61701i 0 0 0 3.27295 + 0.876983i 0 −2.22943 + 2.00739i 0
293.5 0 0.620711 + 1.61701i 0 0 0 −3.27295 − 0.876983i 0 −2.22943 + 2.00739i 0
293.6 0 1.34606 − 1.09001i 0 0 0 −2.23895 − 0.599926i 0 0.623735 − 2.93444i 0
293.7 0 1.45262 + 0.943342i 0 0 0 0.349759 + 0.0937177i 0 1.22021 + 2.74064i 0
293.8 0 1.72968 − 0.0906485i 0 0 0 3.05678 + 0.819062i 0 2.98357 − 0.313585i 0
857.1 0 −1.72968 − 0.0906485i 0 0 0 −3.05678 + 0.819062i 0 2.98357 + 0.313585i 0
857.2 0 −1.45262 + 0.943342i 0 0 0 −0.349759 + 0.0937177i 0 1.22021 − 2.74064i 0
857.3 0 −1.34606 − 1.09001i 0 0 0 2.23895 − 0.599926i 0 0.623735 + 2.93444i 0
857.4 0 −0.620711 + 1.61701i 0 0 0 3.27295 − 0.876983i 0 −2.22943 − 2.00739i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
9.d odd 6 1 inner
45.h odd 6 1 inner
45.l even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.be.f ✓ 32
3.b odd 2 1 2700.2.bf.f 32
5.b even 2 1 inner 900.2.be.f ✓ 32
5.c odd 4 2 inner 900.2.be.f ✓ 32
9.c even 3 1 2700.2.bf.f 32
9.d odd 6 1 inner 900.2.be.f ✓ 32
15.d odd 2 1 2700.2.bf.f 32
15.e even 4 2 2700.2.bf.f 32
45.h odd 6 1 inner 900.2.be.f ✓ 32
45.j even 6 1 2700.2.bf.f 32
45.k odd 12 2 2700.2.bf.f 32
45.l even 12 2 inner 900.2.be.f ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.2.be.f ✓ 32 1.a even 1 1 trivial
900.2.be.f ✓ 32 5.b even 2 1 inner
900.2.be.f ✓ 32 5.c odd 4 2 inner
900.2.be.f ✓ 32 9.d odd 6 1 inner
900.2.be.f ✓ 32 45.h odd 6 1 inner
900.2.be.f ✓ 32 45.l even 12 2 inner
2700.2.bf.f 32 3.b odd 2 1
2700.2.bf.f 32 9.c even 3 1
2700.2.bf.f 32 15.d odd 2 1
2700.2.bf.f 32 15.e even 4 2
2700.2.bf.f 32 45.j even 6 1
2700.2.bf.f 32 45.k odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{32} - 261 T_{7}^{28} + 48195 T_{7}^{24} - 4436694 T_{7}^{20} + 297337959 T_{7}^{16} + \cdots + 43046721 \) acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\). Copy content Toggle raw display