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Properties

Label 588.2.y.a
Level $588$
Weight $2$
Character orbit 588.y
Analytic conductor $4.695$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [588,2,Mod(25,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 588.y (of order \(21\), degree \(12\), 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: \(4.69520363885\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(4\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{3} + 4 q^{9} - 7 q^{11} + 7 q^{15} - 5 q^{17} + 16 q^{19} + 7 q^{21} - 2 q^{23} - 52 q^{25} - 8 q^{27} + 4 q^{29} + 2 q^{31} + 14 q^{33} - 7 q^{35} + 44 q^{37} + 14 q^{39} + 7 q^{41} + 14 q^{45} + 9 q^{47} + 28 q^{49} + 2 q^{51} + 34 q^{53} + 10 q^{57} + 7 q^{59} + 59 q^{61} + 21 q^{63} - 33 q^{65} + 40 q^{67} + 4 q^{69} + 14 q^{71} - 77 q^{73} - 10 q^{75} - 70 q^{77} + 44 q^{79} + 4 q^{81} - 74 q^{83} + 68 q^{85} + 33 q^{87} - 37 q^{89} - 42 q^{91} + 9 q^{93} - 166 q^{95} + 28 q^{97} - 28 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 0 0.365341 + 0.930874i 0 −2.34039 − 0.352757i 0 1.66419 − 2.05681i 0 −0.733052 + 0.680173i 0
25.2 0 0.365341 + 0.930874i 0 −1.64617 − 0.248120i 0 −2.53209 + 0.767160i 0 −0.733052 + 0.680173i 0
25.3 0 0.365341 + 0.930874i 0 −0.972393 − 0.146565i 0 −1.95268 − 1.78523i 0 −0.733052 + 0.680173i 0
25.4 0 0.365341 + 0.930874i 0 1.87521 + 0.282642i 0 1.13195 + 2.39138i 0 −0.733052 + 0.680173i 0
37.1 0 −0.733052 + 0.680173i 0 −2.74977 − 0.848191i 0 −0.420961 + 2.61205i 0 0.0747301 − 0.997204i 0
37.2 0 −0.733052 + 0.680173i 0 0.309084 + 0.0953397i 0 −2.55452 + 0.688799i 0 0.0747301 − 0.997204i 0
37.3 0 −0.733052 + 0.680173i 0 0.974181 + 0.300495i 0 2.18314 + 1.49462i 0 0.0747301 − 0.997204i 0
37.4 0 −0.733052 + 0.680173i 0 2.02190 + 0.623674i 0 0.0408295 − 2.64544i 0 0.0747301 − 0.997204i 0
109.1 0 −0.988831 + 0.149042i 0 −1.18488 + 3.01902i 0 2.56998 + 0.628642i 0 0.955573 − 0.294755i 0
109.2 0 −0.988831 + 0.149042i 0 −0.540937 + 1.37828i 0 −0.639280 − 2.56736i 0 0.955573 − 0.294755i 0
109.3 0 −0.988831 + 0.149042i 0 −0.0443805 + 0.113080i 0 −2.19662 + 1.47473i 0 0.955573 − 0.294755i 0
109.4 0 −0.988831 + 0.149042i 0 1.12658 − 2.87048i 0 1.95455 − 1.78318i 0 0.955573 − 0.294755i 0
121.1 0 0.955573 − 0.294755i 0 −0.982481 − 0.911609i 0 2.64545 + 0.0398929i 0 0.826239 − 0.563320i 0
121.2 0 0.955573 − 0.294755i 0 −0.540527 − 0.501535i 0 −2.42222 − 1.06435i 0 0.826239 − 0.563320i 0
121.3 0 0.955573 − 0.294755i 0 1.36760 + 1.26894i 0 2.61506 − 0.401811i 0 0.826239 − 0.563320i 0
121.4 0 0.955573 − 0.294755i 0 2.81649 + 2.61332i 0 −2.08678 + 1.62645i 0 0.826239 − 0.563320i 0
193.1 0 0.0747301 − 0.997204i 0 −2.21815 − 1.51231i 0 0.0252707 + 2.64563i 0 −0.988831 − 0.149042i 0
193.2 0 0.0747301 − 0.997204i 0 −0.0624765 − 0.0425958i 0 −1.77843 − 1.95887i 0 −0.988831 − 0.149042i 0
193.3 0 0.0747301 − 0.997204i 0 0.382755 + 0.260958i 0 2.64310 + 0.118387i 0 −0.988831 − 0.149042i 0
193.4 0 0.0747301 − 0.997204i 0 2.32054 + 1.58212i 0 −2.24411 + 1.40141i 0 −0.988831 − 0.149042i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.2.y.a ✓ 48
49.g even 21 1 inner 588.2.y.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.y.a ✓ 48 1.a even 1 1 trivial
588.2.y.a ✓ 48 49.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{48} + 16 T_{5}^{46} + 38 T_{5}^{45} + 14 T_{5}^{44} + 849 T_{5}^{43} + 1922 T_{5}^{42} + \cdots + 35721 \) acting on \(S_{2}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display