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Properties

Label 588.2.q.a
Level $588$
Weight $2$
Character orbit 588.q
Analytic conductor $4.695$
Analytic rank $0$
Dimension $30$
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(85,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.85");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 588.q (of order \(7\), 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: \(4.69520363885\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(5\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 5 q^{3} - q^{7} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 5 q^{3} - q^{7} - 5 q^{9} + 13 q^{11} - 2 q^{13} - 7 q^{15} + 5 q^{17} - 12 q^{19} - 8 q^{21} + 8 q^{23} - 27 q^{25} - 5 q^{27} - 4 q^{29} - 10 q^{31} - 8 q^{33} + 7 q^{35} + 3 q^{37} + 12 q^{39} + 23 q^{41} + 4 q^{43} + 7 q^{45} + 21 q^{47} + 13 q^{49} - 2 q^{51} + 14 q^{53} + 21 q^{55} - 12 q^{57} + 35 q^{59} + 21 q^{61} + 6 q^{63} + 33 q^{65} - 48 q^{67} + 8 q^{69} - 20 q^{71} + 3 q^{73} - 6 q^{75} + 55 q^{77} - 40 q^{79} - 5 q^{81} - 4 q^{83} - 8 q^{85} + 3 q^{87} + 4 q^{89} - 44 q^{91} - 17 q^{93} - 2 q^{95} - 102 q^{97} - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
85.1 0 0.623490 + 0.781831i 0 −1.98121 − 2.48436i 0 −2.55409 + 0.690370i 0 −0.222521 + 0.974928i 0
85.2 0 0.623490 + 0.781831i 0 −1.74513 − 2.18832i 0 1.36908 + 2.26398i 0 −0.222521 + 0.974928i 0
85.3 0 0.623490 + 0.781831i 0 −0.786253 − 0.985931i 0 1.03073 − 2.43672i 0 −0.222521 + 0.974928i 0
85.4 0 0.623490 + 0.781831i 0 1.77775 + 2.22923i 0 −1.94266 − 1.79612i 0 −0.222521 + 0.974928i 0
85.5 0 0.623490 + 0.781831i 0 1.88883 + 2.36852i 0 1.87443 + 1.86722i 0 −0.222521 + 0.974928i 0
169.1 0 −0.222521 + 0.974928i 0 −0.871953 + 3.82028i 0 −1.90226 + 1.83886i 0 −0.900969 − 0.433884i 0
169.2 0 −0.222521 + 0.974928i 0 −0.797815 + 3.49545i 0 2.64014 − 0.172213i 0 −0.900969 − 0.433884i 0
169.3 0 −0.222521 + 0.974928i 0 0.0407373 − 0.178482i 0 −1.07657 − 2.41681i 0 −0.900969 − 0.433884i 0
169.4 0 −0.222521 + 0.974928i 0 0.414719 − 1.81700i 0 −2.36113 + 1.19377i 0 −0.900969 − 0.433884i 0
169.5 0 −0.222521 + 0.974928i 0 0.535863 − 2.34777i 0 1.79885 + 1.94014i 0 −0.900969 − 0.433884i 0
253.1 0 −0.900969 + 0.433884i 0 −2.11268 + 1.01741i 0 2.59435 − 0.518969i 0 0.623490 − 0.781831i 0
253.2 0 −0.900969 + 0.433884i 0 −1.70073 + 0.819027i 0 −0.311253 + 2.62738i 0 0.623490 − 0.781831i 0
253.3 0 −0.900969 + 0.433884i 0 0.651134 − 0.313570i 0 −2.63793 + 0.203308i 0 0.623490 − 0.781831i 0
253.4 0 −0.900969 + 0.433884i 0 1.32777 − 0.639419i 0 −0.956339 − 2.46686i 0 0.623490 − 0.781831i 0
253.5 0 −0.900969 + 0.433884i 0 3.35896 − 1.61759i 0 1.93466 + 1.80474i 0 0.623490 − 0.781831i 0
337.1 0 −0.900969 − 0.433884i 0 −2.11268 − 1.01741i 0 2.59435 + 0.518969i 0 0.623490 + 0.781831i 0
337.2 0 −0.900969 − 0.433884i 0 −1.70073 − 0.819027i 0 −0.311253 − 2.62738i 0 0.623490 + 0.781831i 0
337.3 0 −0.900969 − 0.433884i 0 0.651134 + 0.313570i 0 −2.63793 − 0.203308i 0 0.623490 + 0.781831i 0
337.4 0 −0.900969 − 0.433884i 0 1.32777 + 0.639419i 0 −0.956339 + 2.46686i 0 0.623490 + 0.781831i 0
337.5 0 −0.900969 − 0.433884i 0 3.35896 + 1.61759i 0 1.93466 − 1.80474i 0 0.623490 + 0.781831i 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 85.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.2.q.a ✓ 30
49.e even 7 1 inner 588.2.q.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.q.a ✓ 30 1.a even 1 1 trivial
588.2.q.a ✓ 30 49.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{30} + 26 T_{5}^{28} - 44 T_{5}^{27} + 368 T_{5}^{26} - 482 T_{5}^{25} + 5407 T_{5}^{24} + \cdots + 386358336 \) acting on \(S_{2}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display