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Properties

Label 588.2.y.b
Level $588$
Weight $2$
Character orbit 588.y
Analytic conductor $4.695$
Analytic rank $0$
Dimension $60$
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: \(60\)
Relative dimension: \(5\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 5 q^{3} - 2 q^{5} - q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 5 q^{3} - 2 q^{5} - q^{7} + 5 q^{9} - 5 q^{11} + 6 q^{13} + 3 q^{15} - q^{17} + 11 q^{19} - 3 q^{21} + 6 q^{23} + 59 q^{25} + 10 q^{27} - 4 q^{29} + q^{31} - 16 q^{33} + 25 q^{35} - 48 q^{37} + 10 q^{39} + 37 q^{41} - 22 q^{43} + 12 q^{45} + 11 q^{47} + 41 q^{49} + 50 q^{51} + 22 q^{53} + 8 q^{55} - 20 q^{57} + 81 q^{59} - 30 q^{61} - 9 q^{63} + 43 q^{65} - 31 q^{67} + 12 q^{69} + 6 q^{71} + 24 q^{73} + 11 q^{75} + 20 q^{77} - 43 q^{79} + 5 q^{81} - 84 q^{83} - 12 q^{85} - 23 q^{87} - 117 q^{89} - 39 q^{91} - 8 q^{93} - 52 q^{95} + 8 q^{97} + 24 q^{99}+O(q^{100}) \) 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 −4.27013 − 0.643619i 0 1.74380 − 1.98977i 0 −0.733052 + 0.680173i 0
25.2 0 −0.365341 − 0.930874i 0 −1.15015 − 0.173358i 0 −2.24065 + 1.40694i 0 −0.733052 + 0.680173i 0
25.3 0 −0.365341 − 0.930874i 0 −0.852105 − 0.128434i 0 0.271991 + 2.63173i 0 −0.733052 + 0.680173i 0
25.4 0 −0.365341 − 0.930874i 0 2.53571 + 0.382197i 0 2.61613 + 0.394816i 0 −0.733052 + 0.680173i 0
25.5 0 −0.365341 − 0.930874i 0 3.55239 + 0.535436i 0 −1.65821 − 2.06163i 0 −0.733052 + 0.680173i 0
37.1 0 0.733052 − 0.680173i 0 −3.66842 − 1.13156i 0 −0.949188 + 2.46962i 0 0.0747301 − 0.997204i 0
37.2 0 0.733052 − 0.680173i 0 −1.11801 − 0.344861i 0 −1.67419 − 2.04868i 0 0.0747301 − 0.997204i 0
37.3 0 0.733052 − 0.680173i 0 −0.497362 − 0.153416i 0 2.60465 − 0.464533i 0 0.0747301 − 0.997204i 0
37.4 0 0.733052 − 0.680173i 0 2.05963 + 0.635310i 0 −2.63665 + 0.219220i 0 0.0747301 − 0.997204i 0
37.5 0 0.733052 − 0.680173i 0 4.06521 + 1.25395i 0 2.58065 − 0.583321i 0 0.0747301 − 0.997204i 0
109.1 0 0.988831 − 0.149042i 0 −0.721349 + 1.83797i 0 2.43229 − 1.04113i 0 0.955573 − 0.294755i 0
109.2 0 0.988831 − 0.149042i 0 −0.508595 + 1.29588i 0 −1.86030 + 1.88130i 0 0.955573 − 0.294755i 0
109.3 0 0.988831 − 0.149042i 0 0.180871 − 0.460853i 0 −1.90755 − 1.83337i 0 0.955573 − 0.294755i 0
109.4 0 0.988831 − 0.149042i 0 0.367680 − 0.936834i 0 2.63679 + 0.217581i 0 0.955573 − 0.294755i 0
109.5 0 0.988831 − 0.149042i 0 1.43422 − 3.65432i 0 −2.25680 − 1.38089i 0 0.955573 − 0.294755i 0
121.1 0 −0.955573 + 0.294755i 0 −2.62097 − 2.43191i 0 −2.25475 − 1.38424i 0 0.826239 − 0.563320i 0
121.2 0 −0.955573 + 0.294755i 0 −0.971312 − 0.901246i 0 −1.32777 + 2.28846i 0 0.826239 − 0.563320i 0
121.3 0 −0.955573 + 0.294755i 0 −0.628179 − 0.582865i 0 1.64985 − 2.06833i 0 0.826239 − 0.563320i 0
121.4 0 −0.955573 + 0.294755i 0 2.19021 + 2.03221i 0 2.00165 + 1.73014i 0 0.826239 − 0.563320i 0
121.5 0 −0.955573 + 0.294755i 0 2.26864 + 2.10499i 0 −0.895221 − 2.48969i 0 0.826239 − 0.563320i 0
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.5
Significant digits:
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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.b ✓ 60
49.g even 21 1 inner 588.2.y.b ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.y.b ✓ 60 1.a even 1 1 trivial
588.2.y.b ✓ 60 49.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{60} + 2 T_{5}^{59} - 40 T_{5}^{58} - 56 T_{5}^{57} + 762 T_{5}^{56} - 333 T_{5}^{55} + \cdots + 816927980130304 \) acting on \(S_{2}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display