↑

Properties

Label 1620.2.e.b
Level $1620$
Weight $2$
Character orbit 1620.e
Analytic conductor $12.936$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,2,Mod(971,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.971");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.e (of order \(2\), degree \(1\), 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: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{25} + 12 q^{34} + 12 q^{40} - 12 q^{46} - 48 q^{49} + 36 q^{52} + 36 q^{58} - 48 q^{64} - 24 q^{73} - 12 q^{76} - 36 q^{82} - 36 q^{94} + 24 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
971.1 −1.41320 − 0.0535467i 0 1.99427 + 0.151344i − 1.00000i 0 − 2.16432i −2.81019 − 0.320666i 0 −0.0535467 + 1.41320i
971.2 −1.41320 + 0.0535467i 0 1.99427 − 0.151344i 1.00000i 0 2.16432i −2.81019 + 0.320666i 0 −0.0535467 − 1.41320i
971.3 −1.37860 − 0.315389i 0 1.80106 + 0.869588i 1.00000i 0 − 2.19567i −2.20868 − 1.76684i 0 0.315389 − 1.37860i
971.4 −1.37860 + 0.315389i 0 1.80106 − 0.869588i − 1.00000i 0 2.19567i −2.20868 + 1.76684i 0 0.315389 + 1.37860i
971.5 −1.32567 − 0.492552i 0 1.51479 + 1.30592i 1.00000i 0 3.84799i −1.36487 − 2.47732i 0 0.492552 − 1.32567i
971.6 −1.32567 + 0.492552i 0 1.51479 − 1.30592i − 1.00000i 0 − 3.84799i −1.36487 + 2.47732i 0 0.492552 + 1.32567i
971.7 −1.26211 − 0.638030i 0 1.18584 + 1.61053i − 1.00000i 0 4.69912i −0.469091 − 2.78926i 0 −0.638030 + 1.26211i
971.8 −1.26211 + 0.638030i 0 1.18584 − 1.61053i 1.00000i 0 − 4.69912i −0.469091 + 2.78926i 0 −0.638030 − 1.26211i
971.9 −1.14547 − 0.829399i 0 0.624193 + 1.90010i − 1.00000i 0 − 2.90505i 0.860949 − 2.69421i 0 −0.829399 + 1.14547i
971.10 −1.14547 + 0.829399i 0 0.624193 − 1.90010i 1.00000i 0 2.90505i 0.860949 + 2.69421i 0 −0.829399 − 1.14547i
971.11 −1.04983 − 0.947554i 0 0.204282 + 1.98954i 1.00000i 0 − 4.17013i 1.67074 − 2.28224i 0 0.947554 − 1.04983i
971.12 −1.04983 + 0.947554i 0 0.204282 − 1.98954i − 1.00000i 0 4.17013i 1.67074 + 2.28224i 0 0.947554 + 1.04983i
971.13 −1.04383 − 0.954154i 0 0.179181 + 1.99196i 1.00000i 0 0.851747i 1.71360 − 2.25024i 0 0.954154 − 1.04383i
971.14 −1.04383 + 0.954154i 0 0.179181 − 1.99196i − 1.00000i 0 − 0.851747i 1.71360 + 2.25024i 0 0.954154 + 1.04383i
971.15 −0.863299 − 1.12014i 0 −0.509428 + 1.93403i − 1.00000i 0 1.02978i 2.60618 − 1.09902i 0 −1.12014 + 0.863299i
971.16 −0.863299 + 1.12014i 0 −0.509428 − 1.93403i 1.00000i 0 − 1.02978i 2.60618 + 1.09902i 0 −1.12014 − 0.863299i
971.17 −0.499529 − 1.32305i 0 −1.50094 + 1.32181i 1.00000i 0 1.20913i 2.49858 + 1.32555i 0 1.32305 − 0.499529i
971.18 −0.499529 + 1.32305i 0 −1.50094 − 1.32181i − 1.00000i 0 − 1.20913i 2.49858 − 1.32555i 0 1.32305 + 0.499529i
971.19 −0.390365 − 1.35927i 0 −1.69523 + 1.06122i − 1.00000i 0 − 1.67328i 2.10425 + 1.89001i 0 −1.35927 + 0.390365i
971.20 −0.390365 + 1.35927i 0 −1.69523 − 1.06122i 1.00000i 0 1.67328i 2.10425 − 1.89001i 0 −1.35927 − 0.390365i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 971.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.e.b 48
3.b odd 2 1 inner 1620.2.e.b 48
4.b odd 2 1 inner 1620.2.e.b 48
9.c even 3 1 180.2.q.a ✓ 48
9.c even 3 1 540.2.q.a 48
9.d odd 6 1 180.2.q.a ✓ 48
9.d odd 6 1 540.2.q.a 48
12.b even 2 1 inner 1620.2.e.b 48
36.f odd 6 1 180.2.q.a ✓ 48
36.f odd 6 1 540.2.q.a 48
36.h even 6 1 180.2.q.a ✓ 48
36.h even 6 1 540.2.q.a 48
45.h odd 6 1 900.2.r.f 48
45.j even 6 1 900.2.r.f 48
45.k odd 12 1 900.2.o.b 48
45.k odd 12 1 900.2.o.c 48
45.l even 12 1 900.2.o.b 48
45.l even 12 1 900.2.o.c 48
180.n even 6 1 900.2.r.f 48
180.p odd 6 1 900.2.r.f 48
180.v odd 12 1 900.2.o.b 48
180.v odd 12 1 900.2.o.c 48
180.x even 12 1 900.2.o.b 48
180.x even 12 1 900.2.o.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.q.a ✓ 48 9.c even 3 1
180.2.q.a ✓ 48 9.d odd 6 1
180.2.q.a ✓ 48 36.f odd 6 1
180.2.q.a ✓ 48 36.h even 6 1
540.2.q.a 48 9.c even 3 1
540.2.q.a 48 9.d odd 6 1
540.2.q.a 48 36.f odd 6 1
540.2.q.a 48 36.h even 6 1
900.2.o.b 48 45.k odd 12 1
900.2.o.b 48 45.l even 12 1
900.2.o.b 48 180.v odd 12 1
900.2.o.b 48 180.x even 12 1
900.2.o.c 48 45.k odd 12 1
900.2.o.c 48 45.l even 12 1
900.2.o.c 48 180.v odd 12 1
900.2.o.c 48 180.x even 12 1
900.2.r.f 48 45.h odd 6 1
900.2.r.f 48 45.j even 6 1
900.2.r.f 48 180.n even 6 1
900.2.r.f 48 180.p odd 6 1
1620.2.e.b 48 1.a even 1 1 trivial
1620.2.e.b 48 3.b odd 2 1 inner
1620.2.e.b 48 4.b odd 2 1 inner
1620.2.e.b 48 12.b even 2 1 inner