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Properties

Label 588.2.q.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 5 q^{3} - 4 q^{5} + q^{7} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 5 q^{3} - 4 q^{5} + q^{7} - 5 q^{9} + 5 q^{11} + 6 q^{13} - 3 q^{15} + 13 q^{17} - 8 q^{19} + 6 q^{21} + 13 q^{25} + 5 q^{27} + 4 q^{29} + 2 q^{31} + 16 q^{33} - 31 q^{35} + 3 q^{37} + 8 q^{39} - 7 q^{41} + 4 q^{43} + 3 q^{45} + 25 q^{47} + 13 q^{49} + 22 q^{51} + 14 q^{53} + 13 q^{55} + 8 q^{57} - 27 q^{59} - 3 q^{61} - 6 q^{63} - 13 q^{65} + 52 q^{67} + 51 q^{73} + 22 q^{75} - 5 q^{77} + 28 q^{79} - 5 q^{81} + 6 q^{83} - 48 q^{85} - 25 q^{87} - 30 q^{89} - 48 q^{91} + 5 q^{93} - 98 q^{95} - 110 q^{97} + 12 q^{99}+O(q^{100}) \) 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 −2.50393 − 3.13983i 0 0.124695 − 2.64281i 0 −0.222521 + 0.974928i 0
85.2 0 −0.623490 − 0.781831i 0 −1.06716 − 1.33818i 0 1.35659 + 2.27149i 0 −0.222521 + 0.974928i 0
85.3 0 −0.623490 − 0.781831i 0 −0.386773 − 0.484999i 0 −2.43042 + 1.04550i 0 −0.222521 + 0.974928i 0
85.4 0 −0.623490 − 0.781831i 0 0.252304 + 0.316380i 0 2.62278 − 0.347875i 0 −0.222521 + 0.974928i 0
85.5 0 −0.623490 − 0.781831i 0 2.30459 + 2.88986i 0 −1.45113 + 2.21229i 0 −0.222521 + 0.974928i 0
169.1 0 0.222521 − 0.974928i 0 −0.526135 + 2.30515i 0 2.10303 − 1.60539i 0 −0.900969 − 0.433884i 0
169.2 0 0.222521 − 0.974928i 0 −0.262912 + 1.15189i 0 −1.99333 + 1.73972i 0 −0.900969 − 0.433884i 0
169.3 0 0.222521 − 0.974928i 0 −0.256660 + 1.12450i 0 0.394673 + 2.61615i 0 −0.900969 − 0.433884i 0
169.4 0 0.222521 − 0.974928i 0 0.428881 − 1.87905i 0 −2.24385 − 1.40183i 0 −0.900969 − 0.433884i 0
169.5 0 0.222521 − 0.974928i 0 0.740316 − 3.24353i 0 2.64045 + 0.167331i 0 −0.900969 − 0.433884i 0
253.1 0 0.900969 − 0.433884i 0 −2.61240 + 1.25807i 0 1.78040 − 1.95709i 0 0.623490 − 0.781831i 0
253.2 0 0.900969 − 0.433884i 0 −1.99363 + 0.960080i 0 −2.34730 − 1.22073i 0 0.623490 − 0.781831i 0
253.3 0 0.900969 − 0.433884i 0 −1.48278 + 0.714067i 0 −0.505798 + 2.59695i 0 0.623490 − 0.781831i 0
253.4 0 0.900969 − 0.433884i 0 1.80936 − 0.871342i 0 2.39340 − 1.12767i 0 0.623490 − 0.781831i 0
253.5 0 0.900969 − 0.433884i 0 3.55692 − 1.71292i 0 −1.94419 + 1.79447i 0 0.623490 − 0.781831i 0
337.1 0 0.900969 + 0.433884i 0 −2.61240 − 1.25807i 0 1.78040 + 1.95709i 0 0.623490 + 0.781831i 0
337.2 0 0.900969 + 0.433884i 0 −1.99363 − 0.960080i 0 −2.34730 + 1.22073i 0 0.623490 + 0.781831i 0
337.3 0 0.900969 + 0.433884i 0 −1.48278 − 0.714067i 0 −0.505798 − 2.59695i 0 0.623490 + 0.781831i 0
337.4 0 0.900969 + 0.433884i 0 1.80936 + 0.871342i 0 2.39340 + 1.12767i 0 0.623490 + 0.781831i 0
337.5 0 0.900969 + 0.433884i 0 3.55692 + 1.71292i 0 −1.94419 − 1.79447i 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.b ✓ 30
49.e even 7 1 inner 588.2.q.b ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.q.b ✓ 30 1.a even 1 1 trivial
588.2.q.b ✓ 30 49.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{30} + 4 T_{5}^{29} + 14 T_{5}^{28} + 30 T_{5}^{27} + 250 T_{5}^{26} + 866 T_{5}^{25} + \cdots + 121705024 \) acting on \(S_{2}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display