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Properties

Label 390.2.be.b
Level $390$
Weight $2$
Character orbit 390.be
Analytic conductor $3.114$
Analytic rank $0$
Dimension $96$
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] = [390,2,Mod(17,390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(390, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("390.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.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: \(3.11416567883\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 4 q^{3} + 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 4 q^{3} + 24 q^{7} + 16 q^{10} - 8 q^{12} + 40 q^{13} + 48 q^{16} - 8 q^{22} + 64 q^{25} - 32 q^{27} + 24 q^{28} - 8 q^{30} - 96 q^{33} - 24 q^{36} + 72 q^{37} - 8 q^{42} + 16 q^{43} - 60 q^{45} + 24 q^{46} - 4 q^{48} - 128 q^{51} - 8 q^{52} + 32 q^{61} - 120 q^{63} - 16 q^{66} - 96 q^{67} - 24 q^{72} - 4 q^{75} - 24 q^{76} - 16 q^{78} - 48 q^{81} + 32 q^{82} + 32 q^{87} + 8 q^{88} - 56 q^{90} - 56 q^{91} - 12 q^{93} + 24 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −0.965926 + 0.258819i −1.69962 + 0.333621i 0.866025 − 0.500000i 0.801239 + 2.08759i 1.55536 − 0.762146i 0.383052 + 0.102638i −0.707107 + 0.707107i 2.77739 − 1.13405i −1.31424 − 1.80908i
17.2 −0.965926 + 0.258819i −1.64333 − 0.547240i 0.866025 − 0.500000i −1.99178 − 1.01627i 1.72897 + 0.103269i −4.24747 − 1.13811i −0.707107 + 0.707107i 2.40106 + 1.79859i 2.18694 + 0.466129i
17.3 −0.965926 + 0.258819i −1.36137 − 1.07083i 0.866025 − 0.500000i 2.19719 − 0.415166i 1.59213 + 0.681996i 2.80725 + 0.752202i −0.707107 + 0.707107i 0.706639 + 2.91559i −2.01487 + 0.969694i
17.4 −0.965926 + 0.258819i −0.793519 − 1.53959i 0.866025 − 0.500000i −2.16310 + 0.566581i 1.16496 + 1.28175i 4.16232 + 1.11529i −0.707107 + 0.707107i −1.74065 + 2.44338i 1.94275 − 1.10713i
17.5 −0.965926 + 0.258819i −0.605124 + 1.62291i 0.866025 − 0.500000i −2.23558 + 0.0464880i 0.164466 − 1.72422i −0.723770 − 0.193934i −0.707107 + 0.707107i −2.26765 − 1.96412i 2.14738 − 0.623516i
17.6 −0.965926 + 0.258819i −0.527343 − 1.64982i 0.866025 − 0.500000i 0.760049 + 2.10293i 0.936380 + 1.45712i −2.97974 − 0.798419i −0.707107 + 0.707107i −2.44382 + 1.74004i −1.27843 − 1.83456i
17.7 −0.965926 + 0.258819i 0.212829 + 1.71893i 0.866025 − 0.500000i 2.14761 − 0.622719i −0.650468 − 1.60527i 2.96096 + 0.793388i −0.707107 + 0.707107i −2.90941 + 0.731676i −1.91326 + 1.15734i
17.8 −0.965926 + 0.258819i 0.856208 − 1.50563i 0.866025 − 0.500000i −0.0474795 − 2.23556i −0.437349 + 1.67593i 1.84149 + 0.493426i −0.707107 + 0.707107i −1.53382 − 2.57826i 0.624468 + 2.14710i
17.9 −0.965926 + 0.258819i 0.927920 + 1.46252i 0.866025 − 0.500000i −1.18354 − 1.89717i −1.27483 − 1.17252i 0.559924 + 0.150031i −0.707107 + 0.707107i −1.27793 + 2.71420i 1.63423 + 1.52620i
17.10 −0.965926 + 0.258819i 1.08686 − 1.34861i 0.866025 − 0.500000i −2.07976 + 0.821350i −0.700781 + 1.58395i −2.17007 − 0.581468i −0.707107 + 0.707107i −0.637473 − 2.93149i 1.79631 − 1.33164i
17.11 −0.965926 + 0.258819i 1.46404 + 0.925520i 0.866025 − 0.500000i −0.839660 + 2.07243i −1.65370 − 0.515063i 3.48859 + 0.934764i −0.707107 + 0.707107i 1.28682 + 2.71000i 0.274664 − 2.21913i
17.12 −0.965926 + 0.258819i 1.71642 + 0.232192i 0.866025 − 0.500000i 1.80639 + 1.31794i −1.71803 + 0.219961i −1.35048 − 0.361861i −0.707107 + 0.707107i 2.89217 + 0.797077i −2.08594 − 0.805507i
17.13 0.965926 − 0.258819i −1.61555 − 0.624497i 0.866025 − 0.500000i 2.07976 − 0.821350i −1.72213 − 0.185082i −2.17007 − 0.581468i 0.707107 − 0.707107i 2.22001 + 2.01781i 1.79631 − 1.33164i
17.14 0.965926 − 0.258819i −1.49431 − 0.875806i 0.866025 − 0.500000i 0.0474795 + 2.23556i −1.67007 − 0.459208i 1.84149 + 0.493426i 0.707107 − 0.707107i 1.46593 + 2.61745i 0.624468 + 2.14710i
17.15 0.965926 − 0.258819i −1.37036 + 1.05929i 0.866025 − 0.500000i −1.80639 − 1.31794i −1.04951 + 1.37787i −1.35048 − 0.361861i 0.707107 − 0.707107i 0.755798 − 2.90323i −2.08594 − 0.805507i
17.16 0.965926 − 0.258819i −0.805135 + 1.53354i 0.866025 − 0.500000i 0.839660 − 2.07243i −0.380791 + 1.68967i 3.48859 + 0.934764i 0.707107 − 0.707107i −1.70351 − 2.46942i 0.274664 − 2.21913i
17.17 0.965926 − 0.258819i −0.368218 − 1.69246i 0.866025 − 0.500000i −0.760049 − 2.10293i −0.793711 − 1.53949i −2.97974 − 0.798419i 0.707107 − 0.707107i −2.72883 + 1.24639i −1.27843 − 1.83456i
17.18 0.965926 − 0.258819i −0.0825856 − 1.73008i 0.866025 − 0.500000i 2.16310 − 0.566581i −0.527549 − 1.64976i 4.16232 + 1.11529i 0.707107 − 0.707107i −2.98636 + 0.285760i 1.94275 − 1.10713i
17.19 0.965926 − 0.258819i −0.0723418 + 1.73054i 0.866025 − 0.500000i 1.18354 + 1.89717i 0.378020 + 1.69030i 0.559924 + 0.150031i 0.707107 − 0.707107i −2.98953 − 0.250381i 1.63423 + 1.52620i
17.20 0.965926 − 0.258819i 0.643562 − 1.60805i 0.866025 − 0.500000i −2.19719 + 0.415166i 0.205439 − 1.71982i 2.80725 + 0.752202i 0.707107 − 0.707107i −2.17166 − 2.06976i −2.01487 + 0.969694i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
13.e even 6 1 inner
15.e even 4 1 inner
39.h odd 6 1 inner
65.r odd 12 1 inner
195.bf even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.be.b ✓ 96
3.b odd 2 1 inner 390.2.be.b ✓ 96
5.c odd 4 1 inner 390.2.be.b ✓ 96
13.e even 6 1 inner 390.2.be.b ✓ 96
15.e even 4 1 inner 390.2.be.b ✓ 96
39.h odd 6 1 inner 390.2.be.b ✓ 96
65.r odd 12 1 inner 390.2.be.b ✓ 96
195.bf even 12 1 inner 390.2.be.b ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.be.b ✓ 96 1.a even 1 1 trivial
390.2.be.b ✓ 96 3.b odd 2 1 inner
390.2.be.b ✓ 96 5.c odd 4 1 inner
390.2.be.b ✓ 96 13.e even 6 1 inner
390.2.be.b ✓ 96 15.e even 4 1 inner
390.2.be.b ✓ 96 39.h odd 6 1 inner
390.2.be.b ✓ 96 65.r odd 12 1 inner
390.2.be.b ✓ 96 195.bf even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{48} - 12 T_{7}^{47} + 72 T_{7}^{46} - 288 T_{7}^{45} + 273 T_{7}^{44} + 4716 T_{7}^{43} + \cdots + 14164684960000 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\). Copy content Toggle raw display