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Properties

Label 387.2.v.a
Level $387$
Weight $2$
Character orbit 387.v
Analytic conductor $3.090$
Analytic rank $0$
Dimension $96$
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] = [387,2,Mod(8,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 387.v (of order \(14\), degree \(6\), 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: \(3.09021055822\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 20 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 20 q^{4} + 16 q^{10} - 4 q^{13} - 36 q^{16} - 16 q^{25} - 48 q^{31} - 104 q^{40} + 28 q^{43} - 28 q^{46} - 160 q^{49} - 44 q^{52} + 84 q^{55} + 20 q^{58} + 52 q^{64} + 40 q^{67} - 140 q^{70} - 28 q^{73} + 112 q^{76} + 64 q^{79} + 168 q^{88} + 56 q^{91} + 112 q^{94} - 108 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −0.599460 + 2.62640i 0 −4.73671 − 2.28108i 2.17893 − 2.73229i 0 3.66130i 5.47120 − 6.86067i 0 5.86991 + 7.36064i
8.2 −0.567069 + 2.48449i 0 −4.04919 − 1.94999i 0.245918 − 0.308371i 0 − 4.20562i 3.96311 − 4.96959i 0 0.626693 + 0.785849i
8.3 −0.527442 + 2.31087i 0 −3.26001 − 1.56994i −2.27780 + 2.85627i 0 − 1.55842i 2.39168 − 2.99907i 0 −5.39907 − 6.77021i
8.4 −0.362744 + 1.58929i 0 −0.592310 − 0.285241i −0.0283185 + 0.0355103i 0 1.92060i −1.36459 + 1.71114i 0 −0.0461637 − 0.0578874i
8.5 −0.325862 + 1.42770i 0 −0.130190 − 0.0626962i 1.65289 − 2.07265i 0 − 4.12391i −1.69416 + 2.12440i 0 2.42050 + 3.03522i
8.6 −0.236651 + 1.03684i 0 0.782911 + 0.377030i −2.02001 + 2.53301i 0 − 1.06119i −1.90236 + 2.38548i 0 −2.14828 − 2.69386i
8.7 −0.184575 + 0.808678i 0 1.18205 + 0.569244i −0.0786303 + 0.0985993i 0 3.94692i −1.71285 + 2.14784i 0 −0.0652218 − 0.0817856i
8.8 −0.0211977 + 0.0928730i 0 1.79376 + 0.863830i −1.55851 + 1.95431i 0 − 1.70699i −0.237039 + 0.297238i 0 −0.148466 − 0.186171i
8.9 0.0211977 − 0.0928730i 0 1.79376 + 0.863830i 1.55851 − 1.95431i 0 − 1.70699i 0.237039 − 0.297238i 0 −0.148466 − 0.186171i
8.10 0.184575 − 0.808678i 0 1.18205 + 0.569244i 0.0786303 − 0.0985993i 0 3.94692i 1.71285 − 2.14784i 0 −0.0652218 − 0.0817856i
8.11 0.236651 − 1.03684i 0 0.782911 + 0.377030i 2.02001 − 2.53301i 0 − 1.06119i 1.90236 − 2.38548i 0 −2.14828 − 2.69386i
8.12 0.325862 − 1.42770i 0 −0.130190 − 0.0626962i −1.65289 + 2.07265i 0 − 4.12391i 1.69416 − 2.12440i 0 2.42050 + 3.03522i
8.13 0.362744 − 1.58929i 0 −0.592310 − 0.285241i 0.0283185 − 0.0355103i 0 1.92060i 1.36459 − 1.71114i 0 −0.0461637 − 0.0578874i
8.14 0.527442 − 2.31087i 0 −3.26001 − 1.56994i 2.27780 − 2.85627i 0 − 1.55842i −2.39168 + 2.99907i 0 −5.39907 − 6.77021i
8.15 0.567069 − 2.48449i 0 −4.04919 − 1.94999i −0.245918 + 0.308371i 0 − 4.20562i −3.96311 + 4.96959i 0 0.626693 + 0.785849i
8.16 0.599460 − 2.62640i 0 −4.73671 − 2.28108i −2.17893 + 2.73229i 0 3.66130i −5.47120 + 6.86067i 0 5.86991 + 7.36064i
125.1 −1.72346 + 2.16115i 0 −1.25521 − 5.49943i −3.74408 − 1.80305i 0 0.520042i 9.06744 + 4.36665i 0 10.3494 − 4.98402i
125.2 −1.63003 + 2.04399i 0 −1.07587 − 4.71370i 1.13828 + 0.548169i 0 − 0.926403i 6.67754 + 3.21574i 0 −2.97589 + 1.43312i
125.3 −1.33859 + 1.67853i 0 −0.580622 − 2.54387i 2.59338 + 1.24891i 0 − 2.12449i 1.17856 + 0.567565i 0 −5.56779 + 2.68131i
125.4 −1.11819 + 1.40217i 0 −0.270679 − 1.18592i −1.71530 − 0.826043i 0 − 2.15937i −1.26613 − 0.609737i 0 3.07628 − 1.48146i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
43.f odd 14 1 inner
129.j even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.2.v.a ✓ 96
3.b odd 2 1 inner 387.2.v.a ✓ 96
43.f odd 14 1 inner 387.2.v.a ✓ 96
129.j even 14 1 inner 387.2.v.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
387.2.v.a ✓ 96 1.a even 1 1 trivial
387.2.v.a ✓ 96 3.b odd 2 1 inner
387.2.v.a ✓ 96 43.f odd 14 1 inner
387.2.v.a ✓ 96 129.j even 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(387, [\chi])\).