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Properties

Label 420.2.bn.a
Level $420$
Weight $2$
Character orbit 420.bn
Analytic conductor $3.354$
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] = [420,2,Mod(89,420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(420, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("420.89");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.bn (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: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{15} + 12 q^{19} - 8 q^{21} + 6 q^{25} - 12 q^{31} + 24 q^{39} + 33 q^{45} - 44 q^{49} - 10 q^{51} - 24 q^{61} + 21 q^{75} - 28 q^{79} - 20 q^{81} - 4 q^{85} + 16 q^{91} - 28 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1 0 −1.73102 − 0.0597684i 0 1.61141 + 1.55028i 0 −2.58416 + 0.567546i 0 2.99286 + 0.206921i 0
89.2 0 −1.55045 + 0.772078i 0 −0.155374 − 2.23066i 0 −0.477214 + 2.60236i 0 1.80779 − 2.39414i 0
89.3 0 −1.48906 − 0.884700i 0 −1.06703 − 1.96506i 0 0.0973684 − 2.64396i 0 1.43461 + 2.63475i 0
89.4 0 −1.44386 + 0.956690i 0 −2.00950 + 0.980773i 0 0.477214 − 2.60236i 0 1.16949 − 2.76266i 0
89.5 0 −1.16448 − 1.28218i 0 1.81384 − 1.30767i 0 2.08204 + 1.63251i 0 −0.287951 + 2.98615i 0
89.6 0 −0.813749 + 1.52899i 0 2.14829 + 0.620379i 0 2.58416 − 0.567546i 0 −1.67563 − 2.48843i 0
89.7 0 −0.528155 − 1.64956i 0 −1.81384 + 1.30767i 0 2.08204 + 1.63251i 0 −2.44211 + 1.74245i 0
89.8 0 −0.0216419 − 1.73192i 0 1.06703 + 1.96506i 0 0.0973684 − 2.64396i 0 −2.99906 + 0.0749640i 0
89.9 0 0.0216419 + 1.73192i 0 −2.23530 + 0.0584556i 0 −0.0973684 + 2.64396i 0 −2.99906 + 0.0749640i 0
89.10 0 0.528155 + 1.64956i 0 −0.225553 + 2.22466i 0 −2.08204 − 1.63251i 0 −2.44211 + 1.74245i 0
89.11 0 0.813749 − 1.52899i 0 −1.61141 − 1.55028i 0 −2.58416 + 0.567546i 0 −1.67563 − 2.48843i 0
89.12 0 1.16448 + 1.28218i 0 0.225553 − 2.22466i 0 −2.08204 − 1.63251i 0 −0.287951 + 2.98615i 0
89.13 0 1.44386 − 0.956690i 0 0.155374 + 2.23066i 0 −0.477214 + 2.60236i 0 1.16949 − 2.76266i 0
89.14 0 1.48906 + 0.884700i 0 2.23530 − 0.0584556i 0 −0.0973684 + 2.64396i 0 1.43461 + 2.63475i 0
89.15 0 1.55045 − 0.772078i 0 2.00950 − 0.980773i 0 0.477214 − 2.60236i 0 1.80779 − 2.39414i 0
89.16 0 1.73102 + 0.0597684i 0 −2.14829 − 0.620379i 0 2.58416 − 0.567546i 0 2.99286 + 0.206921i 0
269.1 0 −1.73102 + 0.0597684i 0 1.61141 − 1.55028i 0 −2.58416 − 0.567546i 0 2.99286 − 0.206921i 0
269.2 0 −1.55045 − 0.772078i 0 −0.155374 + 2.23066i 0 −0.477214 − 2.60236i 0 1.80779 + 2.39414i 0
269.3 0 −1.48906 + 0.884700i 0 −1.06703 + 1.96506i 0 0.0973684 + 2.64396i 0 1.43461 − 2.63475i 0
269.4 0 −1.44386 − 0.956690i 0 −2.00950 − 0.980773i 0 0.477214 + 2.60236i 0 1.16949 + 2.76266i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.d odd 6 1 inner
15.d odd 2 1 inner
21.g even 6 1 inner
35.i odd 6 1 inner
105.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.bn.a ✓ 32
3.b odd 2 1 inner 420.2.bn.a ✓ 32
5.b even 2 1 inner 420.2.bn.a ✓ 32
5.c odd 4 2 2100.2.bi.n 32
7.c even 3 1 2940.2.f.a 32
7.d odd 6 1 inner 420.2.bn.a ✓ 32
7.d odd 6 1 2940.2.f.a 32
15.d odd 2 1 inner 420.2.bn.a ✓ 32
15.e even 4 2 2100.2.bi.n 32
21.g even 6 1 inner 420.2.bn.a ✓ 32
21.g even 6 1 2940.2.f.a 32
21.h odd 6 1 2940.2.f.a 32
35.i odd 6 1 inner 420.2.bn.a ✓ 32
35.i odd 6 1 2940.2.f.a 32
35.j even 6 1 2940.2.f.a 32
35.k even 12 2 2100.2.bi.n 32
105.o odd 6 1 2940.2.f.a 32
105.p even 6 1 inner 420.2.bn.a ✓ 32
105.p even 6 1 2940.2.f.a 32
105.w odd 12 2 2100.2.bi.n 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bn.a ✓ 32 1.a even 1 1 trivial
420.2.bn.a ✓ 32 3.b odd 2 1 inner
420.2.bn.a ✓ 32 5.b even 2 1 inner
420.2.bn.a ✓ 32 7.d odd 6 1 inner
420.2.bn.a ✓ 32 15.d odd 2 1 inner
420.2.bn.a ✓ 32 21.g even 6 1 inner
420.2.bn.a ✓ 32 35.i odd 6 1 inner
420.2.bn.a ✓ 32 105.p even 6 1 inner
2100.2.bi.n 32 5.c odd 4 2
2100.2.bi.n 32 15.e even 4 2
2100.2.bi.n 32 35.k even 12 2
2100.2.bi.n 32 105.w odd 12 2
2940.2.f.a 32 7.c even 3 1
2940.2.f.a 32 7.d odd 6 1
2940.2.f.a 32 21.g even 6 1
2940.2.f.a 32 21.h odd 6 1
2940.2.f.a 32 35.i odd 6 1
2940.2.f.a 32 35.j even 6 1
2940.2.f.a 32 105.o odd 6 1
2940.2.f.a 32 105.p even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(420, [\chi])\).